Terahertz Spectroscopy: Principles and Applications (Optical Science and Engineering Series)

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Terahertz Spectroscopy Principles and Applications

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OPTICAL SCIENCE AND ENGINEERING Founding Editor Brian J. Thompson University of Rochester Rochester, New York

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108. Optics in Magnetic Multilayers and Nanostructures, Sˇtefan Viˇsˇnovsky’ 109. Optical Inspection of Microsystems, edited by Wolfgang Osten 110. Applied Microphotonics, edited by Wes R. Jamroz, Roman Kruzelecky, and Emile I. Haddad 111. Organic Light-Emitting Materials and Devices, edited by Zhigang Li and Hong Meng 112. Silicon Nanoelectronics, edited by Shunri Oda and David Ferry 113. Image Sensors and Signal Processor for Digital Still Cameras, Junichi Nakamura 114. Encyclopedic Handbook of Integrated Circuits, edited by Kenichi Iga and Yasuo Kokubun 115. Quantum Communications and Cryptography, edited by Alexander V. Sergienko 116. Optical Code Division Multiple Access: Fundamentals and Applications, edited by Paul R. Prucnal 117. Polymer Fiber Optics: Materials, Physics, and Applications, Mark G. Kuzyk 118. Smart Biosensor Technology, edited by George K. Knopf and Amarjeet S. Bassi 119. Solid-State Lasers and Applications, edited by Alphan Sennaroglu 120. Optical Waveguides: From Theory to Applied Technologies, edited by Maria L. Calvo and Vasudevan Lakshiminarayanan 121. Gas Lasers, edited by Masamori Endo and Robert F. Walker 122. Lens Design, Fourth Edition, Milton Laikin 123. Photonics: Principles and Practices, Abdul Al-Azzawi 124. Microwave Photonics, edited by Chi H. Lee 125. Physical Properties and Data of Optical Materials, Moriaki Wakaki, Keiei Kudo, and Takehisa Shibuya 126. Microlithography: Science and Technology, Second Edition, edited by Kazuaki Suzuki and Bruce W. Smith 127. Coarse Wavelength Division Multiplexing: Technologies and Applications, edited by Hans Joerg Thiele and Marcus Nebeling 128. Organic Field-Effect Transistors, Zhenan Bao and Jason Locklin 129. Smart CMOS Image Sensors and Applications, Jun Ohta 130. Photonic Signal Processing: Techniques and Applications, Le Nguyen Binh 131. Terahertz Spectroscopy: Principles and Applications, edited by Susan L. Dexheimer

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Terahertz Spectroscopy Principles and Applications

Edited by

Susan L. Dexheimer

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑0‑8493‑7525‑5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Dexheimer, Susan L. Terahertz spectroscopy : principles and applications / Susan L. Dexheimer. p. cm. ‑‑ (Optical science and engineering) Includes bibliographical references and index. ISBN 978‑0‑8493‑7525‑5 (hardcover : alk. paper) 1. Terahertz spectroscopy. I. Title. II. Series. QC454.T47D49 2007 543’.5‑‑dc22

2007024954

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Preface.......................................................................................................................xi Editor..................................................................................................................... xiii Contributors............................................................................................................ xv

Section I Instrumentation and Methods Chapter 1 Terahertz Time-Domain Spectroscopy with Photoconductive Antennas..................1 R. Alan Cheville Chapter 2 Nonlinear Optical Techniques for Terahertz Pulse Generation. and Detection — Optical Rectification and Electrooptic Sampling......................... 41 Ingrid Wilke and Suranjana Sengupta Chapter 3 Time-Resolved Terahertz Spectroscopy and Terahertz Emission. Spectroscopy............................................................................................................. 73 Jason B. Baxter and Charles A. Schmuttenmaer

Section II A pplications in Physics and Materials Science Chapter 4 Time-Resolved Terahertz Studies of Carrier Dynamics in Semiconductors, Superconductors, and Strongly Correlated Electron Materials.............................. 119 Robert A. Kaindl and Richard D. Averitt Chapter 5 Time-Resolved Terahertz Studies of Conductivity Processes. in Novel Electronic Materials................................................................................. 171 Jie Shan and Susan L. Dexheimer

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Chapter 6 Optical Response of Semiconductor Nanostructures in Terahertz Fields. Generated by Electrostatic Free-Electron Lasers...................................................205 Sam G. Carter, John Cerne and Mark S. Sherwin

Section III A pplications in Chemistry and Biomedicine Chapter 7 Terahertz Spectroscopy of Biomolecules............................................................... 269 Edwin J. Heilweil and David F. Plusquellic Chapter 8 Pharmaceutical and Security Applications of Terahertz Spectroscopy................. 299 J. Axel Zeitler, Thomas Rades and Philip F. Taday Index....................................................................................................................... 325

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Preface The development of new sources and methods in the terahertz, or far-infrared, spectral range has generated intense interest in terahertz spectroscopy and its application in a wide range of fields. Terahertz frequencies, broadly defined as 0.1–30 THz, span the range of low-energy excitations in electronic materials, low-frequency vibrational modes of condensed phase media, and vibrational and rotational transitions in molecules, making this a key spectral range for probing fundamental physical interactions as well as for practical applications. Moreover, the short pulse durations achievable with terahertz techniques based on femtosecond laser sources allow timeresolved measurements on femtosecond timescales, which were previously inaccessible in this spectral range. This volume provides an up-to-date reference on state-of-the-art terahertz spectroscopic techniques, focusing in particular on time-domain methods based on femtosecond laser sources, and reviewing important recent applications of terahertz spectroscopy in physics, chemistry, and biology. Chapters are contributed by leading experts in each topic. The focus of this book specifically on spectroscopy and the range of applications presented set it apart from the few other recent volumes in the emerging terahertz field. The reader is referred to Sensing with Terahertz Radiation edited by Daniel Mittleman (Springer, 2003) and to Terahertz Sensing Technology edited by Dwight L. Woolard et al. (World Scientific, 2003) for applications including ranging and imaging, and Terahertz Optoelectronics edited by Kiyomi Sakai (Springer, 2005) for a review focusing specifically on work on generation of terahertz radiation carried out in Japan. The first section of this book is devoted to instrumentation and methods for time-domain and time-resolved terahertz spectroscopy. The first chapter provides a comprehensive discussion of time-domain terahertz spectroscopic measurements using photoconductive antenna sources and detectors, including methods for determining optical constants from time-domain measurements. The second chapter provides a review of time-domain terahertz techniques based on the nonlinear optical processes of optical rectification for terahertz pulse generation and free-space electrooptic sampling for terahertz pulse detection, and the third chapter reviews current state-of-the-art femtosecond time-resolved techniques and terahertz emission spectroscopy using femtosecond laser sources. Measurements using free electron laser sources and the established far-infrared frequency-domain spectroscopic techniques of FTIR and photomixing are also discussed in the context of specific applications in Sections II and III. Applications of terahertz spectroscopy to key areas of current interest, both fundamental and applied, are presented in Sections II and III of the book. The second section is devoted to applications in physics and materials science, including studies of carrier processes in state-of-the-art semiconductor materials, superconductors, novel

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electronic materials, and semiconductor nanostructures. The third and final section of the book is devoted to applications of terahertz spectroscopy to the biological and chemical fields. This first chapter of this section reviews studies of biologically important molecules, and the final chapter reviews applications of terahertz spectroscopy to pharmaceutical analysis and to the detection of security hazards. Susan L. Dexheimer Editor

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Editor Susan L. Dexheimer received an S.B. degree in Physics from the Massachusetts Institute of Technology and a Ph.D. in Physics from the University of California, Berkeley. She has a broad research background at the interfaces of experimental condensed matter physics, chemical physics, and molecular biophysics. She is currently on the faculty at Washington State University, where her research includes the application of femtosecond optical and terahertz spectroscopic techniques to study ultrafast dynamics in electronic materials and molecular systems.

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Contributors Richard D. Averitt Department of Physics Boston University Boston, Massachusetts Jason B. Baxter Department of Chemistry Yale University New Haven, Connecticut Sam G. Carter JILA University of Colorado Boulder, Colorado John Cerne Department of Physics State University of New York at Buffalo Buffalo, New York R. Alan Cheville Department of Electrical and. Computer Engineering Oklahoma State University Stillwater, Oklahoma Susan L. Dexheimer Department of Physics Washington State University Pullman, Washington Edwin J. Heilweil National Institute of Standards. and Technology Gaithersburg, Maryland

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Robert A. Kaindl Lawrence Berkeley National. Laboratory Berkeley, California David F. Plusquellic National Institute of Standards. and Technology Gaithersburg, Maryland Thomas Rades School of Pharmacy University of Otago Dunedin, New Zealand Charles A. Schmuttenmaer Department of Chemistry Yale University New Haven, Connecticut Suranjana Sengupta Department of Physics, Applied. Physics and Astronomy Rensselaer Polytechnic Institute Troy, New York Jie Shan Department of Physics Case Western Reserve University Cleveland, Ohio Mark S. Sherwin Department of Physics University of California, Santa Barbara Santa Barbara, California Philip F. Taday TeraView Ltd. Cambridge, United Kingdom

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Ingrid Wilke Department of Physics, Applied. Physics and Astronomy Rensselaer Polytechnic Institute Troy, New York

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J. Axel Zeitler Department of Chemical. Engineering University of Cambridge Cambridge, United Kingdom

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Section I Instrumentation and Methods

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1

Terahertz Time-Domain Spectroscopy with Photoconductive Antennas R. Alan Cheville Oklahoma State University

Contents 1.1 Introduction to Terahertz Spectral Region........................................................1 1.2 Brief Theoretical Background for Photoconductive Terahertz Generation.........3 1.3 Terahertz-Generation Process........................................................................... 6 1.4 Detecting Terahertz Radiation Using Photoconductive Antennas.................. 10 1.5 Experimental Considerations of Terahertz Spectroscopy............................... 19 1.6 Terahertz Beam Propagation and Optical Systems.........................................24 1.7 Adaptations and Extensions of Terahertz Spectroscopy Systems................... 30 References................................................................................................................. 37

1.1 Introduction to TerahertZ Spectral Region A recent report from the National Academies outlines that electronic and optics are combining to open up a new “Tera-Era.”1 This vision includes computers performing operations at rates of teraflops, terabit-per-second communication systems, and electronic devices on the picosecond time scale. From a spectroscopic point of view, the terahertz (THz) spectral region from 300 GHz (λ = 1 mm) to 10 THz (λ = 30 µm) has not yet seen the technological development of optical or microwave frequencies with the result that commercial spectrometers covering the entire spectral range are not yet widely available. This is primarily because of the difficulty of generating and detecting THz frequencies compared with the better-established technologies of optics and electronics. The so-called “THz gap” arises from the nature of the sources and detectors used in spectroscopy both at the optical (high-frequency) side and electronic (lowfrequency) side of the gap. One terahertz corresponds to a photon energy of 4 meV (33.3 cm–1). These energies are much less than the electronic state transitions of

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Terahertz Spectroscopy: Principles and Applications Wavelength (m) 10

0

10–2

10–1

10–3

+

1010

10–5

10–6

10–7

300 K

– – Rotational Transitions

109

10–4

+

– – Vibrational Transitions 1011

1012

1013

Electronic Transitions 1014

1015

1016

Frequency (Hz)

Figure 1.1 The electromagnetic spectrum from microwave to optical region. The area easily accessible by THz spectroscopy using photoconductive sources is shown shaded in gray. Rotational, vibrational, and electronic transitions are shown along with the blackbody radiation curve at 300 K.

atoms and molecules commonly used in lasers or other light sources (Figure 1.1). Semiconductor materials also have bandgaps on the order of an electron volt, orders of magnitude larger than THz energies. Vibrational frequencies, such as the 10.6 µm CO2 laser, and rotational transition frequencies do fall in the far infrared. Light sources based on these transitions exist, but typically cover limited frequency ranges.2 Thermal emission from incoherent blackbody radiation is one of the most commonly used sources since the thermal energy, kT, is about 25 meV at room temperature (300 K). The blackbody radiation curve and the relative spectral power emitted are shown as a black line in Figure 1.1. Scaling electronics to operate in the THz gap is limited by the frequency response of devices. The high-frequency limit is determined by the time it takes electrons to traverse the device, which is determined by the physical size and carrier mobility. Although feature sizes are continually shrinking, material properties ultimately limit high-frequency response. New semiconductor materials and the ability to confine carriers at the quantum level are leading to high-speed devices pushing into the THz gap. Some exciting recent developments are laser diodes,3 high electron mobility transistors, and quantum cascade lasers.4 The same physical limitations that arise in scaling optical sources down (or electronic sources up) to the THz spectral region also hold for detectors of THz radiation. THz photons cannot excite electrons into the conduction band in a photodiode (without multiphoton absorption, which would require unreasonable intensities) or provide enough energy to cause photoelectric emission of electrons. Although bolometers can detect radiation, they are limited by incoherent thermal background signals unless cooled to very low temperatures. The far infrared spectrum and the points made previously are summarized in Figure 1.1. There is a great deal of interest in this spectral region ranging from creating faster electronic devices to the wealth of spectroscopic information available.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

THz spectroscopy provides information on the basic structure of molecules and is a useful tool of radio astronomy. Rotational frequencies of light molecules fall in this spectral region, as do vibrational modes of large molecules with many functional groupings, including many biologic molecules that have broad resonances at THz frequencies. This chapter provides an overview of how the technologic difficulties inherent to THz generation and detection can be overcome through a synthesis of ultrafast optical and electronic techniques. By gating micron scale antennas on photoconductive substrates with femtosecond pulses, it is possible to generate and detect electromagnetic pulses with THz bandwidths.

1.2 Brief Theoretical Background for Photoconductive TerahertZ Generation Because THz radiation generated and detected through photoconduction arises from the flow of charge, it is theoretically described by Maxwell’s equations. Maxwell’s equations do not, however, describe the processes from which charge flow occurs on a femtosecond scale in semiconductors. This is a topic worthy of a complete book,5 and the description of processes from which charge flow arises are not gone into in detail here. From the perspective of THz generation, the source terms of Maxwell’s equations are written to show time-dependent terms explicitly by adding a (t) to them.   ∂m (t )H (t ) ∇ × E (t ) = ∂t

   ∂e (t )E (t ) ∇ × H (t ) = s (t )E (t ) + ∂t

(1.1)

Here, m(t)H(t) = B(t) is the magnetic flux, H(t)= s(t)E(t) is the current density, and e(t)E(t) = D(t) is the electric displacement. Any process that creates a time-dependent change in the material properties m, s, or e can act as a source term that can result in emission of THz radiation. For THz generation using photoconductive switches, an ultrashort optical pulse incident on a semiconductor causes rapid transient changes to the macroscopic material properties represented by s(t), m(t), and e(t). For the discussion that follows, the major change caused by the optical pulse is assumed to be in the conductivity, s. The rapid, optically induced change of s on a femtosecond time scale is the origin of ultrafast THz pulses generated through photoconduction as well as the physical mechanism by which the THz pulses are detected. It is important to note that many physical processes occur when ultrafast optical pulses are absorbed by semiconductors, and, from an experimental viewpoint, they are not easily separable. Both resonant (absorption of a photon to create charge carriers, s) and nonresonant (nonlinear optical difference frequency generation, e) effects can contribute to THz pulse generation. A common example of a material in which both processes contribute to THz generation is gallium arsenide (GaAs). The driving term, s(t), in Maxwell’s equations results from an ultrafast optical pulse that impulsively excites the semiconductor. The resulting electric and magnetic

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Terahertz Spectroscopy: Principles and Applications

Electric Field

1

0

–1 –2

–1

0 Time (ps)

1

2

Spectral Amplitude

1

0 0.1

1.0

10 Frequency (THz)

100

Figure 1.2 Electric field of a measured THz pulse and the simulated field of the exciting optical pulse are shown in the upper figure, whereas the amplitude spectrum is shown in the lower figure.

fields created through the time-dependent conductivity have fast transients with correspondingly broad spectral bandwidths. The spectral bandwidth can be estimated from the uncertain relation [between the bandwidth, Dw, required to support a transient signal] Dt. Because DtDw ≥ ½, a 1-ps excitation pulse requires, and can create, spectral bandwidths on the order of 0.3 THz, whereas a 100-fs transient has a bandwidth of at least 3 THz. With sub-10-fs pulses readily available from current commercial ultrafast lasers, very broad spectral coverage of the THz region has been achieved. An example of a THz measurement from the author’s laboratory is shown in Figure 1.2. The upper figure compares the measurement of the electric field of a THz pulse in the time domain with the numerically calculated electric field of the 50-fs optical pulse at 800 nm used to generate the THz pulse. The rapidly oscillating carrier frequency of the optical pulse makes it appear to be a filled Gaussian envelope. The spectral amplitude of the THz pulse and optical pulse are shown in the lower figure. Although the actual bandwidth of the optical pulse (10.6 THz) is considerably larger than that of the THz pulse (0.7 THz), the THz pulse spans a greater fraction of the electromagnetic spectrum in terms of Dn/no: 0.7 for the THz pulse compared to 0.003 for the optical pulse. Recent developments in THz generation and detection

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

have extended the frequency range spanned by THz pulses by an additional order of magnitude or more. One of the great advantages of ultrafast THz measurements is that this bandwidth allows a broad range of energy scales (frequencies) to be probed in a single measurement with high temporal resolution. Figure 1.2 highlights some of the differences between short pulses in the THz and optical spectral regions. The electric field of the THz pulse is a nearly single cycle oscillation, whereas the optical pulse is described as a Gaussian pulse envelope superimposed on a sinusoidal carrier frequency. The optical pulse thus meets the slowly varying envelope approximation, whereas the THz pulse does not. Thus many of the simplifying assumptions used in quantum and nonlinear optics do not necessarily apply to THz systems.6,7 From a theoretical perspective, an accurate description of the pulse propagation requires the solution of the coupled Maxwell-Bloch equations. Because THz pulses used for spectroscopy have very low peak powers compared with optical pulses, nonlinearities can generally be neglected and propagation is treated using linear dispersion theory.8 Perhaps the most significant difference between THz pulses and visible, infrared, or ultraviolet light more familiar to most researchers is THz spectroscopy directly measures the electric field of the pulse rather than the intensity. The direct measurement of field is possible simply because of the much slower variation of the field as can be seen in Figure 1.2. Intensity, I, and electric field, E, are related by:

I=

1 m EE * 2 e

(1.2)

Equation 1.2 illustrates the well-known fact that intensity measurements contain no phase information unless techniques such as holography or interferometry are used to measure phase relative to another optical beam. The time-resolved electric field of the THz pulse does, however, contain complete phase information that can be extracted directly from THz measurements. For the more familiar intensity measurements, energy conservation dictates that reflection and transmission coefficients are real and positive with magnitude less than one in almost all commonly encountered cases. Because THz experiments measure the electric field amplitude the Fresnel coefficients can be complex and have values less than zero or greater than one. For example, the amplitude transmission coefficient from air to silicon is t12 = 0.45, whereas that from silicon to air is t21 = 1.55. Examples will be discussed later. The final commonly encountered difference between experimental measurements at optical and THz frequencies is the effect of phase shifts on the measured pulse shape. Small changes in the phase of an optical pulse generally have little to no effect on the measured pulse shape. Larger frequency-dependent phase shifts, such as those caused by dispersive media, do result in pulse broadening. In contrast, small-frequency, independent-phase changes result in significant reshaping of THz pulses, as shown in Figure 1.3. Numeric phase changes of 0, p/16, p/8, p/4, p/2, and p result in pulse reshaping up to a complete inversion of the pulse. Such phase shifts can occur on reflection,9 when the pulse propagates through a caustic,10 or in propagating through materials with a complex index.

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Terahertz Spectroscopy: Principles and Applications

π

0π

π/16

π/8

π/4 π/2

Figure 1.3 Effect of frequency-independent phase shifts from 0 to π/2 radians on the Terahertz pulse shape.

1.3 TerahertZ-Generation Process Generation and detection of THz pulses occur through nonlinear interaction of the driving optical pulse with a material with fast response. There are two major classifications, of which only one, photoconductivity, is discussed in this chapter. Photoconductive THz generation occurs when optical excitation induces conductivity changes in a semiconductor. This is a resonant interaction and the photons are absorbed through interband transitions. The second mechanism is nonresonant interaction arising from nonlinear frequency mixing or optical rectification. Transient photoconductivity is used for both the generation and detection of THz radiation, with THz generation discussed first. A simplified representation of the process of photoconductive THz generation is illustrated in Figure 1.4. The six frames of Figure 1.4 represent different steps in pulse generation and are discussed below. A great deal of research continues to be done on understanding the physics underlying THz pulse generation from transient photoconductivity. This area is highly multidisciplinary, integrating knowledge of optical generation of hot carriers, their subsequent rapid thermalization, ballistic transport on a femtosecond time scale, rapid screening of generated fields, design of high bandwidth antenna structures, and propagation of single cycle pulses in dispersive media. Because this chapter focuses primarily on using the pulses in spectroscopic application, only the basic physical fundamentals of THz generation are discussed.11 The geometry of a basic photoconductive THz source is shown in a top-down view in Figure 1.4a. A coplanar transmission line is fabricated on a semiconductor substrate with high mobility, m, usually GaAs or low temperature-grown GaAs.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

EDC EDC

– –– – –

–

(a)

t = 0++

(d) EDC

J(t)

–+ –+ –+ +– +– +–

t=0

ETHz(t)

t>0

(b)

(e) EDC

–

+

P(t)

t<0

N(t)

+ + +++

– – – ++ + + + – –

–

+

J(t)

(c)

EDC

t=

0+

N(t)

–

–

0

+ –

–+

+

+

+

t >> 0 (f)

Figure 1.4 Illustration of the process of photoconductive Terahertz generation. Frame (a) shows the coplanar transmission lines before excitation. In (b), the optical pulse generates charge carriers which are accelerated in the static field (c), and create an opposing field in the semiconductor (d). The induced polarization creates a transient current element which radiates a Terahertz pulse (e) followed by recombination back to steady state (f).

Because the mobility gives the ratio of carrier drift velocity to applied electric field, carriers in high-mobility compounds accelerate rapidly, leading to fast rise times. The coplanar transmission lines are DC-biased to produce a field on the order of 106 V/m near the breakdown threshold for air. The electric field near the surface of the semiconductor is represented in the figure by parallel arrows between the lines. Energy is stored by this capacitive structure, with a capacitance on the order of several pF. The semiconductor used has a high dark resistivity to minimize current flow and the resultant local heating and device failure due to electromigration.12 The first step in generating THz radiation is optical excitation of the region of high electric field. It is important that the semiconductor bandgap energy, Egap, is less than the photon energy, hν, so that electron-hole pairs are formed. Carrier generation is shown in a cross-sectional view in Figure 1.4b, where dimensions are shown not to scale to better illustrate the physical processes occurring. A focused ultrafast optical pulse incident between the metal lines generates a thin conductive region down to approximately the absorption depth 1/α. Typical absorption coefficients of semiconductors above the band gap range from α = 103 cm–1 to α = 105 cm–1. The optically generated electron-hole pairs form an electrically neutral plasma near the semiconductor surface as illustrated in Figure 1.4b. The time evolution of the charge

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Terahertz Spectroscopy: Principles and Applications

density is given by N(t) with the total carrier density the sum of that of the electrons and holes: N(t) = Ne(t) + Nh(t). At low-excitation fluences, the charge density is proportional to the intensity profile of the optical pulse I(t). The time evolution of the carrier density is given by:11

N (t ) =

∫

t

0

G (t)δt − N 0e

−

t tc

(1.3)

where the first term describes N(t) at time scales of the laser pulse and the second describes the evolution of the carrier density at longer times. Here tc is the carrier lifetime, G(t) is the optical carrier generation rate determined by the optical pulse profile, and No is the total number of carriers generated. The free carriers affect the conductivity by s(t) = N(t)em. The current density is J(t) = s(t)E, which can be written as J(t) = N(t)ev(t) with v(t) the carrier velocity. Immediately after generation of the carriers in the semiconductor the accelerating field is E(t = 0+) = EDC. The time dependence has been written explicitly to illustrate the transient nature of the photoconductivity on subpicosecond time scales. It is the transient current, J(t), that generates the THz pulse. The carriers are accelerated in the applied electric field, EDC, as shown in Figure 1.4c. The time evolution of carrier velocity is determined by the initial acceleration of carriers with effective mass m* and by rapid carrier scattering with a characteristic time ts: δv (t ) v (t ) e =− + E (t ) δt ts m* The carrier scattering time is determined by phonon scattering, carrier scattering, scattering from dopants, and the photon energy, or where in the conduction band carriers are generated. The spatial separation of the photo-induced electron and hole plasma create a macroscopic polarization, P(t), in the semiconductor, shown in Figure 1.4d. The polarization is given by P(t) = N(t)er(t), where r(t) is the spatial separation of charges. The time evolution of the induced polarization is calculated from:

δP (t ) P (t ) =− + N (t )ev (t ) δt tR

(1.4)

where the time derivative of the polarization is N(t)ev(t). As shown in Figure 1.4d, the induced polarization creates an electric field that opposes the static field of the biased coplanar stripline. The electric field responsible for carrier acceleration then evolves as:

E (t ) = E DC +

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P (t ) eg

(1.5)

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

where g is a numerical factor that depends on the geometry. The four coupled differential equations describing the carrier density N(t), the velocity v(t), the polarization P(t), and the electric field seen by the carriers E(t) describe the time evolution of the current created by photo-excitation of a carrier.11 The THz pulse is generated from the transient current element, J(t), shown in Figure 1.4e on a greatly expanded spatial scale. Generation of a pulse follows directly from Maxwell’s equations because time varying current serves as a source term:

  ∂H THz (t ) ∇ × ETHz (t ) = − m ∂t    ∂e((t )ETHz (t ) ∇ × H THz (t ) = J (t ) + ∂t

(1.6)

where the subscript THz distinguishes the time resolved THz field from the field in the semiconductor. Maxwell’s equations with these terms and those described in the next paragraphs are generally not amenable to analytic solution. Because many experimental techniques of THz generation focus the optical spot to micron diameters to generate large carrier densities, the current element is typically much smaller than one wavelength of the generated THz radiation and thus can be described as a Hertzian dipole.13 The calculations are complicated by the fact that the current element is on the surface of a dielectric half space, but both analytic and numerical solutions exist for dipole antennas on dielectric half spaces.14 The major fraction of the laser-generated burst of THz radiation is emitted into the semiconductor by a factor proportional to n3 where n is the index of refraction of the semiconductor. Because most semiconductors have refractive indices higher than three at THz frequencies, this is a significant enhancement in directivity and aids in collection and collimation of the THz radiation. When the optical excitation spot is not tightly focused so the current distribution is on the same order of a wavelength or larger, the problem is often treated as a summation (array) of small radiating elements. Because the current rises and falls on a time scale of one picosecond, it generates a well-defined pulse front that propagates outward from the optical excitation point. The radiation pattern is changed by the proximity of the dipole to the air–semiconductor interface and will be discussed later. All these processes happen on time scales of picoseconds or less. On longer time scales, carrier recombination within the semiconductor or at the metal electrodes causes the induced polarization to decay and the static bias electric field to reestablish itself, This is illustrated in Figure 1.4f. The recovery of the semiconductor to the state shown in Figure 1.4a typically takes hundreds of picoseconds or longer.11 In summary, a DC-biased semiconductor (a) has an electron-hole plasma created on femtosecond time scales by an optical pulse (b). The rapid acceleration and separation of the electrons and holes create a current transient (c). The macroscopic polarization creates an opposing field that screens the DC field (d). The transient electric field determined (approximately) by the time derivative of the current results in a radiated pulse front (e). In times of hundreds of picoseconds, the semiconductor recovers (f).

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10

Terahertz Spectroscopy: Principles and Applications

Considerable research has gone into development of photoconductive THz emitters for specific applications. The simple coplanar stripline structure of Figure 1.4 can be replaced by an array of antenna structures. Log-spiral, bowtie, and a variety of dipole structures have all been developed. In this case, the photocurrent provides an impulse excitation to the antenna and the radiation pattern and THz pulse are determined both by the response of the semiconductor as well as the antenna. Biased antenna structures are not required to generate photoconductive THz radiation. The electric field that arises from depletion of carriers from recombination centers near the surface will accelerate photo-induced carriers away from or toward the surface. In this case, the induced electric dipole is normal to the surface, rather than in the plane of the surface as with antenna structures, which makes it more difficult to couple radiation out of the substrate. The photo-Dember effect15 can also give rise to THz radiation. The Dember effect arises from the difference in mobility between electrons and holes, which causes the carriers accelerate at different rates.

1.4 Detecting TerahertZ Radiation Using Photoconductive Antennas Detecting THz pulses using photoconductively gated antennas relies on the same physical principles as generation of THz pulses, namely ultrafast pulse excitation in a semiconductor causing rapid changes in conductivity. To detect THz pulses with subpicosecond resolution, a micron-scale dipole antenna is fabricated on a semiconductor substrate with a very short carrier lifetime of 1 ps or less. Both the antenna design and properties of the semiconductor critically affect THz detection. A typical antenna structure is shown in Figure 1.5d from a top-down view. The antenna is a center-excited dipole fabricated between coplanar strip lines with length, h, of 10 to 200 µm. The length is the most critical element of the antenna and will be discussed later. The antenna’s width is tens of microns with a center gap of less than ten microns corresponding to approximately the diameter of the focused gating laser pulse. Typical dark resistance (no optical excitation) of these antenna structures is on the order of megaohms and depends on the substrate resistivity and antenna and coplanar stripline dimensions. The photoconductive detection process is illustrated schematically in Figure 1.5a–c. Figure 1.5a is a side-on view that illustrates the ultrafast laser pulse (not to scale, see Figure 1.2) incident from the left on the antenna of Figure 1.5d. A THz pulse is propagating from the right, but has not yet arrived at the antenna structure. As in generation of THz pulses, the ultrafast laser pulse generates an electron and hole plasma which decreases conductivity of the antenna gap. The time resolved change in conductivity is s(t) = N(t)em, where N(t) is given by Equation 1.3 and m is the semiconductor mobility. The arrival of the laser pulse is analogous to closing a switch, which allows the antenna gap to conduct. The short recombination time tc of the semiconductor causes the gap resistance to change from nearly insulating to conducting (closing the switch) then back to insulating (opening the switch) on a picosecond time scale. The coplanar strip line is connected to a high-sensitivity current amplifier that detects any current flow through the antenna gap with subpicoamp resolution. If the

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11

Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

ETHz

tB = 0–

tA< 0

(b)

w

h d

E(t) ~ I(t)

(a)

–4 (d)

tC = 0+ (c) tB

tA tC 0 Time (ps)

4

(e)

Figure 1.5 Illustration of the process of THz detection. The THz antenna of frame (d) is shown in cross section in frames (a–c). The electric field measurement is determined by the relative arrival times of the optical and THz pulse. In (a), the optical pulse arrives before the THz pulse. By delaying arrival of the optical pulse to coincide with the THz pulse as shown in (b) and (c), the measured current (e) maps out the THz field in time.

resistance of the metal lines is negligible the resistance of the antenna is determined by the antenna gap. The gap resistance is R(t) = w/s(t)A, where the cross-sectional area, A, is approximately 2da with a the absorption depth in the semiconductor; typically d >> a. The current flow from Ohm’s law is I(t) = V(t)/R(t), where the antenna bias voltage is provided by the THz pulse: V(t) ≅ ETHz(t)h. In Figure 1.5a, the THz pulse has not yet arrived at the antenna, the bias voltage is zero, and there is no net current flow through the antenna. This is illustrated in Figure 1.5e at the point labeled tA, which occurs at t < 0. Figure 1.5b shows a point later in time; the time axis is determined by the delay between the laser pulse that generates the THz pulse and the pulse that gates the THz antenna. The positive gating peak of THz pulse is incident on the antenna at the same time the optical gating pulse arrives. In this case, ETHz > 0, and current flows from the lower half of the antenna to the upper are determined by the electric field direction. The measured time average current at point tB in Figure 1.5e is proportional to the THz electric field. After the current is measured, the delay between the laser generating and gating pulses is again changed to advance the THz pulse in time. Figure 1.5c and point tC in Figure 1.5e both correspond to a point on the pulse where the electric field is opposite in sign to Figure 1.5b. The time resolved

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12

Terahertz Spectroscopy: Principles and Applications

measurement of the THz pulse electric field is thus made by changing the delay between the optical gating pulse and generating pulse. From this discussion, it is clear that, to measure the electric field with high temporal resolution, the response time of the semiconductor tC must be short compared with the rate of change of the THz field. Because to a first approximation σ ~ µ, semiconductors with both high mobility and short carrier lifetimes are optimal. The two semiconductor materials most commonly used for photoconductive THz detectors are low temperature-grown GaAs and ion implanted silicon on sapphire. Low temperature-grown GaAs has a thin layer grown on a GaAs substrate with the substrate temperature determining the recombination time.16 Recombination times of hundreds of femtoseconds or less have been reported. Silicon on sapphire is a thin (~1 µm) layer of silicon grown on a sapphire wafer with a reduction of carrier lifetime achieved through ion implantation. The carrier lifetime is dependent on implantation dosage to a point at which it saturates at about 500 fs.17 The response time can be estimated by the convolution of the THz pulse electric field, ETHz, with the detector response time determined by s(t). With the optical gating pulse arriving at t = 0 and the THz pulse at t = to, the time average current is:

I (t ) =

2 hr α w

∫

∞

s (t )ETHz (t − t 0 − t)δt

−∞

(1.7)

where the antenna dimensions have been used to estimate the resistance. Equation 1.7 indicates that photoconductors with infinitely short response times excited by infinitely short optical pulses (i.e., a delta function response) are required to exactly measure the time-resolved THz electric field. Although in general short carrier lifetimes allow the faster THz field transients to be measured, there is a tradeoff in actual performance because short lifetimes arise from a high density of trap or defect states that reduce the semiconductor mobility. Additionally, the average current is proportional to the time integral of s(t) so that fast recombination times can lead to lower average current measurements for a given THz field strength. In practice, measuring THz pulses with high-frequency components is more dependent on the laser excitation pulse than the carrier lifetime. This can be understood qualitatively through Equation 1.7 because both step functions and delta functions result in a non-zero average current. The extraction of accurate material parameters for non-ideal measurements of ETHz(t) is discussed later. This section concludes with a brief discussion of the THz antenna structure. Although Equation 1.7 indicates that an increase in the dipole size, h, allows high average currents, this is true only if the antenna length remains much less than the THz wavelength (Hertzian dipole). Longer antenna lengths have resonances and exhibit strong frequency dependence. Because the incoming THz pulse in most cases is incident through the semiconductor substrate, the wavelength of the THz radiation is reduced compared with free space by λo/n. Silicon, GaAs, and sapphire have high refractive indices in the THz region, so at frequencies near 1 THz, the wavelength at the antenna is on the order of 100 µm. A large number of antenna designs have been used both for generation and detection of THz radiation beyond simple dipoles. These include log-spiral18 and bow-tie19 designs.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

13

1.4.1 THz Spectroscopy Using Photoconductive Antennas This section provides an overview of spectroscopic measurements using photoconductive sources and detectors of THz pulses. Much of the discussion also applies to other ultrafast THz generation and detection techniques such as those based on nonlinear optics covered elsewhere in this book. Photoconductive THz pulses offer unique capabilities for spectroscopic measurements in the far infrared spectral region compared with other techniques such as Fourier transform infrared spectroscopy or techniques such as frequency mixing originally developed at microwave frequencies. This section is intended to provide a brief overview of both the advantages and disadvantages of ultrafast THz spectroscopy, describe the basic theory of extracting spectroscopic information from experimental measurements, and discuss experimental considerations of the technique that can affect the spectroscopic measurements. This section also is intended to provide an overview for the researcher looking to use THz pulses for their own spectroscopic measurements and is thus general rather than specific in the material covered. The first question to ask is whether THz spectroscopy is the appropriate technique for a given measurement. The THz spectral region accessible via ultrafast optical pulse generation and detection extends from less than 0.1 THz to more than 30 THz. Some source and detector techniques have enhanced low-frequency response, whereas other techniques are used to achieve high frequencies. Both the high- and low-frequency limits have associated experimental difficulties. The low-frequency end of this spectral region extends to frequencies that are easily achievable using high-precision electronic instruments. In addition, long temporal scans are required. For the high-frequency limit above about 10 THz, highly mature optical spectroscopy techniques such as Fourier transform spectroscopy exist. In brief, the choice to use time-resolved THz spectroscopy techniques depends on specific applications. The advantages of time-resolved THz spectroscopy are: coverage of more than a decade frequency range with a single measurement, peak spectral response in the 0.3–3.0 THz spectral range, and the ability to measure phenomena with subpicosecond time resolution. The THz spectral region is rich in spectroscopic features. Chapters later in this book provide much more complete reviews of the types of systems that can be probed with such pulses. From the perspective of trying to set up a system to perform measurements, THz spectroscopy can be broadly separated into measurement of gases and that of liquids and solids. At the low number density characteristic of gases, the frequency of collisions between molecules determines dephasing times that occur on time scales generally much longer than 1 ps. Spectral features are then less than 1 THz in width and can be less than 1 MHz at low pressures with Q (Dn/no) of 106 or more. In this limit, time-resolved THz spectroscopy resolves individual transitions over a broad range of energies, but does not have the frequency resolution to measure extremely narrow line widths. Limitations on frequency resolution are discussed in detail later. For liquids and solids resonances are broadened on femtosecond timescales resulting in broad, featureless spectra with corresponding to Q < 1 in most cases. For such systems, the spectral width of the short pulse-based measurement techniques can prove extremely valuable. Free carrier absorption occurs

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14

Terahertz Spectroscopy: Principles and Applications

in semiconductor materials in the THz spectral range and the broad spectral coverage is extremely useful for determining material properties. One other difference between THz spectroscopy and spectroscopic techniques at optical frequencies is that the energies of THz photons correspond to thermally accessible energy states. At room temperature the thermal energy, kT, corresponds to 25 meV, 208 cm–1, or a frequency of 6.25 THz. Thus THz spectroscopy using photoconductive antennas typically covers the region kT > hn where thermal energies are higher than photon energies and can probe states that are populated at room temperature. Using short excitation pulses to achieve high frequencies, it is also possible to simultaneously measure the region hn > kT. The technique of THz time domain spectroscopy measures the change between time-resolved electric fields of THz pulses propagating through a sample and an equivalent length of free space. Depending on the sample geometry, the change in the THz pulse shape permits either analytic or numeric calculation of the complex sample index, permittivity, or conductivity. One assumption in the following treatment of THz time domain spectroscopy is that the interaction of an electric field with a sample is linear. Maxwell’s equation, and Equation 1.1, which describes propagation through a material with constituent parameters given by the permittivity and permeability, is linear for small electric field when higher order terms of the susceptibility can be ignored. THz pulses generated in the laboratory typically have peak powers on the order of 1 µWcm–2 corresponding to an electric field on the order of 1 Vcm–1. Because the assumption of linear interactions in most materials is a good one, linear dispersion theory8 is used to represent the time domain pulse in the frequency domain as a superposition of plane waves of frequency w, all with a k vector along the THz system optical axis, z. The relative amplitude and phase of each plane wave component of the THz pulse are obtained through a Fourier transform of the timeresolved electric field:

E (z , w ) =

1 2p

∫

∞

E (z , t )e − iwt ∂t

−∞

(1.8)

Here E(z,w) is the complex field amplitude and E(z,t) is the experimentally measured THz pulse electric field in the time domain. A typical measurement of two THz pulses is shown in Figure 1.6. The three frames on the left correspond to the reference pulse that has propagated through free space—in this case, nearly dry air. The three frames on the right correspond to the pulse that has propagated through the sample—humid air containing approximately 1 percent water vapor. The experimentally measured pulses are in the top two frames. The effect of water vapor on the THz pulse in the measurements of Figure 1.6a are visible as oscillations after the pulse. Figure 1.6b shows the magnitude of the complex spectral amplitude obtained through a numeric Fourier transform. Comparing the two spectral amplitudes, the discrete rotational transitions of H2O can be seen as distinct absorption lines. The relative phase of the plane wave components that make up the two THz pulses is displayed as modulo 2p in Figure 1.6c. As can be seen from

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15

Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

E(t) (nA)

4 2

(a)

Esamp(t)

Eref(t)

0 –2 0

10

20

0

10

Time (ps)

20

30

Time (ps)

|E(ω)|

1.00

(b)

0.75 0.50

|Esamp(ω)|

|Eref(ω)|

0.25

∠E(ω)

0.00 π

(c)

0 –π 0.0

0.5

1.0

1.5

2.0

0.5

Frequency (THz)

1.0

1.5

2.0

2.5

Frequency (THz)

Figure 1.6 The reference and sample pulses discussed in the text are shown in (a) for dry and humid air. Numerical Fourier transforms of the amplitude and phase, (b) and (c), respectively, illustrate the effect of the sample on the plane wave components that comprise the THz pulse.

the figure, the phase changes linearly except near the resonance of the water vapor rotational frequencies. The gray regions on the plot correspond to invalid phase data caused by the reduced spectral amplitudes beyond 2 THz. Obtaining the complex electric field amplitude spectrum via a Fourier transform permits determination of the complex amplitude after propagating a distance Dz through a sample. Given the measured spectrum at the input of the sample, z = 0, the field exiting the sample is given by:

E ( Dz , w ) = E (0, w )e − ik (w ) Dz

(1.9)

The complex k vector, k(w), expresses complete information about the THz pulse interaction with the sample material. The wave vector k(w) can be expressed several ways, most commonly through the complex index, n(w) = n′(w)-jn″(w) with k(w) = wn(w)/c. The complex index is related to the permittivity through er(w) = n(w)2. The imaginary term is related to the power absorption coefficient by ap(w) = 2wn″(w)/c.

