Controllable Quantum States: Mesoscopic Superconductivity & Spintronics (MS+S2006), Procedings of the International Symposium, NTT Basic Res Labatories, Japan 27 February - 2 Marc

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MS+S2006

CONTROLLABLE QUANTUM Mesoscopic

STATES

Superconductivity and S p i n t r o n i c s

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NTT Basic Research Laboratories, Japan

27 February-2 March 2006

MS+S2M6

CONTROLLABLE QUANTUM Mesoscopic

STATES

Superconductivity and S p i n t r o n i c s

Proceedings of the International Symposium

Editors

Hideaki

Takayanagi

Tokyo University of Science, Japan

Junsaku

Nitta

Tohoku University Japan

Hayato

Nakano

NTT Basic Research Laboratories, Japan

'World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI

• H O N G K O N G • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

CONTROLLABLE QUANTUM STATES Mesoscopic Superconductivity and Spintronics (MS+S2006) Proceedings of the International Symposium Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-461-6 ISBN-10 981-281-461-2

Printed in Singapore.

Benjamin - Controllable Quantum.pmd

1

8/12/2008, 1:40 PM

v

PREFACE

This issue contains the Proceedings of the fourth International Symposium on Mesoscopic Superconductivity and Spintronics (MS+S2006) which was held from February 27th to March 2nd 2006 at NTT Atsugi R&D Center. The first International Symposium on Mesoscopic Superconductivity 2000 (MS2000) was held in March 2000. The main topic of the first symposium was the Andreev-reflection physics at superconductor/semiconductor and superconductor/normal metal interfaces. The scope of the second symposium was extended to include Spintronics, and it was decided to name the second International Symposium on Mesoscopic Superconductivity and Spintronics 2002 (MS+S2002). The third symposium on Mesoscopic Superconductivity and Spintronics 2004 (MS+S2004) was held to highlight a rapidly growing field of quantum computations by adding the subtitle “In the light of quantum computation”. The leading scientists of these research fields participated in the first MS200, the second MS+S2002, and third MS+S2004. From the MS+S2004, the quantum coherence and manipulation have become important and common topics in the fields of Mesoscopic Superconductivity and Spintronics. The fourth MS+S2006 symposium was organized since many researchers looked forward to the discussions of the progress in these fields. The extensively discussed topics in the fourth MS+S2006 were “Control and readout of quantum states in superconducting qubits” and “Spin coherence and manipulation in nano-scale semiconductors” in addition to “Novel phenomena in mesoscopic superconductors”. We believe that the MS+S symposium series have played an important role in the progress of mesoscopic superconductivity, spintronics, and quantum computations. A total of 131 papers were presented at the symposium, including 26 invited talks. The number of participants was 194; 141 from Japan and 53 from 18 foreign countries. This Proceedings contain 64 papers out of those presented at the symposium. We would like to thank all reviewers for their careful reading of the submitted papers. It is our hope that the Proceedings will be useful for many researchers interested in mesoscopic superconductivity, spintronics, and quantum computations. Finally, we would like to thank all participants for their fruitful and exciting discussion throughout the symposium. The symposium was sponsored by JST (Japan Science and Technology) and NTT Basic Research Laboratories. The Organizing Committee would like to express its sincere gratitude to them for their support.

June 2008 Hideaki Takayanagi (Tokyo University of Science, MANA-NIMS ) Junsaku Nitta (Tohoku University) Hayato Nakano (NTT Basic Research Laboratories)

INTERNATIONAL SYMPOSIUM ON MESOSCOPIC SUPERCONDUCTIVITY AND SPINTRONICS (MS+S2006) Date : February 27 (Mon.) – March 2 (Thu.), 2006 Site : NTT Basic Research Laboratories, Atsugi-shi, Kanagawa 243-0198, JAPAN

Organizing Committee H. Takayanagi J. Nitta T. Akazaki Y. Nakamura H. Nakano S. Nomura K. Semba Y. Shimazu H. Tamura M. Ueda

(NTT Basic Research Laboratories), Chair (Tohoku University), Co-Chair (NTT Basic Research Laboratories) (NEC Corporation) (NTT Basic Research Laboratories) (Tsukuba University) (NTT Basic Research Laboratories) (Yokohama National University) (NTT Basic Research Laboratories) (Tokyo Institute of Technology)

International Advisory Committee A. Andreev T. Claeson J. Clarke K. Kajimura J. Mooij A. Leggett E. Rashba H. Sakaki A. Andreev T. Claeson

(P. L. Kapiza Institute of Physical Problems) (Chalmers University of Technology) (University of California) (Japan Society for the Promotion of Machine Industry) (Delft University of Technology) (University of Illinois at Urbana-Champaign) (University of Buffalo) (University of Tokyo) (P. L. Kapiza Institute of Physical Problems) (Chalmers University of Technology)

Sponsors Japan Science and Technology Agency (JST) NTT Basic Research Laboratories

vii

CONTENTS Preface

v

Mesoscopic Effects in Superconductors

1

Tunneling measurements of charge imbalance of non-equilibrium superconductors R. Yagi

3

Influence of magnetic impurities on Josephson current in SNS junctions T. Yokoyama

9

Nonlinear response and observable signatures of equilibrium entanglement A. M. Zagoskin

15

Stimulated Raman adiabatic passage with a Cooper pair box Giuseppe Falci

21

Crossed Andreev reflection-induced giant negative magnetoresistance Francesco Giazotto

27

Quantum Modulation of Superconducting Junctions

33

Adiabatic pumping through a Josephson weak link Fabio Taddei

35

Squeezing of superconducting qubits Kazutomu Shiokawa

41

Detection of Berry’s phases in flux qubits with coherent pulses D. N. Zheng

47

Probing entanglement in the system of coupled Josephson qubits A. S. Kiyko

53

Josephson junction with tunable damping using quasi-particle injection Ryuta Yagi

59

Macroscopic quantum coherence in rf-SQUIDs Alexey V. Ustinov

65

Bloch oscillations in a Josephson circuit D. Esteve

71

Manipulation of magnetization in nonequilibrium superconducting nanostructures F. Giazotto

77

viii Superconducting Qubits

83

Decoherence and Rabi oscillations in a qubit coupled to a quantum two-level system Sahel Ashhab

85

Phase-coupled flux qubits: CNOT operation, controllable coupling and entanglement Mun Dae Kim

91

Characteristics of a switchable superconducting flux transformer with a DC-SQUID Yoshihiro Shimazu

97

Characterization of adiabatic noise in charge-based coherent nanodevices E. Paladino

103

Unconventional Superconductors

109

Threshold temperatures of zero-bias conductance peak and zero-bias conductance dip in diffusive normal metal/superconductor junctions Iduru Shigeta

111

Tunneling conductance in 2DEG/S junctions in the presence of Rashba spin-orbit coupling T. Yokoyama

118

Theory of charge transport in diffusive ferromagnet/p-wave superconductor junctions T. Yokoyama

124

Theory of enhanced proximity effect by the exchange field in FS bilayers T. Yokoyama

130

Theory of Josephson effect in diffusive d-wave junctions T. Yokoyama

136

Quantum dissipation due to the zero energy bound states in high-Tc superconductor junctions Shiro Kawabata

143

Spin-polarized heat transport in ferromagnet/unconventional superconductor junctions T. Yokoyama

149

Little-Parks oscillations in chiral p-wave superconducting rings Mitsuaki Takigawa

155

ix Theoretical study of synergy effect between proximity effect and Andreev interface resonant states in triplet p-wave superconductors Yasunari Tanuma

161

Theory of proximity effect in unconventional superconductor junctions Y. Tanaka

168

Quantum Information

175

Analyzing the effectiveness of the quantum repeater Kenichiro Furuta

177

Architecture-dependent execution time of Shor’s algorithm Rodney Van Meter

183

Quantum Dots and Kondo Effects

189

Coulomb blockade properties of 4-gated quantum dot Shinichi Amaha

191

Order-N electronic structure calculation of n-type GaAs quantum dots Shintaro Nomura

197

Transport through double-dots coupled to normal and superconducting leads Yoichi Tanaka

203

A study of the quantum dot in application to terahertz single photon counting Vladimir Antonov

209

Electron transport through laterally coupled double quantum dots T. Kubo

215

Dephasing in Kondo systems: comparison between theory and experiment F. Mallet

221

Kondo effect in quantum dots coupled with noncollinear ferromagnetic leads Daisuke Matsubayashi

227

Non-crossing approximation study of multi-orbital Kondo effect in quantum dot systems Tomoko Kita

233

Theoretical study of electronic states and spin operation in coupled quantum dots Mikio Eto

239

Spin correlation in a double quantum dot-quantum wire coupled system S. Sasaki

245

x Kondo-assisted transport through a multiorbital quantum dot Rui Sakano

251

Spin decay in a quantum dot coupled to a quantum point contact Massoud Borhani

256

Quantum Wires, Low-Dimensional Electrons

263

Control of the electron density and electric field with front and back gates Masumi Yamaguchi

265

Effect of the array distance on the magnetization configuration of submicron-sized ferromagnetic rings Tetsuya Miyawaki

271

A wide GaAs/GaAlAs quantum well simultaneously containing two dimensional electrons and holes Ane Jensen

277

Simulation of the photon-spin quantum state transfer process Yoshiaki Rikitake

282

Magnetotransport in two-dimensional electron gases on cylindrical surface Friedland Klaus-Juergen

288

Full counting statistics for a single-electron transistor at intermediate conductance Yasuhiro Utsumi

295

Creation of spin-polarized current using quantum point contacts and its detection Mikio Eto

301

Density dependent electron effective mass in a back-gated quantum well S. Nomura

307

The supersymmetric sigma formula and metal-insulator transition in diluted magnetic semiconductors I. Kanazawa

312

Spin-photovoltaic effect in quantum wires A. Fedorov

318

Quantum Interference

325

Nonequilibrium transport in Aharonov-Bohm interferometer with electron-phonon interaction Akiko Ueda

327

xi Fano resonance and its breakdown in AB ring embedded with a molecule Shigeo Fujimoto, Yuhei Natsume

333

Quantum resonance above a barrier in the presence of dissipation Kohkichi Konno

339

Ensemble averaging in metallic quantum networks F. Mallet

345

Coherence and Order in Exotic Materials

351

Progress towards an electronic array on liquid helium David Rees

353

Measuring noise and cross correlations at high frequencies in nanophysics T. Martin

359

Single wall carbon nanotube weak links K. Grove-Rasmussen

365

Optical preparation of nuclear spins coupled to a localized electron spin Guido Burkard

371

Topological effects in charge density wave dynamics Toru Matsuura

377

Studies on nanoscale charge-density-wave systems: fabrication technique and transport phenomena Katsuhiko Inagaki

383

Anisotropic behavior of hysteresis induced by the in-plane field in the ν = 2/3 quantum Hall state Kazuki Iwata

389

Phase diagram of the ν = 2 bilayer quantum Hall state Akira Fukuda

395

Trapped Ions (Special Talk)

401

Quantum computation with trapped ions Hartmut Häffner

403

List of Participants

409

Mesoscopic Effects in Superconductors

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TUNNELING MEASUREMENTS OF CHARGE IMBALANCE OF NON-EQUILIBRIUM SUPERCONDUCTORS R. YAGI, K. UTSUNOMIYA, K. TSUBOI, T. KUBOTA, Y. TERAO AND Y. IKEBUCHI Graduate School of Advanced Sciences of Matter (ADSM), Hiroshima University, Higashi-Hiroshima, 739-8530, Japan

We have observed excess current due to charge imbalance in the voltage-current characteristics of a superconductor-insulator-normal (SIN) tunnel junction connected to a non-equilibrium superconductor. It was found that that the excess current was unchanged against the bias voltage as expected from the theory of charge imbalance. The estimated excess current approximately agreed with the estimation from one-dimensional diffusion model of charge imbalance transport.

1. Introduction Charge accumulation effect is one of the most interesting phenomena in non-equilibrium superconductivity. In normal metal, the numbers of electron and hole excitations are identical because electron excitation inevitably excites holes. This is a consequence of strict charge neutrality. On the other hand, in superconductor, those numbers can be different owing to the superconducting condensate moving without dissipation to satisfy the charge neutrality. This is called a charge imbalance effect discovered theoretically and experimentally by Clarke and Tinkham1-3. Since the discovery, many experimental and theoretical studies have been done in 1970s and 1980s4-6. Past experimental studies have focused on the chemical potential difference between quasi-particle system and superconducting condensate, which is associated with generation of the charge imbalance. The signal voltage to be measured is, in general, so small that they must use superconducting interferometer device (SQUID) voltmeter to detect it. Recent development of experimental technology has realized the experiments that could not have done in the past. In this study we show direct measurement of charge imbalance with voltage-current (V-I) characteristic measurements. The character of the quasi-particles is determined by relative magnitude of the wave vectors: if |k| is larger than the Fermi wave vector kF, it is electron like and in the opposite case, it is hole like. Charge imbalance is defined by the net charge density of quasiparticle system as ∞

∫

Q* = 2 N (0) f > ( E ) − f < ( E )dE , ∆

(1)

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where f>(E) and f<(E) are quasi-particle distribution functions for electron and hole branches. Quasi-particle injection using superconductor-insulator-normal (SIN) tunnel junction creates both the electron and hole excitations are created. If the injecting voltage is considerably larger than the gap voltage, charge imbalance with the same polarity as the injected carrier is created. The tunneling current of a SIN junction connected to the non-equilibrium superconductor has two major components. One is the excess current δI due to charge imbalance. For a arbitrary bias voltage, δI is given as

δI =

Gnn Q* , 2 N (0)

(2)

where N(0) is the density of states and Gnn is the conductance of the junction. The excess current does not depend on the applied bias voltage, and it does not depend on the electron distribution in normal metal electrode, either. The other component of the tunneling current is current In , which is dependent on the bias voltage, and is similar to the tunneling current of SIN junction connected to equilibrium superconductors. In is given by ∞

G  I n =  nn  π ( E ) ( g ( E − eV ) − g ( E + eV ) )dE .  e 

∫

(3)

∆

Here, π(E) is the normalized BCS density of states, g(E) is the electron distribution in normal metal electrode of a detector junction, and Gnn is the tunnel conductance of the junction. These two contributing currents are characteristic to the tunneling current of SIN junction connected to the non-equilibrium superconductors. They have striking difference in their dependence of the bias voltages applied to the SIN junction. 2. Experimental Fig. 1 is a SEM micrograph of our device showing the structure of the device. The device consisted of 0.15 µm width aluminum wire and several Au/AlOx/Al tunnel junctions. Film thickness of aluminum was about 20 nm. The area of the junction was 0.15 x 0.15 µm2 and tunnel resistance was set to be between 5 and 10 kΩ. The distance between the junctions was varied from 2 µm to 40 µm. The device was fabricated by using standard electron beam lithography and angle oblique evaporation technique. Tunnel barrier was formed by exposing the substrate to low pressure oxygen after evaporating first aluminum electrode. Then gold was deposited to form counter electrodes of the tunnel junctions. To extract non-equilibrium quasi-particles, one should reduce the population of thermally excited quasi-particles. We cooled the device down to 70 mK using dilution refrigerator. The device was enclosed in a copper box thermally connected to the mixing chamber. We reduced external noise with filters placed in room temperature part and low temperature part.

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S/I/N Junction

0.5µm Superconducting Narrow Wire

S/I/N Junction 10µm

Fig. 1. Scaning electron micrograph of fabricated multi-terminal small junction device.

3. Result and discussion Fig. 2 is the result of V-I characteristics measurement using two tunnel junctions: one of which served as a detector and other served as an injector. The distance between these two junctions was about 2mm. We varied injection current from −56 nA to 60 nA in 4 nA steps. A curve crossing the origin is for zero injection current. With injection, the curves shifted upwards or downwards depending on the polarity of the injection current. The shift is apparently due to injection and possibly the excess current due to charge imbalance. 200

)A 100 p( I no it 0 ce te d -100

-200 -300

-200

-100

0

100

200

300

detection V (μV) Fig. 2. V-I characteristics of detector junction. Injection is done with junction 2 µm away from the detector junction. T = 70 mK. The injection current was varied from −56 nA to 60 nA in 4 nA steps.

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Observed current was expected to be a sum of In and δI. We can separate these components using different response to the injection current. The shape of the curve of VI characteristics is seen to depend on the magnitude of the injection current, and did not seem to depend on the polarity. This would be because the heating effect due to energy dissipation of the injection current governed the change in the distribution function g(E) in normal metal electrode, which results in voltage dependence of In. Thus In is symmetric in injection current. On the other hand, the excess current is an anti-symmetric function in injection current. The polarity of the excess current is the same as that of the injection current while the magnitude of the excess is determined by the magnitude of the injection current. Considering these fundamental difference in the symmetry against injection current, we extracted the two components using following equations,

2δI = I(V, Iinj) − I(V, −Iinj),

(4)

2In = I(V, Iinj) +I(V, −Iinj).

(5)

Fig. 3 shows the bias dependence of the excess current. It was seen that the excess current of SIN junction was unchanged against bias voltage as expected from Eq.(2). 30501-30514

200.0

inj 2 det 3 T=70mK

150.0

)A p( I 100.0 Δ 50.0

0.0

-100

-50

0

V (µV)

50

100

Fig. 3. Bias voltage dependence of excess current. ∆I = 2δI.

In Fig. 3, although the injection current was varied in equal current steps (4 nA), it was discernible that the excess current was not equally spaced especially in low injections. This indicates that the excess current is suppressed in low injection currents, which is one of the characteristic features of the charge imbalance. When the electrons are injected from normal metal electrode to the superconductor, the excitations are created in both the electron and hole branches with a probability dependent on the bias voltage. The

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efficiency F* to create charge imbalance associated with the probability is given by the formula, ∞

∫ π (E) F* =

−1

( f ( E − eVinj ) − f ( E + eVinj ) )dE

∆ ∞

,

∫ π (E ) ( f ( E − eV

inj ) −

(6)

f ( E + eVinj ) )dE

∆

F* is a monotonic increasing function of Vinj. At Vinj = ∆/e, F* is zero. This is because at that energy, quasi-particles have both the electron like character and hole like character equally and do not contribute to the charge imbalance. For large |Vinj| values, F* approaches 1 monotonically. The observed suppression of the excess current is considered to be due to factor F*, because at those injection current actual bias voltage was near gap voltage. Next we discuss the magnitude of the excess current. Charge imbalance is believed to dQ * Q* conduct in the superconductor obeying diffusion equation, D 2 − = 0 , where D is τ Q* dx the diffusion constant, τ Q* is the relaxation time of charge imbalance. If we limit the system to one dimensional superconducting wire with injection carried out at the one of the end of the wire, we have an analytic solution,

Q* =

λQ I inj F * eDS

(

)

exp − x / λQ* .

(7)

Here, S is the cross sectional area of the wire, λQ* is the spatial relaxation length defined as λQ* = Dτ Q* . Using Eq. (1) and (3), we estimated the excess current using experimentally determined relaxation length λQ* . If Iinj = 60 nA and d = 2 µm, the estimated excess current was about 91pA which agrees with measured value of 92 pA. We used N(0) = 1.05 x 1047 states/Jm3 for density of states of aluminum for one spin orientation. Diffusion constant was estimated to be D = 5.2 x 10-3 m2/sec from the resistivity measurement of normal state aluminum at T = 4.2 K. We also calculated F* using Eq. (2) for actual bias voltage. We used λQ* = 3.8 µm which was measured by the spatial decay of excess current using different probe configurations (not shown). These results assured that the measured phenomenon is due to charge imbalance effect. 4. Conclusion We have observed excess current due to charge imbalance in the V-I characteristic measurements in non-equilibrium superconductors. The V-I characteristics have exhibited striking difference in the dependence on the bias voltages. The excess current was

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independent of the bias voltages as expected, while the normal part of the current was voltage dependent, whose shape in V-I characteristics is approximately the same as that for SIN junction connected to the equilibrium superconductors. These results are consistent with the theory of tunneling current for non-equilibrium superconductors. Our study would provide new method to study the charge imbalance in non-equilibrium superconductors. Acknowledgments This work was supported in part by Grant-in Aid for Scientific Research (No. 16510077) and Grant-in-Aid for COE research (No. 13Ce2002) of the Ministry of Education, Culture, Sports, Science and Technology of Japan. References 1. 2. 3. 4. 5. 6.

M. Tinkham, Phys. Rev. B 46, 1033 (1972). M. Tinkham and J. Clarke, Phys. Rev. Lett. 28, 1366 (1972). J. Clarke, Phys. Rev. Lett. 28, 1363 (1972). J. Clarke and J. L. Peterson, J. Low Temp. Phys. 15, 491 (1974). H. J. Mamin, J. Clarke, and D. J. Van Haringen, Phys. Rev. B 29, 3881 (1984). S. B. Kaplan, C. C. Chi, D.N. Lagenberg, J. J. Chang, S. Jafarey and D. J. Scalapino, Phys. Rev. B 14, 4854 (1976).

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INFLUENCE OF MAGNETIC IMPURITIES ON JOSEPHSON CURRENT IN SNS JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected] Inﬂuence of magnetic impurities on Josephson current in superconductor / insulator / diﬀusive normal metal / insulator / superconductor (S/DN/S) junctions is studied where magnetic impurities are in the DN. We have solved the Usadel equation and calculated Josephson current by changing magnetic scattering rate in DN. It is shown that Josephson current is suppressed by magnetic impurity scattering. When DN is replaced by a diﬀusive ferromagnet, magnetic impurity scattering causes a 0-π transition.

1. Introduction Josephson predicted that supercurrent ﬂows between two superconductors separated by a thin insulating layer (SIS junctions) without bias voltage1 . Until now the Josephson current has been studied in many situations2 . In SIS or superconductor / diﬀusive normal metal / superconductor (S/DN/S) junctions the critical current increases monotonically with decreasing temperature3,4,5 . In S/DN/S junctions Josephson current ﬂows by Cooper pairs tunneling through the DN. This phenomenon is interpreted as a result of proximity eﬀect. To study the proximity eﬀect, the quasiclassical Green’s function theory 6,7,8 has been widely used because of its convenience and broad applicability. Based on this formalism, theory of charge transport in DN/S junctions was formulated by Volkov, Zaitsev and Klapwijk (VZK) 9 . This work was based on the boundary conditions for the Keldysh-Nambu Green’s function at the DN/S interface derived by Kupriyanov and Lukichev (KL) 4 . Applying the VZK theory, proximity eﬀect in DN/S junctions with magnetic impurities is studied in 10,11 . It is shown that

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proximity eﬀect is suppressed by magnetic impurities. In diﬀusive ferromagnet / superconductor (DF/S) junctions Cooper pairs penetrate into the DF layer from the S layer and have a nonzero momentum by the exchange ﬁeld12,13 . This property induces many interesting phenomena14,15,16 . One interesting consequence is an appearance of the π junctions in S/DF/S junctions2 . Although, in several works, Josephson eﬀect is studied in S/DN/S or S/DF/S junctions with magnetic impurities in the quasiclassical scheme, there are oversimpliﬁed treatments of boundary conditions and the Usadel equation17 in the previous theories18,19 . Up to now, there are no works without simpliﬁcation, e.g. linearization, concerning this problem. In this paper we try this problem using the KL boundary conditions in order to obtain more quantitative results. We solve equations fully numerically in order to avoid simpliﬁcation. This is important because recently magnetic semiconductor has received much attention and our study may be useful to understand the Josephson eﬀect in superconducting junctions with magnetic semiconductor. Since a phase of superconductors is one of the most important features in the superconductivity, our result may contribute to not only a fundamental physics but also future technologies, e.g., quantum computing. In the present paper we study the inﬂuence of magnetic impurities on Josephson current in superconductor / insulator / diﬀusive normal metal / insulator / superconductor (S/DN/S) junctions where magnetic impurities are in the DN. We have solved the Usadel equation and calculated Josephson current by changing magnetic scattering rate in DN. It is shown that Josephson current is suppressed by magnetic impurity scattering. When DN is replaced by a diﬀusive ferromagnet, magnetic impurity scattering causes a 0-π transition. 2. Formulation We consider a junction consisting of s-wave superconducting reservoirs (S) connected by a quasi-one-dimensional DN (or DF) with a resistance Rd and a length L much larger than the mean free path. The DN/S interface located at x = 0 has the resistance Rb , while the DN/S interface located at x = L has the resistance Rb . We parametrize the quasiclassical Green’s functions G and F with function Φ:2 ω Φω Gω = 2 , Fω = 2 ∗ ω + Φω Φ−ω ω + Φω Φ∗−ω with Matsubara frequency ω. Then Usadel equation reads17 2 πTC ∂ 2 ∂ ξ Gω Φω − (ω ∓ ih) Φω − γGω Φω = 0 Gω ∂x ∂x for majority (minority) spin with ξ = D/2πTC , diﬀusion constant D, exchange ﬁeld h, the magnetic scattering rate γ and critical temperature TC . The KL boundary conditions are given by4 Gω ∂ Rd Φω −iϕ Φω = − g + ie f − ω ∂x Rb L ω

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ω g= 2 ω + Δ2

Δ f= 2 ω + Δ2

at x = 0 with pair potential Δ = Δ(T ) at a temperature T and Gω ∂ Rd Φω Φω = g + if − ω ∂x Rb L ω at x = L. Here ϕ is the external phase diﬀerence across the junctions. Josephson current is given by the expression eIR ∂ ∗ ∂ RT L G2ω ∗ =i Φω Φ−ω − Φ−ω Φω πTC 4RdTC ω2 ∂x ∂x ↑,↓,ω

with R ≡ Rd + Rb +

Rb .

3. Results Below we ﬁx parameters as Rd /Rb = Rd /Rb = 0.1 and ET h /Δ(0) = 0.1. Let us ﬁrst study the S/DN/S junctions. We show current-phase relation for T /TC = 0.1 in Fig. 1 (a) and critical current as a function of temperature in Fig. 1(b) where IC denotes the critical current. In this case current-phase relation has a sinusoidal form. The Josephson current is suppressed as increasing γ because of the suppression of proximity eﬀect10,11 . The corresponding plots for the S/DF/S junctions with h/Δ(0) = 0.5 are shown in Fig. 2. We show current-phase relation for T /TC = 0.3 in (a). The Josephson current has a negative value at γ/Δ(0) = 0 due to the 0-π transition by the exchange ﬁeld2 . With the increase of γ/Δ(0), additional π-0 transition occurs and hence Josephson current has a positive value19 . The critical current also shows a 0-π transition as shown in (b). The crossover temperature is decreased as increasing γ. This is because we can regard the term γGω as an eﬀective Matsubara frequency in the Usadel equation. Therefore introducing γ corresponds to the shift of the temperature. This explains why the crossover temperature is decreased as increasing γ. Note that magnetic impurity scattering causes a 0-π transition only for nonzero exchange ﬁeld. 4. Conclusion In the present paper, we studied the inﬂuence of magnetic impurities on Josephson current in superconductor / insulator / diﬀusive normal metal / insulator / superconductor (S/DN/S) junctions where magnetic impurities are in the DN. We solved the Usadel equation and calculated Josephson current by changing magnetic scattering rate in DN. It is shown that Josephson current is suppressed by magnetic impurity scattering. When DN is replaced by a diﬀusive ferromagnet, magnetic impurity scattering causes a 0-π transition. We note that an application of the weak magnetic ﬁeld (H) parallel to the junction plane can be treated as spin ﬂip (magnetic impurity) scattering. In this

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)

C

π T

( / R I e

(a ) 0.006

0.004

γγΔ γΔ Δ

0.002

0 0

/ / /

0.2

0.4

(0)=0 (0)=0.01 (0)=0.1 0.6

0.8

ϕ /π

(b)

0.02

)

C

π T

γΔ γΔ γΔ

( / 0.01 R IC e 0

Figure 1. tions.

1

0

0.1

0.2

0.3

T TC

/

(0)=0

/

(0)=0.01

/

(0)=0.1

0.4

0.5

(a) Current-phase relation with T /TC = 0.1 and (b) critical current in S/DN/S junc-

case the pair-breaking rate γ is given by e2 w2 DH 2 /6, where w is the transverse size of the DN20 . Assuming w = 10−5 m, D = 10−3 m2 /s, Δ(0) = 10−3 eV , and H = 10−2 T , we can estimate the pair-breaking rate γ/Δ(0) ∼ 1. An interesting problem is a calculation of Josephson current in difusive unconventional superconducting junctions with magnetic impurities, similar to the present paper because interensting phenemena are predicted in these junctions21,22,23 .

Acknowledgments The authors appreciate useful and fruitful discussions with J. Inoue, Yu. Nazarov and H. Itoh. This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science” . The computational aspect of this work has been performed at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center.

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(a)

)

C

π T

/( R I e

0

γγ ΔΔ γΔ

-1

/ / /

×10

-2

0

(0)=0 (0)=0.1 (0)=0.3

-4

0.2

ϕ /π

0.4

0.6

0.8

0.0015

(b)

)

C

π T

γγ ΔΔ γΔ

0.001

(/ R C I 0.0005 e 0

Figure 2. tions.

0

1

/ / /

0.1

0.2

0.3

T TC

0.4

(0)=0 (0)=0.1 (0)=0.3

0.5

0.6

(a) Current-phase relation with T /TC = 0.3 and (b) critical current in S/DF/S junc-

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

B. D. Josephson, Phys. Lett. 1, 251 (1962). A. A. Golubov, M. Yu. Kupriyanov, and E. Il ichev Rev. Mod. Phys. 76, 411 (2004). V. Ambegaokar and A. Baratoﬀ, Phys. Rev. Lett. 10, 486 (1963). M. Yu. Kupriyanov and V. F. Lukichev, Sov. Phys. JETP 67, 1163 (1988). A. V. Zaitsev, Physica C 185-189, 2539 (1991). G.Eilenberger,Z.Phys.214,195 (1968) G.M. Eliashberg, Sov. Phys. JETP 34, 668 (1972) A.I. Larkin and Yu. V. Ovchinnikov, Sov. Phys. JETP 41, 960 (1975) A.F. Volkov, A.V. Zaitsev and T.M. Klapwijk, Physica C 210, 21 (1993). S. Yip, Phys. Rev. B 52, 15504 (1995). T. Yokoyama, Y. Tanaka, A. A. Golubov, J. Inoue, and Y. Asano, Phys. Rev. B 71, 094506 (2005). A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, JETP Lett. 35, 178 (1982). A.I. Buzdin and M.Yu. Kupriyanov, JETP Lett. 53, 321 (1991). A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005); Phys. Rev. B 73, 094501 (2006). K.D. Usadel Phys. Rev. Lett. 25 (1970) 507. A. I. Makeev, Yu. N. Mitsai and N. V. Shakhova, Sov. J. Low Temp. Phys. 6 (1980)

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203; O. Narikiyo and H. Fukuyama: J. Phys. Soc. Jpn. 58 (1989) 4569. 19. M. Faure, A. I. Buzdin, A. A. Golubov, and M. Yu. Kupriyanov, Phys. Rev. B 73, 064505 (2006). 20. W. Belzig, C. Bruder, and G. Sch¨ on, Phys. Rev. B 54 9443 (1996). 21. Y. Tanaka, Yu. V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003; Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 69 (2004) 144519. 22. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 (2004) 012507; Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B 71 (2005) 094513. 23. T. Yokoyama, Y. Tanaka, A. A. Golubov, and Y. Asano, unpublished.

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NONLINEAR RESPONSE AND OBSERVABLE SIGNATURES OF EQUILIBRIUM ENTANGLEMENT

A. M. ZAGOSKIN∗ Frontier Research System, The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama, Japan and Department of Physics and Astronomy, The University of British Columbia, Vancouver, B.C., Canada E-mail: [email protected]

I investigate how equilibrium N -partite entanglement is manifested in the nonlinear response of the system. In particular, I calculate quadratic magnetic susceptibility of a multiqubit system and show that it provides an observable signature of the formation of 3-tangled states. Generalized higher-order entanglement signatures are introduced.

1. Introduction The “equilibrium”, or “thermodynamical”, entanglement in spin systems quantiﬁes the degree of “quantumness” of the system (e.g. 1,2,3,4,5 ); e.g. it can play the role of order parameter at quantum phase transitions 6,7 . Both classical and quantum correlations are present in irreducible multi-point Greens’ functions of the system, but there is no simple recipe for their separation; moreover, only a measure of bipartite entanglement is currently known for a mixed state. A speciﬁc measure of entanglement was proposed for the BCS ground state3 ; diﬀerent known entanglement measures were expressed through spin correlators8, and observable entanglement witnesses 9 or entanglement estimators 10 were proposed. Equilibrium entanglement is particularly important in the context of adiabatic quantum computing (AQC) 11 . From the equivalence of AQC and ”standard” quantum computing 12 and the polynomial equivalence of quantum and classical computing in case when there is no global entanglement13 it can be inferred that global entanglement of the ground state is necessary for an eﬃcient AQC. Therefore the equilibrium state of any system, ﬁt to be an adiabatic quantum computer, must be a mixture of globally entangled eigenstates. The investigation of equilibrium entanglement allows to shift the focus from the state to the Hamiltonian14 . This opens a new venue of investigation. If the ∗ Work

partially supported by Discovery Grant of the Natural Sciences and Engineering Research Council of Canada.

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Hamiltonian of a quantum system can be restored without free parameters from a series of measurements, then all the eigenstates of the system can be determined, and their entanglement measures can be computed. The experimental investigation of equilibrium entanglement along these lines was performed for the sytems of two15 and four16,17 ﬂux qubits. The qubits were controlled locally; the global response was measured, and all the operations were slow on the scale of characteristic qubit times. In both cases a high degree of global entanglement for the ground and ﬁrst excited states was established. 2. Directly observable signatures of equilibrium entanglement The experiment15 produced a directly observable 2-qubit entanglement signature. The linear susceptibility of the system is proportional to p|σiz |qq|σjz |p, p = q. (1) Rpq ∼ i,j

Here |p, |q are the eigenstates of the Hamiltonian, and σjz is proportional to the magnetic moment of the jth qubit in z-direction. If the eigenstates are not entangled, the contributions with i = j disappear. This eﬀect shows on experiment as a diﬀerence between the sum of signals from two qubits driven through their degeneracy points separately, and the signal when they pass the co-degeneracy point simultaneously. The linear response measurements will not provide a similar signature of N tanglement with N > 2. The expressions like (1), arising in the lowest-order perturbation theory, are quadratic in qubit operators and therefore catch the absence of 2-tangled components, but are insensitive to the higher-order entanglement. This is why the appearance of globally entangled eigenstates (| ↑↓↑↓ ± | ↓↑↓↑) in 16 could be only inferred from the post mortem analysis. In order to obtain information about (N > 2)-tanglement without reverse-engineering of the Hamiltonian, one requires N -point correlators. They can be obtained from the nonlinear response of a multiqubit system. 3. Equilibrium multipartite entanglement and irreducible higher-order correlators Consider ﬁrst an N -point correlator of single-qubit (spin) operators Sk (tk ), acting on qubit k at the moment tk , CN ({k, tk }, |p) = p|S1 (t1 )S2 (t2 ) . . . SN (tN )|p, k = 1. . . . N,

(2)

where |p is some eigenstate of the Hamiltonian. The sequence of operators in (2) generally ﬂips N spins in the state |p. Therefore the correlator should be suppressed unless |p is a superposition of components diﬀering by N spins. To check this intuition, let us ﬁrst deﬁne the ”equilibrium N -tanglement”. We will say that the system of M > N qubits (M denoting the set of all qubits) is not

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N-tangled, if (1) all its eigenstates can be presented as products of states deﬁned on (p) qubit clusters Ωj of less than N qubits each, (p) |pΩ(p) , Ωj = M, (3) |p = (p)

Ωj

j

j

and (2) the clusters do not overlap. The latter condition means that, if there exists (p) (p) an eigenstate |p such that qubits a, b ∈ Ωs , while qubit c ∈ / Ωs , then there is no (q) (q) (q) / Ωr , while b, c ∈ Ωr . such eigenstate |q and cluster Ωr , that a ∈ Together, the conditions (1) and (2) ensure that in equilibrium the qubit system can be uniquely split into mutually non-entangled clusters, M = j Ωj , where Ωj = (p)

maxp (Ωj ), each containing less than N qubits. This is a reasonable deﬁnition of the absence of equilibrium N-partite entanglement. In the presence of N -tanglement, at least one cluster contains at least N entangled qubits. This deﬁnition is actually the (restricted by the plausible non-overlapping condition) k-producibility recently introduced in 18 (with k = N ). Returning to (2), one can easily express the correlator as a sum over the internal labels p2 , p3 , . . . pN of the correlators (p1 ≡ p) cN ({k, |pk }) = p1 |S1 |p2 p2 |S2 |p3 . . . pN |SN |p1 , k = 1. . . . N,

(4)

with the appropriate energy denominators, similar to (1). Now consider irreducible correlators, that is, the ones where no two states |pi , |pj , i = j coincide. (It is obvious from the deﬁnition (4) that the correlator is cyclic invariant. In the presence of coinciding states it can be further reduced to a product of cyclic invariant terms.) For such a correlator cirr N ({k, p}) there are two possibilities: (A) all N qubits belong to the same entangled cluster Ω, or (B) they belong to several clusters. In case (A) the states of the system can be written as |n = |nΩ ⊗ |nΩ¯ etc, ¯ is the complement of Ω. We therefore obtain where Ω ¯ ¯ ¯ (5) cirr N ({k, p}) = (· · · )δp1 p2 (Ω)δp2 p3 (Ω) · · · δpN p1 (Ω); (the Kronecker symbol δpq (Ω) means that the states p and q coincide on the set Ω). We see from (5), that the states of the qubits outside Ω must be the same, and can only diﬀer on Ω. Therefore cirr N is generally nonzero, if there are at least log2 N qubits in Ω (which is not a serious limitation). In case (B) the string of indices S = j1 j2 . . . jN , labeling the qubit operators, consists of 1 < R ≤ N substrings, S = s1 s2 . . . sR . In each substring the indices belong to the qubits in the same entangled cluster Ω1 , Ω2 , . . . ΩR . The only restriction is that clusters corresponding to adjacent index substrings, e.g. Ωr and Ωr±1 , must be diﬀerent, or the substrings would merge. r r , and jN + 1 ≡ j1r+1 ) The rth substring (the indices in which run from j1r to jN r r ¯ r (the complement yields the product of the Kronecker symbols, ensuring that on Ω of Ωr ) r +1 ≡ |p r+1 . |pj1r = |pj1r +1 = · · · = |pjN j r 1

(6)

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Let us call the substrings mutual, if they contain qubit indices from the same entangled cluster; a substring, which has no mutuals on S, is called unique. Then we deﬁne a joint (disjoint) correlator as the one, the index string of which does (does not) contain at least one unique substring. For a joint correlator with the unique substring su : S = s1 s2 . . . su−1 su su+1 . . . sR−1 sR , we obtain ¯ 1 · · · δp u−1 p u Ω ¯ u−1 δp u p u+1 Ω ¯u cjoint = (· · · ) δp1 pj2 Ω j j N j j 1 1 1 1 1 ¯ u+1 · · · δp p1 Ω ¯R = ×δp u+1 p u+2 Ω jR j1 j1 1 ¯1 ∩ Ω ¯u ¯2 · · · ∩ Ω ¯ u−1 δp u p u+1 Ω (· · · ) δp1 pju Ω j 1 1 j1 ¯ u+1 ∩ Ω ¯ u+2 · · · ∩ Ω ¯R = ×δpju+1 p1 Ω 1 ¯u ∪ Ω ¯1 · · · ∩ Ω ¯ u−1 ∩ Ω ¯ u+1 · · · ∩ Ω ¯R . (7) (· · · ) δpju p u+1 Ω 1

j1

¯u ∪ Since su is unique, that is, Ωu ∩ Ω1,...,u−1,u+1,...,R = ∅, the set Ω ¯ ¯ ¯ ¯ Ω1 · · · ∩ Ωu−1 ∩ Ωu+1 · · · ∩ ΩR = M. In other words, the states |pj u+1 and |pj1u 1 coincide on all clusters. Therefore joint correlators (7) cannot be irreducible, sice they contain at least two coinciding pairs of internal labels. ¯1 · · · ∩ Ω ¯R = ¯u ∪ Ω For a disjoint correlator this does not hold, since then Ω ¯ u ⊂ M. (The only exception is the case when s1 and sR , the substrings at the Ω beginning and the end of S, are mutual, and don’t have other mutuals. Then, from the cyclic invariance of the irreducible correlator, these substrings eﬀectively merge into a unique substring.) We found that certain irreducible N-point correlators (correlators with repeating qubit indices and disjoint correlators) do not disappear in the absence of N-partite entanglement. Now we demonstrate that their contribution to the correlators of extensive variables nevertheless disappears in the thermodynamical limit, which allows to use the latter as directly observable entanglement signatures. The global response functions of the system contain the irreducible correlators M of average operators, i.e. operators of the type S = M −1 j=1 Sj : CNirr;p1 p2 ...pN = M −N

M

cirr N (k1 , k2 , . . . , kN ; p1 , p2 , . . . , pN ).

(8)

k1 ,k2 ,...,kN =1

Assume that both the number of qubits M and the number of entangled clusters Q exceeds N . Then we shall see that as M, Q → ∞ the contributions to (8) from the correlators with repeating indices and disjoint correlators disappear as O(1/M ), O(1/Q) respectively. Repeating indices. The number of all possible combinations of N indices, each running from 1 to M , is M N . The number of combinations of diﬀerent indices is M (M − 1)...(M − N + 1), and the number of combinations with repeating indices is M N − M (M − 1)...(M − N + 1) = O(M N −1 ). Therefore in the expression (8) the contribution from the correlators with repeating indices scales as O(1/M ).

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N −1 Disjoint correlators. There are ways to split a string of N inR−1 dices in R nonzero substrings. For each substring there are Q entangled clusters we can choose from, and there are on average (M/Q)N/R possible combinations of indices within the cluster. Therefore the total number of diﬀerent combina N N −1 N tions of indices, counted this way, is (M/Q) QR . On the other R=1 R−1 hand, the number corresponding to joint correlators is no less than of combinations

N N −1 N R−1 (M/Q) (Q−(R−1)). Therefore the number of disjoint comQ R=1 R−1

N N −1 R N binations does not exceed M Q − QR−1 (Q − (R − 1)) = R=1 Q R−1

N N − 1 N M d Q dQ QR−1 =M N Q−1 (N − 1)(1 + Q−1 )N −2 , and the R=1 Q R−1 disjoint contribution to the average irreducible correlator (8) asymptotically disappears at least as fast as O(1/Q) ∼ O(1/M ). Therefore CNirr can be used as an N -tanglement indicator: it is nonzero in the limit M, Q → ∞ only if the system contains a ﬁnite proportion of entangled N -qubit (or larger) clusters.

4. Nonlinear susceptibility and equlibrium multipartite entanglement The correlators like (8) enter the expressions for (nonlinear) susceptibility. Therefore the latter could be used as directly observable entanglement signatures. The unperturbed Hamiltonian of M ﬂux qubits and the perturbation due to their coupling to an external ﬁeld are M M z z 1 x z z z Jij σi σj ; H1 (t) = −h(t) Δj σj + εj σj − λj σj + λxj σjx . (9) H0 = − 2 j=1 i<j j=1

The equilibrium entanglement we are probing is created by the unperturbed Hamiltonian, H0 . Iterating the equation of motion for the density matrix, ρ(t), one obtains the standard expansion of the density matrix and the expansion for

corresponding the magnetic moment of the system, μ(t) = tr ρ(t) j σj . In particular, the 2M (0) 2M quadratic susceptibility, χzz (ω, ω ) = n=1 ρn p,q=1 Cn;pq fn;pq (ω, ω ) , is ex z z z pressed through the the weight factors Cn;pq = ijk n|σi |pp|σj |qq|σk |n ≡ 1 z z z n|μ |pp|μ |qq|μ |n, and the formfactors fn;pq (ω, ω ) = ω+ω −(Eq −Ep )+i0 ×

En +Eq −2Ep En +Ep −2Eq − (ω −(Eq −En )+i0)(ω+ω −(Ep −En )+i0) (ω −(En −Ep )+i0)(ω+ω −(En −Eq )+i0) . In the experimentally relevant low-frequency limit, χn0 = limω→0 pq Cnpq fn;pq (ω, ω),

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where χn0 =3

means that terms with Ep = Eq = En are excluded, we ﬁnd:

|μz nq |2 (μz nn − μz qq )

(Eq − En

q

)2

−6

p

(Cnpq ) ≡ χn0A + χn0B . (10) (E − E )(E − E ) p n q n q
For a system of noninteracting qubits χn0A is the sum of single-qubit susceptibilities, while χn0B is exactly zero. The summation restrictions in (10) mean that the coeﬃcients Cnpq can be replaced by the irreducible correlators (8) of the operators σiz : irr Cnpq → M 3 C3;npq = cirr (11) 3 (i, j, k; n, p, q). i,j,k 3

Therefore the function χ0B /(M ) disappears in the thermodynamical limit in the absence of entangled 3-qubit clusters and can be used as an observable signature of 3-tanglement. We can also introduce the N -th order signatures, M −N μz np1 μz p1 p2 · · · μz pN −1 n .(12) ··· ZNn = (Ep1 − En )(Ep2 − En ) · · · (EpN −1 − En ) p 1

p2 =p1

pN −1 =pN −2 ,pN −1 ,...p1

These functions are related to directly observable global nonlinear response of the system. They are asymptotically zero in the thermodynamic limit, unless the system contains a ﬁnite proportion of entangled clusters of at least N qubits each. Therefore the functions (12) can indeed serve as observable entanglement signatures. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

T. J. Osborne and M. A. Nielsen, Phys. Rev. A66, 32110 (2002). A. Osterloh et al., Nature (London) 416, 608 (2002). M. A. Martin-Delgado, quant-ph/0207026. S. Ghosh et al., Nature (London) 425, 48 (2003). G. Vidal et al., Phys. Rev. Lett. 90, 227902 (2003). F. Verstraete, M. A. Martin-Delgado and J.I. Cirac, Phys. Rev. Lett. 92, 087201 (2004). F. G. S. L. Brand˜ ao, New J. Phys. 7, 254 (2005). U. Glaser, H. B¨ uttner and H. Fehske, Phys. Rev. A68, 032318 (2003). L.-A. Wu et al., Phys. Rev. A72, 032309 (2005). T. Roscilde et al., Phys. Rev. Lett. 93, 167203 (2004). E. Farhi et al., quant-ph/0001106. D. Aharonov et al., quant-ph/0405098. G. Vidal, Phys. Rev. Lett. 91, 147902 (2003). S. Bandyopadhyay and D. Lidar, Phys. Rev. A70, 010301(R) (2004). A. Izmalkov et al., Phys. Rev. Lett. 93, 037003 (2004). M. Grajcar et al., Phys. Rev. Lett. 96, 047006 (2006) . Peter J. Love et al., quant-ph/0602143. O. G¨ uhne, G. T´ oth and H.J. Briegel, New J. Phys. 7, 229 (2005).

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stirap

21

STIMULATED RAMAN ADIABATIC PASSAGE WITH A COOPER PAIR BOX

J. SIEWERT, G. FALCI∗, G. MANGANO AND E. PALADINO MATIS-INFM, Consiglio Nazionale delle Ricerche, and Dipartimento di Metodologie Fisiche e Chimiche per l’Ingegneria, Universita di Catania, I-95125 Catania, Italy T. BRANDES Department of Physics, The University of Manchester, Manchester, United Kingdom

The rapid experimental progress in the ﬁeld of superconducting nanocircuits gives rise to an increasing quest for advanced quantum-control techniques for these macroscopically coherent systems. Here we demonstrate theoretically that stimulated Raman adiabatic passage (STIRAP) should be possible with the quantronium setup of a Cooper-pair box. The scheme appears to be robust against decoherence and should be realizable even with the existing technology. STIRAP can be applied to generate single-phonon states of a resonator by vacuum-stimulated adiabatic passage with the superconducting nanocircuit coupled to the resonator.

1. Introduction Recent experiments have opened new perspectives to study quantum phenomena in solid-state devices that traditionally have been part of nuclear magnetic resonance and quantum optics 1,2 . One of the strong motivations of these studies is to explore advanced control of nanodevices and understand how ﬂexibility in the design of devices and protocols may improve robustness of coherence against noise in the solid state. There exist already a number of theoretical proposals, e.g., the detection of geometric phases 3 , which is a key step for the implementation of new computational odinger cat states in electrical and nanomeparadigms 4 , the preparation of Schr¨ chanical resonators 5 , the implementation of cooling techniques 6 , an analogue of electromagnetically induced transparency 7 , adiabatic passage in superconducting nanocircuits 8 . Advanced control requires speciﬁc tasks to be accomplished, one of the challenges being the preparation of Fock states in a resonating nanocircuit. An key step is operated in quantum optics by applying adiabatic passage to the dark state of a three-level atom using two laser ﬁelds in Λ conﬁguration which couple two hyperﬁne ground states |g and |u to an excited state |e. This protocol named STIRAP 9 ∗ [email protected].

Work partially supported by grant EU-EuroSQIP and PRIN 2005022977.

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has been demonstrated to allow for single photon generation, by replacing one of the external drives by the quantum ﬁeld of a cavity 10 . STIRAP is also a building block of non-abelian geometric quantum computation procedures. Compared to the quantum optics realm nanodevices require investigation of entirely new regimes of stronger coupling between the system elements and of stronger inﬂuence of low frequency noise. In particular for STIRAP the macroscopic nature of the system and the eﬀects of solid state noise impose obstacles requiring more than a mere translation of the protocol from quantum optics to solid state. In the following we will analize STIRAP in a single charge-phase Cooper-pair box (the so-called Quantronium) 11 operated as a three level atom. This circuit is appropriate for the substitution of one of the classical driving ﬁelds by the quantum ﬁeld of a coupled resonator without changing its functionality. Veriﬁcation of the vacuum-assisted adiabatic passage completes the analogue of the atom-cavity system in Refs. 10 . In principle this program can be carried out for diﬀerent regimes and setups of superconducting nanocircuits 12 . The main problem in nanodevices is the tradeoﬀ between eﬃcient coupling to the driving ﬁeld and protecton against low-frequency noise. In this respect the Quantronium oﬀers a uniquely convenient design, where tunability is exploited to obtain selective and relatively strong coupling to the ﬁelds allowing to perform STIRAP before decoherence takes place. 2. STIRAP in the Quantronium The Λ conﬁguration needed for STIRAP is relized using two classical corotating a.c. ﬁelds Ag cos ωg t, Au cos ωu t, with equal detunings Δ = Ee − Eg − ωg = Ee − Eu − ωu (Ei are the unperturbed eigenenergies). In the rotating frame the three-level Hamiltonian reads 9,13 1 Hrot.f. = Δ|ee| + (Au |eu| + Ag |eg| + h.c.) (1) 2 This Hamiltonian has an eigenvector called dark state 1 (Ag |u − Au |g) . (2) |D = 2 |Au | + |Ag |2 In the counterintuitive scheme, the system prepared in |g with the couplings Ag = 0 and Au = 0. is rotated in the two-dimensional subspace spanned by |u and |g by slowly varying in time the coupling strengths Au , Ag . By switching Au oﬀ while Ag is switched on, the population is transferred from state |g to state |u. Adiabaticity requires |A˙j /Aj | < ωj (j = u, g). The Hamiltonian (1) can be implemented in the Quantronium. In the laboratory frame the Cooper pair box is described by EJ (|nn + 1| + h.c.) (3) EC [n − ng (t)]2 |nn| − HQ = 2 n where {|n} are eigenstates of the number of extra Cooper pairs in the island, EC is the charging energy and EJ is the Josephson coupling energy (see Fig. 1a). For

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Cg

a)

C/2 111 000 000 111 000 111 n 000 111 000 111E J /2

3

energy E/EC

C/2 111 000 000 111 000 111 000 111 111 E J /2000

Φ

e

2

ωg ωu

1

u

0

Vg

b)

g 0

0.2

0.4

0.6

0.8

1

gate charge n g

Figure 1. a) The Quantronium is a superconducting island of total capacitance C coupled to a superconducting lead via two Josephson junctions. The charging energy EC = 4e2 /(2C) sets the scale of the electrostatic energy, which is controlled by the gate charge ng = Cg Vg /(2e), where Vg is the gate voltage and Cg C is the gate capacitance. For the magnetic ﬂux we choose Φ = 0. b) The lowest four energy levels of the quantronium with EC = EJ as a function of the gate charge ng . At the working point ng0 the three lowest levels can be used as a Λ scheme |g, |u, |e with resonance frequencies ωg = Ee − Eg and ωu = Ee − Eu .

simplicity we assume EC = EJ . The gate charge has a d.c. bias part ng0 and an a.c. part nac g = A cos ωt with amplitude |A| 1/2 allowing for controlled evolution. Eigenstates of the undriven system are superpositions of charge states. In particular if the system is biased at the symmetry point ng0 = 1/2 the microwave ﬁeld couples oﬀ-diagonally in the basis of the eigenstates, therefore STIRAP can be carried out between the three lowest levels (see Fig. 1b). Moreover at ng0 = 1/2 the lowest decoherence rates are obtained. However in this case selection rules impede the operation of the Λ scheme 8 . This problem is circumvented by working slightly away, e.g. at ng0 = 0.45. Here the Hamiltonian (3) is more complicated than the ideal Eq.(1) but still two a.c. signals with slightly detuned frequencies ωg , ωu are applied to the gate (see Fig. 1b) produce nearly complete population transfer |g → |u. In Fig. 2b–d the numerical solution of the Schr¨ odinger equation for the Hamiltonian (3) is reported (solid lines). Note that there are many parameters that may be used to optimize the eﬃciency such as duration, delay, relative height and over-all shape of the pulses, the detunings etc. 9 . 3. Eﬀects of Decoherence The functionality of coherent nanodevices is sensitive to various (device-dependent) decoherence sources. In the quantronium, high-frequency noise and low-frequency noise coexist. The former is mainly responsible for unwanted transitions which determine exponential decay of the signal in time, whereas the latter mainly makes unstable the calibration of the device and determines the initial power-law decay 15 . Based on the above picture we discuss the feasibility of the protocol. The key observation is that strong unwanted processes involving the level |e have negligible eﬀects on the protocol. Decoherence is mainly determined by processes involving |g and |u, which have been well characterized in the Quantronium and, as a matter of fact, allow for decoherence times as large as τR 300 ns.

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High-frequency noise is analized by the quantum-optical master equation ρ˙ = − Γρ where ρ is the density matrix and H is the Hamiltonian (3) in the rotating frame 16 . At low temperature the dissipator Γρ includes spontaneous decay and environment-assisted absorption between eigenstates in the presence of the laser coupling and in the basis {|g, |u, |e} reads i [ρ, H ]

(Γρ)ij =

γi + γj ρij − (1 − δij )˜ γ ρij − δij ρkk γk→i 2

(4)

k

where γi = k=i γi→k . The dissipator is taken time-independent (which overestimates decoherence) and includes all transitions as well dephasing rates γ˜ accounting phenomenologically for low-frequency noise. For the second excited state we assume γe = γe→u + γe→g = 2γu where γu and γ˜u not smaller than those observed in the experiments 15 are used. As in quantum-optical systems most of the decay rates act on depopulated states and are ineﬀective. In the Quantronium there is the extra rate γu→g which only changes the population during the waiting time after completion of the pulse sequence (dashed lines in Figs. 2b,2c,2d). Low-frequency noise is due to impurities which can be considered as static during each run of the protocol but switch on a longer time scale, leading to statistically distributed level separations. In protocols as Ramsey interference they result in a distribution of oscillating frequencies whose average determines defocusing of the 1.0

a)

0.03 0.02 0.01 0.00

Ag (t)

A u(t) 40002000

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2000 4000

−4

ee

0.02 0.00

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0.06

ee

ρ (t) gg

0.8

uu

uu

0.04

ρ (t)

pulse amplitude

0.05

40002000

0

2000 4000

time t

Figure 2. Population transfer by STIRAP in the Quantronium. Eq. (3) for ng0 = 0.45 is truncated to eight charge states. a) Gaussian pulses ng (t) = ng0 + Ag (t) cos ωg t + Au (t) cos ωu t with zero detuning are applied in the counterintuitive scheme. The maximum gate charge of the microwave ﬁelds are max (Au (t), Ag (t)) = 0.05 1/2. For a charging energy of EC = 50 μeV the time unit corresponds to about 1.3 × 10−11 s. b)–d) Time evolution of the populations ρgg , ρuu , ρee for the isolated system (solid lines) for initial state |g. The arrows denote the ﬁnal populations which shows that selection rules and the presence of more than three levels, and of diagonal and counterrotating drives still allow complete population transfer. We include quantum noise ˜u = 2.6 × 10−4 in EC (dashed lines) using decay rates γu = 4.4 × 10−5 and a dephasing rate γ units, corresponding to a dephasing time of about 50 ns. The only remarkable eﬀect is extra decay |u → |g at the end of the protocol.

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(b)

E

(a)

1

E

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-0.5

-0.5

ρgg

-50

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100

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t

ρee

0.2 -150 -100

-50

50

100

150

t

Figure 3. Eﬀect of static impurities on population transfer by STIRAP. (a) Moderate noise amplitude produce crossings of the instantaneous levels (upper panel) however Zener tunneling still allows almost complete population transfer. (b) Larger noise amplitude produces level repulsion and the ﬁnal state is a superposition of |g and |u.

signal 15 . For STIRAP the eﬀect of low-frequency noise is quite diﬀerent since on one hand it does not produce defocusing but on the other hand it may change drastically the structure of the instantaneous levels. We now analize the eﬀect of averaging over static impurities conﬁgurations, corresponding to statistically distributed δEi . They translate in the rotating frame in ﬂuctuations of the detunings. Even large δEe hardly aﬀect STIRAP since they still leave equal detunings of both microwave ﬁelds. Therefore changes determined in the Hamiltonian Eq.(1) do not aﬀect the structure of the spectrum and the STIRAP protocol can be carried out as in the ideal case. On the other hand, ﬂuctuations of Eu − Eg are potentially detrimental since they translate in ﬂuctuations of the diﬀerence of detunings. As a result instantaneous eigenstates cross during the procedure. However as long as the system is well protected from noise Zener tunneling still allows almost complete population transfer, whereas a noise level typical of charge qubits determines repullsion of the instantaneous level leading eventually the system to a wrong ﬁnal state (Fig. 3). 4. Conclusion The above picture leads to a quantitative analysis showing that STIRAP should be well within reach of present-day technology for superconducting nanocircuits. Moreover eﬃciency can be improved also in the presence of ﬂuctuations, for instance by using larger ﬁeld amplitudes. An important advantage of STIRAP is that the eﬃciency almost does not depend on details of the procedure and on the timing, which makes it robust against

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moderate ﬂuctuations of the solid-state environment. Another advantage is its ﬂexibility, for intance it is possible to change the driving frequencies 9 instead of changing the ﬁeld amplitudes. One might hope to apply this technique for the preparation of peculiar quantum states. One such application is the generation of Fock states in a cavity coupled to a three-level atom 10 . To this end the Cooper-pair box has to be coupled to a harmonic oscillator degree of freedom. This Hamiltonian can be implemented using electrical resonators 17 , transmission lines 2 , and nanomechanical resonators 5,6 . References 1. E. Collin et al., Phys. Rev. Lett. 93, 157005 (2004); I. Chiorescu et al., Nature 431, 159 (2004). 2. A. Wallraﬀ et al., Nature 431, 162 (2004). 3. G. Falci et al., Nature 407, 355 (2000); L. Faoro, J. Siewert, and R. Fazio, Phys. Rev. Lett. 90, 028301 (2003); M. Cholascinski, Phys. Rev. B 69, 134516 (2004). 4. J. Jones, V. Vedral, A.K. Ekert, C. Castagnoli, Nature, 403, 869 (2000); P. Zanardi, and M. Rasetti, Phys. Lett. A, 264, 94 (1999). 5. F. Marquardt and C. Bruder, Phys. Rev. B 63, 054514 (2001); A.D. Armour, M.P. Blencowe, and K.C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). 6. I. Martin, A. Shnirman, L. Tian, and P. Zoller, Phys. Rev. B 69, 125339 (2004); P. Zhang, Y.D. Wang, and C.P. Sun, Phys. Rev. Lett. 95, 097204 (2005). 7. K.V.R.M. Murali et al., Phys. Rev. Lett. 93, 087003 (2004). 8. M.H.S. Amin, A.Y. Smirnov, and A. Maassen van den Brink, Phys. Rev. B 67, 100508(R) (2003). J. Siewert and T. Brandes, Adv. Solid State Phys. 44, 181 (2004). E. Paspalakis and N.J. Kylstra, J. Mod. Optics 51, 1679 (2004); Y.-X. Liu, J.Q. You, L.F. Wei, C.P. Sun, and F. Nori, Phys. Rev. Lett. 95, 087001 (2005). 9. K. Bergmann, H. Theuer, and B.W. Shore, Rev. Mod. Phys. 70, 1003 (1998); N.V. Vitanov, T. Halfmann, B.W. Shore, and K. Bergmann, Annu. Rev. Phys. Chem. 52, 763 (2001). 10. A.S. Parkins, P. Marte, P. Zoller, H.J. Kimble, Phys. Rev. Lett. 71, 3095 (1993); M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, Phys. Rev. Lett. 85, 4872 (2000). 11. D. Vion et al., Science 296, 886 (2002). 12. Y. Makhlin, G. Schoen, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001); M. Mariantoni et al., e-print cond-mat/0509737 (2005). 13. M.O. Scully and M.S. Zubairy: Quantum Optics (Cambridge Univ. Press, Cambridge 1997). 14. M.A. Kmetic, R.A. Thuraisingham, and W.J. Meath, Phys. Rev. A 33, 1688 (1986). 15. E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett. 88, 228304 (2002); G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino , Phys. Rev. Lett. 94, 167002 (2005); G. Ithier et al., e-print condmat/0508588 (2005). 16. A. Kuhn, M. Hennrich, T. Bondo, and G. Rempe, Appl. Phys. B 69, 373 (1999). 17. F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003).

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CROSSED ANDREEV REFLECTION-INDUCED GIANT NEGATIVE MAGNETORESISTANCE∗

F. GIAZOTTO, F. TADDEI, AND F. BELTRAM NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy E-mail: [email protected] R. FAZIO NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy and International School for Advanced Studies (SISSA), Trieste, I-34014, Italy

We show that very large negative magnetoresistance can be obtained in magnetic trilayers in a current-in-plane geometry owing to the existence of crossed Andreev reﬂection. This spin-valve consists of a thin superconducting ﬁlm sandwiched between two ferromagnetic layers whose magnetization is allowed to be either parallelly or antiparallelly aligned. For a suitable choice of structure parameters and nearly fully spin-polarized ferromagnets the magnetoresistance approaches −80%. Our results are relevant for the design and implementation of spintronic devices exploiting ferromagnet-superconductor structures.

1. Introduction Giant Magneto Resistance (GMR) is the pronounced response in the resistance of magnetic multilayers to an applied magnetic ﬁeld 1,2,3,4,5 . A magnetic multilayer consists of an alternating sequence of ferromagnetic (F) and non-magnetic layers (N). The relative orientation of magnetic moments in the F layers can be driven from antiparallel (AP), in the absence of external ﬁeld, to parallel (P), with a small (up to some hundreds of Oe) magnetic ﬁeld. GMR was originally demonstrated 5 in Fe/Cr multilayers with current ﬂowing parallel to the planes, the so-called current-inplane (CIP) conﬁguration. In the CIP measurement the magneto-resistance (MR) ratio, deﬁned as the maximum relative change in resistance resulting from applying the external ﬁeld, is typically around 10% for a number of layers of the order of 50 − 100 5 . These values can be increased up to ∼ 100% in the case of current ﬂow perpendicular to the multilayer plane (CPP conﬁguration) 6 . Here we show that the limitations of the CIP conﬁguration can be overcome by employing a superconductor (S) in the non-magnetic portion of the multilayer. ∗ This

work was partially supported by MIUR under FIRB “Nanotechnologies and Nanodevices for Information Society”, contract RBNE01FSWY.

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tF

lead

tS tF

insulator lead

y z

F1

S

lead insulator

F2

lead

w

L x

F1

S

(a)

2Δ

P

2Δ

AP

F2

F1

(b)

S

F2

Figure 1. Sketch of the FSF spin-valve. The schematic representation of the spin-valve eﬀect for half-metallic ferromagnets, showing the diagrams of the superconducting density of states, is displayed in (a) and (b). (a) In the P alignment, the lack of quasiparticles with opposite spin hinders the condensation of two electrons injected from the ferromagnets in a Cooper pair in S. As a consequence, the electric transport is conﬁned within the F layers. (b) In the AP conﬁguration, two electrons with opposite spin injected from the F layers can form a Cooper pair within the superconductor thanks to crossed Andreev reﬂection, thus “shunting” the current through the whole structure (see text).

2. Operation and discussion The structure we envision (see Fig. 1) consists of two identical diﬀusive ferromagnetic layers (F1 and F2 ), of thickness tF , separated by a (s-wave) superconducting layer of thickness tS . The layers are assumed to be in good metallic contact and have length L and width w. The magnetization of the two ferromagnets (white arrows) is allowed to be aligned either in a parallel or an antiparallel conﬁguration. The trilayer is connected to ferromagnetic leads separated by an insulating layer (light-gray regions in Fig. 1) of the same thickness as the S layer. The magnetization of the upper F leads is equal to the one relative to layer F1 , and analogously for the lower F leads. In the CIP conﬁguration, charge transport in the system is dominated by crossed Andreev reﬂection (CAR) leading to a dramatic enhancement of the magnetoresistance. Let us ﬁrst describe qualitatively the principle of operation of the present spinvalve. For the sake of clarity, let us ﬁrst consider a half-metallic (i.e., with only one spin species) ferromagnet 7 in good metallic contact with a S layer. Quasiparticles with energy below the superconductor gap can be transferred into the superconductor as Cooper pairs only through an Andreev reﬂection (AR) process 8 . The latter consists of a coherent scattering event in which a spin-up(down) electron-like quasiparticle, originating from the F layer, is retroreﬂected at the interface with the superconductor as a spin-down(up) hole-like quasiparticle into the ferromagnet. Since only quasiparticles (electron- and hole-like) of one spin type exist in the fer-

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romagnet, no current can ﬂow between the F and S layers 9 . Similarly, in the case of the FSF trilayer in the P conﬁguration (see Fig. 1(a)), the two F layers cannot transfer charge into the superconductor. Current is conﬁned to the F layers and it consists of fully-polarized quasiparticles. If the S layer is thin enough quasiparticles can also tunnel through it (this will occur for tS values up to some superconductor coherence lengths (ξ0 )). In the AP conﬁguration (see Fig. 1(b)), each of the two F layers can contribute separately to the quasiparticle current through the structure just like in the P conﬁguration. More importantly CAR does take place. In this case a Cooper pair is formed in the superconductor by a spin-up electron originating from the F1 layer and a spin-down electron from the F2 layer. In the AR language, this can be described as the transmission of a spin-up electron-like quasiparticle from one of the F layers to a spin-down hole-like quasiparticle in the other F layer. This is now possible since the quasiparticles involved belong to the majority spin species in each of the two layers. A charge current can therefore ﬂow through the S layer as a supercurrent, thereby shunting the conduction channels in the ferromagnets. This contribution to the current will dominate at least when the structure is long enough and the quasiparticle contribution in the F layers becomes negligible (note that the conductance of each F layer in the diﬀusive regime is proportional to /L, where L is the mean free path). As a result, one can expect the conductance GAP of the AP conﬁguration to be much larger than the conductance GP of the P conﬁguration. This can give rise to a large, negative value of the MR ratio, deﬁned as: GP − GAP . (1) MR = GP A simple expression for the MR ratio for half-metallic ferromagnets in the diffusive regime can be derived as follows. In the P conﬁguration, the conductance is approximately given by 9 e2 (2) N↑ , h L i.e., it is proportional to the number N↑ of open channels for spin-up electrons of each F layer, and inversely proportional to L. In the AP conﬁguration, the conductance can be roughly separated in two contributions. One (G∗ ), due to CAR, is virtually independent of L. The other comes from the direct transmission of quasiparticles (proportional to (2e2 /h)(/L)N↑ ): GP 2

e2 N↑ , hL with α being a numerical factor ∼ 1. As a result, GAP G∗ + α 2

MR 1 − α − G∗

h L 1 , 2e2 N↑

(3)

(4)

negative and large for L . This is in contrast to what expected in a FNF trilayer, where the MR value is positive 5 since the AP conﬁguration yields a reduction of

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the structure conductance. For non half-metallic ferromagnets, the charge current will still be dominated by CAR, but the eﬀect will be reduced. This qualitative understanding of the eﬀect can be validated by a numerical calculation of the conductance, which was performed within the Landauer-B¨ uttiker scattering approach. In the presence of superconductivity, the zero-temperature and zero-bias conductance can be written as 10 G = G↑ + G↓ , where e2 Rσ Rσ − Taσ Taσ Gσ = (5) T σ + Taσ + 2 σ a aσ h Ra + Ra + Taσ + Taσ is the spin-dependent conductance. In Eq. 5, T σ (Taσ ) is the spin-dependent normal (Andreev) transmission probability for quasi-particles injected from the left lead and arriving on the right lead, while Rσa is the Andreev reﬂection probability for quasi-particles injected from the left lead. Similarly, Taσ and Rσa are the Andreev scattering probabilities for quasiparticles injected from the right lead. e is the electron charge and h is the Planck constant. The scattering amplitudes were evaluated numerically by making use of a recursive Green’s function technique based on a tight-binding version 11 of the Bogoliubov-de Gennes equations u H Δ u =E , (6) ∗ ∗ v Δ −H v where H is the single-particle Hamiltonian, and u (v) is the coherence factor for electron- (hole)-like excitations of energy E, measured from the condensate chemical potential μ. Within the tight-binding description, H and Δ are matrices with elements (H)ij = i δij − γδ{i,j} and (Δ)ij = Δi δij , where i is the on-site energy at site i, γ is the hopping potential and Δi is the superconducting gap ( {...} stand for ﬁrst-nearest-neighbor sites). In particular, i = S in the S region, i = I in the insulating barrier, and i = F = S ∓ hexc in the F layers, hexc denoting the ferromagnetic exchange energy, with upper (lower) sign referring to majority (minority) spin species. Δi is assumed to be constant and equal to zero-temperature gap (Δ0 ) in the S region, and zero everywhere else. Note that this is realistic when the S layer thickness is larger than ξ0 12 . Furthermore, disorder due both to impurities and lattice imperfections is introduced by the Anderson model, i.e., by adding to each on-site energy a random number chosen in the range [−U/2, U/2], being U a fraction of the Fermi energy. In what follows we shall indicate energies in units of Δ0 , and lengths in units of the lattice constant a (of the order of the Fermi wavelength). In order to analyze the behavior of conductances and MR as a function of the various parameters we used a two-dimensional (2D) model of the structure so that the F layers are in the diﬀusive regime 13 . To avoid a self-consistent calculation of the superconducting gap, we limited our analysis to values of tS ≥ 30 (corresponding to 3.3ξ0 ) 12 . In addition, the conductance was calculated performing an ensemble average over 100 realizations of disorder.

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Figure 2. (a) Conductance in the P (circles) and AP (triangles) conﬁgurations versus tS with tF = 5. (b) Resulting MR ratio. Data were obtained assuming L = 150, P = 100%, and U = 8 (see text). In (a) the error bars correspond to the standard error over all disorder conﬁgurations. Lines are guides to the eye.

The conductance and MR dependence on S layer thickness is shown in Fig. 2. Here we chose the ferromagnetic thickness tF = 5 and hexc = 20 (the mean free path turns out to be 21). For this latter value the ferromagnet polarization (P) 14 is equal to 100%. Figure 2(a) shows that in the P conﬁguration the conductance GP is initially slightly decreasing and roughly constant for tS ≥ 5.5ξ0 . This is due to the fact that quasiparticles in the two F layers (for large enough tS values ≥ 5.5ξ0 ) are decoupled, but some direct tunneling can occur through thinner S layers. In the AP conﬁguration, the conductance GAP decreases until the value tS 8.5ξ0 is reached, and thereafter remains almost constant. Such a behavior is expected since, on the one hand, for tS of the order of some ξ0 the conductance is dominated by the supercurrent (mediated by CAR between the F1 and F2 layers). On the other hand, by increasing tS , the two F layers tend to decouple and the current through the structure is only due to quasiparticles ﬂowing separately through them, independently of tS . The resulting MR ratio is shown in Fig. 2(b) and exhibits very large negative values around −70% for tS 3.5ξ0 and about −25% for tS 6.5ξ0 . It is noteworthy to mention that when the S layer is in the normal state (i.e., a FNF trilayer) MR (0.7 ± 1.8)% for tS = 4.5ξ0 . The role the F layers thickness on the conductance and magnetoresistance is analyzed in Fig. 3, for ﬁxed tS = 40 and P = 100%. Figure 3(a) shows that the conductance in the P alignment increases linearly with tF according to the estimate in Eq. 2. In the AP conﬁguration the conductance is again linear in tF with the same slope, but it is shifted upwards as compared to GP . This is in agreement with Eq. (3): the diﬀerence GP − GAP ∼ G∗ . As a consequence, the MR ratio (see Fig. 3(b)) starts from −70% at tF 0.5ξ0 and thereafter decreases by increasing the value of tF .

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Figure 3. Conductance in the P (circles) and AP (triangles) conﬁgurations versus tF with tS = 40. (b) Resulting MR ratio. The same parameters as for Fig. 2 were used.

3. Conclusion In conclusion, we have investigated theoretically spin transport in a ferromagnetsuperconductor-ferromagnet trilayer in the current-in-plane geometry. We showed that very large and negative magnetoresistance values (around −80%) can be achieved. Such an eﬀect relies entirely on the existence of crossed Andreev reﬂection. The results presented here are promising in light of the implementation of novel-concept magnetoresistive devices such as, for instance, spin-switches as well as magnetoresistive memory elements. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

G. A. Prinz, Science 282, 1660 (1998). J.-Ph. Ansermet, J. Phys.: Condens. Matter 10, 6027 (1998). S. A. Wolf et al., Science 294, 1488 (2001). ˇ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). I. Zuti´ M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988). W. P. Pratt et al., Phys. Rev. Lett. 66, 3060 (1991). J. M. D. Coey and M. Venkatesan, J. Appl. Phys. 91, 8345 (2002). A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964). M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett. 74, 1657 (1995). C. J. Lambert, J. Phys.: Condens. Matter 5, 707 (1993); C. J. Lambert and R. Raimondi J. Phys.: Condens. Matter 10, 901 (1995). S. Sanvito et al., Phys. Rev. B59, 11936 (1999). C.-Y. You et al., Phys. Rev. B70, 014505 (2004). For further details, see F. Giazotto, F. Taddei, F. Beltram, and R. Fazio, condmat/0512090. The ferromagnet bulk polarization is defined as P = hexc /(4γ − S ).

Quantum Modulation of Superconducting Junctions

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ADIABATIC PUMPING THROUGH A JOSEPHSON WEAK LINK

F. TADDEI1 , M. GOVERNALE2 , ROSARIO FAZIO1,3 , AND F. W. J. HEKKING4 1

NEST CNR-INFM & Scuola Normale Superiore, I-56126 Pisa, Italy 2 Institut f¨ ur Theoretische Physik III, Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany 3 International School for Advanced Studies (SISSA) via Beirut 2-4, I-34014 Trieste, Italy 4 LPMMC, CNRS & Universit´e Joseph Fourier, BP 166, 38042 Grenoble CEDEX 9, France We present a formalism to study adiabatic pumping through a superconductor - normal - superconductor weak link. At zero temperature, the pumped charge is related to the Berry phase accumulated, in a pumping cycle, by the Andreev bound states. We analyze in detail the case when the normal region is short compared to the superconducting coherence length. The pumped charge turns out to be an even function of the superconducting phase diﬀerence. Hence, it can be distinguished from the charge transferred due to the standard Josephson eﬀect.

1. Introduction In a mesoscopic conductor a dc charge current can be obtained, in the absence of applied voltages, by cycling in time two parameters which characterize the system 1,2 . This transport mechanism is called pumping. If the time scale over which the time-dependent parameters vary is large compared to the typical electron dwell time of the system, then pumping is adiabatic, and the pumped charge does not depend on the detailed timing of the cycle, but only on its geometrical properties. Diﬀerent formulations have been developed to describe adiabatic pumping, for example, based on scattering theory in Refs. 3,4,5 or Green’s function methods in Refs. 6,7 . In the scattering approach the pumped charge per cycle can be expressed in terms of derivatives of the scattering amplitudes with respect to the pumping parameters (Brouwer’s formula 3 ). This formulation requires the presence of terminals which provide propagating channels. The scattering approach has been later extended to hybrid systems containing superconducting (S) terminals. In Refs. 8,9 a two terminal structure comprising one superconducting lead was considered. Subsequently, this approach was generalized to multiple-superconducting-lead systems, where at least one normal lead is present 10 . The presence of a normal lead is essential for generalizing Brouwer’s formula to these hybrid structures. If only superconducting leads are present, at low enough temperature, pumping is due to the adiabatic transport of Cooper pairs. Besides the dependence of the pumped charge on the cycle, in the superconducting pumps there is a dependence on

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Figure 1. Schematic setup for the SNS pump. The weak link is composed of a scattering region contacted to the S terminals through two ﬁctitious N leads. The transport properties of the scattering region (characterized by the matrix S0 ) are cycled in time via two external gates which control the parameters X1 and X2 . The superconducting order parameters of the S terminals on the left-hand-side and on the right-hand-side have equal magnitude Δ0 , but diﬀer by a phase ϕ.

the superconducting phase diﬀerence(s) (the overall process is coherent). Moreover, in addition to Cooper-pair pumping, in equilibrium, there is a contribution due to the Josephson eﬀect if the phase diﬀerence between the two superconducting leads is diﬀerent from zero. In this work we consider adiabatic pumping between two superconducting terminals connected through a normal (N) region. We focus on the regime of an open structure (SNS weak link), where charging eﬀects are negligible. The derivation of a formula for the pumped charge makes use of the connection between Berry’s phase 11 and the pumped charge 12,13,14 , which we prove to be valid also for the SNS weak link. The resulting expression for the charge pumped in a period can be written in terms of derivatives of the Andreev-boundstate wavefunctions with respect to the pumping parameters. We point out that there is a close analogy of the problem studied here with that of pumping in a Aharonov-Bohm ring 15 . A diﬀerent mechanism for charge pumping in a voltage-biased superconductor/insulator/normal-metal/insulator/superconductor long junction which exploits both the Andreev bound-state dynamics and the abovegap continuum of quasi-particle states has been proposed in Ref. 16 . 2. SNS pump The system under investigation (depicted in Fig. 1) consists of a SNS junction, with the weak link occupying the region −W/2 < x < W/2. The superconducting order parameter is given by Δ0 e−iϕ/2 and Δ0 eiϕ/2 for the superconductor on the left-hand-side and right-hand-side, respectively. The properties of the weak link can be varied, for example, by realizing it with a semiconductor and operating on two independent external gates, indicated by X1 (t) and X2 (t) in the ﬁgure. The state of such a hybrid structure |ψ(t) is the solution of the time-dependent Bogoliubov-de Gennes equation: i∂t |ψ(t) = H(t)|ψ(t),

(1)

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where the Hamiltonian H(t) =

2

2 ∇ + U (r, t) − μ − 2m Δ(r)e−iφ(r)

Δ(r)eiφ(r) 2 2 r , t) + μ 2m ∇ − U (

(2)

depends on time through the two parameters: H(t) = H[X1 (t), X2 (t)]. In Eq. (2) U (r, t) is the potential that takes into account the eﬀect of the time-varying external gate voltages, φ(r) = ϕ/2[θ(x − W/2) − θ(−x − W/2)], Δ(r) = Δ0 /2[θ(x − W/2) + θ(−x − W/2)] and μ is the superconductor chemical potential (equal for the two S leads). We now assume that the state |ψ(t) evolves adiabatically and that at any time t it is in an instantaneous eigenstate of the Hamiltonian. The instantaneous solutions are deﬁned by the equation: ˜ ˜ H(t)|ψ(t) = (t)|ψ(t),

(3)

whereby t plays the role of a parameter. After a cycle of period τ , the states returns to the initial one, but with an added phase factor Φ: |ψ(τ ) = eiΦ |ψ(0).

(4)

The phase Φ contains both a geometrical (Berry’s) and a dynamical contribution τ ˜ ψ ˜ Φ = γB − γD . The dynamical phase is simply given by γD = 1 0 dt ψ|H| For the SNS system, where Andreev bound states are formed, the condition of validity of the adiabatic approximation is that the frequency ω of the timedependent parameters be much smaller than the energy diﬀerence between any pair of Andreev bound states, or between any Andreev bound states and the continuum of states above the gap. This implies that the pumping frequency needs at least to be smaller than the superconducting gap Δ0 . It is possible to show explicitly that the charge current Jch carried by a ˜ is given by the expectation value of the derivaBogoliubov-de Gennes eigenstate |ψ tive of the Hamiltonian H with respect to the superconducting phase diﬀerence 17 : Jch =

2e ˜ ∂H ˜ ψ| |ψ. ∂ϕ

(5)

∂ (γD − γB ). ∂ϕ

(6)

τ The charge transferred per cycle is then deﬁned as Q = 0 Jch (t) dt. By assuming adiabatic evolution of the state, and making use of Eq. (5), the following relation between the accumulated phase and the charge transferred in a cycle can be written: Q = 2e

The ﬁrst term corresponds to the instantaneous Josephson current integrated over one period, while the second represents the pumped charge. Using Green’s theorem, γB can be written in terms of derivatives with respect to the pumping parameter of the instantaneous eigenfunctions:

∂ ψ˜ ∂ ψ˜ , (7) γB = −2 dX1 dX2 Im ∂X1 ∂X2 S

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S being the area in the parameter space spanned by the parameters over one cycle. In Eq.† (7) the notation · |· stands for a space integration deﬁned by A |B = dr A (r)B(r), A and B being vectors in the Nambu space. In the short junction limit (i.e. when the distance W between the two superconducting interfaces is much smaller than the superconducting coherence length) only the superconducting regions contribute to the space integration in Eq. (7). Making use of the Andreev approximation (for details see Ref. 18 ) the pumped charge reduces to

β† 1 ∂bν ∂bβν ∂ dX1 dX2 Im Qp = 4e + ∂φ S 2Imk + ∂X1 ∂X2 β=+,− ν=L,R

β + + ∂bβ† 1 ν β† ∂bν ∂k β ∂k Re b + b ν ν (2Imk + )2 ∂X2 ∂X1 ∂X1 ∂X2

,

(8)

where we have assumed a single Andreev bound state and a single open channel with particle-like wave vector k + . This is the central result of this work, and a few comments are in order. We have succeeded in expressing the pumped charge as a function of the instantaneous Andreev-bound-state eigenfunction. The vectors ± in the left b± L(R) are the amplitudes of the decaying waves with wave vectors k (right) superconductor. They depend only on the parameters of the system, such as the normal region scattering matrix S0 , the superconducting gap, and the superconducting phase diﬀerence. It is clear that the pumped current can be written in terms of the elements of the normal-region scattering matrix S0 . Now let us consider the following parametrization for the normal-conductor scattering matrix iα √ √ e 1−g i g √ S0 = , (9) √ e−iα 1 − g i g choosing g and α as pumping ﬁelds (X1 = g, and X2 = α). The instantaneous Andreev-bound-state energy 0 is simply related to the transmission probability g by 19 ϕ (10) 0 (t) = Δ0 1 − g(t) sin2 ( ), 2 τ so that γD = 1/ 0 0 (t). The charge transfered due to the Josephson current (in the following named also Josephson charge) reads ϕ ϕ 2π ω Δ0 g(t) sin 2 cos 2 dt (11) QJos = −2e . 0 2 1 − g(t) sin2 ϕ 2

Notice that it depends on the pumping frequency ω. On the other hand, the pumped charge does not depend on the pumping frequency, but only on the geometrical

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properties of the pumping cycle, and it reads

∂ ψ˜ ∂ ψ˜ ∂ . Qp = 4e dgdα Im ∂ϕ ∂g ∂α S

(12)

Interestingly, the integrand of Eq. (12) turns out to be independent of α. We now consider the following sinusoidal pumping cycle: g(t) = g¯ + Δg sin(ωt) and α(t) = α ¯ + Δα sin(ωt + φ0 ). In the weak pumping limit we assume that Δg/¯ g 1 so that the integrand of Eqs. (11) and (12) vary negligibly during the cycle. As far as the frequency is concerned, the maximum value of ω in order for

the adiabatic hypothesis to hold is ω < Δ0 − ¯0 , with ¯0 = Δ0 1 − g¯ sin2 (ϕ/2). In order to compare Qp with QJos , we choose ω = 0.1(Δ0 − ˜0 ), ˜0 being equal to the value of ¯0 at ϕ = π/2. Note that the adiabatic condition breaks down for ϕ = 0 or g¯ = 0, when the Andreev bound state is at the gap boundary. Thus in our analysis we shall avoid small values of those variables. Figure 2 shows the pumped charge as a function of the superconducting phase diﬀerence ϕ for diﬀerent values of g¯. Qp is a non-monotonous function of ϕ exhibiting a maximum at ϕ = ±π. For comparison, in the inset of Fig. 2, we plot the transferred charge due to the Josephson current, whose absolute value is larger, with respect to the pumped current, by a factor of order ω/Δ0 . However, while Qp (ϕ) is an even function of ϕ, QJos (ϕ) is odd, so that a measure of [Q(ϕ) + Q(−ϕ)]/2 will single out only the pumped contribution. The particular parity of Qp is due to the fact that a time-reversal operation implies not only the reversal of phase but also of the pumping trajectory in parameter space. As far as detection is concerned, we wish to stress that for realistic parameters, using Al as superconductor, one can attain sizable pumped currents of the order of 5 nA. A sensitive setup to currents of even smaller size can be realized by inserting the SNS pump in a arm of a SQUID. 3. Conclusion In conclusion, we have presented a formalism to study adiabatic charge pumping in a SNS weak link. The pumped charge is related to the Berry’s phase accumulated in a pumping cycle by the Andreev bound-state wavefunctions, which can be written as a function of the scattering matrix of the normal region. In the short junction limit, the pumped charge is even with respect to the superconducting phase diﬀerence. Hence, it can be easily distinguished from the charge transferred by the Josephson current. Acknowledgments We acknowledge support from Institut Universitaire de France (F.W.J.H.) and from EC through grants EC-RTN Nano, EC-RTN Spintronics and EC-IST-SQUIBIT2 (M.G., F.T. and R.F.).

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8

QJos (2e)

50

Qp (2e)

6

4

0 -50 -π

2

0

-π/2

ϕ

π/2

π

0

-2-π

-3/4π

-π/2

-π/4

0

ϕ

π/4

π/2

3/4π

π

Figure 2. Pumped charge, in units of 2e, as a function of superconducting phase diﬀerence ϕ computed for a sinusoidal weak-pumping cycle. Diﬀerent lines refer to diﬀerent values of average transmission: solid line g¯ = 0.5; dashed line g¯ = 0.7; and dot-dashed line g¯ = 0.9. For the pumping cycle we have chosen Δg = 0.1, Δα = 2π and ω = 0.1Δ0 (1 − 1 − g¯/2). The pumped charge is not plotted for very small values of ϕ where the adiabatic condition is not fulﬁlled. In the inset the Josephson charge is plotted as a function of ϕ for the same parameters of the main panel.

References D.J. Thouless, Phys. Rev. B27, 6083 (1983). M. Switkes, C.M. Marcus, K. Campman, and A.C. Gossard, Science 283, 1905 (1999). P.W. Brouwer, Phys. Rev. B58, R10135 (1998). Yu. Makhlin and A.D. Mirlin, Phys. Rev. Lett. 87, 276803 (2001). M. Moskalets, and M. B¨ uttiker, Phys. Rev. B66, 035306 (2002). F. Zhou, B. Spivak, and B. Altshuler, Phys. Rev. Lett. 82, 608 (1999). O. Entin-Wohlman, A. Aharony, and Y. Levinson, Phys. Rev. B65, 195411 (2002). J. Wang, Y. Wei, B. Wang, and H. Guo, Appl. Phys. Lett. 79, 3977 (2001). M. Blaauboer, Phys. Rev. B65, 235318 (2002) F. Taddei, M. Governale, and R. Fazio, Phys. Rev. B70, 052510 (2004). M.V. Berry, Proc. R. Soc. London A392, 45 (1984). J.E. Avron, A. Elgart, G.M. Graf, and L. Sadun, Phys. Rev. B62 R10618 (2000). M. Aunola and J.J. Toppari, Phys. Rev. B68, 020502 (2003). A. Bender, F.W.J. Hekking, and Yu. Gefen, (unpublished). M. Moskalets, and M. B¨ uttiker, Phys. Rev. B68, 075303 (2003). N. B. Kopnin, A. S. Mel’nikov, and V. M. Vinokur, Phys. Rev. Lett. 96, 146802 (2006). N. I. Ludin, L. Y. Gorelik, R. I. Shekhter, M. Jonson, and V. S. Shumeiko, Superlattices and Microstructures 25, 937 (1999). 18. M. Governale, F. Taddei, Rosario Fazio, and F. W. J. Hekking, Phys. Rev. Lett. 95, 256801 (2005). 19. C.W.J. Beenakker, in Transport Phenomena in Mesoscopic Systems, Eds. H. Fukuyama, and T. Ando, Springer Series in Solid-State Sciences, vol. 109 (SpringerVerlag, Berlin Heidelberg 1992), p. 235.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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SQUEEZING OF SUPERCONDUCTING QUBITS

K. SHIOKAWA1,2 AND F. NORI1,3 1

Digital Material Laboratory, Frontier Research System, The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama 351-0198, Japan 2 Department of Physics, University of Maryland, College Park, MD 20742, USA 3 Center for Theoretical Physics, Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA We investigate squeezing of a multiqubit system made up of Josephson juctions. We extend the concept of atom-cavity QED squeezing to an array of qubits. Squeezing allows probing multipartite entanglement shared among qubits in a fully dynamical setting, which is a crucial step toward precision measurement in quantum metrology and future implementations of quantum computing. We describe how to achieve this within a quantum computing scheme based on superconducting qubits in a cavity.

1. Squeezing of a collective spin Squeezing has been a common technique frequently used in atom optics in order to improve the precision limited by quantum mechanics by reducing quantum noise in one of the conjugate variables less than the value restricted by the standard quantum limit1 . Squeezing has been studied in various applications from optical communications to gravitational wave detection. Atomic spin squeezing has been used to reduce the noise and improve the sensitivity in Ramsey interferometer crucial for atom clocks2 . It is also shown to improve the sensitivity in magnetometry measurement3 . In the ﬁeld of quantum computing, a two-level quantum system, a qubit is used as a building block of computation. Universality argument implies realization of one or two qubit gates are essential for quantum computing. In addition, realistic consideration taking into account decoherence urges us to make a whole computation time as short as possible by performing multiqubit gates. Realization of entangled state among multiqubit system is one crucial step for the implementation of quantum gates for quantum computing. The spin squeezing parameter can be deﬁned similarly to the optical squeezing case2,4 . This is essentially the degree of precision improvement in measurement process compared to value given by the standard quantum limit. In particular, it is directly measurable experimentally. It has been shown that a spin squeezed state is quantum mechanically an entangled state. The squeezing parameter gives the suﬃcient condition for entanglement within many spins. In particular for a

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pairwise entanglement for symmetric states, it gives the necessary and suﬃcient condition. Let Jj be spin operators that satisfy the commutation relation for SU (2) spin: states [Jx , Jy ] = iJz and permutation among these variables. The coherent spin with average spin pointing along z-direction have uncertainties ΔJx = ΔJy = J/2 where J is a total spin number. This will set the standard quantum limit for collective spin measurements. Now we introduce the spin squeezing parameter as the performance factor indicating how much improvement in sensitivity we achieve by taking a ratio between the minimum uncertainty achieved on the plane perpendicular to the direction of the average spin (say along x-axis) and the uncertainty given by the standard quantum limit. ζ2 ≡

2(ΔJx )2 J

(1)

ζ2 ≡

2(ΔJx )2 , J

(2)

or more generally,

where the vector x points in the direction of minimum uncertainty normal to the mean spin. We deﬁne a spin-squeezed state as a state that satisﬁes ζ < 1 for some x. By construction, ζ = 1 for spin-coherent states. The concept of spin-squeezing can be applied to the collective spin state made up of N independent microscopic spins. In this case, J = N/2. It has been shown that there is a direct relation between the entanglement among N spins and the total N degree of squeezing within a collective spin state through the relation J = i si , where si is a microscopic spin variable. ζ is related to another deﬁnition of spin-squeezing parameter ξ discussed in 2 by ζ = ξJ/J. The latter is observable in the Ramsey interferometer relevant for precision measurements for an atom clock2 . It can be shown that ζ ≥ 1 for all the separable states. Thus ζ < 1 gives a suﬃcient condition for entanglement. For a symmetric 2-qubit system, spin squeezing and bipartite entanglement are shown to be equivalent. The necessary and suﬃcient condition for entanglement in the context of spin-squeezing is also discussed5 . For a symmetric state, it can be shown that Jx2 = N/4 + N (N − 1)Ji Jj /4 for i = j and ζ 2 = 1 + (N − 1)Ji Jj /4. Therefore ζ < 1 implies spins have negative correlation and entanglement7 . 2. Squeezing of superconducting qubits In this work, we apply the concept of spin squeezing to squeezing of quantum bits, a building block for quantum computing. Viewing each qubit as a two-level spin, it is possible to consider spin squeezing of an entire qubit system. As a particular implementation, we describe a realization of qubit-squeezing based on a superconductive qubit in a cavity8,9,10,11,12 .

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When we adjust circuit parameters and set Δi = 0 for all i, the standard form of the Hamiltonian for the qubits-cavity system is equivalent to the one for the following atom-cavity system: H =−

N i σ i

z

i

2

+

N gi i

2

(σ −i a† + σ +i a) + Ωa† a.

(3)

We can modify the collective spin state in the presence of nonuniform device paN 2 2 N i rameters and deﬁne J = i is , where satisﬁes = i i . We can show that, ΔJx = ΔJy = J/2 when each microscopic spin is a coherent spin state similarly to the case with uniform device parameters. The squeezing parameter ζ can be deﬁned as in (2). ζ < 1 in this case also implies the bipartite entanglement. If we assume a uniform value gi = g for all qubits, the Hamiltonian (3) becomes H = −Jz + g(J − a† + J + a) + Ωa† a

(4)

For a collection of qubits initially all pointing down along z-direction and a cavity mode initially in a coherent state with amplitude α, the expectation value of the collective spin variable can be calculated13 . Upto the second order in α, they are Jx = 0

√ Jy = 2α sin( 2t)

√ Jz = −1 + α2 sin2 ( 2t)

(5)

and their uncertainties are √ √ √ Δ2 Jx = 1 + α2 (sin2 ( 2t) − 2 sin2 ( 2t)/3 / 2 √ √ √ √ Δ2 Jy = 1 − α2 (sin2 ( 2t) 1 + α2 (− sin2 ( 2t)/2 + 2 sin2 ( 6t)/3 / 2 √ Δ2 Jz = α sin( 2t) (6) If we assume slight nonuniform device parameter distribution for multi-qubits, i.e. gi = g + δi with δi << 1 to be a random variable with a Gaussian probability 2 2 √ distribution: P (δ) = σe−δ /σ / π. Taking into account randomness, for a short time, σ 2 t2 << 1, √ √ √ √ α2 σ 2 α2 t2 α2 sin2 ( 2gt) − (1 − cos( 6gt)) + (cos(2 2gt) − cos( 2gt))(7) ζ¯ = 1 + 2 3 2 In Fig. 1, the temporal evolution of a squeezing parameter ζ from JaynesCummings Hamiltonian (4) is plotted. The squeezed state is realized periodically. The state is robust under small randomness in a qubit-cavity coupling g. In Fig. 2-4, the temporal evolution of an uncertainty ellipsoid is plotted. The center of the ellipsoid corresponds to J and the lengths of each axis along x,y,z direction are ΔJx , ΔJy , ΔJz , respectively. We set an initial state as both qubits pointing downward along z-direction. This corresponds to the spin coherent state as shown in Fig. 2. There is no entanglement between two qubits in this state. The time

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evolution due to Jaynes-Cummings Hamiltonian brings the state into the spinsqueezed state shown in Fig. 3 and 4. Note that both at t = 2 and t = 4 the squeezing parameter ζ becomes negative indicating that the state is spin-squeezed at these moments. It is shown that, for a triplet state, the squeezing parameter is directly related to concurrence C as C = 1 − ζ 2 14 and provides a direct evidence of quantum entanglement without depending on quantum state tomography. Thus we conclude that a squeezing measurement simpliﬁes an experimental setting for the veriﬁcation of quantum entanglement signiﬁcantly. Acknowledgments This work is supported by the Frontier Research System at RIKEN, Japan; the US AFOSR, ARDA, NSA; and the US National Science Foundation. References 1. D. Walls and G. J. Milburn,Quantum Optics (Springer, Berlin, 1995); H. J. Carmicheal, Statistical Methods in Quantum Optics (Springer, Berlin, 1999). 2. D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L. Moore, D. J. Heinzen, Phys. Rev. A 50, 67 (1994). 3. J. M. Geremia, J. K. Stockton, and H. Mabuchi, Science 304, 270 (2004). 4. M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993). 5. J. K. Korbicz, J. I. Cirac, and M. Lewenstein, Phys. Rev. Lett. 95, 120502 (2005). 6. A. Sorensen, L-M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001). 7. X. Wang and B. C. Sanders, Phys. Rev. A 68, 012101 (2003). 8. C-P Yang, S-I Chu, and S. Han, Phys. Rev. A 67, 042311 (2003). 9. J. Q. You and F. Nori, Phys. Rev. B68, 064509 (2003). 10. A. Blais, R-S Huang, A. Wallraﬀ, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A69, 062320 (2004). 11. I. Chlorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Science 431, 159 (2004). 12. S. Saito, M. Thorwart, H. Tanaka, M. Ueda, H. Nakano, K. Semba, and H. Takayanagi, Phys. Rev. Lett. 93, 037001 (2004); Kutsuzawa, H. Tanaka, S. Saito, H. Nakano, K. Semba, and H. Takayanagi, App. Phys. Lett. 87, 073501 (2005). 13. C. Genes, P. R. Berman, and A. G. Rojo, Phys. Rev. A 68, 043809 (2003). 14. D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64, 052106 (2001).

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Ζ 1.2 1.1 4

2

6

10

8

t

0.9 0.8 0.7 0.6 Figure 1. Time evolution of a squeezing parameter. g = 1, α = 1. The dark curve assumes uniform qubit parameters. The grey curve includes the randomness in g with σ = 0.1. We see that the squeezing is robust under small nonuniform device parameter distribution.

-2 -1

y 0 1 2 2

1

0 z -1 -2 -1 0 1

x

2 Figure 2. Time evolution of uncertainty ellipsoid at an initial spin coherent state is shown. Parameter values are the same as in Fig. 1.

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-2 -1

y 0 1 2 2

1

0 z -1 -2 -1 0 1

x

2 Figure 3.

Time evolution of uncertainty ellipsoid at t = 2 is shown.

-2 -1

y 0 1 2 2

1

0 z -1 -2 -1 0 1

x

2 Figure 4.

Time evolution of uncertainty ellipsoid at t = 4 is shown.

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DETECTION OF BERRY’S PHASES IN FLUX QUBITS WITH COHERENT PULSES∗

Z. H. PENG, M. J. ZHANG, S. LI AND D. N. ZHENG National Laboratory for Superconductivity, Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100080, PR China E-mail: [email protected]

We propose a feasible experimental scheme to demonstrate the geometric phase in ﬂux qubits by means of a detuning coherent microwave pulse technique. Furthermore, controlled phase shift gates can be implemented based on the geometric phases by inductance coupling of two ﬂux qubits. There are limitations under the adiabatic condition, but due to its geometric nature, the scheme might have an important application in precise preparation quantum state such as the initial state of decoherence-free subspace (DFS).

1. Introduction It is believed that the geometric quantum computation approach oﬀers the potential of fault tolerance with control parameters because of the global geometric nature 1 . Recently, a number of schemes of geometric quantum computation based on Berry’s phase were proposed in NMR 2 , trapped ions 3 , and superconducting qubits 4,5 . Especially, experimental realization of the two qubits conditional adiabatic phase gates have been reported in NMR 2 and trapped ions 6 . Solid state qubits, such as superconducting qubits due to their scability were regarded as an important candidate for the realization quantum computing 7,8 . However, there is no experimental report about detection of geometric phase in these systems, to our knowledge. We have proposed a feasible experimental scheme to demonstrate the Berry phase in ﬂux qubits by means of a detuning coherent microwave pulse technique. Furthermore, we showed that controlled phase shift gates can be implemented based on the geometric phases 5 . 1.1. Coherent pulse sequence technique NMR techniques are often used in coherent control quantum systems now being considered for the implementation of quantum computers 9 . Collin et al. 10 have ∗ This

work is supported by the national nature science foundation of china (10534060 and 10574154), the ministry of science and technology of china through the 973 program (2006cb601007) and the chinese academy of sciences.

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demonstrated NMR-like control of a superconducting qubit circuit with multipulse sequences. More recently, Kutsuzawa et al. 11 have observed Ramsey fringes by applying a pair of phase-shifted microwave pulses without introducing detuning. A clear advantage of this method over conventional detuning method is that it provides much faster operation. The coherent pulse technique is one of the techniques widely used in NMR 12 . The oscillating radio-frequency ﬁeld produced in a coil by a sequence of pulses occurring at times t1 , t2 , . . . , tk and of durations τ1 , τ2 , . . . , τk can be represented by the function Vk (t) cos(ωrf t + ϕk ), (1) V = k

where Vk (t) = 0 outside of the interval tk ≤ t ≤ tk + τk and is approximately constant inside. The pulses are called incoherent if their phases ϕk are randomly distributed, and coherent if their values can be controlled. The orientation of the rotating ﬁeld with respect to the rotating frame varies at random between pulses in the ﬁrst method but is well deﬁned in the second (moreover, it can be ﬁxed). Coherent pulses have another advantage besides controlling the phase of the rotating ﬁeld: the signal-to-noise ratio is considerably improved. 1.2. Flux qubit and the rotational framework A ﬂux qubit can be described as a two-state system or pseudo spin 1/2 particle. The two classical states(state |1 and state |0, associated with minima in the energy landscape) correspond to clockwise and counter-clockwise circulating currents. The ˆ = ((εf σ ˆz + Δˆ σx ) + V (t)ˆ σz ), where qubit is described by the Hamiltonian H 2 Δ is tunnel splitting, εf is the DC energy bias, V (t) = 2ν cos(ωrf t + ϕ) is the perturbation microwave signal. If we write the total Hamiltonian in the rotating ˆ −i(ωrf t/2)ˆσz and obtain ˆ eﬀ = (ωrf /2)ˆ σz + ei(ωrf t/2)ˆσz He frame approximation 12 , H ωrf − ω0 −νeiϕ sin η ˆ Heﬀ = , (2) 2 −νe−iϕ sin η ω0 − ωrf where η = arctan(Δ/εf ) is the mixed angle, ω0 = εf 2 + Δ2 is the qubit Larmor ˆ eﬀ = frequency at the measured ﬂux bias point. The eﬀective Hamiltonian reads H −(1/2)B · σ ˆ , so we have deﬁned ﬁctitious ﬁeld B ≡ (ν cos ϕ sin η, ν sin ϕ sin η, δω) , where δω = ω0 − ωrf . The ﬂux qubit thus behaves like a spin 1/2 particle in a magnetic ﬁeld. Under the action of coherent pulse sequences, the eﬀective qubit Hamiltonian describes a cone in the parameter space {B}. 2. Creating and detecting the Berry phase in a single flux qubit As shown in Figure 1, analogous to a spin 1/2 particle, the Hamiltonian lies close to the z -axis when |ν| |δω|, while the Hamiltonian lies close to the x − y plane when |ν| |δω|. If the applied radio-frequency radiation is far from resonance, the Hamiltonian is quantized along the z -axis, and if the radio frequency is swept

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Z

,E

(a)

0ZOLQH

(b) H eff

T Y

¶H[W

M X

Figure 1. Schematic design of the system. (a) The ﬂux qubit, pierced by a magnetic ﬂux φext , and microwaves (MW) are applied to the qubit from the on-chip strip line. The quantum state of the ﬂux qubit can be read out by the switching probability of the biased DC-SQUID. (b) Evolution of the eﬀective Hamiltonian in parameter space by adiabatically varying the phase of the radio frequency. Each pulse makes the Hamiltonian rotate around the z -axis at a ﬁxed cone angle θ. After the circular evolution, the Berry phase acquired in ﬂux qubits is ±π(1 − cos θ), where η is the cone angle. θ = arctan ν sin δω

towards resonance (δω = 0), the Hamiltonian rotates from the z-axis towards the x − y plane. The Hamiltonian ends at some angle to the z -axis and so arbitrary cone angles can be realized if one does not sweep the radio frequency all the way to resonance. If the resonance is approached suﬃciently slowly, the spin will follow the Hamiltonian in the rotating frame according to the adiabatic theorem. Next, by adiabatically varying the phase of the radio frequency, a circular motion can be performed. We now describe a procedure to measure the Berry phase in experiments. First, the ﬂux qubit is prepared in the ground state of the Hamiltonian at the degeneracy point and then a hard pulse (a short and strong pulse) is applied to excite the ﬂux qubit in a linear superposition of the energy eigenstates. Next, appropriate amplitude adiabatic coherent pulse sequences are applied. Besides the geometric phases acquired in this way, we will have an additional dynamical component, which depends on experimental details. This can be eliminated using a conventional spin echo approach 12 . The ﬁnal step in the procedure is to measure the persistent current state of the qubit. The probability of measuring the persistent current state |1 in the ﬂux qubit at the end of this procedure is given by P (1) = sin2 (2γ),

(3)

independent of the elapsed time, where γ is the Berry phase that is acquired in the ﬁrst adiabatic loop process. If the initial state of the ﬂux qubit is not prepared at the degeneracy point, we have to deduce the Berry phase value γ from the interference pattern of the persistent current state |1 measuring probability. For a quantum mechanical system, the acquired geometric phase is equal to half of the solid angle subtended by the area in the parameter space enclosed by the

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closed evolution loop of a ﬁctitious magnetic ﬁeld, δω ). γ = π(1 − 2 δω + (ν sin η)2

(4)

If we ﬁx the bias point at the phases to √ degeneracy point, sin η = 1, geometric √ achieve a NOT gate for ν = 37 δω and Hadamard gate for ν = 715 δω. We desicribe an example to illustrate the detection of the Berry phase in a single ﬂux qubit by applying coherent pulse sequences. The parameters were as follows: split energy Δ = 6GHz, detuning frequency δω=150MHz, and phase sweeps implemented using 100 linear steps of about 3ns, giving a total pulse sequence length dH of about 600ns. These parameters satisfy the adiabatic threshold Tν dtef f ν 2 12 , where T is the decoherence time of the ﬂux qubit. It is especially interesting that a decoherence time of 1.9μs has already been achieved by a ﬂux qubit consisting of 4 Josephson junctions 13 . The adiabatic manipulation time must not be longer than the performance of currently realized ﬂux qubit and the required coherent pulse techniques are mature 9,12,13 . These preconditions indicate that we can successfully detect the geometric phase in a ﬂux qubit with coherent pulses in experiments.

qubit a

qubit b

Figure 2. Schematic design of the two-bit gate. The two qubits can be coupled via their mutual inductance. The Berry phase of the target qubit (qubit a) can be controlled by the control qubit (qubit b).

