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/4>o and Ch = (n, m) is the chiral vector. On the other hand, because of the tion (5) then predicts a splitting of AAB W existence of the periodic potential along the 41 meV at 45 T.
236 2.2. Magnetic
alignment
1.0
-l—r-rTTTTn
1—m
Carbon nanotubes have anisotropic magnetic 0.8 properties. 11 An interesting consequence of this is magnetic alignment. In fact, it is 0.6 important to understand the angular distriCO bution of nanotubes in the solution at each 0.4 magnetic field value in order to extract the true AB splitting values from data. 0.2 While the splitting rate v [Eq. (5)] is defined for a nanotube parallel to the applied 0.0 field, experimentally we observe an apparent 1000 10 100 Magnetic Field (T) splitting rate of an ensemble of nanotubes with an angular distribution characterized by Fig. 1. Calculated steady state nematic order pathe probability of finding a nanotube in an rameter, S. The magnetic susceptibility anisotropy 4 5 angular range relative to the B direction. At (1.4 X 10~ emu/mol), average length (300 nm), diameter (1 nm), and temperature (300 K) were used 0 T, the nanotubes are randomized in solu- in calculation. tion. In a magnetic field, the nanotubes align due to their magnetic anisotropy. The angu- 3. Experimental lar probability distribution is given (in spherThe samples studied in the present work were ical coordinates) by: aqueous suspensions of individualized HiPco v> (a\AO exp(-u 2 sin 2 0)sin (6) nanotubes produced at Rice. The samples / 0 exp(-u 2 sin 0) sin 9d0 were prepared by high sheer mixing of nanotubes in a 1 wt. % solution of surfactant where (sodium dodecyl sulfate, sodium benzosulIB2N(X\\-X±) fonic acid, or sodium cholate) in D2O, son(7) kBT ication, and centrifugation. The resulting supernatant was enriched in individualized where x± and X|| are the diamagnetic susSWNTs surrounded by surfactant micelles.15 ceptibilities per mole of carbon atoms in a nanotube along its two principal axes, and 8 The nanotubes are thus unbundled and preis defined as the angle between the nanotube vented from interacting with each other, which leads to chirality-dependent peaks in long axis and the magnetic field. absorption and PL. 1 5 ' 1 6 Using Eq. (6), one can calculate the We used a variety of high-field magnets expectation value of any physical quantity that depends on the angle 6. One use- for performing magneto-optical experiments. ful quantity is the so-called nematic order For DC field experiments a 10 T superconducting magnet (at Rice), a 33 T waterparameter, 12 ' 13 ' 14 defined as cooled magnet, and the 45 T hybrid magS = (3 (cos 2 6) - l ) / 2 . (8) net [both in Tallahassee, Florida] were used. This parameter is 0 when the nanotubes are Pulsed field experiments were conducted uscompletely randomly oriented and 1 when all ing capacitor bank driven magnets of 67 T the nanotubes in the solution are pointing in (in Los Alamos, New Mexico), 54 T, and the 17 the magnetic field direction. Figure 1 shows 75 T ARMS magnet (in Toulouse, France). S calculated for DC fields. If a DC 100 T Samples were mounted on a specially magnet existed, it would be ideal for aligning designed stick with integrated optic fibers, our small diameter SWNTs. lenses, mirrors, and polarizers. The nani
i i 1111
237 otube solution was contained in a quartz cell with an effective optical length of 0.5 cm for absorption (0.1 cm for PL). Absorption spectra were normalized to a 1 wt. % surfactant in D2O solution. We used a Si charge coupled device (in the visible) and an InGaAs array detector (in the near-infrared). A quartz-tungstenhalogen lamp was used for the absorption measurements, and a Ti:Sapphire laser was used in the PL measurements. Experiments in pulsed fields were recorded with an exposure plus readout time of ~1.5 ms, this resulted in spectra taken in ~.5 T increments. We utilized Voigt geometry with two different polarizations (P) for these experiments, B || P and B ± P. Due to the samples being in aqueous suspensions, all measurements were done at room temperature. 4. R e s u l t s
0.9
Absorption spectra in the near-infrared range are shown in Fig. 2 at various DC fields up to 45 T. The peaks in the specra corre-
1.0 1.1 Energy (eV)
Fig. 2. Magnetic field induced anisotropy. Room temperature absorption of the SWNT sample was measured in Voigt geometry for two polarizations of the probe light: B \\ P (B J. P). Dashed lines represent results of our model based on AB effect and t h e magnetic alignment. No intentional offset.
1.0 Energy ( e V )
Fig. 3. Absorption spectra at various magnetic fields taken in the Voigt geometry with light polarization parallel to t h e field. The traces are offset for clarity. The inset shows relative change in absorption at 2.3 eV during a 67 T shot.
spond to the lowest-energy transitions (En) in semiconducting nanotubes with different chiralities. As the field increases, the B \\ P (B -L P) absorption increases (decreases) as a result of magnetic alignment; note that the E\\ transitions are allowed only for the light polarized parallel to the tube axis. At 45 T, while the B _L P absorption (sensitive to the nanotubes lying more perpendicular to the B) shows only peak broadening, the B \\ P absorption (sensitive to the nanotubes lying more parallel to B) shows more pronounced spectral changes. This can be successfully explained using our model (see the dotted lines, explained later) that takes into account magnetic alignment and AB splitting. However, clear absorption peak splittings cannot be seen even at 45 T.
238 I .
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Time (s) Fig. 4. Dynamic alignment. Transmission anisotropy in the second subband during a 54 T shot (a). S, calculated from anisotropy, plotted on a semilog scale shows characteristic relaxation time for nanotubes in solution.
Fig. 5. PL spectra at 0 T and 45 T showing magnetic-field-induced band-gap shrinkage due to the Aharonov-Bohm effect. The data was taking with 790 nm excitation, and different peaks correspond to different chirality nanotubes.
changes in PL data compared to absorption data. Namely, all peaks shift to lower energies as B is increased. The dotted lines are simulations based on our model taking into account the AB effect and magnetic alignThe use of pulsed magnets allowed us to ment and explain the observations very well. access higher fields. Figure 3 shows B \\ P absorption spectra taken during a 67 T magnetic pulse. At fields above 54 T, clear ab- 5. Discussion sorption peak splittings are observed. In ad- In a magnetic field parallel to a nanotube dition, time-dependent polarized transmis- axis, absorption and PL peaks are predicted sion allows for the observation of nanotube to split by an amount proportional to the apsolution dynamics. Figure 4 shows a de- plied field [see Eq. (5)]. The absorption peaks crease (increase) in transmission for B \\ P are predicted to split into two equal height (B ± P) as field is applied. As the field in- peaks with the separation proportional to B. creases the nanotubes begin to align reach- PL peaks will split with the same splitting ing a maximum order parameter of ~0.35 at rate, but with the relative size of the two 54 T. The 45 T DC case reaches a maximum peaks determined by the Boltzmann factor. order parameter of ~0.5, indicating that the The 0 T PL spectrum was fit using tubes do not completely align on fractional Lorentzian peaks that correspond to the chisecond time scales. This is due to many fac- ralities present in the sample. The 45 T PL tors, including nanotube inertia (dependant spectrum was then simulated by varying two on length and diameter) and environment parameters for each Lorentzian, u and v. The (depending on solution viscosity and prox- first parameter, u, describes the angular disimity of other nanotubes). tribution of the nanotubes at a given B field Figure 5 shows PL spectra at 0 T and [see Eq. (7)]. The second, v, is the rate of 45 T. A 790 nm beam selectively excited peak splitting with the field applied paralfour chiralities in the sample. This selec- lel to the nanotube [see Eq. (5)]. For any tivity allows us to see much clearer spectral given 6 this rate was multiplied by cose? to
239 account only for the flux component threading the nanotube. The different intensities of the split PL peaks were taken into account through a Boltzmann factor (with temperature T = 300 K). The best 45 T fit is shown as dotted lines in Fig. 5. The average splitting rate obtained in this way is v = 0.9 meV/T. We use a\f'm\hu,B) and a^'m) {hw, B) to denote the intrinsic absorption spectra (in c m - 1 mole - 1 ) for (n,m) nanotubes in parallel magnetic field B when the light is polarized parallel and perpendicular to the long axis of the nanotube, respectively (where hw is the photon energy). Theoretically, a±'m (hw, B) is predicted to be zero for E\\ and 2?22 transitions due to the depolarization effect,18 but we retain this quantity in the following formal derivation for the sake of completeness. The intrinsic absorption spectra defined above are not directly obtainable from experiments due to the angular distribution of nanotubes. For example, the measured B || P absorption for (n, m) tubes is an average over the angular distribution of nanotubes given by
Eq. (9) becomes:
aj n , m ) (hw, o r ) = o[,n,TO) (hw, or) (cos2 e) = ^-a(^m)(hw,Vr)-N^m\
(11)
which relates a(^'m)(hw,B cos 9) • JV("-m) at 0 T with the measured 0 T absorption. The 67 T absorption can then be simulated using Eq. (9) by fitting parameters u^n'm^ and v(n,m) a g j n j.jj e P L simulations. When fitting the 0 T absorption data, several closely spaced (n, m) peaks might be fitted as a wider absorption peak. Nevertheless, this simulation yields data that agrees well with the measured data [see Fig. (3)] and yielded (n, m)-averaged u = 1.9 and v = 0.7 meV/T. The same model was then applied to the 45 T data [see Fig. (2)]. This treatment again yielded an average splitting rate of 0.7 meV/T. Furthermore, keeping the same fitting parameters, the B A. P data was successfully reproduced at 45 T (see Fig. 2).
The inset in Fig. 3, as well as the data in Fig. 4(a), shows that the absorption change does not exactly follow the applied pulse, i.e., there is a lag between the peak field and peak alignment. Thus, while the pulsed field data in Fig. 3 are explained well using the ana\^m)(hw,B) = gular distribution given by Eq. 6, u is no (aj" ,m) r 2 + <4n'm V +
•*•
(n,m)
( -0|,
•
2/>
,
(ii">)/i
sin 0 + a\
= 6. Conclusions and Future work
• 2 /i\ \
(1 — sin 6) ) xN(n,m)
(1Q)
We used Eqs. (9) and (10) (with a("'m) = 0) to simulate our absorption data. At 0 T,
We have observed the modification of states of Bloch electrons by an Aharonov-Bohm phase in SWNTs by magneto-optical spectroscopy. We also investigated both DC and dynamic magnetic alignment of SWNT so-
240 lutions. There are still many questions t h a t need to be explored such as solution dynamics, t e m p e r a t u r e , other geometries and polarizations, a n d higher field dependences. W h e n analyzing dynamic alignment d a t a by plotting on a semilog scale it becomes apparent t h a t this is a form of pump-probe spectroscopy. T h e short field pulse prepares the n a n o t u b e s in an aligned state (magnetic p u m p ) , while t h e transmission probes the relaxation dynamics (optical probe). This d a t a can be analyzed t o enlighten the inertial a n d environmental effects on t h e thermal motion of the n a n o t u b e s in solution. Analysis of d a t a on samples with different lengths and bundling states is underway. We are also expanding this work with destructuve pulsed magnets up to 150 T in Los Alamos a n d u p to 300 T at Humboldt University in Berlin. We have already completed single wavelength solution dynamics work in Berlin, 1 9 and magneto-absorption measurements are currently in progress. Aligned n a n o t u b e gel films, fabricated by severalgroups, are ideal for ultrahigh field and low t e m p e r a t u r e magneto-optics studies. Film samples can be aligned, cooled, are solid, and are very thin. These characteristics bypass alignment convolution, t e m p e r a t u r e restrictions, allow for easy handling and smaller, simpler sample holders.
Acknowledgements This work was supported in p a r t by the Robert A. Welch Foundation (through Grant No. C-1509) and the National Science Foundation (through G r a n t Nos. DMR-0134058, DMR-0325474, a n d INT-0437342). A portion of this work was performed a t t h e National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-0084173 and by the State of Florida. We t h a n k V. I. Klimov for the use of his I n G a A s detector for measurements in Los Alamos. We also t h a n k
G. N. Ostojic, V. C. Moore, and M. Furis for technical assistance. References 1. D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976). 2. H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993). 3. S. Zaric, G. N. Ostojic, J. Kono, J. Shaver, V. C. Moore, M. S. Strano, R. H. Hauge, R. E. Smalley, and X. Wei, Science 304, 1129 (2004). 4. S. Zaric, G. N. Ostojic, J. Kono, J. Shaver, V. C. Moore, R. H. Hauge, R. E. Smalley, and X. Wei, Nano Lett. 4, 2219 (2004). 5. S. Zaric, G. N. Ostojic, J. Shaver, J. Kono, O. Portugall, P. H. Frings, G. L. J. A. Rikken, M. Furis, S. A. Crooker, X. Wei, V. C. Moore, R. H. Hauge, and R. E. Smalley, Phys. Rev. Lett., to appear (condmat/0509429). 6. T. Ando, Semicond. Sci. Technol. 15, R13 (2000). 7. T. Ando, J. Phys. Soc. Jpn. 73, 3351 (2004). 8. T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005). 9. S. Roche, G. Dresselhaus, M. S. Dresselhaus, and R. Saito, Phys. Rev. B 62, 16092 (2000). 10. F. L. Shyu, C. P. Chang, R. B. Chen, C. W. Chiu, and M. F. Lin, Phys. Rev. B 67, 045405 (2003). 11. J. Kono and S. Roche, "Magnetic Properties," in Carbon Nanotubes: Properties and Applications, ed. M. J. O'Connell (CRC Press, Boca Raton, 2006, to be published). 12. M. Fujiwara, N. Fukui, and Y. Tanimoto, J. Phys. Chem. B 103, 2627 (1999). 13. M. F. Islam, D. E. Milkie, C. L. Kane, A. G. Yodh, and J. M. Kikkawa, Phys. Rev. Lett. 93, 037404 (2004). 14. Y. Murakami, E. Einarsson, T. Edamura, and S. Maruyama, Phys. Rev. Lett. 94, 087402 (2005). 15. M. J. O'Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weisman, and R. E. Smalley, Science 297, 593 (2002). 16. S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, and R. B. Weisman, Science 298, 2361 (2002). 17. H. Jones, P. H. Frings, M. von Ortenberg,
A. Lagutin, L. V. Bockstal, 0 . Portugall, and F. Herlach, Physica B 346-347, 553 (2004). 18. H. Ajiki and T. Ando, Physica B 201, 349 (1994). 19. J. Shaver, J. Kono, S. Hansel, A. Kirste, M. von Ortenberg, C. H. Mielke, O. Portugall, R. H. Hauge, and R. E. Smalley, in: Narrow Gap Semiconductors 2005, Proceedings of the Twelfth International Conference on Narrow Gap Semiconductors, eds. J. Kono and J. Leotin (Institute of Physics, Bristol, 2005, to be published).
242 N O N - E Q U I L I B R I U M T R A N S P O R T T H R O U G H A SINGLE-WALLED C A R B O N N A N O T U B E W I T H HIGHLY T R A N S P A R E N T COUPLING TO RESERVOIRS
P. R E C H E R 1 , 2 ' * , N. Y . K I M 1 A N D Y . Y A M A M O T O 1 ' 3 1 Quantum Entanglement Project, SORST, JST, E.L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan * E-mail: [email protected]
We consider transport through a single-walled carbon nanotube (CNT) in the ballistic regime where weak backscattering at the two contacts gives rise to Fabry-Perot (FP) oscillations in conductance and shot noise. We include the electron-electron interaction and the finite length L of the CNT within the inhomogeneous Tomonaga-Luttinger liquid (TLL) model appropriate for the CNT and treat the non-equilibrium effects due to an applied bias voltage within the Keldysh approach. At low frequencies, the shot noise is S = 2e\Is\, where e is the elementary charge and J g is the backscattered current. Interaction effects are apparent via a characteristic power-law and FP-oscillation suppression of Ig at large bias voltage in agreement with our low-frequency shot noise experiments. The extracted TLL parameter g agrees well with theoretical predictions. At frequencies above vp/gL, with vp the Fermi velocity, the theoretical impurity noise as a function of bias voltage and frequency shows clear and distinct signatures of the two velocities of collective modes present in the C N T which distinguish spin from charge degrees of freedom (spin-charge separation).
Keywords: Carbon nanotubes, Tomonaga-Luttinger liquids; shot noise
1. Introduction In one dimension (ID) electron-electron interaction cannot be neglected anymore. Different from the higher-dimensional counterparts described successfully by the Fermi liquid theory, these ID systems lack the picture of independent quasiparticles and the low energy excitations are collective charge and spin modes travelling with different velocities. The class of systems showing such low-energy behavior is named TomonagaLuttinger liquids (TLL) 1 . Metallic single-walled carbon nanotubes (CNT) have been predicted to behave as a TLL 2 which has been verified in several experiments 3 . These experiments, however, have been restricted to the tunneling regime where the conductance is much smaller than
the theoretically possible upper limit of 2Go where Go = 2e2/h is the spin-degenerate quantum unit of conductance and 2 comes from subband degeneracy in the CNT. More recently, conductance close to 4e 2 //i has been measured in CNTs although only at low bias voltage 4 . In this so called Fabry-Perot (FP) regime the low-energy transport shows interference oscillations in conductance due to weak backscattering at the two contacts. As the bias voltage is increased, the TLL effect becomes pronounced and, as a consequence, the conductance oscillations reduce in size. This has been investigated theoretically at zero temperature in the absence of experimental results 5 . In this work we will extend this theory to include a discussion of the shot noise properties. We compare the theoretical predictions of this model with our low-
243 frequency shot noise measurements in CNTs at 4 K. Good agreement between experiment and theory is observed in terms of power-law scaling of shot noise and Fano factor as well as decreased FP-oscillation amplitudes when the bias voltage is increased. The deduced TLL-parameter g agrees well with theoretical predictions. At higher frequency co, the shot noise becomes sensitive to the current fragmentation at the inhomogeneity of g6. The frequency response of shot noise shows pronounced oscillations with frequency vp/Lg = ULI9 which are dominant over the ordinary FP-oscillations exhibited by all four modes 5 . On the other hand, the bias response of shot noise shows FP-oscillations dominated by the non-interacting modes with frequency w^(due to subband degeneracy and spin). Therefore, the investigation of bias-voltage and frequency dependent shot noise brings out important information about the different velocities of CNT-collective modes. 2. System and Formalism
total(relative) charge a = 1(3) and tctal(relative) spin a = 2(4) of the two bands that cross at the Fermi energy 2 . The longranged Coulomb interaction between electrons couples primarily to the total charge, leading to
a J
x exp
<-ijdt>[H+s(t')
«oi
-iJM'j dx'r){x',t')6\(x',t')
(2)
where SQ denotes the Keldysh action of HeNT- The interfaces between the CNT and the two contacts produce the small backscattering of electrons described by Hbs(t) = u^exp{iA^ijs(t)], where m denotes the two barriers. The phase operator is A^ijs{t) = 6fm - VAst - Vgxm +
We consider a CNT where source and drain contacts are deposited on top defining a cav- Sij)(4>fm + s 4>fm)- The strength of backscat1 ity of length L. Electrons flow from the tering is parametrized by the amplitudes u ^ source contact via the CNT to the drain con- where i,j label the two bands of the CNT. Vds and gate voltage Vg tact due to an applied bias voltage Vas. In The bias voltage _1 addition, a back gate voltage Vg can tune (in units of L ) 5are incorporated as phase the density of electrons in the CNT. The factors in H^s(t) . Finally, s = ± distinelectron-electron interaction is supposed to guishes spin-up and spin-down electrons, rebe strong in the CNT but weak in the higher spectively involved in the scattering event dimensional electron reservoirs. We there- (assumed to be spin-independent and spinconserving). The current operator I(x,t) = fore model the contacts as TLLs with g = 1, —e(2/ir)9i(x,t) couples to the real-valued whereas g < 1 in the CNT. The Hamiltonian source field r](x,t) in Eq. (2). It holds that 5 of such a system is 0l = (0+ +6^)/2 where ± denotes the for4 ward and backward time-path of the Keldysh 1 f HtCNT contour, respectively. a=l
--{dxeaf} 9a
(1)
3. Current Noise
where 6a and na(x) = —(l/Tr)dx
(3)
244 with S(x,t) = (l/2)({SI{x,t),SI(x,0)}) (b = 2 in Eq. (6)) lead to the FP-contribution where {...} denotes the anticommutator, whereas terms with m = m' (b = 1 in Sl(x,t) = I(x,t) - (I) and (...) = T r ^ o - Eq. (6)) contribute to the incoherent part of with po the initial density matrix before Hba the backscattered current. The dimensionis switched on. The noise, to leading order in less effective backscattering amplitudes U\ 1 the backscattering amplitudes u ^ , can also and U2 depend on the amplitudes u ^. In ad5 be written in terms of the Keldysh generat- dition, U2 is modulated by the gate voltage . We also introduced the ideal current in the ing functional denned in Eq. (2) as absence of backscattering, IQ = (4e2//i)Vds 16 c, \ 2 f. / PZ" | . and the dimensionless voltage v = e V d s / ^ i S{x,U) = ~~e JcL, ^. ( a . w ) & ? ( a . w / ) U Due to the non-interacting reservoirs a TLL (4) induced renormalization of the backscattered charge is absent in agreement with recent results on shot noise in TLLs 6 ' 8 . However, 4. Low-Frequency Noise the voltage dependence of shot noise or Fano We evaluate Eq. (4) for low frequencies w factor F = S/2eI with I = I0 + IB shows and to leading order in the backscattering. a pronounced TLL effect. We calculate the At low temperatures or equivalently for high large voltage behavior IB OC |Vd s | 1+a with bias voltage we obtain the shot noise a = -(1/2)(1 - g)/(l + g) up to small F P S = 2e\IB\, (5) oscillations coming from the C/2-term. F shows a similar behavior with 1 + a —> a. which is independent of the measurement point x. IB is the backscattered current 4.1. Theory vs 6=1,2
x / dre
Clb(r)
sin[R 16 (T)/2] sin(wr).(6)
In Eq. (6) we introduced the retarded functions R
W
( T ) = JJJ^T)
+
^mm'(T)
where I and F refer to interacting and free, respectively. The interacting (free) functions describe the propagation of the total charge (free modes) between the two barriers. They are defined as, e.g. for the charge mode, R1mm,{r) = -i&{r)({elm{r),9lm>{Q)})The corresponding correlation function is < w t o = <{0imM,<W(O)}>/2 and C w ( r ) = Clm,(r) + 3C%m,(T). This states that every backscattering event involves the backscattering of four effective modes of the CNT where one is interacting and three are non-interacting. These correlation functions have been given in Ref.5 for zero temperature. The generalization to finite temperatures will be discussed elsewhere7. The terms with m ^ m'
Experiment
We compare the theoretical predictions with shot-noise experiments on a three-terminal geometry consisting of source, drain and back gate at 4 K. The shot noise measurements were performed by placing the CNT device and a weakly coupled light emitting diode (LED) and photodiode (PD) pair, serving as a standard full shot-noise source, in parallel. The input-referred voltage noise of the circuit was approximately 2.2 nV/x/Hz at 4 K with resonant frequency ~15 MHz. Experimentally, the Fano factor F was obtained by F{I) =
SCNT{I)/SPD{I)
for various dc current values J. In Fig. 1 we show a log-log plot of the Fano factor measured for a CNT of length L ~ 360 nm as a function of bias voltage Vds at a particular gate voltage Vg = —7.9 V. The average value of the power exponent in F for this sample over many different gate voltages is ~ —0.34, and the inferred g value is ~ 0.19, which is close to theoretical predictions 2 . In
Fig. 1. Experimental F (diamonds) is compared with theory using Eq. (6) for g = 0.25 and T = 0 K (dotted line). The full line is a fit to the experiment that follows the power-law F oc V£s with a ~ —0.35. From this we extract a TLL-parameter of g ~ 0.18. The highest voltage is V^s = 40 mV. For comparison we plotted the theoretical curve for g = 1 (triangles) which has a = 0 (full line). We used U\ = 0.14 and XJl = 0.10 for the theoretical curves.
Fig. 2. 3D plot of the excess noise Se in units of fio>x,2Go at T = 4 K and g = 0.23 as a function of dimensionless voltage v and frequency ui (in units of U>L) for a measurement point at one of the two barriers. We used U\ = U2 = 0.12.
sponse of shot noise would therefore allow us to determine g without fitting to a power-law and without referring to any system parameters. In addition, the clear distinction of the two velocities of collective modes brings out valuable information about spin charge separation.
Fig. 1 we also clearly see that FP-oscillation become damped at high bias voltage and that the oscillation frequency is close to the theoretical curve (dotted line) ~ vp/L with vp = 8 x 10 5 m/s. We also analyzed SPD and SCNT Acknowledgments separately 9 and found average power-law be- We thank the ARO-MURI grant DAAD19haviors with a ~ 0 for SPD and a ~ —0.31 99-1-0215 for financial support. for SCNT leading to g ~ 0.25.
References 5. High-Frequency Noise In Fig. 2 we present the excess noise Se{V,co) = S(V,u) - 5(0, w). We calculated Eq. (4) for general frequency w, temperature T and position of measurement x7. In addition to the ordinary FP-oscillations induced by the interference of backscattered collective modes at the two barriers we find much stronger oscillations in Se as a function of frequency at high bias voltage and with time period 2g/cji. This is due to momentumconserving reflections of partial charges due to the inhomogeneity of g when traversing the CNT-contact interfaces 10 . Comparing FP-oscillations in the bias response of Se (or F) dominated by U>L with the frequency re-
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246 T R A N S P O R T PROPERTIES IN LOW D I M E N S I O N A L ARTIFICIAL LATTICE OF GOLD NANO-PARTICLES
S. S A I T O * , T . A R M , H. F U K U D A , D. H I S A M O T O , S. K I M U R A , a n d T . O N A I Central Research
Laboratory, Hitachi, Ltd., Kokubunji, Tokyo 185-8601, *E-mail: [email protected]
Japan.