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16

Terahertz Spectroscopy: Principles and Applications

The choice of sign for the imaginary term of n(w) is made so that plane waves are attenuated and not amplified. The complex, frequency-dependent k vector is then:

k (w ) =

w w[n ′(w ) − 1] i α p (w ) + − 2 c c

(1.10)

where the first term, k0 = w/c, corresponds to the phase change of the THz pulse through a length of free space equivalent to the sample thickness. The reason for expressing the wave vector in this form will be made clear later. To measure the complex refractive index of the sample, a measurement is first done without the sample in place and with the spectrometer purged with dry air to ensure n = 1 along the THz pulse path. The measured THz pulse, Eref (t) in Figure 1.6a, is numerically Fourier transformed to obtain Eref (ω). The measured THz pulse in a general sense can be written as:

E ref (w ) = E gen (w )T (w )H (w ) = E gen (w )T (w )e − ik 0 Dz

(1.11)

where Egen(ω) is the THz pulse generated by the source, T(ω) is the frequency response of the THz spectrometer including the THz source and detector, and H(ω) is the response function of the sample. For the reference sample of length ∆z of free space, H(w) = exp(-ikoDz). The concept of a transfer function is drawn from engineering and describes the effect of a system on a complex input signal. The terms Egen(ω) and T(ω) are not directly measured and depend on numerous experimental parameters and the THz source and detector used. After the sample is placed into the system, the measured pulse that has propagated through the sample is

E samp (w ) = E gen (w )T (w )H (w ) = E gen (w )T (w )e − ik (w ) Dz

(1.12)

The material response is determined by dividing the complex amplitude spectrum of the sample pulse by the free space pulse

E samp (w ) E gen (w )T (w )e − ik (w ) Dz = = e − i ( k (w )− k 0 ) Dz E ref (w ) E gen (w )T (w )e − ik 0 Dz

(1.13)

As can be seen from the k(ω)-ko term of Equation 13, division of the sample spectrum by the reference spectrum effectively removes the effect of an equivalent length of free space. The absorption coefficient and index of refraction can be determined from the magnitude and phase of the spectral ratio using Equation 1.10. The refractive index and absorption coefficient of the time domain measurements of humid air, Figure 1.6, are shown in Figure 1.7. The complex index of the rotational transitions have the form of the quasi-Lorentzian line shape with phase and magnitude similar to that of a harmonic oscillator.20 Besides permitting direct measurement of the

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas 2

17

(b)

n(ω)z

1 0 –1 –2 6

(a)

α(ω)z

4 2 0 0.0

0.5

1.0 Frequency (THz)

1.5

2.0

Figure 1.7 Refractive index and absorption coefficient extracted from the data of Figure 1.6.

index and absorption, dividing the Fourier transforms of the measured pulses in the frequency domain is mathematically equivalent to a time domain deconvolution to remove the frequency response of the generated THz pulse and optical system from the sample response function. The effect of this deconvolution is one reason that the response time of the semiconductor does not need to be much shorter than the rate of change of the THz field to perform high-resolution THz spectroscopy. The formalism of the transfer function in the derivation above is useful because most samples measured with THz time domain spectroscopy do not have simple transfer functions. The derivation previously considered propagation in which the Fresnel reflection and transmission coefficients can be ignored. A more general case is the measurement of a finite thickness of material with index n2(w) surrounded by a material of index n1(w), usually air, so n1 = 1. In this case the Fresnel coefficients need to be taken into account and the transfer function H(ω) is then defined as:21

H (w ) =

t12t 21e ib (1 − r12r21e i 2b

)

(1.14)

where t12(w), t21(w), r11(w), and r 21(w) are the Fresnel transmission and reflection coefficients21 and depend on the index of the sample, n2(w). b = n2 kodcos q is the phase delay associated with multiple reflections inside the slab where q is the angle of propagation inside the slab determined from Snell’s law. Although there are no

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18

Terahertz Spectroscopy: Principles and Applications

exact analytical solutions for the index and absorption coefficient, the equation can be solved numerically either using an iterative guessing approach or piecewise in frequency trying to minimize phase and magnitude errors.22 Analytical approximations also serve in many cases. Although Equation 1.14 describes propagation through a dielectric slab, any other sample geometry can be represented equally well with the transfer function formalism. Assuming that the system being measured is one for which linear dispersion theory applies, after the system transfer function is determined, the THz measurements can compared with theory with equal ease in either the frequency or time domain. For a known H(ω) the time domain THz pulse is calculated after propagating through the sample from the inverse Fourier transform:

E samp (t ) =

∫

∞

E ref (w )H (w )e − i (wt )∂w

−∞

(1.15)

To summarize the process for extracting the index and absorption from measurements of the THz pulse, the THz pulse is Fourier transformed to express it in terms of a plane wave expansion. Taking the ratio of a THz pulse transmitted through a sample to one through a reference sample with known index deconvolves the response of the THz antenna and optical system and returns the sample transfer function. The index and absorption data are extracted from an analytic expression for the sample transfer function H(w). The complex index is measured over the system bandwidth without resorting to Kramers-Kronig analysis. As with any experimental technique, there are several assumptions inherent to the treatment discussed previously that can affect the results. The most common source of error arises from fluctuations of the THz pulse shape over the course of the measurement. Because two THz pulses—a reference pulse and a sample pulse—are measured to extract the complex index, any changes in the THz pulse shape over the duration of the measurement result in errors of the measured index. Relatively long-term experimental system drifts can be difficult to control so due experimental diligence and careful experimental design are required to avoid long-term drifts from thermal or mechanical fluctuations. Linear dispersion theory requires that the THz pulse can be expanded into a superposition of plane waves propagating along the z axis. Unlike laser beams at optical frequencies, this is not always true for THz pulses. Typical THz beam diameters are approximately 1 cm or less with a wavelength from millimeters to tens of microns. The beam diameter:wavelength ratio then ranges from 3 to 100 compared with values of several thousand for optical beams. Thus THz beams are more correctly represented as a summation of plane waves over a range of angles to the z axis. The angular spread of the k vectors can be estimated from Dq = 2Dk x/ko with ko = w/c, Dk x ≅ (2Dx)–1 and ∆x ≅ 1 cm. The angular spread is then on the order 0.3° at 1 THz and 0.9° at 0.3 THz. The angular spread of the beam can affect the measured transmission of samples angled to the beam.23 If the spatial profile of the THz pulse is measured, a spatial Fourier transform permits the angular-dependent transmission of the sample to be resolved.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

19

A final assumption is that the shape of the THz beam is not changed by passing through the sample. Referring to Equation 1.13, it was assumed that transmission through the sample does not affect T(w). Again, this is usually not an issue with optical beams, and is generally true for thin, low index samples that do not act as optical elements. The larger effect of diffraction at THz frequencies can result in beam reshaping for thick samples of high index. Techniques to calculate the effect of the sample on the THz beam will be discussed in the following section on experimental considerations of THz spectroscopy.

1.5 Experimental Considerations of TerahertZ Spectroscopy The basic technique of THz time domain spectroscopy using photoconductive switches has been outlined previously from the viewpoints of the fundamental physics of the THz generation and detection process and also from the mathematical treatment of extracting complex index from measurements. This section introduces the third element required for successful THz spectroscopic measurement: the experimental measurement system. This section is divided into three parts that attempt to provide practical considerations for performing THz spectroscopy: the optical pulses used to generate and detect the THz pulse, the propagation and coupling of the THz beam from source to detector, and miscellaneous experimental considerations required for successful measurements. A simplified schematic representation of a simple THz spectroscopy setup is shown in Figure 1.8. On the left side of the figure, a femtosecond laser provides a pulse train of femtosecond pulses. A beamsplitter separates the pulse train into beams that go to the THz source and detectors with nominally equivalent power. In Figure 1.8, the optical pulse going to the THz source (a) passes through an adjustable delay. The THz pulse created by the THz source (b) goes through several beamforming optics, passes through the sample, which changes the pulse shape (c), and is incident on the THz detector. The second optical pulse (d) arrives at nominally the

fs Laser Beamsplitter

Optical Delay (a)

THz Source

THz Detector

Sample

(b)

(c) (d)

Figure 1.8 Schematic illustration of a THz spectrometer showing the laser excitation source, optical delay line, THz source and detector, and THz coupling optics.

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20

Terahertz Spectroscopy: Principles and Applications

same time as the THz pulse and gates the THz detector. Changes in the delay permit the THz pulse shape to be mapped out as discussed in previous sections. A key experimental issue is ensuring the time of arrival of the THz pulse on the detecting dipole antenna coincides with the arrival of the optical pulse. The laser source and associated optics are shown on the left side of Figure 1.8. THz spectroscopy using photoconductive switches can be done successfully using a wide range of laser excitation sources. There are a large number of ways to specify or characterize laser sources, some of which have a strong impact on THz spectroscopic measurements and others that are much less important. The characteristics of the laser source are discussed briefly in decreasing order of importance. The ranking in importance is rather arbitrary and other experimental data can certainly change the required laser source characteristics. Arguably the most important characteristic of the laser source is stability, defined as constant power output and beam pointing over periods longer than several seconds. The importance of this for THz spectroscopic measurements arises from the need to measure two data scans for the reference and sample. Because the complex index is determined by the ratio of the two data scans, any change in the laser power or beam position can introduce large errors in measurements. Experimentally, measurements usually are performed by taking a reference scan before and after a scan with the sample in the beam in order to determine if there has been any drift of the system during the measurement period. Temperature fluctuations in the laboratory or problems with optical beam pointing stability can also degrade measurements. Figure 1.9a illustrates differences between two commercial laser sources determined by measuring the peak of the THz signal—point tB in Figure 1.5—over 30 minutes. As can be seen from the figure, there is a significant difference in stability between different sources. Of nearly equal importance to medium-term stability is the optical noise of the laser source. The primary noise source on THz experiments is noise from the laser coupled onto the THz beam.24 The noise specification of interest depends on the data acquisition technique. For the majority of THz experiments, phase-sensitive or lock-in detection is used and the noise figure of interest is the noise at the modulation frequency of the THz time domain spectroscopy system. For data acquisition systems that use rapid scanning and signal averaging, the noise at lower frequencies is of more importance. Figure 1.9b shows the effect of the stability and noise of the laser sources on the spectral representation of data used to determine the complex index through THz spectroscopy. The figures show ratios of the magnitude of the spectral amplitudes of five sequential reference scans, |EN|/|EM| where M ≠ N, performed in dry air. The variation between separate ratios indicates approximately the accuracy of the transfer function which is used to determine the complex index. Laser 1, with less noise and drift, results in a transfer function accurate to about ±1% over a large bandwidth. The noisier laser 2 has a factor of three less accuracy. The experimental artifact near 1.6 THz is due to small changes in the water vapor content in the experimental system on the order of less than 2%. The increasing error of the ratios at low and high frequencies is due to the finite bandwidth of the THz system.25 A third important, but not necessarily vital, specification of the laser source is the optical pulse width, which under some experimental circumstances, has a critical effect on the THz signal bandwidth.

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21

Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

Normalized Peak Signal (%)

+1

Commercial Laser #1

0 –1 +3

Commercial Laser #2

+2 +1 0 –1 –2

0

10

20

Time (minutes)

30

Ratio of Spectral Amplitudes (%)

(a) 105 100 Commercial Laser #1

95 110

Commercial Laser #2 100

90

|EN| |EM| 0

0.4

0.8

1.2

1.6

2

Frequency (THz) (b)

Figure 1.9 The upper figure, (a), compares the peak THz signals measured for two commercial laser sources over a 30-minute period to illustrate the impact of the excitation source on system signal-to-noise ratio. The lower figure, (b), compares five consecutive scans with the two laser sources illustrating how short-term stability impacts measurement accuracy.

For experiments using photoconductive switches over the spectral range 0.1–3 THz, the laser pulse width is not generally the limiting factor to the system bandwidth. Rather, the optical system and dipole response limit the bandwidth. However, to reach higher THz frequencies using relatively short dipole antennas on semiconductor substrates with fast response time, the bandwidth is critically dependent on the laser pulse width because the rise time of the conductivity limits the system spectral response. As a rule of thumb, pulse widths lower than 100 fs are more than adequate for most THz experiments and noise considerations can have a much larger effect on the effective system bandwidth than the laser pulse width. This is illustrated in Figure 1.10, in which the Fourier transforms of the five consecutive scans of Figure 1.9 are shown on a logarithmic scale for the two commercial lasers. The pulse width of the first laser, on the upper half of the figure, is specified by the manufacturer as <150 fs, and has an autocorrelation width of 110 fs. The second laser produces

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22

Terahertz Spectroscopy: Principles and Applications Commercial Laser #1

104 103

Relative Amplitude

102 101 100 Commercial Laser #2

104 103 102 101 100

0

1

2

3

4

Frequency (THz)

Figure 1.10 Comparison of the measured amplitude spectra of the two commercial lasers of Figure 1.9 illustrating the effect of laser amplitude noise on the THz spectral amplitudes. Laser 1 has a longer pulse width than laser 2.

much shorter pulses with an autocorrelation width of 55 fs. Although water lines are visible out to 4 THz for the longer pulse laser, the noise makes them impossible to distinguish for the second laser with shorter pulse widths. Of relatively less importance is the output power of the laser source, because most femtosecond sources will operate at higher than 100 mW of output power, whereas the typical THz system built around photoconductive switches only needs about 50 mW of total optical power to generate good experimental data. Using optical beams focused to about 5 µm, the semiconductor on which the detector is fabricated saturates at tens of milliwatts of optical power. Increasing optical power beyond the point of saturation has different effects on the THz source and detector. As the optical power on the source reaches saturation, the laser noise coupled in on the optical gating beam rises faster than the detector signals response resulting in decreased signal to noise of the system. At the THz source, the generated THz signal increases with optical power sublinearly following saturation. Of greater practical importance, the combination of high electric field and optical power can lead to electromigration12 and catastrophic failure of the THz source. The effect on saturation on the system dynamic range is shown in Figure 1.11. The dynamic range is defined in this chapter to be the RMS noise of the system before the arrival of the THz pulse compared with the signal at the peak of the THz pulse. The dynamic range of the system rises to a peak at 30 mW of excitation power then begins to decrease. This is mirrored in the inset that shows the measured THz signal increase linearly with excitation power until saturation at a power of tens of milliwatts.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

23

Dynamic Range (unitless)

20,000

Measured Signal

1000 30,000

100

30 µm dipole

10 1 0.1 0.01

0.01 0.1 1 10 100 Probe Average Power (mW)

10,000

0 0.1

1.0

10

100

Receiver Excitation Optical Power (mW)

Figure 1.11 Effect of optical excitation on the THz detector dynamic range. Increasing power beyond 30 mW decreases the dynamic range because of an increase of noise in conjunction with saturation of the detector as shown in the inset.

Assuming the laser operates at a photon energy above the bandgap of the semiconductor material used for the photoconductive antennas, the wavelength has secondary importance for THz spectroscopy. There have been reports in the literature, however, of improved THz emission efficiency near the bandgap.26 For most semiconductor materials, the need to excite above the bandgap rules out fiber lasers operating in the 1,550-nm range, although frequency-doubled fiber lasers are routinely used for THz spectroscopy. The pulse repetition frequency of the laser does affect spectroscopy using photoconductive switches, but pulse repetition frequencies of tens to hundreds of megahertz have been demonstrated. Lower repetition rate sources, such as those from amplified lasers, are more problematic because the pulse energy can cause saturation. In this case, large-aperture sources are used.27 The optical delay line shown in Figure 1.8 permits the relative delay of the THz pulse and optical gating pulse on the THz detector to be changed to map out the THz pulse shape. Because the Fourier transform used to convert the pulse to a spectral amplitude assumes the temporal spacing between the points is uniform, numeric linearization of the points is required if the delay line does not move linearly. The properties of the Fourier transform also show the measurement bandwidth is numerically limited by the Nyquist frequency, (2Dt)–1, where Dt is the time step of the delay line. Typically, Dt is chosen so that the Nyquist frequency is two or three times larger than the system bandwidth. This is done both to avoid wraparound in the frequency domain,28 but also to oversample the THz pulse shape so that noise fluctuations have less effect on the data. The frequency resolution of time domain THz measurements depends critically on the delay line parameters. The frequency resolution, Dn, is inversely proportional

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24

Terahertz Spectroscopy: Principles and Applications

to the number of data points acquired for a fixed time step and given by Dn = (2NDt)–1, where N is the number of data points. The resolution is usually not an issue for solid or liquid samples that have broad resonances, but for gases, the measured line width is often experimentally limited. As a rule of thumb, the experimentally resolvable line width is given by 0.6(NDt)–1. For a 100-ps measurement scan, this corresponds to 6 GHz spectral resolution. Because the overall noise measured increases as the square root of the size of the measurement window, it is clear that time domain THz techniques are generally not suitable for high-frequency resolution measurements.

1.6 TerahertZ Beam Propagation and Optical Systems In performing THz time domain spectroscopy measurements, a second critical element is ensuring there is good coupling between the THz source and THz detector. The effects of propagation and coupling considerations on experimental measurements are much more critical for THz experiments than for similar experiments at optical frequencies. There are several reasons for the increased care that needs to be taken in ensuring good coupling. First, the requirement that the photoconductive THz antenna is much smaller than a wavelength means the antenna is not highly directional. Additional coupling optics are required to produce a well collimated THz beam. Second, because the wavelength-to-beam diameter ratio is much less for THz pulses than for optical pulses, diffraction plays a larger role in beam propagation and the THz beam can quickly exceed the size of optics. Third, because the frequency components that make up the THz pulse typically cover more than a decade in frequency, different frequencies of the pulse have very different propagation characteristics. Finally, the THz detector measures the field rather than intensity of the THz pulse so it is not sufficient simply to get power to the detector. The spatial profile of the field pattern needs to match that of the detector. These considerations are discussed in the remainder of this section. Figure 1.12 shows a silicon—or other high-index material—lens attached to the back of the semiconductor wafer used for the THz source for the two orthogonal planes corresponding to S and P polarization of the THz radiation from the dipole antenna. As illustrated in the figure on the right, the silicon lens is centered on the dipole so that radiation from the antenna is collimated. The radiation of a dipole on a surface has a radiation pattern considerably different from a dipole in free space. The radiation patterns for S and P polarization are shown as shaded regions in Figure 1.12 on both sides of the dielectric interface of the THz antenna substrate. The reduced size of the antenna pattern in free space corresponds to the n3 reduction of the THz power radiated into space compared to that radiated into the dielectric as discussed previously.29 The radiation pattern of the Hertzian dipole on the surface leads to a complex radiation pattern on the surface of the lens.30 This radiation then diffracts to form an approximately Gaussian distribution in the far field.11,30 The antenna pattern inside the lens is, to first order, independent of frequency so that the field pattern at the lens surface is also frequency-independent. Because the more localized field patterns on the lens surface diffract strongly and do not contribute in a major way to the signal in the far field, a good first-order approximation is to assume there is a frequency-independent Gaussian beam waist at the coupling lens.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

25

Figure 1.12 Illustration of the antenna patterns for a dipole on a surface for P polarization (left) and S polarization (right). The rays in the figure are calculated from a geometrical optics approximation and illustrate the effect of spherical aberration resulting from the strongly curved lens surface.

The assumption of a frequency-independent Gaussian beam waist at the coupling lens allows the use of simple Gaussian beam theory to be useful for predicting THz beam propagation. Although other methods of calculating propagation of THz beams can also be used, the Gaussian treatment provides a compromise that is more precise than simple ray approximation, and less computationally intensive than finite difference time domain methods. For a Gaussian beam with waist wo at z = 0 and frequency ω, the field pattern at a distance z is given by:31

E (w , r , z ) = E 0 (w )

w0  r2  − 2  w (w , z )  w (w , z ) 

  z w  z     r 2w  − tan −1  × exp − j    exp  − j   c  z 0 (w )     2cR (w , z )   

(1.16)

where r is the radial distance in the x-y plane and the Rayleigh range, zo, is given by:

Z0 =

ww 02 2c

(1.17)

The waist at distance z for frequency ω is written in terms of the Rayleigh range as:   2 z w (z ) = w 1 +      z0     2 0

(1.18)

This is the standard theoretical treatment used for description of laser beam propagation, but unlike the more familiar propagation of a single-frequency beam, the

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Terahertz Spectroscopy: Principles and Applications

THz pulse is a superposition of a wide range of frequencies. As a result, the Rayleigh range and waist size are not single-valued at a given distance and the different frequency components of the THz beam have different propagation characteristics. The following paragraphs provide an overview of some of the propagation characteristics of THz beam; for a more complete treatment, the reader is referred to previous work.32 The Rayleigh range dictates the distance from the waist at which the propagating beam goes from propagating with minimal divergence and planar phase fronts to divergent propagation at angle θ = 2c/pnwo with spherical phase fronts. Alternatively it roughly demarcates the regions of near-field and far-field propagation. From Equation 1.17, using commonly encountered lens radii of several millimeters, the Rayleigh range is given approximately as zo ≅ 10ν (cm) where ν is the frequency in THz. Considering that typical propagation distances in a THz spectrometer are less than 1 m, both near-field and far-field behavior is exhibited simultaneously in the system by different frequency components. These variations in the Rayleigh range across the bandwidth of the THz pulse are reflected in the divergence full angle of the beam in the far field, θ, and the corresponding spread of the 1/e beam waist of the THz beam w(z). In the far field where z >> zo, the divergence full angle scales approximately as θ(n) ≅ .06/n (rad), whereas the beam waist scales as w(z) ≅ 0.03z/n (cm). This is illustrated graphically in Figure 1.13 that shows propagation of 0.1, 0.3, 1.0, 3.0, and 10.0 THz components of a THz pulse propagating a distance of 1 m from a lenscoupled THz source as calculated from Equation 1.16. At 100 cm distance, the beam at 0.3 THz has a radius of 10 cm; in practice, freely propagating THz beams cover much smaller distances and diffraction is controlled using optics. Because of the differing propagation characteristics of different frequency components of the THz beam, the propagation of THz beams through optical systems

100

Waist Radius, w(z) (cm)

0.1 THz 0.3 THz

10

1 THz 3 THz

1.0

10 THz 0.1

0

20

40

60

80

100

Distance from Source, z (cm)

Figure 1.13 Variation of the THz beam waist radius (1/e amplitude point) with distance from the lens-coupled, photoconductive THz source.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

27

exhibits effects that are not commonly seen at optical frequencies. A good first-order approximation of THz beam propagation through optical systems that assumes paraxial propagation and ignores aberrations and aperturing effects of mirrors can be made using Gaussian beam formalism combined with the ABCD matrix method for describing optical systems.31 To calculate propagation through optical systems, both the amplitude and phase terms, w(w,z) and R(w,z), respectively, are represented by the complex q parameter: 2c / w 1 1 ≡ −j 2 q ( w , z ) R( w , z ) w (w, z )

(1.19)

As the THz beam propagates through an optical system with a given ABCD matrix, the phase and amplitude profiles are changed, which transforms the q parameter according to:

1 = qout

  C +D 1   q in    A + B 1   q in 

(1.20)

where qin is the q parameter at the input plane of the optical system, which is usually defined as the THz source coupling lens and qout is the value at the system output, usually the coupling lens of the THz detector. To achieve optimal power transfer, the beam that is detected at the receiver must have an identical phase and amplitude profile to the emitted THz beam at all component frequencies.32 It can be shown that this occurs if the phase and amplitude profiles of the beam from the THz source and a hypothetical beam from the THz detector fully overlap at some point in the system. The coupling efficiency is determined by integrating the normalized field profiles of the beam from the source and the hypothetical detector beam over a plane at some point in the system.32 The normalized power coupling is then given by:

Cp =

 

∫∫

∫∫

E s (w , z )E D (w , z ) * dS

E s (w , z )ES (w , z ) * dS  ×   

∫∫

2

E D (w , z )E D (w , z ) * dS  

(1.21)

where for TEM0,0 modes the source and detector beams are given by ES (w,z) and ED (w,z), respectively. For ideal coupling, CP must be in unity for all the frequencies comprising the THz pulse. Assuming the THz source and THz detector are identical, unity power coupling occurs when the beam at the output of the optical system has identical phase and amplitude profiles to that at the input so that qout = qin. From Equation 1.20, this condition requires that the elements along the main diagonal of the ABCD matrix,

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28

Terahertz Spectroscopy: Principles and Applications 0.1 THz ƒ1

ƒ1

0.3 THz 1 THz

3 THz 10 THz ƒ1

ƒ1

2ƒ1 (a)

ƒ2

ƒ1

ƒ1

ƒ 1 + ƒ2

ƒ2

2ƒ2

ƒ1

ƒ1 + ƒ2

ƒ1

(b)

Figure 1.14 Two common configurations of THz spectrometers. The configuration shown in (a) is a confocal system with a frequency-dependent beam waist at the midpoint of the system. The configuration of (b) is commonly used in THz imaging systems and creates a frequency-independent waist between the short focal length lenses, ƒ2.

A and D, have values ±1 and the off-diagonal elements; B and C are zero—in other words, a unity transfer function. T(w), with ideal coupling for all THz frequencies is achieved for optical systems whose ABCD matrix is the identity matrix, I. Two separate optical systems with unity coupling and an identity transfer matrix are shown in Figure 1.14 along with the calculated beam diameters (1/e amplitude point) at 0.1 THz, 0.3 THz, 1.0 THz, 3.0 THz, and 10 THz. In both systems, lenses are spaced from each other by the sum of their focal lengths. Figure 1.14a illustrates the most common system for THz spectroscopy. Because the field pattern at the back focal plane of a lens is the spatial Fourier transform of the pattern at the front focal plane, the frequency-independent Gaussian beam waist at the coupling lens forms overlapping frequency dependent waists midway between the two lenses in Figure 1.14a. This frequency-dependent waist pattern is then transformed back to a frequency independent waist at the THz detector. The fact that most samples are placed midway between the two lenses for THz spectroscopy needs to be accounted for in data analysis because experimental artifacts from the frequency-dependent waist size can be introduced if the sample is not uniform over large areas. One method of

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

∆z = 10%

1 Power Coupling Efficiency, CP

29

∆z = 20%

0.8 0.6

∆z = 10%

0.4 Figure 14(a) Figure 14(b)

0.2 0

0

2

∆z = 20%

6 4 Frequency (THz)

8

10

Figure 1.15 Effect of changing the spacing of focusing optics by 10% and 20% on the frequency-dependent THz power coupling efficiency for the optical configurations shown in Figure 1.14. The amplitude coupling coefficient can be determined by the square root of the plotted values.

measuring nonuniform samples or those with small area is shown in Figure 1.14b. Here, ƒ2 = ¼ƒ1 and the THz beam midway between the two short focal length lenses has a frequency-independent beam waist that is four times smaller than that created by the THz source. This configuration is often used in THz imaging applications33 in which a sample is raster-scanned at the beam waist. THz beam waist diameters smaller than 1 mm can be routinely achieved in this configuration. For both configurations shown in Figure 1.14, the spacing between the lenses affects the frequency-dependent coupling efficiency. The coupling efficiency is shown in Figure 1.15 as a function of frequency at lens spacings corresponding to 80% and 90% of the ideal distances in Figure 1.14, 2ƒ1 or ƒ1 + ƒ2, respectively. As can be seen from the figure, the optical transfer function is an important parameter for the design of any terahertz system. For nonideal optical configurations, it is possible to choose corrective optics to ensure the system has a unity transfer by simply finding the matrix inverse of the transfer function. Because T(w)T(w)–1 = ±I where I is the identity matrix, for nonideal system with transfer matrix T(w), the matrix T(w)–1 gives the transfer function of the corrective optics required to achieve unity coupling. For complex THz optical systems, it is good practice to build THz beam optics in blocks so that the transfer function of each block is the identity matrix. The general treatment outlined previously in this chapter can also be used when coupling THz pulses into waveguides or other photonic devices.34 In this case, the overlap integral is done between the waveguide or device modes and the free-space THz beam. Two final considerations are important when designing THz spectroscopy systems. First, as mentioned previously, the TEM0,0 Gaussian beam formalism is a useful approximation, but it suffers from the fact that the antenna pattern at the lens surface is not a true Gaussian. The propagation can be more precisely, but not

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Terahertz Spectroscopy: Principles and Applications

uniquely, described in terms of a summation higher order Laguerre-Gauss modes.30 The coupling theory still applies in this case for each of the orthogonal modes. For most situations involving beam propagation, the TEM0,0 approximation has acceptable limits of error and the increased complexity of a full modal expansion is unwarranted. There are, however, several situations where the more complex description of the beam profile may need to be accounted for. The first of these is in the near-field region of the coupling lens, where z << zo. A second situation is when coupling to waveguides or other photonic devices. Last, a brief mention will be made of optical materials at THz frequencies. Most optical glasses are, for practical purposes, opaque at THz frequencies. Exceptions are fused silica, crystalline quartz, and sapphire, although sapphire is highly birefringent.35 In addition, the refractive indices of most materials at THz frequencies are much higher than the indices at optical frequencies. For example, sapphire has an index of approximately 1.7 at optical frequencies and ordinary and extraordinary indices of 3.1 and 3.4 at THz frequencies, whereas fused silica has an index near 2.0. All these materials are somewhat lossy—with the absorption generally increasing with frequency—and highly dispersive compared with their properties at optical frequencies. High-resistivity silicon of greater than 1 kΩ cm has extremely low loss compared with other optical materials and is nearly dispersionless and not birefringent. The relatively high index, n = 3.42, leads to large losses because of Fresnel reflections; the amplitude transmission coefficient of a silicon slab is 0.7. Standard quarter-wave index matching methods for creating antireflection coatings do not work for THz pulses because of the extremely broad bandwidth. As a result, many THz systems use reflective optics, particularly off-axis parabolic mirrors. The reflectivity of most metals is effectively 100% across the THz band and, because of the long wavelength, there are not strict requirements for surface polishing. For example, l/20 surface smoothness at 3 THz is 5 µm, which can be achieved in metals using medium grit sandpaper. Any commercially available metal optics generally work well for THz applications.

1.7 Adaptations and Extensions of TerahertZ Spectroscopy Systems The basic technique of THz spectroscopy outlined previously has been adapted by many investigators to extend the capabilities of THz spectroscopy to measure a wide range of samples. There are too many variations to go into in detail, and many of these are covered more completely in later chapters of this book. The variations of THz spectroscopy can be broadly broken down into two categories. The first is techniques that measure the phase-coherent THz pulse shape to extract the optical properties of a sample across a broad range of wavelengths in the far infrared spectral region. Although the experiment measures the electric field with subpicosecond resolution, the information that is obtained has no inherent time dependence. In these measurements, the sample itself is in equilibrium; it does not change on a fast time scale. The second category is measurements that create an optically induced time-dependent change to the sample and can provide fundamentally different spectroscopic information.

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Terahertz Time-Domain Spectroscopy with Photoconductive Antennas

31

Among the first set of variations of the basic spectroscopic technique are those developed to measure samples with small index and absorption such as thin films. Because THz spectroscopy measures the changes of the transmitted spectra, any noise on the THz signal contributes to measurement errors. The dynamic range— defined as the RMS noise of the THz detector compared to the peak THz signal—of THz spectroscopy using photoconductive antennas is extremely high, typically more than 10,000. However, the precision with which spectroscopic data can be measured is limited by the noise on the laser source, which is typically 0.1 to 1%. For low values of the sample absorbance, e–ad ≅ 1-ad, and for values of αd ≤ 10 –2 the change in transmission approaches the measurement accuracy. The sample refractive index can be measured with better accuracy because the index causes a temporal shift of the THz pulse by an amount Dt ≅ nd/c and jitter noise is generally much lower than amplitude noise. Time shifts of tens of femtoseconds can generally be resolved so that index thickness values of nd > 10 –3 cm can be accurately measured. To improve the accuracy for thin film or low absorbance samples several techniques have been developed. The most widely used is known as differential time domain spectroscopy.36 In differential time domain spectroscopy, the sample position is spatially modulated as shown in Figure 1.16a so that the sample moves in and out of the THz beam, which is focused to a small spot size. The film on a substrate is attached to a shaker operating at frequency fs. The difference between the sample and reference signals, Eref (t)-Esamp(t), is measured by a phase-sensitive detector that detects only the modulated signal. Because the THz pulse is not modulated, the noise on the THz background spectrum is effectively eliminated. The measurement of a difference between the reference and sample scans rather than a ratio complicates data analysis. Approximate analytic expressions for the index and absorption have been developed,35 and spectroscopic information can also be extracted numerically.

Shaker fs Beamsplitter

Sample

Al Film

nfilm nair

nsub Iamp

(a)

LIA Silicon Prisms fs

(b)

Figure 1.16 Two experimental configurations used for measuring thin film samples with thickness much less than one wavelength. The differential time domain spectroscopy (DTDS) system is shown in (a) while a system for interferometric THz measurements is shown in (b).

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Terahertz Spectroscopy: Principles and Applications

Using combinations of lock-in amplifiers, it is possible to measure samples with thicknesses of less than 100 nm and simultaneously extract the THz reference pulse using a single measurement.37,38 Extremely high-power dynamic ranges down to –80 dB have also been reported using this technique.39 An alternative technique to measure thin films is to create destructive interference between two THz pulses by splitting the THz pulse train into two equal parts and providing a near-π, frequency-independent phase shift to one of the pulses.40,41 Experimentally, this can be accomplished using a Michelson or Mach-Zender interferometer. Destructive interference between the pulses eliminates the background signal, drift, and some sources. Because of the broad spectral bandwidth, adjusting the phase through path length changes is not possible so the phase shift is accomplished from a Gouy phase shift,10 total internal reflection,9 or geometrical rotation of the electric field vector with a roof mirror.42 An experimental configuration for a THz interferometer is shown in Figure 1.16b. The Michaelson configuration achieves a near-π phase shift through total internal reflection in two silicon prisms, of which one face has been aluminized. The film thickness resolution of THz interferometric measurements is determined by the Sparrow criterion43; for typical THz photoconductive sources and detectors, time resolution of 1 fs is resolvable corresponding to measurements of films on the order of hundreds of nanometers of thickness. A second modification of THz spectroscopy systems is to measure the THz signal reflected or scattered from a sample rather than the transmitted THz pulse. A typical configuration for reflection measurements is shown in Figure 1.17. The THz pulse from the source can be scattered from the sample as shown in Figure 1.17a or reflected specularly as shown in Figure 1.17b. The detector used to measure the scattered pulse is fiber-coupled so the scattering angle β can be varied. Fiber-coupling photoconductive antennas are accomplished by prechirping the optical pulse so that the pulse is recompressed at the output of the fiber.

θ θ

β Optical Fiber Scattered THz Pulse

Fiber Coupled Detector (a)

(b)

Figure 1.17 THz scattering (a) and reflection (b) experimental configurations.