3. The adiabatic conditional geometric phase gate For the implementation of a universal two-qubit gate, we consider a system of two coupled ﬂux qubits with magnetic inductance. The two qubits are described by the total Hamiltonian: ˆ ˆ = ˆ z(a) σ ˆ z(b) . (5) H i=a,b Hi + J σ Two qubits can be coupled via mutual inductance or a large Josephson junction to achieved the desired coupled strength.

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600

700

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900

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0

0

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Figure 3. Calculated results of the controlled phase shift in the target qubit. (a) The switching probability of the persistent current state |1 in target qubit (qubit a) vs. amplitude of coherent pulses. (b) Berry phase γ0 , γ1 and the controlled Berry phase diﬀerence Δγ as a function of the amplitude of the microwave pulse. Open triangles, circles and stars denote γ1 , γ0 and Δγ respectively.

We consider that the Δ of the two qubits is diﬀerent and the Larmor frequency of the two qubits is diﬀerent at the degeneracy point, and only qubit a is close to resonance. The two transitions of qubit a (corresponding to the two current states of qubit b) will be split by ±πJ , and so will have diﬀerent resonance oﬀsets. With the coherent radio-frequency microwave pulses, the eﬀective Hamiltonian depends on the resonance oﬀset, and so the cone angle (and hence the Berry phase acquired) will depend on the state of qubit a. This permits a conditional Berry phase to be applied to qubit a, where the size of the phase shift is controlled by qubit b. If the frequency is applied at a frequency δω away from the resonance frequency of qubit a when qubit b (the control qubit) is in state 0, and ν is the amplitude of radio-frequency pulses, then the diﬀerential Berry phase shift Δγ = ±π(

δω δω + J − ), 2 2 2 (δω + J) + (ν sin η) δω + (ν sin η)2

(6)

depends only on δω, ν and J; it does not depend on the details of how the process is carried out, as the time needed for the adiabatic manipulation is shorter than decoherence times. A range of controlled Berry Phases can be obtained by choosing appropriate values of δω and ν (J is ﬁxed by the two coupled ﬂux qubits with mutual inductance.) If we ﬁx the value of δω/J, the controlled Berry phase will rise and then fall as ν is increased. It is very robust that the desired Δγ occurs at the maximum in this curve. Then the dependence on the δω is reduced to second order 2 . To achieve the controlled-NOT gate, we ﬁx the bias point at the degeneracy point sin η = 1, and we can design a controlled π shift at δω = 0.309J and ν = 0.9511J, as shown in Figure 3. Here, the pulse sequence is applied twice in order to eliminate the dynamics phase. The split energy of the two coupled ﬂux qubits is chosen from Ref.14 , J=500MHz, δω=154.5MHz. At low amplitude, the switching probability of the target qubit when the control qubit at state |0

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oscillates very quickly. Therefore, one should choose an appropriate amplitude of pulses in an experiment to obtain high readout resolution. 4. Conclusion We have proposed an experimental scheme to detect geometric phases with coherent microwave pulses in ﬂux qubits. Furthermore, we have designed a one-qubit quantum logic gate and a two-qubit controlled phase shift gate based on the Berry phase. The idea of the geometric phase shift gate demonstrated by the ﬂux qubits system here should, in principle, also work for other superconducting qubits. Lastly, we emphasize that there are limitations under the adiabatic condition, but due to its geometric nature, the scheme might have an important application in precise preparation quantum state such as the initial state of decoherence-free subspace (DFS) 15 . References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15.

M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). J. Jones, V. Vedral, A. Ekert and G. Castagnoli, Nature (London) 403, 869 (2000). L. M. Duan, J. I. Cirac and P. Zoller, Science 292, 1695 (2001). G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral, Nature (London) 407, 355 (2000). Z. H. Peng, M. J. Zhang, D. N. Zheng, Phys. Rev. B 73, 020502 (2006) D. Lelbfried, B. Demarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, D. J. Wineland, Nature (London) 422, 412 (2003). M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). Yu. Makhlin, G. Sch¨ on and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. 76, 1037 (2004). E. Collin, G. Ithier, A. Aassime, P. Joyez, D. Vion, and D. Esteve, Phys. Rev. Lett. 93, 157005 (2004). T. Kutsuzawa, H. Tanaka, S. Saito, H. Nakano, K. Semba, and H. Takayanagi, Appl. Phys. Lett. 87, 073501 (2005). A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961). Y. Nakamura et al. (unpublished). J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. Lett. 94, 090501 (2005). D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).

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PROBING ENTANGLEMENT IN THE SYSTEM OF COUPLED JOSEPHSON QUBITS

A.S. KIYKO, A.N. OMELYANCHOUK, S.N. SHEVCHENKO B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103 Kharkov, Ukraine E-mail: [email protected] We study theoretically the quantum state of the system of coupled superconducting Josephson qubits created by an electromagnetic pulse and discuss its observability. The possibility of controlling the system state via the rectangular pulse is investigated. We calculate both time-dependent levels occupation probabilities and concurrence which is the measure of formed entangled state. Also we calculated the phase shift in the resonant circuit weakly coupled to the qubits which can be measured experimentally.

1. Introduction There are three basic types of Josephson-junction circuits that behave quantum mechanically at low temperature. They are charge 1 , phase 2 , and ﬂux 3 qubits. All of them can be fabricated with high precision with the help of modern lithography and can be the basis of the quantum computer. The most promising for quantum computations is the ﬂux qubit that consists of the superconducting loop with three Josephson junctions 3 . This type of qubit is insensitive to the charge noise and it was shown that it has a high quality factor 4 . Such qubit can be integrated in the circuit via the inductive coupling. In this work we study the dynamics of the single ﬂux qubit and of two coupled ﬂux qubits subjected to the rectangular electromagnetic pulse. In particular we study the transitions between the states of such systems and both formation and evolution of the quantum entangled state. 2. The model Our aim is to study the quantum state of the system of superconducting qubits created by the rectangular electromagnetic pulse and to discuss its observability. For concreteness we study the system of ﬂux qubits interacting via zz-coupling. The Hamiltonian of the single ﬂux qubit has the form 5 H = −EJ cos(2π

Q2 Φ (Φ − Φx )2 + , )+ Φ0 2L 2C

(1)

where L is the self-inductance of the loop, EJ is the Josephson energy, C is the capacitance of the junction, Q = −i∂/∂Φ is the charge operator, canonically

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conjugated to the magnetic ﬂux Φ, Φx is the externally applied ﬂux, Φ0 = h/2e is the ﬂux quantum. The potential proﬁle has two minima which correspond to the opposite current ﬂows. At low temperature only the two lowest states in the two wells are essential. The reduced two-level Hamiltonian has the form 5 = −ε H σ z − Δ σx , (2) 1 x where diagonal term is the bias ε(Φx ) = 2π 6(βL − 1)EJ ( Φ Φ0 − 2 ), βL = 2 2 EJ /(Φ0 /4π L) and the oﬀ-diagonal term Δ is the tunneling amplitude between the wells. For the system of coupled qubits the eﬀective Hamiltonian is 3, 6 : (i) =− H εi σ + Jσ (1) z + Δi σ (i) (2) (3) x z σ z , i

where J is the coupling energy between qubits, and σ x , σ z are Pauli matrices in (i) the basis {|↓ , |↑} of the current operator in the i-th qubit: Ii = Ii σ z , where (i) 1 1 − (2β)2 is the absolute value of the current in the i-th qubit, the Ii I0

σ z |↓ = − |↓) and counterclockwise eigenstates of σ z correspond to the clockwise ( ( σ z |↑ = |↑) current in the i-th qubit. Here β is the parameter of the qubit; for all calculations below we have taken β = 0.8. The tunneling amplitudes Δi are assumed to be constants. The biases 1 (i) εi = I0 Φ0 fi − (4) 2 are controlled by the dimensionless magnetic ﬂuxes fi = Φi /Φ0 through i-th qubit. The ﬂux consists of three components: (i)

fi = fx + fpulse (t) + fshif t ,

(5)

which describe the adiabatically changing magnetic ﬂux (from a common coil), fx , the time-dependent component (from the same coil), fpulse (t), and the additional ﬂux (created by the additional wires, which individually control the qubits, and (i) from other qubits, via mutual inductances), fshif t 7 . We will study the possibility to control the system state via the rectangular pulse with the amplitude f0 and duration from t = t1 to t = t2 : fpulse (t) = f0 (θ(t − t1 ) − θ(t − t2 )) , where θ(t) stands for the theta-function. The eﬀect of the pulse is in changing the level occupation probabilities and to make them oscillating functions of time during the pulse. It must be noted that in the basis {|↓ , |↑} of the current operator, which is not the eigenstates of the Hamiltonian, the probabilities oscillate both during and after pulse. We will assume that the relaxation time is longer than the characteristic measurement time (which is of the order of ωT−1 ∼ 0.1 μs 7 , ωT is the characteristic frequency of the tank circuit), so that the simpedance measurement technique 9, 10

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can probe not only the ground state of the system but rather the coherent superposition created by the pulse. Then we describe the system’s evolution with the Liouville equation for the density matrix ρ ( = 1):

d ρ = −i H, ρ . (6) dt The impedance measurement technique consists in that the tank circuit probes the eﬀective inductance of the system via measuring the phase shift α between the voltage and current in the tank circuit. The argument from Ref. 12, generalized for the system of several qubits, with the assumption that the rings’ inductances of all qubits are identical, Li = L, as well as their mutual with the tank inductances, Mi = M , leads to the following expression: tan α = k 2 Q ·

i

L (i) LJ

+L

,

(7)

∂ Ii 1 1 ∂

(i) LJ Ii Sp . (8) = = ρσ (i) z Φ0 ∂fi Φ0 ∂fi √ Here Q = (ωT CT RT )−1 , ωT = 1/ LT CT , k 2 = M 2 /(L · LT ). In the following we will neglect the ring’s inductance L in the denominators in Eq. (7). In particular, for a single qubit: Sp ( ρσ z ) = 1 − 2PL , where PL(R) is the probability to ﬁnd the system in the left (right) well correspondingly. For one qubit in the ground state it results in the following:

−1

tan α −k 2 Q · Lq

(ε2

Δ2 I 2 . + Δ2 )3/2

(9)

3. Excitation of the single flux qubit In this section we study the excitation of the single ﬂux qubit with the rectangular pulse. We start from the general 1-qubit Hamiltonian that has the form Eq. (2) in the basis of states {|↓ , |↑}. For a ﬂux qubit these states correspond to a deﬁnite direction of the current circulating in the ring. First the time-independent Hamiltonian is diagonalized in the basis of eigenstates {|− , |+} with the rotation matrix We want to have the density matrix in the energy representation {|− , |+}, S. where its diagonal components are equal to the probability of the system to be in the ground |− or excited state |+. We next introduce the time-dependent terms S, we get (t) = S−1 H(t) into the Hamiltonian. Making use of the transformation H´ 13 (t) in energy representation : the Hamiltonian H´ (t) = − ΔE τ z − I0 Φ0 fpulse (t) (2ε H´ τ z − 2Δ τ x ) /ΔE, ΔE = 2

Δ2 + ε2 .

The time evolution of the density matrix, which can be taken in the form 1 τx + Y τ y + Z τ z , is described by the equation of motion (6). Initial ρ = 2 1 + X

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P (t)

1 t2

+

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a

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L

b

α,rad

−0.5 0.495 0.5 0 −1 0.495

0.5

0.505

c

0.5

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0.505

0

Figure 1. The time-dependence of the upper level occupation probability P+ (t) (a); probability of finding the system in the left well PL in the ground state before (solid line) and after the pulse (dashed line) (b); phase shift α before (solid line) and after the pulse (dashed line) (c).

condition for the density matrix in {|− , |+} basis is X(0) = Y (0) = 0, Z(0) = 1, which corresponds to the ground state of the system. Solving the system of equations for X(t), Y (t), Z(t) we obtain the probability of occupation of the upper level |+ (the excited state) P+ (t) = ρ22 (t) = 12 (1 − Z(t)). After this we calculate the ρS−1 density matrix in the ﬂux basis making use of the transformation ρf lux = S and obtain the probability of ﬁnding the system in the left (right) well PL (PR ). Then we obtain: 1 εZ(t) ΔX(t) −√ . (10) PL (t) = 1+ √ 2 Δ2 + ε2 Δ2 + ε2 We make such transformation because we are interested in the phase shift α between the voltage and current in the tank circuit and it is calculated in the ﬂux basis. The parameters are taken to compare with the results of Ref. 11: Δ = 0.45 GHz, I0 Φ0 = 990 GHz. Then in Fig. 1(a) we plot the time dependence of the upper level occupation probability P+ (t). The probability PL as a function of the bias fx is plotted in Fig. 1(b) in the ground state both in the absence (solid line) and in the presence of excitation (dashed line). The phase shift α as a function of the bias fx is plotted in Fig. 1(d). The pulse’s parameters are: t1 = 0.2 ns, t2 = 1.2 ns, f0 = 5 · 10−4 . Here it must be noted that in the presence of the excitation we plot time-averaged phase shift α since the probability of ﬁnding the system in the left well PL oscillates after switching oﬀ the pulse. The phase shift α calculated for the ground state according to Eq. (7) for Δ = 0.25 GHz coincides with Fig. 2 from Ref. 11.

4. Excitation of two coupled flux qubits In this section the system of two coupled ﬂux qubits excited by the rectangular pulse is studied. We investigate the energy levels populations 8 and the phase shift

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α as in the previous section. The parameters are taken as in Ref. 7: Δ1 = 0.55 GHz, Δ2 = 0.45 GHz, I0 Φ0 = 990 GHz, J = 0.42 GHz. During the pulse the transitions between the energy levels occurs. After the excitation is switched oﬀ we obtain an entangled state which appears due to these transitions. Besides we study the concurrence C( ρ) which deﬁnes the entanglement ε(C) 14 : ε(C) = −

1+

√

1 − C2 1+ log 2 2

√ √ √ 1 − 1 − C2 1 − C2 1 − 1 − C2 − log2 . (11) 2 2 2

For the case of two qubits C( ρ) = max{0, λ1 − λ2 − λ3 − λ4 }, and the λi -s are the eigenvalues, in decreasing order, of the Hermetian matrix R ≡ ρρ ˜ ρ. Alternatively, λi -s are the square roots of the eigenvalues of the non-Hermetian y ) ρ∗ ( σy ⊗ σ y ). matrix ρρ ˜. Here the spin-ﬂipped state ˜ρ is deﬁned as ˜ρ = ( σy ⊗ σ We make calculation in the same way as in the previous section - we solve the Liouville equation in the basis where the time-independent Hamiltonian is diagonal. Then we recalculate the density matrix to the ﬂux basis and calculate the phase shift α according to Eq. (7). The time-dependence of the levels occupation probabilities (the density matrix diagonal elements ρii ) and the entanglement are plotted in Fig. 2(a) and 2(b) with parameters: Δ1 = 0.55 GHz, Δ2 = 0.45 GHz, I0 Φ0 = 990 GHz, J = 0.42 GHz, t1 = 0.1 ns, t2 = 1.6 ns, f0 = 0.01. We obtain the same results as in Ref. 15. After the pulse we obtain the new initial condition. Due to this condition the nondiagonal terms in the density matrix appear and this results in the oscillations of the entanglement. We plot the phase shift α for the ground state of two coupled qubits as function of the bias fx in Fig. 2(c). It must be noted that the height of the barrier between the two wells depends on the coupling constant J and grows with the J increasing. We calculate the time-averaged levels occupation probabilities as function of the dimensionless ﬂux fx both before and after pulse in Fig. 2(d) and 2(e).

5. Conclusion The dynamics of the single ﬂux qubit and two coupled ﬂux qubits subjected to the rectangular electromagnetic pulse have been studied. We investigated the change of the tank circuit phase shift α for the single qubit that appears due to the excitation by the pulse. Also for two coupled ﬂux qubits we study the time dependence of entanglement that is the important value for the quantum computations. The entanglement can be changed and controlled in two ways: in the ground state (without excitation) we can change the ﬂux Φx and for excited state after the switching oﬀ the pulse we can start the manipulations in the deﬁnite moment of time because the entanglement oscillates in time.

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ρ

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Figure 2. Time-dependence of the levels occupation porbabilities (a) and entanglement ε (b). The dependence of the phase shift α for the system of two coupled qubits on the bias fx in the ground state: solid - fshif t1 = 0.005, fshif t2 = 0.0016; dashed - fshif t1 = fshif t2 = 0 (c). Timeaveraged diagonal elements ρii of the density matrix versus dimensionless flux fx before (d) and after pulse (e). For these figures dashdot line - i = 1, dashed line - i = 2, dotted line - i = 3, solid line - i = 4.

S.N.S. acknowledges the ﬁnancial support of INTAS under Conference Individual Grant (No. 05-1000002-5637) and INTAS Fellowship Grant for Young Scientists (No. 05-109-4479). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Y. Nakamura, Y.A. Pashkin, and J.S. Tsai, Nature (London) 398, 786 (1999). J.R. Friedman et al., Nature (London) 406, 43 (2000). J.E. Mooij et al., Science 285, 1036 (1999). C.H. van der Wal, F.K. Wilhelm, C.J.P.M. Harmans, J.E. Mooij, Eur. Phys. J. B. 31, 111 (2003); L. Tian, S. Lloyd, T.P. Orlando, Phys. Rev. B. 65, 144516 (2002). Y.Makhlin, G.Sch¨ on, A.Shnirman, Rev. Mod. Phys. 73, 357 (2001). M. Grajcar et al., Phys. Rev. Lett. 96, 047006 (2006). A. Izmalkov et al., Phys. Rev. Lett. 93, 037003 (2004). J.B. Majer et al., Phys. Rev. Lett. 94, 090501 (2005). R. Rifkin and B.S. Deaver, Jr., Phys. Rev. B 13, 3894 (1976). E. Il’ichev et al., Rev. Sci. Instr. 72, 1882 (2001). M. Grajcar et al., Phys. Rev. B. 69, 060501 (2004). V.I. Shnyrkov et al., Phys. Rev. B. 73, 024506 (2006). S.N. Shevchenko, A.S. Kiyko, A.N. Omelyanchouk, W. Krech, Fiz. Nizk. Temp. 31, 752 (2005) [Low Temp. Phys. 31, 564 (2005)]. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). T. Mouri, H. Nakano, H. Takayanagi, arXiv:cond-mat/0501581.

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JOSEPHSON JUNCTION WITH TUNABLE DAMPING USING QUASI-PARTICLE INJECTION

K. UTSUNOMIYA, K. TSUBOI AND R. YAGI∗ Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8530, Japan ∗ E-mail: [email protected] We demonstrated that the dumping of Josephson junctions, one of characteristic parameters of actual Josephson junctions, could be varied by quasi-particle injections by performing injection experiments using multi-terminal Josephson junction devices fabricated by using electron beam lithography. Current-voltage (I-V) characteristics of a Josephson junction exhibited a transition from hysteretic to non-hysteretic behavior with increasing injection current of a control junction placed near the Josephson junction. McCumber Stewart parameter of the junction was varied about two orders of magnitude by injection. This can be used for tunable energy dissipation in Josepshon junction devices. Keywords: Josephson junction; Tunable dissipation; Quasi-particles.

1. Introduction Recently, Josephson devices are extensively studied with interests in fundamental physics and practical applications. It is known that dynamics of Josephson junctions, such as current-voltage (I-V) characteristics, can be formulated by the resistively shunted Josephson junction (RSJ) model, in which a Josephson junction are described by a Josephson tunneling, a dissipative resistor and a capacitor. Magnitude of these parameters determines hysteretic behavior in current-voltage (I-V) characteristics. The damping is the most important factor: with increasing the magnitude of damping, hysteretic behavior disappears at critical magnitude of damping. The damping resulting from the energy dissipation is diﬃcult to control in actual experimental setup. One of the eﬀorts is done by attaching a metallic Ohmic resistor in the vicinity of a Josephson junction1,2 . However, resistance of the metallic resistor is ﬁxed in actual devices and could not be varied after fabrication. In this work, we wish to demonstrate controllable energy dissipation using quasi-particle injection. Quasi-particle injection from an injector junction increases quasi-particle density, which results in dissipative quasi-particle tunneling in a target junction. 2. Experimental Method Figure 1 illustrates schematic of device structure and experimental setup. Our device consisted of an aluminum thin wire which was connected to two Josephson junctions locating 2 μm apart. We measured I-V characteristics of one of the junction (target

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Target Junction

Control Junction

2μm

V

Quasi-particle Injection Figure 1.

Schematic diagram of device structure and experimental setup.

junction) while injecting quasi-particles with the other (control junction). Injection creates non-equilibrium quasi-particles in the superconductor so that quasi-particle tunnel resistance of the target junction was varied. The device was fabricated by using standard electron beam lithography and angle oblique evaporation technique. The area of the target junctions was about 0.10 μm2 and the tunnel resistance was about 1.8kΩ at T = 4.2 K. Those for control junction were 0.05 μm2 and 4.4kΩ, respectively. Measurements were done at low temperature by cooling the devices down to about 0.1 K with a dilution refrigerator. By reducing population of thermally excited quasi-particles, one can extract eﬀect of quasi-particle injection. Superconducting energy gap Δ/e measured at lowest temperature was about 210 μeV which agreed with an estimation from superconducting transition temperature TC =1.3 K. Current applied at a ﬁnite bias voltage larger than twice the superconducting gap energy create quasi-particles in the superconductors. Injected quasi-particles do not recombine to Cooper pairs immediately, but remain in a non-equilibrium state for a short time. Recombination time or relaxation length of the non-equilibrium quasi-particles is an important parameter. By performing preliminary experiments to determine the relaxation length, the recombination length was turned out to be about 10 μm using a similar aluminum wire and with injection of 1 μA. Therefore, we could vary the quasi-particle density of a superconducting electrode of the target junction by injecting quasi-particles with the control junction placed about 2 μm away. 3. Result and Discussion Figure 2(a) shows I-V characteristics of the target junction for diﬀerent injection current Iinj . From bottom to top, Iinj was varied from 0.0 to 1.0 μA in 0.1 μA steps. It is seen clearly that quasi-particle injection reduced the hysteretic behavior in I-V

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Injection (μA) 1.0

(a)

0.9 0.8 0.7 0.6

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C

A

B

0.0

0

0

-500 -40

-20

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20

Current (nA)

40

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0.1 Injection Current (μA)

Figure 2. (a) I − V characteristics of Josephson junction with injection. Injection current was varied from 0.0 to 1.0 μA. Arrows indicate direction of current sweep. (b) Retrapping induced by quasi-particle injection for diﬀerent bias current. Bias current for A,B and C are 0.07nA, 0.08nA and 0.1nA.

characteristics. At Iinj = 0.9 μA, the hysteretic behavior was lost. The reduction of hysteresis is partly due to increase in retrapping current Ir . The retrapping current, which was about 7 nA without injection, increased to about 30 nA with increasing injection to 0.8 μA. Retrapping current is associated with energy dissipation of Josephson junction, and is roughly proportional to 1/R where R is resistance of a parallel ohmic resistor in RSJ model. This is because power supplied to the Josepshon junction in a voltage state in unit time is IΔ/e while the energy should be balanced with dissipation RI 2 of the parallel ohmic resistor. Thus, the change in retrapping current reﬂected the variation of R due to quasi-particle tunneling. One of other reasons for the change in hysteretic behavior can be the variation of critical current Ic : Ic reduced with increasing injection current. We think this is relevant to heating eﬀect.

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In Fig. 2(a), a voltage state at about half of 2Δ/e is discernible. The voltage state could be observed both after switching and before retrapping. It would be possible that the step was due to the Shapiro step for some characteristic frequency of external electromagnetic noise. It would be also possible that some resonance of the junction and electromagnetic environment resulted in the steps. Figure 2(b) shows a result of retrapping induced by the dissipation. Bias current was swept up to switch the Josephson junction to the voltage state. Then the current was reduced to some diﬀerent values less than the switching current but larger than the retrapping current so that the junction was still in a running state near V = 2Δ/e. Then, keeping the bias current constant, we swept up the injection and measured the voltage of the Josephson junction. In Fig. 2(b), it is seen that, at a critical injection current dependent on the bias current, the junction switched to the zero voltage state. At those injection currents at which the junction was retrapped to zero voltage state, the bias current equals the retrapping current. These facts resulted from the retrapping controlled by energy dissipation due to quasi-particle tunnel resistance: a particle moving in a tilted Josephson potential has trapped at a local potential minimum owing to the friction given by the quasiparticle injection. The abrupt changes in the voltage may be used for some reset mechanism of Josephson logic circuits. From our experimental result, we estimated McCumber Stewart parameter βc , a measure of damping of Josephson junction, which is deﬁned by √ 4 2 1 − α2 , (1) βc = π α2 where α is a retrapping current normalized by the switching current ( α = Ir /Ic ). We ﬁtted our data to Eq.(1) as shown in Fig. 3 using capacitance C = 0.2 fF as a ﬁtting parameter. We could found good agreement between the theory and exper-

βc

100 10 1 Fitting 0.1 0

0.5

1.0

Ir / Ic Figure 3. McCumber Setwart parameter βc vs. normalized retrapping current. Solid line is ﬁtting to theory. Squares are experiment.

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iment. McCumber Stewart parameter was varied about two orders of magnitude by the injection. Junction capacitance approximately agreed with the estimation from the speciﬁc capacitance of Al/AlOx /Al junction. Deviations of the experiment from the ﬁtting at Ir /Ic > 0.5 can arise from the estimation of quasi-particle resistance. Quasi-particle injection could raise system temperature because it inevitably gave energy to the superconductors. Indeed, the switching current seen in Fig. 2(a) reduced with increasing injection current. Using Ambegaokar-Baratoﬀ’s formula, the relation between the switching current and superconducting energy gap, we can roughly estimate the reduction of superconducting energy gap to be about 66 Recalling that a superconducting order parameter depercent if Iinj = 0.9 μA. pends on temperature as 1 − (T /Tc )2 , the change in the critical current could be interpreted that the system temperature has raised close to the superconducting transition temperature. However, we think that the temperature was overestimated. Since energy gap parameter satisfy the BCS gap equations, considerable number of non-equilibrium quasi-particles can reduce the gap energy of superconducting condensate. We estimated the eﬀective temperature from zero bias resistance due to phase diﬀusion branch. The resistance due to phase diﬀusion by thermal activation is given by R = (h/4e2 )(hωp /kB T )exp(−EB /kB T ), where, ωp is the plasma frequency and EB is the potential barrier approximated by twice the Josephson energy EJ .3,4 The rise in the temperature for maximum injection current (Iinj = 0.9 μA) was estimated to be about 50 mK. We speculate that high energy phonons created in the relaxation process and recombination process escaped the superconductor immediately after they were created, considering that size of the substrate was much larger than that of highly non-equilibrium region in the superconductor. 4. Conclusion We have demonstrated that damping of a Josephson junction could be controlled with quasi-particle injection using multi-terminal Josephson device. We observed that degree of the hysteresis appearing in the I − V characteristics, gradually reduced with increasing injection current. At the largest injection current, I − V characteristics showed no hysteresis. McCumber Stewart parameter was varied by about two orders of magnitude by quasi-particle injection. We also demonstrated retrapping induced by dissipation. This method described here would be applied for studying dissipative tunneling phenomena. Acknowledgments This is work was supported in part by Grant-in-Aid for Scientiﬁc Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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References 1. 2. 3. 4.

R. Yagi, S. Kobayasi and Y. Ootuka, J. Phys. Soc. Jpn. 66, 3722 (1997). J. S. Penttila, U. Parts and P. J. Hakonen, Phys. Rev. Lett. 82, 1004 (1999). J. M. Martinis and R. L. Kautz, Phys. Rev. Lett. 63, 1507 (1989). R. L. Kautz and J. M. Martinis, Phys. Rev. B 42, 9903 (1990).

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MACROSCOPIC QUANTUM COHERENCE IN rf-SQUIDs

J. LISENFELD, A. LUKASHENKO, AND A. V. USTINOV Physics Institute, University of Erlangen-Nuremberg Erwin-Rommel-Str. 1, 91058 Erlangen, Germany e-mail: [email protected] Superconducting circuits display quantum mechanical peculiarities at macroscopic scales. The recent experiments to tailor and control these circuits gave rise to great hopes towards realizing a solid-state quantum computing chip. We outline the basic principles of rf-SQUID operation as a phase qubit, and present our experimental results demonstrating coherent quantum dynamics in a commercially fabricated sample.

1. Introduction In the year 2000, Friedmann et al.1 and van der Wal et al.2 for the ﬁrst time experimentally demonstrated the existence of quantum mechanical superpositions of macroscopic states in rf-SQUIDs. An rf-SQUID consists of a superconducting loop which contains a single Josephson junction. Applying an external magnetic ﬁeld to this loop gives rise to a persistent screening current circulating in a way as to drag the total ﬂux threatening the loop towards an integer number of ﬂux quanta, minimizing the system’s free energy. In particular, when the enclosed ﬂux is close to half a ﬂux quantum, a Schr¨ odinger’s Cat state emerges where the circulating current ﬂows at the same time clockwise and counterclockwise. Since the experimental demonstration of this eﬀect, two major types of quantum bits based on SQUIDs have been developed, namely the ﬂux qubit3 , in which the geometrical loop inductance is replaced by the nonlinear Josephson inductances of additional junctions, and the phase qubit5 , for which the eigenstates of the Josephson phase rather than diﬀerent ﬂux states are used as logical qubit states. 2. The phase qubit principles The potential energy of an rf-SQUID is composed of the Josephson junction energy plus the inductive contribution of the current circulating in the loop with inductance L: 2 1 Φ0 IC 2πΦext 1 − cos ϕ + U (ϕ) = , ϕ− 2π 2βL Φ0 where ϕ is the phase diﬀerence across the junction, IC is its critical current, Φext the externally applied ﬂux through the qubit loop, Φ0 = h/2e is the magnetic

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(a)

(b)

U

E [GHz]

4

Φext Φ

3

Φext

=0

Φ

Φext

=0.5

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=0.8

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15

2

L

>

R

>

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1 0 0

5

10

15

20

25

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ϕ

Phase difference ϕ Figure 1. (a) Potential energy of an rf-SQUID with βL =4.5 and Ic = 10 μA for external magnetic ﬂuxes of Φext =0, 0.5, and 0.8 Φ0 (from left to right, each curve oﬀset by ϕ=6 for better visibility), normalized to EJ = Ic Φ0 /2π. (b) Zoom into the shallow well showing the discrete energy levels and the wavefunctions of the corresponding quantum states |0, |1 and |2.

ﬂux quantum with 2e being the Cooper pair charge, and βL ≡ 2πLq IC /Φ0 . For βL ≈ 2π, the potential has the form of a double well which is tilted proportional to the external magnetic ﬂux, see Fig. 1(a). States associated to two neighboring minima, like the states |L and |R in Fig. 1(a), correspond to opposite directions of circulating current. 3. Quantum processes in the rf-SQUID As indicated in Fig. 1(b), the conﬁnement of the phase variable within one potential well leads to the formation of discrete energy states (energy quantization). The two states of lowest energy in the shallow potential well can then be used as the logical states |0 and |1 for quantum computation. Since the depth of the potential well depends on the strength of the applied magnetic ﬁeld, it is possible to tune the energy diﬀerence separating these states in situ. Transitions between the states can be driven by resonant photon absorption and emission, which is experimentally realized by inducing alternating currents of microwave frequencies in the rf-SQUID loop. The anharmonicity of the potential hereby assures that the transition frequencies between adjacent energy levels are diﬀerent enough to avoid undesired population of higher levels5 . To distinguish in which state the qubit is, one makes use of quantum tunneling through the potential barrier separating the wells. The rate of this process depends exponentially on the barrier height, and therefore tunneling from the excited state occurs much faster than from the ground state. Whether tunneling occurred is determined by monitoring the magnetic ﬂux through the qubit loop. This is realized by measuring the maximum supercurrent of a dc-SQUID which is inductively coupled to the rf-SQUID loop, as illustrated in Fig. 2(a).

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(a)

I

Control flux bias

L Φ X

X

Φext

Rb

Qubit rf-SQUID

sq

X

μw

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Readout dc-SQUID

(b) DC-pulse, measurement

Φ prepare 0

reset

I sq

μW-pulse sequence

reset

freeze

time

readout

0

1

0

>

>

time

Figure 2. (a) Schematic of the phase qubit circuit. Dc- and microwave currents control the ﬂux Φ in the qubit loop by means of an on-chip superconducting coil. For readout, this ﬂux is measured by a dc-SQUID which is coupled inductively to the qubit. (b) Timing proﬁle of the magnetic ﬂux Φ applied to the qubit (top) and the current through the readout SQUID (bottom). The latter is ramped up from zero only after completion of the qubit operation.

4. Qubit operation The qubit is operated in a way illustrated in Fig. 2(b). The qubit state is ﬁrst reset to the left well by switching the applied magnetic ﬂux to zero. Then, the ﬂux is increased to Φ ≈ Φ0 until the left well becomes shallow enough to contain only a small number of energy levels. After the microwave pulse was applied, a small readout ﬂux pulse of nanosecond length is applied for generating an additional small tilt of the potential, which causes immediate tunneling only if the qubit is in its excited state. The applied magnetic ﬁeld is then reduced in order to increase the barrier height, avoiding further inter-well transitions. Finally, a current-ramp measurement of the critical current of the readout dc-SQUID decides wether the qubit is in the left or right potential well, and hence wether it was in state |0 or |1 at the time when the readout pulse was applied. 5. Sample Our sample has been fabricated at a commercial foundry provided by Hypres4 using standard Nb/Al-AlOx /Nb-trilayer circuits of 30 A/cm2 current density and minimum 3 × 3 μm2 Josephson junction size. The critical current of the smallest

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junction of Ic ≈ 2.7 μA together with the requirement βL = 2πLIC /Φ0 ≈ 4 to 5 determines suitable loop inductances L ≈ 400 to 500 pH. In our sample, following Ref.5 , this inductance is realized by a two-turn coil of about 70 μm diameter. The mutual inductance between the qubit and readout dc-SQUID, M ≈ 15 pH, has been chosen large enough to permit single-shot readout of the qubit state. Measurements were performed in a dilution refrigerator at a base temperature of 20 mK. 6. Rabi oscillations Coherent qubit operation can be demonstrated by observing Rabi ﬂopping of the qubit state. This mechanism leads to oscillating state populations during the application of a resonant microwave ﬁeld, which eﬀectively couples states |0 and |1. The strength of this coupling grows with the rf-ﬁeld amplitude, which in consequence also determines the Rabi oscillation frequency fRabi . Experimentally, this oscillation is made visible in the time domain by applying a microwave pulse of varying duration which is followed by the dc-readout pulse. Figure 3(a) shows Rabi oscillation in the probability P (|1) to measure the excited qubit state for three values of applied microwave power and a frequency of frf = 14.2 GHz. We ﬁnd the oscillation amplitude to decay exponentially with a characteristic time of typically 6 ns. Particularly, for certain values of ﬂux bias, we observed an additional (a)

(b)

P( 1 )

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0.2

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2

3

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6

microwave amplitude [ mW ]

microwave pulse duration [ns]

Figure 3. (a) Rabi oscillation in the probability P (|1) to measure the excited qubit state. Each curve corresponds to the indicated microwave power P . (b) Measured dependence of the oscillation frequency on microwave amplitude (dots). For small amplitudes, saturation of the Rabi frequency detuning Δfrf of the rf frequency from the resonance between the levels. fRabi indicates residual

Line is a ﬁt to fRabi =

2 + αP , where α is the ﬁtting parameter. Δfrf

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modulation (beating) in the oscillation amplitude. We attribute this behavior to coherent coupling of the qubit to parasitic two-level ﬂuctuators, which were proven to exist in the silicon oxide used to isolate the junction electrodes6 . In Fig. 3(b), we plot the measured Rabi frequency on microwave amplitude, which displays a linear dependence at high amplitudes as expected for a two-level quantum system. 7. Inhomogeneous broadening An obvious source of decoherence are ﬂuctuations in the qubit parameters between individual repetitions of a measurement, which are then averaged to obtain the statistical state probability. This eﬀect can be distinguished from other decoherence mechanisms by spectroscopic measurements, in which a resonant enhancement of the probability to ﬁnd the qubit in the excited state occurs in the form of a Lorentzian peak. The width of this peak is increased by ﬂuctuations in the energy level spacing, an eﬀect called inhomogeneous broadening a . According to theory7 , the full width at half maximum of the resonance peak in the strong driving limit is given by 1 T1 1 , + ωRabi Δf ≈ π T2 T2 where T1 is the energy relaxation time, T2 is an intrinsic dephasing time and T2 is the contribution to dephasing due to inhomogeneous broadening. Plotting the linewidth vs. microwave amplitude, T2 results from a linear extrapolation to zero microwave amplitude. Figure 4 shows the result of such an experiment. The extrapolated resonance width of Δf =60 MHz for this data corresponds to T2 = 1/(π Δf ) ≈ 5.3 ns, in close agreement with the lifetime of Rabi oscillations observed in this sample. This suggests that coherence in our qubits is limited by the same mechanisms which give rise to inhomogeneous broadening. Its origin can be ﬂuctuations of the junction critical current as well as magnetic ﬁeld instabilities due to motion of trapped vortices. 8. Conclusion In conclusion, we described operation of a commercially-fabricated rf-SQUID in the coherent quantum regime as a phase qubit. The rich variety of quantum eﬀects accessible experimentally in superconducting circuits based on Josephson junctions bears great potential in the study of origins and characteristics of decoherence in macroscopic objects and the possibility of realizing solid-state quantum computing.

a This term originates in NMR experiments, where an inhomogeneous magnetic ﬁeld leads to diﬀerent energy separations in spatially separated nuclei, which constitute the individual qubits.

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(a)

(b)

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Resonance width Δ f [GHz]

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(a) Spectroscopic resonance peaks in the excited state population probability vs. applied external ﬂux. Both width and amplitude of the Lorentzian peak increase with increasing microwave power. (b) Full width at half maximum of the resonance peaks shows a linear dependence on microwave amplitude. For small amplitudes, saturation of the width above zero is observed due to inhomogeneous resonance broadening.

Figure 4.

References 1. J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo and J.E. Lukens, Nature 406, 43 (2000). 2. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J. Harmans, T.P. Orlando, Seth Lloyd, J.E. Mooij, Science 290, 773 (2000). 3. J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, and S. Lloyd, Science 285, 1036 (1999); I. Chiorescu, Y. Nakamura, C.J. Harmans, and J.E. Mooij, Science 299, 1869 (2003); B.L.T. Plourde, T.L. Robertson, P.A. Reichardt, T. Hime, S. Linzen, C.-E. Wu, and J. Clarke, Phys. Rev. B 72, 060506(R) (2005). 4. Hypres Inc., Elmsford, NY, USA. 5. R.W. Simmonds, K.M. Lang, D.A. Hite, D.P. Pappas, and J.M. Martinis, Phys. Rev. Lett. 93, 077003 (2004); K. B. Cooper, M. Steﬀen, R. McDermott, R. W. Simmonds, S. Oh, D. A. Hite, D. P. Pappas, and J.M. Martinis, Phys. Rev. Lett. 93, 180401 (2004). 6. J.M. Martinis, K.B. Cooper, R. McDermott, M. Steﬀen, M. Ansmann, K. Osborn, K. Cicak, S. Oh, D.P. Pappas. R.W. Simmonds and C.C. Yu, Phys. Rev. Lett. 95, 210503 (2005). 7. T. L. Robertson, Fundamentals of Flux-based Quantum Computing, Ph.D. Thesis, University of California, Berkeley (2005).

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BLOCH OSCILLATIONS IN A JOSEPHSON CIRCUIT N. BOULANT, G. ITHIER, F. NGUYEN, P. BERTET, H. POTHIER, D. VION, C. URBINA, AND D. ESTEVE Quantronics, Service de Physique de l’Etat Condensé, URA 2464 CEA-Saclay, 91191 Gif-sur-Yvette, France

Bloch oscillations predicted to occur in current-biased single Josephson junctions have eluded direct observation up to now. Here, we demonstrate similar Bloch oscillations in a slightly richer Josephson circuit, the quantronium. The quantronium is a Bloch transistor with two small junctions in series, defining an island, in parallel with a larger junction. In the ground state, the microwave impedance of the device is modulated 2e periodically with the charge on the gate capacitor coupled to the transistor island. When a current I flows across this capacitor, the impedance modulation occurs at the Bloch frequency f = I /(2e ) , which yields Bloch sidebands in the spectrum of a reflected continuous microwave signal. We have measured this spectrum, and compared it to predictions based on a simple model for the circuit. We discuss the interest of this experiment for metrology and for mesoscopic physics.

1. Bloch oscillations in Josephson junctions The phenomenon of Bloch oscillations [1], was first considered for a quantum particle moving in a periodic potential and subject to a constant driving force (see [2] for a review). When the particle stays in the first Bloch band, its quasi-momentum changes linearly with time until it reaches the boundary of the first Brillouin zone, where it is Bragg-reflected to the symmetric opposite band-edge and increases again. The velocity of the particle oscillates during the motion. This phenomenon pertains to many physical situations where a quantum system with a periodic potential is subject to a constant drive [2]. Experimental evidence of Bloch oscillations can be found in particular for electrons in solid state superlattices [3], and for ultracold atoms in optical lattices [4]. Bloch oscillations are also predicted to occur in a single Josephson junction biased by a dc current I [5]. The variables describing the system are the charge Q on the junction capacitance and the phase difference δ across the junction. They form a set of conjugated variables: [Q, δ ] = i (2e) , with e the electron charge. The Hamiltonian writes:

H = Qˆ 2 /(2C ) − E J cos(δˆ ) − I δˆ /(2e) ,

(1.1)

with C the junction capacitance, and E J the Josephson energy. The Bloch oscillations should manifest here as a periodic oscillation of the voltage across the junction at the Bloch frequency f = I /(2e) . The locking of these oscillations by an external microwave signal applied to the junction is predicted to induce voltage steps at constant current in the current-voltage characteristic. These steps would be dual of Shapiro steps in voltage

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72 72 biased Josephson junctions, and provide an appealing solution for the metrology of currents. This experiment is however difficult because a source impedance large compared to the resistance quantum RQ = h /(2e) 2 6.5 k Ω is required over a wide frequency range. Although a Bloch-nose feature related to Bloch oscillations and an effect of microwaves have been observed [6-7], the locking regime has never been reached. In this work, we have performed a related experiment which shows a phenomenon analogous to Bloch oscillations in a multi-junction Josephson circuit, the quantronium [8], which is a recently developed quantum bit circuit.

Ng

δ

V

∆n Carrier

0

1/f

t

S (f) 2∆nf

L (Ng,δ=π) 0

1

2

3

4

5

N

ν

Fig (1). When a triangular gate charge modulation (top left) is applied to a quantronium (top right), its inductance varies periodically, which modulates the reflection coefficient of the device at microwave frequencies close to the plasma resonance of the larger junction on the right. When the extrema of the gate sweep correspond to integer or half-integer values of N g , the modulation of the reflection factor is the same as in the case of a linear sweep of the gate charge (continuing dashed line). The spectrum of the reflected signal then presents sidebands shifted from the carrier by multiples of the Bloch frequency 2 ∆nf . For an arbitrary periodic sweep of the gate charge, other sidebands are predicted at all frequency shifts kf .

2. Bloch oscillations with the quantronium The quantronium circuit, schematically shown in Fig. (1), can be figured as a Bloch transistor (i.e. two small junctions in series) in parallel with a larger junction. The circuit is connected to a microwave transmission line used for probing its impedance. The transistor Hamiltonian is controlled by the reduced charge on the gate electrode N g = C g V /(2e) , with C g the gate capacitance, and by the reduced flux δ = φ / ϕ 0 , with φ the flux threading the loop, and ϕ 0 = / 2e :

Hˆ = EC ( Nˆ − N g ) 2 − E J cos(δ / 2)cos θˆ

(1.2)

Here, the conjugate variables are the island pair number N and phase θ . All the circuit properties vary periodically with N g (period 1) and with δ (period 2π ). We consider here the case of a quantronium in which the large junction is loaded by an on-chip capacitor in order to place its plasma resonance frequency in the convenient 1 − 2GHz frequency range. Assuming the circuit stays in the ground state of Hamiltonian (1.2) with

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73 energy ε ( N g , δ ) , the effective inductance of the transistor L = ϕ 02 /(∂ 2ε / ∂δ 2 ) varies with N g as shown in Fig. (1), which modifies the reflection factor of the circuit. The calculations were performed for the parameters of the sample: EC 0.5 K, E J 1.0 K . If the gate charge N g could increase linearly in time, i.e. in the case of a constant gate current I , the circuit properties, such as the island potential or the impedance, would be modulated at the Bloch frequency f B = I / 2e . When a cw signal is injected on the microwave line, the spectrum of the reflected signal is expected to develop sidebands shifted from the carrier by the Bloch frequency. Harmonics are also expected since the modulation of the reflection coefficient is not purely sinusoidal. Of course it is impossible to maintain a constant current through the gate capacitor for a long time. It is nevertheless possible to carry out an experiment showing the Bloch oscillations predicted for a constant current, as now explained. Let us consider the case of a triangular modulation with frequency f and span ∆n of N g between two exactly integer or half-integer values, as shown in Fig. (1). In this situation, the modulation of the inductance, and thus of the reflection coefficient, is expected to be exactly the same as in the case of a constant gate current i = 2∆n f (2 e ) because the inductance modulation is periodic, and symmetric around integer and half-integer values of N g . The corresponding Bloch frequency is f B = 2∆nf . With the inductance deduced from the ground-state energy [8], one easily calculates for an arbitrary periodic gate modulation the Fourier series giving the reflection factor

R(t ) = ∑ k rk exp(2iπ kft ) .

(1.3)

The coefficients rk provide the reflected amplitudes at all frequencies kf shifted from the carrier frequency. The Bloch sidebands shifted by ± f B dominate when the triangular gate modulation is properly tuned, as predicted by the simple physical picture presented above. The spectrum S (ν ) calculated following this procedure is in excellent agreement with a full calculation of the reflected signal in the linear excitation regime of the circuit.

3. Observation of Bloch oscillations A quantronium sample, with an Al/AlOx/Al on-chip capacitor, was fabricated using electron-beam lithography, placed in a sample-holder fitted with microwave lines, and cooled down to 30 mK. The gate was connected to a 250 MHz bandwidth rf line, and the quantronium was connected both to a microwave injection line and to a measuring line through a circulator. The injected microwave power at the circuit level was chosen to maintain the phase dynamics in the harmonic oscillator regime. The reflected signal was sent through two decoupling circulators to a cryogenic amplifier at 4.2 K, and was finally amplified at room temperature. The effective gain of the measuring line was 76 dB. The signal was then either demodulated with the input cw signal in order to directly observe Bloch oscillations, which is possible at low Bloch frequencies, or sent to a spectrum analyzer. A series of spectra showing the Bloch lines is shown in Fig. (2) for a properly

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74 74 tuned triangular gate modulation with ∆n = 10 and zero offset at different sweep frequencies. The Bloch lines have a narrow sub-Hz linewidth. We have checked the selection rules for the offset and the amplitude predicted from the triangular modulation patterns mimicking a linear evolution of the gate charge. When the offset is detuned, other non-Bloch sidebands appear, as calculated using Eq. (1.3). This experiment also revealed that the reflected signal randomly alternates between two values corresponding to gate charges shifted by one electron. This phenomenon is due to spurious quasiparticle states lying in the superconducting gap. This la ck of robustness of the parity, commonly observed in Cooper pair devices, may be further enhanced when the gate charge is swept fast and over a wide range.

-30

S(ν) dBm -50

f 4 kHz 3 kHz 2 kHz 1 kHz 0.5 kHz

-70 -90 160992 0

161000 0 ν (kHz)

161008 0

Fig (2). Spectrum of the reflected signal showing Bloch lines when a triangular gate charge modulation with zero offset and amplitude ∆n = 10 is applied to the gate, for different frequencies f. Curves have been shifted vertically for clarity. The arrow indicates the predicted position of the Bloch lines at fC ± 2 ∆n f , with fC = 1.61GHz the carrier frequency. The second Bloch harmonic is visible on some traces.

The dependence of the sideband amplitude on ∆n is compared to predictions based on Eq. (1.3) on Fig. (3), for three frequencies of the triangular modulation. The agreement with the spectrum calculated with the inductance modulation deduced from the sample parameters is satisfactory at low frequency. The amplitude of an even sideband is maximum when it corresponds to a Bloch line. The dependence on ∆n was checked up to k=40. The agreement strongly degrades when the sweep frequency increases f ≥ 1MHz . This discrepancy might arise from parity changes during the sweeps and from Zener transitions. High frequency operation of the quantronium should indeed be ultimately limited by Zener tunneling towards upper Bloch bands. The gap with the first excited band normally vanishes at the points {N g = 1/ 2 mod(1), δ = π } . In order to avoid this zero gap point, an asymmetry was introduced between the transistor junctions, yielding a designed gap frequency of 2 GHz. This allowed to operate the device close to δ = π where the gate-charge modulation of the inductance is maximum. The calculated Zener transition rate for this gap does not account however for the observed sweep frequency dependence of the spectrum in the present experiment.

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4. Perspectives for metrology and for mesoscopic physics The observation of Bloch oscillations in a Josephson circuit is a first step towards their use for the metrology of currents. A significant effort is presently devoted to close the quantum metrology triangle, which relates time, voltage and current units. The production of a current directly related to a frequency, or reversely, would allow to check consistency with the Josephson and Quantum Hall Effects. The currents presently produced by single electron pumps, in the pA range, are however too small compared to the currents needed in Quantum Hall Effect experiments, even when using cryogenic current comparators based on superconducting transformers. Metrological current sources in the 100 pA are clearly needed to improve the accuracy, presently limited at about 10 −6 .

experiment

f = 0.1 MHz f = 1 MHz f = 10 MHz

-60

theory

sideband at 2f

-80 -100

0

1

2

3

4

level (dBm)

5

6 0

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

sideband at 4f

-60 -80 -100

0

1

2

3

4

5

6 0

1

sideband at 6f

-60 -80 -100

0

1

2

3

4

5

6 0

1

sideband at 8f

-60 -80 -100

0

1

2

3

∆n

4

5

6 0

1

∆n

Fig (3). Comparaison of the measured sideband amplitude dependence (left) with the predicted one (right), for the even sidebands at three different sweep frequencies. A good agreement is found only at the lowest frequency, with a maximum when the sideband corresponds to a Bloch line. The amplitude dependence departs from the predicted one for sweep frequencies ≥ 1 MHz .

The demonstration of Bloch oscillations at larger frequencies than achieved in this work, with the injection of a dc current, is thus an important goal. In order to inject such a current, the gate capacitor has to be replaced by an impedance large compared to the resistance quantum over a wide frequency range. Large chromium resistors, and linear arrays of junctions have already been used to achieve this goal, and the successful

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76 76 observation of a Bloch nose in the current-voltage characteristics of a small Josephson junction already proves that the Bloch oscillation regime is attainable [7]. The phenomenon of quantum phase slippage in an ultrasmall superconducting wire was also recently proposed to reach this regime [9]. Our experiment is also related to the electron counting experiment [10], which probes the voltage oscillations of a small electrode in a tunnel junction array with a radiofrequency single electron transistor (RFSET). In our experiment, the passage of a Cooper pair through the device induces a cyclic evolution of the microwave reflection coefficient. This time dependence results in Bloch lines in the spectrum, but its direct observation in the time domain is also possible, and was performed in the present experiment up to Bloch frequencies of a few kHz. Counting the number of periods completed in a given time, which could be achieved by processing the demodulation quadrature signals, would provide a direct measure of the current. The transposition of the injected current into the frequency domain may also be useful in mesoscopic physics since the shape of the Bloch lines is determined by the fluctuations of the injected current. The measured spectrum is thus related to statistical properties of the injected current, such as the third moment of its Full Counting Statistics. In conclusion, we have observed Bloch oscillations in the quantronium circuit by microwave reflectometry when suitable signals are applied to the gate electrode of the device. The Bloch sidebands observed in the spectrum vary as predicted theoretically with the amplitude and offset of the gate modulation at small sweep frequency.

Acknowledgments We acknowledge M. Devoret for discussions and for providing us with a sample for preliminary experiments, H. Mooij, D. Haviland, H. Grabert, and F. Hekking for discussions, and P.F. Orfila, P. Sénat, and M. Juignet for technical help. This work was supported by the european project Eurosqip.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

F. Bloch, Z. Phys. 52, 555 (1928). T. Hartmann et al., New J. Phys. 6 2 (2004). C. Waschke et al., Phys. Rev. Lett. 70 3319 (1993). M.G. Raizen, C. Salomon and Q. Niu, Phys. Today 50 30 (1997). K.K. Likharev and A.B. Zorin, J. Low. Temp. Phys. 59, 347 (1985). L.S. Kuzmin and D.B. Haviland, Phys. Rev. Lett. 67, 2890 (1991). M. Watanabe and D.B. Haviland, Phys. Rev. B 67, 094505 (2003). D. Vion et al., Science 296, 886 (2002). J.E. Mooij and Yu. V. Nazarov, cond-mat/0511535. J. Bylander, T. Duty, and P. Delsing, Nature 434, 361 (2005).

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MANIPULATION OF MAGNETIZATION IN NONEQUILIBRIUM SUPERCONDUCTING NANOSTRUCTURES∗

F. GIAZOTTO, F. TADDEI, AND F. BELTRAM NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy E-mail: [email protected] R. FAZIO NEST CNR-INFM and Scuola Normale Superiore, Pisa, I-56126, Italy and International School for Advanced Studies (SISSA), Trieste, I-34014, Italy

Electrostatic control of the magnetization of a normal mesoscopic conductor is analyzed in a hybrid superconductor-normal-superconductor system. This eﬀect stems from the interplay between the non-equilibrium condition in the normal region and the Zeeman splitting of the quasiparticle density of states of the superconductor subjected to a static in-plane magnetic ﬁeld. Unexpected spin-dependent eﬀects such as magnetization suppression, diamagnetic-like response of the susceptibility as well as spin-polarized current generation are the most remarkable features presented. The impact of scattering events is evaluated and let us show that this eﬀect is compatible with realistic material properties and fabrication techniques.

1. Introduction The interplay between out-of-equilibrium transport and superconductivity was recently successfully exploited in a number of systems in order to implement Josephson transistors 1,2,3 , π junctions 4 and electron microrefrigerators 5 , just to mention a few relevant examples. In this work we explore its potential in the area of magnetism 6 and spintronics 7 and present a novel approach to control the magnetization and spin-dependent properties of a mesoscopic normal conductor 8 . In particular, we show that manipulation of the (nonequilibrium) distribution of a normal metal through an applied voltage can lead to the control of a number of spin-dependent phenomena. The key ingredients are superconductor electrodes (with energy gap Δ) and a weak external magnetic ﬁeld. As we shall argue, the interplay between Zeeman splitting and nonequilibrium yields dramatic consequences on quasiparticle dynamics stemming from the peculiar shape of the superconductor DOS whose energy gap compares well with magnetic ﬁelds readily accessible experimentally. ∗ This

work was supported in part by MIUR under FIRB “Nanotechnologies and Nanodevices for Information Society”, contract RBNE01FSWY and by RTN-Spintronics.

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H RI

(a)

N

S

RI

S

VC

(b)

N

S

S

VC Figure 1. Scheme of the structure investigated. An in-plane static magnetic ﬁeld H is applied across the whole SINIS system ((a), a-type setup) or localized at the S electrodes ((b), b-type setup). A ﬁnite voltage bias VC drives the normal metal out-of-equilibrium allowing to control its magnetization. The N wire is assumed quasi-one-dimensional.

2. Operation and discussion Let us consider the system sketched in Fig. 1. It consists of two superconducting reservoirs (S) connected by a mesoscopic normal metal wire (N) through tunnel contacts (I) of resistance RI . The structure is biased at a voltage VC and in the presence of a static in-plane magnetic ﬁeld H, applied either across the whole structure (Fig. 1(a), in the following referred to as a-type setup) or localized at the superconductors (Fig. 1(b), b-type setup). For the sake of simplicity let us assume a symmetric structure (a resistance asymmetry would not change the overall physical picture). As for the superconductors we focus on conventional low criticaltemperature thin (< 10 nm) ﬁlms. In this case the eﬀect of H on the electron spin becomes dominant and, assuming negligible spin-orbit interaction 9 , the superconductor DOS per spin is BCS-like but shifted by the Zeeman energy (EH = μB H), NσS (ε) = NFN |Re[(ε + σEH )/2 (ε + σEH )2 − Δ2 ]| 10 , where ε is the quasiparticle excitation energy measured from the Fermi energy (εF ), NFN is the DOS in the normal state at εF (2 spin directions), μB is the Bohr magneton, and σ = ±1 refers to spin parallel(antiparallel) to the ﬁeld. At a ﬁnite bias VC , in the presence of H and in the limit of negligible inelastic collisions, the steady-state distribution functions in the metal wire are spin-dependent and are given by 11 fσ (ε, VC , H) =

NσL F L + NσR F R , NσL + NσR

(1)

where F L(R) = f0 (ε±eVC /2), NσL = NσS (ε+eVC /2), NσR = NσS (ε−eVC /2), f0 (ε) is the Fermi distribution at lattice temperature T and e is the electron charge. Owing to the nonequilibrium regime driven by the applied electric ﬁeld, the quasiparticle distributions corresponding to diﬀerent spin species behave diﬀerently, f+(−) being shifted towards lower(higher) energy. The magnetic properties of the N region are entirely determined by its (spin-dependent) quasiparticle distribution functions.

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μB H/Δ = 0.4

6

6

T = 0.1Tc

0 0

(a)

ε

0.9

+

0.6 0.4

0.1 1

2

eVC /Δ

m-3) -1

(JT

T = 0.1Tc

ε

+

_

0.4 0.5

1.0

eVC /Δ

0.4

1.5

μ-

μ+

= Pauli

0.3

(b)

ε

2μB H N(ε)

2.0 = 0

+

εF

N(ε)

_

=0

+

_

-1

0

eVC /Δ

1

2 μB H

N(ε)

= - Pauli

(c)

0.0 -2

N(ε)

μμ+

0

0.2

2 μB H

ε

(b’)

0.2

-12 0.0

μB H/Δ

εF

_

3

μB H/Δ = 0.1 -6

Nonequilibrium

(a’)

0.8

3

0.2 0.1

0 0

T /Tc = 0.95 Equilibrium

(JT-1m-3)

0.3

(JT

-1

m-3)

12

-0.5 -1.0

χ/χPauli

2

Figure 2. (a) Magnetization density M vs bias voltage VC at T = 0.1 Tc for diﬀerent magnetic ﬁelds (H) for a-type setup (see Fig. 1(a)). Inset: M vs VC for diﬀerent temperatures at H = 0.2 Δ/μB . (b) The same as in (a) for b-type setup. (a’) Schematic diagrams of the N region density of states and quasiparticle occupation both at equilibrium (left) and nonequilibrium (right) for atype setup. (b’) The same as in (a’) for b-type setup. (c) Contour plot of the normalized magnetic susceptibility χ/χP auli vs VC and H at T = 0.1 Tc for b-type setup.

The magnetization density in the wire is indeed given by M(VC , H) = μB dε [N+N (ε)f+ (ε) − N−N (ε)f− (ε)],

(2)

where NσN (ε) = 12 N N (εF + ε + σμB H) and N N (ε) is the N region DOS in the absence of magnetic ﬁeld. The function M(VC , H) vs VC is displayed in Fig. 2(a,b) for diﬀerent magnetic-ﬁeld values. We assumed a silver (Ag) N region (with NFN = 1.03 × 1047 J−1 m−3 ) at temperature T = 0.1 Tc , where Tc = (1.76 kB )−1 Δ = 1.196 K is the critical temperature of bulk aluminum (Al, the material forming the S regions) and kB is the Boltzmann constant. When H is applied across the whole SINIS structure (a-type setup), M decreases upon increasing VC starting from its equilibrium value MP auli = μ2B NFN H typical of a Pauli paramagnet 6 (see Fig. 2(a)). M shows a complete suppression for VC Δ/e, i.e. the N region is demagnetized. The inset of Fig. 2(a) shows how M(VC ) is weakly dependent on the lattice temperature up to T = 0.4Tc owing to the BCS Δ(T ) dependence together with the temperature-induced broadening of f0 (ε). Conversely, when the magnetic ﬁeld is localized at the S electrodes (b-type setup) a negative magnetization is induced in the wire (see Fig. 2(b)). Note that M is antiparallel to H. Therefore, the N region behaves as a “diamagnet”. For eVC Δ the wire susceptibility χ (shown in Fig. 1(c) at T = 0.1 Tc ) reaches the Pauli value but with opposite sign χ = ∂M/∂H = −μ2B NFN = −χP auli . This gives rise to a sort of “artiﬁcial” Pauli diamagnetism. Insight into the physical origin of this superconductivity-controlled magnetism

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can be qualitatively gained by considering the (zero-temperature) steady-state DOS diagrams of Fig. 2(a’,b’), where the normal metal is described by parabolic subbands typical of a free electronlike paramagnetic conductor such as silver. At equilibrium the occupation of quasiparticle states is identical for both spin species leading to M = MP auli and M = 0 for a-type and b-type setups, respectively. When VC = 0, electron distributions for the two spin populations are characterized by distinct chemical potentials μσ . Since μ+ < μ− the occupation of spin states antiparallel to the magnetic ﬁeld is favored with respect to the parallel one, owing to the opposite energy shift of the superconductor spin-dependent DOS in the external magnetic ﬁeld. This leads to a reduction of M for the a-type setup and to negative magnetization for the b-type setup. In particular, at VC ∼ Δ/e the chemical potential separation is δμ = μ+ − μ− ∼ −2μB H for both setups. This shows a full electrostatic control of the magnetization, a unique feature of the present system. The experimental accessibility of this operational principle must be carefully assessed. Electrons in metals experience both elastic and inelastic collisions. The latter drive the system to equilibrium and can be expected to hinder the observation of the phenomena discussed here. Our analysis will show a remarkable robustness of these eﬀects. At low temperatures (typically below 1 K) electron-electron scattering 12 , and scattering with magnetic impurities 13,14 are the dominant sources of inelastic collisions 15,14,16 . The eﬀect of electron-electron scattering due to direct Coulomb interaction on the spin-dependent distributions can be accounted for by solving a pair of coupled stationary kinetic equations. This can be done by generalizing the method outlined in Ref. 17 to a spin-dependent system.√The strength of the screened electron-electron interaction 18 is given by Kcoll = (L/ 2)(RI /RK ) Δ/D 19,20,17 , where RK = h/2e2 , L is length and D the wire diﬀusion constant. We stress that Kcoll is linear both in the wire length and in the tunnel contact resistance. We analyzed quantitatively a realistic Ag/Al SINIS microstructure 5 with L = 1 μm, wire cross-section A = 0.2 × 0.02 μm2, and RI = 103 Ω. Figure 3(a) illustrates the eﬀect of electron-electron scattering. We solved the kinetic equations with H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc for several Kcoll values from negligible (Kcoll = 0, square), to moderate (Kcoll = 1, circle) and extreme (Kcoll = 100, solid line) 18,19 . As expected, electron-electron interactions have virtually no impact. By increasing the strength of Coulomb interaction the quasiparticle distribution of each spin species relaxes toward spin-dependent Fermi functions still characterized by diﬀerent chemical potentials (a similar eﬀect is expected in the presence of interaction with the lattice phonons 21 ). As a result the nonequilibrium magnetization in the normal wire here presented is virtually unaﬀected. The situation drastically changes if we assume the presence of magnetic impurities in the N region, due to the resulting spin-ﬂip processes. Above the Kondo temperature (TK ), the distribution functions can be calculated including in the kinetic equations an additional term derived by generalizing the theory developed by G¨ oppert and Grabert in Ref. 22 . It is noteworthy to mention that its strength

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coll

fσ (ε)

VC = 1.8 Δ/e H = 0.4 Δ/μB T = 0.1Tc

(a) -0.3

ε /Δ

0.3

0.4

0 -0.8

0.5

cm

0 0.1 1 20

0.0

ε /Δ

0.0

0.4

0.4

ε /Δ

0.8

1 -4

-1

fσ (ε)

0 0.1 1

(c) -0.4

-0.4

0

1

0 -0.8

cm(ppm)

VC = 1.8 Δ/e H = 0.4 Δ/μB T = 0.1Tc

(b)

-0.4

-3

-0.5

(JT m )

0

1

0 1 100

fσ (ε)

1

0.8

-8

cm= 0

0.001

T = 0.1Tc H = 0.4 Δ/μB (d)

-12 0

1

0.1 0.01

eVC /Δ

2

3

Figure 3. (a) Spin-dependent distribution functions fσ (ε) vs energy ε calculated for three Kcoll values at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc . Solid(dashed) lines correspond to antiparallel(parallel) spin species. (b) fσ (ε) vs ε calculated for various cm values at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc for a-type setup. (c) The same as in (b) for b-type setup. (d) Magnetization density M vs VC at H = 0.4 Δ/μB and T = 0.1 Tc for diﬀerent magnetic impurity concentration. Open circles refer to b-type setup, ﬁlled triangles to a-type setup for cm = 0.001. The latter were shifted by −MP auli . Data in (b)-(d) were obtained assuming D = 0.02 m2 s−1 , TK = 40 mK and S = 12 .

turns out to be proportional, apart from the electron and magnetic impurity spin coupling constant, to the total number of magnetic impurities present within the wire volume (i.e., to the product cm LA, with cm the impurity concentration) and RI . The resulting distribution functions relative to the a-type setup are shown in Fig. 3(b) at H = 0.4 Δ/μB , VC = 1.8 Δ/e and T = 0.1 Tc ≈ 120 mK for various cm values expressed in parts per million (ppm). We assumed D = 0.02 m2 s−1 (typical of high-purity Ag), magnetic impurities with spin S = 12 , and TK = 40 mK (as appropriate, for example, for Mn impurities in Ag) 24 . By increasing cm , examination of the ﬁgure immediately shows that spin-dependent distributions are marginally aﬀected even for impurity concentrations as large as 20 ppm. This shows that in the a-type setup the nonequilibrium M is relatively insensitive to large amounts of magnetic impurities. Figure 3(c) shows the fσ (ε) calculated for various cm values for the b-type setup. In such a case, by contrast, the spin-dependent distribution functions tend to merge for much lower values of cm thus suppressing the induced magnetization. In the presence of a magnetic ﬁeld across the N region (a-type setup) impurity spins tend to polarize yielding a suppression of spin-ﬂip relaxation processes for the ﬁeld intensities of interest here 13,22,23,14 . This does not occur in the b-type setup and makes magnetic impurities more eﬀective in mixing spins. The full behavior of M(VC ) for b-type setup at T = 0.1 Tc and H = 0.4 Δ/μB is

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displayed in Fig. 3(d) for several cm values (open circles). For comparison, M(VC ) for a-type setup (ﬁlled triangles) is shown at low impurity concentration. We wish to underline the robustness of the induced magnetization, M being suppressed only for rather large concentrations: the latter can in fact be limited to less than 0.01 ppm in currently available high-purity metals 24 . 3. Conclusions In conclusion, we have presented a scheme to control the magnetic properties of a mesoscopic metal. Magnetism suppression as well as artiﬁcial Pauli diamagnetism can be accessed in metal-superconductor microstructures thus making available a number of characteristics of much relevance in light of possible applications: (1) Full-electrostatic control of magnetization over complex nanostructured metallic arrays for enhanced performance and optimized device geometries; (2) reduced power dissipation (10−14 ÷ 10−11 W depending on the control voltage) owing to the very small driving currents intrinsic to SIN junctions; (3) high magnetization switching frequencies up to 1011 Hz 17 ; (4) ease of fabrication that can take advantage of the well-established metal-based tunnel junction technology. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

A.F. Morpurgo, T. M. Klapwijk, and B. J. van Wees, Appl. Phys. Lett. 72, 966 (1998). A. M. Savin et al., Appl. Phys. Lett. 84, 4179 (2004). F. Giazotto et al., Appl. Phys. Lett. 83, 2877 (2003). J. J. A. Baselmans et al., Nature 397, 43 (1999). J. P. Pekola et al., Phys. Rev. Lett. 92, 056804 (2004). K. Yosida, Theory of Magnetism, Springer Series in Solid-State Sciences, Vol. 122 (Springer-Verlag, Berlin, 1996). I. Zutic, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). F. Giazotto, F. Taddei, R. Fazio, and F. Beltram, Phys. Rev. Lett. 95, 066804 (2005). P. M. Tedrow and R. Meservey, Phys. Rev. Lett. 27, 919 (1971). R. Meservey, P. M. Tedrow, and P. Fulde, Phys. Rev. Lett. 25, 1270 (1970). D. R. Heslinga and T. M. Klapwijk, Phys. Rev. B47, 5157 (1993). B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Systems, edited by A. L. Efros and M. Pollak (Elsevier, Amsterdam, 1985). A. Kaminski and L. I. Glazman, Phys. Rev. Lett. 86, 2400 (2001). A. Anthore et al., Phys. Rev. Lett. 90, 076806 (2003). H. Pothier et al., Phys. Rev. Lett. 79, 3490 (1997). K. E. Nagaev, Phys. Rev. B52, 4740 (1995). F. Giazotto et al., Phys. Rev. Lett. 92, 137001 (2004). B. L. Altshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 75, 1610 (1978) [Sov. Phys. JETP 48, 812 (1978)]. A. Kamenev and A. Andreev, Phys. Rev. B60, 2218 (1999). B. Huard et al., Solid State Commun. 131, 599 (2004). M. L. Roukes et al., Phys. Rev. Lett. 55, 422 (1985). G. G¨ oppert and H. Grabert, Phys. Rev. B68, 193301 (2003). G. G¨ oppert et al., Phys. Rev. B66, 195328 (2002). F. Pierre et. al., Phys. Rev. B68, 085413 (2003).

Superconducting Qubits

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DECOHERENCE AND RABI OSCILLATIONS IN A QUBIT COUPLED TO A QUANTUM TWO-LEVEL SYSTEM

S. ASHHAB1, J. R. JOHANSSON1 AND FRANCO NORI1,2 1

Frontier Research System, The Institute of Physical and Chemical Research (RIKEN) Wako-shi, Saitama, Japan 2 Center for Theoretical Physics, CSCS, Department of Physics, University of Michigan Ann Arbor, Michigan, USA

In this paper we review some of our recent results on the problem of a qubit coupled to a quantum two-level system. We consider both the decoherence dynamics and the qubit’s response to an oscillating external ﬁeld.

1. Introduction Signiﬁcant advances in the ﬁeld of superconductor-based quantum information processing have been made in recent years1 . However, one of the major problems that need to be treated before a quantum computer can be realized is the problem of decoherence. Recent experiments on the sources of qubit decoherence saw evidence that the qubit was strongly coupled to quantum two-level systems (TLSs) with long decoherence times2 . Furthermore, it is well known that the qubit decoherence dynamics can depend on the exact nature of the noise causing the decoherence. Therefore, an environment comprised of a large number of TLSs that are all weakly coupled to the qubit will generally cause non-Markovian decoherence dynamics in the qubit. The two above observations comprise our main motivation to study the decoherence dynamics of a qubit coupled to a quantum TLS. A related problem in the context of the present study is that of Rabi oscillations in a qubit coupled to a TLS. That problem is of great importance because of the ubiquitous use of Rabi oscillations as a qubit manipulation technique. We perform a systematic analysis with the aim of understanding various aspects of this phenomenon and seeking useful applications of it. Note that the results of this analysis are also relevant to the problem of Rabi oscillations in a qubit that is interacting with other surrounding qubits. This paper is organized as follows: in Sec. 2 we introduce the model system and Hamiltonian. In Sec. 3 we analyze the problem of qubit decoherence in the absence of an external driving ﬁeld. In Sec. 4 we discuss the Rabi-oscillation dynamics of the qubit-TLS system. We ﬁnally conclude our discussion in Sec. 5.

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2. Model system and Hamiltonian The model system that we shall study in this paper is comprised of a qubit that can generally be driven by an harmonically oscillating external ﬁeld, a quantum TLS and their weakly-coupled environment3 . The Hamiltonian of the system is given by: ˆ ˆ q (t) + H ˆ TLS + H ˆI + H ˆ Env , H(t) =H

(1)

ˆ q and H ˆ TLS are the qubit and TLS Hamiltonians, respectively, H ˆ I describes where H ˆ the coupling between the qubit and the TLS, and HEnv describes all the degrees of freedom in the environment and their coupling to the qubit and TLS. The (generally time-dependent) qubit Hamiltonian is given by: ˆ q (t) = − Eq sin θq σ ˆx(q) + cos θq σ ˆz(q) + F cos(ωt) sin θf σ ˆx(q) + cos θf σ ˆz(q) , (2) H 2 (q)

where Eq and θq are the adjustable control parameters of the qubit, σ ˆα are the Pauli spin matrices of the qubit, F and ω are the amplitude (in energy units) and frequency, respectively, of the driving ﬁeld, and θf is an angle that describes the orientation of the external ﬁeld relative to the qubit σ ˆz axis. We assume that the TLS is not coupled to the external driving ﬁeld, and its Hamiltonian is given by: ˆ TLS = − ETLS sin θTLS σ (3) H ˆx(TLS) + cos θTLS σ ˆz(TLS) , 2 where the parameters and operators are deﬁned similarly to those of the qubit, except that the parameters are uncontrollable. The qubit-TLS interaction Hamiltonian is given by: ˆI = − λ σ H ˆ (q) ⊗ σ ˆz(TLS) , 2 z

(4)

where λ is the (uncontrollable) qubit-TLS coupling strength. Note that, with an ˆ I .3 appropriate basis transformation, this is a rather general form for H 3. Qubit decoherence in the absence of a driving field We start by studying the eﬀects of a single quantum TLS on the qubit decoherence ˆ Env is small enough that its eﬀect on the dynamics dynamics. We shall assume that H of the qubit+TLS system can be treated within the framework of the Markovian Bloch-Redﬁeld master equation approach. The quantity that we need to study is therefore the 4 × 4 density matrix describing the qubit-TLS combined system. Following standard procedures we can write a master equation that describes the time-evolution of that density matrix. We shall not include that master equation explicitly here. Once we ﬁnd the dynamics of the combined system, we can trace out the TLS degree of freedom to ﬁnd the dynamics of the reduced 2 × 2 density matrix describing the qubit alone. From that dynamics we can infer the eﬀect of the TLS on the qubit decoherence dynamics.

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3.1. Analytic results for the weak-coupling limit If we take the limit where λ is much smaller than any other energy scale in the problem6 , and we take the TLS decoherence rates to be substantially larger than those of the qubit, we can perform a perturbative calculation on the master equation and obtain the following approximate expressions for the leading-order corrections (q) (q) to be added to the background relaxation and dephasing rates, Γ1 and Γ2 ( = 1): (q)

δΓ1 ≈ (q)

δΓ2 ≈

(TLS)

(q)

(TLS)

(q)

(q)

Γ2 + Γ2 − Γ1 1 2 2 λ sin θq sin2 θTLS 2 2 (TLS) (q) (q) 2 Γ2 + Γ2 − Γ1 + (Eq − ETLS ) Γ2 − Γ2 1 2 2 λ sin θq sin2 θTLS . 2 4 (TLS) (q) 2 Γ2 − Γ2 + (Eq − ETLS )

(5)

The above expressions can be considered a generalization of the well-known results of the traditional weak-coupling approximation. Those results are obtained if we take the qubit decoherence rates to be much smaller than those of the TLS. We shall discuss shortly, however, that our expressions have a wider range of validity. 3.2. Numerical solution of the master equation Given the large number of parameters that can be varied, we restrict ourselves to certain special cases that we ﬁnd most interesting to analyze6 . Since the TLS eﬀects on the qubit dynamics are largest when the two are resonant with each other, we set Eq = ETLS . We are therefore left with the background decoherence rates and the coupling strength as free parameters that we can vary in order to study the diﬀerent possible types of behaviour in the qubit dynamics. We ﬁrst consider the weak-coupling regimes. Here we only discuss the relaxation dynamics (see Ref.4 for full analysis). Figure 1 shows the relative correction to the qubit relaxation rate as a function of time for three diﬀerent sets of parameters diﬀering by the relation between the qubit and TLS decoherence rates, maintaining (q) (q) (TLS) (TLS) /Γ1 = 2. As a general simple rule, which is the relation Γ2 /Γ1 = Γ2 inspired by Fig. 1(a), we ﬁnd that the relaxation rate starts at its unperturbed value and follows an exponential decay function with a characteristic time given by (TLS) (q) (q) + Γ2 − Γ1 )−1 , after which it saturates at a steady-state value given by (Γ2 Eq. (5) (with Eq = ETLS ): dPex (t)/dt (q) (q) (TLS) (q) (q) ≈ −Γ1 − δΓ1 1 − exp − Γ2 + Γ2 − Γ1 t . (6) Pex (t) − Pex (∞) It turns out that all the curves shown in Fig. 1 agree very well with Eq. (6). In the limit when the TLS decoherence rates are much larger than those of the qubit, the qubit decoherence rate saturates quickly to a value that includes the correction given in Eq. (5). In the opposite limit, i.e. when the TLS decoherence rates are much smaller than those of the qubit, the contribution of the TLS to the qubit relaxation dynamics is a Gaussian decay function.

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x 10

−2

x 10

6

1.2

4

0 0

0.5

(q)

Γ1 t

1

1.5

−1

1

0.8

1

(q)

0.4

x 10

(c) δΓ(q)/Γ(q)

(q)

δΓ1 /Γ1

δΓ(q) /Γ(q) 1 1

0.8

1.2

(b)

(a)

2 0 0

0.5

Γ(q)t

1

1.5

1

0.4

0 0

0.5

(q)

Γ1 t

1

1.5

Figure 1. Relative corrections to qubit relaxation rate as a function of scaled time in the case of (TLS) (q) (a) strongly, (b) moderately and (c) weakly dissipative TLS. The ratio Γ1 /Γ1 is 10 in (a), 1.5 (q)

in (b) and 0.1 in (c). The solid, dashed, dotted and dash-dotted lines correspond to λ/Γ1 0.3, 0.6 and 0.9, respectively. θq = π/3 and θTLS = 3π/8.

= 0,

In the strong-coupling regime corresponding to large values of λ, one cannot simply speak of a TLS contribution to qubit decoherence. We therefore do not discuss that case here. Instead, we discuss the transition from weak to strong coupling. We use the criterion of visible deviations in the qubit dynamics from exponential decay as a measure of how strongly coupled a TLS is. The results of our calculations can be summarized as follows: a given TLS can be considered to interact weakly with the qubit if the coupling strength λ is smaller than the largest background decoherence rate in the problem. We have also checked the boundary beyond which the numerical results disagree with our analytic expressions given in Eq. (5), and we found that the boundary is similar to the one given above. That result conﬁrms the statement made in Sec. 3.2 that our analytic expressions describing the contribution of the TLS to the decoherence rates have a wider range of validity than those of the traditional weak-coupling approximation. 4. Dynamics under the influence of a driving field We now include the oscillating external ﬁeld in the qubit Hamiltonian (Eq. 2). Furthermore, since decoherence does not have any qualitative eﬀect on the main ideas discussed here, we neglect decoherence completely for most of this section. 4.1. Intuitive picture If we take the experimentally relevant limit λ Eq , we easily ﬁnd the energy levels to be given by: λcc ETLS + Eq 1 λcc 2 − ; E2,3 = ∓ , (7) E1,4 = ∓ (ETLS − Eq ) + λ2ss + 2 2 2 2 where λcc = λ cos θq cos θTLS , and λss = λ sin θq sin θTLS . If a qubit with energy splitting Eq is driven by a harmonically oscillating ﬁeld with a frequency ω close to its energy splitting as described by Eq. (2), one obtains

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the well-known Rabi oscillation peak in the frequency domain with on-resonance Rabi frequency Ω0 = F | sin(θf − θq )|/2 and full g ↔ e conversion probability on resonance. Note that the width of the Rabi peak in the frequency domain is also given by Ω0 . We can now combine the above arguments as follows: The driving ﬁeld tries to ﬂip the state of the qubit alone, with a typical time scale of Ω−1 0 , whereas the TLS . Therefore if Ω0 λss , can respond to the qubit dynamics on a time scale of λ−1 ss we expect the TLS to have a negligible eﬀect on the Rabi oscillations. If, on the other hand, Ω0 is comparable to or smaller than λss , the driving ﬁeld becomes a probe of the four-level spectrum of the combined qubit-TLS system. 4.2. Numerical results (q)

In our numerical analysis, we focus on the quantity P↑,max , which is deﬁned as the maximum probability for the qubit to be found in the excited state between (q) times t = 0 and t = 20π/Ω0 . Figure 2 shows P↑,max as a function of detuning (δω ≡ ω − Eq ) for diﬀerent values of coupling strength λ. In addition to the splitting of the Rabi peak into two peaks, we see an additional sharp peak at zero detuning and some additional dips. The peak can be explained as a two-photon transition where both qubit and TLS are excited from their ground states to their excited states (note that Eq = ETLS ). The dips can be explained as “accidental” suppressions of the oscillation amplitude when one energy splitting in the four-level spectrum is a multiple of another energy splitting in the spectrum. 1

1 (a)

0.8 0.6

0.8 0.6

0.2

0

0

0 δω/Ω0

2

↑,max

0.4

0.2 −2

P(q)

P(q)

0.4

(c)

0.8 0.6

↑,max

↑,max

P(q)

1

(b)

0.4 0.2

−2

0

δω/Ω0

2

4

0

−2

0

2 4 δω/Ω

6

0

(q)

Figure 2. Maximum qubit excitation probability P↑,max between t = 0 and t = 20π/Ω0 for λ/Ω0 = 0.5 (a), 2 (b) and 5 (c). θq = π/4, and θTLS = π/6.

4.3. Experimental considerations In the early experiments on phase qubits coupled to TLSs2 , the qubit relaxation (q) rate Γ1 (∼40 MHz) was comparable to the splitting between the two Rabi peaks λss (∼20-70 MHz). The constraint that Ω0 cannot be reduced to values much lower than the decoherence rate made the strong-coupling regime, where Ω0 λss , inaccessible. Although the intermediate-coupling regime was accessible, observation of the additional features in Fig. 2 discussed above would have required a time at

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least comparable to the qubit relaxation time. With the new qubit design7 , the qubit relaxation time has been increased by a factor of 20. Therefore, all the eﬀects that were discussed above should be observable. We ﬁnally consider how our results can be applied to the problem of characterizing an environment comprised of TLSs. Since measurement of the locations of the three peaks in Fig. 2 provides complete information about the four-level spectrum, both λcc and λss can be extracted from such results. One can therefore obtain the distribution of values of both ETLS and θTLS for all the TLSs in the environment. Note that in some cases, e.g. a phase qubit coupled to the TLSs through the operator of charge across the junction, we ﬁnd that θq = π/2, and therefore λcc vanishes for all the TLSs. In that case the two-photon peak would always appear at the midpoint (to a good approximation) between the two main Rabi peaks. Although that would prevent the determination of the values of ETLS and θTLS separately, it would provide information about the qubit-TLS coupling mechanism. 5. Conclusion We have studied the problem of a qubit that is coupled to an uncontrollable twolevel system and a background environment. We have derived analytic expressions describing the contribution of a quantum TLS to the qubit decoherence dynamics, and we have used numerical calculations to test the validity of those expressions. Our results can be considered a generalization of the well-known results of the traditional weak-coupling approximation. Furthermore, our results concerning the qubit’s response to an oscillating external ﬁeld can be used in experimental attempts to characterize the TLSs surrounding a qubit, which can then be used to reduce or eliminate the TLS’s detrimental eﬀects on the qubit operation. Acknowledgments This work was supported in part by the NSA and ARDA under AFOSR contract number F49620-02-1-0334; and also by the NSF grant No. EIA-0130383. One of us (S. A.) was supported by a fellowship from JSPS. References See e.g. J. Q. You and F. Nori, Phys. Today 58 (11), 42 (2005). R. W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004). For a more detailed discussion of our assumptions, see Refs.4,5 . S. Ashhab, J. R. Johansson, and F. Nori, cond-mat/0512677. S. Ashhab, J. R. Johansson, and F. Nori, cond-mat/0602577. Note that in this paper we shall only consider the zero temperature case. For a treatment of the ﬁnite temperature case, see Ref.4 . Furthermore, we take the energy splitting, which is the largest energy scale in the problem, to be much larger than all other energy scales, such that its exact value does not aﬀect any of our results. 7. J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005).

1. 2. 3. 4. 5. 6.

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PHASE-COUPLED FLUX QUBITS: CNOT OPERATION, CONTROLLABLE COUPLING AND ENTANGLEMENT

MUN DAE KIM Korea Institute for Advanced Study, Seoul 130-722, Korea E-mail: [email protected]

Coupling and entanglement of two ﬂux qubits are studied. The three-Josephson junction qubits with threading ﬂuxes are coupled by a connecting loop interrupted by Josephson junctions, but not by mutual inductance. By coupling the phase diﬀerences of the ﬂux qubits using the connecting loop instead of the weak induced ﬂuxes, present scheme oﬀers the advantages of a large and tunable level splitting in implementing the controlled-NOT (CNOT) operation. We obtain the coupling strength as a function of the coupling energy of the Josephson junctions in connecting loop. The Josephson junction in the connecting loop can be replaced by dc-SQUID loop and, by varying the magnetic ﬂuxes threading the dc-SQUID’s, we can control the coupling strength of the coupled qubits. We also suggest a scheme to achieve the maximum entanglement of two phase-coupled ﬂux qubits. Bell states are obtained at the ground and excited state of the coupled qubits system, as the energy levels are split and the two-qubit tunnelling channel is opened.

1. Introduction The qubits using Josephson junction device have been proposed as promising candidates of quantum computer owing to the advantages of relatively long decoherence time and possible scalability. These devices are named as the charge, phase and ﬂux qubit. The ﬂux qubit has a long decoherence time compared to the charge qubit that could be aﬀected severely by the background charge ﬂuctuation. The magnetic background, on the other hand, is relatively clean and stable. The ﬂux qubits have shown quantum superposition between two eigenstates.1 The observation of Rabi oscillation has been reported recently for one using three-junction qubit.2 Therefore, the feasibility of the ﬂux qubit as a practical quantum computer increases as time goes by. One of universal logic gates is composed of the controlled-NOT operation and special single-qubit rotations. However, previous coupling schemes which use mutual inductance between nearest neighbor qubits3,4 cannot give strong coupling strength enough to implement two-qubit coherent oscillations and CNOT gate operation. Hence we propose a design of two-qubit coupling connected by a loop carrying a persistent current (Fig. 1) whose coupling can be as large as the coupling energy of the Josephson junctions.5

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2. Phase-Coupled Flux Qubits The boundary condition of a qubit loop becomes approximately 2πn + 2πft − (φ1 + φ2 + φ3 ) = 0,

(1)

where the total ﬂux ft is the sum of external and induced one, ft = f +find = f + LΦs0I , where f ≡ Φext /Φ0 with the external ﬂux Φext and the superconducting unit ﬂux quantum Φ0 = h/2e and Ls is the self-inductance of the qubit loop. Thus we can obtain the relation for the current in the qubit loop given by φR(L)1 + φR(L)2 + φR(L)3 Φ0 nR(L) + fR(L) − IR(L) = − . (2) Ls 2π From this expression we can get for the eﬀective potential Ueﬀ (φ), R(L) Ueﬀ (φR(L)1 , φR(L)2 , φR(L)3 )

=

3

EJi (1 − cos φR(L)i ) + EnR(L) ,

(3)

i=1

where EJi = EJRi = EJLi is the Josephson coupling energy and the induced energy R(L) 2 = IR(L) /2Ls can be represented as5,6 En φR(L)1 + φR(L)2 + φR(L)3 2 Φ20 R(L) En = . (4) nR(L) + fR(L) − 2Ls 2π In order to obtain the eﬀective potential of connecting loop, we use the periodic boundary condition over the circumference of connecting loop with self inductance Ls and obtain the relation for the current in connecting loop such that Φ0 φ + φR + φR1 − φL1 I = − r − L , (5) Ls 2π where r is an integer. The eﬀective potential of connecting loop with the Josephson coupling energy EJ can be represented by Ueﬀ (φL , φR , φR1 , φL1 ) = En + EJ (1 − cos φL ) + EJ (1 − cos φR ),

(6)

Figure 1. Coupled ﬂux qubits with penetrating ﬂuxes of opposite directions, where φ’s are phase diﬀerences across the Josephson junctions with the Josephson coupling energies EJ i and EJ in qubit loops and connecting loop, respectively. Here, for example, we show the state, |↑↓, out of four current states. Thick lines show the diamagnetic (paramagnetic) current of left (right) qubit and thin line the current in connecting loop.

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where En

2 Φ20 φL + φR + φR1 − φL1 = . r− 2Ls 2π

(7)

tot The total eﬀective potential, Ueﬀ , of the coupled qubit system is given by tot L R Ueﬀ (φ) = Ueﬀ (φL1 , φL2 , φL3 ) + Ueﬀ (φR1 , φR2 , φR3 ) + Ueﬀ (φL , φR , φR1 , φL1 ), (8)

where φ = (φL1 , φL2 , φL3 , φR1 , φR2 , φR3 , φL , φR ). The local minima of the total tot (φ) eﬀective potential can be obtained by minimizing the eﬀective potential Ueﬀ tot with respect to φi , i.e., ∂Ueﬀ /∂φi = 0, which results in the current relations at the Josephson junctions as follows: 2πEJ1 sin φR1 Φ0 2πEJ1 −I + IL + sin φL1 Φ0 2πEJi sin φR(L)i IR(L) + Φ0 2πEJ I + sin φR(L) Φ0 I + IR +

= 0,

(9)

= 0,

(10)

= 0,

(11)

= 0,

(12)

where i=2,3. From these coupled equations, we can calculate four energy levels, Ess , corresponding to the two-qubit current states and show the lowest energy level as a function of the external ﬂuxes in Fig. 2(a), where s, s = 1/2 and −1/2 stand for

(a)

(b)

Figure 2. (a) Ground state energy diagram for coupled current qubits. A,B,C, and D denote the resonance lines where two energy levels of 1st excited as well as ground states of two-qubit states |s, s are degenerated. The external ﬂux diﬀerence, δfg , between the straight resonance lines originates from the coupling energy. We consider EJ = EJ . (b) Time evolutions of the coupled qubits with t1 = t2 =0.8GHz for EJ = EJ , fR = 0.457 and fL = 0.4837 (upper panel) and EJ = 0.005EJ , fR = 0.4995 and fL = 0.4793 (lower panel). The solid (dashed) lines display the probabilities that the target (control) qubits occupy the corresponding states.

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paramagnetic (spin up, ↑) and diamagnetic (spin down, ↓) current state, respectively. The Hamiltonian for the coupled qubits then can be written as H= (Ess |s, s s, s | + t1 |s, s −s, s | + t2 |s, s s, −s |) , (13) s,s =± 12

where t1 (t2 ) is the single-qubit tunnelling matrix element related with the left (right) qubit. The charging energies of the Josephson junctions in the qubit loop arising from the quantum ﬂuctuation of the Josephson junction phase give rise to the tunnelling between the qubit current states. Figure 2(b) shows the CNOT gate operation. Writing the two-qubit state as |Ψ(t) = c1 | ↓↓+c2 | ↑↓+c3 | ↓↑+c4 | ↑↑ with the initial condition, c1 = c2 = 1 and odinger c3 = c4 = 0, at t = 0, the time evolution of this state is obtained by the Schr¨ equation, HΨ(t) = i∂Ψ(t)/∂t, with the Hamiltonian in Eq. (13). At time Ωt ≈ π with Ω ≡ 2t1 = 2t2 we can see that for the strong coupling case in the upper panel the state | ↑↓ evolves into the state | ↑↑, while the states, | ↓↓ and | ↓↑ do not respond to this operation, which is the CNOT gate operation. However, for the weak coupling strength in the lower panel which corresponds to the usual inductive coupling, the states, | ↓↓ and | ↓↑, also evolve during the operation, which means that weak coupling cannot eﬀectively implement discriminating operation. 3. Controllable Coupling and Entanglement of Phase-Coupled Flux Qubits In real quantum computing the coupling strength should be controllable as well as strong. In Fig. 3(a) the coupling between two qubits can be tunable by varying the control ﬂuxes threading the dc-SQUID’s in the connecting loop,7 where the periodic boundary conditions become φL(R)1 + φL(R)2 + φL(R)3 = 2π(nL(R) + ft,L(R) ),

(14)

φ1

(15)

+

−φ1

φ3 +

= 2π(r + find )+ φ2 = 2π(fL + p),

(φL1 − φR1 ), −

φ3

+

φ4

=

−2π(fR

+ q).

(16)

If we introduce a rotated coordinate such as φp ≡ (φL3 + φR3 )/2, φm ≡ (φL3 − φR3 )/2, φp ≡ (φ1 + φ3 )/2 and φm ≡ (φ1 − φ3 )/2, the energies of two qubit loops, Uqubit , and connecting loop, Uconn , can be given by Uqubit (φm , φp ) = 2EJ1 cos 2φp cos 2φm − 4EJ cos φp cos φm + 2EJ1 + 4EJ , (17) ˜ cos(φ + πf ) cos 2φm , Uconn(φm , φm ) = 4EJ − 4E (18) J m ˜ ≡ E cos πf and f = f = f . where E J J L R Since the induced energy can be negligible, the total eﬀective potential Ueﬀ is given by the sum of the energies in Eqs. (17) and (18) such as Ueﬀ (φm , φp , φm ) = Uqubit (φm , φp ) + Uconn (φm , φm ) and we can calculate the energies, Ess , of the coupled two-qubit states. Then, since the coupling constant J of the coupled qubits

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can be represented as J = (1/4)(E↓↑ + E↑↓ − E↓↓ − E↑↑ ),6 we get J=

EJ2 EJ cos πf EJ1 EJ1 + 2EJ cos πf

(19)

as shown in Fig. 3(b). For f = 0, J is of the order of EJ so that we can obtain suﬃciently strong coupling by using Josephson junctions with high coupling energy. By increasing f we can control the coupling strength from the maximum value to zero at f = 0.5. Here the directions of control ﬂuxes, fL and fR , threading the dcSQUID’s in the connecting loop play the key role for achieving controllable coupling. If two control ﬂuxes are in opposite directions, they can be cancelled. However, if two control ﬂuxes are in the same direction or there is only one dc-SQUID loop in the connecting loop, the qubit states become disturbed so that the two-qubit Hamiltonian cannot be described solely by the change of coupling strength. Another important key to quantum information science is the quantum entanglement but, for ﬂux qubits, direct entanglements have not yet been observed. The main reason, we think, is the weak coupling strength between two inductively coupled qubits.8 In this study we show how we can obtain maximal entanglement and thus Bell states using phase-coupled ﬂux qubits. We, for simplicity, consider the coupled qubits in Fig. 1 with one Josephson junction instead of two in connecting loop.9 Then the eﬀective potential Ueﬀ (φˆp , φˆm ) is given by Ueﬀ (φp , φm ) = 2EJ1 (1 + cos 2φp cos 2φm ) + 4EJ (1 − cos φp cos φm ) + EJ (1 − cos 4φm ).

(20)

As increasing the coupling strength, we have found that the energy levels of diﬀerent spin states are lifted and thus two-qubit tunnelling processes become dominant over the single qubit tunnelling. Thus we introduced the two-qubit tunnelling Hamilto-

(a)

(b)

Figure 3. (a) Phase-coupled ﬂux qubits with a connecting loop interrupted by two dc-SQUID’s. Gray squares denote Josephson junctions with Josephson coupling energy EJ i for qubit loops and EJ for connecting loop. (b)Energies of coupled qubit states and coupling constant J in Eq. (19), where the coupled qubits have two dc-SQUID’s with control ﬂuxes in opposite directions and we set EJ = 0.1EJ .

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Figure 4.

Concurrences for various EJ with fm = 0.

nian in addition to Hamiltonian in Eq. (13) as follows, H2 = (t3 |s, s−s, −s| + t4 |s, −s−s, s|),

(21)

s=± 12

where t3 (t4 ) is two-qubit tunnelling matrix element related with coupled qubits of the same (diﬀerent) spins. Then the entanglement of the eigenstates can be obtained by calculating the concurrence C(ρ) = max{0, λ1 − λ2 − λ3 − λ4 }, where λ s are √ √ ρ˜ ρ ρ in decreasing order, ρ is the eigenvalues of the Hermitian matrix R ≡ the density matrix of an eigenstate of the total Hamiltonian H and ρ˜ is deﬁned as ρ˜ ≡ (σy ⊗ σy )ρ∗ (σy ⊗ σy ).10 In Fig. 4 we can see that maximal entanglements can be observed for strong coupling but, for weak coupling case of EJ /EJ = 0.005 which corresponds to the inductive coupling, we can expect just partial entanglement. We actually calculated the coeﬃcients of eigenstates and found that the Bell states, √ |Φ+ = √12 (| ↓↓ + | ↑↑) and |Φ− = (1/ 2)(| ↓↓ − | ↑↑), are formed at ground state and 1st excited states at the co-resonance point fR = fL = 0.5, repectively. We believe that present coupling scheme is experimentally achievable and thus experimental implementation is now invoked. References 1. J. E. Mooij et al., Science 285, 1036 (1999); Caspar H. van der Wal et al., Science 290, 773 (2000). 2. I. Chiorescu et al., Science 299, 1869 (2003). 3. J. B. Majer et al., Phys. Rev. Lett. 94, 090501 (2005). 4. Y.-x. Liu, L. F. Wei, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 96, 067003 (2006); P. Bertet, C. J. P. M. Harmans, J. E. Mooij, Phys. Rev. B 73, 064512 (2006); B. L. T. Plourde et al., Phys. Rev. B 70, 140501(R) (2004). 5. M. D. Kim and J. Hong, Phys. Rev. B 70, 184525 (2004). 6. M. D. Kim, D. Shin, and J. Hong, Phys. Rev. B 68, 134513 (2003). 7. M. D. Kim, Phys. Rev. B 74, 184501 (2006). 8. A. Izmalkov et al.,Phys. Rev. Lett. 93, 037003 (2004). 9. M. D. Kim and S. Y. Cho, Phys. Rev. B, to be published. 10. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

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CHARACTERISTICS OF A SWITCHABLE SUPERCONDUCTING FLUX TRANSFORMER WITH A DC-SQUID Y. SHIMAZU† AND T. NIIZEKI Yokohama National University, Hodogaya-ku, Yokohama 240-8501, Japan CREST, Japan Science and Technology Agency, Kawaguchi, 332-0012, Japan

We have investigated the flux transfer characteristics of a switchable flux transformer comprising a superconducting loop and a DC-SQUID. This system can be used to couple multiple flux qubits with a controllable coupling strength. Its characteristics were measured by using a flux input coil and a DC-SQUID for readout coupled to the transformer loop in a dilution refrigerator. The observed characteristics were in good agreement with the calculation results. We have demonstrated the reversal of the slope of the characteristics and complete switching-off of the transformer, which are useful features for its application as a controllable coupler.

1. Introduction Superconducting qubits are a promising candidate for the implementation of a scalable quantum computer [1]. Coupling of qubits is necessary for constructing a multiple qubit gate. A flux qubit, which is a superconducting loop interrupted by ultrasmall Josephson junctions [2], can be coupled inductively by means of the flux generated by the circulating currents. Previous experiments on coupled flux qubits employed fixed coupling through mutual inductances [3]. It is very desirable for the coupling to be switchable with a fast switching time in order to realize efficient operation on multiple qubits. Mooij et al. have presented a scheme for a switchable flux transformer to meet this requirement [2]. This transformer is a closed superconducting loop that contains two Josephson junctions in parallel, the structure of a DC-SQUID. The coupling strength can be varied by changing the magnetic flux in the SQUID loop, which is applied by the current in the control coil adjacent to the SQUID loop. We have investigated this original scheme of the switchable flux transformer theoretically and experimentally. A switchable flux transformer using a DC-SQUID with a different configuration has recently been studied by Castellano et al. [4]. Instead of using a control coil, we injected a current in the segment of the DC-SQUID to change the effective magnetic flux in the SQUID loop. This method is advantageous in that the control current can be small owing to the large kinetic inductance associated with a superconducting wire [5], thereby minimizing the influence of the control current on the qubits. The observed flux transfer characteristics will be compared with the calculation results. We note that for the circuit that is the same as the

†

E-mail: [email protected]

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Figure 1. Schematic of the switchable flux transformer. The crosses represent Josephson junctions. An input coil and a DC-SQUID for readout are coupled through mutual inductances Min and Mout in order to measure the characteristics of the transformer. The transformer involves a controlling DC-SQUID. The flux transfer characteristic is controlled by the magnetic flux Φc in the SQUID loop, which is varied by the control current Icont injected in the loop segment of the SQUID. The magnetic flux produced by the circulating current J is detected by the readout SQUID.

present system under investigation, a quantum superposition of two magnetic flux-states was experimentally demonstrated [6]. 2. Theoretical analysis and calculation results Figure 1 shows a schematic of the switchable flux transformer. An input coil and a DCSQUID for readout, which are used to measure the characteristics of the transformer, are also shown. We will present experimental results for the sample described by this schematic. The macroscopic variables that describe the switchable flux transformer are the fluxes Φ in the main loop (inductance L) and Φc in the controlling SQUID loop (inductance l) [7]. The externally applied fluxes for the loops, Φx and Φcx, are given by the currents Icoil and Icont in the input coil and control line, respectively. We assume that the critical currents I0 of the junctions in the SQUID are equal. The 2D potential describing the system is given by U (ϕ , ϕc ) =

Φ 02  1 1 ϕc ϕc  . 2 2  (ϕ − ϕ x ) + γ (ϕc − ϕcx ) − β 0 cos(ϕ + ) cos( )  4π 2 L  2 2 2 2 

(1)

where γ = L / l , β 0 = 4π I 0 L / Φ 0 , ϕ = 2πΦ / Φ 0 , ϕ x = 2πΦ x / Φ 0 , ϕc = 2πΦ c / Φ 0 , ϕcx = 2πΦcx / Φ0 , and Φ0 is the flux quantum. In the case of γ >> 1 similar to our sample, ϕc is frozen to the equilibrium value ϕcx. Then, the effective 1D potential for ϕ is given by

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99 U (ϕ ) =

Φ 02  1 ϕcx ϕ  2 ) cos( cx )  .  (ϕ − ϕ x ) − β 0 cos(ϕ + 4π 2 L  2 2 2 

(2)

In the classical ground state of the system, ϕ is fixed as the minimum of this 1D potential. The circulating current in the main loop is given by J=

Φ0 (ϕ − ϕ x ) . 2π L

(3)

The equation that determines the equilibrium value of J as a function of ϕx and ϕcx is expressed as follows: j + β 0 sin( j + ϕ x +

ϕcx 2

) cos

ϕcx 2

= 0,

(4)

where j = 2π LJ / Φ0 . This equation implies that the current J is a periodic function of Φx and Φcx with a period of Φ0. The variation in J can be detected by the readout DC-SQUID coupled to the main loop of the transformer. The overall flux transfer function is determined by the response of J to Icoil. This is also a periodic function of Icont, which is proportional to Φcx, and can be calculated using Eq. (4).

Figure 2. Calculated circulating currents in the transformer main loop as a function of the normalized magnetic flux fx externally applied to the loop. The results for various values of fcx, which is the normalized magnetic flux in the control SQUID loop, are compared. The curves are offset for better visibility. The range of fcx is (a) from 0.2 to 1.8 and (b) from 0.47 to 0.53.

Figures 2 (a) and (b) indicate the calculated circulating currents j in the classical ground state as a function of the normalized input flux fx for various values of the control flux fcx, where fx = Φx/Φ0 and fcx = Φcx/Φ0. The phase of the oscillation of j as a function of fx gradually changes with increasing fcx until fcx is a half-integer. It suddenly jumps as fcx crosses the half-integer value, at which point j is zero and the input flux is not

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100 100 transferred. When fcx is in the vicinity of the half-integer value, the amplitude of the oscillation is small, and the phase change is π as fcx crosses the half-integer value, as shown in Fig. 2 (b). 3. Experimental results and discussion The sample shown in Fig. 1 was fabricated by e-beam lithography. All the superconducting wires and Josephson junctions were made of Al. The junctions were fabricated using the shadow deposition technique [8]; their areas were about 0.06 (µm)2. The critical current of each junction in the DC-SQUID was found to be about 500 nA. The area of the readout SQUID loop is 5700 (µm)2. The readout SQUID was made relatively large in order to obtain a large output flux signal by increasing the coupling between the readout SQUID and the transformer. The experiment was carried out in a dilution refrigerator at a base temperature of 25 mK. We measured the switching current ISW of the readout DC-SQUID as a function of the input current Icoil, the control current Icont, and an external magnetic field. It should be noted that the change in ISW is proportional to the change in the circulating current in the transformer loop.

Figure 3. The switching current ISW of the readout DC-SQUID as a function of the control current Icont. The current Icoil in the input coil was zero, and the external magnetic field was fixed. ISW is modulated periodically with a period of Φ0 with respect to the flux in the SQUID loop of the transformer.