Transport properties were examined in monolayer and sub-monolayer films composed of Au nanoparticles covered with organic thiol ligands. In the monolayer film, the observed linear conductance scaled exponentially with the ligand chain length. On the other hand, in the sub-monolayer film, nonlinear current-voltage characteristics were observed as expected for one-dimensional meandering paths induced by structural defects. Keywords: nanoparticle; artificial lattice; low dimensionality.
1. Introduction
2D monolayer films of Au nanoparticles using the Langmuir-Blodgett method. The linStrongly correlated electron systems ear I-V characteristics were observed for four are potential candidates for future nanoorders of magnitude, and the sheet conducelectronic devices. The basic prerequisite for tance was limited by the tunneling between the application is to control inherent parameadjacent nanoparticles. By contrast, in a ters, such as, transfer (t) and interaction (U) sub-monolayer film, prepared by introducing energies in the Hubbard type model, carrier structural defects, we observed nonlinear I-V density (n), and dimensionality (D) 1. characteristics. We discuss the origin of this However, these parameters are difficult nonlinearity in the context of a Tomonagato control in d-band materials like cuprates Luttinger liquid 5 in ID percolation paths. or manganites, because a gate control of high n (~ 1015 c m - 2 ) or a high pressure (~ 1 GPa) is required for the Mott transition. By 2. Experiments contrast, energy scales and lattice structure A Si substrate with 200-nm-thick Si02 was may be significantly controlled in an artificial prepared using thermal oxidation at 1000 °C. lattice, where lattice sites are quantum dots We avoided using S13N4 dielectrics, which instead of single atoms or molecules. may well have a quenched charge disorder Arrays of nanoparticles are considered to that induces threshold behaviors in I-V charbelong to artificial strongly correlated sys- acteristics. In fact, according to our estitems 2 . A breakthrough in chemical synthe- mate in a state-of-the-art Si process line, sis 3 enables the mass production of nanopar- SisN4 film typically contains positive fixed ticles, whose size (< 10 nm) is enough small charges with the density of ne x > 5 x 1012 to observe single electron charging at room c m - 2 , which roughly corresponds to one temperature. However, the basic transport fixed charge per every Au nanoparticle. In mechanisms of monolayer films are not com- contrast, nfix of Si02 is more than two orders pletely understood. of magnitude less, so the quenched charge In this paper, we examined the impacts density is estimated to be less than 1% of of low dimensionality and many-body effects the total nanoparticle density. on transport properties in an artificial latAu electrodes were patterned using contice of Au nanoparticles. We prepared the ventional lithography and lift-off. The ac:
247 tive channel length, L, separated by adjacent Au electrodes, was varied from 5 to 100 /tm, and the width, W, of each electrode was 200 //m. After the formation of electrodes, the monolayer film of An nanoparticles (Figs. 1 and 2) was prepared on the substrate using Langmuir-Blodgctt (LB) techniques 2 . To examine the impact of structural defects, we also prepared the sub-monolayer LB film by reducing the pressure during the film formation in the trough. In this way, we intentionally introduced voids in a controlled manner.
depended linearly on L using two-terminal measurements and that the contact resistance was small using four-terminal measurements. The Ohm's law was also confirmed for all the monolayer films of nanoparticles with other ligands. Then, we obtained the sheet conductance, G, which decreases exponentially with I, as shown in Fig. 4. This tendency and the magnitude of G in our 2D films are consistent with previous reports on 3D films, prepared by casting droplets and evaporating solvent.
3. Results and Discussion
3.2. Estimation energy
film
Figure 3 shows the I-V characteristics of the monolayer film of Au nanoparticles with C4 thiols, where the linear I-V characteristics were maintained at least for four orders of magnitude at temperatures (T) from 300 K to 50 K. We found that the resistivity
transfer
To understand the dependence of G on I, we analytically calculated the transfer en• I I [ Mill E - .WOK I "-250K
I I Mill
I I Mill
I I IIJJ ^JL^0 •
F-2Q0K IMiK r- IOOK :-*>K SIO
-
T
u
3 . 1 . Monolayer
of
r
^&
^*^^
UMUitn
Organic l i g a n d
10
II)
II)
Vollagc(V)
Fig. 3. Linearity of I-V characteristics for 2D monolayer film of Au nanoparticles with C4 ligands.
Fig. 1. Au nanoparticle used as lattice site. Organic chain length, I, determines lattice constant, d ~ 21.
— 10nm Fig. 2. Transmission electron microscope image of self-organized Au nanoparticles covered with C12 thiols.
0 0.5 1 1.5 2 2.5 Length (nm) Fig. 4. Observed sheet conductance (G) exponentially decreased with the ligand chain length. G is proportional t o the calculated transfer energies (t) for both electrons and holes.
248 ergy for electrons (te) and holes (th) using an instanton method. Note that by electron (hole), we mean one extra electron added to (removed from) an Au nanoparticle. Although conduction-band electrons with the number comparable to that of Au atoms are present in each nanoparticle, the state with (without) a single extra charge can be clearly distinguished due to single electron charging, as confirmed by voltammetry measurements. We assumed a simple double-well potential, V(x) = m0uj2(x2 - l2)2/(8l2), for the tunneling along the ligand chain direction, x, where mo is the effective mass and w is the attempt frequency. We used mo = 0.25me with the bare electron mass, m e . We obtained w = 2y/2AE/mo/l, where AE is barrier height. We estimated AE as AEe =
The activation energy, Eact = 9.7 meV, is much smaller than EQ. This means that the chemical potential, /i, in the artificial lattice, is moved from EF. In such cases, minority carriers are negligible compared to majority ones, because their ratio is estimated as exp(—2|/u — EF\/(kBT)) ~ exp(-2(£b - E&ct)/(kBT)) ~ 1 0 ' 1 6 < 1 at T = 300 K, where fee is the Boltzmann constant. Now, we consider a Hamiltonian, H, to describe arrays of Au nanoparticles. Double occupancy is strictly forbidden, because the interaction energy is estimated as U = 4E0 ~ 1.93 eV » te ~ 2.81 meV for the nanoparticles with C4 ligands, so the system is in the strong coupling limit. The exchange interaction energy, J ~ t2/U ~ 4.10 /xeV, •ELUMO — {EF + Eo) = 4.25 eV for electrons is not relevant in the temperature range we and as AEh = {EF - E0) - £HOMO = 3.65 considered. Therefore, the system can be deeV for holes, where .EHOMO = —9.20 eV scribed by a spinless fermion, whose creation (-ELUMO = —0.334 eV) is the energy level (annihilation) operator, c\ (CJ), describes eiof the highest occupied (lowest unoccupied) ther an electron or a hole at the i-th lattice molecular orbital of C4, Ep = —£AU — site. Assuming the Fermi anticommutation —5.07 eV is the Fermi level of Au estimated relations, the double 2occupancy is automatically forbidden : (c?) = 0. The relevant enfrom the work function of Au (111) -EAU, and ergy competed with t is the long-range part of the Coulomb repulsion, V(\*i — Xj|), between charges located at x, and Xj. Finally, = 0.482 eV (2) we obtain as is the single electron charging energy. Then, by using the bounce solution, XB(T) = itanh(wr), along the imaginary time, r, we obtain the bounce action SB = / ^ d T { r n o ± B ( T ) 7 2 + V(a:B(T))} = 5m0u)l2/6. Finally, the transfer energy is estimated as f = hu)exp(—S&/K) = hujexp(—l/X), where A is the penetration depth : A = m/b/^2m0AE ~ 1 A. The calculated dependence of te and th on I agrees reasonably well with the observed G, which means that G is limited by the tunneling between adjacent nanopartides. The temperature dependence of G obtained from Fig. 3 exhibits an activation behavior from 300 K down to 100 K.
H=-t^2
(C\CJ + h.c.) - fi^2m (ij)
i
where () means the sum is restricted to neighboring sites, h.c. stands for the Hermitian conjugate, and m is the number operator. The carrier hoppings would be significantly limited once the structural defects are introduced, since they corresponds to the local absence of the lattice sites. Although structural defects due to disorders are not explicitly written in H, they are implicitly considered by limiting the sum to the available lattice site. We estimated V(|x» — Xj|) from
249 the repulsion energy between charges embedded in a dielectric medium of permittivity K as e2/(47r/teo)/|xj — Xj|, where eo is the vacuum dielectric constant. Although H is simple, the properties of the system include highly nontrivial results in low-dimensional systems, as shown below.
0 0-
.0 100 200 300 400 500
3.3. Sub-monolayer
film
Next, we examine the sub-monolayer film of Au nanoparticles with C4 ligands, whose nonlinear I-V characteristics are shown in Fig. 5. We attributed this nonlinearity to the lower-dimensional characteristics caused by the formation of I D meandering paths, as schematically shown in Fig. 6. In fact, the atomic force microscope (AFM) image in Fig. 7 shows that the coverage of the film was 69 %, and significant amounts of voids can be observed. These voids limited the conduction paths, and lower dimensional characteristics are expected.
Position (nm) Fig. 7- Atomic force microscope image of submonolayer film.
for ID Tomonaga-Luttinger liquid system connected with Fermi liquid electrodes. In fact, we actually confirmed the power law behaviors of the differential conductance with the same a = 2.5 for both V and T. We also confirmed the universal scaling 5 behaviors of (dJ/dV)/Ta against eV/kBT. 4. Conclusion
Then, we compared our data with the In summary, we observed that the linear contheoretical predictions by Kane and Fisher 8 ductance scales exponentially with the ligand chain length in the monolayer film. In the sub-monolayer film, we observed non-linear current-voltage characteristics, which can be explained by the realization of TomonagaLuttinger liquids in a one dimensional meandering path. However, we cannot rule out alternative scenarios, such as Coulomb blockade transport, and further works are necessary to completely understand the transport Voliage (V) mechanism in the nano-particle arrays. Fig. 5. Nonlinear I-V characteristics of submonolayer films of Obligated Au nanoparticles.
References
Fig. 6. I D meandering paths induced by structural defects.
1. M. Imada et al., Rev. Mod. Phys. 70, 1039 (1998). 2. C. P. Collier et al., Science 277, 1978 (1997). 3. M. Brust et al., Chem. Commun. , 801 (1994). 4. T. Arai et al., Jpn. J. Appl. Phys. , (to be published). 5. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992).
250 FIRST P R I N C I P L E S S T U D Y OF D I H Y D R I D E - C H A I N S T R U C T U R E S O N H - T E R M I N A T E D Si(100) SURFACE
Y U J I SUWA, M. F U J I M O R I , S. H E I K E , Y. T E R A D A a n d T . H A S H I Z U M E Advanced
Res. Lab., Hitachi,
Ltd., Hatoyama,
Saitama
350-0395,
Japan
On a H-terminated Si(100)-2xl surface, a dihydride-chain structure parallel to the step edge is often found by Scanning Tunneling Microscopy. The chain is located near an S B step edge, but away from it by more than one Si-dimer's distance. We performed first principles calculations to determine the mechanism of the chain formation. As a result, we found that the rebonded step edge of the clean Si surface turns into a combination of a dihydride chain and a non-rebonded step edge after hydrogen termination. We also found that the chain adjacent to the step edge is energetically less stable than that at one Si-dimer distance away from the step. Keywords: dihydride, hydrogen termination, rebonded structure, non-rebonded structure, scanning tunneling microscopy, first principles calculation
Self-assemblage/self-organization is a key to realize nanodevices through a bottomup approach. When the size of the circuit is smaller than the limit of photolithography technique, one has to find an alternative method to place each part as designed. Although the atom manipulation by scanning tunneling microscope (STM) 1,2 ' 3 ' 4 provides possibilities for nano-device realization, speed limitation for atom manipulation will be a problem in practical use. For mass production, utilization of self-organization phenomena is necessary. Fujimori et al.5 showed that there is a dihydride chain in each terrace on a hydrogenterminated Si(100)-2xl surface (see Fig. 1). These are on the terraces below the S B step edges and parallel to those edges. These chains are located at a few Si-dimer's distance away from the SB step edges. These self-assembled atomic-scale chains have a possibility of applications in nano-scale devices. When the fabrication of a semiconductor chain or a metal chain based on this structure becomes possible, one can use it as a wiring of nano-scale circuits and nanodevices. In order to explore such a possibility, we investigate how theses structures are created and what kind of experimental con-
ditions are important in creating these structures. To obtain these structures, samples were prepared as follows: First, a standard in situ cleaning process was performed in the chamber. The cleaning consisted of degassing of the sample at 1070 K for 2 h, followed by flashing the sample several times for 10 s. The temperature was increased up to 1470 K in the final flashing process. Next, hydrogen termination was performed by supplying atomic hydrogen to the clean surface of the sample. During this process (1-10 min), the temperature of the sample was kept constant between 550 K and 610 K. Finally, sample heating and atomic hydrogen supply were terminated simultaneously and the surface structure were quenched. ,' When hydrogen termination process was carried out below 520 K, many dihydride chains were produced to form striped patterns with monohydride Si-dimer rows, i.e., the 3 x 1 phase, was formed. On the other hand, when the temperature was above 610 K during the process, no dihydride chains were observed on the surface. Several questions arise from these experimental results; (1) Why only one dihydride chain exists in each terrace below the S B
251
Fig. 2, Atomic structures of the S B step, (a) Rebonded structures of the clean surface, (b) Hterminated rebonded structures, (c) H-terminated non-rebonded structures and an adjacent dihydride chain, (d) H-terminated non-rebonded structures and a distant dihydride chain.
eralized gradient approximation (GGA) 6 with plane-wave-based ultrasoft pseudopotentials 7,8 . The energy cutoff was set at 12.25 Rydberg. The convergence criterion of the geometry optimization was as follows: All forces acting on atoms were less than 1 x 10~ 3 Hartree/a.u. Slab models of five step? (2) Why most of the chains are located (partly six) silicon atomic layers were used near the S B step edges but not adjacent to for the calculation of H-terminated Si(100) them? (3) Why the temperature of the sam- surfaces. Based on our calculation, we explain the ple preparation should be between 520 K and 610 K to produce dihydride chain structure? formation of the dihydride chain structure: In order to clarify these points, we performed Figures 2(a)-2(d) show atomic structures of first principles calculations of H-terminated the SB step edges during the sample preparation process. First, in the cleaning process of Si(100) surfaces and S B step structures. Calculations in this paper were based the Si surface, the SB step edge on a Si(100) on density functional theory applying gen- clean surface takes a "rebonded" structure as Fig. 1. Filled state STM images of dihydride chains on Si(100)-2xl-H surface, obtained at room temperature with a sample bias voltage of —2.0V and a constant tunneling current of 10 pA. Dihydride chains are (a) marked by arrows and (b) accentuated by colored lines.
252 shown in Fig. 2(a). Oshiyama 9 showed that this structure is more stable than the other, "non-rebonded", structure for a Si(100) surface without hydrogen. While the SB step edges form rebonded structures, the terraces below the SB step edges take the width of an even number of Si sites, so that the terraces are filled with Si-dimer rows parallel to the steps without unstable Si-monomer rows. Re-formation of the width of the terrace is possible because the surface migration of Si atoms is possible at the sample temperature of the cleaning process. Figure 2(b) shows the H-terminated rebonded structures of the SB step edges. Here all the dangling bonds of Si atoms in Fig. 2(a) are H-terminated and buckling Sidimer structures are relaxed. We expect that rebonded SB steps remain until all the dangling bonds in terrace are hydrogenated to form this structure because the termination of the dangling bonds is energetically favorable. Figure 2(c) shows the H-terminated nonrebonded structures of the SB step edges and adjacent dihydride Si-monomer chain. These structures are formed by acquiring two more hydrogen atoms at each step-edge structure of Fig. 2(b). Total energy for each atomic row perpendicular to the non-rebonded step edge is 0.45 eV lower than that of the rebonded step edge (Fig. 2(b)) with a hydrogen molecule. Therefore, at low temperatures, the non-rebonded structure is more stable than the rebonded structure for Hterminated surface, in contrast to that for clean surface. The energy gain of the nonrebonded structure comes from the relaxation of the deformation energy of the rebonded structure by decoupling Si-Si bonding at the edge. In order to discuss relative stability of structures with different numbers of hydrogen atoms under the condition of finite temperature and atomic hydrogen gas pressure,
one must compare free energies by introducing chemical potential. Higher temperature makes the chemical potential lower and stability of the non-rebonded structure weaker. Above a certain critical temperature, the rebonded structure becomes more stable. We do not calculate the critical temperature here; however, this explains the experimental results that hydrogen termination process above 610 K produces no dihydride chains (Fig. 2(b)). Jeong and Oshiyama 10 discussed relative stability of the rebonded and the non-rebonded structures on H-terminated surfaces by treating the chemical potential of hydrogen atoms as a parameter. Although they have not relaxed the geometry of the terrace below the step, their result is qualitatively consistent with ours. Figure 2(d) shows the H-terminated nonrebonded structures of the SB step edges and a dihydride chain with a monohydride Sidimer row in between them. Total energy of this structure is 0.09 eV lower than that of Fig. 2(c) for each non-rebonded structure. This means that the structure in Fig. 2(c) still has deformation energy due to a repulsion between too close hydrogen atoms at the edge, and it is relaxed when the dihydride chain is not adjacent to the step edges. We also calculated the structure with two Sidimer rows between the step edges and dihydride chain, and found that its total energy is the same as that of Fig. 2(d) within the accuracy of the calculation. We concluded that once a dihydride chain is formed, its original position adjacent to the step edge is no longer stable, and the dihydride chain takes at least one Si-dimer row distance from the step edge. Figure 3 shows simulated filled-state STM images of the SB step structures of the H-terminated surface. Figures 3(a)-3(c) correspond to the structure shown in Figs. 2(b)2(d), respectively. By comparing these STM images with experimental images (Fig. 1),
253
(a)
step structures at H-termination. Because the dihydride chain at its original position adjacent to the step edge is not stable energetically, it exchanges its position with that of the neighboring Si-dimer row.
rebonded
i
Acknowledgments
(b) non-rebonded dihydride
This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government. References
(c) non-rebonded dihydride Ml]
ill
Fig. 8. Simulated filled-state STM images of Hterminated S B steps for (a) rebonded structures, (b) non-rebonded structures and a dihydride chain adjacent to it, and (c) non-rebonded structures and a dihydride chain with a Si-dimer row between, (a-c) correspond to Figs. 2(b-d), respectively.
we can determine which structures are experimentally observed. For this purpose, we must look at positions of nionohydride Sidimer rows adjacent to the SB step. Since those in Fig. 1 are located very close to the step edge, they correspond to the structure in Fig. 2(d) and the image in Fig. 3(c). In summary, we have clarified the origin of the existence of a dihydride-chain structure parallel to the SB step edge. The rebonded step structures stable on a clean surface are transformed into the combination of the dihydride chain and the non-rebonded
1. D. M. Eigler and E. K. Schweizer, Nature 344, 524 (1990). 2. T. C. Shen, C. Wang, G. C. Abeln, J. R. Tucker, J. W. Lyding, P. Avouris, and R. E. Walkup, Science 268, 1590 (1995). 3. T. Hashizume, S. Heike, M. I. Lutwyche, S. Watanabe, K. Nakajima, T. Nishi, and Y. Wada, Jpn. J. Appl. Phys. 35, L1085 (1996). 4. T. Hitosugi, S. Heike, T. Onogi, T. Hashizume, S. Watanabe, Z.-Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and K. Kitazawa, Phys. Rev. Lett. 82, 4034 (1999). 5. M. Fujimori, S. Heike, Y. Suwa, and T. Hashizume, Jpn. J. Appl. Phys. 42, L1387 (2003). 6. J.P. Perdew, K. Brake and A. Wang, Phys. Rev. B 54, 16533(1996). 7. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 8. K. Laasonen, A. Pasquarelio, R. Car, C. Lee and D. Vanderbilt, Phys. Rev. B 47, 10142 (1993). 9. A. Oshiyama, Phys. Rev. Lett. 74, 130 (1995). 10. S. Jeong and A. Oshiyama, Phys. Rev. Lett. 81, 5366 (1998). 11. J. E. Northrap, Phys. Rev. B 44, 1419 (1991).
254 ELECTRICAL P R O P E R T Y OF AG N A N O W I R E S FABRICATED O N H Y D R O G E N - T E R M I N A T E D SI(IOO) SURFACE
M A S A A K I F U J I M O R I 1 ' * , S. H E I K E 1 , A N D T . H A S H I Z U M E 1 ' 2 Advanced Research Department of Physics,
Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan Tokyo Institute of Technology, Oh-okayama, Tokyo 152-8551, Japan *E-mail: [email protected]
Current-voltage characteristics were measured of atomic-scale Ag wires on the hydrogenated surface of Si(100) using four-probe fine electrodes fabricated with scanning-probe nanofabrication technique. Carrier transport was observed even at the initial stage of Ag deposition and conductivity increased as a function of supplied Ag atoms. Ballistic and diffusive transport are discussed as possible mechanisms of carrier transport. Keywords: scanning probe nanofabrication; nanowire; conductivity measurement.
1. Introduction An atomic wire can be regarded as an "ultimate" small wire. Electrical conduction through such a wire is an attractive phenomenon, because we can expect manifestation of quantum phenomena such as quantized conduction, or even emergence of unknown phenomena. One of the milestones in a series of experiments concerning this issue was made by Takayanagi and his colleagues [1]. They reported clear quantum conductance in an isolated gold wire between gold electrodes. In addition to isolated atomic-scale wires, atomic-scale wires fabricated on a substrate surface are attractive objects. Because a bulk substrate causes significant changes on electronic states of the atomic-scale wires on the substrate due to coupling of electronics states. Further, we can arrange wire structures. We can control electrical properties of the wire, even constructing artificial structures. Several types of atomic-scale wires have been fabricated on substrate surfaces. Lyding and colleagues [2] demonstrated atomicscale dangling-bond wires on a hydrogenterminated surface of Si(100), followed by Ga wires [3], C 6u wires [4], Ag wires [5], and P
wires [6]. Although interconnects for isolable nanowires can be made by placing the nanowires on the pre-fabricated fine electrodes, the interconnects for the surface-modified nanowires must be made on the same surface of the substrate on which the nanowires are made. In that case, the surface should be maintained atomically clean to manipulate atoms or molecules. This is usually accompanied by serious difficulty because clean surface is damaged by conventional lithography technique. Therefore, only few groups have succeeded in preparing electrodes on the clean surface [7,8]. In order to measure the electrical properties of the surface-modified nanowires, we have fabricated four-probe fine electrodes on a Si(100) surface. In this paper, we report result of current-voltage measurement for an atomic-scale Ag wire fabricated on the hydrogenated surface. 2. Experiment We used a commercial Si(100) wafer (Pdoped, n-type, resistivity of 0.5 to 1 fi-cm) as the substrate. We applied a conventional photolithography for fabricating large part of electrodes and used a scanning-probe (SP)
255 nanofabrication technique [9] for making fine structure of the electrodes. Thin films of SiN, W and Ti with thicknesses of 20, 10 and 5 nm, respectively, were first sputter-deposited on the substrate. By photolithography, patterns for 1mm x lmm contact pads and 20/xm-wide intermediate electrodes were transferred to a resist film. A set of patterns for four-probe fine electrodes were then drawn on the resist film with the SP nanofabrication technique. For process details, see reference [10].
averaging 1024 measured values. When a bias voltage was applied to the electrodes, a leak current flowed through paths except the atomic-scale Ag wires. To avoid influence of the leak current, we measured the leak current before fabricating the atomic-scale Ag wires and subtracted it from the current obtained after fabricating the atomic-scale Ag wires.
The substrate was placed in a ultra high vacuum (UHV) chamber. Then, high temperature heating for surface cleaning and atomic-hydrogen exposure for hydrogen termination of a surface [3] were carried out. We confirmed by scanning tunneling microscope (STM) observation that an atomically flat Si(100)-2xl-H surface was obtained. All STM images in this report were taken at room temperature with a sample bias voltage of —2.0 V and a constant tunneling current of 10 pA.
The fabricated fine electrodes were made of grains (Fig. 1), formed during the hightemperature cleaning process. We observed the metallic feature of the fine electrodes by STM. The center region of the four-probe fine electrodes showed an atomically flat terrace, where an atomic-scale Ag wires were formed. An example of the four-probe interconnects is shown in Fig. 2 (a). We observed that the Ag clusters were densely formed on the interconnects by selective adsorption of Ag atoms on the dangling-bond wire. The larger clusters of Ag atoms were observed at the side of the interconnects. Because the larger clusters were sparsely distributed and spatially separated, they were electrically separated too. The STM data showed that step-bunched surfaces near the fine electrodes were also terminated with hydrogen atoms. Figure 2(b) shows a formed dangling-bond wire (Fig. 2 (b)).
For the four-probe interconnects between the pre-fabricated fine electrodes and atomic-scale Ag wires, we fabricated 10 to 50-nm-wide dangling-bond wires. Selective adsorption of Ag atoms, supplied from a tungsten coil-heater surrounded by a thin layer of Ag, was used to metalize the dangling-bond wire [5]. Finally, atomicscale Ag wires were fabricated in the similar way starting with one to two atomic-size dangling-bond wires connecting through the four-probe interconnects. The amount of the deposited Ag atoms on the atomic-scale Ag wires was controlled through the evaporation time. When the sample holder was set on the STM stage, electrical contact with the 10mm x 10mm pads on the sample was realized with four Ta wires. All the measurements were performed by the four-probe method in UHV at room temperature. All data at sampling points were obtained by
3. Results and discussion
Fig. 1. STM image of the four-probe fine electrodes. The inset shows a cross-sectional profile along the dotted line.