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It is more difficult to extract spectroscopic information from scattered or reflected pulses than it is from THz transmission measurements. For the specularly reflected pulse, the reflection coefficient is primarily determined by the nonresonant index. Rarely are there large enough changes in the real part of the index to result in large changes in the reflection coefficient. Additionally, there is no temporal shift of the THz pulse as there is in transmission measurements. As a rough estimate, the normalized change in reflectivity, DR/R, is approximately equal to the normalized change in index, Dn/n, with larger reflectivity changes for lower index materials. The complex index n″ is related to the absorption coefficient by n″ = a/2ko, where ko is the wave vector in free space with a value of approximately 2 × 10 4 m–1 at 1 THz. Except for extremely lossy materials, the real part of the index dominates the imaginary component and contributes negligibly to the reflection coefficient. It is possible to characterize a sample using THz ellipsometry44 and also to make spectroscopic measurements near Brewster’s angle 45 where the reflection coefficient has large changes. For nonspecular reflection, the scattered THz pulse depends to a much larger extent on the sample geometry than the sample composition. Various wave front reconstruction techniques can be used to reconstruct the sample from the scattered wave front; these are discussed in subsequent chapters. Besides adapting the basic technique of photoconductive THz spectroscopy through sample modulation or to measure reflection, it is possible to use photoconductive THz spectroscopy to measure the time-dependent change of a sample after external optical perturbation. Two common ways to accomplish this are optical pump THz probe spectroscopy and THz emission spectroscopy. This chapter concludes with an overview of these techniques with selected experimental results. Optical pump THz probe spectroscopy optically excites a sample with an ultrafast pulse and measures the dynamic processes that occur in the sample by probing with a THz pulse. The experimental modification required for optical pump THz probe spectroscopy shown in Figure 1.18 permits measurements of an optically excited

THz Source τTHz THz Detector fs Laser τopt

Figure 1.18 Schematic experimental configuration of optical pump THz probe spectroscopy described in the text.

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Terahertz Spectroscopy: Principles and Applications

sample in the far infrared spectral region with picosecond time resolution. To minimize timing jitter between the optical excitation of the sample and the THz probe, the excitation pulse typically comes from the same THz source used to generate and detect the THz pulses. For this reason, an amplified laser system is often required to ensure there is sufficient excitation pulse energy to cause a measurable change in the sample. The timing between optical and THz pulses is controlled using two delay lines as shown in the figure. The delay line labeled τTHz is used to delay the optical gating pulse for the THz detector, so that the THz electric field can be mapped out in time. The delay line labeled τOPT changes the delay between the arrival of the optical and THz pulses on the sample. The THz optical system has also been modified by adding additional focusing optics to create a small, frequency-independent beam waist to maximize overlap of the THz and optical pulses. Ultrafast optical beams with a TEM0,0 Gaussian profile and a center wavelength of λ ideally have spot size of approximately wf = λƒ/πd where wf is the radius of the focused beam waist, ƒ is the lens focal length, and d is the laser spot diameter. Because the optical wavelengths are submicron size, whereas THz wavelengths are submillimeter size, without careful selection of optics there is poor overlap which can result in data with experimental artifacts. The optical pump THz probe experiment measures a time-resolved THz pulse for different delays between the THz probe and optical pump pulses, E(tTHz,topt). When the optical delay is zero, topt = 0, the THz and optical pulses arrive at the sample at the same time. As in simple THz spectroscopy (Figure 1.8), a reference pulse is needed, and the THz pulse measured before the arrival of the optical pulse is used as the reference pulse, Eref (t, topt < 0). The THz pulse measured after arrival of the optical pulse constitutes the sample pulse, Esamp(t, topt > 0). The dynamic response of the sample to optical excitation, H(w,topt), can then be mapped out as a function of time. Inherent in the analysis given previously was the assumption that the properties of the sample are time invariant during transmission of the THz pulse. Because the THz pulse may be of longer temporal duration than the sample properties being measured this assumption may not be true in optical pump THz probe measurements. In this case, determining the dynamic response of the sample is done by deconvolving the THz pulse from the time-dependent complex refractive index. There are many reports in the literature of optical pump THz probe spectroscopy that have been used to make measurements that are difficult using other spectroscopic techniques; for example, the optical pump pulse is able to create charge carriers in normally insulating materials in order to measure time-resolved carrier dynamics. The acceleration of the created charge carriers by the THz electric field attenuates the THz probe pulse allowing the complex conductivity as a function of THz frequency and as a function of time after optical excitation, topt, to be experimentally mapped. The wide range photon energies available from current ultrashort pulse sources can excite carriers from the valence band to the conduction band of semiconductors as well as materials that are insulators at room temperature. A good example of these measurements is optical pump THz probe spectroscopy of sapphire,46 a material typically considered an outstanding insulator with a dielectric strength of nearly 5 × 106 Vm–1, resistivity over 1014 Ω-cm and a bandgap of 8.7 eV. To excite carriers, an amplified Ti:Al2O3 laser pulse with photon energy of 1.5 eV was frequency tripled

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to 4.5 eV and two photon absorption moved electrons from the valence to the conduction band. The transmitted THz pulse showed the carriers behaved according to the Drude theory and had strongly temperature-dependent mobility that could be as high as µ = 3 × 104 cm2v–1s–1 at low temperatures when the scattering of the carriers by phonons (lattice vibrations) was reduced. In chemistry, optical pump THz probe measurements have been used to measure solvation dynamics in liquids, critical to many aspects of liquid-phase chemistry including the growth and self-assembly of nanostructures. Because the reorientation of the solvent occurs on time scales of 0.1–10 ps, THz frequencies are useful for probing these systems. In one set of measurements,47 the optical pulse was used to excite a solvated dye molecule and the reorientation of the solvent molecule dipole moments surrounding the optically excited dye molecule was measured with the THz pulse. Another example of optical pump THz probe spectroscopy is measurement in superconducting materials. In the previous example, the optical pulse caused conductivity changes by creating charge carriers; in the prototypical high Tc superconductor, YBCO short-pulse optical excitation breaks superconducting pairs.48 The pair breaking causes transient changes in the complex conductivity and the recovery time of the superconducting pairs can be determined by using a THz probe to measure the conductivity as a function of delay between the optical excitation pulse and the THz pulse. These measurements are only a representative fraction of the physical systems that have been examined to date with optical pump THz probe spectroscopy; more detailed discussion is found in later chapters. It is not necessary to use photoconductively generated THz pulses in reflection or transmission for spectroscopic measurements. Because the dynamics of the photogenerated carriers in the THz source determine the temporal profile of THz pulse, the generated pulse itself can provide information on carrier dynamics on a subpicosecond time scale and can be used for spectroscopic measurements. Using the generated pulse itself for spectroscopic measurement is known as THz emission spectroscopy and a simplified experimental configuration is illustrated in Figure 1.19. THz emission spectroscopy provides fundamentally different information than optical pump-probe spectroscopy. Optical techniques generally give insight into the dynamics of how carriers evolve in energy, the vertical axis on an E versus k diagram.

Delay

Sample

fs Laser

Figure 1.19 Experimental configuration of THz emission spectroscopy.

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Terahertz Spectroscopy: Principles and Applications

THz emission spectroscopy, in contrast, probes the velocity, or momentum, evolution of carriers in time. Photoconductively generated THz pulses have electric fields proportional to the rate of change of the current, E(t) ∝ δJ/δt. Because the current is proportional to carrier density and velocity, J = N(t)ev(t), the THz field strength is proportional to the dynamic velocity ensemble of the carriers. For example, the rising edge of the THz pulse arises from carrier acceleration in conjunction with an increasing number of carriers because of optical excitation. The peak of the THz pulse occurs when the carrier ensemble is undergoing the maximum acceleration. The THz field then decreases and has a zero crossing when the ensemble velocity is a maximum (δv/δt = 0). Accurately extracting relevant information on carrier dynamics using THz emission spectroscopy requires the ability to accurately measure the electric field profile. Because the pulse shape is affected by the detector bandwidth, dispersion effects, and propagation, it is important to account for these effects of the measurement system to avoid experimental artifacts. Because THz emission spectroscopy is sensitive to the motion carriers with excess energy from photoexcitation in accelerating fields, it provides information on the dynamics of hot carriers. The ability to study hot carrier transport processes in high fields has impact on the development of high speed semiconductor devices such as high-electron mobility transistors. The THz pulse shape depends on both acceleration and collisions of carriers, so the pulse profile provides insight into carrier scattering processes including collisions with impurities and dopants, interaction with phonons (quantized lattice vibrations), and scattering from other carriers. It is possible to generate THz radiation from most semiconductors so that THz emission spectroscopy can measure the hot carrier dynamics in semiconductor materials without requiring direct electrical contact. It is straightforward to create heterostructures or lithographically define structures so that conditions such as temperature, the electrical bias, or the optical excitation density can be probed to map out carrier dynamics over a large parameter space. Similarly, by tuning the optical excitation wavelength, the dynamics of carriers created at different energies in the conduction band can be experimentally probed by looking at how the shape of the generated THz pulse changes. Even states below the conduction band, the so-called Urbach tail, can be investigated. THz emission spectroscopy directly measure Bloch oscillations in semiconductors. Bloch oscillations are changes in electron energy along with an oscillatory motion in real space that result from motion in the periodic crystal potential and were first predicted by Zener in 1934 based on Bloch’s 1928 theory of electrons in a periodic band structure.49 Much of the interest in Bloch oscillations arises from their potential to produce powerful, tunable sources of radiation from the oscillatory motion of the electrons. Because electron scattering processes are usually much faster than the period of the Bloch oscillation, dephasing processes need to be controlled. The development of controllable band structures in semiconductor superlattices allowed materials to be created in which Bloch oscillations could be observed. THz emission spectroscopy directly observes Bloch oscillations as oscillations following the initial THz pulse. As the bias on the sample increases, so does the frequency of the Bloch oscillations, as does the spectrum of the emitted THz radiation.

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Any other process that gives emits a THz pulse is a candidate for THz emission spectroscopy; examples are carrier acceleration in the surface depletion field of a semiconductor50 and emission from metal-air surfaces from optical rectification.51 THz emission spectroscopy is not limited to crystalline semiconductors because many materials produce a transient polarization after optical excitation. Emission from semiconducting polymer materials has helped to clarify the process of charge generation in these materials. Work has also looked at magnetic nonlinearities as well as electronic nonlinearities.48 Some of the more novel work on THz emission spectroscopy has been done on polar molecules in liquids. In these experiments, a static electric field is placed across the liquid, and the polar molecules orient in the field. Orientation of the ensemble is required to generate THz radiation because the net dipole moment of an ensemble of randomly aligned molecules will be zero. When the molecules are photoexcited, they undergo intermolecular charge transfer on a picosecond time scale, which causes a change in the dipole moment and a transient polarization of the ensemble of molecules that emits a THz pulse. These measurements provide information on the time scale and direction of intermolecular charge transfer.47

References 1. Harnessing light: optical science and engineering in the 21st century. Washington, D. C.: National Academy Press, 1998. 2. M. Inguscio et al., A review of frequency measurements of optically pumped lasers from 0.1 to 8 Thz, J. Appl. Phys., 60, R161, 1986. 3. D. S. Kurtz, et al., Submillimeter-wave sideband generation using Varactor Schottky diodes, IEEE Trans. Microwave Theory Tech., 50, 2610, 2002. 4. A. Tredicucci, et al., Terahertz quantum cascade lasers—first demonstration and novel concepts, Semicond Sci Technol, 20, S222, 2005. 5. S. D. Ganichev and W. Prettle, Intense THz excitation of semiconductors. New York: Oxford University Press, 2006. 6. C. Meystre and M. Sargent III, Elements of quantum optics. Berlin: Springer-Verlag, 1991. 7. Y. R. Shen, The principles of nonlinear optics. Hoboken, N. J.: John Wiley & Sons, 2003. 8. L. Allen and J. H. Eberly, Optical resonance and two-level atoms. New York: John Wiley & Sons, 1975. 9. S. R. Keiding and D. Grischkowsky, Measurements of the phase shift and reshaping of terahertz pulses due to total internal reflection, Opt. Lett., 15, 48, 1990. 10. S. Feng, H. G. Winful, and R. W. Hellwarth, Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses, Opt. Lett., 23, 385, 1998. 11. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, Generation and detection of terahertz pulses from biased semiconductor antennas, J. Opt. Soc. Am. B, 13, 2424, 1996. 12. K. Bock, H. L. Hartnagel, and J. Dumas, Surface induced electromigration in GaAs devices, in Reliability of gallium arsenide MMICs, A. Christou, ed. New York: John Wiley & Sons, 1992, 192. 13. C. A. Balanis, Advanced engineering electromagnetics. New York: John Wiley & Sons, 1989. 14. R. W. P. King and G. S. Smith, Antennas in matter. Cambridge: MIT Press, 1981. 15. K. Liu, et al., Terahertz radiation from InAs induced by carrier diffusion and drift, Phys. Rev. B, 73, 2006.

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Terahertz Spectroscopy: Principles and Applications 16. S. D. Benjamin, H. S. Loka, and P. W. E. Smith, Tailoring of low-temperature MBE-grown GaAs for ultrafast photonic devices, Can. J. Phys., 74, S68, 1996. 17. F. E. Doany, D. Grischkowsky, and C.-C. Chi, Carrier lifetime versus ion-implantation dose in silicon on sapphire, Appl. Phys. Lett., 50, 462, 1987. 18. D. R. Dykaar, et al., Log-periodic antennas for pulsed terahertz radiation, Appl. Phys. Lett., 59, 262, 1991. 19. H. Murakami, et al., Tetrahertz pulse radiation properties of a Bi2Sr2CaCu2O8 + delta bowtie antenna by optical pulse illumination, Jap. J. Appl. Phys. Part 1, 41, 1992, 2002. 20. H. Harde, R. A. Cheville, and D. Grischkowsky, Terahertz studies of collision-broadened rotational lines, J. Phys. Chem. A, 101, 3646, 1997. 21. M. Born and E. Wolf, Principles of optics, ed 7. New York: Cambridge University Press, 1999. 22. L. Duvillaret, F. Garet, and J. L. Coutaz, A reliable method for extraction of material parameters in terahertz time-domain spectroscopy, IEEE J. Selected Topics Quantum Electr., 2, 739, 1996. 23. M. T. Reiten, et al. Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling, Opt. Lett., 26, 1900, 2001. 24. L. Duvillaret, F. Garet, and J. L. Coutaz, Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy, Appl. Opt., 38, 409, 1999. 25. P. U. Jepsen and B. M. Fischer, Dynamic range in terahertz time-domain transmission and reflection spectroscopy, Opt. Lett., 30, 29, 2005. 26. S. C. Howells, S. D. Herrera, and L. A. Schlie, Infrared wavelength and temperature dependence of optically induced terahertz radiation from InSb, Appl. Phys. Lett., 65, 2946, 1994. 27. G. Zhao, et al., Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter, Rev. Sci. Instr., 73, 1715, 2002. 28. W. H. Press, et al., Numerical recipes in Fortran, 2nd ed., New York: Cambridge University Press, 1993. 29. W. Lukosz and Z. S. Eth, Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation, J. Opt. Soc. Am., 69, 1495, 1979. 30. M. T. Reiten, S. A. Harmon, and R. A. Cheville, Terahertz beam propagation measured through three-dimensional amplitude profile determination, J. Opt. Soc. Am. B, 20, 2003. 31. A. E. Siegman, Lasers. Mill Valley: University Science Books, 1986. 32. J. C. G. LeSurf, Millimetre-wave optics, devices, and systems. Bristol: Adam Hilger, 1990. 33. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, T-ray imaging, IEEE J. Sel. Topics Quantum Electr, 2, 679, 1996. 34. G. Gallot, et al., Terahertz waveguides, J. Opt. Soc. Am. B, 17, 851, 2000. 35. D. Grischkowsky, et al., Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors, J. Opt. Soc. Am. B, 7, 2006, 1990. 36. Z. P. Jiang, M. Li, and X. C. Zhang, Dielectric constant measurement of thin films by differential time-domain spectroscopy, Appl. Phys. Lett., 76, 3221, 2000. 37. K. S. Lee, T. M. Lu, and X. C. Zhang, The measurement of the dielectric and optical properties of nano thin films by THz differential time-domain spectroscopy, Microelectr J, 34, 63, 2003. 38. S. P. Mickan, et al., Double modulated differential THz-TDS for thin film dielectric characterization, Microelectr J, 33, 1033, 2002.

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39. M. Brucherseifer, P. H. Bolivar, and H. Kurz, Combined optical and spatial modulation THz-spectroscopy for the analysis of thin-layered systems, Appl. Phys. Lett., 81, 1791, 2002. 40. J. L. Johnson, T. D. Dorney, and D. M. Mittleman, Enhanced depth resolution is terahertz imaging using phase shift interferometry, Appl. Phys. Lett., 78, 835, 2001. 41. S. Krishnamurthy, et al., Characterization of thin polymer films using terahertz time-domain interferometry, Appl. Phys. Lett., 79, 875, 2001. 42. J. A. Small and R. A. Cheville, Measurement and noise characterization of optically induced index changes using terahertz interferometry, Appl. Phys. Lett., 84, 4328, 2004. 43. D. Guenther, Modern optics, 2nd ed., New York: John Wiley & Sons, 1990. 44. T. Nagashima and M. Hangyo, Measurement of complex optical constants of a highly doped Si wafer using terahertz ellipsometry, Appl. Phys. Lett., 79, 3917, 2001. 45. M. Li, et al., Time-domain dielectric constant measurement of thin film in GHz-THz frequency range near the Brewster angle, Appl. Phys. Lett., 74, 2113, 1999. 46. J. Shan, et al., Measurement of the frequency-dependent conductivity in sapphire, Phys. Rev. Lett., 90, 247401, 2003. 47. M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, Terahertz spectroscopy, J. Phys. Chem. B, 106, 7146, 2002. 48. D. J. Hilton, e. al., On photo-induced phenomena in complex materials: probing quasi- particle dynamics using infrared and far-infrared pulses, J. Phys. Soc. Japan, 75, 011006, 2006. 49. K. Leo, Interband optical investigation of Bloch oscillations in semiconductor superlattices, Semicond Sci Technol, 13, 249, 1998. 50. J. Darmo, et al., Surface-modified GaAs terahertz plasmon emitter, Appl. Phys. Lett., 81, 871, 2002. 51. F. Kadlec, P. Kuzel, and J. L. Coutaz, Study of terahertz radiation generated by optical rectification on thin gold films, Opt. Lett., 30, 1402, 2005.

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2

Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection—Optical Rectification and Electrooptic Sampling Ingrid Wilke and Suranjana Sengupta Rensselaer Polytechnic Institute

Contents 2.1

Introduction.................................................................................................... 42 2.1.1 Terahertz Time-Domain Spectroscopy: An Overview....................... 42 2.2 Optical Rectification and Linear Electrooptic Effect....................................44 2.3 Experimental Results of Terahertz-Frequency Radiation Generation by Optical Rectification of Femtosecond Laser Pulses................................. 49 2.3.1 Materials............................................................................................. 50 2.3.1.1 Semiconductors..................................................................... 50 2.3.1.2 Inorganic Electrooptic Crystals............................................ 52 2.3.1.3 Organic Electrooptic Crystals............................................... 52 2.3.2 Recent Developments.......................................................................... 56 2.4 Experimental Results of Terahertz Electrooptic Detection........................... 57 2.4.1 Materials............................................................................................. 57 2.4.1.1 Semiconductors and Inorganic Crystals................................ 57 2.4.1.2 Organic Crystals....................................................................64 2.5 Application of Electrooptic Sampling of Terahertz Electric Field Transients....................................................................................................... 65 References................................................................................................................. 69

41

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Terahertz Spectroscopy: Principles and Applications

2.1 Introduction The generation and detection of short pulses of terahertz (THz) frequency electromagnetic radiation are attracting great interest. Research activities in this area are driven by applications of THz-frequency radiation pulses in time-domain THz spectroscopy and time-domain THz imaging.1,2 Recent examples of major scientific advancements in THz wave research include the detection of single-base pair differences in femtomolar concentrations of DNA,3 the observation of the temporal evolution of exciton formation in semiconductors4 and understanding of carrier dynamics in high-temperature superconductors.5 Real-world applications of time-domain THz spectroscopy and imaging address important problems such as nondestructive testing,6 as well as new approaches to medical diagnostics and rapid screening in drug development.7 One of the research directions in THz science and technology is to increase the bandwidth of short THz-frequency radiation pulses. Nonlinear optical phenomena such as optical rectification and the linear electrooptic effect (Pockels’ effect) are attractive options for broadband generation and detection of THz radiation. This chapter provides an overview of the generation and detection of freely propagating THz pulses based on nonlinear optical techniques. In Section 2.1.1 of this chapter, the principles of time-domain THz spectroscopy and time-domain THz imaging using nonlinear optical techniques for THz pulse generation and detection are briefly discussed. Section 2.2 describes the principles of generation and detection of THz-frequency radiation by optical rectification of femtosecond laser pulses and femtosecond electrooptic sampling. THz radiation emission has been reported from a variety of nonlinear materials such as lithium niobate (LiNbO3),8,9 lithium tantalate (LiTaO3),9,10 zinc telluride (ZnTe),11,12 indium phosphide (InP),13 gallium arsenide (GaAs),14 gallium selenide (GaSe),15,16 cadmium telluride (CdTe),17 cadmium zinc telluride (CdZnTe),18 DAST,10,19 and metals.20,21 Section 2.3 summarizes the performance of THz emitters based on these materials in terms of bandwidth and signal strength of the emitted THz radiation. Materials for the detection of THz radiation by femtosecond electrooptic sampling are the subjects of Section 2.4. In Section 2.5, an application of electrooptic detection of THz-frequency electromagnetic transients is discussed.

2.1.1 Terahertz Time-Domain Spectroscopy: An Overview The basic ideas of time-domain THz spectroscopy and a time-domain THz imaging system is illustrated in Figure 2.1. A subpicosecond pulse of THz-frequency electromagnetic radiation passes through a sample placed in the THz beam and its time profile is compared to a reference pulse. The latter can be a freely propagating pulse or a pulse propagating through a medium with known properties. The frequency spectra of the transmitted and reference THz radiation pulses are obtained by Fourier transformation. Analysis of the frequency spectra yields spectroscopic information on the material under investigation. In the case of time-domain THz imaging the THz radiation focal spot is scanned across the sample and a THz image of the object under investigation is obtained. The time-domain THz spectroscopy system shown in Figure 2.1 is powered by a laser, which emits a train of femtosecond duration pulses at near-infrared frequencies.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 43 Delay Stage M1

Laser

M2

Probe Beam Beam Splitter

M5

PM I

PM II

Pump Beam

THz Beam PBS

Chopper

Emitter

L2

Detector

L1 M3

M6

Quarter Wave Plate

M4

M7 Data Acquisition Software

Lock-in Amplifier

Diodes

Wollaston Prism

Figure 2.1 Experimental arrangements for time-domain THz spectroscopy measurements using nonlinear optical techniques for the generation and detection of freely propagating THz-radiation pulses. (M1–M7 optical beam steering mirrors, PMI, PMII parabolical THzbeam steering mirrors, L1, L2 lenses).

The initial laser beam after passing through a beam splitter is split into two parts: the pump and the probe beam. The pump beam after being modulated by an optical chopper is focused on the THz emitter (a second-order nonlinear optical medium), which releases a subpicosecond pulse of THz radiation in response to the incident femtosecond near infrared laser pulse. The generated THz radiation is focused onto a detector using two off-axis parabolic mirrors. The detector is an electrooptic crystal. The probe beam gates the detector, whose response is proportional to the amplitude and sign of the electric field of the THz pulse. By changing the time delay between the pump and the probe beams by means of an optical delay stage, the entire time profile of the THz transient can be traced. Electrooptic detection of THz transients is possible when the THz radiation pulse and the probe beams coincide in a copropagating geometry inside the electrooptic crystal. A pellicle beam splitter is put in the THz beam line for this purpose. As the THz pulse and probe beam copropagate through the electrooptic crystal, a phase modulation is induced on the probe beam which depends on the electric field of the THz radiation. The phase modulation of the probe beam is analyzed by a quarter wave (l/4) plate and the beam is then split into two beams of orthogonal

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Terahertz Spectroscopy: Principles and Applications

polarizations by a Wollaston prism. At this point, the phase modulation of the probe beam is converted to an intensity modulation of the two orthogonal polarizations of the probe beam, which are then steered into a pair of photodiodes. A lock-in amplifier subsequently detects the difference in probe laser light intensities measured by the photodiodes. Next, the key aspects of the nonlinear optical processes involved in the generation and detection of short pulses of THz-frequency radiation are discussed in depth.

2.2 Optical Rectification and Linear Electrooptic Effect Optical rectification and the linear electrooptic effect (Pockels’ effect) are nonlinear optical techniques for the generation and detection of freely propagating subpicosecond THz-frequency radiation pulses. Generally, optical rectification refers to the development of a DC or low- frequency polarization when intense laser beams propagate through a crystal. The linear electrooptic effect describes a change of polarization of a crystal from an applied electric field. Optical rectification and the linear electrooptic effect occur only in crystals that are not centrosymmetric. However, optical rectification of laser light by centrosymmetric crystals is possible if the symmetry is broken by a strong electric field. Furthermore, generation and detection of THz-radiation pulses by optical rectification and the Pockels’ effect require that the crystals are sufficiently transparent at THz and optical frequencies. Freely propagating subpicosecond THz-radiation pulses are generated by optical rectification of femtosecond (fs) near infrared laser pulses in crystals with appropriate nonlinear optical properties for this process. The detection of freely propagating THz-radiation pulses is performed by measuring the phase modulation of an fs near infrared laser pulse propagating through an electrooptic crystal simultaneously with a THz-radiation pulse. The electric field of the THz radiation induces a phase modulation of the fs laser pulse through the linear electrooptic effect. In the following section, the theory of optical rectification and the linear electrooptic effect is discussed. The discussion of theory is limited to material we consider useful for the reader to get a general idea of the underlying physics of THz pulse generation and detection by nonlinear optical techniques and to equations that are relevant to the implementation of the techniques in the laboratory. Our discussion mainly follows the original work by pioneers in the field. First, DC optical rectification of 694 nm continuous-wave laser radiation in potassium dihydrogen phosphate and potassium dideuterium phosphate was demonstrated experimentally by Bass et al. in 1962.22 After that, monochromatic THz- radiation generation at 3 THz by low-difference frequency mixing of near-infrared (nir) laser radiation (lnir = 1.059–1.073 Nm) in quartz was achieved in 1965 by Zernike and Berman.23 Yang et al. demonstrated generation of broadband (0.06–0.36 THz) freely propagating THz-radiation pulses by optical rectification of picosecond Nd: glass laser pulses in LiNbO3 in 1971.8 Eventually, Hu et al. produced free-space THz-frequency radiation with a bandwidth of 1 THz by optical rectification of femtosecond CPM dye laser pulses in LiNbO3 in 1990.10

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 45

In 1893, Friedrich Pockels discovered the linear electrooptic effect.24 Subsequently, the phase modulation of optical laser light at microwave frequencies using the linear electrooptic effect was demonstrated in 1962 by Harris et al.25 Then, Valdmanis and coworkers built the first electrooptic sampling system with picosecond resolution for the measurements of ultrafast electrical transients in 1982.26 Finally, electrooptic sampling of freely propagating THz-radiation pulses was demonstrated in 1995 by Wu and Zhang27 and in 1996 by Jepsen et al.28 and Nahata et al.29 An excellent description of laser light modulation based on the linear electrooptic effect is given by Yariv.30 Rigorous quantum mechanical calculations of optical rectification were carried out by Armstrong et al.31 Previous reviews of nonlinear generation and detection of sub-picosecond THz-radiation pulses were published by Bonvalent and Joffre,32 Shen et al.,33 and Jiang and Zhang.34 The discussions of optical rectification and the Pockels’ effect begin by considering scalar relationships of polarization P, electric susceptibility, and electric field E.35 P = c( E ) E

(2.1)

The electric polarization P of a material is proportional to the applied electric field E. Here, c(E) is the electric susceptibility. The nonlinear optical properties of the material are described by expanding c(E) in powers of the field E.

P = ( c1 + c 2 E + c3 E 2 + c 4 E 3 + ) E

(2.2)

Optical rectification and the linear electrooptic effect are second order nonlinear optical effects P2nl and described by the P2nl = c2 E2 term in the expansion. For example, consider an optical electric field E described by E = E 0 cos wt. In this case, the second-order nonlinear polarization P2nl consists of a dc polarization c2 E 02/2 and a polarization with a cos 2wt dependence.

P2nl = c 2 E 2 = c 2

E0 2 (1 + cos 2 w t ) 2

(2.3)

The DC polarization results from the rectification of the incident optical electric field by the second-order nonlinear electric susceptibility of the material. The polarization with the cos 2wt dependence describes second harmonic generation. This nonlinear optical process is not relevant to the generation and detection of THz radiation by nonlinear optical techniques and therefore not discussed further. Similarly, consider two optical fields oscillating at frequencies E1 = E 0cos w1t and E2 = E 0cos w2t. P2nl = c 2 E1 E2 = c 2

E0 2 [cos(w1 - w 2 )t + cos(w1 + w 2 )t ] 2

(2.4)

this case, the DC second-order nonlinear polarization P2nl consists of a term In P2w1-w2 proportional to the difference frequency w1 + w2 and a term P2w1+w2 proportional to the sum frequency w1 + w2. The production of THz radiation by optical rectification relies on low difference frequency generation and is described by the

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46

Terahertz Spectroscopy: Principles and Applications

term P2w1 - w2. Again, sum frequency generation P2w1 + w2 is not relevant to generation of THz radiation by nonlinear optical techniques. In the same way, generation of THz-radiation pulses from optical rectification of fs laser pulses is based on difference frequency mixing of all frequencies within the bandwidth Dw of an fs near infrared laser pulse. Specifically, an fs pulse ∆t is characterized by a large frequency bandwidth Dw. The bandwidth of the laser pulse is described by a Gauss function of width 1/4G.36  ( w - w 0 )2  E (w ) ∝ exp   4G  

(2.5)

In the time domain, the fs laser pulse is described by an optical field oscillating at frequency w0 with a time dependence described by a Gauss function.

E (t ) = E0 exp (iw 0t ) exp( - Gt 2 )

(2.6)

The bandwidth of the THz-radiation pulse is determined by difference frequency generation by all frequencies within the bandwidth of the fs laser pulse. The time profile of the radiated THz pulse from optical rectification of the fs laser pulse is proportional to the second time derivative of the difference frequency term P2w1 - w2. The time dependence of P2w1 - w2 is determined by the Gaussian time profile of the optical laser pulse. The physics of the linear electrooptic effect is described by the same term P2n1 = 2 c2 E in the expansion of c(E) as optical rectification. To understand the relationship between optical rectification and the Pockels’ effect, it is now necessary to describe polarization P and electric field E by vectors and the susceptibility c by a third rank tensor. Quantum mechanical calculations of the polarization for the case of two laser beams with frequencies w2 , w1 present in a crystal demonstrate that the ith component of the second-order nonlinear polarization piw1-w2 is related to the components j and k of the electrical fields Ejw1 and Ekw2 via the susceptibility tensor component cijkw1 - w2 35 w1 - w 2

w1

w2

pi = c ijk E j Ek (2.7) DC optical rectification and the linear electrooptic effect result from Equation 2.7 considering the limits of w2 - > w1 and w2 - > 0, respectively.

w1 - w 2

w 2 → w1

pi0 = c ijk0 E wj 1 Ekw1

(2.8)

w2 → 0

piw1 = c ijkw1 E wj 1 Ek0

(2.9)

Equations 2.8 and 2.9 demonstrate that a strong electric field at frequency w1 gives rise to a dc polarization pi0 and a DC electric field Ek 0 changes the polarization piw1 at frequency w1.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 47

Further calculations demonstrate that c 0ijk is equal to c w1jik. Therefore, the third rank tensors are identical under simple interchange of indices as well as frequencies. This demonstrates the identity of optical rectification and the linear electrooptic effect. As a result of this identity, it is possible to estimate the magnitude of optical rectification from the linear electrooptic coefficients of a material. The most widely employed crystals for nonlinear optical generation and detection of THz radiation are ZnTe, gallium phosphide (GaP), and GaSe. ZnTe and GaP exhibit zincblende structure and 43m point group symmetry. GaSe is a hexagonal crystal with 62m point group symmetry. All three crystals are uniaxial. Uniaxial crystals have only one axis of rotational symmetry referred to as the c-axis or optical axis of the crystal. The efficacy of ZnTe, GaP, and GaSe for nonlinear optical THz generation and detection is rated by the linear electrooptic coefficient of the materials. The linear electroptic coefficients for ZnTe, GaP, and GaSe are listed in Table 2.1. The relationship between susceptibility tensor elements describing optical rectification cijk 0 and linear electrooptic effect c jik w is: cijk 0 + cikj 0 =

1 c jik w 2

(2.10)

and the relationship between the linear electrooptic coefficient rjki and the linear electrooptic effect tensor element c jik w is:  4p  rjki = -  2 2  c jik w  n0 ne 

(2.11)

Efficient generation and detection of THz radiation by optical rectification and the Pockels’ effect require single crystals with high second-order nonlinearity or large electrooptic coefficients, proper crystal thickness and proper crystal orientation with respect to the linear polarization of the THz radiation. The surfaces of the crystals should be optically flat at the laser excitation wavelengths and of high crystalline quality (e.g., low levels of impurities, structural defects, and intrinsic stress). The bandwidth of an electrooptic crystal for THz generation and detection is determined by the coherence lengths and optical phonon resonances in the material.

Table 2.1 Electrooptic Coefficients of Some Commonly Used Terahertz Emitters and Detectors Material ZnTe GaP GaSe

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Structure Zincblende(30) Zincblende(30) Hexagonal (53)

Point Group Symmetry ( 30 )

43m 43m( 30 ) 62 m( 30 )

Electrooptic Coefficient (pm/V) r41 = 3.9(30) r41 = 0.97(30) r41 = 14.4(76)

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48

Terahertz Spectroscopy: Principles and Applications

Infrared active optical phonon resonances result in strong absorption of electromagnetic radiation at the absorption frequencies. The phonon absorption bands (optical phonons) of a material in the THz frequency range are not available for THz generation and detection. If the refractive indices of the near infrared laser excitation frequency and the THz radiation are identical, then the bandwidths of the THz radiation depend only on the pulse width of the incident femtosecond near infrared laser beam. Furthermore, the strength of emission and sensitivity of THz detection would increase similarly for all frequencies within the bandwidth with increasing crystal thickness. However, the refractive indices for the near infrared laser frequency and THz frequency are generally not identical. Consequently, electromagnetic waves at THz and near infrared frequencies travel at slightly different speeds through the crystal. The efficacy of nonlinear optical THz generation and detection decreases if the mismatch between the velocities becomes too large. The distance over which the slight velocity mismatch can be tolerated is called the coherence length. As a result, efficient THz generation and detection at a given frequency only occur for crystals that are equal in thickness or thinner than the coherence length for this frequency. The coherence length in case of pulsed near-infrared laser radiation is defined as:11 lc (w THz ) =

w THz

pc | nopt eff (w 0 ) - nTHz (w THz ) |

(2.12)

with  ∂nopt  nopt eff = nopt (w ) - l opt   ∂l  l

(2.13) opt

In Equations 2.12 and 2.13, c is the velocity of light, w = 2p/u the THz freTHz THz quency, w0 is the near infrared excitation frequency nTHz the refractive index at THz frequencies, nopt the index of refraction at near infrared wavelength lopt, and nopt eff the group velocity refractive indices of the femtosecond near infrared laser pulse. THz emission strength of a crystal and sensitivity of an electrooptic THz detector are proportional to the thickness of the crystal. However, the bandwidth of the THz emitter crystal and bandwidth of the THz detector crystal also depend on the crystal thickness. Generally THz emission strength and THz emission bandwidth for a crystal have a reciprocal relationship. The THz emission bandwidth increases when the crystal becomes thinner. The THz emission strength decreases when the crystal becomes thinner. The same rule applies to the sensitivity of an electrooptic THz detector and the detector bandwidth. Thinner crystals have a higher bandwidth but lower sensitivity than thicker crystals and vice versa. For efficient generation of THz radiation by optical rectification of femtosecond laser pulses from zincblende structure crystals, it is important to select the proper orientation of the crystal with respect to the linear polarization of the laser beam. The preferred crystal orientations of selected materials are discussed in Sections 2.3 and 2.4.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 49

In case of the longitudinal linear electrooptic effect, the electric field is applied parallel to the direction of propagation of the optical probe laser beam. In case of the transverse linear electrooptic effect, the electric field is applied perpendicular to the direction of propagation of the optical probe beam. Detection of THz radiation pulses is generally performed using the geometry of the transverse electrooptic effect. The amplitude and phase of the subpicosecond THz-radiation pulse are measured by recording the phase change of femtosecond near infrared laser pulses traveling collinearly and simultaneously with the THz radiation pulses through the electrooptic crystal. The electric field of the THz radiation induces a change of the index of refraction of the crystal via the linear electrooptic effect such that the material becomes birefringent. The phase retardation G between the ordinary and extraordinary ray after propagating through the birefringent crystal is proportional to the amplitude and phase of the THz electric field E, the crystal thickness l, the linear electrooptic coefficient r14, the index of refraction of the crystal at the near infrared laser frequency no, and the near-infrared wavelength l.30 G∝

1 3 ln0 r41ETHz l

(2.14)

Furthermore, the phase retardation also depends on the orientation of the crystal and directions of polarization of the THz radiation pulse and the near infrared laser pulse.37 For 43m zincblende structure crystals, preferred directions of the THz electric field ETHz, optical field Eopt, and crystal orientation are discussed in Sections 2.3 and 2.4. The temporal profile of the THz radiation is mapped out in time by delaying the much shorter femtosecond near infrared laser pulse with respect to the THz radiation pulse and measuring the phase retardation as a function of the delay between THz radiation pulse and optical probe pulse.