Figure 3 shows the switching current ISW as a function of Icont at a fixed magnetic field with Icoil = 0, while the dependence of ISW on Icoil for various control currents is presented in Figs. 4 (a) and (b). The periodic modulations shown in these figures exhibit the periodic behavior of the circulating current in the transformer as a function of both Φx and Φcx, which was explained in the previous section. The linear background in the figures is due to the spurious coupling of the input coil and control line to the readout SQUID. The opposite tendencies of the backgrounds of Figs. 3 and 4 can be attributed to the directions of the currents Icoil and Icont shown in Fig. 1.

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Figure 4. The switching current ISW of the readout SQUID as a function of Icoil for various control currents Icont. The periodicity of the signal corresponds to the flux change of Φ0 in the main loop of the transformer. The range of Icont is (a) from -5 to 7 µA and (b) from -5 to -3 µA. The abrupt change in the phase of the oscillation is observed for Icont between -4 and -3 µA and that between 6 and 7 µA.

From the period of the modulation, the mutual inductance between the input coil and transformer main loop is estimated to be 180 pH, while that between the transformer SQUID loop and control current line is determined to be 220 pH. These mutual inductances reasonably agree with the values estimated from the geometry of the sample and the kinetic inductance. It should be noted that the mutual inductance between the transformer SQUID loop and control current line, which share the same line of length 40 µm, is dominated by the kinetic inductance associated with the superconducting wire. We have verified the estimate of the kinetic inductance associated with the sample by directly measuring the effective mutual inductance in a separate sample [9]. In Fig. 4 (a) the gradual change and sudden jump of the phase of the oscillation with increasing Icont is indicated, which is in agreement with the features shown in Fig. 2 (a). Discontinuities of the circulating current, as shown in the calculation, are not observed experimentally. This can be attributed to the thermal excitation and quantum rounding. The experimental data for the case when the normalized magnetic flux in the control SQUID loop fcx is near the half-integer value is shown in Fig. 4 (b). The decrease in the amplitude of the oscillation and the change in the phase, which agree with the calculation shown in Fig. 2 (b), are also observed in this figure. In the system under investigation, the sign of the slope of the flux transfer characteristics can be inverted at a particular value of Icoil by varing Icont, as clearly shown in Fig. 4 (b). This is an attractive feature of the present system because complete switching-off of the transformer, that is, zero response for a small finite change in the input flux, can be achieved at a particular operation point. In contrast, for a similar

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102 102 switchable flux transformer, shown in Fig. 5 (a), complete switching-off is not realized since the slope of the flux transfer characteristics is always positive, as shown in Fig. 5 (b) [4]. This system can be applied to couple not only multiple qubits but also a single qubit to a readout SQUID. The influence of the SQUID readout on the coherence in a flux qubit might be investigated by means of a controllable coupling between the qubit and the readout SQUID.

Figure 5. (a) Schematic of the switchable flux transformer with a different configuration. (b) Calculated circulating currents in the transformer main loop as a function of fx (normalized magnetic flux externally applied to the main loop) for various applied flux Φcx for the control SQUID. The curves are for fcx = Φcx/Φ0 = 0, 0.2, 0.4, 0.46, 0.48, 0.5, 0.52, 0.54, 0.6, 0.8, and 1.

References 1. Y. Makhlin, G. Schön and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 2. J. E. Mooij, T.P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal and Seth Lloyd, Science 285, 1036 (1999). 3. J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans and J. E. Mooij, Phys. Rev. Lett. 94, 090501 (2005). 4. M. G. Castellano, F. Chiarello, R. Leoni, D. Simeone, G. Torrioli, C. Cosmelli and P. Carelli, Appl. Phys. Lett. 86, 152504 (2005). 5. J. B. Majer, J.R. Butcher and J. E. Mooij, Appl. Phys. Lett. 80, 3638 (2002). 6. J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Nature 406, 43 (2000). 7. S. Han, J. Lapointe and J. E. Lukens, Phys. Rev. Lett. 63, 1712 (1989). 8. G. J. Dolan, Appl. Phys. Lett. 31, 337 (1977). 9. Y. Shimazu and T. Niizeki, Physica C in press.

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CHARACTERIZATION OF ADIABATIC NOISE IN CHARGE-BASED COHERENT NANODEVICES

A. D’ARRIGO, G. FALCI, A. MASTELLONE AND E. PALADINO∗ MATIS CNR -INFM, Catania (Italy) & Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universit` a di Catania. Viale A. Doria 6, 95125 Catania (Italy)

Low-frequency noise, often with 1/f spectrum, has been recognized as the main mechanism of decoherence in present-day solid state coherent nanodevices. The responsible degrees of freedom are almost static during the coherent time evolution of the device leading to eﬀects analogous to inhomogeneous broadening in NMR. Here we present a characterization of the eﬀects of adiabatic noise exploiting the tunability of nanodevices.

1. Introduction Over the last years impressive progress has been achieved in quantum control of coherent nanodevices. Many early proposals of implementation of quantum bits with superconductors and semiconductors have been demonstrated to exhibit coherent properties in the time domain 1−5 . Solid state nanodevices are inﬂuenced by noise sources with broadband spectrum and a variety of statistical features. However they often operate in regimes of limited sensitivity to details of the noise nature, either because of limited control on protocols (responsible for instance of inhomogeneous broadening) or because protocols eﬀectively decouple part of noise sources (echo protocol). This induces to develop approximation schemes including systematically only the relevant information about noise, focusing on the eﬀects of the environment on the controlled dynamics rather than on the speciﬁc nature of the noise sources. This is the appropriate point of view in the perspective of using nanodevices as processors for quantum information. The following classes of noise can be identiﬁed and studied with “ad hoc” tools: quantum noise responsible for spontaneous decay, adiabatic noise whose main eﬀect is analogous to inhomogeneous broadening in NMR and strongly coupled noise, producing the analogue of uncontrollable chemical shifts 6 . In this paper we focus on low frequency noise which has been recognized as the main mechanism of decoherence for present-day experiments 1,2,7 . It originates from microscopic degrees of freedom in the device with dynamics slow compared to its typical frequencies Ω0 , acting as adiabatic uncontrollable extra driving ﬁelds 6 . Examples are charged switching impurities in the oxides or in the substrates 8,9 , or trapped ﬂux tubes in persistent current qubits or nuclear magnetic moments for spin ∗ e-mail:

[email protected]

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qubits. Often low frequency noise has 1/f spectrum 10 , the corresponding degrees of freedom being almost static during the coherent time evolution of the device. Here we discuss a way to characterize the eﬀects of low-frequency adiabatic noise, which can be achieved by exploiting the tunability of nanodevices. Since they may work at diﬀerent operating points also the resilience of the device to external noise is tunable and may be expressed in a very transparent way in terms of response functions of the system to external perturbations. In this work we will explicitly work out the case of adiabatic charge noise in the Cooper Pair Box (CPB). 2. Cooper Pair Box based nanodevices Design of charge devices presents at least an island of metallic or semiconducting material, connected to the rest of the system by tunnel junctions. If the system is superconducting, the excess charge in the island may be controlled at the level of a single Cooper pair. The one-island circuit is the CPB described by 11 HBOX = EC (ˆ q − qx )2 − EJ cos ϕˆ

[ϕ, ˆ qˆ] = i ,

(1)

qˆ is the excess number of Cooper pairs in the island which may change due to Josephson tunneling, e±iϕ |q = |q ± 1. EC is the charging energy, preventing states with large q − qx to enter the dynamics, and EJ is the Josephson energy. If the dynamics involves only the two lowest energy states, the CPB implements a qubit. The computational states may be superpositions of two or more charge states |q, depending on the ratio EJ /EC . For charge qubits EJ /EC 1 1 , whereas for charge-phase qubits EJ /EC ∼ 1 2 . Control is eﬃciently operated with electric ﬁelds, via a gate electrode capacitively coupled to the island modulating the control parameter qx (t) = Cg Vg (t)/(2e) a . The spectrum of the device depends periodically on the operating point ﬁxed by qx . By operating at the symmetry point qx = 1/2 where the splitting has a minimum, the decoherence time is maximized. The environment couples to the device via the same ports allowing driving 12 . At the quantum level sources of charge noise are accounted for by adding to ˜ = qˆX ˆ + HR . Here HR describes the reserEq.(1) the environment Hamiltonian H ˆ is an environment operator, whose ﬂuctuations correspond to voir alone and X ˆ ↔ −2EC δqx . The environment is characterﬂuctuations of the gate charge X ized statistically, the gross information being contained in the power spectrum ∞ ˆ X(0) ˆ ˆ X(t) ˆ + X(0) . In solid state devices S(ω) extends S(ω) = −∞ dt eiωt 21 X(t) over several decades, often exhibiting 1/ω behavior at low frequencies. We may ˆ f , where X ˆ f describes fast modes of the environment (frequencies ˆ →X ˆs + X split X ≥ Ω0 ), responsible for quantum noise whose main eﬀect at a low enough temperˆ s corresponds to degrees of freedom slow atures is spontaneous decay. Instead X on the scale of the system dynamics, responsible for adiabatic noise which can be ˆ s → X(t) of frequencies modeled as an additional stray adiabatic classical ﬁeld X a Resorting

to various SQUID geometries it is also possible to modulate EJ . Since coupling via the EJ port is less eﬀective we consider it as a ﬁxed quantity.

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ω < γM < Ω0 . The reduced density matrix (RDM) of the system can be written as ρˆ(t) = DX(s) P [X(s)] ρˆf [t|X(s)], where ρˆf [t|X(s)] is the RDM obtained after elimination of quantum noise 13 for a system driven adiabatically by the classical stochastic process X(s), with probability amplitude P [X(s)]. If the stochastic process is strongly correlated over time scales ∼ t, the path integral can be evaluated in the static-path approximation, X(s) = X. Finally if we consider only errors accumulating in time we may write the coherences as ρ(t) = e−iΩ0 (qx )t e−iΦ(qx ,t) , where the average phase shift is deﬁned as − [ 12 Γf (qx ,X) + i δΩ(qx ,X)] t Φ(qx , t) = i ln dXp(X)e . (2) The exponential in the integrand describes the phase of a device biased at the working point qx − X/(2EC ), where the splitting is Ω(qx , X) = Ω0 (qx ) + δΩ(qx , X) and Γf (qx , X) is the relaxation rate due to quantum noise. Since the stray polarization X is a sum of many independent variables b , then p(X) is a Gaussian distribu γ M dω 2 = γ ∗ π S(ω), depending on the total measurement time tion with variance σX ∗ tmeas = 1/γ . Despite of its simplicity, this approximation proves very eﬀective in describing defocusing due to 1/f noise observed in experiments 6 . 3. Defocusing for charge and charge-phase qubits We now calculate the decay function D(qx , t) = Im[Φ(qx , t)] for the CPB. Since the main part of decay comes from defocusing of the signal, at this stage we neglect spontaneous decay in Eq.(2). For charge qubits D(qx , t) has been calculated in Ref. 6 . The splitting is found by truncating the Hamiltonian (1) to the lowest

2 (q − 1 )2 + E 2 . The eigenvectors two charge states |0 and |1, Ω0 (qx ) = 4EC x J 2 are superpositions of the two charge states with mixing angle given by tan θ = EJ /[2EC (qx − 12 )]. Including the eﬀect of the stray ﬁeld X and expressing all the variables in units of Ω0 (qx ) we ﬁnd Ω0 (qx , X) = Ω0 (qx ) (c + x)2 + s2 , where c (s) is the cosine (sine) of the mixing angle θ. Assuming σX Ω(qx ) we may expand with respect to x the quantity δΩ0 (qx , X), and the integral Eq.(2) yields

D(τ ) = −

(cστ )2 1 1 ln[1 + (s2 σ 2 τ )2 ] − , 4 2 1 + (s2 σ 2 τ )2

(3)

where σ = σX /Ω0 (qx ) and τ = Ω0 (qx )t. In the charge-phase regime characterization of spectral properties requires diagonalizing the CPB Hamiltonian (1) with several charge states. As long as gate charge ﬂuctuations are small, they can be easily converted in energy ﬂuctuations. The CPB Hamiltonian plus ﬂuctuations can be ex pressed in the eigenbasis for bias qx as HBOX = i |φi Ei φi | + X ij |φi qij φj | and δΩ(qx , X) can be approximated by second order perturbation theory in X. As b Although

the stray polarization X is supposed to be constant during each coherent time evolution it can ﬂuctuate during the whole experiment. If the experiment measures the average signal of an ensemble of a large number N coherent time evolutions taking place during the time intervals [ti , ti + t], then p(X) is the distribution of X(t) sampled at times ti .

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1

(b)

0.4

Ω(qx)

1.5 1

*

0.6

(c)

2

(√2)/στ

(√2)/στ*

0.8

0.2

0.1 0.5

(a) 0.4

0.5

0.6 0

qx

0

0.2

0.4

qx

0.6

-1

0.8

-0.8 1

-0.6

-0.4

-0.2

0

c

0.2

0.4

0.6

0.8

0 1

√ Figure 1. Dephasing rate √2/στ ∗ as a function of qx (a), and of √c (b) for charge qubits, EJ /EC = 0.11, and σX /EC = 0.06/ 2 (gray), 0.06 (thick black), 0.06 2 (thin black). Continuous lines correspond to Eq.(3), symbols are obtained by exact numerical integration of D(qx , t). (c) Exact splittings E1 − E0 = Ω0 (qx ) (black dashed) and E2 − E0 = Ω0 (qx ) (black) obtained diagonalizing the CPB Hamiltonian (in a subspace of dimension Ns = 14). The ﬁrst splitting Ω0 (qx ) obtained by truncation of the Hamiltonian to the lowest energy states at the optimal point is also shown (gray).

a diﬀerence with the charge qubit, the curvature of the energy spectrum also depends on higher energy states |φm with m ≥ 2. This eﬀects Ω(qx , X) both via Ω0 (qx ) and δΩ(qx , X) c . In fact, diagonalizing exactly the Hamiltonian in a Ns > 2 subspace is not equivalent to estimating Ω0 (qx ) by truncating the Hamiltonian to the two lowest states at the optimal point. Discrepancies are more evident with increasing EJ , thus variations of qx , or equivalently ﬂuctuations of X, determine leakage from the Hilbert space spanned by the qubit computational states at the optimal point. Due to the smooth variation of Ω0 (qx ) with gate charge ﬂuctuations, Eq.(3) with the perturbative expansion of Ω0 (qx ) is a good approximation of the exact expression obtained numerically by diagonalization (in a basis with Ns 1 charge states) and direct evaluation of D(qx , t). Dephasing time: The dephasing time τ ∗ (D(τ ∗ ) = −1) depends on the bias qx via θ and Ω0 (qx ). We notice that the only parameter characterizing the environment is σX , whose actual value has a weak dependence on the total duration of the experiment. Figures 1 and 2 show also a comparison between Eq.(3) and the result obtained by numerically integrating Eq.(2) with the exact band-shape. Eq.(3) is √ ∗ −1 2 accurate close to the optimal point c ≈ 0 where [στ (qx )] ≈ s σ/ 2. Away from it small discrepancies are observed with increasing values of the stray polarization distribution width σ. Close to the “pure dephasing” point √ s → 0 the dephasing rate does not depend on σ and reads [στ ∗ (qx )]−1 → |c|/ 2. Because of the band bending by approaching s = 0, this is a region of greater sensitivity to the width σ. Discrepancies with increasing σ when qx = 1/2 are due to contributions to the integral both from the operating frequency Ω0 (qx ) and from the “most stationary” frequency Ω0 ( 12 ), the latter being suppressed for smaller σ. However in this regime Eq.(3) is still a good interpolation formula. This agreement is even better in the c

2

δΩ(qx , X) = c(qx )X +s2 (qx ) 2ΩX(q 0

x)

; c(qx ) = q11 −q00 ;

s2 (qx ) 2

2 − = 2q10

m≥2

2 q1m Ω0 Em −E1

q2 Ω − E 0m−E0 . m

0

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(b)

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(a)

2

1.5

0.2

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qx

0.8

1 0

1

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-0.15

qx

0.6

0.8

1

-0.1

-0.05

0

c

0.05

0.1

0.01 0.15

√ Figure 2. Dephasing rate 2/στ ∗ as a function of qx (a), √ and of c (b) for charge-phase qubits EJ /EC = 1.27 and widths σX /EC = 0.06 (gray), 0.06 2 (black). Continuous line result from diagonalizing exactly HBOX in a Ns = 14 subspace and numerical integration of D(qx , t); dots are obtained by inserting s(qx ) and c(qx ) given in footnote c (parameters obtained diagonalizing HBOX ) into Eq.(3); squares follow the approximate Ω(qx ) resulting from truncating the Hamiltonian to the lowest eigenstates at the optimal point and exact numerical integration of D(qx , t). (c) Exact level splittings and approximate Ω0 (qx ) obtained as in Fig.(1).

charge-phase regime, since the splitting Ω0 (qx ) is smoother, Fig.2. Therefore the series expansion analysis is valid at least for devices exhibiting coherent dynam√ ics 15 . We stress that the simple result [στ ∗ (qx )]−1 = |c|/ 2 diﬀers in two ways from the standard result for quantum noise obtained from the Master Equation 13 . In this latter case the dephasing rate is proportional to the power spectrum at low √ frequencies S(0), whereas for slow noise it is proportional to σX ∼ S. Moreover for quantum noise 1/τ ∗ ∝ c2 therefore the eﬀects of slow and high frequency noise can be to a certain extent discriminated. Indeed, both c(qx ) and s(qx ) have a clear physical interpretation, c = q11 − q00 (qij = φi | qˆ |φj ) representing the charge sensitivity of the device and s2 (qx ) being directly related to the quantum capacitance of the CPB 14 . Interestingly enough we found that characterization of adiabatic noise involves operating regimes where leakage from the two-state Hilbert space of the computational degree of freedom has to be taken into account. Line-shapes: Some characteristic features of low frequency noise may also be inferred from the analysis in frequency domain 7 . Close to the optimal point the asymmetric line-shape peaked at Ω(qx ) is indicative of the form of the distribution p(x), this information being progressively washed out with increasing qx , as shown in Fig.3(a). This is easily understood from the Fourier transform of ρ(t) in the static path approximation, close to the optimal point for Γf → 0, √ p(−a+k 2(w − 1)), where w = ω/EJ , a = 2EC (qx −1/2)/EJ . ρ˜(w) ∝ Θ(w−1) w−1 k=±1

At the optimal point charge noise can only increase the eﬀective level splitting, leading to the high frequency tail 7 . The static path approximation (SPA) has been proven 6,7 to quantitatively explain the observed initial reduction of the signal in the experiment of Ref 2 . On the other side the behavior of the line-shape close to Ω0 (qx ) also reﬂects the signal decay at longer time scales, thus it is sensitive also to the dynamics of faster degrees of freedom in the environment. These include higher frequency components in the spectrum both slow compared to Ω0 (qx ), like

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2e+05

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2e+04

qx=0.42

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qx=0.44

8e+04

0 1

qx=0.46

2e+03

qx=0.48

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qx=0.50

0

0 1

1.005

1.01

1.015

ω/Δ

1.02

1.025

1

1.00005

1.0001

ω/Δ

Figure 3. Real part of the Fourier transform of ρ(t) (Δ = 1.03 · 1011 rad/s, σ = 1.92 · 10−2 ): (a) close to the optimal point qx = 1/2 (black), for Γf /2Δ = 6 · 10−5 (curves for the indicated values of qx are shifted upwards for visibility). (b) static path approximation Eq.(3) with s = 1, and c = a(1/2) = 0 (black), and stochastic Schr¨ odinger simulations for 1/f noise with γmin /Δ = 10−11 and γmax /Δ = 10−5 (gray) or γmax /Δ = 10−3 (dashed). Inset: convolution with a Lorentzian due to relaxation with Γf /2Δ = 10−5 .

tails of 1/f noise with ω ≈ 106 rad/s Ω0 (qx ), and those responsible for relaxation ω ≈ Ω0 (qx ). Dynamical impurities responsible for 1/f noise broaden the line and slightly shift the maximum to higher frequencies (gray and dashed lines in Fig.(3)(b)). Convolution with a Lorentzian due to relaxation leads to a similar qualitative eﬀect. In experiments where typically both “slow dynamical” ﬂuctuators and quantum noise are present, we expect that the corresponding eﬀects might be hardly distinguishable, whereas the observed asymmetric line-shapes are clear signatures of adiabatic noise components. References 1. Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature 398, 786 (1999). 2. D. Vion et al., Science 296, 886 (2002). 3. Y. Yu et al., Science 296, 889 (2002); J. M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002); I. Chiorescu et al., Science 299, 1869 (2003); T. Duty et al., Phys. Rev. B 69, 140503(R) (2004); O. Astafiev et al., Phys. Rev. B 69, 180507(R) (2004). 4. T. Hayashi et al., Phys. Rev. Lett. 91, 226804 (2003); J. Gorman, D. G. Hasko and D. A. Williams, Phys. Rev. Lett. 95, 090502 (2005). 5. T. Yamamoto et al., Nature 425, 941 (2003); E. Collin et al., Phys. Rev. Lett. 93, 157005 (2004). 6. G. Falci et al., Phys. Rev. Lett. 94, 167002 (2005). 7. G. Ithier et al., Phys. Rev. B 72, 134519 (2005). 8. E. Paladino et al., Phys. Rev. Lett. 88, 228304 (2002); E. Paladino, L. Faoro and G. Falci, Adv. Solid State Phys. 43, 747 (2003). 9. M. Galperin et al., Phys. Rev. Lett. 96, 097009 (2006). 10. M. B. Weissmann, Rev. Mod. Phys. 60, 537 (1988). 11. Y. Makhlin, G. Sh¨ on and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 12. G. Falci, E. Paladino, R. Fazio, in “Quantum Phenomena of Mesoscopic Systems”, V. Tognetti and B. Altshuler eds., IOS Press, 2003. 13. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom-Photon Interactions, Wiley-Interscience, New York (1993). 14. T. Duty et al., Phys. Rev. Lett. 95, 206807 (2005). 15. G. Falci et al., in preparation.

Unconventional Superconductors

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THRESHOLD TEMPERATURE OF ZERO-BIAS CONDUCTANCE PEAK AND ZERO-BIAS CONDUCTANCE DIP IN DIFFUSIVE NORMAL METAL / SUPERCONDUCTOR JUNCTIONS

IDURU SHIGETA1,∗, TAKEHITO YOKOYAMA2, YASUHIRO ASANO3 , FUSAO ICHIKAWA4 , AND YUKIO TANAKA2,5 1

Department of General Education, Kumamoto National College of Technology, Kumamoto 861-1102, Japan 2 Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 3 Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan 4 Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan 5 CREST, Japan Science and Technology Agency (JST), Nagoya 464-8603, Japan

We have studied how zero-bias conductance dip (ZBCD) and zero-bias conductance peak (ZBCP) are smeared by increasing temperature in the diﬀusive normal metal (DN) / s-wave or d-wave superconductor junctions using the theory based on the KeldyshNambu quasiclassical Green’s function formalism. Tunneling conductance is calculated by changing the magnitudes of the resistance in DN, the Thouless energy in DN, the transparency of the insulating barrier, and the angle between the normal to the interface and the crystal axis of d-wave superconductors. We present a threshold temperature from a possible observation of the ZBCD and ZBCP in line shapes of tunneling conductance.

1. Introduction The low-energy transport in the mesoscopic system is governed by Andreev reﬂection.1 For diﬀusive normal metal / superconductor (DN/S) junctions, the phase coherence between incoming electrons and outgoing Andreev holes persists in DN at a mesoscopic length scale. These results in strong interference eﬀects on the probability of Andreev reﬂection through the proximity eﬀect. The theory based on the Keldysh-Nambu quasiclassical Green’s function formalism has recently explained that the interplay between diﬀusive and interface scattering produces a wide variety of line shapes of the tunneling conductance, not only for d-wave junctions, but also for s-wave junctions: zero-bias conductance dip (ZBCD), zero-bias conductance peak (ZBCP), gap-like, and rounded bottom structures.2–7 Here, we focus on the temperature dependence of the ZBCD and ZBCP due to the proximity eﬀect, and discuss the thermal broadening eﬀect on line shapes of the tunneling conductance. Then, we give a threshold temperature TTh , which is deﬁned by disappearing temperature of the structure of the ZBCD or ZBCP. ∗ Present

address: Department of Physics, Kagoshima University, Kagoshima 890-0065, Japan.

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2. Formulation We consider a junction consisting of a normal reservoir (N) and a superconducting one connected by a quasi-one-dimensional DN conductor with a resistance Rd and a length L, much larger than the mean-free path . The interface between the DN conductor and the superconductor electrode has a resistance Rb , while the N/DN interface has zero resistance. The positions of the N/DN interface and the DN/S interface are denoted as x = −L and x = 0, respectively. The scattering zone (x = 0) is modeled as an insulating δ-function barrier with the transparency T (φ) = 4 cos2 φ/(4 cos2 φ + Z 2 ), where Z is a dimensionless constant, and φ is the injection angle measured from the interface normal to the junction. The Thouless energy ETh can be expressed by ETh = D/L2 with the diﬀusive constant D in the DN and the length L of the DN. Here, ETh is determined by the properties of a normal metal as a counterelectrode in tunnel junctions. In the present paper, we adapt the units with kB = = 1. From the retarded or advanced component of the Usadel equation,8 the spatial dependence of θ(x) in the DN is determined by D

∂ 2 θ(x) + 2iE sin [θ(x)] = 0. ∂x2

(1)

Taking account of the Nazarov’s boundary condition at the DN/S interface,3 we obtain L ∂θ(x) F = . (2) Rd ∂x x=0− Rb The average over the angle of injected particles at the interface is deﬁned by π/2 π/2 F = F cos φ dφ T (φ) cos φ dφ, −π/2

(3)

−π/2

and F =

2T (φ) [Γ1 cos θ0 − Γ2 sin θ0 ] , [2 − T (φ)] Γ3 + T (φ) [Γ2 cos θ0 + Γ1 sin θ0 ]

(4)

Γ1 = f + + f − , Γ2 = g + + g − , Γ3 = 1 + g + g − + f + f − , (5) 2 with θ0 = θ(x = 0− ), f± = Δ± (φ) Δ± (φ) − E 2 and g± = E E 2 − Δ2± (φ). E denotes the energy of the quasiparticles measured from the Fermi energy EF . Δ+ (φ) (or Δ− (φ)) is the pair potential felt by the outgoing (or incoming) quasiparticles. The total resistance R in the superconducting state is written by Rd 0 dx Rb + R= , (6) Ib0 L −L cosh2 θIm (x) where Ib0 is obtained from the Keldysh component of the matrix current Iˇ and is a complex function of g± , f± , θ0 , and T (φ).6 By using the relations of σS (E) = 1/R

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and σN (E) = 1/(Rd + Rb ), the normalized tunneling conductance σT (eV ) in ﬁnite temperatures is given by ∞ E + eV σS (E) sech2 dE 2T −∞ σT (eV ) = ∞ . (7) E + eV σN (E) sech2 dE 2T −∞ For the equations, we assume Δ(T ) and Δ(T ) cos[2(φ ± α)] as temperature dependent pair potentials of s-wave and d-wave superconductors, where Δ(T ) is the maximum amplitude of the pair potential at a temperature T , Δ0 = Δ(0), and α denotes the angle between the normal to the interface and the crystal axis of d-wave superconductors. It is presumed that Δ(T ) changes on temperatures according to the Bardeen-Cooper-Schrieﬀer (BCS) theory. 3. Results 3.1. Line shapes of tunneling conductance Figure 1 shows temperature dependence of line shapes of tunneling conductance σT (eV ), which is normalized by its value in the normal state for s-wave superconductors. Here, we choose high transparent junctions with Z = 0, Rd /Rb = 1. The Thouless energy is ETh /Δ0 = 10−4 in Fig. 1(a), ETh /Δ0 = 10−3 in Fig. 1(b), ETh /Δ0 = 10−2 in Fig. 1(c), and ETh /Δ0 = 10−1 in Fig. 1(d), respectively. As shown in Fig. 1, the tunneling conductance has a ZBCD enhancement at T /Tc = 0, 1.5 -4

Z=0 Rd/Rb=1 ETh/Δ0=10

-3

Z=0 Rd/Rb=1 ETh/Δ0=10

1.3 T/Tc=0.3 T/Tc=0.2 T/Tc=0.1 T/Tc=0.05 T/Tc=0.0

㩷

σT(eV/Δ0)

1.4

1.2 1.1

(a) 㩷

(b)

s-wave

1.5 -1

-2

㩷

1.3 㩷

σT(eV/Δ0)

Z=0 Rd/Rb=1 ETh/Δ0=10

Z=0 Rd/Rb=1 ETh/Δ0=10

1.4

1.2 1.1 1.0

(c) -0.4

(d) -0.2

0.0

eV/Δ0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

eV/Δ0

Figure 1. Temperature dependence of the normalized tunneling conductance for s-wave superconductor with Z = 0 and Rd /Rb = 1. (a) ETh /Δ0 = 10−4 , (b) ETh /Δ0 = 10−3 , (c) ETh /Δ0 = 10−2 and (d) ETh /Δ0 = 10−1 , respectively.

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where Tc is the transition temperature of superconductors. At zero temperature, the zero-bias conductance σT (0) is independent of ETh and the ZBCD has a width with E ETh . As a result, the ZBCD width at zero temperature widens gradually with increasing ETh . In high transparent interface, the proximity eﬀect involving the coherent Andreev reﬂection (CAR) suppresses the tunneling conductance because the DN plays a role of the insulating barrier. In this case the probability of the Andreev reﬂection decreases around zero-bias voltage. This is the reason that the ZBCD appears in line shapes of tunneling conductance for high transparent junctions around zero temperature. The amplitude of pair potentials Δ(T ) for d-wave superconductors is the same as that for s-wave superconductors. We can choose 0 α π/4. It is known that quasiparticles with π/4 − α < φ < π/4 + α can contribute to the midgap Andreev resonant state (MARS) at the interface and are responsible for the ZBCP enhancement in low transparent junctions. It was shown that the proximity eﬀect and MARS do not coexist in the d-wave symmetry.6 In fact, at α/π = 0, the MARS does not exist while the proximity eﬀect is possible. Thus, we can expect similar results of the d-wave symmetry in the case of α/π = 0 to the s-wave symmetry. Therefore, we choose α/π = 0 for d-wave superconductors. In Fig. 2, we plot line shapes of the tunneling conductance as a function of temperatures for the low transparent junction of d-wave superconductors, where Z = 10, Rd /Rb = 1 and α/π = 0. The Thouless energy is ETh /Δ0 = 10−4 in Fig. 2(a), ETh /Δ0 = 10−3 in Fig. 2(b), ETh /Δ0 = 10−2 in Fig. 2(c), and ETh /Δ0 = 10−1 in Fig. 2(d), respectively. The zero-bias conductance σT (0) is independent of ETh at 1.0 -4

-3

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

㩷

0.6 㩷

σT(eV/Δ0)

0.8

0.4 0.2

(a)

(b)

d-wave

㩷

1.0 㩷

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

㩷

-1

Z=10 Rd/Rb=1 ETh/Δ0=10 α/π=0

0.6 T/Tc=0.3 T/Tc=0.2 T/Tc=0.1 T/Tc=0.05 T/Tc=0.0

㩷

σT(eV/Δ0)

0.8

-2

0.4 0.2 0.0

(c) -0.4

(d) -0.2

0.0

eV/Δ0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

eV/Δ0

Figure 2. Temperature dependence of the normalized tunneling conductance for d-wave superconductor with Z = 10, Rd /Rb = 1 and α/π = 0. (a) ETh /Δ0 = 10−4 , (b) ETh /Δ0 = 10−3 , (c) ETh /Δ0 = 10−2 and (d) ETh /Δ0 = 10−1 , respectively.

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zero temperature. The ZBCP also has a width with E ETh . Hence, the ZBCP width for d-wave superconductors increases with increasing ETh , as well as the ZBCD width for s-wave superconductors. Contrary to the high transparent junctions, the probability of the Andreev reﬂection increases around zero-bias voltage in low transparent interface.6 Consequently, the ZBCP appears in line shapes of the tunneling conductance for low transparent junctions around zero temperature. 3.2. Tunneling conductance at zero-bias voltage Figures 3(a) and 3(b) show temperature dependence of the normalized tunneling conductance σT (0) at zero-bias voltage in Figs. 1 and 2, respectively. σT (0)’s have 㩷

㩷

1.2

1.4

(a)

(b) Z=10 Rd/Rb=1 α/π=0

Z=0 Rd/Rb=1

0.6

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1.2

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ETh/Δ0=10

σT(0)

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1.3

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s-wave 1.0 0.0

0.2

ETh/Δ0=10

d-wave

0.4

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T/Tc

0.8

1.0

0.2

0.3

-4

ETh/Δ0=10

0.4

0.6

0.8

0.0 1.0

T/Tc

Figure 3. Normalized tunneling conductance σT (0) at zero-bias voltage as a function of T /Tc for DN/S tunnel junctions. (a) Z = 0 and Rd /Rb = 1 for s-wave superconductors. (b) Z = 10, Rd /Rb = 1 and α/π = 0 for d-wave superconductors.

a dip around zero temperature as shown in Fig. 3(a) because the probability of the Andreev reﬂection decreases for high transparent interfaces. On the other hand, the probability of the Andreev reﬂection increases for low transparent interfaces. As a consequence, σT (0)’s have a peak around zero temperature as shown in Fig. 3(b). Corresponding to the sharp ZBCD and ZBCP in Figs. 1 and 2, σT (0)’s with small ETh tend to be aﬀected by the thermal broadening and quickly change around zero temperature. Thus, σT (0)’s are a decreasing (or increasing) function of ETh /Δ0 for the high (or low) transparent junction especially in low temperatures except for T /Tc = 0. 3.3. Threshold temperature We can give the threshold temperature TTh on the basis of calculated results for temperature dependence of line shapes of tunneling conductance. TTh is deﬁned by disappearing temperature of the structure of the ZBCD or ZBCP with the width of E ETh . Figure 4 shows the TTh plots as a function of ETh , which are obtained from Figs. 1 and 2. As shown in Fig. 4, we found that TTh increases monotonically with increasing ETh in the small ETh region and has a maximum TTh /Tc 0.2

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TTh/Tc

0.2

0.1

ZBCD ZBCP

0.0 -4

10

-3

10

-2

10

-1

10

0

10

ETh/Δ0 Figure 4. Threshold temperature of the ZBCD for s-wave superconductors and the ZBCP for d-wave superconductors in the DN/S junctions. The solid and broken lines are a guide for eyes.

around ETh /Δ0 = 0.01–0.1 for both of the s-wave and d-wave symmetries. Thereafter, TTh becomes zero above ETh /Δ0 = 1 because the ZBCD and ZBCP do not exist inside the coherent peaks of superconductors. TTh has a maximum in the region of ETh /Δ0 0.01–0.1 also for any other parameter values. The result suggests that it is easy to observe the ZBCD and ZBCP due to the proximity eﬀect on the condition of ETh /Δ0 = 0.01–0.1 also in actual measurements. Therefore, we ﬁnally estimate several actual TTh ’s from our results. Using the typical values of D = 45 cm2 /s and L = 1 μm for a Ag ﬁlm as a DN conductor,9 we calculated the value of ETh /Δ0 = 1.0 × 10−4 and TTh = 4.5 K for hightemperature oxide superconductor Bi2 Sr2 CaCu2 O8+δ from Tc = 90 K and Δ0 = 30 meV. Similarly, we obtained the value of ETh /Δ0 = 2.0 × 10−4 and TTh = 1.4 K for conventional superconductor Nb from Tc = 7.2 K and Δ0 = 1.5 meV. 4. Conclusions We have studied the tunneling conductance in DN/S junctions in ﬁnite temperatures, and have given the threshold temperature TTh of the ZBCD and ZBCP, due to the proximity eﬀect. The proximity eﬀect decreases σT (0) for high transparent junctions, while increases for low transparent junctions. The ZBCD and ZBCP have a width of E ETh and TTh /Tc has a maximum at ETh /Δ0 = 0.01–0.1. The facts indicate a possibility of distinguishing electronic transport of DN/S junctions by careful comparison of the present calculations and experimental results.

References 1. 2. 3. 4. 5.

A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov. Phys. JETP 19, 1228 (1964)]. A. F. Volkov, A. Z. Zaitsev and T. M. Klapwijk, Physica C210, 21 (1993). Yu. V. Nazarov, Superlattices Microstruct. 25, 1221 (1999). Y. Tanaka, Yu. V. Nazaro and S. Kashiwaya Phys. Rev. Lett. 90, 167003 (2003). Y. Tanaka, A. A. Golubov and S. Kashiwaya Phys. Rev. B68, 054513 (2003).

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6. Y. Tanaka, Yu. V. Nazarov, A. A. Golubov and S. Kashiwaya, Phys. Rev. B69, 144519 (2004). 7. Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B71, 094513 (2005). 8. K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970). 9. V. T. Petrashov, V. N. Antonov, P. Delsing and T. Claeson, Phys. Rev. Lett. 74, 5268 (1995).

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TUNNELING CONDUCTANCE IN 2DEG/S JUNCTIONS IN THE PRESENCE OF RASHBA SPIN-ORBIT COUPLING

T. YOKOYAMA, Y. TANAKA AND J. INOUE Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected]

We have studied the tunneling conductance in two dimensional electron gas (2DEG) / insulator / superconductor junctions in the presence of Rashba spin-orbit coupling in 2DEG. It is clarified how the tunneling conductance is influenced by the Rashba spin-orbit coupling and found that for low insulating barrier the tunneling conductance is suppressed by the Rashba spin-orbit coupling while for high insulating barrier the tunneling conductance is slightly enhanced by the Rashba spin-orbit coupling in contrast to ferromagnet / superconductor junctions. The results give a possibility to control the Andreev reflection probability by a gate voltage.

1. Introduction In normal metal / supercunductor (N/S) junctions Andreev reﬂection (AR)1 is one of the most important process for low energy transport. Taking the AR into account, Blonder, Tinkham and Klapwijk (BTK) proposed the formula for the calculation of the tunneling conductance in N/S junctions2 . This method was extended to the ferromagnet / superconductor (F/S) junctions and used to estimate the spin polarization of the F layer experimentally3,4 . In F/S junctions, AR is suppressed because the retro-reﬂectivity is broken by the exchange ﬁeld in the F layer5, which causes many interesting phenomena6,7,8 . Spin dependent transport can be important to fabricate a new device manipulating electron’s spin. This ﬁeld is called spintronics. Spintronics has recently received much attention because of its potential impact on electric devices and quantum computing. Among recent works, many eﬀorts have been devoted to study the eﬀect of spin-orbit coupling on transport properties of two dimensional electron gas (2DEG)9,10,11,12 . The pioneering work by Datta and Das suggested the way to control the precession of the spins of electrons by the Rashba spin-orbit coupling13 in F/2DEG/F junctions14 . This spin-orbit coupling depends on the applied ﬁeld and can be tuned by a gate voltage. On the other hand spin dependent transport based only on spin-orbit coupling without ferromagnet is also a hot topic15,16 . The Rashba spin-orbit coupling induces an energy splitting, but the energy splitting doesn’t break the time reversal symmetry unlike an exchange splitting in ferromagnet. Therefore transport properties in 2DEG/S junctions may be quali-

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tatively diﬀerent from those in F/S junctions. However, in 2DEG/S junctions the eﬀect of Rashba spin-orbit coupling on transport phenomena is not studied well. Recent experimental and theoretical advances in spintronics drives us to attack this problem. In the present paper we calculate the tunneling conductance in 2DEG/S junctions and clarify how Rashba spin-orbit interaction aﬀects it. We think the obtained results are useful for the design of mesoscopic 2DEG/S junctions and for a better understanding of related experiments. In the present paper we conﬁne ourselves to zero temperature. 2. Formulation We consider a ballistic 2DEG / superconductor junctions. The 2DEG/S interface located at x = 0 (along the y-axis) has an inﬁnitely narrow insulating barrier described by the delta function U (x) = U δ(x). The eﬀective Hamiltonian with Rashba spin-orbit coupling is given by ⎛ ⎞ ξk iλk− θ (−x) 0 Δθ (x) ⎜ −iλk+ θ (−x) ⎟ ξk −Δθ (x) 0 ⎟ H=⎜ (1) ⎝ −iλk+ θ (−x) ⎠ 0 −Δθ (x) −ξk −ξk Δθ (x) 0 iλk− θ (−x) 2 2 with k± = kx ± iky , the energy gap Δ, ξk = 2m k − kF2 , Fermi wave number kF , Rashba coupling constant λ, and step function θ(x). Velocity operator in the x-direction is given by17 ⎛ ⎞ ∂ iλ 0 0 mi ∂x θ (−x) ⎜ − iλ θ (−x) ∂ ⎟ ∂H 0 0 mi ∂x ⎟. vx = =⎜ (2) ∂ iλ ⎠ − θ (−x) 0 0 − mi ∂kx ⎝ ∂x iλ ∂ 0 0 θ (−x) − mi ∂x Wave function ψ(x) for x ≤ 0 (2DEG region) is represented using eigenfunctions of the Hamiltonian: ⎡

⎛

⎢ ⎜ ⎢ ⎜ eiky y ⎢ √12 eik1(2) cos θ1(2) x ⎜ ⎣ ⎝ ⎛ ⎜ c1(2) −ik1 cos θ1 x ⎜ e + √ ⎜ 2 ⎝

−i kk1+ 1 1 0 0

⎛

⎞ 0 ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎟ a√ 1 eik1 cos θ1 x ⎜ k1+ ⎟ + ⎟ + 1(2) 2 ⎝ i k1 ⎠ ⎠ 0 1 ⎛ k ⎞⎤ ⎞ 0 i k2+ ⎜ 2 ⎟⎥ ⎟ ⎟ d√ −ik2 cos θ2 x ⎜ 1 ⎟⎥ e ⎜ ⎟ + 1(2) ⎟⎥ 2 ⎝ 0 ⎠⎦ ⎠ 0

(−) i

k1(2)− k1(2)

⎞

⎛

0 ⎜ 0 b1(2) ik2 cos θ2 x ⎜ √ e ⎜ k2+ 2 ⎝ −i k2 1

⎞ ⎟ ⎟ ⎟ ⎠

mλ 2 for an injection wave with wave number k1(2) where k1 = − mλ + + kF2 , k2 = 2 2 mλ mλ 2 + kF2 and k1(2)± = k1(2) e±iθ1(2) . a1(2) and b1(2) are AR coeﬃcients. 2 + 2

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c1(2) and d1(2) are normal reﬂection coeﬃcients. θ1(2) is an angle of the wave with wave number k1(2) with respect to the interface normal. Similarly for x ≥ 0 ψ(x) (S region) is given by a linear combination of the eigenfunctions. Note that since the translational symmetry holds for the y-direction, the momenta parallel to the interface are conserved: ky = kF sin θ = k1 sin θ1 = k2 sin θ2 . The wave function follows the boundary conditions17 : ψ (x)|x=+0 = ψ (x)|x=−0 vx ψ (x)|x=+0 − vx ψ (x)|x=−0 =

⎛ 2mU mi 2

10 ⎜0 1 ⎜ ⎝0 0 00

⎞ 0 0 0 0 ⎟ ⎟ ψ (0) −1 0 ⎠ 0 −1

Applying BTK theory to our calculation, we obtain the dimensionless conductance represented in the form: θ 2 2 2 2 σs = N1 −θCC 12 K21 + |a1 | K21 + |b1 | K12 λ21 − |c1 | K21 − |d1 | K12 λ21 cos θdθ π 2 2 2 2 +N2 −2 π Re 12 K12 + |a2 | K21 λ12 + |b2 | K12 − |c2 | K21 λ12 − |d2 | K12 cos θdθ 2

with K12 = 1 + λ12 =

k1 k2 , K21

k1 cos θ1 k2 cos θ2

=1+

k2 k1

λ21 =

k2 cos θ2 k1 cos θ1

and N1 =

1 1+

mλ 2 k1

N2 =

1 1−

mλ 2 k2

.

N1 and N2 are normalized density of states for wave number k1 and k2 respectively. The critical angle θC is deﬁned as cos θC = 2mλ 2k . 1 σN is given by the conductance for normal states, i.e., σS for Δ = 0. We deﬁne and Z = 2mU normalized conductance as σT = σS /σN and parameters as β = 2mλ 2k 2k . F F 3. Results First we study the normalized tunneling conduntace σT as a function of bias voltage V in Fig. 1. For Z = 3 where the AR probability is low, σT is almost zero within the enregy gap and independent of β. For Z = 0.3 the dependence of σT on β is also weak. For Z = 0.1 where the AR probability is very high, σT becomes nearly two for β = 0 within the energy gap. It is reduced by the increase of β within the energy gap. This is because the increase of β induces a mismatch of wave number between 2DEG and S, which plays a role of insulating barrier. Next we focus on the dependence of σS on β at zero voltage. Fig. 2 shows the dependence of σS on β at zero voltage for various Z. For Z = 3 it has an exponential dependence on β but its magnitude is small. This is an essentially diﬀerent feature from the one in F/S junctions where the tunneling conductance is always reduced as increasing exchange ﬁeld. For Z = 0.1 and Z = 0 it decreases linearly as a function of β.

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Figure 1.

Normalized tunneling conductance.

The eﬀect of the Rashba splitting on conductance is essentially diﬀerent from that of Zeeman splitting on conductance. This can be explained as follows. The Zeeman splitting gives unbalance of populations of up and down spin electrons. Thus it suppresses the AR where pairs of spin-up and spin-down electrons are transmitted to S. On the other hand, the Rashba splitting never causes such an unbalance. Thus it cannot suppresses the AR, which results in various β dependence of the conductance. 4. Conclusions In the present paper we have studied the tunneling conductance in two dimensional electron gas / insulator / superconductor junctions with Rashba spin-orbit coupling. We have extended the BTK formula and calculated the tunneling conductance. It is found that for low insulating barrier the tunneling conductance is suppressed by the Rashba spin-orbit coupling while for high insulating barrier the tunneling conductance is slightly enhanced by the Rashba spin-orbit coupling in contrast to F/S junctions.

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Figure 2. Tunneling conductance for superconducting states at zero voltage. Here the inequality 0 ≤ σS ≤ 8 holds from the definition.

The results give a possibility to control the AR probability by a gate voltage. We believe that the obtained results are useful for the design of mesoscopic 2DEG/S junctions and for a better understanding of related experiments. Here we have considered the N/S junctions where Rashba spin-orbit coupling exists in the N region. The N/S junctions with Rashba spin-orbit coupling in the S region also give an interesting result as shown in Ref.18 . Acknowledgments The authors appreciate useful and fruitful discussions with A. Golubov. This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science” . The computational aspect of this work has been performed at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center.

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References 1. A. F. Andreev, Sov. Phys. JETP 19 (1964) 1228. 2. G. E. Blonder, M. Tinkham, T. M. Klapwijk, Phys. Rev. B 25 (1982) 4515. 3. P. M. Tedrow, R. Meservey, Phys. Rev. Lett. 26 (1971) 192 ; Phys. Rev. B 7 (1973) 318 ; R. Meservey, P. M. Tedrow, Phys. Rep. 238 (1994) 173 . 4. S. K. Upadhyay, A. Palanisami, R. N. Louie, R. A. Buhrman Phys. Rev. Lett. 81 (1998) 3247. 5. M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. Lett. 74 (1995) 1657. 6. A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 7. F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). 8. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005); Phys. Rev. B 73, 094501 (2006). 9. J. E. Hirsch, Phys. Rev. Lett. 83 (1999) 1834. 10. P. Streda, P. Seba, Phys. Rev. Lett. 90 (2003) 256601. 11. J. Schliemann, D. Loss, Phys. Rev. B 68 (2003) 165311. 12. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, A. H. MacDonald, Phys. Rev. Lett. 92 (2004) 126603. 13. E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2 (1960) 1224; [Sov. Phys. Solid State 2(1960) 1109 ]; Yu. A. Bychkov, E. I. Rashba, J. Phys. C 17 (1984) 6039. 14. S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. 15. V. M. Edelstein, Solid State Commun. 73 (1990) 233. 16. J. Inoue, G. E. W. Bauer, L. W. Molenkamp, Phys. Rev. B 67 (2003) 033104; 70 (2004) 041303. 17. L. W. Molenkamp, G. Schmidt, G. E. W. Bauer, Phys. Rev. B 64 (2001) 121202. 18. T. Yokoyama, Y. Tanaka, and J. Inoue Phys. Rev. B 72, 220504(R) (2005)

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THEORY OF CHARGE TRANSPORT IN DIFFUSIVE FERROMAGNET/p-WAVE SUPERCONDUCTOR JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Tunneling conductance and density of states (DOS) in the diﬀusive ferromagnet (DF) / insulator / p-wave superconductor junctions are calculated by changing the exchange ﬁeld in DF and the transparencies of the insulating barriers. It is shown that zero-energy peak in DOS and zero bias conductance peak split into two peaks as increasing exchange ﬁeld.

1. Introduction There is a continuously growing attraction in charge and spin transport in ferromagnet(F) / superconductor(S) junctions. In this junctions Cooper pairs penetrate into the F layer from the S layer and have a nonzero momentum by the exchange ﬁeld1,2,3 . This property produces many interesting phenomena. For example, a strong enhancement of the proximity eﬀect occurs by the exchange ﬁeld4 . This resonant proximity eﬀect can inﬂuence various physical quantities5 . To study the proximity eﬀect in F/S junctions, a quasiclassical Green functions theory with Nazarov’s boundary condition is often used6,7 . The original Nazarov’s boundary condition is generalized and applied to diﬀusive normal metal (DN) / unconventional superconductor junctions8,9 . The formation of the midgap Andreev resonant states (MARS) at the interface of unconventional superconductors10 is naturally taken into account in this approach. It was demonstrated that the formation of MARS in DN/p-wave superconductor junctions coexists with the proximity eﬀect, which produces a giant zero bias conductance peak (ZBCP)9 . This theory treat spin independent charge transport. Calculations of tunneling conductance in the presence of the magnetic impurities in DN /S junctions were performed in Ref11,12,13 . Spin dependent transport is one of the most interesting topics in this ﬁeld and also realized in F/S junctions. In the present paper, we study the tunneling conductance in normal metal / insulator / diﬀusive ferromagnet

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/ insulator / p-wave superconductor junctions for various parameters such as the heights of the insulating barriers at the interfaces, resistance Rd in DF, the exchange ﬁeld h in DF, the Thouless energy ET h in DF. We consider the px -wave junctions because in this case proximity eﬀect is strongly enhanced. The conductance σT (eV ) as a function of the bias voltage V is given by σT (eV ) = σS (eV )/σN (eV ) where σS(N ) (eV ) is the tunneling conductance in the superconducting (normal) state at a bias voltage V . In the present paper we conﬁne ourselves to zero temperature and put kB = = 1. 2. Formulation In this section we introduce the model and the formalism. The formulation is the same as the one in Ref.5 except for treating the p-wave superconductors. We consider a junction consisting of normal and p-wave superconducting reservoirs connected by a quasi-one-dimensional diﬀusive ferromagnet conductor (DF) with a length L much larger than the mean free path. The interface between the DF conductor and the S electrode has a resistance Rb while the DF/N interface has a resistance Rb . The positions of the DF/N interface and the DF/S interface are denoted as x = 0 and x = L, respectively. We model inﬁnitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting trans are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) parencies of the junctions Tm and Tm 2 2 2 and Tm = 4 cos φ/(4 cos φ + Z ), where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We apply the quasiclassical Keldysh formalism in the following calculation of the tunneling conductance. The 4 × 4 Green’s functions in N, DF and S are ˇ 1 (x) and G ˇ 2 (x) where the Keldysh component K ˆ 0,1,2 (x) is ˇ 0 (x), G denoted by G ˆ ˆ ˆ ˆ ˆ ˆ given by Ki (x) = Ri (x)fi (x)− fi (x)Ai (x) with retarded component Ri (x), advanced ˆ ∗ (x) using distribution function fˆi (x)(i = 0, 1, 2). In the component Aˆi (x) = −R i ˆ 0 (x) = τˆ3 and fˆ0 (x) = fl0 + τˆ3 ft0 . R ˆ 2 (x) is expressed ˆ 0 (x) is expressed by R above, R √ √ ˆ 2 (x) = (gˆ τ3 +f τˆ2 ) with g = / 2 − Δ2 and f = Δ/ Δ2 − 2 , where denotes by R the quasiparticle energy measured from the Fermi energy and fˆ2 (x) = tanh[/(2T)] in thermal equilibrium with temperature T . We put the electrical potential zero in ˇ 1 (x) in DF is determined the S-electrode. In this case the spatial dependence of G by the static Usadel equation 14 . ˇ 1 (x) at the DF/S interface The Nazarov’s generalized boundary condition for G 9 ˇ 1 (x) is given by Ref. . We also use Nazarov’s generalized boundary condition for G at the DF/N interface: ˇ1 L ˇ ∂G −1 )|x=0+ = −Rb (G1 < B > , (1) Rd ∂x B=

ˇ ˇ 1 (0+ )] 2Tm [G0 (0− ), G . ([G ˇ 0 (0− ), G ˇ 1 (0+ )]+ − 2) 4 + Tm

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The average over the various angles of injected particles at the interface is deﬁned as π/2 dφ cos φB(φ) −π/2 () < B(φ) > = π/2 () (φ) cos φ −π/2 dφT ()

with B(φ) = B and T () (φ) = Tm . The resistance of the interface Rb is given by ()

()

Rb = R0 π/2 −π/2

2 dφT () (φ) cos φ

.

()

()−1

= Here R0 is Sharvin resistance, which in three-dimensional case is given by R0 () () e2 kF2 Sc /(4π 2 ), where kF is the Fermi wave-vector and Sc is the constriction area. ˇ 1 (x) as The electric current per one spin is expressed using G ∞ ˇ −L ˇ 1 (x) ∂ G1 (x) )K ], Iel = dTr[τˆ3 (G (2) 8eRd 0 ∂x ˇ1 (x) K ˇ1 (x) where (Gˇ1 (x) ∂ G∂x ) denotes the Keldysh component of (Gˇ1 (x) ∂ G∂x ). In the actual calculation it is convenient to use the standard θ-parameterization where ˆ 1 (x) = τˆ3 cos θ(x) + τˆ1 sin θ(x). The parameter θ(x) ˆ 1 (x) is expressed as R function R is a measure of the proximity eﬀect in DF. Following the method in Ref.5 , the diﬀerential resistance R per one spin projection at zero temperature is given by 2Rd L 2Rb dx 2Rb + + (3) R= < Ib0 > L 0 cosh2 θim (x) < Ib1 >

with Ib1 =

2 Λ1 + 2Tm (2 − Tm )Real{cos θ0 } Tm ) + T cos θ |2 | (2 − Tm 0 m

Λ1 = (1+ | cos θ0 |2 + | sin θ0 |2 ) where Ib0 is given in Ref.9 . In the above θim (x), θ0 and θL denote the imaginary part of θ(x), θ(0+ ) and θ(L− ) respectively. Then the total tunneling conductance in the superconducting state σS (eV ) is given by σS (eV ) = ↑,↓ 1/R. The local density of states (DOS) normalized by its normal states value in the DF layer is given by 1 Re cos θ(x). (4) 2 ↑,↓

In the following section, we will discuss the normalized DOS at x = 0 and the normalized tunneling conductance σT (eV ) = σS (eV )/σN (eV ) where σN (eV ) is the tunneling conductance in the normal state given by σN (eV ) = σN = 1/(Rd + Rb + Rb ).

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3. Results Let us ﬁrst choose relatively large barier Z = Z = 3. In this case a giant ZBCP and zero-energy peak (ZEP) in the DOS appear due to the enhanced proximity eﬀect. Figure 1 (a) and (b) show the tunneling conductance and the corresponding DOS for Z = 3, Rd /Rb = 0.1, Z = 3, Rd /Rb = 0.1, ET h /Δ = 0.01 and various h/Δ. The ZBCP splits into two peaks at eV ∼ h with increasing h/Δ. The corresponding DOS also splits with increasing h/Δ. In (c) and (d), we plot a tunneling conduntance for majority and minority spins respectively. In (c), the spin-resolved ZBCP is shifted to lower energy as increasing exchange ﬁeld while it is shifted to higher energy as increasing exchange ﬁeld as shown in (d). Therefore the ZBCP splits into two peaks with increasing h/Δ as shown in (a). Similar discussion about DOS also holds.

Z = 3 Rd / Rb = 0.1 Z ′ = 3 Rd / Rb′ = 0.1 ETh / Δ = 0.01 20

20

(a)

h/ h/ h/

σ

T10

0

-0.1

0

h/

8

S O D

h/ h/

6

Δ Δ Δ

(c )

=0 =0.01 =0.1

σ

h/ h/

10

0

-0.1

ΔΔ Δ

0

=0.01 =0.1

σ

(d )

h/ h/ h/

T

=0 =0.01 =0.1

0.1

eV Δ

20 =0

ΔΔ Δ

=0 =0.01 =0.1

10

(b)

4

h/

T

0.1

eV Δ

10

ΔΔ Δ

2 0

0

0.05

ε

Δ

0.1

0.15

0

-0.1

0

eV Δ

0.1

Figure 1. (a) normalized tunneling conductance and (b) DOS. (c) and (d) spin-resolved tunneling conductance.

Figure 2 displays the tunneling conductance and the corresponding DOS for Z = 0, Rd /Rb = 0.1, Z = 0, Rd /Rb = 0.1, ET h /Δ = 0.01 and various h/Δ. The ZBCP and ZEP in the DOS also appear in this case and but thier magnitudes are small compared to those in Fig. 1. This is because the resonant states are formed only for large Z. The ZBCP and ZEP in the DOS split with an increase in h/Δ as in Fig. 1.

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Z = 0 Rd / Rb = 0.1 Z ′ = 0 Rd / Rb′ = 0.1 ETh / Δ = 0.01 4

(a)

h/ h/ h/

σ T

3

Δ Δ Δ

=0 =0.01 =0.1

2

-0.1

0

0.1

eV Δ

10

(b)

8

S O D

6 h/ h/

4

h/

Δ Δ Δ

=0 =0.01 =0.1

2 0

0

0.05

ε Figure 2.

Δ

0.1

0.15

(a) normalized tunneling conductance and (b) DOS.

4. Conclusions In the present paper, based on Nazarov’s boundary condition, we have found a splitting of peak of the conductance and the DOS in N/DF/p-wave superconductor junctions with increasing exchange ﬁeld. This stems from the fact that the peak for majority (minority) spin is shifted to lower (higher) energy as increasing exchange ﬁeld.

Acknowledgments The authors appreciate useful and fruitful discussions with Yu. Nazarov and H. Itoh. This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science” . The computational aspect of this work has been performed at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center.

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References 1. A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, Pis’ma Zh. Eksp. Teor. Phys. 35, 147, (1982) [JETP Lett. 35, 178 (1982)]. 2. A.I. Buzdin and M.Yu. Kupriyanov,, Pis’ma Zh. Eksp. Teor. Phys. 53, 308 (1991) [JETP Lett. 53, 321 (1991)]. 3. E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev. B 55, 15 174 (1997). 4. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005). 5. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 73, 094501 (2006). 6. Yu. V. Nazarov, Superlattices and Microstructuctures 25, 1221 (1999). 7. Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 68 (2003) 054513. 8. Y. Tanaka, Yu. V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003; Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 69 (2004) 144519. 9. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 012507 (2004); Y. Tanaka, S. Kashiwaya, and T. Yokoyama Phys. Rev. B 71, 094513 (2005). 10. C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526;Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451; S. Kashiwaya, Y. Tanaka, M. Koyanagi, K. Kajimura, Phys. Rev. B 53 (1996) 2667; Y. Tanuma, Y. Tanaka, S. Kashiwaya Phys. Rev. B 64 (2001) 214519;S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641 and references therein. 11. A.F. Volkov, A.V. Zaitsev and T.M. Klapwijk, Physica C 210, 21 (1993). 12. S. Yip, Phys. Rev. B 52, 15504 (1995). 13. T. Yokoyama, Y. Tanaka, A. A. Golubov, J. Inoue, and Y. Asano, Phys. Rev. B 71, 094506 (2005). 14. K.D. Usadel Phys. Rev. Lett. 25 (1970) 507.

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THEORY OF ENHANCED PROXIMITY EFFECT BY THE EXCHANGE FIELD IN FS BILAYERS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Enhanced proximitiy eﬀect in normal metal / insulator / diﬀusive ferromagnet / insulator / superconductor (N/I/DF/I/S) junctions is studied for various situations solving the Usadel equation under the Nazarov’s generalized boundary condition. Conductance of the juntion and density of states of the DF are calculated by changing the magnitude of the resistance, Thouless energy and the exchange ﬁeld in DF. We heve clariﬁed that due to the enhanced proximity eﬀect, a sharp zero bias conductance peak (ZBCP) appears for small Thouless energy while a broad ZBCP appears for large Thouless energy. The magnitude of this ZBCP can exceed its value for normal states in contrast to the ZBCP observed in diﬀusive normal metal / superconductor junctions. We ﬁnd structures similar to the conductance in the density of states.

1. Introduction In diﬀusive ferromagnet / superconductor (DF/S) junctions Cooper pairs penetrate into the DF layer from the S layer and have a nonzero momentum by the exchange ﬁeld1,2,3 . This property induces many interesting phenomena. The exchange ﬁeld in the ferromagnet usually breaks Cooper pairs. But for a weak exchange ﬁeld the pair amplitude can be enhanced and the energy dependent DOS can have a zero-energy peak4 . Although the DOS has been studied extensively4,5,6,7 , the condition for the appearance of the DOS peak was not studied systematically. We studied the conditions for the appearance of such anomaly, i.e., strong enhancement of the proximity eﬀect and found two conditions corresponding to weak proximity eﬀect and strong one8 . Since DOS is a fundamental quantity, this resonant proximity eﬀect can inﬂuence various transport phenomena. The purpose of the present paper is to study the inﬂuence of the resonant proximity eﬀect by the exchange ﬁeld on the tunneling conductance and the DOS in DF/ S junctions with Nazarov’s boundary conditions. A weak exchange ﬁeld compared to the Fermi energy can be realized in ex-

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periments with , e.g., Ni doped Pd7 . Therefore it is interesting to predict new phenomena in DF/S junctions for a future experiment. In the present paper, we study the tunneling conductance in normal metal / insulator / diﬀusive ferromagnet / insulator / superconductor junctions for various parameters such as resistance Rd in DF, the exchange ﬁeld h in DF, the Thouless energy ET h in DF. In the present paper we conﬁne ourselves to zero temperature and put kB = = 1. 2. Formulation In this section we introduce the model and the formalism. The formulation is the same as the one in Ref.9 . We consider a junction consisting of normal and superconducting reservoirs connected by a quasi-one-dimensional diﬀusive ferromagnet conductor (DF) with a length L much larger than the mean free path. The interface between the DF conductor and the S electrode has a resistance Rb while the DF/N interface has a resistance Rb . The positions of the DF/N interface and the DF/S interface are denoted as x = 0 and x = L, respectively. We model inﬁnitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) transparency of the junctions Tm and Tm 2 2 2 and Tm = 4 cos φ/(4 cos φ + Z ), where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We apply the quasiclassical Keldysh formalism in the following calculation of the tunneling conductance. The 4 × 4 Green’s functions in N, DF and S are ˇ 1 (x) and G ˇ 2 (x) where the Keldysh component K ˆ 0,1,2 (x) is ˇ 0 (x), G denoted by G ˆ i (x)fˆi (x)− fˆi (x)Aˆi (x) with retarded component R ˆ i (x), advanced ˆ i (x) = R given by K ∗ ˆ ˆ ˆ component Ai (x) = −Ri (x) using distribution function fi (x)(i = 0, 1, 2). In the ˆ 0 (x) = τˆ3 and fˆ0 (x) = fl0 + τˆ3 ft0 . R ˆ 2 (x) is expressed ˆ 0 (x) is expressed by R above, R √ √ 2 2 2 2 ˆ τ3 +f τˆ2 ) with g = / − Δ and f = Δ/ Δ − , where denotes by R2 (x) = (gˆ the quasiparticle energy measured from the Fermi energy and fˆ2 (x) = tanh[/(2T)] in thermal equilibrium with temperature T . We put the electrical potential zero in the S-electrode. ˇ 1 (x) at the DF/S interface The Nazarov’s generalized boundary condition for G 11,12,13 and the one at the DF/N interface is given in Ref.8 . is given in Ref. The resistance of the interface Rb is given by 2 () () Rb = R0 π/2 . () (φ) cos φ −π/2 dφT ()

()−1

= Here R0 is Sharvin resistance, which in three-dimensional case is given by R0 () 2 2 () 2 e kF Sc /(4π ), where kF is the Fermi wave-vector and Sc is the constriction area. ˇ 1 (x) as The electric current per one spin is expressed using G ∞ ˇ −L ˇ 1 (x) ∂ G1 (x) )K ], dTr[τˆ3 (G (1) Iel = 8eRd 0 ∂x

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132 ˇ1 (x) K ˇ1 (x) where (Gˇ1 (x) ∂ G∂x ) denotes the Keldysh component of (Gˇ1 (x) ∂ G∂x ). In the actual calculation it is convenient to use the standard θ-parameterization where ˆ 1 (x) = τˆ3 cos θ(x) + τˆ2 sin θ(x). The parameter ˆ 1 (x) is expressed as R function R θ(x) is a measure of the proximity eﬀect in DF. The spatial dependence of θ(x) is determined by the following equation

D

∂2 θ(x) + 2i( − (+)h) sin[θ(x)] = 0 ∂x2

(2)

with the diﬀusion constant D in DF for minority (majority) spin where h denotes the exchange ﬁeld. Note that we assume a weak ferromagnet and neglect the diﬀerence of Fermi velocity between majority spin and minority spin. Following the method in Ref.9 , the diﬀerential resistance R per one spin projection at zero temperature is given by 2Rd L 2Rb 2Rb dx + + R= (3) < Ib0 > L 0 cosh2 θim (x) < Ib1 > with Ib1 =

2 Tm Λ1 + 2Tm (2 − Tm )Real{cos θ0 } ) + T cos θ |2 | (2 − Tm 0 m

Λ1 = (1+ | cos θ0 |2 + | sin θ0 |2 ) where Ib0 is given in Ref.12 . In the above θim (x), θ0 and θL denote the imaginary part of θ(x), θ(0+ ) and θ(L− ) respectively. Then the total tunneling conductance in the superconducting state σS (eV ) is given by σS (eV ) = ↑,↓ 1/R. The local density of states (DOS) normalized by its normal states value in the DF layer, N , is given by 1 Re cos θ(x). (4) N= 2 ↑,↓

In the following section, we will discuss the normalized DOS at x = 0 and the normalized tunneling conductance σT (eV ) = σS (eV )/σN (eV ) where σN (eV ) is the tunneling conductance in the normal state given by σN (eV ) = σN = 1/(Rd + Rb + Rb ). Below we ﬁx Z = Z = 3 and Rd /Rb = 0.1. 3. Results In this section, we study the inﬂuence of the resonant proximity eﬀect on tunneling conductance as well as the DOS in the DF region. The resonant proximity eﬀect is characterized as follows. When the proximity eﬀect is weak (Rd /Rb 1), the condition is given by Rd /Rb ∼ 2h/ET h . When the proximity eﬀect is strong (Rd /Rb 1), the condition is given by ET h ∼ h. We choose Rd /Rb = 0.2 as a typical value to study the weak proximity regime. We also choose Rd /Rb = 4 to

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study the strong one. Let us ﬁrst study the tunneling conductance. In the following we show that a zero bias conductance peak (ZBCP) appears due to the enhanced proximity eﬀect when resonant condition is satisﬁed. Figure. 1 shows the tunneling conductance for Z = 3, Z = 3, Rd /Rb = 0.1 and various ET h /Δ with (a) Rd /Rb = 0.2 and h/Δ = 0.1, and (b) Rd /Rb = 4 and h/Δ = 1. In Fig. 1 (a) dip structures at eV ∼ 0, ±h appear at small ET h /Δ. When resonant condition is satisﬁed (ET h /Δ = 1), a ZBCP emerges and it is suppressed with further increasing ET h /Δ. In Fig. 1 (b) it is shown that the tunneling spectrum is ﬂat at small ET h /Δ. When resonant condition is satisﬁed (ET h /Δ = 1), a broad ZBCP appears and it disappears by increasing ET h /Δ. The magnitude of the ZBCP exceeds unity. Contrary, in diﬀusive normal metal / superconductor junctions, the normalized conductance never exceed unity12 . (a)

1.4

Rd / Rb = 0.2 h / Δ = 0.1 E Th /

1.2

E Th / E Th /

1

Δ Δ Δ

=0.2 =1 =2

0.8

σ

T 0.6 (b) 2

Rd / Rb = 4 h / Δ = 1 E Th / E Th / E Th /

Δ Δ Δ

=0.2 =1 =2

1

-1

Figure 1.

0

eV Δ

1

Normalized tunneling conductance.

In Figs. 2 and 3 we show the DOS as a function of ε and x with corresponding parameters to those of Fig. 1 (a) and (b) respectively. We can see the similar structures to those of the conductances. Especially the zero energy peak in the DOS appears when the resonant conditions are satisﬁed. 4. Conclusion In the present paper we calculated the conductance and the DOS in the DF/S junctions by solving the Usadel equation and found that due to the enhanced proximity eﬀect, a sharp ZBCP appears for small Thouless energy while a broad ZBCP ap-

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(a )

ETh / Δ = 0.2

(b)

ETh / Δ = 1

(c )

ETh / Δ = 2

Figure 2.

Normalized DOS with Rd /Rb = 0.2 and h/Δ = 0.1.

pears for large Thouless energy. The magnitude of this ZBCP can exceed its value for normal states in contrast to the ZBCP observed in diﬀusive normal metal / superconductor junctions. We found structures similar to the conducance in the density of states. References 1. 2. 3. 4. 5.

A.I. Buzdin, L.N. Bulaevskii, and S.V. Panyukov, JETP Lett. 35, 178 (1982). A.I. Buzdin and M.Yu. Kupriyanov, JETP Lett. 53, 321 (1991). E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev. B 55, 15 174 (1997). A. A. Golubov, M. Yu. Kupriyanov, and Ya. V. Fominov, JETP Lett. 75, 223 (2002). A. Buzdin, Phys. Rev. B 62, 11 377 (2000).

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(a)

ETh / Δ = 0.2

(b)

ETh / Δ = 1

(c )

ETh / Δ = 2

Figure 3.

Normalized DOS with Rd /Rb = 4 and h/Δ = 1.

6. M. Zareyan, W. Belzig, and Yu. V. Nazarov, Phys. Rev. Lett. 86, 308 (2001); Phys. Rev. B 65, 184505 (2002). 7. T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys. Rev. Lett. 86, 304 (2001); T. Kontos, M. Aprili, J. Lesueur, X. Grison, and L. Dumoulin, Phys. Rev. Lett. 93, 137001 (2004). 8. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005). 9. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 73, 094501 (2006). 10. K.D. Usadel Phys. Rev. Lett. 25 (1970) 507. 11. Yu. V. Nazarov, Superlattices and Microstructuctures 25, 1221 (1999). 12. Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 68 (2003) 054513. 13. T. Yokoyama, Y. Tanaka, A. A. Golubov, J. Inoue, and Y. Asano, Phys. Rev. B 71, 094506 (2005).

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THEORY OF JOSEPHSON EFFECT IN DIFFUSIVE d-WAVE JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected] We study the Josephson eﬀect in d-wave superconductor / diﬀusive normal metal / dwave superconductor junctions by solving the Usadel equation, changing the magnitude of the resistance and Thouless energy in the diﬀusive normal metal, the transparencies of the insulating barriers at the interfaces and the angles between the normal to the interfaces and the crystal axis of d-wave superconductors. We ﬁnd that, in contrast to the case of conventional s-wave junctions, the product of the Josephson current and the normal state resistance is enhanced with the decrease of transparency of the interface. In the presence of midgap Andreev resonant states (MARS), the Josephson current has a nonmonotonic temperature dependence due to the competition between the proximity eﬀect and the MARS.

1. Introduction Since the discovery of Josephson eﬀect1 in superconductor / insulator / superconductor (SIS) junctions, it has been studied in various situations2,3 . In SIS and superconductor / diﬀusive normal metal / superconductor (S/DN/S) junctions the critical current increases monotonically with decreasing temperature4,5,6 . In S/DN/S junctions Josephson current is carried by Cooper pairs penetrating into the DN as a result of the proximity eﬀect. On the other hand, Josephson eﬀect depends strongly on pairing symmetries of superconductors. In d-wave superconductor / insulator / d-wave superconductor (DID) junctions, nonmonotonic dependence of critical current on temperature7,8,9,10 occurs due to the formation of midgap Andreev resonant states (MARS) at the interface11 . The MARS stem from sign change of pair potentials of d-wave superconductors 12 .

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In order to study the combined eﬀect of the proximity eﬀect and MARS, Tanaka et al. 13,14,15 have recently extended the circuit theory16 to the systems with unconventional superconductors. The circuit theory provides the boundary conditions for the Usadel equations 17 widely used in diﬀusive superconducting junctions. These boundary conditions generalize the Kupriyanov-Lukichev conditions 5 for an arbitrary type of connector which couples the diﬀusive nodes. An application of the extended circuit theory to the DN/d-wave superconductor junctions has revealed a strong competition (destructive interference) of the MARS with the proximity eﬀect in DN13,14 . However, this competition has not yet been tested experimentally, thus it’s important to propose the way to verify this prediction. In the present paper, we show that Josephson eﬀect is a suitable tool to observe the above competition. We calculate Josephson current in d-wave superconductor / diﬀusive normal metal / d-wave superconductor (D/DN/D) junctions, solving the Usadel equations with new boundary conditions derived in Ref.18 . This allows us to study simultaneously the inﬂuence of the proximity eﬀect and the formation of MARS on the Josephson current. We ﬁnd that the competition between the proximity eﬀect and the formation of MARS provides a new mechanism for a nonmonotonic temperature dependence of the maximum Josephson current.

2. Formulation We consider a junction consisting of d-wave superconducting reservoirs (D) connected by a quasi-one-dimensional diﬀusive conductor (DN) with a resistance Rd and a length L much larger than the mean free path. The DN/D interface located at x = 0 has a resistance Rb , while the DN/D interface located at x = L has a resistance Rb . We model inﬁnitely narrow insulating barriers by the delta function U (x) = Hδ(x − L) + H δ(x). The resulting transparencies of the junctions Tm and 2 are given by Tm = 4 cos2 φ/(4 cos2 φ + Z 2 ) and Tm = 4 cos2 φ/(4 cos2 φ + Z ), Tm where Z = 2H/vF and Z = 2H /vF are dimensionless constants and φ is the injection angle measured from the interface normal to the junction and vF is Fermi velocity. We parameterize the quasiclassical Green’s functions G and F with a function Φω 2,3 : ω Φω , Fω = 2 Gω = 2 ∗ ω + Φω Φ−ω ω + Φω Φ∗−ω

(1)

where ω is the Matsubara frequency. Then Usadel equation reads17 2 πTC ∂ 2 ∂ ξ (2) Gω Φω − Φω = 0 ωGω ∂x ∂x with the coherence length ξ = D/2πTC , the diﬀusion constant D and the transition temperature TC . We use the boundary conditions in Ref.18

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The Josephson current is given by the expression eIR ∂ ∗ ∂ RT L G2ω ∗ =i Φω Φ−ω − Φ−ω Φω πTC 2Rd TC ω ω 2 ∂x ∂x

(3)

with temperature T and R ≡ Rd + Rb + Rb , respectively. In the following we focus on the IC R value where IC denotes the magnitude of the maximum Josephson current. Below α and β denote the angles between the normal to the interface and the crystal axes of d-wave superconductors for x ≤ 0 and x ≥ L respectively, and ϕ is a phase diﬀerence between d-wave superconductors. 3. Results We consider symmetric barriers with Rb = Rb and Z = Z for simplicity. Figure 1 (a) shows IC R value as a function of temperature for Rd /Rb = 1, ET h /Δ(0) = 0.5 and (α, β) = (0, 0) with various Z. For α = β = 0, IC R increases with the decrease of the temperature as in the case of s-wave junction. The magnitude of IC R is enhanced with increasing Z in contrast to the s-wave junctions where IC R is enhanced with decreasing the magnitude of Z as shown in (b). We can explain the increase of IC R with Z by calculating anomalous Green’s functions F which corresponds to the degree of the proximity eﬀect. We choose x = L/2 and ϕ = π/2 with corresponding parameters in (a) at T /TC = 0.1. As the value of Z increases, both the magnitudes of ReF and ImF increase as shown in (c) and (d). This results in the larger magnitude of the IC R. In d-wave junctions, as shown in our previous papers 14 , the magnitude of the proximity eﬀect is reduced with the decrease of the value of Z, i.e., the increase of the transparency of the junctions. In d-wave junctions with α = β = 0, quasiparticles with injection angles φ in π/4 <| φ |< π/2 and 0 <| φ |< π/4 feel diﬀerent signs of d-wave pair potential. For high transparent junctions, the contribution from π/4 <| φ |< π/2 cannot be negligible, and the cancellation due to the sign change of the pair potential reduces the proximity eﬀect. On the other hand for low transparent junctions, only the quasiparticles with 0 <| φ |< π/4 has the dominant contribution. Consequently, the resulting magnitude of F is enhanced. This is the reason of the enhancement of IC R product for low transparent junctions with large magnitude of Z. Next we focus on the case with nonzero values of α and β. Figure 2 (a) displays IC R value as a function of temperature for Z = 10, ET h /Δ(0) = 0.5 and Rd /Rb = 1. For (α, β) = (0, 0), IC R value increases monotonically with decreasing temperature as already discussed in Fig. 2. In contrast, for (α, β) = (π/8, 0) or (π/8, π/8), it has a nonmonotonic temperature dependence. It is important to notice that the transparencies at the interfaces greatly change the peak structure as shown in (b). In (b), temperature dependence of IC R is plotted with various Z for (α, β) = (π/8, 0). With decreasing Z, the peak structure is smeared. This nonmonotonic temperature dependence can be explained in terms of the competition between the proximity eﬀect and the formation of MARS. It is known

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0.2

α β)=(

(

,

0.4

0, 0 )

)

0.1

Fe R

π

,

0, 0 )

Z=10 Z=3 Z=1

0.3 0.2

(a )

C

α β)=(

(

Z=10 Z=3 Z=1

T

(c )

0.1

( / 0 R C I e

0 S-wave

(

Z=10 Z=3 Z=1

0.3

,

0, 0 )

F m I

-0.2

0.2 (b)

0.1

0

α β)=(

-0.1

0.2

0.4

0.6

T TC

0.8

Z=10 Z=3 Z=1 (d )

-0.3

1

-0.4 0

0.2

0.4

0.6

ω / π TC

0.8

1

Figure 1. (a) Temperature dependence of IC R value for Rd /Rb = 1, ET h /Δ(0) = 0.5 and (α, β) = (0, 0) with various Z. (b) Same as (a) for s-wave junctions. Real (c) and imaginary (d) part of F with corresponding parameters in (a) at T /TC = 0.1.

from the previous studies that for α = β = 0 the proximity eﬀect exists but MARS is absent at the interface. On the other hand, for α = β = π/4, only MARS exists and the proximity eﬀect is absent 13,14 . In other cases, both the proximity eﬀect and MARS are present. With the decrease of the temperature, the formation of MARS strongly suppresses the proximity eﬀect. This results in the suppression of the Josephson current at low temperatures. Therefore, a nonmonotonic temperature dependence appears when both the proximity eﬀect and MARS exist ((α, β) = (π/8, 0) or (π/8, π/8)). We can conﬁrm it by calculating F as a function of Matsubara frequency ω as shown in Figs. 2 (c) and (d) at x = L/2 and ϕ = π/2 for (α, β) = (π/8, 0). At low temperatures (T /TC = 0.01) the magnitude of F is suppressed at low energy in contrast to the case of high temperature (T /TC = 0.2 and 0.3). This demonstrates strong suppression of the proximity eﬀect by the formation of MARS at low temperatures, which leads to the nonmonotonic temperature dependence. The inﬂuence of MARS on the proximity eﬀect is strongest when the interface barrier is high, i.e., when the value of Z is large. Therefore the resulting nonmonotonic temperature dependence becomes much pronounced for large Z as shown in Fig. 2 (b). To study the nonmonotonic temperature dependence of IC R in more detail, we will calculate IC R for various Rd /Rb and ET h /Δ(0) as shown in Fig. 3. As the ratios Rd /Rb and ET h /Δ(0) increase, the temperature at the peak also increases monotonically. In order to observe the peak structure experimentally, large magnitudes of Z, Rd /Rb and ET h /Δ(0) are needed. This means the high insulating barrier, small cross section area of the DN and thin DN. Note that, according to the circuit theory16 , we can change the cross section area independently of the

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0.2 (

(

α β) =( ,

0, 0)

π / 8, 0)

0.1

(a)

)

C

π T

(/ 0.1 R C I 0.08 e

Fe R

0.3 T/T C=0.01 T/T C=0.1 T/T C=0.2

0.2

(c )

0.1

( π / 8, π / 8)

0 (

α β ) = (π ,

-0.05

Z=10 Z=3

0.06

F m I

Z=1

0.04

0.2

0.4

0.6

0.8

(d )

-0.1

(b)

0.02

0

/ 8, 0 )

T/T C =0.01 T/T C =0.1

-0.15

-0.2

1

T TC

T/T C =0.2

0

0.2

0.4

0.6

0.8

ω / π TC

Figure 2. (a) Temperature dependence of IC R value for Z = 10, ET h /Δ(0) = 0.5 and Rd /Rb = 1 in (a). (b) Same as (a) for (α, β) = (π/8, 0) with various Z. Real (c) and imaginary (d) parts of F for various T in the case (α, β) = (π/8, 0).

constriction area. Rd = 0 .5 Rb

ETh / Δ(0) = 1 0.1

)

C

0.15

0.08

E Th /

R d/R b=0.5

/( R 0.04 C I e 0.02

(a)

0.2

0.4

0.6

T TC

0.8

E Th / E Th /

0.1

R d/R b=0.1

0.06

0

ΔΔ Δ

R d/R b=1

π T

(0)=1 (0)=0.5 (0)=0.1

(b)

0.05

1

0

0.2

0.4

0.6

0.8

1

T TC

Figure 3. Temperature dependence of IC R value for Z = 10 and (α, β) = (π/8, 0) with various (a) Rd /Rb and (b) various ET h /Δ(0).

Finally, we compare our results with those of previous studies in ballistic regime. In DID junctions, the nonmonotonic temperature dependence was predicted for the mirror type junctions (α = −β), and for symmetric junctions (α = β) the nonmonotonic temperature dependence never occurs7,8 . In this case, the magnitude of the Josephson current ﬁrst increases, then decreases and ﬁnally increases again. The Josephson junctions can have a 0-π transition with the decrease of temperature. These predictions were conﬁrmed in recent experiments with sub-micron grain-boundary d-wave Josephson junctions 9,10 . On the other hand, in D/DN/D

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junctions both mirror type and symmetric junctions give the same result, especially the nonmonotonic temperature dependence, where the resulting Josephson current increases and decreases with the decrease of temperature. There is no enhancement at low temperatures as in the case of DID junctions with mirror type symmetry. The 0-π transition never appears in D/DN/D junctions in contrast to DID junctions. 4. Conclusions In the present paper, we have studied the Josephson eﬀect in d -wave superconductor / diﬀusive normal metal / d -wave superconductor junctions by solving the Usadel equation. By calculating the Josephson current for various parameters, we have clariﬁed the following points. (1)The magnitude of IC R is enhanced with the increase of the insulating barriers at the interfaces in contrast to the conventional s-wave junctions. (2)The IC R value can have a nonmonotonic temperature dependence due to the competition between the proximity eﬀect and the formation of the MARS. The origin of such a nonmonotonic behavior is completely diﬀerent from that in the established case of DID junctions 7,9,10 , where 0-π transition occurs due to the MARS. In order to observe experimentally the predicted nonmonotonic temperature dependence of IC in D/DN/D junctions, large magnitudes of Z, Rd /Rb and ET h /Δ(0) are required. It follows from the present approach that similar nonmonotonic temperature dependence is also expected for s-wave superconductor / diﬀusive normal metal / d -wave superconductor junctions. There exists one report in literature about the observation of nonmonotonic temperature dependence of IC in YBaCuO/Pb junctions 19 . Though there is a possibility that this experiment is relevant to our results, more detailed discussion is needed to check whether it is the case using more experimental data. An interesting future problem is calculation of the Josephson current in p-wave superconductor / diﬀusive normal metal / p-wave superconductor junctions because interesting phenomena were recently predicted in diﬀusive normal metal / p-wave superconductor bilayers15 . References B. D. Josephson, Phys. Lett. 1, 251 (1962). K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979). A. A. Golubov, M. Yu. Kupriyanov, and E. Il ichev Rev. Mod. Phys. 76, 411 (2004). V. Ambegaokar and A. Baratoﬀ, Phys. Rev. Lett. 10, 486 (1963). M. Yu. Kupriyanov and V. F. Lukichev, Zh. Exp. Teor. Fiz. 94 (1988) 139 [Sov. Phys. JETP 67, (1988) 1163]. 6. A. V. Zaitsev, Physica C 185-189, 2539 (1991). 7. Yu. S. Barash, H. Burkhardt, and D. Rainer, Phys. Rev. Lett. 77, 4070 (1996); Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, R11957 (1996); Y. Tanaka and S. Kashiwaya, Phys. Rev. B 56, 892 (1997). 8. A. A. Golubov and M. Yu. Kupriyanov, JETP Lett. 69, 262 (1999). 1. 2. 3. 4. 5.

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9. E. Il’ichev, M. Grajcar, R. Hlubina, et al., Phys. Rev. Lett. 86, 5369 (2001). 10. G. Testa, E. Sarnelli, A. Monaco, E. Esposito, M. Ejrnaes, D.-J. Kang, S. H. Mennema, E. J. Tarte, and M. G. Blamire Phys. Rev. B 71, 134520 (2005) 11. C. Bruder, Phys. Rev. B 41, (1990) 4017; C.R. Hu, Phys. Rev. Lett. 72, (1994) 1526; S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, (2000) 1641; Y. Asano, Y. Tanaka and S. Kashiwaya, Phys. Rev.B 69, (2004) 134501. 12. Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, (1995) 3451. 13. Y. Tanaka, Y.V. Nazarov and S. Kashiwaya, Phys. Rev. Lett. 90, 167003 (2003). 14. Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, and S. Kashiwaya, Phys. Rev. B 69, 144519 (2004). 15. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70, 012507 (2004); Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B 71, 094513 (2005); Y. Tanaka, Y. Asano, A. A. Golubov and S. Kashiwaya, Phys. Rev. B 72, 140503(R) (2005). 16. Yu. V. Nazarov, Superlattices and Microstructuctures 25, 1221 (1999). 17. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970). 18. T. Yokoyama, Y. Tanaka, A. A. Golubov and Y. Asano, unpublished. 19. I. Iguchi and Z. Wen, Phys. Rev. B 49, R12388 (1994).

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QUANTUM DISSIPATION DUE TO THE ZERO ENERGY BOUND STATES IN HIGH-TC SUPERCONDUCTOR JUNCTIONS∗

S. KAWABATA Nanotechnology Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan E-mail: [email protected] S. KASHIWAYA Nanoelectronics Research Institute (NeRI), AIST, Tsukuba, Ibaraki, 305-8568, Japan Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan T. KATO The Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa, 277-8581, Japan A. A. GOLUBOV Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

The macroscopic quantum tunneling (MQT) in the in-plane high-Tc superconductor Josephson junctions and the inﬂuence of the zero energy bound states (ZES) on the MQT are investigated theoretically. We obtained the analytical formula of the MQT rate and showed that the presence of the ZES at the insulator/superconductor interface leads to a strong Ohmic quasiparticle dissipation. Therefore, the MQT rate is noticeably inhibited in compared with the c-axis junctions in which the ZES are completely absent.

1. Introduction Great attention has been attracted to theoretical and experimental studies of the eﬀect of the dissipation on the macroscopic quantum tunneling (MQT) in superconductor Josephson junctions. In current-biased Josephson junctions, the macroscopic ∗ This

work is supported by NEDO-SYNAF and JSPS Grants-in-Aid for Scientiﬁc Research.

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variable is the phase diﬀerence φ between two superconductors and measurements of the MQT are performed by switching the junction from its metastable zero-voltage state to a non-zero voltage state (see Fig. 1 (d)). Heretofore, experimental tests of the MQT have been focused on s-wave (low-Tc) junctions. This fact is due to the naive preconception that the existence of the low energy quasiparticle in the d-wave order parameter of a high-Tc cuprate superconductor1 may preclude the possibility of observing the MQT. Recently we have theoretically investigated the eﬀect of the nodal-quasiparticle on the MQT in the d-wave c-axis junctions.2 We have shown that the eﬀect of the nodal-quasiparticle gives rise to a super-Ohmic dissipation3,4 and the suppression of the MQT due to the nodal-quasiparticle is very weak. First experimental observation of the MQT in the high-Tc Josephson junction is realized by Bauch et al. using a YBCO grain boundary bi-epitaxial junction.5,6 Recently, Inomata et al.,7 Jin et al.,8 and Kashiwaya et al.9 have experimentally observed the MQT in the c-axis (Bi2212 intrinsic) junctions. They have reported that the eﬀect of the nodal-quasiparticle on the MQT is negligibly small and the thermal-to-quantum crossover temperature is relatively high (0.3∼1K) compared with the case of the low-Tc junctions. In this paper, we will investigate the MQT in the d-wave in-plane junctions parallel to the ab-plane (see Fig. 1).10,11 In such junctions, the zero energy bound states (ZES)12 are formed near the interface between superconductor and the insulating barrier. The ZES are generated by the combined eﬀect of multiple Andreev reﬂections and the sign change of the d-wave order parameter symmetry, and are bound states for the quasiparticle at the Fermi energy. Below, we will show that the ZES give rise to the Ohmic type strong dissipation so the MQT is considerably suppressed in compared with the c-axis and the d0 /d0 junction cases.

a

y

a

d0

dπ/4

dπ/4

(d)

ZES

a

I

b

dπ/4 I ZES

b

(b)

(c)

b

b

a

d0

b

b

d0

a

a

(a)

I

U0 x

ZES

ωp Figure 1. Schematic drawing of the in-plane d-wave Josephson junction. (a) d0 /d0 , (b) d0 /dπ/4 , and (c) dπ/4 /dπ/4 junction. In the case of the d0 /dπ/4 and dπ/4 /dπ/4 junctions, the ZES are formed near the boundary between superconductor dπ/4 and insulating barrier I. (d) Potential U (φ) v.s. the phase diﬀerence φ between two superconductors. U0 is the barrier height and ωp is the Josephson plasma frequency.

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2. Eﬀective Action By using the method developed by Eckern et. al.,13 the partition function of the system can be described by a functional integral over the macroscopic variable (the phase diﬀerence φ), Z = Dφ(τ ) exp (−Sef f [φ]/), where the eﬀective action Sef f in the high barrier limit is given by 2 β β M ∂φ(τ ) φ(τ ) − φ(τ ) . dτ + U (φ) − dτ dτ α(τ − τ ) cos Sef f [φ] = 2 ∂τ 2 0 0 (1) In this equation, β = 1/kB T , M = C (/2e)2 is the mass (C is the junction capacitance) and the potential U (φ) is described by 1 dλ φIJ (λφ) − φ Iext , (2) U (φ) = 2e 0 where IJ is the Josephson current and Iext is the external bias current, respectively. The dissipation kernel α(τ ) is related to the quasiparticle current Iqp under constant bias voltage V by ω ∞ dω −ωτ e Iqp V = α(τ ) = , (3) e 0 2π e at the zero temperature. Below, we will derive the eﬀective action for the three types of the d-wave junction (d0 /d0 , d0 /dπ/4 , and dπ/4 /dπ/4 ) in order to investigate the eﬀect of the ZES on the MQT. In the case of the d0 /d0 junction, the node-to-node quasiparticle tunneling can contribute to the dissipative quasiparticle current. However, the ZES are completely absent. These behaviors are qualitatively identical with that for the c-axis Josephson junctions.2 On the other hand, in the case of the d0 /dπ/4 and dπ/4 /dπ/4 junction, the ZES are formed around the surface of the right superconductor dπ/4 . Therefore the node to the ZES quasiparticle tunneling becomes possible. Firstly, we will calculate the potential energy U in the eﬀective action (1). As mentioned above, U can be described by the Josephson current through the junction in the high barrier limit. In order to obtain the Josephson current, we solve the Bogoliubov-de Gennes equation together with the appropriate boundary conditions.12 By substituting the Josephson current into eq. (2), we can obtain the analytical expression of the potential U, i.e., ⎧ I1 Iext ⎪ ⎪ − φ for d0 /d0 cos φ + ⎪ ⎪ ⎪ 2e I1 ⎪ ⎨ I Iext 2 φ for d0 /dπ/4 . (4) U (φ) ≈ − − cos 2φ + 2 ⎪ 4e I2 ⎪ ⎪ ⎪ I3 φ 1 Iext ⎪ ⎪ φ for dπ/4 /dπ/4 cos + ⎩ − e 2 2 I3

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146 2 where I1 ≡ 3πΔ0 /10eRN , I2 ≡ π 2 βΔ20 /35e3 Nc RN , and I3 = 3πz0 Δ0 /4eRN (RN is the normal state resistance of the junction, Nc is the number of channel at the Fermi energy, and z0 is the normalized barrier height for insulating barrier I, respectively). In calculation, we have assumed that the amplitude of the pair potential is given by Δ0 cos 2θ ≡ Δd0 (θ) for d0 and Δ0 sin 2θ ≡ Δdπ/4 (θ) for dπ/4 . As in the case of the s-wave and the c-axis junctions,2 U can be expressed as the tilted washboard potential (see Fig. 1(d)).

3. Dissipation due to ZES Next we will calculate the dissipation kernel α(τ ) in the eﬀective action (1). In the high barrier limit, the quasiparticle current is given by12 ∞ 2e Iqp (V ) = |tN |2 dENL (E, θ)NR (E + eV, θ) [f (E) − f (E + eV )] , (5) h p −∞ where tN ≈ cos θ/z0 is the transmission coeﬃcient of the barrier I, NL(R) (E, θ) is the quasiparticle surface density of states (L = d0 and R = d0 or dπ/4 ), and f (E) is the Fermi-Dirac distribution function. The explicit expression of the surface density of states is obtained by Matsumoto and Shiba.14 In the case of d0 , there are no ZES. Therefore the angle θ dependence of Nd0 (E, θ) is the same as the bulk and is given by |E| . Nd0 (E, θ) = Re E 2 − Δd0 (θ)2 On the other hand, Ndπ/4 (E, θ) is given by E 2 −Δdπ/4 (θ)2 +π|Δdπ/4 (θ)|δ(E). Ndπ/4 (E, θ) = Re |E|

(6)

(7)

The delta function peak at E = 0 corresponds to the ZES. Because of the bound state at E = 0, the quasiparticle current for the d0 /dπ/4 junctions is drastically diﬀerent from that for the d0 /d0 junctions in which no ZES are formed. By substituting Eqs. (6) and (7) into Eq. (5), we can obtain the analytical expression of the quasiparticle current Iqp (V ). In the limit of low bias voltages (eV Δ0 ) and low temperatures (β −1 Δ0 ), this can be approximated as ⎧ 32 π 2 eV 2 ⎪ ⎪ √ for d0 /d0 ⎪ ⎪ ⎪ 28 2 Δ0 RN ⎪ ⎨ 2 3π V √ for d0 /dπ/4 . Iqp (V ) ≈ (8) 4 2 RN ⎪ 2 ⎪ ⎪ 2 5 ⎪ ⎪ 2 π Δ0 V ⎪ ⎩ for dπ/4 /dπ/4 35 RN In the calculation of Iqp for the dπ/4 /dπ/4 junctions, we have replaced the ZES delta function δ(E) in Eq. (7) with the Lorentz function ( /π)/( 2 + E 2 ) in order

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to avoid a mathematical diﬃculty and model the real systems (e.g. the disorder eﬀects and the many body eﬀects). It is apparent from Eq. (8) that, in the case of d0 /d0 junctions, the dissipation is the super-Ohmic type as in the case of the c-axis junctions.2 This can be attributed to the eﬀect of the node-to-node quasiparticle tunneling. Thus the quasiparticle dissipation is very weak. On the other hand, in the case of the d0 /dπ/4 junctions, the node-to-ZES quasiparticle tunneling gives the Ohmic dissipation which is similar to that in normal junctions.13 Therefore the dissipation for the d0 /dπ/4 junctions is stronger than that for the d0 /d0 junctions. Moreover, in the case of the dπ/4 /dπ/4 junctions, the ZES-to-ZES quasiparticle tunneling dominates the quasiparticle dissipation. The broadening of the ZES peak is typically one order magnitude smaller than Δ0 . Therefore, due to the prefactor Δ0 / in Eq. (8), the quasiparticle dissipation in the dπ/4 /dπ/4 junctions becomes enormously stronger than that for the d0 /d0 and d0 /dπ/4 cases. From Eq. (3), the asymptotic form of the dissipation kernel is given by ⎧ 32 2 RQ 1 ⎪ ⎪ √ for d0 /d0 ⎪ ⎪ 7 ⎪ 2 Δ0 RN |τ |3 ⎪ ⎨ 2 3 RQ 1 √ for d0 /dπ/4 . (9) α(τ ) ≈ 4 R ⎪ 2 2 N |τ |2 ⎪ ⎪ 2 ⎪ 25 Δ0 RQ 1 ⎪ ⎪ ⎩ for dπ/4 /dπ/4 35π RN |τ |2 4. MQT in In-plane d-wave Junctions Let us move to the calculation of the MQT rate Γ for the d-wave Josephson junctions based on the standard Caldeira and Leggett theory.15 At the zero temperature Γ is given by Γ ≈ A exp (−SB /), where SB ≡ Sef f [φB ] and φB is the bounce solution. Following the procedures as above, we obtain the analytical formulae of the MQT rate for the in-plane d-wave junctions as ⎧ U0 π η 18 δM 35√ ⎪ exp − c for d0 /d0 + ⎪ 0 5 Mωp ⎪ ⎨ 27 42 Δ0 Γ 3 ζ(3) ≈ exp − 25 √2π2 η(1 − x2 ) for d0 /dπ/4 , (10) Γ0 ⎪ 8 3 ⎪ 2 ⎪ 3 ζ(3) Δ0 ⎩ exp − 2 35π η(1 − x2 ) for dπ/4 /dπ/4 3 ∞ where c0 = 0 dy y 4 ln(1 + 1/y 2 )/ sinh2 (πy) ≈ 0.0135, ζ(3) is the Riemann zeta function, η = RQ /RN is the dissipation parameter, U0 is the barrier height of the potential U , ωp is the Josephson plasma frequency, x = Iext /Ii (i=1,2,3), and Γ0 = 12ωp 3U0 /2πωp exp (−36U0 /5ωp ) is the MQT rate without the dissipation. In Eq. (10) δM is the renormalized mass due to the high frequency components (ω ≥ ωp ) of the quasiparticle dissipation. In order to compare the inﬂuence of the ZES and the nodal-quasiparticle on the MQT more clearly, we will estimate the MQT rate (10) numerically. For a mesoscopic bicrystal YBCO Josephson junction16 (Δ0 = 17.8 meV, C = 20 ×

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10−15 F, RN = 100 Ω, x = 0.95), the MQT ⎧ ⎨ 83% for Γ ≈ 25% for ⎩ Γ0 0% for

rate is estimated as d0 /d0 d0 /dπ/4 . dπ/4 /dπ/4

(11)

As expected, the node-to-ZES and ZES-to-ZES quasiparticle tunneling in the d0 /dπ/4 and dπ/4 /dπ/4 junctions give strong suppression of the MQT rate compared with the d0 /d0 junction cases. 5. Summary In conclusion, the MQT in the in-plane high-Tc superconductors has been theoretically investigate and analytically obtained the formulae of the MQT rate which can be used to analyze experiments. The node-to-node quasiparticle tunneling in the d0 /d0 junctions gives rise to the weak super-Ohmic dissipation as in the case of the c-axis junctions.2 For the d0 /dπ/4 junctions, on the other hand, we have found that the node-to-ZES quasiparticle tunneling leads to the Ohmic dissipation. Moreover, in the case of the dπ/4 /dπ/4 junctions, the ZES-to-ZES quasiparticle tunneling gives very strong Ohmic dissipation so the MQT is considerably suppressed. References 1. C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000). 2. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys. Rev. B 70, 132505 (2004). 3. Y. V. Fominov, A. A. Golubov, and M. Kupriyanov, JETP Lett. 77, 587 (2003). 4. M. H. S. Amin and A. Yu. Smirnov, Phys. Rev. Lett. 92, 017001 (2004). 5. T. Bauch, F. Lombardi, F. Tafuri, A. Barone, G. Rotoli, P. Delsing, and T. Claeson, Phys. Rev. Lett. 94, 087003 (2005). 6. T. Bauch, T. Lindstr¨ om, F. Tafuri, G. Rotoli, P. Delsing, T. Claeson, and F. Lombardi, Science 311, 57 (2006). 7. K. Inomata, S. Sato, K. Nakajima, A. Tanaka, Y. Takano, H. B. Wang, M. Nagao, T. Hatano, and S. Kawabata, Phys. Rev. Lett. 95, 107005 (2005). 8. X.Y. Jin, J. Lisenfeld, Y. Koval, A. Lukashenko, A. V. Ustinov and P. M¨ uller, condmat/0603445 (2006). 9. H. Kashiwaya, T. Matsumoto, H. Shibata, H. Eisaki, S. Kawabata, S. Kashiwaya, and Y. Tanaka, to appear in MS+S2006 proceedings (World Scientific, 2006). 10. S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka, Phys. Rev. B 72, 052506 (2005). 11. S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, T. Kato and A. A. Golubov, in preparation. 12. S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). 13. U. Eckern, G. Sch¨ on, and V. Ambegaokar, Phys. Rev. B 30, 6419 (1984). 14. M. Matsumoto and H. Shiba, J. Phys. Soc. Jpn. 64, 1703 (1995). 15. A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981). 16. A. Y. Tzalenchuk, T. Lindstr¨ om, S. A. Charlebois, E. A. Stepantsov, Z. Ivanov, and A. M. Zagoskin, Phys. Rev. B 68, 100501(R) (2003).

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SPIN-POLARIZED HEAT TRANSPORT IN FERROMAGNET/ UNCONVENTIONAL SUPERCONDUCTOR JUNCTIONS

T. YOKOYAMA AND Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected]

Heat transport in ballistic ferromagnet / insulator / d-wave and p-wave superconductor junctions is studied by solving the Bogoliubov-de Gennes equation under the quasiclassical approximation. Thermal conductance of the junctions is calculated by changing the spin-polarization and the angle between the interface normal and the crystal axis of d- or p-wave superconductors. We have obtained various line shapes of the thermal conductance of the juntions. We ﬁnd a qualitative diﬀerence of dependence of thermal conductance on spin-polarization between d-wave and p-wave superconductor junctions. The dependence of thermal conductance on the spin-polarization is very weak compared to the conductance.

1. Introduction In normal metal / supercunductor (N/S) junctions Andreev reﬂection (AR)1 is one of the most important process for low energy transport. The AR is a process that an electron injected from N with energy below the energy gap Δ is converted into a reﬂected hole. Taking the AR into account, Blonder, Tinkham and Klapwijk (BTK) proposed the formula for the calculation of the tunneling conductance2 . It revealed the gap like structure or the douling of tunnelilng conductance due to the AR. This method was extended to normal metal / unconventional superconductor (US) junctions3 . It is shown that a zero bias conductance peak (ZBCP) appears when the mid gap Andreev resonant state (MARS) is formed, which causes resonant transsmission of quasiparticles. The BTK theory was also extended to the ferromagnet / superconductor (F/S) or the ferromagnet / unconventional superconductor (F/US) junctions4 and used to estimate the spin polarization of the F layer experimentally5 . In F/S junctions, it is found that AR is suppressed because the retro-reﬂectivity is broken by the spin-polarization in the F layer6. As a result, many interesting phenomena have been predicted7,8,9 . Contrary to the charge transport, thermal transport has not been studied well in F/S junctions so far. In the present paper we study heat transport in ballistic ferromagnet / insulator / d-wave and p-wave superconductor junctions by solving the Bogoliubov-de Gennes equation under the quasiclassical approximation. Thermal conductance of the junctions is calculated by changing

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the spin-polarization and the angle between the interface normal and the crystal axis of d- or p-wave superconductors. Recently experiments for thermal transport are perfomed in order to study the property of USs. Our calculation may serve as a tool for determination of pairing symmetry of USs. 2. Formulation We consider F/I/US junctions, where I denotes an insulator. We use the same method as in Ref. 4 and the same notations. In the following ↑ (↓) denotes majority (minority) spin. The F/US interface located at x = 0 (the y-axis) has an inﬁnitely narrow insulating barrier described by the delta function U (x) = Hδ(x). As a model of the ferromagnet we apply the Stoner model with exchange potential U . The pair potential matrix we consider is given by 0 Δ↑↓ (θ) Δ (θ) = 0 Δ↓↑ (θ) where θ denotes the direction of motions of quasiparticles measured from the normal to the interface. Below we consider d-wave superconductors, and p-wave superconductors in Eu (1) state and Eu (2) state. The Eu (2) state is realized in Sr2 RuO4 . We take the Eu (1) state as a reference. For d-wave superconductors, the pair potentials are given by Δ↓↑ = −Δ↑↓ = Δ cos[2(θ − α)] where α denotes the angle between the normal to the interface and the crystal axis of d -wave superconductors. For p-wave superconductors in Eu (1) state, the matrix elements are given by Δ↓↑ = Δ↑↓ = Δ(cos[(θ − α)] + sin[(θ − α)]) where α denotes the angle between the normal to the interface and the crystal axis of p-wave superconductors, and in the Eu (2) state, the matrix elements are given by Δ↓↑ = Δ↑↓ = Δ exp(iθ). Applying the BTK method2,4 , we obtain the angle and energy resolved thermal conductance κS↑(↓) for up (down) spin quasiparticle represented in the form: 1−|Γ+ Γ− |2 (1−κN ↓ )−κN ↓ |Γ+ |2

√ √ 2 Θ (θC − |θ|) |1−Γ+ Γ− 1−κN ↑ 1−κN ↓ exp[i(ϕ↓ −ϕ↑ )]| 2 √ 1−|Γ+ Γ− | +κN ↑ 2 Θ (|θ| − θC ) |1−Γ+ Γ− 1−κN ↑ exp[i(ϕ↓ −ϕ↑ )]| 2 1−|Γ+ Γ− | (1−κN ↑ )−κN ↑ |Γ+ |2 √ √ κS↓ = κN ↓ 2 Θ (θC − |θ|) |1−Γ+ Γ− 1−κN ↑ 1−κN ↓ exp[i(ϕ↑ −ϕ↓ )]|

κS↑ = κN ↑

with 1−λ− +iZθ , 1−κN ↓ (1+λ− −iZθ ) Δ± (θ)

exp (iϕ↓ ) = √ Γ± =

E+

√

E 2 −|Δ± (θ)|2

+ −iZθ exp (−iϕ↑ ) = √1−κ1−λ(1+λ , +iZ ) N↑

+

θ

,

Z 2mH U −1 Zθ = cos θ , Z = 2 kF and θC = cos EF with quasiparticle energy E, eﬀective mass m, Fermi wavenumber kF and Fermi energy EF . In the above Θ(x) is the Heaviside step function and κN ↑(↓) denotes the angle and energy resolved thermal

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conductance for up (down) spin quasiparticle in the normal state: κN ↑ =

4λ+

2

(1 + λ+ ) +

Zθ2

,

κN ↓ =

4λ−

2

(1 + λ− ) + Zθ2

Θ (θC − |θ|)

U with λ± = 1 ± EF cos 2θ. Normalized thermal conductance at zero voltage is expressed as ∞ π2 2 dθ cos θ (κS↑ + κS↓ ) coshE2 ( E ) dE 0 −π 2 2T κT = ∞ π E2 2 dθ cos θ (κ + κ ) N↑ N ↓ cosh2 ( E ) dE 0 −π 2

2T

with temperature T . We deﬁne X as X = U/EF and use it as a parameter in the following calculation. 3. Results Below we ﬁx Z = 0 because X dependence is weak for large Z. We study the normalized thermal conduntace κT as a function of temperature in Fig. 1. It shows the thermal conductance of d-wave superconductor junctions for (a) α = 0, (b) α = π/8 and (c) α = π/4. In Fig. 1 (a) it is slightly suppressed with the increase of the spin-polarizarion X. On the other hand, it is enhanced with increasing X as shown in Fig. 1(b) and (c). As a reference for the Eu (2) state, we will check the Eu (1) state. Fig. 2 displays the thermal conductance of p-wave superconductors in Eu (1) state junctions for (a) α = 0, (b) α = π/4 and (c) α = π/2. In this case κT is enhanced with an increase of X for any α. Fig. 3 displays the thermal conductance of p-wave superconductors in Eu (2) state junctions. In this case κT is also enhanced as increasing X. In contrast to the line shapes of the Figs. 1 and 2, κT has an expontential dependence for small T . This diﬀerence stems from the existence of the line nodes of the pair potential. It is important to note that for d-wave superconductors the thermal conductance is suppressed with increasing X for small α, while it is always enhanced with the increase of X for p-wave superconductors. This may be used to distinguish the pairing symmetry. The dependence of thermal conductance on the spin-polarization is very weak compared to the conductance4 . This is because heat cannot be carried in the process of AR. The exchange potential strongly suppresses the AR. Thus thermal oonductance has a weak dependence on the spin-polarization. Note that AR facilitates charge transport. 4. Conclusions In the present paper we have studied heat transport in ballistic ferromagnet / insulator / d-wave and p-wave superconductor junctions by solving the Bogoliubov-de

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X=0.5 X=0.9

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κ

X=0.9 0.6

(b)

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X=0.5 X=0.9

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(c)

Z = 0 α =π / 4

0.2

0 0

Figure 1.