256 An example of an atomic-scale Ag wire is shown in Fig. 3. We measured electrical conductance of atomic-scale Ag wires as a function of a number of deposited Ag atoms per dangling bond estimated from deposition conditions. No electrical conduction was observed for a dangling bond wire before Agatom deposition. The conductance was obviously changed after Ag atom deposition, even at the initial stage of the deposition. Thus we concluded that we properly detected electrical conduction through the atomic-scale Ag wires. Conductance in unit of quantum conductance, Go = 2e2/h, is summarized in Fig. 4. Quantized conduction is expected for an atomic-scale metallic wire but clear steps
Fig. 2. (a) STM image of four-probe interconnect running from the upper left to the lower right. White protrusions are Ag clusters. Small Ag clusters are densely located on the interconnect and larger clusters are sparsely distributed near the interconnect, (b) A dangling-bond wire formed on a step-bunched surface.
were not observed. We, however, obtained a weak smeared-out step-like feature. Under the ballistic conduction condition, the steplike feature of quantized steps should appear. Therefore, we conjecture that the broadening is probably due to the influence of temperature or the bulk substrate. Since our atomic-scale Ag wires consisted of aggregation of Ag clusters, strong scattering of the carriers is expected at cluster boundaries and rough surfaces of the atomic-scale Ag wires. In this way, ballistic conduction is prevented. The STM observation showed that the atomic-scale Ag wire developed in three steps as Ag was deposited : (I) nucleation and growth of clusters, (II) growth and coalescence of the clusters, (III) formation of cluster boundaries. Thus the atomic-scale Ag wire consisted of a chain of atomic-scale Ag clusters. At the step (II), the clusters began to touch with the neighboring clusters. Cluster boundaries were densely formed along the atomic-scale Ag wire. Since spacing of neighboring boundaries was less than a typical mean free path of carriers in a metal (~10 nm), the carriers passing through the atomic-scale Ag wire were scattered and flowed diffusively. In addition, the surface of the atomic-scale Ag wire acts as scattering site of the carriers. Therefore, a ballistic conduction in the atomic-scale Ag wire was
5
Fig. 3. Example of an atomic-scale Ag wire. The Ag wire runs from the upper left to the lower right shown by arrows. White protrusions surrounding the Ag nanowire are isolated Ag-atom clusters.
i — i — i — i — i — i — i — i — i
Fig. 4. Conductance of an atomic-scale Ag wire as a function of the estimated number of deposited Ag atoms per dangling bond. Conductance is normalized by unit of 2e2/h.
257 suppressed and the diffusive conduction appeared. We observed the electrical conduction even at the initial stage of the Ag deposition. The Ag clusters on the dangling-bond wire are not in touch with each other at this stage. This reveals that the carriers were supplied from Ag atoms to electronic states of the dangling-bond wire. According to the result by Henzler and his colleague [11], electronic states of a Ag thin film changes from localized to metallic state at the border of the film thickness being 2 monolayers. We conjecture that the bent in the conductance curve observed in our experiments at the number of deposited Ag atom being 2.5 is originated from a metallic transition in the atomic-scale wire. Since the number of deposited Ag atom means a deposited number of Ag atoms per dangling bond, the value of 2.5 means virtually 2 monolayers. If diffusive metallic conduction assumed, the observed conductance should be smaller than the value obtained by setting typical values of the atomic-scale Ag wire size (width w, height h and length / are 5, 1 and 100 nm, respectively) in the simple estimation, g = awh/l, where g and a are conductance and the conductivity of bulk Ag, respectively. This can be explained by the weak localization of carriers due to strong scattering by surfaces and cluster boundaries of the atomic-scale wires and low dimensionality. Indeed, similar localization was observed in a thin Ag film [11]. Further investigation such as temperature dependence measurements will be indispensable to clarify detailed mechanisms of observed conduction behavior.
4. Summary Electrical conduction of an atomic-scale Ag wire was measured by using the four-probe method. The conductance showed the following features: the conduction appears at
the initial stage of Ag deposition, and the bent in the conductance curve. Possible mechanisms of the carrier transport characteristics are diffusive conduction with strong scattering at cluster boundaries and surfaces of the atomic-scale Ag wire. Acknowledgment s This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government. References 1. H. Ohnishi, Y. Kondo, K. Takayanagi, Nature 395, 1998 (780). 2. J. W. Lyding, T. C. Shen, J. S. Hubacek, J. R. Tucker and G. C. Abeln, Appl. Phys. Lett. 64, 2010 (1994). 3. T. Hashizume, S. Heike, M. I. Lutwyche, S. Watanabe, K. Nakajima, T. Nishi and Y. Wada, Jpn. J. Appl. Phys. 35, L1085 (1996). 4. A. W. Dunn, P. H. Beton and P. Moriarty, J. Vac. Sci. Technol. B14, 1596 (1996). 5. M. Sakurai, C. Thirstrup and M. Aono, Phys. Rev. B62, 16167 (2000). 6. S. R. Schofield, N. J. Curson, M. Y. Simmons, F. J. Ruess, T. Hallam, L. Oberbeck and R. G. Clark, comd-mat/0307599. 7. O. V. Hul'ko, R. Boukherroub, and G. P. Lopinski, J. Appl. Phys. 90, 1655 (2001). 8. F. J. Ruess, L. Oberbeck, M. Y. Simmons, K. E. J. Goh, A. R. Hamilton, T. Hallam, S. R. Schofield, N. J. Curson, and R. G. Clark, Nano Lett. 4, 1969 (2004). 9. M. Ishibashi, S. Heike, H. Kajiyama, Y. Wada and T. Hashizume, Appl. Phys. Lett. 72, 1581 (1998). 10. M. Fujimori, S. Heike, Y. Terada and T. Hashizume, Nanotechnology 15, S333 (2004). 11. R. Schad, S. Heun, T. Heidenblut and M. Henzler, Phys. Rev. B45, 11430 (1992).
258
EFFECT OF ENVIRONMENT ON IONIZATION OF EXCITED ATOMS EMBEDDED IN A SOLID-STATE CAVITY *M. ANDO, 2C. LEE, and 3 S. SAITO 'ARL, 2HM„ and3CRL, Hitachi, Ltd., 'Omika 7-1-1, Hitachi, Ibaraki 319-1292, Japan *E-mail: anmasa@rd. hitachi. co.jp 4
Y. A. ONO
4
School of Engineering, Univ. Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan E-mail: yaono@t-adm. t. n-tokyo. ac.jp
By increasing the lifetime of the excited atoms embedded in a solid-state cavity, the field-ionization rate was enhanced approximately three times over that without the cavity effect. The characteristic temperature dependence of the enhanced field-ionization rate suggested the generation process was dissipative quantum tunneling induced by the strong electron-phonon coupling between localized electrons in the excited atoms and the surrounding lattice vibrations of the host in the cavity. Keywords: Cavity QED; Electroluminescence; Dissipative quantum tunneling.
Introduction Cavity quantum electrodynamics illuminates the most fundamental aspects of coherence and decoherence in quantum mechanics1, and governs quantum efficiency of optical devices such as light emitting diodes, and solar cells2, where two metallic electrodes sandwiching the active layer act as a cavity. In these solid-state cavities, dynamics of excited atoms is affected by environmental factors such as electric field, temperature, and inevitably electron-phonon coupling with the surrounding atoms as well as the electron-photon coupling, i.e. cavity effect. This paper describes the effect of environment on the ionization process of a photo-excited atoms embedded in a solid-state cavity. The observation is made possible by inhibiting the visible spontaneous emission of the excited atoms. This phenomena was successfully observed using thin-film electroluminescent (TFEL) devices with a lightemitting active layer composed of a CaS:Eu2+ thin film3 (Fig.l). The TFEL device was developed for emissive flat-panel displays and CaS:Eu2+ is a red-emitting phosphor material showing a broad-band emission with a peak at
around 660 nm. The emission is due to a partially allowed 4f -5d intra-shell transition of divalent Eu2+ ions embedded in an ionic host crystal CaS. We stress that this system is a new class of materials for exploring dissipative quantum tunneling in solids at low temperature where strong electron-phonon coupling is theoretically predicted to enhance or suppress the tunneling probability4,5.
Electron-phonon coupling (dissipation) Fig.l. Solid state cavity of CaS:Eu2+ TFEL device.
259 1. Experimental An 500 nm-thick active layer composed of the rare-earth ion doped ionic crystal CaS:Eu2+ is sandwiched between 500 nm-thick highdielectric Ta 2 0 5 insulating layers and these stacks are further sandwiched between a transparent front electrode (200 nm-thick ITO) and a rear electrode composed of 200 nm-thick ITO or Al with reflectivity of 30% and 98%, respectively. The two insulating layers prevent carrier injection from the electrodes to the active layer and high electric field up to 1 MV/cm can be applied to induce the field ionization. The amount of emitted electrons by ionization was measured by photo-capacitance method6. CaS:Eu2+ has a unique energy configuration, where the ground and the excited states of Eu2+ are situated within the bandgap of CaS, and the 5d excited state is located just below the conduction band edge. Gated spectral hole burning due to this specific energy configuration was reported7. Eu2+ ions are excited by a 488 nm continuous wave Ar laser with an output power of 5mW/cm2. The excitation wavelength corresponds to the peak wavelength of the direct excitation spectrum of Eu2+ in CaS. The sample size is 2 mm2 and the photoluminescent (PL) emission with the peak wavelength at 660 nm is guided to a photo-
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.
-—^
,
— 10- excitatiw! emission.* ^-y
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OLV
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Applied Voltage (V) Fig.3. Field-modulate PL and photoionization.
multiplier (PMT) tube using an optical-flexible fiber bundle of 2 mm in diameter. A red color filter, which only transmits light with wavelengths longer than 600 ran, is inserted between the optical fiber and the PMT in order to detect only the PL emission from CaS:Eu2+. The applied pulsed voltage used for the modulation has 50 ms duration with rise/fall times of 1 ms of alternating polarity at 100 Hz. 2. Results and Discussion
Wavelength (nm) Fig.2. Photo-emission, excitation, and ionization spectrum.
Figure 2 shows the effect of the rear electrode material on the PL emission and excitation spectra, and the photo-ionization spectra; the latter is defined as the number of photo-ionized Eu2+ ions as a function of the excitation wavelength with an applied voltage at 10V. The sample with ITO transparent rear electrode shows a PL emission spectrum almost identical to that for a CaS:Eu2+ thin film deposited on a glass substrate, peaking at 660 nm and full width at half maximum of 60 nm. This indicates there is no noticeable cavity effect taking place in this case. On the other hand,
260 the PL emission intensity of the sample with an Al reflective rear electrode is partially suppressed at the peak at 660 nm, while two peaks, one at shorter, and the other at longer, wavelengths appear. This modulated PL intensity is confirmed to be the cavity effect; the modulation pattern is well reproduced by multiplying the CaS:Eu2+ emission spectrum data with the modulated intensity calculated based on a simple cavity model and normal incidence.
samples is quantitatively in good agreement with that obtained through the field-induced inhibition of the PL emission.
g80-
As shown in Fig.3(a), with cavity effect (an Al rear electrode), PL intensity of direct excited Eu2+ was reduced to 97% with applied voltage at 120V (IMV/cm), approximately three times larger than that without cavity effect (an ITO rear electrode). The inhibited spontaneous emission of Eu2+ coincides with the same effect observed in nanospheres8. Inhibition of the PL emission is larger in the sample with the cavity effect because the field-ionization rate of the excited Eu2+ ions is larger in this sample, as shown in Fig.3(b), which indicates the voltage dependence of the number of field-ionized Eu2+ ions. In both samples, the number of the field-ionized Eu2+ ions increases with applied voltage. However, the number for the sample without the cavity effect is 1/3 of the value for the sample with the cavity effect. The difference in the number of field-ionized Eu2+ ions between the two
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^ 60-
2+
Eu ions are photo-ionized with the excitation wavelength ranging from 400 to 600 nm and the profile of the photo-ionization spectra corresponds to that of the direct excitation spectrum for the PL. The number of the ionized Eu2+ ions of the sample with the cavity effect (an Al rear electrode) is approximately three times larger than that without the cavity effect (an ITO rear electrode). These results strongly suggest that the photoelectrons are emitted from the directly excited Eu2+ ions and the field ionization of the excited Eu2+ ions is enhanced by the cavity effect.
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100
1
200
1
300
Temperature (K)
1
400
500
Fig.4. Temperature effect on PL and photoionization.
PL emission quenching accompanying Eu2+ ionization begins at -200 K with the activation energy of 160 meV (Fig.4(a)). This activation energy of ionization is larger than the LO phonon energy in CaS (42meV) and prevents complete thermal ionization at room temperature (26meV). Field-ionization rate defined by reduction rate of PL intensity with applied field at 0.IMV/cm increases with temperature, reaches the maximum at -300K, and decreases with higher temperature (Fig.4(b)). In contrast to the usual cavity-atom systems, where atoms are freely placed in the cavity, Eu2+ ions are embedded in a solid-state host composed of ionic-crystal, CaS, in the present system schematically shown in Fig.l. Therefore, a strong electron-phonon coupling exists between the localized electrons in Eu2+ ions and the vibrational mode of the surrounding ions constituting the host lattice.
261 This electron-phonon coupling makes possible energy transfer between the electrons in the localized ions and the surrounding ions during the excitation, the emission, and the ionization processes. These processes accompany the displacement or relaxation of the relevant ions, usually observed in ionic crystals. The strong interaction between the excited electron and the lattice vibration is enhanced with increasing temperature. It should change the fieldionization mechanism and cause the sensitive temperature dependence of the observed fieldionization rate.
inverse process of self-trapping of excitons . Therefore, the ionization is enhanced by an electric field, thermal energy, and the life time of the excited state. Since the spontaneous emission is a bottleneck of for the lifetime, the ionization rate was enhanced by inhibiting spontaneous emission using the cavity effect. With increasing temperature, the strong interaction between the atom (Eu2+) and the surrounding environment (CaS) should destroy the coherence of the system and may induce the transition of the field-ionization mechanism from quantum to classical10. The fieldionization rate was increased by the thermally assisted quantum tunneling with lower temperature, but was decreased by thermallysuppressed classical diffusion with higher temperature. References
*-Q Fig.5. Configuration coordinate of the field-ionization.
Based on the above argument, these experimental results are explained using one configuration coordinate for the CaS:Eu2+ system under the cavity effect (Fig.5). Curves G and E correspond to the total system energy with a localized electron in either the ground state or the excited state of Eu2+. Curve I corresponds to that with the electron delocalized into the conduction band of CaS by perturbation of electric field or lattice vibration with temperature. It is critical for this diagram that during derealization from E to I the system experience relatively large lattice displacement generally observed in ionic crystal. Ionization of the excited Eu2+ occurs via the thermally assisted quantum tunneling across the potential barrier at the E-I crossing point and it may be theoretically treated as an
1. H. Mabuchi et al., Science 298, 1372 (2002). 2. H. Becker et al., Phys. Rev. B56, 1893 (1997). 3. M. Ando et al., Phys. Rev. B58, 13580 (1998). 4. A. J. Leggett, Proc. 4th ISQM(Tokyo, 1992), p.10. 5. K. Fujikawa et al., Phys. Rev. B46, 10295 (1992). 6. M. Segal et al., Phys. Rev. B68, 075211 (2003). 7. S. A. Basun et al., Phys. Rev. B56, 12992 (1997). 8. H. Schniepp et al., Phys. Rev. Lett. 89, 257403 (2002). 9. Y. Toyozawa, Optical Processes in Solids (Cambridge Univ. Press, 2003). 10. W. H. Zurek, Phys. Today 44(10), 36 (1991).
262 DEVELOPMENT OF UNIVERSAL VIRTUAL SPECTRSCOPE FOR OPTOELECTRONICS RESEARCH: FIRST PRINCIPLES SOFTWARE REPLACING DIELECTRIC CONSTANT MEASUREMENTS T. HAMADA*" , T. YAMAMOTO*, H. MOMIDA", T. UDA4 and T. OHNO"c N. TAJIMA1, S. HASAKA'', ML INOUE'', and N. KOBAYSAHf "Institute of Industrial Science, University of Tokyo, 4-6-1,Komaba, Meguro, Tokyo, 153-8605, Japan h Advancesoft Corporation, 4-6-1, Komaba, Meguro, Tokyo, 153-8601, Japan 'National Institute of Materials Research, 1-2-1, Sengen, Tsukuba, Ibaraki, 305-0047, Japan Taiyo-Nippon Sanso Corporation, 1-3-26, Oyama, Shinagawa, Tokyo, 142-8558, Japan "Semiconductor Leading Edge Technologies, 16-1, Onogawa, Tsukuba, Ibaraki, 305-8569, Japan E-mail: hamada@fsis. iis. u-tokyo. ac.jp We have developed a first principles software, universal virtual spectroscope for opto-electronics research (UVSOR), which can calculate dielectric functions of materials at atomistic levels on the basis of the density functional pseudopotential method in the frequency range from the static to ultra-violet region. The UVSOR can calculate separately electronic and lattice components of the dielectric function, by using the random phase approximation and the Berry phase polarization theory, respectively. This makes the UVSOR unique "virtual spectroscope" on computer, covering all frequencies interested in materials science; i. e., radio, Tera-Hz, infrared, visible, and ultra-violet frequencies. Since the UVSOR can quantitatively calculate the dielectric constant of materials, it is quite effective for studying new dielectrics such as high-k and low-k materials discussed in nano-scale semiconductor technologies. The UVSOR is free software and can be downloaded from our web site: http://www.fsis.iis.u-tokyo.ac.jp. Keywords: First Principles Calculation; Software; Dielectric Function.
1. Introduction 1.1. Dielectric function of materials Accurate measurement of dielectric constant e is a subject of great concern in physics as well as in chemistry because the measurement provides detailed information on the quantum mechanical structure of materials. The measurement in the visible region provides information on the electronic structure of materials, and in the infrared (IR) region provides information on the phononic structure of materials. The measurement is also important in electronics and opto-electronics, because most of electronic and optical devices, such as transistors, lasers, and optical fibers, are made of dielectrics. Recently, nano-scale dielectrics have attracted much attention in electronics, because the recent drastic scaledown of electronics devices reduces the size of the devices to the nanometer (10"9m) scale. In
silicon transistor devices, Si0 2 has been used as core insulating dielectrics since the born of the devices. However, the scale-down of the devices makes Si0 2 useless because of the following reason. If Si0 2 is used for the insulating layers, the layer thickness should be reduced to several nanometers in nano-scale transistors. Then Si0 2 layers are too thin to be insulating, and the direct tunneling current through the layers becomes substantially large when the transistors are operated. It is currently recognized that the development of new dielectrics replacing Si0 2 is necessary to overcome this problem, and that quantum mechanical study of dielectrics is effective for this purpose. We have developed first principles software, universal virtual spectroscope for opto-electronics research (UVSOR) [1], enabling to calculate quantum mechanically dielectric functions of dielectrics. In this paper, we describe the theory of
263
UVSOR and its performance, showing e calculation results for new dielectrics for the next generation electronics. 2. Dielectric Phenomena Dielectric constant e of materials is described by complex numbers and is frequency dependent. Figure 1 shows frequency dispersion of the real part of e. If the local field is neglected, s is given by the sum of the orientation £p", lattice vibration dat, and electronic etk dielectric constants. The £p" is due to the orientation of molecules or atomic groups to an external electric field, and is observed in liquids or polymers having polar molecules or atomic groups at frequencies lower than 1 tera (1012) Hz. The dat is due to the lattice polarizations induced by lattice vibrations and is observed in infrared (IR) active materials at frequencies lower than 10 tera Hz. The dh is due to the electronic structure of materials and is observed in all materials at frequencies lower than 10 peta (1016) Hz. The UVSOR can calculate separately £'* and s!"' of insulators, taking into account frequency dependences, and also can calculate s of non-molecular solids. The calculation is based on the first principles density functional (DFT) pseudopotential (PP) method using the local density approximation (LDA) and generalized gradient approximation (GGA) exchange-correlation functionals. The norm-conserving (NC) as well as ultra-soft
calculations. 3. Theory 3.1 Electronic dielectric constant The d'e is calculated by using the time dependent perturbation theory for the optical property calculation of materials [2]. Imaginary part oftfle,Im[£ffe], of an insulator is calculated from electronic transition moments between its valence and conduction bands at k-points in the Brillouin zone (BZ), using the random phase approximation (Eq. (1)). Im[£efc] = ^ I J 8 Z | f e | u r | ^ ) | 2 < J ( £ J - ^ - ^ ) d k ( l )
Here, (p'\s the Kohn-Sham wavefunction, k is the wave vector and E is the eigenenergy of cp, c and v are conduction and valence band indices, respectively, and r is the position operator. In addition, u and co are polarization vector and frequency of an external electromagnetic field, respectively. The BZ means the integration is done in the Brillouin zone. Real part of «"', R e ^ ] , is calculated from \m[i?'e] by using the Kramers-Kronig transformation. The PPs have electron-moment dependent, non-local potentials for electrons and thus influence electronic transition moments, interacting with an external electric field. The UVSOR can calculate ile in the same way as all-electron calculations, compensating the PP effects on electronic transition moments. 3.2 Lattice dielectric constant Lattice dielectric constant d"' (real part) of insulators is given by [3]. E%(a>)--
107
10'°
10 13
10"
(US) PP can be used in the £
and
2
(2)
O)
Here, a and /? are the Cartesian indices, O is the unit cell volume, and a>x is the frequency of the A-th lattice vibration mode of an insulator. The Zx is the transition moment for the A-th mode given by
Frequency (Hz) Fig. 1 F r e q u e n c y d e p e n d e n c e of dielectric constant le
*e yZxa Zxp Q
1
i
a
Zxa~
'P\rrii
ZiftP^is iA0-
(3)
264 Here, z'and w, are the Born effective charge and mass of the i-th atom, respectively, and £j is the -l-th mode eigenvector. The z* is a second-rank tensor describing the lattice polarization AP, of the insulator induced by the i-th atom infinitesimal displacement Am as AP/ = ~ ^ Z * A u / ,
(4)
where AP, is calculated from the Berry phase of valence electrons of the insulator, using socalled the Berry phase polarization theory[4],
Here, u is the periodic part of valence band wavefunction
classical molecular dynamics (MD) and the first principles methods: initial guess structures were generated by using the melt and quench MD method and then the generated structure were energy-optimized by using the first principles method [5]. It was characteristic of amorphous A1203 models that they had Al sites with several O-coordination numbers. Table I shows calculated i?le, J"', and s (=/ fe +£ /a ') for these high-k materials, along with experimental values for comparison. The calculated ^'e and fi'"' values for the high-k materials are in good agreement with the experimental £k and d"1 values, respectively. It is remarkable that the UVSOR calculations well reproduces the £le and dal of amorphous A1203. The results show that UVSOR is effective for theoretical studies of crystalline and amorphous high-k materials, which have a large dal and a moderate ^le.
Table 1. Calculated dielectric constants of high-k materials. Experimental values are shown in parentheses for comparison. ?kc
Material
£**
alpha Al203(hexagonal) e=em 2.9(3.1) 6.1(5.8) w
A1,0, 2
3
«zz amorphous
g=g""c + e"' 9.0(8.9)
2.8(3.0) ~3
8.4(8.1) 5.9-8.8
11.2(11.1) 8.9-11.8
(2.5-2.8)
(5.7-8.2)
(8.2-11.0)
Hf0 2 (monoclinic)
4.7(~5)
Ce02
7.5(6)
11.4
15.9(16-25)
16.8(17)
24.8(23)
4.2. Low-k materials Low-k materials are a group of insulating materials having an e smaller than 3.9, and are used as the insulator for three-dimensional circuits of semiconductor devices. The materials are required to have a lower e to avoid the signal delay in the circuits and high mechanical strength to tolerate the mechanical stress imposed during the circuit fabrication process. Model structures of a typical amorphous SiOCH low-k material were generated and its IR absorption spectra and static e were calculated by using the UVSOR
265 [6]. The Si-O network topology of generated models was determined by using a network structure modeling technique in classical MD calculations, and then the network structure was energy-optimized by using the first principles GGA DFT PP method. Figure 2 shows the structure of a model most likely to represent the low-k material with the chemical composition of Si10154Co67H2i4. Figure 3 shows calculated IR absorption spectra for the model structure, in comparison with experimental spectra of the material. The theoretical and experimental IR spectra have similar peak positions, and the similarity suggests the validity of the material modeling
2.9 for the low-k material. The model structure had an t?,e of 2.14 and an ial of 0.98; /. e., the electronic contribution to e was dominant contrary to the high-k material cases. The difference in e component between high- and low-k materials suggests that the material design strategy would differ, though they belong to the same group of materials, dielectric insulators. 5. Conclusions We developed first principle software UVSOR, which can accurately calculate the dielectric function of materials. The UVSOR can work as "virtual spectroscope" on computer, and was shown to be effective for the theoretical study of new dielectrics. Acknowledgments The UVSOR software was developed in "Frontier Simulation Software for Industrial Science (FSIS)" project supported by IT program of Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
Fig.2 Model structure of low-k SiiOi 54C067H2.M(2X2X2 super-cell structure)
material
wavenumber fcm ')
Fig.3 (a)Experimental infrared (IR) spectra for a low-k material and (b)theoretical IR spectra for model structure of the low-k material (Fie.2). and £lat calculation. Calculated static e for the model structure was
3.12, and was close to an experimental value of
References 1. T. Hamada, T, Yamamoto, H. Momida, T. Uda, and T. Ohno, UVSOR ver. 1.0, Univ. Tokyo (2005). 2. G. Harbeke, in Optical Properties of Solids, ed. F. Abeles (North-Holland, Amsterdam, 1972); p21. 3. X. Zhao and D. Vanderbilt, Phys. Rev. B, 65, 233106(2992). 4. R. Resta, Rev. Mod. Phys. 66, 899 (1994). 5. T. Hamada, H. Momida, T. Yamamoto, T. Uda, and T. Ohno, 1ECE Technical Report, SDM200558, 1 (2005) (in Japanese). 6. N. Tajima, T. Hamada, T. Ohno, K. Yoneda, N. Kobayashi, T. Hasaka, N. Inoue, Proceedings of the IEEE 2005 International Interconnect Technology Conference, 66 (2005).