2.3 ExperimenTAL Results of Terahertz-Frequency Radiation Generation by Optical Rectification of Femtosecond Laser Pulses In this chapter, we review the experimental results for generation of THz radiation pulses based on optical rectification. Several nonlinear optical materials have been reported to generate THz radiation by optical rectification of a femtosecond near infrared laser pulse. Table 2.2 lists the various crystals investigated for generation of THz radiation by optical rectification. For the sake of organization, the experimental results will be divided into three broad material categories: semiconductors, inorganic electrooptic crystals and organic electrooptic crystals, and the characteristics of the emitted THz radiation will be discussed in brief under each material category. The performance of various emitters will be discussed on the basis of bandwidth and amplitude of the emitted THz radiation. We conclude our discussion by presenting recent experimental developments in THz emitter research.

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50

Terahertz Spectroscopy: Principles and Applications

Table 2.2 Materials Used for Generation of Terahertz Optical Rectification Semiconductors

Inorganic Electrooptic Crystals

Organic Electrooptic Crystals

LiNbO3 LiTaO3

4-N-methylstilbazolium tosylate (DAST) N-benzyl-2-methyl-4-nitroaniline (BNA) (-) 2-(a-methylbenzyl-amino)- 5-nitropyridine (MBANP) electrooptic polymers

GaAs InP CdTe InAs InSb GaP ZnTe ZnCdTe GaSe

2.3.1 Materials 2.3.1.1 Semiconductors A significant amount of research dating back as early as the 1970s has explored the generation of THz radiation by optical rectification of pico- and femtosecond optical pulses in nonlinear optical crystals. Experimental work in the early 1990s reported that THz-frequency radiation could also be generated by femtosecond optical pulses incident on a semiconductor surface.38,39 This observation was initially explained as the dipole radiation of a time-varying current resulting from photo-excited charge carriers in the depletion field near the semiconductor surface.39–41 Although the ultrafast photocarrier transport model successfully explained the transverse magnetic polarization of the emitted THz radiation and the observed emission maxima for Brewster angle incidence, the intensity modulation of the emitted THz radiation observed when the crystal was rotated about its surface normal indicated that there was more than one mechanism responsible for the generation of THz radiation. In 1992, Chuang et al.42,43 proposed a theoretical model based on optical rectification of femtosecond laser pulses at semiconductor surfaces, which successfully explained all of the earlier experimental observations. The investigation of the physics of optical rectification indicates that depending on the optical fluence, the THz radiation generation by optical rectification is either a second-order nonlinear optical process governed by the bulk second-order susceptibility tensor c2, or a third-order nonlinear optical process whereby a second-order nonlinear susceptibility results from the mixing of a static surface depletion field and the third-order nonlinear susceptibility tensor c3.41,44 THz radiation generation from optical rectification of femtosecond laser pulses has been reported from <100>, <110>, and <111>-oriented crystals with zincblende structure commonly displayed by most III–V (and some II–VI) semiconductors. These crystals have a cubic structure with 43m point group symmetry and only one independent nonlinear optical coefficient, namely r41 = r52 = r63.30 Theory predicts that for <100>-oriented zincblende crystals, there is no optical rectification field at normal incidence (as the nonvanishing second-order optical coefficients r41, r52, and

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 51

r63 are not involved), whereas a three-fold rotation symmetry about the surface normal may be observed for <110> and <111> orientations. Experimental studies conducted on <100>, <110>, and <111> oriented GaAs, CdTe, and indium phosphide (InP) crystals13,17,41 and <110> GaP crystal45 are in excellent agreement with the theoretical results providing evidence that second-order bulk optical rectification is the major nonlinear process under the condition of moderate optical fluence (~nJ/cm2) and normal incidence on unbiased semiconductors. A dramatic variation in the radiated THz signal along with polarity reversal is observed in <110> and <111> CdTe and GaAs crystals, as the incident photon energy is tuned near the bandgap.13,17,41 This phenomenon can be explained as the dispersion of the nonlinear susceptibility tensor near the electron resonance state.13,41 THz generation by optical rectification has recently also been reported from narrow bandgap semiconductors such as indium arsenide (InAs) and indium antimonide (InSb) at high optical fluence (1–2 mJ/cm2) and off-normal incidence.44,46–50 Experimental results of THz emission from <100> InSb crystal and <100>, <110>, and <111> InAs crystal for both n- and p-type doping were examined.44,49 A pronounced angular dependence of the emitted THz radiation on the crystallographic orientation of the samples as well as on the optical pump beam polarization was observed. The experimental results are in good agreement with the theory, which predicts that for high excitation fluence, THz emission from narrow band gap semiconductors results primarily from surface electric field-induced optical rectification with small bulk contribution.50 This observation is in contrast with the earlier results obtained for wide bandgap semiconductors such as GaAs, in which the THz emission is primarily dominated by strong bulk contribution. The anomaly can be explained in the light of the theoretical calculations by Ching and Huang51 whereby narrow bandgap III–V semiconductors (InAs and InSb) are predicted to have third-order nonlinear susceptibilities (c3) several orders of magnitude higher than those of larger bandgap semiconductors. Among the zincblende crystals, perhaps the most popular candidate for generation of THz radiation by optical rectification is ZnTe. Nahata et al.11 first reported THz generation in ZnTe in an experiment that used a pair of ZnTe crystals for both generation and detection of THz radiation. In their work, a <110> ZnTe crystal was pumped by 130 femtosecond pulses at 800 nm from a mode-locked Ti:sapphire laser. The Fourier spectrum of the temporal waveform demonstrated a broad bandwidth with useful spectral information beyond 3 THz. Subsequently, Han et al.12 reported a higher frequency response of 17 THz limited only by a strong phonon absorption at 5.3 THz. Systematic studies have also been conducted on THz emitter application of cadmium zinc telluride ternary crystals with varying Cd composition. It has been reported18,52 that the optimum Cd composition x = 0.05 enhances THz radiation as well as crystal quality of ZnTe. Apart from the zincblende crystals described earlier, GaSe is a promising semiconductor crystal that has been exploited recently for THz generation with an extremely large bandwidth of up to 41 THz.15,53 GaSe is a negative uniaxial layered semiconductor with a hexagonal structure characteristic of 62m point group and a direct bandgap of 2.2 eV at 300 K. The crystal has a large nonlinear optical coefficient (54 pm/V),53 high damage threshold, suitable transparent range, and low absorption coefficient, which make it an attractive option for generation of

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52

Terahertz Spectroscopy: Principles and Applications

broadband mid infrared electromagnetic waves. For broadband THz generation and detection using a sub-20 fs laser source, GaSe emitter-detector system performance is comparable to or better than the use of thin ZnTe crystals. Using GaSe crystals of appropriate thickness as emitter and detector it is also possible to obtain a frequencyselective THz wave generation and detection system. The disadvantage of GaSe lies in the softness of the material that makes it fragile during operation. 2.3.1.2 Inorganic Electrooptic Crystals In 1971 Yang et al. first demonstrated the generation of THz radiation by optical rectification of picosecond optical pulses in inorganic electrooptic materials. In their experiment they observed THz radiation from a LiNbO3 crystal illuminated by picosecond optical pulses from a mode-locked Nd:glass laser.8 In the late 1980s, Auston and coworkers reported THz radiation generation by optical rectification in LiTaO3 with spectral bandwidth as high as 5 THz.54 A major drawback in this experiment was the difficulty in extracting the THz pulses from the generating crystal because of total internal reflection owing to small critical angle (and high static dielectric constant) in LiTaO3. In their later publications Auston et al. reported an approach that minimized the reflection loss and permitted the extraction of the THz radiation into free space.10 Zhang et al.9 later measured and calculated optical rectification from LiNbO3 and y-cut LiTaO3 under normal incident optical excitation. The spatiotemporal shape of the radiated THz pulse was found to be similar for both materials with LiNbO3 emitting a stronger signal because of a smaller dielectric constant and hence smaller reflection losses at the crystal boundary. The maximum signal obtained from LiTaO3 was found to be 185 times smaller than that emitted by a 4-Nmethylstilbazolium tosylate (DAST) organic crystal.19 2.3.1.3 Organic Electrooptic Crystals Organic crystals have been a recent source of interest as THz emitters as they have been reported to generate stronger THz signals than commonly used semiconductors or inorganic electrooptic emitters owing to their large second-order nonlinear electric susceptibility. Zhang et al.19 first reported THz optical rectification from an organic crystal, dimethyl amino DAST, which is a member of the stilbazolium salt family. Electrooptic measurements at 820 nm have reported a high electrooptic coefficient (>400 pm/V).19 In their work on THz emission by optical rectification of femtosecond laser pulses in DAST, Zhang et al. observed a strong dependence of the radiated THz field amplitude on sample rotation about the surface normal, for both parallel and perpendicular orientations of the incident optical pump beam. The angular dependence was in excellent agreement with the predicted theory, with the maximum signal occurring for the angle at which the optical polarization, the crystal polar “a” axis and the detector dipole axis, were aligned in the same direction.19 Zhang and coworkers19 also reported that with a 180-mW optical pump beam focused to a 200-µm diameter spot, the best DAST sample provided a detected THz electric field that was 185 times larger than that obtained from an LiTaO3 crystal and 42 times larger than GaAs and InP crystals under the same experimental conditions. DAST has also been shown to

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 53

perform well at higher frequencies with an observable bandwidth up to 20 THz from a 100-µm DAST crystal showing a six-fold increase over that reported from a 30-µm ZnTe crystal under similar experimental conditions.55 The frequency spectrum of DAST shows a characteristic strong absorption line at 1.1 THz (from TO phonon resonance) along with some additional absorption lines between 3 and 5 THz. The latter (with the exception of the line at 5 THz) are significantly weaker than the absorption line at 1.1 THz, such that the THz amplitude at these frequencies is still well above the noise level.56 Nahata et al.57 advocated the use of organic compounds in polymeric forms for THz-frequency radiation generation instead of single crystals as these compounds offer significant advantages over single crystals. In particular they are easily processable, can be poled to introduce noncentrosymmetry and have higher nonlinear coefficients than inorganic materials. In their experiment Nahata and coworkers used a copolymer of 4-N-ethyl-N- (2-methacryloxyethyl) amine-4′-nitro-azobenzene (MA1) and methyl methacrylate (MMA) (commonly known as MA1:MMA) for generation of THz radiation via optical rectification. The radiated field amplitude from a 16-mm thick sample (electrooptic coefficient ~11 pm/V) was found to be four times smaller than that observed from a 1-mm thick y-cut LiNbO3 crystal. However, the coherence length of the polymer was reported to be 20 times larger. Nahata and coworkers suggested that the conversion efficacy relative to LiNbO3 could be substantially improved using available polymers with six times the nonlinear coefficient and millimeter-thick films poled at high fields. This could be easily created by a dielectric stack of thin-poled films.58 After the success of DAST and MA1:MMA as THz emitters, Hashimoto et al. reported THz emission via optical rectification from organic single crystals of N-substituted (N-benzyl, N-diphenylmethyl, and N-2 naphthylmethyl) derivatives of 2-methyl-4-nitroaniline.59 While no substantial emission was observed from N-diphenylmethyl and N-2 naphthylmethyl derivatives (owing to the strong phonon modes existing in the range 0 to 2.5 THz), the integrated intensity of the THz radiation emitted by N-benzyl-2-methyl-4-nitroaniline (BNA) was as intense as that by the DAST crystal under similar experimental conditions. The dynamic range for BNA is limited to 2.1 THz because of a strong phonon mode that exists around 2.3 THz. Kuroyanagi and coworkers60 later reported improved signal intensity (about three times greater than that generated by ZnTe) and increased bandwidth (up to 4 THz) from highly purified BNA crystals. Although DAST and BNA have been reported to perform better than ZnTe when used for THz generation by optical rectification, the THz electric fields generated by these crystals are more complicated in both time and frequency domains. Phonon bands in these crystals also result in the production of smaller range of frequencies than ZnTe. Recent investigations of a range of organic molecular crystals revealed that the crystal (–) 2-(a-methylbenzyl-amino)-5-nitropyridine (MBANP) is superior to ZnTe when used for optical rectification at 800 nm.61 MBANP crystallizes in P21 monoclinic62 and has the advantage of being relatively easy to grow in single crystals, probably because of its L-shaped structure and strong H bonding in certain directions as well as a suitable habit.61 Optical rectification of 800-nm pulses in <001>-oriented MBANP resulted in THz pulses that are very similar to those

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Terahertz Spectroscopy: Principles and Applications

produced by ZnTe both in frequency and time domain. However, MBANP produces a signal that is an order of magnitude higher because of its larger electrooptic coefficient (18.2 pm/V at 632.8 nm).63 Several other crystals similar to MBANP have also been studied.61 It was found that 4-(N, N-dimethylamino)-3-acetamidonitrobenzene and 4‑nitro-4′-methylbenzylidene aniline (NMBA) produced THz pulses through optical rectification of 800-nm pulses of comparable strength to MBANP. While NMBA is easy to grow, 4-(N, N-dimethylamino)-3-acetamidonitrobenzene is much more difficult owing to its needle-like habit, and obtainable crystal size is limited. Strong phonon band absorption also limits the power in the range 1.6 to 3 THz, which makes these crystals unsuitable for high-frequency applications. We have thus reviewed THz optical rectification from a variety of materials including semiconductors and inorganic and organic electrooptic crystals. The performance of these emitters in terms of bandwidth has been summarized in Table 2.3. ZnTe is the most popular choice for THz generation by optical rectification because it can generate extremely short and high-quality THz pulses. Organic crystals have been of recent interest owing to their higher electrooptic coefficient and enhanced capacity to generate stronger signals than the commonly used ZnTe. However, the THz fields generated by organic crystals such as DAST and more recently BNA are more complicated in both time and frequency domain. Low-frequency phonon bands also limit the bandwidth in these crystals than compared with ZnTe. Organic crystals also have a large naturally occurring birefringence, which complicates their application.55 This has resulted in ZnTe remaining the THz generator of choice. The comparative performance of a ZnTe THz emitter pumped by an amplified titanium sapphire laser system is shown in Figure 2.2.64 The choice of a suitable THz emitter also largely depends on the operating conditions. As mentioned previously, to achieve reasonable efficacy for nonlinear optical processes, long interaction length and appropriate phase matching conditions must be met. The phase matching condition is satisfied when the phase of the THz wave travels at the group velocity of the optical pulse. In the case of ZnTe, the phase matching condition and the subsequent enhancement of coherence length are achieved at 800 nm, making it the most suitable electrooptic crystal for THz wave emission and detection using a Ti:sapphire laser system with a center wavelength of 800 nm.

Table 2.3 Bandwidth Performances of Some Common Terahertz Emitters

Material GaAs GaSe ZnTe GaP DAST BNA

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Experimental Bandwidth (THz) 40(12) 41(15) 17(12) 3.5(45) 20(55) 2.1(59)

Lowest Optical Phonon Resonance (THz) at 300 K 8.02(71) 7.1(15) 5.4(12) 10.96(71) 1.1(56) 2.3(59)

Incident Pulse Wavelength (nm)

Incident Pulse Width (fs)

Detection Scheme

800(12) 780(15) 800(12) 1055(45) 800(55) 800(59)

12(12) 10(15) 12(12) 300(45) 15(55) 100(59)

Electrooptic Electrooptic Electrooptic PC antenna Electrooptic Electrooptic

11/15/07 10:47:11 AM

1000

Estimated based on measured saturation properties of ZnTe GaAs 10.7 kV/cm (You at al.) Fit based on (Darrow et al.)

(a)

G

0.1 1000

d

m a2 n

ae

sm

Pla

P

m

Ha

a(

m las

r ste

et

al.)

ETHz ∝ Jopt

Estimated based on measured saturation properties of ZnTe Fit based on (Darrow et al.) /cm 1kV s A Ga Te al.) Zn et m eld er t fi m s . 2 m xt Ha ae a( sm a m l s P Pla

10 1

on

ic

0.1

nd

ha

rm

0.01

JTHz ∝ Jopt2

as

m

a2

1E–3

Pl

1E–4 10–2 Pulse Energy Conversion (η)

eld

Fi xt.

GaAs 10.7 kV/cm (You et al.)

(b)

100

10

Te

Zn

as

m 2m

Pl

1

/cm

kV

ha

1 aAs

on ic

10

rm

100

GaAs 10.7 kV/cm (You et al.)

(c)

Estimated based on measured saturation properties of ZnTe

–3

Fit based on (Darrow et al.)

10–4 10–5

η∝

GaAs 1 kV/cm

10–6

m

10–7

2m

–8

1×10

1×10–9 1

10

J opt η∝

e ZnT Pla Ha sma 2 n rm d on ic

THz Pulse Energy (nJ)

Maximum Focused Field Amplitude (kV/cm)

Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 55

100

d

. Fiel

a ext

Plasm

ma

Plas

ter ams

(H

1000

l.)

et a

J opt

η ∝ J op

t

10000

100000

Optical Pulse Energy (µJ)

Figure 2.2 Comparison of the THz emission efficacy of a 2-mm ZnTe crystal with other types of THz emitters: (a) the focused peak electric field, (b) the THz pulse energy, and (c) the energy conversion efficacy are displayed as a function of the energy per pulse of the femtosecond near infrared pump laser. (From Loeffler T., Semicond. Sci. Technol. 20, S140, 2005, IOP Publishing. With permission.)

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Terahertz Spectroscopy: Principles and Applications

However, the large size of these lasers prevents them from being used in portable THz systems. To build a compact, integrated, and sensitive THz spectroscopy or THz imaging system, it has thus become crucial to develop a THz wave generation and detection technique coupled to an optical fiber cable. The erbium (Er)- and ytterbium (Yb)-doped fiber lasers operating at 1,550 nm and 1,000 nm, respectively, are especially attractive from this viewpoint. The large velocity mismatch and consequently a small coherent length, however, make ZnTe unsuitable for application at these wavelengths. Although GaP and CdTe have been suggested as potentially compatibles emitters for Yb-doped laser systems operating at 1,000 nm, GaAs is an excellent candidate for THz wave emission with erbium-doped fiber lasers. Calculations17 show that the phase matching condition is satisfied at 1,050 nm and 1 THz in CdTe crystals resulting in an enhancement of coherence length, whereas GaAs crystal phase matching occurs at 1,330 nm. This optical wavelength in GaAs is the longest among all binary semiconductors.14 Even at 1,550 nm, the coherence length still retains a large value, thus enabling GaAs to be used a suitable emitter in this wavelength range. GaAs also has the added advantage of having its lowest phonon resonance lying at higher THz frequencies (10 THz) and consequently increased bandwidth capability. The coherence length of some organic crystals such as BNA is also quite large in the long wavelength regime; however, because of the difficulties highlighted previously, organic crystals have not gained popularity as THz emitters.

2.3.2 Recent Developments The increasing interest in the development of novel THz sources has stimulated indepth studies of microscopic mechanisms of THz field generation in conventional semiconductors,38–53 electrooptic materials,55–63 and an extensive search for new materials to be employed in THz generation and detection. We conclude our discussion on THz emitters by highlighting some recent developments in THz emitter research. The THz emission from nonresonant optical rectification discussed in earlier sections resulted from dipolar excitations from such materials as GaAs, ZnTe, and LiNbO3. THz emission may also result from nondipolar excitations. Emission via optical rectification of femtosecond laser pulses in YBa2Cu3O7,65 single crystals of iron (Fe),66 and films consisting of nanosized graphite crystallites67 was observed and was found to have resulted from quadrupole magnetic dipole nonlinearities. For Fe, which crystallizes in a body-centered cubic lattice, optical rectification is forbidden by lattice symmetry. A nonvanishing second-order optical nonlinearity can, however, result from an electric quadrupole magnetic dipole contribution, surface nonlinearity or sample magnetization. Each of these contributions to the second-order nonlinear electric susceptibility has a characteristic dependence on sample azimuth and optical pump pulse polarization for a given crystal orientation. The THz emission reported from Fe was found to be approximately three orders of magnitude weaker than that from a 1-mm thick <110> ZnTe crystal under the same optical excitation conditions. THz emission from metals such as gold and silver has also come to light recently.20,21 Although second harmonic generation had been reported from metal surfaces as early as the 1960s, optical rectification of laser light had not been reported until recently. Kadlec et al.20,21 generated intense THz emission by optical

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 57

rectification of p‑polarized 810-nm laser pulses at a fluence of 6 mJ/cm2 in gold films. The THz amplitude was found to scale quadratically with optical fluence for values up to 2 mJ/cm2. The s-polarized optical pump beam resulted in THz radiation emission, which was approximately three times weaker than the signal obtained from a p-polarized optical pump beam. THz emission from silver showed a different behavior. At the highest optical pump fluence, the emitted THz radiation was 15 times weaker than for gold and showed a linear dependence on incident pump fluence.20 These recent experimental findings reveal the possibility of accessing a wealth of new information about nonlinear optical properties and electronic structures of metal surfaces via femtosecond optically excited THz emission measurements.

2.4 Experimental Results of Terahertz Electrooptic Detection Much research activity has been devoted to increasing the sensitivity and bandwidth of electrooptic detection of THz-frequency radiation pulses. Other than the duration of the optical probe pulse, the bandwidth of free-space electrooptic sampling is only limited by the dielectric properties of the electrooptic crystal. Electrooptic sampling of THz-frequency radiation is a powerful tool for detection of electromagnetic radiation pulses in and beyond the mid-infrared range. A variety of electrooptic materials, including semiconductors and inorganic and organic electrooptic crystals, have been tested for detector applications in THz spectroscopy. In this section, we present a brief overview of the experimental results of broadband THz electrooptic detection with various materials.

2.4.1 Materials 2.4.1.1 Semiconductors and Inorganic Crystals In 1995, Wu and Zhang27 first demonstrated free-space electrooptic sampling of short THz radiation pulses. In their experiment, a GaAs photoconductive emitter triggered by 150-fs optical pulses at 820 nm radiated electromagnetic waves of THz frequency. A 500-µm thick LiTaO3 crystal, with its c axis parallel to the electric field polarization, was used as the electrooptic sampling element. To improve detection efficacy, the THz radiation beam was focused onto the detector crystal by a highresistivity silicon lens. The temporal resolution of the detected THz transient was limited by the velocity mismatch between optical and THz frequencies. Later work by Wu and Zhang68,69 addressed this shortcoming by careful geometric design of the radiation emitter and a special cut detector crystal. A signal-to-noise ratio of about 170:1 was obtained with a 0.3-second lock-in integration time constant. Wu and coworkers also tested ZnTe for detector applications in a novel experimental geometry in which the optical and THz beams were made to propagate collinearly in the electrooptic crystal.68 In this setup, a 1.5-mm thick, <110>-oriented ZnTe crystal was used as the detector with the optical probe and THz beam set parallel to the <110> edge of the ZnTe crystal for optimal electrooptic phase modulation. A signal-to-noise ratio of about 10,000:1 was obtained for a single scan using again a lock-in amplifier with time constant of 0.3 seconds. Nahata et al.11 demonstrated a

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Terahertz Spectroscopy: Principles and Applications

broadband time domain THz spectroscopy system using <110>-oriented ZnTe crystals for generation of THz radiation by optical rectification and detection of THz radiation by electrooptic sampling. An important condition for generation and detection of THz radiation by nonlinear optical techniques is the appropriate phase matching between the optical and THz pulses in the nonlinear optical crystal. For a medium with dispersion at optical frequencies, phase matching is achieved when the phase velocity of THz wave is equal to the velocity of optical pulse envelope (or the optical group velocity).11 The data in Figure 2.8 show the effective index of refraction of ZnTe at optical wavelengths nopt eff and THz frequencies nTHz. The coherence length lc of ZnTe at THz frequencies for an optical wavelength λ = 780 nm decreases as illustrated in Figure 2.3. The temporal waveform of the measured THz electric field and its Fourier amplitude and Fourier phase are displayed in Figures 2.3, 2.4, and 2.5, respectively. The peak of the THz pulse amplitude shows a three-fold rotational symmetry when the ZnTe detector crystal is rotated by 360° about an axis normal to the <110> surface. This result is inherent to all <110> and <111> zincblende crystals.40 The broadband detection capability of ZnTe is obvious from the Fourier amplitude spectrum (Figure 2.4), which extends well beyond 2 THz. The rolloff in sensitivity for higher frequencies from phase mismatch and the shorter coherence length are in agreement with data presented in Figures 2.6, 2.7, and 2.9. A thinner crystal detects higher frequencies better than a thicker crystal. However, the detector sensitivity of a thinner crystal is reduced compared to a thicker crystal. In addition to limitations imposed by

Electro-optic Signal (arb. units)

4 2 0 –2 –4 2

4

6

8

10

Time (ps)

Figure 2.3 Time domain measurement of a THz pulse by electrooptic sampling with a ZnTe crystal 10 mm × 10mm × 1 mm in size. The THz pulse and the optical probe pulse are incident onto the surface of the ZnTe crystal parallel to the <110> plane. The optical axis of the crystal is in the <001> direction. The linear polarization of the THz radiation pulse is perpendicular to the optical axis. The linear polarization of the incident optical probe pulse is parallel to the optical axis. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7,2000. With permission.)

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 59

Fourier-amplitude (arb. units)

104

103

102

101

100

0.5

1.0

1.5

2.0

2.5

3.0

Frequency f (THz)

Figure 2.4 Fourier amplitude of the time domain measurement displayed in Figure 2.3. The bandwidth of the signal is approximately 2.75 THz. The dynamic range of the measurement is 500:1. (Adapted from Selig, Hanns, Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

phase mismatch, the absorption caused by a transverse optical phonon resonance at 5.4 THz11 in ZnTe is expected to attenuate the high-frequency components of THz radiation. Later experiments by Han and Zhang70 reported a detection bandwidth up to 40 THz using a 27-µm ZnTe detector crystal. 2.0×104

Fourier-phase (arb. units)

1.9×104 1.8×104 1.7×104 1.6×104 1.5×104 1.4×104 1.3×104 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Frequency f (THz)

Figure 2.5 Fourier phase of the time domain measurement displayed in Figure 2.3. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

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Terahertz Spectroscopy: Principles and Applications

Electro-optic Signal (arb. units)

6 4 2 0 –2 ZnTe 0.5 mm ZnTe 1.0 mm

–4 –6

0

1

2

3

4

5

6

7

8

Time (ps)

Figure 2.6 Time domain measurements of THz pulses by electrooptic sampling with ZnTe crystals of different thickness. For both measurements, the optical probe beam and the THz pulse both incident onto the surface of the crystals parallel to the <110> plane. The surface of the crystals is 10 mm × 10 mm. The linear polarizations of the incident THz-pulse and probe pulse are perpendicular to each other. The polarization of the probe beam is parallel to the optical axis (<001> direction). The ratio of the peak-to-peak amplitude of the THz pulses measured by the 1-mm crystal and the 0.5-mm crystal is 2.3 and scales approximately with the thickness of the two crystals (1 mm/0.5 mm = 2). The thicker ZnTe crystal exhibits a larger signal than the thinner crystal because the phase retardation G ∝ l is linearly proportional to the thickness l of the electrooptic crystal. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 35,2000. With permission.)

Although ZnTe delivers excellent performance in the 800-nm wavelength regime, its efficacy is inherently limited by large group velocity mismatch (GVM) of 1 ps/mm68 between optical and THz frequencies. An ideal material would be one with larger electrooptic coefficient and lower GVM. At 886 nm, GaAs has been found to possess a moderate electrooptic coefficient (25 pm/V), but a low GVM of 15 fs/mm. The electrooptic coefficient of LiTaO3 was found to be comparable to that of ZnTe, but LiTaO3 has a large GVM (>14 ps/mm), whereas for organic crystals such as DAST, the electrooptic coefficient is very large but GVM is comparable to ZnTe.68 A substantial improvement in bandwidth detection and sensitivity may be obtained by using materials with phonon resonance occurring at higher THz frequencies. Among inorganic media with comparable nonlinear optical properties zinc selenide (lowest TO phonon resonance at 6 THz),11 GaAs (TO phonon resonance at 8 THz), and GaP (TO phonon resonance at 11 THz)71 are excellent candidates. To verify the performance of electrooptic sampling with GaAs, Vosseburger et al.72 performed time-resolved experiments of THz radiation detection using bulk GaAs as the electrooptic sampling element. An excitation wavelength of 900 nm was chosen to avoid photoexcitation of carriers by interband absorption in the GaAs. The measured THz radiation pulse had a complex

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 61 104

Fourier Amplitude (arb. units)

ZnTe 0.5 mm ZnTe 1.0 mm 103

102

101

100 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Frequency (THz)

Figure 2.7 Fourier amplitudes of the time domain measurements displayed in Figure 2.6. The thinner ZnTe crystal has a slightly larger bandwidth than the thicker ZnTe crystal. The calculated bandwidths of a 0.5-mm and a 1.0-mm ZnTe crystal are 2.86 THz and 2.62 THz, respectively. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 35,2000. With permission.)

Optical Wavelength (Nm) 6.0

0.5

0.6

0.7

0.8

0.9

1.0 6.0

Index of Refraction n

5.5

5.5 nTHz

nopt eff

5.0

5.0

4.5

4.5

4.0

4.0

3.5

3.5

3.0

3.0

2.5

2.5

2.0

0

1

2

3

4

5

2.0

Frequency f (THz)

Figure 2.8 Index of refraction of ZnTe at optical wavelength (nopt eff ) and THz frequencies (nTHz). Optical radiation at 760-µm wavelength propagates at the same speed as 2.6 THz- frequency radiation in ZnTe. The effective index of refraction at λ = 780 µm is nopt eff = 3.27. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 33, 2000. With permission.)

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Terahertz Spectroscopy: Principles and Applications

Coherence Length lc (mm)

10

1

0.1

0.01

0

1

2

3

4

5

Frequency (THz)

Figure 2.9 Calculated coherence length lc of ZnTe as a function of THz frequency for λ = 780 µm. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 7, 2000. With permission.)

temporal structure and was smaller by a factor of approximately 20 compared with the same pulse measured by a ZnTe crystal. The reduction in amplitude had its origins in the smaller electrooptic coefficient of GaAs than that of ZnTe and poor phase matching of GaAs at 900 mm and 1 THz. Among zincblende crystals, GaP is an excellent alternative to ZnTe for electrooptic sampling of THz-frequency radiation because of its high-frequency phonon band at 11 THz,71 the highest available among all zincblende crystals. The electrooptic coefficient of GaP (r41 ≈ 0.97 pm/V30), however, is one fourth that of ZnTe. In 1997, Wu and Zhang73 first demonstrated electrooptic sampling of THz radiation with a GaP crystal. Using a GaAs emitter excited by 50-fs pulses and a 150-µm thick <110> GaP crystal, they demonstrated a bandwidth resolution up to 7 THz. Broadband THz electrooptic sampling with GaP crystals was further developed by Leitenstorfer et al.74 They carefully evaluated the frequency dependence of the refractive index, the electrooptic coefficient and the response function of the GaP electrooptic THz detector. Using a 13-µm thick crystal and 12-fs titanium–sapphire laser pulses the detected spectral bandwidth of GaP was estimated to be ~70 THz. One of the recently reported materials used for free-space electrooptic sampling of THz radiation is ZnSe in crystalline and polycrystalline form.75 Although the electrooptic coefficient of ZnSe (r41 ≈ 2pm/V75) is only half of that of ZnTe, the TO phonon resonance frequency (6 THz)11 is higher than that of ZnTe (5.4 THz),11 thus promising a higher detection bandwidth potential for the former. The GVM for ZnSe (0.96 pm/V75) is comparable to that for ZnTe. Using a 10-fs titanium–sapphire laser pulse to excite a 100-µm GaAs photoconducting antenna and a 0.5-mm, <111>-oriented ZnSe single crystal as detector, a spectral bandwidth of 3 THz was

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 63 90 120

1.0

φ (°)

60

Electro-optic Signal (arb. units)

0.8 30

150

0.6 0.4 0.2 0.0

180

0

0.2 0.4 0.6

330

210

0.8 1.0

240

300 270

Figure 2.10 Measurements of the peak electrooptic signal as a function of the angle Φ between the <001> direction of a <110> ZnTe crystal and the direction of linear polarization of the optical probe beam. The polarization of the THz pulse is perpendicular to the polarization of the optical probe pulse. During the measurements, the orientations of the polarization of the optical and the THz pulses are fixed and the crystal is turned in steps of 10°. An angle Θ = 90° corresponds to the situation in which the polarization of the optical probe beam is parallel to the <001> direction of the ZnTe. Two major maxima at Θ = 90° and Θ = 270° and four minor maxima at Θ = 30°, Θ = 150°, Θ = 210°, and Θ = 330° are observed. The amplitude of the peak THz electric field is estimated to be 5 V/m. (Adapted from Selig, H., Elektro-optisches Sampling von Terahertz Pulsen, Hochschulschrift: Hamburg, Universitaet, Fachbereich Physik, Diplomarbeit, 47, 2000. With permission.)

measured. In this experiment, the bandwidth detected by the ZnSe crystal was limited only by the generation of the THz radiation. It was found that for thicker (1 mm) polycrystalline electrooptic ZnSe samples, the random nature of crystallographic orientations within the interaction length distorted the phase of THz waveform. However, a reduction of the thickness to 0.15 mm minimized the distortion and extended the bandwidth up to 4 THz. Because polycrystalline semiconductors offer practical advantages in terms of ease of fabrication75 over their single crystal counterparts, the successful application of polycrystalline materials for free-space electrooptic sampling permits the possibility of utilizing non-lattice matched thin film integrated electrooptic detectors of THz radiation. As discussed previously, the mismatch between THz phase velocity and the group velocity of the optical probe pulse limits the detection bandwidth of zincblende electrooptic crystals such as ZnTe and GaP. However, this limitation can be eliminated by using a novel detection scheme, taking advantage of the type II phase matching in a naturally birefringent crystal such as GaSe.53,76 The phase matching condition can be satisfied by angle tuning whereby the electrooptic crystal

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Terahertz Spectroscopy: Principles and Applications

Table 2.4 Properties of Typical Electrooptic Sensor Materials*

Material

Crystal Structure

Point Group

ZnTe GaP ZnSe LiTaO3

Zincblende(30) Zincblende(30) Zincblende(30) Trigonal(30)

43m( 30 ) 43m( 30 ) 43m( 30 ) 3m(30)

GaSe DAST Poled polymers

Hexagonal(53) — —

62 m( 53) — —

Electrooptic Coefficient (pm/V)

Group Velocity Mismatch (ps/mm)

Surface Orientation

1.1(68) — 0.96(75) 14.1(68)

110(11) 110(73) 111(75) —

2.5(11) 7(73) 3(75) —

0.10(76) 1.22(68) —

— — —

120(76) 6.7(77) 33(78)

r41 = 3.90(30) r41 = 0.97(30) r41 = 2.00(30) r33 = 30.3(30) r13 = 5.70 r22 = 14.4(76) r11 = 160(68) —

Experimental Detection Bandwidth (THz)†

* A more comprehensive listing may be found in reference 68. † The bandwidth reported here is subject to experimental conditions and not the absolute bandwidth for a particular emitter/sensor material.

(z-cut GaSe crystal) is tilted by an angle qdet (the phase matching angle) about a horizontal axis perpendicular to the direction of time-delayed probe beam.76 In the experiments conducted by Kubler and coworkers,76 a THz radiation pulse was generated by optical rectification of 10-fs titanium–sapphire laser pulses in a 20-µm thick GaSe crystal (qem = 57°) and detected with a GaSe sensor (qdet = 60°) of thickness 30 µm. qdet/em = 0 signifies normal incidence. The amplitude spectrum of the THz pulse peaked at 33.8 THz and extended from 7 THz to beyond 120 THz. Compared with a 12-µm thick ZnTe crystal that displayed a local minimum at 34 THz under the same excitation, the GaSe spectrum showed a nearly flat response from 10 THz up to 105 THz. The enhanced performance of GaSe crystals over ZnTe crystals is attributed mainly to a greater interaction length because of better phase matching and a larger electrooptic coefficient of GaSe (Table 2.4). 2.4.1.2 Organic Crystals In the quest for novel materials for electrooptic sampling of THz radiation pulses, organic electrooptic crystals have attracted attention owing to their very high electrooptic coefficients. Han and coworkers55 first demonstrated the application of the organic ionic salt crystal DAST as a free space electrooptic detector of THz radiation. The electrooptic coefficient for DAST is 160 pm/V at 820 nm68 and is almost two orders higher than that found in ZnTe (Table 2.4). However, DAST exhibits two confirmed phonon absorption peaks at 1.1 THz and 3.05 THz.77 Recently, nonlinear optical generation and detection of THz radiation pulses was investigated theoretically and experimentally by Schneider et al.77 They demonstrated that the THz radiation spectrum generated and detected using DAST crystals extended from 0.4 THz to 6.7 THz, depending on the laser excitation wavelength in the 700-nm

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 65

to 1,600-nm wavelength range. Furthermore, they found that the technologically important wavelengths around 1,500 nm are velocity matched to THz frequencies between 1.5 THz and 2.7 THz. Among other organic materials, poled polymers have been reported to exhibit excellent broadband electrooptic detection capability.78 These materials demonstrate large electrooptic coefficients,79 low dispersion between the THz and optical refractive indices,57,80 ease of fabrication in large area thin films, and easily modifiable chemical structures for better THz and optical properties because of their organic nature. Using poled polymers for free-space electrooptic detection, Nahata and coworkers78 recently measured an amplitude spectrum up to frequencies of 33 Hz.

2.5 Application of Electrooptic Sampling of THz Electric Field Transients Nonlinear techniques for the generation and detection of short pulses of THz- frequency radiation have found important applications such as time domain THz spectroscopy and time domain THz imaging.34 Here, we discuss briefly the application of electrooptic sampling of THz radiation pulses to longitudinal electron bunch length measurements. The electrooptic detection of the local nonradiative electric field that travels with a relativistic electron bunch has recently emerged as a powerful new technique for subpicosecond electron bunch length measurements.81–84 The method makes use of the fact that the local electric field of a highly relativistic electron bunch that moves in a straight line is almost entirely concentrated perpendicular to its direction of motion. Consequently, the Pockels’ effect induced by the electric field of the passing electron bunch can be used to produce birefringence in an electrooptic crystal placed in the vicinity of the beam. Specifically, the birefringence induced by a single electron bunch is measured by monitoring the change of polarization of the wavelength components of a chirped, synchronized titanium–sapphire laser pulse. When the electric field of an electron bunch and the chirped optical pulse copropagate in the electrooptic crystal, the polarizations of the various wavelength components of the chirped pulse that passes through the crystal are rotated by different amounts that correspond to different portions of the local electric field. The direction and degree of rotation are proportional to the amplitude and the phase of the electric field. Thus the time profile of the local electric field of the electron bunch is linearly encoded to the wavelength spectrum of the optical probe beam. An analyzer converts the modulation of the polarization of the chirped optical pulse into an amplitude modulation of its spectrum. The time profile of the electric field of the electron bunch is measured as the difference of the spectrum with and without a copropagating electron bunch. The width of the temporal profile corresponds directly to the electron bunch length, and the shape of the temporal profile is proportional to the longitudinal electron distribution within the electron bunch. The length and the shape of individual electron bunches are determined by measuring the spectra of single chirped laser pulses with an optical multichannel analyzer equipped with a nanosecond shutter. The method allows direct in situ electron bunch diagnostics with a high signal‑to‑noise ratio and subpicosecond time resolution.