0.2

0.4

T / TC

0.6

0.8

1

Thermal conductance for d-wave superconductors with various X.

Gennes equation under the quasiclassical approximation. Thermal conductance of the junctions is calculated by changing the spin-polarization and the angle between the interface normal and the crystal axis of d-, and p-wave superconductor. We found a qualitative diﬀerence of dependence of thermal conductance on spinpolarization between d-wave and p-wave superconductor junctions. The dependence of thermal conductance on the spin-polarization is very weak compared to the conductance. Acknowledgments This work was supported by NAREGI Nanoscience Project, the Ministry of Education, Culture, Sports, Science and Technology, Japan, the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technol-

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T

κ

X=0.5 X=0.9

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0.2

1 X=0.1 0.8

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(c)

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Z = 0 α =π /2

0.2

0

Figure 2.

0

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0.6

T / TC

0.8

1

Thermal conductance for p-wave superconductors in Eu (1) state with various X.

1

Z =0

0.8

T

κ

0.6

0.4 X=0.1 X=0.5

0.2

X=0.9 0

Figure 3.

0

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0.6

T / TC

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1

Thermal conductance for p-wave superconductors in Eu (2) state with various X.

ogy Corporation (JST) and a Grant-in-Aid for the 21st Century COE “Frontiers of Computational Science”. The computational aspect of this work has been performed

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at the Research Center for Computational Science, Okazaki National Research Institutes and the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo and the Computer Center. References 1. A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964). 2. G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982). 3. Y. Tanaka, S. Kashiwaya, Phys. Rev. Lett. 74 (1995) 3451; S. Kashiwaya, Y. Tanaka, M. Koyanagi, K. Kajimura, Phys. Rev. B 53 (1996) 2667; Y. Tanuma, Y. Tanaka, S. Kashiwaya Phys. Rev. B 64 (2001) 214519; S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641 and references therein. 4. N. Yoshida, Y. Tanaka, J. Inoue, and S. Kashiwaya, J. Phys. Soc. Jpn. 68, 1071 (1999); S. Kashiwaya, Y. Tanaka, N. Yoshida, and M.R. Beasley, Phys. Rev. B 60, 3572 (1999). 5. P. M. Tedrow and R. Meservey, Phys. Rev. Lett. 26, 192 (1971); Phys. Rev. B 7, 318 (1973); R. Meservey and P. M. Tedrow, Phys. Rep. 238, 173 (1994). 6. M.J.M. de Jong and C.W.J. Beenakker, Phys. Rev. Lett. 74, 1657 (1995). 7. A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 8. F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). 9. T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 72, 052512 (2005); Phys. Rev. B 73, 094501 (2006).

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LITTLE-PARKS OSCILLATIONS IN CHIRAL p-WAVE SUPERCONDUCTING RINGS ∗

M. TAKIGAWA AND Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected] M. ICHIOKA Department of Physics, Okayama University, Okayama 700-8530, Japan

Oscillations of critical temperature Tc of superconducting rings under the applied magnetic field are studied based on the time-dependent Ginzburg-Landau theory. We analyze the Little-Parks oscillations in s-wave and chiral p-wave superconductors. In s-wave superconductor, we confirm clear oscillations in Tc . When phase winding numbers around a hole change, Tc takes local minimum. Period of oscillation behaviors depends on a shape of rings. The conventional Little-Parks oscillations do not appear in chiral p-wave superconductor because of the unusual flux penetration in superconducting ring.

1. Introduction Much attention has been focused on vortex physics in type II superconductors. A vortex which is the topological defect is characterized by winding numbers in phase of order parameter and zero amplitude of order parameter at its core. The continuous deformation of the order parameter make various interesting phenomena. The transition temperature Tc of a thin-walled superconducting cylinder oscillates as a function of an applied magnetic ﬁeld, i.e., so-called Little-Parks oscillations.1,2,3 The Little-Parks oscillations have been observed in s-wave superconductors,2,3 where superconductivity can be described by one-component order parameter in the Ginzburg-Landau (GL) theory. Spin-triplet chiral p-wave superconductivity is mainly suggested in Sr2 RuO4 . 5,6,7,8,9,10 In the chiral p-wave superconductor, p+ and p− states are degenerate at the ground state. Therefore chiral p-wave superconductivity is described by two-components of order parameters. In this paper, we study eﬀects of two-component order parameters on the Little-Parks oscillations on the basis of the time-dependent Ginzburg-Landau (TDGL) theory.11 By solving the TDGL ∗ This

work is supported by a Grant-in-Aid the 21st Century COE “Topology Science and Technology” in Hokkaido University.

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equation, we compare characteristic behavior of the Little-Parks oscillations in the chiral p-wave superconducting rings with those in the s-wave rings. 2. TDGL Theory with Two Components Order Parameter To obtain the two-component GL equation for the chiral p-wave superconductor,12 a pair potential is decomposed as Δ(r, p) = η+ (r)φ+ (p) + η− (r)φ− (p) with the order parameters η± , where r is the center-of-mass coordinate of the Cooper pair. The pairing functions φ± (p), depending on the relative momentum p of the pair, are given by px ±ipy for the chiral p-wave symmetry. The GL free-energy density is written as 1 1 T f˜ = −(1 − )(|η+ |2 + |η− |2 ) + |η+ |4 + |η− |4 + 2|η+ |2 |η− |2 Tc 2 2 ∗2 2 ∗2 ∗ ∗ +C1 (η− η+ + η+ η− ) + η+ (qx2 + qy2 )η+ + η− (qx2 + qy2 )η− ∗ 2 ∗ 2 ∗ 2 ∗ 2 +C2 (η+ q− η− + η− q+ η+ ) + C3 (η+ q+ η− + η− q− η+ )

(1)

in the dimensionless form, where q± = qx ±iqy , q = (/i)∇ + 2π φ0 A. The vector 1 potential is ﬁxed as A = 2 H × r in the symmetric gauge. The coeﬃcients, C1 , C2 , C3 , are related to the pairing functions and the Fermi surface structure as 2 ∗ φ+ φ− > < φ∗2 < v+ − φ+ > = , C , 2 4 2 < |φ+ | > 2 < v+ v− |φ+ |2 > 2 ∗ φ+ φ− > < v− , C3 = 2 < v+ v− |φ+ |2 >

C1 =

(2)

where v± = (vx ±ivy )/2 with a Fermi velocity (vx , vy ), and < · · · > indicates the average on p along the Fermi surface. The parameters C1 and C3 represent the anisotropy of Fermi surface. They have ﬁnite values when the pairing functions on the Fermi surface have the fourfold symmetric structure. For isotropic Fermi surface, C1 and C3 are taken to be zero. In this paper, we treat C1 ,C2 ,C3 as parameters because detailed forms of φ± have not been established yet. In what follows, we consider that C1 = C3 = 0 and C2 = 1. The GL free energy for the s-wave symmetry is obtained by η− (r) = 0.0 and η+ (r) = η(r) in Eq. (1). In our simulations, we use the TDGL equation ∂ 1 ∂ f˜ η± = − ∗ , ∂t 12 ∂η±

(3)

as shown in Ref. 12. First, we solve the TDGL equation at a ﬁxed magnetic ﬁeld H at a certain temperature T slightly above Tc . Temperatures is then decreased until suﬃcient amplitudes of order parameters appears after solving the TDGL equation, which deﬁne the superconducting transition temperature Tc (H) at H (0 < H < Hc2 ). The calculations are performed in a two-dimensional square shaped superconductor with area being 25ξ0 ×25ξ0 where ξ0 is the coherence length at T = 0 in the GL theory. Sizes of the square hole are chosen as 5ξ0 ×5ξ0 , 10ξ0 ×10ξ0 , and

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15ξ0 ×15ξ0 . For circular shaped superconductor with diameter being 25ξ0 , hole diameters are 5ξ0 , 10ξ0 , and 15ξ0 . Outside of the open boundary and inside of the hole, we set η+ (r)=η− (r)=0 and B(r)=H with an applied ﬁeld H. 3. Result 3.1. S-wave (One Component) First, we study the Little Parks oscillation for the s-wave pairing in our formulation. Figures 1(a) and 1(b) show the transition lines Tc (H) in H-T -plane for the square 0 = φ0 /πξ02 . and circular shaped superconducting rings, respectively. We set Hc2 The critical temperature in the absence of a hole is denoted by Tc0 . It is an eﬀect of the ﬁnite-size superconductor that Tc (H) is suppressed even at H=0. When H increases along the Tc (H)-line, vortices penetrate into the hole from outside of the superconductor. Throughout this process, spatial structures of the order parameter changes continuously. While the vortices goes across superconductor, Tc (H) takes a local minimum and the phase winding number of order parameter around the hole changes. Figure 1 shows that behavior of the Little-Parks oscillations depends on size of hole when H increases from zero to Hc2 . In order to analyze a period of the oscillations, we show the Fourier component of Tc (H) which is given by F (θH ) = 1 −iH θH 0 T (H )e dH , H =H/Hc2 in Figs. 1(c) and 1(d). The characteristic peaks c 0 appear in F (θH ). Amplitudes of the peaks in circular case in Fig. 1(d) are larger than those in square case in Fig. 1(c). In circular shaped rings, the winding number changes by 4 at each period of oscillations in Tc (H), this number appears in the structure of ArgΔ(r). We note that high symmetry of ring is responsible for this behavior. The winding number can change by one when a hole position deviates from the center of rings. The oscillation behavior depends on the hole size and the winding number around the hole. The shape of rings is very important for the Tc (H) oscillations. 3.2. Chiral P -wave (Two Components) We show the transition line Tc (H) in H-T -plane for the chiral p-wave symmetry in Fig. 2. When H increases along the Tc (H)-line, periodic penetration of vortex into a hole makes oscillations of Tc (H) in one-component case. For the chiral pwave, however, the oscillations of the transition line does not appear. We conﬁrmed that the phase winding number around a hole increases with increasing H. In order to understand why the Little-Parks oscillations do not appear, we plot spatial 0 , η− structures of the two order parameters in Fig. 3 for several H. At H = 0.1Hc2 in (d) and η+ in (a) are dominant and subdominant component, respectively. At 0 , amplitude relation is switched with each other, η+ in (b) and η− in (e) H = 0.2Hc2 are dominant and subdominant component, respectively. With increasing H, two order parameters are continuously transformed and η+ or η− appears alternatively as a dominant component. It is noted that there appears vortex only in the minor

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component, while dominant component does not have vortex. Since amplitudes of order parameters correspond to the condensation energy, the dominant component mainly keeps the condensation energy and, at the same time, the subdominant component has vortices. In this way, order parameters change their phase winding number with saving of the condensation energy. Thus conventional Little-Parks oscillations disappear in two-component superconducting states such as the chiral p-wave symmetry. The vortices appearing in the subdominant component may have fractional ﬂux quantum. 12 4. Summary In summary, solving the two-component TDGL equation on the two dimensional superconductor with a hole, we study the Little-Parks oscillations both in the s-wave and the chiral p-wave pairing symmetries. In the s-wave pairing, the clear oscillation of Tc (H) appears in the H-T phase diagram. The characteristic behavior of the oscillations are sensitive to size and shape of the hole. The chiral p-wave symmetries are described by two components of order parameters (i.e.,η+ and η− ). With increasing magnetic ﬁelds, η+ and η− become the dominant component alternately and vortex of the minor component is able to change phase winding numbers without loss of condensation energy. Due to these eﬀects, the conventional Little-Parks oscillation does not appear in the chiral p-wave superconductor. References 1. 2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12.

W.A. Little and R.D. Parks, Phys. Rev. Lett. 9, 9 (1962). R.P. Groﬀ and R.D. Parks, Phys. Rev. 176, 567 (1968). L. Meyers and R. Meservey, Phys. Rev. B 4, 824 (1971). J. Cayssol, T. Kontos and G Montambaux, Phys. Rev. B 67, 184508 (2003). Y. Maeno, H. Hashimono, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz and F. Litenberg, Nature 372, 532 (1994). T.M. Rice and M. Sigrist, J. Phys.: Condens Matter 7, L643 (1995). K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z.Q. Mao, Y. Mori and Y. Maeno, Nature 396, 658 (1998). G.M. Luke, Y. Fudamoto, K.M. Kojima, M.I. Larkin, J. Merrin, B. Nachumi, Y.J. Uemura, Y. Maeno, Z.Q. Mao, Y. Mori, H. Nakamura and M. Sigrist, Nature 394, 558 (1998). P.G. Kealey, T.M. Riseman, E.M. Forgan, L.M. Galvin, A.P. Machenzie, S.L. Lee, D.McK. Paul, R. Cubitt, D.F. Agterberg, R. Heeb, Z.Q. Mao and Y. Maeno, Phys. Rev. Lett. 84, 6094 (2000). Y. Hasegawa, K. Machida and M. Ozaki, J. Phys. Soc. Jpn. 69 336 (2000). V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). M. Ichioka, Y. Matsunaga and K. Machida, Phys. Rev. B 71, 172510 (2005).

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1

1

l/ξ0=5 l/ξ0=10 l/ξ0=15

.9 .8

.8

¼¾

.7 .6

.7 .6

.5

.5

.4

.4

.3

.3

.2

.2

.1

.1

0

0 0

0.4 0.2

´ µ

r/ξ0=5 r/ξ0=10 r/ξ0=15

.9

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

0

.1

.2

.3

.4

¼

l/ξ0=5 l/ξ0=10 l/ξ0=15

0.4 0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

.5

.6

.7

.8

.9

1

¼

r/ξ0=5 r/ξ0=10 r/ξ0=15

-1 .1

.2

.3

.4

.5

.6

´¾ µ

.7

.8

.9

1

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

´¾ µ

Figure 1. The transition lines Tc (H) of the s-wave symmetry are shown for (a) square shaped ring and (b) circular shaped ring, when diameters of the hole are 5ξ0 , 10ξ0 , and 15ξ0 . Fourier components of Tc (H) in (a) and (b) are shown in (c) and (d), respectively.

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.8

.8

.6

.5

.5

.4

.4

.3

.3 .2

.1

.7

.6

.2

r/ξ0=5 r/ξ0=10 r/ξ0=15

.9

.7

¼¾

1

l/ξ0=5 l/ξ0=10 l/ξ0=15

.9

.1

0

0 0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

0

.1

.2

.3

.4

.5

¼

.6

.7

.8

.9

1

¼

Figure 2. The transition lines Tc (H) of the chiral p-wave symmetry are shown for (a) square shaped ring and (b) circular shaped ring.

·

¼¾

Figure 3. Spatial structures of order parameters in the circular shaped chiral p-wave rings are shown for several magnetic fields H/Hc2 = 0.1,0.2 and 0.3, where the diameter of the hole is fixed at 5ξ0 and a white ring in (d) corresponds to the superconductor. We present |η+ | and |η− | as density plots, where brighter regions indicate larger amplitudes. The subdominant components are located at the dark region within the superconductor.

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THEORETICAL STUDY OF SYNERGY EFFECT BETWEEN PROXIMITY EFFECT AND ANDREEV INTERFACE RESONANT STATES IN TRIPLET p-WAVE SUPERCONDUCTORS

Y. TANUMA1∗, Y. TANAKA2 AND S. KASHIWAYA3 1

Institute of Physics, Kanagawa University, Rokkakubashi, Yokohama, 221-8686, Japan 2 Department of Applied Physics, Nagoya University, Nagoya 464-8063, Japan 3 National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8568, Japan

The interplay between Andreev interface resonant states and proximity eﬀect in normal metal/ triplet p-wave superconductor junctions is investigated within quasiclassical Green’s function methods. We show numerical results for self-consistently determined pair potentials both in the normal metal and triplet p-wave superconductor sides. Moreover, the quasiparticle local density of states at the interface is calculated in detail by changing the transparency of junctions. The resulting local density of states at the interface depends on the transparency of the junctions as well as pairing symmetries of the pair potentials.

1. Introduction Nowadays spin-triple superconductivity has much attention as an important problem in solid states physics. The existence of triplet superconductivity has become promising after a series of experiments in Sr2 RuO4 1 . Tunneling spectroscopy using Andreev interface resonant states (AIRS) is essentially phase sensitive for unconventional superconductors 2,3,4 . The most dramatic eﬀect is the emergence of a zero-bias conductance peak (ZBCP) in tunneling spectra between normal metal and unconventional superconductors. The AIRS, which is originated from the interference eﬀect in the eﬀective pair potential through the reﬂection at the interface, plays an important role in order to determine the pairing symmetry of unconventional superconductors. One open question is how the resonant states and proximity eﬀect multiplies in normal metal and unconventional superconductor junctions. Actually, the ZBCP reﬂecting on the existence of the AIRS is observed in several tunneling experiments of Sr2 RuO4 5,6,7 . On the other hands, tunneling experiments in Sr2 RuO4 with Ru-metals by Mao et al. 6 suggest that the overall-line shape of tunneling spectra has a broad ZBCP with a sharp peak. This feature is diﬀerent from those in high-Tc cuprate junctions where only sharp ZBCP is reported 8,9 . Since these resonant states are expected for singlet d-wave pairing as well as triplet ∗ E-mail:

[email protected]

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pairings 10 , the diﬃculty may arise in determining the pairing symmetry only from conventional tunneling spectroscopy. This is why we focus on the proximity eﬀect between the normal metal (N) and triplet p-wave superconductor (P) junctions. From theoretical view points, one of the remarkable features of triplet p-wave junctions, which is diﬀerent from d-wave ones, is that injected quasiparticles perpendicular to the interface form the AIRS 11 . Therefore, the perpendicular injection contributes most dominantly to the tunneling conductance. Beside this, it is revealed that the proximity eﬀect is enhanced by the AIRS in diﬀusive N/P junctions 12,13,14 . However, it is not clariﬁed how these resonant states near the interface in the P side with broken-time reversal symmetry states (BTRSS) aﬀect on the induced s-wave component by the proximity eﬀect in the N side. The study along this direction may serve as a guide to understand novel interface phenomena expected in Ru/Sr2 RuO4 junctions 15 . For these reasons, it is one of challenging issues to study the proximity eﬀect in the N/P junctions in the presence of the induced pair potential in the N side. 2. Formulation We introduce the generalized Eilenberger equations 16 for spin-triplet superconductors σσ ∂ σσ ˇ α , x) gαβ i|vFx | gαβ (φ, x) = −α iωm τˇ3 + Δ(φ (φ, x) ∂x σσ ˇ β , x) , + βgαβ (φ, x) iωm τˇ3 + Δ(φ (1) ˇ α , x) = Δ(φ

iΔ(φα , x) · σ ˆσ ˆy ˆ0 iˆ σy Δ(φα , x)∗ · σ ˆ ˆ0

,

(2)

where vFx = vF cos θ and τˆi (i = 1, 2, 3) stand for the x component of the Fermi velocity and the Pauli matrices, respectively. Here ωm = πT (2m + 1) (m: integer) is the Matsubara frequency. The Pauli matrix τˇ3 in Eq.(1) is a 4 × 4 matrix. The σσ ˇ α , x) quasi-classical Green’s function gαβ (φ, x) and 4 × 4 pair potential matrix Δ(φ have diﬀerent spin dependencies. The triplet pair potential Δ(φα , x), is represented by d-vector as Δ = i(d· σ ˆ )ˆ σy . In the present study, we consider triplet p-wave states ˇ α , x) is reduced to be by choosing dx = dy = 0. The 4 × 4 pair potential matrix Δ(φ ˆ α , x). the 2 × 2 matrix Δ(φ Next, we consider the N/P junctions in order to determine the spatial variations of pair potentials self-consistently. Considering a semi-inﬁnite geometry, where the N is located at x < 0 and the P extended elsewhere, the pair potential in P [N] side will tend to the bulk value [zero] ΔP (φα , ∞) [ΔN (φα , −∞)] at suﬃciently large x. For simplicity, since we do not consider spin-ﬂip scattering at the interface, we ↑↑ ↓↓ ↑↓ ↓↑ = gαβ = gαβ and gαβ = gαβ = 0. can put the quasi-classical Green’s functions gαβ Therefore, we only have to deal with the Eilenberger equations constructed out of gαβ . However, we need to pay attention to sign of 21 oﬀ-diagonal element as the odd parity in Δ(φα , x) is maintained.

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163 l The resulting quasi-classical Green’s function gˆαα (φα , x) for l(= N,P) regions is given by 17 N i (x)F−N (x) 2iF−N (x) 1 + D− N gˆ−− (φ− , x) = , (3) N N N (x)F N (x) (x) −1 − D− (x)F−N (x) 2iD− 1 − D− − P i (x)F+P (x) 2iF+P (x) 1 + D+ P (φ+ , x) = gˆ++ , (4) P P P (x)F P (x) (x) −1 − D+ (x)F+P (x) 2iD+ 1 − D+ + l l (φ− , x) = −ˆ g++ (−φ+ , x)† . The quantities Dαl (x) and Dαl (x) obey the with gˆ−− following Riccati type equations:

∂ l Dα (x) =α 2ωm Dαl (x) + Δl (φα , x)Dαl (x)2 − Δl (φα , x)∗ , ∂x ∂ |vFx | Fαl (x) = − α 2ωm Fαl (x) − Δl (φα , x)∗ Fαl (x)2 + Δl (φα , x) . ∂x

|vFx |

(5) (6)

Moreover, the boundary condition of the Dαl (0) and Fαl (0) at the interface x = 0 are 18,19,20 F−N = F+P =

P P N − D+ + (1 − R)D+ RD− , P P DP − RD+ ] − (1 − R)D− +

(7)

N N P RD+ − D− + (1 − R)D− , P [D N − RD N ] − (1 − R)D N D N D− + − + −

(8)

N [D P D+ −

with reﬂection probability R. The pair potentials for both N and P sides are given by 21 π/2 l 1 l l Δ (φ, x) = dφ V l (φ, φα ) [ˆ gαα (φα , x)]12 − [ˆ gαα (φα , x)† ]12 , 4π −π/2 α 0≤m<ωc /2πT

(9) l l gαα (φα , x)]12 means the 12 element of gˆαα (φα , x). where ωc is the cutoﬀ energy and [ˆ Here V l (φ, φα ) is the eﬀective inter-electron potential of the Cooper pair in the l l (φα , x) are obtained using side. In our numerical calculations, new Δl (φα , x) and gˆαα Eqs.(3)-(6) and (9). We iterate this process until the convergence is suﬃciently obtained. Based on the self-consistently determined pair potentials, the local density of states (LDOS) can be calculated as 22 N N0 π/2 N NN (E, x) = dφIm Tr gˆ−− (φ− , x) − gˆ++ (φ+ , x) τˆ3 iωm →E+iδ , (10) 4π −π/2 P N0 π/2 P NP (E, x) = dφIm Tr gˆ++ (φ+ , x) − gˆ−− (φ− , x) τˆ3 iωm →E+iδ , (11) 4π −π/2

where N0 means the density of states in normal states, and δ is an inﬁnitesimal number.

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3. Results In this section, the spatial variations of the self-consistently determined pair potential in the N/P junctions and the LDOS at the interface are calculated by changing the reﬂection probability R. The spatial variation of the pair potentials in the N side is expressed as ΔN (φ, x) = ΔN (x), where ΔN (x) denotes the amplitude of s-wave component in the N side. On the other hands, in the P side, in order to consider three kinds of pair potentials with px -, py -, and px +ipy -wave symmetries, we choose the pair potentials ΔP (φ, x) = Δcos (x) cos φ + Δsin (x) sin φ, where Δcos (x) and Δsin (x) denote the amplitude of px -wave, and py -wave components, respectively. The attractive potential V P (φ, φ ) is given by ⎧ (px -wave), ⎨ 2VP cos φ cos φα P V (φ, φα ) = 2VP sin φ sin φα (py -wave), ⎩ 2VP (cos φ cos φα + sin φ sin φα ) (px +ipy -wave), 2πkB T VP = , 1 T ln + Tp m + 1/2

(12)

(13)

0≤m<ωc /2πT

where Δcos (x) and Δsin (x) mean the amplitude of px -wave and py -wave components, respectively. We choose the temperature T as T /Tp = 0.05, where Tp is the critical temperature of the bulk p-wave superconductors.

1.0

ΔN,P(x) / Δ p0

(a)

TN / Tp = 0.2

0.5

0.0 -8

Re[ Δ cos ]

-4 0

x / ξp

1.0

Re[ Δ N ] 0.5

R

4 8

0.0

(c)

(b) 10

2

R 1.0

NP (E,0) / N0

NN(E,0) / N0

4

5

R 1.0

0.5 0

-1

0 E / Δ p0

1

0.0

0.5 0

-1

0 E / Δ p0

1

0.0

Figure 1. (a) Spatial dependence of N/px -wave junction for various R and TN /Td = 0.2. The corresponding LDOS at the interface in the N and P sides are plotted in (b) and (c), respectively.

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In the N/px -wave junctions (Fig. 1), the reduction of Δcos (x) originates from a depairing eﬀect that the eﬀective pair potentials ΔP (φ+ , 0) and ΔP (φ− , 0) have reversed contribution to the pairing interaction. Thus, Δcos (x) is suppressed at the interface in the P side. On the other hands, for the N side, we can see that the pair potential ΔN (x) is induced except for R = 1. This indicates that the superconducting s-wave pair potential penetrates into the N side due to the proximity eﬀect. This situation is signiﬁcantly diﬀerent from the corresponding case of d-wave superconductors. Turning to our results of corresponding LDOS [see Figs. 1(b) and 1(c)], these have zero-energy peaks (ZEP). The origin of the ZEP for high barrier cases is the formation of the AIRS in the P side of the interface. And then, we focus on the N side. We can see that the LDOS has the ZEP due to the induced pair potential ΔN (x). This ZEP originates from the fact that the injected and Andreev reﬂected quasiparticles in the N side can feel diﬀerent sign of the pair potential. Because quasiparticles with up spin in the N side feel positive sign of the pair potential, and those with down spin inevitably feel negative sign of the pair potential. On the other hand, in the P side, the pair potential felt by quasiparticles does not have sign change by the spin inversion. Consequently, at least quasiparticles with up and down spins feel diﬀerent sign of the pair potentials between the N and P sides. This is the origin of the ZEP due to the proximity eﬀect as in the case of d-wave (100) junction case. However, in this case, the ZEP in the N side does not disappear even for the nonzero R. This is maintained by the existence of the strong resonance induced by the AIRS in the P side. Because every quasiparticle feels these resonant states independent of the injection angles at the interface. Therefore, these special situations can induce the ZEP in the N side even for nonzero R. And then, the height of the ZEP decreases with the increase of the magnitude of R. At the same time, the line shape of the LDOS at the P side of the interface can be expressed by a summation of the broad ZEP by the interface resonant states and sharp ZEP by the proximity eﬀect. These unusual features are speciﬁc to px -wave junctions, where both proximity eﬀect and AIRS can coexist each other 12,13,14 . For our results of N/py -wave junctions, the s-wave component in the N side is not induced near the interface even by changing R. The corresponding LDOS is equivalent to that of isolated py -wave superconductors. Finally, let us look at the case of N/px +ipy -wave junctions. The spatial dependencies of pair potentials are shown in Fig. 2(a). In low transparent case, the magnitude of the pair potential Re[Δcos (x)] is suppressed and the Im[Δsin (x)] are slightly enhanced as they approach the interface. These results are consistent with those by the previous works on B phase of 3 He. 2,23,24 For high transparent case, the pair potential Re[ΔN (x)] is induced near the interface due to the proximity eﬀect. This situation is similar to the case of N/px -wave junctions. Here, Im[Δcos (x)], Re[Δcos (x)], and Im[ΔN (x)] are absent. The corresponding LDOS at the interface for the N and P sides are plotted in Figs. 2(b) and 2(c), respectively. For R = 1,

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1.5

ΔN,P(x) / Δ p0

(a) 1.0

TN / Tp = 0.2

0.5 0.0 -8

Im[ Δ sin ]

0 x / ξp

(b)

0.5

Re[ Δ N ]

R

4 8

4

0.0

(c)

6

R 2

1.0 0.5

0

-1

0 E / Δ p0

1

0.0

NP (E,0) / N0

NN(E,0) / N0

1.0

Re[ Δ cos ]

-4

4

R 1.0

2 0.5 0

-1

0 E / Δ p0

1

0.0

Figure 2. Spatial dependence of N/px +ipy -wave junction for various R and TN /Td = 0.2. The corresponding LDOS at the interface in the N and P sides are plotted in (b) and (c), respectively.

the ZEP originates from the formation of AIRS with sign change of the pair potentials between ΔP (φ+ , 0) and ΔP (φ− , 0) in the P side. With the decrease of the magnitude of R, the height of the ZEP is reduced and the resulting LDOS has a gap like structure. On the other hand, in the N side, the line shape of the LDOS has a minigap around zero energy due to the presence of ReΔN (x). 4. Conclusions In this paper, we have shown the tunneling spectra in the N/P junctions in the presence of the proximity eﬀect. The spatial dependence of pair potentials in the N/P junctions is self-consistently determined based on the quasiclassical Green’s function methods. We assume that the attractive interelectron potentials which induce subdominant s-wave components in the N side. The LDOS at the interface of the N/P junctions is studied by varying the transparency of the junction. We conclude that the LDOS at the interface has the ZEP or line shape depending on the transparency of the interface and pairing symmetries of pair potentials. References 1. 2. 3. 4.

A.P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). L.J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 (1981). Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995). S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000).

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5. F. Laube, G. Goll, H.v. L¨ ohneysen, M. Fogelstr¨ om, and F. Lichtenberg, Phys. Rev. Lett. 84, 1595 (2000). 6. Z.Q. Mao, K.D. Nelson, R. Jin, Y. Liu, and Y. Maeno, Phys. Rev. Lett. 87, 037003 (2001). 7. M. Kawamura, H. Yaguchi, N. Kikugawa, Y. Maeno, H. Takayanagi, J. Phys. Soc. Jpn. 74, 531 (2005). 8. Y. Tanuma, Y. Tanaka, M. Yamashiro, and S. Kashiwaya, Phys. Rev. B57, 7997 (1998). 9. Y. Tanuma, Y. Tanaka, M. Ogata, and S. Kashiwaya, Phys. Rev. B60, 9817 (1999). 10. Y. Tanuma, K. Kuroki, Y. Tanaka, R. Arita, S. Kashiwaya, and H. Aoki, Phys. Rev. B66, 094507 (2002). 11. Y. Tanaka, Y. Tanuma K. Kuroki, and S. Kashiwaya, J. Phys. Soc. Jpn. 71, 2102 (2002). 12. Y. Tanaka and S. Kashiwaya, Phys. Rev. B70 012507 (2004) 13. Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B71, 094513 (2005), 14. Y. Tanaka, Y. Asano, A.A. Golubov and S. Kashiwaya, Phys. Rev. B72 140503, (2005). 15. M. Sigrist and H. Monien, J. Phys. Soc. Jpn. 70, 2409 (2001). 16. G. Eilenberger, Z. Phys. 214, 195 (1968). 17. M. Ashida, S. Aoyama, J. Hara, and K. Nagai, Phys. Rev. B40, 8673 (1989). ´ 18. A.V. Zaitsev, Zh. Eksp. Teor. Fiz. 86, 1742 (1984) [Sov. Phys. JETP 59, 1015 (1984)]. 19. A. Shelankov and M. Ozana, Phys. Rev. B61, 7077 (2000). 20. M. Ozana and A. Shelankov, J. Low Temp. Phys. 124, 223 (2001). 21. C. Bruder, Phys. Rev. B41, 4017 (1990). 22. J.A.X. Alexander, T.P. Orlando, D. Rainer, P.M. Tedrow, Phys. Rev. B31, 5811 (1985). 23. J. Kurkij¨ arvi and D. Rainer, Helium Three, edited by W. P. Halperin and L. P. Pitaevskii (Elsevier, Amsterdam, 1990). 24. E.V. Thuneberg, Phys. Rev. B33, 5124 (1986).

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THEORY OF PROXIMITY EFFECT IN UNCONVENTIONAL SUPERCONDUCTOR JUNCTIONS

Y.TANAKA AND T. YOKOYAMA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan and CREST Japan Science and Technology Cooporation (JST), Nagoya 464-8603, Japan E-mail: [email protected] S. KASHIWAYA National Institute of Advanced Industrial Science and Technology (AIST), Japan E-mail: [email protected] A. A. GOLUBOV Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands E-mail: [email protected] Y. ASANO Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan E-mail: [email protected]

We clarify the remarkable features of the proximity eﬀect in diﬀusive normal metal (DN) / unconventional superconductor junctions. For spin singlet d-wave superconductor junctions, mid gap Andreev state competes with proximity eﬀect. In this case, local density of states (LDOS) of quasiparticle in DN has a gap like or almost constant structure. The pair potential f (ε) satisﬁes the conventional relation f (ε) = f ∗ (−ε), where ε is a quasiparticle energy measured from the Fermi energy. On the other hand, for spin triplet p-wave junctions, the resulting LDOS has a zero energy peak. At the same time, pair amplitude f (ε) satisﬁes f (ε) = −f ∗ (−ε).

1. Introduction Almost twenty years have passed since the discovery of high-TC cuparte. Stimulated by the long debate of the mechanism of cuprate, nowadays, so called physics of unconventional superconductor has a great activity. In unconventional superconductors, to avoid the strong on site Coulomb repulsion, the resulting angular momentum of the Cooper pair should be larger than zero 1 . To clarify the phase sensitive phenomena speciﬁc to unconventional superconductors is very important 2 . In the light of this aspect, physics of superconducting junctions with unconventional superconductor becomes a signiﬁcant topic both from the view point of the

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fundamental physics and future device applications. As regards the ballistic metal / unconventional superconductor junctions, up to 1995, there was no available theory about tunneling spectroscopy. Two of the present authors, YT and SK, developed a general theory of tunneling spectroscopy of unconventional superconductors in 1995. This theory clariﬁed that the origin of the zero bias conductance peak (ZBCP) in actual tunneling spectroscopy in cuprate is the mid gap Andreev resonant state (MARS) formed at the interface 3 . As shown in Fig. 1, MARS is formed when the sign of the pair potential felt by an electron like quasiparticle and a hole like quasiparticle is diﬀerent from each other. Up to

a-axis

Mid gap Andreev resonant state (MARS)

Normal metal

β

Δ+ Δ - < 0

electron

electron like quasiparticel Δ + (=Δ 0 cos[ 2(θ−β) ])

hole θ

electron

hole like quasiparticel

Δ - (=Δ 0 cos[ 2(θ+β) ])

Unconventional superconductor

Figure 1. Schematic illustration of an elementary process at the interface of normal metal / unconventional superconductor junctions. We have chosen a d-wave superconductor for example. For Δ+ Δ− < 0, MARS is formed at the interface.

now, there are many tunneling experiments which report ZBCP in high-TC cuprate 4 , Sr2 RuO4 9 , UBe13 10 , CeCoIn5 11 , and two dimensional organic superconductor κ-(BEDT-TTF)2 Cu[N(CN)2 ]Br 12 . However, until 2003, almost all of the theories are limited to the ballistic transport regime and there is no theory of proximity eﬀect in diﬀusive normal metal/ unconventional superconductor (DN/US) junctions. On the other hand, it is well known in conventional superconductor junctions, proximity eﬀect inﬂuences crucially the charge transport, when the size of the normal metal attached to the superconductor is much smaller than phase coherence length but larger than the mean free path 5 . In these systems, various quantum interference phenomena occur due to the proximity eﬀect. It is an interesting problem to make a theory of proximity eﬀect available in unconventional superconductor junctions. To reply this situation, YT and SK made a theory of diﬀusive normal metal /unconventional superconductor junctions by extending the circuit theory6 . It was revealed that the proximity eﬀect is suppressed in the presence of the MARS in d-wave superconductor junctions 7 . On the other hand, proximity eﬀect is enhanced in triplet superconductor junctions, like p-wave ones 8 . In the present article, we will discuss in detail the remarkable features about proximity eﬀect in unconventional superconductor junctions.

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2. Formulation and results We consider a junction consisting of a normal metal and a superconducting reservoirs connected by a quasi-one-dimensional diﬀusive conductor (DN) with a length L much larger than the mean free path. The interface between the DN and the superconductor has a resistance Rb while the DN/N interface has a resistance Rb . We assume ﬂat interfaces and the insulating barriers at the two interfaces are modeled by the delta-function 7,8 . As regards the spin structure of the Cooper pair, we choose Sz = 0. The positions of the DN/N interface and the DN/superconductor (S) interface are deﬁned as x = 0 and x = L, respectively. To calculate the LDOS in DN, we focus on the retarded part of the Nambu-Keldysh Green’s functions. In the present theory quasiclassical Green’s function in DN can be expressed by the so called proximity parameter θ which satisﬁes the Usadel equation13 : ∂2θ + 2iε sin θ = 0 ∂x2 where D is the diﬀusion constant in DN. The boundary condition of θ is D

L ∂θ < F1 > 2T (fS cos θL − gS sin θL ) ( ) |x=L = , F1 = Rd ∂x Rb 2 − T + T (cos θL gS + sin θL fS )

(1)

(2)

with θL = θ(x = L). In the above, gS is given by gS = (g+ + g− )/(1 + g+ g− + f+ f− ), fS = (f+ + f− )/(1 + g+ g− + f+ f− ) for spin singlet pairing and gS = (g+ + g− )/(1 + g+ g− + f+ f− ), fS = i(f+ g− − f− g+ )/(1 + g+ g− + f+ f− ) for spin triplet pairing. For the N/DN interface, L ∂θ F2 2T sin θ0 |x=0+ = , F2 = , Rd ∂x Rb 2 − T + T cos θ0

(3)

with θ0 = θ(x = 0) and T = Tφ . Here . . . denotes the average over angle φ deﬁned as follows π/2 4 cos2 φ 4 cos2 φ −π/2 dφ cos φF1(2) (φ) , Tφ, , = 2 < F1(2) (φ) >= π/2 , Tφ = 2 2 ( ) Z + 4 cos φ Z + 4 cos2 φ dφT cos φ φ −π/2 (4) where Z and Z denote the barrier parameters at the interfaces. In the present = 0.25Δ0 . calculation, we ﬁx Z = 1,Z = 1, Rd /Rb = 1, R d /Rb = 0.01 and ET h

In the above, g± and f± are given by g± = ε/ 2 − Δ2± f± = Δ± / Δ2± − 2 , respectively. Here, Δ(φ+ ) and Δ(φ− ) are the pair potentials felt by quasiparticles with an injection angle φ and π − φ in US. It is possible to ﬁx the quasiclassical Green’s function in US as g± τˆ3 +f± τˆ2 . The resulting quasiclassical Green’s function τ3 + f τˆ1 for spin for DN can be given by gˆ τ3 + f τˆ2 for spin singlet junctions and gˆ triplet ones with g = cos θ and f = sin θ. Here, we choose a d-wave superconductor with Δ± = Δ0 cos(2φ ∓ 2β) as an example of a singlet superconductor, where β

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denotes the angle between the normal to the interface and the lobe direction of the d-wave pair potential. As an example of a triplet superconductor, we choose a pwave superconductor with Δ± = ±Δ0 cos(φ∓α) where α denotes the angle between the normal to the interface and the lobe direction of the p-wave pair potential. In the above, Δ0 is the magnitude of the maximum value of the pair potentials. In the following, we concentrate on the quasiparticle local density of state (LDOS) ρ(ε) as a function of quasiparticle energy ε. First, we discuss the local density of states and pair amplitude in DN for conventional s-wave superconductor junctions. As shown in Fig. 2, the resulting LDOS has a gap like structure. On the other hand, pair amplitude f (ε) satisﬁes f (ε) = f ∗ (−ε). Since f (ε) is a pure real number at ε = 0, the resulting amplitude of real part of g(ε) is smaller than unity. Next, we focus on the d-wave pair potential case. We s-wave Electrode

DN

0

L spin singlet s−wave

spin singelt s −wave ρ(ε)

1 f(ε)

1

0

−1 0 −0.3

0

ε/Δ 0

0.3 −0.3

0

ε/Δ 0

0.3

Figure 2. Left panel: LDOS of DN in s-wave superconductor junctions. Right panel: corresponding pair amplitude f (ε). Solid line, real part. Dotted line, imaginary part.

choose β = 0, i.e., dx2 −y2 -wave and β = π/4, i.e., dxy -wave case. For the latter case, quasiparticles feel MARS at the interface of d-wave superconductor independent of the direction of their motions. As shown in Fig.3, for dx2 −y2 -wave case, LDOS has a gap like structure similar to the s-wave superconductor case. However, for dxy -wave case, LDOS becomes always unity. This means the absence of the proximity eﬀect for dxy -wave case. In the present case, MARS and proximity eﬀect compete with each other. Due to this competition, anomalous temperature dependence of the Josephson current emerges in d-wave junctions14 . As regards the ε dependence of the pair amplitude, f (ε) = f ∗ (−ε) is satisﬁed as in the case of s-wave superconductor. To understand the relation between the MARS and the proximity eﬀect in detail, we then focus on the triplet superconductor junctions. We choose p-wave pairing

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ρ(ε) Electrode

DN

dxy−wave

L

0

Electrode

dx2−y2−wave

1

DN

0 −0.3

Figure 3.

0

ε/Δ 0

0.3

LDOS for d-wave superconductor. dx2 −y 2 and dxy pairing.

spin singelt d−wave 1 f (ε) 0

−1 −0.3

0

ε/Δ 0

0.3

Figure 4. The corresponding pair amplitude f (ε) for dx2 −y 2 -wave pairing in Fig. 3. Solid line, real part. Dotted line, imaginary part.

for α = 0, i.e. px -wave pairing, and α = π/2, i.e. py -wave pairing. For the former case, quasiparticles feel MARS at the interface independent of their directions of motions. For px -pairing, LDOS has a zero energy peak (ZEP). On the other hand, for py -wave case, due to the absence of the proximity eﬀect, LDOS becomes unity as shown in Fig. 5. If we choose a general value of α within 0 < α < π/2, the ZEP always emerges 8 . Thus the existence of ZEP of LDOS in DN is a signiﬁcantly robust features. It is possible to propose an experimental setup which discriminates the spin triplet superconducting state from spin singlet one by the presence or absence of ZEP in DN by using scanning tunneling spectroscopy. It is also interesting to look at the ε dependence of pair amplitude f (ε). The resulting f (ε) satisﬁes f (ε) = −f ∗ (−ε). Such an unusual energy dependence can not be realized with the even frequency pairing 15 . In such a case, the resulting pair amplitude at ε = 0 becomes a pure imaginary number. This property is related to the emergence of the ZEP in the LDOS.

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spin triplet p−wave 2 Electrode

-

DN

ρ(ε) 0

DN

-

-L

Figure 5.

py

1 +

-L

Electrode

px

+

0

0 −0.3

0

ε/Δ 0

0.3

LDOS for p-wave superconductor. px pairing and py pairing.

spin triplet p−wave

10 f (ε) 0

−10 −0.1

0

ε/Δ 0

0.1

Figure 6. The corresponding pair amplitude f (ε) of px -wave pairing in Fig. 5. Solid line, real part. Dotted line, imaginary part.

3. Conclusion In the present paper, we have summarized an activity of theoretical works about unconventional superconductor junctions. For spin singlet d-wave superconductor junctions, mid gap Andreev state competes with the proximity eﬀect. In this case, local density of state of quasiparticle in DN has a gap like structure as a function of ε. The pair potential f (ε) satisﬁes conventional relation f (ε) = f ∗ (−ε). On the other hand, for spin triplet p-wave junctions, the resulting LDOS has a zero energy peak. At the same time, pair amplitude f (ε) satisﬁes f (ε) = −f ∗ (−ε). References 1. M. Sigrist and T. M. Rice, Rev. Mod. Phys. 67 (1995) 503. 2. D. J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515, C.C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, (2000) 969.

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3. 4. 5. 6. 7.

8.

9. 10. 11. 12. 13. 14. 15.

Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, (1995) 3451. S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, (2000) 1641. Yu. V. Nazarov, Phys. Rev. Lett. 73 (1994) 1420; Yu. V. Nazarov, Superlattices and Microstructuctures 25, (1999) 1221. Y. Tanaka, Yu. V. Nazarov, S. Kashiwaya, Phys. Rev. Lett. 90 (2003) 167003; Y. Tanaka, Yu. V. Nazarov, A. A. Golubov, S. Kashiwaya, Phys. Rev. B 69 (2004) 144519. Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70 (2004)012507; Y. Tanaka, S. Kashiwaya and T. Yokoyama, Phys. Rev. B 71 (2005) 094513, Y. Tanaka, Y. Asano, A. Golubov, S. Kashiwaya, Phys. Rev. B 72 (2005) 14503. M. Kawamura, H. Yaguchi, N. Kikugawa, Y. Maeno, H. Takayanagi, J. Phys. Soc. Jpn. 74, (2005) 531. Ch. W¨ alti, H.R. Ott, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 84, (2000) 5616. P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y. T. Wei, Phys. Rev. Lett. 94, (2005) 107005. K. Ichimura, S. Higashi, K. Nomura and A. Kawamoto, Synthetic Metals Vol. 153 (2005) 409. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970). T. Yokoyama, Y. Tanaka, A. A. Golubov and Y. Asano, Phys. Rev. B 73, 140504(R) (2006). Y. Tanaka and Golubov, Phys. Rev. Lett. Y. Tanaka and A. Golubov, Phys. Rev. Lett. 98, 037003 (2007).

Quantum Information

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ANALYZING THE EFFECTIVENESS OF THE QUANTUM REPEATER

KENICHIRO FURUTA AND HIROFUMI MURATANI Toshiba Corporation 1, Komukai-Toshiba-cho, Saiwai-ku, Kawasaki, 212-8582, JAPAN E-mail: [email protected] The range of quantum key distribution is limited by exponential attenuation of photons in optical ﬁbers. It is believed that a quantum repeater can improve the order of the attenuation and can be useful for extending the communication distance of QKD. It is already shown that quantum repeater is eﬀective when not considering the repeater noise. In this paper, we analyze the eﬀectiveness of the quantum repeater when considering the repeater noise. We point out that there is some threshold of length of EPR pairs, under which quantum repeater protocol is valid and over which quantum repeater is not valid. Besides, this threshold depends on the largeness of the repeater noise. So, it is important to suppress the repeater noise in order that the quantum repeater scheme can really improve the bit rate. We also analyze the eﬀectiveness from the viewpoint of the security and show that QKD is secure even if quantum repeater is used.

1. Introduction There is an everlasting threat that a current practical cryptographic scheme whose security is based on computational assumptions will become insecure due to a future improvement of computers. Therefore, quantum key distribution (QKD), e.g. BB84[2] and B92, has been attracting considerable attention because its security is based only on quantum principles and it is unconditionally secure. Due to exponential attenuation of photons in the channel, the naive QKD is valid only in the range of short distance. It is important to extend the communication distance from a practical viewpoint. So far, three approaches have been proposed to extend the range of QKD: 1.Protocol modiﬁcation for multiple photon emission: Some protocol modiﬁcations[7] which make the scheme robust against the photon number splitting(PNS) attack were proposed. However, modiﬁed protocols can extend the range of QKD to only a few times that of the original. Due to this limit, further extensions require introduction of other improvements. 2.Coherent states: Some protocol using coherent states[1] were proposed. It was demonstrated that they are more resistant to noises than the single photon protocols and can achieve high bit rate even in long distance. However, its security has been discussed enthusiastically[6, 8]. In this paper, we do not consider this. 3.Quantum repeater protocol: In order to reduce noises on quantum state transferred through the optical ﬁber, quantum teleportation is used to send the quantum state by using an EPR pair generated by a quantum repeater protocol. We call such a scheme QKD with quantum repeater. The quantum repeater recursively applies

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entanglement swapping(ES) and entanglement purification protocol(EPP) to shortlength-EPR pairs[4]. It was demonstrated in [4] that a quantum repeater protocol can generate the long-length-EPR pair for the quantum teleportation with high ﬁdelity. At a glance, the quantum repeater seems to be the most promising approach of the three approaches. However, the discussion in [4] seems to assume that noises in quantum memory on checkpoints can be negligible. Therefore, we reexamine the practical possibility of the quantum repeater by evaluating the order of the bit rate with considering the repeater noise. Although the repeater noise is noticed in [5], the evaluation is done without the repeater noise. We also examine the security of QKD with quantum repeater. In Section 2, we review the quantum repeater protocol in [4]. In section 3, we categorize noises that occur in the protocol. In Section 4, we evaluate the bit rate without considering the repeater noise. In Section 5, we evaluate the bit rate with considering the repeater noise. In Section 6, we prove the security of QKD with quantum repeater. In Section 7, we provide a summary of this paper. 2. Quantum repeater protocol The quantum repeater is a scheme which extends the length of an EPR pair with high ﬁdelity. In this section, we review the quantum repeater protocol in [4]. 2.1. Abstract specification We explain an abstract speciﬁcation of the quantum repeater protocol. The protocol recursively applies ES and EPP to short-length-EPR pairs and ﬁnally generates a long-length-EPR pair with high ﬁdelity. Let L be the number of EPR pairs which are linked in a single ES execution, N be the number of checkpoints and n be the depth of the recursive executions. These satisfy a relation, N = Ln . In the channel between the sender A and the receiver B, N − 1 checkpoints, denoted C1 , C2 , · · · , CN −1 , are settled. For convenience, A and B are denoted C0 and CN , respectively. The distance between A and B is denoted as D and the distance between two adjacent checkpoints is denoted as d. That is, D = N d. After completing the above protocol, an EPR pair with high ﬁdelity shared between C0 and CN is obtained. 2.2. Entanglement swapping In ES, partners of two EPR pairs are swapped. Here, we provide an explicit realizations of ES based on local measurement. ES can also be realized based on Bell measurement. First, Controlled NOT gate (CNOT) is applied to photons 2 and 3 of |φ+ 1,2 ⊗ + |φ 3,4 . Next, WH(Walsh-Hadamard) transformation is applied and produces 1 + + − − 2 (|02 |03 ⊗ |φ 1,4 + |02 |13 ⊗ |ψ 1,4 + |12 |03 ⊗ |φ 1,4 + |12 |13 ⊗ |ψ 1,4 ). Then, one of four computational bases of photons 2 and 3 is measured and this measurement maps the state of two photons 1 and 4 into a Bell state. Here, the observed basis of photons 2 and 3 indexes the projected Bell state of photons 1 and 4. In the next step, the projected Bell state of photons 1 and 4 is transformed into |φ+ . For this purpose, the measurement results of photons 2 and 3 are sent from

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Algorithm 1 Quantum repeater 1: Initialization: At each checkpoint Ci , i = 0, · · · , N −1, EPR pairs are generated and one photon of each pair is sent to the next checkpoint Ci+1 . 2: for x = 1 to n do 3: ES: Execute ES in each of the checkpoints CkLx−1 ,(k = 1, 2, · · · , N/Lx−1 ) except CLx , C2Lx , · · · , CN −Lx . Then, the EPR pairs of length Lx can be obtained. 4: EPP: Execute EPP for EPR pairs in each of the checkpoints CLx , C2Lx , · · · , CN −Lx . Then, the EPR pairs of length Lx with high ﬁdelity can be obtained. 5: end for a checkpoint having photons 2 and 3 to ckechpoints having photons 1 and 4 with the classical communication. 2.3. Entanglement purification EPP pulls out an EPR pair of high ﬁdelity from multiple EPR pairs of low ﬁdelity. We consider an EPP which is also considered in [4]. The validity of EPP requires that the ﬁdelity of EPR pairs before the puriﬁcation should be in a certain range. It is demonstrated as follows. Let F and F be the ﬁdelity of the EPR pairs before EPP and the ﬁdelity of puriﬁed EPR pair, respectively. In the case that EPP generates the puriﬁed pair from two EPR pairs, F can be expressed in terms of F as follows[3]: F = Φ/Λ, where F¯ = (1 − F )/3, Φ = F 2 + F¯ 2 and Λ = F 2 + 2F F¯ + 5F¯ 2 . (1)

In order that F ≥ F in Eq.(1), F should be in the range of 1/2 ≤ F ≤ 1. 3. Noises Here, we categorize possible noises which occur during an execution of the protocol. 3.1. Noises during the protocol execution Several types of noises can occur during the execution of the quantum repeater protocol. We classify them by their causes. The measurement noise is noises which occur during a measurement of a quantum state. The one-qubit operation noise is noises which occur during a one-qubit operation in the protocol. The two-qubit operation noise is noises which occur during a two-qubit operation in the protocol. The channel noise is noises of a quantum state transferring through the channel. The repeater noise is noises of a quantum state in the repeater devices even in the absence of operation. Here, the one-qubit operation and the two-qubit operation mean a unitary operation on one qubit and a unitary operation on two qubits, respectively. We consider that quantum noises which occur during the classical communication in the execution of EPP or quantum teleportation is an example of the repeater noise. The ﬁrst three noises were modeled and analyzed in [4]. For the channel noise and the repeater noise, we evaluate with the order, such as exponential or polynomial. We review the models and analyses of the ﬁrst three noises in the next subsection.

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3.2. Conventional models and analysis We show models of such noises and the modiﬁcation of Eq.(1) caused by these noises. Let ρ be a density matrix before operations. First, the one-qubit operation noise is ideal 1 are one-qubit modeled as ρ → O1 ρ = p1 O1ideal ρ+ 1−p 2 tr1 ρ⊗I1 , where O1 and O1 operations with and without the noise, respectively, and I1 is the identity operator and p1 is the probability with which the operations are performed without noise. ideal 2 ρ + 1−p The two-qubit operation noise is modeled as ρ → O12 ρ = p2 O12 4 tr12 ρ ⊗ ideal I12 , where O12 and O12 are two-qubit operations with and without the noise, respectively, and I12 is the identity operator and p2 is the probability with which the operations are performed without noise. The measurement noise is modeled as P0η = η|00| + (1 − η)|11|, P1η = η|11| + (1 − η)|00|, where η is the probability with which the measurements are performed correctly and P0η and P1η are POVM |00| and |11|, respectively, with error probability η. Based on the above noise models, the ﬁdelity, FL , after linking L EPR pairs 2 L−1 p p (4η 2 −1) 4F −1 L . Similarly, by ES executions is expressed as FL = 14 + 34 1 2 3 3 based on the above noise models, the change of the ﬁdelity by EPP is expressed as follows: (2) F = {ΘΦ + 2η η¯Ξ + Π}/{ΘΛ + 4(2η η¯Ξ + Π)}, 2

1−p where η¯ = 1 − η, Θ = η 2 + η¯2 , Ξ = F F¯ + F¯ 2 and Π = 8p22 . F and F have three 2 intersections. Let two intersections except 0.25 be Fmin andFmax , where Fmin < Fmax . Then, in order that F ≥ F in Eq.(2), F should be in the range of Fmin ≤ F ≤ Fmax . The range of F where the quantum repeater is valid become narrow as noises become large. 4. Bit rate in absence of repeater noise In [4], it was demonstrated that the required amount of resources of the quantum repeater increases as a polynomial function of the distance between A and B, D. This leads to the conclusion that the bit rate of the quantum repeater decreases as an inverse of a polynomial function of the distance. This result was derived under the condition that only the channel noise, the measurement noise, the one-qubit operation noise and the two-qubit operation noise are considered. Theorem 4.1. Consider QKD with quantum repeater. If only the channel noise, the measurement noise, the one-qubit operation noise and the two-qubit operation noise are considered, there exists a polynomial function p(·) such that the bit rate of the QKD decreases as Ω(p(D)−1 ). Here, g(n) = Ω(f (n)) means ∃c > 0 ∃N ∈ N ∀n > N g(n) ≥ cf (n).

Proof. The bit rate is estimated by considering both the merit of quantum repeater, keeping high ﬁdelity, and the demerit, increase of resource. Let M be the number of EPR pairs consumed by a single execution of EPP. Then, the number of EPR pairs, R, in the whole execution of the quantum repeater, is R = (LM )n = Ln M n = N M n = N (LlogL M )n = N logL M+1 , where N is proportional to the communication distance D, and L and M do not depend on D. Thus, R is a polynomial function of D.

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In this scheme, when not considering the repeater noise, the ﬁdelity of the EPR pair generated by the quantum repeater protocol stays constant even if D increases. So, the bit rate of the QKD decreases as Ω(p(D)−1 ). In contrast to exponential damping in the absence of the quantum repeater, the bit rate of QKD in the presence of the quantum repeater decreases as an inverse of a polynomial function of the communication distance. Although Theorem 4.1 does not indicate whether the exact value of the bit rate is really improved by the quantum repeater, it can be expected to be eﬀective for suﬃciently large D. 5. Bit rate in presence of repeater noise We next consider the case the repeater noise is taken into account. In ES and EPP in the quantum repeater protocol, classical communications between repeater devices are needed. In addition, quantum teleportation sends classical information from A to B. During these classical communications, the quantum states in the repeater devices lose their ﬁdelity. We assume the repeater noise as follows: The ﬁdelity of a quantum state in a repeater device decreases exponentially with respect to the time length of a classical communication. We call the assumption of this model the exponential damping assumption. Theorem 5.1. Under the exponential damping assumption, the bit rate of QKD with quantum repeater decreases exponentially with respect to the distance D. Proof. The length of EPR pairs increases as the quantum repeater protocol proceeds. As far as the length is small, the ﬁdelity can be recovered by EPP. However, if the length exceeds a threshold, Dth , then the ﬁdelity becomes lower than Fmin and EPP can’t recover the ﬁdelity any more. The reason is that EPR pairs have to stay in quantum memory on ckeckpoints during classical communication and the time of classical communication becomes large as the length of EPR pairs become large. Thus, time of being aﬀected by the repeater noise get larger. After the ﬁdelity goes under the threshold of EPP, the ﬁdelity continues to decrease as the quantum repeater protocol proceeds. So, after crossing the threshold, ΔFES +ΔFEP P +ΔFRN ≥ ΔFRN , where ΔFES , ΔFEP P and ΔFRN are the ﬁdelity decreases due to ES, EPP and the repeater noise, respectively, during an iteration in the recursive execution of the quantum repeater protocol. The dumping due to the repeater noise is exponential according to the exponential dumping assumption. So, the overall dumping is exponential according to the equation above. Of course, there may be many other quantum repeater protocols. However, in general, our result holds for protocols provided the classical communication whose distance is proportional to that between a sender and a receiver is used in the protocols. 6. Security The security proof can be done with simple idea. Let IE be the amount of eavesdropper’s information and Pcont be the person who controls the quantum repeater unit and Eve be an eavesdropper. The following relationship holds for IE . (IE in

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original QKD)≥ (IE in QKD with quantum repeater, where Pcont is Eve)≥ (IE in QKD with quantum repeater, where Pcont is except Eve). The reason is as follows. For QKD with quantum repeater, Eve can get more information (or equal at least) when he controls repeater unit more than when he does not. So, (IE in QKD with quantum repeater, where Pcont is Eve)≥ (IE in QKD with quantum repeater, where Pcont is except Eve). Thus, it is suﬃcient to prove the security when repeater unit is controlled by Eve. Here we deal with QKD protocols where operations for quantum repeater protocol can be done within attacks allowed for Eve in original QKD protocol. Unconditionally secure protocols, such as BB84[B84], belong to this category because Eve is allowed to do almost every quantum operations as attacks. So, when repeater unit is controlled by Eve, operations for quantum repeater can be considered as a part of Eve’s attacks allowed in original QKD. Then, (IE in original QKD)≥ (IE in QKD with quantum repeater, where Pcont is Eve). Thus, we can turn the proof of QKD with quantum repeater into the proof of original QKD. 7. Summary We demonstrated that the bit rate of QKD with quantum repeater decreases asymptotically exponentially with respect to the communication distance when the repeater noise is taken into account. This is because EPP can not work when the length of EPR pairs exceed the threshold. In contrast, quantum repeater protocol works when the length of EPR pairs does not exceed the threshold. This threshold depends on the largeness of the repeater noise. So, it is important to suppress the repeater noise in order to enlarge the range where quantum repeater is eﬀective. Besides, we showed abstract of proof that QKD with quantum repeater is secure. References 1. G. Barbosa, E. Corndorf, P. Kumar, and H. Yuen. Secure Communication Using Mesoscopic Coherenet States. Phys. Rev. Lett., Vol. 90, p. 227901, 2003. 2. C.H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. Proc. of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE New York), pp. 175-179, 1984. 3. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters. Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels. Phys. Rev. Lett., Vol. 76, No. 5, pp. 722-725, 1996. 4. H.J. Briegel, W. D¨ ur, J.I. Cirac, and P. Zoller. Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication. Phys. Rev. Lett., Vol. 81, No. 26, pp. 5932–5935, 1998. 5. L. Childress, J.M. Taylor, A.S. Sørensen, and M.D. Lukin. Fault-tolerant quantum repeaters with minimal physical resources and implementations based on single photon emitters. quant-ph/0502112, 2005. 6. T. Nishioka, T. Hasegawa, H. Ishizuka, K. Imafuku, and H. Imai. How much security does Y-00 protocol provide us? Phys. Lett. A, Vol. 327, pp. 2832, 2004. 7. V. Scarani, A. Acin, G. Ribordy, and N. Gisin. Quantum Cryptography Protocols Robust against Photon Number Splitting Attacks for Weak Laser Pulse Implementations. Phys. Rev. Lett., Vol. 92, No. 5, p. 057901, 2004. 8. H. Yuen, E. Corndorf, G. Barbosa, and P. Kumar. Barbosa et al. Reply:. Phys. Rev. Lett., Vol. 94, No. 4, p. 048902, 2005.

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ARCHITECTURE-DEPENDENT EXECUTION TIME OF SHOR’S ALGORITHM

RODNEY VAN METER1∗, KOHEI M. ITOH1 AND THADDEUS D. LADD2 1

Graduate School of Science and Technology, Keio University and CREST-JST 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522, Japan 2 Edward L. Ginzton Laboratory Stanford University, Stanford, CA, 94305-4085, USA

We show how the execution time of algorithms on quantum computers depends on the architecture of the quantum computer, the choice of algorithms (including subroutines such as arithmetic), and the “clock speed” of the quantum computer. The primary architectural features of interest are the ability to execute multiple gates concurrently, the number of application-level qubits available, and the interconnection network of qubits. We analyze Shor’s algorithm for factoring large numbers in this context. Our results show that, if arbitrary interconnection of qubits is possible, a machine with an application-level clock speed of as low as one-third of a (possibly encoded) gate per second could factor a 576-bit number in under one month, potentially outperforming a large network of classical computers. For nearest-neighbor-only architectures, a clock speed of around twenty-seven gates per second is required.

1. Introduction Quantum computers are currently being designed that will take advantage of quantum mechanical effects to perform certain computations much faster than can be achieved using current (“classical”) computers 1 . Many technological approaches have been proposed, some of which are being investigated experimentally. DiVincenzo proposed five criteria which must be met by any useful quantum computing technology 2 . In addition to these criteria, a useful quantum computing technology must also support a quantum computer system architecture which can run one or more quantum algorithms in a usefully short time. This observation subsumes into one requirement several issues which, while not strictly necessary to build a quantum computer, will have a strong impact on the possibility of engineering a practical system. These include the importance of gate “clock” speed, support for concurrent gate operations, the total number of application-level qubits supportable, and the complexities of the qubit interconnect network 3 . This paper discusses the impact of these architectural elements on algorithm execution time using the example of Shor’s algorithm for factoring large numbers 4 . Shor’s algorithm ignited much of the current interest in quantum computing because of the improvement in computational class it appears to offer on this important problem. Using Shor’s algorithm, a quantum computer can solve the problem in polynomial time, for a superpolynomial speed∗ e-mail

address: [email protected].

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Figure 1. Scaling of number field sieve (NFS) on classical computers and Shor’s algorithm for factoring on a quantum computer, using BCDP modular exponentiation with various clock rates. Both horizontal and vertical axes are log scale. The horizontal axis is the size of the number being factored.

up. Shor’s algorithm is theoretically important, well defined, and utilizes building blocks (arithmetic, the quantum Fourier transform) with broad applicability, making it ideal for our analysis. On a classical computer, or a collection thereof, the time and computing resources to factor a large number, using the fastest known algorithm, scale superpolynomially in the length of the number (in decimal digits or bits). This algorithm is the generalized number field sieve (NFS) 5 . Its asymptotic computational complexity on large numbers is O(e(nk log

2

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where n is the length of the number, in bits, and k = 64 9 log 2. The comparable computational complexity to factor a number N using Shor’s algorithm is dominated by the time to exponentiate a randomly chosen number x, modulo N , for a superposition of all possible exponents. Therefore, efficient arithmetic algorithms for calculating modular exponentiation in the quantum domain are critical. Very often clock speed and other architectural features are ignored as issues in quantum computing devices, assuming that the superpolynomial speed-up will dominate, making the algorithm practical on any experimentally realizable quantum computer. Shor’s algorithm runs in polynomial time, but the details of the polynomial matter: what degree is the polynomial, and what are the constant factors? An immediate comparison of the execution time to factor a number on classical and quantum computers is shown in Figure 1. The performance of Shor’s algorithm on a quantum computer using the Beckman-Chari-Devabhaktuni-Preskill (BCDP) modular exponentiation algorithm 6 is compared to classical computers running the general Number Field Sieve (NFS). The steep curves are for NFS on a set of classical computers. The left curve is extrapolated performance based on a previous world record, factoring a 530-bit number

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in one month, established using 104 PCs and workstations made in 2003 7 . The right curve is speculative performance using 1,000 times as much computing power. This could be 100,000 PCs in 2003, or, based on Moore’s law, 100 PCs in 2018. From these curves it is easy to see that Moore’s law has only a modest effect on our ability to factor large numbers. The shallower curves on the figure are predictions of the performance of a quantum computer running Shor’s algorithm, using the BCDP modular exponentiation routine, which uses 5n qubits to factor an n-bit number, requiring ∼ 54n3 gate times to run the algorithm on large numbers. The four curves are for different clock rates from 1 Hz to 1 GHz. The performance scales linearly with clock speed. Factoring a 576-bit number in one month of calendar time requires a clock rate of 4 kHz. A 1 MHz clock will solve the problem in about three hours. If the clock rate is only 1 Hz, the same factoring problem will take more than three hundred years. The performance of the BCDP modular exponentiation algorithm is almost independent of architecture. However, the performance of most polynomial-time algorithms varies noticeably depending on the system architecture 8,9 . The main objective of this paper is to show how we can improve the execution time shown in Figure 1 by understanding the relationship of architecture and algorithm. 2. Results We have analyzed two separate architectures, still technology independent but with some important features that help us understand performance. The AC (abstract concurrent) architecture is our abstract model, akin to what is commonly used when drawing quantum circuits. It supports arbitrary concurrency and gate operands any distance apart without penalty. The second architecture, NTC (neighbor-only, two-qubit gate, concurrent) , assumes the qubits are laid out in a one-dimensional line, and only neighboring qubits can interact. This is a reasonable description of several important experimental approaches, including a one-dimensional chain of quantum dots 10 , the original Kane proposal 11 , and the all-silicon NMR device 12 . Above the architecture resides the choice of algorithm, especially for basic arithmetic operations. The computational complexity of an algorithm can be calculated for total cost, or for latency or circuit depth, if the dependencies of variables allow multiple parts of a computation to be conducted concurrently. Fundamentally, the computational complexity of quantum modular exponentiation is O(n3 ) 13,6 , that is, the execution cost grows as the cube of the number of qubits. It consists of 2n modular multiplications of n-bit numbers, each of which consists of O(n) additions, each of which requires O(n) operations. However, O(n3 ) operations do not necessarily require O(n3 ) time steps; the circuit depth can be made shallower than O(n3 ) by performing portions of the calculation concurrently. On an abstract machine, we can reduce the running time of each of the three layers (addition, multiplication, exponentiation) to O(log n) time steps by running some of the gates in parallel, giving a total running time of O(log3 n). This requires O(n3 ) qubits and the ability to execute an arbitrary number of gates on separate qubits. Such large numbers of qubits are not expected to be practical for the foreseeable future, so interesting engineering

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lies in optimizing for a given set of architectural constraints. Addition forms the basis of multiplication, and hence of exponentiation. Classically, many forms of adders have been used in computer hardware 14 . The most basic type of adder, variants of which are used in both VBE and BCDP (as well as our algorithm F, below), is the carry-ripple adder, in which the carry portion of the addition is done linearly from the low-order bits to the high-order. This form of adder is O(n) in both circuit depth and complexity; it is the only efficient type for NTC linear architectures, in which the time to propagate the low-order carry is inherently constrained to O(n). When long-distance gates are available, as in AC architectures, the use of faster adders such as conditional-sum, carry-lookahead, or carry-save adders can result in O(log n) latency, though the complexity remains O(n) 15,16,17 . We have composed several algorithm variants, A through F, as well as investigated concurrent and parallel versions of the original Vedral-Barenco-Ekert (VBE) 13 and BCDP algorithms 15 ; only the fastest for our AC and NTC architectures are presented here. Four parameters control the behavior of the algorithm variants and how well they match a particular architecture. These parameters include the choice of type of adder and the amount of space required. Algorithm variant D is tuned for AC using the conditional-sum adder, and F is tuned for NTC using the Cuccaro-Draper-Kutin-Moulton (CDKM) carry-ripple adder 18 . We have optimized the parameter settings for each individual data point, though the differences are just barely visible on our log-log plot. The values reported here for both algorithms are calculated using 2n2 qubits of storage to exponentiate an n-bit number, the largest number of qubits our algorithms can effectively use. The primary characteristics of the algorithms shown in Figure 2 are summarized in Table 1. The table lists the number of multiplication units executing concurrently, the space, measured in number of logical qubits, the concurrency, or number of logical operations taking place at the same time, and the overall circuit depth, or time, measured in gates. Table 1. Composition of our algorithms. algorithm conc. BCDP algorithm D algorithm F

adder BCDP cond. sum CDKM

multipliers (s) 1 ∼ n/4 ∼ n/4

space 5n + 3 2n2 2n2

concurrency 2 ∼ n2 ∼ 3n/4

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Figure 2 shows our results for our faster algorithms. We have kept the 1 Hz and 1 MHz lines for BCDP, and added matching lines for our fastest algorithms on the AC and NTC architectures. For AC, our algorithm D requires a clock rate of only about 0.3 Hz to factor the same 576-bit number in one month. For NTC, using our algorithm F, a clock rate of around 27 Hz is necessary. The graph shows that, for problem sizes larger than 6,000 bits, our algorithm D is one million times faster than the basic BCDP algorithm, and algorithm F is one thousand times faster. For very large n, the latency of D is ∼ 9n log22 (n). The latency of F is ∼ 20n2 log2 (n). This relationship of architecture and algorithm has obvious architectural implications: concurrency is critical, and support for long-distance gates is important.

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n (bits)

Figure 2. Scaling of number field sieve (NFS) and Shor’s algorithms for factoring, using faster modular exponentiation algorithms.

3. Discussion A fast clock speed is obviously also important for a fast algorithm; however, it remains an open question whether those quantum computing technologies which feature naturally fast physical quantum gates will have the fastest overall algorithm speed. All quantum computing technologies feature some level of decoherence, requiring resources for quantum error correction 19,20,1 . As an example, quantum computers based on Josephson junctions are likely to have extremely fast single-qubit and two-qubit gates, with a physical clock rate at the gigahertz level, as demonstrated in recent experiments 21 . However, the single-qubit decoherence time is only about 1 µs for the most coherent superconducting qubits 22 . Although “fast,” the difficulty in long-term qubit storage and the needed resources for fault tolerant operation may be quite large, so these implementations might make excellent processors with poor memories. In sharp contrast, NMR-based approaches 11,12 are quite slow, with nuclear-nuclear interactions in the kilohertz range. However, the much longer coherence times of nuclei 23 make the use of NMR-based qubits as memory substantially easier 24 . Ion trap implementations have the benefit of faster single-qubit-gate, two-qubit-gate, and qubit-measurement speeds with longer coherence times, but the added complication of moving ionic qubits from trap to trap physically 25 or exchanging their values optically 26 complicates the picture for the application-level clock rate. New physical proposals for overcoming speed and scalability obstacles continue to be developed, leaving the ultimate hardware limitations on clock speed and its relation to algorithm execution time uncertain. 4. Conclusions We have shown that the actual execution time of Shor’s algorithm is dependent on the important features of concurrent gate execution, available number of qubits, interconnect topology, and clock speed, as well as the critical choice of an architecture-appropriate

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arithmetic algorithm. Our algorithms have shown a speed-up factor ranging from nearly 13,000 for factoring a 576-bit number to one million for a 6,000-bit number. Acknowledgments The authors wish to thank Eisuke Abe, Kevin Binkley, Fumiko Yamaguchi, Seth Lloyd, Kae Nemoto, and W. J. Munro for helpful discussions. References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). 2. D. P. DiVincenzo, Science 270, 255 (1995). 3. R. Van Meter and M. Oskin. J. Emerging Tech. in Comp. Sys., 2(1), Jan. 2006. 4. P. W. Shor, in Proc. 35th Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994), pp. 124–134. 5. D. E. Knuth, The Art of Computer Programming, volume 2 / Seminumerical Algorithms, 3rd ed. (Addison-Wesley, Reading, MA, 1998). 6. D. Beckman, A. N. Chari, S. Devabhaktuni, and J. Preskill, Phys. Rev. A 54, 1034 (1996). 7. RSA Security Inc., web page, 2004, http://www.rsasecurity.com/rsalabs/node.asp?id=2096. 8. N. Kunihiro, IEICE Trans. Fundamentals, E88-A(1):105–111, (2005). 9. A. G. Fowler, S. J. Devitt, and L. C. Hollenberg, Quantum Information and Computation 4, 237 (2004). 10. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 11. B. E. Kane, Nature 393, 133 (1998). 12. T. D. Ladd et al., Phys. Rev. Lett. 89, 17901 (2002). 13. V. Vedral, A. Barenco, and A. Ekert, Phys. Rev. A 54, 147 (1996). 14. M. D. Ercegovac and T. Lang, Digital Arithmetic (Morgan Kaufmann, San Francisco, CA, 2004). 15. R. Van Meter and K. M. Itoh, Phys. Rev. A 71, 052320 (2005). 16. T. G. Draper, S. A. Kutin, E. M. Rains, and K. M. Svore, A Logarithmic-Depth Quantum CarryLookahead Adder, http://arXiv.org/quant-ph/0406142 (2004). 17. P. Gossett, Quantum Carry-Save Arithmetic, http://arXiv.org/quant-ph/9808061 (1998). 18. S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, A new quantum ripple-carry addition circuit, http://arXiv.org/quant-ph/0410184, 2004. 19. A. M. Steane, Phys. Rev. A 68, 042322 (2003). 20. S. J. Devitt, A. G. Fowler, and L. C. Hollenberg, Simulations of Shor’s algorithm with implications to scaling and quantum error correction, http://arXiv.org/quant-ph/0408081, 2004. 21. T. Yamamoto, et al., Nature 425, 941 (2003). 22. D. Vion, et al., Science 296, 886 (2002). 23. T. D. Ladd et al., Phys. Rev. B 71, 014401 (2005). 24. K. M. Itoh, Solid State Comm. 133, 747 (2005). 25. D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002). 26. B.B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, Nature 428, 153 (2004). 27. L. M. Duan, Phys. Rev. Lett. 93, 100502 (2004).

Quantum Dots and Kondo Effects

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COULOMB BLOCKADE PROPERTIES OF 4-GATED QUANTUM DOT SHINICHI AMAHA1†, TSUYOSHI HATANO1, SATOSHI SASAKI2, TOSHIHIRO KUBO1, YASUHIRO TOKURA1, 2, SEIGO TARUCHA1, 3 1

ICORP-JST, 3-1 Morinosato-Wakamiya, Atsugi-shi,Kanagawa, 243-0198, Japan 2NTT BRL, 3-1 Morinosato-Wakamiya, Atsugi-shi,Kanagawa, 243-0198, Japan 3 Graduate School of Applied Phys., Univ. Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

We fabricated a few electron vertical quantum dot (QD) device having four separate gate electrodes and characterized the electronic properties. In this device, geometrical symmetry in the confining potential for the QD, that is, orbital degeneracy for the QD electronic states can be tuned by adjusting the voltages applied to the four gates. It is then necessary to precisely characterize the performance of each gate. We use a nonlinear single electron tunneling spectroscopy technique to characterize the gate performances and apply a capacitance network model for this device to reproduce the observed gate performances.

1. Introduction Circular vertical quantum dots (QDs) are often referred as to “artificial atoms” because of atom-like features, such as shell filling and Hund’s rule [1]. The shell filling is well understood by considering electronic states confined by a two-dimensional (2D) harmonic potential with a high degree of rotational symmetry [2]. This brings about a novel concept to manipulate the degree of shell filling or orbital degeneracy by introducing anisotropy to the 2D harmonic potential in a controlled manner. We use a four-gate tuning technique to manipulate anisotropy in the lateral confinement potential for a vertical QD (See Fig. 1) [3]. The electron number in this QD can be varied one-by-one, starting from zero up to ~20, however, the tunability of the confinement anisotropy is significantly different from device to device, probably because the way of metal electrode attachment to the mesa is different from gate to gate. This causes a variation of gate performance to control the confinement anisotropy among the four gates. Here we introduce a capacitance model to deal with the difference in the gate performance among the four gates, and compare with the experimental data of Coulomb diamonds. We finally discuss the ability of our device to manipulate the degree of shell filling in the QD.

†

Email: [email protected].

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Fig.1. Schematic diagram of the four-gate vertical QD.

2. Capacitance network model for the quantum dot having four gates Figure 2 shows a capacitance network model for the four-gate QD [4], that is capacitively coupled to the four gate electrodes labeled i=1 to 4 and also connected to the source (S) and drain (D) contact electrodes (source and drain contacts).

Fig.2. Capacitance network of the four-gate QD.

We employ the constant interaction (CI) model and write the total charge Q on the QD as sum of the charges on all capacitors (source, drain and four gates) 4

Q = CS (V − VS ) + C D (V − VD ) + ∑ C gi (V − Vgi ) ,

(1)

i =1

where CS (CD) is the capacitance between the QD and the source (drain) electrode and Cgi is that between the QD and the gate i. VS (VD) is the electronic potential of the source (drain) electrode and Vgi is that of the gate i. The electrochemical potential (N) for the N-electron QD is defined as an energy cost for adding the Nth electron to the (N-1)-electron QD. Using Eq. (1), (N) is presented as

μ

µ ( N ) = ( N − N 0 − 1 / 2)

μ

4 e2 e 4 − ∑ C giVgi ≡ − e ∑ α iVgi + β N , C C i =1 i =1

(2)

α≡

where N0 is electron number in the QD at VS, VD, and Vgi = 0, C=∑Ci+CS+CD, i Cgi/C and βN ≡(N-N0-1/2) e2/C. The condition of Coulomb blockade for the N-th electron with respect to the source-drain voltage, Vsd, can be written as:

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µ ( N ) < µ0 −

1 1 e Vsd < µ 0 + eVsd < µ ( N + 1) , 2 2

(3)

where µ0 is the Fermi energy of the source and drain electrodes at Vsd=0. The condition (3) is also written as

β N − µ0 +

e Vsd 2

4

< e ∑ α iVgi < β N + i =1

e Vsd e2 . − µ0 − 2 C

(4)

Figure 3 shows the schematic diagram of Coulomb diamonds for sweeping Vg1 with the other gate voltages constant (a) and with the other gate voltages altogether, (Vg1=Vg2=Vg3=Vg4 ≡Vg) (b), respectively. The vertical sizes of the Coulomb diamonds, ∆Vgi , and ∆Vg , for (a), and (b), respectively, are presented as:

 4   ∑ C gi  ∆Vg = C gi ∆Vgi = e .   i =1

(5)

We use Eqs. (5) to analyze the measured Coulomb diamonds, and derive the capacitance ratio for each gate electrode. It is ideal that all gate capacitances are the same in order to create complete circular symmetry in the confining potential. However, this is not the case for real devices, as described bellow.

Fig.3. Schematic diagrams of Coulomb diamonds for the four-gate QD (a) Coulomb blockade region plotted in Vg v.s. Vsd plane with setting all gate voltages the same (Vg=Vg1=Vg2=Vg3=Vg4). (b) Coulomb blockade region plotted in Vg1 v.s. Vsd plane with setting other gate voltages (Vg2, Vg3, Vg4) fixed.

3. Experiments and discussion 3.1. Device fabrication and measurement technique The starting material for device fabrication is a specially designed AlGaAs/GaAs/AlGaAs double barrier structure (DBS) with an n-AlGaAs contact layer above and below. In this DBS the well and contacts are so strongly coupled that the Kondo effect is observed in the conductance measurement at low temperatures. Details of the Kondo effect observed in this device will be discussed elsewhere. We first made a electrode metal pattern on the

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194 194 material surface: a small square (~0.35µm x 0.35µm) with a narrow line extending from each corner outwards, etched the material using the metal pattern as a self-aligned mask to make a cross-shaped mesa (Fig. 1), and evaporated gate metal in the surrounding of the mesa. The line mesa is narrow enough that current only flows in the metal. This current then flows in the cross-shaped mesa downwards and reaches the bottom electrode. We use a standard DC technique to measure the current, I, flowing through the QD in response to a voltage Vsd applied to the bottom electrode. We apply voltage Vgi to each gate electrode Gi independently. The measurement temperature is 1.5K. 3.2. Coulomb diamonds with sweeping voltage for all gates common Figure 4 shows the measured dI/dVsd in the plane of Vsd and Vi (=Vg1=Vg2=Vg3=Vg4), which is the case for Fig. 3(a). A series of Coulomb diamonds are observed in the center for Vg > -1.06 V. However, for the more negative gate voltage there are no more closed Coulomb diamonds observed, indicating N = 0 for Vg < -1.06 V. Note no Coulomb oscillations are observed in this gate voltage range, either. So the number of electrons is increased for each Coulomb diamond one-by-one to the top, starting from N=0. The Coulomb diamond with N=2 is larger than that with N=1 corresponding to the complete shell filling of the1s state. Edges of each Coulomb diamond are not clear due to high sample temperature (~1.5K) and the strong coupling between the well and source drain electrodes.

Fig.4. Measured conductance plotted in source drain voltage (Vsd) v.s. gate voltage applied to all gate electrodes as common (Vg1=Vg2=Vg3=Vg4=Vg).

3.3. Coulomb diamonds with sweeping voltage just to one gate Figure 5 shows the Coulomb diamonds measured in the same as those in Fig. 4 but by sweeping just one gate voltage: Vgi =Vg1 (a), Vg2 (b), Vg3 (c), and Vg4 (d), respectively. In

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195 195 Fig. 4 we see the N=6 Coulomb diamond for Vg1=Vg2=Vg3=Vg4=-0.60V. So we can compare the gate performance for this Coulomb diamond among the four gates in the setup for Fig. 3(b). Actually in Fig. 5 we can compare the N =5 Coulomb diamonds as well. The shape of N=6 Coulomb diamond looks similar among (a) to (d). However, the threshold lines forming this Coulomb diamond have the slopes different among (a) to (d). In Fig.5 the N=6 Coulomb diamond width is evaluated as ∆Vg1= 0.34V, ∆Vg2=0.44V, ∆Vg3=0.28V and ∆Vg4=0.22(V), respectively. In addition, this width is evaluated as ∆Vg=0.097V in Fig.4. Using these values, we derive α1=0.28, α2=0.22, α3=0.34, and α4=0.45, respectively. From these αi (i=1, 2, 3, 4), we can also evaluate capacitance ratio as C1:C2:C3:C4~2.5:2:3:4. So that, only two diamonds are clearly distinguished in (a) and (b), whereas three, and four diamonds are distinguished in (c), and (d), respectively. Thus obtained α value is so significantly different among the four gates, but still in the relevant range for tuning the degree of symmetry with voltages applied to the four gates. The capacitances of gate 1 and 2 are estimated to be slightly lower than that of gate 3 and 4. The Ti/Au metals of gate 1 and 2 electrodes are expected to lower attachment to the cross-shaped mesa than that of gate 3 and 4 due to the tilt on evaporation process of gate metal.

Fig.5. Measured conductance plotted in each gate voltage v.s. source drain voltage by fixed other gate voltages at -0.60V. (a) Vg1 v.s. Vsd (Vg2=Vg3=Vg4=-0.60V). (b) Vg2 v.s. Vsd (Vg1=Vg3=Vg4=-0.60V). (c) Vg3 v.s. Vsd (Vg1=Vg2=Vg4=-0.60V). (d) Vg4 v.s. Vsd (Vg1=Vg2=Vg3=-0.60V).

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196 196 4. Conclusion Gate performance is studied for a few-electron vertical QD having four gates. The capacitance of each gate to the QD is evaluated using a capacitance network model and from measurements of Coulomb diamonds by sweeping only the corresponding gate voltage. The derived capacitance values are significantly different among the four gates but still in the relevant range for tuning the degree of symmetry in the QD confining potential with voltages applied to the four gates. Acknowledgments The authors thank K. Ono, T. Maruyama, W. G. van der Wiel, D.G. Austing, R. Sakano, T. Kita, M. Eto, S. Suga and N. Kawakami for fruitful discussion. Part of this work is financially supported by the Grant-in-Aid for Scientific Research A (No. 40302799) and by CREST-JST, and by IT Program, MEXT and by the DARPA-QUIST program (DAAD19-01-1-0659). References 1. S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hoge and L.P. Kowenhoven, Phys. Rev. Lett. 77, 3613(1996). 2. P. Matagne and J.P. Leburton, Phys. Rev. B 65, 235323(2002). 3. D.G. Austing, T. Honda and S. Tarucha, Jpn. J. Appl. Phys. 36, 4151 (1997). 4. W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha and L.P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2003).

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ORDER-N ELECTRONIC STRUCTURE CALCULATION OF n-TYPE GaAs QUANTUM DOTS

S. NOMURA Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan E-mail: [email protected] T. IITAKA RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan E-mail: [email protected] A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested in calculating the electronic structure of a model n-type GaAs quantum dot.

1. Introduction Linear scale methods for calculating the electronic structures have been actively investigated in the last decade because of increasing demand for predicting properties of large and complex systems with computational cost linear scale with respect to the system size N . One of such methods is the Chebyshev polynomial expansion method.1 The electron density is obtained by using a matrix representation of the Fermi-operator, which is expanded in the Chebyshev matrix polynomials. With a combination of the Chebyshev polynomial expansion method and time-dependent method2 (CPE-TDM), the electron density of states (DOS) is obtained without calculating eigenenergies and eigenstates. The computational time of CPE-TDM scales as O(N ), as compared with that of the conventional method such as conjugate gradient method (CGM), which grows as O(N 2 ). Thus CPE-TDM enables us to calculate large systems which require prohibitively large computational time by CGM. CPE-TDM was applied to calculate the optical properties of hydrogenated Si nanocrystals containing atoms more than 10,000 within the empirical pseudopotential formalism3 and the ESR spectrum of s = 1/2 antiferromagnet Cu benzoate,4 which have proved the advantage of CPE-TDM. However, CPE-TDM has not been applied to calculation of the electronic structure within a LDA. In this paper, we report on an implementation of CPE-TDM for a calculation of the electronic struc-

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ture of n-type GaAs quantum dot (QD) within a LDA and compare the results with a CGM.

2. Method of calculation The model structure is a 20 nm-wide GaAs quantum well sandwiched by undoped Alx Ga1−x As(x = 0.3) barriers. The electrons are assumed to be supplied from 5 nm-thick Si-doped Alx Ga1−x As layer, located 20 nm above the GaAs quantum well layer. The Fermi-energy (EF ) is taken as the origin of the energy. The Fermi-level pinning model is assumed. The number of the electrons in a QD is not ﬁxed to an integer number and is determined by EF and the potential energy. The lateral conﬁning potential is ﬁxed to a parabolic potential, which may be created by a surface gate structure in experiments. The model Hamiltonian of the system within the LDA is H=

1 p2 + m∗ ω02 r 2 + Vc (r) + VH (r) + Vx (r) ∗ 2m 2

(1)

where m∗ is the eﬀective mass of the electrons, ω0 = 3 meV, and, Vc (r), VH (r), and Vx (r) are the vertical conﬁning potential, the Hartree potential, the exchange potential, respectively. A 3D mesh of 64 × 64 × 8 is used for the calculation of the electron density, and 64 × 64 × 16 is used for the calculation of the potentials. The axis perpendicular to the quantum well layer is taken to be z-direction. The Hamiltonian is discretized in real space by the higher-order ﬁnite diﬀerence method.5 N A random phase vector as deﬁned by |Φ ≡ n=1 |nξn , where {|n} is a basis set and ξn are a set of random phase variables, is used as an initial state. This was shown to give results with the smallest statistical error.6 Here Φ is a Nx × Ny × Nz column vector for a system deﬁned by a real-space uniform grid of Nx ×Ny ×Nz . The 1 electron density n(r) is extracted by the Fermi operator function f (H) = eβ(H−E F ) +1 as n(r) = |Φ|f (H )|r|2 .

(2)

The Fermi operator is evaluated by the Chebyshev polynomial expansion, f (H)|Φ =

ak (β)Tk (H )|Φ.

(3)

k

Actually the electron density is calculated with jmax sets of |Φ as n(r) = j =jmax | r|f (H )|Φj |2 /jmax , where · stands for statistical average. The j =1 Hartree and exchange potentials are calculated using n(r). The new solution of the potential VHnew (r) is combined with the solution obtained for the previous iteration old (r) + αVHnew (r). Similarly, in order to reduce the statistical by VH (r) = (1 − α)VH ﬂuctuation, n(r) is combined with the density obtained for the previous iteration

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by n(r) = (1 − γ)n old (r) + γn new (r). The parameter α is varied between 0.10 to 0.05. The parameter γ is varied between 0.30 and 0.025 for the CPE-TDM method. For the CGM calculations, γ is ﬁxed to 1. The DOS is calculated by a time evolution method as given by 1 (4) ρ(ω) = − Im( Φ|G(ω + iη)|Φ ), π where G(ω + iη) is a real-time Green’s function. The DOS is calculated with kmax sets of |Φ. The energy resolution η is chosen to be 0.5 meV. It should be noted that kmax used for calculating the DOS can be independently chosen from jmax for each self-consistent iteration procedure. 3. Results and discussions Model calculations are performed for a GaAs QD containing about 77 electrons. The number of the self-consistent iterations is ﬁxed to 100 for both the CGM and CPETDM calculations. The potential is converged to |VH (r) − VHnew (r)| < 0.003meV for the CGM calculation. The electron density distributions are shown in Fig. 1 for CPE-TDM with jmax = 64 and for CGM. The calculated electron density distribution reasonably agrees with the result by a CGM within the statistical ﬂuctuations. The Friedel-type spatial oscillations of the electron density7 are reproduced in both the results by the CPE-TDM and CGM.

*+,

*?, A<

A<

:@;9

A<

A<

B@;9

-./0123

-./0123

:@;9

A<

=@;9

A<

<@;9

A<

A<

=@;9

A<

<@;9

A<

;@;9

99

B@;9

;@;9 :9

;99

4 6/;:9<99<:9 78 =99

9 ;99 :9 <99 ;:9 8 =99 <:9 7 / 6 >

9

:9

4;99 6/;:9 7<99 8 <:9=99

9 ;99 :9 <99 ;:9 78 <:9 / 6 =99 >

Figure 1. (a) The electron density distribution obtained by CPE-TDM. 64 sets of random vectors are used at each self-consistent iteration procedure. (b) The electron density distribution obtained by CGM.

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,*

=>?@ABC EF @BGB>@ H+IJ>KL

+*

*

*

*

*

*

HAL HAAL HAAAL HAPL HPL HPAL

* .+*

X

Y

Z[\ ./

*

M?>NOC HJ>KL

/

Figure 2. Density of states (ρ(ω)) obtained by CPE-TDM with (i) 8, (ii) 16, (iii) 32, and (iv) 64 sets of random phase vectors for extracting n(r) at each self consistent iteration procedure and for evaluating ρ(ω). (v) ρ(ω) obtained by CPE-TDM with 8 sets of random phase vectors for extracting n(r) and 64 sets of random phase vectors for evaluating ρ(ω). (vi) ρ(ω) obtained by CGM. The vertical dashed lines are guide to the eye.

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The calculated DOS are shown in Fig. 2. For CPE-TDM, the self-consistent iteration procedures are performed with jmax = 8, 16, 32, and 64. The same number of random phase vectors are used for evaluating ρ(ω) except for the case of jmax = 8 where ρ(ω) is evaluated with kmax = 8 and 64. It can be seen that the statistical ﬂuctuations decrease with increase in jmax in calculating ρ(ω). There are two types of the ﬂuctuations observed in Fig. 2. One is the ﬂuctuations in the peak energy positions, and the other is the ﬂuctuations in the peak heights. The former can be reduced by increasing jmax and decreasing γ. The latter also depends on kmax . In fact, the ﬂuctuations in the peak heights are reduced by increasing kmax from 8 to 64 with small changes in the peak energy positions in the case of jmax = 8. Finally we note that the statistical ﬂuctuation of the total number of electrons (Ne ) is smaller than that of DOS because of the self-averaging eﬀect. Ne is calculated to be 76.8, 76.7, 77.1, 76.8 for the cases of jmax = 8, 16, 32, and 64, respectively, which can be compared with Ne = 77.1 by CGM. 4. Conclusions In this paper, it has been demonstrated that CPE-TDM can be applied to a LDA calculation of a model QD system despite the presence of the statistical ﬂuctuations of the calculated quantities originated from the random phase vectors. It has been shown that the random phase vectors are useful if the statistical ﬂuctuations are controlled by carefully choosing the parameters for the calculations. Our linear scale method opens up possibilities for calculating the electronic and optical properties of large and complex systems, such as QD arrays with interaction between QDs and devices employing the Rashba type spin-orbit interaction.8 It should also be possible to calculate the electronic structure of hydrogenated Si nanocrystals within LDA. Finally, because the Green’s function can be eﬀectively estimated by CPE-TDM, the properties of the electronic system such as the DC and Hall conductivities, and the optical absorption spectra, are obtained within O(N ) computational costs. Acknowledgments This work was partly supported by the Grant-in-Aid for Scientiﬁc Research from Japan Society for the Promotion of Science. Computational support by RIKEN Super Combined Cluster System is gratefully acknowledged. References 1. R. Kosloﬀ and H. Tal-Ezer, Chem. Phys. Lett. 127, 223 (1986); S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999). 2. T. Iitaka, Phys. Rev. E 49, 4684 (1994). 3. S. Nomura et al., Phys. Rev. B 56, R4348 (1997); T. Iitaka et al., Phys. Rev. E 56, 1222 (1997). 4. T. Iitaka and T. Ebisuzaki, Phys. Rev. Lett. 90, 047203 (2003). 5. S. Nomura and Y. Aoyagi, Phys. Rev. Lett. 93, 096803 (2004).

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6. T. Iitaka and T. Ebisuzaki, Phys. Rev. E 69, 057701 (2004). 7. J.H. Luscombe, A.M. Bouchard, and M. Luban, Phys. Rev. B 46, 10262 (1992). 8. T. Koga, J. Nitta, and M. van Veenhuizen, Phys. Rev. B 70, 161302 (2004).

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TRANSPORT THROUGH DOUBLE-DOTS COUPLED TO NORMAL AND SUPERCONDUCTING LEADS

Y. TANAKA AND N. KAWAKAMI Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan E-mail: [email protected]

We study transport through double quantum dots coupled to normal and superconducting leads, where the Andreev reflection plays a key role in determining characteristic transport properties. We shall discuss two typical cases, i.e. double dots with serial or parallel geometry. For the parallel geometry, the interference of electrons via multiple paths is induced, so that the transmission probability has Fano-type dip structures which are symmetric with respect to the Fermi energy. We also investigate the AharonovBohm(AB) effect for the parallel geometry. In some particular situations, we find that the general AB period for double dots, 4π, is reduced to 2π.

1. Introduction Recent advances in nanotechnology have enabled us to realize mesoscopic normalmetal/superconductor hybrid systems. In these systems, the Andreev reflection plays an important role for quantum transport. In particular, the Andreev reflection for a quantum dot coupled to normal and superconducting leads gives rise to characteristic transport properties due to the discreteness of energy levels in a dot.1,2,3,4,5,6 Moreover, the interplay between the Andreev reflection and the Kondo effect in quantum dot systems has been investigated intensively.7,8,9,10,11,12,13 In this work14 , we study transport through double quantum dots (DQD’s) coupled to normal and superconducting leads. Here, we concentrate on transport due to the Andreev reflection, which is referred to as the Andreev tunneling in the following. We shall discuss two typical cases: the DQD is connected in series or parallel. In particular, we focus on the interference effect, which is caused via multiple paths of electron propagation, on the Andreev reflection in the parallel DQD.14,15,16 Since the interference effect is sensitive to the magnetic flux, we also investigate the influence of the Aharonov-Bohm(AB) effect on the Andreev tunneling. In the following, we first give a brief explanation of the model, and then describe the results for the differential conductance due to the Andreev tunneling in Sec. 3. A brief summary is given in the last section.

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2. Model We consider a DQD system coupled to normal and superconducting leads (N/DQD/S) shown in Fig. 1. In this figure, two dots are coupled via the inter-dot

dot1

Γ1

Γ1

S

N

tc

S-lead

Γ2S

N-lead

Γ2N dot2

Figure 1. DQD system coupled to normal(N) and superconducting(S) leads. tc is the inter-dot N (S) tunneling, and Γi (i = 1, 2) represents the resonance width due to the transfer between the dot-i S and normal(superconducting) lead. Note that two dots are connected in series when Γ N 1 = Γ2 = 0 (the dashed arrows), which corresponds to α = 0 (see text).

tunneling tc , and the dot-i is connected to the normal (superconducting) lead via N (S) the tunneling, which causes the resonance width of Γi (i = 1, 2). Introducing the N S S ratio α = ΓN /Γ = Γ /Γ in the same notation as in Refs. 17-19, we discuss two 1 2 2 1 typical cases α = 0 and α ∼ 1, namely, two dots are connected in series or parallel. We assume that the superconducting lead is well described by the BCS theory with a superconducting gap ∆. In addition, the intra-dot Coulomb interaction is ignored for simplicity. Since we are interested in the Andreev tunneling, we concentrate on the zero temperature case (T = 0) in the region of small bias voltage V (i.e. |V | < ∆). We calculate the differential conductance dI/dV (I: current) and the density of states (DOS) of the dots by using the Keldysh Green functions.

3. Numerical Results 3.1. Andreev tunneling in serial and parallel DQD systems We first discuss the Andreev tunneling in the serial and parallel DQD systems. For simplicity, we fix the energy level of the dot-i (εi ) at the Fermi energy (ε1 = ε2 = 0), and use the gap ∆ as the unit of energy. Figure 2 shows the differential conductance as a function of the bias voltage V for (a) serial (α = 0), (b) parallel (α = 0.7) DQD system. In both DQD systems, the conductance has four peaks in its voltage dependence, which are symmetric with respect to the Fermi energy. We shall discuss characteristic aspects of these peaks in terms of the DOS of the dots. Note first that the interdot coupling tc forms

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(a)

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1

2

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205

−0.2

0 V

0.2

(b)

0.5

0

−0.2

0 V

0.2

Figure 2. The conductance as a function of the bias voltage V for the N/DQD/S system. (a) S serial DQD (α = 0), (b) parallel DQD (α = 0.7). We set tc = 0.2, ΓN 2 = 0.01, Γ1 = 0.1 (∆=1).

the bonding and anti-bonding states for electrons in dots, whereas those for holes are obtained by inverting the DOS profile with respect to the Fermi energy. The Andreev reflection at the DQD/S-lead interface mixes these states for electrons and holes, giving rise to the four Andreev bound states in the dots. Therefore, the DOS of the dots has the four peaks, where the width of these peaks originates from the resonance width ΓN i . In the serial DQD system, these peaks are indeed observed in the profile of the conductance as shown in Fig. 2(a). On the other hand, in the parallel DQD, Fano-type dip structures (V ' ±0.2) appear in addition to the four peaks, as shown in Fig. 2(b). Here, it is instructive to recall the interference effect in the parallel DQD system coupled to two normal leads (N/parallel-DQD/N). In this system, the DOS for electrons in the dots consists of sharp and broad resonance peaks, as shown by the thick line in Fig. 3(b). When an electron transports via these resonances, its transmission probability acquires a Fano-type dip structure around the position where the sharp peak in the DOS is located.17,18,19 We note that the DOS for holes is obtained by inverting that for electrons with respect to the Fermi energy (thin line in Fig. 3(b)). Coming back

(b)

DOS( ε )

(a) 20 10 0

−0.2

0

ε

0.2

0 ε

Figure 3. (a) DOS of the dot for the N/parallel-DQD/S system. The solid (dashed) line is for the dot-1(2). The parameters are the same as in Fig. 2(b). (b) Sketch of DOS for the N/parallelDQD/N system. The thick (thin) line is for electrons (holes). ε = 0 corresponds to the Fermi energy.

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to the N/parallel-DQD/S system, we now see that mixing of the electron and hole states induced by the Andreev reflection gives rise to four Andreev bound states in the dots, similarly to the serial case. In contrast to the latter case, however, the resonances in the parallel case have two different widths: two sharp peaks and two broad peaks shown in Fig. 3(a). The interference between the distinct transport channels via these resonances gives rise to the Fano-type dip structures in the conductance, which are symmetric with respect to the Fermi energy (Fig. 2(b)). Here, we make a brief comment on the special case α = 1, i.e. the symmetric couplings with leads. In this case, the sharp peaks in Fig. 3(a) become of deltafunction type, which means that the corresponding local states in the dots are completely decoupled from the leads. Therefore, the remaining states with the broad resonances in Fig. 3(a) only contribute to the Andreev reflection, so that the transport shows similar behavior to the case of a single dot coupled to normal and superconducting leads. 3.2. AB effect in N/DQD/S system

2

conductance (4e /h)

We next discuss how the Andreev tunneling changes its character when the magnetic flux is added in the parallel DQD system. Here, we assume that the magnetic flux equally pierces the two subrings formed by the interdot tunneling tc , so that the effect of the magnetic flux is symmetrically incorporated in the tunneling between the dot and the lead. Before considering the N/DQD/S system, we briefly mention the AB effect in the N/DQD/N system. As noted in the recent literature,20,21,22,23 the interdot coupling between the dots forms two-subring structure, so that the AB period in the N/DQD/N system becomes 4π instead of the normal period of 2π. Similarly, the AB period in the N/DQD/S system is expected to be 4π in general, as pointed out by Zhang et al..15 We find, however, that the AB period is reduced to 2π in some particular situations, as will be explicitly shown below.

1

φ=0 (2π)

φ=0.5π (1.5π)

φ=π

0.5

0

−0.2

0 V

0.2

−0.2

0 V

0.2

−0.2

0 V

0.2

Figure 4. The conductance as a function of V for several choices of the magnetic flux φ. The other parameters are the same as in Fig. 2(b).