266 Q U A N T U M N E R N S T EFFECT
HIROAKI NAKAMURA Theory
and Computer
Simulation Center, National Institute for Fusion Oroshi-cho, Toki, Gifu 509-5292, Japan E-mail: [email protected]
Science,
NAOMICHI HATANO Institute
of Industrial
Science,
University E-mail:
of Tokyo, Komaba, Meguro, [email protected]
Tokyo 153-8505,
Japan
RYOEN SHIRASAKI Department Tokiwadai,
of Physics, Yokohama National University, Hodogaya-ku, Yokohama 240-8501, Japan E-mail: sirasaki@phys. ynu. ac.jp
We report our recent predictions on the quantum Nernst effect, a novel thermomagnetic effect in the quantum Hall regime. We assume that, when the chemical potential is located between a pair of neighboring Landau levels, edge currents convect around the system. This yields theoretical predictions that the Nernst coefficient is strongly suppressed and the thermal conductance is quantized. The present system is a physical realization of the non-equilibrium steady state. Keywords: Nernst effect; Nernst coefficient; edge current; quantum Hall effect; thermoelectric power; thermomagnetic effect; non-equilibrium steady state
1. Introduction
the temperature bias such that AT > 0 if the temperature is higher in the heat bath The adiabatic Nernst effect arises in a conon the —x surface than that on the +x surductor bar under a magnetic field B in the face. We also define the Nernst voltage such z direction and a temperature bias AT in that VN > 0 if the voltage is higher on the +y the x direction. The conductor is electrically and thermally insulated on all surfaces, ex- surface than on the —y surface. (We always cept that heat baths are attached to the sur- put B > 0 here and hereafter.) The above faces facing the +x and — x directions, which classical-mechanical consideration, where no produces the temperature bias. A classical- scattering is taken into account, gives a posimechanical consideration gives the following: tive Nernst coefficient. In fact, electron scatcoefficient both electrons carrying the heat current in the x tering can make the Nernst 1 direction are deflected to the y direction be- positive and negative. In a recent article, 2 we considered the cause of the Lorentz force generated by the magnetic field in the z direction, and thereby Nernst effect in the quantum Hall regime, produce a voltage difference (the Nernst volt- that is, the Nernst effect of the twoage) VN in the y direction. The Nernst coef- dimensional electron gas in semiconductor heterojunctions under a strong magnetic ficient is defined by field (namely a Hall bar) at low temperatures, low enough for the mean free path to ( ~ WBAT' ' be greater than the system size. We theowhere W and L are the width and the length retically predicted that, when the chemical of the conductor bar, respectively. We define
267 potential is located between a pair of neighboring Landau levels:
a temperature T_ and is equilibrated to the Fermi distribution / ( T _ , /x_), arriving at the corner C3. The edge current along the lower (i) the Nernst coefficient is strongly supedge runs ballistically from the corner C3 to pressed; the corner C4, maintaining the Fermi distri(ii) the thermal conductance in the x direcbution / ( T _ , ^ _ ) all the way. The Nernst tion is quantized. voltage is then given by Hasegawa and Machida are planning an Afx _ jj,+ - fi. VN = -^ (2) experiment 3 in the setup of the present theory. In what follows, we review our predicfor the temperature bias AT = T+—T- > 0, tions and numerical demonstration. where e(< 0) is the charge of the electron. Incidentally, a convecting edge current 2. Convection of edge currents constitutes the non-equilibrium steady state Our argument is based on a simple assump- (NESS), a new concept that attracts much tion that edge currents 4 convect around the attention in the field of non-equilibrium 56 system. We here explain our idea (Fig. 1). statistical physics. ' The non-equilibrium Since the Hall bar is electrically insulated, steady state is almost the first statistical edge currents circulate along the edges of state of a quantum system far from equilibthe Hall bar when the chemical potential is rium that can be handled analytically. It between a pair of neighboring Landau lev- consists of a pair of independent currents els. An edge current on the left end of the running in different directions with different Hall bar is, while running from the corner temperatures and different chemical potenC4 to the corner Ci, heated up by the heat tials. In most studies, the non-equilibrium bath with a temperature T+ and is equili- steady state has been considered in a onedimensional non-interacting electron system brated to the Fermi distribution f(T+,fi+) and hence has been an almost purely mathwith the temperature T+ and a chemical poematical concept. The pair of the upper and tential /i+ around the corner Ci. Since the lower edge currents in Fig. 1, however, can be upper edge is electrically and thermally insuregarded as a non-equilibrium steady state. lated, the edge current runs ballistically from We consider it valuable to give a physical rethe corner Ci to the corner C2, maintaining alization to the mathematical-physical conthe Fermi distribution /(T + ,/x + ) all the way. cept. The edge current, upon arriving at the corner C2, encounters the other heat bath with 3. Nernst coefficient and heat conductance
Ci
(T+.\U) T. C4^-
(T-,\L-)
->-x Fig. 1. A schematic view of the convection of an edge current in a Hall bar under the setup for the Nernst effect.
Let us describe our calculation briefly. We define the electric current and the heat current in the form Ie = {ev)
and
IQ = ((E - /j)v),
(3)
where the thermal average is given by
(A) = -Y, 7T
n=0'
AfnM{T{yk), »{yk))dk l-lc
(4)
268 with yk = hk/\e\B. The subscript n stands for the channel of the edge current. The integration limits ±fcm are the maximum and minimum possible momenta. The function /n
S
£ .
(6)
For details of the calculation, refer to the original article, Ref. 2. 4. Predictions We can understand our predictions based on the convection of edge currents. First, the number of the conduction electrons is conserved during the convection. Hence the difference in the chemical potential of the upper edge current and that of the lower edge current is of a higher order of the difference in the temperature of the upper and lower edge currents; that is, A/x = o(AT). The Nernst coefficient (5) hence vanishes as a linear response, or N = 0 in the limit A T —> 0. Second, the heat current IQ in the x direction is carried by the ballistic edge currents along the upper and lower edges; the
total heat current is the difference in the heat carried by the upper edge currents and the lower edge currents. The edge currents are quantized as long as the chemical potential remains between a pair of neighboring Landau levels. The heat current hence has quantized steps as a function of B. The heat current is M times a unit current when there are M channels of the edge current. After some algebra, we arrive at the conclusion that the heat conductance GQ is quantized as 2
^
~
M
( M = 1,2,3,
(7)
when the chemical potential is located between the M t h and (M + l)th Landau levels. 5. Numerical demonstration We now present a numerical demonstration of our predictions using a confining potential in the y direction in the form7 V(y) =
0
for
^ (|y| - f Y for
\y\ <
f<|y|
(8) We used the following parameter values: the effective mass is m = 0.067mo for GaAs, where mo is the bare mass of the electron; the size of the Hall bar is L = 20/zm and W = 20/xm (less than the mean free path at low temperatures 8 ) with w = 16^m; the confining potential is given by V(±W/2) = 5.0eV, the work function of GaAs; the chemical potential at equilibrium is n = 15meV, or the carrier density ns = 4.24 x 10 1 5 m - 2 . We thus evaluated the Nernst coefficient N and the thermal conductance GQ as in Fig. 2. Our predictions N = 0 and Eq. (7) are indeed realized at low temperatures and when the chemical potential is located between a pair of neighboring Landau levels. (The gray curves in Fig. 2 are our new results of the self-consistent Born approximation in the case where we took account of impurity scattering of strength h/r = 1.0 x 10~4eV. See Ref. 9 for details.) We also note that the
269 -70
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1
2
3
4
5
6
7
8
Inverse Dimensionless Magnetic Field
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II I 2
T
&'
*
H 1 0.
(b)
0
1
2
3
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Inverse Dimensionless Magnetic Field
Fig. 2. Scaling plots of (a) N x B and (b) GQ/T X 3H/nk#2, both against mn/h\e\B at T = 1, 5,10 and 20K for IT < B < 20T. The gray curves in each panel indicate results that take account of impurity scattering at T = 1 and 5K; see text. Nernst coefficient is generally negative in t h e present calculation.
on the basis of t h e Fermi liquid theory, general expressions of t h e Nernst coefficient a n d the thermal conductivity of strongly correlated electron systems such as high-T c materials. Akera a n d Suzuura 1 2 considered with t h e use of thermohydrodynamics, t h e Ettingshausen effect, the reciprocal of t h e Nernst effect. T h e q u a n t u m behavior predicted here, however, was not reported in either studies. Acknowledgments T h e authors express sincere gratitude to Dr. Y. Hasegawa a n d Dr. T. Machida for useful comments on experiments of t h e Nernst effect a n d t h e q u a n t u m Hall effect. This research was partially supported by the Ministry of Education, Culture, Sports, Science a n d Technology, Grant-in-Aid for Exploratory Research, 2005, No. 17654073 as well as by t h e M u r a t a Science Foundation. N.H. gratefully acknowledges the financial support by the Sumitomo Foundation. References
6.
Summary
We predicted a novel q u a n t u m effect of t h e two-dimensional electron gas, in close analogy to t h e q u a n t u m Hall effect. W h e n t h e chemical potential is between a pair of Land a u levels, t h e edge currents suppress t h e Nernst coefficient and quantize t h e thermal conductance. T h e system is a physical realization of t h e non-equilibrium steady state. T h e precise forms of t h e peaks and t h e steps in Fig. 2 can be different when we take account of electron scattering. T h e electronic states extend over t h e system when the chemical potential is close to a L a n d a u level, namely when m/x/fi,|e|B = n+ \. T h e n the heat current is carried mainly by t h e bulk states a n d we have to take account of impurities as well as electron interactions possibly. 9 We comment on other approaches to t h e q u a n t u m Nernst effect. K o n t a n i derived 1 0 ' 1 1
1. H. Nakamura, K. Ikeda, Y. Ishikawa, A. Suzuki and H. Shirai, Jpn. J. Appl. Phys. 38, 5745 (1999). 2. H. Nakamura, N. Hatano and R. Shirasaki, Solid State Commun. 135, 510 (2005). 3. Y. Hasegawa and T. Machida, private communication. 4. B. I. Halperin, Phys. Rev. B 25, 2185 (1982). 5. D. Ruelle, J. Stat. Phys. 98, 57 (2000). 6. Y. Ogata, Phys. Rev. E 66, 016135 (2004). 7. S. Komiyama, H. Hirai, M. Ohsawa, Y. Matsuda, S. Sasa, and T. Fujii, Phys. Rev. B 45, 11085 (1992). 8. S. Tarucha, T. Saku, Y. Hirayama, and Y. Horikoshi, Phys. Rev. B 45, 13465 (1992). 9. H. Nakamura, N. Hatano, and R. Shirasaki, in preparation. 10. H. Kontani, Phys. Rev. Lett. 89, 237003 (2002). 11. H. Kontani, Phys. Rev. £ 6 7 , 014408 (2003). 12. H. Akera and H. Suzuura, cond-mat/0409498.
270 QUANTUM PHENOMENA VISUALIZED USING ELECTRON WAVES
Akira TONOMURA Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395 Japan Frontier Research System, Riken, Wako, Saitama 350-0198 Japan
Phase information of bright and monochromatic field-emission electron beams, which we had been repeatedly developing for the past 30 years, was used to visualize quantum phenomena, which have recently begun to crop up in many microscopic regions. Recent examples include microscopic quantum phenomena we directly observed in superconductors such as quantized vortices that exhibited strange behaviors peculiar to anisotropic layered high-rc superconductors.
1.
Introduction
Quantum mechanics, which was born as a law describing the behavior of electrons inside atoms, now provides the basis for nearly all physical theories. Quantum mechanics has explained many of the microscopic mechanisms not only of natural phenomena but also of electronic devices in fields where practical applications are sought, and has thus helped open up the semiconductor and other industries. Furthermore, recent technological developments since the 1980s have made it possible to experimentally test the fundamentals of quantum mechanics, and all the results obtained up to now have agreed well with quantum mechanics. Quantum phenomena are thus attracting attention from two aspects: as obstacles to further improvements in device performances and as pathfinders in the development of future devices. We have been continuously developing electron microscopes equipped with field-emission guns that produce ever-more-coherent electron beams with the aim of directly observing such quantum phenomena [1], so that we can precisely measure not only the intensity but also the phase of the transmitted electrons. In this paper, we describe our direct observation of the behaviors of quantum objects
using the wave nature of electrons after briefly reviewing the development of coherent electron beams. 2. Historical Developments Electron Beams
of Coherent
We started our research on electron holography in 1967 as a way of overcoming the saturated state of electron microscopy technology and confirmed after our first year of experiments that optical images could be reconstructed from in-line electron holograms [2]. We intended to overcome the resolution limit, which was restricted by the inevitable aberrations found in axially-symmetric electron lenses, by applying electron holography. However, we clarified from the experiments that bright electron beams, like laser beams, would be needed to practically apply such holographic techniques. We then began developing coherent electron beams that were field-emitted from pointed tips, and we have continued to do so up to now. However, since the electron source is extremely small, typically 50 A in diameter, and it has to be immobile even within a fraction of the source diameter, we had to overcome such technical difficulties to prevent even the slightest mechanical vibration of the tip, the accelerating tube, and the microscope column, or a deflection of the
271 fine beam due to stray ac magnetic fields. Otherwise, the inherent brightness, or luminosity, of the electron beam would deteriorate. After a decade of work, we developed an 80-kV field-emission electron beam [3], which was two orders of magnitude brighter than the thermal beams then used (see Table 1), and we became the first to observe electron interference patterns directly on the fluorescent screen and to record up to 3000 interference fringes on film. Thanks to this development, information that had previously been inaccessible by conventional electron microscopy could be obtained through using both electron holography and this bright electron beam. In fact, magnetic lines of force inside and outside ferromagnetic samples were directly and quantitatively observed in hie flux units [4] in interference micrographs, which could be obtained in the optical reconstruction stage of off-axis electron holography formed with an electron biprism [5]. In 1978, the existence of the Aharonov-Bohm (AB) effect [6] had been questioned and controversy about it has again
Year
Electron microscope
Brightness (A/cm2- Ster)
arisen [7]. We attempted to provide definitive experimental evidence, since the AB effect was also the fundamental principle behind our method of observing magnetic lines offeree. At that time, Prof. C. N. Yang of State University of New York kindly visited our laboratory [8] to discuss our proposal to experimentally verify the AB effect, and since then he has continued to show interest in our work on electron interference and to advise us in our experiments. We had to conduct a series of experiments on the AB effect up to 1986 [9-11], since repeated objections arose about our results during the controversy. In 1992, we observed magnetic vortices in metal superconductors using phase information of electrons transmitted through thin film samples [12], and later vortices in high-Tc superconductors [13-15]. 3. Experimental Results New advanced technologies, such as coherent electron beams, sensitive detectors, and lithography, have made it possible to carry out fundamental experiments in quantum
Applications
(Thermionic electrons)
1 x106
Experimental feasibility of electron holography [2]
1978
80 - kV FEEM
1 x108
Direct observation of magnetic lines of force [4]
1982
250-kVFEEM
4x108
1989
35-kV FEEM
5x109
2000
1 - MV FEEM
2x10
1968
100-kVEM
10
Conclusive experiments ofAB effect [11] Dynamic observation of vortices in metal superconductors [12] Observation of unusual behaviors of vortices in high - Tc superconductors [13-15] FEEM: field-emission electron microscope
Table 1
History of developments of bright electron beams
272
mechanics that once belonged to the realm of "thought experiments", such as the single-electron build-up of an interference pattern [16], conclusive experiments verifying the existence of the AB effect [9-11], and the observation of static and dynamic behaviors of quantized vortices in superconductors. Here, we report results that concern the unusual behaviors of vortices peculiar to anisotropic, layered high-rc superconductors observed with our newly developed 1-MV microscope [17], where the beam was four orders of magnitude brighter than that of a conventional 100-kV thermal beam, and the number of biprism interference fringes that could be recorded on film increased from 300 to 11,000 [18]. An incident electron wave is phase-shifted or deflected by the magnetic fields of vortices when passing through them, and a vortex appears as a spot consisting of black-and-white contrast produced by image defocusing (Lorentz microscopy). Vortices usually form a closely packed triangular lattice. This is the case even for anisotropic high-r c superconductors, as long as the magnetic field is directed along the anisotropy c-axis. When the magnetic field is strongly tilted away from the c-axis, however, Bitter images reveal that the vortices no longer form a triangular lattice. Instead, in YBaCu 3 0 7 8 (YBCO) [19], they form arrays of linear chains along the direction of the tilting field, and in i
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Bi-2212 alternating domains of chains and triangular lattices [20,21] (see Fig. 1). While the chain state in YBCO can be explained by the tilting of vortex lines within the framework of anisotropic London theory [22], the chain-lattice state in Bi-2212 has long been an object of discussion. It has been attributed to two sets of vortex lines perpendicular to each other [23], one set forming chains and the other forming triangular lattices. Koshelev [23] proposed an interesting model for the chain lattice state; Josephson vortices penetrate between the layer planes in Bi-2212 and vortices that perpendicularly intersect the Josephson vortices form chains; the rest of the vortices form triangular lattices. No direct evidence for such mechanisms, however, was given through experiments because there were few methods for observing the arrangements of vortex lines inside superconductors. In addition, this model was difficult to accept, since no interaction took place between the two perpendicular magnetic fields. However, Koshelev took second-order approximation into consideration and determined there was an energy reduction in this vortex arrangement. As a result of interaction, a vertical vortex line winds slightly in opposite directions, above and below the crossing Josephson vortex, thus forming a stable state, since the circulating supercurrent of the Josephson vortex exerts Lorentz forces onto the vertical vortex line. Lorentz microscopy with our 1-MV electron microscope has been used to determine whether the vortex lines in the chain states inside high-rc superconductors are tilted or not. In the case of YBCO, we found that vortices tilted together in the direction the applied magnetic field was tilted. This conclusion is evident from the Lorentz micrographs in Fig. 2, in which the vortex images become more elongated and form linear chains together as the tilting angle of the magnetic field increases. In the case of our Bi-2212, Lorentz microscopy observations under various defocusing conditions clarified
273 that neither chain nor lattice vortices tilted, but both stood perpendicular to the layer plane [24]. If vortex lines are strongly tilted at an angle comparable to that of the applied magnetic field, the vortex images in Fig. 3(a) should be elongated. Our findings that both chain and lattice vortices stand straight perpendicular to the c-plane and do not tilt is clear evidence of the Koshelev mechanism. Although the clearest evidence for this model would of course be the direct observation of Josephson vortices, the magnetic field of a Josephson vortex extends widely between the layers to several tens of urn and, therefore, makes it difficult to detect with our method. Although we could not observe Josephson vortices directly, we had supporting evidence for their existence. For example, we could observe vertical vortices always beginning to penetrateg into the sample along some straight lines.
7- ' (a)
'••".' (b)
Josephson vortex Fig.3. Chain-lattice state of vortices at tilted magnetic field in Bi-2212: (a) Lorentz micrograph (T = 50 K and B = 50 G). (b) Schematic. The chains are indicated by the white arrows in (a). If vortex lines are tilted at an angle comparable to the tilt angle of the magnetic field, i.e., 85°, the vortex images must be elongated and weak in contrast under this defocusing condition. Since the images of chain vortices as well as those of lattices vortices are not elongated but circular, all the vortex lines are not tilted, but perpendicular to the layer plane. If tilted, die vortex line images become elongated as the vortex images indicated by the arrows in the inset in (a)
(c)
Tilted vortex
(d) Fig.2. Lorentz micrographs of vortices in YBCO film sample at tilted magnetic fields (T= 30 K; B, = 3 G): (a) 6 -75? (b) 0 =82? (c) 0 =83? (d) Schematic of tilted vortex lines. When the tilt angle becomes larger than 75? the vortex images begin to elongate and, at the same time, to form arrays of linear chains. This implies that chain vortices in YBCO are produced by some attractive force between tilted vortex lines.
Fig.4. Series of Lorentz micrographs of vortices in field-cooled Bi-2212 film sample when magnetic field B p perpendicular to layer plane begins to be applied and increases at fixed in-plane magnetic field of 50 G at T= 50 K: (a) Bp = 0; (b) B, = 0.2 G; (c) B p = 1 G; (d) Schematic of vortex lines consisting of vertical and Josephson vortices.
274 These straight lines must be determined by Josephson vortices. There is an example in Fig. 4. When we apply an in-plane magnetic field, no vortex images can be seen in the Lorentz micrograph [see Fig. 4(a)]. However, when a perpendicular magnetic field is applied and slowly increased, images of vertical vortices appear in this field of view. When a magnetic field is applied parallel to the layer plane as in (a), no vertical vortices are produced. Therefore, no vortex images can be seen. Although there should be Josephson vortices parallel to the layer plane, these cannot be observed by Lorentz microscopy because of the wide distribution of the vortex magnetic field. When the vertical magnetic field increases, vertical vortices begin to appear along straight lines, indicated by the white lines in (b), which are considered to be determined by Josephson vortices. Since vortices are arranged along straight lines even when there are large intervals between vortices, we could find no reason why chain vortices were produced other than assuming that vertical vortices crossing Josephson vortices formed chains as illustrated in Fig. 3 (b). Above 6 P (magnetic field perpendicular to the layer plane) = 1 G, vertical vortices appeared also between chain vortices as shown in (c). We also found that only images of chain vortices and not of lattice ones in Bi-2212 began to disappear at temperatures much lower than Tc. Figure 5 is a Lorentz micrograph of such an example.
•I t b
I nr»
3(im
Fig.5. Lorentz micrograph of disappearing chain vortices inBi-2212(r=57K,B p = 8G, 6=80° )
We attribute the disappearance of vortex images to the synchronous oscillation of vortices between "node vortices" A and C along the chain direction, just like a coupled oscillation. This does not mean that the vortices themselves disappear, since vortex images gradually fade away with rising temperatures and also partly do so along the chain. In Fig. 5, vortices A and C in the chain can clearly be seen, but vortices that are far away from these vortices cannot. Vortices A and C are located "stably" in the midst of six surrounding vortices (indicated by the red points in the figure), while vortex B is sandwiched "unstably" between two vortices above and below. This vortex arrangement may be stable at low temperatures, but at high temperatures where vortices vibrate thermally, vortex B begins to oscillate back and forth, like pinballs connected by springs on incommensurate periodic potentials, as in the Frenkel-Kontorova model. We attribute the disappearance of chain vortices to such longitudinal oscillations of vortices along chains. 4. Conclusion Thanks to recent developments in advanced technologies, such as coherent electron beams, highly sensitive electron detectors, and photolithography, some experiments that were once regarded as "Gedanken" experiments can now be carried out in practice. In addition, the wave nature of electrons can now be utilized to observe microscopic objects that were previously not able to be observed. Examples are the quantitative observations of both the microscopic distribution of magnetic lines of force in hie units and the dynamics of quantized vortices in superconductors. These measurement and observation techniques are expected to play important roles in future research on and developments in nano-science and nano-technology.
275 References 1)
2)
3)
4)
5) 6) 7) 8)
9)
10)
11)
12)
13)
14)
A. Tonomura. (1999) Electron Holography (2nd edition, Springer, Heidelberg). A. Tonomura, A. Fukuhara, H. Watanabe, & T. Komoda, Jpn. J. Appl. Phys. 7 (1968) 295. A. Tonomura, T. Matsuda, J. Endo, H. Todokoro & T. Komoda, J. Electron Microsc. 28 (1979) 1-11. A. Tonomura, T. Matsuda, J. Endo, T. Arii & K. Mihama, Phys. Rev. Lett. 44 (1980) 1430-1433. G. MoUenstedt & H. DUker, Z. Physik 145(1956) 377-397. Y. Aharonov & D. Bohm, Phys. Rev. 115 (1959)485-491. P. Bocchieri & A. Loinger, Nuovo Cimento ^(1978)475-482. C. N. Yang, Foreword, Foundations of Quantum Mechanics in the Light of New Technology, eds. S. Nakajima, C. N. Yang, Y. Murayama & A. Tonomura, Advanced Series in Applied Physics Vol. 4 (World Scientific, Singapore, 1996) p. v. A. Tonomura, H. Umezaki, T. Matsuda, N. Osakabe, J. Endo & Y Sugita, Phys. Rev. Lett. 51(1983)331-334. A. Tonomura, T. Matsuda, R. Suzuki, A. Fukuhara, N. Osakabe, H. Umezaki, J. Endo, K. Shinagawa, Y Sugita & H. Fujiwara, Phys. Rev. Lett. 48(1982) 1443-1446. A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, & H. Yamada. Phys. Rev. Lett. 56(1986) 792-795. K. Harada, T. Matsuda, J. Bonevich, M. Igarashi, S. Kondo, G. Pozzi, U. Kawabe & A. Tonomura, Nature 360 (1992) 51-53. A. Tonomura, H. Kasai, O. Kamimura, T. Matsuda, K. Harada, Y. Nakayama, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, M. Sasase & S. Okayasu, Nature 412(2001) 620-622. T. Matsuda, O. Kamimura, H. Kasai, K.
15)
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21) 22) 23) 24)
Harada, T. Yoshida, T. Akashi, A. Tonomura, Y. Nakayama, J. Shimoyama, K. Kishio, T. Hanaguri & K. Kitazawa, Science 294(2001) 2136-2138. A. Tonomura, H. Kasai, T. Matsuda, K. Harada, T. Yoshida, T. Akashi, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, T. Masui, S. Tajima, N. Koshizuka, P. L. Gammel, D. Bishop, M. Sasase & S. Okayasu, Phys. Rev. Lett. 88 (2002)237001. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki & H. Ezawa, Amer. J. Phys. 57(1989)117-120. T. Kawasaki, T. Yoshida, T. Matsuda, N. Osakabe, A. Tonomura, I. Matsui & K. Kitazawa, Appl. Phys. Lett. 76(2000) 1342-1344. T. Akashi, K. Harada, T. Matsuda, H. Kasai, A. Tonomura, T. Furutsu, T. Moriya, T. Yoshida, T. Kawasaki, K. Kitazawa & H. Koinuma, Appl. Phys. Lett. 81(2002) 1922-1924. P. L. Gammel, D. J. Bishop, J. P. Rice & D. M. Ginsberg, Phys. Rev. Lett. 68 (1992) 3343-3346. A. Bolle, P. L. Gammel, D. G. Grier, C. A. Murray, D. J. Bishop, D. B. Mitzi & A. Kapitulnik, Phys. Rev. Lett. 66 (1991) 112-115. I. V. Grigorieva & J. W. Steeds, Phys. Rev. B51 (1995) 3765-3771. A. I. Buzdin & A. Y. Simonov, JETP Lett. 51 (1990) 191-195. A. E. Koshelev, Phys. Rev. Lett. 83 (1999) 187-190. A. Tonomura, H. Kasai, O. Kamimura, T. Matsuda, K. Harada, T. Yoshida, T. Akashi, J. Shimoyama, K. Kishio, T. Hanaguri, K. Kitazawa, T. Matsui, S. Tajima, N. Koshizuka, P. L. Gammel, D. J. Bishop, M. Sasase & S. Okayasu, Phys. Rev. Lett. 88 (2002)237001.