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Terahertz Spectroscopy: Principles and Applications

Grating f + ∆z

Lens

Mirror

f

Pulse Generator ccd Intensifier Spectrometer

Signal

(a) Optical Stretcher

Reference Ti: Sapphire Amplifier

Optical Stretcher (a)

Fiber Analyzer

Polarizer FEL Cavity

FEL

rf-clock ZnTe Accelerator

e-beam

Undulator Dump

Figure 2.11 Experimental setup for electron bunch length measurements by electrooptic sampling with chirped optical pulses. The electron bunch length is measured by using a ZnTe crystal placed inside the vacuum pipe at the entrance of the undulator. The femtosecond near infrared laser repetition rate is synchronized with the repetition rate of the electron bunches. The inset (a) exhibits a two-dimensional optical stretcher for laser pulse chirping. (Adapted figure with permission from I. Wilke, A. M. McLeod, W. A. Gillespie, G. Berden, G. M. H. Knippels, A. F. G. van der Meer, Phys. Rev. Lett. 88, 124801-1 (2002). Copyright 2002 by the American Physical Society.)

The experimental arrangements for the electrooptic electron bunch length measurements are schematically illustrated in Figure 2.11. The electron bunch source is the radiofrequency linear accelerator at the FELIX free electron laser facility in the Netherlands.85 The electron beam energy of FELIX was set at 46 MeV, and its charge per bunch at around 200 pC. The micropulse repetition rate was 25 MHz, and the macropulse duration was around 8 µ with a repetition rate of 5 Hz. A titanium– sapphire amplifier, producing 30-fs FWHM pulses at 800 nm with a repetition rate of 1 kHz, was used as a probe beam. The 30-fs optical laser pulses are chirped to pulses of up to 20 ps (FWHM) duration with an optical stretcher which consists of a grating, a lens, and a plane mirror.86 The duration of the chirped pulses has been measured with an optical autocorrelator based on second harmonic generation in a BBO crystal. The electron bunch length is measured inside the accelerator beam pipe at the entrance of the undulator. A 0.5-mm thick <110> ZnTe crystal is used as an

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 67

electrooptic sensor and is placed with its 4 × 4 mm 2 front face perpendicular to the propagation direction of the electron beam. The incoming chirped laser beam is linearly polarized. The outgoing chirped beam is split into a signal beam and a reference beam used to monitor possible laser power fluctuations. The signal beam passes through an analyzer (a second polarizer), which is crossed with respect to the first polarizer. Subsequently, the spectra of the chirped laser pulses are dispersed with a grating spectrometer and the line spectra are focused onto a charge-coupled device (CCD) camera. The CCD camera is equipped with an intensifier, which acts as a nanosecond shutter with a gate width of 100 ns. Single-shot measurements are performed by actively synchronizing the titanium–sapphire amplifier87 with both the repetition rate of the electron beam and the gate of the CCD camera, and therefore recording only the spectrum of one chirped laser pulse at a time. The temporal overlap between the electron bunch and chirped laser pulse is controlled by an electronic delay. Figure 2.12a shows measurements of the chirped laser pulse spectra with and without a copropagating electron bunch. The signal spectra are labeled by S and S′. The crossed polarizers exhibit a finite transmission of 1.7% if the electron bunch does not overlap with the chirped laser pulse. This is attributed to a small intrinsic stress birefringence of the ZnTe. The transmission of the crossed polarizers changes significantly when the electron bunch and the laser pulse copropagate: in these circumstances, a large peak, which corresponds to a strong enhancement of the transmission, is observed in the center of the spectrum. The strong change of the spectrum is attributed to the wave length-dependent change in polarization of the chirped laser pulse because of the electric field of the electron bunch. The length and shape of the electron bunch is obtained by subtracting the spectrum without copropagating electron bunch, which has been corrected for laser power fluctuations by multiplication with the ratio of the reference spectra R/R′ from the spectrum with copropagating electron bunch. This difference S-(S′R/R′) is corrected for the wavelength-dependent variations in intensity in the spectrum by dividing by the spectrum S′R/R′. The pixels are converted to time by measuring the length of the chirp t, the spectral resolution of the spectrometer and CCD setup, Dl/pixel, and the bandwidth of the chirped laser pulses Dlbw. Then, the time interval per pixel is given by (Dt/pixel) = (Dl/pixel)/(t/Dlbw). For the spectra of Figure 2.12a, the chirp was 4.48 ps FWHM, which results in an electron bunch measurement as displayed in Figure 2.12b. The width of the electron bunch is (1.72 ± 0.05) ps FWHM. The signal-to-noise ratio depends on the position in the spectrum; in the center of the spectrum, it is better than 200:1. The measured width and shape of a single electron bunch agree very well with the electron bunch measurements averaged over more than 8,000 electron bunches88 and CTR measurements.89 Linear accelerators used as drivers for new femtosecond x‑ray free‑electron lasers or employed in new teraelectron volt linear electronpositron colliders for high-energy physics, require dense, relativistic electron bunches with bunch lengths shorter than a picosecond. Precise measurements of the electron bunch length and its longitudinal charge distribution are important to monitor the preservation of the beam quality, whereas the electron bunch train travels through the beam pipe, as well as to tune and to operate a linear collider or a FEL.

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S S´

Counts

3×104 2×104 1×104 0 100

200

300

400

Pixel (a)

Norm. Electro-optic Signal (a.u.)

1.0 0.8 0.6 0.4 0.2 0.0 –0.2

–4

–2

0

2

4

Time (ps) (b)

Figure 2.12 Fig. 2.12 (a) Measurements of the chirped laser pulse spectra with(s) and without (s′) a copropagating electron bunch. (b) single shot electron bunch length measurements. The width of the electron bunch is (1.72 ± 0.05) ps FWHM.

References 1. Mittleman, D., Ed., Sensing with terahertz radiation, Springer Optical Science 85. Berlin: Springer, 2003. 2. Nuss, M. C., and Orenstein, J., Terahertz time-domain spectroscopy, in Millimeter and sub-millimeter-wave spectroscopy of solids, Grüner, G., ed., Topics in Applied Physics 74. Berlin: Springer, 1998.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 69 3. Brucherseifer, M., et al., Label free probing of binding state of DNA by time-domain terahertz sensing, Appl. Phys. Lett., 77, 4049, 2000. 4. Kaindl, R. L., et al., Ultrafast terahertz probes of transient conducting and insulating phases in an electron-hole gas, Nature, 423, 73, 2003. 5. Corson, J., et al., Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+d Nature, 398, 221, 1999. 6. Karpowicz, N., et al., Compact continuous-wave terahertz system for inspection applications, Appl. Phys. Lett., 86, 54105, 2005. 7. Wallace, V. P. et al., Terahertz pulsed imaging and spectroscopy for biomedical and pharmaceutical applications, Faraday Discuss., 126, 255, 2004. 8. Yang, K. H., Richards, P. L., and Shen, Y. R., Generation of far-infrared radiation by picosecond light pulses in LiNbO3, Appl. Phys. Lett., 19, 320, 1971. 9. Zhang, X. C., Jin, Y., and Ma, X. F., Coherent measurement of terahertz optical rectification from electrooptic crystals, Appl. Phys. Lett., 61, 2764, 1992. 10. Hu, B. B., et al., Free-space radiation from electrooptic crystals, Appl. Phys. Lett., 56, 506, 1990. 11. Nahata, A., Weling, A. S., and Heinz, T. F., A wideband coherent terahertz spectroscopy system using optical rectification and electrooptic sampling, Appl. Phys. Lett., 69, 2321, 1996. 12. Han, P. Y., and Zhang, X. C., Free-space coherent broadband terahertz time-domain spectroscopy, Meas. Sci. Technol., 12, 2001, 1747. 13. Rice, A., et al., Terahertz optical rectification from <110> zinc-blende crystals, Appl. Phys. Lett., 64, 1324, 1994. 14. Nagai, M. et al., Generation and detection of terahertz radiation by electrooptical process in GaAs using 1.56 µm fiber laser pulses, Appl. Phys. Lett., 85, 3974, 2004. 15. Huber, R., et al., Generation and field resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz, Appl. Phys. Lett., 76, 3191, 2000 16. Shi, W. et al., Efficient, tunable and coherent 0.18–5.27-THz source based on GaSe crystal, Opt. Lett., 27, 1454, 2002. 17. Xie, X., Xu, J., and Zhang, X.C., Terahertz wave generation and detection from a CdTe crystal characterized by different excitation wavelengths, Opt. Lett., 31, 978, 2006. 18. Liu, K. et al., Study of ZnCdTe crystals as THz wave emitters and detectors, Appl. Phys. Lett., 81, 4115, 2002. 19. Zhang, X.C. et al., Terahertz optical rectification from a nonlinear organic crystal, Appl. Phys. Lett., 61, 3080, 1992. 20. Kadlec, F., Kuzel, P., and Coutaz J.L., Optical rectification at metal surfaces, Opt. Lett., 29, 2674, 2004. 21. Kadlec, F., Kuzel, P., and Coutaz J.L., Study of terahertz radiation generated by optical rectification on thin gold films, Opt. Lett., 30, 1402, 2005. 22. Bass, M., et al., Optical rectification, Phys. Rev. Lett., 9, 446, 1962. 23. Zernike, F., Jr., and Berman, P. R., Generation of far infrared as a difference frequency, Phys. Rev. Lett., 15, 999, 1965. 24. Pockels, F., Lehrbuch der Kristalloptik, Bibliotheca Mathematica Teubneriana. Leipzig: Band 39, 1906. 25. Harris, S. E., McMurtry, B. J., and Siegman, A. E., Modulation and direct demodulation of coherent and incoherent light at microwave frequency, Appl. Phys. Lett., 1, 37, 1962. 26. Valdmanis, J. A., Mourou, G., and Gabel, C. W., Picosecond electrooptic sampling system, Appl. Phys. Lett., 41, 211, 1982. 27. Wu, Q., and Zhang, X. C., Free-space electrooptic sampling of terahertz beams, Appl. Phys. Lett., 67, 3523, 1995.

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Terahertz Spectroscopy: Principles and Applications 28. Jepsen, P. U., et al., Detection of THz pulses by phase retardation in lithium tantalate, Phys. Rev. E, 53, 3052, 1996. 29. Nahata, A., et al., Coherent detection of freely propagating terahertz radiation by electrooptic sampling, Appl. Phys. Lett., 68, 150, 1996. 30. Yariv, A., Modulation of optical radiation, in quantum electronics, 3rd ed., Hoboken, NJ. John Wiley & Sons, 1988. 31. Armstrong, J. A., et al., Interactions between light waves in a non-linear dielectric, Phys. Rev., 127, 1918, 1962. 32. Bonvalet, A. and Joffre, M., Terahertz femtosecond pulses, in Femtosecond laser pulses: principles and experiments, Rullière, C., ed., Berlin: Springer, 1998. 33. Shen, J., Nahata, A., and Heinz, T. F., Terahertz time-domain spectroscopy based on nonlinear optics, J. Nonlinear Opt. Phys., 11, 31, 2002. 34. Jiang, Z., and Zhang X. C., Free space electrooptic techniques, in Sensing with terahertz radiation, Optical Science 85, Mittleman, D., ed., Berlin: Springer, 2003, 155. 35. Franken, P. A., and Ward, J. F., Optical harmonics and nonlinear phenomena, Rev. Mod. Phys., 35, 23, 1963. 36. Hirlimann, C., Further methods for the generation of ultrashort optical pulses, in Femtosecond laser pulses: principles and experiments, Rullière, C., ed., Berlin: Springer, 1998. 37. Planken, P. C. M., et al., Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe, J. Opt. Soc. Am. B., 18, 313, 2001. 38. Zhang, X. C. et al., Generation of femtosecond electromagnetic pulses from semiconductor surfaces, Appl. Phys. Lett., 56, 1011, 1990. 39. Greene, B. I., et al., Interferometric characterization of 160 fs far-infrared light pulses, Appl. Phys. Lett., 59, 893, 1991. 40. Zhang, X. C., and Auston, D. H., Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics, J. Appl. Phys., 71, 326, 1992. 41. Jin, Y. H., and Zhang, X. C., Terahertz optical rectification, J. Nonlinear Opt. Phys., 4, 459, 1995. 42. Chuang, S. L. et al., Optical rectification at semiconductor surfaces, Phys. Rev. Lett., 68, 102, 1992. 43. Greene, B. I., et al., Far-infrared light generation at semiconductor surfaces and its spectroscopic applications, IEEE J. Quantum Electron, 28, 2302, 1992. 44. Reid, M., Cravetchi, I. V., and Fedosejevs, R., Terahertz radiation and second harmonic generation from InAs: bulk versus surface electric field induced contributions, Phys. Rev. B., 72, 035201-1, 2005. 45. Chang, G., et al., Power scalable compact THz system based on an ultrafast Ybdoped fiber amplifier, Opt. Express, 14, 7909, 2006. 46. Gu, P., et al., Study of THz radiation from InAs and InSb, J. Appl. Phys., 91, 5533, 2002. 47. Reid, M., and Fedosejevs, R., Terahertz emission from (100) InAs surface at high excitation fluences, Appl. Phys. Lett., 86, 011906-1, 2005. 48. Lewis, R. A., et al., Terahertz generation in InAs, Physica B, 376, 618, 2006. 49. Adomavicius, R., et al., Terahertz emission from p-InAs due to instantaneous polarization, Appl. Phys. Lett., 85, 2463, 2004. 50. Reid, M. and Fedosejevs, R., Terahertz emission from surface optical rectification in n-InAs, Proc. of SPIE, 5577, 659, 2004. 51. Ching, W. Y., and Huang, M. Z., Calculation of optical excitations in cubic semiconductors. III. Third-harmonic generation, Phys. Rev. B, 47, 9479, 1993. 52. Kang, H. S., et al., Technical digest of Pacific Rim conference on lasers and electro‑ optics. New York: IEEE, 1107, 1999.

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Nonlinear Optical Techniques for Terahertz Pulse Generation and Detection 71 53. Liu, K., Xu, J., and Zhang, X. C., GaSe crystals for broadband terahertz wave detection, Appl. Phys. Lett., 85, 863, 2004. 54. Auston, D. H. and Cheung, K. P, Coherent time domain far-infrared spectroscopy, J. Opt. Soc. Am. B, 2, 606, 1985. 55. Han, P. Y., et al., Use of organic crystal DAST for terahertz beam applications, Opt. Lett., 25, 675, 2000. 56. Schneider, A., and Gunter, P., Spectrum of terahertz pulses from organic DAST crystals, Ferroelectrics, 318, 83, 2005. 57. Nahata, A., et al., Generation of terahertz radiation from a poled polymer, Appl. Phys. Lett., 67, 1358, 1995. 58. Khanarian, G., Mortazavi, M. A., and East, A. J., Phase matched second-harmonic generation from free standing periodically stacked polymer films, Appl. Phys. Lett., 63, 1462, 1993. 59. Hashimoto, H., et al., Characteristics of terahertz radiation form single crystals of N‑substituted 2-methyl-4-nitroaniline, J. Phys.: Condens. Matter, 13, L529,

2001. 60. Kuroyanagi, K., et al., All organic terahertz electromagnetic wave emission and detection using highly purified N-benzyl-2-methyl-4-nitroaniline crystals, Jpn. J. Appl. Phys., 45, 4068, 2006. 61. Carey, J. J., et al., Terahertz pulse generation in an organic crystal by optical rectification and resonant excitation of molecular charge transfer, Appl. Phys. Lett., 81, 4335, 2002. 62. Bailey, R. T., et al., The linear optical properties of the organic molecular crystal (+) 2(a-methylbenzylamino)-5-nitropyridine (MBA-NP), Mol. Cryst. Liq. Cryst., 231, 223, 1993. 63. Bailey, R. T., et al., Linear electrooptic dispersion in (-)-2-(a-methylbenzylamino)5‑nitropyridine single crystals, J. Appl. Phys., 75, 489, 1994. 64. Loffler, T., et al., Comparative performance of terahertz emitters in amplifier-laserbased systems, Semicond. Sci. Technol., 20, S134, 2005. 65. Siders, J. L. W., et al., Terahertz emission from YBa2Cu3O7-d thin films via bulk electric‑quadrupole-magnetic-dipole optical rectification, Phys. Rev. B, 61, 13633, 2000. 66. Hilton, D. J., et al., Terahertz emission via ultrashort-pulse excitation of magnetic metal films, Opt. Lett., 29, 1805, 2004. 67. Mikheev, G. M., et al., Giant optical rectification effect in nanocarbon films, Appl. Phys. Lett., 84, 4854, 2004. 68. Wu, Q., and Zhang, X. C., Ultrafast electrooptic field sensors, Appl. Phys. Lett., 68, 1604, 1996. 69. Wu, Q., and Zhang, X. C., Electrooptic sampling of freely propagating terahertz fields, Opt. Quantum Electron., 28, 945, 1996. 70. Han, P. Y., and Zhang, X. C., Coherent, broadband midinfrared terahertz beam sensors, Appl. Phys. Lett., 73, 3049, 1998. 71. Handbook series on semiconductor parameters, Levinstein, M., Rumyantsev, S., and Shur, eds., London: World Scientific, 1996, 1999. Available online at http://www. ioffe.rssi.ru/SVA/NSM/Semicond/. 72. Vosseburger, M., et al., Propagation effects in electrooptic sampling of THz pulses in GaAs., Appl. Opt., 37, 3368, 1998. 73. Wu, Q., and Zhang, X. C., Seven terahertz broadband GaP electrooptic sensor, Appl. Phys. Lett., 70, 1784, 1997. 74. Leitenstorfer, A., et al., Detectors and sources for ultrabroadband electrooptic sampling: experiment and theory, Appl. Phys. Lett., 74, 1516, 1999.

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Terahertz Spectroscopy: Principles and Applications 75. Holzman, J. F., et al., Free-space detection of terahertz radiation using crystalline and polycrystalline ZnSe electrooptic sensors, Appl. Phys. Lett., 81, 2294, 2002. 76. Kubler, C., Huber, R., and Leitenstorfer, A., Ultrabroadband terahertz pulses: generation and field-resolved detection, Semicond. Sci. Technol., 20, S128, 2005. 77. Schneider, A., et al., Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment, J. Opt. Soc. Am. B, 23, 1822, 2006. 78. Nahata, A., and Cao, H., Broadband phase-matched generation and detection of terahertz radiation, Proc. SPIE, 5411, 151, 2004. 79. Shi, Y. Q., et al, Electrooptic polymer modulators with 0.8 V half-wave voltage, Appl. Phys. Lett., 77, 1, 2000. 80. Sinyukov, A. M., and Hayden, L. M., Generation and detection of terahertz radiation with multilayered electrooptic polymer films, Opt. Lett., 27, 55, 2002. 81. Van Tilborg, J., et al., Terahertz radiation as a bunch diagnostic for laser‑wakefield‑ accelerated electron bunches, Phys. Plasmas, 13, 56704–1, 2006. 82. Berden, G., et al., Electrooptic technique with improved time resolution for real time, nondestructive, single shot measurements of femtosecond electron bunch profiles, Phys. Rev. Lett., 93, 114802/1, 2004. 83. Wilke, I., Single-shot electron-beam bunch length measurements, Phys. Rev. Lett., 88, 124801/1, 2002. 84. Tsang, T., et al., Electrooptical measurements of picosecond bunch length if 45 MeV electron beam, J. Appl. Phys., 89, 4921, 2001. 85. Oepts, D., van der Meer, A. F. G., van Amersfoort, P. W., The free-electron-laser user facility FELIX, Infrared Phys. Techn., 36, 297, 1995. 86. Knippels, G. M. H., et al., Generation of frequency-chirped pulses in the far-infrared by means of a sub-picosecond free-electron laser and an external pulse shape, Opt. Commun., 118, 546, 1995. 87. Knippels, G. M. H., et al., Two-color facility based on a broadly tunable infrared free‑electron laser and a subpicosecond-synchronized 10-fs Ti:sapphire laser, Opt. Lett., 23, 1754, 1998. 88. Yan, X., et al., Subpicosecond electrooptic measurement of relativistic electron pulses, Phys. Rev. Lett., 85, 3404, 2000. 89. Ding, M., Weits, H. H., and Oepts, D., Coherent transition radiation diagnostic for electron bunch shape measurement at FELIX, Nucl. Instrum. Meth. A, 393, 504, 1997.

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Time-Resolved Terahertz Spectroscopy and Terahertz Emission Spectroscopy Jason B. Baxter and Charles A. Schmuttenmaer Yale University

Contents 3.1 3.2

3.3

Introduction.................................................................................................... 74 Time-Resolved Terahertz Spectroscopy........................................................ 75 3.2.1 Introduction 3.2.2 History and Examples......................................................................... 77 3.2.3 Experimental Setup and Data Collection........................................... 78 3.2.3.1 Basic Requirements............................................................... 78 3.2.3.2 Detailed Description of Spectrometer................................... 79 3.2.3.3 Data Collection...................................................................... 81 3.2.3.4 Importance of Spot Size........................................................ 83 3.2.4 Data Workup.......................................................................................84 3.2.4.1 Calculating Conductivity......................................................84 3.2.4.2 Use of Complex Transmission Coefficients.......................... 88 3.2.4.3 Special Treatment at Short Pump-Delay Times....................92 3.2.4.4 Treatment of Porous Media................................................... 95 Terahertz Emission Spectroscopy..................................................................96 3.3.1 Experimental.......................................................................................97 3.3.1.1 Far Field versus Near Field................................................... 98 3.3.1.2 Terahertz Focusing Optics....................................................99 3.3.2 Data Analysis......................................................................................99 3.3.2.1 Sample Orientation.............................................................. 100 3.3.2.2 Excitation Polarization........................................................ 100 3.3.2.3 Emitted Waveform.............................................................. 100 3.3.3 Specific Examples............................................................................. 103 3.3.3.1 Photoconductive Switches................................................... 103 3.3.3.2. Shift Currents and Optical Rectification............................. 103

73

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3.3.3.3 Intramolecular Charge Transfer in Orienting Field............ 107 3.3.3.4 Demagnetization Dynamics................................................ 112 3.3.4 General Formalism........................................................................... 113 3.4 Conclusions.................................................................................................. 115 References............................................................................................................... 115

3.1 Introduction This chapter explores time-resolved aspects of THz spectroscopy. THz spectroscopy involves generating and detecting subpicosecond far-infrared (IR) pulses with an ultrafast visible or near-IR laser, as has been discussed in previous chapters. THz spectroscopy covers the region from 0.3 to 20 THz (10–600 cm–1), with most of the work being done between 0.5 and 3 THz. When converted to other units, 1 THz is equivalent to 33.3 cm–1 (wave numbers) or 0.004 eV photon energy or 300 µm wave length. For example, the THz region is ideal for probing semiconductors because the frequency range closely matches typical carrier scattering rates of 1012 to 1014 s–1, allowing for more accurate modeling of the photoconductivity data. In conventional benchtop experiments, THz pulses are created and detected using short-pulsed Ti:sapphire lasers with pulsewidths ranging from ~100 fs to ~10 fs. The pulsed light allows time-resolved THz studies, in contrast to continuous sources of far-IR radiation such as arc lamps or globars. Pulsed sources such as free electron lasers or synchrotrons have also recently achieved subpicosecond pulse duration, but these are large, expensive user facilities and are less accessible than a Ti:sapphire laser in an individual laboratory. THz spectroscopy also has the advantage that it measures the transient electric field, not simply its intensity.1 Coherent detection allows direct determination of both the amplitude and the phase of each of the spectral components that make up the pulse. The absorption coefficient and refractive index of the sample are computed from the amplitude and phase. Thus the complex-valued permittivity of the sample is obtained without requiring a Kramers-Kronig analysis. The great majority of the results reported in the literature that use conventional far-IR sources and detectors present the frequency-dependent absorption coefficient, but not the refractive index. In this respect, THz spectroscopy provides a convenient method for determining the complex permittivity, even for studies that are not time-resolved. There are three categories of commonly performed experiments: THz time domain spectroscopy (THz-TDS), time-resolved THz spectroscopy (TRTS), and THz emission spectroscopy (TES). The information obtained in a THz-TDS experiment is equivalent to that obtained from a frequency domain linear absorption spectrometer, for example, an FTIR. THz-TDS made its debut in 1988–1989, and the majority of THz work done to date has been along these lines. There are distinct advantages of the THz-TDS method over frequency domain methods, as discussed in detail in Chapter 1. However, the present chapter concerns time-resolved experiments, and THz-TDS need not be considered further. Aside from a small number of initial studies,2–8 most of the development of TRTS has taken place since the year 2000. In contrast to a standard THz-TDS experiment,

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this type of experiment is somewhat more difficult to perform and analyze. In TRTS, the sample is typically photoexcited with a laser pulse (with wavelength ranging from near-IR to ultraviolet) and the time-dependent THz spectrum is obtained with the THz probe pulse. The THz spectrum can be obtained at times ranging from less than 100 fs to more than 1 ns after photoexcitation. This method is sometimes referred to as optical pump/THz probe spectroscopy, but that designation is too limiting. One need not pump optically to make a time-resolved measurement, although the examples given in this chapter all employ optical excitation. TRTS differs from THz-TDS at a very fundamental level. THz-TDS provides information about the static properties of the sample, but TRTS probes the dynamic, evolving properties of the material. For example, the THz transmission of a material depends strongly on how well it conducts electricity. In fact, TRTS is a noncontact electrical probe with subpicosecond temporal resolution, and the behavior of undoped semiconductors on photoexcitation has been studied in great detail as they switch from insulating to conducting. THz emission spectroscopy, TES, has been carried out for as long as THz-TDS has. To perform THz-TDS, it is necessary to generate THz pulses before sending them through the sample under study. However, in that capacity, one is not trying to learn about the material that generates the THz pulses, but is simply using it as a source of THz radiation. In contrast, TES refers to the situation where one is trying to understand the properties of the material being studied by analyzing the shape and amplitude of the THz waveform emitted. It is basically a THz-TDS setup in which the sample itself is the THz emitter. Over the years, semiconductors, superconductors, heterostructures, oriented molecules in solution, and magnetic films have all been studied with this method. All of these methods (THz-TDS, TRTS, and TES) take advantage of one or more of the unique characteristics of THz spectroscopy. The time-dependent electric field is measured (rather than its intensity). These are very bright far-IR sources, and the detectors employed have certain advantages over standard far-IR detectors as well. Most important, however, is the ability to carry out time-resolved experiments in the far-IR region of the spectrum with subpicosecond time resolution.

3.2 Time-Resolved Terahertz Spectroscopy 3.2.1 Introduction TRTS is a noncontact electrical probe capable of determining the complex-valued, frequency-dependent, far-IR permittivity with a temporal resolution of better than 200 fs. TRTS is used to study the changes in a material’s permittivity upon photoexcitation with an optical pump pulse. Changes in the permittivity arise from phenomena such as the photoinduced creation of mobile electrons and holes, polarizable excitons, and polarons. We will refer to “photoconductivity” as the change in sample permittivity upon photoexcitation, although conductivity does not necessarily arise from electronic conduction. Knowledge of a material’s frequency-dependent photoconductivity is important for its use in electronic and optoelectronic devices. Furthermore, the ability to characterize electrical properties in a noncontact fashion

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Sample

Figure 3.1 Schematic illustration of time-resolved THz spectroscopy experiment. The sample is photoexcited with an optical pump pulse and probed at a time, τpp, later with a THz pulse. The sample attenuates and delays the THz pulse according to its complex permittivity.

is critical in the study of nanomaterials, where it can be difficult or impossible to make electrical measurements using conventional probes. The intensity of the THz transmission change after photoexcitation is, in most cases, proportional to both the photoexcited carrier density and mobility. This is in contrast to methods such as transient absorption or time-resolved luminescence, which are sensitive to either the sum or the product of the electron and hole distribution functions, respectively. Additionally, luminescence methods are limited to direct gap semiconductors, whereas TRTS can be used to study carrier dynamics in both direct and indirect gap materials. Even methods that are sensitive to the diffusion of carriers, such as transient grating or four-wave mixing, are not able to determine the frequency-dependent conductivity. TRTS has high sensitivity to mobile electrons. For example, it can measure photoexcited carrier column densities as low as 5 × 1010 cm–2 in GaAs (equivalent to a carrier density of 5 × 1014 cm–3 with a skin depth of 1 µm). Because TRTS measures the sample permittivity, it is sensitive not only to mobile electrons and holes, but also to quasiparticles such as excitons and polarons. The different signatures of these particles in the frequency domain allow one to distinguish the source of permittivity changes using a combination of TRTS and appropriate models. Several different types of experiments make use of the subpicosecond pulses in the far-IR region of the spectrum. In each case, the sample is excited with an optical pulse and then probed with a terahertz pulse as shown in Figure 3.1. So-called “pump scans” monitor a single point on the THz waveform (or the integrated power using a bolometer) and vary the pump probe delay time to measure the average response of the material. In this way, it is possible to characterize the average far-IR response to an optical perturbation. One can measure the overall change in transmission as well as the time scales for the material to respond and to return to its equilibrium state. A second technique, often called a probe scan, is used to record the entire THz wave form at a fixed delay time after photoexcitation. This method obtains the frequencydependent, complex-valued photoconductivity. Finally, the most comprehensive type of study is to create a two-dimensional (2D) grid that maps out the THz waveform at many pump delay times. Although average response and fixed pump delay studies

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are quite useful, the maximum amount of information is obtained from a 2D study in which the photoconductivity can be determined as a function of both frequency and delay time. This method is often referred to as optical pump/THz probe spectroscopy or time-resolved THz spectroscopy. Many different materials systems have been investigated using TRTS. Carrier dynamics in bulk and nanostructured semiconductors, superconductors, and strongly correlated electron systems have been topics of recent research. A very brief accounting of some initial and more recent studies is given below to demonstrate some of the different phenomena that can be observed with TRTS, and other examples will be discussed in detail in subsequent chapters. The remainder of this section describes the experimental apparatus, collection of data, and analysis of data. It points out the common assumptions made during the interpretation of data as well as methods of dealing with possible pitfalls or artifacts.

3.2.2 History and Examples The methods of data collection and analysis used in TRTS were first extensively tested on the well-characterized GaAs system in the late 1990s. A <100> gallium arsenide (GaAs) wafer was shown to have frequency-dependent photoconductivity that follows the Drude model and mobilities that match well with those determined by other experimental techniques.9 Carrier injection time was a function of optical pump wavelength. High-energy photons can excite electrons into low mobility valleys in the conduction band. Carriers scatter and relax into lower energy, high mobility regions of momentum space, resulting in an increase in THz absorbance. In low temperature-grown GaAs (LT-GaAs) thin films, the electron lifetime was found to be only 1.1 ps,10 compared with hundreds of picoseconds in GaAs,9 because of the increased recombination rates arising from higher defect densities. Mobility in lowtemperature-GaAs is also a factor of two lower than in GaAs single crystals. The mobilities determined in these studies could be compared with those from electrical measurements, with TRTS contributing previously unobtainable information at very short time scales. After the techniques of TRTS had been benchmarked using bulk samples with well-characterized electrical properties, its noncontact nature could be exploited to study semiconductor nanomaterials such as nanowire arrays and nanoparticle films. Although it is cumbersome to make contacts to and electrically characterize individual nanowires, TRTS allows characterization of a nanowire array without having to remove the nanowires from the substrate. In cases such as these, an appropriate effective medium theory must be taken into account. Conductivities of InP,11 CdSe,12,13 TiO2,14,15 and ZnO nanoparticle films16 and in ZnO nanowire arrays16 have been studied. In addition to studying carriers generated by bulk absorption of photons with energy larger than the semiconductor bandgap, TRTS is also sensitive to electrons injected into the semiconductor from adsorbed dye molecules.14,15 TRTS can be combined with transient absorption spectroscopy to gain detailed insight into interfacial electron transfer processes. In addition to detecting mobile electrons as in the above systems, TRTS is also sensitive to polarizable excitons. TRTS has given unprecedented information on the

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(a)

4

(b)

2 0

0 –4

4

–2 τ = 15 ps 0.5

τ = 200 ps 0.5

1.0

–4 1.0

σ0 (ω, τ = 200 ps) (10–1 S/m)

σ0 (ω, τ = 15 ps) (S/m)

Terahertz Spectroscopy: Principles and Applications

ω (THz)

Figure 3.2 Conductivity of a ZnO wafer (a) 15 ps and (b) 200 ps after 400 nm photoexcitation. Open and closed circles are the real and imaginary components, respectively, of the complex conductivity. In (a), the response is dominated by free electrons and holes and is well-described by the Drude model (lines). In (b), excitons dominate the response, with lines modeling exciton polarizability. (Figure reprinted with permission from Hendry, E., Charge dynamics in novel semiconductors, Ph.D. thesis, 2005.)

time scales for exciton formation from free electrons and holes in systems such as single crystal ZnO wafers,17 carbon nanotubes,18 and semiconducting polymers.19–21 A dominant positive real conductivity indicates the presence of mobile free carriers, whereas a strong negative imaginary conductivity can be indicative of excitons, as shown in Figure 3.2. Simultaneous detection of both free carriers and bound excitons is another advantage of TRTS.

3.2.3 Experimental Setup and Data Collection 3.2.3.1 Basic Requirements The fundamental requirement of TRTS is the generation of synchronized optical pump and THz probe pulses. Figure 3.1 shows a sample being photoexcited with an optical pulse and subsequently probed with a THz pulse. Varying the delay time between the pump and probe pulses allows study of many different processes including carrier injection, cooling, decay, and trapping. Depending on the pulse energy and wavelength desired, these pulses could be provided from sources such as unamplified or amplified Ti:sapphire lasers or synchrotrons. Amplified Ti:sapphire laser systems are the most commonly used, as they can provide high energy, sub-100-fs pulses at kHz repetition rates with wavelengths tunable by nonlinear optics and optical parametric amplifiers. A detailed description of the amplified Ti:sapphire TRTS spectrometer used in the Schmuttenmaer lab is given in the following section. For samples that exhibit a large change in THz transmission per fundamental Ti:sapphire photon and whose longest time constants are on the order of nanoseconds, it is possible to perform TRTS with an 82-MHz Ti:sapphire oscillator alone. Samples requiring very high photon flux can be investigated using free electron lasers or synchrotron sources.

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Time-Resolved Terahertz Spectroscopy Ti: Sapphire Regenerative Amplifier 1 kHz, 100 fs, 1W Average Power

(a)

(c)

(b)

(f )

(p) (o) (l)

Chopper Pos. 1 (h)

(n)

(i)

Pump Beam

2

(m)

1

(q)

(k)

THz Beam

Lock-in Amplifier

(j) (e)

Chopper Pos. 2 (d)

(g)

Detector Beam

Oscilloscope

3

Computer

Figure 3.3 Time-resolved THz spectrometer. (a) Galilean telescope reduces the laser beam waist, (b) beam splitter, (c) optional frequency doubling crystal, (d) pump beam delay table, (e) optional lens, (f) dielectric mirror (99/1 beam splitter), (g) generator delay table, (h) 800-nm λ/2 waveplate, (i) 1-mm <110> ZnTe crystal, (j) four off-axis parabolic mirrors, (k) sample position, (l) 0.5-mm <110> ZnTe crystal, (m) detector delay table or galvanometer, (n) polarizer, (o) 800-nm λ/4 waveplate, (p) Wollaston polarizing beam splitter, and (q) balanced photodiodes. The balanced photodiodes are output to a lock-in amplifier.

3.2.3.2 Detailed Description of Spectrometer Figure 3.3 displays a detailed schematic of a typical TRTS spectrometer. This particular design has been employed in the Schmuttenmaer laboratory at Yale University, but many variations are possible. A Spectra Physics regenerative amplifier system (Millennia-Tsunami-Merlin-Spitfire) produces a 1-kHz pulse train of 800 µJ, 800 nm pulses of 100-fs duration (full width at half maximum). There are three arms of the spectrometer: THz generation, THz detection, and optical photoexcitation. Roughly two thirds of the light is used as the optical pump beam, whereas the other third generates and detects the THz radiation. The beam for the THz probe is split into a

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collimated 2-mm diameter near-IR beam of 250 µJ/pulse for THz generation and a focused 200-µm diameter near-IR beam of 5 nJ/pulse for THz detection. The intensity of the optical pump beam can be varied to a maximum of ~500 µJ/pulse for 800-nm excitation, ~200 µJ/pulse for 400-nm light, or lower energies for other wavelengths generated by an optical parametric amplifier. The pump spot size is usually between 5–10 mm diameter, but can be adjusted depending on the experimental requirements. The following list describes each component of the TRTS spectrometer shown in Figure 3.3. (a) A Galilean telescope (one focusing and one diverging lens) used to reduce the laser beam waist by a factor of two and collimate the beam. (b) 70:30 beam splitter (Melles Griot). Pump arm (c) 1-mm thick β-BaB2O4 crystal (Quantum Technology Inc.) for experiments requiring second harmonic generation to produce 400-nm excitation. (d) Optical delay table (Parker-Daedal) for pump arm of spectrometer. (e) Optional lens for expanding the pump-beam diameter. THz generation arm (f) A dielectric mirror that acts as a 99:1 beam splitter. The reflected beam generates the THz pulse and the transmitted beam is used to detect the THz pulse. (g) Optical delay table (Parker-Daedal) for THz generation arm of spectrometer. (h) A λ/2 waveplate (Melles Griot) that rotates the vertically polarized 800-nm beam before optical rectification; the output THz beam is polarized horizontally. (i) THz generator crystal, usually a 1-mm thick <110> ZnTe single crystal (eV Products) mounted in a rotation stage. The crystal axis is rotated relative to the polarization axis of the visible pulse to maximize THz generation. (j) Four off-axis parabolic mirrors, (Melles Griot O2 POA 017). Mirror 1 collimates the diverging THz beam, Mirror 2 refocuses the beam onto the sample, Mirror 3 recollimates it, and Mirror 4 focuses it onto the detector crystal. Mirror 4 has a shorter focal length (38.1 mm) than the others (119 mm). Mirrors 2 and 4 have small holes drilled in them to allow transmission of the pump and detector beams, respectively. (k) Sample position. (l) A 0.5-mm thick (110) ZnTe crystal (eV Products) used to detect the THz pulses by free space electrooptic sampling. THz detection arm (m) Galvanometer (General Scanning) for detector arm of spectrometer. (n) 800-nm polarizer (Polacor). (o) A λ/4 waveplate (Melles Griot) to balance the detectors while THz is blocked.