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Figure 4 shows the conductance for several values of φ, where φ represents the normalized value of the magnetic flux Φ as φ = 2πΦ/Φ0 (Φ0 = h/e). The other parameters are the same as in Fig. 2(b). It should be noted that the conductance is symmetric with respect to the Fermi energy for any value of φ and has the AB period of 2π (not 4π). More precisely, the conductance at the magnetic flux φ, G(φ), satisfies G(φ) = G(2π − φ). Including the situation for Fig. 4, we obtain the general condition that reduces the AB period to 2π, (i) ε1 = ε2 = 0, (0 < α ≤ 1) (ii) ε1 = −ε2 (6= 0) and α = 1. The condition for the energy levels of the dots, ε1 = −ε2 , means that the bonding state and the q anti-bonding state in dots are symmetric with respect to the Fermi 2 energy (ε = ± ε1(2) + t2c ). Then, the DOS of the dot-1(2) at the magnetic flux φ, ρ1(2) (φ) has the AB period of 2π (ρ1(2) (φ) = ρ1(2) (φ + 2π)) for the case (i). On the other hand, ρ1(2) (φ) for the case (ii) has the periodicity of ρ1(2) (φ) = ρ2(1) (φ + 2π), namely, the DOS of the dot-1(2) in the absence of the magnetic flux is the same as that of the dot-2(1) for φ = 2π. Note here that the N/parallel-DQD/S system we treat here remains unchanged even if we exchange the dot-1 and the dot-2 because of the high symmetry of the couplings between the dots and the leads (α = 1). Taking this into account, we can see that the total DOS of the dots has the period of 2π. Since the DOS with this periodicity directly determines electron transport, the conductance changes in the AB period of 2π. In this connection, we stress the difference in the AB period between our result and the related work by Zhang et al.15 , who treated the same DQD system. We have found here that the AB period is reduced to 2π in the case (ii), although they claimed that it still remains 4π in the same condition. Concerning the calculational method, the conductance (the transmission probability) in both studies is formulated by the Keldysh Green functions and the Landauer formula. We thus think that the difference may possibly come from an error in numerical calculations in ref. 15. 4. Summary We have studied transport properties through the DQD coupled to normal and superconducting leads. It has been discussed that the four-peak structure in the DOS is formed by the bonding and antibonding states in the dots coupled to the superconducting lead. This structure in the DOS indeed determines the characteristic properties in the differential conductance, in accordance with the results of Zhang et al.15 In particular, in the parallel DQD system, the interference effect due to the multiple paths gives rise to the Fano-type dip structures in the differential conductance, which are symmetric with respect to the Fermi energy. We have also found the interesting fact that the AB period is reduced to 2π in some particular situations, which is contrasted to the AB period 4π expected generally for DQD systems.

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References 1. C. W. J. Beenakker, Phys. Rev. B 46, 12841 (1992). 2. H. -K. Zhao and G. v. Gehlen, Phys. Rev. B 58, 13660 (1998). 3. Q. -f. Sun, J. Wang, and T. -h. Lin, Phys. Rev. B 59, 3831 (1999); Phys. Rev. B 59, 13126 (1999); Phys. Rev. B 62, 648 (2000). 4. P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. B 63, 165314 (2001). 5. H. -K. Zhao and J. Wang, Phys. Rev. B 64, 094505 (2001). 6. Z. Chen, J. Wang, B. Wang, and D. Y. Xing, Phys. Lett. A 334, 436 (2005). 7. R. Fazio and R. Raimondi, Phys. Rev. Lett. 80, 2913 (1998). 8. P. Schwab and R. Raimondi, Phys. Rev. B 59, 1637 (1999). 9. A. A. Clerk, V. Ambegaokar, and S. Hershfield, Phys. Rev. B 61, 3555 (2000). 10. J. C. Cuevas, A. L. Yeyati, and A. Martin-Rodero, Phys. Rev. B 63, 094515 (2001). 11. Q. -f. Sun, H. Guo, and T. -h. Lin, Phys. Rev. Lett. 87, 176601 (2001). 12. A. Golub and Y. Avishai, Phys. Rev. B 69, 165325 (2004). 13. M. R. Gr¨ aber, T. Nussbaumer, W. Belzig, and C. Sch¨ onenberger, Nanotechnology 15, S479 (2004). 14. Our work was also presented at the Physical Society of Japan meeting (September 19, 2005, Kyoto, Japan), where we discussed the Andreev tunneling in both of serial and parallel DQD systems. We note that the parallel DQD case was already discussed by Zhang et al. in Ref. 15. Our results are in accordance with theirs in this case, except for the magnetic-flux dependence of the conductance. 15. Y. -P. Zhang, H. Yu, Y. -F. Gao, and J. -Q. Liang, Phys. Rev. B 72, 205310 (2005). 16. J. Peng, B. Wang, and D. Y. Xing, Phys. Rev. B 71, 214523 (2005). 17. M. L. L. de Guevara, F. Claro, and P. A. Orellana, Phys. Rev. B 67, 195335 (2003). 18. Y. Tanaka and N. Kawakami, Phys. Rev. B 72, 085304 (2005). 19. Y. Tanaka and N. Kawakami, “REALIZING CONTROLLABLE QUANTUM STATES”, Proc. of the International Symposium on Mesoscopic Superconductivity and Spintronics 2004, pp.433-438 (World Scientific, Singapore, 2005). 20. Z. T. Jiang, J. Q. You, S. B. Bian, and H. Z. Zheng, Phys. Rev. B 66, 205306 (2002). 21. P. A. Orellana, M. L. Ladr´ on de Guevara, and F. Claro, Phys. Rev. B 70, 233315 (2004). 22. K. Kang and S. Y. Cho, J. Phys.: Condens. Matter 16, 117 (2004). 23. Z. -M. Bai, M. -F. Yang, and Y. -C. Chen, J. Phys.: Condens. Matter 16, 2053 (2004).

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A STUDY OF THE QUANTUM DOT IN APPLICATION TO TERAHERTZ SINGLE PHOTON COUNTING H. HASHIBA, V. ANTONOV Physics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK L. KULIK Institute of Solid State Physics, RAS, Chernogolovka, 142432 Russia S. KOMIYAMA Department of Basic Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan

We present an experimental study of a quantum dot for application to a single photon terahertz detector. A GaAs/AlGaAs quantum dot of ~1 micrometer size is defined by mesa patterning and metal gates. The charge polarization of the QD induced by the absorption of individual photons is detected by a metallic single electron transistor. We have investigated the operation of the QD at different rates of the external terahertz photon flux and at various temperatures. It was found that a state of the nearly isolated QD can potentially be used for counting individual terahertz photons. The operation of the detector is however strongly affected by the exact composition of the QD excitation states.

1. Introduction Recently we suggested new single-photon detectors for the terahertz range [1-3]. The detectors employ an effect of photoconductive gain in quantum dots (QD) in a regime of single-electron transistors (SET) where long lived non-equilibrium states exist. When a photo-excited electron leaves the QD the resulting electric potential changes the conductivity of the SET source-drain channel. Thus a single photoelectron can govern the current of millions of electrons, creating a huge photoconductive gain. Earlier detectors however lacked the possibility of application, which was hampered by the necessity for ultra-low temperature, high magnetic field or fine adjustment of electrostatic gates for operation. The need to tune for the highest sensitivity to the microwave radiation, and the highest amplification gain simultaneously, is another disadvantage of these detectors. To overcome these limitations we studied a system where a nearly isolated QD is capacitively coupled to an external SET electrometer, see Figure 1. The SET works as a non-invasive probe of the QD charge state. It records events of the photon-assisted electron tunneling from the QD to the surrounding electron reservoir, thus enabling single photon counting. The photo-assisted excitation is mediated by plasma oscillations in the QD with resonance frequency determined by the shape of the confining potential and the size of the QD [4].

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Figure 1. Scanning electron micrograph of the device and equivalent circuit diagram are shown at the left. Operation of the QD as a terahertz detector is shown on the right: photon-assisted excitation of an electron from the QD is sensed by the metallic SET. The SET conductance has a telegraph-type switch every time when an electron escapes the QD. The conductance switches back when the charge excitation of the QD is relaxed.

In this paper we examine the operation of the detector depending on the charge states of the QD, its coupling to charge reservoirs, the gate potentials and illumination with terahertz radiation. 2. Model and experiment In a classical picture, the QD-SET system is modeled as a network of tunnel resistors and capacitors, see Figure 1. The number of electrons on the QD and SET islands are ND and NS. The islands are coupled to each other by a capacitance CC and the cross gates with voltages VC1 and VC2 through capacitances CCD1 , CCS1 , CCD2 and CCS 2 . The gate voltages VC1 and VC2 govern the QD tunnel barriers: increasing the negative voltage bias on the gates reduces the QD tunnel capacitances and increases the tunnel resistances. The offset charge on the SET is expressed as:

(

)

QSET = VC1CCS1 + VC 2CCS 2 +

CC VC1CCD1 + VC 2CCD2 − eN D CΣ

(

)

(1)

where CΣ = CC + CCD1 + CCD2 + C1 + C2 . When VC1 and VC2 changes, the SET shows Coulomb blockade peaks in conductance (CBP) each time when QSET − N S = 1 . The charge state of the QD, which is hidden in the second term of the equation, can be worked out from analysis of the CBPs. The effect of the QD however is only seen when the QD is weakly coupled to the 2DEG reservoirs and CΣ is limited. An intensity graph of the CBPs in coordinates VC1 - VC2 is shown in Figure 2a. The graph is a compilation of the successive scans of VC2 from -1.8V to -2.3V at constant VC1. Two regions can be identified in the graph. In the region I the QD is strongly coupled to the 2DEG, the QD charge is not quantized, and the capacitances C1 and C2 are so large that the second term of the equation is negligible. Close to the I- II boundary the growing

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Figure 2. Intensity plot of the SET conductance against biases on cross gates: (a) without external radiation, and (b) under external radiation. Thin white lines indicate pinch-off boundary and the “leaking” area boundary. An excited state of the QD IIph shows up under external radiation in the map (b).

potential barriers isolate the QD from the 2DEG, so that the they are coupled by the tunneling junctions. Correspondingly capacitances C1 and C2 decrease and the second term in Eqn. (1) starts to contribute to the charge QSET . This contribution is however nonlinear. The number of electrons in the QD ND decreases by one every time that VC1CCD1 + VC 2CCD2 − eN D reaches 0.5. This results in short-period oscillations superimposed on the CBPs of the SET. The overall slope of the CBP lines in Figure 2a is almost unaffected. Further biasing of the gates makes the QD to be completely isolated from the 2DEG, region IIS. In the absence of any external radiation ND is constant. The CBP lines change slope reflecting the increased contribution to QSET from the second term of Eqn. (1). The IIS region is a meta-stable region of isolated QD, where the number of electrons is constant [5]. The potential barriers and electro-chemical potential of electrons µQD increase in this region almost linearly, with the largest growth rate for the barrier next to the biasing gate and the smallest growth rate for the opposite barrier. Close to IIS-IIC boundary µQD reaches the height of the opposite barrier and electrons start to leak from the QD. The electron drip is seen as abrupt switches of the SET conductance. The position of the boundary dividing the regions IIC and IIS is dependent on the number of electrons trapped in the QD when the I-IIS boundary is crossed. It thus depends on the history of the gate voltages. The slope of the CBP lines changes again at the IIS-IIC boundary reflecting leak of the electrons. The first two regions in Figure 2a are important for finding the system parameters (capacitances to the gates, shunt and total capacitances of the QD and the SET). Analysis of the third state (the 'leaky' area shape) provides a direct experimental probe for potential barrier formation in the QD. Assuming the number of non-equilibrium electrons trapped in the QD is linearly proportional to the electro-chemical potential one can draw the barrier height as a function of the corresponding gate bias [6]. We modeled the intensity graph of Figure 2 using equation (1) and found capacitances of the system: CCS1 = 1.1 aF, CCS 2 = 1.2 aF, CCD1 = 50 aF, CCD2 = 30 aF, CC = 15 aF, C1 + C 2 = 12 aF.

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212 212 To study the operation of the system under terahertz radiation, we used filtered blackbody radiation generated by passing a current through a 5 kΩ resistor. The resistor is sealed in a metal box at the 1 K stage of the refrigerator, 50 cm away from the device. A black polyethylene and Si crystal filter act as low-pass filters, which cut out the highfrequency part with wave length larger than 500 cm-1. Radiation is fed to the device by a stainless steel tube waveguide. When illuminated by the terahertz radiation, the system develops a new region denoted as IIph in Figure 2b. This photo-sensitive region starts at the I-II boundary and spans for ~100 mV. The photo-response vanishes at the gate voltages, where the difference between the height of the barriers forming the QD and µQD exceeds the photon energy hν . The CBPs in IIph have a unique pattern, which reflects the excitations of the QD.

Figure 3. (a) Time traces of SET conductance at different power applied to emitter; (b) Number of counts per second at different power applied to emitter at constant VC2 =-2.34 V.

The time traces of the SET conductance at the fixed gate voltage are shown in Figure 3a. Each telegraph type switch corresponds to an excitation of the electron from the QD. Two types of excitations can be envisaged: those caused by thermal noise and caused by absorption of the terahertz photons. The thermal excitations are present close to the I-IIS boundary even in the absence of external radiation. The rate of switching can be as large as 10 s-1, but they quickly vanish when moving away from the I-IIS boundary, as the difference between µQD and the height of the potential barriers exceeds the thermal energy kBT. This happens within ~20 mV from the boundary. The switches of the second type, caused by the photon absorption, have a comparable maximum rate but they extend for a longer range of the gate voltages, ~ 100 mV. Consequently one has a high photoinduced/dark switching ratio, more than 50, in a large region of the gate voltages. It is important to note that the photo-response expands further to the more negative gate voltages when a stronger photon flux is applied: see Figure 3b. This can be ascribed to multi-electron excitation of the QD at high photon flux. The maximum counting rate however remains the same, limited by the time resolution of the measurement set up, which is ~ 10 ms in this experiment. It is worth noting also that the shape of the photocounting curve correlates with the excitation states of the QD of Figure 2b. The photon

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213 213 counting rate has peaks and valleys reflecting the excitation probability of the QD and the position of the CBPs of the SET. The photo-sensitive operation of the device extends to higher temperatures, see Figure 4. As one can expect, the dark switches are more frequent at higher temperatures because of the increased population of the high-energy electrons. The maximum rate of dark-switching increases from ~ 10 counts/sec at 0.3K to almost 20 counts/sec at 0.7K. But away from the I-IIS boundary the dark rate is well below ~5 counts/sec. In this region the system has photo-induced/dark switching ratio better than 20. The photosensitive region expands as the temperature increases. It covers almost twice as large a range of the gate voltages at T=0.7K as it does at 0.3K. The effect can be attributed to the increased population of thermally excited electrons which need less extra energy to escape the QD.

Figure 4. Counting of dark switches (open circles) and photo-stimulated switches (filled circles) at different temperatures. Photo-sensitive region expands at higher temperatures.

3. Conclusions In summary, we studied a quantum dot in application to single-photon terahertz detection. It was found that a state of the nearly isolated QD can potentially be used for counting individual terahertz photons. The photo-sensitive operation of the detector is affected by the temperature, composition of the excited states in the QD, and amplitude of the external photon flux. At the optimum operation point, the ratio of photon induced to dark counts can exceed 50. We found that the photo-response expands to a larger region of gate voltages as the photon flux is increased. This can be used for development of the detector with high dynamical range.

Acknowledgments This work is supported by the Solution Oriented Research for Science and Technology (SORST) from Japan Science and Technology (JST).

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References 1. S. Komiyama, O. Astafiev, V. Antonov, T. Kutsuwa, and H. Hirai, Nature 403, 405 (2000). 2. H. Hashiba, V. Antonov, L. Kulik, S. Komiyama, and C. Stanley, Appl. Phys. Lett. 85, 6036 (2004). 3. H. Hashiba, V. Antonov, L. Kulik, A. Tzalenchuk, P. Kleinschmid, S. Giblin, and S. Komiyama, Phys. Rev. B 73, 081310(R) (2006). 4. O. Astafiev, S. Komiyama, T. Kutsuwa, V. Antonov, Y. Kawaguchi, and K. Hirakawa, Appl. Phys. Lett. 80, 4250 (2002). 5. J.C. Chen, Zhenghua An, T. Ueda, S. Komiyama, K. Hirakawa, V. Antonov, submitted to Phys. Rev. B (2006). 6. J. Martorell, D. W. L. Sprung, P. Machado, and C.G. Smith, Phys. Rev. B 63, 045325 (2001).

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ELECTRON TRANSPORT THROUGH LATERALLY COUPLED DOUBLE QUANTUM DOTS

T. KUBO1∗, Y. TOKURA1,2 , T. HATANO1, AND S. TARUCHA1,3 1

ICORP, JST, 3-1, Atsugi, Kanagawa 243-0198, Japan NTT BRL, NTT Corporation, 3-1, Atsugi, Kanagawa 243-0198, Japan 3 Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 2

We study electron transport through laterally coupled double quantum dots. We introduce the coupling parameter α, which characterizes the strength of the coupling via the reservoirs between quantum dots. We ﬁnd that the visibility decreases as |α| increases. We also examine the pseudo-spin Kondo eﬀect in the laterally coupled double quantum dots. The diﬀerential conductance shows the additional structure in the spilt peak by the inter-dot tunnel coupling due to the asymmetry of the spectral density of states.

1. Introduction It is already well known that in interference experiments with an Aharonov-Bohm (AB) ring containing a quantum dot (QD), the current is modulated as a function of magnetic ﬂux threading through the ring. This reﬂects that phase coherence is maintained in the tunneling through the QD1 . Recently AB oscillations have been observed for a tunneling current passing through a laterally coupled double quantum dot (DQD) system by Holleitner et al.2 . Such oscillations has also been observed by two of the present authors3 for laterally coupled DQD but in the vertical conﬁguration. In this paper, we consider electron transport in laterally coupled DQDs. We introduce the coupling parameter α, which characterizes the strength of the coupling via the reservoirs between two QDs4,5 . A system with a maximum coupling |α| = 1 has already been widely studied theoretically6 . In actual situations, however, |α| = 1 is just a special case, |α| < 1 will be more relevant. For |α| = 1, only a single mode contributes to the transport5 , whereas two conduction modes exist for |α| < 1. Note that the situation with α = 0 has also been explored in the context of the pseudospin Kondo problem7 . For 0 < |α| < 1, we examine the pseudo-spin dependent linewidth functions which producing the exotic pseudo-spin Kondo eﬀect.

∗ e-mail:

[email protected]

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2. Model We consider a DQD coupled to two reservoirs as shown in Fig. 1. We assume only a single energy level in each QD and ignore the spin degree of freedom. The two reservoirs of Fermi seas of electrons are described by νk cνk † cνk . (1) HR = ν∈{U,L} k

Here νk is the electron energy with wave number k in the reservoir ν and the operator cνk (cνk † ) annihilates (creates) an electron in the reservoirs. The isolated DQDs are described by HDQD =

2

0 di † di + tc d1 † d2 + h.c. + Vinter n1 n2 .

(2)

i=1

Here 0 is the single-particle energy level of two QDs and di (di † ) annihilates (creates) an electron in the ith QD. The second term represents direct tunneling between two QDs. The energy levels of the QDs split to form symmetric and anti-symmetric states because of the inter-dot tunnel coupling. Their energy levels are s = 0 + tc and a = 0 − tc , where we chose the gauge such that tc is real and negative. The third term describes the inter-dot Coulomb interaction and ni = di † di is the number operator for the ith QD. The tunneling between the reservoirs and QDs is modeled by φ φ φ φ tUk ei 4 cUk † d1 + tUk e−i 4 cUk † d2 + tLk ei 4 cLk † d2 + tLk e−i 4 cLk † d1 + h.c. HT = k

=

2 (i) tνk cνk † di + h.c. ,

ν∈{U,L} k

(3)

i=1

φ

where tνk is real. The factors e±i 4 with φ = 2πΦ/Φ0 indicate the phase eﬀect due to the magnetic ﬂux Φ, where Φ0 = h/e is the magnetic ﬂux quantum. 3. AB oscillations 3.1. Formulation In this section, we neglect the inter-dot Coulomb interaction, namely Vinter = 0. The tunneling current can be written as8 e d [fU () − fL ()] Tr Gr ()ΓU ()Ga ()ΓL () . I= (4) h Here the boldface notations indicate 2 × 2 matrices, where Gr(a) () is the retarded (advanced) Green’s function of a DQD and Γν is the linewidth function deﬁned by ∗ (j) tνk (i) tνk δ( − νk ). (5) Γνij ≡ 2π k

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In this paper, we restrict our discussion to a symmetrically coupled QD system, (1) (2) |tνk | = |tνk | = |tk |. We choose the Fermi level of the reservoirs as the origin of energy. Then, in terms of a basis of localized states, the linewidth functions at the Fermi level are

−i φ iφ 2 2 γ αγe γ αγe , ΓL (0) = , (6) ΓU (0) = φ φ αγei 2 γ αγe−i 2 γ where γ = 2πρF |tk |2 , where ρF is the density of states (DOS) at the Fermi level of the reservoirs and · · · represents the average on the Fermi surface. α characterizes the strength of the coupling via the reservoirs between two QDs and its expression is given as a function of the propagation length s12 in Refs. [4] and [5]. In this 2 section, we discuss the linear conductance at zero temperature: G = eh T (0), where T (0) is the transmission probability at the Fermi level. The expression of T (0) is given in Ref. [5]. 3.2. Zero coupling First we discuss the situation of tc = 0. Figure 2(a) shows the ﬂux dependence of the linear conductance for various α values. When α = 1, the linear conductance shows AB oscillations with the visibility of 1 and the period of 2π because of the interference between the probability amplitude associated with one path through QD 1 and that associated with another path through QD 2. The visibility decreases as |α| decreases and ﬁnally becomes zero at α = 0 since the linear conductance is independent of the ﬂux. 3.3. Finite coupling Here we consider the ﬁnite coupling regime, in particular, |0 | > |tc | = 0. The interference is completely destructive for |α| = 1. The AB oscillation period becomes 4π since the linear conductance has two kinds of peak structures as shown in Fig.2(b). The peak heights of the conductance via the symmetric state are higher than those via the anti-symmetric state since the energy of the latter is further from Fermi energy than that of the former. The visibility decreases as |α| decreases. 4. Pseudo-spin Kondo eﬀect In this section, we study the eﬀect of inter-dot Coulomb interaction. In particular, we assume that Vinter → ∞, which prevents double occupancy of a DQD, and study the pseudo-spin Kondo eﬀect by mapping the occupancy of the symmetric (antisymmetric) state in the DQDs to the pseudo-spin up (down) state. We set the AB ﬂux to be zero, namely Φ = 0 and consider only |α| = 1a . The Hamiltonian in the a The pseudo-spin Kondo eﬀect does not appear when |α| = 1 since only a single mode contributes to the transport.

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symmetric and anti-symmetric modes is given by νk cνk † cνk + n dn † dn + Vinter ns na H= ν∈{U,L} k

+

n∈{s,a}

(n) tνk cνk † dn + h.c. ,

(7)

ν∈{U,L} k n∈{s,a}

where dn (dn † ) is an annihilation (creation) operator of the symmetric and antioperator of the symmetric and symmetric states, and nn = dn † dn is the number (s) (1) (2) √ (a) (1) (2) √ anti-symmetric states, and tνk = (tνk + tνk )/ 2 and tνk = (tνk − tνk )/ 2. Using the coupling parameter α, the linewidth functions in this basis are

1+α 0 U L Γ =Γ =γ . (8) 0 1−α To study this model, we adopt the slave-boson (SB) mean-ﬁeld theory (MFT)9 . In the SB representation, operators of a DQD are replaced by dn = b† fn , where the SB operator b† creates an empty state and the pseudo-fermion operator fn annihilates the singly occupied state with the pseudo-spin n. Moreover, the following constraint has to be imposed on these operators: b† b + fn † fn = 1. (9) n∈{s,a}

In the MFT, the SB operator is replaced by a constant c-number ¯b. From the constraint and the equation of motion of SB operator, the self-consistent equations are described as d d < Gn () + ¯b2 − 1 = 0, G< ()( − ˜n ) + 2λ¯b2 = 0, (10) 2πi 2πi n n∈{s,a}

where λ is a Lagrange multiplier introduced by taking into account the constraint (9). Here ˜n = n + λ and the non-equilibrium Green’s function of the pseudo-spin state n in the DQD is ˜ n [fU () + fL ()] iΓ G< n () = −˜ n 2 ˜ n )2 + (Γ

(11)

L (−μν )/kB T ˜ n = ¯b2 (ΓU ] is the Fermi-Dirac distriwhere Γ n + Γn ) and fν () = 1/[1 + e bution function with μν being the chemical potential of the reservoir ν. From the solutions of Eqs. (10), we can calculate the spectral DOS 1 1 Im {Grn ()} , Grn () = −˜ , (12) ρ() = − i˜ n π + 2 Γn n∈{s,a}

using the retarded Green’s function of the pseudo-spin state n in the DQD. We solve numerically Eqs. (10) for D/γ = 100, where D is the width of conduction band in the reservoirs. When 0 /γ = −3 and α = 0.5, the spectral DOS is shown in Fig. 3 (a). The DOS has two peaks since tc = 0, which are

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Figure 1. Aharonov-Bohm interferometer containing a laterally coupled DQD. Both magnetic ﬂuxes threading through the upper and lower subcircuits are Φ/2, and cause the AB eﬀect. s12 is the propagation length of electrons in the reservoirs.

Figure 2. AB oscillations of the linear conductance. (a)0 /γ = 0.5 and tc = 0. (b)0 /γ = 2 and tc /γ = −1.

Figure 3. (a)The Kondo resonant peaks in the spectral DOS for α = 0.5. (b)The diﬀerential conductance as a function of applied source-drain bias voltage VSD for α = 0.5.

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asymmetric for 0 < |α| < 1, reﬂecting that the linewidth functions depend on the pseudo-spin. The full-width-at-half-maximums of two Kondo resonant peaks shown in Fig. 3 (a) are proportional to 1 + α and 1 − α. In Fig. 3 (b), we plot the diﬀerential conductance dI/dVSD when 0 /γ = −3 and α = 0.5. The diﬀerential conductance at zero temperature is given by dI/dVSD = 12 ρ( eV2SD ) + ρ(− eV2SD ) in terms of DOS. The zero-bias Kondo peak splits due to the inter-dot tunnel coupling. For the larger inter-dot tunnel coupling, there exists the additional structure in the split peaks as shown in Fig. 3 (b). The subpeak at the smaller |VSD | and the subpeak at the larger |VSD | in the major split structures reﬂect the sharp peak at the low energy and the broad peak at the high energy in the asymmetric spectral DOS, respectively. 5. Conclusions We investigated the electron transport through a DQD with a coupling parameter α. In the non-interacting transport, we found that the visibility of the AB oscillations decreases as |α| decreases and reaches zero when α = 0. In the strong inter-dot interaction limit and for 0 < |α| < 1, we examined the pseudo-spin Kondo eﬀect. The DOS is asymmetric for ﬁnite inter-dot tunnel coupling since the linewidth functions depend on the pseudo-spin. As a result, we found that the diﬀerential conductance curve shows the additional structure in the split peaks due to the asymmetry of DOS. References 1. A. Yacoby et al., Phys. Rev. Lett. 74, 4047 (1995); R. Schuster et al., Nature 385, 417 (1997); Y. Ji et al., Science 290, 779 (2000). 2. A. W. Holleitner et al., Phys. Rev. Lett. 87, 256806 (2002). 3. T. Hatano et al., Physica E 22, 534 (2004). 4. T. V. Shahbazyan and M. E. Raikh, Phys. Rev. B 49, 17123 (1994). 5. T. Kubo et al., cond-mat/0602539. 6. B. Kubala and J. K¨ onig, Phys. Rev. B 65, 245301 (2002); K. Kang and S. Y. Cho, J. Phys. Condens. Matter 16, 117 (2004); Z.-M. Bai et al., J. Phys. Condens. Matter 16, 2053 (2004). 7. T. Pohjola et al., Europhys. Lett. 54, 241 (2001); U. Wilhelm et al., Physica E14, 385 (2002). 8. Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992). 9. P. Coleman, Phys. Rev. B 35, 3035 (1984); N. Read and D. M. Newns, J. Phys. C 16, L1055 (1983); N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987).

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DEPHASING IN KONDO SYSTEMS: COMPARISON BETWEEN THEORY AND EXPERIMENT

´ FRANC ¸ OIS MALLET, FELICIEN SCHOPFER† , JERRY ERICSSON, ∗ ¨ LAURENT SAMINADAYAR AND CHRISTOPHER BAUERLE Institut N´eel and Universit´e Joseph Fourier, B.P. 166, 38042 Grenoble, France DOMINIQUE MAILLY Laboratoire de Photonique et Nanostructures, route de Nozay, 91460 Marcoussis, France G. ESKA Physikalisches Institut, Universit¨ at Bayreuth, Universit¨ atsstrasse 30, 95440 Bayreuth, Germany

We report on the phase coherence time τφ in quasi one-dimensional Au/Fe Kondo wires and compare the temperature dependence of τφ with a recent theory of inelastic scattering from magnetic impurities. A very good agreement is obtained for temperatures down to 0.2 TK . In particular, we show that the inverse of the phase coherence time varies basically linearly with temperature below TK over almost one decade in temperature.

1. Introduction Recently, the presence of a small amount of magnetic impurities have been invoked to explain the often observed saturation of the phase coherence time1 in metallic quantum wires at low temperatures2 . In order to verify this scenario, a theory is necessary to account for the inﬂuence of magnetic impurities on the phase coherence time. At present, no exact solution is available for the phase coherence time in the presence of Kondo impurities. Only a high temperature expansion, the NagaokaSuhl (NS) expression3 , was able to describe the experimental data at temperatures T TK . In the opposite limit (T TK ), the Fermi liquid theory predicts a T 2 dependence of the inelastic scattering rate4 . Very recently, Zar´ and and coworkers5 have been able to obtain an exact solution for the inelastic scattering time in Kondo metals using NRG techniques6 . Here we compare this new theory with existing experimental data and show that the NRG calculation gives indeed a very good description of the temperature dependence of the phase coherence time τφ in metallic quantum wires containing magnetic impurities7,8 . ∗ This

work is supported by ACI grants # 02 2 0222 and # NN/02 2 0112, and the European Commission FP6 NMP-3 project 505457-1 Ultra-1D. Mail to: [email protected]

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2. Results and discussion Samplesa are fabricated by standard nano-fabrication using lift-oﬀ technique. A scanning tunneling microscope image of the sample is shown in the inset of ﬁgure 2a. The electrical as well as geometrical parameters are listed in Table 1. Table 1. Sample characteristics: w, t, l, R, correspond to the width, thickness, length and electrical resistance, respectively. D is the diﬀusion coeﬃcient and cimp is the impurity concentration extracted from the NRG ﬁts. Sample

w(nm)

t(nm)

L(μm)

R(Ω)

D(cm2 /s)

cimp (ppm)

AuF e1

180

40

155

393

200

3.3

AuF e2

150

45

450

4662

56

45

Au1

120

50

450

1218

241

< 0.015

Ag1

120

50

485

2997

105

< 0.015

The phase coherence time of sample AuFe1 and AuFe2 as extracted from from standard weak localization are shown in ﬁgure 1. Both samples display a distinct plateau at a temperature above T ≥ 0.3 K. This plateau originates from spin ﬂip scattering due to the Kondo eﬀect. Decreasing the temperature, the magnetic impurity spin is screened and as a result, the phase coherence time increases again. The phase coherence time τφ can be written in the following way9 , 1 1 2 1 = + + τφ τe−e τe−ph τm

(1)

where 1/τe−e = a T 2/3 is the contribution due to the electron-electron interaction10,11 , 1/τe−ph = b T 3 to the electron-phonon interaction, while 1/τm correspond to the contribution due to the scattering oﬀ magnetic impurities. To account for the latter process, the common Nagaoka-Suhl expression2,3,9 was usually used, which is, however, only valid at temperatures T TK . Instead, we will use the inelastic scattering cross section as calculated by NRG for the contribution of the magnetic impurities τm , 1 σ(ω)inel 1 = =A cimp τm τN RG σ0

(2)

where σ(ω)inel is the inelastic cross section at ﬁnite energy ω, σ0 = 4π/kf2 the elastic cross section at zero temperature and A a constant we ﬁx in order that the NS and NRG expressions coincide at T TK . The dotted line in the inset of ﬁgure 1, denoted as AAK, corresponds the assumption that the only mechanisms for decoherence are electron-electron and electronphonon interactions10 . For the two other dotted lines, denoted as NS1 (for sample a sample Au1 (Ag1) has been evaporated from a 5N5 gold source (6N silver source). We acknowledge R.A. Webb for the gold evaporation and the Quantronics group for the silver evaporation.

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Figure 1. Phase coherence rate as a function of temperature for AuFe2 (sample A of Ref. 7) on a linear scale. The dotted line (denoted NS2) corresponds to the Nagaoka-Suhl expression and the solid line to the NRG calculation. The inset shows τφ versus T on a log-log scale for both samples AuFe1 (Ref. 12) and AuFe2 (Ref.7).

AuFe1) and NS2 (for sample AuFe2) we also include the contribution due to magnetic impurities via the NS expression for S=1/2. We see that the NS expression describes relatively well the observed temperature dependence of τφ at temperatures above TK but fails to describe the desaturation of τφ due to the screening of the magnetic impurities at temperatures below TK . This is of course not surprising as the NS expression is based on a perturbative calculation in (T /TK ) and breaks down when approaching TK . Instead, taking the inelastic scattering rate obtained by NRG as shown by the solid lines denoted NRG1 and NRG2 for sample AuFe1 and AuFe2, respectively, we obtain a very good agreement with the experimental data in basically the entire temperature range. From the NRG ﬁtting, we extract an impurity concentration of approximately 3.3 ppm (45 ppm) and TK = 0.4 K ±0.05 K (TK = 0.9 K ±0.05 K) for sample AuFe1 (for sample AuFe2)13,14 , in good agreement with TK values for Au/Fe found in the literature15 . At the lowest temperatures we observe deviations from the NRG theory. A probable explanation is that the NRG calculations have been done in the one channel limit for a S=1/2 impurity spin. Indeed, it is quite surprising that the NRG model for S=1/2 ﬁts this well the experimental data for the Kondo system Au/Fe as the real spin of iron is probably very diﬀerent from S=1/2. 3. Phase coherence time in very clean metallic quantum wires We have shown that the NRG theory describes satisfactorily the temperature dependence of τφ in metallic quantum wires containing magnetic impurities. In the

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following we will reexamine the temperature dependence of τφ in extremely pure gold and silver wires to see whether the deviations from the standard AAK prediction often observed experimentally can be explained by the presence of a tiny amount of magnetic impurities. The aim is not to give any deﬁnite conclusion, but simply to state in a very objective way the experimental facts which argue for or against the presence of magnetic impurities. In ﬁgure 2 we present measurements of the phase coherence time of two extremely pure metallic quantum wires (Au 1 and Ag 1) down to temperatures of 10 mK b . In order to ensure that the electrons are cooled to these low temperatures we measure the Altshuler-Aronov correction to the resistivity. For both samples we √ observe the theoretically expected 1/ T temperature dependence down to the lowest temperatures 8 . The phase coherence length for sample Au1 (Ag1) is larger than 20 μ m (15 μ m) at 10 mK which is an indication of their high purity.

Figure 2. Left: phase coherence rate as a function of temperature for sample Au1 (◦). The dotted line corresponds to the AAK prediction, the solid lines correspond to the NRG calculation assuming TK = 40 mK, TK = 10 mK, and TK = 5 mK and TK = 2 mK. The inset shows a SEM picture of a typical quantum wire. Right: phase coherence rate as a function of temperature for sample Ag1 (◦). The dotted line corresponds to the AAK prediction, the solid lines correspond to the NRG calculation assuming TK = 5 mK, TK = 1 mK, and TK = 0.5 mK and TK = 0.1 mK.

For sample Au1, we ﬁt the experimental data with the AAK expression such that an almost perfect agreement is obtained at high temperatures (T > 300 mK), as shown by the dotted line. At temperatures below 300 mK our data deviate substantially from the AAK prediction. To see whether these deviations can be explained by the presence of a very small amount of magnetic impurities, we simulate the temperature dependence of τφ for the presence of a small amount of magnetic impurities using the NRG calculations. The (a), (b) (c) and (d) solid lines correspond to a b the

temperature dependence of our samples is almost identical to the ones of Ref. 2, with the only diﬀerence that our data extend to lower temperature.

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simulation assuming TK = 40 mK, TK = 10 mK, TK = 5 mK and TK = 2 mK, with an impurity concentration of cimp = 0.008 ppm, cimp = 0.013 ppm, cimp = 0.015 ppm and cimp = 0.024 ppm, respectively. It is clear from our simulations that only magnetic impurities with a Kondo temperature between 5mK to 10mK and with a concentration smaller than 0.015 ppm describe satisfactorily the experimental data. A possible magnetic impurity with a Kondo temperature in this temperature range is Mn (TK 3mK)16 . We note however, that it is curious that the observed temperature dependence of τφ for the Au sample can only be described satisfactorily by assuming the presence of one speciﬁc magnetic impurity with a Kondo temperature below the measuring temperature T ≤ 10mK, whereas it is known that the dominant magnetic impurity in gold is iron. If we assume an additional iron concentration of the same order, the temperature dependence of τφ does not satisfactorily describe the experimental data8 . The temperature dependence of τφ of sample Ag1 is in very good agreement with the AAK prediction down to temperatures of 30mK. At lower temperatures, we see again deviations from the AAK picture. To account for these deviations we have to assume magnetic impurities with an impurity concentration less than 0.01 ppm and a Kondo temperature below 1mK. The question is whether these numbers are reasonable. To our knowledge the lowest known Kondo temperature for the host metal Ag is obtained with Mn impurities and is on the order of 20-40 mK 15 . This, however, seems to be in contradiction with our analysis. It is clear from our analysis that assuming magnetic impurities with a Kondo temperature below the measuring temperature leads to an almost temperature independent scattering rate for T ≥ TK . Any experimentally observed saturation of τφ can therefore always be assigned to a tiny amount of magnetic impurities with an unmeasurably low Kondo temperature, but does not rule out the predictions of Ref. 17. In conclusion we have shown that the NRG calculation of the inelastic scattering rate describe very well the experimentally observed temperature dependence of τφ caused by the presence of magnetic impurities. Below TK the inverse of the phase coherence time varies basically linearly with temperature over almost one decade in temperature. Using the NRG results to analyse the temperature dependence of very clean metallic quantum wires shows that is is not clear at present that magnetic impurities are responsible for the observed deviation of τφ from the AAK predictions at very low temperatures. note added: more recent measurement of the phase coherence time in AgFe Kondo wires 18 come to the conclusion that the screening at low temperatures is almost perfect, however, such a screening may involve more than one channel and a spin larger than 1/2.

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Acknowledgments We acknowledge helpful discussions with P. Simon, S. Kettemann, G. Zar´ and, A. Rosch, G. Montambaux, C. Texier, H. Bouchiat and L.P. L´evy. In addition, we would like to thank G. Zar´ and for providing us with the numerical data of the NRG calculation of σinel . We are also indebted to R.A. Webb and Salcay Quantronics Group for the gold and silver evaporation of sample Au1 and sample Ag1, respectively. References †. 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13. 14.

15. 16. 17. 18.

Present address: Laboratoire National de M´etrologie et d’Essais, 29 avenue Roger Hennequin, 78197 Trappes, France. P. Mohanty, E.M.Q Jariwala, and R.A. Webb, Phys. Rev. Lett. 78, 3366 (1997). F. Pierre, A.B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N.O. Birge, Phys. Rev. B 68, 085413 (2003). for a review see G. Gr¨ uner and A. Zawadowski, Rep. Prog. Phys. 37, 1497 (1974). P. Nozi`eres, J. Low Temp. Phys. 17, 31 (1974). G. Zar´ and, L. Borda, J.v. Delft, and N. Andrei, Phys. Rev. Lett. 93, 107204 (2004). (2004). more recently, after submission of this manuscript, a NRG calculation of τφ has been achieved: T. Micklitz, T. A. Costi, A. Altland, and A. Rosch, Phys. Rev. Lett. 96, 226601 (2006). F. Schopfer, C. B¨ auerle, W. Rabaud, and L. Saminadayar, Phys. Rev. Lett, 90, 056801 (2003). C. B¨ auerle, F. Mallet, F. Schopfer, D. Mailly, G.Eska, L. Saminadayar, Phys. Rev. Lett 95, 266805 (2005). C. Van Haesendonck, J. Vranken, and Y. Bruynseraede, Phys. Rev. Lett. 58 1968 (1987). B.L. Altshuler et al. J. Phys. C 15, 7367 (1982); E. Akkermans and G. Montambaux, in Physique m´esoscopique des ´electrons et des photons, EDP Sciences (2004); Mesoscopic physics of electrons and photons, Cambridge University Press, (2007). P. Mohanty and R.A. Webb, Phys. Rev. Lett. 84, 4481 (2000). R.P. Peters, G. Bergmann and R.M. Mueller, Phys. Rev. Lett. 58,1964 (1987) The fact that the extracted Kondo temperatures are diﬀerent for the two samples is somewhat surprising. However, these results are consistent with the maxima observed previously in the spin ﬂip rate7,13 . C. Rizutto, Rep. Prog. Phys. 37, 147-229 (1974). G. Eska et al., J. Low Temp. Phys 28, 551 (1977). D.S. Golubev and A.D. Zaikin, Phys. Rev. Lett. 81, 1074 (1998). ¨ ubayir, D. Reuter, A. Melnikov, A.D. Wieck, F. Mallet, J. Ericsson, D. Mailly, S. Unl¨ T. Licklitz, A. Rosch, T.A. Costi, L. Saminadayar and C. B¨ auerle, Phys. Rev. Lett. 97, 226804 (2006).

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KONDO EFFECT IN QUANTUM DOTS COUPLED WITH NONCOLLINEAR FERROMAGNETIC LEADS

D. MATSUBAYASHI Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan E-mail: [email protected] M. ETO Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected]

We study the Kondo eﬀect in a quantum dot coupled to two noncollinear ferromagnetic leads. First, we study the spin splitting δ = ↓ − ↑ in the quantum dot by the tunnel coupling to the ferromagnetic leads, using the Poor man’s scaling method. The spin splitting takes place in an intermediate direction between the polarization directions in the two leads. δ ∝ cos(θ/2), where θ is the angle between the polarization directions; spin splitting is maximal in the parallel alignment of two ferromagnets (θ = 0) and disappears in the antiparallel alignment (θ = π). Second, we calculate the Kondo temperature TK . The scaling calculation yields an analytical expression of TK as a function of θ and polarization ratio p in the ferromagnets, TK (θ, p), when δ TK . When δ is relevant, we evaluate TK (δ, θ, p) using the slave-boson mean ﬁeld method. The Kondo resonance is split into two by ﬁnite δ, which results in the suppression of Kondo eﬀect.

1. Introduction The Kondo eﬀect in quantum dots (QDs) coupled to ferromagnetic leads has been studied theoretically with motives elucidating how the ferromagnetism of leads inﬂuences the ﬂuctuation of localized spins. One of the theoretical predictions is the spin splitting δ = ↓ − ↑ in the QD by the tunnel coupling to the ferromagnets, which weakens the Kondo eﬀect [1-5]. This has been demonstrated in experiments as the suppression of zero-bias anomaly of the diﬀerential conductance [6, 7]. However, previous theoretical works have been limited to the collinear cases of two ferromagnets, parallel (P) or antiparallel (AP) alignments. In this article, we extend the discussion based on the scaling approach in Ref. [1] to an arbitrary alignment of ferromagnetic leads and evaluate the spin splitting δ and Kondo temperature. We also consider the dependence of the Kondo temperature on δ using the slave-boson mean ﬁeld approximation.

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Figure 1. A quantum dot connected to ferromagnetic leads, L and R. The angle of polarization direction in lead L(R) is θ/2 (−θ/2) from z axis in the z-x plane.

2. Model We consider a single level QD coupled to noncollinear ferromagnetic leads as shown in Fig.1. The polarization direction l(r) in the lead L(R) is tilted by θ/2 (−θ/2) from the z axis in the z-x plane. This system is represented by the Anderson model, H= kσ c†Lk,lσ cLk,lσ + kσ c†Rk,rσ cRk,rσ + 0 d†zσ dzσ + U d†z↑ dz↑ d†z↓ dz↓ kσ

+

σ

kσ

(VL c†Lk,lσ dlσ

+

VR c†Rk,rσ drσ

+ h.c.),

(1)

kσ

where cL(R)k,l(r)σ are fermion operators for electrons with wavenumber k and spin l(r)σ in the lead L(R), and dzσ are those with spin zσ at the QD level. We denote l(r) ↑ /l(r) ↓ for majority/minority spin in lead L(R) and z ↑ /z ↓ for spin-up/down in z direction in the QD. U is the Coulomb interaction for double occupancy in the QD. For simplicity, we assume that the two ferromagnets are identical and barriers are symmetric, VL = VR ≡ V . The density of states in the ferromagnets is constant, ρ↑ and ρ↓ for majority and minority spins, respectively (ρ↑ ≥ ρ↓ ), in the band of −D0 ≤ ω ≤ D0 . The polarization strength is given by p = (ρ↑ − ρ↓ )/(ρ↑ + ρ↓ ) with 0 ≤ p ≤ 1. From the symmetry of the system, we can consider the z axis as a well-deﬁned quantization axis in the QD. In this perspective, it is convenient to rewrite the tunnel Hamiltonian HT [the last term in Eq. (1)] as √ √ θ θ HT = 2V (cos c†kσ − sin c¯†k¯σ )dzσ + h.c. ≡ 2V a†kσ dzσ + h.c., (2) 4 4 kσ

kσ

by the unitary transformation ck↓ = ck↑ = √12 (cLk,l↑ + cRk,r↑ ) , c¯k↑ = √12 (−cLk,l↑ + cRk,r↑ ) , c¯k↓ =

dl↑ = cos θ4 dz↑ + sin θ4 dz↓ , dl↓ = − sin θ4 dz↑ + cos θ4 dz↓ ,

√1 2 √1 2

(cLk,l↓ + cRk,r↓ ) , (cLk,l↓ − cRk,r↓ ) ,

dr↑ = cos θ4 dz↑ − sin θ4 dz↓ , dr↓ = sin θ4 dz↑ + cos θ4 dz↓ .

(3)

(4)

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Note that index σ =↑ / ↓ of akσ does not any more mean majority/minority spins since akσ consists of ckσ and c¯k¯σ . The anticommutation relation {ak↑ , a†k ↓ } = 0 ensures that spins z ↑ and z ↓ in the QD couple to diﬀerent “channels” in the leads. The extension to the case of asymmetric barriers, VL = VR , is straightforward by choosing an appropriate quantization axis for the QD spin. 3. Scaling Calculations 3.1. Scaling for the Anderson Model We examine the energy region, −D0 0 −Δ < μ = 0 < Δ D0 0 + U , where μ is the Fermi energy of conduction electrons in the leads and Δ = 2π(ρ↑ + ρ↓ )|V |2 . The Coulomb interaction U is assumed to be strong enough (U → ∞) to forbid double occupancy in the QD. The number of electrons in the QD ﬂuctuates between zero and one through HT . We denote |0 for the empty state and |1zσ = d†zσ |0 for singly occupied state with spin zσ. They have the energies E0 and E1zσ , respectively. Under the instruction of the Poor man’s scaling [8], we renormalize the energies E0 and E1zσ with integrating out the high energy excitations in the conduction band, within the second order perturbation with respect to HT . The renormalized ˜1zσ − E ˜0 depends on the spin state, reﬂecting diﬀerent density energy level ˜zσ = E of states for majority and minority spins in the leads. When half of the band width D |˜ zσ |, we obtain the scaling equations for ˜zσ Γσ¯ d˜ zσ =− , d ln D π

(5)

θ where Γ↑/↓ = Δ 2 (1 ± p cos 2 ) are the level broadening for spin z ↑ /z ↓ in the QD. Note that the renormalization of ˜zσ depends on the coupling strength Γσ¯ for the opposite spin. Because Γ↑ ≥ Γ↓ , we observe the spin splitting of the energy level, δ˜ ≡ ˜z↓ − ˜z↑ , even in the absence of magnetic ﬁeld. The scaling from D0 to D1 yields

δ˜ =

Δ θ D0 , p cos ln π 2 D1

(6)

z↑ |D=D1 at which the perturbation theory breaks down. where D1 ≈ −˜ When p = 0, δ˜ becomes maximal in the P alignment (θ = 0) and zero in the AP alignment (θ = π). The spin splitting decreases with increasing θ from 0 to π. 3.2. Scaling for the Kondo Model In this subsection, we assume that the spin splitting δ˜ is negligibly small. (With ﬁnite δ˜ , we can make such a situation by tuning a magnetic ﬁeld along the z axis [1].) Therefore, ˜z↓ = ˜z↑ ≡ ˜. Now we proceed the scaling calculations to D < D1 to study the Kondo eﬀect.

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In the energy scale of D D1 , the empty state |0 is irrelevant. Since the charge ﬂuctuation is quenched, the number of electrons in the QD is ﬁxed to be nd 1 (Kondo regime). To restrict the QD states to |1z ↑ or |1z ↓ , we apply the Schrieﬀer-Wolﬀ transformation to the Anderson model with renormalized level ≡ ˜|D=D1 = −D1 and obtain the Kondo model, HKondo = H0 + Hsd , H0 = d†zσ dzσ + kσ (c†kσ ckσ + c¯†kσ c¯kσ ), (7) Hsd =

kk

σ

J+ S + a†k↓ ak ↑

kσ

+

J− S − a†k↑ ak ↓

+ Sz (Jz↑ a†k↑ ak ↑ − Jz↓ a†k↓ ak ↓ ) .

(8)

Here, we have neglected the level shift, −D1 (ρ↑ + ρ↓ ) σ Jzσ (1 + σp cos θ2 )d†zσ dzσ , and potential scattering terms, but the former can be included in the ﬁrst term in H0 . J+ = J− ≡ J± and Jσ are exchange coupling constants. The initial values are given by J± = Jz↑ = Jz↓ = 2|V |2 /|| ≡ J. Applying the Poor man’s scaling to HKondo , we obtain the scaling equations to the second order with respect to Hsd , dk˜± dk˜σ 2 = −(k˜↑ + k˜↓ )k˜± , = −2k˜± , (9) d ln D d ln D where k˜σ and k˜± are eﬀective coupling constants, k˜± = 12 (ρ↑ + ρ↓ )J˜± 1 − p2 cos2 θ2

and k˜σ = 12 (ρ↑ + ρ↓ )J˜σ 1 + σp cos θ2 . The Eqs. (9) yield the Kondo temperature

arctanh p cos θ2 1 . (10) TK (θ, p) = D1 exp − (ρ↑ + ρ↓ )J p cos θ2 Equation (10) describes the dependence of the Kondo temperature on arbitrary polarization strength (0 ≤ p ≤ 1) and relative angles (0 ≤ θ ≤ π) between the polarization directions in the ferromagnetic leads. With an increase in p cos θ2 , TK decreases because of the suppression of spin ﬂuctuation in the QD. When the leads are completely polarized (p = 1) and in the P alignment (θ = 0), the Kondo eﬀect vanishes owing to the quench of the spin ﬂuctuation. In the AP alignment (θ = π), theexpressionof TK agrees with that for non-magnetic leads (p = 0), 1 . TK = D1 exp − (ρ↑ +ρ ↓ )J

4. Mean Field Calculations When the spin splitting δ is relevant, we estimate the Kondo temperature using the slave-boson mean ﬁeld (SBMF) theory [10]. We treat δ as an arbitrary parameter in the Anderson model. Introducing the boson operator b† (b) which creates (annihilates) a hole in the QD, we rewrite the Hamiltonian as H= kσ c†Lk,lσ cLk,lσ + kσ c†Rk,rσ cRk,rσ + zσ d†zσ dzσ kσ

kσ

σ

1 +√ (V c†Lk,lσ dlσ b† + V c†Rk,rσ drσ b† + h.c.) + λ(b† b + d†zσ dzσ − 1), (11) 2 kσ σ

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θ=π (a) θ=π/2 θ=0

1

TK

0.8

θ=π (b) θ=π/2 θ=0

0.6 0.4 0.2 0 0

0.5

1 δ

1.5

2

0

0.5

1 δ

1.5

2

Figure 2. The Kondo temperature TK as a function of the spin splitting δ for three diﬀerent polarization angles θ. The polarization strength in the ferromagnets is (a) p = 0.5 and (b) 0.9. Both TK and δ are normalized by the Kondo temperature at δ = 0 and θ = 0 (AP alignment).

where z↑/↓ = ∓ δ/2. The constraint on the number of electrons or holes in the QD is considered in the last term with λ being a Lagrange multiplier. In the SBMF theory, √ the boson operator b is assumed to be constant and replaced by its average b ≡ 2r. Minimizing the expectation value of the Hamiltonian, we determine λ and r. In the Kondo regime of nd 1, we observe the spin-split ˜z↑/↓ = ∓ δ 1 ± p cos θ with the Kondo resonant levels, which are located at E 2 2 2 ˜ resonant widths Γ↑/↓ = r Γ↑/↓ . They are located below and above the Fermi level μ = 0 in an asymmetric way with respect to μ = 0 except in the AP alignment (θ = π). We deﬁne the Kondo temperature TK as the geometric mean of the ˜ ˜ resonant widths Γ↑ Γ↓ . It is given by TK (δ, θ, p) =

˜z↑ E ˜z↓ |, TK (0, θ, p)2 − |E

(12)

and

θ θ 2π|| − p cos arctanh(p cos ) . TK (0, θ, p) = D1 exp − Δ 2 2

(13)

Although TK (0, θ, p) in Eq. (13) is not exactly the same as that obtained by the scaling calculations [Eq. (10)], they agree to each other qualitatively, e.g., TK = 0 when p = 1 and θ = 0. In Fig.2, TK (δ, θ, p) is plotted as a function of δ. The Kondo temperature decreases with an increase in δ monotonically. This is because δ separates the Kondo resonant levels for spins z ↑ and z ↓ and in consequence suppresses the spin ﬂuctuation. The suppression of the spin ﬂuctuation can be also understood in terms of the spin accumulation, δnd = nz↑ − nz↓ . Using δnd , the Kondo temperature can be written as TK (δ) = TK (0) sin(π nz↑ ) sin(π nz↓ ) TK (0) cos2 ( π2 δnd ). TK (δ) vanishes when δnd = 1, or nz↑ = 1 and nz↓ = 0. In this situation, the Kondo eﬀect does not take place because spin z ↓ cannot exist in the QD.

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5. Conclusions We have theoretically investigated the Kondo eﬀect in a QD coupled to noncollinear ferromagnetic leads, using the Poor man’s scaling method and the SBMF approximation. Using the scaling method, we have shown that the spin splitting δ = ↓ − ↑ ∝ p cos θ2 in the QD arises by the spin-dependent charge ﬂuctuation. The spin splitting takes place in an intermediate direction between the polarization directions in the two ferromagnets. When δ is irrelevant, we have derived an analytical expression of the Kondo temperature TK (θ, p) by the scaling calculation. With increasing p cos θ2 , TK decreases owing to the suppression of spin ﬂuctuation in the QD. When δ is relevant, we have estimated the Kondo temperature using the SBMF theory. We have found that the spin-split Kondo resonance results in the spin accumulation in the QD and weakens the Kondo eﬀect. Acknowledgments The authors gratefully acknowledge discussions with Y. Utsumi. This work was partially supported by a Grant-in-Aid for Scientiﬁc Research in Priority Areas “Semiconductor Nanospintronics” (No. 14076216) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. References 1. J. Martinek, Y. Utsumi, H. Imamura, J. Barna´s, S. Maekawa, J. K¨ onig, and G. Sch¨ on, Phys. Rev. Lett. 91, 127203 (2003). 2. B. Dong, H. L. Cui, S. Y. Liu, and X. L. Lei, J. Phys.: Condens. Matter 15, 8435 (2003). 3. J. Martinek, M. Sindel, L. Borda, J. Barna´s, J. K¨ onig, G. Sch¨ on, and J. von Delft, Phys. Rev. Lett. 91, 247202 (2003). 4. M.-S. Choi, D. S´ anchez, and R. L´ opez, Phys. Rev. Lett. 92, 056601 (2004). 5. Y. Utsumi, J. Martinek, G. Sch¨ on, H. Imamura, S. Maekawa, Phys. Rev. B 71, 245116 (2005). 6. A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E. Grose, L. A. K. Donev, P. L. McEuen, and D. C. Ralph, Science 306, 85 (2004). 7. J. Nyg˚ ard, W. F. Koehl, N. Mason, L. DiCarlo, and C. M. Marcus, cond-mat/0410467. 8. P. W. Anderson, J. Phys. C 3, 2439 (1970). 9. F. D. M. Haldane, Phys. Rev. Lett. 40, 416 (1978). 10. P. Coleman, Phys. Rev. B 29, 3035 (1984).

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NON-CROSSING APPROXIMATION STUDY OF MULTI-ORBITAL KONDO EFFECT IN QUANTUM DOT SYSTEMS

T. KITA, R. SAKANO, T. OHASHI∗, AND S. SUGA Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan E-mail: [email protected]

We study the three-orbital Kondo eﬀect in quantum dot (QD) systems by applying the non-crossing approximation to the three-orbital Anderson impurity model. By investigating the tunneling conductance through a QD, we show that the competition between the Hund-coupling and the orbital level-splitting gives rise to characteristic behavior in transport properties. It is found that the Hund-coupling becomes more important in the three-orbital case than in the two-orbital case. We also show that the enhancement of Kondo temperature due to the singlet-triplet mechanism suggested for the two-orbital model tends to be suppressed by the existence of the third orbital.

1. Introduction Electron transport properties in nanoscale systems have been studied extensively. In particular, recent progress in nanofabrication enables us to observe the correlation eﬀect due to the orbital degrees of freedom in highly-symmetric quantum dot (QD) systems. In these systems, not only ordinary spin Kondo eﬀect but also various orbital Kondo eﬀects, such as SU(4) Kondo eﬀect in a carbon nanotube QD,1 singlet-triplet Kondo eﬀect,2 etc., have been observed. These observations of orbital Kondo eﬀects in QD systems have activated theoretical works for orbital Kondo eﬀects. Theoretical studies using the two-orbital Anderson impurity model (AIM) 3,4,5,6,7 have pointed out the importance of orbital degeneracy; (i) When one electron occupies two nearly-degenerate orbital-levels, the Kondo temperature TK gets enhanced as the system approaches to the SU(4) symmetric point where two orbitals are degenerate. (ii) When two electrons occupy two orbital-levels, the ground state of the isolated Anderson impurity from conduction electrons is the triplet (spin S = 1) for degenerate energy levels of two orbitals. When the level-splitting Δε becomes larger, the ground state changes into the singlet. As Δε increases, TK takes a maximum at the point where the energy levels of the two states are degenerate. ∗ Present

Japan.

address: Condensed-Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198,

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In contrast to the detailed investigation of the two-orbital case, the three-orbital Kondo eﬀect in QD systems has not been suﬃciently understood yet. Experimentally, the three-orbital Kondo eﬀect has been realized in the vertical QD,8 where the orbital degeneracy is well controlled by an external magnetic ﬁeld or deformation of the QD.9 Therefore, it is desirable to study transport properties via the three-orbital Kondo eﬀect by systematically changing orbital degeneracy in the three-orbital AIM. In this paper, we study the three-orbital Kondo eﬀect by applying the noncrossing approximation (NCA)10 to the three-orbital AIM. We focus on the Kondo eﬀect for integer ﬁlling; two or three electrons occupy three orbitals. By investigating the tunneling conductance through a QD, we show that the competition between the Hund-coupling and the orbital level-splitting gives rise to characteristic transport properties. It is found that the Hund-coupling becomes more important in the three-orbital case than in the two-orbital case. We also show that the enhancement of TK due to the singlet-triplet mechanism suggested for the two-orbital model tends to be suppressed by the existence of the third orbital. This paper is organized as follows. In the next section, we brieﬂy mention the model and method. In Sec. 3, we show the numerical results, and discuss the characteristic transport properties due to the three-orbital Kondo eﬀect in comparison with the two-orbital case. Brief summary is given in Sec. 4. 2. Model and Method We study the three-orbital Kondo eﬀect by exploiting the three-orbital AIM, H = Hc + Hloc + Hmix , Hc = εk c†kiσ ckiσ , kiσ

Hloc =

iσ

Hmix =

Edi d†iσ diσ + U

(1)

i

ndi↑ ndi↓ + U

Vki c†kiσ diσ + H.c. ,

(i=j)σσ

ndiσ ndjσ − J

Sdi · Sdj ,

(i=j)

ki

where Hc , Hloc , and Hmix describe a part of conduction electrons in the leads, a QD, and mixing between the leads and the QD, respectively. Here, ckiσ (diσ ) annihilates a conduction electron (localized electron in the QD) with spin σ in the orbital i, and ndiσ = d†iσ diσ . Sdi is the spin operator for a localized electron in the orbital i. Edi denotes the local level of the orbital i, U (U ) the intraorbital (interorbital) Coulomb interaction and J denotes the Hund-coupling among orbitals. To analyze our model (1), we use the NCA.10 By calculating the local density of states ρiσ (ω), we obtain the tunneling conductance G through the QD,11 e2 Γ df (ω) G= ρiσ (ω), (2) dω − dω iσ

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where Γ denotes the strength of the hybridization between conduction electrons and localized electrons in the QD, and f (ω) is the Fermi distribution function. 3. Numerical Results We address the case where three electrons occupy three orbitals in the QD (referred as the case of three electrons in three orbitals), and the case where two electrons occupy three orbitals in the QD (referred as the case of two electrons in three orbitals). We especially focus on the competition between the Hund-coupling J and the orbital level-splitting Δε. We set U = U in the following calculation and use Γ in units of the energy.

Ed1 Ed1=Ed2

Δε

Ed3 (a)

Δε/2

Ed2

Δε/2

Ed3 (b)

Ed1

Δε

Ed2=Ed3

(c)

Figure 1. Three types of the orbital splitting. (a) Ed1 = Ed2 = Ed3 + Δε, (b) Ed1 = Ed2 + Δε/2, Ed3 = Ed2 − Δε/2, (c) Ed1 = Ed2 + Δε = Ed3 + Δε. The electron ﬁlling is assumed to be unchanged due to Δε.

We ﬁrst investigate the orbital Kondo eﬀect for the case of three electrons in three orbitals. We consider three types of the orbital splitting shown in Fig. 1. In large Δε limit for each type of the orbital splitting, it is expected that (a) SU(4) Kondo eﬀect, (b) SU(2) Kondo eﬀect, and (c) SU(4) Kondo eﬀect with three electrons are realized, respectively. Let us start our discussion for the results without Hund-coupling. In Fig. 2(a), we plot the tunneling conductance G as a function of Δε for each type of the orbital splitting in Fig. 1. The conductance of (a) and (c) types results in the same behavior because of particle-hole symmetry. For each type of the orbital splitting, G monotonically decreases as Δε increases. The reduction of the orbital degeneracy due to Δε lowers TK , which yields the decrease in G. For large Δε (Δε/Γ > 0.4), G of (a) and (c) types is larger than G of (b) type. This behavior is consistent with the fact that TK in SU(4) Kondo eﬀect is higher than TK in SU(2) Kondo eﬀect, although G in both cases take the same value at zero temperature.12 We next show the results including the eﬀects of the Hund-coupling J in Fig. 2 (b). For comparison, we plot the result for the case of two electrons in two orbitals. Each conductance exhibits a maximum structure, which is due to the competition between the Hund-coupling and the level-splitting. For detailed explanation, we describe the ground state of Hloc . For the ﬁnite Hund-coupling J, the ground state is the spin S = 3/2 state (quartet state) at Δε = 0. When the orbital splitting is introduced as shown in Fig. 1(a), the ground state changes into the quartet state

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Fig.1(a),(c) Fig.1(b)

1.5 G[e2/h]

2

G[e /h]

1.5

1

1

0.5

0.5 0 0

3o3e:Fig.1(a),(c) 3o3e:Fig.1(b) 2o2e

(a) J=0 0.1

0.2

0.3 Δε/Γ

0.4

0.5

0.6

0 0

(b) J/Γ=0.5 0.1 0.2

0.3 Δε/Γ

0.4

0.5

0.6

Figure 2. The conductance G as a function of the level-splitting Δε for the case of three electrons in three orbitals (a) without Hund-coupling J = 0, (b) with Hund-coupling J/Γ = 0.5. Parameters are set as U/Γ = U /Γ = 10 and T /Γ = 0.05. The types of the orbital splitting as shown in Fig. 1 (a) and (c) (dashed line), and Fig. 1 (b) (thick solid line) are considered. For comparison, we plot the result for the case of two electrons in two orbitals for the same parameters (thin solid line).

with fourfold degeneracy of the spin and orbital degrees of freedom at large Δε, where the SU(4) Kondo eﬀect is induced, as mentioned above. On the other hand, when the orbital splitting is introduced as shown in Fig. 1(b), the ground state changes from the S = 3/2 quartet state to the doublet state with the spin ↑ and ↓ degrees of freedom, which leads to the ordinary SU(2) Kondo eﬀect. At the critical point where the ground state changes as shown in Fig. 3, the degeneracy of the ground state is enlarged, which gives rise to the enhancement of the Kondo temperature. Actually, as Δε increases, the conductance G in Fig. 2(b) take a maximum near the critical point. Note that G for the case of Fig. 1(a) is somewhat larger than that of Fig. 1(b) near the critical point, because the degeneracy for the case of Fig. 1(a) is larger than that of Fig. 1(b). The mechanism of the maximum structure of G is similar to the singlet-triplet Kondo eﬀect, which occurs in the case of two electrons in two orbitals. The diﬀerence appears at the point where TK is enhanced; In the three-orbital case, TK is enhanced at Δε ∼ 34 J, while TK is enhanced at Δε ∼ 14 J in the two-orbital case. This indicates that the eﬀect of J remains even at larger Δε in the three-orbital case, because three electrons (b)

(c)

S=3/2

quartet

SU(4)

quartet Δε = 34 J

energy

energy

energy

(a)

S=3/2

quartet

SU(2)

doublet Δε = 34 J

S=1

triplet

singlet Δε =J4

Figure 3. Schematic diagram for the change of the ground state of Hloc with ﬁnite J and Δε. (a) Orbital splitting is introduced as shown in Fig. 1 (a). (b) Orbital splitting is introduced as shown in Fig. 1 (b). (c) The case of two electrons in two orbitals.

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gain more Hund-coupling energy than two electrons. Namely, the Hund-coupling becomes more important in the three-orbital case than in the two-orbital case.

Ed1

Δε

Ed2

Ed1

Δε

Ed3

Δε

Ed2

(a)

(b)

Figure 4. (a) Level-splitting of three orbitals. Ed1 = Ed2 + Δε, Ed3 = Ed2 − Δε. (b) Levelsplitting of two orbitals. Ed1 = Ed2 + Δε. The electron ﬁlling is assumed to be unchanged due to Δε.

We now turn to the case of two electrons in three orbitals. Here, we discuss how the third orbital aﬀects the transport properties, by comparing the results to those for the case of two electrons in two orbitals. We consider the level-splitting Δε as shown in Fig. 4. In Fig. 5(a), we plot the tunneling conductance G as a function of the level-splitting Δε for the case of two electrons in three orbitals and the case of two electrons in two orbitals without Hund-coupling. As Δε increases, G in both cases decreases monotonically, and approaches the same value. At large Δε (Δε/Γ > 0.4), the orbital with the highest energy-level does not contribute to the conductance so that G results in the same value. For the ﬁnite Hund-coupling J/Γ = 0.5, we ﬁnd the noticeable diﬀerence between the two- and three-orbital cases. As shown in Fig. 5(b), for the two-orbital case, the Hund-coupling J leads to a hump structure with a maximum around Δε/Γ ∼ 0.12, which is due to the singlet-triplet Kondo eﬀect. On the other hand, for the threeorbital case, the conductance G is not enhanced but monotonically decreases, as

G[e2/h]

3o2e 2o2e

1

0 0

2

3o2e 2o2e

G[e2/h]

2

1

(a) J=0 0.1

0.2

0.3 0.4 Δε/Γ

0.5

0 0.6 0

(b) J/Γ=0.5 0.1 0.2

0.3 0.4 Δε/Γ

0.5

0.6

Figure 5. The conductance G for the case of two electrons in three orbitals (3o2e: thick solid line) and the case of two electrons in two orbitals (2o2e: thin solid line) as a function of level-splitting Δε for U/Γ = U /Γ = 10, T /Γ = 0.05. We show the results (a) without Hund-coupling (J = 0) and (b) with ﬁnite Hund-coupling J/Γ = 0.5.

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Δε increases. For small Δε, the ground state of Hloc is the triplet state, and then it changes into the singlet state. Therefore, it is expected that G gets enhanced due to the singlet-triplet Kondo eﬀect at intermediate Δε. However, for the threeorbital case without Δε, the ground state of Hloc has ninefold degeneracy, which gives very high Kondo temperature. In this case, for small Δε, the naively expected S = 1 Kondo eﬀect is not realized but the Kondo eﬀect with high TK similar to that without Δε occurs. Therefore, the enhancement of TK due to the singlet-triplet mechanism merges into decrease of TK due to the collapse of the ninefold degenerate Kondo eﬀect, which makes diﬃcult to see the singlet-triplet Kondo eﬀect. 4. Summary We have studied the three-orbital Kondo eﬀect in QD systems by exploiting the three-orbital AIM. By means of NCA, we have calculated the tunneling conductance through the QD. We have found that the Hund-coupling becomes more important in the three-orbital case than in the two-orbital case. We have also shown that the enhancement of TK due to the singlet-triplet mechanism tends to be suppressed by the existence of the third orbital. It is expected that the characteristic behavior of the tunneling conductance obtained here will be observed experimentally in the near future. Acknowledgments The authors thank S. Amaha for valuable discussions. This work was partly supported by a Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan. References 1. P. Jarillo-Herrero, J. Kong, H. S. J. van der Zant, C. Dekker, L. P. Kouwenhoven, and S. De Franceschi, Nature (London) 434, 484 (2005). 2. S. Sasaki, S. De Franceschi, J. M. Elzerman, W. G. van der Wiel, M. Eto, S. Tarucha, and L. P. Kouwenhoven, Nature (London) 405, 764 (2000). 3. M. Eto, J. Phys. Soc. Jpn. 74, 95 (2005). 4. M. Pustilnik and L. I. Glazman, Phys. Rev. B 64, 045328 (2001). 5. W. Hofstetter and H. Schoeller, Phys. Rev. Lett. 88, 016803 (2002). 6. W. Izumida, O. Sakai, and S. Tarucha, Phys. Rev. Lett. 87, 216803 (2001). 7. M. S. Choi, R. Lopez, and R. Aguado, Phys. Rev. Lett. 95, 067204 (2005). 8. S. Tarucha, D. G. Austing, S. Sasaki, Y. Tokura, W. van der Wiel, and L. P. Kouwenhoven, Appl. Phys. A 71, 367 (2000). 9. Y. Tokura, S. Sasaki, D. G. Austing, and S. Tarucha, Physica B 298, 260 (2001). 10. N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987). 11. Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992). 12. R. Sakano and N. Kawakami, Phys. Rev. B 73, 155332 (2006).

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THEORETICAL STUDY OF ELECTRONIC STATES AND SPIN OPERATION IN COUPLED QUANTUM DOTS

D. GOTO AND M. ETO Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected]

We theoretically study the electronic states and spin operation in coupled quantum dots, based on the calculations of many-body wavefunctions. We adopt a tight-binding model on a square lattice with a smooth tunnel barrier around its center. Taking into account the electron-electron interaction by the exact diagonalization method, we evaluate the spin coupling J between two electron spins, as a function of magnetic ﬁeld perpendicular to the quantum dots, and show a transition from antiferromagnetic coupling (J > 0) to ferromagnetic coupling (J < 0). The coupling J is not seriously inﬂuenced by the size diﬀerence between the dots if the energy levels are tuned to match each other using the gate voltage. Next, we simulate SWAP gate operation by calculating the time development of two electron spins. A nonadiabatic change of the tunnel barrier between the quantum dots may cause operation errors, due to the contribution from high energy states. The complete exchange of the spin states could be also blocked by the spin-orbit interaction.

1. Introduction Quantum information processing in solid-state systems such as electrons in semiconductor quantum dots,1,2 nuclear spins,3 and superconducting circuits,4,5 has a great advantage in the integration. Although the decoherence by the coupling to the environment is a major obstacle for the processing in the systems, imperfection of the gate architectures may also cause some problems. In this paper, we study two-qubit operations in the spin-based quantum computers utilizing quantum dots. The operations between two spins s1 and s2 localized in adjacent quantum dots can be performed by changing the spin coupling J in HHeis = Js1 · s2 , using the central gate between the dots.1 It is not obvious, however, whether the orbital degrees of freedom of electrons can be disregarded in the operations. Imperfection of fabricating quantum dots, e.g., size ﬂuctuation, should also be checked. To investigate the realistic operations in double quantum dots, we adopt a tight-binding model and consider the electron-electron interaction by the exact diagonalization method. First, we evaluate the spin coupling J as a function of magnetic ﬁeld perpendicular to the quantum dots. Second, we simulate the SWAP gate operation by calculating the time development of two electron spins in

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Figure 1. Model for double quantum dots. We adopt a tight-binding model on a square lattice with a smooth tunnel barrier. Two electrons are conﬁned in the area of Lx × Ly with hard-wall boundary condition. Lx = 12a and Ly = 6a with lattice constant a. εL and εR denote the lowest energy levels in quantum dot L (deﬁned by the area of x < xc , where xc is the center of tunnel barrier) and in quantum dot R (x > xc ), respectively.

double quantum dots. The SWAP gate requires the change in the height of tunnel barrier between the quantum dots.6,7 We investigate a nonadiabatic operation in which the potential barrier is lowered suddenly at a certain time and discuss the operation errors caused by the sudden change. We also examine the eﬀect of spin-orbit interaction on the SWAP gate operation. 2. Model and calculation method 2.1. Model Our model for double quantum dots is shown in Fig. 1. We adopt a tight-binding model of 11 × 5 sites. Two electrons are conﬁned in the area of Lx × Ly with hard-wall boundary condition; Lx = 12a and Ly = 6a with lattice constant a. The Hamiltonian is given by H = H0 + Hint , where −t(e−iΔΦ(l) c†k+1,l,σ ck,l,σ + c†k,l+1,σ ck,l,σ + h.c.) + V (k, l) c†k,l,σ ck,l,σ H0 = σ

k,l

(1) and Hint represents the Coulomb interaction between electrons. In Eq. (1), c†i,j,σ , ci,j,σ are creation and annihilation operators of electron at site (i, j) with spin σ. The hopping parameter is given by t = 2 /2m∗e a2 , where m∗e is the eﬀective mass. The magnetic ﬁeld of B = (0, 0, B) is taken into account by the phase factor of ΔΦ(l) = −2πlBa2 /Φ0 , with Φ0 = h/e being the magnetic ﬂux quantum. As an energy unit, we deﬁne Eunit by the ﬁrst excitation energy in the conﬁnement of 2 3π 2 3π 2 length Lx , Eunit = 2m ∗ L2 = 122 t. e x V (i, j) is the electrostatic potential at site (i, j), which represents the tunnel barrier. It is assumed to be given by8 2 Vc Vc y − y± (x) π(x − xc ) V (x, y) = θ(±[y − y± (x)]), + 1 + cos 2 Lw 0.7 ± 0.25Ly Ly π(x − xc ) y± = ± , (2) 1 − cos 4 Lw

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where (x, y) = (0, 0) is the center of the rectangle in Fig. 1. Vc and Lw indicate the height and width of tunnel barrier, respectively. In this paper, we ﬁx Lw = Lx /4. We deﬁne quantum dot L (R) by the area of x < xc (x > xc ). When an electron is conﬁned in quantum dot L (R), the lowest energy level is given by εL (εR ). When two quantum dots are equivalent in size (xc = 0), εL = εR . When the dots diﬀer in size (xc > 0), εL < εR . In the latter, we can match the energy levels by adding the electrostatic potential εL − εR (x > xc ) (3) Vﬁx (x, y) = 0 (x < xc ) to V (i, j). In Hint , the electron-electron interaction between sites (i, j) and (i , j ) is given by U0 , Uij,i j = 2 (i − i ) + (j − j )2 + δii δjj · (0.5)2

(4)

where U0 = e2 /(4πa) with dielectric constant . The value of on-site Coulomb repulsion (i = i , j = j ) reﬂects the ﬁnite size of Wannier function (∼ a). We choose U0 so that U = Eunit when the distance between the sites is Lx /2. 2.2. Calculation methods We calculate the many-body wavefunctions and energies by the exact diagonalization of the Hamiltonian; H|n = En |n. We obtain not only the ground state (n = 1) but also excited states (n > 1), with changing the magnetic ﬁeld. From the energies of lowest spin-singlet (|S1 ) and -triplet states (|T1 ), we evaluate the exchange coupling between the electron spins as J = ET1 − ES1 .

(5)

Next, we examine the time development of many-body state |Ψ. It is given by |Ψ(τ ) =

N max n=1

En

τ |nn|Ψ(0). exp − i

(6)

The eigenstates with n > Nmax are truncated. The accuracy of this approximation Nmax Ψ(0)|nn|Ψ(0) > 0.998. To simulate SWAP gate operation, we change is n=1 (0) the height of potential barrier suddenly from Vc to Vc at time τ = 0. The initial (0) state |Ψ(0) is the eigenstate of the Hamiltonian with Vc , in the limited Hilbert space where up-spin is in dot L (x < xc ) and down-spin is in dot R (x > xc ); |L ↑, R ↓. We examine the number of up-spin electrons in dot L, NL,up, as a function of time. If the system can be described by the Heisenberg Hamiltonian, HHeis = Js1 · s2 , the ideal SWAP operation is possible. |Ψ(0) = |1 ↑, 2 ↓ can be expressed by

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0.05

(ii)

J [ Eunit ]

0.04 0.03

(i) 0.02

(iii)

0.01 0 0

5 Φ/Φ0

10

Figure 2. Spin coupling J as a function of magnetic ﬁeld. Φ = BLx Ly is the magnetic ﬂux penetrating the system. Φ0 = h/e. The tunnel barrier height is Vc = 2Eunit . Two quantum dots are the same in size [xc = 0; solid line (i)], or diﬀerent in size [xc = 0.1Lx ; broken lines (ii) and (iii)]. The tuning potential Vfix is (ii) absent or (iii) present to match the energy levels in the two dots.

a linear combination of√the lowest spin-singlet and -triplet states only; |S1 /T1 = (|1 ↑, 2 ↓ ∓ |1 ↓, 2 ↑)/ 2. Then ET

1 J 1 |Ψ(τ ) = √ e−i τ ei τ |S1 + |T1 , 2 J 1 1 2πJ

NL,up (τ ) = cos2 τ = + cos τ . 2 2 2 h

(7) (8)

The spin exchange between the quantum dots is completed at time τ = h/2J. Generally, the time development by Eq. (6) is more complicated owing to the contribution from higher energy states. 3. Calculated results 3.1. Spin coupling J We examine the spin coupling J deﬁned by Eq. (5), using the lowest energies of spinsinglet and triplet states obtained by the exact diagonalization method. Figure 2 presents the spin coupling as a function of magnetic ﬁeld. The double quantum dots are the same in size [solid line (i)] or diﬀerent in size [broken lines (ii) and (iii)]. In the absence of magnetic ﬁeld, the ground state is a singlet while the ﬁrst excited state is a triplet, which indicates an antiferromagnetic coupling between the two spins (J > 0). With an increase in magnetic ﬁeld, J decreases and becomes negative (ferromagnetic spin coupling) when the magnetic ﬂux penetrating the system is Φ 4.5Φ0 . It corresponds to the magnetic ﬁeld of B 3.1 T in the case of Lx 110 nm. If the magnetic ﬁeld is increased further, J changes to be positive again although B dependence of J is weak in large magnetic ﬁelds.

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When the quantum dots are not equivalent in size, the energy levels in dots L and R are diﬀerent without the tuning potential of Vﬁx in Eq. (3) [dotted line (ii)]. A small size diﬀerence (xc = 0.1Lx) signiﬁcantly enhances the spin coupling J by stabilizing the spin-singlet state. When Vﬁx is turned on to match the energy levels [dotted line (iii)], the spin coupling J is almost identical to that in the case of equivalent dots [solid line (i)]. We conclude that the size diﬀerence between the quantum dots hardly inﬂuences the spin coupling unless the size diﬀerence is too large or potential barrier is too high. 3.2. SWAP gate operation We calculate the time dependence of NL,up with two equivalent quantum dots (xc = 0) in the absence of magnetic ﬁeld. In Fig. 3(a), we plot NL,up (τ ) by solid line (broken line) after the sudden change of tunnel barrier height from Vc /Eunit = 10 to 4 (6) at τ = 0. NL,up (0) = 1 in the initial state; an up-spin electron is in dot L (x < xc ) and a down-spin electron is in dot R (x > xc ). We observe ﬁne structures in addition to the ideal SWAP gate operation, Eq. (8). The ﬁne structures, which may seriously lead to gate errors, are ascribable to the contribution from excited states besides the lowest spin-singlet and -triplet states. The periods of ﬁne structures are determined by the excitation energies larger than J. The amplitudes are much smaller on broken line than on solid line. Hence the gate errors produced by the nonadiabatic operations can be reduced by optimizing the amount of change of tunnel barrier height between the quantum dots. 3.3. SWAP gate with spin-orbit interaction Finally, we consider the eﬀect of Rashba spin-orbit (SO) interaction, α(σx py − σy px )/, on the SWAP gate operation. To this end, we add α † HSO = −ck+1,l,↓ ck,l,↑ + c†k+1,l,↑ ck,l,↓ a k,l

− ic†k,l+1,↑ ck,l,↑ − ic†k,l+1,↑ ck,l+1,↓ + h.c. (9) to the Hamiltonian (1). The strength of the SO interaction is characterized by a dimensionless parameter kα /kF , where kα = m∗e α/2 and kF is the Fermi wavenumber. It is 0.005 in GaAs and 0.05 in InGaAs. Figure 3(b) shows the time dependence of NL,up after the SO interaction is suddenly turned on at τ = 0, whereas the tunnel barrier is ﬁxed at Vc /Eunit = 6.9 The initial condition is NL,up (0) = 1. The SO interaction strength is a, kα /kF = 0; b, 0.005; and c, 0.05. With SO interaction in GaAs (curve b), NL,up is almost the same as that without SO interaction (curve a). The strong SO interaction in InGaAs, however, spoils the SWAP gate (curve c). We also see the ﬁne structures on curve c owing to the nonadiabatic change of the SO interaction.

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1

c

NL,up

NL,up

1

b 0

(a) 0

1 time [ h/(2J) ]

2

0

a 0

1 time [ h/(2J) ]

(b) 2

Figure 3. Time development of number of up-spin electrons in quantum dot L, NL,up (τ ), in the absence of magnetic ﬁeld. The initial condition is NL,up (0) = 1. Two quantum dots are the same in size (xc = 0). (a) Solid (broken) line indicates the case in which the tunnel barrier height is suddenly changed from Vc /Eunit = 10 to 4 (6) at τ = 0. (b) The spin-orbit (SO) interaction is suddenly turned on at τ = 0, whereas the tunnel barrier is ﬁxed at Vc /Eunit = 6. The strength of SO interaction is a, kα /kF = 0; b, 0.005; and c, 0.05. J is the value calculated in case a.

4. Conclusions We have theoretically studied the electronic states and spin operations in double quantum dots, from a viewpoint of spin-based quantum computation. We have evaluated the spin coupling J by the energy diﬀerence between the spin-singlet and -triplet states, as a function of magnetic ﬁeld. We have observed the transition from antiferromagnetic coupling (J > 0) to ferromagnetic one (J < 0), which is not seriously inﬂuenced by the size diﬀerence between the dots if the energy levels are tuned to match each other. We have calculated the time development of manybody states to simulate the SWAP gate operation. We have found that the sudden change of tunnel barrier height may result in the gate errors and that the strong spin-orbit interaction could block the complete spin exchange. References D. Loss and D. P. DiVincenzo: Phys. Rev. A 57 (1998) 120. X. Hu and S. Das Sarma: Phys. Rev. A 61 (2000) 062301. B. E. Kane: Nature (London) 393 (1998) 133. A. Shnirman, G. Sch¨ on and Z. Hermon: Phys. Rev. Lett. 79 (1997) 2371. Y. Nakamura, Y. A. Pashkin and J. S. Tsai: Nature (London) 398 (1999) 786. X. Hu and S. Das Sarma: Phys. Rev. A 66 (2002) 012312. J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson and A. C. Gossard: Science 309 (2005) 2180. 8. A similar potential has been used in T. Ando: Phys. Rev. B 44 (1991) 8717. 9. We assume that the SO interaction is turned on at τ = 0 since the initial state of |L ↑, R ↓ cannot be obtained by diagonalizing the Hamiltonian in its presence. Although this is not a realistic situation, this enables us to clarify the eﬀect on SWAP gate operation, in comparison with the sudden change of tunnel barrier height.

1. 2. 3. 4. 5. 6. 7.

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SPIN CORRELATION IN A DOUBLE QUANTUM DOT-QUANTUM WIRE COUPLED SYSTEM

S. SASAKI1,∗, S. KANG2,3 , S. MIYASHITA4, T. MARUYAMA4, H. TAMURA1,3 , T. AKAZAKI1,3, Y. HIRAYAMA1,5 AND H. TAKAYANAGI1,3 1

NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198, Japan 2 Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan 3 CREST-JST, Honmachi, Kawaguchi, Saitama 331-0012, Japan 4 NTT Advanced Technology Corporation, Atsugi, Kanagawa 243-0198, Japan 5 SORST-JST, Honmachi, Kawaguchi, Saitama 331-0012, Japan

Low temperature transport measurement has been performed on a double quantum dotquantum wire coupled device fabricated from a GaAs/AlGaAs heterostructure. Nonlocal control of the Kondo eﬀect in one dot is realized by manipulating the spin state of the other dot. The modulation of the local density of states in the wire region due to the Fano-Kondo antiresonance, and the Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction are the two possible mechanisms underlying the observed features.

1. Introduction Recent advances in fabrication of semiconductor nano-structures have opened up possibilities to investigate various quantum correlation eﬀects. A quantum dot (QD) is an important building block for constructing functional devices because of the wide tunability of its electronic states. A QD behaves as a single magnetic impurity with spin S = 1/2 when it holds an odd number, N , of electrons, since the uppermost orbital state is occupied by an unpaired electron. This magnetic aspect is clearly revealed in the Kondo eﬀect, where the spin in the QD forms a singlet state with another spin in the lead electrodes at low temperatures 1,2,3,4 . When a QD is attached to the side of another quantum wire (QW), the Fano resonance is observed in the conductance of the QW due to an interference between a localized state of the QD and a continuum of the QW 5,6 . Moreover, in the odd electron number regions between two Fano resonances, the electrons in the QW experience enhanced scattering by the spin-carrying QD, which is referred to as the Fano-Kondo antiresonance 7,8 . By adding second QD to this dot-wire coupled system, one can study the indirect correlation between two localized spins mediated by conduction electrons in the interceding QW, which is known as Ruderman-Kittel-Kasuya-Yoshida ∗ E-mail:

[email protected]

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(a)

IL

400

IR (pA)

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Vsd

VmL

VdL VpL VsL

IW wire

dot

dot

200 nm

VsR VpR VdR

VmR

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100 0 -0.9

-0.8

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(c)

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-0.7

-0.70 -0.8

-0.72 -0.5

0.0

0.5

Vdc (meV)

0

1

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B (T)

Figure 1. (a) The current through the right QD, IR , in response to excitation source-drain voltage Vac = 3µV (Vdc = 0 V) as a function of VpR . The left QD is not formed yet. A solid triangle denotes a Kondo valley with an odd NR . The inset shows a scanning electron micrograph of the double QD-QW coupled device together with a schematic of the measurement setup. Dotted lines indicate the positions of the QDs and QW. (b) Gray-scale plot of the conductance through the right QD as a function of Vdc and VpR . Black corresponds to 15 µS, and white to 65 µS. Dotted lines denote edges of the Coulomb diamonds, and a solid triangle denotes a Kondo ridge found in the odd NR Coulomb blockade region. (c) Gray-scale plot of the conductance through the right QD as a function of magnetic ﬁeld and VpR (Vdc = 0 V). Black corresponds to 0 S, and white to 70 µS. A solid triangle denotes the Kondo valley.

(RKKY) interaction 9,10,11,12,13,14 . In this paper, we study transport characteristics of the two QDs coupled to a common QW, and demonstrate non-local control of the Kondo eﬀect in one QD by tuning the spin states of the other. 2. Experiment The inset to Fig. 1(a) shows scanning electron micrograph of our device fabricated by depositing Ti/Au Schottky gates on the surface of a GaAs/Alx Ga1−x As heterostructure. Two QDs and a QW between them are formed (shown by dotted lines) by biasing the eight gates. Transport measurement is performed with a stan-

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dard lock-in technique at temperature T ∼100 mK in a dilution refrigerator. The source-drain bias Vsd = Vdc +Vac , where Vdc is a DC oﬀset, and AC excitation (about 13 Hz) Vac = 3 μV. Magnetic ﬁeld is applied perpendicular to the two-dimensional electron gas (2DEG) plane. 3. Results and discussion Figure 1(a) shows the current through the right QD, IR , as a function of VpR when the left QD is not formed yet. Pairs of Coulomb oscillation peaks are observed suggesting that the electron number in the right QD, NR , is odd in the Coulomb blockade regions between paired peaks at VpR = −0.81 V, −0.75 V, and −0.69 V. In particular, the conductance at the valley marked with a solid triangle is significantly enhanced due to the Kondo eﬀect. Figure 1(b) shows Coulomb diamond characteristics where the conductance is plotted as a function of the source-drain voltage, Vdc , and the plunger gate voltage VpR . The conductance is enhanced along Vdc = 0 only within the odd NR diamond forming a ridge structure (solid triangle), which corroborates the Kondo eﬀect. Figure 1(c) shows magnetic ﬁeld dependence of the Coulomb oscillation characteristics. The Kondo eﬀect is suppressed owing to the Zeeman splitting and the reduced dot-lead tunnel coupling as magnetic ﬁeld increases. Consequently, the conductance at the Kondo valley marked with a solid triangle gradually decreases. At the same time, the alternating high and low conductance regions are observed at more positive VpR forming a pattern like a chessboard 15 . Next, we form another QD by biasing the other gates. Figure 2(a) shows a gray-scale plot of the conductance through the right QD as a function of VpL and VpR . The gate voltage that controls the coupling of the left QD with the rest of the system, VsL , is adjusted in the weak-coupling regime. The two dotted lines designate Coulomb oscillation peaks for the right QD, and the region between them is the same Kondo valley as the one marked with a triangle in Fig. 1. A series of almost vertical ridges appear as VpL is swept. When the conductance through the QW is independently measured as a function of VpL with the same gate voltage conditions, the Fano resonances associated with the left QD are observed as slightly asymmetric peaks 16 . Since these Fano resonances are almost in-phase with the above ridges found in the right QD conductance, we assign the ridges to the resonances of the left QD states with the chemical potential of the lead. The ridges are not observed when VsL is smaller than about −1.2 V because the left QD is decoupled from the rest of the system. Please note that the four ridges marked with arrows suggest spin pairs. Based on this, we deduce that narrow gaps within the pairs correspond to the Coulomb blockade regions where the number of electrons in the left QD, NL , is odd. Let us increase the coupling of the left QD by increasing VsL . Figure 2(b) shows evolution of the conductance proﬁles taken along the mid Kondo valley of the right QD when VsL is changed from −1.04 V (weak-coupling regime) to −0.9 V (strong-

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VpR (V)

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60

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V sL = -0.9V

-0.5

-0.4

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VpL (V) Figure 2. (a) Gray-scale plot of the conductance through the right QD as a function of VpL and VpR in the weak-coupling regime. Black corresponds to 10 µS, and white to 42 µS. (b) Conductance proﬁles taken along the mid Kondo valley of the right QD with the coupling gate voltage VsL changed between -0.9 and -1.04 V in 20 mV steps. For clarity, curves are oﬀset vertically. The scales on the axes apply to the lowermost VsL = −0.9 V trace. Odd NL regions are marked with solid triangles, and one even NL region on the VsL = −0.9 V trace is marked with an open triangle. A gray solid line connects the same NL regions in VsL = −0.9 V and VsL = −1.04 V traces.

coupling regime). The resonances associated with the left QD states appear as peaks in the weak-coupling regime. The spin-pair peaks and the odd NL regions are marked with dash-dotted lines and solid triangles, respectively. We found that the resonances appearing as conductance maxima in the weak-coupling regime change into minima in the strong-coupling regime, just as in the previous report 8 . Figure 3(a) shows a plot similar to Fig. 2(a), but with VsL = −0.9 V, in the strong-coupling regime. Here, in contrast to Fig. 2(a), the resonances of the left QD states produce conductance minima. The intensity of the several conductance maxima corresponding to ﬁxed NL seems to alternate; the maxima at VpL = −90 mV corresponding to the even NL peak (marked with an open triangle in Fig. 2(b)), is more prominent than that at VpL = −102 mV and −73 mV corresponding to the odd NL ones. As shown in Fig. 3(b), the zero-bias Kondo peak still remains in the dIR /dVdc vs Vdc characteristics for even NL . The Kondo eﬀect is also conﬁrmed in the Coulomb diamond characteristics shown in Fig. 3(c). On the other hand,

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Figure 3. (a) Gray-scale plot of the conductance through the right QD as a function of VpL and VpR with VsL = −0.9 V. Black corresponds to 5 µS, and white to 35 µS. The open and solid triangles denote even and odd NL regions, respectively. (b) dIR /dVdc vs Vdc with VpR ﬁxed at −775 mV in the middle of the odd NR Coulomb blockade valley. (c) Conductance of the right QD as a function of Vdc and VpR for even NL . Black corresponds to 20 µS, and white to 40 µS. A triangle denotes a Kondo ridge observed in the odd NR Coulomb blockade region.

no Kondo peak is found in Fig. 3(b) for odd NL . Thus, a non-local control of the Kondo eﬀect in the right QD is realized. When the Fano resonance involving the left QD modulates the conductance of the QW, it also indirectly modulates the dot current IR since the QW acts as one lead for the right QD. As coupling of the left QD increases, the Fano-Kondo antiresonance suppresses the local density of states of the QW at the odd NL regions 8 . Because TK ∝ exp(−1/ρJ), where ρ is the density of states and J is the coupling constant 17 , TK decreases when ρ decreases. This may be one mechanism for the suppressed Kondo eﬀect for the odd NL . Another possible mechanism is the RKKY interaction. When both NR and NL are odd, the RKKY interaction couples two spins in the two QDs via interceding conduction electrons either ferromagnetically or anti-ferromagnetically depending on the distance between them. The eﬀective inter-dot distance in our device is considered to be close to zero since they couple to the opposing sides of the quasi-onedimensional QW with a small relative distance along the wire direction 9,10 . Hence, the ferromagnetic coupling may be realized, and relatively large interaction strength

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is expected. If this is the case, the localized spin in one QD is ferromagnetically locked by the remote spin in the other QD, and the Kondo eﬀect is suppressed. However, this RKKY mechanism is inseparable from the Fano-Kondo-modulated density of states scenario described above, and presumably both mechanisms are in action. 4. Summary We have performed transport measurement on the double QD-QW coupled system and demonstrated non-local control of the Kondo eﬀect in one QD by manipulating the spin states of the other. The Kondo temperature in one QD may be modulated via the local density of states modulation in the QW due to the Fano-Kondo antiresonance involving the other QD. As another mechanism, the RKKY interaction between the two QDs is considered. Acknowledgments This work is ﬁnancially supported by Grant-in-Aid for Scientiﬁc Research from the Japan Society for the Promotion of Science. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

D. Goldhaber-Gordon et al., Nature 391, 156 (1998). S. M. Cronenwett et al., Science 281, 540 (1998). J. Schmid et al., Physica (Amsterdam) 256B-258B, 182 (1998). W. G. van der Wiel et al., Science 289, 2105 (2000). K. Kobayashi et al., Phys. Rev. Lett. 88, 256806 (2002). K. Kobayashi et al., Phys. Rev. B70, 035319 (2004). K. Kang et al., Phys. Rev. B63, 113304 (2001). M. Sato et al., Phys. Rev. Lett. 95, 066801 (2005). H. Tamura et al., Jpn. J. Appl. Phys. 43, L691 (2004). H. Tamura and L. Glazman, Phys. Rev. B72, 121308(R) (2005). N. J. Craig et al., Science 304, 565 (2004). P. Simon, Phys. Rev. B 71, 155319 (2005). P. Simon et al., Phys. Rev. Lett. 94, 086602 (2005). M. G. Vavilov and L. I. Glazman, Phys. Rev. Lett. 94, 086805 (2005). M. Stopa et al., Phys. Rev. Lett. 91, 046601 (2003). S. Sasaki et al., Phys. Rev. B 73, 161303(R) (2006). D. L. Cox and A. Zawadowski, Adv. Phys. 47, 599 (1998).

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KONDO-ASSISTED TRANSPORT THROUGH A MULTIORBITAL QUANTUM DOT

R. SAKANO AND N. KAWAKAMI Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan E-mail: [email protected] We study the SU(6) orbital Kondo eﬀect which emerges in a vertical quantum dot system with three degenerate orbital states. A diagramatic technique with the noncrossing approximation is exploited to calculate the conductance at ﬁnite temperatures as a function of the magnetic ﬁeld. By examining the eﬀect of orbital splittings on the SU(6) Anderson model, we ﬁnd that the introduction of the zero-ﬁeld splitting smears the maximum structure of the ﬁeld-dependent conductance and gives rise to a terrace-like structure in accordance with the recent experimental observations.

1. Introduction Quantum transport in mesoscopic systems is one of exciting topics in condensed matter physics. In particular, recent progress in technology of semiconductor processing has made it possible to fabricate such mesoscopic systems where quantum eﬀects become signiﬁcant. These artiﬁcial systems with a lot of tunable quantum parameters have been stimulating systematic investigations of electron correlations such as the Kondo eﬀect.1,2 Among others, a vertical quantum dot (QD) system,3,4 which is a typical example having tunable orbital parameters, allows us to observe the “orbital” Kondo eﬀect.5,6 This is one of the remarkable progress in experimental studies since the ﬁrst observation of the spin Kondo eﬀect in the QD system.7 Recently we studied transport properties via the multiorbital Kondo eﬀect and found a conductance maximum due to the three- or more-orbital Kondo eﬀect in ﬁnite magnetic ﬁelds.8 Although we predicted the essential properties of the conductance due to the orbital Kondo eﬀect in the vertical QD, the model exploited there was too simpliﬁed to compare the computed conductance with the experimental observations: we assumed that three orbital states are all degenerate at zero magnetic ﬁeld. However, such orbital degeneracy may be partly lifted by Coulomb interaction eﬀects, even if the vertical QD has a symmetric shape.4,9 In this paper, we address this problem and investigate how the orbital splitting caused by the interactions affects the conductance due to the SU(6) three-orbital Kondo eﬀect. We demonstrate that the introduction of the zero-ﬁeld splitting smears the maximum and changes it into a terrace-like structure in the ﬁeld-dependent conductance, which is consistent with the recent experimental observations.

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2. Model and calculation 2.1. SU(6) three-orbital Anderson Hamiltonian Let us consider a single QD system with three degenerate orbitals in the equilibrium state: for example, a vertical QD with seven electrons, where six electrons are in the inner shells and the extra electron occupies one of three degenerate orbital states, which gives rise to the orbital Kondo eﬀect. The orbital level splitting δE due to the magnetic ﬁeld is supposed to be proportional to the ﬁeld. For simplicity we ignore the Zeeman splitting since it is much smaller than the orbital one so that the degenerate levels are split into three spin doublets. In the previous study,8 we assumed that three orbital states are degenerate in the absence of magnetic ﬁelds. In realistic situations, however, there may appear orbital splittings even at zero ﬁeld. This naturally motivates us to consider the energy level splitting due to the interactions; an orbital state localized in the restricted area around the center of the QD feels the Coulomb interaction more strongly, thereby causing the orbital splitting. We thus consider the energy levels speciﬁed as εσl = ε + lδE + Δδl,0

(l = −1, 0, 1)

(1)

with Kronecker’s delta δl,l , where l(σ) represents orbital(spin) index, ε denotes the centre of the energy levels at Δ = 0, and δE is the energy splitting due to the magnetic ﬁeld. Here we have introduced the additional energy shift Δ for the orbital l = 0, which is caused by the interaction eﬀects. In Fig. 1, we show the energy levels in our QD system schematically.

l=0 l=+1

ε

l=-1 Figure 1.

Δ δE δE

Energy-level scheme in the three-orbital QD.

We model the above QD system by assuming that the Coulomb repulsion U between electrons is suﬃciently strong at the QD, so that we eﬀectively put U → ∞. We focus on the three degenerate orbital states in the QD, which are assumed to hybridize with the corresponding conduction channels in the leads for simplicity. It is known that this assumption is valid for the vertical QD.5 In these assumptions, our system can be described by an SU(6) three-orbital extension of the impurity Anderson model in the strong correlation limit (U → ∞). The Hamiltonian reads

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H = H0 + H I , H0 = εσl |σlσl| + εk c†kσl ckσl , σ,l

HI =

(2) (3)

kσl

(Vk |σl0|ckσl + h.c.),

(4)

kσl

where c†kσl (ckσl ) creates (annihilates) a conduction electron with wavenumber k, spin σ(= ±1/2) and orbital l. An electron state at the dot is expressed as |σl, which is supplemented by the unoccupied state |0. Note that the Hamiltonian has SU(6) symmetry in the absence of the splittings. 2.2. Noncrossing approximation method We make use of the noncrossing approximation (NCA) method to analyze the above Hamiltonian at ﬁnite temperatures.2,10 The NCA is a self-consistent perturbation theory, which summarizes a speciﬁc series of expansions in the hybridization V . This method is known to give physically sensible results at temperatures around or higher than the Kondo temperature. In fact, it was successfully applied to the Ce and Yb impurity systems, for which orbital degrees of freedom play an important role. 11 The NCA basic equations can be obtained in terms of coupled equations for two types of self-energies Σ0 (z) and Σσl (z) of the resolvents Rα (z) = 1/(z − εα − Σα (z)), 2Γ D f (ε) Σ0 (z) = , (5) dε π z − ε + ε − Σσl (z + ε) σl D l Γ D 1 − f (ε) , (6) Σσl (z) = dε π D z − ε0 + ε − Σ0 (z + ε) for the ﬂat conduction band of width 2D. At ﬁnite temperatures, the conductance is given as, 2e2 df (ε) Γ G= (7) dε − Aσl (ε), h dε l

where f (ε) is the Fermi distribution function and Aσl (ε) is the one-particle spectral function including the eﬀect of the above self-energies. By evaluating the selfenergies and the spectral function self-consistently, we obtain the conductance at a given temperature in the Kondo regime. 3. Results and discussion To see how the zero-ﬁeld splitting Δ aﬀects transport properties via the SU(6) three-orbital Kondo eﬀect, we show the conductance as a function of the orbital splitting δE due to the magnetic ﬁeld for several choices of Δ in Fig. 2. As found

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previously,8 the conductance shows a clear maximum at Δ = 0, i.e. the SU(6) case. It is seen that the maximum structure at Δ = 0 gradually changes into a terrace-like structure as Δ increases. This ﬁnding is the main result in this paper. We note that the SU(6) Kondo eﬀect appears at Δ = 0, while for large Δ it is replaced by the eﬀective SU(4) Kondo eﬀect within l = ±1 quartets. In the latter limit, the conductance monotonically decreases in magnetic ﬁelds similarly to the ordinary spin Kondo eﬀect. Therefore we can say that partial lifting of the three-fold orbital degeneracy causes the terrace-like structure in the conductance. It is also seen that the conductance at δE = 0 gets large as Δ increases. Since at absolute zero the unitary limit of the conductance for the SU(4) two-orbital Kondo eﬀect, 2e2 /h, is larger than that for the SU(6) three-orbital case, 3e2 /2h, the above result implies that the temperatures considered here are smaller than the eﬀective SU(4) Kondo temperature. On the other hand, in the region of larger δE, the conductance monotonically decreases for all Δ chosen here, indicating that the eﬀective SU(2) Kondo temperature deﬁned in this region is lower than the temperatures shown in the ﬁgure. Experimentally, such a terrace-like structure in the ﬁeld-dependent conductance has been found recently in the vertical QD system.12 Therefore, we conclude that the experimentally observed behavior can be ascribed to the SU(6) Kondo eﬀect, which is subjected to the small zero-ﬁeld orbital splitting due to the Coulomb interactions.

2

conductance G/(2e /h)

0.5 0.4 0.3 Δ=0 Δ=0.5Γ/π Δ=1.0Γ/π Δ=1.5Γ/π Δ=2.0Γ/π

0.2 0.1 0 0

0.2

0.4

0.6

δE/Γ Figure 2. Conductance as a function of the orbital splitting δE due to magnetic ﬁeld for several choices of the zero-ﬁeld splitting Δ. We take the temperature of the system, kB T = 0.1Γ/π and SU (6) ∼ 0.2Γ. the energy level of the QD ε = −25Γ, which gives TK

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4. Summary We have extended our previous studies on the SU(6) three-orbital Kondo eﬀect 8 to more realistic situations which may be suitable for experimental observations. We have found the formation of the terrace-like structure in the ﬁeld-dependent conductance, which indicates that the degeneracy of orbitals is partly lifted by the interaction eﬀects. Even in such cases, the orbital Kondo eﬀect still plays a crucial role in controlling transport properties for the vertical QD systems. More detailed discussions on this issue will be reported elsewhere. Acknowledgments We would like to express our sincere thanks to S. Amaha and S. Tarucha for valuable discussions. References 1. J. Kondo, Prog. Theor. Phys. 32, 37 (1964). 2. A. C. Hewson, The Kondo Problem to Heavy Fermions. (Cambridge University Press, Cambridge, 1997). 3. S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996). 4. S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 (2002). 5. S. Sasaki, S. Amaha, N. Asakawa, M. Eto, and S. Tarucha, Phys. Rev. Lett. 93, 017205 (2004). 6. P. Jarillo-Herrero, J. Kong, H. S.J. van der Zant, C. Dekker, L. P. Kouwenhoven, and S. De Franceschi, Nature 434, 484 (2005). 7. D. Goldhaber-Gordon, J. G¨ ores and M. A. Kastner, Nature (London) 391, 156 (1998); Phys. Rev. Lett. 81, 5225 (1998). 8. R. Sakano and N. Kawakami, Phys. Rev. B 73, 155332 (2006). 9. P. Matagne, J. P. Leburton, D. G. Austing, and S. Tarucha, Phys. Rev. B 65, 85325 (2002). 10. N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987). 11. Y. Kuramoto, Z. Phys. B 53, 37 (1980); F. C. Zhang and T. K. Lee, Phys. Rev. B 28, 33 (1983); P. Coleman, Phys. Rev. B 28 5255 (1983); S. Maekawa, S. Takahashi, S. Kashiba, and M. Tachiki, J. Phys. Soc. Jpn. 54, 1955 (1985). 12. S. Amaha, T. Hatano, S. Sasaki, T. Kubo, Y. Tokura, International Symposium on Mesoscopic Superconductivity and Spintronics 2006, PMo-4.