276 A N OPTICAL LATTICE CLOCK: ULTRASTABLE ATOMIC CLOCK WITH ENGINEERED PERTURBATION
H. KATORI1'2'*, M. TAKAMOTO1, R. HIGASHI1, AND F.-L. HONG3 Department of Applied Physics, Graduate school of Engineering, The University of Tokyo, PRESTO, Japan Science and Technology Agency, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8563, Japan *E-mail: [email protected] An optical lattice clock was realized for 8 7 S r atoms and its absolute frequency has been measured. Projected uncertainty in the lattice clock is illustrated for the case of 8 7 Sr. We discuss application of the clock scheme to other atom species, in search for better lattice clocks as well as for the variation of fine-structure constant. Keywords: atomic clock; optical lattice; frequency measurement.
1. Introduction A singly trapped ion in Paul traps embodies an ideal quantum absorber for atomic clocks, which led to the most accurate optical atomic-clocks1 with fractional uncertainty at 1 0 - 1 5 level2 operated at the quantum projection noise (QPN) limit. One of the important features of Paul traps that make them attractive for atomic clocks is that they provide smallest possible perturbations for interrogated ions 1 , while they precisely control the ions' motion. Throughout the history of atomic clocks, indeed, such careful removal of electromagnetic perturbations has been considered to be the heart of the atomic clocks. Recent experiments, 2 however, have witnessed that conventional approaches, such as Cs fountain clocks, neutral-atom-based optical clocks interrogated in their free fall, and single-ion-based clocks, are confronted with serious limitations in their fractional accuracy or stability at a level of 1 0 - 1 6 . In order to break through this limitation, we proposed an "optical lattice clock"3 that utilizes engineered perturbation to enhance the clock stability without degrading its accuracy. Figure 1 depicts the scheme: Sub-wavelength lo-
calization of a single atom in each optical lattice site suppresses the first order Doppler shift4 as well as the collisional frequency shift, which simulates millions of single-ion clocks operated simultaneously. Although this lattice clock interrogates atoms while they are strongly perturbed by an external field, we have shown that this perturbation can be cancelled out to below 1 0 - 1 7 precision level5 by designing the light shift trap so as to adjust dipole polarizabilities of probed electronic states. In this contribution, we describe our recent experiments and future prospects for the optical lattice clock. 2. Light shift cancellation The transition frequency v of atoms exposed to a lattice-trap laser with intensity / is given by the sum of the unperturbed transition frequency Z/Q and the light shift fac, "(Az,,e) = VQ + V& VQ
Aa(\L,e) I + 0(I2) 2eQch
(1)
where Aa(XL,e) = ae(XL,e) - a g (Ai,,e) is the difference of ac polarizabilities of the upper and lower states that depend both on
277 3
[n[0
Detection
461 nm
yg|
sfl ^
Fig. 1. An optical lattice clock. Atoms localized in a sub-optical wavelength region offer Doppler-free spectrum of the clock transition, and simulate millions of single-ion clocks operated in parallel.
the trap laser wavelength A/, and its polarization e. By adjusting both of the polarizabilities so as to satisfy Aa(Az,,e) = 0, the observed atomic transition frequency v will be equal to v§ independent of the trap laser intensity I,6'7 as long as the higher order corrections 0{I2) are negligible. In search for a clock transition in which the Stark shift cancellation condition Aa(Ai,,e) = 0 can be given by a well-definable parameter of XL, the use of J = 0 state that exhibits a scalar light shift is preferable. Specifically, we adopt the 5s 2 ^So -> 5s5p 3Po forbidden transition of 87 Sr with nuclear spin of 9/2 as a "clock" transition (Fig. 2), in which hyperfine mixing of the 3 P 0 (F = 9/2) state with the 1,3 Pi states provides a finite lifetime of rV 1 = 150s.
3. E x p e r i m e n t 3.1. The magic wavelength spectroscopy
and
The schematics of our experiment 8 is shown in Fig. 3. 87 Sr atoms were laser-cooled and trapped on the :So — 3Pi transition at a few /xK and loaded into a ID optical lattice that was formed by a standing wave of TiSapphire laser. For atoms trapped in the optical lattice, we irradiated a clock laser at Ao = 698 nm, with its wave-vector parallel
III PI
5,
0.5
1
Position / XL 1.5 2 ^
' •' ' I I I
I
I
I
Fig. 2. The ^ o and 3PQ states of Sr are coupled to the upper respective spin states by an off-resonant standing wave light field to produce equal light shifts in the clock transition, where atoms are excited on the pSo) <8> \n) —» | 3 Po) ® \n) electronic-vibrational transitions.
to the fast axis of the ID lattice potential, for 10 — 40 ms to excite atoms to the 3Po state. The excitation of the clock transition was monitored by the population of atoms in the ground state by driving the xSn — ^ l cyclic transition. We measured the light shift ^ as a function of the lattice laser intensity of / to determine the differential dipole polarizability Aa(Ax,) of Eq. (1), which is given by 6Vac(\L,I)/6I = -Aa(XL)/(2s0ch). Figure 4 plotted the normalized light shift Svac/SI as a function of the lattice laser wavelength Az, to determine the magic wavelength, where the light shift ^ac(Az,,-0 due to the lattice laser is cancelled out, to be 813.420(7) nm. 9 The inset of Fig. 5 shows a clock spectrum (empty circles) measured at the magic wavelength with an interrogation time of 40 ms and a cycle time of 1 s for each point. Nearly Fourier-limited linewidth of 27 Hz was obtained, which demonstrated an order of magnitude reduction of the linewidth measured for neutral-atom-based optical clocks so far. 10 ' 12 This measurement inferred the clock stability of
278 in addition to the normalization of the clock excitation rate. 12 Optical /1L fibre
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j i
Nd:YAG/I2 (NMIJ-Y1)
ftep
/
4L
in Table 1. Recently, the optical lattice clock based on 87 Sr was also realized by JILA group and its absolute frequency has been reported. 13
&4 f. s
Optical frequency I | comb
813.0
Fig. 3. a. Ultracold Sr atoms were trapped in a one-dimensional optical lattice formed by a standing wave of a Ti-sapphire laser tuned t o a magic wavelength. The clock laser was introduced along the fast axis of the lattice potential to guarantee the LambDicke confinement, b. The frequency of the clock laser was measured by an optical frequency comb referenced to the SI second.
813.2
813.4
813.6
813.8
Lattice laser wavelength (nm)
Fig. 4. a. Light shifts on t h e clock transition were measured as a function of the lattice laser wavelength to determine the magic wavelength to be AL = 813.420(7) nm.
3.2. Absolute frequency measurement The frequency of the lattice clock was measured with an optical frequency comb referenced to a commercial Cs clock (Agilent 5071A) as depicted in Fig. 3b. 9 - 14 The frequency of the Cs clock was monitored by the GPS time during the period of the frequency measurement of over one month, which determined the frequency offset between the Cs clock and the GPS time 15 to be -1.04(8) x
io- 13 . Figure 5 summarizes the absolute frequency measurements referenced to the Cs clock. Each filled symbol represents a frequency measurement of typically 104 s to reduce the fractional instability of the Cs clock down to 6.5 x 10~ 14 corresponding to an uncertainty of 28 Hz. With total averaging time of T « 9.4 x 104 s over 9 days, we determined the clock transition to be 429,228,004,229,952(15) Hz, 9 including the systematic correction of —45.7 Hz as given
Clock laser frequency (Hz)
i 1
2
,
*•-*•
3
4
5
ff
i 6
7
Data number
Fig. 5. Absolute frequency measurement of the 1 So — 3Po transition of 8 7 Sr atoms in an optical lattice. Filled symbols correspond to a frequency measurement over 10 4 s. The inset shows the typical clock transition resolving a linewidth of 27 Hz.
3.3. Uncertainty
budget
Table 1 summarizes the uncertainties for this experiment as well as those expected for future experiments. The first-order Doppler shift may be introduced by the relative motion between the clock laser and the lattice potential. An active phase-noise cancellation by referring
279 Table 1. Systematic corrections and uncertainties for the absolute frequency measurement of the 1So—3Po transition o f 8 ' S r atoms in an optical lattice. A typical lattice laser intensity of Io = 10 k W / c m is assumed for attainable uncertainties associated with the light shifts. Frequency shifts are given in the unit of Hz. Effect 1st order Doppler 2nd order Doppler recoil shift 1st order Zeeman collision shift blackbody shift probe laser light shift Scalar light shift Vector light shift Tensor light shift 4th order light shift Cs clock offset Frequency measurement Systematic total Total uncertainty 8v
Correction 0 0 0 0 0.6 2.4 0.1 -3.8 0 0 0 -45 0
Achieved uncertainty
Attainable uncertainty
3 X 10~ 2 2 x IO"6 0 10 2.4 0.1 0.01 4 IO"3 10~ 3 10~ 3 3 9
< 10~ 3 < 2 X 10~ 6 0 10~ 3 10~ 5 3 x 10~ 3
15 Hz
4 x 10~ 3 Hz
io-3 10~ 3
io-3 10~ 3 IO-3
- 4 5 . 7 Hz
to a reflection from a mirror surface that forms a standing wave for the lattice potential could reduce this Doppler shift to less than 1 mHz. 16 The first-order Zeeman shift arises due to the hyperfine mixing in the 3Pn state, which is m x 106 Hz/G for Am = 0 transitions. The reduction or control of the magnetic field at /zG level would allow 1 mHz uncertainty. The collisional frequency shift may exist in the ID lattice configuration, since tens of unpolarized atoms were typically trapped in a single lattice potential with an atom density of ~ 10 1 2 cm - 3 . This collision shift would ultimately be removed by spinpolarizing 87 Sr atoms to activate Fermi suppression of collisions or by applying a 3D lattice with less than unity occupation. For the latter case, a frequency shift due to the resonant dipole-dipole interaction at an interatomic distance of A/2 « 400 nm is given. The Blackbody shift scales as VB « 3 • 10" 1 0 x (T/K) 4 Hz, which would introduce a 3 mHz uncertainty for the temperature imhomogeneity of AT = 0.1 K at T = 293 K. The vector light shift can be problem-
atic in 3D optical lattices. A phase stable optical lattice, where the 3D laser intersection is formed by a single linearly-polarized standing-wave with folded mirrors, 17 could be applied to provide linearly polarized trapping light fields. The forth-order light shift in Table 1 is estimated 5 without including resonant effects, which require accurate information on the magic wavelength and dipole moments of associated transitions. While the energy difference between the 5s7p1P\ state and the clock upper state 5s5p3P0 (813.359 nm x 2) nearly coincides with the magic wavelength, this transition is only weakly allowed by hyperfine-mixing, hence its contribution to the 4th order shifts should be small. Next closely lying states are the 5s4/ 3 i r ; f and 5s7p3P§ connected by 2-photons with wavelengths of 818.55 nm and 796.81 nm, respectively. 4. Outlook 4.1. Lattice
clock
families
The optical lattice clock scheme is applicable to odd isotopes of other divalent atoms 1 8 ' 1 9
280 Table 2. The lattice clock scheme is applicable to divalent atoms with nuclear spin / . The ^ o - 3 P 0 clock transition at Xp may be interrogated in optical lattices tuned to the magic wavelength A^,. The sensitivity q on a variation and the fractional frequency variation Su/uo for A a / a = 1 0 - 1 6 are listed. 87
43
Nuclear spin (/) Abundance (%) Clock transition, Ap (nm) Magic wavelength, A^, (nm) Light shift, vu ( k H z / k W c m " 2 ) Atom temperature, Ti (/iK) Catching power, Pi ( k W c m - 2 ) Min. power, Po ( k W c m - 2 ) Photon scattering rate (Hz)
7/2 0.14 660 735.5 11 5.6 106 65.3 0.4
9/2 7.0 698 813.42 13 1 16 26.7 0.03
1/2 14 578 752 20 20 209 16 0.01
1/2 17 266 358 1.4 28 4480 995 1
Sensitivity q Sv/vo x l O " 1 6
125 0.017
443 0.062
2714 0.31
15299 0.81
Ca
that have hyperfine-mixing induced J = 0 —¥ J = 0 transition between long-lived states. The candidate atoms include, Be, Mg, Ca 18 , Sr, Yb 1 9 ' 2 0 , Hg 21 and rare gases such as Ne, Kr, Xe. Some of them are listed in Table 2. Furthermore, a multi-photon excitation of the clock transition 22 may allow using even isotopes of these candidates that exhibit purely scalar nature of J = 0 state. A lattice clock with ultimate performance needs to be experimentally explored among these possible candidates because of difficulties in predicting some of uncertainties associated with higher order light field perturbations; such as resonant contribution to the 4th order light shifts as mentioned previously and (multi-)photon ionization processes. Once the far-off-resonance condition is satisfied for the magic wavelength, in order to minimize such uncertainties, it is essential to operate the lattice clock with lowest possible lattice laser intensity Po, as long as the tunneling of atoms among lattice potentials does not affect the clock accuracy. 21 Table 2 summarizes laser parameters for lattice clocks. Light intensities Pi required to catch laser cooled atoms are given assuming the lattice depth of Ui = lOksTi with atom temperatures Ti reported so far (Doppler temperature is used for Hg). Once
Sr
171
199Hg
Species
Yb
21
Ne
3/2 0.27 74 613 0.03 100 697 • 10 3 2.1 • 10 8 350• 10 3
atoms are trapped, they can be, in principle, sideband-cooled down to their vibrational ground states using narrow "clock" transitions. The lattice potential depth that confines these atoms with the Lamb-Dicke parameter 77 = y/h/(2Mvt)/\p is given by,
•W^Vi.
(2)
where we assumed the lattice potential periodicity of A L / 2 . ER represents the photon recoil energy associated with the clock transition Ap. M and Vt are the mass of trapped atom and its trapping frequency, respectively. Assuming rj = 0.26, the minimum lattice laser intensities Po that provide Uo and associated photon scattering rates are calculated as shown in the Table. These estimates suggest that Sr and Yb atoms would be favorable candidates for the lattice clock, unless the multi-photon resonant effect is not encountered. 4.2. Search for a
variation
The potential high precision and wide applicability of the lattice clock scheme to other atom species would allow exploring a possible variation of fine structure constant a. 23 So far frequencies of several atom clocks operated at 1 0 - 1 5 uncertainty level were com-
281 pared over a few years to verify t h e constancy a t t h e level of | d / a | = 2.0 • 1 0 " 1 5 / y r - 2 4 Assuming A a / a = 1 0 - 1 6 , at which level per year astrophysical determinations gave controversial results, 2 3 a fractional change of the clock frequency 2 5 5V/VQ is listed in Table 2. Heavier atoms such as Yb and Hg tend to show higher sensitivity as t h e relativistic corrections are proportional to a t o m number as Z2. In order t o verify Aa/a at 10~ 1 6 , one may take Sr lattice clock as an anchor to detect t h e fractional frequency change of Y b or Hg based lattice clock at 8V/I>Q = 1 0 - 1 6 level. Such experiment would be feasible in view of t h e current laser-stabilization technologies 2 6 and frequency measurement based on t h e optical frequency comb technique. 1 1
Acknowledgement This work received support from t h e Strategic Information and Communications R & D Promotion P r o g r a m m e ( S C O P E ) of t h e Ministry of Internal Affairs and Communications of J a p a n .
References 1. H. Dehmelt, IEEE Trans. Instrum. Meas. 31, 83 (1982). 2. See references in P. Gill, Metrologia 42, S125 (2005). 3. H. Katori, Proceedings of the 6th Symposium Frequency Standards and Metrology, edited by P. Gill (World Scientific, Singapore, 2002) p323. 4. T. Ido and H. Katori, Phys. Rev. Lett. 9 1 , 053001 (2003). 5. H. Katori, M. Takamoto, V. G. Pal'chikov, and V. D. Ovsiannikov, Phys. Rev. Lett. 9 1 , 173005 (2003). 6. H. Katori, T. Ido, and M. Kuwata-Gonokami, J. Phys. Soc. Jpn., 68, 2479 (1999). 7. H. J. Kimble et al., Laser Spectroscopy, Proceedings of the XIV International Conference, edited by R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler (World Scientific, Singapore, 2000) p. 80. 8. M. Takamoto and H. Katori, Phys. Rev. Lett. 91, 223001 (2003).
9. M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, Nature 435, 321 (2005). 10. F. Ruschewitz et al, Phys. Rev. Lett. 80, 3173 (1998). 11. Th. Udem et al, Phys. Rev. Lett. 86, 4996 (2001). 12. G. Wilpers et al, Phys. Rev. Lett. 89, 230801 (2002). 13. A. D. Ludlow et al, arXive:physics/ 0508041. 14. F.-L. Hong et al, Opt. Express 13, 5253 (2005). 15. The agreement between the GPS time and the TAI was about a few parts in 10 during the period of the frequency measurement. 16. L.-S. Ma, P. Jungner, J. Ye and J. L. Hall, Opt. Lett. 19 1777 (1994). 17. A. Rauschenbeutel, H. Schadwinkel, V. Gomer and D. Meschede, Opt. Commun. 148 45 (1998). 18. C. Degenhardt et al, Phys. Rev. A. 70, 023414 (2004). 19. S. G. Porsev, A. Derevianko, and E. N. Fortson, Phys. Rev. A 69, 021403 (2004). 20. C. W. Hoyt et al, Phys. Rev. Lett. 95, 083003 (2005). 21. P. Lemonde and P. Wolf, Phys. Rev. A 72, 033409 (2005). 22. T. Hong, C. Cramer, W. Nagourney, and E. N. Fortson, Phys. Rev. Lett. 94, 050801 (2005); R. Santra et al, Phys. Rev. Lett. 94, 173002 (2005). 23. J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003). 24. E. Peik et al, Phys. Rev. Lett. 9 3 , 170801 (2004). 25. In Table 2, sensitivities q is defined as follows: the clock transition frequency varies as v = i/o + q(a2/aQ — 1) + 0(a ), where ao is the present-day value of a. See, E. J. Angstmann, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. A 70, 014102 (2004). 26. M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, Opt. Lett. 30, 1815 (2005).
282
DEVELOPMENT OF MACH-ZEHNDER INTERFEROMETER AND "COHERENT BEAM STEERING" TECHNIQUE FOR COLD NEUTRONS K. Taketani.*-, H. Funahashi and Y. Seki Department of Physics, Kyoto University, Kyoto 606-8502, Japan . -E-mail:[email protected] M. Hino, M. Kitaguchi and R. Maruyama* Research Reactor Institute, Kyoto University, Kumatori, Osaka 590-0494, Japan Y. Otake and H. M. Shimizu RIKEN, Wako, Saitama 351-0198, Japan We are developing multilayer interferometer of Mach-Zehnder type for cold neutrons. To fulfill severe requirements for aligning the four mirrors, we utilize six solid-etalon-plates and a high precision flat base plate. New devices making use of magnetic birefringence have been developed and demonstrated to be useful for fine adjustment of superposition between the two paths labeled with neutron spin. Keywords: Neutron Interferometry, Cold Neutron, Mach-Zehnder Interferometer
1. Introduction One of the most important applications of interferometer is precision measurements of the energy difference. When there is an interaction energy difference AE between two paths of a neutron interferometer, the relative phase is given by the expression
At = 2x^AE h2
(1)
where m is the neutron mass, I is the neutron wavelength, and L is the interaction path length. A large interferometer for long wavelength neutrons enables us to measure small interaction energy difference. The multilayer mirror is one of the most useful devices in cold-neutron optics. A multilayer of two materials with different potentials is understood as a one-dimensional crystal that is suitable for Bragg reflection of long-wavelength neutrons. Since the first successful interferogram of cold neutrons using multilayer mirrors [1], some remarkable experiments [2] have been performed. The aim of our development is to increase
the spatial beam separation of multilayer interferometer in order to broaden the applicability of neutron interferometry and carry out high precision measurements. 2. Jamin interferometer The first interferometer using four independent multilayer mirrors was achieved in 2002 [3]. Interference fringes with a contrast of 60% were observed. The interferometer is equivalent to the Jamin type based on a pair of air-spaced etalons. An etalon consists of two parallel planes with high-precision surface finish. The new devices named "beam-splitting etalons" (BSE) in ref[3], which are etalons deposited multilayer mirrors on their parallel planes, splits a neutron beam into two parallel paths spatially. The two waves are superposed each other on the second BSE. The success of Jamin type has qualified the flatness and roughness of etalon plates. They are good enough for neutron-mirror substrates, which cause no serious distortion of wave front to compose an interferometer.
t Present address JAERI, Tokai, Ibaraki 319-1195, Japan
283
3. Mach-Zehnder interferometer In order to enlarge an area enclosed by the two paths of interferometer, we are developing Mach-Zehnder interferometer (Fig. 1). To fulfill the severe requirement for aligning the four mirrors, we utilize six solid-etalon-plates. Each of them has a pair of precision parallel planes and an equal thickness to all of the six. We deposit neutron mirrors on four of the six solid etalons, and place the six on a high precision flat base plate using optical contact technique. To determine the tolerance of etalons and base plate, we have examined the coherence lengths of our neutron beam and also examined moire fringes arising from finite crossing angle of the superposed two waves. The coherence length is here defined as the optical path-difference which gives interferograms with the contrast of e _1 of the maximum. All our experiments are performed using the cold neutron beam line "MINE2" at the JRR-3M reactor in JAERI. The mean wavelength A is 0.88nm and the bandwidth is 2.7% in FWHM. The longitudinal coherence length is evaluated at 17nm which is consistent with neutron spin-echo spectra [3]. The transverse coherence length lT can be approximately expressed as I
=JL2l±.
(2)
In #div where
L
(3)
e
where 0 is the finite crossing angle of the two beams. The contrast C of interferograms obtained using a neutron counter with the narrow slit of w wide is given by C = sin
nw
L
V moire / /
7CW
L
(4)
When w//moire is larger than 0.7, the contrasts are reduced to smaller than e" 1 . Therefore, the crossing angle of the two beams should be smaller than ±44nrad vertically and +1.8urad horizontally. Relative inclination or displacement, or both between the two beams arise from dimensional errors of the six etalons and the flat base plate. For example, irregularity in thickness of the six etalons causes both horizontal and longitudinal displacements, and the finite nonparallelism of etalon plates makes the relative inclination. The imperfection of the flat base also causes the relative inclination and displacement. The etalon plates must match in thickness to±0.15um , and must be parallel within 6nm over the clear aperture of 20mm diameter. The surface of base plate must be flat better than 30nm over the clear aperture of 480mm long by 80mm wide. Using etalons and base plate satisfying tolerances on each degree Analyzer mirror
*/2
Phase shifter coil
Top view 424mm
10mm Magnetic Non Magnetic' mirror Magnetic m j r r 0 r mirror
SOO'ifif"--
Fig. 1 Schematic view and photograph of the present neutron multilayer interferometer of Mach-Zehnder type
284
of freedom, we have constructed a pilot unit of Mach-Zehnder interferometer. The beam separation is 10mm and the distance from the splitter mirror to the analyzer mirror is 424mm. We have deposited magnetic mirrors as splitting mirrors and non-magnetic mirrors as reflection mirrors. We performed a trial run, but the contrast of observed interferograms was not significant enough. Each dimension of the etalons and the base was surely within each individual tolerance in the accuracy of measurements, however, it was impossible to measure exactly the value of actual errors, and more than two simultaneous errors would happen in reality. To adjust the superposition and improve the contrast of interferograms, we have developed "coherent beam steering" technique. 4. "Coherent Beam technique for cold neutrons
Steering"
4.1. Moire"fringes
"Coherent beam steering" is a new technique to correct a crossing angle and parallel shift of the two paths using magnetic birefringence. It is possible to label the two paths with spin divided by the magnetic-multilayer splitter of our Mach-Zehnder interferometer. A quadrupole magnet generates a gradient of magnetic field, which deflects a polarized neutron beam along the direction of gradient. The deflection angle is dBr A<9 = //(5) A T ax rnv where m , ju, v are mass, magnetic moment and velocity of neutron respectively. BQ is the strength of magnetic field. L is the length of magnetic field along beam. A triangle solenoid coil bends a polarized neutron beam in the plane perpendicular to the magnetic field just like an optical prism. The bending angle is given by A
mjuBrA
Plank's constant. To avoid the spin depolarization, we usually apply a weak magnetic field of several Gauss in the vertical direction to the whole of setup. When we operate the steering devices introduced above, we have to arrange their field for the neutron beam not to pass through zero magnetic field regions. For adjustment of the horizontal crossing angle and parallel shift, we use triangle solenoids of vertical axis, and for the vertical steering, quadrupole magnets with narrow collimation slits as shown in Fig.2. The crossing angle and parallel shift of our pilot unit were estimated from the flatness data of the base and the specifications of etalons. As the error estimation of horizontal parallel shift is less than half of its tolerance, there is no need to adjust it. We have demonstrated the feasibility of adjustment using the present steering devices in the other degrees of freedom as follows.
a
(6) he = 2-^-~—tan — 2 h 2 where BT is magnetic field inside the solenoid. a is the vertex angle of triangle, h is the
Optical devices of the MINE2 were arranged in a form of neutron spin-interferometer (NSI) [5]. When the two spin components between the polarizer and analyzer are deflected in the opposite direction slightly even tens of nano radians, an image of moire' fringe appears in the superposition as the evidence of deflection. We used a quadrupole magnet shown in
ALU.