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(p) Wollaston polarizing beam splitter (Karl Lambrecht) to send horizontal and vertical polarizations to different photodiodes. (q) Balanced large-area silicon photodiodes (ThorLabs) for signal detection. The voltages from the photodiodes are directed to a lock-in amplifier which measures the difference in their voltages. Coherent detection allows pulses below the blackbody radiation level to be measured without the use of specialized detectors. A computer interfaces with the experiment by controlling the delay tables and the lock-in amplifier. 3.2.3.3 Data Collection

Electric Field (a.u.)

Time domain scans are collected by scanning the probe delay line (delay line 2 in Figure 3.3) with the pump beam blocked and the chopper in position 1 in Figure 3.3. Figure 3.4 shows THz time domain scans with and without a sample in the THz beam path. Introducing a sample delays, disperses, and attenuates the THz wave form. Methods of extracting the absorbance and refractive index from the THz wave forms are described in the Section 3.2.4. Photoexcitation causes a change in transmission of the sample. For example, photoexcitation of many semiconductors generates free carriers that absorb THz radiation, resulting in a decrease in transmission. The transmission change is a function of both the delay time between the optical pump and THz probe and the time point on the THz waveform. Figure 3.5 shows a 2D grid of the transmission change as a function THz delay time and pump probe delay time.22 These data can be visualized either as a three-dimensional plot or as contour lines projected onto the pump delay–THz delay plane. Taking projections of the 2D grid allows us to measure the dynamics of an experiment. A cut parallel to the pump-delay axis is a 1D pump scan, and a cut parallel to the THz delay axis is a 1D probe scan. The 1D pump scans give the average THz absorption as a function of pump delay time. By Fourier transforming the 1D probe scans, we obtain the frequency-dependent optical constants of the photoexcited material at a given time after photoexcitation.

10 8 6 4 2 0 –2 –4 –6

Reference Sample

–4

–2

0

2

4

6

Time (ps)

Figure 3.4 Time domain scans of a terahertz pulse with and without a sample, in this case a ZnO wafer, at the focus of the terahertz beam. The sample delays, disperses, and attenuates the pulse.

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Terahertz Spectroscopy: Principles and Applications 1 D Pump Scan

Pump-Delay (ps)

3

1D Probe Scan

2 1 0 –1 1

2 3 THz Delay (ps)

4

Figure 3.5 Contour plot of THz difference scan of low-temperature-GaAs photoexcited at 400 nm solid lines represent negative values, whereas dashed lines correspond to positive values. Because the far-infrared response is slower when photoexcitation occurs at 400 nm compared with 800 nm, there is no need for detector deconvolution. One-dimensional (1D) cuts can be taken from the data: 1D pump scans are parallel to the pump delay axis and 1D probe scans are parallel the THz delay axis. (Figure reprinted with permission from Beard, M.C. et al., J. Appl. Phys., EPAPS, 90, 5915, 2001. Copyright 2000 by the American Institute of Physics.)

We can obtain these 1D cuts experimentally by fixing one delay line and scanning the other. We collect the 2D grid by collecting a series of 1D probe scans at a series of pump delay times. Pump scans are taken by fixing the THz delay (line 2) at the peak of the photoexcited THz waveform and scanning the pump delay (line 3) while chopping the optical pump beam (position 2 in Figure 3.3). By chopping the pump beam, we measure the difference between the photoexcited and non-photoexcited THz transmission through the sample. Figure 3.6a shows an 800-ps pump scan for a ZnO thin film, with the inset showing a 16-ps pump scan. Pump scans taken over the time immediately bracketing the optical pump are useful in determining the time constants of electron photoinjection, whereas long-time pump scans can be used to measure the decay dynamics of the photoexcited electron population. In this example, there is a fast injection component in the subpicosecond regime and a slower injection component on the 5-ps time scale. The long-time pump scan is very flat, indicating that the photoexcited electron population and mobility remain constant at higher than 800 ps. “Difference scans” are probe scans taken by fixing the pump delay (line 3) and scanning the THz delay (line 2) while chopping the optical pump beam. Figure 3.6b shows a difference scan for a ZnO wafer, the reference and sample THz scans of which are shown in Figure 3.4. The magnitude of the difference waveform is less than 10% of the sample THz waveform, allowing the photoexcitation to be treated as a small perturbation to the system. In many semiconductors, photoexcitation generates free carriers that absorb THz radiation. Therefore the transmission decreases on photoexcitation and the shape of the difference waveform resembles the negative THz waveform. Because the photoconductivity is determined by the difference between the photoexcited and nonphotoexcited waveforms, it is most accurate to measure this difference directly with the chopper in the pump beam path rather than

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Norm. Transmission

Time-Resolved Terahertz Spectroscopy 0.0

0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –4

–0.2 –0.4 –0.6 –0.8

0

4

8

12

–1.0 0

200

400

800

600

Time (ps)

Electric Field (a.u.)

(a) 0.2 0.0 –0.2 –0.4 –0.6 –6

–4

–2

0

2

4

6

Time (ps) (b)

Figure 3.6 (a) Pump scans for a ZnO thin film. The inset shows injection at short pump– probe delay times and the main panel shows that the signal does not decay over 800 ps. (b) Difference scan, ∆ETHz (pump-on minus pump-off), taken 20 ps after photoexcitation of a ZnO wafer. Note the similar shape, but opposite sign and smaller magnitude, to the associated non-photoexcited THz waveform shown in Figure 3.4.

to measure the THz waveform with the chopper in position 1 both with and without pumping. 3.2.3.4 Importance of Spot Size To avoid frequency-dependent artifacts, it is important that the THz probe samples a region of uniformly photoexcited material. Therefore, the spot size of the visible (pump) beam should be least two times larger than that of the THz (probe) beam. The parabolic mirrors focus the THz probe at the sample to a spot size of approximately 3 mm, but higher THz frequencies are focused more tightly than lower frequencies. As a result, the extracted spectrum is skewed to higher frequencies when the spot size (the diameter at which the intensity of a Gaussian beam falls to 1/e2 of its value at the beam center) of the pump is the same size or smaller than the THz probe, as shown in Figure 3.7. If higher pump intensities are needed, then a smaller diameter pump beam can be used in conjunction with an iris at the sample. The iris blocks THz radiation that would have passed through nonphotoexcited regions of the sample. Additionally, the THz and visible beams should be copropagated to minimize temporal smearing.

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∆(OD) (arb. units)

Terahertz Spectroscopy: Principles and Applications

0.3 0.2 0.1 0.0

0

1

2

Frequency (THz)

Figure 3.7 Change in optical density spectra of photoexcited GaAs as a function of visible (pump) spot size at a 20-ps pump–probe delay time. The visible spot sizes are 6.0 mm (solid line), 1.7 mm (dashed line), and 1.1 mm (dot dashed line), whereas the THz spot size is 2.3 mm. The spectrum becomes skewed toward higher frequencies as the visible spot size becomes smaller than the THz spot size. (Figure reprinted with permission from Beard, M.C. et al., Phys. Rev. B, 62, 15766, 2000. Copyright 2000 by the American Physical Society.)

3.2.4 Data Workup 3.2.4.1 Calculating Conductivity A THz waveform can be Fourier transformed to give the frequency-dependent, complex-valued permittivity. The frequency-dependent, complex-valued photoconductivity, σˆ , is obtained from

ˆ ˆ (w ) = eˆ (w ) + iσ(w ) , η e 0w

(3.1)

ˆ is the photoexcited permittivity, eˆ is the nonphotoexcited permittivity, w where η is the angular frequency, e0 is the free-space permittivity, and i is the unit imaginary. The photoconductivity refers to the response induced by the photoexcitation. This response could arise from a variety of sources such as mobile electrons (holes) excited into the conduction (valence) band of a semiconductor, polarizable excitons, polaritons, or other quasiparticles. Prior knowledge of the sample properties combined with careful experiments can determine the source of the conductivity. Determination of the conductivity from the data does not require any model, although models can later be fit to the data to extract relevant parameters about the system. Depending on the degree of accuracy required and the validity of certain assumptions, calculating permittivity and conductivity from THz waveforms can range from fairly simple to rather complex. If we assume that no THz radiation is reflected at any of the sample interfaces such that all THz is either transmitted or absorbed, the sample permittivity can be calculated in a straightforward way from the Fourier transform of the reference and sample scans shown in Figure 3.4. The power and phase, calculated from the Fourier transform of the time domain scans, are shown as a function of frequency in Figure 3.8. Because the phase is determined using the arctan, it can only be determined in the range (–π, π), resulting in a saw

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Power (a.u.)

Time-Resolved Terahertz Spectroscopy 10–3 10–4 10–5 10–6 10–7 10–8 10–9 10–10 10–11

Ref Sample

0

1

2

3

4

3

4

Frequency (THz) (a)

Phase (rad)

200

π

150

0 –π

100

Sample 1

2 Ref

50 0

0

1

2 Frequency (THz) (b)

Figure 3.8 (a) Power and (b) phase of a reference and sample pulse calculated from the time domain scans in Figure 3.4. The inset in (b) shows the reference phase before correcting for discontinuities.

tooth behavior with increasing frequency. The discontinuities should be removed to facilitate later calculations and for ease of visualization by adding 2π at each point of discontinuity. This process smooths the phase to a continuously increasing function without loss of information and allows ∆φ to be determined properly. Additional factors of 2π may also need to be added to the sample phase at all frequencies for thick or high index samples. The absorbance and refractive index of the sample are related to the power, P, and phase, f, of the Fourier transform of the sample and reference scans by a=

1  P ln d  P0 

(3.2)

and

n = 1+

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c (φ - φ 0 ). 2 pwd

(3.3)

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Terahertz Spectroscopy: Principles and Applications

Absorbance (1/cm)

50 40 30 20 10

Refractive Index

0 0.2

0.4

0.6

0.8 1.0 Frequency (THz) (a)

0.4

0.6

1.2

1.4

1.2

1.4

3.4 3.2 3.0 2.8 2.6 0.2

0.8

1.0

Frequency (THz) (b)

Figure 3.9 (a) Absorbance and (b) refractive index of a ZnO wafer calculated using Equations 3.2 and 3.3 and the power and phase data in Figure 3.8. In this case, transmission losses are actually the result of reflection, but they appear as absorbance because of the assumptions in this simplistic procedure. Using complex transmission coefficients as described in Section 3.2.4.2 will remove this “absorbance offset.”

Figure 3.9 shows the absorbance and refractive index calculated using Equations 3.2 and 3.3 along with the power and phase data from Figure 3.8. The complex- valued permittivity and the complex-valued refractive index are related through eˆ = nˆ 2 where nˆ = n + ik , and k = λa / 4 p = c a / 2w. Thus the real and imaginary components of the permittivity are determined from a and n through

e′ = n2 - k 2

(3.4)

e ′′ = 2nk .

(3.5)

and

Photoconductivity can then be calculated from Equation 3.1 by using the photoexcited permittivity as the generalized permittivity, ηˆ , and the non-photoexcited permittivity as the static permittivity, eˆ . The real and imaginary parts of the

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photoconductivity are given by

σ ′ = e 0w ( η′′ - e ′′)

σ ′′ = e 0w (e ′ - η′).

(3.6) (3.7)

Example photoconductivities are plotted in Figure 3.2 for a ZnO wafer at 30 K at two pump delay times.17 At short times, the photoconductivity is well fit by the Drude model for free electrons, whereas at long times, the photoconductivity has the signature of excitons. The free electrons and holes become bound as excitons on the 50-ps timescale. ˆ is the photoIt is important to note that the measured change in conductivity, σ, conductivity and not the overall conductivity of the sample when using this method. In the case in which the change on photoexcitation is small, it is useful to measure the “difference scan” directly by placing the chopper in the pump beam as discussed in the experimental methods section. Hence, ∆E(t) = Ep* (t) – Enp(t) is measured and added to the measured nonphotoexcited THz waveform Enp(t) to yield the photoexcited waveform Ep* (t). An alternative method has been presented by Heinz et al. for samples with ∆E(t) << Enp(t).23 This method neglects multiple reflections, whose influence is slight in most cases. From Maxwell’s equations, they derive

 wd eˆ eˆ - eˆ ext Deˆ = 2eˆ  i c  eˆ + eˆ ext

-1

 DE (w )  × E (w ) 

(3.8)

for the photoinduced change in permittivity, where ∆E(w) and E(w) are the Fourier transforms of the difference scan and the nonphotoexcited THz scan, respectively. eˆ and eˆ ext are the frequency-dependent, complex permittivities of the nonphotoexcited sample and its surrounding material, respectively, and d is the thickness of the photoexcited material. Again, both of these methods ignore reflections at the sample interfaces. To treat such reflections requires the introduction of complex transmission coefficients. Two mathematically convenient methods of dealing with complex transmission coefficients are outlined in Section 3.2.4.2. These methods are useful for dealing with reflections from a single layer, from a multilayered sample, or from a sample that has been partitioned into slabs of varying conductivity that arise due to absorbance that varies according to Beer’s law. When the sample thickness is small compared with the optical skin depth, one can assume uniform absorption throughout the sample. However, when the optical skin depth is smaller than the sample thickness, it is appropriate either to define a photoexcited layer and a nonphotoexcited layer or to define many layers of exponentially decreasing conductivity.

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These methods also assume that the conductivity changes slowly over the time that the THz probe interacts with the sample, which is approximately 1 ps. This constraint is often met for pump–probe delay times greater than 3 ps. Additionally, because the optical pump pulse is approximately of 100-fs duration and the probe pulse is approximately 1 ps, at short times the beginning of the THz wave probes nonphotoexcited material, whereas the end of the THz wave probes photoexcited material. In this case, the beginning of the THz wave sees different sample properties than the end, leading to data that can be misleading if not treated appropriately. Scanning a 2D grid of pump–probe data and deconvolving the detector response function allow correct interpretation of data at short pump delay times, as described in detail in Section 3.2.4.3. 3.2.4.2 Use of Complex Transmission Coefficients Many systems studied with TRTS are layered stacks of dielectric materials. The dielectric stack could consist of semi-infinite layers of air that surround a thin film on a substrate, or, in the case of a photoexcited semiconductor whose optical skin depth is less than its own thickness, could consist of photoexcited and non- photoexcited semiconductor. The frequency-dependent absorption coefficient ap(w) and refractive index np(w) of the photoexcited sample can be extracted directly from the measured data without assuming a model to describe the medium. Extraction of ap(w) and np(w) requires knowing the photoexcited path length d and the reflectivity losses at all the interfaces, but reflectivity is not known a priori when the material permittivity is unknown. Furthermore, the photoexcited path length is not well defined because the photoexcited medium gradually transforms into the nonphotoexcited medium. In strongly absorbing materials, the photoexcited path length d is small compared with the spatial extent of the THz pulse. This can result in interface effects that, if not treated correctly, can lead to large errors in determining the optical properties. The interface effects are most pronounced when photoexcitation causes large changes in np(w) and can even cause apparent superluminal propagation through photoexcited materials.2, 9 Photoexcitation creates an exponentially decreasing concentration of carriers as a function of distance into the sample. The spatial distribution of carriers evolves over time as the excited carriers diffuse and recombine at the surface and in the bulk, and diffusion and recombination can be modeled to determine the carrier spatial distribution at any given pump delay time.9 In most cases, diffusion is only important at longer pump delay times when t >do2/D, where D is the diffusion coefficient and do is the initial photoexcited path length. As a first approximation, the photoexcited distribution can be treated as a uniform slab with thickness d which is chosen to be the optical skin depth for exponential carrier distributions. For nonexponential distributions, the slab thickness is determined by normalizing the distribution such that its maximum is unity with the integral of the normalized distribution giving the appropriate thickness, d. Beard and Schmuttenmaer have performed finite difference time domain (FDTD) simulations of TRTS experiments using different non-uniform spatial distributions to understand conditions under which the slab approximation holds24 and found that it is valid when the spatial extent of the distribution is less than approximately 5 µm.

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After a scheme for relating the conductivity profile to the absorbance of the optical pump has been developed, we can then address the formalism of treating the dielectric stack. The basic premise is to determine the permittivity of each layer such that when THz light is propagated through the entire dielectric stack, the calculated transmittance equals the measured transmittance. Measured TRTS data can be directly transformed to determine the complex ratio of the THz field with pump on (photoexcited) versus pump off (nonphotoexcited):

qmeas = E p* (w , d ) / E np (w , d )

(3.9)

where Ep* (w,d) is the complex field measured after transmission through the photoexcited material, and Enp(w,d) is the field measured after transmission through the nonphotoexcited material. Enp(w,d) and Ep* (w,d) are the Fourier transforms of the time domain data, where the THz difference pulse ∆E(t,d) = Ep* (t,d) – Enp(t,d) is measured and added to Enp(t,d) to yield Ep* (t,d). Treating the interfaces in the dielectric stack is accomplished by relating qmeas to a complex transmission amplitude coefficient z(k), which accounts for the reflectivity of each interface as well as propagation through the structure. The transmission coefficient relates the complex transmitted electric field amplitude A tr(w,d) to the complex initial electric field amplitude25 Ao(w,0) as

Atr (w, d ) = z( k ) A0 (w, 0 ).

(3.10)

To be general, E = A exp(ikz – iwt) where E is the full complex electric field, A is the complex amplitude, and k is the complex wave vector given by k (w ) =

2 pn(w ) a (w ) +i . λ 2

(3.11)

The far-IR optical properties of the photoexcited material can be calculated directly from qmeas and the nonphotoexcited optical properties via the complex transmission amplitude coefficients of the photoexcited zp* (k p*) and nonphotoexcited znp(knp) media. From Equations 3.9 and 3.10, qmeas =

z*p (kp* ) . znp ( knp )

(3.12)

Separate non–time-resolved measurement of the far-IR optical properties of the nonphotoexcited material give the nonphotoexcited transmission coefficient znp(k np). By inverting Equation 3.12 and combining it with an appropriate equation for zp* (k p*), discussed later in this chapter, we can obtain the wave vector for the photoexcited medium k p*, and thus the photoexcited optical constants a(w) and n(w). The inversion can be done numerically by generating a grid of |q(aguess, nguess) – qmeas | and finding the minimum on the grid or by other numerical procedures for solving these

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Terahertz Spectroscopy: Principles and Applications

equations, such as the Newton-Raphson method. The photoconductivity can be found from a and n for the photoexcited and nonphotoexcited material using the procedure in Section 3.3.4.1. Several methods exist for the calculation of the transmission coefficient for a dielectric stack. Transfer matrices provide an intuitive approach to developing the equations to describe propagation through a dielectric stack and thereby calculate the transmission coefficient z(k).26 First, consider a single layer with refractive index n1 sandwiched between two semi-infinite media with refractive indices of n 0 and nT, respectively. The equation describing propagation through this layer single layer is given in matrix form by26

  1   1  A0′  cos kd =  +   n0   - n0  A0  - in sin kd  1

 -i sin kd  n1  cos kd 

 1  AT ,    nT  A0

(3.13)

where d is the layer thickness, and k is the wavevector in the layer, k = 2pn1/l0. Letting the reflection and transmission coefficients be r = A0′ / A0 and z = AT / A0 , respectively, allows Equation 3.13 to be written more compactly as 1  1  r =M  +  n0   - n0 

1  z ,  nT 

(3.14)

where M is the transfer matrix. For multiple layers, the overall transfer matrix M becomes the product of the individual layer’s transfer matrices, such that

A M 1M 2M 3 ... M N = M =  C

B . D

(3.15)

For the situation shown in Figure 3.10a, where there is a photoexcited layer and a nonphotoexcited layer, we have   cos k1d1 M = M 1M 2 =   - in1 sin k1d1

 -i sin k1d1  n1  cos k1d1 

  cos k 2d 2   - in2 sin k 2d 2

 -i sin k 2d 2  n2 . cos k 2d 2 

(3.16)

Combining Equations 3.14 and 3.16, and solving for the transmission coefficient yields26 z=

2n0 . An0 + BnT n0 + C + DnT

(3.17)

Finally, the complex refractive index of the photoexcited layer, n1, which provides a transmission coefficient that satisfies Equation 3.12 is determined iteratively.

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Time-Resolved Terahertz Spectroscopy

Eo

d

Eo

Air

0

Sample*

1

Sample

2

Air

T d

ET

(a)

Air

n

Sample*

n–1

Sample

2

Air

1

(b)

Figure 3.10 Schematic of layers in a dielectric stack with labeling conventions for (a) matrix method and (b) complex impedance method. Here the skin depth of the optical pump is much smaller than the sample thickness, and the sample is divided into photoexcited and non-photoexcited layers.

Alternatively, the transmission coefficient can be defined in terms of the impedances of each layer in the stack by25

1 = z( k )

j = n -1

∏ j =1

Z j +1 + Z Zj + Z

( j) in ( j) in

exp ( - i φ j )

(3.18)

where z(k) is the transmission coefficient for a wave incident on layer n of an nlayered stack as shown in Figure 3.10b, and fj = kj(w)dj. The propagation distance through the jth layer dj is defined as zero for the semi-infinite layers j = 1 and j = n. The impedance of the jth layer is defined as Zj = w/ckj(w), and the input impedance of the jth interface is defined as Z in( j ) =

Z inj -1 - iZ j tan φ j Zj Z j - iZ inj -1 tan φ j

(3.19)

where Zin(0) = Z1. It is worth noting that this procedure for extracting the photoexcited optical constants of a single layer in the dielectric stack assumes that none of the other layers’ optical constants change on photoexcitation. When multiple layers are excited, it is mathematically impossible to determine all of the individual optical constants. In these cases, TRTS still measures the changes in optical density and phase of the complete stack, but extraction of material parameters for each individual layer requires additional information about the system.

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3.2.4.3 Special Treatment at Short Pump-Delay Times When the response of the material is fast compared with the temporal duration of the THz pulse, the data collection and analysis described previously are not adequate. For most samples, this is only important when the pump pulse arrives whereas the THz pulse is propagating through the sample. In these cases, proper analysis requires a full 2D grid of THz difference scans, DETHz(t,t″), along with subsequent deconvolution of the detector response function. DETHz(t,t″) corresponds to pump-on minus pump-off, with t referring to the THz-delay time and t″ marking the arrival of the pump pulse. Figure 3.11 displays such a 2D TRTS data set, where a series of THz difference scans are taken at a variety of pump delay intervals by scanning the detector delay line.22 It is clear from Figure 3.11 that the individual THz difference scans are strongly influenced by the arrival of the pump pulse (shown as arrows). Schmuttenmaer et al. have shown that the data parallel to the 45° axis of this plot (shown with a bold line) consist of THz difference scans that have experienced a constant delay from the pump pulse.9,27 These scans can be obtained by a numerical projection of the original data set. Alternatively, they can be collected directly by fixing the relative time between the detector and the pump delays, either by synchronously scanning the THz detector and pump delay lines together with fixed THz 4

(w)

2

(x)

t´´ (ps)

3

1

(y) (z) 0

1

2

3

4

0

THz Delay (ps) (w) (x) (y) (z)

Figure 3.11 Typical time-resolved THz spectrometer two-dimensional dataset. Contour plot of ∆ETHz (pump-on minus pump-off). The solid lines are negative values and the dashed lines are positive. The bold solid diagonal line represents the arrival of the pump pulse. The pump pulse affects only that part of the THz pulse that comes after it. These are individual scans corresponding to the arrows on the grid; the arrows represent the arrival of the visible pulse. (Figure reprinted with permission from Beard, M.C. et al., J. Appl. Phys., EPAPS, 90, 5915, 2001. Copyright 2000 by the American Institute of Physics.)

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2

2

1

t-t´´(ps)

3

1

0 1

2

3

4

–1

t-t´´(ps)

93

Time-Resolved Terahertz Spectroscopy

0 1

2

3

t(ps)

t(ps)

(a)

(b)

4

–1

Figure 3.12 (a) Contour plot of ∆ETHz (pump-on minus pump-off) as a function of constant reference time. The solid lines correspond to positive ∆ETHz values and the dashed lines correspond to negative values. The 45° component seen in Figure 3.11 has been removed as explained in the text. However, there is a component near pump delay t″ = 0, which curves to the left and appears to come before the pump pulse has arrived. (b) After deconvolution, these features have essentially been removed. (Figure reprinted with permission from Beard, M.C. et al., J. Appl. Phys., EPAPS, 90, 5915, 2001. Copyright 2000 by the American Institute of Physics.)

generator delay, or by scanning the THz generator delay with the pump and THz detector delays fixed. Any of these methods will result in a dataset that consists of THz scans that have experienced a constant delay from the pump pulse. Figure 3.12a displays a data set for GaAs which has been collected by scanning the THz generator delay line.22 This method removes the 45° component in Figure 3.11, but a new component curving to the left at early pump delay times has appeared. This new feature appears to grow in before the arrival of the pump pulse. This apparent superluminal phenomenon arises from the convolution of the detector response function with the true THz waveform. Deconvolution of the detector response function removes this feature, as shown in Figure 3.12b. The detector response must be deconvolved because its finite bandwidth acts as a low-pass filter, broadening the true pulse. To correct for this artifact we deconvolve the detector response from the measured signal using the procedure shown in Figure 3.13.22 Figure 3.13a shows a hypothetical reference pulse (solid line) after passing through the sample, but before propagating through the ZnTe detector crystal. Overlaid is a pulse that has experienced photoexcitation while the THz pulse was in the sample (dashed line), such that only the trailing portion of the pulse is affected by the photoexcitation. The vertical line indicates the moment of photoexcitation. After propagation through the detector, the photoexcited output pulse (Figure 3.13b) has features that appear to differ from the reference pulse before the pump pulse arrives. After numerical deconvolution, the reference and the photoexcited scans (Figure 3.13c) are largely restored to their shapes before passing though the detector, and the features occurring before the arrival of the pump pulse have essentially disappeared. Figure 3.13d shows the photoexcited scans before and after passing through the detector and after deconvolution.

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Terahertz Spectroscopy: Principles and Applications (a) (b)

ZnTe Detector (d)

5

6

(c)

7

THz Propagation (ps)

1 1

2

3

2

3

THz Propagation (ps)

THz Propagation (ps)

Figure 3.13 Illustration of the detector filtering effect on a THz pulse transmitted through a photoexcited medium. In (a), the solid line is the reference scan (no photoexcitation), the dashed line represents the transmitted pulse when there is photoexcitation, and the dotted line represents a decrease in sample transmission due to the photoexcitation pulse. The reference and photoexcited pulses are identical until the moment of photoexcitation, represented by the vertical line. (b) Propagation through the detector (ZnTe) distorts the waveforms. (c) Numerical deconvolution of the detector response removes the effect of propagation and nearly restores the scans to their shape before propagation. The differences between the reference and photoexcited scans are shown in (d) to further illustrate this point. The thin solid line is the difference before propagation, the bold solid line is the difference after propagation and without deconvolution, and the dashed line (hardly distinguishable from the thin solid line) is the difference after deconvolution. (Figure reprinted with permission from Beard, M.C. et al. J. Appl. Phys., EPAPS, 90, 5915, 2001. Copyright 2000 by the American Institute of Physics.)

To perform the detector deconvolution, Beard et al. have used a Fourier transform method that includes optimal filtering.10,28 The electrooptic detection of a THz pulse has been modeled by Bakker et al.29 and is given by DI EO (τ) ∝ r41

l

∫ ∫ dz

0

∞

-∞

dt I p (z ,t - τ) E (z ,t )

(3.20)

with E ( z, t ) =

∫

∞

-∞

dwE ( 0, w ) exp (izkznTe - iwt )

(3.21)

where DIEO (t) is the detected signal, E(z,t) is the propagated THz pulse at a distance z into the detector crystal, Ip(z,t-t) represents the propagating visible pulse (assumed to be Gaussian), t is the temporal delay between the two pulses, kZnTe is the complex propagation vector of the THz pulse in the ZnTe crystal, and r41 is the electrooptic coefficient for ZnTe. Equation 3.20 can be regarded as a simple convolution of the detector response with the THz pulse, given by

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DI EO ( τ ) ∝

∫

t

-∞

E ( 0, t ) χ d (t - τ ) dt

(3.22)

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Time-Resolved Terahertz Spectroscopy

95

with χ d being the response function of the detector. If χ d is known, a Fourier deconvolution can be used to obtain E(0,t), the THz pulse before propagation through the detector. The detector response function, χ d, is most easily generated by numerically propagating a Dirac delta function through the detector using Equations 3.20 and 3.21. The complex propagation vector kZnTe, given in Equation 3.11, can be measured in a separate THz-TDS measurement. The deconvolution procedure requires data collected in the conventional fashion as shown in Figure 3.11, where the pump delay line is fixed while scanning the THz (generator) delay line. Each THz difference scan is then deconvolved from the detector. Finally, the data are numerically projected to obtain deconvolved difference scans with constant pump delay times. This process removes most of the features that result from convolution with the detector, as seen in Figure 3.12b. The frequency-dependent optical parameters can be then be extracted from the deconvolved and numerically projected data as discussed in Section 3.3.4.1. Deconvolution is not required when the response of the material is slow compared with the detector response time. For example, Figure 3.5 is a 2D plot of the response of low-temperature-GaAs collected when scanning the THz generator.22 Photoexcitation at 400 nm produces electrons that immediately (~10 fs) scatter to the low mobility L and X valleys. The far-IR response is then governed by the return of the electrons to the high mobility G valley.9,30–32 As a result, the rise time of the THz absorption is slow and deconvolution is not required. Figure 3.5 shows the collected data without deconvolution, and no artifacts are present. The group velocity mismatch of the pump and probe pulses in the sample can typically be neglected, because they traverse only a few microns of photoexcited material in most samples. The temporal resolution of TRTS is determined by the approximate 200-fs detector response function and any geometrical limitations such as spatial overlap of the pump and probe beams, or noncollinear propagation of the beams.9 3.2.4.4 Treatment of Porous Media That TRTS is a noncontact probe is a great advantage for studying nanomaterials. Making contacts to very small objects is difficult and the contacts themselves can sometimes dominate resistance measurements. When investigating porous nanomaterials such as nanoparticle films or nanowire arrays, where the wavelength of the THz probe radiation (150 µm–1.5 mm) is much larger than the characteristic particle dimensions, it is appropriate to use effective medium theory.33 Effective medium theory is frequently used to calculate the permittivity of a composite material given the permittivity and volume fraction of each of the individual components. However, it can also be used in reverse to calculate the permittivity of one component when the permittivities of the other component and the composite are known.16,34 The latter is often the more useful method, for example when calculating the permittivity of the semiconductor in an air–semiconductor composite where the permittivity of air is known and that of the composite is measured. The Bruggeman effective medium approximation (EMA)35 and MaxwellGarnett theory (MGT),36 both based on the Lorentz-Lorenz (Clausius-Mossotti) equation, are two of the most commonly used effective medium theories. Each

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makes slightly different assumptions, and each has advantages and limitations.33 For example, Hendry et al. have used MGT to compare the measured composite permittivity of a mesoporous TiO2 nanoparticle film with that calculated using the permittivities of bulk TiO2 and air.15 However, singularities in reverse MGT occur for large semiconductor volume fractions and large contrast ratios of the individual component permittivities. Therefore, using reverse MGT to calculate nanostructure permittivities from composite measurements does not give robust and reliable results.16 Conversely, the reverse Bruggeman EMA does not exhibit these singularities. The Bruggeman EMA is given by

 e -e   em - e  f  p = 0,  + (1 - f )   e m + K e   ep + Ke 

(3.23)

where f is the particle volume fraction, e is the composite permittivity, em is the matrix permittivity, ep is the particle permittivity, and K is a geometric factor. K is 1 for an array of cylinders with axes collinear with the incident radiation and 2 for spherical particles.34 The semiconductor absorption coefficient and refractive index are calculated from the composite absorption coefficient and refractive index by using the reverse Bruggeman EMA. Solving for the particle permittivity in Equation 3.23, with em equal to unity for air, gives

ep =

e [f (1 + K e ) + (1 - f ) K (e - 1)] . f (1 + K e ) + (1 - f ) (1 - e )

(3.24)

Using this procedure, permittivities of nanomaterials can be calculated and compared with those of bulk materials. Additionally, nanomaterials of different shapes and sizes can be compared with each other to examine the influence of morphology on conductivity without having to make contacts to individual nanostructures.

3.3 TerahertZ Emission Spectroscopy An overview of types of materials studied over the years using THz emission spectroscopy (TES) can be found in previous work.1 Examples include bulk semiconductors, semiconductor heterostructures such as quantum wells and superlattices, superconductors, magnetoresistive manganite thin films, organic molecules, either crystalline or in solution, and relativistic electron beam bunches. More recently, ultrafast demagnetization in thin magnetic films has been characterized with TES,37,38 and THz emission from surface plasmons in a metal grating has been observed.39 In general, a sample is photoexcited and radiates a THz pulse from an induced current or polarization (either electric or magnetic), and the radiated THz waveform is analyzed to uncover the dynamics of the underlying process. TES differs from TRTS and THz-TDS in that a THz probe pulse is not used to interrogate the far-IR optical properties of the sample, either with or without prior photoexcitation, respectively.

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Time-Resolved Terahertz Spectroscopy

A THz pulse is emitted due to a time-dependent polarization (or current) of the sample. In the near-field it is given by

E x (t ) ∝

∂2Px (t ) , ∂t 2

(3.25)

E x (t ) ∝

∂J x (t ) , ∂t

(3.26)

or equivalently,

where Px(t) and Jx(t) are the time-dependent polarization and current, respectively. Furthermore, if the underlying mechanism is optical rectification, a nonresonant, nonlinear process, Jx(t) in Equation 3.26 is given by the “rectification current”

J rect (t ) = 2e o χ( 2)

∂ E (t )E * (t ), ∂t

(3.27)

where c(2) is the second-order nonlinear susceptibility. That is, it is the time derivative of the excitation pulse. Thus by carefully measuring the emitted waveform, it is possible to determine the origin of the signal (i.e., a nonresonant process such as rectification versus a resonant process in which there is a real current). The waveform for a nonresonant process is determined solely by the excitation laser pulse shape and duration. However, in the case of a real current, the time-dependent polarization or current can be extracted from the waveform, assuming it is slower than the duration of the excitation pulse. Similarly, if the magnetization of the sample is time-dependent, it generates a current, and the emitted THz pulse is given by

E y (t ) ∝

∂2M x (t ) , ∂t 2

(3.28)

where Mx(t) is the time-dependent magnetization. Note that the electric field of the emitted pulse is orthogonal to the direction of the changing magnetization. These equations, and Maxwell’s equations themselves for that matter, are independent of any particular timescale. That is, picosecond dynamics lead to THz waveforms, nanosecond dynamics lead to GHz waveforms, and so on. However, such a wide variety of important chemical and physical processes occur on a picosecond timescale that TES is an indispensable technique.

3.3.1 Experimental A typical TES setup is shown schematically in Figure 3.14. A regeneratively amplified titanium–sapphire laser produces an approximate 100-fs laser pulses centered at a wavelength of 800 nm with a repetition rate of 1 kHz. The beam is split into

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Terahertz Spectroscopy: Principles and Applications θy θx

Regeneratively Ampliﬁed Ti: Sapphire Laser 1 mJ/pulse, 1 kHz Rep Rate, 100 fs Pulsewidth Variable Time Delay

θz Visible Beam Block

(a)

Nonpolarizing ZnTe Beam Splitter

Sample

Polarizing Beam Splitter λ/4

Vertical Polarizer

Si Photodiodes

Figure 3.14 Experimental schematic for THz emission spectroscopy. “Transmission” mode is indicated. It is also possible to operate in “reflection” mode wherein the excitation beam travels away from the detector, and the THz pulse emitted in the reflected direction is detected. Inset (a) displays a coordinate system (z is the direction of propagation of the excitation laser).

two parts, pump and readout, with about 99.9% of the energy being used to excite the sample after traveling along a variable delay line. A paper or polystyrene visible beam block ensures that any visible power not absorbed by the sample does not reach the detector. The other portion of the visible pulse reflects off a beamsplitter and travels through a vertical polarizer before reflecting off a second non-polarizing beamsplitter. It is used to detect the EM transient via free space electrooptic sampling40 in a 0.5-mm thick <110> ZnTe crystal. The detector is polarization-sensitive, but a wire-grid polarizer that rejects orthogonally polarized THz radiation is placed in front of the detector crystal to ensure maximum fidelity. The sample can be rotated azimuthally about the surface normal, qz, or it can be rotated about the vertical, qy, and/or horizontal axes, qx, if oblique incidence is required (see inset in Figure 3.14 for coordinate system). 3.3.1.1 Far Field versus Near Field The far-field pulse shape is related to the near‑field pulse by a time derivative.41 This relationship can be verified by using ZnTe as both the generator and detector. When using the highly efficient ZnTe generator, the beam waist can be expanded such that the near‑field regime is obtainable. A comparison of the near- and far‑field signals is shown in Figure 3.15. The far field is defined as d >> r 2 / ct0 , where d is the distance of the detector from the sample, r is the transverse radius of the visible pump beam, and t0 is the full width half maximum of the transient. For large bandwidth nearly signal cycle pulses, the pulse length, ct0, is a better metric for the near‑ and far‑field regimes than is the wavelength, λ. Typically, r 2 /ct0 ≈ 1 cm, and it is best to ensure that the far field has

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EΩ(arb. units)

Time-Resolved Terahertz Spectroscopy Near-Field d << r2/(ct0)

1

Sample Detector

0 Far-Field d >> r2/(ct0)

–1 –2

–2

0 THz Delay (ps)

d 2

Figure 3.15 THz generation is produced from ZnTe via optical rectification to illustrate near- versus far-field regimes. The open circles are the first derivatives of the near-field signal, whereas the lines are the measured near-field and far-field signals. The detector configuration is shown to the right. (Figure reprinted with permission from Beard, M.C. et al., J. Phys. Chem. A 106, 878, 2002. Copyright 2002 by the American Chemical Society.)

been achieved by increasing d until the pulse shape no longer changes shape with increasing distance, at which point d is usually about 3 cm or more. In rare cases, the signal can be collected in the near-field regime (that is, when it is possible to position the detector essentially in contact with the sample), which would allow the pulse shape to be measured directly. However, because of the relatively small excitation spot size and because it is often impossible to place the detector close enough to the photoexcited medium to be in the near field, the signal is typically collected in the far‑field regime. 3.3.1.2 Terahertz Focusing Optics Regardless of whether one collects data in the far field or near field, one of the most important considerations is that focusing optics not be used. The detector is placed adjacent to the sample as shown in Figure 3.14. Collecting the data in this manner allows one to obtain the true underlying dynamics, but at the cost of a lower signal-to-noise ratio compared with when THz focusing optics are used. However, one avoids difficulties of astigmatism, Guoy phase shift, and other diffraction‑induced pulse distortions that occur with large bandwidth pulses. The pitfall with using focusing optics is that variations in the THz waveform will be attributed to time-dependent changes in the polarization, when in fact it is nothing more than experimental misalignment. In addition, one must be absolutely certain that there are not other systematic artifacts. This is especially true when rotating the laser polarization, or if the sample is inside a cryostat. For extremely weak signals, it is imperative that every possible source of generation is ruled out; we have measured very weak THz generation from neutral density filters for example.