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SPIN DECAY IN A QUANTUM DOT COUPLED TO A QUANTUM POINT CONTACT

MASSOUD BORHANI Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland E-mail: [email protected] VITALY N. GOLOVACH Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland E-mail: [email protected] DANIEL LOSS Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland E-mail: [email protected] We consider a mechanism of spin decay for an electron spin in a quantum dot due to coupling to a nearby quantum point contact (QPC) with and without an applied bias voltage. The coupling of spin to charge is induced by the spin-orbit interaction in the presence of a magnetic ﬁeld. We perform a microscopic calculation of the eﬀective Hamiltonian coupling constants to obtain the QPC-induced spin relaxation and decoherence rates in a realistic system. This rate is shown to be proportional to the shot noise of the QPC in the regime of large bias voltage and scales as a−6 where a is the distance between the quantum dot and the QPC. We ﬁnd that, for some speciﬁc orientations of the setup with respect to the crystallographic axes, the QPC-induced spin relaxation and decoherence rates vanish, while the charge sensitivity of the QPC is not changed. This result can be used in experiments to minimize QPC-induced spin decay in read-out schemes.1

1. Introduction Recent progress in nanotechnology has enabled access to the electron spin in semiconductors in unprecedented ways, 2,3 with the electron spin in quantum dots being a promising candidate for a qubit due to the potentially long decoherence time of the spin.4 Full understanding of the decoherence processes of the electron spin is thus crucial. On the other hand, as a part of a quantum computer, read-out systems play an essential role in determining the ﬁnal result of a quantum computation.

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However, read-out devices, in general, aﬀect the spin state of the system in an undesired way. Quantum point contacts (QPCs) which are used as charge detectors,5,6 in particular, couple to the spin via the spin-orbit interaction. Motivated by these recent experiments, we study here the eﬀect of the QPC on spin relaxation and decoherence in the quantum dot. For this, we ﬁrst derive an eﬀective Hamiltonian for the spin dynamics in the quantum dot and ﬁnd a transverse (with respect to the external magnetic ﬁeld) ﬂuctuating magnetic ﬁeld. We calculate microscopically the coupling constants of the eﬀective Hamiltonian by modeling the QPC as a one-dimensional channel with a tunnel barrier. We show that this read-out system speeds up the spin decay and derive an expression for the spin relaxation time T1 . However, there are some regimes in which this eﬀect vanishes, in the ﬁrst order of spin-orbit interaction. The relaxation time will turn out to be strongly dependent on the QPC orientation on the substrate, the distance between the QPC and the quantum dot, the direction of the applied magnetic ﬁeld, the Zeeman splitting EZ , the QPC transmission coeﬃcient τ , and the screening length λsc (see Fig. 1). Although this eﬀect is, generally, not larger than other spin decay mechanisms (e.g. coupling of spin to phonons7 ), it is still measurable with the current setups under certain conditions. The following results could be of interest to experimentalists to minimize spin decay induced by QPC-based charge detectors. 2 λsc

y

Y

L

a

I

R

QPC

R

x

θ

QD

r

λd

X

Schematic of the quantum dot (QD) coupled to a QPC. The (X, Y ) frame gives the setup orientation, left (L) and right (R) leads, with respect to the crystallographic 110]. The dot has a radius λd and is located at a distance directions x ≡ [110] and y ≡ [¯ a from the QPC. The vector R describes the QPC electrons and r refers to the coordinate of the electron in the dot. The noise of the QPC current I perturbs the electron spin on the dot via the spin-orbit interaction. Figure 1.

2. The Eﬀective Hamiltonian The quantum dot electron spin couples to charge ﬂuctuations in the QPC via the spin-orbit (Rashba and Dresselhaus) Hamiltonian. The charge ﬂuctuations are caused by electrons passing through the QPC. To derive an eﬀective Hamiltonian for

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the coupling of spin to charge ﬂuctuations, we perform a Schrieﬀer-Wolﬀ transfor˜ = exp(S)H exp(−S), and remove the spin-orbit Hamiltonian in leading mation, H order. We thus require that [Hd + HZ , S] = HSO , under the condition λd λSO , where λd is the quantum dot size and λSO = /m∗ (|β| + |α|) is the shortest spinorbit length. The transformed Hamiltonian is then given by 1 ˜ = Hd + HZ + HQ + HQd + [S, HQd ] , H p2 Hd = + U (r), 2m∗ 1 1 HZ = gμB B · σ = EZ n · σ, 2 2 HSO = β(−px σx + py σy ) + α(px σy − py σx ), † ¯ Clkσ , k C¯lkσ HQ =

(1) (2) (3) (4) (5)

lkσ

HQd =

ll kk σ

S=

† ¯ Cl k σ . ηll (r)C¯lkσ

(6)

m ∞ 1 1 1 HSO = HSO , −LZ Ld + LZ Ld m=0 Ld

HSO = iLd (σ · ξ),

(7) (8)

where L is Liouville superoperator for a given Hamiltonian deﬁned by LA ≡ [H, A] and ξ is a vector √ in the 2DEG plane √ and has a simple form in the coordinate frame x = (x + y)/ 2, y = (y − x)/ 2, z = z, namely, ξ = (y /λ− , x /λ+ , 0), where λ± = /m∗ (β ± α) are the spin-orbit lengths. The transformation matrix S (to ﬁrst order in spin-orbit interaction) can be derived as following 1 −iS = ξ · σ + [n × ξ 1 ] · σ − [n × [n × ξ 2 ]] · σ,

ξ 1 = ((α1 py + α2 x )/λ− , (α1 px − α2 y )/λ+ , 0) ,

(9) (10)

ξ 2 = ((β1 px + β2 y )/λ− , (−β1 py + β2 x )/λ+ , 0) ,

(11)

α1 =

(12)

α2 = β1 = β2 =

EZ [EZ2 − (ω0 )2 ] 2 )(E 2 − E 2 ) , ∗ m (EZ2 − E+ − Z EZ ωc (ω0 )2 2 )(E 2 − E 2 ) , (EZ2 − E+ − Z EZ2 ωc 2 )(E 2 − E 2 ) , m∗ (EZ2 − E+ − Z 2 2 (ωc ) + (ω0 ) − EZ2 EZ2 2 2 )(E 2 − E 2 ) , (EZ − E+ − Z

(13) (14) (15)

where E± = ω ± ωc /2, with ω = ω02 + ωc2 /4 and ωc = eBz /m∗ c. Here, we assume E± − |EZ | |EZ λd /λSO |, which ensures that the lowest two levels in the quantum dot have spin nature. Below, we consider low temperatures T and bias Δμ, such that T, Δμ E± −|EZ |, (hence only the orbital ground state is populated

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so that its Zeeman sublevels constitute a two level system) and average over the dot ground state in Eq. (1). We obtain the following eﬀective spin Hamiltonian 1 Heﬀ = gμB [B + δB(t)] · σ, (16) 2 and the eﬀective ﬂuctuating magnetic ﬁeld δB(t) is then given by the operator δB(t) = 2B × [Ω1 (t) + n × Ω2 (t)] , e2 γ1 −1 Ω1 = (λ− Ey , λ−1 + Ex , 0), m∗ e2 γ2 −1 (−λ−1 Ω2 = − Ex , λ+ Ey , 0), m∗ m∗ EZ2 − (ω0 )2 γ1 = α1 = 2 − E 2 )(E 2 − E 2 ) , EZ (E+ − Z Z m∗ EZ ωc γ2 = β1 = 2 − E 2 )(E 2 − E 2 ) . EZ (E+ − Z Z

(17)

3. Spin Relaxation Time For the ﬂuctuating ﬁeld δB(t), we use the Born-Markov approximation8 and obtain the spin relaxation rate 1 = 4πν 2 (MLL + MRR )F (EZ ) T1 (18) +4πν 2 MLR [F (EZ + Δμ) + F (EZ − Δμ)], F (x) = x coth(x/2kB T ),

(19)

where ν = 1/2πvF is the density of states per spin and mode in the leads and the coeﬃcients Mll read n · ωl l , (20) Mll = ω ll · ωl l − n · ω ll

ll ωll = Ωll 1 + n × Ω2 , eγ1 EZ −1 ll −1 ll λ− εy , λ+ εx , 0 , Ωll 1 = m∗ eγ2 EZ −1 ll −1 ll Ωll −λ− εx , λ+ εy , 0 , 2 = m∗

where Ωll i (i = 1, 2 and l, l = L, R) are matrix elements of the operators Ωi with respect to the leads. In addition, in deriving Eq. (19) we assumed T, Δμ EF . Note that, if the transmission coeﬃcient of the QPC is zero or one (τ = 0, 1), then Eq. (19) reduces to 1 = 4πν 2 (MLL + MRR )EZ , T EZ . (21) T1 On the other hand, the equilibrium part of the relaxation time is obtained by assuming Δμ = 0, 1 = 4πν 2 (MLL + MRR + 2MLR )EZ , T EZ . (22) T1

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Therefore, even with zero (or one) transmission coeﬃcient or in the absence of the bias, the spin decay rate is non-zero due to the equilibrium charge ﬂuctuations in the leads. Another case of interest is the large bias regime EZ Δμ ω0 , which simply means that only the second term in Eq. (19) appreciably contributes to the relaxation rate. Therefore, the non-equilibrium part of Eq. (19) is given by 1 ≈ 8πν 2 MLR Δμ, T1

EZ , T |Δμ ± EZ | ω0 .

(23)

To estimate the relaxation time, we use typical experimental parameters for GaAs quantum dots.5 ) We consider an in-plane magnetic ﬁeld B which leads to Ω2 = 0 (γ2 = 0) and, for simplicity, assume that B is directed along one of the spin-orbit axes (say x , see Fig. 1). In this special case we obtain the following expression for kF−1 λsc a, MLR

EZ2 cos2 θ e4 2 λ4sc τ (1 − τ ), 2 2 m∗ κ2 λ+ a6 (2 ω02 − EZ2 )2

(24)

or equivalently, the relaxation rate is given in terms of the QPC shot noise 1 8π 2 e2 4 ν 2 λ4sc EZ2 cos2 θ ≈ SLL , T1 m∗ 2 κ2 a6 λ2+ (2 ω02 − EZ2 )2 SLL =

e2 Δμ τ (1 − τ ), π

(25) (26)

where SLL is the current shot noise in the left lead of the QPC, and due to current conservation, SLL = SRR = −SLR = −SRL . We note that Eq. (25) is the nonequilibrium part of the relaxation rate. Thus, even if the constant equilibrium part (∼ MLL , MRR in Eq. (19)) is of comparable magnitude, the non-equilibrium part can still be separated, owing to its bias dependence. Moreover, at low temperatures and large bias voltages, the relaxation rate is linear in the bias Δμ and proportional to the current shot noise in the QPC, 1/T1 ∝ τ (1 − τ )Δμ. The lifetime T1 of the quantum dot spin strongly depends on the distance a to the QPC. The non-equilibrium part of 1/T1 depends on a as follows, 1/T1 ∝ a−6 . On the other hand, the charge sensitivity of the QPC scales as a−1 , which allows one to tune the QPC into an optimal regime with reduced spin decoherence but still suﬃcient charge sensitivity. The spin lifetime T1 strongly depends on the QPC orientation on the substrate (the angle θ between the axes x and X in Fig. 1). For example, the non-equilibrium part of the relaxation rate vanishes at θ = π/2, for an in-plane magnetic ﬁeld B along x . Finally, we remark that, for a perpendicular magnetic ﬁeld (B = (0, 0, B)), we have

Mll = ω ll · ω l l ,

n = ez ,

(27)

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and the relaxation rate can be calculated analogously. The only diﬀerence is that Ω2 is no longer zero and the matrix elements Mll are given by more complicated expressions. 4. Concluding Remarks In conclusion, we have shown that charge read-out devices (e.g. a QPC charge detector) induces spin decay in quantum dots due to the spin-orbit interaction (both Rashba and Dresselhaus). Due to the transverse nature of the ﬂuctuating quantum ﬁeld δB(t), we found that pure dephasing is absent and the spin decoherence time T2 becomes twice the relaxation time T1 , i.e. T2 = 2T1 . Finally, we showed that the spin decay rate is proportional to the shot noise of the QPC in the regime of large bias (Δμ EZ ) and scales as a−6 (see Fig. 1). Moreover, we have shown that this rate can be minimized by tuning certain geometrical parameters of the setup. Our results should also be useful for designing experimental setups such that the spin decoherence can be made negligibly small while charge detection with the QPC is still eﬃcient. Acknowledgments We thank J. Lehmann, W. A. Coish, T. Heikkil¨ a, H. Gassmann and S. Erlingsson for helpful discussions. This work was supported by the Swiss NSF, the NCCR Nanoscience, EU RTN Spintronics, DARPA, and ONR. References 1. M. Borhani, V. N. Golovach, and D. Loss, Phys. Rev. B 73, 155311 (2006). 2. Semiconductor Spintronics and Quantum Computation, D.D. Awschalom, D. Loss, and N. Samarth (eds.), (Springer, Berlin, 2002). 3. V. Cerletti, W. A. Coish, O. Gywat, and D. Loss, Nanotechnology 16, R27 (2005). 4. D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 5. J.M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwenhoven, Nature 430, 431 (2004). 6. J.R. Petta, A.C. Johnson, C.M. Marcus, M.P. Hanson, and A.C. Gossard, Phys. Rev. Lett. 93, 186802 (2004). 7. V.N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004). 8. C.P. Slichter, Principles of Magnetic Resonance, (Springer-Verlag, Berlin, 1980).

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CONTROL OF THE ELECTRON DENSITY AND ELECTRIC FIELD WITH FRONT AND BACK GATES

M. YAMAGUCHI1,2, S. NOMURA1,2,3, K. MIYAKOSHI2,4, H. TAMURA1,2 , T. AKAZAKI1,2, AND H. TAKAYANAGI1,2,4 1 NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa, 243-0198, Japan. 2 CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi-shi, Saitama, 332-0012, Japan 3 Institute of Physics, University of Tsukuba, 1-1-1 Tennodai Tsukuba-shi, Ibaraki 305-8571, Japan 4 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan E-mail: [email protected]

The electron density and electric field in a GaAs quantum well can be controlled independently using front and back gates. We observed Landau level splitting of the photoluminescence (PL) spectrum of a two-dimensional electron system with holes at the δ-doped acceptors with changing the electron density and the PL energy shift due to the gate electric field. We compared the observed PL energy shift with the simulation with the Schr¨ odinger-Poisson coupled equation and got a reasonable agreement.

1. Introduction Good controllability of the electron density is advantageous for studying the twodimensional electron system (2DES) because most of the interesting properties of the 2DES strongly depend on electron density. A modulation doped quantum well (QW) with the gate bias has been widely used to study the 2DES while controlling the electron density. In this structure, an electric ﬁeld induced by the gate bias inevitably changes the physical properties in changing electron density. An undoped QW oﬀers good controllability of the electron density as well as the electric ﬁeld. We can control the electron density and the electric ﬁeld independently by attaching gates to both sides of the QW (double-gate structure). The advantage of the doublegate undoped QW is that the electric ﬁeld induced by both gates is not screened by the doping layer. In this paper, we report on the experimental demonstration of the control of the electron density and electric ﬁeld with front and back gates and compare the observed PL energy shift with a numerical simulation.

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2. Experiment

Vf

Ohmic metal

The layer structure and the electrical connections of our sample are shown in Fig. 1. The 80-nm GaAs QW is separated from the metal front gate and back gate by Al0.3 Ga0.7 As layers and GaAs/AlAs superlattice. Be atoms of 2×1010 cm−2 are δ-doped 30 nm from the bottom of the channel as acceptors. The sample was processed into a 1 mm × 1 mm mesa and an AuGeNi ohmic contact was connected to the QW. The front gate is 15 nm of semi-transparent Ti/Au. Gate biases Vf and Vb were applied to the front gate and to the back gate, respectively. The sample was cooled in a dilution refrigerator to about 100 mK and magnetic ﬁelds induced with a superconducting magnet were applied parallel to the growth axis. A continuous wave 532 nm YAG laser was used to irradiate the sample through an optical ﬁber. The excitation power density was about 100 μW/cm2 . The PL signal was collected via the same ﬁber and detected by a 1-m spectrometer with a nitrogen-cooled charge-coupled device detector.

un GaAs

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un Al0.3Ga0.7As (20 nm) AlAs/GaAs barrier (600 nm) n-GaAs back-gate

EcBe

ΔBe Be

Figure 1. Right panel shows the layer structure and electrical connections. Left panel shows the schematic band diagram at the QW.

2.1. Electron accumulation by gate operation Figure 2 shows the Vb dependence of the PL spectra at 1 T at Vf =0.7 V. The back-gate voltage Vb is increased from 0 V (bottom) to 2.0 V (top). Around Vb = 0.3 V, the conduction electrons start to accumulate, where the PL peak jumps to higher energy. Below the threshold voltage (Vb =0.3 V), the PL peak originates from the radiative recombination of the photo-excited electron and hole, which is bound to an ionized acceptor (Be− X0 ). Above the threshold (Vb =0.3 V), the PL peak corresponds to the radiative recombination of holes in the acceptor sites with the 2DES (2DES-Be). When the conduction electrons accumulate in the QW, the oscillator strength of Be− X0 decreases because the free electrons screen the Coulomb interaction between the electron and hole. The energy diﬀerence between the Be− X0

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b

Figure 2. The back-gate voltage dependence of the PL spectra at Vf = 0.7 V in 1 T. LL0, LL1, LL2, LL3 indicate the N=0,1,2,3 Landau level peaks, respectively.

and the 2DES-Be is about 3 meV, which corresponds to the binding energy of the electron in Be− X0 . Above the threshold, the 2DES-Be spectra split into Landau levels as the electron density increases. The Vb dependence of the electron density which is obtained from the Landau level is in good agreement with the expected value estimated by an electrostatic model calculation 1 . 2.2. The PL energy shift by the front-gate bias Figure 3 shows the front-gate bias Vf dependence of PL spectra at 1.5 T at Vb =1.0 V, where Vf decreases from 0.7 V (bottom) to 0.3 V (top). The electron density decreases as Vf decreases. The threshold of conduction electron accumulation is indicated by the arrow, although the energy jump from Be− X0 to 2DES-Be is not clear compared to that in Fig. 2. In this case, the PL peak energy shift for 2DES-Be is not monotonic compared with the case of the Vb sweep. The Vf dependence of the PL spectra can be divided into three regimes: the peak does not shift between 0.7 V to 0.65 V (regime A), it shows a red shift between 0.65 and 0.45 V (regime B), and the slope of the shift changes below the threshold 0.45 V (regime C). This nonmonotonical energy shift is due to the electric ﬁeld and the change of the electron distribution in the QW. In the next section, the calculated energy shift is compared with the experiment.

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f

Figure 3. The front-gate voltage dependence of the PL spectra at 1.5 T at Vb = 1.0 V. The dotted line traces the peak position as a guide of the eye.

(a) Vb= 0 V 1.0 V 2.0 V

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Figure 4. The back-gate bias dependence of the electron density and the conduction band profile for constant Vf (a),(b), and the front-gate bias dependence of them for constant Vb (c),(d).

3. Simulation We calculated the conduction band proﬁle of the QW and the single particle energy and wave function of the electron states using the Schr¨ odinger-Poisson coupled equation. The exchange energy terms were not taken into account, then many-

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0.3 C

Vf [V]

0.4 0.5 B

0.6 0.7

-30

A

-20 -10 Energy shift [meV]

0

Figure 5. The Vf dependence of energy shift of the 0 − EcBe (solid line) and the observed PL energy shift measured in Fig. 3 (open circles).

body eﬀects such as band gap renormalization were neglected. We used the same layer structure (Fig. 1) excluding the Be-δ-doped layer. The parameters were the eﬀective mass of the electron m∗ = 0.067m0 , dielectric constant ε = 13.1ε0 and Schottky barrier height φ0 =0.7 V for the GaAs, and m∗ = 0.092m0, ε = 12.2ε0 for Al0.3 Ga0.7 As, and ε = 11.6ε0 for AlAs/GaAs superlattice, where m0 and ε0 are the bare electron mass and dielectric constant in a vacuum, respectively. The conduction band oﬀset relative to GaAs was set to be 0.232 eV and 0.337 eV for the Al0.3 Ga0.7 As and AlAs/GaAs superlattice, respectively. We also assumed that the electrons accumulated only around the QW. We performed simulations for two cases, i.e. Vb swept with Vf constant and Vf swept with Vb constant. Figure 4 shows the electron density distribution and conduction band proﬁle of the QW. In the case of the Vb sweep [Fig. 4 (a,b)], the conduction band changes only in the region extending about 20 nm from the back side. The front half of the QW shows almost no change against Vb because of the screening by the accumulated electrons. On the other hand, in the case of the Vf sweep [Fig. 4 (c,d)] for ﬁxed Vb =1.0 V, the wider region of the conduction band proﬁle changes with Vf . For small Vf [See 0.4 V in Fig. 4 (c,d)], the QW is empty since the conduction band edge at the back side of the QW is higher than EF , which corresponds to regime C in Fig. 3, where the slope of the conduction band changes simply as F ∝ (Vf − Vb ). When the conduction band becomes lower than EF by increasing Vf (See 0.6 V), the conduction electrons accumulate at the back side of the QW, which corresponds to regime B, where the slope of the front side of the QW changes with Vf . When Vf is further increased (See 0.8 V), which corresponds to regime A, the electrons start

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ﬁlling the ﬁrst excited state which has a large probability amplitude in the front side of the QW. In this case, the electric ﬁeld in the QW is screened by the electrons in the ground state as well as by those in the ﬁrst excited state and the slope of the potential near the center of the QW does not change when Vf is changed. The transition energy for Be− X0 and 2DES-B can be given by 0 − EcBe +ΔBe − Ebin and 0 − EcBe +ΔBe , respectively, where Ebin is the binding energy of the electron in Be− X0 , EcBe is the conduction band edge at the Be δ doped position (30 nm from the back side of the QW), and ΔBe =1.4915 eV is the acceptor level measured from the conduction band edge2 (The deﬁnition of 0 , EcBe , and ΔBe are schematically shown in the right panel of Fig. 1). The PL peak energy shift from ΔBe =1.4915 eV is plotted in Fig. 5 together with the calculated 0 − EcBe . We found that most of the observed energy shift can be explained by the change in 0 − EcBe . Since the shape of the wave function along the z direction is not very sensitive to the perpendicular magnetic ﬁeld, the application of 1.5 T only shifts the calculated curve in Fig. 5 to the higher energy by the cyclotron energy of 1.3 meV in A and B regime. The band-gap renormalization slightly changes the calculated curve to lower energy in A and B regime, which is estimated to be -2.7 meV at 7.5 × 1010 cm−2 in the 2D limit3 . The curve of the C regime should be lower by Ebin ∼ 3 meV. The discrepancy between the calculated curve and the experimental data can not be explained by the contributions of the magnetic ﬁeld and the band-gap renormalization. This discrepancy may come from the eﬀect of Be--doping, which induces an additional potential in the QW, and/or from the drop of the gate electric ﬁeld because of the leakage current. 4. Summary We demonstrated the control of the electron density and electric ﬁeld of an undoped QW with front and back gates through PL measurements. The observed electron density and energy shift were in good agreement with a simulation based on the Schr¨ odinger-Poisson coupled equation. The double-gate quantum well provides for independent control of the electron density and electric ﬁeld, which is advantageous for studying the properties of the 2DES with and without electric ﬁelds. References 1. Y. Hirayama, K. Muraki and T. Saku, Appl. Phys. Lett. 72, 1745 (1998). 2. Properties of Gallium Arsenide, 3rd ed., edited by M. R. Bruzel and G. E. Stillman, Inspec, Herts, UK, 1996. 3. Hock-Kee Sim, R. Tao and F. Y. Wu, Phys. Rev. B 34 7123 (1986) and references therein.

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EFFECT OF THE ARRAY DISTANCE ON THE MAGNETIZATION CONFIGURATION OF SUBMICORN-SIZED FERROMAGNETIC RINGS T. MIYAWAKI†, K. TOYODA†, M. KOHDA†, A. FUJITA† AND J. NITTA†,‡ †

Department of Materials Science, Tohoku University, 6-6-02 Aramaki-aza Aoba, Aoba-ku, Sendai 980-8579, Japan ‡ CREST, 4-1-8 Honmachi, Kawaguchi, 331-0012, Japan * [email protected]

Magnetization characteristics of one dimensionally arrayed submicron-sized ferromagnetic rings were investigated by MOKE varying inter-ring distance lx. Changing of onion-to-vortex and vortexto-onion transition fields, HOV and HVO, respectively, due to magnetostatic interaction were found to be proportional to the power of lx. The interaction was found to cause significant change of magnetic configuration and enhance the lx-dependence of HOV and HVO than considering only the decay of magnetostatic energy of uniform-magnetized ring array. Energies of the ring array were also calculated and lx-dependence of HOV and HVO was discussed qualitatively.

1. Introduction Recently, submicron-sized ferromagnets have attracted much attention, because they will offer potential applications for such as magnetic random access memory (MRAM)1. In the number of various proposed shapes2,3 of the ferromagnetic elements towards the submicron-sized memory elements, ring-shaped ferromagnets4,5 are of great interest because two characteristic magnetic configurations called the “vortex” state and the “onion” state have been confirmed6. Especially, in the vortex state, the magnetization oriented circularly so that there is almost no stray field, which will provide a potential for the high density memory1. To achieve these applications, it is necessary to reduce the inter-ring distance in the ferromagnetic ring array. On the other hand, influence of magnetostatic interaction between neighboring rings on bit switching becomes significant when the inter-ring distance becomes smaller. Several groups have investigated the interring distance of two dimensional (2D) ring array7,8 using magneto-optical Kerr effect (MOKE)7 and magnetic force microscopy (MFM) 8. However, quantitative discussion on the variation of transition field as a function of inter-ring distances has been very limited. In the present study, we fabricated one dimensional (1D) ferromagnetic ring array and hysteresis loops were measured using MOKE. Micromagnetic simulation was also carried out to calculate the energy and magnetic configurations. The inter-ring dependence of the onion (vortex)-to-vortex(onion) transition fields and its mechanism were discussed by the results of MOKE experiment and micromagnetic simulation9.

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Figure 1. An SEM image of fabricated Fe ring arrays with outer diameter dout = 800 nm, inner diameter din = 500 nm, thickness t = 80 nm, inter-ring distance lx = 100 nm. An external field was applied along x-direction, and the MOKE measurements were carried out along the same direction.

2. Experimental Various Fe ring arrays with outer diameter dout = 800 nm, inner diameter din = 500 nm and thickness t = 80 nm have been fabricated on thermally oxidized Si substrates by electron beam lithography and lift-off techniques. Inter-ring distance for x-direction lx was varied from 50 nm to 1000 nm and the inter-ring distance for y-direction ly was fixed to 1000 nm. Figure 1 shows a scanning electron microscopy (SEM) image of fabricated Fe ring array. An external magnetic field was applied along x-direction and hysteresis loops were measured using MOKE technique. The spot size of the laser was about 10 µm. 3. Results and discussions Figure 2(a) – (c) shows normalized Kerr rotation signals of the ring arrays with lx = 50, 100 and 700 nm. The arrows in Fig. 2 indicate the transition field from the onion-tovortex state, HOV, and the vortex-to-onion state, HVO. Both HOV and HVO shifted and the onion state became stable with reducing lx. Especially the ring array of lx = 50 nm takes onion state even after the external magnetic field decreased to zero, and transition into the vortex state is observed when magnetic field is applied to reverse direction. The open triangles and squares in Fig. 2(d) represent the onion-to-vortex (HOV) and vortex-to-onion (HVO) transition fields, respectively, in each inter-ring distance. HOV and HVO exhibit drastic variation below lx = 200 nm, while almost constant above lx = 700 nm. From the reported results for dot array10, it is expected that the interaction between adjacent rings is negligibly small when the inter-ring distance becomes larger than the diameter of the rings. The inter-ring distance for y-direction is designed to be 1000 nm in this study, and the interaction for y-direction is negligible. Therefore, the change of the transition fields shown in Fig. 2(d) is attributed to the inter-ring interaction along xdirection. By assuming asymptotic variation of transition field Htr (=HOV, HVO) to Htr0 of isolated ring, the variation of Htr - Htr0 has been analyzed by plotting to a double logarithm chart (the inset in Fig. 2(d)).

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Figure 2. (a)-(c) Normalized Kerr rotation signals of various inter-ring distance for dout / din = 800 nm / 500 nm Fe ring arrays. HOV and HVO indicate the onion-to-vortex and vortex-to-onion transition fields, respectively. (d) HOV and HVO variation as a function of lx.

For both HOV and HVO, the plotted data obviously exhibit linear variation. The shift from Htr0 to Htr is described as H’, then Htr is described as follows: Htr = Htr0 – H’ = Htr0 – h / (lx)n

(1)

where h is a constant. The values of n obtained from the gradient of the lines of the double logarithm plot are evaluated to be 1.36 and 0.79 for HOV and HVO, respectively. Since the difference of the magnetization before and after the transition ∆M is about the same, change of the transition fields mainly comes from the change of internal magnetic energy of the ring array. It is well known that the magnetostatic interaction between two magnetic dipoles is proportional to l-3, where l is the distance between them. Therefore, the value of n is expected to be n = 3 in the equation (1) when the diameter of the single-domain ferromagnetic rings are negligibly small compared to the inter-ring distance, i.e. the rings can be regarded as magnetic dipoles. However, the values of n obtained by the present MOKE experiments are much smaller than n = 3. It should be noticed that magnetic charges distributed on two finite facing surfaces bring about the value of n for the potential energy (= magnetostatic energy) less than n = 1. For the ferromagnetic rings, it is difficult to obtain the analytical formula for variation of n, accordingly, we have carried out numerical simulation9 to investigate the decay of the magnetostatic energy against the inter-ring distance lx. All magnetic moments in the ring were fixed to x-direction, and magnetostatic energy was calculated with changing lx. From this simulation, the magnetostatic energy of the ring array was found to be proportional to lx-0.6 in contrast to the lx-dependence of the transition fields obtained in the MOKE experiment. Consequently, variation of magnetic configuration must be taken into account to explain the difference between the MOKE experiment and numerical simulation. It has been reported that the transition from the onion state to the vortex state involves another local magnetic configuration4, and here we call them local vortices.

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Figure 3. Magnetic configurations of (a) isolated ring at external magnetic field Hext = 3500 Oe, (b) isolated ring at Hext = 200 Oe, (c) a ring array with lx = 31 nm at Hext = 3500 Oe, (d) a ring array with lx = 31 nm at Hext = 200 Oe.

The local vortex appears around the edge of ring where magnetic charge appears in the onion state4. This structure is metastable, and its configuration is strongly affected by external configuration. The local vortex close to perfect disclosure configuration results in few magnetic flux leakage to out side of ring, while one with halfway closure structure gives partial leakage of fluxes. Therefore, the variation of the inter-ring interaction is possible to change the configuration of the local vortices. What has to be noticed is that aperture of local vortex affects to not only the magnetostatic but also the exchange energy. To investigate the variation of the magnetic configuration and energies of the ring array, it is necessary to carry out micromagnetic simulation. Magnetic configuration and energy of 1D ring array was calculated by using OOMMF package9. Isolated ring and 1D ring array of rings of outer and inner diameters are settled to be dout = 500 nm and din = 310 nm, respectively. Inter-ring distance lx of the ring array was 31 nm. An isolated ring was also calculated. The ratio between dout and din is the same as the Fe ring arrays of dout = 800 nm and din = 500 nm used for MOKE experiment. Saturation magnetization Ms = 1.6×106 A/m, exchange stiffness constant A = 2.1×10-11 J/m, crystalline anisotropy constant K1 = 4.7×104 J/m3, damping constant α = 0.5 and cell size = 6 nm were used in the micromagnetic simulation. As shown in Fig. 4, in the external magnetic field Hext = 3500 Oe applied along x-direction, isolated ring is in the onion state (Fig. 4a). Curling structures where the magnetic moments follow the perimeter of the ring to reduce stray field are observed in the vicinity of the ring edge where the magnetic flux leaks. While sweeping down Hext, the curling structures change to the local vortices. Figure. 4b shows the magnetic configuration of an isolated ring at Hext = 200 Oe. Continuously reducing Hext, one of the vortices travels around the ring resulting in the transition into the vortex state. In contrast, arrayed rings exhibit the onion state at Hext = 3500 Oe (Fig. 4c) without the curling magnetic configuration observed in the isolated ring. Local vortices are not formed even the Hext is reduced to 200 Oe (Fig. 4d). It is obviously seen that the configuration of curling states affect the inter-ring interaction. Figure 4 shows the calculated energy variation of an isolated ring (dashed lines) and a ring array (solid lines) as a function of Hext. The magnetic configuration of a ferromagnetic ring is determined by Zeeman, magnetostatic, exchange and anisotropy energies. Anisotropy energy can be neglected when Fe rings are polycrystalline.

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Figure 4. Magnetostatic energy (Em), exchange energy (Eex) and Zeeman energy (Eh) variation of an isolated ring (dashed lines) and an arrayed rings with lx = 31 nm (solid lines) as a function of external magnetic field.

Isolated ring takes the onion state when large external magnetic field is applied. Magnetostatic energy, Em (Fig. 4(a)), decreases with decreasing external magnetic field (ab in Fig 4(a)), while exchange energy, Eex (Fig. 4(b)), increases continuously (cd in Fig 4(b)). It is obviously indicating that the formation of local vortices to reduce stray field. Subsequently reducing Hext, rapid decrease of Em and Eex around Hext = 0 Oe due to the transition into the vortex state is observed. Increasing Hext for the opposite direction, a little increase of Em is observed. The little decrease in Em and the increase in Eex indicates that there is stray field slightly from the ring edge even in the vortex state. Subsequently the transition into the onion state for large gain of Zeeman energy, and sharp increase of Em and Eex is occurred around 1.7 kOe. The increase of Em is due to the generation of large stray field and the increase of Eex is due to the formation of local vortices. For the ring array, Em in the onion state is lower than that of the isolated ring due to the magnetostatic interaction. Eex is also reduced because of the absence of the curling structure observed in the isolated ring. While reducing Hext, behaviors of Em and Eex are similar to those of the isolated ring. The stabilization of the onion state due to the reduction of Em and Eex causes the shift of onion-to-vortex transition field. While increasing Hext along the opposite direction, gradual increase of Em, the evidence for the existence of stray field, is observed. Although inter-ring interaction does not exist in the ideal vortex state, there is measurable interaction because of the existence of stray field from the ring edge and it causes the transition field HVO to shift from that of the isolated ring. Additionally the different values of n between the MOKE experiment and the singledomain simulation can be explained as follows. Magnetostatic interaction of onedimensionally arrayed single-domain rings decays proportional to lx-0.6 as described. The rings used in the MOKE experiments show change of magnetic configuration which is confirmed by OOMMF. The magnetic configuration is found to have a tendency that curling structure become prominent with increasing lx, which reduces the magnetic flux leaking from ring edge resulting in the enhancement of the change of interaction. It results

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276 276 in the larger value of n obtained from the MOKE results than that of single-domain simulation. The difference of the lx-dependence between HOV and HVO reflects the difference of the strength of interaction between adjacent rings. Although magnetic flux leaks exist in the vortex state before the vortex-to-onion transition, the amount of the flux leakage in the vortex state is less than that in the onion state or the local vortex structure before onion-to-vortex transition. 4. Conclusions We have studied the inter-ring interaction of 1D ring array quantitatively. The transition fields (HOV and HVO) were found to be fitted with linear lines when plotted to a double logarithm chart against the inter-ring distance lx. The shift of HOV and HVO while decreasing inter-ring distance lx was larger than that of considering only magnetostatic energy. Magnetic configuration was changed with varying lx and exchange energy was also varied with varying lx. The lx dependence of HOV and HVO can be explained by combining contributions of magnetostatic and exchange energies. Acknowledgments The authors gratefully acknowledge H. Shima for useful discussions. References 1. J-G. Zhu, Y. Zheng, G. A. Prinz, J. Appl. Phys. 87, 6668 (2000) 2. J. K. Ha, R. Hertel, J. Kirshner, Phys. Rev. B 67, 224432 (2003) 3. K. J. Kirk, J. N. Chapman, S. McVitie, P. R. Aitchison, C. D. W. Wilkinson, Appl. Phys. Lett. 75, 3683 (1999) 4. T. A. Moore, T. J. Hayward, D. H. Y. Tse, J. A. C. Bland, F. J. Castano, C. A. Ross, J Appl. Phys. 97, 063910 (2005) 5. M. Steiner, J. Nitta, Appl. Phys. Lett. 84, 939 (2004) 6. S. P. D. Li, D. Peyrade, M. Natali, A. Lebib, Y. Chen, U. Ebels, L. D. Buda, K. Ounadjela, Phys. Rev. Lett. 86, 1102 (2001) 7. M. Klaui, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman, Appl. Phys. Lett. 86, 032504 (2005) 8. J. Wang, A. O. Adeyeye, N. Singh, Appl. Phys. Lett. 87, 262508 (2005) 9. M. Donahue et al., see http://math.nist.gov/oommf/. 10. K. Y. Guslienko, V. Novosad, Y. Otani, H. Shima, K. Fukamichi, Phys. Rev. B 65, 024414 (2001)

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A WIDE GaAs/GaAlAs QUANTUM WELL SIMULTANEOUSLY CONTAINING TWO DIMENSIONAL ELECTRONS AND HOLES

ANE JENSEN, MAGDALENA UTKO, AND POUL ERIK LINDELOF Niels Bohr Institute, Nano-Science Center, University of Copenhagen, Universitetsparken 5 D, 2100 Copenhagen, Denmark E-mail: [email protected] KEI TAKASHINA, AND YOSHIRO HIRAYAMA NTT Basic Research Laboratories, 3-1, Monrinosato-Wakamiya, Atsugi, 243-0198, Japan

We report on a novel 200 nm wide GaAs/GaAlAs quantum well, which can contain either electrons or holes. The structure has a p-doped layer above the quantum well, and a superlattice (AlAs/GaAs) barrier below shielding the quantum well from the highly ndoped 311A GaAs substrate, which acts as a back-gate. p-Hall bars are fabricated with ohmic Au/Zn p-contacts, and n-Hall bars are fabricated by etching away the p-doped top layer and applying ohmic Au/Ge/Ni contacts. There is an overlapping region of the back-gate voltage where a two dimensional electron gas is induced in the n-Hall bars and a two dimensional hole gas still exists in the p-Hall bars. This unique system allows for devices, where coupling between high quality electron and hole gases is controlled by the back-gate.

1. Motivation In this paper we focus on the properties of a two dimensional hole gas (2DHG) and a two dimensional electron gas (2DEG), which are present in the same quantum well (QW) structure in p- and n-type devices, respectively. Such a structure is promising for the realization of a lateral light emitting pn-diode. The lateral light emitting diode, in contrast to conventional vertical diodes, has proven to be a technological challenge, as reﬂected by the sparse experimental examples.1,2,3,4 Yet, applications may be manifold. The two dimensional nature of the device allows for controllably small area and capacitance of the diode. Furthermore, the lateral design enable the incorporation with other lateral devices. A single-photon source has been proposed by combining a single-electron surface acoustic wave pump, and a lateral light emitting pn-diode.5 Our structure is an MBE grown lattice matched heterostructure. It is build on the concept of a gate induced 2DEG, and a 2DHG generated by modulation doping. The advantages of this solution are fairly high carrier mobilities, and high structural quality of the wafer, which are important for both optical and electronic devices.

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Figure 1. (a) A schematic outline of the Hall bar conﬁgurations on our semiconductor heterostructure. To the left is the p-Hall bar in which 2DHG is formed, and to the right the n-Hall bar with the 2DEG. (b) The band edge diagram for the wafer as-grown, i.e. p-type. Vg = 6 V. (c) The band edge diagram for the n-type version of the wafer, the p-dopants are removed. Vg = 6 V. The insets show the change of the valence and conduction band edges as a function of Vg .

2. Device design and fabrication The structure is grown on a highly n-doped GaAs (311A) substrate, which serves as a global back-gate. An AlAs/GaAs (2nm/2nm×200) superlattice ensures an insulating barrier between the back-gate and the 200 nm wide GaAs QW. The Al0.3 Ga0.7 As layer above the QW consists of 20 nm undoped spacer and a 70 nm thick modulation doped p-type layer, with the Si dopant concentration of 4 · 1018 cm−3 . The as-grown QW has only a 2DHG but a 2DEG may be induced, when the pdoped Al0.3 Ga0.7 As layer at the top of the wafer is removed. Band edge diagrams for these conﬁgurations are calculated by a 1D Poisson-Schr¨odinger solver.a Figure 1b and c show the results for the back-gate voltage: Vg = 6 V. The insets show a a The calculations are performed with the programme 1D Poisson, written by G. Snider (www.nd.edu/∼gsnider). This programme solves the 1D Poisson and the Schr¨ odinger equations self-consistently.

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blow up of the change in the valence and conduction band edges, respectively, as Vg is varied. A 2DHG is formed at the upper interface of the QW in the p-type conﬁguration for low Vg . It is cutoﬀ for Vg 8 V. In the n-type conﬁguration, Vg can induce a 2DEG at the lower interface of the QW, the threshold Vg for the 2DEG is about 4 V. Most signiﬁcantly is that a range of Vg exists, where both the 2DHG and the 2DEG are present in the QW. A simple capacitor model implies that Vg controls the charge carrier densities. A decrease of Vg implies an increase in hole concentration and a reduction of the electrons, and an increase of Vg creates more electrons and fewer holes. p- and n-type Hall bars are fabricated in one process run by use of UV lithography. These are schematically shown in ﬁgure 1a. The Hall bar mesas, comprising the QW, are deﬁned by wet etching, the width of the Hall bars are 50 µm. The acceptor layer is removed by subsequently wet etching the n-Hall bars. Contact to the back-gate (not shown) is realized from the top, by wet etching through the superlattice barrier, and hereby reaching the n-doped substrate. On the p-Hall bar, p-type contacts are applied as a Au/Zn/Au (30nm/50nm/60nm) sequence, and annealed for three minutes at 420◦ C in a forming gas atmosphere (N2 :90%, H2 :10%). n-type contacts of Ni/AuGe(12%)/Ni (6nm/160nm/16nm) are fabricated, to n-Hall bars and the back-gate. They are annealed at 420◦C for one minute in a forming gas. 3. Transport measurements of the 2DEG and 2DHG Carrier transport measurements are performed at liquid He temperature, T = 4.2 K. For all measurements presented here Vg is between −2 and 12 V, where the leakage current to the gate is negligible (less than 1 nA). The linear Hall-eﬀect is measured in the p- and n-Hall bars in a magnetic ﬁeld up to 2 T with standard lock-in technique. Shubnikov de Haas oscillations in the 2DEG have also been observed. We discuss the carrier densities, n and p, as plotted in ﬁgure 2a as a function on Vg . The carrier densities cross for Vg ≈ 8.5 V, for lower Vg the 2DHG is dominant and for higher Vg the 2DEG is the dominant. There are linear relations between carrier densities and Vg , as there must be due to the ﬁeld eﬀect, this is indicated by the lines in Fig. 2a. The eﬃciency of the capacitative coupling between the gate and the 2D carriers is indicated by the slope of the these lines: −2.3 · 1010 cm−2 /V for holes, and 3.0 · 1010 cm−2 /V for the electrons. The control of the 2DHG and the 2DEG with Vg qualitatively agrees with the simulations of the band edge diagram for p- and n-type devices from the wafer, ﬁgure 1b and c. The electron and hole mobilities increase with the carrier concentrations in the region of interest. They are determined to be µp = 0.5 → 3.4 · 105 cm2 /Vs in the p-Hall bar for Vg < 12 V and µn = 1.8 → 4.0 · 105 cm2 /Vs in the n-Hall bar for Vg > 7 V. In the interval of Vg where both the 2DHG and the 2DEG exist the mobilities are as high as 2 · 105 cm2 /Vs with carrier densities in the order of 1011 cm−2 .

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Figure 2. From measurements of the Hall eﬀect, the carrier densities and mobilities are determined in p- and n-Hall bars. (a) The carrier densities for holes () and electrons (•) as a function of Vg . The lines are linear ﬁts, with slopes of −2.3 · 1010 cm−2 /V and 3.0 · 1010 cm−2 /V for holes and electrons, respectively. (b) The mobilities for holes () and electrons (•) are plotted as a function of Vg .

4. Summary We have demonstrated that it is possible to simultaneously have a 2DHG and a 2DEG in the very same QW as controlled by Vg . We are about to explore the light emission from the lateral pn diode, formed by making a junction between an n- and a p-type conﬁguration of the QW. Our results are an important step towards the development of various types of photon sources both coherent and incoherent. Acknowledgments Useful discussions with Søren Stobbe and Pawel Utko are greatly appreciated. This work is supported by NTT Joint Research Grant and by the Danish Research Council (FTP).

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References 1. T. Saiki, S. Mononobe, M. Ohtsu, N. Saito, and J. Kusano, Appl. Phys. Lett. 67, 2191 (1995). 2. Y. Hirayama, Jpn. J. Appl. Phys. 35, L1245 (1996). 3. P. O. Vaccaro, H. Ohnishi, and K. Fujita, Appl. Phys. Lett. 72, 818 (1998). 4. B. Kaestner, J. Wunderlich, D. G. Hasko, and D. A. Williams, Microelectronics J. 34, 423 (2003). 5. C. L. Foden, V. I. Talyanskii, G. J. Milburn, M. L. Leadbeater, and M. Pepper, Phys. Rev. A 62, 11803(R) (2000).

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SIMULATION OF THE PHOTON-SPIN QUANTUM STATE TRANSFER PROCESS

Y. RIKITAKE1,2 , H. IMAMURA1,2 AND H. KOSAKA1,3 1

CREST-JST, 4-1-8 Honcho, Kawaguchi, Saitama 322-0012, Japan Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan 3 Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan Email: [email protected] 2

We analyze the yield and ﬁdelity of the quantum state transfer(QST) from a photon polarization qubit to an electron spin qubit in a semiconductor quantum dot. We use the modiﬁed one-dimensional light-atom interaction model which includes the parameters such as photon-dot interaction, tunneling rate of the hole and bandwidth of the incident photon. We ﬁnd the optimal conditions where both the high-yield and high-ﬁdelity QST is realized.

1. Introduction Quantum state transfer(QST) has drawn enormous attention as one of the key concepts in quantum information science 1,2,3,4 . Because the quantum information or qubit can take several diﬀerent forms, such as photon, nuclear spin of atoms, and electron-spin of quantum dots, we have to choose the right qubit for each process considering its merits and demerits. Photon is the most convenient medium for sharing quantum information between distant locations5,6 . On the other hand, electron spin in a semiconductor quantum dot is the most convenient medium for quantum gate and quantum memory 7,8,9,10 since the coupling among the qubits can be easily controlled by applying a gate voltage. It is then natural to study the QST from a photon qubit to an spin qubit to construct eﬃcient quantum information processing devices. In 2001, Vrijen and Yablonovitch proposed a spin-coherent semiconductor photodetector which transfer the quantum information from a photon-polarization (photon qubit) to an electron-spin (spin qubit)1 . Such a quantum spin-coherent photodetector is a basic element of a quantum repeater11,12,13 which enables us to expand the distance of quantum key distribution drastically. They showed that the wellknow optical orientation in semiconductor heterostructure can be used for QST with help of g-factor engineering 14,15,16,17,18,19,20,21,22 and strain engineering23,24,25 . The photo-detector has a quantum dot whose energy levels are shown in Fig. 1(a). The g-factor of the electron spin is tuned to be zero (ge = 0). By using the strain

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(a)

(b) ge ∼ 0

hh−

σ+

σ−

lh−

lh+

dot

continuum

ωJ glh μB

γh

photon Γd

hh+

x

x=0

Figure 1. (a)Energy levels of the quantum dot. From |lh+ state, the electron with |↑ (|↓) spin state is optically excited by the righthanded (lefthanded) circularly polarized photon |σ+ (|σ− ) (c)The theoretical model we consider. The model includes the photon-dot interaction Γd , the electron-hole exchange interaction in the dot ωJ and the tunneling rate of the hole γh .

engineering and applying a magnetic ﬁeld, the light hole state |lh+ becomes the topmost level of the valence band. According to the selection rule, the electron with spin up(down) state |↑ (|↓) is excited by the righthanded (lefthanded) circularly polarized photon |σ + (|σ − ) from |lh+ state. After elimination of the hole in |lh+ state, the quantum state of photon qubit α+ |σ + + α− |σ − is transferred to the state of the spin qubit α+ |↑ + α− |↓. We analyze the quantum dynamics of the QST process from the photon qubit to the spin qubit in such a spin-coherent semiconductor photo-detector, and then reveal the condition for the high-yield and high-ﬁdelity QST. 2. Model In order to describe the whole dynamics of the QST from the photon qubit to the spin qubit, we consider the modiﬁed one-dimensional light-atom interaction model26 shown in Fig. 1 (b). The incident photon propagates along the x-axis and interacts with the quantum dot (in a bad cavity) at the position x = 0. The interaction between the photon and the dot is expressed by the dipole relaxation rate Γd . The quantum dot is connected with a continuum of the hole through the tunneling barrier. Once the electron-hole pair is excited in the quantum dot, the hole escapes to the continuum with the rate γh . The electron-hole exchange interaction ωJ modiﬁes the orientation of the electron-spin while the hole is in the dot and therefore reduces the ﬁdelity of the QST. The wave function of the system is written as ϕEs (t)|Es + ψls (t)|ls, (1) dxφσ (x; t)|xσ + |Ψ(t) = σ=±

s=↑↓

s=↑↓

l

where |xσ denotes the photon state with the position x and polarization σ = ±, |Es the state of the electron-hole pair in the dot with the electron spin s =↑, ↓, and |ls the state that the electron with spin state s is in the dot and the hole is in the continuum state l. Here φσ (x; t), ϕEs (t) and ψls (t) are coeﬃcients to be determined by solving the Schr¨ odinger equation. For later use, we name the states

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represented by ﬁrst, second and last terms in Eq. (1), photon state, exciton state and spin state, respectively. The Hamiltonian of the system is expressed as H = Hph + Hph-dot + HE + HT + Hspin . Here Hph , HE , and Hspin are the Hamiltonians for the photon state, the exciton state and the spin state, respectively. They are deﬁned as (hereafter we set = 1) ck|kσkσ|, (2) Hph = σ=± k

HE =

(ωe + ωh )|EsEs| + ωJ {|E ↑E ↓| + |E ↓E ↑|},

(3)

s=↑↓

Hspin =

s=↑↓

(ωe + ωl )|lsls|,

(4)

l

where ωe , ωh , and ωl are the energy levels of the electron in the dot, the hole in the dot, and the hole in the continuum state l, respectively. The speed of light is c and we use the wavenumber representation k in Hph instead of the position representation x. We note that HE includes the electron-hole exchange interaction ωJ . Hph-dot is the Hamiltonian of the interaction between the photon and the dot, and HT is the tunneling Hamiltonian of the hole between the dot and the continuum. cΓd Hph-dot = {|kσEs(σ)| − |Es(σ)kσ|}, (5) i π σ=± k HT = iWl {|Esls| − |lsEs|}, (6) s=↑↓

l

where Wl is the tunneling coupling of the hole between the dot and the continuum state l. The spin index s(σ) represents the selection rule : s(+) =↑, s(−) =↓. The tunneling rate of the hole from the dot to the continuum is deﬁned as γh ≡ 2 l |Wl | δ(ω − ωl )/π. For simplicity, we assume that γh is constant. 3. Yield and Fidelity The dynamics of the wave function of the system |Ψ(t) (Eq. (1)) can be calculated by solving the Schr¨ odinger equation with the proper initial condition at the initial time t = ti . We consider the incident photon with the Gaussian wave packet as the initial state : φσ (r; ti ) = ασ

e−iωph r/c e−(r−r0 ) √ π 1/4 L

2

/2L2

,

(7)

where r0 (< 0) is the center position of the wave packet at t = ti , L the coherence length of the packet (L |r0 |) and ωph the center frequency of the photon. We assume ωph = ωe + ωh . The probability amplitudes α+ and α− characterize the superposition state of the photon qubit. The bandwidth of the incident photon is given by Δωph = c/L.

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Let us consider the ﬁnal state of the system for t → ∞ so that we will discuss the yield and the ﬁdelity of QST. We deﬁne the reduced density matrix of the electron state in the dot by tracing out the degree of freedom for the photon state and the hole state, ⎞ ⎛ ρ↑↑ ρ↑↓ 0 (8) ρ ≡ trph,hole |Ψ(∞)Ψ(∞)| = ⎝ρ↑↓ ρ↓↓ 0 ⎠ . 0 0 ρ00 The reduced density matrix ρ is 3 × 3 matrix, because there are three states for the electron in the dot, i.e., spin up state ↑, spin down state ↓, and no spin states 0. The eﬃciency of the QST is represented by the yield deﬁned as P = ρ↑↑ + ρ↓↓ = 1 − ρ00 . Then the reduced density matrix of the generated spin is deﬁned as

1 ρ↑↑ ρ↑↓ ρspin ≡ . (9) P ρ↑↓ ρ↓↓ Using ρspin , we obtain the ﬁdelity of the QST as F = ΨI |ρspin |ΨI , where |ΨI = α+ |↑ + α− |↓ is the ideal spin state which we desire. Figure 2 (a) shows the contour plots of the yield P as a function of Γd /Δωph and γh /Δωph . The left, middle and right panels show the plot for ωJ = 0, 0.5Δωph and 2.0Δωph , respectively. The quantum state of the incident photon qubit is |σ + , i.e., |ΨI = |↑. Given that we ﬁx Γd , the yield P is not a monotonic function of γh but it has a maxima at γh = Γd . If we increase γh and Γd independently, we can not always obtain higher yield. The matching condition γh = Γd is important for the high-yield QST. As shown in Fig. 2 (a), the condition Δωph Γd , γh is equally important for high yield. This condition is needed for the eﬀective energy transfer from the incident photon to the exciton. Although the plots of the yield for the diﬀerent ωJ are qualitatively similar, they are quantitatively diﬀerent from each other. We can say that the yield becomes lower with increasing ωJ . If the electron-hole exchange interaction is taken into account, the excitation energy of the exciton in the dot splits into ωph + ωJ and ωph − ωJ . Therefore the large ωJ leads to the suppression of the yield due to the large detuning. From the above discussion, the condition for the high-yield QST is summarized as ωJ Δωph Γd ∼ γh .

(10)

Next we discuss the ﬁdelity of the QST. Figure 2 (b) shows a contour plots of the ﬁdelity for the case ωJ = 0 (left panel), case ωJ = 0.5Δωph (middle panel), and ωJ = 2.0Δωph (right panel). If there is no electron-hole exchange interaction (left panel), the spin state of the created electron is preserved, then the ﬁdelity F is unity for the all range of Γd and γh . In the case of ωJ Δωph (middle panel), we can see that the ﬁdelity F is reduced from unity for Γd + γh < ωJ . From the analysis of the dynamics, we can estimate the life time of the exciton state ϕEs (t) as (Γh + γh )−1 for the impulse like incident photon wave packet. In order to avoid the eﬀect of the

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(a) Yield P

(b) Fidelity F

Figure 2. Contour plots of (a)yield and (b)ﬁdelity. The horizontal axes represents the dipole relaxation rate Γd and vertical axes represents the tunneling rate of the hole γh . Both axes are normalized by the bandwidth of input photon Δωph . The left, middle and right panels show the plot for ωJ = 0, 0.5Δωph and 2.0Δωph , respectively.

electron-hole exchange interaction on the ﬁdelity, the life time of the exciton state should be much shorter than the spin ﬂip time ωJ−1 . Therefore Γh + γh ωJ is needed for the high-ﬁdelity QST. In the case of ωJ Δωph , the spin orientation is modiﬁed by the electron-hole exchange interaction while the photon wave packet passes through the dot. Therefore the ﬁdelity becomes low for the large range of Γd and γh as shown in the right panel. The condition for the high-ﬁdelity QST is summarized as ωJ Δph , Γd + γh .

(11)

When the condition for the high yield (Eq. (10)) is satisﬁed, the condition for the high yield (Eq. (11)) is automatically satisﬁed. Thus the ideal condition where we can realize the both high-yield and high-ﬁdelity QST is given by Eq. (10). 4. Summary We have discussed the yield and ﬁdelity of the quantum state transfer from the photon qubit to the spin qubit in the spin-coherent semiconductor photo-detector. By considering the modiﬁed one-dimensional light-atom interaction model, we analyze the quantum dynamics of the QST and then clarify the conditions for the high-yield and the high-ﬁdelity QST. It is found that the condition ωJ Δωph Γd ∼ γh should be satisﬁed for the ideal QST.

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Acknowledgments We acknowledge T. Takagahara for valuable discussions. This work was supported by CREST, MEXT.KAKENHI (No. 16710061), NAREGI Nanoscience Project, and NEDO Grant. References 1. R. Vrijen and E. Yablonovitch, Physica E10, 569 (2001). 2. L. Duan, M. Lukin, J. Cirac, and P. Zoller, Nature, 414, 413 (2001). 3. H. Kosaka, D. Rao, H. Robinson, P. Bandaru, K. Makita, and E. Yablonovitch, Phys. Rev. B67, 045104 (2003). 4. S. Muto, S. Adachi, T. Yokoi, H. Sasakura, and I. Suemune, Appl. Phys. Lett. 87, 112506 (2005). 5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002). 6. C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004). 7. D. Loss and D. DiVincenzo, Phys. Rev. A57, 120 (1998). 8. J. Taylor, C. Marcus, and M. Lukin, Phys. Rev. Lett. 90, 206803 (2003). 9. J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossord, Science 309, 2180 (2005). 10. J. Petta, A. Johnson, A. Yacoby, C. Marcus, M. Hanson, and A. Gossard, Phys. Rev. B72, 161301 (2005). 11. H. Briegel, W. Dur, J. Cirac, and P. Zoller, Phys. Rev. Lett., 81, 5932 (1998). 12. L. Childress, J. Taylor, A. Sorensen, and M. Lukin, Phys. Rev. A72, 052330 (2005). 13. J. Taylor, W. Dur, P. Zoller, A. Yacoby, C. Marcus, and M. Lukin, Phys. Rev. Lett. 94, 236803 (2005). 14. A. Kiselev, E. Ivchenko, and U. Rossler, Phys. Rev. B58, 16353 (1998). 15. E. Ivchenko and A. Kiselev, Jetp Lett. 67, 43 (1998). 16. K. Matveev, L. Glazman, and A. Larkin, Phys. Rev. Lett. 85, 2789 (2000). 17. H. Kosaka, A. Kiselev, F. Baron, K. Kim, and E. Yablonovitch, Electron. Lett. 37, 464 (2001). 18. G. Salis, Y. Kato, K. Ensslin, D. Driscoll, A. Gossard, and D. Awschalom, Nature 414, 619 (2001). 19. G. Salis, Y. Kato, K. Ensslin, D. Driscoll, A. Gossard, and D. Awschalom, Physica E16, 99 (2003). 20. J. Nitta, Y. Lin, T. Akazaki, and T. Koga, Appl. Phys. Lett. 83, 4565–4567 (2003). 21. Y. Lin, J. Nitta, T. Koga, and T. Akazaki, Physica E21, 656 (2004). 22. J. Nitta, Y. Lin, T. Koga, and T. Akazaki, Physica E20, 429 (2004). 23. S. Lin, H. Wei, D. Tsui, J. Klem, and S. Allen, Phys. Rev. B43, 12110 (1991). 24. T. Nakaoka, T. Saito, J. Tatebayashi, and Y. Arakawa, Phys. Rev. B70, 235337 (2004). 25. T. Nakaoka, T. Saito, J. Tatebayashi, S. Hirose, T. Usuki, N. Yokoyama, and Y. Arakawa, Phys. Rev. B71, 205301 (2005). 26. K. Kojima, H. Hofmann, S. Takeuchi, and K. Sasaki, Phys. Rev. A68, 013803 (2003).

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MAGNETOTRANSPORT IN TWO-DIMENSIONAL ELECTRON GASES ON CYLINDRICAL SURFACES K.-J. FRIEDLAND, R. HEY, H. KOSTIAL, U. JAHN, E. WIEBICKE, K. H. PLOOG Paul-Drude-Institute for Solid State Electronics, Hausvogteiplatz 5-7, 10117 Berlin, Germany, A. VOROB’EV*1, JU. YUKECHEVA1, AND V. PRINZ1 Institute of Semiconductor Physics, Russian Academy of Science, Acad. Lavrentyev Ave. 13, 630090 Novosibirsk, Russia

We have fabricated high-mobility, two-dimensional electron gases in a GaAs quantum well on cylindrical surfaces, which allows to investigate the magnetotransport behavior under varying magnetic fields along the current path. A strong asymmetry in the quantum Hall effect appears for measurements on both sides of the conductive path. We determined the strain at the position of the quantum well. We observe ballistic transport in 8 µm wide collimating structures.

1. Introduction Free-standing strained semiconductors are of considerable interest for new fundamental research fields, such as current-induced spin polarization [1], micromechanics with semiconductor cantilevers [2] and optics in microtube ring resonators [3]. One important new experimental approach is the self-rolling of thin pseudomorphically strained semiconductor bilayer systems as proposed by Prinz and coworkers [4]. This method allows the fabrication of free-standing lamellas or tubes of cylindrical shape with a constant radius of curvature based on epitaxial heterostructures grown by molecular beam epitaxy (MBE). A promising application is the realization of heterostructures containing a two-dimensional electron gas (2DEG). Recently, tubes with a laterally structured 2DEG, e.g. in the form of a conventional Hall-bar with Ohmic and Schottky contacts on the unrolled part of the 2DEG were successfully fabricated [5,6]. This will allow to realize various structures with low-dimensional transport behavior on evenly curved surfaces. In the electrons’ coordinate system, this is translated in a motion of the electrons along different and/or varying magnetic field lines with a given spatial dependence along the rolling direction of the semiconductor heterostructure according to the angle ϕ between the surface normal and the magnetic field B⊥ = B0 cos (ϕ). An important parameter here is the low-temperature mean free path of the electrons lS = vF τ, where vF is the Fermi ∗

A.V. acknowledges support by INTAS (Grant No. 04-83-2575) and the kind hospitality of the Paul-Drude-Institute 1 Work partially supported by grant 04-02-16910 of the Russian Foundation for Basic research

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289 289 velocity and τ the scattering time. Ballistic transport occurs, when lS is comparable to a typical system size L. In addition, to observe new physical phenomena which are related to the special motion of electrons, or motion of electrons in varying magnetic fields, lS should be comparable or even larger than the radius of the tube r, on which the electron is moving. For high mobilities heterostructures based on (Al,Ga)As systems are usually used. However, a significant draw-back for free-standing heterostructures on the basis of (Al,Ga)As systems with the formation of an inherent new surface is the Fermi-level pinning in the middle of the gap. This results in a significant depletion of the 2DEG, which is simultaneously accompanied by enhanced fluctuations of the potential and, therefore, by a reduced mobility in the 2DEG. Here we are faced with an obstacle which could not be circumvented so far. On the one hand, we need a large distance of the new surface from the 2DEG in order to avoid degradation of the electron mobility due to surface states. On the other hand, the heterostructure packets should be as thin as possible to realize a minimized rolling radius of the tube. Both requirements are contrary to each other, therefore ballistic transport on curved surfaces was not yet observed. 2. GaAs based high mobility 2DEG’s on cylindrical surfaces 2.1. Screened high mobility 2DEG

1.0 0.8 0.6

δ

δ

0.4 0.2

EΓ1

0.0

EΓ0

EX0

-0.2 -0.4

4

X

X Γ

2

Density (10nm)

Potential, VΓ,X-Ef (eV)

We use a particular heterostructure, which was proven to significantly reduce the influence of the second surface on the mobility of the 2DEG in hybrid structures fabricated by epitaxial-lift-off and wafer-bonding [7].

-3

0 -30 -20 -10

0

10

20

Position (nm) Figure 1. Potential and charge distribution in a GaAs SQW cladded by GaAs/AlAs SPSLs, calculated from a self-consistent solution of the Poisson and Schrödinger equations, including Γ and X conduction band states. δ marks the positions of the δ-doped layers.

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290 290 The high-mobility 2DEG (HM2DEG) is located in a 13-nm-wide single quantum well (SQW) with barriers consisting of short period AlAs/GaAs superlattices (SPSL), grown by (MBE) [8]. The superlattice period is chosen sufficiently short in order to ensure, that the X-like conduction-band states are lowest in energy in the AlAs compound of the SPSL. At sufficiently high doping concentration, these states are occupied with heavy mass X electrons, which are located close to the doping layer (Figure 1). The heavy mass of the carriers provides a high screening capability. Additionally, their Bohr-radius aB as well as their nominal distance from the doping layer is smaller or nearly equal to the average distance between the Si-atoms in the δ-like doping plane. Therefore, the X electrons can be very easily localized at the minima of the fluctuating potential, thus smoothing the fluctuations of the scattering potential (FSP). As a result, the mobility of the electrons in the GaAs SQW with X electrons in the SPSL can be considerably increased. The new concept for rolled-up films is that these X electrons also screen the high-mobility 2DEG from new surface states and its related FSP. The HM2DEG package with an overall thickness of 192 nm was pseudomorphically grown on top of a 20 nm thick In0.15Ga0.85As stressor layer forming the strained multilayered films (SMLF). An additional 50-nm-thick AlAs sacrificial layer is introduced below the SMLF in order to release the SMLF from the substrate. Conventional Hall-bar structures with current flow along the [1 0 0] direction were fabricated photolithographically on the original substrate prior to the detachment process of the SMLF by shallow mesa etching and alloying of the Ohmic contacts. The magnetotransport measurements for a flat structure at a temperature of 0.3 K reveal a parallel conductivity in the GaAs and (In,Ga)As quantum wells. For the GaAs SQW we estimate an electron density of nGaAs = 7.1 ×1015 m-2 with a mobility of about 120 and 50 m2/Vs along the [1 1 0] and [1 0 0] directions respectively. The anisotropy of the mobility results from an anisotropic surface corrugation on the substrate [9]. 2.2. Rolled-up 2DEG with contacts for magnetotransport measurements Part of the SMLF containing the Hall-bar was detached by selective etching with an HF acid/water solution and directionally rolled up along the [100] direction in a tube with a visible outer radius of about 20 µm. Details of the fabrication process of the tubes are described in [10]. As a result, we obtain high-mobility 2DEG Hall-bar devices on rolled tubes. At low temperatures, we observe only one conductive channel, namely the one in the GaAs SQW. The electron densities and mobilities range from (5 - 7) × 1015 m-2 and (18 - 57) m2/Vs respectively. These values correspond to a mean free path length lS between 1 - 3 µm. The (In,Ga)As QW is fully depleted due to the close distance to the newly created surface.

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291 291 2.2.1. Ballistic transport in free-standing high-mobility 2DEG For a device with the longest lS, we chose transversal Hall-terminals for a nonlocal resistance measurement, when the electrons move ballistically, which resembles electronbeam collimation at two opposite quantum point contacts [11]. The rolled tube is oriented in such a way that the magnetic field is parallel to the surface normal at the Hall-cross position. The layout is schematically shown as an inset in Figure 2. Terminal 1 and 3 are source and collector contacts, respectively. A current of 50 nA is driven along the terminals 1 and 2, the voltage is measured between the terminals 4 and 3. The distance between source and collector is 8 µm, while the width of terminals 1 and 3 is 2 µm.

Figure 2. Bend resistance as a function of magnetic field at T = 0.3 K. The left inset displays the experimental layout. Right inset: Cross-sectional scanning electron micrograph of a rolled-up film. The position of the Hall-cross is schematically shown.

Due to the long lS, electrons are collected at the opposite terminal at zero magnetic fields, causing a negative bend resistance (NBR). A small magnetic field bend off the ‘beam’ from the opposite terminal, thereby reducing the NBR [12]. At moderate magnetic fields, rebound trajectories of ballistic electrons cause a positive resistance, which disappears when all electrons are bent into the current terminal by high fields. This experiment demonstrates only the high quality of the 2DEG rather then carrying information about the ballistic transport in curved 2DEG structures. The ratio of the length of the mean free path to the rolling radius R is still rather low with a value of about 0.25.

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292 292 2.2.2. Quantum Hall effect in high-mobility 2DEG’s with varying magnetic field along the current direction In a 16-µm-wide Hall-bar structure, we do not observe ballistic transport behaviour anymore. By orienting the Hall-bar with a current along the rolling direction, the measurement of the Hall resistances RH1 and RH2 at two adjacent transversal terminal pairs allow to derive the rolling radius r through estimates of the local angles ϕΗi between the field and the surface normals from RHi = R0 cos(ϕΗi). R0 is the Hall resistance at a perpendicular magnetic field. If the magnetic field is perpendicular to the surface at a position within the adjacent transversal terminal pairs, the rolling radius r can be determined from ϕΗ1 - ϕΗ2 = l/r, where l is the distance between the Hall-terminals. For our tubes, we estimate r to be about 18 µm. The rolling radius rTh can also be predicted from continuum strain theory using [13]

h 4 + 4 χ h13 h2 + 6 χ h12 h22 + 4 χ h1 h23 + χ 2 h24 rTh = 1 6 εχ ( 1 + υ )h1 h2 ( h1 + h2 )

(1)

For the present layer system with the thicknesses of the stressor layer and the stack h1 = 18.7 nm and h2 = 156 nm, respectively, the ratio of Young’s module χ, the Poisson ratio υ and the strain ε, we calculate rTh = 23.9 µm, which is close to, but systematically larger than the experimentally observed rolling radius r. The reason for this deviation is not clear yet, a more systematic study is necessary. From strain theory, we can estimate the strain at the position of the GaAs SQW, which is compressive with a value ε = 5·10−4. At the original surface, the compressive strain is about one order of magnitude larger.

1.5

Hall effect 1 - ϕH1=24°

1

2 - ϕH2=26°

2

12 10

1.0

3 6

ρxy (kΩ)

ρ xx (kΩ )

8

Magnetoresistivity 3 - right side 4 - left side

4 0.5

4

2 0

0.0 0

2

4

6

8

10

12

14

magnetic field (T) Figure 3. Transversal (1,2) and longitudinal (3,4) magnetoresistivities for the right and left sides of the Hall-bar on a cylindrical surface. The current and magnetic field directions are described in the text. T = 0.3 K.

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293 293 Despite the rather large strain and the varying magnetic field projection over the current path, spin-split integer quantum Hall states remain observable for a perpendicular magnetic field at a position near the middle of adjacent transversal terminals, see Figure 3. Even the fractional Hall effect starts to develop at fields as high as 12 T. In this case, the difference between the Hall voltages remains sufficiently low and does not deform the equipotential lines too much across the conductor width. Nevertheless, even for this case, a strong asymmetry occurs in the magnetoresistivy for measurements on both sides of the Hall-bar, which we refer to as the right and the left sides in Figure 3. This asymmetry was interpreted in terms of current path bunching towards one of the Hall-bar edge [10] at classical magnetic fields, while for quantizing nonuniform magnetic fields an explanation was given [14], which takes into account the bending of the Landau edge states into the bulk. References 1. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett 93, 176601 (2004). 2. H. Yamaguchi, S. Miyashita, and Y. Hirayama, Physica E 24, 70, (2004). 3. T. Kipp, H. Welsch, Ch. Strelow, Ch. Heyn, and D. Heitmann, Phys. Rev. Lett 96, 077403 (2006). 4. V. Ya. Prinz, V. A. Seleznev and A. K. Gutakovsky, Proc. 24th ICPS, Jerusalem, 1998, World Scientific, Singapore, 1998, pp. Th3-D5; V. Ya Prinz, V. A. Seleznev, A. K. Gutakovsky, A. V. Chehovskiy, V. V. Preobrazhenskii, M. A. Putyato and T. A. Gavrilova. Physica E 6, 828 (2000). 5. A. B. Vorob’ev, V. Ya. Prinz, Yu. S. Yukecheva, and A. I. Toropov, Physica E 23, 171 (2004). 6. S. Mendach, O. Schumacher, Ch. Heyn, S. Schnüll, H. Welsch, and W. Hansen, Physica E 23, 274 (2004). 7. K.-J. Friedland, A. Riedel, H. Kostial, M. Höricke, R. Hey, and K. H. Ploog, J. of Electronic Materials 90, 817 (2001). 8. K.-J. Friedland, R. Hey, H. Kostial, R. Klann, K. Ploog, Phys. Rev. Lett. 77, 4616 (1996). 9. K.-J. Friedland, R. Hey, O. Bierwagen, H. Kostial, Y. Hirayama, and K. H. Ploog, Physica E 13 642 (2002). 10. A. B. Vorob’ev, K.-J. Friedland, H. Kostial, R. Hey, Yu. S. Yukecheva, U. Jahn, E. Wiebicke, and V. Ya. Prinz, will be published elsewhere. 11. A. S. Heindrichs, H. Buhmann, S. F. Godijn, amd L. W. Molenkamp, Phys. Rev. B57, 3961 (1998). 12. Y. Takagaki, G. Gamo, S. Namba, S. Ishida, S. Takaoko, K. Murase, K. Ishibashi, and Y. Aoyagi, Solid state Commun. 68, 1051 (1988).

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294 294 13. M. Grundmann, Appl. Phys. Lett. 83, 2444 (2003). 14. S. Mendach, O. Schumacher, H. Welsch, C. Heyn, M. Holz, and W. Hansen, Appl. Phys. Lett., in press (2006); Stefan Mendach, Dissertation, Fachbereichs Physik, Universität Hamburg (2005).

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FULL COUNTING STATISTICS FOR A SINGLE-ELECTRON TRANSISTOR AT INTERMEDIATE CONDUCTANCE

¨ 2,3 YASUHIRO UTSUMI1,2 , DMITRI S. GOLUBEV3,4 , AND GERD SCHON 1

3

Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan 2 Institut f¨ ur Theoretische Festk¨ operphysik and DFG-Center for Functional Nanostructures (CFN), Universit¨ at Karlsruhe, 76128 Karlsruhe, Germany Forschungszentrum Karlsruhe, Institut f¨ ur Nanotechnologie, 76021 Karlsruhe, Germany 4 I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physics Institute, 119991 Moscow, Russia

We evaluate the full counting statistics for a single-electron transistor with intermediate strength of the tunnel conductance. Using the Schwinger-Keldysh approach and the Majorana representation we develop an approximation, exact up to the second order in the tunnel conductance, which accounts for the renormalization of system parameters consistent with the renormalization group analysis. Out of equilibrium, the quantum ﬂuctuations of charge induce a lifetime broadening of charge-state levels, which suppress large current ﬂuctuations.

1. Introduction Noise measurements out of equilibrium are a powerful tool to characterize the mesoscopic quantum transport beyond the the linear response regime. Recently, signiﬁcant progress has been made in measuring the third cummulant for tunneling current 1,2 . Properties of the distribution of current ﬂuctuations through a quantum dot (QD), measured by using a quantum point contact detector, have also been studied 3,4 . The driving force behind these experiments is the theory of the ‘full counting statistics’ (FCS), pioneered by Levitov and Lesovik 5 and further developed during the last decade 6,7 . The measurement setup of the FCS scheme consists of a mesoscopic conductor (G), a voltage source (V) and an Amperemeter (A) [Fig. 1 (a)]. A typical signal is schematically plotted in Fig. 1 (b). From a single measurement, the number t /2 of transmitted electrons q = −t0 0 /2 dtI(t)/e through G during the measurement time is obtained. From repeated measurements the distribution P (q) [Fig. 1 (c)] is obtained. The FCS theory deals with the cumulant generating function (CGF) or the characteristic function, the discrete Fourier transformation of the distribution P (q), W (λ) =

∞

P (q)eiqλ =

q=−∞

∞ 1 δq n (iλ)n . n! n=0

(1)

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Most studies in the past have been devoted to the average current I = eδq/t0 and the zero-frequency noise SII = 2e2 δq 2 /t0 , related to the peak position and width of the distribution [Fig. 1 (c)]. Higher cumulants with further information about the distribution, such as the asymmetry, the sharpness, etc. have not been studied. More recently, it has been recognized that the FCS of interacting mesoscopic systems provides valuable information about nonequilibrium current ﬂuctuations. Early eﬀorts in this direction were based on solvable models using Bosonization techniques. One system studied is an open QD coupled to two reservoirs of spinless fermions by single-channel point contacts 8 . The model was used for modeling an adiabatic quantum pump 9 . The same model with one reﬂectionless contact was analyzed 10 to deal with the FCS theory of the dynamical Coulomb blockade system 11 . Another system studied is the generalized two-channel Kondo model modeling a QD in the Kondo regime. The exactly solvable Toulouse point 12 and corrections around the point 13 have been analyzed. The FCS has also been adopted to characterize the statistical property of particles with fractional charges 5/3 e 14 predicted for the Kondo dot in the unitary limit 15 . (a) )

(b)

I(t) 8

(c)

δ q4 δ q2

P(q)

#

t0

δ

t

q3

δq

q

Figure 1. (a) A schematic picture of measurement setup consisting of a mesoscopic conductor G, a voltage source V and an Amperemeter A. (b) The time evolution of measured current signal. (c) The distribution of the number of transmit electrons q.

Despite the mentioned attempts, the understanding of the FCS for interacting QDs is far from complete. In what follows we discuss the FCS of a ‘single-electron transistor’ (SET), a small metallic island coupled to source and drain electrodes through tunnel junctions [Fig. 2 (a)] for an intermediate strength conductance regime. 2. Bose-Kondo Model and Cumulant Generating Function The SET consists of a metallic island coupled to left (drain), right (source) and gate electrodes [Fig. 2(a)]. The island exchanges electrons with left, right electrodes through tunnel junctions with resistance RL and RR . It also capacitively couples with the three electrodes with small total capacitance CΣ = CL +CR +CG . The resulting single-electron charging energy EC = e2 /(2CΣ ) can exceed the temperature. Figure 2(b) shows the charging energy for charge neutral state |0 and with excess charge ±e, |±1 as a function of the gate-induced charge QG . For most values of QG , there is an unique ground state and at zero temperature the island is insulating. But around QG = 1/2, if the energy diﬀerence between the charge-number

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states |1 and |0, Δ0 = EC (1−2QG /e), reaches the condition μR < Δ0 < μL where μr = κr eV is the voltage drop in the junction r [and κL/R = ±CR/L (CL + CR )−1 ], electrons can tunnel through the island. TL e

CL V/2 VG

i κRϕ

TR e

CG C R

IR

(b)

(c)

1

0

0.5

−1

-V/2 0

-1

Δ0 0

QG/e

G

i κLϕ

IL

E/EC

(a)

∝ 1/ ln(T / TK )

1 1

T

Figure 2. (a) The equivalent circuit of a SET transistor. (b) The charging energy as a function of the gate charge QG . (c) A schematic plot of the temperature dependent conductance at Δ0 = 0.

The physics characteristic for strong electron-correlations emerges when the (effectively parallel) tunneling resistance RT = RL RR /(RL +RR ) is reduced to a value of the order of the resistance quantum RK = h/e2 and quantum ﬂuctuations of the charge get strong16 . Main consequences are a logarithmic renormalization of the charging energy and of the conductance, governed by the dimensionless conductance α0 ≡ RK /(2π)2 RT . If the inverse RC time is smaller than the charging energy, (EC RT CΣ )−1 ∝ α0 1, the charge-state levels are well resolved and two charge states |0 and |1 are dominant. Then the SET is mapped onto the anisotropicmultichannel Kondo model 17 . The perturbative renormalization group (RG) theory predicted the renormalization factor z0 = 1/{1 + α0 ln(EC /Λ)}17 where Λ is the energy cutoﬀ. The logarithmic temperature dependence of the linear conductance [Fig. 2 (c)] is conﬁrmed experimentally 18 . The theory for the average current accounting for the renormalization was developed using the real-time diagrammatic technique 19 . An extension of this technique to the FCS theory has been provided in Ref. 20 up to the second order in α0 . In order to correctly account for the logarithmic renormalization consistent with the RG analysis, ﬁnite-order perturbation theory is not enough. In the following, we develop an approximation using the Schwinger-Keldysh generating functional approach24,25,26. The starting point is the Bose-Kondo model with anisotropic coupling and the ‘magnetic ﬁeld’ derived from the Kondo model for lage number of channels Nch counting spin and transverse modes. The charging energy and the tunneling term read in the Majorana representation21,22,23,27,28 i Sch = dt{c∗ (i∂t −Δ0 )c + φ∂t φ}, St ≈ − dtdt c∗ (t)φ(t) α(t, t ) φ(t )c(t ). 2 C C Here c and φ are Dirac and Majorana fermions. C is the closed time-path shown in Fig. 3(a). The particle-hole Green function α = αL +αR describes the tunneling of an electron from either electrode into the island. In the Keldysh space, they are

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expressed by 2×2 matrices, α ˜ r (ω) = −iπαr0

2 (ω −μr ) EC 2 (ω −μr )2 + EC

0 −1 r 1 2 coth ω−μ 2T

, αr0 =

RK , 4π 2 Rr

(2)

where the high-energy cutoﬀ EC is introduced to regularize the UV divergence. To obtain CGF, ﬁrst the counting ﬁeld λ is introduced α ˜ λr (ω) = iκr λτ 1 /2 −iκr λτ 1 /2 α ˜r (ω)e where [τ 1 ]ij = 1 − δij is the Pauli matrix. Then the e CGF is logarithm of the ‘partition function’. The path-integral is performed as n

(−1)n δ W (λ) ∗ i S(λ) λ δ Tr gc α = D[c , c, φ] e = exp − e n δJ δJ n −1 i d1d2 J(1)gφ (1, 2)J(2) eTr[ln gc ] , (3) × exp 2 C J=0 where J is a Grassmann number and αλ is the particle-hole Green function containing the counting ﬁeld. gc and gφ are Dirac and Majorana Green functions, respectively. Further calculations are based on the diagrammatic expansion 28 . The diagrams shown in Fig. 3(b) are taken into account in our approximation 27 , t0 W (λ) = dω ln{1 + T F(ω)f (ω − μL )[1 − f (ω − μR )](eiλ −1) 2π + T F(ω)f (ω − μR )[1 − f (ω − μL )](e−iλ −1)}, f (ω) = 1/(eω/T + 1). (4) It looks similar to the Levitov-Lesovik formula for noninteracting systems 5 , but the eﬀective transmission probability T F (ω) accounts for the strong quantum ﬂuctuations of charge, T F(ω) = (2π)2

ω − μR αL0 αR ω − μL 0 (ω − μL )(ω − μR ) coth , coth 2 |ω − Δ0 − r=L,R ΣR (ω)| 2T 2T r ω −μr r (ω − μr ) coth 2T . ΣR (ω) = α dω r 0 ω − ω + i0

(5) (6)

Equation (4) is fully consistent with former results obtained by other techniques: (i) The expression for the average current obtained by the real-time diagrammatic technique 19 can be reproduced. So the approximation properly accounts for the parameter renormalization consistent with the RG result 17 . (ii) The ﬁrst order expansion in α0 reproduces the result by the Master equation approach by Bagrets and Nazarov 29 . (iii) The exact result up to the second order in α0 by Braggio et al. can be also reproduced. 3. Renormalization and Lifetime Broadening Figure 4(a) shows the current (I = eq/t0 ) distribution for a symmetric SET. The eﬀect of the renormalization can be seen in the panel (a). Since z0 decreases with increasing α0 , the mean value of the current, i.e. the position of the peak of the distribution, shifts to lower values. Actually, the CGF is approximated by that

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−∞

C−

Cτ −∞ − iβ

+∞

t

(b) − 13

− 12

−

(a)

−

Σ

=

Figure 3. (a) The closed time-path going from −∞ to ∞ (C+ ), going back to −∞ (C− ), connecting the imaginary time path Cτ and closing at t = −∞ − i/T . (b) Diagrammatic expansion of W (λ). Solid, dashed and wavy lines are for Dirac, Majorana and particle-hole Green functions.

for ‘correlated’ Poissonian with renormalized conductance, W (λ) ≈ 2 q¯ (eiλ/2 − 1) where e¯ q /t0 = z0 G0 V /2 and G0 = 1/(RL +RR ). The renormalization eﬀect can be absorbed if we re-plot the same data with the vertical and horizontal axes normalized by the average current I rather than by G0 V /2 [panel (b)]. However, even after this procedure the three curves do not completely collapse to a single one. The remaining diﬀerences can be attributed to the non-Markovian eﬀect of the broadening of the charge-state levels due to their ﬁnite life time, which is described by the imaginary part of the self-energy, Eq. (6). The trend observed for large α0 is the suppression of the tail of the distribution representing the probability for current much larger than the average value. Then the distribution approaches Gaussian like proﬁle.

(a)

-1

-2

(b)

α 0 =5 ×10 -2 10 -2 10 -4 eV/E C =10 -6

1

I/ (G 0V/2)

-1

α 0 =10 -1 5×10 -2 10 -4 eV/E C =0.1 Δ 0 =0

Δ 0 =0

0

2

0

0

ln P (e/ Ι t 0)

ln P (2e/G0Vt 0)

0

-2 1

Ι/ Ι

2

Figure 4. The current distribution for various α0 at zero temperature. The axes are normalized by G0 V /2 for panel (a) and by the average current I for panel (b).

4. Summary Performing a partial resummation of diagrams of the Schwinger-Keldysh generating function we calculated the distribution of the current through a SET in the intermediate conductance regime. The approximation covers previous theories and results and extends them. The strong quantum ﬂuctuations cause a conductance renormalization. Moreover, out of equilibrium, they induce a lifetime broadening of the charge-state levels, thus suppressing large current ﬂuctuations in the tail of the current distribution. We thank D. Bagrets, A. Braggio, Y. Gefen, J. K¨onig, A. Shnirman for valuable discussions. YU was supported by the DFG “Center for Functional Nanostructures” and RIKEN Special Postdoctoral Research Program.

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References 1. B. Reulet, J. Senzier, and D. E. Prober, Phys. Rev. Lett. 91, 196601 (2003). 2. Yu. Bomze, G. Gershon, D. Shovkun, L. S. Levitov, and M. Reznikov, Phys. Rev. Lett. 95, 176601 (2005). 3. S. Gustavsson, R. Leturcq, B. Simovic, R. Schleser, T. Ihn, P. Studerus, K. Ensslin, D. C. Driscoll, and A. C. Gossard, Phys. Rev. Lett. 96, 076605 (2006). 4. T. Fujisawa, T. Hayashi, R. Tomita, Y. Hirayama (unpublished). 5. L. S. Levitov and G. B. Lesovik, JETP Lett. 58, 230 (1993); L. S. Levitov, H.-W. Lee, and G. B. Lesovik, Journal of Mathematical Physics, 37, 4845 (1996) . 6. Quantum Noise in Mesoscopic Physics, Vol. 97 of NATO Science Series II: Mathematics, Physics and Chemistry edited by Yu. V. Nazarov (Kluwer Academic Publishers, Dordrecht/Boston/London, 2003). 7. W. Belzig in CFN Lecture Notes on Functional Nanostructures Vol. 1, eds. K. Busch, A. Powell, C. R¨ othig, G. Sch¨ on, and J. Weissm¨ uller, Lecture Notes in Physics, Vol. 658 (Springer-Verlag, Berlin, 2004). 8. A. Furusaki and K. A. Matveev, Phys. Rev. Lett. 75, 709 (1995). 9. A. V. Andreev and E. G. Mishchenko, Phys. Rev. B 64, 23316 (2001). 10. M. Kindermann and B. Trauzettel, Phys. Rev. Lett. 94, 166803 (2005). 11. I. Saﬁ and H. Saleur, Phys. Rev. Lett. 93, 126602 (2004). 12. A. Komnik and A.O. Gogolin, Phys. Rev. Lett. 94, 216601 (2005). 13. A.O. Gogolin and A. Komnik, Phys. Rev. B 73, 195301 (2006). 14. A.O. Gogolin and A. Komnik, cond-mat/0604287. 15. E. Sela, Y. Oreg, F. von Oppen and J. Koch, cond-mat/0603442. 16. D. S. Golubev, J. K¨ onig, H. Schoeller, and G. Sch¨ on, Phys. Rev. B 56, 15782 (1997) and references therein. 17. K. A. Matveev, Sov. Phys. JETP. 72, 892 (1991). 18. P. Joyez, V. Bouchiat, D. Esteve, C. Urbina and M. H. Devoret, Phys. Rev. Lett. 79, 1349 (1997). 19. H. Schoeller, and G. Sch¨ on, Phys. Rev. B 50, 18436 (1994). 20. A. Braggio, J. K¨ onig, and R. Fazio, Phys. Rev. Lett. 96, 026805 (2006). 21. H. J. Spencer and S. Doniach, Phys. Rev. Lett. 18, 23 (1967) 22. P. Coleman, L. B. Ioﬀe, and A. M. Tsvelik, Phys. Rev. B 52, 6611 (1995). 23. A. Shnirman and Yu. Makhlin, Phys. Rev. Lett. 91, 207204 (2003). 24. K.-C. Chou, Z.-B. Su, B.-L. Hao, and L. Yu, Phys. Rep. 118, 1 (1985). 25. A. Kamenev in Nanophysics: Coherence and Transport, (Les Houches, Volume Session LXXXI) eds. H. Bouchiat, Y. Gefen, S. Gu´eron, G. Montambaux, and J. Dalibard, NATO ASI (Elsevier, Amsterdam, 2005). 26. A. Kamenev in Strongly Correlated Fermions and Bosons in Low-Dimensional Disordered Systems, eds. I. V. Lerner et al., NATO Science Ser. II, Vol. 72 (Kluwer, Dordrecht, 2002). 27. Y. Utsumi, D. S. Golubev, and G. Sch¨ on, Phys. Rev. Lett. 96, 086803 (2006). 28. Y. Utsumi, H. Imamura, M. Hayashi, and H. Ebisawa, Phys. Rev. B 67, 035317 (2003). 29. D. A. Bagrets, and Yu. V. Nazarov, Phys. Rev. B 67, 085316 (2003).

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CREATION OF SPIN-POLARIZED CURRENT USING QUANTUM POINT CONTACTS AND ITS DETECTION

M. ETO, T. HAYASHI, Y. KUROTANI AND H. YOKOUCHI Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected]

We theoretically study the creation of spin-polarized current using quantum point contacts (QPCs) in the presence of Rashba spin-orbit interaction. When the conductance is quantized in units of 2e2 /h, the current is spin-polarized in the transverse direction to the current ﬂow. This is due to the transition between subbands of diﬀerent spins during the transport through QPCs. No magnetic ﬁeld is required. To detect the spin polarization of the current, we examine the transport properties of a QPC connected to a ferromagnetic lead. We observe the maximum (minimum) conductance when the magnetization in the lead is parallel or antiparallel (perpendicular) to the spin-polarization direction of the current from QPC.

1. Introduction Injection of spin-polarized current into semiconductors is an important issue for the development of spin-based electronics, “spintronics.” To manipulate electron spins, the Rashba spin-orbit (SO) interaction is useful since its strength is controllable by applying an electric ﬁeld.1,2 Several spin-ﬁltering devices for producing the spin current have been proposed utilizing the SO interaction.3,4,5,6,7 In our previous paper,8 we have theoretically studied the ballistic transport through a quantum point contact (QPC) in the presence of Rashba SO interaction and shown that QPC can be a useful tool for the spin injection. No magnetic ﬁeld is required. We have found that the conductance is quantized in units of 2e2 /h unless the SO interaction is too strong and that the current is spin-polarized in the transverse direction. The spin-polarization ratio of more than 50% can be realized in InGaAs heterostructures. In this paper, we discuss the detection of the spin-polarized current when the QPC is connected to a ferromagnetic lead. We show that the conductance is maximal (minimal) when the magnetization in the lead is parallel or antiparallel (perpendicular) to the spin-polarization direction of the current. The same conductance in the parallel and antiparallel alignments can be explained by the Onsager relation in the two-terminal measurement.

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2. Spin-polarized current using QPC 2.1. Numerical calculations Before the connection to a ferromagnetic lead, we show the spin-polarized current out of a QPC, by numerical calculations. We consider a two-dimensional electron gas conﬁned in the z direction. The electric ﬁeld in the z direction results in the Rashba SO interaction, HSO =

α (py σx − px σy ),

(1)

where σx and σy are Pauli matrices. The strength of the SO interaction is characterized by a dimensionless parameter, kα /kF , where kα = mα/2 with m being the eﬀective mass and kF is the Fermi wavenumber. In InGaAs heterostructures, α = (3 ∼ 4)×10−11 eVm and kα /kF ≈ 0.1.9,10 The electron-electron interaction and impurity scattering are neglected. Electrons propagate in a quantum wire along the x direction, with width W0 in the y direction (−W0 /2 < y < W0 /2 with hard-wall boundary condition). For a narrow constriction around x = y = 0, we consider a potential at −L1 < x < L2 , which is analogous to that adopted by Ando:11 V0 V (x, y) = 2

y − y± (x) 2 πx θ(±[y − y± (x)]), + EF 1 + cos Lx Δ ±

(2)

πx , 1 − cos Lx

(3)

with y± (x) = ±

W0 4

where Lx = L1 (−L1 < x < 0) and Lx = L2 (0 < x < L2 ). EF is the Fermi energy and θ(t) is a step function. [θ(t) = 1 for t > 0, 0 for t < 0]. We ﬁx W0 = 4λF and Δ = λF , where λF is the Fermi wavelength (λF = 2π/kF ). Numerical calculations are performed using a tight-binding model on a square lattice. The transmission coeﬃcients are evaluated for incident electrons from the left side of the constriction (x < −L1 ) to the right (L2 < x), using the Green function’s recursion method.11 They yield the conductance G through the Landauer formula. We ﬁnd that the conductance G is quantized in units of 2e2 /h when the QPC changes gradually in space and the SO interaction is not too strong (kα /kF < 0.5).8 We ﬁx V0 = 0.7EF in V (x, y), where G = 2e2 /h. We divide the output current into two components, one carried by spin-up electrons in the y direction (Sy = 1/2) and the other by spin-down electrons (Sy = −1/2). In Fig. 1(a), we plot the conductance by each component, G± (G = G+ + G− ), as a function of kα /kF . The spin polarization in the y direction, P = (G+ − G− )/(G+ + G− ), increases with an increase in kα /kF . It is 30% at kα /kF ≈ 0.1 in the case of L1 = L2 = 4λF (curve a). It is as large as 60% in the case of L1 = 4λF and L2 = 12λF (curve b).

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G (2e2 /h)

(a)

1

(b)

a

b

A

0.5

B

0

C

a

b 0

E

0.1 0.2 kα / k F

A BC

k

Figure 1. (a) Conductance G± through QPC for electrons with Sy = ±1/2 in the output current, as a function of kα /kF (solid lines for G+ and broken lines for G− ). V0 = 0.7EF in V (x, y), where G = G+ + G− = 2e2 /h. L1 = L2 = 4λF for curve a, whereas L1 = 4λF and L2 = 12λF for curve b. (b) Schematic drawings of subbands, En,+ (k) (solid lines) and En,− (k) (broken lines) with n = 1, 2, 3, in a quantum wire with a component of SO interaction, −(α/)px σy . Horizontal lines indicate EF (relative to the subbands) corresponding to three positions of electrons around QPC.

2.2. Spin polarization mechanism We explain the mechanism of spin polarization at QPCs. We divide the Hamiltonian into two parts: H = H0 + H , 1 2 α (p + p2y ) − px σy + V (x, y), 2m x α H = py σx ,

H0 =

(4) (5)

and treat H as a perturbation. In the quantum wire outside of the constriction, the eigenvalues of H0 are given by En,± (k) =

2 2 2 (k ∓ kα )2 − k + εn , 2m 2m α

(6)

where εn = (πn)2 /(2mW02 ) (n = 1, 2, 3, · · · ). The index ± denotes the spinup or -down in the y direction. The dispersion relations of En,± (k) (subbands) are schematically shown in Fig. 1(b). Unless the SO interaction is too weak, some of the intersections between En,− (k) and En ,+ (k) (n < n ) appear below EF (horizontal line A). This is the condition to create the spin-polarized current. In the transport through QPC, we assume an adiabatic motion of electrons except in the vicinities of the intersections: The wavefunction changes gradually retaining the quantum numbers of transverse motion n and of spin ±. The wavenumber k changes with x, which is determined by an intersection between the subband and EF ; positive k’s for incident electrons. As electrons propagate from the wire to a narrow constriction (x < 0), the subbands shift upwards. Alternatively, we move EF downwards, as horizontal lines B and C in Fig. 1(b). The conduction modes

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are (1, ±), (2, ±), (3, ±) at position A, (1, ±), (2, ±) at position B, and (1, ±) at position C. Similarly, at x > 0, we move EF upwards. Around the intersections between En,− (k) and En ,+ (k), the subbands are mixed by the perturbation H . Hence the modes change to each other with a transition probability P . The electrons pass by the intersections twice, once at x < 0 and once at x > 0. Let us look at an intersection between modes (1, −) and (2, +), surrounded by a square in Fig. 1(b). Around the ﬁrst pass (x < 0), both modes are occupied by electrons just before the modes cross each other. Then spin-up electrons are ﬂipped to spin-down with probability P , while spin-down electrons are ﬂipped to spin-up with the same probability. Accordingly, no spin polarization takes place. Around the second pass (x > 0), on the other hand, mode (2, +) is empty while mode (1, −) is full of electrons just before the modes cross. Then spindown electrons in the latter mode are spin-ﬂipped to spin-up in the former mode with probability P . The spin-up electrons in mode (1, +) are transmitted through the constriction without passing by any mode crossing. Consequently we obtain the spin-polarization ratio of [(1 + P ) − (1 − P )]/2 = P . Note that the total conductance is not aﬀected by the spin polarization unless the SO interaction is too strong.8 In Fig. 1(b), both the derivatives of E1,− (k) and E2,+ (k) are positive at their intersection. Since the group velocities have the same sign, the transition from one mode to the other is not accompanied by a reﬂection (forward scattering). The transition probability P is evaluated using the Landau-Zener theory:12,13 P = 1 − exp(−2πλ), where λ = J 2 /(|v|) represents the degree of adiabaticity. ∂ [E(2, +) − E(1, −)], “velocity” of the change of level J = |2, +|H |1, −| and v = ∂t spacing. P = 1 for λ = ∞ in the adiabatic limit, whereas P = 0 for λ = 0 in the sudden-change limit. If V (x, y) is a hard-wall potential with width W (x) in the y direction, λ is estimated as 1 dW −1 , (7) λ = kα · W dx apart from a numerical factor. W and its derivative should be evaluated at x where electrons pass by the intersection. In our model, λ increases with L2 . In Fig. 1(a), a larger spin-polarization is observed with larger L2 , in agreement with our theory. The polarization is independent of L1 (not shown here),8 in accordance with the previous discussion. 3. Detection of spin polarization We discuss the detection of spin polarization of the current from QPC. The electron spins have to be directly measured since the conductance is not inﬂuenced by the polarization. Possible experiments are an optical experiment using circularly polarized light (Kerr rotation),14 dynamical nuclear spin polarization through the hyperﬁne interaction, an injection into spin detectors of magneto-focusing15 or

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(b) θ

y z

x

G (2e2/h)

(a)

0.8

P=0.5

0.7 0.6

P=0.1

0.5 0

θ

π

Figure 2. (a) QPC connected to a ferromagnetic lead. The lead is a half-metal, the magnetization of which is directed to (sin θ sin ϕ, cos θ, sin θ cos ϕ); tilted by θ from spin polarization of the current from QPC (y axis). (b) The conductance G through the QPC and ferromagnetic lead, as a function of θ. G is averaged over ϕ, with the standard deviation indicated by error bars. The spin-polarization ratio at QPC is P = 0.1 and 0.5.

quantum dots,16 etc. Now we examine the possibility to detect the spin polarization by the connection to ferromagnetic leads. We study the transport through a complex system of a QPC and a ferromagnetic lead, shown in Fig. 2(a). We assume that the lead is a half-metal for simplicity. The magnetization direction is tilted by θ from spin polarization of the current from QPC (y axis) and given by (sin θ sin ϕ, cos θ, sin θ cos ϕ). We calculate the conductance by the S-matrix method, considering the lowest two subbands in the quantum wire. Electrons pass by the intersection between the subbands, the narrowest part of QPC, and the intersection again before they inject the ferromagnetic lead. The probabilities of inter-subband transition at the intersection are treated as parameters in the S-matrix formalism. The spin polarization of the current is determined by the probability P at the second pass through the intersection. We show the calculated results in Fig. 2(b). The conductance averaged over ϕ, G, is shown as a function of θ. The standard deviation is indicated by error bars. We choose P = 10% or 50%. When the magnetization in the lead is parallel or antiparallel to the spin-polarization direction of the current (θ = 0, π), G is the maximal. Then G is always given by G = (e2 /h)(1+P ). When the magnetization is perpendicular to the spin-polarization direction (θ = π/2), G is the minimal. This is ascribable to the interference of reﬂected waves between the QPC and ferromagnetic lead. Indeed the conductance largely depends on ϕ (or length between the QPC and lead, where the spin precession takes place around y axis). The same conductance in parallel (θ = 0) and antiparallel (θ = π) alignments can be understood by the Onsager relation in the two-terminal measurement because the two alignments are transformed to each other by the reverse of time.

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4. Conclusions We have examined the ballistic transport through a QPC in the presence of Rashba SO interaction and shown the creation of spin-polarized current. The spinpolarization ratio is determined by the adiabaticity of the transition between subbands of diﬀerent spins during the transport from a narrow region to a wide region. The polarization ratio can be as large as 60% with the SO interaction strength in InGaAs heterostructures. To detect the spin polarization of the current, we have examined the transport properties of a QPC connected to a ferromagnetic lead. The conductance is the maximal (minimal) when the magnetization in the lead is parallel or antiparallel (perpendicular) to the spin-polarization direction of the current from QPC. Acknowledgments The authors gratefully acknowledge discussions with G. E. W. Bauer. This work was partially supported by a Grant-in-Aid for Scientiﬁc Research in Priority Areas “Semiconductor Nanospintronics” (No. 14076216) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. References 1. E. I. Rashba, Fiz. Tverd. Tela (Lenningrad) 2, 1224 (1960) [Solid State Ion. 2, 1109 (1960)]. 2. J. Nitta, T. Akazaki, H. Takayanagi and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). 3. A. A. Kiselev and K. W. Kim, Appl. Phys. Lett. 78, 775 (2001). 4. T. Koga, J. Nitta, H. Takayanagi and S. Datta, Phys. Rev. Lett. 88, 126601 (2002). ˇ 5. Y. Stˇreda and P. Seba, Phys. Rev. Lett. 90, 256601 (2003). 6. T. P. Pareek, Phys. Rev. Lett. 92, 76601 (2004). 7. J. Ohe, M. Yamamoto, T. Ohtsuki and J. Nitta, Phys. Rev. B 72, 41308(R) (2005). 8. M. Eto, T. Hayashi, and Y. Kurotani, J. Phys. Soc. Jpn. 74, 1934 (2005). 9. D. Grundler, Phys. Rev. Lett. 84, 6074 (2000). 10. Y. Sato, T. Kita, S. Gozu and S. Yamada, J. Appl. Phys. 89, 8017 (2001). 11. T. Ando, Phys. Rev. B 44, 8017 (1991). 12. L. Landau, Phys. Z. Sowjetunion 2, 46 (1932). 13. C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932). 14. Y. K. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Nature 427, 50 (2003). 15. J. A. Folk, R. M. Potok, C. M. Marcus and V. Umansky, Science 299, 679 (2003). 16. J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen and L. P. Kouwenhoven, Nature 430, 431 (2004).

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DENSITY DEPENDENT ELECTRON EFFECTIVE MASS IN A BACK-GATED QUANTUM WELL

S. NOMURA,1,2,4, M. YAMAGUCHI,2,4 T. AKAZAKI,2,4 K. MIYAKOSHI,3,4 H. TAMURA,2,4 H. TAKAYANAGI,2−4 AND Y. HIRAYAMA2,5 1

Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, 305-8571, Japan 2 NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, 243-0198, Japan 3 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan 4 CREST-JST, 4-1-8 Honcho, Kawaguchi, 332-0012, Japan 5 SORST-JST, 4-1-8 Honcho, Kawaguchi, 332-0012, Japan E-mail: [email protected] The electron density dependent eﬀective mass of an electron in a 2-dimensional electron system is reported in a back-gated undoped GaAs/AlGaAs quantum well. The enhancement of the eﬀective masses is observed with decrease in the electron density by the transport and the photoluminescence measurements.

1. Introduction Two-dimensional electron systems (2DES) in the low electron density regime provides an excellent system to investigate many-body phenomena induced by electronelectron interaction. With decrease in the electron density (ns ), transition of the electron ground state to a ferromagnetic ﬂuid phase was predicted before transition to the Wigner crystal phase. 1 Currently, it is still a challenge to detect these transitions experimentally in GaAs heterostructures, however, there have been continuous eﬀorts to realize a high quality 2DES at low electron densities utilizing gated-undoped structures. 2,3 Precursors to the transitions such as the enhancement of the Lande g-factor, 4 and the electron eﬀective mass have been observed. 6 A back-gated undoped quantum well structure has advantages in investigating the properties of 2DES in the low electron density regime because of the high tunability of the electron density and the low degradation of the electron mobility at low ns , 3 as evidenced by the observation of jump of the optical transition energies and the peak photoluminescence (PL) intensities correlated with fractional ﬁlling factors ν at magnetic ﬁelds below 5 T. 5 In this paper, we report results of systematic measurements of the electron eﬀective mass (m∗e ) and the electron-hole reduced mass (μ∗ ) as functions of ns by the Shubnikov-de Haas (SdH) oscillation and the magneto-PL, respectively.