Top view of the triangle solenoid
100mm 50mm
Neutron
Down
Side view of the quadrupole magnet
i0omm
Fig. 2 Schematic view and photograph of the quadrupole magnet and the triangle solenoid
285 Fig.2. The gradient of magnetic field was 3.2G/cm at 2A. The size of magnet along beam was 50 mm. Clear moire fringes were recorded using imaging plate (Fuji film BAS1800) with exposure time of 60min. The interval of them was roughly 10mm. The shrinkage of moire fringes was also observed by increasing the current of the quadrupole magnets. The stronger gradient of magnetic field makes the larger crossing angle between the two spin components. Over the range of about ± 90nrad, we have demonstrated the control of vertical crossing angle. We prepared a triangle solenoid shown in Fig.2. This triangle solenoid generated 50G inside the triangle when it was applied 4A. The vertex angle was 127deg. Using a scanning detector with a fine slit of 0.25mm wide, we observed clear moire fringes with the interval of approximately 1mm at the current of 4A. This result demonstrates that we can control horizontal crossing angle over the range of about ±lurad.
We are developing multilayer interferometer of Mach-Zehnder type to carry out high precision measurements, and have developed "coherent beam steering" technique. The experimental results in this paper show that when we apply the present technique to our Mach-Zehnder interferometer, we can realize the adjustment of superposition accurately enough to achieve clear interferograms.
4.2. Recovery of contrast
Acknowledgments
We tested the adjustment of vertical parallel shift using a pair of quadrupole magnets. We installed a pair of BSE with the gap thickness of 9.75 um [4] and placed the pair of quadrupole magnets at the downstream of the second BSE. The inter distance of the 200 150
1 1 5°0° U °
******
0A
36 ±2% • V• T.*
|i.56nm QQ
0
« 200 § 150 0 100 " 50
1 °
5 200 a> 150
****
59±2^ + +
1.75A
31 ±2%
3.5A
V* \/ * vHHI ••*•.
-M~
100 V ••»•• *•.•** 50 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Phase shifter coil current [A]
Fig. 3 Demonstration of the recovery of NSI interferograms using a pair of quadrupole magnets
pair of quadrupole magnets was 500mm. The maximum contrast of the NSI interferograms was 60% without tilting the second BSE. When we tilted the second BSE in 0.16deg, the contrast of interferograms decreased from 60% to 36% due to the intentional parallel shift of the two paths caused by twisting the pair of BSEs [4]. Although a quantitative explanation has yet to be produced for the optimum current, we have succeeded in recovering the contrast from 36% to 60% using the pair of quadrupole magnets. 5. Summary
This work was supported by the interuniversity program for common use JAERI and KUR, and financially by Special Coordination Funds for Promoting Science and Technology of the Ministry of Education of Japanese Government. One of the authors (K.T.) was supported by JSPS Research Fellowships for Young Scientists. References 1. H. Funahashi, et al., Phys. Rev. A54 (1996) 649. 2. Y. Otake, et al., Proc. of ISQM, Tokyo, 98 (1999)323. 3. M. Kitaguchi, et al., Phys. Rev. A67 (2003) 033609.; M. Kitaguchi, PhD thesis, Kyoto University (2004). 4. M. Kitaguchi, et al., JPSJ 72 (2003) 3079. 5. D. Yamazaki, Nucl. Instr. Meth. A488 (2002) 623.
286 SURFACE P O T E N T I A L M E A S U R E M E N T B Y ATOMIC FORCE M I C R O S C O P Y U S I N G A QUARTZ RESONATOR
SEIJI HEIKE* AND T. HASHIZUME Advanced
Research
Laboratory, Hitachi, Ltd., Hatoyama, Saitama * E-mail: [email protected]
350-0395,
Japan
Scanning Kelvin probe microscopy is demonstrated by using a noncontact atomic force microscope with a 1 MHz quartz length-extensional resonator as a force sensor. A tungsten probe tip glued onto the end of the quartz rod is electrically connected to the electrode of the resonator. To detect Coulomb force, frequency shift signal is used instead of oscillation amplitude of the resonator. A surface potential on a A u ( l l l ) single crystal with nanoscale carbon dots is measured. The potential difference is observed between the Au surface and the carbon dots. Keywords: Kelvin probe; atomic force microscopy; surface potential; work function.
1. Introduction
ever, it is quite difficult to realize Kelvin Scanning tunneling microscope (STM) [1] probe measurement using quartz resonator, because the resonator has a low sensitivity have been expected to be a powerful tool for to Coulomb force because of its high force miniaturization of future electronic devices. constant. In this paper, we report developAlthough recent developments have made it ment of the SKPM measurement system uspossible to manipulate individual atoms and ing a quartz resonator and demonstration of molecules with STM probe tips [2-4], the aca potential mapping on a Au surface using 1 tual devices should include insulating mateMHz quartz length-extensional resonator. rials which cannot be observed with an STM. As a scanning probe method with atomic resolution, noncontact atomic force microscopy 2. E x p e r i m e n t s (NC-AFM) has been spotlighted in recent Figure 1(a) shows a photograph of the resyears. In particular, the frequency modulaonator consisting of a rod shaped quartz tion (FM) detection technique [5] realized to (2.75 mm x 85 /mi x 85 /*m, force constant achieve AFM observation with a true atomic k~ 4 x 105 N/m) and two electrodes on both resolution [6-9]. However, in the process of sides of the rod as introduced in [10]. Figsurface modification where electric field is ofure 1(b) shows an electrochemically etched ten used, a sudden change in electrostatic tungsten probe tip (100 to 200 /mi in length) force causes bends in cantilever. To avoid glued onto the end of the rod and electrithis problem, a quartz resonator with a high cally connected to one of the electrodes by force constant was used as a resonator for using silver epoxy. Bias voltages are applied NC-AFM [10-13]. Moreover, a quartz resbetween the tip and the surface for SKPM onator can sense the deflection electrically by observation. itself. A schematic diagram of the SKPM deTo investigate electrical characteristics tection system is shown in Fig. 2. An ac voltof nanoscale electronic devices, scanning age signal tuned to the resonant frequency /o Kelvin probe microscopy (SKPM) based on of the resonator is applied to one of the elecNC-AFM is a useful method because it can trodes, and the output current from another map the surface potential [14, 15]. Howelectrode is converted to a voltage signal by
287
phase shift
voltage J1
Fig. 1. (a) 1 MHz quartz resonator, (b) Magnified photograph of end of the resonator. Tungsten tip is glued by silver epoxy to electrically contact with one of the resonator electrodes.
a pre-amplifier. The phase shift data for the AFM feedback are obtained by comparing the phases of the input voltage and the output current. The phase shift can easily be converted to a frequency shift A / by using a frequency-phase curve. The oscillation amplitude AQ is 0.4 to 0.6 nm, estimated from the displacement of the tip height when the amplitude was changed in the STM feedback mode. The bias voltage is applied to the sample and the tunneling current through the sample is detected for STM operation. For Kelvin probe measurement, an ac voltage fy (~ 100 Hz, 2 V) was added to the bias voltage. The amplitude of the fy component in the phase shift signal is detected with a lock-in amplifier and used as an electrostatic force signal. The surface potential is determined by adjusting the bias voltage so as to make the electrostatic force zero. In potential mapping measurement, the Kelvin probe feedback loop works only when the tip comes to data acquisition points for mapping. The tip stays at each point for 100 ms until the feedback is completed. The AFM used in this experiment was based on a commercial Omicron ultra high vacuum (UHV) STM and the whole scanner head was replaced by a homemade one. We used a polished sapphire sample, a Si(lll) clean suface sample and a A u ( l l l ) single crystal sample. The silicon wafer (n-type,
Fig. 2. Schematic diagram of the SKPM detection system. An ac Yoltage /Q (~ 1 mV) is applied to one of the electrodes, and the output current from another electrode is detected by a pre-amplifier. The phase shift data are obtained by comparing the input and output signals. For Kelvin probe measurement, an ac voltage fy is added to the sample bias voltage. The amplitude of the fy component in the phase shift signal is detected with a lock-in amplifier as an electrostatic force signal.
0.005 Ocm) was flushed in UHV to obtain a 7x7 structure. The A u ( l l l ) sample was sputtered by Ar to make small hole structures on the surface and was annealed at around 300 °C. The chamber pressure was maintained below IxlO"" 8 Pa during the SKPM measurements. An electrochemically etched tungsten tip was cut and glued onto the resonator, and then was sputtered by Ar to remove the overlaying oxide layer formed during etching. 3* R e s u l t s a n d discussions To confirm AFM/STM observation, we measured topographic images of polished sapphire surface and Si(lll)-7x7 clean surface. Figure 3 (a) shows an AFM image over the 500 nm x 500 nm area taken with Af = —0.5 Hz and AQ = 0.4 nm. Single atomic steps of 0.2 nm high and terraces of 150 nm wide are clearly observed. A magnified image with 100 nm x 100 nm area is shown in Fig. 3 (b). The terraces are not atomi-
288 cally l a t with a roughness of about 0.1 nm. Next, AFM/STM observation was demonstrated on a Si(lll)-7x7 surface. Figure 3 (c) shows an AFM image of Si(lll)-7x 7 surface with 8.4 nm x 8.4 nm area obtained with imaging parameters, Af = —0.5 Hz, Vsampie = 0 V and A0 = 0-55 nm. An STM image obtained with a tunneling current of 0.5 nA and Vsampie = 2.0 V is shown in Fig. 3 (d). The scan size is the same as that of Fig. 3 (c), though the imaged region is not the same. Individual Si atoms and corner holes are resolved in both images. However, the resolution of the AFM image is a little bit lower than that of the STM image. These results indicate that sub-nanometer resolution is achieved both on insulating and conducting surface.
Fig. 3. Images of A F M / S T M measurements, (a) AFM image of polished sapphire surface observed in 500 nm X 500 n m area. Parameters are Af = —0.5 Hz and AQ = 0.4 nm. (b) Magniied image with 100 nm x 100 n m area, (c) AFM image of S i ( l l l ) - 7 x 7 surface obtained with Af = —0.5 Hz, V3ampie = 0 V and AQ = 0.55 nm. (d) STM image observed using the same tip as (c). Itunnei = 0.5 nA and F 5 0 m p l e = 2.0 V.
age. The imaging parameters were Af = -0.2 Hz and Vup = 0 V. Flat terraces with atomic steps are clearly observed. The hole structures, 5 to 20 nm in diameter, may be caused by Ar sputtering. Hillocks with approximately 1 nm in height and 10 to 20 nm in diameter are also observed. Since these structures have conically shaped terraces and seem to pin the steps, they probably consist of hydrocarbon not removed by sputtering. Figure 4 (b) is a simultaneously obtained surface potential image for the same area as (a). The brightness corresponds to the work function. Bright spots indicate approximately 100 mV higher work function than that of the dark region. Because the position of these spots agree well with the position of the hillocks in Fig. 4 (a), the spots can be attributed to the hillock structures.
Fig. 4. (a) 200 nm x 200 nm AFM image of A u ( l l l ) single crystal surface after Ar sputtering. Parameters are Af = —0.2 Hz and AQ = 0.6 nm. Atomic steps and hole structures caused by sputtering are clearly observed. Hillocks, around 1 nm in height, 20 n m in diameter, are also observed, which may originate from carbon contamination remaining after sputtering, (b) Simultaneously obtained surface potential mapping of the same area as (a). The bright spots correspond to the hillock structures in (a), and their work function is 100 mV higher than that of the A u ( l l l ) surface.
Because of the enhanced contrast, the noise signals appear as small dark spots not related to actual surface structures. From the minimum size of the spots corresponding The surface potential was measured on to the hillocks, the lateral resolution of pothe Au surface with 200 nm x 200 nm area. tential mapping is estimated to be less than Figure 4 (a) shows the AFM topographic im- 5 nm. From the potential difference between
289 the Au surface and hillocks, the energy resolution is estimated to be less than 100 mV. Although surface potential data are averaged in the actual mapping measurement, there still remain noises of around 50 mV, estimated from a cross section of Fig. 4 (b). Most presumable origin of the noise is intermittent feedback signals for potential measurement. At every data acquisition point, the Kelvin probe feedback loop has to experience an unstable transition period where the bias voltage is fluctuated until the feedback in completed. To remove this problem, a continuous feedback system should be introduced.
Acknowledgments This study was performed through Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government. References
1. G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Phys. Rev. Lett. 49, 57 (1982). 2. D. M. Eigler, and E. K. Schweizer, Nature 344, 524 (1990). 3. H. Uchida, D. Huang, J. Yoshinobu, and M. Aono, Surf. Sci. 2 8 7 / 2 8 8 , 1036 (1993). 4. T. Hitosugi, S. Heike, T. Onogi, T. Hashizume, S. Watanabe, Z.-Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and K. 4. Summary Kitazawa, Phys. Rev. Lett. 82, 4034 (1999). 5. T. R. Albrecht, P. Griitter, D. Home, and Surface potential measurement was perD. Ruger, J. Appl. Phys. 69, 668 (1991). formed by SKPM based on the NC-AFM 6. F. J. Giessibl, Science 267, 68 (1995). using a 1 MHz quartz length extension res- 7. S. Kitamura, and M. Iwatsuki, Jpn. J. Appl. Rhys. 34, 145 (1995). onator as a force sensor. A tungsten probe 8. H. Ueyama, M. Ohta, Y. Sugawara, and S. tip glued onto the end of the quartz rod was Morita, Jpn. J. Appl. Phys. 34, 1086 (1995). electrically connected to the electrode of the 9. T. Eguchi, and Y. Hasegawa, Phys. Rev. resonator for the detection of Coulomb force Lett. 89, 266105 (2002). between the tip and the surface. The fre- 10. U. Grunewald, K. Bartzke, and T. Antrack, quency shift signal was used for Coulomb Thin Solid Films 264, 169 (1995). force detection instead of oscillation am- 11. A. Michels, F. Meinen, E. Bechmann, T. Murdfield, W. Gohde, U. C. Fischer, and H. plitude of the resonator used in the conFuchs, Thin Solid Films 264, 172 (1995). ventional SKPM. A surface potential was 12. F. J. Giessibl, Appl. Phys. Lett. 76, 1470 measured on a A u ( l l l ) single crystal with (2000). nanoscale carbon dots. The potential dif- 13. S. Heike, and T. Hashizume, Appl. Phys. Lett. 83, 3620 (2003). ference of 100 mV were observed between Au surface and the carbon dots. The lat- 14. A. Kikukawa, S. Hosaka, and R. Imura, Appl. Phys. Lett. 66, 3510 (1995). eral resolution was less than 5 nm, and the 15. L. Biirgi, H. Sirringhaus, and R. H. energy resolution was less than 100 mV. It Friend, Appl. Phys. Lett. 80, 2913 (2002). was shown that the resolution of potential Rhys.4126152002.
measurement was high enough to evaluate nanoscale electronic devices.
290 BERRY'S P H A S E S A N D TOPOLOGICAL P R O P E R T I E S IN T H E BORN-OPPENHEIMER APPROXIMATION
KAZUO FUJIKAWA Institute of Quantum Science, College of Science and Nihon University, Chiyoda-ku, Tokyo 101-8308, E-mail: [email protected]
Technology Japan
The level crossing problem is neatly formulated by the second quantized formulation, which exhibits a hidden local gauge symmetry. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval T. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. The crucial difference between the AharonovBohm phase and the geometric phase is explained. It is also noted that the gauge symmetries involved in the adaibatic and non-adiabatic geometric phases are quite different.
1. Introduction The geometric phases revealed the importance of hitherto un-recognized phase factors in the adiabatic approximation 1 ' 2 ' 3 ' 4 ' 5 ' 6 . It may then be interesting to investigate how those phases behave in the exact formulation. We formulate the level crossing problem by using the second quantization technique, which works both in the path integral and operator formulations 7 ' 8 ' 9 . In this formulation, the analysis of geometric phases is reduced to the familiar diagonalization of the Hamiltonian. Also, a hidden local gauge symmetry replaces the notions of parallel transport and holonomy. When one diagonalizes the Hamiltonian in a very specific limit, one recovers the conventional geometric phases defined in the adiabatic approximation. If one diagonalizes the Hamiltonian in the other extreme limit, namely, in the infinitesimal neighborhood of level crossing for any fixed finite time interval T, one can show that the geometric phases become trivial and thus no monopole-like singularity. At the level crossing point, the conventional energy eigenvalues become degenerate but the degeneracy is lifted if one diag-
onalizes the geometric terms. Our analysis shows that the topological interpretation 3 ' 1 of geometric phases such as the topological proof of the Longuet-Higgins' phase-change rule 4 fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. This analysis shows that the topological properties of the geometric phase and the Aharonov-Bohm phase are quite different. Also, the difference between gauge symmetries for adiabatic phase and "nonadiabatic phase" by Aharonov-Anandan 10 becomes quite clear in this formulation.
2. Second quantized formulation We start with the generic hermitian Hamiltonian H = H(p,x,X(t)) for a single particle theory in a slowly varying background variable X(t) = {X1{t),X2{t),...). The path integral for this theory for the time interval 0 < t < T, which is taken to be the period of the slower background system, in the second quantized formulation is given by
fvip*Vipexp{?r
J
dtd3x[C}}
291 where
When we define the Schrodinger picture %eff{t)by replacing all bn(t) ->• bn(0) in Heff(t) we can show 7,8
8 i/>*(t,x)ih—ip(t,x)
C =
-r(t,S)H(^,x,X(t)Mt,x).(l)
We then define a complete set of eigenfunctions H(^ —
= (n\T*e:X P {
4/
H&2,X(t))dt}\n(0)) Heff(t)dt}\n).
(7)
,x,X(t))vn(x,X(t))
Both-hand sides of this formula are exact, but the difference is that the geometric = £n(X(t))vn(x,X(t)), terms, both of diagonal and off-diagonal, are Jd3xv*n(x,X(t))vm(x,X(t)) = <5„,m (2) explicit in the second quantized formulation on the right-hand side. The state vectors and expand in the second quantization are defined by \n) = 6jj(0)|0), and the state vectors in the iP(t,x) = J2bn{t)vn(x,X(0)). (3) first quantized states by (2). If one retains n only the diagonal elements in this formula The path integral for the time interval 0 < (7), one recovers the conventional adiabatic t < T in the second quantized formulation is formula5 given by
Z=
exp{~ J
[l[-Db*n-Dbn J
n
xexp{^ f
dt[J^b*n(t)ih^-tbn(t)
u
n
-X>*(*)£«(A-(t))M*)]}
(4)
n
where the second term in the action stands for the term commonly referred to as Berry's phase 1 and its off-diagonal generalization. The second term is defined by (n\ih—\m)
=
. fl d'xv*n(x,X(t))ih-vm(x,X(t)).
h
=
Y/bt(t)£n(X(t))bn(t) -Y,biit)(n\ih^-\m)bm{t).{&)
-
(n\ih—\n)]}.{8)
The above formula (7) represents the essence of geometric phases: If one performs an exact evaluation one does not obtain a clear physical picture of what is going on. On the other hand, if one makes an adiabatic approximation one obtains a clear universal picture. The path integral formula (4) is based on the expansion (3), and the starting theory depends only on the field variable ip(t, x), not on {bn(t)} and {vn(x,X(t))} separately. This fact shows that our formulation contains a hidden local gauge symmetry v'n(x,X(t))=eia^vn(x,X(t))
(5)
In the operator formulation of the second quantized theory, we thus obtain the effective Hamiltonian Heff(t)
dt[£n(X(t))
b'n(t)=e-^%n(t)
(9)
where the gauge parameter an(t) is a general function of t. By using this gauge freedom, one can choose the phase convention of the basis set {vn(x,X(t))} at one's will such that the analysis of geometric phases becomes simplest. Prom the view point of hidden local symmetry, the formula (8) is a result of the specific choice of eigenfunctions
292 v„(x,X(0)) — v„(x,X(T)) variant expression
in the gauge in-
vn(x;X(0))*vn(x;X(T))x exp{-^ J
(10)
[Sn(X(t)) -
The effective Hamiltonian (6) is then given by Heff(t)
=
(E(t)+gr(t))blb+
+ (E(t) -
(n\ih-\n)\dt}.
gr(t))blb-
k
This hidden local symmetry replaces the notions of paralell transport and holonomy in the analyses of geometric phases, and it works not only for cyclic but also for noncyclic evolutions 9 . 3. Level crossing problem For a simplified two-level problem, the Hamiltonian is defined by the matrix in the neighborhood of level crossing 7
h(X(t))=^]
0 l E it^+ga m(t)(U)
- hY,blA mn{y)ykbn.
with r(t) = y/yl + y\ + y\. The point r(t) = 0 corresponds to the level crossing. In the adiabatic approximation, one neglects the off-diagonal terms in the last geometric terms, which is justified for Tgr(t) ;§> hit, where tm stands for the magnitude of the geometric term times T. The adiabatic formula (8) then gives the familiar result exp{i7r(l — cos#)} xexp{-1-
after a suitable re-definition of the parameters by taking linear combinations of Xk (t). Here yi (t) stands for the background variable and a1 for the Pauli matrices, and g is a suitable (positive) coupling constant. The eigenfunctions in the present case are given by
-<"-(*£?)
(12)
(14)
m,n
[ n
dt[E(t) -
gr(t)]ll5)
JC(Q->T)
for a 2ir rotation in ip with fixed 0, for example. To analyze the behavior near the level crossing point, we perform a unitary transformation bm = J2n U(0(t))mncn where m,n run over ± with
"<»<«»-(E|"c#)'
(16)
which diagonalizes the geometric terms and the above effective Hamiltonian (13) is writ-
by using the polar coordinates, yi = rsin9cos(p, 3/2 = rsmdsiaip, y$ = rcos9. Note that, by using hidden local symmetry, our eigenfunctions are chosen to be periodic under a 2n rotation around 3-axis, which is quite different from a 2TT rotation of a spin1/2 wave function. If one defines «m(y)*^«n(l/) =
Akmn{y)yk
where m and n run over ± , we have .k . , . (1 +cos6>) . Ak++(y)yk = g *
Ak+_(y)yk = AUy)n
S
^
+l-6 = (Ak_+(y)yky,
=^ ^ V
(13)
Fig. 1. The path 1 gives the conventional geometric phase for a fixed finite T, whereas the path 2 gives a trivial phase for a fixed finite T. Note that both of the paths cover the same solid angle 27r(l — cos8).
293 5. Discussion
ten as Heff(t)
a (E(t) + gr cos 6)c\c+ + (E(t) — grcos6)c_£-
(17) — tupc'+c+
in the infinitesimal neighborhood of the level crossing point, namely, for sufficiently close to the origin of the parameter space (Vi(t),y2(t),y3(t)) but (yi(t),y2{t),y3(t)) ? (0,0,0). To be precise, for any given fixed time interval T, we can choose in the infinitesimal neighborhood of level crossing Tgr(t)
(18)
and thus no physical effects. In the infinitesimal neighborhood of level crossing, the states spanned by (6 + ,6_) are transformed to a linear combination of the states spanned by (c+,C-), which give no non-trivial geometric phase. We emphasize that this topological property is quite different from the familiar Aharonov-Bohm effect 10 , which is topological^ exact for any finite time interval T. Besides, the setting of the Aharonov-Bohm effect differs from the present level crossing problem in the fact that the space is not simply connected in the case of the AharonovBohm effect. 4. Non-adiabatic phase We comment that the non-adiabatic phase by Aharonov and Anandan 10 is based on the equivalence class (or gauge symmetry) which identifies all the Schrodinger amplitudes of the form { e - ^ M } .
The notion of Berry's phase is known to be useful in various physical contexts 11 . Our analysis however shows that the topological interpretation of Berry's phase associated with level crossing generally fails in the practical Born-Oppenheimer approximation where T is identified with the period of the slower system. The notion of "approximate topology" has no rigorous meaning, and it is important to keep this approximate topological property of geometric phases associated with level crossing in mind when one applies the notion of geometric phases to concrete physical processes.
(19)
This gauge symmetry is quite different from our hidden local symmetry which is related to an arbitrariness of the choice of coordinates in the functional space.
I thank Prof. N. Nagaosa for stimulating discussions. References 1. M.V. Berry, Proc. Roy. Soc. A392, 45 (1984). 2. B. Simon, Phys. Rev. Lett. 51, 2167 (1983). 3. A.J. Stone, Proc. Roy. Soc. A351, 141 (1976). 4. H. Longuet-Higgins, Proc. Roy. Soc. A344, 147 (1975). 5. H. Kuratsuji and S. Iida, Prog. Theor. Phys. 74, 439 (1985). 6. M.V. Berry, Proc. Roy. Soc. A414, 31 (1987). 7. K. Fujikawa, Mod. Phys. Lett. A20, 335 (2005). 8. S. Deguchi and K. Fujikawa, Phys. Rev. A72, 012111 (2005). 9. K. Fujikawa, Phys. Rev. D72, 025009 (2005). 10. Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987). 11. For the review of the subject see, for example, A. Shapere and F. Wilczek, ed., Geometric Phases in Physics (World Scientific, Singapore, 1989); D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics (Birkhauser, Berlin, 2004).
294 S E L F - T R A P P I N G OF BOSE-EINSTEIN C O N D E N S A T E S B Y OSCILLATING I N T E R A C T I O N S
HIROKI SAITO1 and MASAHITO TJEDA1'2 Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan CREST, Japan Science and Technology Corporation (JST), Saitama 332-0012, Japan We show that the self-trapped condensate mensional free space by periodic oscillation produces an effective potential that prevents the effect of dissipation to simulate realistic
can be dynamically stabilized in two and three diof the scattering length. The oscillating interaction the condensate from collapsing. We take into account situations.