3.3.2 Data Analysis After the data are collected, useful information must be extracted. The data analysis begins with observing the dependence of the THz emission on sample rotation/

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Terahertz Spectroscopy: Principles and Applications

orientation and excitation polarization. In addition, one can analyze the emitted waveforms for additional information. 3.3.2.1 Sample Orientation The sample may be rotated azimuthally (about the surface normal, qz) or obliquely (about the axes perpendicular to the surface normal, qy or qx). For example, several well-known generation mechanisms in semiconductors rely on a current surge along the surface normal. These are the surface depletion field and the photo-Dember effect. The surface depletion field occurs when the Fermi level is pinned to surface states or trap states thereby causing “band bending” at the surface. On photoexcitation, electrons are accelerated toward or way from the surface, depending on the sign of the field.42 The photo-Dember effect occurs even when band bending is not present because electrons and holes have different mobility. On photoexcitation, the carriers diffuse, but at different rates. Thus a net current normal to the surface results.43 If the sample is at or near normal incidence, THz pulses from either of these mechanisms will be very small. The only reason they are measurable at all is because either the sample is not perfectly normal to the excitation beam and detector, or more likely, because a focusing lens is being used.44 So, if either of these mechanisms is responsible, it will become apparent by a strong dependence on rotation of the sample about the x or y axis, but not on azimuthal rotation. If the current is along a well defined direction in the surface plane, then the THz emission will decrease as the sample is rotated about the axis perpendicular to the direction of the current surge, but not if it is rotated about the axis in the direction of the current surge (neglecting any effects due to differences in reflectivity as a function of angle). It will also depend (co)sinusoidally on the azimuthal angle. 3.3.2.2 Excitation Polarization Another important consideration is the dependence of the waveform on the polarization of the excitation beam (linear, either vertical or horizontal, or some of both, elliptical, or circular). For example, the THz emission in the x and y directions (lab-fixed) as a function of rotation of a linearly polarized excitation beam for a system of partially oriented dye molecules is shown in Figure 3.16. Conversely, the emission from a magnetic thin film is independent of excitation laser polarization since the fundamental mechanism is ultrafast thermal heating of the electrons: it only depends on sample orientation as seen in Figure 3.17.37 Similarly, the emission from a photoconductive switch is independent of the polarization direction (or ellipticity) of the excitation laser because the fundamental mechanism is simply generation of conduction band electrons, which are then accelerated by the applied bias voltage. Thus the underlying mechanism of the THz emission dictates whether it depends on the excitation laser polarization or not. 3.3.2.3 Emitted Waveform The shape of the emitted pulse reveals additional information about the underlying process. For example, in GaAs or other semiconductors, one can produce THz

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2e-5 1e-5 8e-6 4e-6 0 –4e-6 –8e-6

0

50

100

150

200

Polarization Angle (ζ) (b)

Figure 3.16 Dependence of the generated signal amplitude as a function of visible polarization angle, ζ, detecting the x component (filled circles) and y component (open circles) of the generated pulse amplitude. The static electric field is in the x direction. (Figure reprinted with permission from Beard, M.C. et al., J. Phys. Chem. A 106, 878, 2002. Copyright 2002 by the American Chemical Society.)

THz Amplitude (arb. units)

radiation (at normal incidence and without a bias voltage) from either optical rectification, if the photon energy is below the bandgap, or a shift current if the photon energy is above the bandgap.45 By analyzing the waveforms, it is possible to distinguish these two mechanisms, even though the amplitude for THz emission for both mechanisms depends on excitation polarization in the same fashion. As will be seen in Section 3.3.3.2, the two waveforms are related through a time derivative. With sample orientation and excitation polarization information in hand, the analysis of the waveform is guided by the putative mechanism of basis for THz pulse generation. That is, if one suspects that a current surge is the mechanism, either increasing or decreasing, he will then invoke a model that treats time-varying

1e-5 5e-6

M

0

y

–5e-6 –1e-5

0

90

180

270

360

z

x

Sample Rotation (Degrees)

Figure 3.17 (Left) THz emission from demagnetization as a function of sample orientation as it is rotated about z axis. (Right) Relevant coordinate system. A change in magnetization along the y axis produces a THz pulse polarized along the x axis. The THz emission is independent of excitation laser polarization.

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THz Amplitude (arb. units)

Terahertz Spectroscopy: Principles and Applications 1e-6 8e-7 6e-7 4e-7 2e-7 0

–2e-7

0

2

4 Time (ps)

6

8

(a)

0

1

2

3

4

5

6

7

8

6

7

8

Time (ps) (b)

0

1

2

3

4

5

Time (ps) (c)

Figure 3.18 Experimentally measured THz emission shown with solid line (a). The sample is a 40-nm thick in-plane magnetized Ni film. (b) The “basis set” of Gaussian functions whose amplitudes were then arbitrarily varied in a nonlinear least-squares loop to produce the change in magnetization shown with the thick line (c). One of the Gaussians was randomly chosen and shown with a thick line (b). The THz emission shown with the dashed line (a) is calculated by taking its first derivative and convoluting with the detector response function.

currents. Similarly, if a change in polarization or magnetization is suspected, then it is analyzed as such. It is also possible to extract the underlying change in current, polarization, or magnetization without assuming any model. For example, the solid line in Figure 3.18a is the emitted near-field THz pulse from a 40-nm thick magnetic Ni thin film. One generates (numerically) a series of Gaussians in the time domain as shown in Figure 3.18b, then uses a least-squares loop to vary their amplitudes, and convolute the underlying change in current, polarization, or magnetization with the

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103

detector response until agreement with the data is achieved. Figure 3.18c shows the underlying change in magnetization before convolution with the detector response function and before taking its first derivative. The dashed line in Figure 3.18a is the calculated THz pulse based on the time-dependent magnetization shown in Figure 3.18c (i.e., taking the first derivative of the magnetization shown in Figure 3.18c), and convoluting it with the detector response function. It might also be necessary to investigate both the first and second derivative of the underlying change if the source is not known.

3.3.3 Specific Examples In all cases, the goal is to determine Jx(t), Px(t), or Mx(t) from the measured THz waveform. Typically, a model is chosen to describe the underlying process, which is then convoluted it with an instrument response function. The parameters of the model are then varied using a nonlinear least squares fitting program until agreement between the calculated and measured waveforms is achieved, at which point one has obtained information about the underlying generation mechanism. 3.3.3.1 Photoconductive Switches Some of the earliest TES studies were to understand the mechanism behind THz generation from a photoconductive switch, the first type of source used for THzTDS. This required understanding the current surge because of the acceleration of photoexcited electrons by an applied bias field. That is, Jx(t) in Equation 3.26 is given by:

J x (t ) = -enf (t )v (t ),

(3.29)

where e is the magnitude of the electron charge, nf (t) is the time-dependent density of free (i.e., photogenerated) electrons, and v(t) is their time-dependent mean velocity. The time-dependent carrier density is related to the excitation pulsewidth as a generation mechanism, and any scattering, damping, or trapping that may be present. The time-dependent velocity depends on the effective mass of the electrons which depends on their location in the (nonparabolic) conduction band, as well as the local field, which results from the applied field as well as any screening effects. These effects can be treated either analytically,46–48 or numerically via a Monte Carlo approach.49 3.3.3.2 Shift Currents and Optical Rectification Although biased semiconductor photoconductive switches produce THz radiation because of a current surge, there exist other mechanisms as well. Perhaps most well known is optical rectification (the mechanism for THz generation in ZnTe emitters, which are popular sources of THz radiation for THz-TDS and TRTS experiments). Shift current and injection current are second-order mechanisms that do not require a bias field and can produce a pulse when the sample surface is normal to the direction of excitation.

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Sipe and coworkers have developed a detailed mechanistic treatment of secondorder responses in semiconductors by deriving an expression for the second-order susceptibility, χ(2) which is valid for below, above, and at bandgap conditions.45 The most salient results of their work are summarized below. For semiconductors, the second-order susceptibility may be written in the following form (in the frequency domain): χ( 2) (- W ;w , -w + W) = χ( 2)′ (- W; w , - w + W) +

σ ( 2) (- W; w , -w + W) iW

η( 2) (- W; w , -w + W) + (- i W) 2

(3.30)

The first term describes optical rectification, the second shift current, whereas the last term accounts for injection currents. W is a THz frequency, and W is an optical frequency. For zincblende materials such as GaAs, the injection current term vanishes because of symmetry. The current density has contributions due to rectification and shift currents:

J (t ) = J rect (t ) + J shift (t )

(3.31)

These current densities can be related to the applied electric field:

J rect (t ) = 2e o χ( 2)

∂ E (t )E * (t ) ∂t

J shift (t ) = 2e o σ ( 2)E (t )E * (t )

(3.32)

(3.33)

when detecting in the far field, where the signal appears as its first derivative with respect to time,41 the THz waveform should have the following profile with respect to the optical pulse:

rect (t ) = 2e χ ( 2 ) E THz o

∂2 E (t )E * (t ) ∂t 2

(3.34)

shift (t ) = 2e σ ( 2 ) ETHz o

∂ E (t ) E * (t ) ∂t

(3.35)

3.3.3.2.1 Laser Polarization Dependence of Second-Order Response The THz pulse generated from optical rectification or shift currents depends on the polarization state of the laser (linear, elliptical, circular) and the orientation of the sample. We employ the Jones matrix formalism to account for the various optics

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and sample orientation.50 In this formalism, the elements of the vectors and matrices below are complex quantities. The initial polarization of the optical beam is written as

 E x ,opt    Elab =  E y,opt   Ez,opt   

(3.36)

where x, y, and z are lab fixed coordinates. The z direction is the direction of propagation, and we can arbitrarily choose the x direction as horizontal and y vertical. Thus a horizontally, linearly polarized beam emerging from the laser is represented as

Elab

1    = 0  . 0   

(3.37)

When the beam passes through a waveplate, the polarization state of the emerging beam is expressed as:

 E x ′,opt   E x ,opt      Elab ′ =  E y′, opt  = W  E y, opt  = WElab E  E   z′, opt   z, opt 

(3.38)

where W is the product of the matrices:

W = R (-q) W0 R (q)

(3.39)

with:

e - i G / 2  W0 = e i φ  0  0 

0 e - iG / 2 0

0  0 1 

 cos q  R(q) =  - sin q  0 

sin q cos q 0

0  0. 1 

(3.40)

and

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(3.41)

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Terahertz Spectroscopy: Principles and Applications

W0 is the matrix describing the phase retardation placed on the optical beam with Γ equal to p for a half-wave plate or p/2 for a quarter-wave plate. R(θ) is a rotation matrix that transforms the incident optical beam onto the fast and slow axes of the wave plate, and θ is the angle of the optical beam with respect to the fast axes. The emerging beam is then transformed back into lab coordinates using R(-θ) and carries the polarization induced by the waveplate. The optical field is then projected onto the crystallographic axis of the sample:

Extal

 Ei    = Ej  = T   Ek 

 Ex′    ′  E y ′  = TElab    Ez ′ 

(3.42)

where T is the transformation matrix projecting the electric field in lab coordinates onto the i, j, k crystallographic axes of the GaAs <111> sample:    T =  

2 3 1

3

1

6 1 6

2

-

   1  3  1  3   1

0

1 2

(3.43)

For zincblende crystals such as GaAs, the induced polarization, or “current”, in the ith direction is proportional to the product of the optical fields polarized along the jth and kth direction: Pi ∝ E j E k* , where i, j, and k are mutually orthogonal Cartesian coordinates, and E* is the complex conjugate of E.51,52

0  P = Ek 0 

0 0

Ei

Ej   0 0 

 E i*   0  *   E j  = Ek  *  0 Ek  

0 0 Ei

Ej   * 0  Extal 0 

(3.44)

This time-dependent polarization leads to the observed THz emission, which is transformed back into lab coordinates:

ETHz

 E x ,THz    =  E y ,THz  = T -1P    E z ,THz 

(3.45)

where T–1 is the inverse of the transformation matrix T. The calculated signal is obtained by taking the real part of ETHz in Equation 3.45.

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Putting this together, we have:

ETHz = T-1 P TR ( - q)W 0 R(q) Elab .

(3.46)

On the other hand, if we were to rotate the sample instead of the waveplate, it would be

ETHz = R( - q) T -1 P TR (q)W 0 Elab

(3.47)

(with the optical axis of the waveplate along the polarization of the optical beam), or perhaps more simply, to eliminate the waveplate:

ETHz = R ( - q) T -1 P TR (q) Elab .

(3.48)

In summary, the Jones matrix formalism50 is a well-known and efficient way to describe and understand the effects of essentially any optical element as well as the samples themselves. 3.3.3.3 Intramolecular Charge Transfer in Orienting Field It is possible to generate THz emission from a sample of oriented dye molecules that undergo intramolecular charge transfer on photoexcitation. This provides a way to understand this process without relying on secondary effects such as a change in absorption or fluorescence properties. The signal is simply because of the change in polarization of the sample. It is possible to orient the molecules several ways. If a single crystal can be grown, and if it has the appropriate symmetry, then that will lead to a net orientation. Also, in some cases it is possible to make monolayers or multilayers of oriented molecules. In the work discussed here, however, an applied electric field is used to partially orient the molecules. Any TES experiment can be described as a nonlinear process. However, it is sometimes useful to think more intuitively about the underlying physical process. For the purpose of illustration, we will compare and contrast TES from oriented dye molecules in terms of the considerations of intramolecular charge transfer in partially oriented molecules versus a description based on the third-order nonlinear susceptibility. In both cases, the x axis is defined as the direction of the applied electric field. The angle of a molecular ground state dipole moment relative to the x axis is denoted q. The angle of the linearly polarized photoexcitation beam relative to the x axis is denoted z. The angle of the polarization axis of the THz detector relative to the x axis is denoted f. 3.3.3.3.1 Intuitive Description The interaction of the static electric field with a dipolar molecule in solution results in a fractional orientation of the molecules along the field direction, which provides the underlying physical basis for this method. This fractional orientation can be calculated by considering the interaction energy of a dipole with an applied field, μgE 0 in comparison to kBT, where kB is Boltzmann’s constant and T is the temperature

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in Kelvin. It is expressed as a function of angle, θ, between the applied field and molecular dipole moment by53,54

f (q) =

1 e -V / k BT ≈ -V / k BT ∫ e d q 2p 2p 0

 mE 0 cos q  1+ , k T   B

(3.49)

where the interaction potential, V, is given by V = –mE 0cosq. Terms resulting from polarizability anisotropy have been neglected. Given the inherent azimuthal symmetry, we work in polar coordinates. Thus interaction with the static field results in a linear combination of an isotropic distribution and one that varies as cosθ. The visible pulse introduces a cos2(θ – ζ) distribution of excited molecules because of projection of the transition dipole moment onto its polarization. An additional factor of cos(θ – φ) results from a projection of the emitted polarization onto an axis at angle φ. Thus the emitted amplitude at a given polarization angle φ varies as a function of visible polarization angle, ζ, as

E φW (z) ∝

∫

2p

0

(

)

cos(q) cos2 (q - z) cos (q - φ) d q = A z, φ ,

(3.50)

where the integral evaluates to

A(z, φ) =

p [cos φ (3 cos2 z + sin 2 z) + 2 sin φ cos z sin z]. 4

(3.51)

3.3.3.3.2 Third-Order Nonlinear Susceptibility Description It is also possible to treat this process in terms of a third-order nonlinear susceptibility:55

3) w w 0 E iW ∝ χ(ijkl E j Ek El ,

(3.52)

where E W is the THz electric field (which arises from the time-dependent, third-order polarization, Pi(3)), E w is the optical electric field, E 0 is the static electric field, and i, j, k, and l are one of the( 3x, y, z Cartesian coordinates. The third-order nonlinear ) susceptibility is denoted χijkl . As defined previously, the applied field (E 0) is in the x direction (therefore, l = x). Because z is the direction of propagation, i, j, and k must be x or y. For isotropic media there are only three independent tensor elements and they are related by52

3) = χ ( 3) + χ ( 3) + χ ( 3) . χ(xxxx xyyx yxyx yyxx

(3.53)

3) = χ ( 3) = χ ( 3) Also, because Klienman symmetry52 holds in this case, χ(xyyx yyxx yxyx ( 3 ) ( 3 ) and χ xxxx = 3χ xyyx . From these relationships, we can determine the amplitude and

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3) . direction of the induced third‑order polarization, P(3), in terms of χ(xyyx The x and y components of P(3) are

3 ) E w E - w E 0 + χ ( 3 ) E w E - w E 0 = χ ( 3 ) I E 0 [3 cos 2 (z ) + sin 2 (z)], Px( 3) (z) = χ(xxx x x x xyyx y y x xyyx w x

(3.54)

and 3 ) E w E - w E 0 + χ ( 3 ) E w E - w E 0 = χ ( 3 ) I E 0 [2 cos(z )sin (z)] Py( 3) (z) = χ(yyxx y x x yxyx x y x xyyx w x

(3.55)

where I wjk is the intensity of the visible pulse with polarization angle ζ relative to the x-direction. The visible polarization vector, eˆ jk , is given by eˆ jk = eˆx cos z + eˆy sin z. The emitted polarization is at an angle φ with respect to the static field and is obtained from the x and y components P (φ3)= Px( 3) xˆ + Py( 3) yˆ,

(3.56)

where φ = arctan[2 cos(z)sin(z)/(3 cos 2 (z) + sin 2 (z))]. It can easily be verified that Equations 3.54 through 3.56 are equivalent to the angular dependence of Equation 3.51, and the two formalisms are complementary. The x and y components of the emitted field are shown in Figure 3.16 and compared with Equations 3.54 and 3.55 (shown with solid lines). 3.3.3.3.3 Propagation Effects and Solvent Response If propagation effects of the fields through the solvent can be neglected, then the emitted amplitude in the near-field regime is equal to the second derivative of the time-dependent polarization (see Equation 3.25). However, nonnegligible propagation effects such as group velocity mismatch between the visible and generated THz pulses, absorption of the visible pulse by the solution, and dispersion of the generated pulse by the solvent must be accounted for to correctly obtain the charge transfer dynamics. We do so by numerically solving Maxwell’s equations in the time domain56 coupled with the phenomenologic model below for the time-dependent polarization. We then perform a nonlinear least-squares fit of the model to the data to obtain the charge transfer dynamics. It should be noted that an alternative approach is described in Section 3.3.4 and in previous studies.57 Consider a delta function excitation pulse of a single dye molecule at z = 0 and t′ = 0. The pulse induces an electron transfer with rate kET, and subsequent back electron transfer with rate kBET, and the change in polarization is given by Dp (t ) =

7525_C003.indd 109

k ET [exp(- k ET t ] - exp(- k BET t )] D ′m, ( k BET - k ET )

(3.57)

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Terahertz Spectroscopy: Principles and Applications

where ∆′µ is the change in dipole moment along the ground state dipole. We average the contributions from individual molecules over an anisotropic distribution that is created by the visible and static fields. Because reorientation of the excited molecules occurs on a longer timescale than these measurements, the orientational average is given by D ′m = m ′e

∫

2p 0

f (q) cos 2 (q - z) cos (q - φ ) dq

E0 = A (zz, φ ) (m ′e - m g ) m g , 8kBT

(3.58)

where µg is the ground state dipole moment and m e′ is the projection of the excited state dipole moment along the ground state dipole moment. Thus, the change in polarization is obtained by replacing D′m with 〈D′m〉 in Equation 3.57. The solvent molecules also affect the measured change in polarization, so we must account for their reaction field. The electrostatic potential of a dipole in solution is modified from its value in vacuum by the reaction potential of the polarized surrounding dielectric medium. On photoexcitation, the electrostatic potential changes abruptly and therefore a repolarization of the solvent occurs. If the change in dipole is fast compared with the solvent motions, then the measured change in polarization will reflect this solvent repolarization. We describe the solvent response as an impulse response function to a delta function change in the solute charge configuration. We treat the solvent response as a single exponential, and the combined solvent–solute polarization ps response to an impulse excitation pulse is given by

ps (t ) =

t

∫ dt ′ Dp (t ′) Φ (t - t ′),

(3.59)

0

where Φ(t) = exp(-kst) represents the response of the solvent with rate constant ks. If ks << kET the rise of the signal will be governed by the polarization of the solvent. The FDTD method56 is used to numerically solve Maxwell’s equations with these constraints. The advantage of using the FDTD method is that the slowly varying envelope approximation and rotating wave approximation are not assumed. Although these approximations are valid for the optical pulse, they are not applicable for the generated EM pulse. Furthermore, the FDTD method completely accounts for the generation term and the dispersive term, and group velocity mismatch is automatically included in the simulation. The FDTD calculation provides the generated field in the near‑field regime, and a near-field to far-field transformation is performed by taking the first derivative of the calculated field. The ZnTe detector further distorts the measured signal, and a numerical propagation through the detector is also included in the simulations.9,29 A nonlinear least-squares fit is performed to extract the best fit values of the forward and back electron transfer rates. The results of the fits for Betaine-30 (Reichardt’s dye) in chloroform and DMANS in toluene are shown in Figure 3.19 and the rate constants are provided in Table 3.1.

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EΩ(t) (arb. units)

2 DMANS

1 0

Betaine-30

–1 –2 1

2

3 Time (ps)

4

5

DMANS

P(t)

1

Betaine-30 0 0

2

4

6

8

Time (ps)

Figure 3.19 Results of the nonlinear least-squares fit of the charge transfer model (including solvent repolarization) for Betaine-30 in chloroform and DMANS in toluene. The data are shown with solid lines, and the results of the fit with dashed lines. The lower panel displays the underlying time-dependent polarization (Figure reprinted with permission from Beard, M.C. et al., J. Phys. Chem. A 106, 878, 2002. Copyright 2002 by the American Chemical Society.)

Table 3.1 Optimized Parameters Resulting from Nonlinear Least-Squares Fits for Betaine-30 in Chloroform and DMANS in Toluene ∆w (fs)

ks

τs (fs)

(ps ) -1

kET

kBET

(ps )

(ps )

-1

τBET (ps)

-1

Betaine-30

150*

2.86

350

>100*

0.53

1.9

DMANS

150*

1.37

730

>100*

<0.01*

>100*

Note: The corresponding time constants are included for convenience. The Gaussian width is given by ∆w. The asterisk denotes that the values were held fixed during the fit.

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3.3.3.4 Demagnetization Dynamics Finally, it has been shown recently that a sample with magnetization that changes on a picosecond to subpicosecond timescale (as achieved by heating a magnetic thin film with an ultrashort, ~100-fs, laser pulse) will emit THz radiation.37,38 Although initially surprising, it is completely understood through Maxwell’s equations (see Equation 3.28). This implies that the measurement of transient electric field emitted by the sample is related to the second time derivative of the magnetization. Figure 3.20a shows the THz emission from a 40-nm thick, in-plane magnetized Ni film on heating with an ultrafast laser pulse (~100-fs duration, 800-nm wavelength). The THz emission was calculated from a model of the temporal variation of the magnetization (Figure 3.20b) similar to the one previously used in optical pump–probe experiments of Kerr rotation in the same type of sample.58

DM (t ) = {-Θ (t )[k1 (1 - exp ( -t / τth )) exp ( -t / τ ep ) + k2 (1 - exp)( -t / τ ep ))]} ⊗ G (t )

(3.60)

where t is time, ∆M(t) is the time-dependent change in magnetization, Θ(t) is the Heaviside step function centered at t = 0, k1 and k2 are constants depicting the relative

THz Amplitude or Magnetization (arb. units)

1.0

0.5 (a) 0.0

–0.5 (b) –1.0

0

2

4

6

8

Time (ps)

Figure 3.20 The solid line (a) is the near-field THz pulse generated upon ultrafast laser heating of 40-nm thick in-plane magnetized Ni film (in “transmission” mode). (b) The timedependent magnetization as described by Equation 3.60. The dashed line in part (a) is the calculated emission obtained by convoluting the first derivative of the time-dependent magnetization of part (b) with the ZnTe <110> detector response function.

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amount of transient response versus long-term values. In the present context, tth and tep may be viewed as phenomenologic constants describing the electron thermalization time and electron–phonon coupling time.58–60 ⊗G(t) represents convolution with a Gaussian instrument response function. The best fit of the second derivative of the magnetization after convolution with the detector response function is plotted as the dashed line in Figure 3.20a. Here we find that tth = 0.37 (0.01) ps, tep = 1.30 (0.04) ps, k1/k2 = 2.0 (held fixed), and the Gaussian instrument response function full width at half maximum is 640. (6.) fs.

3.3.4 General Formalism Recently, Wynne and Carey have developed an integrated description of THz generation from optical rectification, charge transfer, and current surge.61 Each of these three processes is described using an analytical formula in the frequency domain followed by a numerical Fourier transform into the time domain. They account for absorption and dispersion in the detector crystal, as well as the effect of propagation into the far field. Although their treatment avoids explicit FDTD calculations, it cannot describe a situation wherein the response of a sample changes on a timescale fast compared to the inverse of the THz spectral bandwidth. However, there are many scenarios where it is perfectly valid. They provide a recipe. First the THz signal in the near field is calculated and then converted to the far field.

f eikz ( n - nVIS ) - 1 ikzn VIS E THz, NF ( z, w ) = e 2 n nVIS - n E THz ( z, w ) = -iw E THz, NF ( z, w ).

(3.61) (3.62)

In the time domain, the far-field signal is simply the time derivative of the near-field signal,41 which in the frequency domain corresponds to multiplying it by –iw. Here z is the propagation distance, w is the THz frequency, ñ is the complex-valued refractive index of the sample, nVIS is the group refractive index of the material at the excitation wavelength, determined through vgroup = c/nVIS, where c is the speed of light in vacuum, and k is the free-space wavevector of the THz field, k = w/c. Response functions, f(w), are provided for four different scenarios: optical rectification, direct charge transfer, indirect charge transfer, and the current surge model. Optical rectification:

f (w ) = χ( 2) (w ; W, W - w ) Ipump (w ),

(3.63)

where χ(2) is the second-order nonlinear response of the medium, and W represents the range of frequencies contained in the ultrashort pump pulse. Note: Ipump (w ) is the Fourier transform of the intensity envelope of the excitation pulse rather than its spectrum.

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Direct charge transfer: 3e (w ) N Dm f (w ) = I (w ) 2e (w ) + 1 e 0 ( g - i w ) pump

(3.64)

where e (w ) is the continuum dielectric of the sample, N is the number density of photoexcited molecules, ∆m is the change in molecular dipole moment on photoexcitation, e0 is the permittivity of free space. In the case of direct charge transfer, g is the rate of decay back to the ground state with the original dipole moment. Indirect charge transfer: f (w ) =

3e (w ) N Dm  1 1   I pump (w ). 2 e (w ) + 1 e 0  G - iw g - iw 

(3.65)

In the case of indirect charge transfer, g is the rate of decay back to an intermediate state, and Γ is the rate of decay back to the ground state with the original dipole moment. Current surge model:

3e (w ) 1 -e 2Enf 1  1 1   I f (w ) = (w ) 2e (w ) + 1 e 0 g s m* - i w  G - i w g s - i w  pump

(3.66)

where E is a DC external electric field which accelerates the electrons, gs is the momentum relaxation rate, m* is the carrier effective mass, nf is the carrier density, and Γ is the carrier recombination rate (it is assumed that Γ << gs). Note that all these response functions are of the form f (w ) = r (w )Ipump (w ),where r (w ) is a property of the medium in question. The process of electrooptic sampling affects the measured signal, but in a well understood manner.29,62,63 Wynne and Carey account for it and provide an expression for the measured signal:

∫

DI EOS ( τ ) ∝ dw e - iwτ Ipump (w ) Igate (w )[iw ]z gen (w ) r (w )z det (w ),

where

eikL1 ,( n - nVIS ) - 1 , z gen (w ) = 2 n ( nVIS - n )

(3.67)

(3.68)

and

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e ikL2 ,( n - nVIS ) - 1 z det (w ) = . ik (n - nVIS )

(3.69)

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Time-Resolved Terahertz Spectroscopy

115

3.4 Conclusions TRTS and TES can provide information not obtainable from THz-TDS studies (and certainly not from conventional far-IR studies). Both of these methods are sensitive to subpicosecond dynamics manifested in the far-IR region of the spectrum. TRTS is an excellent way to probe the evolution of a material’s complex-valued, time-dependent, far-IR spectrum as a function of time after a perturbation such as a photoexcitation pulse. Section 3.2 describes many considerations that must be taken into account when performing these experiments. TES is complementary in the sense that a full spectrum as a function of time is not obtained. However, that the signal is generated by the sample directly, rather than requiring a THz probe pulse has distinct advantages in some cases. It is evident from Sections 3.3.2 through 3.3.4 that there are many complementary ways of extracting information from a TES experiment.

References 1. Schmuttenmaer, C.A., Exploring dynamics in the far-infrared with terahertz spectroscopy, Chem. Rev., 104, 1759, (2004). 2. Schall, M. and Jepsen, P.U., Photoexcited GaAs surfaces studied by transient terahertz time- domain spectroscopy, Opt. Lett., 25, 13, 2000. 3. Ralph, S.E., et al., Subpicosecond photoconductivity of In0.53Ga0.47As: Intervalley scattering rates observed via THz spectroscopy, Phys. Rev. B, 54, 5568, 1996. 4. Flanders, B.N., Arnett, D.C., and Scherer, N.F., Optical pump-terahertz probe spectroscopy utilizing a cavity-dumped oscillator-driven terahertz spectrometer, IEEE J. Sel. Top. Quantum Electron., 4, 353, 1998. 5. Prabhu, S.S., et al., Carrier dynamics of low-temperature-grown GaAs observed via THz spectroscopy, Appl. Phys. Lett., 70, 2419, 1997. 6. Groeneveld, R.H.M. and Grischkowsky, D., Picosecond time-resolved far-infrared experiments on carriers and excitons in GaAs-AlGaAs multiple-quantum wells, J. Opt. Soc. Am. B-Opt. Phys., 11, 2502, 1994. 7. Saeta, P.N., et al., Intervalley scattering in GaAs and InP probed by pulsed far- infrared transmission spectroscopy, Appl. Phys. Lett., 60, 1477, 1992. 8. Greene, B.I., et al., Picosecond pump and probe spectroscopy utilizing freely propagating terahertz radiation, Opt. Lett., 16, 48, 1991. 9. Beard, M.C., Turner, G.M., and Schmuttenmaer, C.A., Transient photoconductivity in GaAs as measured by time-resolved terahertz spectroscopy, Phys. Rev. B, 62, 15764, 2000. 10. Beard, M.C., Turner, G.M., and Schmuttenmaer, C.A., Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy, J. Appl. Phys., 90, 5915, 2001. 11. Beard, M.C., et al., Electronic coupling in InP nanoparticle arrays, Nano Lett., 3, 1695, 2003. 12. Beard, M.C., Turner, G.M., and Schmuttenmaer, C.A., Size-dependent photoconductivity in CdSe nanoparticles as measured by time-resolved terahertz spectroscopy, Nano Lett., 2, 983, 2002. 13. Hendry, E., et al., Direct observation of electron-to-hole energy transfer in CdSe quantum dots, Phys. Rev. Lett., 96, 057408, 2006. 14. Turner, G.M., Beard, M.C., and Schmuttenmaer, C.A., Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide, J. Phys. Chem. B, 106, 11716, 2002.

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15. Hendry, E., et al., Local field effects on electron transport in nanostructured TiO2 revealed by terahertz spectroscopy, Nano Lett., 6, 755, 2006. 16. Baxter, J.B. and Schmuttenmaer, C.A., Conductivity of ZnO nanowires, nanoparticles, and thin films using time-resolved terahertz spectroscopy, J. Phys. Chem. B, 110, 25229, 200). 17. Hendry, E., Charge dynamics in novel semiconductors. Amsterdam: University of Amsterdam, 2005. 18. Perfetti, L., et al., Time-resolved THz spectroscopy: ultrafast charge carrier dynamics in low-dimensional solids, Abs. Ultrafast Phenomena, 2006. 19. Hendry, E., et al., Ultrafast charge generation in a semiconducting polymer studied with THz emission spectroscopy, Phys. Rev. B, 70, 033202, 2004. 20. Hendry, E., et al., Efficiency of exciton and charge carrier photogeneration in a semiconducting polymer, Phys. Rev. Lett., 92, 2004. 21. Hendry, E., et al., Interchain effects in the ultrafast photophysics of a semiconducting polymer: THz time-domain spectroscopy of thin films and isolated chains in solution, Phys. Rev. B, 71, 2005. 22. Beard, M.C., Turner, G.M., and Schmuttenmaer, C.A., Supplementary information for: Sub-picosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved THz spectroscopy, J. Appl. Phys. EPAPS, 90, E, 2001. 23. Knoesel, E., et al., Charge transport and carrier dynamics in liquids probed by THz time-domain spectroscopy, Phys. Rev. Lett., 86, 340, 200). 24. Beard, M.C. and Schmuttenmaer, C.A., Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments, J. Chem. Phys., 114, 2903, 2001. 25. Brekhovskikh, L.M., Waves in layered media. London: Academic Press, Inc., 1960. 26. Fowles, G.R., Introduction to modern optics. New York: Dover, 1989. 27. Kindt, J.T. and Schmuttenmaer, C.A., Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy, J. Chem. Phys., 110, 8589, 1999. 28. Press, W.H., et al., Numerical recipes in Fortran, 2nd ed., New York: Cambridge University Press, 1986. 29. Bakker, H.J., et al., Distortion of terahertz pulses in electro-optic sampling, J. Opt. Soc. Am. B-Opt. Phys., 15, 1795, 1998. 30. Bailey, D.W., Stanton, C.J., and Hess, K., Numerical-studies of femtosecond carrier dynamics in GaAs, Phys. Rev. B, 42, 3423, 1990. 31. Stanton, C.J. and Bailey, D.W., Rate-equations for the study of femtosecond intervalley scattering in compound semiconductors, Phys. Rev. B, 45, 8369, 1992. 32. Nuss, M.C., Auston, D.H., and Capasso, F., Direct subpicosecond measurement of carrier mobility of photoexcited electrons in gallium-arsenide, Phys. Rev. Lett., 58, 2355, 1987. 33. Choy, T.C., Effective medium theory: principles and applications. Oxford: Clarendon Press, International Series of Monographs on Physics, 1999. 34. Black, M.R., et al., Infrared absorption in bismuth nanowires resulting from quantum confinement, Phys. Rev. B, 65, 195417, 2002. 35. Bruggeman, D.A.G., Calculation of various physics constants in heterogenous substances I: Dielectricity constants and conductivity of mixed bodies from isotropic substances, Ann. Phys., 24, 636, 1935. 36. Maxwell-Garnett, J.C., Colours in metal glasses and in metallic films, Philos. Trans. R. Soc. London, Ser. A, 203, 385, 1904. 37. Beaurepaire, E., et al., Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses, Appl. Phys. Lett., 84, 3465, 2004.

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38. Hilton, D.J., et al., Terahertz emission via ultrashort-pulse excitation of magnetic metal films, Opt. Lett., 29, 1805, 2004. 39. Welsh, G.H., Hunt, N.T., and Wynne, K., Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating, Phys. Rev. Lett., 98, 2007. 40. Han, P.Y. and Zhang, X.C., Coherent, broadband midinfrared terahertz beam sensors, Appl. Phys. Lett., 73, 3049, 1998. 41. Kaplan, A.E., Diffraction-induced transformation of near-cycle and subcycle pulses, J. Opt. Soc. Am. B-Opt. Phys., 15, 951, 1998. 42. Han, P.Y., Huang, X.G., and Zhang, X.C., Direct characterization of terahertz radiation from the dynamics of the semiconductor surface field, Appl. Phys. Lett., 77, 2864, 2000. 43. Johnston, M.B., et al., Simulation of terahertz generation at semiconductor surfaces, Phys. Rev. B, 65, 165301, 2002. 44. Pedersen, J.E., et al., Terahertz pulses from semiconductor-air interfaces, Appl. Phys. Lett., 61, 1372, 1992. 45. Sipe, J.E. and Shkrebtii, A.I., Second-order optical response in semiconductors, Phys. Rev. B, 61, 5337, 2000. 46. Jepsen, P.U., Jacobsen, R.H., and Keiding, S.R., Generation and detection of terahertz pulses from biased semiconductor antennas, J. Opt. Soc. Am. B-Opt. Phys., 13, 2424, 1996. 47. Leitenstorfer, A., et al., Femtosecond charge transport in polar semiconductors, Phys. Rev. Lett., 82, 5140, 1999. 48. Leitenstorfer, A., et al., Femtosecond high-field transport in compound semiconductors, Phys. Rev. B, 61, 16642, 2000. 49. Castro-Camus, E., et al., Polarization-sensitive terahertz detection by multicontact photoconductive receivers, Appl. Phys. Lett., 86, 2005. 50. Yariv, Y. and Yeh, P., Optical Waves in crystals: propagation and control of laser radiation. New York: Wiley, 1984. 51. Rice, A., et al., Terahertz optical rectification from (110) zincblende crystals, Appl. Phys. Lett., 64, 1324, 1994. 52. Butcher, P.N. and Cotter, D., The elements of nonlinear optics. Cambridge: Cambridge University Press, 1990. 53. Smirnov, S.N. and Braun, C.L., Advances in the transient dc photocurrent technique for excited state dipole moment measurements, Rev. Sci. Instrum., 69, 2875, 1998. 54. Atkins, P., Physical chemistry. ed. 6. New York: W. H. Freeman, 1997. 55. Beard, M.C., Turner, G.M., and Schmuttenmaer, C.A., Measuring intramolecular charge transfer via coherent generation of THz radiation, J. Phys. Chem. A, 106, 878, 2002. 56. Beard, M.C., Schmuttenmaer, C.A., Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments, J. Chem. Phys., 114, 2903, 2001. 57. Cote, D., Sipe, J.E., and van Driel, H.M., Simple method for calculating the propagation of terahertz radiation in experimental geometries, J. Opt. Soc. Am. B-Opt. Phys., 20, 1374, 2003. 58. Guidoni, L., Beaurepaire, E., and Bigot, J.Y., Magneto-optics in the ultrafast regime: thermalization of spin populations in ferromagnetic films, Phys. Rev. Lett., 89, (2002). 59. Koopmans, B., et al., Ultrafast magneto-optics in nickel: Magnetism or optics?, Phys. Rev. Lett., 85, 844, 2000. 60. Beaurepaire, E., et al., Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett., 76, 4250, 1996.