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2. Experimental The sample layer consists of n-GaAs, superlattice barrier, 20-nm Al0.33 Ga0.67 As layer, 50-nm GaAs well, and 250-nm Al0.33 Ga0.67 As layer. The sample has peak mobility of 3 × 106 cm2 /Vs. Au-Ge-Ni annealed contacts were patterned as Corbino disk geometry. The back-gate bias voltage was applied between the annealed contacts and the conducting n-GaAs layer. The resistivity ρxx of the Corbino device was measured over temperature below 1 K. The PL spectra of a 1 mm square mesa structure processed from the same wafer were measured at about 100 mK in perpendicular magnetic ﬁelds. 3. Results and discussions The oscillating part of the magnetoresistivity is obtained by ﬁtting the SdH curve to sinusoidal curve in the low magnetic ﬁeld regime, avoiding the quantum-Hall regime. The electron eﬀective masses are obtained by a log(Δρxx /T ) vs T plot as described in the literature. 6 Figure 1 shows the obtained m∗e as a function of ns . The obtained m∗e increases with decrease in ns . Although the structure and the peak mobility of our sample diﬀer form that in the literature, 6 the obtained m∗e agrees well each other. This shows that the increase in m∗e is due to the electron-electron interaction, ruling out the role of disorder. While the cyclotron resonance measurement is often applied to measure the eﬀective mass, Kohn’s theorem states that the only single-particle electron mass is obtained by this method. Here, as an alternative method, the eﬀective mass is estimated by the peaks in the PL due to the Landau-levels. As grown undoped QW sample exhibits PL due to a charged exciton and a neutral exciton. Both a charged exciton and a neutral exciton show diamagnetic shift in the magnetic ﬁelds. By applying the back-gate voltage, 2DES is starting to be formed. With increase in ns , a transition to the PL due to 2DES occurs. The PL shows linear dependence on magnetic ﬁeld above this transition. In this region, the electron-hole reduced mass μ∗ = ( m1∗ + m1∗ )−1 , where m∗h is the hole eﬀective mass, is obtained by the e h Landau-fan diagram in the magneto-PL spectra at ﬁlling factors ν > 2, typically below 1 T. At ν > 2, the dispersion is known to linear and the eﬀective mass is given by the slope of the dispersion. 7 Because the energy dispersion deviates to lower energies at ν < 2, this region is not used for estimating μ∗ . The obtained μ∗ increases with decrease in ns as shown in Fig. 2. For high densities since several Landau-levels are observed, m∗e and m∗h are obtained independently. For low densities m∗e are obtained by assuming that m∗h is independent of ns . The obtained m∗e increases with decrease in ns as also plotted in Fig. 2. Gekhtman et al. measured the electron eﬀective mass 8 by tuning ns by illuminating modulation doped QW samples above barrier band-gap. Their result is also plotted in Fig. 2. While the overall tendency of the increase of the m∗e with decrease in ns agrees each other, their estimated values are larger than our values.

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0.1 Tan et al., fit to whole amplitude

0.095

Ours, fit to whole amplitude

m* (m0)

0.09 0.085 0.08 0.075 0.07 0.065 0.06 1

ns (1010 cm-2)

10

Figure 1. The electron density dependence of the electron eﬀective mass obtained by SdH measurements (ﬁlled circles). The values in the literature are also shown (open squares) [6].

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µ* (m ) 0

0.17

m * (m ) Gekhtman et al. e

0

e

0

m * (m )

Effective mass (m0 )

0.15

0.13

0.11

0.09

0.07

0.05

bulk : µ* ~ 0.58 m 1

0

ns (1010 cm-2)

10

Figure 2. The electron density dependence of the reduced eﬀective mass (ﬁlled circles) and the electron eﬀective mass (open circles) obtained by magneto-PL measurements (ﬁlled circles). The values in the literature are also shown (open squares) [8].

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The source of the disagreement is not clear but may be attributed to the diﬀerence in the method of tuning ns . The m∗e obtained by PL measurements is found to be systematically larger than those obtained by SdH measurements in the low electron density regime of ns < 3.3 × 1010cm−2 . Further investigations may be necessary on the role of the optically created hole to the renormalized eﬀective mass. 4. Conclusions In this paper, the electron density dependence of the electron eﬀective mass and the reduced mass is investigated by the transport and the photoluminescence measurements. The m∗e obtained by PL measurements is found to be systematically larger than those obtained by SdH measurements in the low electron density regime. This result points out that the electron-hole interaction needs to be taken into account on the density-dependent electron eﬀective mass. Acknowledgments This work was partly supported by the Grant-in-Aid for Scientiﬁc Research from Japan Society for the Promotion of Science. References 1. C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G.B. Bachelet, Phys. Rev. Lett. 88, 256601 (2002). 2. B.E. Kane, L.N. Pfeiﬀer, K.W. West, and C.K. Harnett Appl. Phys. Lett. 63, 2132 (1993). 3. Y. Hirayama, K. Muraki, and T. Saku, Appl. Phys. Lett. 72, 1745 (1998). 4. E. Tutuc, S. Melinte, and M. Shayegan Phys. Rev. Lett. 88, 036805 (2002). 5. S. Nomura, M. Yamaguchi, T. Akazaki, H. Tamura, H. Takayanagi, and Y. Hirayama, (in press). 6. Y.-W. Tan, J. Zhu, H.L. Stormer, L.N. Pfeiﬀer, K.W. Baldwin, and K.W. West, Phys. Rev. Lett. 94, 016405 (2005). 7. I. Bar-Joseph, Chem. Phys. 318, 99 (2005). 8. D. Gekhtman, E. Cohen, A. Ron, and L.N. Pfeiﬀer, Phys. Rev. B 54, 10320 (1996).

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THE SUPERSYMMETRIC SIGMA FORMULA AND METAL-INSULATOR TRANSITION IN DILUTED MAGNETIC SEMICONDUCTORS

I. KANAZAWA Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan E-mail: [email protected] The localization and freezing mechanisms of hole-induced magnetic solitons in the randomly distributed system have been proposed, by using the eﬀective Lagrangian of diffusion modes, in diluted magnetic semiconductors.

1. Introduction Detailed measurement of transport very close to two metal-insulator critical points and large magnetoresistance (MR) eﬀects in a diluted magnetic semiconductor (Ga, Mn)As have been reported [1]. The ﬁrst metal-insulator (MIT) at the dilute-side of the metallic phase is associated with the appearance of the ferromagnetism. On the other hand, the second MIT at higher concentration is purely driven by disorder and disorder-modiﬁed Coulomb interaction. It has been strongly suggested that holes can change the strength of spin exchange among Mn ions. The key to clarify the microscopic mechanism for carrier-induced ferromagnetism is to understand the nature of the doped hole carriers as well as the exchange interaction between the holes in host valence band and localized orbitals of the magnetic ions, so called p-d exchange interaction. In addition, interesting phenomena such as the photoinduced magnetic polaron in diluted magnetic semiconductors have been discovered [2]. These works stimulated us to the study of the carrier-induced magnetic soliton, which is an interesting and challenging subject. Recently, the present author has discussed the localization mechanism in the MIT, using the gauge-invariant Lagrangian density for the hole-induced magnetic solitons [3,4]. In this study, we have discussed the freezing mechanism of magnetic solitons, extending the previous formula [3,4]. 2. A model system Ferromagnetic properties, in hole-doped diluted magnetic semiconductors, suggest strongly creating of the hole-induced ferromagnetic solitons, which might be induced by the p-d exchange interaction between Mn spins and balence band holes. It has

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been suggested that the ferromagnetic interaction induced by the hole seems to be cooperative and non-linear. In order to argue in the gauge-invariant formula, we shall introduce the non-linear gauge ﬁelds (Yang-Mills ﬁelds) Aaμ , which mediate the eﬀective ferromagnetic interaction induced by the hole. It has been proposed that the hedgehog-like soliton in three-dimensional system is speciﬁed by rigidbody rotation, which is related to gauge ﬁelds of SO(4) symmetry for S3 [5-8]. Thus it is thought that the non-linear gauge ﬁelds Aaμ introduced by the hole have a local SO(4) symmetry. Then we have assumed that the SO(4) quadruplet ﬁelds, Aaμ , are spontaneously broken around the doped hole through the Anderson-Higgs mechanism, in the III-V-based DMS with magnetic manganese ion-doping.We set the symmetry breaking 0|φa |0 = 0, 0, 0, μ of the Bose ﬁeld φa in the Lagrangian density as follows, After the symmetry breaking 0|φa |0 = 0, 0, 0, μ, we can obtain the eﬀective Lagrangian density, Lef f =

2 1 ∂i S j − g1 εiˆjk Abi S k 2 +ψ + (i∂0 − g2 Ta Aa0 ) ψ 2 1 + − ψ i∇ − g2 Ta Aa(μ=0) ψ 2m 2 1 − ∂ν Aaμ − ∂μ Aaν + g3 εabc Abμ Acν 4 2 1 + ∂μ φa − g4 εabc Abμ φc 2 1 + m21 (A1μ )2 + (A2μ )2 + (A3μ )2 2 +m1 A1μ ∂μ φ2 − A2μ ∂μ φ1 +m1 A2μ ∂μ φ3 − A3μ ∂μ φ2 +m1 A3μ ∂μ φ1 − A1μ ∂μ φ3 +g4 m1 φ4 (A1μ )2 + (A2μ )2 + (A3μ )2 −g4 m1 A4μ φ1 A1μ + φ2 A2μ + φ3 A3μ −

m2 g 4 m2 g 2 m22 (φ4 )2 − 2 φ4 (φa )2 − 2 24 (φa φa )2 , 2 2m1 8m1 (1)

where S j is the spin of Mn, ψ is the Fermi ﬁeld of the hole, m1 = μ · g4 , m2 = 2(2)1/2 λ · μ. Recent study [9] shows that carriers of the hole seem to be coupled to Mn spins by an antiferromagnetic Heisenberg exchange interaction. Thus ˆj corresponds to the reverse direction of the spin one of the hole. The eﬀective Lagrangian describes three massive gauge ﬁelds A1μ , A2μ , and A3μ , and one massless gauge ﬁeld A4μ . The generation function Z[J] for Green functions is shown as

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follows,

Z[J] =

+ ¯ DADBDSDCDCDψ DψDφ

· exp i d4 x (Lef f + LGF+FP + J · Φ) ,

1 LDF+FP = B a ∂ μ Aaμ + αB a B a + iC¯ a ∂ μ Dμ C a , 2

(2)

(3)

where B a and C a are the Nakanishi-Lautrup(NL) ﬁelds and Faddeev-Popov ﬁctitious ﬁelds, respectively. a a J · Φ ≡ J aμ Aaμ + JB B + JS · S + J¯Ca · C a + JCa¯ C¯ a

+ η¯ψ + ηψ + + Jφa φa

(4)

BRS-quartet [10, 11] in the present theoretical formula are (φ1 , B 1 , C 1 , C¯1 ), (φ2 , B 2 , C 2 , C¯ 2 ), (φ3 , B 3 , C 3 , C¯3 ), and (A4L,μ , B 4 , C 4 , C¯ 4 ). Where A4L,μ is the longitudinal component of A4μ . Thus we need these ﬁelds for the unitality condition, although these ﬁelds are unobservable and ﬁctitious ones. Because masses of A1μ , A2μ and A3μ are created through the Anderson-Higgs mechanism by introducing the hole, the ﬁelds A1μ , A2μ amd A3μ exist around the hole within the length of ∼ 1/m1 ≡ RC . From the ﬁrst term in Eq. (1), the spins S of Mn atoms are induced in the ferromagnet state, where the average spin is parallel to ˆj direction, within the length of ∼ RC around the hole. That is, the eﬀective Lagrangian represents that the ferromagnetically aligned Mn spins form clusters, in which the hole is trapped, with the radius, RC ∼ 1/m1 . Especially Katsumoto et. al. [1] indicated that the ﬁnite localization length lc of the wave functions of holes plays a crucial role in MIT in (Ga,Mn)As. It looks like that the lc might correspond to RC ∼ 1/m1 . 3. Localization and freezing of magnetic solitons We shall consider the transport property in the randomly distributed system of the hole-induced magnetic solitons in DMS, by using the eﬀective Lagrangian of diﬀusion modes. In terms of the four-component supervector ψ[12], the Lagrangian in this system takes the form,

1 (5) L = i ψ(r)(−H0 − V (r) + (ω + iδ)Λ)ψ(r)dr 2 1 Δ + μ. (6) H0 = ε + 2m where V(r) is the random potential, and Λ is the diagonal supermatrix 1 0 Λ= 0 −1

(7)

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with 1 the 4 x 4 unity matrix. The potential will be regarded as a random quantity with a Gaussian δ-correlation distribution V (r)V (r ) =

ImΣ(kF , F ) δ(r − r ), 2πν

(8)

where Σ(kF , F ) of the self energy of the the hedgehog-like solitons [3], and ν is the state density per spin. After averaging, eq(5) can be rewritten in the form,

¯ 2 − i(ω + iδ) ψΛψ]dr ¯ ¯ 0 ψ + ImΣ(kF , F ) (ψψ) Lef f = [−iψH 4πν 2

¯ 0 + i (ω + iδ)Λ + ImΣ(kF , F ) Q)ψ]dr ∼ [−iψ(H (9) 2 2 with the 8 x 8 supermatrix Q satisfying the following self-consistency equation,

2 2 ¯ ¯ ef f . Q= ψ ψexp(−L ψ ψ (10) ef f )Dψ ≡ πν πν Then the free energy F (Q) can be written as

ImΣ(kF , F ) i 1 Q) F (Q) = [− str ln(−iH0 − (ω + iδ)Λ + 2 2 2 πνImΣ(kF , F ) + strQ2 ]dr. (11) 8 In the case of low values of ω, the free energy for all Q’s satisﬁng the condition Q2 = 1 diﬀers little from the minimum value. All such zero-trace matrices can be written in the form, Q = V ΛV¯ , where V is an arbitrary unitary supermatriv, V V¯ = 1. Because longitudinal variations of Q alter greatly the free energy, we can ignore the longitudinal ﬂuctuations. Homogeneous transverse ﬂuctuations do not alter the free energy in the limit of low frequencies. In the case of small gradients Q, we obtain

πν (12) str [Def f (∇Q)2 + 2i(ω + iδ)ΛQ]dr, F (Q) ∼ 8 where the eﬀective diﬀusivity Deﬀ ∼

vF2 3ImΣ(kF , F )

and vF is the Fermi velocity. When

high magnetic ﬁeld imposes on this system, the hedgehog-like ferromagnetic solitons, in which the hole is trapped, will be broken and ImΣ(kF , F ) is reduced remarkably. Thus high magnetic ﬁeld induces strongly the increase of the eﬀective diﬀusivity Lef f of the hole. In order to discuss the spin dynamics, we envisage an eﬀective hamiltonian, H, for the magnetic-soliton, O(r˜i ), which is introduced in eq.(1), H = −J

S˜i · S˜j O(r˜i ) · O(r˜j ) |S˜i ||S˜j |

<˜i,˜ j>

1 O(r˜i ) · O(r˜j ) + K 2 |r˜i − r˜j | ˜ i=˜ j

(13)

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with J > K > 0 and the ﬁrst sum taken only over nearest neighbor(the distance between each magnetic soliton is ≤ 2Rc ) and the second taken over all pair(˜i = ˜j means |r˜i − r˜j | 2Rc ). S˜i ≡ Si . That is, S˜i is the summation of the i∈(4/3)πR3c

ferromagnetic spin, Si , of Mn within ∼ (4/3)πRc3 (˜i) around the photo-induced hole at the site r˜i . S˜i represents the eﬀective spin of the soliton O(r˜i ). The ﬁrst term corresponds to short-range ferromagnetic ordering interaction and the second corresponds to long-range frustration. Although the ﬁrst term of the eﬀective Hamiltonian in eq.(13) cannot be derived immediately from the eﬀective Lagrangian in eq.(1), this term can be introduced approximately as follows. When the magnetic soliton,O(r˜i ), with the eﬀective spin S˜i is located in the nearest neighors of the magnetic soliton,O(r˜j ), with the eﬀective spin S˜j , holes are hopping between two solitons O(r˜i ) and O(r˜j ). If S˜i is parallel to S˜j , p-d exchange interaction induces much reduction of the kinetic energy. This introduces the ferromagnetic short-range interaction such as the ﬁrst term of eq.(13). In the present theoretical formula, we deﬁne the topological number, q, for the hedgehog-like soliton (the magnetic soliton) as follows [13],

1 dSμ εμαβγ T r(A4α A4β A4γ ), (14) q= 24π 2 S 3 where S 3 is a sphere, whose radius is much larger than Rc ≡ 1/m1 . If a sphere S 3 completely surrounds one soliton, whose center position is r˜i , its topological number is represented as q ≡ q˜i = 1. When the hedgehog-like solition (monopolelike cluster) is located at the position r˜j , and the condition |r˜i − r˜j | > 1/m1 is assumed, the gauge ﬁeld A4μ (r˜i , r˜j ) at the position r˜i is represented as follows [14, 15], A4μ (r˜i , r˜j ) ∝

q˜j |r˜i − r˜j |

.

(15)

Thus the long-range interaction, V˜i˜j , ( |r˜i − r˜j | > 1/m1 ) between two hedgehog-like q˜·q˜ solitions at the position r˜i and r˜j is ∝ |r˜i−rj˜ | . If g3 in eq.(2) is assumed to be i j equal to π/K, the second term of eq (9) is introduced approximately. For the mean-ﬁeld approximate, it is assumed that V˜i·˜j describes N hedgehog-like soliton’s interaction, which mediated by the massless U(1) A4μ ﬁelds in pairs (˜i, ˜j) via inﬁniterange Gaussian-random interaction for simplifying discussion [16], −V˜i2˜j 1 (16) exp P (V˜i˜j ) = 2V˜i2˜j (2πV˜i2˜j )1/2 Now we can get the freezing behavior from the analogy of the Sherrington-Kirkpatrik (SK) formula by using the replica method. We can deﬁne the order parameter 1 ¯ G ≡ 0 dxG(x), where G(x) is the Parisi order parameter and is derived from Gαβ = q˜iα q˜iβ , in Parisi’s theoretical formula. α and β are the replica indices. In

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1/2 the temperature region below Tf = N V˜i˜j 2 /kB , we got the phase of the α ¯ order parameter G = 0 and q˜i = 0, which corresponds to the freezing phase of magnetic solitons, within the mean-ﬁeld approximate. 4. Conclusion The localization of hole-induced magnetic solitons in the randomly distributed system has been discussed in the supersymmetric sigma formula. We have proposed the freezing mechanism of magnetic solitons in diluted magnetic semiconductors. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

S. Katsumoto et al., Mater. Sci. Eng. B 84, 88 (2001). S. Koshihara et al., Phy.Rev.Lett 78, 4617 (1997). I. Kanazawa, Phase Transition 75, 1013 (2001). I. Kanazawa, Physica E 29, 647 (2005). D.R. Nelson, Phys. Rev. Lett.50, 982 (1983). J.P. Sethna, Phys. Rev. Lett. 51, 2198 (1983). S. Sachdev and D.R. Nelson, Phys. Rev. Lett. 53, 1947 (1984). I. Kanazawa, J. Non-Cryst. Solids 293, 615 (2001). M. Berciu and R.N. Bhatt, Phys. Rev. Lett. 87, 107203 (2001). C. Becchi, A. Rouet, and R. Stora, Commun. Math. Phys. 42, 127 (1975). T. Kugo, and I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979). K.B. Efetov, Sov. Phys. JETP 55, 514 (1982). R. Rajaraman, Soliton and Instantons, (Elsevier Science, 1987). A.M. Polyakov, Gauge Fields and Strings, (Harwood Academic, 1987). E. Fradkin, Field Theories of Condensed Matter System, (Addison-Wesley, 1991). I. Kanazawa, Prog. Theor. Phys. Suppl. 126, 393 (1997).

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SPIN-PHOTOVOLTAIC EFFECT IN QUANTUM WIRES

A. FEDOROV

∗

Center for Quantum Device Technology, Department of Physics and Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13699-5721, USA E-mail: [email protected] YU. V. PERSHIN AND C. PIERMAROCCHI Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320, USA E-mails: [email protected] and [email protected]

We consider theoretically spin and charge currents induced in a quantum wire by an external electromagnetic radiation. The mechanism of photocurrent generation is related to the interplay of spin-orbit interaction (Rashba and Dresselhaus terms) and an external in-plane magnetic ﬁeld and also to the fact that the spin-orbit interaction constants are diﬀerent for diﬀerent subbands. The currents are calculated using a Wigner functions approach, taking into account radiation-induced transitions between transverse subbands. The important role in the photovoltaic eﬀect is played by transitions in which the direction of electron velocity changes. The photocurrent sensitivity to the direction of the in-plane magnetic ﬁeld can be used to extract spin-orbit interaction parameters from transport measurements.

The photovoltaic eﬀect is a physical process through which the energy of photons is converted into electricity.1 In the present paper we overview our recent results on a new kind of photovoltaic eﬀect - spin-photovoltaic eﬀect in quantum wires.2,3 The spin-photovoltaic eﬀect is based on peculiarities of electronic subbands of a quantum wire with spin-orbit interactions in the presence of an in-plane magnetic ﬁeld. In the past, the photovoltaic eﬀect and photoconductance in quantum wires due to spin-independent mechanisms have been investigated.4,5,6,7,8,9,10,11,12,13,14,15,16 The main requirement for photovoltaic eﬀect is an assymetry of quantum wire properties with respect to the left- and right-moving electrons. Indeed, in a perfect quantum wire without a spin-orbit interaction, intersubband excitation probabilities are equal for the states with opposite momentum. Therefore, there is no change in current associated with external radiation. The idea of spin-photovoltaic eﬀect is to use the in-plane magnetic ﬁeld to break the symmetry of the system. In the ballistic ∗ Present

address: Institut f¨ ur Theoretische Festk¨ orperphysik, Universit¨ at Karlsruhe, 76128 Karlsruhe Germany

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Figure 1. Schematics of the system: quantum wire with an applied magnetic ﬁeld in (x, y) plane, and irradiated by a electro-magnetic wave linearly polarized in the y-direction.

transport regime, the total transmission probability for the left- and right-moving electrons at the same energy will be diﬀerent what can result in non-zero current through the wire at zero applied bias voltage. The system under investigation is a semiconductor-based quantum wire with spin-orbit interactions. Fig. 1 shows a possible geometry of the experiment where linearly-polarized radiation propagates perpendicularly to the plane of twodimensional electron gas with a quantum wire created by split-gate technique. Due to the quantum conﬁnement in y and z directions, the energy spectrum in the wire consists of a set of transverse subbands which can be labeled by indexes m and n, corresponding to the conﬁnement along y and z respectively. There are two types of spin-orbit interactions existing in such system. The Rashba spin-orbit interaction is due to the inversion-assymetry of the conﬁning potential and has the form 17 α HR = − px σy

(1)

in 1D, where the interaction parameter α depends on the subband index n.18 The Dresselhaus spin-orbit interaction 19 is present in semiconductors lacking bulk inversion symmetry, such as zinc-blend semiconductors. When restricted to a onedimensional geometry, this coupling is of the form β HD = − px σx ,

(2)

where β depends both on n and m. Thus, without electro-magnetic radiation the system is governed by the Hamiltonian H0 =

p2 g ∗ μB σB. + HR + HD + V (y) + U (z) + ∗ 2m 2

(3)

Here, p is the momentum of the electron, m∗ is the eﬀective mass, V (y) is the lateral conﬁnement potential due to the gates, U (z) is the conﬁnement potential in

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Figure 2. Dispersion relations (two lowest spin-splitted subbands) calculated for (a) Bx = By = 0T and (b) Bx = 2T, By = 2T. These plots were obtained using the parameters values: m∗ = 0.067me , g ∗ = −0.44, α0 = 0.5 × 10−11 eV m, β0,0 = 0.84 × 10−11 eV m, β1,0 = 0.36 × 10−11 eV m, 1 − 0 = 3.64meV.

z-direction, μB and g ∗ are the Bohr magneton and eﬀective g-factor, σ is the vector of the Pauli matrices and B is constant external magnetic ﬁeld in (x,y) plane. The energy spectrum of two lowest spin-splitted subbands in the quantum wire subjected to zero (a) and ﬁnite (b) external in-plane magnetic ﬁeld is depicted in Fig. 2. Dependence of α and β in Eqs. (1,2) on subband indexes results in diﬀerent shapes of zero and ﬁrst subbands in Fig. 2. Fig. 2(b) also clearly demonstrates that the in-plane magnetic ﬁeld breaks the symmetry of the spectrum with respect to k = 0. We note that due to the typically strong conﬁnement in z direction the orbital eﬀects of the in-plane magnetic ﬁeld can be considered as negligible. We can identify the following groups of intersubband transitions that lead to a photovoltaic eﬀect in quantum wires: (i) transitions between spin-splitted subbands with the same conﬁnement quantum numbers3 and (ii) transitions between subbands with diﬀerent conﬁnement quantum numbers.2 The main diﬀerence between these two groups is that the ﬁrst type of transitions is generated by the magnetic ﬁeld component of electromagnetic radiation, while transitions from the second group are due to the electric ﬁeld component. We found that the photocurrent due to the second type of transitions is signiﬁcantly stronger.2 The interaction of an electron with electric component of the radiation is described by the Hamiltonian HI = −

e eEy py Ap = − ∗ cos(ωt), m∗ m ω

(4)

where, Ey and ω are the amplitude and frequency of electric ﬁeld of the polarized electro-magnetic wave. The interaction causes inter-subband transitions with conservation of the wave vector k. However, the electron velocity, deﬁned as v = (1/) ∂E/∂k, is not conserved. Moreover, there are situations when the electron velocity direction is changed in the excited state. Two vertical lines in Fig.

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2 represent such a situation when the velocity direction of right-moving electrons (left vertical line) changes to the opposite while the velocity direction of left-moving electrons (right vertical line) remains the same. Therefore, a part of right-moving electrons, due to the interaction with radiation, scatters back and a current imbalance appears. This is the core of the spin-photovoltaic eﬀect. The current as a function of radiation frequency was calculated numerically from coupled equations involving Wigner functions. Among the advantages of the Wigner function formalism we mention the phase-space nature of Wigner functions which are similar to the classical Boltzmann distribution functions. This feature makes possible to separate the incoming and outgoing components of the electron distribution at the boundaries which, in turn, facilitates the modeling of an ideal contact. The commonly used assumptions are: the distribution of electrons emitted in the quantum wire can be described by the equilibrium distribution function of the leads reservoirs, and all electron are collected by the leads reservoirs without reﬂection. We extended the description of the transport dynamics to include intersubband transitions due to electro-magnetic wave excitation. The Liouville-von Neumann equation for the density operator of the electron is given by iρ˙ = [H, ρ],

(5)

where H = H0 + HI is the total Hamiltonian. The Wigner function can be obtained by integrating 20 W (R, k, t) = ρ(R, Δr, t) exp(−ikΔr)dΔr, (6) where the density operator is written in the new spatial variables R = (r +r )/2 and Δr = r − r and k is the electron wave vector. Following the standard procedure (see, e.g., Ref. 21) we derive a set of transport equations and neglect non-local correlations Wigner functions. Further details of calculations can be found in Ref. 2. A typical result of our calculations, the photo-induced current as a function of the photon energy, is presented in Fig. 3. Fig. 3 clearly shows a number of current peaks corresponding to diﬀerent transitions. The peaks are related to resonances between radiation and inter-subband transition frequencies which occur for speciﬁc values of k depending on the frequency value. Among these resonances the most important are those with the change of direction of electron motion. If such conditions are fullﬁled an electron can stay in the wire for much longer time than other electrons due to numerous backscattering events excited by the radiation. As a results the eﬀect of interaction with electromagnetic ﬁeld is larger and induced current imbalance grows. The location of these peaks depend on the magnetic ﬁeld strength and direction, as a consequence of the magnetic ﬁeld dependence of the energy spectrum, as well as on other system parameters such as spin-orbit coupling constants. Therefore, the current can be used to determine material parameters.

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Figure 3. Current through the wire as a function of the photon energy calculated for Bx = 2T, By = −2T.

The calculated current strength for a realistic set of parameters is of the order of 0.1 nA. Consequently, it can be measured using standard experimental techniques. Acknowledgments We gratefully acknowledge useful discussions with M. Cheng, L. Fedichkin and V. Privman. This research was supported by the National Science Foundation, Grants DMR-0121146 and DMR-0312491. References 1. A. Luque and S. Hegedus, Handbook of Photovoltaic Science and Engineering (John Wiley & Sons, 2003). 2. A. Fedorov, Yu. V. Pershin and C. Piermarocchi, Phys. Rev. B 72, 245327 (2005). 3. Yu. V. Pershin and C. Piermarocchi, Appl. Phys. Lett. 86, 212107 (2005). 4. F. Hekking, Yu. V. Nazarov Phys. Rev. B 44, 11506 (1991) 5. S. Feng, Q. Hu, Phys. Rev. B 48, 5354 (1993) 6. L. Fedichkin, V. Ryzhii and V. Vyurkov, J. Phys.: Cond. Matter 5, 6091 (1993). 7. A. Grincwajg, L.Y. Gorelik, V.Z. Kleiner, and R.I. Shekhter, Phys. Rev. B 52, 12 168 (1995). 8. F.A. Maao and L.Y. Gorelik, Phys. Rev. B 53, 15 885 (1996). 9. C. S. Tang and C. S. Chu, Phys. Rev. B 53, 4838 (1996). 10. S. Blom, L.Y. Gorelik, M. Jonson, R.I. Shekhter, A.G. Scherbakov, E.N. Bogachek, and U. Landman, Phys. Rev. B 58, 16 305 (1998). 11. Y. Levinson and P. W¨ olfle, Phys. Rev. Lett. 83, 1399 (1999). 12. C. Niu and D. L. Lin, Phys. Rev. B 62, 4578 (2000). 13. S. Blom and L. Y. Gorelik, Phys. Rev. B 64, 045320 (2001).

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14. G. Platero and R. Aguado, Phys. Rep. 395, 1 (2004). 15. N. G. Galkin, V. A. Margulis, and A. V. Shorokhov, Phys. Rev. B 69, 113312 (2004). 16. H. Kosaka, D.S. Rao, H.D. Robinson, P. Bandaru, T. Sakamoto, and E. Yablonovitch, Phys. Rev. B 65, 201307(R) (2002). 17. Yu. A. Bychkov and E. I. Rashba, JETP 39, 78 (1984). 18. E. Shafir, M. Shen, and S. Saikin, Phys. Rev. B 70, 241302 (2004). 19. G. Dresselhaus, Phys. Rev. 100, 580 (1955). 20. E. Wigner, Phys. Rev. 40, 749 (1932). 21. S. Saikin, J. Phys.: Condens. Matter 16, 5071 (2004).

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NONEQUILIBRIUM TRANSPORT IN AHARONOV-BOHM INTERFEROMETER WITH ELECTRON-PHONON INTERACTION

AKIKO UEDA AND MIKIO ETO Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected]

We theoretically investigate an eﬀect of electron-phonon (e-ph) interaction on transport properties in an Aharonov-Bohm (AB) ring with an embedded quantum dot. The current under ﬁnite bias is calculated using Keldysh Green function method. The e-ph interaction is taken into account by self-consistent Born approximation. We ﬁnd that the amplitude of the Fano resonance diminishes and the asymmetric shape grows like a symmetric one with an increase in the bias voltage. This is in qualitative agreement with experimental results. We also study the e-ph interaction in an interferometer of double quantum dots.

1. Introduction In semiconductor quantum dots, preservation of the quantum coherence is an important issue for the application to the quantum information processing. To examine the coherence, transport measurements have been reported using an AharonovBohm (AB) ring with an embedded quantum dot, as an interferometer.1,2,3 In such a system, the AB oscillation is observed as a function of magnetic ﬂux penetrating the ring, which indicates that the wave of electrons passing through the quantum dot keeps the coherence and interferes with the wave of the electrons passing through the other arm of the ring (reference arm). Kobayashi et al. observed an asymmetric shape of the conductance with peak and dip, as a function of the gate voltage attached to the quantum dot, so-called Fano resonance.3 The Fano resonance is ascribable to the higher-order interference between a discrete level in the quantum dot and continuum spectrum in the reference arm. They found that the conductance changes to a symmetric shape when the bias voltage increases. This implies that the ﬁnite bias signiﬁcantly reduces the coherence. As a decoherence mechanism under ﬁnite bias, we examine the eﬀect of electronphonon interaction on the Fano resonance. We consider acoustic phonons with continuous spectrum in the present paper. With increasing the bias voltage, a larger number of acoustic phonons interact with electrons in a quantum dot, which should enhance the decoherence. Although the e-ph interaction is present in the

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We

μL

iϕ

μR

ε0 tL

tR

Figure 1. Model that we adopt for an AB ring with an embedded quantum dot in the presence of electron-phonon interaction inside the quantum dot. The phase ϕ represents the magnetic ﬂux penetrating through the ring.

quantum dot only in our model, our calculated results agree with the experimental results qualitatively. To discuss the eﬀect of e-ph interaction in the reference arm, we also examine a simple situation of the interferometer of double quantum dots. 2. Model Our model for the AB interferometer is shown in Fig. 1. There are two paths between leads L and R, one path connects the leads through a quantum dot by tL and tR , and the other path connects the leads directly by W eiϕ (reference arm). The phase ϕ represents the magnetic ﬂux in the ring. The bias voltage between the leads is given by eV = μL − μR , where μL (μR ) is the chemical potential in lead L (R). We ﬁx μL = eV and μR = 0. Omitting the spin indices, the Hamiltonian for electrons is written as Hel = HL + HR + HT + HD , HL(R) = εk c†L(R)k cL(R)k ,

(1) (2)

k

HD = ε0 d† d, (3) † † iϕ † HT = (tL cLk d + H.c.) (tR cRk d + H.c.) + (We cRk cLk + H.c.), (4) k

k

kk

where c†L(R)k and cL(R)k denote the creation and annihilation operators of electron with momentum k in lead L (R), respectively. We assume a single energy level ε0 in the quantum dot, with creation and annihilation operators being d† , d. The electron-electron interaction is neglected. The strength of the tunnel coupling between the quantum dot and leads is characterized by the level broadening, Γ = ΓL + ΓR with ΓL(R) = 2πt2L(R) ν, where ν is the density of states in the leads. We deﬁne ξ = π 2 ν 2 W 2 for the direct tunnel coupling between the leads. The transmission probability through the reference arm is given by Tr = 4ξ/(1 + ξ)2 . We consider the e-ph interaction inside the quantum dot. The Hamiltonian for

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the interaction and that for the phonons are given by He−ph = Mq (aq + a†−q )d† d,

(5)

q

Hph =

q

ωq a†q aq ,

(6)

respectively, where a†q (aq ) creates (annihilates) a phonon with momentum q. Unless otherwise noted, we will be setting to unity. The coupling coeﬃcient Mq is written as Mq = λq d|eiq·r |d,

(7)

with λq being the amplitude of the e-ph interaction in the two dimensional electron gas. |d represents the envelope function of electrons in the quantum dot. It should be noted that the electron-phonon interaction is negligibly small when |q| 2π/L, where L is the size of the dot, owing to an oscillating factor of d|eiq·r |d. Hence we can restrict the summation over q to be |q| 2π/L in Eqs. (5) and (6). The current is expressed as4,5 2e I = dω fL (ω) − fR (ω)] h ˜ cos ϕReGr (ω) × Tr + αTr (1 − Tr )Γ dd

1 2 r ˜ (8) − α 1 − Tr cos ϕ) − Tr ΓImGdd (ω) , 2 where Grdd (ω) is the Fourier transform of the retarded Green function of the quantum dot, Grdd (t − t ) = −iθ(t − t ){d(t), d† (t )}.

(9)

fL (ω) [fR (ω)] is the Fermi distribution function in lead L [R]. The level broadening ˜ = Γ/(1 + ξ). α = 4ΓL ΓR /Γ2 is the asymmetric factor of the is renormalized as Γ quantum dot. We assume that the temperature is T = 0 hereafter. We consider the e-ph interaction by self-consistent Born approximation. The self energy is written as6 i Σγ (ω) = |Mq |2 dω Gγdd (ω − ω )Dγ (q, ω ), (10) 2π q where γ stands for t, t , <, and > for four kinds of Keldysh Green functions. Dγ (ω) are the phonon Green functions. For the acoustic phonons, |λq |2 = g

π 2 c2s , V |q|

(11)

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1

dI/dV /(2e /h)

(b)

2

2

dI/dV /(2e /h)

(a) 0.5

0 -4

-2

0

2

0.5

0

4

-4

(ε0-μL)/Γ

-2

0

2

4

(ε0-μL)/Γ

Figure 2. Diﬀerential conductance as a function of the dot level ε0 . The asymmetric factor is α = 1. The coupling constant of e-ph interaction is g = 0.05, whereas the dot size is L = cs /2Γ. (a) Case of resonant tunneling in the absence of reference arm (ξ = 0, Tr = 0), and (b) case of Fano resonance in the presence of reference arm (ξ = 0.05, Tr = 0.18). The bias voltage is eV = 0.5Γ (dashed line) and 2Γ (solid line), respectively. A dotted line indicates the conductance in the absence of e-ph interaction.

where g is a coupling constant and cs is the sound velocity.7 We set g = 0.05 and cs = 5000m/s, which are the values in GaAs.8,9 The dispersion relation of the phonons is given by ωq = cs |q|.

(12)

For the convenience of the calculation, we assume that |d|eiq·r |d|2 is given by √ 2 1 iq·r 2 (13) |d|e |d| = 1/2 2 2 π L q + (1/L)2 in Eq. (7). 3. Calculated results Before discussing the Fano resonance, we examine the case of resonant tunneling in the absence of reference arm, by setting ξ = 0 (Tr = 0). We ﬁx L = cs /(2Γ). (When Γ = 0.1meV, the dot size is L = 0.015μm, which is smaller than in Ref. 3.) The diﬀerential conductance is plotted in Fig. 2(a), as a function of the energy level ε0 in the quantum dot. The bias voltage is eV = 0.5Γ (dashed line) or 2Γ (solid line). A dotted line indicates the conductance in the absence of e-ph interaction, which indicates the Breit-Wigner resonance with symmetric Lorentzian shape. The e-ph interaction decreases the peak height of the resonance and increases the peak width. These eﬀects are more prominent with larger bias, reﬂecting the larger decoherence by the e-ph interaction. Figure 2(b) presents the Fano resonance in the presence of reference arm (ξ = 0.05, Tr = 0.18). As in the case of Breit-Wigner resonance, the amplitude of the Fano resonance becomes smaller and the width is broadened with an increase in the bias voltage.

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wR

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tR

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(a) dI/dV /(2e2/h)

wL

(b)

0.5

0 -2

-1

0

1

2

3

(ε0-μL)/Γ Figure 3. (a) Model for an interferometer of double quantum dots. (b) Diﬀerential conductance as a function of the dot level ε1 (ε1 − ε2 = Γ). The coupling constant of e-ph interaction is g = 0.01, whereas the size of both quantum dots is L = cs /(2Γ). The bias voltage is 2Γ. The e-ph interaction is taken into account only in quantum dot 1 (dashed line), or in both quantum dots 1 and 2 (solid line). The dotted line indicates the conductance in the absence of e-ph interaction.

The resonant dip becomes almost invisible with high bias and as a result, the asymmetric shape of the Fano resonance grows like a symmetric one. Although this is in qualitative accordance with the experimental results,3 the resonant peak is still in an asymmetric form even in a smaller size of quantum dots (with stronger e-ph interaction) than in the experiment. To explain the experimental results quantitatively, we would have to consider the e-ph interaction in the reference arm as well as inside the quantum dot.

4. Case of interferometer of double quantum dots To examine the e-ph interaction in the reference arm, we consider a simple situation of the interferometer of double quantum dots, shown in Fig. 3(a). Quantum dot 1 (2) has a single energy level ε1 (ε2 ), which connects to the leads through the tunnel barriers by tL and tR (wL and wR ). The coupling constant of e-ph interaction is smaller than in the previous section (g = 0.01). Figure 3(b) shows the diﬀerential conductance as a function of energy level ε1 . The energy level ε2 is changed simultaneously in such a way as ε1 − ε2 = Γ. The tunnel couplings are tL = tR = wL = wR . The dotted line indicates the conductance in the absence of e-ph interaction. We observe two peaks at ε1 and ε2 and a dip between them. The conductance is 2e2 /h at the peaks and becomes zero at the dip, which is due to the interference between the electron waves passing through the quantum dots. When the e-ph interaction is present in the quantum dot 1 only (dashed line), it lowers one of the peak height and decreases the dip depth. In the presence of e-ph interaction in both quantum dots (solid line), both peaks decrease in height and the dip diminishes further.

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This implies that the Fano resonance would be suppressed more strongly if the e-ph interaction worked not only in the quantum dot but also in the reference arm of AB ring. 5. Conclusion We have investigated the Fano resonance in quantum dots with e-ph interaction. Using Keldysh Green function method and self-consistent Born approximation method, we have obtained the bias-voltage dependence of the decoherence. The e-ph interaction with acoustic phonons suppresses the amplitude of the Fano resonance and broadens the width. The dip structure of the resonance becomes almost invisible under high bias voltages. The asymmetric resonant shape becomes like a symmetric one, in qualitative accordance with the experimental results.3 However, the model with e-ph interaction in the quantum dot only is not suﬃcient for the quantitative explanation of the experiment. We have also examined the e-ph interaction in an interferometer of double quantum dots. The decoherence is more eﬀective when the e-ph interaction is taken into account in both quantum dots than in one of the quantum dots. Acknowledgments The authors gratefully acknowledge discussions with F. Marquardt. This work was partially supported by a Grant-in-Aid for Scientiﬁc Research in Priority Areas “Semiconductor Nanospintronics” (No. 14076216) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. References 1. A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman, Phys. Rev. Lett. 74, 4047 (1995). 2. R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky, and H. Shtrikman, Nature (London) 385, 417 (1997). 3. K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, Phys. Rev. Lett. 88, 256806 (2002). 4. A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50, 5528 (1994). 5. W. Hofstetter, J. K¨ onig, and H. Schoeller, Phys. Rev. Lett. 87, 156803 (2001). 6. A. Ueda and M. Eto, cond-mat/0601327. 7. M. Keil and H. Schoeller, Phys. Rev. B 66, 155314 (2002). 8. H. Bruus, K. Flensberg, and H. Smith, Phys. Rev. B 48, 11144 (1993). 9. T. Brandes and B. Kramer, Phys. Rev. Lett. 83, 3021 (1999).

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FANO RESONANCE AND ITS BREAKDOWN IN AB RING EMBEDDED WITH A MOLECULE

SHIGEO FUJIMOTO AND YUHEI NATSUME Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan E-mail: [email protected] To investigate Fano resonances in the system of quantum dots, the conduction spectra for the circuit containing a C60 is calculated in consideration of vibrational eﬀects for some characteristic modes. The contribution of a referential path of an Aharonov-Bohm ring to structures in spectra is also studied. In mesoscopic systems, the Fano eﬀect is mainly found for the AB ring system with a quantum dot. In addition, the broadening of the peak for each electronic level becomes signiﬁcant by the inter-level coupling via the ungerade mode for the case where gerade and ungerade electronic levels are close to the other. On the other hand, we would like to stress that the referential path of the AB ring is not required to obtain the Fano patterns as for the breathing mode. We would like to emphasize that the interference between narrow peak and long tail of strong peak takes place in this molecule; The former plays the role of the discrete level, the latter the continuum state.

1. Introduction Recent development of nanoscale technology has been enabled us to investigate various types of quantum mechanical systems, such as quantum dots (QDs) and an Aharonov-Bohm (AB) ring etc. In recent several years, to investigate interference eﬀects in quantum systems, transport measurements by the use of an AB ring embedded with a QD in the one branch has been carried out. In fact, an asymmetric shape of the Fano resonance was reported 1 . Here, it should be noted that Fano eﬀect is caused by the quantum interference between continuum states and discrete levels 2 . In addition to the experimental work, Fano-type interference through a QD without external interference circuit has been observed in the latest experiments in mesoscopic semiconductors, in which semi-open QD and T-shaped QD etc. are used 3,4 . 2. Present model Since Fano eﬀect is interference phenomena, multiple transmission channels should be play an important role to the phenomena of the conduction. For this motivation, we discuss theoretically on a molecular conductor, C60 , as schematically illustrated in Fig. 1. In fact, it exhibits the characteristic Fano patterns in conductance as shown in Fig. 2 (The formalism of the calculation is discussed in the next section.),

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tr ˜ t

v˜L

α

β

Lead L

v˜R

˜ t

Lead R

γ

vl vr r l

Figure 1. Schematic illustration of the AB ring system which has a C60 in its lower branch. The AB ring is connected with carbon sites of the C60 symmetrically.

which is caused by the interference between electrons through the upper and lower of the AB ring. Because the C60 is high symmetric molecule, it has many levels with multiple-degeneracy in electronic states. Furthermore, such levels are essentially coupled with intramolecular vibrations. When the degeneracy is resolved, the other transmission channel is expected. Here, we take account for two types of vibrations which have rotational symmetry for the transmission direction; (1) ungerade mode and (2) gerade mode. Because the deviation of transfer energy between carbon sites is treated by the second order perturbation, vibrational eﬀects on stationary electronic states in conduction process are considered as the averaged feature at ﬁnite temperature. We calculate conductance through a C60 without referential path, as well as an AB ring system with above-mentioned molecule. 3. Calculation and results 3.1. Calculation method Here, we show the calculation method used in the present work. The calculation method is based on the Green’s function method using the tight-binding approximation 5,6 . The total Hamiltonian is H = HC + Hleads , where the Hamiltonian of the two semi-inﬁnite leads is † † ci+1 ci + h.c. − t˜ ci+1 ci + h.c. . (1) Hleads = −t˜ i∈Lead L

i∈Lead R

On the other hand, the Hamiltonian of a C60 , HC = HC0 + HC , is expressed as

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1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Ag

Hg

Gg+Hg

Gu

-3

-2

Hg

Hu

Tg

Gg

Hu

Gu

-1 0 1 2 gate voltage [eVg/t]

3

Figure 2. Conductance through the AB ring system. In this calculation, molecular vibrations are not considered: δ = 0.0.

follows: HC0 = −t

† ci cj + h.c. − eV g ni , ∈ all bonds

HC = −δ

† ci cj + h.c. ,

(2)

i

(3)

where δ = t /t − 1.0. In other words, δ is the deviation of the transfer energy between carbon sites. It should be noted that we give a transfer energy t to bonds according to patterns of vibrational modes in order to investigate the eﬀect of the molecular vibration in a C60 . In the present report, gerade (i.e. breathing) and ungerade modes are adopted to the calculation. In Eq. (2), Vg is gate potential. Furthermore, t˜ and t are transfer energies between sites for semi-inﬁnite leads and a C60 , respectively. The present conduction process is expressed by Green’s functions. First of all, the Green’s function of a C60 is described as ⎤ ⎡ |m| H |n|2 l |n n |r C ⎦, ⎣1 + GC γ→l,γ→r (, eV g) = l + Γr ) /2 2 − + eV g + i (Γ ( − ) n n m n n n m=n

(4) where Hc treated as the second order perturbation. Here, n and |n are the eigenvalue and eigenstate of Eq. (2). Terms Γin (i ∈ l, r) in the denominator of the Eq. (4)

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1.6

2

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1.4 1.2 1 0.8 0.6 0.4 0.2 0 -3

-2

-1

0

1

2

3

gate voltage [eVg/t] Figure 3. Conductance through the AB ring system with the ungerade vibrational mode. In this calculation, we have δ = 0.375. The broadening of the peak for each electronic level becomes signiﬁcant by the inter-level coupling via the ungerade mode.

shows peak width due to the coupling between leads and a C60 . The Green’s function for the total conduction process through the AB ring system is written as the following form; r0 r r () + vl vr gαγ−1 ()GC Gr (, eVg ) = gαβ γγ (, eVg )gγ+1β (),

(5)

where vl and vr are the coupling constants between lower leads (left/right) of the r0 r () and gij () are the Green’s function for AB ring and sites of a C60 . In Eq. (5) gij the upper and lower leads of the AB ring, respectively. By the use of these Green’s functions, we can calculate the conductance through the AB ring system G(, eVg ) =

2e2 T (, eVg ), h

(6)

where the transmittance T (, eVg ) is expressed as follows: 2

2 2 vL v˜R |Gr (, eVg )| ImgL () ImgR (), T (, eVg ) = 4˜

(7)

where v˜L and v˜R are the coupling constants between leads and the AB ring. In Eq. (7), gL and gR are the Green’s function of left and right leads, respectively. In this procedure of the calculation, we obtain following results.

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2

conductance [2e /h]

0.35 Gg+Hg

0.3 0.25 0.2 0.15

Gu

0.1 0.05 0 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 gate voltage [eVg/t]

Figure 4. Conductance through a C60 without the referential path of the AB ring. The contribution from gerade (breathing) vibrational mode is taken into account. Here, the value of δ is 0.375. In order to show the distinctive eﬀect of interference, we draw two lines; While solid line shows conductance with interference eﬀect, dashed line shows conductance without interference eﬀect; simple summation of T2u , Gu and Gg+Hg spectra.

3.2. The case of ungerade mode for systems of AB ring Broadening by inter-level coupling between gerade and ungerade electronic levels play an essential role in features of spectra. In fact, the peaks of the Fano resonance distinctly spread as shown in Fig. 3. In particular, the broadening is signiﬁcant at the Gu and Tg level because these levels are closely mixed by the ungerade mode. 3.3. The case of gerade mode for systems without the referential path By the broadenings and splittings for electronic levels by intra- and inter-level couplings, interference eﬀects appear even for cases without the referential path. As clearly seen in Fig. 4, increasing the coupling between electronic states and intramolecular vibrations, we can ﬁnd close overlapping between weak peak (e.g. Gu with four-fold degeneracy) and the tail of strong peak (e.g. Gg+Hg with nine-fold degeneracy) as clearly seen in Fig.4. In this case, two transport channels are opened simultaneously. While the phase of the transmission through Gu peak changes by π, the phase via the tail of the Gg+Hg peak is unchanged. In this situation, the interference eﬀect play an important role in conduction spectra. In fact, spectra with and without interference terms embody quite diﬀerent features as shown by

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solid and dashed line in Fig. 4. In detail, signiﬁcant dips are found for Gu lines in spectra with interference terms. We can recognize that these characteristic shapes are caused by Fano eﬀect. In fact, this situation satisﬁes the condition of Fano eﬀect; Gu peak behaves discrete level, whereas the tail of Hg+Gg peak plays a role of continuum states. As a result, asymmetric peaks are obtained without referential path. We would like to emphasize that the C60 with the gerade vibrational mode is the interferometer by itself. 4. Summary In this work, we discuss characteristic two kinds of conductive paths satisfying the condition of Fano resonance, which are prepared in some molecules. We point out that the referential path is not required for the interference, if we prepare the condition for Fano resonance in the molecule by itself. References K. Kobayashi, H. Aikawa, S. Katsumoto and Y. Iye, Phys. Rev. Lett. 88, 256806 (2002). U. Fano, Phys. Rev. 124, 1866 (1961). K. Kobayashi, H. Aikawa, S. Katsumoto and Y. Iye, Phys. Rev. B70, 035319 (2004). H. Aikawa, K. Kobayashi, A. Sano, S. Katsumoto and Y. Iye, J. Phys. Soc. Jpn. 73, 3225 (2004). 5. S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, (1995). 6. Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).

1. 2. 3. 4.

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QUANTUM RESONANCE ABOVE A BARRIER IN THE PRESENCE OF DISSIPATION

K. KONNO1, M. NISHIDA2, S. TANDA1, AND N. HATAKENAKA2,3 1

Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan 2 Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima, 739-8530, Japan 3 Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan E-mail: [email protected] Quantum transmissions of a free particle passing through a rectangular potential barrier with dissipation are studied by using path decomposition technique. Dissipative processes strongly suppress the transmission probability at resonance just above the barrier. A promising experimental candidate for testing the applicability of quantum mechanics at macroscopic scales is also discussed.

1. Introduction Quantum tunneling is an attractive phenomenon in quantum mechanics, which cannot be understood in classical sense. Therefore, the quantum tunneling is eagerly investigated on one hand with respect to the pursuit of quantum mechanics on macroscopic scale, on the other hand with respect to applications of quantum mechanics such as quantum computation. In such situations, dissipation associated with larger degrees of freedom for environment is inevitable. However, not only the tunneling but also resonance are intrinsic quantum feature that cannot be found in classical phenomena. In this study, we investigate quantum resonance in the presence of dissipation by considering a particle passing through a rectangular potential barrier. The resonance occurs when integer multiple of the half of particle’s wavelength coincides with the width of the potential barrier. In a classical sense, if the particle energy is larger than the potential height, then the particle successfully passes through the barrier and is never reﬂected. However, in quantum mechanics, quantum reﬂection and resonance occurs, depending on the relation between the wavelength and the width of the potential barrier. The Wentzel-Kramers-Brillouin (WKB) method,1 which is usually utilized for theoretical analyses, breaks down in the situation mentioned above. Thus we developed another method to incorporate dissipation into quantum resonance, based on a path decomposition method,2,3 whose formulation is given in Sec. 2. Furthermore, we numerically estimate the transmission probability through the potential

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barrier in the presence of dissipation in Sec. 3. The results are explained in terms of the traversal time through the region of the potential barrier. Finally we give a summary in Sec. 4, in which the experimental setup to verify the results is also discussed. 2. Formulation Based on the Path Decomposition Method We brieﬂy review calculations of a transmission probability of a particle through a rectangular potential as shown in Fig. 1 using the path decomposition method. In the path decomposition method, the summation of possible paths is decomposed into certain groups as shown in Fig. 2. The ﬁrst group is composed of all possible paths which straightly pass over regions I, II, and III in order. The next group includes all paths which go from region I to region II, then return to region I once, and go from region II to III. In the same way, one can consider all other groups. Each group is completely expressed by the propagators deﬁned in the restricted regions I, II and III. Since all possible paths are taken for summation, any approximation is not used in this technique. This point prevails against the WKB method. Thus, the propagator K(xT , x0 ; T ) from x0 (< a) at t = 0 to xT (> b) at t = T is decomposed as T T −t1 dt1 dt2 iK (I) (a, x0 ; t1 )Σa iK (II) (b, x; t2 ) iK(xT , x0 ; T ) = 0

0

×Σb iK (III) (xT , x; T − t1 − t2 ) + · · · ,

(1)

where Σa (· · · ) ≡ /2m · ∂x (· · · ) |x=a , and K (X) is the propagator deﬁned in the restricted region X (X = I, II or III). The ﬁrst term on the right-hand side corresponds to the left-hand ﬁgure in Fig. 2. The above method enables us to deﬁne traversal time through the barrier. Traversal time in quantum-mechanical sense diﬀers from the time in classical picture speciﬁed by a single value. Traversal time distributes quantum-mechanically owing

Figure 1. Schematic of a particle passing through a rectangular potential barrier and three decomposed regions.

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Figure 2. Illustration of the path decomposition technique. First several groups in summation of all possible paths are shown (see also text). Each group is classiﬁed according to which regions the paths have passed over in terms of temporal development. Two examples out of possible paths are shown by solid and dashed curves in each ﬁgure.

to diﬀerent amount of time spent on each path in a huge number of diﬀerent paths. The probability amplitude for a particle to spend time τ in region II is deﬁned by3 iS(path)/ δ(τ − τII ) paths e F (τ ) ≡ , (2) iS(path)/ paths e where S denotes the action, and τII is the time for the particle on each path to spend in region II. The denominator is equivalent with the propagator K(xT , x0 ; T ) m in deﬁnition. The numerator can be obtained by inserting a factor of δ(τ − i=1 t2i ) in term of the right-hand side in Eq. (1). By utilizing the formula δ(t) = ∞each iωt dω/2π and by considering an energy eigenstate in the limit of both x0 → −∞ e −∞ and xT → ∞, one can obtain the form ∞ 1 dω −iωτ e F (τ ) = w(E, V0 − ω), (3) w(E, V0 ) −∞ 2π where w(E, V0 ) denotes the transmission amplitude for an incident particle with energy E. Here we used the propagator in that limit expressed as K(xT , x0 ; T ) = eikxT w(E, V0 )K0 (0, x0 ; T ),

(4)

where K0 is the propagator in the absence of the barrier. Now let us discuss the eﬀect of dissipation within the potential barrier on quantum transmissions. In the presence of dissipation, we recall the work by Caldeira & Leggett,4 in which quantum Brownian motion is studied by considering a set of harmonic oscillators coupled to a quantum system of interest. Based on the work, the time evolution of the density matrix at t = τ can be calculated as | x|Ψp (τ ) |2 =

σeγτ ≡ f (τ )2 , σ cosh στ + γ sinh στ

(5)

where |Ψp (t) denotes p initially, γ is the relaxthe state vector having momentum ation rate, and σ ≡ γ 2 + Δω 2 with being Δω ≡ 4γΩ/π, where Ω corresponds to the cutoﬀ frequency of harmonic oscillators. We assume the constant relaxation

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even though the relaxation rate γ may depend on an incident energy of the particle in general. We also assume an ohmic dissipation. The function f (τ ) remains at unity when γ = 0. For non-zero γ, f (τ ) decreases monotonically as a function of τ . The probability density in region II is reduced by a factor of f (τ )2 (< 1), which depends on the dwell time τ in that region, compared with the non-dissipative case. Therefore, we can approximate the propagator in region II as f K (II) in the presence of dissipation when forces acting on the system at diﬀerent times are uncorrelated. In this case, coupling between distant space-time points via external environment can be neglected. Thus, this approximation takes into account certain dephasing within region II partially. In the dissipative case, therefore, the full propagator K from region I to region III can be obtained by inserting the factor f ( i t2i ) in each term in Eq. (1), where i t2i denotes the total dwell time in region II. In the calculation, we utilize the relation ∞ ∞ ∞ 1 dτ f (τ ) δ (τ − Σi t2i ) = dτ f (τ ) dωe−iω(τ −Σi t2i ) . (6) f (Σi t2i ) = 2π −∞ 0 0 The Fourier transform ofthe full propagator K (xT , x0 ; T ) deﬁnes the Green func∞ tion as G (xT , x0 ; E) ≡ 0 dT iK (xT , x0 ; T ) eiET / . The Green function in the presence of dissipation is then given as G (xT , x0 ; E) = wD (E, V0 ) G0 (xT , x0 ; E) ,

(7)

where G0 (xT , x0 ; E) is the Green function for a free particle in the absence of the potential barrier, and the transmission amplitude wD (E, V0 ) for the dissipative case is written as ∞ ∞ dω −iωτ e dτ f (τ ) w (E, V0 − ω) . (8) wD (E, V0 ) = 0 −∞ 2π This is our main result. The transmission probability through the rectangular potential is then given by |wD (E, V0 ) |2 . 3. Numerical Estimate of Transmission Probability Two dimensionless parameters characterize the transmission processes, i.e., the particle energy normalized by the potential height, E/V0 , and the potential width nor√ malized by the typical scale of length, d/λ0 , where d = b − a and λ0 = / 2mV0 . Figure 3 shows the transmission probability for diﬀerent γ values. A dashed line is the transmission probability for non-dissipative case seen in the textbook. Below E/V0 < 1, transmission is owing to tunneling. The oscillatory structure above E/V0 > 1 is due to quantum interference regarded as quantum reﬂection that cannot be explained in terms of classical mechanics. Perfect transmission occurs whenever the barrier contains an integer times of half wavelength, i.e., κd = nπ (n = 1, 2, · · · ) where κ = 2m(E − V0 )/. This is nothing but quantum resonance. In the presence of dissipation, the transmission probabilities are always suppressed. The blurred oscillatory structure implies that the suppression is not uniform on energy even in the energy-independent relaxation rate γ. In fact, signiﬁcant

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suppression occurs at resonance. The particle in region II at resonance experiences longer traversal time due to multiple quantum reﬂections at the boundaries, leading to eﬀectively larger dissipation.

Figure 3. Transmission probability as afunction of E/V0 . Three cases of γτ∗ = 3 × 10−3 , 1 × 10−3 and 0, are plotted, where τ∗ = md2 /2V0 . In the calculations, we adopted d/λ0 = 5 and Ωτ∗ = 100.

The mean traversal time in the absence ∞ of dissipation is evaluated using traversal-time distribution F (τ ) as τ = 0 τ F (τ )dτ . For E > V0 , the mean traversal time is analytically calculated as3 m 2k Aκd − B sin κd cos κd m Bκd cos κd − A sin κd B +i sin κd, (9) κ B 2 sin2 κd + 4k 2 κ2 B 2 sin2 κd + 4k 2 κ2 κ2 √ where A = k 2 + κ2 , B = k 2 − κ2 , and k = 2mE/. At resonances, Eq. (9) reduces to the expression τ = mdA/2k 2 κ2 that is always larger than the classical traversal √ time τcl deﬁned by τcl ≡ md/κ from the inequality (α+β)/2 ≥ αβ (∀α, β > 0). In addition, the imaginary part of the traversal time vanishes. On the other hand, the destructive interference occurs at κd = (n+1/2)π, where Re[τ ] = 2mkd/A ≤ τcl . The real part of τ is always smaller than the classical traversal time. Therefore, the most signiﬁcant reduction of transmission probability at around the ﬁrst resonance point is due to the longer traversal time. τ =

4. Summary We have studied the eﬀect of dissipation on the quantum transmissions through a rectangular potential barrier. For this purpose, we have extended the path decomposition technique to incorporate dissipation into the system. The transmission probability is always suppressed due to dissipation, especially at the ﬁrst resonance owing to the longer traversal time. Finally we discuss an experimental setup to test the situation discussed in this study. A promising candidate is a ﬂuxon in a long Josephson junction. The ﬂuxon is a topological soliton excitation with a quantum unit of magnetic ﬂux and can be

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regarded as a free particle characterized by a macroscopic degree of freedom. In a mesoscopic Josephson junction with small capacitance per unit area, the ﬂuxon behaves like a quantum particle.5 In fact, quantum tunneling of a ﬂuxon in annular long Josephson junction supported by an external magnetic ﬁeld has been observed.6 Fluxon transmissions through a barrier as we treated, however, diﬀer from these previous works that studied a decay from a metastable state by quantum tunneling. The potential barrier should be introduced by a microshort, in which the width of insulator is thinner than that of the other part.7 Recently, a qubit using superposition states of ﬂuxons or breathers are considered.8 Thus, ﬂuxon transmissions also provide a important basis for implementing quantum computer in superconducting nanocircuits. Acknowledgments This work was supported in part by a Grant-in-Aid for Scientiﬁc Research of The 21st Century COE Program “Topological Science and Technology”. References 1. 2. 3. 4. 5. 6. 7. 8.

R. Bruinsma and P. Bak, Phys. Rev. Lett. 56, 420 (1986). A. Auerbach and S. Kivelson, Nucl. Phys. B257, 799 (1985). H. A. Fertig, Phys. Rev. Lett. 65, 2321 (1990); Phys. Rev. B 47, 1346 (1993). A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983); Erratum ibid. 130A, 374 (1985). T. Kato and M. Imada, J. Phys. Soc. Jpn. 65, 2963 (1996). A. Wallraﬀ et al., Nature (London) 425, 155 (2003). Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989). T. Fujii, et al., Proceeding of Foundation of Quantum Mechanics in the Light of New Technology, 88-91 (2006).

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ENSEMBLE AVERAGING IN METALLIC QUANTUM NETWORKS

´ FRANC ¸ OIS MALLET∗, FELICIEN SCHOPFER†, JERRY ERICSSON, ‡ § ¨ LAURENT SAMINADAYAR AND CHRISTOPHER BAUERLE Institut N´eel and Universit´e Joseph Fourier, B.P. 166, 38042 Grenoble, France DOMINIQUE MAILLY Laboratoire de Photonique et Nanostructures, route de Nozay, 91460 Marcoussis, France CHRISTOPHE TEXIER AND GILLES MONTAMBAUX Laboratoire de Physique des Solides, Universit´e Paris-Sud, 91405 Orsay Cedex, France

We report on the size dependence of the amplitudes of Aharonov-Bohm (AB) as well as Altshuler-Aronov-Spivak (AAS) magnetoconductance oscillations in silver networks with anisotropic aspect ratio and for various sizes ranging from 10 to 106 plaquettes. We show that the amplitude of both AB and AAS oscillations exhibit an unexpected dependence as a function of number of plaquettes N when the smallest dimension of the network becomes smaller than the phase coherence length: in this case, the network can be considered as a fully coherent object (mesoscopic) in one direction, whereas macroscopic in the other.

1. Introduction The Aharonov-Bohm eﬀect1 in a small metallic ring is one of the most spectacular manifestation of quantum interference of electrons in a disordered conductor. By applying a magnetic ﬂux through the ring, the conductance oscillates with a periodicity φ0 = h/e, the ﬂux quantum, h being the Planck constant and e the charge of the electron2 . Such magnetoconductance oscillations are a direct consequence of the coupling of the electron charge to the vector potential, and is thus the most direct evidence of the quantum nature of the conduction in mesoscopic systems3 . An important point is the understanding of how such quantum eﬀects disappear when going from mesoscopic to macroscopic conductors4 . If one considers lines of N mesoscopic metallic rings, the Aharonov-Bohm (AB) conductance oscillations ∗ Present

´ address: Service de Physique de l’Etat Condens´e, dsm/drecam/spec, cea Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette cedex, France. † Present address: Laboratoire National de M´ etrologie et d’Essais, 29 avenue Roger Hennequin, 78197 Trappes, France. ‡ Member of the Institut Universitaire de France, 103 boulevard Saint Michel, 75005 Paris, France. § This work is supported by ACI grants # 02 2 0222 and # NN/02 2 0112, and the European Commission FP6 NMP-3 project 505457-1 Ultra-1D. Mail to: [email protected].

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√ δGAB /G vanishes to zero as 1/ N . This has been demonstrated in a pioneering experiment by Umbach el al.5 where they studied lines of N silver rings with N varying from 1 to 30. The authors showed that the amplitude of the AB oscillations decrease when increasing the number of rings N . On the other hand, there exist a second class of magnetoconductance oscillations. They are due to interferences between time reversed trajectories and therefore do survive such an ensemble averaging. These oscillations have a period of φ0 /26,7 and are known as Altshuler-Aronov-Spivak (AAS) oscillations. The robustness of these oscillations has been shown in several experiments 5,8,9 . We would like to stress that all these experiments have been carried out in a regime where the phase coherence length Lφ is much smaller than the system size. One then deals with the simple case of an ensemble averaging consisting in a summation of uncorrelated contributions from phase coherent regions. In what follows, we will address the issue of ensemble averaging when the system size decreases and becomes of the order or smaller than the phase coherence length Lφ . 2. Sample Fabrication Samples are fabricated on a silicon substrate using electron beam lithography on polymethyl-methacrylate resist. Silver is deposited from a 99.9999 % purity source using an electron gun evaporator and lift-oﬀ technique without any additional adherence layer. All samples have been evaporated in a single run to ensure that the sample characteristics (elastic mean free path le ≈ 15 nm and phase coherence length Lφ ) are similar. In this work, two diﬀerent topologies have been studied: the square lattice and the so-called T3 lattice10 . The wires forming the networks are 60 nm wide, 50 nm thick and 640 nm (690 nm) long for the square (T3 ) lattice. The size of the plaquettes (square or diamond) is chosen such that the magnetic ﬁeld corresponding to one ﬂux quantum φ0 per plaquette is B = 100 G. All networks with number of plaquettes N

Figure 1. Scanning Electron Micrograph of several samples of various sizes; two contacts are visible for the small sample.

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varying from 10 to 106 have the same aspect ratio Lx /Ly = 10 (see ﬁgure 1). As a consequence, their resistances are similar and of the order of 100 Ω. Measurements have been performed at a temperature of 400 mK at which the signal to noise ratio is optimum. At this temperature, the phase coherence length, determined from standard weak localization measurements on a wire fabricated on the same wafer is about Lφ 6 μm11 . 3. Experimental Results and Discussion In ﬁgure 2a we show typical data for the magnetoresistance of a square network with 3000 plaquettes at low ﬁelds. Clear magnetoconductance oscillations are observed around zero ﬁeld. These oscillations have a period of B = 50 G, as emphasized in the Fourier spectrum (ﬁgures 2b), which corrsponds to φ0 /2 per plaquette and are identiﬁed as the AAS oscillations.

Figure 2. Magnetoresistance of a square network containing 3000 plaquettes at low ﬁeld data and its Fourier amplitude.

At ﬁelds typically higher than the ﬁeld which suppresses weak localization, we observe a diﬀerent type of oscillations. They are shown in ﬁgure 3 for networks with diﬀerent number of plaquettes as well as their Fourier spectra. These oscillations have a periodicity of B = 100 G and correspond to AB oscillations. Let us now look at the size dependence of the AB as well as AAS oscillations versus the number of plaquettes N . To measure the AB oscillations we sweep the magnetic ﬁeld from 7000 G to 13000 G, whereas for the AAS oscillations we cover a ﬁeld range of ±1200 G. The amplitude of the AB oscillation is obtained by integration of the 1st peak of the Fourier spectrum of the magneto resistance curve. The AAS amplitude is obtained in a similar way. Note however, that the second harmonic (φ0 /2) of the AB oscillation has the same frequency as the ﬁrst harmonic (φ0 /2) of the AAS oscillation. For small networks (typically N ≤ 100) this contribution cannot be completely neglected. In order to extract the AAS signal, we therefore determine ﬁrst the amplitude of the second harmonic of the AB

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Figure 3. Magnetoresistance for several networks at high magnetic ﬁelds as well as their Fourier spectra. The average resistance is ≈ 100 Ω for the three samples.

oscillations at high ﬁeld and then subtract this amplitude from the ﬁrst harmonic of the oscillations measured at low ﬁeld. In Figure 4 we display the amplitude of magnetoconductance oscillations (AAS and AB) extracted from the Fourier spectra as a function of the number N of plaquettes for both, square and T3 lattices. For large√networks (N 300), the amplitude of the AB oscillations clearly decreases as 1/ N , whereas the amplitude of the AAS oscillations are independent on the number of plaquettes as naively expected. More surprising is the behavior observed for small networks: when they contain typically less than √ N 300 plaquettes, the amplitude of the AB oscillations varies faster than 1/ N . At the same time the AAS amplitude now depends on N (ﬁgure 4). In the following, we will show that this new behavior results from a dimensional crossover when the transverse size of the network becomes smaller than the phase coherence length, a new regime where the transport properties are eﬀectively one-dimensional on the two-dimensional network. For our discussion we start with the following equation which relates the AAS conductance oscillation ΔgAAS and the AB conductance oscillation δgAB 12,13,14,15 . 2 = δgAB

2πL2T ΔgAAS 3L2x

(1)

where LT is the thermal length and Lφ the phase coherence length. As we are working at constant temperature, both quantities are ﬁxed parameters. In the following we consider a network of dimensions Lx × Ly (see ﬁgure 1). Note however, that experiments presented here are performed on several networks of diﬀerent sizes,

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but of constant aspect ratio Lx /Ly = 10. The length and width of the networks √ thus scale with the number of plaquettes N as Lx = 10Ly ∝ N . Let us ﬁrst consider large networks with both dimensions larger than the phase coherence length: Lx , Ly Lφ . In this case the weak localisation correction Δσ which is directly related to the AAS oscillation is size independent as interfering time reversed trajectories extend over a typical size Lφ and therefore do not feel the boundaries of the system. The AAS amplitude varies as ΔgAAS ∝ Ly /Lx and is hence is independent on N : ΔgAAS ∝ N 0

(2)

2 ∝ Ly /L3x , which leads to: In this regime we also see from eq. (1) that δgAB

δgAB ∝ N −1/2

(3)

This is exactly what is observed for large networks: when the number of plaquettes is larger than 300, electrons diﬀuse on what they feel as a two-dimensional network.

Figure 4. AAS amplitude ΔgAAS () and AB amplitude δgAB () as a function of the number of plaquettes N for diﬀerent networks of various sizes.

For smaller networks, the transverse dimension Ly eventually becomes smaller than the phase coherence length: we enter a regime where the network becomes transversally coherent whereas it remains longitudinally incoherent: Ly Lφ Lx . In this case, we have the usual quasi-1D scaling ΔσAAS ∝ Lφ /Ly . Therefore 2 ∝ 1/L3x, which leads to: we ﬁnd ΔgAAS ∝ 1/Lx and δgAB ΔgAAS ∝ N −1/2 δgAB ∝ N

−3/4

(4) (5)

Indeed, this is exactly what is observed for small networks (see ﬁgure 4). The crossover hence appears when the phase coherence length is of the order of the smallest extension of the network. In our case it occurs at a network size of N 300 which corresponds to Ly 3.8 μm. This length has to be compared with the phase

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coherence length Lφ 6 μm measured at T = 400 mK on a wire fabricated on the same wafer. This is in relatively good agreement and hence supports our analysis. Finally let us compare these N dependences with the case of a 1D chain where the number N of rings scales linearly with the length Lx of the chain. In this case 2 ∝ 1/L3x ∝ 1/N 3 and ΔgAAS ∝ 1/N . Since the conductance g scales we have δgAB √as 1/N , this yields for the relative ﬂuctuations ΔgAAS /g ∝ N 0 and δgAB /g ∝ 1/ N as was observed experimentally5 . 4. Conclusions In conclusion, we have measured both Aharonov-Bohm φ0 periodic oscillations and Altshuler-Aronov-Spivak φ0 /2 periodic oscillations in metallic networks containing 10 to 106 plaquettes. Ensemble averaging leads to diﬀerent size dependences for small and large networks. The crossover takes place when the width of the network is of the order of the phase coherence length; this behavior does correspond to a dimensional crossover between eﬀectively one- and two-dimensional networks. In this new regime, we have shown that the amplitude of the AB oscillations varies as N −3/4 and the AAS oscillations as N −1/2 , a behavior which has never been observed up to now. References 1. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). 2. Y. Gefen, Y. Imry, and M. Ya. Azbel, Phys. Rev. Lett. 52, 129 (1984); M. B¨ uttiker, Y. Imry, and M. Ya. Azbel, Phys. Rev. A 30, 1982 (1984). 3. S. Washburn and R. A. Webb, Adv. Phys. 35, 375 (1986). 4. F. Schopfer, F. Mallet, D. Mailly, C. Texier, G. Montambaux, C. B¨ auerle and L. Saminadayar, Phys. Rev. Lett. 98, 026807 (2007). 5. C. P. Umbach, C. Van Haesendonck, R. B. Laibowitz, S. Washburn, and R. A. Webb, Phys. Rev. Lett. 56, 386 (1986). 6. B. L. Al’tshuler, A. G. Aronov, and B. A. Spivak, JETP Lett. 33, 94 (1981). 7. D. Yu. Sharvin and Yu. V. Sharvin, JETP Lett. 34, 272 (1981). 8. B. Pannetier, J. Chaussy, R. Rammal, and P. Gandit, Phys. Rev. Lett. 53, 718 (1984); B. Dou¸cot and R. Rammal, Phys. Rev. Lett. 55, 1148 (1985). 9. G. J. Dolan, J. C. Licini, and D. J. Bishop, Phys. Rev. Lett. 56, 1493 (1986). 10. J. Vidal, R. Mosseri, and B. Dou¸cot, Phys. Rev. Lett. 81, 5888 (1998); C. Naud, G. Faini, and D. Mailly, Phys. Rev. Lett. 86, 5104 (2001). 11. F. Mallet, F. Schopfer, D. Mailly, C. Texier, G. Montambaux, C. B¨ auerle, and L. Saminadayar, to be published. 12. I. L. Aleiner and Ya. M. Blanter, Phys. Rev. B 65, 115317 (2002). 13. T. Ludwig and A. D. Mirlin, Phys. Rev. B 69, 193306 (2004). 14. C. Texier and G. Montambaux, Phys. Rev. B 72, 115327 (2005). ´ Akkermans and G. Montambaux, Physique m´esoscopique des ´electrons et des pho15. E. tons, EDP Sciences (2004); Mesoscopic physics of electrons and photons, Cambridge University Press, Cambridge (2007).

Coherence and Order in Exotic Materials

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PROGRESS TOWARDS AN ELECTRONIC ARRAY ON LIQUID HELIUM D.G. REES, P. GLASSON, P. J. MEESON, L.R. SIMKINS, V. ANTONOV, P.G. FRAYNE, M. J. LEA Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 OEX, United Kingdom Email: [email protected] I.D. BAIKIE KP Technology, Milton House, Thurso Road, Wick, Caithness, KW1 5LE, Scotland Email: [email protected] Surface state electrons on liquid helium have been proposed as condensed matter qubits. The trapping of a single electron above the surface of superfluid liquid helium in an electrostatic trap and its detection using a single electron transistor have been performed previously1. A prototype design for a quantum information processor involving a linear array of electron traps on liquid helium has been fabricated. Here we present preliminary results showing the trapping, control and detection of small numbers of electrons with the device. It is found that a large contact potential difference between different elements of the nanostructure strongly affects the performance of the device.

1. ELECTRONS ON HELIUM AS QUBITS Surface state electrons on helium are attracted by a weak positive image charge in the liquid. The resulting Coulomb potential well U(z) = −Ze2/4πε0z gives rise to a series of hydrogen-like energy states perpendicular to the liquid surface, with Rydberg levels Em = −Re/m2 where Z = (ε−1)/4(ε+1) = 0.007, Re = 0.67 meV, m (≥ 1) is the quantum number and ε is the dielectric constant of liquid helium. The electrons are prevented from entering the liquid by a 1 eV potential barrier at the surface. Below 2 K the electrons are in the quantum ground state, floating at = 11 nm above the liquid surface. The transition to the first excited state ( = 45nm) may be induced by applying resonant microwave radiation2. External electric fields may be used to Stark shift this transition frequency3 from ~125 GHz at E = 0 to ~ 200 GHz for E = 11 kV/m. Platzman and Dykman3 first proposed that the two lowest energy states of an electron above liquid 4He may form the |0> and |1> basis states of a qubit. Neighbouring qubits at distances z1 and z2 above the helium surface respectively and separated by a distance a interact via a dipolar interaction Ve-e = (z1z2)e2/4πε0a3 which, for qubits in resonance, permits coherent energy exchange. By allowing the qubit system to evolve over appropriate timescales, quantum gate operations may be performed4. At millikelvin temperatures the electrons interact only weakly with the environment via coupling to

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354 354 thermal vibrations, or ripplons, on the atomically smooth superfluid surface. The effect of these interactions on the relaxation and dephasing of the qubit system has been calculated5 and has been shown to be small, making electrons on helium excellent candidates for qubits, with long coherence times. An essential requirement is the trapping and detection of individual localized electrons, which has been performed for a single electron1. The work reported here aims to develop a linear electron array suitable for quantum information processing.

Fig. 1. A linear array of electron traps has been designed and fabricated. (a) Electrons on the surface of liquid helium may be moved from a reservoir into a microchannel and detected using an SET. (b) Modeling of the electrostatic potential shows that control electrodes submerged in the microchannel form an array of electron traps. (c) The device has been fabricated by multi-layer electron beam lithography. A SET at the mouth of the microchannel is used as an electrometer to measure charge in and out of the microchannel as the reservoir electrode bias is swept.

2. A LINEAR ELECTRON TRAP ARRAY The new device was fabricated using standard electron beam lithography techniques and is shown in Fig. 1. Independently biased control electrodes 1.4 µm apart are fabricated at the base of a microchannel 0.5 µm deep and 4 µm wide. The microchannel is filled with liquid helium by the capillary action of the superfluid liquid 4He, once a small quantity is introduced into the experimental cell. Adjacent to the microchannel is a larger helium pool which acts as a reservoir of surface state electrons. These can be manipulated in and out of the channel according to the dc bias potential applied to an electrode at the base of

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355 355 the reservoir. The potential energy of the surface-state electrons on the reservoir will then be E ≈ −eVres + 2nd/εε0, where n m−2 is the number density of the electrons on the reservoir and d is the depth of the helium. An SET is fabricated in aluminium by the standard shadow evaporation technique6 between the reservoir electrode and the control electrodes. This acts as a sensitive electrometer to count individual electrons as they pass in and out of the channel. This arrangement of electrodes uses seven layers of electron beam lithography on a Si substrate with all the structures fabricated in gold (Au), except for the aluminium (Al) SET. Electrostatic simulations (Fig. 1b) show that, with appropriate bias voltages applied to the SET, control electrodes and the surrounding guard electrode, potential wells form above the SET and the control electrodes. Electrons are deposited on the surface of the helium in the reservoir by thermionic emission from a pulsed filament in the experimental cell. The SET fabricated in this device exhibited a resistance of R = 99.7 kΩ at 300 K. Experiments were conducted in a dilution refrigerator at 130 mK. Coulomb blockade oscillations (CBO) were observed in the dc current of the SET as the reservoir, guard or control electrode voltages were swept at low source-drain bias. For this SET, the observed CBO were more complicated than the usual sinusoidal or Lorentzian shape oscillations (Fig. 2.). The oscillations were highly charge stable and repeatable; random phase shifting or two-level fluctuator noise was rarely observed. A noise level of 1 x 10-4 e/√Hz was achieved, allowing highly sensitive monitoring of charge movement close to the SET. The aperiodic oscillations meant that absolute charge shifts could be measured, without the 2π uncertainties of sinusoidal oscillations. 3. CHARGING/DISCHARGING THE TRAP The CBO before and after the sample was charged with electrons is shown in Fig. 2c. After firing the filament, shifts in the phase of the oscillation occur which may be measured in relation to the uncharged reference data. When sweeping the reservoir electrode, shifts typically of Vres = 2 mV were measured, corresponding to electrons entering the potential well above the SET during a negative sweep of the reservoir bias and to the subsequent discharging of the potential well during a positive bias sweep. This charging/discharging cycle could be repeated typically two or three times before the charge motion ceased indicating that electrons were slowly lost from the system. A clear mechanism for the leak of electrons from the system is not yet understood. The charge motion reappeared once the sample was recharged with electrons by another filament pulse.

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Fig. 2. (a) An unusual structure of the CBO in our voltage-biased SET is observed. (b) This structure is reproduced by modelling the SET as having two capacitively coupled islands each with tunnel junctions to the source and drain of the SET. (c) The CBO after firing (black) show discrete shifts in phase as Vres is swept, each corresponding to a single electron entering or leaving the trap above the SET. This shift can be measured by comparison with the uncharged reference oscillation (grey).

A similar experiment in which the potential of the SET source and drain electrodes were swept negative with respect to all other electrodes is shown in Fig. 3a. Although electrostatic modelling shows that the well above the SET should disappear at VSET ≈ −300 mV the discharging of the well continues up to the maximum applied voltage of −1.2 V. Some 60 electrons are observed to leave the trap over this range indicating that the potential well is deeper than calculated due to a positive potential on the SET. This offset is probably due to the contact potential Vc formed between the gold (Au) electrodes of the device and the small aluminium (Al) electrodes which make up the SET. As Vc = − (ΦB − ΦA)/e where ΦA and ΦB are the work functions of the two metals, a ~1 V positive contact potential is formed on an Al structure in contact with a Au electrode at zero voltage bias. Our results indicate that this contact potential is not strongly screened by the oxidisation of the Al surface or by the adhesion of mobile charged particles to the material. The work function was measured directly on a grounded test sample by KP

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357 357 Technology (http://www.kelvinprobe.com) using their off-null SKP5050 system7 with a sub 50 micron diameter tip and a work function resolution < 5 mV. Fig. 3b. shows ambient scanning Kelvin probe measurements of the electrostatic potential across the surface of a typical section of Si wafer with Au electrodes leading to a central area in which a 100 µm x 100 µm square of Al of thickness 70 nm was fabricated using our electron beam lithography techniques. A positive potential offset is visible above the Al section with Vc ≥ 800 mV.

Fig. 3. (a) As VSET is swept negative electrons leave the trap one by one. Discharging events are observed up to the maximum applied voltage, indicating that the potential well is deeper than expected. (b) Kelvin Probe scans of a small Al square fabricated at the centre of an array of Au electrodes reveal a potential well above the Al due to the contact potential difference between the two materials.

In conclusion, we have shown that small numbers of electrons can be moved in and out of a microchannel of liquid helium using submerged electrodes and detected with a SET. A single electron has not yet been isolated in this new device. It is anticipated that fabrication of similar devices in niobium rather than gold will significantly reduce the contact potential difference between the elements of the device leading to the controllable manipulation of a single electron. A further aim is to trap a linear array of electrons and to ionise electrons from the traps. Subsequently, the coherence time of a single qubit will be measured and the control one and two coupled qubits with microwave radiation investigated. Selective ionisation of qubits in the excited |1> state before passing the remaining |0> state electrons through the channel would allow the use of the SET to readout the resulting shift register in classical time. A 200 GHz pulsed microwave system is also being installed in our dilution refrigerator in order to facilitate these experiments. Acknowledgments We thank M.I. Dykman, Y. Mukharsky, E. Rousseau, S.A. Lyon, P.M. Platzman, H. Hashiba and J. Saunders for discussions and F. Greenough, A.K. Betts, J. Taylor and others for technical support. The work was supported by the EPSRC and by Royal Holloway, University of London.

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358 358 References 1. Papageorgiou G, et al. Counting Individual Trapped Electrons on Liquid Helium. Appl. Phys. Lett. 2005; 86: 153106; E. Rousseau and Y. Mukharsky; to be published. 2. Collin E et al. Microwave Saturation of the Rydberg States of Electrons on Helium. Phys. Rev. Lett. 2002; 89: 245301. 3. Platzman P, Dykman MI. Quantum Computing with Electrons Floating on Liquid Helium. Science 1999; 284: 1967–1969. 4. Lea MJ, Frayne PG, Mukharsky Y. Could we Quantum Compute with Electrons on Helium? Fortschritte der Physik 2000; 48: 1109–1124. 5. Dykman MI, Platzman PM, Seddighrad P. Qubits with electrons on liquid helium. Physical Review B 2003; 67: 155402. 6. Fulton TA, Dolan GJ. Observation of single-electron charging effects in small tunnel junctions. Phys. Rev. Lett. 1987; 59: 109–112. 7. Baikie ID, Estrup PJ. Low cost PC based scanning Kelvin probe. Rev. Sci. Instrum. 1998; 69: 3902-3907.

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MEASURING NOISE AND CROSS CORRELATIONS AT HIGH FREQUENCIES IN NANOPHYSICS

´ M. CREUX, A. ZAZUNOV, T.K.T. NGUYEN, A. CREPIEUX, AND T. MARTIN Centre de Physique Th´eorique et Universit´e de la M´editerran´ee Case 907, 13288 Marseille, France

The purpose of the present paper is to propose two scenarios for measuring high frequency noise. The ﬁrst one uses inductive coupling to an LC circuit, and describes the measurement of noise cross-correlations as in an Hanbury-Brown and Twiss geometry. The second one uses the photo-assisted transfer of two electrons in a normal metal– superconductor circuit, which is capacitively coupled to the mesoscopic circuit to be measured.

1. Introduction The measurement of ﬁnite frequency noise in mesoscopic systems provides a crucial diagnosis of the carriers involved in the transport1,2 . Here, we are interested in 2 types of systems where noise is detected via a measuring device – coupled to the mesoscopic device which should pick up (via repeated time measurements) the noise contribution at a speciﬁc frequency, without the manipulation of a time series. Such proposals have been put forth and some have been implemented experimentally within the last decade3,4,5,6,7,8. Low frequency noise cross-correlations measurements have also been performed9,10 , which showed that a fully degenerate electron gas has negative noise correlations. Yet, ﬁnite frequency noise cross-correlations are useful in the study of electronic entanglement in mesoscopic devices 11 , and in the identiﬁcation of anomalous charges in Luttinger liquid wires12 . The purpose of the present paper is twofold: to suggest a way to measure noise cross-correlation with inductive coupling, and to suggest a way to measure noise via photo-assisted Andreev reﬂection in a capacitive coupling setup.

2. Measurement of noise cross-correlations with inducting coupling For measuring cross-correlations with an LC circuit, two inductances (L1 and L2 ) and a single capacitor (C) are needed. The two inductors, with coupling constants α1 and α2 , are placed next to the two outgoing arms of the three terminal mesoscopic device (Fig. 1), and is placed in series. Depending on the wiring of these inductances (Fig. 1a or Fig. 1b), the two inductances “see” the outgoing currents with the opposite sign or with the same sign.

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360 0 1 1 0 0 1 0 1

α1

C

α2

α1

C

α2

000000000000000000000000000 000000000000000000000000000 000000000000111111111111111 111111111111 111111111111111111111111111 000000000000000111111111111 000000000000 111111111111111 000000000000000111111111111 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 111111111111111 111111111111111 000000000000000 000000000000000 10 0 1 10 0 1 1010 0 1 0 1 10 0 1 10 0 1

M

a)

M

b)

Figure 1. Schematic description of the noise cross-correlation setup. M is the mesoscopic circuit to be measured, C is the capacitor and there are two inductors with coupling constants α1 and α2 to the mesoscopic circuit. The electrical components of the detector are in series and they “see” the current with the same sign (a) or with the opposite sign (b).

It was shown4 that ﬁnite frequency noise can be detected by coupling inductively an LC circuit to the wires connected to an arbitrary mesoscopic device. Here we address two issues: a) we generalize this proposal to the measurement of noise cross correlations; b) we brieﬂy mention the role played by dissipation in this measurement scheme. In the initial proposal4 , the noise is measured at the resonant frequency of the LC circuit, by performing repeated time measurements of the charge ﬂuctuations on the capacitor plates. The measured noise turns out to be a mixture of the two unsymmetrized noise correlators, which correspond to emission and absorption of radiation from the mesoscopic device. For the case of cross correlations, one measures the charge ﬂuctuations x2 (0)2+ with the geometry of Fig. 1a and subsequently one can switch the wiring and measure such ﬂuctuations x2 (0)2− with the circuit of Fig. 1b. By subtracting the two signals 13 : 1 x2 (0)2 = x2 (0)2+ − x2 (0)2− , (1) 2 one isolates the contribution of cross-correlations, which is proportional to α1 α2 . The following non-symmetrized current correlators are introduced in Fourier space: dt iωt dt iωt + − e Ii (0)Ij (t) , e Ii (t)Ij (0) . (ω) = Sij (ω) = (2) Sij 2π 2π with i, j = 1, 2 corresponding to the leads of the mesoscopic circuit. The charge ﬂuctuations take the ﬁnal form: πα1 α2 + − Re (N (Ω) + 1)S12 (Ω) − N (Ω)S12 (Ω) (3) x2 (0) = 2 η(2M ) with N (Ω) = 1/(eβΩ + 1) is the Bose Einstein distribution at the detector circuit temperature, which is not necessarily the mesoscopic device temperature, and M = L1 + L2 is the “mass” of the detector circuit. Note the presence of the adiabatic parameter η, which suggests that the measured noise diverges! If one includes dissipation in the LC circuit due to a ﬁnite resitance of the circuit using the CaldeiraLegett model14 , it turns out15 that η is replaced by the linewidth of the LC circuit. Next, this proposal is tested on a system of three terminals (Fig. 2), a so called “Y junction”, described by scattering theory. We consider the case where the voltage biases satisfy μ13 > μ12 > μ23 > 0 (μij ≡ μi −μj ). At a low temperature, the charge

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ﬂuctuations stay negative and singularities are present at frequencies equal to μ23 , μ12 , μ13 . When the temperature becomes larger than the bias voltages, the charge ﬂuctuations become positive. However, when the temperature goes to zero, the + (ω). measured charge ﬂuctuations become equal to S23 0

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3. Noise measurement via photo-assisted Andreev reflection A high frequency noise measurement setup was proposed, which used capacitive coupling5 between a double quantum dot and a nearby mesoscopic conductor. This idea was implemented recently6 using a superconductor-insulator-superconductor (SIS) junction for the detector circuit: a DC measurement of quasiparticle tunneling provided information on high frequency noise. The purpose of the present work is to analyze a similar situation, except that the SIS junction is replaced by a circuit which transfers two electrons via Andreev reﬂection. Here, Andreev reﬂection processes occur through a quantum dot, allowing to ﬁlter electron energies: two a) DETECTOR CIRCUIT

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Figure 3. a) Schematic description of the set up: the mesoscopic device to be measured is coupled capacitively to the detector circuit. The latter consists of a normal metal lead–quantum dot– superconductor circuit with a DC bias. b), c), d) Photo-assisted Andreev reﬂection: b), c) Emission of a Cooper pair in a normal metal in the case of a “photon” is provided to or provided by a neighboring environment. d) Case of absorption of a Cooper pair with photo-assisted Andreev reﬂection, where a “photon” is provided by a neighboring environment. Notice that the tunneling of electrons is sequential.

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inelastic electron jumps are required for a current to pass through the measurement circuit, the energy being injected/emitted by a “photon” provided/absorbed by the mesoscopic circuit to be tested. The circuit is depicted in Fig. 3a. The charging energy of the quantum dot is large enough that double occupancy is prohibited. This dot is in tunneling contact with a superconductor. Two capacitors are placed between each side of the mesoscopic device and each side of the quantum dot– superconductor junction. Current ﬂuctuations in the mesoscopic device generate voltage ﬂuctuations across the dot–superconductor tunnel junction, which translate into phase ﬂuctuations at the junction, in the spirit of the dynamical Coulomb blockade16 . The Hamiltonian which describes the decoupled normal metal lead–dot– superconductor–environment (mesoscopic circuit) system reads + k c+ D c+ Eq γq,σ γq,σ +μS NS +Henv , (4) H0 = D,σ cD,σ +U n↑ n↓ + k,σ ck,σ + k,σ

σ

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which correspond to the normal metal lead, the quantum dot (single occupancy only), and the (BCS) superconductor, described in terms of quasiparticle operators (γq,σ ) which are related to cq,σ by the Bogoliubov transformation. The tunneling Hamiltonian reads: −iφ TD,q c+ + Tk,D c+ (5) HT = D,σ cq,σ e k,σ cD,σ . q,σ

k,σ

The ﬂuctuating phase factor represents the coupling to the mesoscopic circuit. All the information about the environment is contained in the autocorrelation of the phase operators J(t) t = [φ(t) − φ(0)]φ(0). The phase operator is related to the voltage by φ(t) = e −∞dt V (t ). Indeed, because of the capacitive coupling between the sides of the dot-superconductor junction and the mesoscopic circuit, current ﬂuctuations of the mesoscopic device translate into voltage ﬂuctuations of the dotsuperconductor junction across the trans-impedance5 δV (ω) = Z(ω)δI(ω). The current through the normal lead/quantum dot/superconductor (NDS) junction, due to the tunneling of two electrons, is calculated using the Fermi Golden Rule generalized with the T–matrix. Here one needs to carry out calculations of the matrix element to fourth order in the tunneling Hamiltonian in order to describe electron transfer. The inelastic current through the NDS detector reads: ∞ Δ2 16π 2 e 4 4 4 ∞ NN T1 T2 dE dE √ I inel RK Δ Δ E 2 − Δ2 E 2 − Δ2

eV eV |Z( + )|2 S ( + ) I × d d )2 ( + Dinel −∞ −∞ ∞ ∞ |Z(−( + ))|2 SI (−( + )) , (6) − d d ( + )2 Dinel eV eV where Dinel is an energy denominator product17 , RK = 2π/e2 is the quantum of resistance, Z(ω) is the trans-impedance of the circuit and SI (ω) = S + (−ω), with

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S + (ω) = dteiωt I(0)I(t). The ﬁrst term in Eq. (6) describes the tunneling of a Cooper pair from the normal lead to the superconductor via the quantum dot: as depicted in Figs. 3b,c the tunneling between the dot and the superconductor involves the emission/absorption of a photon. The second term describes the inverse tunneling process: a Cooper pair absorbing energy from the neighboring device (Figs. 3d), its constituent electrons then tunneling from the superconductor to the normal lead. We illustrate the present results by considering a point contact for the mesoscopic circuit. We compute the photon assisted tunneling (PAT) current through the detector due to the high frequency current ﬂuctuations of the device, as a function of detector bias voltage IP AT (eV ) = I(eVd = 0, eV ) − I(eVd = 0, eV ) (excess current), at zero temperature for simplicity. The left panel of Fig. 4 depicts the PAT current as a function of detector bias voltage for a ﬁxed dot energy level position, but for diﬀerent values of eVd . When the bias of the mesoscopic device increases, the noise increases, energy emitted or absorbed also increases, which triggers the PAT current. From eVd = D , if we decrease eVd , the PAT current is reduced, approaching 0 and the step disappears hereafter. This is realized at the point when the spectral density of noise of the mesoscopic device (point contact) contains a singularity in its derivative. In the right panel of Fig. 4, we plot the dependence of the PAT current on the dot energy level for several values of the bias voltage of the detector circuit, which is chosen smaller than the device bias voltage eVd = 0.2, specifying eV > eVd /2. By choosing this range of eV , only the current from the normal metal to the superconductor contributes: the dominant contribution comes from emission processes. The PAT current decreases when the dot level is raised, however, it displays a step at D = eV , provided that eV < eVd . At the step location, the PAT current drops fast, then it decreases more slowly and saturates. For eV > eVd /2, if

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we increase eV , the step height decreases and eventually vanishes when eV = eVd . Hereafter, if one continues to increase eV , the PAT current is still the same as it with eV = eVd because the energy emission to the mesoscopic device reaches saturation. 4. Conclusion We have shown how a careful wiring of the detector circuit can isolate the noise cross-correlations signal. This promises to have applications in the detection of fractional charges in carbon nanotubes12 as well as in the detection of electronic entanglement18 . For capacitive coupling to an NS circuit, by controlling the detector circuit (varying eV and D ), we can make a mapping of the spectral density of noise. Here the frequency ω = eVd corresponds to the point where the excess noise of the point contact contains a singularity in its derivative. This constitutes an encouraging scenario for detecting speciﬁc features in the noise spectrum. References 1. C. Chamon, D. E. Freed, and X. G. Wen, Phys. Rev. B 53, 4033 (1996). 2. T. Martin in Les Houches Summer School session LXXXI, edited by E. Akkermans, H. Bouchiat, S. Gueron, and G. Montambaux (Springer, 2005). 3. R. J. Schoelkopf et al., Phys. Rev. Lett. 78, 3370 (1997). J. Gabelli et al., Phys. Rev. Lett. 93, 056801 (2004). ´ 4. G. B. Lesovik and R. Loosen, Pis’ma Zh. Eksp. Teor. Fiz. 65, 280 (1997) [JETP Lett. 65, 295, (1997)]; Y. Gavish, I. Imry, and Y. Levinson, Phys. Rev. B 62, 10637 (2000). 5. R. Aguado and L. Kouwenhoven, Phys. Rev. Lett. 84, 1986 (2000). 6. R. Deblock et al., Science 301, 203 (2003); A. Onac et al.,Phys. Rev. Lett. 96, 026803 (2006); P.-M. Billangeon et al., cond-mat/0508676. 7. R. Deblock, E. Onac, L. Gurevich, and L. P. Kouvenhoven, Science 301, 203 (2003). 8. P.-M. Billangeon, F.Pierre, H. Bouchiat, and R. Deblock, cond-mat/0508676. 9. T. Martin and R. Landauer, Phys. Rev. B 45, 1742 (1992); M. B¨ uttiker, Phys. Rev. B 45, 3807 (1992). 10. M. Henny , Science 284, 296 (1999); W. D. Oliver et al., Science 284, 299 (1999). 11. G. B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B. 24, 287 (2001); P. Recher, E.V. Sukhorukov, and D. Loss, Phys. Rev. B 63, 165314 (2001). 12. A. V. Lebedev, A. Cr´epieux, and T. Martin, Phys. Rev. B 71, 075416 (2005). 13. M. Creux, A. Cr´epieux, and T. Martin, cond-mat/0507708. 14. A.O. Caldeira and A.J. Leggett, Phys. Rev. A 31 1059 (1985) 15. M. Creux, A. Zazunov, E. Paladino, A. Cr´epieux, and T. Martin, in preparation. 16. G. L. Ingold and Yu. V. Nazarov, in Single Charge Tunneling, H. Grabert and M.H. Devoret eds. (Plenum, New York 1992). 17. T. Nguyen et al., cond-mat/0602408. 18. N. Chtchelkatchev et al., Phys. Rev. B 66, 161320 (2002).

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SINGLE WALL CARBON NANOTUBE WEAK LINKS

K. GROVE-RASMUSSEN† AND H. I. JØRGENSEN† AND P. E. LINDELOF

†

Nano-Science Center Niels Bohr Institute Universitetsparken 5, 2100 Copenhagen O, Denmark E-mail: k [email protected] These authors contributed equally to this work.

We have reproducibly contacted gated single wall carbon nanotubes (SWCNT) to superconducting leads based on niobium. The devices are identiﬁed to belong to two transparency regimes: The Coulomb blockade and the Kondo regime. Clear signature of the superconducting leads is observed in both regimes and in the Kondo regime a narrow zero bias peak interpreted as a proximity induced supercurrent persist in Coulomb blockade diamonds with Kondo resonances.

1. Introduction Carbon nanotubes have been under intense investigation for more than a decade due to their unique mechanical and electrical properties. Single wall carbon nanotubes (SWCNT) are truly one-dimensional systems and in contrast to the semi-1D channels deﬁned in a two-dimensional electron gas they can easily be contacted to materials with interesting properties such as superconductors (S). A normal region between two superconductors acts as a weak link1 and eﬀects as proximity induced supercurrent and sub-gap structure due multiple Andreev reﬂections (MAR) can be seen. In small low capacitance junctions (e.g. SWCNT) the above eﬀects are modiﬁed by size and charge quantization. Size quantization leads to a discrete density of states in the normal region with levels of width Γ separated by ΔE. The supercurrent will be maximum in resonance (aligned with a level) and decreases to a minimum oﬀ resonance.2,3,4 For weakly coupled SWCNTs, charge quantization gives rise to Coulomb blockade6 which generally suppresses the supercurrent.7,2 However, when the number of electrons on the dot is odd (net spin 1/2 on the dot) and relative good coupling is achieved, the Kondo eﬀect becomes important. The Kondo eﬀect eﬀectively sets up a resonance with width ∼ kB TK at the Fermi energy of the leads and in some sense turns oﬀ Coulomb blockade. Thus a resonance exists despite Coulomb blockade to carry the supercurrent (kB TK Δ). The interplay between Kondo/Coulomb blockade and superconductivity is under intense interested.5,7,8,9,10 We here present measurements on SWCNT devices with Nb contacts showing

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Figure 1. (a) Top view of the alignment marks used to align the EBL mask for the catalyst islands and electrodes. In the black rectangle, catalyst islands have subsequently been deﬁned. (b) Zoom-in on the catalyst islands in (a). (c) SEM Micrograph of a catalyst island after CVDgrowth. Only one or a few SWCNTs reach further than 1 µm from the catalyst island which makes it likely to have only one SWCNT between the electrodes to be deﬁned at some distance from the catalyst island. (d) Electrodes are deﬁned with gap sizes of 300 to 700 nm (here 300 and 500 nm).

high quality SWCNT quantum dots and address the topic of the supercurrent carried through Kondo resonances. From a technical point of view it is not the ﬁrst time niobium has been used as superconducting contacts for SWCNT.11,12 However, to the best of our knowledge niobium based devices have not been presented clearly revealing Kondo resonances and Coulomb blockade before. Several groups have made week links based on carbon nanotubes with other superconductors such as tantalum, rhenium13 and aluminum.3,4,5,10,12,14 The advantage of using niobium is its high critical temperature ∼ 9 K, which should allow for measurements above 4 K (see below). Furthermore, a high critical temperature gives a high superconducting energy gap Δ which increases the critical current2 as Ic ∼ Δ. 2. Sample processing The two terminal SWCNT devices presented in this proceeding is made with two types of superconducting contacts based on niobium: Pd/Nb/Pd and Ti/Nb/Ti. The lower metallic layer (Ti or Pd) ensures relatively good contact to the SWCNT, while Nb has a very high critical temperature ∼ 9 K. The top layer protects the Nb from oxidation. However, we experienced that the superconducting proximity eﬀect from the Nb layer into the lower Ti or Pd layer is very weak resulting in relative low critical temperatures of Tc ∼ 2 K. The Nb ﬁlm still have high critical temperature ∼ 9 K measured on a four terminal device. The substrate consist of 500 nm SiO2 on a highly doped silicon wafer (acceptors Sb, resistivity ρ < 1 − 3 mΩ cm), which is used to electrostatically change the potential of the SWCNT. First alignment marks are deﬁned by electron beam lithography (EBL) as shown in Fig. 1a. The resist used is double layered consisting of 6% copolymer followed by 2% PMMA.