Keywords: Bose-Einstein condensate; Feshbach resonance; soliton
1. Introduction A raindrop keeps its shape without the help of an external confinement potential because of the interaction between water molecules. The attractive part of the interaction potential between molecules prevents a droplet from expanding and the repulsive part prevents the collapse. Is it possible to make a droplet in the gas phase using the ultracold gas? In one dimension (ID), the answer is yes. Recently, the ENS [1] and Rice [2] groups have succeeded in creating the matter-wave bright solitons using a Bose Einstein condensate (BEC). The soliton in a quasi-lD trap keeps its shape, since the attractive interaction counterbalances the quantum kinetic pressure. The effective potential for the characteristic size of the condensate R consists of the interaction and kinetic energies, which are proportional to —R~d and R~2 in d dimensions. Hence the effective potential always has a minimum for d = 1. In 2D and 3D, however, the effective potential for R has no minimum, and the matter-wave droplet is always unstable against collapse or expansion, as long as the interaction is constant in time. We try to stabilize the matter-wave droplet in 2D and 3D by oscillating the interaction using the Feshbach resonance. This
idea comes from the "inverted pendulum" in classical mechanics [3,4], in which rapid oscillation of the pivot stabilizes the pendulum in an upturned position. Rapid modulation of the motion of the pendulum effectively produces a barrier against falling down. In a similar manner, we will show that rapid oscillation of the interaction between repulsive and attractive effectively produces a barrier that prevents the condensate from collapsing and stabilizes the "gaseous droplet." [5-7] 2. Gross-Pitaevskii equation with oscillating interactions We consider the Gross-Pitaevskii (GP) equation given by at
2m
m
-H'7/it/>,
(1)
where m is the mass of the atom and a(t) is the time-dependent s-wave scattering length. We assume that a(t) oscillates as a(t) = do + a\ sin
fit.
(2)
The factor 7 on the left-hand side of Eq. (1) describes the energy dissipation, and the last term on the right-hand side assures the conservation of the norm, where fi is the chemical potential. This type of phenomenological dissipative equation has been used to
295 study the damping of collective modes [8] and vortex nucleation [9]. When 7 = 0, Eq. (1) reduces the usual GP equation in free space. In order to reach the dynamically stabilized state, we start from the noninteracting ground state in a trapping potential. The trapping potential is gradually turned off as v{t)=(V(0)(l-t/Ttr) K
'
\ 0
t
v ;
At the same time, the oscillating scattering length is gradually turned on as Or(t)
a(t){t/T&) a(t)
t
(4)
T
where a(t) is given by Eq. (2). This gradual change of parameters avoids initial nonadiabatic disturbances and makes the condensate more stable. 3. Two-dimensional case We first consider a tight pancake-shaped Fig. 1. (a) Time evolution of the central dentrap, in which the axial frequency co± is sity }MT_= 0)| 2 and the mean radius (r) = 2 2 85 much larger than the radial frequency u>±_ f y/xi + y \ilj\ dxdy of a R b in 2D with 7 = 0.03. The radial frequency of the trap is u>± = 27r X 10 and other characteristic frequencies. In this Hz, and the number of atoms is 10 3 . The scattering case, the axial dynamics is frozen and the length oscillates as — 0.6 + 2.4sin(30ui±t). The ramp system behaves as 2D with an effective scat- functions in Eqs. (3) and (4) with T t r = T a = 0.3 s are used, (b) The density profiles at t = 0 (dashed tering length y/mu!z/(2Trh)a. line) and ( = 1 s (solid line). The inset shows the We numerically solved the GP equation semilogarithmic plots, where the dotted line is pro18r in 2D and obtained the time evolution of the portional to e~ . system as shown in Fig. 1 (a). The initial state is the noninteracting ground state with COZ/LJ± = 50, and the ra- same as the initial Gaussian profile, but the dial trapping potential uj2L(x2+y2)/2 is grad- peak density is higher and the tail is slightly ually turned off while the oscillating inter- longer. The inset shows the semilogarithmic action is gradually turned on according to plots of the density profiles. We find that the Eqs. (3) and (4). We find that even af- tail of the droplet is approximately propor_18r . It is interesting to note that ter the radial confinement is completely re- tional to e the Townes soliton also decays exponentially moved, the condensate does not expand, nor at infinity [10]. does it collapse, and thus the BEC "droplet" We note that the amplitude of the slow is stabilized. The condensate undergoes the rapid oscillation due to the oscillating inter- oscillation gradually decays, while the ampliaction and the slow oscillation due to the tude is almost constant for 7 = 0 in Ref. 5. initial nonadiabaticity. Figure 1 (b) show The decay of the slow oscillation is therefore the density profiles at t = 0 and t = 1 s. due to the energy dissipation. For large t, The "droplet" state at t = 1 s is almost the the slow oscillation decays to vanish, and the
296
2
4
6
8
10
Fig. 2. (a) Time evolution of the central density \tl>(r = 0)| 2 and the mean radius (r) = fr\ip\2dr of a 8 5 R b in 3D with 7 = 0.03. The frequency of the isotropic trap is u> — 2n X 10 Hz, and the number of atoms is 5 x 10 3 . The scattering length oscillates as —0.7 + 1.5sin(30a;t). The ramp functions in Eqs. (3) and (4) with Tti-0.3 s and T a = 0.2 s are used, (b) The density profiles at t = 0 and t = l s .
isotropic with the radial distribution shown in Fig. 2 (b). We find from Fig. 2 (b) that the density profile of the droplet strongly deviates from the Gaussian. Thus, we have demonstrated that the BEC droplet can be stabilized in 3D in the presence of dissipation with 7 = 0.03. However, we have not succeeded in stabilizing it for 7 = 0 yet. Even when we start from the stable state in Fig. 2 and adiabatically decrease 7, unstable oscillations grow and destroy the droplet. The dissipation therefore suppresses the dynamical instabilities and stabilizes the droplet. We have not understood the origin of the dynamical instabilities yet. The stabilization in 3D is much more difficult than in 2D; still, we cannot exclude the possibility that there is a parameter regime in which 3D droplets are stabilized.
5. Stabilization mechanism
The physical mechanism of the stabilization of BEC droplets is very similar to that in the inverted pendulum [3,4]. Here we understand it by a simple argument. As mentioned in Sec. 1, the interaction energy is proportional to —R~d in d dimensions. Hence effecd oc time evolution becomes "stationary" with tive "force" for R is given by -d(R~ )/dR d 1 R~ ~ , which rapidly oscillates. According the rapid oscillation. This state is the Floto the textbook [3], an effective potential proquet solution of Eq. (1). duced by rapid oscillation of the force is proportional to the square of the amplitude of 4. Three-dimensional case the oscillating force, and then oc R~2d~2. by We numerically solve the 3D GP equation. In 2D, the effective potential produced 6 the oscillating interaction is oc R~ , which The initial state is the noninteracting ground the attractive interaction state confined in an isotropic trapping po- can counterbalance -2 2 2 oc — J R , and then stabilizes the droplet. In tential mw r /2 with w = 2n x 10 Hz. The 3D, the effective potential is oc R~8, which trapping potential is linearly turned off from the attractive intert = 0 to 0.3 s, while the oscillating scatter- can also counterbalance 3 action oc — R~ . Thus, the oscillating intering length — 0.7+ 1.5sin(30u>£) nm is linearly action prevents the condensate from collapsturned on from t = 0 to 0.3 s. Even after ing and stabilizes the droplet. the trapping potential vanishes, the condenThe quantitative analysis of the above sate neither expands nor collapses, as shown argument has been done in Ref. 5, 6 using the in Fig. 2 (a). The density profile of the 3D droplet is variational method with Gaussian trial wave
297 functions. T h e Gaussian variational method can predict the parameter dependences of the size of the droplet a n d the collective frequencies in 2D [5]; however, in 3D, it fails t o predict even the stability [6]. Reference 11 has pointed out t h a t the variational and other approximations used so far have considerable deviations from t h e exact numerical results. We must therefore find more reliable approximations t o t r e a t the rapidly oscillating atomic cloud.
6. C o n c l u s i o n s a n d d i s c u s s i o n We have shown t h a t matter-wave droplets can be stabilized in 2D [5] and 3D [6] free space by the oscillating interactions. T h e oscillating interaction produces the effective potential t h a t prevents the droplet from collapsing. One m a y think t h a t t h e rapid oscillation o f t h e interaction induces thermal excitations and the condensate is destroyed. However, this is not the case as long as the oscillation frequency is nonresonant with the excitation frequencies. In fact, numerically solving the full 3D G P equation, we find t h a t no coherent excitations above t h e droplet state grow. We therefore expect t h a t incoherent excitations are also suppressed a n d the B E C is retained. T h e stabilization of the 2D condensate has also been studied in Ref. 12. Reference 13 has considered the multi-component condensate, and shown t h a t the multi-component B E C droplet can also be stabilized by oscillating interactions, which is called a "vector soliton." Reference 14 has studied the case of 3D space with I D optical lattice. For strong lattice potential, the condensate is confined in each lattice site as 2D confinement, a n d a train of droplets is stabilized by oscillating interactions. T h e stabilization by oscillating interactions will be applied to various systems, and the application o f t h e matter-wave droplet will be explored.
Acknowledgments This work was supported by Grant-in-Aids for Scientific Research (Grant No. 17740263, No. 17071005 a n d No. 15340129) and by a 21st Century C O E program at Tokyo Tech "Nanometer-Scale Q u a n t u m Physics," from the Ministry of Education, Culture, Sports, Science and Technology of J a p a n .
References 1. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 (2002). 2. K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature (London) 417, 150 (2002). 3. L. D. Landau and E. M. Lifshitz, Mechanics, (Pergamon, Oxford, 1960). 4. An illuminating account of this stabilizing mechanism is found in: R. P. Feynman, R. B. Leighton, and M. L. Sands, The Feynman Lectures on Physics, (Addison-Wesley, Reading, MA, 1964), Vol. II, Chap. 29. 5. H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403 (2003). 6. H. Saito and M. Ueda, Phys. Rev. A70, 053610 (2004). 7. F. Kh. Abdullaev, J. G. Cupto, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A67, 013605 (2003). 8. S. Choi, S. A. Morgan, and K. Burnett, Phys. Rev. A57, 4057 (1998). 9. M. Tsubota, K. Kasamatsu, and M. Ueda, Phys. Rev. A 6 5 , 023603 (2002). 10. C. Sulem and P. L. Sulem, The Nonlinear Schrodinger Equation, (Springer, New York, 1999). 11. A. Itin, S. Watanabe, and T. Morishita, cond-mat/0506472. 12. G. D. Montesinos, V. M. Perez-Garcia, and P. J. Torres, Physica D 1 9 1 , 193 (2004). 13. G. D. Montesinos, V. M. Perez-Garcia, and H. Michinel, Phys. Rev. Lett. 92, 133901 (2004). 14. M. Trippenbach, M. Matuszewski, and B. A. Malomed, Europhys. Lett. 70, 8 (2005).
298 S P I N O R SOLITONS IN BOSE-EINSTEIN C O N D E N S A T E S — ATOMIC SPIN T R A N S P O R T
J. IEDA1'2 1 CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan E-mail: [email protected] Matter-wave solitons in Bose-Einstein condensates of ultracold gaseous atoms with spin degrees of freedom are investigated based on the exact solution method. Combining some theoretical and experimental results, we present the undiscovered spin dependent soliton dynamics such as macroscopic spin precession and spin switching. Keywords: Bose-Einstein Condensate; Spin Dynamics; Exact Soliton Solution.
1. Introduction
mer is attributed to the inter-atomic interactions while the latter comes from the kinetic BBC of Atomic Gases. — For a decade, Boseenergy. Either dark or bright solitons are alEinstein Condensation (BEC) of atomic lowable depending on the positive or negative gases has renewed theoretical and experisign of the inter-atomic coupling constants a, mental interests in quantum many body sysrespectively, and indeed have been observed tems at extremely low temperatures 1 , which in a quasi-one dimensional (qlD) optically stem from two main features: (1) Almost constructed waveguide3. Such matter-wave all the parameters of the system, such as solitons are expected in atom optics for apthe shape, dimensionality, internal states of plications in atom laser, atom interferometry, the condensates and even the strength of and coherent atom transport 4 . the inter-atomic interactions, are controlSpinor Condensates.— When one exlable. (2) Due to the diluteness, no? -C 1 plores those future applications, atomic BEC (n the particle number density and a the has another advantage. That is, atoms have s-wave scattering length), mean-field thespin degrees of freedom liberated under opory explains experiments. In particular, tical traps 5 ' 6 ' 7 , giving rise to a multiplicity the Gross-Pitaevskii (GP) equation demonas signals. Here, "spin" means the hyperstrates its validity as a basic equation for fine spin of atoms; the hyperfine spin / is dethe dynamics of the condensates. The GP fined by f = s + i, where s and i denote the equation is a counterpart of the nonlinear electronic and nuclear spins. The so-called Schrodinger (NLS) equation in nonlinear opspinor condensates, being a candidate media tics. Thus, a study based on nonlinear analfor multicomponent solitons, have nontrivial ysis acquires importance. nonlinear terms reflecting the SU{2) symMatter-Wave Solitons.— Among the metry of the spins. The spin-exchange innonlinear physics, soliton is a remarkable ob- teractions are the sources of the spin-mixing ject not only for the fact that the exact so- within condensates 8 and there is no analogue lutions can be obtained but also for the use- in conventional nonlinear optics. fulness as communication tools due to the We combine the two fascinating properrobustness 2 . In general, solitons are formed ties: matter-wave soliton and spin dynamics under the balance between nonlinearity and of the condensates, considering alkali atoms dispersion. For atomic condensates, the for-
299 of / = 1 with i = 3/2, e.g., 7 Li, 8 7 Rb, and 23 Na, confined in the qlD space by purely optical means. Without external magnetic fields, the substates m/ = 0, ± 1 (mj the magnetic quantum number) are degenerate. Based on the multicomponent GP equations, the spin dependent soliton dynamics is analyzed.
this, we obtain a set of equations: h2 i/id t $±i = - ^ a * $ ± i + (co " c 2 ) | $ T i | 2 $ ± i + (So + c 2 )(|$±i| 2 + |$o| 2 )$±i + C2*5=i*o. h2 d ihdt$0 = ~^ x®o + c 0 |$o| 2 $o + (co + c 2 ) 2 2 X (|$!| + | $ _ i | ) $ 0 + 2C2*5*i*_i(3)
Based on Refs. 10, 11, we go into the case c 0 = c2 = — c < 0 which corresponds to the 2. Integrable Model integrable condition of Eqs. (3). Note that The assembly of atoms in the F = 1 state c~o < 0 means that bright soliton is stable and is characterized by a vectorial order param- dark soliton in the opposite case c"o = c2 = eter: &(x,t) = [$i(a:,i),$oOM),$-i(a;,t)] T c > 0 was analyzed in Ref. 12. The effective with the components subject to the hyper- interactions between atoms in the qlD BEC fine spin space. The system is quasi-one di- can be tuned with the so-called confinement mensional: the trap is elongated in the x induced resonance 9 , which do not affect the direction such that the transverse dynamics rotational symmetry of the spin states, by m Eq. (2) is factorized from the longitudinal and stays setting oj_ = 3Caoa 2 /(2oo + a 2 ) 9 in the transverse ground state . The Gross- when ao > a 2 > 0 or a 2 > 0 > aoPitaevskii energy functional is expressed as 6 By the dimensionless form $ —> (
idtQ = -d2xQ-
2QQtQ,
«-(££)•
(4)
(5
>
Since the matrix NLS equation (4) is integrable 13 , the dynamical problems of this system can be solved exactly. Here we list 2 some conserved quantities of this system: _ 4ft aF 1 number, NT = J dxn(x,t), n(x,t) = <&1-# = 9F V ; ma\ (l-CaF/a±y ti{Q^Q}; spin, FT = f dxf(x,t), f(x,t) = where ap are the s-wave scattering lengths $ t f $ = ti{Q^a-Q}, (
300
IT0I
0.1, 0.05, 0,
0.1,
0.05, 0,
20
20
V 1
20 -20
l* ± 0.1 0.05 0
•T"
0.1, 0.05,
/ - V'
20
4 20
ii
•
. .
%-
i
Fig. 1. Density plots of |<^o|2 (top) and |<£±i| 2 (bottom) for a polar-polar collision. Soliton 1 (left mover) carries only 0 component and soliton 2 (right mover) consists of ± 1 components.
Fig. 2. Density plots of \4>o\2 (top) and |<)!>±i|2 (bottom) for a polar-ferromagnetic collision. Soliton 1 (left mover) is a polar soliton and soliton 2 (right mover) is a ferromagnetic soliton.
NT, and the polar |FT-| = 0. The ferromagnetic solitons carry the macroscopic spin and the polar solitons do not. According to the classification of the soliton solutions, we analyze two-soliton collisions in the following three cases: 1) Polar-polar (PP) solitons collision. 2) Polar-ferromagnetic (PF) solitons collision. 3) Ferromagnetic-ferromagnetic (FF) solitons collision. PP solitons collision.— In Figs. 1, we show a P P solitons collision. These figures correspond to each component of the exact two-soliton solution 11 for one collisional run. For simplicity, we choose the parameters to have |(/>i | = |-i |. Collisional effects appear only in the position shift and the phase shifts (not shown). In this sense, the polar-polar collision is basically the same as that of the single-component NLS equation. PF solitons collision. — • In Figs. 2, we have density plots of a P F collision. Again
we set |0i | = | 0 - i |. The polar soliton (soliton 1) initially prepared in mp — ± 1 are switched into a soliton with a large population in rriF = 0 and the remnant of mp = ± 1 after the collision. Through the collision, the ferromagnetic soliton (soliton 2) plays only a switcher, showing no mixing in the internal state of itself outside the collisional region. FF solitons collision.— In Figs. 3, we give examples of this type of collisions for different soliton velocities: Figs. 3 (a) high speed, Figs. 3 (b) middle speed, Figs. 3 (c) low speed, with the other conditions fixed to illustrate the velocity dependence. The internal shift 0i —+ 0_i, and vice versa, gradually increase by slowing down the velocity of the solitons. We can gain a better understanding of the FF solitons collision by recasting it in terms of the spin dynamics. The total spin conservation restricts the motion of the spin of each soliton on a circumference
301
In conclusion, t h e spin dependent soliton dynamics predicted in this work should b e observed in experiments a n d open u p variety of applications in coherent a t o m transport. References 1. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 7 1 , 463 (1999); A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001); R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Rev. Mod. Phys. 77, 187 (2005). 2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981); G. Agrawal, Nonlinear -10 0 10 -10 0 10 -10 0 10 Fiber Optics, 3rd ed. (Academic, 2001). x x x 2 Fig. 3. Time evolution of |^o| (left column), \
302 S P I N D E C O H E R E N C E IN A GRAVITATIONAL FIELD
HIROAKI TERASHIMA* and MASAHITO UEDA Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan CREST, Japan Science and Technology Corporation (JST), Saitama 332-0012, Japan * E-mail: [email protected]. ac.jp
We discuss a mechanism of spin decoherence in gravitation within the framework of general relativity. The spin state of a particle moving in a gravitational field is shown to decohere due to the curvature of spacetime. As an example, we analyze a particle going around a static sphericallysymmetric object. Keywords: spin entropy; general relativity; quantum information.
1. Introduction The spin of a particle is an interesting degree of freedom in quantum theory. Recently, Peres, Scudo, and Terno 1 have shown that in special relativity the spin entropy (i.e., the von Neumann entropy of a spin state) of a particle is not invariant under Lorentz transformations unless the particle is in a momentum eigenstate. Namely, even if the spin state is pure in one frame of reference, it may become mixed in another frame of reference. The origin of this spin decoherence is that the Lorentz transformation entangles the spin and momentum via the Wigner rotation 2 . The entanglement then produces spin entropy by a partial trace over the momentum. In this paper, we study the spin state of a particle moving in a gravitational field to show its decoherence by the effects of general relativity 3 . Our result implies that even if the spin state is pure at one spacetime point, it may become mixed at another spacetime point. This spin decoherence is derived from the curvature of spacetime caused by the gravitational field. Such a spacetime curvature entails a local description of spin by local Lorentz frame due to a breakdown of the global rotational symmetry. The motion of a particle is thus accompanied by the change of frame, which can in-
crease the spin entropy analogous to the case of special relativity. As an example, we consider a particle in a circular orbit around a static spherically-symmetric object using the Schwarzschild spacetime. 2. Formulation Consider a wave packet of a spin-1/2 particle with mass m in a gravitational field. The gravitational field is described by a curved spacetime with metric in general relativity. Nevertheless, despite the spacetime curvature, we can locally describe this wave packet as if it were in a flat spacetime, since a curved spacetime locally looks like flat. More precisely, for any spacetime point we can find a coordinate system in which the metric becomes the Minkowski one. The coordinate transformation from a general coordinate system {x11} to this local Lorentz frame {xa} can be carried out using a vierbein (or a tetrad) ea'i(a;) and its inverse ea^(x) defined by 4 ^(x)eb'/(x)gllv(x)
:
e%(x)eb"(x)=6°b,
Vab,
(1)
where g^ix) is the metric in the general coordinate system and rjab = diag(—1,1,1,1) is the Minkowski metric with a, b = 0,1,2,3. The vierbein then transforms a tensor in the general coordinate system into that in the
303 local Lorentz frame, and vice versa. For example, momentum p^(x) in the general coordinate system can be transformed into that in the local Lorentz frame via the relation pa(x) = eall(x)p»(x). Therefore, we describe the wave packet as in the case of special relativity 1 using a local Lorentz frame at the spacetime point X1* where the centroid of the wave packet is located. Since a momentum eigenstate of the particle is labeled by four-momentum pa = 2 2 ( \ / | P 1 2 +Tn c ,p) and by the ^-component a (=T> I) of spin 5 as \pa,a), the wave packet can be expressed by a linear combination M = E
/
d3 N
a
P (P )
a
C{P ,
where *PN{jr)
mc
= #p
(3)
2 2
is a Lorentz-invariant volume element. From the normalization condition (p'^a'\pa,a)
= J^-)S3(P-p)S^,
the coefficient C(pa,a)
(4)
with a being the Pauli matrices. Note that this spin is not Dirac spin but Wigner one 6 . As is well known, Dirac spin, which corresponds to the index of 4-component Dirac spinor, is not a conserved quantity in a relativistic regime and thus is not suitable degree of freedom for labelling one-particle states. In contrast, Wigner spin is a conserved quantity suitable for labelling one-particle states, because it is defined using the particle's rest frame. 3. Decoherence Suppose that the centroid of the wave packet is moving with four-velocity u^{x) normalized as u' i (x)u /i (x) = —c2; this motion is not necessarily geodesic in the presence of an external force. After an infinitesimal proper time dr, the centroid moves to a new point J, i and then the wave packet x>n _ xi +u' (x)dT is described by the local Lorentz frame at the new point. This change in the local Lorentz frame is represented by a Lorentz transformation A.ab(x) = Sab + x°b(x)dT, where a X b(x)
must satisfy
= u»(x)[eb»(x)Vlie\(x)].
(9)
In addition to this change, the acceleration by an external force is also interpreted as a Lorentz transformation. Thus, the motion of To obtain the spin state of this wave the wave packet is equivalent to a Lorentz packet, we take the trace of the density matransformation Aa6(a;) = Sab + Xab(x)dT, trix p = \i/))(if>\ over the momentum, where 7 fd3pN(pa)\C(pa,a)\2
£
= l.
(5)
d3pN(pa)(pa,a'\p\pa,a).
pr(a';a) = J
(6) The spin entropy is then given by the von Neumann entropy of this reduced density matrix: S=-Tr[Pl(a';a)log2Pl(a';a)}.
(7)
Moreover, the spin operator for the wave packet is defined by 3
a
a
a
^= l E ^ [d pN(p )\p ,a)(p ,P\, a,0
J
(8)
A°fc(x) = - ^
x\(.x) [ aa(x) qb(x) - qa(x) ab(x)}
(10)
using the momentum and acceleration of the centroid in the local Lorentz frame qa(x) = e\(x)[mu»(x)}, a
a (x) = e^ix)
v
[u {x)Vvu»(x)
(11) ].
(12)
Note that even if the wave packet moves as straight as possible along a geodesic curve, this Lorentz transformation may be nonzero in general relativity because of the first term.