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Both layers are spun at 4000 rpm, 45 s and baked on a hotplate at 185 ◦ C for 90 s. The EBL is done on a JEOL JSM-6320F scanning electron microscope (SEM) with Elphy software at 30 keV with a sensitivity of 200 μC/cm2 and currents of ∼ 40 pA and ∼ 150 pA for small and big patterns, respectively. The pattern is developed in MIBK:IPA (1:3) for 60 s stopped by 30 s in IPA. The sample is ashed for 20-40 s in an oxygen plasma prior to deposition of 60-70 nm Cr. Lift-oﬀ is done in acetone. The same EBL procedure is used to make the pattern for the catalyst islands. The catalyst liquid is spun on the sample at 4000 rpm for 150 s and baked at 185 ◦ C for 3 min. to ensure that the catalyst sticks to the substrate. Lift-oﬀ is made in acetone and ultrasound if necessary. Figure 1b shows an optical image of the catalyst islands. The sample is now transferred to a chemical vapor deposition (CVD) tube furnace, where SWCNTs are grown in a controlled mixture of argon, hydrogen and methane (Ar: 1 L/min, H2 : 0.1 L/min, CH4 : 0.5 L/min). The optimal growth temperature depends on the thickness and the area of the catalyst islands. Typically, the temperature used is in the range of 850 − 960 ◦ C. After each growth the sample is examined in a SEM on a test area, which might appear similar to the picture shown in Fig. 1c. Sometimes several trials are needed changing the temperature to achieve an appropriate density of tubes. The test area ensures that the SWCNTs in the regions for devices are not exposed to the electron beam, which might damage the SWCNT. On the catalyst island and within a micron, the nanotubes grow rather densely. However, one or a few tubes ”escape” to distances of several microns. The contacts are aligned relative to the catalyst island and placed some microns away by EBL (no oxygen plasma step). Thus it is likely that only one SWCNT lies in the gap between the two electrodes. The Pd or Ti is evaporated by thermal evaporation, while Nb is DC-sputtered in an Ar atmosphere. Typical layers used are 4 nm Ti (or Pd) followed by 60 nm Nb and capped by 10 nm Ti (or Pd). Finally bonding pads of Au/Cr are deﬁned by optical lithography. 3. Coulomb blockade regime We ﬁrst present measurements of a Pd/Nb/Pd contacted SWCNT with poor transparency G e2 /h that leads to Coulomb blockade at low temperatures. Figure 2a shows a bias spectroscopy plot of such a device where clear Coulomb blockade diamonds are seen with charging energies of Uc ∼ 2 − 3 meV. At low bias a dip in the diﬀerential conductance is observed due to the tunnel-like behavior of the device reﬂecting the density of states of the superconducting leads.16 The green arrows point to the onset of quasiparticle tunneling eVsd = 2Δ ∼ ±500 μeV. In Fig. 2b a zoom of another slightly more conducting gate region of the device is shown. The onset of quasiparticle tunneling is clearly revealed (green arrows) and some conductance in the resonances is seen below the gap which is attributed to Andreev reﬂections. Such regular behavior has been observed in several devices with niobium based contacts and poor transparency. The onset of quasiparticle tunneling corresponds to a critical temperature of the Tc ∼ 1.5 K consistent with the features

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vanishing above Tc . This critical temperature is much lower than the critical temperature measured in the niobium ﬁlm and is properly due to poor proximity eﬀect into the lower palladium layer in contact with the nanotube. 4. Supercurrent and the Kondo eﬀect The next device is more conducting and thus belong to a transparency regime where the Kondo eﬀect can be observed, when the SWCNT quantum dot has a net spin.15 The contacts are made of a Ti/Nb/Ti trilayer. Figure 3a shows a bias spectroscopy plot in a gate region with an even-odd ﬁlling of single particle levels marked by the letters E/O. The Kondo resonances are identiﬁed by the clear change in magnitude of the features for the onset of quasiparticle tunneling (eVsd = ±2Δ) and one Andreev reﬂection (eVsd = ±Δ). For even electron ﬁlling, the onset of quasiparticle tunneling is more pronounced because quasiparticle tunneling is a lower order process than one Andreev reﬂection and transport happens via cotunneling in the Coulomb blockade region. However, when the Kondo eﬀect is present (odd ﬁlling), a resonance exists at the Fermi energy of each contact leading to resonances for one Andreev reﬂection processes at eVsd = ±Δ. This qualitative explanation depends on the relative magnitude of kB TK and Δ. We also note that this eﬀect has been observed in other Kondo devices.5,17 Figure 3b shows the Kondo resonance K1 at low bias voltages where the high

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Figure 3. (a) Bias spectroscopy plot of three Kondo resonances in a Ti/Nb/Ti sample in the superconducting state. O/E denote odd/even electron ﬁlling on the dot. Conductance peaks at Vsd = ±2Δ/e are most pronounced for even ﬁlling (red arrows) while ±Δ/e conductance peaks are more pronounced for odd ﬁlling (green arrows) due to the Kondo resonances (see text). (c) Detailed plot of Kondo resonance K1 from (a). The green arrow points to a zero bias peak interpreted as a noise smeared supercurrent present throughout the gate region shown. (c) Another Kondo resonance with a similar less pronounced zero bias peak. (d) Zero bias peak area versus gate voltage extracted from (b). The zero bias peak is suppressed in Coulomb blockade regions, while it exists in both Coulomb blockade- and Kondo-resonances. (e) Similar plot as (d). The zero bias peak area has maximum in the resonances and is also ﬁnite in the Kondo resonance, while being suppressed in Coulomb blockade regions. Note, that the data of (a) and (c) have been interpolated and smoothened.

conductive black region (green arrow) reﬂects a zero bias peak in dI/dV versus bias.5,4 We interpret this peak as a noise smear proximity induced supercurrent with an estimated magnitude given by the area of the peak (not the whole area under the peak). This interpretation is supported by the successful analysis of an analogous zero bias peak interpreted as a supercurrent in an open quantum dot.4 We do not believe the peak to be a narrow Kondo resonance since it is present in the Coulomb blockade regions with even ﬁlling as well. Figure 3d shows the zero bias peak area versus gate voltage across the Kondo resonance. The peak is maximum in each Coulomb blockade resonance and also present in the Kondo resonance, while it is heavily suppressed in the Coulomb blockade regions. Figures 3c and 3e show another Kondo resonance (K2) with a smaller zero

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bias peak. Similar behavior is seen in this Kondo resonance: Maxima of the zero bias peak area in the Coulomb blockade resonances and a small ﬁnite value in the Kondo resonance region, while the zero bias peak area is fully suppressed entering Coulomb blockade. These behaviors are qualitatively expected for a supercurrent and could be consistent with K2 having a lower Kondo temperature than K1. We are unfortunately not able to turn oﬀ the superconductivity in the leads in the setup used and more experimental work is needed to determine the relation between the zero bias peak area and kB TK /Δ. 5. Conclusion We have succeeded in making weak links between niobium based leads, where the link is a high quality SWCNT. At low temperature the devices behave as quantum dots with observation of Coulomb blockade and Kondo eﬀect. In the case of closed dots the onset of quasiparticle tunneling is clearly revealed. For more open dots a zero bias peak interpreted as a supercurrent is enhanced in Coulomb blockaded diamonds with an odd number of electron compared to diamonds with an even number due to the presence of a Kondo resonance. Acknowledgments We like to thank Jørn Bindslev Hansen and Inge Rasmussen for assistance and use of their niobium sputtering machine. This work is supported by the Danish Technical Research Council (The Nanomagnetism framework program), EU-STREP Ultra-1D program and the Nano-Science Center, University of Copenhagen, Denmark. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979). C. W. J. Beenakker, cond-mat 0406127 (2004). P. Jarillo-Herrero, J. A. van Dam, and L. P. Kouwenhoven, Nature 439, 953 (2006). H. I. Ingerslev, K. Grove-Rasmussen, T. Novotn´ y et. al., cond-mat 0510200 (2005). K. Grove-Rasmussen, H. I. Ingerslev, and P. E. Lindelof, cond-mat 0601371 (2006). M. Bockrath, D. H. Cobden, P. L. McEuen et. al., Science, 275, 1922 (1997). L. I. Glazman and K. A. Matveev, JETP 49, 659 (1989). M.-S. Choi, M. Lee, K. Kang, and W. Belzig, Phys. Rev. B 70, 020502(R) (2004). F. Siano and R. Egger, Phys. Rev. Lett. 93, 047002 (2004). M. R. Buitelaar, T. Nussbaumer and C. Sch¨ onenberger, Phys. Rev. Lett. 89, 256801 (2002) A. F. Morpurgo, J. Kong, C. M. Marcus, and H. Dai, Science, 286, 263 (1999) J. Haruyama, A. Tokita, N. Kobayashi et. al., Appl. Phys. Lett. 84, 4714 (2004) A. Yu. Kasumov, R. Deblock, M. Kociak, B. Reulet et. al., Science, 284, 1508 (1999) M. R. Buitelaar, W. Belzig, T. Nussbaumer et. al., Phys. Rev. Lett. 91, 057005 (2003) J. Nyg˚ ard, David H. Cobden, P. E. Lindelof, Nature 408, 342 (2000) V. Krsti´c, S. Roth, M. Burghard et. al., Phys. Rev. B 68, 205402 (2003) T. S. Jespersen M. Aagesen, C. B. Sørensen P. E. Lindelof, and J. Nyg˚ ard, to be published.

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OPTICAL PREPARATION OF NUCLEAR SPINS COUPLED TO A LOCALIZED ELECTRON SPIN

DIMITRIJE STEPANENKO AND GUIDO BURKARD Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland We discuss a scheme for optically narrowing the Overhauser ﬁeld distribution in an ensemble of nuclear spins in a quantum dot, based on the detection of light scattered from an electronic three-level system which is optically driven by a pair of lasers satisfying a two-photon Raman resonance. The frequency and power of the lasers can be adjusted depending on the measurement record. Repeating the cycle of detection and adjustment narrows the nuclear state distribution to the limit set by the width of the two-photon resonance. Here, we describe realistic limitations on the eﬃciency of the scheme due to a ﬁnite detector eﬃciency and a ﬁnite time allowed for the nuclear spin preparation.

1. Introduction A single electron spin trapped in a quantum dot (QD) provides a suitable arena for the study of basic quantum dynamics and potential applications in quantum information processing 1,2 . Ideally, the spin can be described as a two-level quantum system interacting with controllable external ﬁelds, but decoupled from the environment except during measurement. Some of the important steps for realizing such a setup have already been demonstrated experimentally 3,4,5,6 . One important deviation from the ideal of a completely controllable quantum system is the coupling to the nuclear spins in a QD. The electron in a typical QD is localized within tens of nanometers, therefore the electron wave function overlaps with many (∼ 105 − 106 ) nuclei, so that the electron spin couples to many nuclear spins via a Fermi-type contact interaction. This coupling is an important mechanism for the loss of localized spin coherence 7 . This mechanism has been studied extensively 8,9,10,11,12,13 in the context of electron spins as quantum bits (qubits) for quantum information processing 1,2 . The hyperﬁne coupling of the electron spin to the nuclear ensemble and the possibility to control the electron spin leads to the idea of using the electron-spin control to inﬂuence the nuclear ensemble. Such an indirect nuclear control is used in all-optical nuclear magnetic resonance 14 , imprinting of the nuclear spin polarization 15,16,17 , as well as in dynamic nuclear polarization 18 . Here, we investigate the nuclear spin control mechanism based on the optical excitation of the electronic two-photon resonance and photon detection as proposed in an earlier work19 . The scheme for nuclear spin preparation relies on a monitored

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Figure 1. (a) Three-level system in a quantum dot. States 1 (|↑) and 2 (|↓) are single conductionband electron states split by gμB Btot + δhz where δhz is the z-component of the nuclear (Overhauser) ﬁeld ﬂuctuations. State 3 (|X) is a trion with Jz = 3/2. Driving lasers of frequencies ωp and ωc near the 13 and 23 resonances are detuned by Δ1,2 . The 12 transition is forbidden, while both 13 and 23 are allowed for Btot canted with respect to the structural axis. (b) Two-photon resonance. Occupancy of the exciton state ρXX as a function of the nuclear ﬁeld ﬂuctuation δhz for various control laser powers Ωc . The probe laser power is Ωp , with Ωp = 0.2 ns−1 . (c) Electron coherence time t0 at the various times t since the beginning of the preparation, t = 0. The results are obtained by numerical integration of the Lindblad equation (4) and subsequent evolution of the electron spin in the resulting nuclear ﬁeld. The preparation eﬃciency depends on the driving lasers strengths Ωp and Ωc .

two-photon resonance in a driven three-level system (TLS), analogous to electromagnetically induced transparency (EIT) in atoms 20 . The TLS in a QD is formed by a Zeeman-split electron level, |↑ and |↓, and an excited charged exciton state |X comprising a two-electron spin singlet and a Jz = +3/2 heavy hole. Under excitation by σ + circularly polarized light, the transitions from the | ↑ and | ↓ states to the |X are allowed, while all other transitions are forbidden (Fig. 1a). A pair of σ + polarized lasers with frequencies satisfying the two-photon resonance between spin up |↑ and down |↓ states drives the TLS into a dark state with no amplitude for being in |X. Since all the photon emission from TLS comes from |X, there is no light scattering from the system in a dark state. The position of the resonance depends on the interaction with the nuclei through the eﬀective magnetic (Overhauser) ﬁeld h, and the quantum ﬂuctuations δh cause light scattering from the TLS. Adaptation of the driving laser frequency when a scattered photon from the TLS is detected enables a narrowing of the nuclear spin state and therefore an enhanced spin coherence 19 . The ﬁnal nuclear distribution width is limited by the width of the EIT resonance. The eﬃciency of our scheme that leads to this narrow distribution is limited by the nuclear spin evolution during the preparation, setting the limit τN on the available

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time for preparation. Here we show that the preparation using imperfect photon detection prolongs the time needed to achieve the nuclear ﬁeld narrowing. We also show that the choice of the driving laser powers allows for a tradeoﬀ between the prepared nuclear state width and the time of preparation.

2. Model The driven TLS in the Overhauser ﬁeld of the QD nuclei is described with theHamil σi 0 1 tonian H = H0 + Hint + Hhf , where H0 = − 2 ωz Σz + ωX PX , with Σi = 0 0 and PX = |XX| = (001)T (001), describes the TLS in the eﬀective magnetic ﬁeld including the average nuclear ﬁeld, Hint includes the driving lasers, and Hhf the hyperﬁne coupling to the nuclear ﬁeld ﬂuctuations. The part H0 contains the electron levels | ↑ and | ↓, split by the external magnetic and the mean nuclear ﬁeld, ωz = gμB Btot = |gμB B + h|. The nuclear (Overhauser) ﬁeld operaN 2 tor is h = i=1 Ai Ii , where Ai = ai v0 |Ψ(ri )| , and Ψ(ri ) is the electron wave function at the position ri of the ith atomic nucleus and v0 is the volume of the unit cell. In the rotating wave approximation and rotating frame, the probe (control) laser of frequency ωp(c) and intensity measured by the Rabi frequency Ωp(c) causes transitions between |X and | ↑(| ↓). The hyperﬁne coupling to the average Overhauser ﬁeld is included in H0 , and the coupling to its ﬂuctuations is Hhf = − 21 δh · Σ, where δh = h − h. The resulting Hamiltonian is block-diagonal H = diag(H1 , H2 , . . . , HK ), with 19 ⎞ ⎛ k δh + δ 0 Ωp ⎝ z Hk = − 0 −δhkz − δ Ωc ⎠ , 2 Ωp Ωc −Δ

(1)

where δhkz are the eigenvalues of the operator δhz , and Δ = Δ1 + Δ2 . The time evolution of the total density matrix ρ describing the state of the TLS and the nuclei is governed by the generalized master equation 20 ρ˙ = Lρ ≡ −i [H, ρ]+ W ρ. The exciton decay into the states α =↑, ↓ with a rate ΓXα , accompanied by a photon emission, and the dephasing of |↓ and |X with respect to |↑ at the rates γβ are described by the dissipative term W ρ = α=↑,↓ ΓXα (2σαX ρσXα − σXX ρ − ρσXX )/2 + β=↓,X γβ (2σββ ρσββ − σββ ρ− ρσββ )/2, where σij = σij ⊗ 1 = |ij| acts on the TLS only. The master equations for the 3-by-3 blocks ρkk that correspond to the diﬀerent elements of the reduced nuclear density matrix decouple due to the block form of H, and we obtain ρ˙ kk = −i (Hk ρkk − ρkk Hk ) + W ρkk .

(2)

In the diagonal (k = k ) blocks, Eq. (2) reduces to the Lindblad equation, ρ˙kk = Lk ρkk , where Lk ρ = −i[Hk , ρ] + W ρ.

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3. Manipulation of the nuclear spin ensemble Due to the hyperﬁne coupling between the TLS and the nuclei, the two systems become entangled as the stationary state of the driven evolution (2) is reached. The stationary density matrix ρ¯ has the general form ρ¯ = kk ρ¯kk ⊗ |δhkz δhkz |. We derived an analytical expression for the diagonal 3-by-3 blocks ρ¯kk of ρ¯ as a function of all parameters, including δhkz . The oﬀ-diagonal blocks ρkk with k = k are in general oscillatory, but are decoupled from the diagonal (k = k ) blocks in (2). For nuclear spin preparation, our goal is to narrow the Overhauser ﬁeld distribution, given by the diagonal elements νkk of the reduced nuclear density matrix ν = TrΛ ρ, where TrΛ is the partial trace over the TLS. A narrow nuclear distribution can be prepared by monitoring the photon emission from the QD. The longer the period t during which no photon is emitted, the higher is the probability for δhz to be at the two-photon resonance, δhz = δ. The state of the system conditional on a certain measurement record is the conditional density matrix ρc . In the absence of photon emission, the blocks of ρc obey Eq. (2) with Lk replaced by Lk − S where the collapse operator is deﬁned as S ρ = α=↑,↓ ΓXα σαX ρσXα 21 . We have numerically calculated ρc conditional on the absence of emitted photons for a given duration t. From ρc , the updated nuclear probability distribution νkk is extracted using ν = TrΛ ρc . We ﬁnd that the a posteriori probability is concentrated around the two-photon resonance. The growth of nuclear spin population on resonance will eventually be stopped by the emission of a photon. The stationary emission rate is Γem = TrS ρ¯(t) = Γ k (ρkk )XX νkk , where Γ = ΓX↑ + ΓX↓ 21 . The waiting time distribution for photon emissions is 19 t = Γ−1 pwait (t) = Γ−1 em exp(−Γem t) with a mean waiting time em . The progressive narrowing of the Overhauser ﬁeld distribution leads to a decreasing photon emission rate Γem , and therefore to an increasing average waiting time t. We ﬁrst assume that every emitted photon is detected and later generalize to the case of imperfect detection. The update rule of the nuclear density matrix upon photon emission is 19 ν = TrΛ S ρc /TrS ρc . The Overhauser ﬁeld distribution after the emission is νkk (ρkk )XX = , νkk j νjj (ρjj )XX

(3)

where νkk and (ρkk )XX = X|ρkk |X are taken before the emission. According to Eq. (3), the population in the Overhauser ﬁeld δhz corresponding to the two photon resonance δhz = δ is depleted by the photon emission. The emission of a photon burns a hole in the Overhauser ﬁeld distribution of the shape determined by the ρXX in the stationary state of TLS, as illustrated in Fig. (1). 4. Conditional evolution for imperfect detectors The time evolution of the conditional density matrix ρc as we have described it so far applies only under the assumption that the photon detectors are perfect, i.e., that every emitted photon results in a detector click. Here, we explain how one

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arrives at a more realistic model that reﬂects the fact that real detectors have an eﬃciency e < 1. Our model for a imperfect detector is the following: The detector is sensitive (“on”) with probability e and insensitive (“oﬀ”) with probability 1 − e. In the “on” state of the detector, the conditional density matrix ρc evolves according to the “perfect detector” conditional master equation, ρ˙ con = (L − S)ρc where L is the Lindblad superoperator and S the “quantum jump” superoperator corresponding to the measurement performed by the detector 21 . In the “oﬀ” state of the detector, the detector has no eﬀect on the system, therefore the system evolves according to the usual Lindblad equation, ρ˙ coﬀ = Lρc . The time evolution for an imperfect detector is now given by the weighted average of these two equations with the probabilities e and 1 − e for the two states of the detector, ρ˙ c = eρ˙ con + (1 − e)ρ˙ coﬀ , from which we obtain the conditional evolution for imperfect detectors, ρ˙ c = (L − eS)ρc .

(4)

In the special cases of a perfect (e = 1) and a disabled (e = 0) detector, our conditional master equation (4) reduces to the known cases. Applied to the photon detection in the TLS studied here, we obtain ρ˙ ckk = (Lk − eS)ρckk . Integrating this conditional master equation numerically, we have analyzed the case e = 10% and ﬁnd 19 for the frequency-adaptive technique an electron coherence time of t0 ≈ 460 ns after a preparation time of t = 50 μs, while for e = 100% we obtain t0 ≈ 500 ns after the preparation time t = 50 μs (with all other parameters unchanged). The reason for the enhanced optical preparation time is the reduction of the photon detection rate eΓem < Γem . 5. Adaptive techniques The stationary, isolated TLS at the two-photon resonance is in a dark state, but the coupling to the nuclei causes a nonzero amplitude for the population of |X and thus for photon emission. Detection of a photon thus provides information about δhz . We describe two techniques for using this information to manipulate the EIT setup. If the EIT setup was not adapted after photon detection events, then the repeated photon detection and intermittent evolution conditional on no photon emission would reproduce the initial νkk , according to the quantum trajectory method. However, if we adapt the EIT setup after photon detection, the nuclear density matrix is driven into a new state with a narrow nuclear distribution. In the ﬁrst setup19 , we use the photon detection as an input for the adjustment , so that the new two-photon resonance lies on of the laser frequencies, ωp(c) → ωp(c) max the maximum δhz of the Overhauser ﬁeld distribution. The driven TLS relaxes into the stationary state corresponding to the new nuclear distribution and the photon emission from the QD can again be monitored, leading to an enhanced nuclear population at the new resonance. This procedure can be repeated many times, further narrowing the nuclear distribution. The EIT line width w is a natural limit to the nuclear state narrowing in the fre-

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quency adaptation regime. The narrowing comes from the decay of the probability νkk for ﬁelds δhkz far from resonance in the absence of photon emission and from the depletion of the population at the resonance on photon emission. The narrowing stops when the entire nuclear distribution is within the EIT line. By adjusting the power of the driving lasers, Ωp and Ωc , one can inﬂuence the width w. The width of a dip in ρXX (δhz ) in the stationary state, as shown in Fig. (1b), is a measure of w. It turns out that reducing w comes at a cost: for weaker laser ﬁelds where w is smaller 20 , the preparation time becomes longer. Furthermore, a narrower w is also shallower, i.e., the dip in ρXX (δhz ) is not as pronounced as in the case of wide w. The photon emission rate Γem is reduced due to the small ρXX near the resonance (Fig. 1b), in the area with signiﬁcant population, and the preparation is slower. Due to nuclear state decay, the optical preparation via the electron spin system (TLS) has to be completed in a ﬁnite time. This constraint becomes more restrictive with the use of ineﬃcient detectors. Numerically, we ﬁnd that the preparation time decreases from 10 μs in a setup with a narrow EIT line (Ωp = Ωc = 0.2 ns−1 ) to approximately 6 μs when Ωp = 0.2 ns−1 and Ωc = 0.4 ns−1 , as shown in Fig. (1c). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). A. Imamoglu et al., Phys. Rev. Lett. 83, 4204 (1999). J. M. Elzerman et al., Phys. Rev. B 67, 161308 (2003). A. C. Johnson et al., Nature 435, 925 (2005). F. H. L. Koppens et al., Science 309, 1346 (2005). J. R. Petta et al., Science 309, 2180 (2005). V. Cerletti, W. A. Coish, O. Gywat, and D. Loss, Nanotechnology 16, R27 (2005); cond-mat/0412028. G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999). A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B 65, 205309 (2002). A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B 67, 195329 (2003). W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004). W. A. Coish and D. Loss, Phys. Rev. B 72, 125337 (2005). J. M. Kikkawa and D. D. Awschalom, Science 287, 5452 (2000). R. K. Kawakami et al., Science 294, 131 (2001). R. J. Epstein et al., Phys. Rev. B 68, 041305(R) (2003). C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu, Phys. Rev. Lett. 96, 167403 (2006). C. P. Slichter, Principles of Magnetic Resonance (Springer, Berlin, 1989). D. Stepanenko, G. Burkard, G. Giedke, and A. Imamoglu, Phys. Rev. Lett. 96, 136401 (2006). M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).

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TOPOLOGICAL EFFECTS IN CHARGE DENSITY WAVE DYNAMICS

TORU MATSUURA, KATSUHIKO INAGAKI, AND SATOSHI TANDA Department of Applied Physics, Hokkaido University, Kita-ku, Sapporo 060-8628, Japan E-mail: [email protected] TAKU TSUNETA Low Temperature Laboratory, Helsinki University of Technology, Otakaari 3A, Espoo, Finland We investigate charge density wave (CDW) dynamics in ring-shaped crystals of NbSe3 . We performed a measurement of the frequency dependence of resistance for a closed-ring topology (ring-shaped crystal without edge) and an open-ring topology (cut ring-shaped crystal with two edges) in the 300 kHz to 1.3 GHz frequency range. Comparison of the frequency dependence of the resistance indicated that the pinning potential in the closed-ring topology was smaller than that in the open-ring topology. This result suggests existence of a CDW circulating current in NbSe3 ring-shaped crystals.

1. Introduction The discovery of topological crystals of quasi one-dimensional conductor, niobium triselenide (NbSe3 ), has provided new experimental systems for investigation of topological eﬀects in charge density waves (CDWs).1,2,3 As well known, topology is a subject in mathematics that treats invariant properties of objects under continuous transformation, and neglects other details.4 At the topological viewpoint, a coﬀee cup is identiﬁed with a doughnut, because the number of hole is preserved under continuous transformation. This concept is also powerful in physics because physicists always try to extract intrinsic properties from nature. For some physical phenomena, such as topological defects, quantum Hall eﬀects, and Berry’s phase.5 , the parameter space topology or k-space topology dominates their physical properties. Moreover, real space topology of physical systems is also important since it determines boundary conditions. If real space topology were changed to a nontrivial topology, new physical phenomena would be emerged. The CDWs are the best systems to investigate topological eﬀects of the real space topology because there is an interaction between CDWs and a real space, in this case the real space is a crystal, through the electron-phonon coupling. A CDW state is a ground state of low dimensional electron systems. At high temperatures, the electron system is in a metallic state. The electron density is spatially constant, and the ion positions are periodic with a lattice constant. Below a CDW transition

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Figure 1. (a) Scanning ion beam microscopy (SIM) image of a typical ring crystal. (b) SIM image of the ring crystal after focused ion beam (FIB) surgery. The FIB surgery provided the two edges into the ring crystal. The forms in (a) and (b) are referred to in the text as a closed-ring topology and an open-ring topology, respectively. FIB surgery is a suitable tool for investigation of topological eﬀects because physical properties in diﬀerent topology can be compared without sample dependence.

temperature, the electrons density is modulated to a standing wave with the wave vector 2kF , and the ion positions are also periodically modulated with 2kF . This phenomenon was predicted by Fr¨ohlich as a model of supercurrent before BCS theory.6 He argued that a CDW could ﬂow collectively without dissipation. In reality, the CDW supercurrent has not been observed because the CDW is pinned by impurities or edges of the crystal. If one edge can be connected to another edge, it becomes a CDW in the closed-ring topology. Since there is no edge along its one-dimensional axis, realization of the CDW supercurrent is expected. The main purpose of our research is to explore the CDW supercurrent using the topological CDW systems. We have investigated CDWs in the topological crystals of NbSe3 .1,2,3 NbSe3 crystals are a well-known material as a typical quasione dimensional metal that exhibits two metal-CDW transitions at TC1 = 144 K and TC2 = 59 K.7 In the topological crystals, the CDW wave vector heads toward the circumference direction. And the order of CDW phase correlation length is about 200 μm, which is comparable to the crystal size. Since there is no edge in the one-dimensional axis, the topological crystals are the candidates for observation of the CDW supercurrent. We compared the frequency dependent resistance in diﬀerent topologies. The ﬁrst one is a closed-ring topology that has no edges in its one-dimensional axis, and the second one is an open-ring topology that has two edges. The open-ring topology was realized by topology-change surgery8 using a focused ion beam (shown in ﬁgure 1). We found that the pinning potential of the closed-ring topology was smaller than that of the open-ring topology. This result suggests that an ac circulation current mode exists in ring-shaped crystals.

2. Experimental The CDW dynamical property in the ring-shaped crystals was investigated using an ac resistance measurement. The CDW dynamics is expressed in term of the

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damped oscillation.9,10 The single coordinate model is dφ nπe d2 φ + ω02 sin φ = − ∗ E(t), +Γ (1) 2 dt dt m where φ is the CDW phase, Γ the damping constant, and ω02 the pinning potential. The ﬁrst term in the left-hand side is the inertial term. The right-hand side indicates an external force, where E(t) is electric ﬁeld, n number of electrons in one wavelength of the CDW, m∗ mass, and e electron charge. The CDW current is expressed as ICDW = − πe dφ dt . The CDW contributes electrical conduction as a displacement current in an ac electric ﬁeld. When the amplitude of the applied voltage is small and the frequency f is much smaller than ω0 /2π, the inertial term d2 φ/dt2 can be neglected. Furthermore, sin φ ≈ φ. Then conductivity follows the Drude law, σCDW (ω) = σ∞

iω/ωC . 1 + iω/ωC

(2)

Here, the characteristic frequency ωC = ω02 /Γ is introduced. The CDW dynamical parameters can be investigated by measuring the frequency dependence of conductivity. Since the ac resistance measurement can be performed without current injection that accompanies conversion between condensed electrons and quasi particles, it is an advantageous method for investigating the CDW dynamics in the closed-ring topology. The ring-shaped crystals of NbSe3 are synthesized by chemical vapor transition method.1,2,3 A NbSe3 ring crystal, with diameter of 40 μm and both width and thickness of less than 1 μm, was ﬁxed on a sapphire insulator plate. We attached two electrical contacts to the sample to enable to supply ac current. The plate was placed on a cupper block in a vacuum chamber of a cryostat. The sample was cooled to 77 K by liquid nitrogen and measured the temperature dependence of the resistance while heating the crystal from 77 K to room temperature. A conventional transmission method was used for the frequency dependent resistance measurement.11 The measurement was performed by a Hewlett-Packard HP8712ET network analyzer in a frequency range of f = 300 kHz to 1.3 GHz. The voltage amplitude was kept below 10 mV that is enough smaller than the threshold voltage of the sample (∼ 1 V). After measuring of the closed-ring topology, we performed FIB surgery. Then measurement was performed again for the open-ring topology. 3. Results and Discussions Figure 2 is the temperature dependence of the ac resistance of the ring crystal at 300 kHz, 300 MHz, and 600 MHz. The increase of the resistance at 145 K indicates a metal-CDW transition. When the electrons condense into a CDW state, they do not contribute conductivity because the CDW is pinned. The peak of resistance is suppressed at high frequency because the CDW oscillates around a pinning center

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Figure 2. Temperature dependence of resistance in a closed-ring topology and in an open-ring topology. The line thickness indicates the frequencies, and the dashed lines are linear ﬁtting lines for metallic region; R(T ) = α T + R0 , which were ﬁtted to data points from 160 to 200 K. The inset shows the temperature dependence of the ˜ normalized resistance, R(ω, T ) = [R(ω, T ) − R0 ]/α − T . The ﬁrst CDW transition temperature TC1 is ∼ 145 K in both topologies. The resistance peak below TC1 is maximum at the same temperature.

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ω/2π [Hz] Figure 3. The frequency dependence of the ˜ normalized resistance, R(ω, T ) = [R(ω, T ) − R0 ]/α − T , at T = 100, 110, and 120 K below the ﬁrst CDW transition temperature TC1 = 145 K. The black dots are experimental data for the closed-ring topology, and the gray dots are data for the open-ring topology. The solid and dashed lines indicate the ﬁtting curves given in the text for the closedring and open-ring topologies, respectively. The ﬁtting parameter β is set at 0.8 in both topologies and for all temperatures. The characteristic frequency ωC /2π is 0.85 GHz in the closed-ring topology and 1.2 GHz in the open-ring topology at all temperatures.

and contributes conduction as a displacement current. These features are consistent with the previous research for straight NbSe3 crystals.9,10 The FIB surgery did not change the CDW transition temperature. In metallic state, the ﬁtting lines are determined by two parameters, the slope α and the intercept R0 . The intercept R0 did not change. R0 is a contact resistance. On the other hand, the slope α doubled. The temperature coeﬃcient α is 0.0208 Ω/K in the closed-ring topology and 0.0407 Ω/K in the open-ring topology. The change of α is consistent with the ˜ elimination of a path by the surgery. Thus, the normalized resistance R(ω, T) = [R(ω, T ) − R0 ]/α − T in the CDW state can be compared. We found a diﬀerence in the frequency dependence. Figure 3 shows the frequency dependences of the normalized resistances at 100, 110, and 120 K. The resistance of the closed-ring topology starts to decrease at lower frequency, at all temperatures. We analyzed

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I

I

(a)

(b)

Figure 4. Model of elastic deformation of the CDWs in the closed-ring topology (a) and the open-ring topology (b) under an electric ﬁeld. (a) Due to the closed-ring topology, the CDW is compressed and expanded near the contacts. If the CDW is a rigid body, the CDW conduction is nothing in this geometry. (b) There are two edges in the opne-ring topology. The edges act as strong pinning centers. The CDW is compressed near one edge and expanded near another edge. The elastic energy is increased in both topologies.

the diﬀerence using an empirical curve, σCDW (ω) = σ∞

(iω/ωC )β , 1 + (iω/ωC )β

(3)

which was used for NbSe3 by Wu et al.12 This expression is a modiﬁed equation of (2) by an empirical parameter β. The parameter represents the random distribution of weak pinning centers by the impurities, 12 which is topology-independent. We assumed that the total conduction is the sum of that of the CDW and quasi-particles. Hence, −1 (ω)] = Re[{σmetal + σCDW (ω)}−1 ]. R(ω) ∝ Re[σtotal

(4)

˜ R(ω = 0) is set to 62, 64, and 60 at T = 100, 110, and 120 K, respectively. In our sample, β = 0.8 in both topologies at 100, 110, and 120 K. These parameters have the same values in both topologies, except for the characteristic frequency ωC /2π. ωC /2π of the closed ring is 0.85 GHz, and that of the open ring is 1.2 GHz. According to this model, the characteristic frequency is proportional to the pinning potential ω02 . Therefore, this result suggests that the pinning potential of the closed ring is eﬀectively smaller than that of the open ring. Why the pinning in the closed-ring topology is small? First, the elastic energy of the CDW in both topologies is considered. When an electric ﬁeld is applied to the open ring, which is shown in ﬁgure 4(b), the edges act as strong pinning centers. Since the CDW is compressed near one edge and expanded near another edge, the pinning eﬀect is enhanced. On the other hand, in the closed ring, which is shown in ﬁgure 4(a), the CDW also is compressed and expanded near the contacts in an electric ﬁeld. The elastic energy in the closed ring also increases. Therefore, this model suggests that the elastic energy is increased in both topologies. However, it is inconsistent with the experimental result. We need another explanation. A possible explanation is that an ac circulation current mode of the CDW is excited by the ac electric ﬁeld in the closed-ring

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topology and reduces the pinning eﬀects. Nevertheless, it can ﬂow in the closedring topology but it does not ﬂow in the open-ring topology. If the circulation current ﬂows in the ring, the phase modulation would be relaxed. The diﬀerence of the pinning potential suggests existence of the ac circulation current mode. 4. Conclusion We found the pinning potential in the closed-ring topology is smaller than that in the open-ring topology. We proposed that an ac circulation current mode would reduce the pinning potential. These topological CDW systems will contribute to a development of quantum devices working at room temperature. Our group discovered the ring crystals of niobium trisulﬁde.13 Its CDW transition temperature is 350 K. Hopefully, the topological CDW devices will be installed laptop computer in the near future. Acknowledgments The authors are grateful to Prof. K. Yamaya, T. Toshima, and K. Shimatake for stimulating discussions. This research was supported by Research Fellowship of the Japan Society for the promotion of Science for Young Scientists and the 21 COE program on “Topological Science and Technology” from the Ministry of Education, Culture, Sport, Science and Technology of Japan. One author (T. M.) thanks to MS+S 2006 committee for ﬁnancial support for students. References 1. S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and N. Hatakenaka, Nature 417, 397 (2002). 2. S. Tanda, H. Kawamoto, M. Shiobara, Y. Sakai, S. Yasuzuka, Y. Okajima, and K. Yamaya, Physica B 284, 1657 (2000). 3. T. Tsuneta and S. Tanda, J. Cryst. Growth 264, 223 (2004) . 4. C. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press (1983). 5. M. V. Berry, Proc. R. Soc. Lond. A392, 45 (1984). 6. H. Fr¨ ohlich, Proc. Roy. Soc. Lond. A232, 296 (1954). 7. J. Chaussy, P. Haen, J. C. Lasjaunias, P. Monceau, G. Waysand, A. Waintal, A. Meerschaut, P. Molini´e, and J. Rouxel, Solid State Commun. 20, 759 (1976). 8. T. Matsuura, S. Tanda, K. Asada, Y. Sakai, T. Tsuneta, K. Inagaki, and K. Yamaya, Physica B 329, 1550 (2003). 9. G. Gr¨ uner, L. C. Tippie, J. Sanny, W. G. Clark, and N. P. Ong, Phys. Rev. Lett. 45, 935 (1980). 10. D. Reagor, S. Sridhar, and G. Gr¨ uner, Phys. Rev. B 34, 2212 (1986). 11. T. Matsuura, T. Tsuneta, K. Inagaki, and S. Tanda, Phy. Rev. B 73, 165118 (2006). 12. Wei-yu Wu, L. Mih´ aly, George Mozurkewich, and G. Gr¨ uner, Phys. Rev. B 33, 2444 (1986). 13. H. Nobukane, M. Nishida, K. Inagaki, and S. Tanda, Proceeding of The International Conference on Topology in Ordered Phases 2005, 76 (2006).

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STUDIES ON NANOSCALE CHARGE-DENSITY-WAVE SYSTEMS: FABRICATION TECHNIQUE AND TRANSPORT PHENOMENA

KATSUHIKO INAGAKI AND SATOSHI TANDA Department of Applied Physics, Graduate School of Engineering, Hokkaido University Kita 13 Nishi 8 Kita-ku, Sapporo 060-8628 Japan E-mail: [email protected] We report studies of o-TaS3 nanocrystals, including sample preparation of nanocrystals, fabrication of electrodes attached to the nanocrystals, and transport properties. Since the correlation length of a charge density wave (CDW) is in the range of several microns, electric properties of CDW in nanocrystals are expected to show signiﬁcant diﬀerence. We have developed a new fabrication technique for electrodes by annealing with electron beam to reduce the contact resistance signiﬁcantly. We applied this technique for the study of nanoscale CDW systems. Temperature dependence of resistance was measured in a o-TaS3 nanocrystal with the electrode distance of several μm. The observed resistance did not show a clear Peierls transition. The resistance was well described in terms of one-dimensional (1D) variable-range-hopping (VRH) conduction over the wide range of temperature. These behaviors are consistent with those of the previously reported nanoscale CDW systems. We also found a ﬁnite-size eﬀect in current-voltage characteristics in nanoscale o-TaS3 crystals with various sample dimensions. These results are consistent with the idea that electric transport phenomenon of the nanoscale o-TaS3 crystals should be attributed to charged solitons in commensurate CDW systems.

1. Introduction The reduction of the system size may lead to new developments in macroscopic quantum systems such as superconductors and charge-density-wave (CDW) systems. CDW occurs on account of macroscopic quantum coherence, accompanied by the lattice deformation of the wavevector 2kF 1 . The investigation of the nanoscale CDW system will be key to the realization of prospective applications, such as ﬁeld eﬀect transistors2, electron pumps3 , and ultrafast memory4 , which exploit the quantum collective motion of CDW. Moreover, the recent discovery of M X3 (M = Nb, Ta; X = Se, S) topological (ring, Mobius, and ﬁgure-of-eight) crystals has created a novel ﬁeld of possible CDW devices that exploit the interference of long-range CDW order through their nontrivial topology5 . Here we report studies of o-TaS3 nanocrystals, including sample preparation of nanocrystals, fabrication of electrodes attached to the nanocrystals, and transport properties. Temperature dependence of resistance was measured in a o-TaS3 nanocrystal with the electrode distance of several μm. The observed resistance did not show a clear Peierls transition. The resistance was well described in terms of one-dimensional (1D) variable-range-hopping (VRH) conduction over the wide

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range of temperature. We also found a ﬁnite-size eﬀect in current-voltage characteristics in nanoscale o-TaS3 crystals with various sample dimensions. These results are consistent with the idea that electric transport phenomenon of the nanoscale o-TaS3 crystals should be attributed to charged solitons in commensurate CDW systems. 2. Fabrication of samples In order to obtain nanoscale o-TaS3 samples, we synthesized single crystals by the chemical vapor transport method. A pure tantalum sheet and sulfur powder were placed in a quartz tube. The quartz tube was evacuated to 1×10−6 Torr and heated in a furnace at 530 ◦ C for ﬁve hours. Since we required rather smaller crystals, the reaction duration that was considerably shorter than the typical condition was chosen. After the growth of the crystals, one end of the quartz tube was quenched in a liquid nitrogen bath in order to prevent the condensation of excess sulfur gas on the crystals. The crystals were sonicated in toluene for 15 min and kept free from perturbation for several hours in order to sediment unwanted larger crystals. Subsequently, the dispersion was deposited on a silicon substrate with a thermal oxide layer of 1 μm. Prior to the deposition on the substrate, 50-nm thick gold markers and contact pads were prepared. After blow-drying, the crystals were examined with an optical microscope. The obtained crystals were typically 0.1–1 μm in width and 10–100 μm in length. Electrodes were fabricated by standard electron beam lithography with a scanning electron microscope (SEM) (JSM-5200, JEOL) equipped with a homemade writing system. Standard bi-layer 950k/495k-polymethylmethacrylate (PMMA) resist was spun to the total thickness of 500 nm. Based on the position of the crystal with respect to the markers, a mask pattern was designed and loaded onto an electron beam writer. The thin gold ﬁlm (50 nm in thickness) was evaporated after the developing process. An adhesive layer such as chromium and titanium was not used in order to prevent the contact resistance layer from degrading. Figure 1 shows the scanning electron micrograph of an o-TaS3 crystal on a silicon substrate before and after it was attached with the gold electrodes. The ﬁgure shows that the gold electrodes were well deﬁned and located precisely on the nanocrystal. The width of the electrode was 400 nm with a separation of 1 μm. Each electrode connects to a bonding pad (100 × 100 μm2 ). We fabricated gold electrodes for several nanocrystals; however, approximately half of them failed. In most cases of failure, the crystal was washed away at the lift-oﬀ stage; this was probably because of the strong aﬃnity between the o-TaS3 surface and PMMA. Another process was required to obtain ohmic contact6 . It has commonly been found in previous studies that ohmic contact was scarcely obtained and several methods were used to reduce the contact resistance. In this study, we heated each electrode locally by the irradiation of an electron beam with an SEM (JSM-820, JEOL). The electron current was monitored at both the sample stage and the

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Figure 1. Scanning electron micrograph of an o-TaS3 nanocrystal (a) before and (b) after being attached with gold electrodes. The gold markers shown in (a) were prepared before the deposition of the nanocrystal, and the locations relative to the markers were used to fabricate the beam pattern of the second lithography.

probe current detector inserted in the beam line. Only a small negligible electron beam current (< 10−9 A) was required to specify the sample. Once the contact to be heated was located on the SEM screen, we increased the beam current to 1 × 10−6 A by using the larger aperture and the high bias. Subsequently, the beam position was ﬁxed at a desirable position using the “SPOT” feature of the SEM for typically 30 s. The appearance of the contact did not change after the electron beam treatment. The resistance was initially larger than 100 GΩ, and it decreased rapidly to 21 kΩ after 30 s of electron beam irradiation at an acceleration voltage of 25 kV and a typical beam current of 1 × 10−6 A. 3. Electric properties We measured three samples (denoted A, B, and C) taken from a same batch of synthesis as shown in Table 1. The sample dimensions were measured by tapping-mode AFM (SPI-3700, Seiko). The observed temperature dependence of the nanoscale oTaS3 crystal was signiﬁcantly diﬀerent from that of bulk crystal. Figure 2 shows that temperature dependence of resistance for Sample A are diﬀerent from that of the bulk samples at the following points: 1) Peierls transition was not clearly observed in the nanocrystal, and 2) resistance was described in terms of VRH conduction7 σ(T ) = σ0 exp[−(T0 /T )1/α ],

(1)

Table 1. Summary of the parameters of the o-TaS3 nanocrystals used for this study. Length l, width w, and, thickness t (thereby cross section S) were measured by tapping mode AFM. Eﬀective cross section S was determined by the room temperature resistivity ρ300K and the bulk resistivity 4 × 10−6 Ω · m. The observed I-V characteristics of the samples are also shown.

name A B C

l × w × t (μm3 ) 3.4 × 0.38 × 0.050 0.47 × 1.04 × 0.05 8.7 × 0.40 × 0.140

S (μm2 ) −2

1.9 × 10 5.2 × 10−2 5.6 × 10−2

ρ300K (Ω · m)

S (μm2 )

1.7 × 10 6.6 × 10−4 2.1 × 10−3

4.5 × 10 3.1 × 10−4 1.1 × 10−4

−4

−4

I-V exp[−(V0 /V )2 ] exp[−(V0 /V )] exp[−(V0 /V )]

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−6

10

10

R (Ω)

G (S)

10

8

−8

10

−10

10

10

−12

10

0.2

6

0.4 1/2

1/T

10

−1/2

(K

)

4

10

0

100

200

300

T (K) Figure 2. Temperature dependence of resistance of the o-TaS3 nanocrystal. Inset shows the linear conductance as the function of T −1/2 . The data ﬁt to a linear line very well, suggesting that the variable-range-hopping conduction: σ ∝ exp[−(T0 /T )1/2 ].

where T0 is a characteristic temperature and α is a characteristic exponent of the system. The inset of Fig. 2 shows in our system α = 2 was observed, as well as in Samples B and C (not shown). In the standard interpretation of VRH for Anderson insulator, α = 1 + d where d is dimensionality of the system, while Coulomb interaction also gives same behavior (1) with α = 2 with no respect to the system dimensionality. Recent studies show similar behavior in nanoscale M X3 systems8 . Zaitsev-zotov et al., performed a systematic study of size dependence on NbSe3 and o-TaS3 . They claimed that by the reduction of the cross section, the sample became closer to the true 1D system rather than quasi-1D, consequently the system exhibited the Luttinger-Liquid (LL) behavior. The absence of clear Peierls transition was also attributed to their LL nature. It is known that in LL the physical properties tend to obey a power law, however, more recently, the possibility of VRH behavior in LL with randomness has been predicted with α = 2 by the Coulomb interaction9 . Figure 3 shows I-V characteristics of Samples A and B measured at 0.39 K. Sample A exhibited I ∝ exp[−(V0 /V )2 ], while B (and C – not shown) showed I ∝ exp[−(V0 /V )]. This is an astonishing result because though all the samples came from a same batch, and could be considered identical. Only diﬀerence was the sample dimension as listed in Table 1. These size eﬀect cannot be explained by the random LL theory which predict a simple power law in I-V characteristics9. Hence rather than the LL theory, we would like to explain the experimental results within the CDW framework10 . Role of soliton nucleation has been an important issue to understand electrical transport phenomena of the CDW systems11,12,13 . Instead of sliding motion of

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Sample B

Sample A

T = 389 mK

−8

T = 389 mK

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I (A)

I (A)

10

10

10

−10

10

−12

10

2

4 6 2 −2 1/V (V )

8

20

−1 40 1/V (V )

Figure 3. I-V characteristics of the samples A and B measured at 0.39 K. Sample A and B obey I ∝ exp[−(V0 /V )2 ] and I ∝ exp[−(V0 /V )], respectively, in the currant ranges over four orders.

the whole CDW, charged solitons of ±2e, which can freely move, are nucleated to transport electric current, as expected from a solution of the sine-Gordon equation. The ﬁrst experimental evidence of the soliton transport was the discrepancy of the temperature dependence and anisotropy of resistivities of o-TaS3 from that of the conventional Arrhenius formulae12 , and such the discrepancy was observed by other groups11 . In particular, the latter case demonstrated that I-V characteristics at low temperatures also obeyed an unexpected behavior. Hatakenaka et al. has proposed a clear explanation how the system shows either exp[−(V0 /V )2 ] or exp[−(V0 /V )] forms, by considering dimensionality of its crosssection13. They predicted the quantum nucleation possibility of the solitons Γ, which is proportional to the current, obeys a dimensionality-dependent formula, Γ ∝ exp(const./E D ),

(2)

where D is the system dimension. According to this, our observation suggests that the system dimension relevant to soliton transport was either D = 1 or D = 2, which depends on the system size. This interpretation is justiﬁed when the system dimension D relates to the eﬀective cross section S = Sρbulk /ρmeasure , where ρbulk = 4.0 × 10−6 Ω · m (as shown in Table 1). It should be noted that the ﬁrst observation of the D = 2 type I-V characteristics was performed on a S = 1.2×10−2 μm2 crystal 11 , and is consistent with our observation. The diﬀerence of the eﬀective cross section between Samples A and B may too small to separate 1D and 2D behaviors. Hence we need to know more precisely the shape of cross section of the samples to complete understanding of the observed size eﬀect. 4. Summary We report studies of o-TaS3 nanocrystals, including sample preparation of nanocrystals, fabrication of electrodes attached to the nanocrystals, and transport properties. The observed resistance did not show a clear Peierls transition. The resistance was well described in terms of one-dimensional (1D) variable-range-hopping (VRH) conduction over the wide range of temperature. We also found a ﬁnite-size eﬀect

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in current-voltage characteristics in nanoscale o-TaS3 crystals with various sample dimensions. These results are consistent with the idea that electric transport phenomenon of the nanoscale o-TaS3 crystals should be attributed to charged solitons in commensurate CDW systems. Acknowledgments The authors thank Prof. Noriyuki Hatakenaka (Hiroshima University) for fruitful discussions. This work was partly supported by the 21st century COE program “Topological Science and Techonology,” the Ministry of Education, Culture, Sports, Technology and Science. References 1. G. Gr¨ uner, Density Waves in Solids, Addison-Wesley Longman, Reading (1994). 2. T. L. Adelman, S. V. Zaitsev-Zotov, and R. E. Thorne, Phys. Rev. Lett. 74 (1995), 5264. 3. M. I. Visscher and G. E. W. Bauer, Appl. Phys. Lett. 75 (2002), 1007. 4. D. Mihailoviˇc, D. Dvorsek, V. V. Kabanov, J. Demsar, L. Forro, and H. Berger, Appl. Phys. Lett. 80 (2002), 871. 5. S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and N. Hatakenaka, Nature 417 (2002), 397. 6. K. Inagaki, T. Toshima, S. Tanda, K. Yamaya, and S. Uji, Appl. Phys. Lett. 86, 073101 (2005). 7. N. F. Mott, Metal-Insulator Transition, 2nd Ed., Tayler & Francis, London (1990). 8. S. V. Zaitsev-Zotov, Microelectronic Engineering 69, 549 (2003). 9. S. N. Artemenko, J. Phys. IV, 131, 175 (2005). 10. K. Inagaki, T. Toshima, and S. Tanda, J. Phys. Chem. Solid. 66, 1563 (2005). 11. S. V. Zaitsev-Zotov, Phys. Rev. Lett. 71, 605 (1993). 12. T. Takoshima, M. Ido, K. Tsutsumi, T. Sambongi, S. Honma, K. Yamaya, and Y.Abe, Solid State Commun. 35, 911 (1980). 13. N. Hatakenaka, M. Shiobara, K. Matsuda, and S. Tanda, Phys. Rev. B 57, 2003 (1998).

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ANISOTROPIC BEHAVIOR OF HYSTERESIS INDUCED BY THE IN-PLANE FIELD IN THE ν = 2/3 QUANTUM HALL STATE

K. IWATA Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo, Kyoto, 606-8502, Japan A. FUKUDA, A. SAWADA Research Center for Low Temperature and Materials Sciences, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo, Kyoto, 606-8502, Japan M. MORINO, Z.F. EZAWA, M, SUZUKI Graduate School of Science, Tohoku University, Aoba, Aramaki-aza, Aoba, Sendai, 980-8578, Japan N. KUMADA, Y. HIRAYAMA NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, 243-0198, Japan We report the observation of an anisotropic transport around a spin transition point in the Landau level ﬁlling factor ν = 2/3 in tilted magnetic ﬁelds. When a direction of the in-plane component of the magnetic ﬁelds (B ) is normal to the current I, a strong hysteretic transport owning to the electron domain of diﬀerent spin conﬁgurations emerges. When B is applied along I, the hysteresis almost disappears. At the same time, a current dependence of the anisotropy is examined. The hysteresis occurs by a nuclear-spin polarization which is generated when electrons pass across domain walls. Therefore the phenomenon suggests that a large number of electrons scatters by domain walls when B is perpendicular to I.

1. Introduction An electron-electron interaction in a two dimensional electron system (2DES) produces an incompressible quantum liquid state of fractional quantum Hall eﬀects (FQHEs) in strong perpendicular magnetic ﬁelds (B⊥ ) 1 . Very strong magnetic ﬁelds make electron spins align parallel to the applied magnetic ﬁeld and the ground state with the spins fully polarized occurs. However, because of the relatively modest eﬀective g factor in GaAs-based 2DESs, the Zeeman splitting is not so large that the occurrence of a spin-unpolarized FQH state is allowed. In fact, transitions from spin-unpolarized to -polarized state are observed at various ﬁlling factors by changing a electron density or a parallel magnetic ﬁeld (B ) 2,3 .

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Recently, a strong hysteresis in the magnetoresistance (Rxx ) has been observed around the spin transition point at the Landau level ﬁlling factor ν = 2/3 4,5 . The involvement of the nuclear spin system in the hysteresis has previously been conﬁrmed by resistively detected nuclear magnetic resonance measurements 6,7 . The occurrence of the polarization and the hysteresis are explained by the following scenario. At the transition point, an electron domain of diﬀerent spin conﬁgurations is formed. When a current passes across domain walls, electron spins ﬂip-ﬂop scatter nuclear spins, producing a nuclear spin polarization 8 . The nuclear polarization aﬀects to the 2DES through the hyperﬁne interaction and modiﬁes the domain structure. The modiﬁcation changes electron scatterings and hence Rxx . Therefore, the polarization is detected by Rxx 6 . In addition, it is known that the nuclear spin polarization and Rxx depend on the sweep direction of the magnetic ﬁeld and thus the hysteresis due to the nuclear spin polarization occurs 9 . Our recent work reveals that the hysteresis has an anisotropy about the direction of B though the origin is not completely understood 10 . Many anisotropic transport measurements have brought out contextures of a ground state of various Landau level ﬁllings and make clear a interaction of electrons in bilayer and monolayer systems 11,12,14,15 . So, it is interesting to investigate the anisotropy for ﬁnding out a interaction of the particles. In this work, we study the current dependence of the anisotropic behavior of the hysteresis around the spin transition point in the ν = 2/3 FQH state by changing the current direction against B in tilted magnetic ﬁelds. The transition point is determined by activation energy measurements. The data reveals that the hysteresis tends to arise when the direction of I is orthogonal to B . This result suggests that the domain structure has a unidirectional structure about B since the nuclear polarization and the hysteresis occurs when currents ﬂow across the domain walls. 2. Experiment The sample used in this experiment consists of a GaAs/AlGaAs quantum well of 20 nm. The sample was processed into the 50 -μm wide Hall bar with the voltage probe distance of 180 μm. The electron density (n) in the 2DES was controlled by the gate voltage. The mobility was 1.5 × 106 cm2 / Vs with n = 0.8 × 1011 cm−2 . Rxx and Hall resistances Rxy were measured by using a standard low-frequency ac lock-in technique (∼ 17Hz). We controlled a tilting angle (θ) and a relative angle (φ) between B and I by using a two-axis goniometer 11 . The deﬁnition of θ and φ are illustrated in the inset of Fig. 1(a). These angles were calibrated by measuring Rxy of the sample and a conventional Hall element attached to the goniometer at a right angle with the sample. The ﬁeld was swept slowly (0.13 T/min) to observe the hysteresis 16 . All measurements were performed n = 0.8 × 1011 cm−2 .

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B|| ⊥ I B|| || I Btotal (φ=0°) (φ=90°) I B|| B I || θ φ B⊥ B|| sample

(a)

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φ=90° (B|| ⊥ I)

6.0 4

0.0 0

10

6 8 1 / T [K-1]

20

φ=0° (B|| || I)

10

30

θ [degs]

40

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Figure 1. Activation energy Δ at ν = 2/3 as a function of the tilting angle θ. Open (closed) circles correspond to Δ at φ = 0◦ (φ = 90◦ ). Solid and dashed lines are guides for eyes. The transition from spin un-polarized to -polarized state occurs near θ = 35◦ . Drawings in the inset (a) indicate the deﬁnition of θ and φ. The inset (b) shows the Arrhenius plot of Rxx at ν = 2/3. Activation energies are extracted from a slope of the Arrhenius plot, according to Rxx ∝ exp(−Δ/2T ).

3. Result and Discussion To ﬁnd out the spin transition point, we determine the θ dependence of the activation energy Δ of ν = 2/3 at B I (φ = 0◦ ) and B ⊥ I (φ = 90◦ ). The activation energy is taken from the slope of the Arrhenius plot (see the inset in Fig. 1(b)), according to Rxx ∝ exp(−Δ/2T ). The Δ is plotted as a function of θ in Fig. 1. Open (closed) circles indicate the energy at φ = 0◦ (φ = 90◦ ). The data reveals the existence of a transition near θ = 35◦ at both conditions. Since the Zeeman energy depends on total magnetic ﬁelds, for smaller θ, the Coulomb energy dominates the Zeeman energy and the ν = 2/3 spin-unpolarized state occurs. On the other hand, for larger θ, the Zeeman energy is large and the spin-polarized state arises. Thus, we conclude that the transition from spin-unpolarized to -polarized states occurs near θ = 35◦ . Now, we present an anisotropic transport around the spin transition point. Figure 2 shows that Rxx as a function of B⊥ while θ is ﬁxed at the spin transition angle of ν = 2/3 (θ = 35.2◦ ). The data are taken at T = 70 mK and I = 10 nA. At the transition point, the two QH states with diﬀerent spin conﬁguration coexist and the hysteresis appears around ν = 2/3 at φ = 90◦ in Fig. 2. Important ﬁndings in Fig. 2 are the disappearance of the hysteresis at φ = 0◦ . To elucidate this anisotropy, we compare the current dependence of the hysteresis between I driven along and across to B at θ = 35.2◦ since the nuclear polarization and the hysteresis depends on the magnitude of the current 6,8 . Figure 3 demonstrates the current dependence of the hysteresis at T = 70 mK. Figure 3(a)

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Figure 2. Rxx as a function of the magnetic ﬁeld (B⊥ ) near the transition angle (θ = 35.2◦ ) at 70 mK. The current is 10 nA. Solid and dashed line correspond to traces for B up sweep and down sweep, respectively. Arrows indicate the direction of the ﬁeld sweeps. The bottom (top) lines are measured at φ = 90◦ (φ = 0◦ ). An oﬀset of Rxx is 5 kΩ for two traces. The data are taken at T = 70mK and I = 10nA. A obvious hysteresis due to the nuclear spin polarization occurs at φ = 90◦ while it almost disappears at φ = 0◦ .

shows the results obtained at φ = 0◦ and the right one shows those obtained at φ = 90◦ . At I = 1 nA, the hysteretic transport is not shown at both side. This results that the rate of the current-pumped nuclear spin polarization is less than the rate of the thermal relaxation of the nuclear spin polarization at this temperature 8 . As current increasing, at I = 5 and 10 nA, the hysteresis emerges at φ = 90◦ alone. This is the most signiﬁcant data in this work. It indicates that the current passes over domain walls only when I is normal to B . On increasing I further (I > 20 nA), the hysteresis emerges at not only φ = 90◦ but also φ = 0◦ . This feature may be understood that the large current disarrays the order of the domain structure and the current crosses over the domain walls both side, and thus the hysteresis occurs at φ = 0◦ as well as φ = 90◦ . Because the hysteresis is owing to the current-pumped nuclear spin polarization, this anisotropy indicates that more electrons ﬂow across domain walls when the current is driven perpendicular to B . Accordingly, we suggest that the longer direction of the domain structure is parallel to B and domain walls are aligned to that. Although the origin of this anisotropy of the domain structure against B is not completely understood yet, we propose the analogy between the charge-density wave (CDW) in half ﬁlled high Landau levels (LLs), called the stripe phase 17 , and the domain structure at ν = 2/3. In the stripe phase, the stripe CDWs are stabilized by the formation of the stripes running normal to the direction of B , so that the easy-transport direction is orthogonal to B . Half of the highest LL is occupied in

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the stripe phase, while in the domain structures at ν = 2/3, electrons also ﬁll half of the state because two QH states with diﬀerent electron spin polarizations are degenerate at the transition point. Namely, the spin transition point at ν = 2/3 appears similar to the stripe phase in half ﬁlling. Hence, these two phenomena may have common features. Note that there is a sharp diﬀerence of the longer direction between the domain structure at ν = 2/3 and the stripe CDW, one aligns parallel to B the other vertical to that. We speculate that this reﬂects a diﬀerence in the origin of two phenomena, the domain structure of ν = 2/3 emerges by the dipole-dipole interaction of the electron spins while the stripe arises by the Coulomb interaction of electrons. 4. Summary We have studied the magnetoresistance anisotropy of a hysteresis by comparing between I ⊥ B and I B at the spin transition point for the ν = 2/3 state. The transition point is found out by the activation energy measurements. The current dependence of the anisotropy shows that the longitudinal direction of the domain structure is along to B and the large currents destroy the order of the domain structure. We suggest that this phenomenon has a similarity to the stripe CDW of half ﬁlled high Landau levels.

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5. Acknowledgment We are grateful to T. Saku for growing the sample used in the present studying and K. Muraki for useful discussion. This work was supported in part by Grants-inAid for the Scientiﬁc Research (No. 14340088) and a 21st Century COE Program Grant of the International COE of Exploring New Science Bridging Particle-Matter Hierarchy from Ministry of Education, Culture, Sports, Science and Technology. A part of this experiment was performed at the clean-room facility of the Center for Interdisciplinary Research of Tohoku University. K. Iwata is grateful to Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. References 1. See, for example, Z.F. Ezawa Quantum Hall Eﬀect, Singapore: World Scientific, 2000 and Perspectives in Quantum Hall Eﬀects, edited by S. Das Sarma and A. Pinczuk, New York: Wiley, 1997. 2. I. V. Kukushkin et al., Phys. Rev. Lett. 82, 3665 (1999). 3. N. Kumada et al., Phys. Rev. Lett. 89, 116802 (2002). 4. H. Cho et al., Phys. Rev. Lett. 81, 2522 (1998). 5. J.H. Smet et al., Phys. Rev. Lett. 86, 2412 (2001). 6. S. Kronm¨ uller et al., Phys. Rev. Lett. 81, 2526 (1998). 7. S. Kronm¨ uller et al., Phys. Rev. Lett. 82, 4070 (1999). 8. S. Kraus et al., Phys. Rev. Lett. 89, 266801 (2002). 9. N. Kumada et al., Phys. Rev. B 69, 155319 (2004). 10. K. Iwata et al., To be published in the AIP Conference Proceedings series of 24th International Conference on Low Temperature Physics. 11. M. Morino et al., Int. J. Mod. Phys. B 18, 3705 (2004). 12. K. Iwata et al., Journal of Physics and Chemistry of Solids 66 1556 (2005). 13. M. Morino et al., To be published proceedings of 16th International Conference on Electronic Properties of Two-Dimensional Systems. 14. M.P. Lilly et al., Phys. Rev. Lett. 83 824 (1999). 15. K. B. Cooper et al., Phys. Rev. Lett 90, 226803 (2003). 16. K. Hashimoto et al., Phys. Rev. Lett. 88, 176601 (2002). 17. K. B. Cooper et al., Phys. Rev. B 65, 241313 (2002).

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PHASE DIAGRAM OF THE ν = 2 BILAYER QUANTUM HALL STATE

A. FUKUDA∗ AND A. SAWADA Research Center for Low Temperature and Materials Sciences, Kyoto University, Kitashirakawa-oiwakecho, Sakyo, 606-8502 Kyoto, Japan ∗ E-mail: [email protected] K. IWATA Graduate School of Science, Kyoto University, Kitashirakawa-oiwakecho, Sakyo, Kyoto, 606-8502, Japan S. KOZUMI, D. TERASAWA, Y. SHIMODA AND Z. F. EZAWA Graduate School of Science, Tohoku University, Aramaki-Aoba, Aoba, 980-8578 Sendai, Japan N. KUMADA AND Y. HIRAYAMA NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, 243-0198 Atsugi, Japan We study the phase diagram of the ν = 2 bilayer quantum Hall (QH) state. We measure the thermal activation energy elaborately as a function of the total electron density nT and the density diﬀerence between the two layers σ. At σ = 0, we clearly ﬁnd three phases as increasing nT - the singlet phase, the canted antiferromagnetic phase and the ferromagnet phase. By making the similar measurements at σ = 0, we construct the phase diagram of the ν = 2 bilayer QH state in the σ - nT plane.

1. Introduction Many attempts have been made both experimentally and theoretically to investigate bilayer quantum Hall (QH) systems [1, 2], which have an additional degree of freedom associated with layer degrees. The layer degree of freedom is represented in terms of ‘pseudospin’. The bilayer QH system at the total Landau level ﬁlling factor ν = 2 has a variety of quantum phases related to the combination of ‘real’ spin and pseudospin. In a one-body picture, a competition between the Zeeman energy ΔZ and the tunneling energy ΔSAS leads to two ground states: one is the spin-ferromagnet and psuedospin-siglet phase (called F phase) and the other is the spin-singlet and pseudospin-ferromagnet phase (called S phase). Interactions between electrons strongly modify the ground state of the ν = 2 QH state. Interlayer interactions energetically favor the S phase, in which interlayer phase coherence is

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expected to be developed [3]. In contrast, intralayer interactions stabilize the F phase. Between these two phases, a novel canted antiferromagnetic phase (called CAF phase) due to the interplay of the interlayer and intralayer interactions has been predicted to emerge [4, 5]. The interlayer and intralayer quantum coherence plays an essential role for the CAF phase. In the CAF phase, spin moments have an antiferromagnetic correlation in the plane perpendicular to the magnetic ﬁeld, and a ferromagnetic correlation parallel to the magnetic ﬁeld. The planar antiferromagnetic correlation breaks spin rotation symmetry. Therefore there is a gapless, linearly dispersing, Goldstone collective mode in its excitation spectrum and a Kosterlitz-Thouless transition is expected at a ﬁnite temperature [5, 6]. The existence of the CAF phase was ﬁrst suggested by Das Sarma et al. [4, 5]. Many theoretical studies tried to improve their work [7–13]. Experimentally, inelastic light scattering [14] and capacitance spectroscopy [15] were carried out to ﬁnd the onset of the CAF phase. Although magnetotransport measurements were also performed [3, 16, 17], only the S and F phases were found. Recently, the Goldstone mode in the CAF phase has been observed by a nuclear spin relaxation measurement [18]. In this report, we elaborately study the magnetotransport properties of the ν = 2 bilayer QH state. Detailed measurements of the activation energy reveal the existence of the three phases, i.e., the S, CAF and F phases. We construct the phase diagram as a function of the total electron density nT and density imbalance between the two layers σ. Here, nT and σ are deﬁned as nT = nf + nb and σ = (nf − nb )/(nf + nb ), respectively. Spin and pseudospin conﬁgurations in the CAF phase are discussed. 2. Experiments We used a sample consisting of two GaAs quantum wells of 20 nm in width separated by a 3.1-nm-thick Al0.33 Ga0.67 As barrier (the center-to-center distance d is 23.1 nm). The sample was grown by molecular-beam epitaxy. Si-modulation doping was carried out only on the front side of the double quantum well. The tunneling energy ΔSAS is 11 K and the low temperature mobility at nT = 1.0 × 1011 cm−2 is 1.0 × 106 cm2 /Vs. The width of the Hall bar is 50 μm and the distance between voltage probes is 180 μm. We can control nf and nb , thus nT and σ, independently by applying the front- and back-gate biases. The sample was mounted in a mixing chamber of a dilution refrigerator with a base temperature of 30 mK. To measure resistances, standard low-frequency ac lock-in techniques were used. 3. Results and Discussions Figure 1 shows the magnetoresistance Rxx as a function of nT and σ. The data were taken with simultaneous sweep of the magnetic ﬁeld, the front-gate bias and the back-gate bias in order to keep the Landau ﬁlling factor ν exactly 2 throughout the

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0.0

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0.2

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0.4

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σ Figure 1. Image plot of the magnetoresistance Rxx as a function of nT and σ. The ﬁlling factor ν is kept at 2 at all points. The temperature is 1.00 K.

measurements. For lower temperature (T ≈ 50 mK), the bilayer ν = 2 QH state is well developed and Rxx ∼ 0 Ω for all sets of (σ, nT ) that we measured. As the temperature is increased, Rxx of weak QH states starts to increase and has a ﬁnite value. Figure 1 shows Rxx for the ν = 2 QH state at T = 1.00 K. The white and black areas correspond to strong and weak QH states, respectively. Roughly, there are two strong QH states: one appears around σ = 0 for higher nT and the other spreads for all σ. The former is the F phase because it consists of two single-layer ν = 1 QH states. The latter corresponds to the S phase, which evolves into the single-layer ν = 2 QH state at large σ limit. Between these two phases, there exist relatively weak QH region. This region should be identiﬁed as the CAF phase. To determine the boundaries between the S, CAF and F phases unambiguously, we measured the activation energy Δ in the ν = 2 QH state from the temperature dependence of Rxx : Rxx ∝ exp(−Δ/2T ). Figure 2(a) is the nT dependence of the activation energy for various ﬁxed σ. When σ = 0, as nT is increased, the activation energy ﬁrst increases, then decreases and increases again. This demonstrates that there are three phases in the balanced conﬁguration. Since the transition from the S or F phase to the CAF phase is expected not to be ﬁrst order (theoretically predicted to be second order [4, 5, 13]), any drastic change in the activation energy cannot be expected. Therefore, we deﬁne nT that gives the local maximum (minimum) of the activation energy as the representative transition point between the S (F) phase and the CAF phase. As σ is increased, both the F-CAF transition point (indicated by solid arrows) and the S-CAF one (indicated by dashed arrows) move to higher nT . This fact indicates that the phases that have the interlayer coherence only can survive as increasing σ = 0.

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nT=0.8 nT=1.7 nT=2.6

0.1 0.2 0.3 0.4 Density Imbalance σ

Figure 2. (a) Activation energy as a function of the total density nT for several density imbalance σ. σ for each data is listed in the graph. The data at σ = 0 has a correct scale and each trace for σ = 0 is shifted by 10 K for easy to see. Solid arrows (dashed arrows) indicate the local maxima (minima) of the activation energy. (b) Activation energy as a function of σ for nT . nT for each data is listed in the graph (×1011 cm−2 ).

Figure 2 (b) shows the σ dependence of the activation energy for three nT ’s, where the S, CAF and F phases occur at σ = 0. In the S phase, at nT = 0.8 × 1011 cm−2 , the activation energy is almost constant or increases as σ is increased. It shows that the S phase is robust against the layer imbalance, implying that the interlayer phase coherence develops there. At nT = 2.6 × 1011 cm−2 near σ = 0, small increment of σ causes a sharp drop in the activation energy, consistent with the fact that the F phase is weak against the layer imbalance. Further increasing σ causes a phase transition to the CAF phase. At nT = 1.7 × 1011 cm−2 , the activation energy slightly decreases with increasing σ. This indicates that the CAF phase around σ = 0 is unstable with respect to the density imbalance. As σ is increased more, the activation energy turns to increase at σ = 0.07. This minimum is in the CAF phase and does not correspond to neither the S-CAF nor F-CAF phase boundaries. The non-monotonous σ dependence of the activation energy in the CAF phase suggests that properties of the CAF phase changes. We name the CAF phase at the left side of the minimum as F-like CAF (FCAF) region because the activation energy decreases with σ like the F phase. The phase at the right side of the minimum is named as S-like CAF (SCAF) region. Finally we construct the phase diagram of the ν = 2 QH state in the σ-nT plane(Fig. 3). As stated before, the S-CAF (F-CAF) phase boundary are determined from the sets of (nT , σ) that gives the local minima (maxima) of the Δ for nT at ﬁxed σ (Fig. 2 (a)). By collecting these points for various nT ’s, we construct the phase diagram including the S, F and CAF phases, where phase boundaries are illustrated with triangles, circles and solid lines in Fig. 3. Furthermore, we can add

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SCAF

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Figure 3. Phase diagram in the σ-nT plane. Triangles (circles) are nT that give the local minima (maxima) of the activation energy at ﬁxed σ, showing the boundary between the F (S) and CAF phases. Diamonds give the SCAF-FCAF region boundary. Solid and dashed lines are guides for eyes.

another region boundary manifested by the minima of Δ for σ at ﬁxed nT (Fig. 2 (b)). This region boundary between the SCAF and FCAF regions is shown as the diamonds and dashed line in Fig. 3. The physical concepts of the SCAF and FCAF are totally new. We shortly discuss the spin and pseudospin conﬁgurations in the SCAF and FCAF phases. Here, it is important to note that σ dependence of the activation energy is mainly determined by the pseudospin conﬁguration. When pseudospins are unpolarized like the F phase, electrons can not be transferred from one layer to the other without destroying the QH state, and thus the activation energy decreases rapidly as σ is increased. On the other hand, when pseudospins are polarized like the S phase, the QH state is stable in any σ. Accordingly, the evolution from the FCAF to SCAF phases suggests that pseudospins are gradually polarized and, at the same time, spins are canted more with increasing σ. There are only few papers [12] concerning on eﬀects of density imbalance on the CAF phase, and further theoretical investigations are required to conclude the problem. 4. Conclusion Magnetotransport measurements were carried out in the ν = 2 bilayer QH system. Elaborate measurements of the activation energy as a function of the total density nT and the density imbalance σ revealed the existence of the CAF phase. The phase diagram in the σ - nT plane was constructed. The CAF phase can be divided into two regions having diﬀerent properties.

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Acknowledgment We are grateful to T. Saku for growing the heterostructures, and K. Muraki for fruitful discussions. We also thank to A. Ueda, M. Sako and H. Sugiura for the initial experimental setup at Kyoto University. This research was supported in part by Grants-in-Aid for the Scientiﬁc Research and Technology of Japan (Nos. 14010839,14340088) and a 21st Century COE Program Grant of the International COE of Exploring New Science Bridging Particle-Matter Hierarchy from the Ministry of Education, Culture, Sports, Science. Some of the authors (D.T. and K.I.) are grateful to Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. References 1. Z. F. Ezawa, Quantum Hall Eﬀects, Field Theoretical Approach and Related Topics (World Scientiﬁc, Singapore, 2000). 2. A. Pinczuk and S. Das Sarma (eds.), Perspectives in Quantum Hall Eﬀects (Wiley, New York, 1997). 3. A. Sawada, Z. F. Ezawa, H. Ohno, Y. Horikoshi, Y. Ohno, S. Kishimoto, F. Matsukura, M. Yasumoto and A. Urayama, Phys. Rev. Lett. 80, p. 4534 (1998). 4. L. Zheng, R. Radtke and S. Das Sarma, Phys. Rev. Lett. 78, p. 2453 (1997). 5. S. Das Sarma, S. Sachdev and L. Zheng, Phys. Rev. Lett. 79, 917 (1997) ; Phys. Rev. B 58, 4672 (1998). 6. M. Yang and M. Chang, Phys. Rev. B 61, p. R2429 (2000). 7. E. Demler and S. Das Sarma, Phys. Rev. Lett. 82, p. 3895 (1999). 8. K. Yang, Phys. Rev. B 60, p. 15578 (1999). 9. Y. Shimoda, T. Nakajima and A. Sawada, Physica E 22, 56 (2004) ; Int. J. Mod. Phys. B 18, 3713; Mod. Phys. Lett. B 19, 539 (2005). 10. J. Schliemann and A. H. MacDonald, Phys. Rev. Lett. 84, p. 4437 (2000). 11. A. H. MacDonald, R. Rajaraman and T. Jungwirth, Phys. Rev. B 60, p. 8817 (1999). 12. L. Brey, E. Demler and S. Das Sarma, Phys. Rev. Lett. 83, p. 168 (1999). 13. Z. F. Ezawa, M. Eliashvili and G. Tsitsishvili, Phys. Rev. B 83, p. 168 (2005). 14. V. Pellegrini, A. Pinczuk, B. S. Dennis, A. S. Plaut, L. N. Pfeiﬀer and K. W. West, Phys. Rev. Lett. 78, 310 (1997); Science 281, 799 (1998). 15. V. S. Khrapai, E. V. Deviatov, A. A. Shashkin, V. T. Dolgopolov, F. Hastreiter, A. Wixforth, K. L. Campman and A. C. Gossard, Phys. Rev. Lett. 84, p. 725 (2000). 16. A. Sawada, Z. F. Ezawa, H. Ohno, Y. Horikoshi, A. Urayama, Y.Ohno, S. Kishimoto, F. Matsukura and N. Kumada, Phys. Rev. B 59, p. 14888 (1999). 17. N. Kumada, D. Terasawa, M. Morino, K. Tagashira, A. Sawada, Z. F. Ezawa, K. Muraki, Y. Hirayama and T. Saku, Phys. Rev. B 69, p. 155319 (2004). 18. N. Kumada, K. Muraki and Y. Hirayama, to be published elsewhere.

Trapped Ions (Special Talk)

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QUANTUM COMPUTATION WITH TRAPPED IONS

1,2 1 ¨ ¨ H. HAFFNER , W. HANSEL , C. F. ROOS1,2 , P. O. SCHMIDT1 , M. RIEBE1 , M. CHWALLA1 , D. CHEK–AL–KAR1 J. BENHELM2 , U. D. RAPOL2 , 1 2 ¨ ¨ 2,3 , AND R. BLATT1,2 ¨ , C. BECHER1,† , O. GUHNE , W. DUR T. KORBER 1

2

Institut f¨ ur Experimentalphysik, Innsbruck, Austria ¨ Institut f¨ ur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften, Austria 3

†

Institut f¨ ur Theoretische Physik, Innsbruck, Austria

Present address: Fachrichtung Technische Physik, D-66041 Saarbr¨ ucken, Germany Trapped ions can be prepared, manipulated and analyzed with high fidelities. In addition, scalable ion trap architectures have been proposed (Kielpinski et al., Nature 417, 709 (2001).). Therefore trapped ions represent a promising approach to large scale quantum computing. Here we concentrate on the recent advancements of generating entangled states with small ion trap quantum computers. In particular, the creation of W–states with up to eight qubits and their characterization via state tomography is discussed. Keywords: Entanglement; Quantum information; Ion traps.

1. Introduction Among the proposals for constructing a quantum computer, the approach based on trapped ions is currently one of the most advanced 1 . Here, the internal electronic states of trapped ions implement the qubits2 . The state of the ions can be initialized, manipulated and read out with very high ﬁdelities2 . In addition, quantum information can be stored for up to 10 minutes as demonstrated by Bollinger and co–workers3. Thus, trapped ions are an ideal quantum memory. For processing this quantum information, however, a controllable quantum interaction between the qubits is also required. In 1995, Cirac and Zoller realized that a string of ions trapped in a linear Paul trap provides a system where such an interaction may be realized4 . Addressing individual ions with cleverly chosen laser pulses causes the conditional evolution of physically separated qubits. Thus, this is a system in which at one hand, the carriers of information are well separated from each other and the environment and on the other hand the interaction can be turned on and oﬀ at will. In addition, the size of resources necessary to control this system does not increase exponentially with the number of qubits4 . This scalability is a necessary condition for a useful quantum computer.

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Soon after the proposal by Cirac and Zoller in 1995, the ion storage group at NIST around David Wineland realized the key idea of the proposal —a controlledNOT operation— 5 with a single Be+ –ion. Furthermore they demonstrated a few other two–qubit gates6,7,8 , entangled up to four ions in 20006 , realized a so–called decoherence free subspace9 and simulated a nonlinear beam–splitter10 . In Rainer Blatt‘s group in Innsbruck the Deutsch–Josza algorithm was demonstrated in 2003 on a single Ca+ –ion11 , followed by the ﬁrst implementation of a set of universal gates on a two–ion string12 . Further milestones in ion–trap quantum computing were then experiments on quantum teleportation by both groups13,14 , an error correction protocol by the NIST–group15 and entanglement of six ions16 . Here we now concentrate on recent experiments17 where up to eight ions were entangled in a so–called W–state18,19 . Entanglement properties of two and three particles have been studied extensively and are very well understood. However, both creation and characterization of entanglement becomes exceedingly diﬃcult for multi–particle systems. To entangle numerous qubits, it is quite advantageous to use methods where the eﬃciency (e.g. the creation time) scales polynomially in the number of qubits. While this can be achieved, the full characterization of the entangled states still requires an exponentially increasing number of measurements. In spite of this, the availability of the density matrix such multi–particle entangled states together with the full information on these states in form of their density matrices is an important a test-bed for theoretical studies of multi–particle entanglement. Here we obtain the maximum possible information on our entangled states by performing full characterization via state tomography20. With the density matrix at hand, we prove in a detailed analysis that they carry genuine four-, ﬁve-, six-, seven- and eight–particle entanglement, respectively. 2. Experiment The generation of such W–states is performed in an ion–trap quantum processor21. We trap strings of up to eight 40 Ca+ ions in a linear Paul trap. Superpositions of the S1/2 ground state and the metastable D5/2 state of the Ca+ ions (lifetime of the |D–level: τ ≈ 1.16 s) represent the qubits. Each ion–qubit in the linear string is individually addressed by a series of tightly focused laser pulses on the |S ≡ S1/2 (mj = −1/2) ←→ |D ≡ D5/2 (mj = −1/2) quadrupole transition employing narrow-band laser radiation near 729 nm. Doppler cooling and subsequent sideband cooling prepare the ion string in the ground state of the center–of–mass vibrational mode. Optical pumping initializes the ions’ electronic qubit states in the |S state. After preparing an entangled state with a series of laser pulses, the quantum state is read out with a CCD camera.

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The entangled states prepared in the experiments are W–states18,19 . An N – particle W–state |WN = (|D · · · DDS + |D · · · DSD √ +|D · · · DSDD + · · · + |SD · · · D) / N

(1)

consists of a superposition of N states where exactly one particle is in the |S–state while all other particles are in |D. The W–states are eﬃciently generated by sharing one motional quantum between the ions with partial swap–operations17. The time needed for entangling procedure is about 1 ms, while the cooling and read–out take 14 ms and 4 ms, respectively. 2.1. State tomography Full information on the N –ion entangled state is obtained via quantum state reconstruction. For this we expand the density matrix in a basis of observables23 and measure the corresponding expectation values. We employ additional laser pulses to rotate the measurement basis prior to state detection to accomplish the required basis rotation20 . We use 3N diﬀerent bases and repeat the experiment 100 times for each basis. For N = 8, this amounts to 656 100 experiments and a total measurement time of 10 hours. To obtain a positive semi–deﬁnite density matrix ρ, we follow the iterative procedure outlined by Hradil et al.24 for performing a maximum–likelihood estimation of ρ. The reconstructed density matrix for N = 8 is displayed in Fig. 1. To retrieve the ﬁdelity F = WN |ρ|WN , we adjust the local phases such that F is maximized. The local character of those transformations implies that the amount of the entanglement present in the system is not changed. We obtain ﬁdelities F4 = 0.85, F5 = 0.76, F6 = 0.79, F7 = 0.76 and F8 = 0.72 for the 4,5,6,7 and 8–ion W–states, respectively. One important issue is to estimate the uncertainty the density matrix elements and of quantities derived from it. This is achieved with a Monte Carlo simulation: Starting from the reconstructed density matrix, we simulate up to 100 test data sets taking into account the major experimental uncertainty, i.e. quantum projection noise. Then the test sets are analyzed and we can extract probability distributions for all observables from the resulting density matrices. Table 1 lists the expectation values for witness operators. An entanglement witness25,26 is an operator which has for all separable states a positive expectation value. Thus a negative expectation value proves that genuine multi–partite entanglement is present. We have constructed such entanglement witnesses for our produced states and thus veriﬁed that they are genuinely N –partite entangled. Details can be found in Ref. 17 .

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Figure 1. Absolute values of the reconstructed density matrix of a |W7 –state as obtained from quantum state tomography. Ideally, the dark entries have all the same height of 17 , the bright bars indicate imperfections. The state at the bottom corner corresponds to |DDDDDDD. Table 1. Entanglement properties of ρN . First row: Fidelity after properly adjusting local phases. ˜ N (for N = 8 we used additionally local filters). Second row: Expectation value of the witnesses W For completeness we also analyzed the data published previously in Ref. 22 for N = 3. F ˜ N ρN ) tr(W

N=3

N=4

N =5

N=6

N =7

N =8

0.824

0.846 (11)

0.759 (7)

0.788(5)

0.763 (3)

0.722 (1)

−0.532

−0.460 (31)

−0.202 (27)

−0.271 (31)

−0.071 (32)

−0.029 (8)

3. Experimental Imperfections For an investigation of the experimental imperfections and scalability, we simulate the preparation procedure by solving the Schr¨ odinger equation with the relevant imperfections. We identify four major sources of deviations from the ideal W–states: addressing errors, imperfect optical pumping, non–resonant excitations, and laser frequency noise. The trap frequency inﬂuences these experimental imperfections diametrically: for example, to keep the addressing error reasonably low [i.e. less than 5%, where the addressing error is deﬁned as the ratio of the Rabi–frequencies between the addressed ion and the neighboring ion(s)], we adjust the trap frequency such that the inter–ion distance in the center of the ion string is about 5 μm. However, for large N the required trap relaxation implies that the sideband transition frequency moves closer to the carrier transition frequency. Thus the strong laser pulses driving the weak sideband transition cause more oﬀ–resonant excitations on the carrier transition, which in turn spoil the obtainable ﬁdelity. Therefore we reduce the laser power for driving the sideband, which then results in longer preparation times and

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leads to an enhanced susceptibility to laser frequency noise. For N = 8 we used 0.813 MHz for the center-of-mass frequency and 380 μs for a 2π–pulse on the blue sideband. For the simulations we approximate the ions only as two-level systems and include only the ﬁrst three levels of the center-of-mass excitation. For a serious analysis of the imperfections this is by no means suﬃcient as e.g. no environment is included. Still, the simulation time for the generation of a |W8 –state under these idealized conditions is already 20 minutes on a 3 GHZ processor using matlab. As the computational time for the simulations scales with 4N , it is quite demanding to include a reasonable environment or even use a density matrix approach. The ﬁdelity reduction of |W6 for the diﬀerent imperfections are as follows: 0.1 (addressing error), 0.07 (oﬀ–resonant excitations), 0.04 [laser frequency noise (200 Hz rms)]. Another possible error source is imperfect ground state cooling. Intensity noise of the 729–laser (ΔImax /I ≈ 0.03) does not contribute signiﬁcantly. Finally, we experimentally observed non–ideal optical pumping which can result in a reduction of 0.02 of the ﬁdelity per ion. For N ≥ 6, we therefore minimize the errors due to optical pumping and a part of the addressing errors by checking the initialization procedure with a detection sequence . In addition, we switched the blue-sideband pulses adiabatically with respect to the trap frequency to minimize oﬀ–resonant excitations. Thus for the |W6 –state used for the analysis in Tab. 1 the expected ﬁdelity should be on the order of 0.91 as only addressing errors (signiﬁcantly reduced by the conditional check after the initialization procedure (0.05)) and laser frequency noise (0.04) contribute to the imperfect ﬁdelity. Even though it is hard to estimate the ﬁdelity for N = 8 it seems that the discrepancy between the estimations and the experiment is even larger for N = 8. If one looks closely at our produced density matrices (they are made available as supplementary material in Ref. 17 , see also Fig. 1) the relatively strong occupation of the |DD · · · D stands out. This is not expected at all from the simulations. Future investigations will reveal whether this discrepancy is due mistakes in the pulse sequence, some unexpected short comings or whether some interesting physics is missing in the simulations. 4. Conclusion In this contribution, we have presented experiments on W–states with up to eight ions. Methods to characterize the states and experimental imperfections are discussed. Even though the experimental system under investigation seems well understood, the quantitative behavior is not reproduced properly by the simulations. Acknowledgments We gratefully acknowledge support by the Austrian Science Fund (FWF), by the European Commission (SCALA, QGATES, CONQUEST PROSECCO, QUPRODIS

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and OLAQUI networks), by the Institut f¨ ur Quanteninformation GmbH, the DFG, ¨ and the OAW through project APART (W.D.). This material is based upon work supported in part by the U. S. Army Research Oﬃce. We thank P. Pham for the pulse modulation programmer and A. Ostermann, M. Thalhammer and M. Jeˇzek for the help with the iterative reconstruction. References 1. Army Research Oﬃce ARDA quantum computation roadmap: http://qist.lanl.gov/qcomp map.shtml 2. D. Wineland, C. Monroe, W. Itano, D. Leibfried, B. King, D. Meekhof, Journal of Research of the National Institute of Standards and Technology 103, 259 (1998). 3. J. Bollinger, D. Heinzen, W. Itano, S. Gilbert, D. Wineland, IEEE Trans. Instrum. Meas. 40, 126 (1991). 4. I. Cirac, P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 5. C. Monroe, D. Meekhof, B. King, W. Itano, D. Wineland Phys. Rev. Lett. 75, 4714 (1995). 6. C.A. Sackett et al., Nature 404, 256 (2000). 7. B. DeMarco et al., Phys. Rev. Lett. 89, 267901 (2002). 8. D. Leibfried et al., Nature 422, 412 (2003). 9. D. Kielpinski, V. Meyer, M.A. Rowe, C.A. Sackett, W.M. Itano, C. Monroe, D. Wineland, Science 291, 1013 (2001). 10. D. Leibfried et al., Phys. Rev. Lett. 89, 247901 (2002). 11. S. Gulde et al. Nature 421, 48 (2003). 12. F. Schmidt-Kaler et al., Nature 422, 408 (2003). 13. M.D. Barrett et al., Nature, 429, 737 (2004). 14. M. Riebe et al., Nature, 429, 734 (2004). 15. J. Chiaverini et al. Nature 432, 602 (2004). 16. D. Leibfried et al. Nature 438, 639 (2005). 17. H. H¨ aﬀner et al. Nature 438, 643 (2005). 18. A. Zeilinger, M.A. Horne, D.M. Greenberger, Higher-order quantum entanglement. NASA Conf. Publ. 3135, pp 73–81 (1992). 19. W. D¨ ur, G. Vidal, J.I. Cirac, Phys. Rev. A 62, 062314 (2000). 20. C.F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004). 21. F. Schmidt–Kaler et al., Appl. Phys. B 77, 789–796 (2003). 22. C.F. Roos et al., Science 304, 1478–1480 (2004). 23. U. Fano, Rev. Mod. Phys. 29, 74–93 (1957). 24. Z. Hradil, J. Reh´ aˇcek, J. Fiur´ aˇsek, M. Jeˇzek, Lect. Notes Phys. 649, 59–112 (2004). 25. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1–8 (1996). 26. M. Bourennane et al., Phys. Rev. Lett. 92, 087902 (2004).

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LIST OF PARTICIPANTS Abdufarrukh Abdumalikov Physikalisches Institut III, Erlangen University Erwin-Rommel-Str. 1, D-91054 Erlangen, Germany

Tatsushi Akazaki NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Boris Altshuler Princeton, Colombia Physics Department Princeton University Princeton NewJersey 08544 USA

Shinichi Amaha ICORP-JST 4S-308S, NTT-BRL, 3-1, Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Vladimir Antonov Department of Physics, Royal Holloway, University of London Egham, Surrey TW20 0EX, U.K.

Masanari Asano Tokyo University of Science 2641 Yamazaki, Noda, Chiba, 278-8510, Japan

Tetsuya Asayama Sony Corporation 6-7-35 kitashinagawa, Shinagawa-ku, Tokyo, 141-0001, Japan

Sahel Ashhab RIKEN Hirosawa 2-1, Wako, Saitama 351-0198, Japan

Dimitri V. Averin Department of Physics and Astronomy, Stony Brook University, SUNY Stony Brook, NY 11794-3800, USA

David D. Awschalom Department of Physics, University of California, Santa Barbara Santa Barbara, CA 93106-9530 USA

Christopher Bauerle CNRS-CRTBT 25 avenue des Martyrs, Grenoble, 38042, France

Hendrik Bluhm Stanford University McCoullough Bldg., Rm 18, 476 Lomita Mall, Stanford, CA, 94305, USA

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410 410 Massoud Borhani Department of Physics, University of Basel Klingelbergstrasse 82, Basel 4056, Switzerland

Denis Bulaev Department of Physics and Astronomy, University of Basel Klingelbergstrasse 82, CH-4056 Basel, Switzerland

Guido Burkard Department of Physics and Astronomy, University of Basel Klingelbergstrasse 82 CH-4056 Basel, Switzerland

Tord Claeson Physics and Engineering Physics, Chalmers University of Technology SE-41296 Göeborg Sweden

Robert G. Clark Australian Research Council Centre for Quantum Computer Technology, University of New South Wales Sydney, 2052, NSW, Australia

John Clarke Department of Physics, University of California, Barkeley 349 Birge 94720-7300 Barkeley CA USA

Per Delsing Department of Microelectronics and Nanoscience, Chalmers University of Technology SE-41296 Goeteborg Sweden

Frank Deppe Walther Meissner Institute, TU of Munich Walther-Meissner-Str. 8, 85748 Garching, Germany

Daniel Esteve Quantronics, SPEC-CEA Saclay CEA-Saclay 91191 Gif-sur-Yvette, France

Mikio Eto Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama Kanagawa 223-8522, Japan

Giuseppe Falci Dipartimento di Metodologie Fisiche e Chimiche (DMFCI) - Universita' di Catania Citta' Universitaria - Viale A. Doria 6 - I95125, Italy

Lara Faoro Rutgers University 136 Frelinghuysen Road, Piscataway 08854, New Jersey, USA

Rozario Fazio Scuola Normale Superiore Scuola Normale Superiore Piazza dei Cavalieri 7 I-56126, Pisa-Italy Italy

Klaus-Juergen Friedland Paul-Drude-Institute Hausvogteiplatz 5-7, 10117 Berlin, Germany

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411 411 Toshimasa Fujisawa NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa, 243-0198, Japan

Akira Fukuda LTM Center, Kyoto University Kitashirakawa-oiwakecho, Sakyo, Kyoto, Japan

Kenichiro Furuta R&D Center, Toshiba Corporation 1, Komukai-Toshiba-cho, Saiwai-ku, Kawasaki, Kanagawa, Japan

Zafer Gedik Sabanci University Faculty of Engineering and Natural Sciences, Tuzla 34956 Istanbul, Turkey

Yuval Gefen Department of Condenced Matter Physics, The Weizmann Institute of Science Rehovot 76100, Israel

Francesco Giazotto NEST CNR-INFM and Scuola Normale Superiore Piazza dei Cavalieri 7, 56126, Pisa, Italy

Paula Giudici NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Leonid Glazman University of Minnesota 116 Church St SE Minneapolis MN 55455, USA

Daisuke Goto Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, Japan

Hartmut Häffner Institut für Quantenoptik und Quanteninformation ICT-Gebäude, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria

Yuichi Harada NTT Europe BRL 3rd Floor, Devon House, 58-60 St. Katharine's Way, London, E1W 1LB, UK

C.J.P.M. Harmans Department of Applied Physics, Delft University of Technology Lorentzweg 1, 2628 CJ, Delft, The Netherlands

Khalil Harrabi CREST - JST Fundamental and Environmental Research Laboratorie, Tsukuba, Ibaraki, 305-8501, Japan

Hideomi Hashiba Physics Department, Royal Holloway University of London Egham, Surrey TW20 0EX, U.K.

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412 412 Tetsuya Hata Department of Applied Physics, Nagoya University Furo-cho, Nagoya, Aichi, Japan

Frank W. J. Hekking Theoretical condensed matter physics, CNRS, Grenoble LPMMC-CNRS Maison des Magristeres B.P.166 38042 Grenoble cedex 9 France

Yoshiro Hirayama NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Xuedong Hu Department of Physics, University at Buffalo, SUNY 239 Fronczak Hall, Buffalo, New York 14260-1500 USA

Katsuhiko Inagaki Department of Applied Physics, Hokkaido Univ Kita 13 Nishi 8 Kita-ku, Sapporo, Hokkaido, Japan

Kazuki Iwata Graduate School of Science, Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Yasuhiro Iye Institute for Solid State Physics, University of Tokyo 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-858, Japan

Ane Jensen Niels Bohr Institute, Nano-Science Center, University of Copenhagen Universitetsparken 5 D, 2100 Copenhagen, Denmark

Pobert Johansson RIKEN 2-1 Hirosawa, Wako-shi, Saitama, 351-0198 Japan

Henrik Ingerslev Jorgensen Niels Bohr Institute, Nano-Science Center, University of Copenhagen Universitetsparken 5, 2100 Copenhage O, Denmark

Hiroyuki Kageshima NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Kosuke Kakuyanagi NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Ikuzo Kanazawa Department of Physics, Tokyo Gakugei University Nukuikitamachi 4-1-1, Koganeishi, Tokyo 184-8501, Japan

Alexander Kasper NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

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413 413 Shiro Kawabata National Institute of Advanced Industrial Science and Technology Umezono 1-1-1, Tsukuba, Ibaraki, Japan

Minoru Kawamura RIKEN 2-1 Hirosawa, Wako, Saitama, Japan

Stefan Kettemann Institute for Theoretical Physics, Hamburg Univ. Jungiusstrasse 9, 20355 Hamburg, Germany

Mun Dae Kim Korea Institute for Advanced Study Cheongryang-ri 2 dong, Dongdaemun-gu, 207-43 Seoul, 130-722, Korea

Tomoko Kita Department of Applied Physics, Osaka University 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan

Andriy Kiyko Institute for Low Temperature Physics and Engineering Linin Ave., 47, Kharkov, 61103, Ukraine

Makoto Kohda Tohoku University 6-6-02 Aramaki-aza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan

Kohkichi Konno Department of Applied Physics, Graduate School of Engineering, Hokkaido University Kita 13, Nishi 8, Kita-ku, Sapporo Hokkaido 060-8628, Japan

Hideo Kosaka Tohoku univ. 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Nicholas C. Koshnick Stanford University 476 Lomita Mall, Stanford, CA 94306, USA

Toshihiro Kubo Quantum Spin Information Project, ICORP-JST 4S-308S, NTT R&D Center, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa, Japan

Norio Kumada NTT Basic Research Laboratories 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan

Watson Kuo National Chung Hsing Uinversity 250, Guo-Guang Rd. 402 Taichung city, Taiwan

Yu-xi Liu The Institute of Physical and Chemical Research (RIKEN) Wako-shi, Saitama 351-0198, Japan

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414 414 Imran Mahboob NTT Basic Research Laboratories 3-1 Morinosato Wakamiya, Atsugi-shi, Kanagawa 243-0198, Japan

Charles Marcus Physics Department, Harvard Lyman 328 17 Oxford Street Cambridge, MA 02138, USA

Thierry Martin Centre de Physique Theorique Marseille and Universite de la Mediterranee case 907, 13288 Marseille, France

Jan Martinek Institute for Materials Research, Tohoku University Sendai 980-8577, Japan

John Martinis Department of Physics, University of California, Santa Barbara Santa Barbara, CA 93106-9530 USA

Koji Maruyama RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Daisuke Matsubayashi Department of Physics, University of Tokyo 14-13 Matsukazedai, Aoba-ku, Yokohama, Kanagawa, 227-0067, Japan

Toru Matsuura Department of Applied Physics, Graduate School of Engineering, Hokkaido University Kita 13, Nishi 8, Kitaku, Sapporo, Hokkaido, 060-8628, Japan

Phil Meeson Royal Holloway, University of London Dept of Physics, Egham Hill, Egham, Surrey, TW20 0EX, UK

Rodney Van Meter Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522, Japan

Yasuyoshi Miyamoto NHK 1-10-11 Kinuta, Setagaya, Tokyo 157-8510 Japan

Tetsuya Miyawaki Tohoku University 6-6-02 Aramaki-aza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan

Hisao Miyazaki University of Tsukuba Ootuka Lab., Inst of Physics, Tennnodai 1-1-1, Tsukuba, 305-8571, Japan

J.E. Mooij Department of Applied Physics, Delft University of Technology Lorentzweg 1, Delft 2612 HD The Netherlands

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Tetsuya Mukai NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Koji Muraki NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Anri Nakajima Hiroshima University 1-4-2 Kagamiyama, Higashi-Hiroshima, 739-8527, Japan

Yasunobu Nakamura NEC Fundamental Research Labs. 34 Miyukigaoka, Tsukuba, Ibaraki 305-8051, Japan

Hayato Nakano NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Yuhei Natsume Graduate School of Science and Technology, Chiba University Inage-ku, Chiba, Japan

Antti Niskanen JST-CREST NEC Fundamental Research Labs, 34 Miyukigaoka, Tsukuba, Ibaraki, 305-8501, Japan

Tadashi Nishikawa NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Junsaku Nitta Tohoku University 6-6-02, Aramaki-Aza, Aoba, Sendai, Miyagi, 980-8579, Japan

Shintaro Nomura Institute of Physic, University of Tsukuba 1-1-1 Tennoudai, Tsukuba, Ibaraki, Japan

Franco Nori RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Hideaki Obuse RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Hideo Ohno Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Satoru Ohno 102, Bisou-house, 3-3-4, Hiyoshi, Kouhoku-ku, Yokohama, Kanagawa 223-0061 Japan

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Keiji Ono Low temperature Lab., RIKEN 2-1 HIrosawa, Wako-shi, Saitama, Japan

Takuya Ono Department of Applied Physics, Osaka University Suita Osaka 565-0871, Japan

Takeshi Ota SORST-JST NTT Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa, Japan

Elisabetta Paladino MATIS CNR-INFM & DMFCI, Universita' di Catania Viale A. Doria 6, Ed.10, 95128 Catania, Italy

Jukka Pekola Low Temperature Laboratory, Helsinki University of Technology P.O. Box 3500, 02015 TKK, Finland

Zhi-Hui Peng National Laboratory for Superconductivity, Institute of Physics, Chinese Academy of Sciences Beijing 100080, China

Yuriy Pershin Michigan State University 4247 BPS Building, East Lansing, 48824, USA

Kasper Grove-Rasmussen Nano-Science Center, Niels Bohr Instititute Universitetsparken 5, Copenhagen O, 2100 Denmark

David Rees Royal Holloway University of London Egham Hill, Egham, TW20 OEX, UK

Yoshiaki Rikitake CREST-JST Tohoku Univ, Aramaki, Aobaku, Sendai, Miyagi, 980-8579, Japan

Keiji Saito University of Tokyo Bunkyo-ku Hongo 7-3-1, Tokyo, 113-0033, Japan

Hiroyuki Sakaki University of Tokyo 4-6-1 Komaba, Meguro, Tokyo, 153-8505, Japan

Rui Sakano Dept. of applied physics, Osaka University 2-1, Yamada, Suita, Osaka, 565-0871, Japan

Laurent Saminadayar Centre de Recherches sur les Trés Basses Températures CRTBT, BP 166 X, Grenoble Cedex 09, 38042, France

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Satoshi Sasaki NTT Basic Research Laboratories 3-1, Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Tetsuo Satoh Asian Technology Information Program 6-15-21, Roppongi, Minato-ku, Tokyo, 106-0032, Japan

Sergey Savel'ev FRS, The Institute of Physical and Chemical Research (RIKEN) 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan

Yoshiaki Sekine NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Kouichi Semba NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Sergiy N. Shevchenko Institute for Low Temperature Physics and Engeneering Lenin Ave. 47, 61103 Kharkov, Ukraine

Iduru Shigeta Department of General Education, Kumamoto National College of Technology 2659-2 Suya, Nishigoshi-Machi, Kikuchi-Gun, Kumamoto 861-1102, Japan

Yoshihiro Shimazu Yokohama National University Tokiwadai 79-5, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan

Gou Shinkai Tokyo institute of technology / NTT-BRL 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Kazutomu Shiokawa RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Alexander Shnirman Institut für Theoretische Festköperphysik, Universität Karlsruhe D-76128 Karlsruhe Germany

Fabio Taddei NEST CNR-INFM & Scuola Normale Superiore, Pisa Piazza dei Cavalieri, 7; I-56126 Pisa, Italy

Masao Takahashi Kanagawa Institute of Technology 1030 Shimo-Ogino, Atsugi, Kanagawa, 243-0292, Japan

Hideaki Takayanagi NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

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Mitsuaki Takigawa Department of Applied Physics, Graduate School of Engineering, Hokkaido University Kita 13, Nishi 8, Kita-ku, Sapporo Hokkaido 060-8628, Japan

Yoichi Tanaka Osaka University 2-1 Yamadaoka, Suita Osaka, 565-0871, Japan

Yukio Tanaka Department of Applied Physics Nagoya Chikusa-ku Furocho, Japan

Hiroyuki Tamura NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Yasunari Tanuma Kanagawa University 3-27-1, Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa, 221-8686, Japan

Seigo Tarucha University of Tokyo Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656, Japan

Isao Tomita NTT Photonics Laboratories 3-1, Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Jaw-Shen Tsai NEC Fundamental Research Labs. 34 Miyukigaoka, Tsukuba, Ibaraki, 305-8501, Japan

Kazuki Tsuboi Hiroshima University Higasihiroshima, Hiroshima, 739-8526, Japan

Kazuyuki Uchida NTT Basic Research Laboratories 3-1, Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Akiko Ueda Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Shinya Uji National Institute for Materials Science 3-13, Sakura, Tsukuba, Ibaraki 305-0003 Japan

Alexey Ustinov Physikalisches Institut, Universität Erlangen-Nürnberg Physikalisches Institut III, Erwin-Rommel-Str. 1, Erlangen, 91058, Germany

Yasuhiro Utsumi Condensed Matter Theory Laboratory: RIKEN 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

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Kazuhiro Utsunomiya Hiroshima University Kagamiyama, Higashihiroshima, Hiroshima, 739-0046, Japan

Shoko Utsunomiya NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Juha Vartiainen Low Temperature Laboratory, Helsinki University of Technology P.O. Box 3500, FIN-02015 TKK, Finland

Andreas Wallraff Department of Applied Physics, Yale University P.O.Box 208284 429 Becton Center, 15 Prospect St. New Heacen, CT06520-8284 USA

Lianfu Wei Frontier Research System, The Institute of Physical and Chemical Research (RIKEN) Wako-shi, Saitama, 351-0198, Japan

Cheng-En Wu Department of Physics, National Tsing Hua University 101 Section 2 Kuang Fu Road, Hsinchu, 30013, Taiwan

Shin Yabuuchi Keio University 3-14-1, Hiyoshi, Kouhoku-ku, Yokohama, Kanagawa, Japan

Ryuta Yagi ADSM, Hiroshima University Kagamiyama 1-3-1, Higashi Hiroshima, 739-8530, Japan

Hiroshi Yaguchi Department of Physics, Kyoto University Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

Hiroshi Yamaguchi NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Masumi Yamaguchi NTT Basic Research Laboratories 3-1, Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Makoto Yamashita NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Itaru Yokohama NTT Basic Research Laboratories 3-1, Morinosato, Wakamiya, Atsugi, Kanagwa 243-0198, Japan

Nobuhiko Yokoshi Waseda University 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

August 12, 2008

10:15

Proceedings Trim Size: 9.75in x 6.5in

participants

420 420

Takehito Yokoyama Department of Applied Physics Nagoya University Huro-cho, Chikusa-ku, Nagoya, Aichi 464-8603, Japan

Fumiki Yoshihara RIKEN Tsukuba Ibaraki 305-850, Japan

Alexandre M. Zagoskin The University of British Columbia 6224 Agricultural Rd, Vancouver, Canada

Alexander Zaitsev Institute of Radioengineering and Electronics, RAS Mokhovaya 11, Moscow, 103907, Russia