304
Since spin entropy is not invariant under a Lorentz transformation 1 , neither is it invariant during the motion of the wave packet. Note that a Lorentz transformation rotates the spin of a particle through an angle that depends on the particle's momentum; this rotation is known as Wigner rotation. The momentum eigenstate |p a ,er) thus transforms under the Lorentz transformation (10) as 5 U(A(x))\pa,cr)
= J2D„>AW(x)) | A(x)pa,a'),
(13)
a'
where Daia(W(x)) is the 2x2 unitary matrix that represents a Wigner rotation given by 7
Fig. 1. A wave packet (small circle) going around a static spherically-symmetric object (large circle). The 1- and 3-axes of the local Lorentz frame are illustrated at the initial and final points.
where f(r) = 1 — (rs/r) with the Schwarzschild radius rs. In the Schwarzschild space\i0(x)pk-Xk0(x)p*dT + time, we introduce an observer at each point v p° + mc ' who is static with respect to the time t using with i,k = 1,2,3. Taking the trace of the a static local Lorentz frame. The vierbein density matrix p' = U(A{x))\ip)(ij\U(A(x)y (1) is then over the momentum, we obtain the spin state 1 e 0 '(x) = ei r (x) = Vftr), p'T((r'; <J) and the spin entropy S' of the wave cy/fir)' packet in the local Lorentz frame at the new 1 point ar'M. However, the spin has been en\x) = (16) e2e(x) = \ , r rsin0 tangled with the momentum by the Lorentz Suppose that the centroid of the wave transformation (10), since the Wigner rotapacket is moving along a circular trajectory tion (14) of spin depends on the momentum. of radius r (> rs) with a constant velocity Due to this entanglement, the new entropy rd(j>/dt = Vy/JJr) on the equatorial plane S" is not, in general, equal to the original one S. This implies that the spin state may de- 0 = 7r/2 (see Fig. 1). The four-velocity of cohere during the motion of the wave packet the centroid is then given by Wik(x) = Sik +
Xik(x)dr
by the effects of general relativity.
cosh£
u*(x) = 4. Example As an example in general relativity, we consider the Schwarzschild spacetime, which is the unique static spherically-symmetric solution of Einstein's equation in vacuum. In the spherical coordinate system (t,r,6,
= -f(r)c2dtz
+
1
-dr'
:*M -= u*(ar)
CSillh
£
(17)
where £ is denned by tanh£ = v/c. We assume that at the initial point the wave packet has the definite values a = | and p1 = p2 = 0 but is Gaussian in p 3 with width w: \C(p",a)\2
=
^ 6 ^ ) 8 ^ ) 5 ^ {p3-q3(x))^
y/nw exp
(18)
tir
f(r) where q3(x) = mc sinh£ is the momentum of 2 + r (d9 + sin 0 # ) , (15) the centroid (11) along the 3-direction. 2
2
2
305 whereas extremely rapid decoherence occurs (r<2 —> 0) near the Schwarzschild radius r —> rs. The spin state does not decohere also at r = 3rs/2, because the first term in Eq. (10) is canceled by the second term. Of course, this spin decoherence is very slow in the gravitational field of the earth rs ~ 1 cm. For example, when a wave packet is at rest in the International Space Station Fig. 2. The spin entropy S at v/c = 0.8, r/rs = 0.9, going around the earth, v ~ 7.7km/s and and w/mc = 0.1 as a function of the proper time T r ~ 6800 km, the characteristic decoherence normalized by T3 = mr3/w. time is Td ~ 2.2 x mc/w years. 5. Conclusion We have shown that spin entropy is generated when a particle moves in a gravitational field. The spin state evolves into a mixed state even if the particle moves as straight as possible along a geodesic curve. This decoherence is due to the spacetime curvature by gravity. Fig. 3. The inverse of the characteristic decoherence time TJ , normalized by TJ" , as a function of rs/r Acknowledgments at v/c = 0.8. This research was supported by a Clearly, the spin entropy of this wave Grant-in-Aid for Scientific Research (Grant packet is zero at the initial point, since the No. 15340129) by the Ministry of Education, spin is separable from the momentum. How- Culture, Sports, Science and Technology of ever, after a proper time r of the parti- Japan. cle, spin entropy is generated by the gravity and acceleration, i.e., by the first and sec- References ond terms in Eq. (10). Figure 2 shows the 1. A. Peres, P. F. Scudo, and D. R. Terno, generated spin entropy S as a function of Phys. Rev. Lett. 88, 230402 (2002). the proper time T in the case of v/c = 0.8, 2. E. P. Wigner, Ann. Math. 40, 149 (1939). r/rs = 0.9, and w/mc = 0.1. The spin state 3. H. Terashima and M. Ueda, J. Phys. A: Math. Gen. 38, 2029 (2005). of the wave packet decoheres to a mixed state 4. N. D. Birrell and P. C. W. Davies, Quantum and becomes maximally mixed (S —> 1) in Fields in Curved Space (Cambridge Univerthe limit of T = oo. The characteristic decosity Press, Cambridge, 1982). herence time is given by the inverse of 5. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, - l _ w(cosh£ — 1) 1995). 12r/(r) 6. D. R. Terno, Phys. Rev. A 67, 014102 mr (19) (2003). Figure 3 shows this value rd- i1 as a function of 7. H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004). rs/r when v/c = 0.8. No decoherence occurs (jd —> oo) at the spatial infinity r —• oo,
vm-
306 BERRY'S PHASE OF ATOMS WITH DIFFERENT SIGN OF THE g-FACTOR IN A CONICAL ROTATING MAGNETIC FIELD OBSERVED BY A TIME-DOMAIN ATOM INTERFEROMETER Atsuo Morinaga , Hirotaka Narui, Akinori Monma and Takatoshi Aoki Department of Physics, Faculty of Science & Tech., Tokyo University ofScience, 2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan E-mail: [email protected] Berry's phase of the atom in the state with a positive or negative g factor for partial cycles of a conical rotating magnetic field was determined using a time-domain atom interferometer. The experimental results show that the solid angle for positive g-factor is $l-cos#) and that for a negative g-factor is ^(l+cos#) addition to the reversing the sign. Keywords: Berry's phase;Atom interferometry; g-factor.
1. Introduction In 1984, Berry predicted that a quantal system slowly transported round a circuit C by varying parameters in its Hamiltonian acquires the geometrical phase in addition to the dynamical phase. So far, numerous experiments on Berry's phase have been carried out and it has become well known in many physical systems [1]. As an example, Berry proposed that for a whole turn of the magnetic field round a cone of semiangle 0, the particle with a magnetic quantum number mF will give a phase shift of [2]
y = -2mnF(\-cos0).
(1)
Recently, the experiment was realized by us using a time-domain atom interferometer [3]. It was comprised of the two long lived states with different magnetic quantum numbers coupled by two two-photon stimulated Raman pulses with a time interval T. Then the phase of the interference fringes was shifted in proportion to the difference between Berry's phases of the two states. First, we measured the phase shifts between F=l, mp=\ and F=2, nif=2 of the sodium ground hyperfine states under the rotating magnetic field on the equator. The results show that the phase is dependent on the magnetic quantum number multiplied by the rotation angle for partial cycles and that the
sense of the phase shift depends on the direction of rotation of the magnetic field. Finally, the phase was shifted with a slope of 2.9±0.2 as a function of rotating angle
307 During the pulse separation, if the amplitude of the magnetic field varies, the dynamical phase shift, which is known as the scalar Aharonov-Bohm effect, occurs [6]. While, during the pulse separation, a conical rotating magnetic field with a different frequency v was applied to the atoms and Berry's phase was observed as a function of rotating angle of the
field ^ =
2nTlv.
*~h$\. Baxial
Laser pulses
Fig. 2. A conical rotating magnetic field with a semiangle
The ground hyperfine states of sodium atom are composed of the F=l state which has a g-factor of-1/2 and F=2 state of+1/2. Their Zeeman splitting is shown in Fig. 1. We could couple only the state of F=l, m^=l and F=2, mp=2 with a circular polarized two-photon Raman transitions.
it , a
F=2, g=l/2
Br,
mF=2
F=l,g=-I/2 mF=l
Fig. 1. Zeeman energy splitting of sodium atom and a two photon Raman transition.
In order to get interference fringes with a visibility of more than 50 %, we carefully reduced the stray magnetic field to less than 0.2 mG and the magnetic field gradient due to zeeman slower was compensated by applying the anti-Helmholz magnetic field. The conical rotating magnetic field was produced as a resultant of the rotating magnetic field Brot and the axial magnetic field B ^ i , which is perpendicular to the rotation plane. The rotation of the magnetic field was made by two mutually orthogonal pairs of Helmholtz
of 8 from the propagation direction of the laser field..
coils which were driven by alternating currents with a relative phase shift of 90°. The axial magnetic field was applied so that the semiangle of the resultant magnetic field was 60°. The strength of the rotation magnetic field was 183 mG, which corresponds to a frequency shift of 385 kHz. The resultant magnetic field was 212 mG, which corresponds to frequency shifts of 445 kHz. At 3 ms after free expansion of the cold ensemble of atoms, they were initialized perfectly by optical pumping to the F=l state. Then a quantization magnetic field was applied to sodium atoms and two-photon Raman pulses with a pulse width of 20 ms were applied to them with a pulse separation of 180 ms to compose the atom interferometer. During two Raman pulses, the resultant magnetic field was rotated from DC to several kHz, while keeping the amplitude of the magnetic field constant. Then we obtained the Ramsey fringes with a period of 5 kHz and a visibility of 50%. The experimental results of 0=60° and 120°, where the direction of the axial magnetic field were reversed, shown in Fig.3. The slopes are -2.8±0.2 and 2.7±0.3, respectively. The experimental detail will be written in another paper [7]. 3. Discussions The reverse of the slope between semiangle of #=60° and #=120° will be understood easily as follows. Even if the axial magnetic field is
308
I—r-
1
1
4. Conclusion
r
Berry's phase of atoms with magnetic quantum number using time-domain atom interferometer was investigated. The dependency of Berry's phase on the sign of the g-factor was discussed. Acknowledgments
[ T -- 0=60° : -( 2.8 + 0.2 X> J I -0=120" :( 2.7 ±0.3)4 _i -2
1 -I
1 0
1 I
1— 2
The authors would like to thank M. Kitano of Kyoto University, F. Riehle of PTB(Germany) and A. Oguchi of Tokyo University of Science for many discussions and comments on Berry's phase.
Rotation Angle
reversed, the sense of the giromagnetic motion is hold, while the sense of the rotation of the magnetic field is reversed for the direction of the axial magnetic field. Consequently, the reverse of the axial magnetic field corresponds to the reverse of the sense of the magnetic field keeping the sense of the axial magnetic field. Namely, the sense of the rotation is defined according to the direction of the axial magnetic field. This fact is easily ascertained by reversing the sense of the rotation magnetic field. The fact that the magnitude of the slope is near the 2.5, not but 1.5 will be more important, since the Eq. (1) for #=60° predicts 1.5 only taking into consideration the sign of the g-factor. The fact will be explained by the idea that a solid angle for a positive g-factor is Q = 0(1-COS 0 ) , while that for a negative g-factor should be
Q = -0(1 + cos<9), including the reverse of the sign. Then, we could get a slope of 2.5 for #=60°. To confirm this idea, further experiments for a few other semiangle will be necessary.
References 1. Geometric Phases in Physics, ed. A. Shapere and F. Wilczek (World Scientific, Singapole, 1989). L. Maiani, Phys. Lett. B62, 183(1976). 2. M. V. Berry, Proa R. Soc. London, Ser. A 392, 451(1984)., M. Colonna, M. Di Toro and V. Greco, Phys. Rev. Lett. 86, 4492 (2001). 3. M. Yasuhara, T. Aoki, H. Narui and A. Morinaga, IEEE Trans. Instrum. Measur. 54, 864(2005). 4. A. Morinaga, T. Aoki and M. Yasuhara, Phys. Rev. A71, 054101(2005). 5. R. Tycko, Phys. Rev. Lett. 58,2281 (1987). 6. K. Shinohara, T. Aoki and A. Morinaga, Phys. Rev. A 66, 042106 (2002). 7. H, Narui, T. Aoki, A. Monma and A. Morinaga, to be published.
LIST OF PARTICIPANTS Aeppli, Gabriel Department of Physics and Astronomy University College London
Ebisawa, Hiromichi Graduate School of Information Sciences Tohoku University
Altshuler, Boris Department of Physics Princeton University
Edamatsu, Keiichi Research Institute of Electrical Communication Tohoku University
Ando, Tsuneya Department of Physics Tokyo Institute of Technology
Ando, Masahiko Advanced Research Laboratory Hitachi, Ltd.
Ashhab, Sahel S. Digital Materials Laboratory Frontier Research System RIKEN
Awschalom, David D. Department of Physics University of California, Santa Barbara
Azuma, Koji Department of Applied Physics The University of Tokyo
Fujii, Toru Imaging Technology Department Nikon Corporation
Fujikawa, Kazuo Institute of Quantum Science College of Science and Technology Nihon University
Fujimori, Masaaki Advanced Research Laboratory Hitachi, Ltd.
Fujiwara, Akio Department of Mathematics Osaka University
Fukuyama, Hidetoshi Institute for Materials Research Tohoku University
Corbett, John V. Division of Information and Communication Sciences Macquarie University
Funahashi, Haruhiko Physics Department Kyoto University
Dietl, Tomasz Institute of Physics Polish Academy of Sciences
Giddings, Devin Physics & Astronomy The University of Nottingham
Gorman, John Cavendish Laboratory The University of Cambridge
Hatano, Naomichi Institute of Industrial Science The University of Tokyo
Hansel, Wolfgang Institute for Experimental Physics Innsbruck University
Hayakawa, Jun Advanced Research Laboratory Hitachi, Ltd.
Haffner, Hartmut Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences
Hayashi, Masahiko Graduate School of Information Sciences Tohoku University
Hamada, Tomoyuki FSIS Center of Collaborative Research Institute of Industrial Science The University of Tokyo
Harada, Ken Advanced Research Laboratory Hitachi, Ltd.
Hasegawa, Shuichi Quantum Engineering and Systems Science The University of Tokyo
Hashimoto, Yoshinori Quantum Engineering and Systems Science The University of Tokyo
Hashizume, Tomihiro Advanced Research Laboratory Hitachi, Ltd.
Hatakenaka, Noriyuki Faculty of Integrated Arts and Sciences Hiroshima University
Heike, Seiji Advanced Research Laboratory Hitachi, Ltd.
Heydari, Hoshang Institute of Quantum Science Nihon University
Hirai, Yasuharu Advanced Research Laboratory Hitachi, Ltd.
Hirayama, Yoichi Advanced Research Laboratory Hitachi, Ltd.
Hitosugi, Taro Department of Chemistry The University of Tokyo
Hofmann, Holger F. Graduate School of Advanced Sciences of Matter Hiroshima University
Ichimura, Masahiko Advanced Research Laboratory Hitachi, Ltd.
Katori, Hidetoshi Department of Applied Physics The University of Tokyo
Ieda, Jun 'ichi Institute for Materials Research Tohoku University
Katsumoto, Shingo Institute for Solid State Physics The University of Tokyo
Iijima, Sumio Department of Materials Science and Engineering Meijo University
Katsura, Hosho Department of Applied Physics The University of Tokyo
Ishioka, Sachio Advanced Research Laboratory Hitachi, Ltd.
Kawabata, Shiro Nanotechnology Research Institute National Institute of Advanced Industrial Science and Technology
lye, Yasuhiro Institute for Solid State Physics The University of Tokyo
Kimble, H. Jeff Norman Bridge Laboratory of Physics California Institute of Technology
Jain, Jainendra K. Department of Physics The Pennsylvania State University
King, Christophers. Physics & Astronomy The University of Nottingham
Johansson, Robert J. Digital Materials Laboratory Frontier Research System WKEN
Kobayashi, Kensuke Institute for Chemical Research Kyoto University
Kanda, Akinobu Institute of Physics and TIMS University of Tsukuba
Kato, Masanori Institute for Solid State Physics The University of Tokyo
Kobayashi, Shun-ichi Tokyo University of Agriculture and Technology
Kohashi, Teruo Central Research Laboratory Hitachi, Ltd.
312 Koizumi, Hideaki Advanced Research Laboratory Hitachi, Ltd.
Miyazaki, Hisao Institute of Physics University of Tsukuba
Kono, Junichiro Department of Electrical and Computer Engineering Rice University
Mooij, J. E. Kavli Institute of Nanoscience Delft University of Technology
Liu, Yu-xi Digital Materials Laboratory Frontier Research System RIKEN
MacDonald, Allan H. Department of Physics The University of Texas at Austin
Marcus, Charles M. Department of Physics Harvard University
Maruyama, Eiichi Center for Intellectual Property Strategies RIKEN
Matsuoka, Hideyuki Advanced Research Laboratory Hitachi, Ltd.
Matsuoka, Leo Quantum Engineering and Systems Science The University of Tokyo
Misko, Vyacheslav R. Digital Materials Laboratory Frontier Research System RIKEN
Morinaga, Atsuo Department of Physics Tokyo University of Science
Moshchalkov, Victor V. Nanoscale Superconductivity and Magnetism Group Katholieke Universiteit Leuven
Murakami, Shuichi Department of Applied Physics The University of Tokyo
Nagaosa, Naoto Department of Applied Physics The University of Tokyo
Nakajima, Sadao Superconductivity Research Laboratory International Superconductivity Technology Center
Nakamura, Michiharu Hitachi, Ltd.
Nakamura, Yasunobu Fundamental and Environmental Research Laboratories NEC Corporation
Nakano, Hayato Superconducting Quantum Physics Group NTT Basic Research Laboratories
Ogawa, Susumu Advanced Research Laboratory Hitachi, Ltd.
Ohnishi, Takashi Department of Applied Physics The University of Tokyo
Ohno, Hideo Research Institute of Electrical Communication Tohoku University
Ohta, Fumio Systems Development Laboratory Hitachi, Ltd.
Ong, Nai-Phuan Department of Physics Princeton University
Otto, Teruo Institute for Chemical Research Kyoto University
Onogi, Toshiyuki Advanced Research Laboratory Hitachi, Ltd.
Ootuka, Youiti Institute of Physics University of Tsukuba
Osakabe, Nobuyuki Advanced Research Laboratory Hitachi, Ltd.
Otani, Yoshichika Institute for Solid State Physics The University of Tokyo
Otsuka, Tomohiro Institute for Solid State Physics The University of Tokyo
Ozeki, Akira Science News Department The Asahi Shimbun
Recher, Patrik Edward L. Ginzton Laboratory Stanford University
Saikawa, Kazuhiko Ono, Yoshimasa A. Center for Innovation in Engineering Education The University of Tokyo
Saito, Hiroki Department of Physics Tokyo Institute of Technology
Onoda, Shigeki Department of Applied Physics The University of Tokyo
Saito, Kazuo Advanced Research Laboratory Hitachi, Ltd.
Saito, Shin-ichi Central Research Laboratory Hitachi, Ltd.
Shiokawa, Tom Frontier Research System RIKEN
Sakurai, Akio Department of Physics Kyoto Sangyo University
Soderholm, Jonas M.A. Institute of Quantum Science Nihon University
Sano, Hirotaka Institute for Solid State Physics The University of Tokyo
Sugano, Ryoko Advanced Research Laboratory Hitachi, Ltd.
Satoh, Tetsuo Tokyo Office Asian Technology Information Program
Suwa, Yuji Advanced Research Laboratory Hitachi, Ltd.
Savel'ev, Sergey Digital Materials Laboratory Frontier Research System RIKEN
Suzuki, Kazuya Institute for Solid State Physics The University of Tokyo
Seki, Yoshichika Graduate School of Science Kyoto University
Seto, Katsunori Central Research Laboratory Hitachi, Ltd.
Takagi, Kazumasa Corporate Technology Group Hitachi Europe, Ltd.
Takagi, Shin Fuji Tokoha University
Takayanagi, Hideaki NTT Basic Research Laboratories Shimazu, Yoshihiro Department of Physics Yokohama National University
Takei, Nobuyuki Department of Applied Physics The University of Tokyo
Shimizu, Fujio The University of Electro-Communications Taketani, Kaoru Department of Physics Kyoko University
Takeuchi, Masayuki Corporate Communications Division Hitachi, Ltd.
Tonomura, Akira Advanced Research Laboratory Hitachi, Ltd.
Tanamoto, Tetsufumi Advanced LSI Technology Laboratory Corporate R & D Center Toshiba Corporation
Tsai, Jaw-Shen Quantum Information Technology Group NEC Corporation
Tarucha, Seigo Department of Applied Physics The University of Tokyo
Tatara, Gen Department of Physics Tokyo Metropolitan University
Terashima, Hiroaki Department of Physics Tokyo Institute of Technology
Teshima, Tatsuya Advanced Research Laboratory Hitachi, Ltd.
Togawa, Yoshihiko Quantum Phenomena Observation Technology Laboratory Frontier Research System RIKEN
Tokura, Yoshinori Department of Applied Physics The University of Tokyo
Tomaru, Tatsuya Advanced Research Laboratory Hitachi, Ltd.
Tsai, Sheng-Yi Institute of Quantum Science Nihon University
Tsujimura, Tatsuya Kyodo News
Uchida, Fumihiko Central Research Laboratory Hitachi, Ltd.
Uchida, Takahiro Institute for Solid State Physics The University of Tokyo
Uchiyama, Fumiyo Institute of Applied Physics University of Tsukuba
Utsunti, Yasuhiro Condensed Matter Theory Laboratory RIKEN
Williams, David A. Hitachi Cambridge Laboratory Hitachi Europe, Ltd.
Wunderlich, Jorg Hitachi Cambridge Laboratory Hitachi Europe, Ltd.
Xu, Xiulai Hitachi Cambridge Laboratory Hitachi Europe, Ltd.
Yamada, Eizaburo Information Science Meisei University
Yamamoto, Yoshihisa Edward L. Ginzton Laboratory Stanford University
Yamashita, Taro Institute for Materials Research Tohoku University
Yang, Chen N. Tsinghua University
Yasuda, Tomooki Science News Department The Asahi Shimbun
Yoshida, Takaho Advanced Research Laboratory Hitachi, Ltd.
Zeilinger, Anton Institute for Experimental Physics Vienna University
Author Index Aikawa, H. Ando, M. Ando, T. Aoki, T. Arai, T. Baelus, B.J. Baskaran, G. Becher, C. Benhelm, J. Blatt, R. Carballeira, C. Cexilemans, A. Chek-al-Kar, D. Chwalla, M. Cleaver, J.R.A. Corbett, J. V. Crooker, S.A. Dietl, T. Ebisawa, H. Edamatsu, K. Endo, A. Fazio, R. Fujii, T. Fujikawa, K. Fujimori, M. Fujita, S. Fujiwara, A. Fukuda, H. Fukuyama, H. Funahashi, H. Furubayashi, Y. Furusawa, A. Gorman, J. Grabecki, G. Haffner, H. Hamada, T. Hansel, W. Harrabi, K. Hasaka, S. Hasegawa, T. Hashizume, T.
99 258 228 306 246 200 183 33 33 33 194 194 33 33 105 38 234 159 204 50 109,212 64 88 290 250,254 76 80 246 1 282 191 29 60 159 33 262 33 54 262 191 250,254,286
Hasko, D.G. Hatakenaka, N. Hatano, N. Hatano,T. Hauge, R.H. Hayashi, M. Heike, S. Heydari, H. Higashi, R. Hino, M. Hirano, K. Hirose, Y. Hisamoto, D. Hitosugi, T. Hofmann, H.F. Home, D. Hong, F.-L. Ichimura, M. Ieda, J. Inaba, K. Inoue, M. Inoue, S. Iye,Y. Jungwirth, T. Kadowaki, K. Kanda, A. Kasai, S. Kaestner, B. Kato, Masanori. Kato, M. Katori, H. Katsumoto, S. Kawabata, S. Kim, N. Y. Kimble, H.J. Kimura, S. Kimura, T. Kinoda, G. Kitaguchi, M. Kobayashi, N. Kohno, H.
60 88 266 92 234 204 250,254,286 42 276 282 46 191 246 191 68 38 276 183,187 298 191 262 46 99,109,212 140 200 200,208,216 165 140 109 204 276 99,109,212 224 242 10 246 171 191 282 262 177
Kono, J. 234 Korber, T. 33 Kuboki, K. 204 Kurimura, S. 46 Lee, C. 258 Lee, W.-L. 121 Liu, Y.-X. 72 MacDonald, A.H. 140,150 Maekawa, S. 84,113,183,187 Mar, J.D. 105 Martinek, J. 113 Maruyama, R. 282 Misko, V. R. 220 Miyazaki, H. 208,216 Momida, H. 262 Monma, A. 306 Mori, S. 46 Morinaga, A. 306 Moshchalkov, V.V. 194 Murakami, S. 146 Nagaosa, N. 134 Nakamura, H. 266 Nakamura, Y. 54 Nakano, H. 64 Narui, H. 306 Nishida, M. 88 Nomura, K. 140 Nori, F. 72,220 Nunez, A.S. 150 Ohno, T. 262 Okamoto, R. 68 Onai, T. 246 Ong, N.P. 121 Ono, T. 165 Ono, Y. A. 258 Oohata, G. 50 Ootuka, Y. 200,208,216 Osakabe, N. 3 Otake, Y. 282 Otani, Y. 171 Peeters, F.M. 200 Portugall, 0. 234 Rapol, U.D. 33 Recher, P. 242 Riebe, M. 33 Rikken, G.L.J.A. 234
Roos, C. Saito, H. Saito, Shin-ichi Saito, Shiro Saitoh, E. Sano, H. Sato, M. Savel'ev, S. Schmidt, P.O. Schmidt-Kaler, Schon, G. Seki, Y. Shaver, J. Shibata, J. Shimada, T. Shimizu, H.M. Shimizu, N. Shimizu, R. Shirasaki, R. Sinova, J. Smalley, RE. Sdderholm, J. Stopa,M. Sun, C.P. Suwa, Y. Suzuki, T. Tadano, K. Taguchi, Y. Tajima, N. Takahashi, S. Takamoto, M. Takayanagi, H. Takei, N. Taketani, K. Takeuchi, S. Tanaka, H. Tanamoto, T. Tanigawa, H. Tanikawa, K. Tarucha, S. Tatara, G. Teniers, G. Terada, Y. Terashima, H. Tokura, Y. Tonomura, A.
Tsai, J.S. Uda, T. Ueda, M. Utsumi, Y. Wei, L.F. Wei, X. Williams, D.A. Wrobel, J. Wunderlich, J. Xu,X. Yamaguchi, A. Yamaguchi, T. Yamamoto, T. Yamamoto, Yoshihisa Yamamoto, Y. Yamashita, T. Yang, C.N. Yano, K. Yoshihara, F. You, J.Q. Zaric, S. Zeilinger, A.
54 262 294,302 113,117 72 234 60,105 159 140 105 165 216 262 16,242 191 84 4 165 54 72 234 24
The goal of the 8th International Symposium on Foundations of Quantum Mechanics in the Light of New Technology was to link recent advances in technology with fundamental problems and issues in
FOUNDATIONS OF OURNTUM MECHANICS IN THE LIGHT OF NEW TECHNOLOGY •
quantum mechanics with an emphasis on quantum coherence, decoherence, and geometrical phase. The papers collected in this volume cover a wide range of quantum physics, including quantum information and entanglement, quantum computing, quantum-dot systems, the anomalous Hall effect and the spinHall effect, spin related phenomena, superconductivity in nano-systems, precise measurements, and fundamental problems. The volume serves both as an excellent reference for experts and a useful introduction for newcomers to the field of quantum coherence and decoherence.
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