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61. Wynne, K. and Carey, J.J., An integrated description of terahertz generation through optical rectification, charge transfer, and current surge, Opt. Commun., 256, 400, 2005. 62. Gallot, G. and Grischkowsky, D., Electro-optic detection of terahertz radiation, J. Opt. Soc. Am. B-Opt. Phys., 16, 1204, 1999. 63. Nahata, A., et al., Coherent detection of freely propagating terahertz radiation by electro-optic sampling, Appl. Phys. Lett., 68, 150, 1996.

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Section II Applications in Physics and Materials Science

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Time-Resolved Terahertz Studies of Carrier Dynamics in Semiconductors, Superconductors, and Strongly Correlated Electron Materials Robert A. Kaindl

Lawrence Berkeley National Laboratory

Richard D. Averitt Boston University

Contents 4.1 4.2

4.3

4.4

Introduction.................................................................................................. 120 Bulk and Nanostructured Semiconductors.................................................. 121 4.2.1 Overview........................................................................................... 121 4.2.2 Free Carrier Dynamics in Bulk Semiconductors............................. 122 4.2.3 Intraexcitonic Spectroscopy.............................................................. 128 4.2.4 Intersubband Transitions................................................................... 134 Superconductors........................................................................................... 136 4.3.1 Overview........................................................................................... 136 4.3.2 Far-IR Spectroscopy of Superconductors......................................... 137 4.3.3 Quasiparticle Dynamics in Conventional Superconductors............. 142 4.3.4 Quasiparticle Dynamics in High-TC Superconductors..................... 145 Half-Metallic Metals: Manganites and Pyrochlores.................................... 150 4.4.1 Overview........................................................................................... 150 4.4.2 Optical Conductivity and Spectral Weight Transfer......................... 152

Written while at Los Alamos National Laboratory.

119

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4.4.3 Dynamic Spectral Weight Transfer in Manganites.......................... 153 4.4.4 Carrier Stabilization in Tl2Mn2O7 through Spatial Inhomogeneity.................................................................................. 155 4.5 Summary and Outlook................................................................................. 158 Acknowledgments................................................................................................... 160 References............................................................................................................... 160

4.1 Introduction Perhaps the most important aspect of contemporary condensed matter physics involves understanding strong Coulomb interactions among the large numbers of electrons in a solid. Electronic correlations lead to the emergence of new system properties, such as metal–insulator transitions, superconductivity, magnetoresistance, Bose-Einstein condensation, the formation of excitonic gases, or the integer and fractional quantum Hall effects. The discovery of high-TC superconductivity in particular was a watershed event, leading to dramatic experimental and theoretical advances in the field of correlated-electron systems.1–10 Such materials often exhibit competition among the charge, lattice, spin, and orbital degrees of freedom, whose cause–effect relationships are difficult to ascertain. Experimental insight into the properties of solids is traditionally obtained by time-averaged probes, which measure, for example, linear optical spectra, electrical conduction properties, or the occupied band structure in thermal equilibrium. Many novel physical properties arise from excitations out of the ground state into energetically higher states by thermal, optical, or electrical means. This leads to fundamental interactions between the system’s constituents, such as electron–phonon and electron–electron interactions, which occur on ultrafast timescales. Although these interactions underlie the physical properties of solids, they are often only indirectly inferred from time-averaged measurements. Time-resolved spectroscopy, consequently, is playing an ever increasing role to provide insight into light–matter interaction, microscopic processes, or cause–effect relationships that determine the physics of complex materials. Experiments using visible and near-infrared (IR) femtosecond pulses have been extensively employed (e.g., to follow relaxation and dephasing processes in metals and semiconductors).11,12 However, many basic excitations in strongly correlated electron systems and nanoscale materials occur at lower energies. The terahertz (THz) regime is particularly rich in such fundamental resonances. This includes ubiquitous lattice vibrations and low-energy collective oscillations of conduction charges. In nanoscale materials, band structure quantization also yields novel IR and THz transitions, including intersubband absorption in quantum wells. The formation of excitons in turn leads to lowenergy excitations analogous to interlevel transitions in atoms. In transition-metal oxides, fundamental excitation gaps arise from charge pairing into superconducting condensates and other correlated states. This motivates the use of ultrafast THz spectroscopy as a powerful tool to study light–matter interactions and microscopic processes in nanoscale and correlated-electron materials. A distinct advantage of coherent THz pulses is that the amplitude and phase of the electric field can be measured directly, because the THz fields are coherent with

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the femtosecond pulses from which they are generated. Using THz time domain spectroscopy (THz-TDS), both the real and imaginary parts of the response functions, such as the dielectric function ε(w) = ε1(w) + iε2(w), are obtained directly without the need for Kramers-Kronig transforms.13 The THz response can also be expressed in terms of absorption a(w) and refractive index n(w), or as the optical conductivity s(w) ≡ s1(w) + is2(w) = iwε0[1-ε(w)]. The conductivity s(w) describes the current response J(w) = s(w)E(w) of a many-body system to an electric field, an ideal tool to study conducting systems. A second important advantage is the ultrafast time resolution that results from the short temporal duration of the THz time domain sources. In particular, optical pump THz probe spectroscopy enables a delicate probe of the transient THz conductivity after optical photoexcitation. These experiments can provide insight into quasiparticle interactions, phase transitions, or nonequilibrium dynamics. In this chapter, we will provide many such examples. Because THz spectroscopy of solids is a quickly expanding field of research, we must define a limited scope to be covered in this chapter. We will review studies of semiconductors, superconductors, and strongly correlated electron systems using few-cycle IR pulses that are accessible to direct field-resolved detection. This entails a photon energy range from below 4 meV (≈1 THz) to about 100 meV. We specifically omit measurements with “conventional” time-averaged techniques such as Fourier transform IR spectroscopy,1,14 and we will skip experiments that are covered elsewhere in this book, such as THz studies of semiconductor nanocrystals or THz sideband generation on intersubband transitions. In Section 4.2, we discuss quasiparticle dynamics in bulk semiconductors and nanostructured materials such as gallium arsenide (GaAs) quantum wells. This includes the free carrier response, the formation of quasiparticles, intraexcitonic spectroscopy, and the nonlinear response of intersubband transitions. In Section 4.3, we consider the THz response and dynamics in conventional superconductors, and in the high-TC superconductors YBCO and Bi-2212. As we will show, with optical pump THz probe spectroscopy, it is possible to simultaneously monitor quasiparticle and superconducting condensate fractions with subpicosecond temporal resolution. Dynamics in half-metallic transition metal oxides La0.7Ca0.3MnO3 and the pyrochlore Tl2Mn2O7 are presented in Section 4.4. Colossal magnetoresistance is observed in both materials, and time-resolved THz spectroscopy provides insight into the relevance of the spin degree of freedom for charge transport and carrier lifetimes. Finally, in Section 4.5, we provide a summary and offer suggestions for future experiments on nanoscale and correlated-electron materials using time-resolved THz spectroscopy.

4.2 Bulk and Nanostructured Semiconductors 4.2.1 Overview In this section, we review low energy excitations and carrier dynamics in bulk and nanostructured semiconductors as determined via coherent THz spectroscopy. The revolutionary impact of semiconductors on modern technology goes hand in hand with an extremely high atomic-scale precision in their manufacture and with an

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immensely detailed knowledge of their physical properties that is presently unattainable for the strongly correlated electron systems discussed in later sections. This makes semiconductors an ideal testing ground for novel physical concepts and experiments and for tailored optoelectronic devices. Femtosecond laser pulses in the visible and near-IR have been extensively used to study photoexcited carriers in semiconductors. Such pulses are resonant to the typically 1–2 eV band gaps. Experiments have progressed from pump–probe and luminescence to modern multidimensional coherent spectroscopies which all provide deep insight into relaxation timescales, inter-band coherences, or many-body correlations.11,12,15 As we will show in this section, semiconductors provide equally interesting and fundamental excitations at energies far below their valence-to-conduction band gaps. Here, time domain THz and optical pump THz probe studies are powerful techniques to discern excitations that remain inaccessible to visible light and to probe conduction processes at frequencies comparable to relaxation processes in these materials.

4.2.2 Free Carrier Dynamics in Bulk Semiconductors The simplest description of free carriers in response to an applied electric field in a doped or photoexcited semiconductor (or other conducting material) is given by the Drude model. In this model, carriers elastically scatter with a rate τ -1. The complex Drude conductivity is given as

s(w ) =

s DC 1 - iwτ

(4.1)

where the DC conductivity sDC is given by

s DC ≡ ne 2 τ/m = ne m = ε 0 w 2p τ.

(4.2)

In this equation, n is the carrier density, t the collision time, m* the effective mass, µ the mobility, ε0 the free space permittivity and wp the bare plasma frequency. While the Drude response can be derived from very simple considerations, it is important to emphasize that its validity is much broader. In fact, the Drude response can be derived from the semi-classical Boltzmann transport equation and even more rigorously from Kubo-Greenwood formalism.16 For example, using a Boltzmann approach, it is easy to show that s DC =

e2 〈 l F 〉 SF 12 p3 

(4.3)

where is the mean free path averaged over the Fermi surface, and SF is the area of the Fermi surface. For a spherical Fermi surface this is equivalent to the Drude result highlighting that the relevant electrons are those at the Fermi surface with velocity vF. The Drude model serves as a useful starting point for understanding the properties and time evolution of many materials, as we will see time and again throughout this chapter. In superconductors, for instance, the spectral weight associated with a

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normal state Drude response downshifts into a zero-frequency d function associated with the superconducting condensate upon cooling below the transition temperature TC. In manganites, the Drude spectral weight is transferred to higher energies as phonon and spin fluctuations transform the coherent free carrier response into a hopping-like polaronic response. In this sense, changes in the Drude response due to thermal or nonequilibrium perturbations serve as a sensitive probe of interactions or the onset of correlations. Deviations from the Drude model are, of course, often observed. They have been treated via a frequency-dependent scattering rate17 or a distribution of scattering times,18 but the Drude model is an essential starting point even in these cases. In the literature (and throughout this chapter) the Drude response is plotted in various ways. Figure 4.1 shows the calculated Drude response as a function of reduced frequency w/ws where ws is the screened plasma frequency (ws = wp /√e∞) for sDC = 4 × 104 W-1 cm-1, t = 200 fs, ∇wp = 0.1 eV (or wp /2p = 24.2 THz), and e∞ = 4. The complex conductivity is plotted in Figure 4.1a, with the real part s1(w) shown as the solid line and the imaginary part s2(w) by the dashed line. Note that s2(w) = s1(0)/2 at 1/t, which provides a direct intuitive measure of the scattering rate. Figure 4.1b shows the corresponding reflectivity R, which drops dramatically at ws where the electrons no

1

3

Reflectivity

σ/104(Ω cm)−1

4

2 1 0

0.8 0.6 0.4 0.2

0

0

0.1 0.2 0.3 0.4 Frequency (ω/ωs)

0

(a)

0.5 1 1.5 Frequency (ω/ωs)

2

0.5 1 1.5 Frequency (ω/ωs)

2

(b)

4 −lm(1/ε), Re(1/ε)

εreal

0 −10 −20 −30

2 0 −2

0.5

1 Frequency (ω/ωs) (c)

1.5

0

(d)

Figure 4.1 Example of the Drude response plotted in terms of (a) the real part (solid line) and imaginary part (dashed) of the complex conductivity σ(ω), (b) the reflectivity R(ω), (c) the real part ε1(ω) of the permittivity, and (d) the loss function shown as –Im(1/ε) (solid line) and Re(1/ε) (dashed). Frequencies are scaled to the screened plasma frequency ωs = ωp /√ε∞.

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longer respond to the applied field. At higher frequencies, R rises again to the value (1 - √e∞)2/(1 + √e∞)2. Quite often, the real part of e(w) is plotted as in Figure 4.1(c), where it is negative below ws increasing to e∞ at higher frequencies. Finally, the Drude response is sometimes displayed as in Figure 4.1(d), where the loss function -Im(1/e(w)) peaks at the screened plasma frequency ws with a line width 1/t. As an example of a Drude-like response, Figure 4.2 shows the results of THz-TDS measurements on lightly doped silicon19, with carrier density n ≈ 1015 cm-3. Figure 4.2a displays the real conductivity σ1(ω) for n-type and p-type samples with the corresponding imaginary conductivity σ2(ω) displayed in (b). This data highlights some specific advantages of THz-TDS. As mentioned earlier, the time domain THz measurement provides the real and imaginary parts of σ(ω) directly. Moreover, the signal-to-noise in THz-TDS is sufficient to test various conductivity models. For

6 Re Conductivity (1/Ω cm)

n-type, 0.21 Ω cm Si p-type, 0.17 Ω cm Si

(a)

5

4-point Probe 4 3 2 1 0

lm Conductivity (1/Ω cm)

3.0

(b)

2.5 2.0 1.5

Drude Cole-Davidson Lattice Scattering Impurity Scattering Lattice - Impurity Scattering

1.0 0.5 0

0

0.5

1

1.5

2

2.5

Frequency (THz)

Figure 4.2 Terahertz conductivity of n-type and p-type doped silicon, showing (a) the real part s1(w) and (b) the imaginary part s2(w). Lines represent fits to various Drude-like conductivity models as indicated. (Reprinted with permission. Copyright 1997 by the American Physical Society.)

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both the n-type and p-type samples, the response cannot be fully explained with the Drude model. Rather, the best experimental fit is obtained with the Cole-Davidson model s(w ) =

s DC . (1 - iwτ )β

(4.4)

Here, β describes a distribution of relaxation times, such that the average scattering time <τ> is given by τβ.19 Thus, β = 1 reduces to the Drude response. For the data in Figure 4.2, a best fit was obtained with β = 0.84 (0.76) for the n-type (p-type) samples. A pure Drude response (i.e. β = 1) was obtained for samples with somewhat higher carrier densities around ≈ 1017 cm-3.19 With optical pump THz probe spectroscopy, measurements of the Drude conductivity upon photoexcitation of electron hole pairs have been performed by several groups.20–22 Some experiments are described in more detail in the chapter by Schmuttenmaer. One example of a photo-induced Drude-like response we would like to highlight is the result on GaAs as reported by Huber and Coworkers in Ref. 23. These results provide a striking example of the ultrafast evolution of the THz response during the development of many-body correlations following photoexcitation. In this experiment, pulses with 1.55-eV photon energy and approximately10 fs duration excited an electron hole (e-h) plasma at a density of 1018 cm-3. Monitoring the dynamics requires THz probe pulses with sufficient temporal resolution and with a spectral bandwidth extending beyond 40 THz. This was achieved using a scheme based on difference frequency generation in GaSe combined with ultrabroadband free space electrooptic sampling.23–25 The experimental results are displayed in Figure 4.3, where the THz spectra of the dynamic loss function 1/ε(ω,τD) are plotted at various delays τD between the optical pump and THz probe pulses. The imaginary part of 1/ε(ω,τD) is plotted in panel (a) and the real part in panel (b). As explained by the authors of Ref. 23, this highlights the evolution of particle interactions from a bare Coulomb potential Vq to a screened interaction potential Wq(ω,τD), where q is the momentum exchange between two particles during a collision: Vq =

Vq 4 pe 2 .  → Wq (w, τ d ) ≡ 2 ε q (w, τ d ) q

(4.5)

In essence, Vq becomes renormalized by the longitudinal dielectric function leading to a retarded response associated with the polarization cloud about the carriers. This is a many-body resonance at the plasma frequency ωs and as described previously in Figure 4.1(d), the loss function peaks at ωs with a width corresponding to the scattering rate. Thus, the results of Figure 4.3 show the evolution at q = 0 from an uncorrelated plasma to a many-body state with a well-defined collective plasmon excitation. This is evident in Figure 4.3(a) where, prior to photoexcitation, there is a well-defined peak at 36 meV corresponding to polar optical phonons. Following photoexcitation, a broad resonance appears at higher energies that evolves on a 100-fs timescale into a narrow plasma resonance centered at 14.5 THz. The response

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0

30

60

90

0.8

Energy (meV) 120

0

30

60

90

Drude

120

Drude

0.6

0.4

tD =

0.3

0

175 fs

0.0

150 fs

–0.3

125 fs

Re(1/(εq=0(ω,tD))

–lm(1/εq=0 (ω, tD))

Energy (meV)

100 fs 75 fs 50 fs 25 fs 0 fs Polar Lattice 0

5

10 15 20 25 30 35 Frequency (THz) (a)

0

5

10 15 20 25 30 35 Frequency (THz) (b)

Figure 4.3 Quasiparticle formation in GaAs after excitation with 10-fs 1.55-eV pulses. The dynamic loss function is plotted as a function of frequency at various delays after photoexcitation. The dynamic response evolves to a coherent Drude response on a timescale of ~175 fs as dressed quasiparticles are formed from an initially uncorrelated state at zero delay. (Reprinted with permission. From Huber, R. et al., Nature 414, 286–289, 2001.)

is described by the Drude model only at late delay times. These results are consistent with quantum kinetic theories describing nonequilibrium Coulomb scattering.23 As a further example related to free carriers in doped semiconductors, we consider the response at high THz electric fields26 with E > 100 kV/cm. Several groups are pursuing high field THz generation using various approaches including large aperture photoconductors,27 polaritons,28 and transition radiation at the plasma- vacuum boundary.29 A recently developed technique based on four-wave-mixing of the fundamental and second harmonic of a Ti:sapphire laser has been used to generate THz pulses with peak field strengths greater than 400 kV/cm. In Ref. 26, Bartel et al., focused amplified near-IR pulses with 25-fs duration into nitrogen with a 100 µm BBO crystal placed before the focus.26 The fundamental and second harmonic interact at the focus through a χ(3) process to efficiently generate THz radiation. This source of THz radiation was subsequently used by Gaal et al., in terms to induce a nonlinear response in n-type GaAs chemically doped to a carrier density of 1017cm-3.30 In this experiment, the transmission of high peak field THz pulses (E > 50 kV/cm) was compared to the transmission of lower field strength THz pulses (<1 kV/cm). Figure 4.4(a) shows the incident and transmitted THz electric field in the time domain, and Figure 4.4(b) displays the corresponding spectra. The sample is placed at the focus of an off-axis

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(a)

(b) Ein(t)

0

50

Electric Field (kV/cm)

Eout(t) 20 kV/cm 10

(c)

0

(d)

5

5

Spectral Power (arb. u.)

Sample

0

0 –5 Eout(t) – Ein(t)

–10 –1

0

1

2

Time (ps)

3 0

–5 |Eout(ν)|2 – |Ein(ν)|2 2

4

–10

Frequency ν (THz)

Figure 4.4 Nonlinear THz transmission through GaAs. (a) Incident and transmitted pulses in the time domain and (b) corresponding spectrum of the pulses. (c) Time domain difference between the incident and transmitted pulses clearly highlight the oscillations following the main transient. In (d), the solid line is the difference in spectral power for low peak field transmission while the diamonds show high peak field results. (Reprinted with permission. Copyright 2006 by the American Physical Society.)

parabola (inset). In Figure 4.4(c), the difference between incident and transmitted pulses reveals a 500-fs oscillation persisting after the main THz transient. This is more striking in the frequency domain, plotted as the difference in spectral power in Figure 4.4(d). The solid line shows the low-field THz response, where the data is negative at all frequencies as expected for a Drude response. In contrast, for the highfield response (diamonds), a strong increase in the spectral power is observed over a narrow frequency range around 2 THz. This result was explained by Gaal et al., in terms of the superradiant decay of optically inverted impurity transitions in bulk GaAs. In this novel process that occurs at high THz fields, the incident field strength is sufficient to ionize a significant fraction of s-like neutral donors through excitation into continuum states. A fraction of the excited carriers are subsequently scattered to p-like impurity states, followed by stimulated emission of dipole-allowed p → s transitions. A calculation based on a single-particle Schrödinger equation with randomly distributed donors is consistent with this interpretation.

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4.2.3 Intraexcitonic Spectroscopy Bound e-h pairs (excitons) bring about strong modifications of the optical and electronic properties of semiconductors. Sharp exciton lines in the visible or near‑IR are often observed around the bandgap, which provides the basis for the widely employed absorption and luminescence studies based on exciton photogeneration and annihilation.12,31 This contrasts sharply with intraexcitonic resonances,32,33 which are low-energy electromagnetic transitions between internal exciton levels as illustrated in Figure 4.5(a). Because exciton binding energies typically range between 1 and 100 meV, intraexcitonic transitions occur at THz frequencies. The sensitivity to transitions within existing exciton populations makes them an ideal tool to probe the absolute pair densities and the low-energy structures of excitonic gases. This differs from near-IR absorption and luminescence studies, which couple strongly to only a subset of states around K ≈ 0 (optical light cone) because of momentum conservation, and are thus dependent on both exciton density and the energy distribution functions in a complex way. Because intraexcitonic transitions are due to changes of the relative momentum of the e-h pair, they can occur at much larger center-of-mass momenta K outside the visible light cone. Moreover, as discussed in the following section,

4

Continuum 2p 1s THz THz

Near-IR

Momentum K

Induced Conductivity Δσ1(ω) (Ω–1 cm–1)

e-h Pair Energy

Photon Energy (meV) 6 8 10 4 6 8

20

10 (b)

10

10

0

0

(c) 0

0

Dielectric Function Change Δε1(ω)

(a)

–10

–10 1

1 2 Frequency (THz)

2

Figure 4.5 Intraexcitonic spectroscopy. (a) Schematic dispersion of excitons and unbound eh pairs along their center‑of‑mass momentum. Arrows indicate near-infrared photogeneration of excitons, as well as intraexcitonic 1s-2p transitions. (b) Measured THz response in GaAs quantum wells (circles) 5 ps after resonant excitation at the 1s heavy hole exciton line, at low temperature (T = 6 K). Lines represent intraexcitonic model curve for density 3 × 1010 cm-2. (c) Response after nonresonant excitation into the continuum at T = 300 K. In the experiments, photoexcitation of free carriers in the substrate was avoided by etching off the substrate, leaving only the quantum well structure and surrounding cladding layer attached to a THz-transmitive MgO substrate.

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so-called “dark excitons” with a forbidden inter-band dipole moment are easily accessible via intraexcitonic spectroscopy. These arguments strongly motivate the use of intraexcitonic THz probes as new tools for exciton spectroscopy. Intraexcitonic spectroscopy is a rising field with an increasing breadth of applications. Early studies took advantage of the long recombination times in indirect semiconductors such as Ge, which enabled measurements using conventional infrared spectrometers.34 The development of modern time-resolved THz spectroscopy gives access to a much larger range of materials, and enables studies of the ultrafast dynamics of excitons via intraexcitonic probes.35–38 Their use to study fundamental exciton physics is strongly supported by recent microscopic calculations, which contrast the information gained to that of luminescence and predict significant changes of the THz spectrum in low-temperature excitonic phases.39–41 We first discuss optical pump THz probe experiments that detect intraexcitonic resonances to directly observe exciton formation and ionization dynamics.37 These experiments were performed on a GaAs multiple-quantum well sample, which at the lowest temperatures (6 K) exhibits a sharp near-IR absorption line of 1s heavy hole excitons followed by higher bound exciton lines and the broadband continuum. A 250-kHz regenerative Ti:sapphire amplifier delivering 150-fs pulses at 800 nm was used to generate and probe THz pulses in the 2–12 meV range via optical rectification and electrooptic sampling in 500‑µm thick ZnTe crystals. The high repetition rate of this setup provides both sensitive detection with 104:1 signal-to-noise ratio in the THz electric field and sufficiently high µJ pulse energies to photoexcite the large THz probe areas. Near-IR pump pulses were spectrally narrowed to 2 meV and tuned to selectively excite either the sharp 1s exciton line or the continuum of unbound e-h pairs. In a first experiment, the THz response at low lattice temperatures was studied after resonant photoexcitation at the 1s–HH exciton line. The THz response is expressed as σ(ω) = σ1(ω) + iωε0[1 - ε1(ω)], where σ1(ω) measures the absorbed power density and ε1(ω) the inductive out-of-phase response. Just after photoexcitation, as shown in Figure 4.5(b), the THz response consists of a strongly asymmetric peak around 1.7 THz in ∆σ1 accompanied by a dispersive dielectric function change ∆ε1 at the same frequency. This response corresponds to the formation of a new, low-energy oscillator that is absent in the semiconductor’s ground state. The THz conductivity peak can be explained by the lowest energy intraexcitonic transition, i.e., between the 1s → 2p exciton levels, in agreement with known level spacings. After excitation into the continuum of unbound states, a very different behavior is observed. Figure 4.5(c) shows data taken with the sample at room temperature to minimize the influence of excitonic correlations. The THz response then exhibits a pure Drude response of a conducting e‑h gas. It is characterized by a large lowfrequency conductivity ∆σ1 and a purely negative dielectric function change ∆ε1. Note in contrast the vanishing low-frequency THz conductivity for the intraexcitonic response in Figure 4.5(b), which results from the insulating, charge-neutral nature of the bound e-h pairs. An important advantage of intraexcitonic spectroscopy is its capability to provide an absolute measure of exciton densities. A quantitative intraexcitonic dielectric function ε(ω) of an exciton gas in the ground state can be derived by summing

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up the dipole transitions between the 1s level and the higher bound and continuum states. The solid lines in Figure 4.5(b) are such calculations, using two-dimensional bound and continuum hydrogenic wave functions corrected for Coulomb screening effects from the finite well width of the quasi–two-dimensional system.42–44 A reduced effective mass µ = 0.054 m 0 and a dielectric constant εs = 13.2 were used for the GaAs quantum well material. The calculation shows that, although the peak is due to the 1s-2p transition, the shoulder stems from transitions into higher bound np levels and the continuum. Within a parabolic band approximation, we can write the “partial oscillator strength sum rule” as WB

∫

s1 (w )dw =

0

p ne 2 2 m

(4.6)

∆t = 0 ps

0

60 ps

10 0 –10 –20

20 hv 10

200 ps 0

400 ps 1,000 ps 4

6

8 10

4

6

8 10

–10 0

200

400

600

800 1,000

Photon Energy (meV)

Time Delay (ps)

(a)

(b)

Dielectric Function Change, ∆ε1, at 4.1 meV

10

∆ε1(ω)

∆σ1(ω) (Ω–1 cm–1)

where n is the e-h pair density. According to Equation 4.6, the intraband spectral weight up to energies WB below the interband transitions is directly determined by the e-h pair density n and reduced mass µ. This renders THz probes an important direct gauge of e-h pair and exciton densities, in strong contrast to more indirect measures of luminescence or absorption spectroscopy at the band gap. Optical pump THz probe experiments have been employed to investigate the formation and ionization dynamics of excitons.37 Figure 4.6 shows the low- temperature transient THz response after nonresonant excitation above the bandgap. Directly after excitation into the unbound e-h continuum (∆t = 0 ps), the THz response is spectrally broad, with a negative dielectric function change ∆ε1 that closely follows a Drude model (solid line, Figure 4.6(a)). The system is thus dominated by a conducting gas of unbound e-h pairs, as also substantiated by the large

Figure 4.6 Exciton formation in GaAs quantum wells at T = 6 K, studied via intraexcitonic spectroscopy. (a) Induced THz conductivity ∆σ1 and dielectric function change ∆ε1 after nonresonant excitation into the continuum, for different pump-probe delays ∆t. Solid line at ∆t = 0 ps: Drude model. (b) Dynamics of ∆ε1 at 1 THz, which illustrates the decay of the Drude-like response of the unbound e-h pairs while excitons continue to form. (Reprinted with permission. From Kaindl, R.A. et al., Nature 423, 734–738, 2003.)

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low-frequency conductivity. However, the induced conductivity ∆σ1 already at this time point deviates significantly from the Drude model, showing a distinct excitonic peak around 8 meV. This peak signifies the quasi-instantaneous appearance of excitonic e-h correlations directly after continuum excitation, within the ≈1-ps time resolution of the experiment. As the system evolves over several hundred picoseconds the exciton peak increases in spectral weight and sharpens, whereas the Drude components of the response decay away. This reveals two very different time scales that exist during exciton formation. Acoustic phonon emission can explain the slow scattering of unbound e-h pairs into excitons on a 100-ps time scale.45–47 In contrast, the quasi-instantaneous exciton peak can be explained by a much faster process, such as ultrafast Coulomb interactions in the photoexcited e-h gas. The opposite process of exciton ionization at increased lattice temperatures can be studied, in which intraexcitonic spectroscopy provides quantitative insight into the dynamics of exciton breakup and into the resulting quasi-equilibrium mixture of excitons and unbound pairs.37,44 Further experiments also studied the intraexcitonic response of a high-density exciton gas.48 Those results show an increasing free carrier fraction as the photoexcitation density is increased, accompanied by a strong renormalization of the 1s-2p resonance to lower energies. Although the previous experiments have studied intraexcitonic absorption, it was suggested early on to reverse this process to induce THz gain from inverted exciton populations.33 The search for intraexcitonic gain is, however, nontrivial because excitons differ strongly from atomic gases. Excitons interact with phonons and they typically exhibit significant many-body interactions such as phase-space filling, screening, and scattering. Also, in stark contrast to atoms, they are initially photogenerated as coherent polarization waves and possess an often short recombination lifetime. Recent calculations propose to obtain population inversion by exciting higher bound s‑like excitons and predict a transfer into p-like excitons via many-body interactions.50 A promising candidate for intraexcitonic stimulated THz emission is Cu2O, which exhibits a particularly well-defined excitonic Rydberg series. These visible lines shown in Figure 4.7(a) are due to p-like excitons,51 which makes this an ideal system to search for p → s intraexcitonic population inversion. Unlike most semiconductors, photoexcitation of s-like excitons in Cu2O is dipole-forbidden because the parity of valence and conduction bands is identical around the Brillouin zone center.51 A recent optical pump THz probe experiment provides the first demonstration of stimulated emission between internal exciton levels.49 In that study, a 330-µm thick Cu2O single crystal was resonantly photoexcited using spectrally shaped pump pulses around λ ≈ 570 nm. Figure 4.7(b) shows the THz response at ∆t = 1 ps. Excitation in the continuum leads to a broad Drude response (top curve), whereas excitation of 2p excitons (bottom curve) induces a broad THz absorption that can be explained by intraexcitonic transitions. A strikingly different response occurs after selective 3p excitation, as shown in the middle curve of Figure 4.7(b): here, a negative absorption change at 6.6 meV is observed, which is surrounded by absorption at higher and lower photon energies. This negative absorption is explained by 3p → 2s stimulated THz emission, commensurate with the known level spacing. It should be noted that Cu2O has no measurable absorption in this range, and that the effect occurs only after resonant 3p excitation. While the overall signals are small, the cross-section

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0

3p

–5

4s 3s

4d 3d

6 5

2s

–10

4

2p

–15

g 2p 2.13

3p

4p

5p

2.14 2.15 2.16 2.17 Photon Energy (eV) (a)

3 2 1

0 Continuum

–2

1.0 ∆t = 1 ps 0.5 0.0

–4

∆α(cm–1)

5p 4p

∆n(10–3)

5

Absorption Coeﬃcient α (100 cm–1)

Energy (meV)

132

3p

2p 4

6

8 10 4 6 8 Photon Energy (meV)

10

(b)

Figure 4.7 Stimulated emission from intraexcitonic transitions. (a) Visible absorption spectrum of Cu2O at T = 6 K (solid line) and pump spectrum (dashed). Inset: exciton levels and their energy difference to the 3p level. (b) THz refractive index change ∆n (left panel) and induced absorption ∆α (right panel) 1 ps after continuum, 3p, or 2p excitation. Arrow indicates 3p-2s stimulated emission at 6.6 meV. (Reprinted with permission. Copyright 2006 by the American Physical Society.)

corresponds to a large value of σ ≈ 10 –14 cm2, which motivates further work toward generation of strong stimulated THz emission from excitons. The material Cu2O has also been of much interest for experiments that attempt to find Bose-Einstein condensation (BEC) of excitons in semiconductors. Exciton BEC was predicted in 1968 by Keldysh and Kozlov,52 but its observation has been elusive until now. The small exciton mass pushes the transition temperature upward into the Kelvin range, about 106 times higher than for atoms. Very intriguing effects were observed in coupled quantum wells,53,54 but Cu2O remains the longest and most extensively studied candidate material. It is a key contender for BEC because its lowest energy 1s exciton is dipole-forbidden and thus has an extremely long lifetime. Because of electron hole exchange interactions, the 1s exciton in Cu2O is split into the lowest energy paraexciton (singlet, S = 0) and a three-fold degenerate orthoexciton (triplet, S = 1) that lies 12 meV higher in energy (Figure 4.8(a). The paraexciton is optically inactive to all orders, and lifetimes from ≈0.1 to 1 ms have been reported. Luminescence experiments provided strongly differing interpretations: an early study suggested evidence for exciton BEC in Cu2O based on the luminescence line shape,55 but the same authors later questioned this56 and concluded that the densities for BEC may not be reached because of fast exciton–exciton Auger recombination above n ≈ 1015 cm–3. However, Auger recombination was shown to contradict recent results, and a fast exciton capture into short-lived biexcitons was proposed instead to explain the fast dynamics.57–59 Here, intraexcitonic spectroscopy promises new insights because of the lack of optical coupling of the 1s exciton in the visible inter-band range. Several intraexcitonic experiments were reported that probe the 1s-2p transitions in Cu2O crystals.38,60–64 Kubouchi et al. studied the picosecond dynamics of the intraexcitonic transitions of the lowest energy exciton species in a 170-µm thick Cu2O single crystal.38,63,64

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16 14 E

para 1s-2p

800 ps

12 2p-ortho

2p-para 1s-ortho

∆α(cm–1)

10

200 ps

8 6

30 ps

4

1s-para K

10 ps

2

–10 ps

0 110

(a)

120 130 Energy (meV) (b)

Figure 4.8 (a) Schematic level diagram of the lowest energy excitons in Cu2O, along with intraexcitonic transitions between the 1s and 2p levels. (b) Transient intraexcitonic absorption at 4.2 K for different time delays after two-photon resonant excitation of orthoexcitons with a 1220-nm wavelength pump beam. (Reprinted with permission. Copyright 2005 by the American Physical Society.)

As illustrated in Figure 4.8(a), the 1s-2p transitions within the paraexcitons occur at higher photon energies than within the orthoexcitons because of the different fine structure splittings of the 1s versus the 2p levels. Orthoexcitons in the 1s level are resonantly photogenerated in those experiments by two-photon excitation with a femtosecond pulse around the 1220-nm wavelength.38 The resulting absorption changes ∆α are followed in transmission between 108 and 140 meV photon energy (≈26–34 THz) on a picosecond timescale. As shown in Figure 4.8(b), the signals directly evidence a decrease of the orthoexciton density (116-meV peak) accompanied by a concomitant rise of the paraexciton density (129-meV peak) on a 100-ps time scale. This intraexcitonic study therefore shows directly that rapid orthoexciton decay arises from ortho-para exciton interconversion. This can be attributed by Kubouchi and coworkers to an electron spin exchange mechanism. Such experiments can provide new insight into the properties of the optically dark 1s excitons to optimize the conditions under which a possible degenerate excitonic quantum state may be generated. Moreover, calculations predict strong changes of the intraexcitonic spectra when the material undergoes a transition into the BEC state,39 which strongly motivates further work on intraexcitonic spectroscopy of low-temperature e-h gases.

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4.2.4 Intersubband Transitions The quantum confinement of carriers in semiconductor heterostructures leads to a splitting of the conduction and valence bands into subbands. New THz transitions can then occur between these subbands within the conduction band (or valence band) when a quasi two-dimensional carrier plasma is introduced into the structure by photoexcitation, electrical injection, or chemical doping. These intersubband transitions represent a totally new character of optoelectonic excitations below the fundamental band gap of a semiconductor, especially when compared with the broad near-IR interband absorption from the valence to conduction bands.65,66 A schematic band structure and a typical intersubband absorption line are shown in Figure 4.9. Because of the identical sign of the curvature of both subbands, the absorption is concentrated into a small interval of photon energies. Ideally, subbands with identical effective mass would result in a vanishing line width except for homogeneous broadening. However, a real semiconductor exhibits a nonparabolic band dispersion; transitions at different in-plane momenta then occur in a finite frequency interval, leading to inhomogeneous broadening. Sample imperfections and Coulomb manybody interactions further complicate intersubband line shapes. Intersubband transitions can be tailored to occur within a large photon energy range from approximately 1 to 100 THz via quantum heterostructure engineering. Technical applications of intersubband transitions are abundant, with unipolar quantum well photodetectors and quantum cascade lasers being particularly relevant devices.67,68 The operating principle of THz and mid-IR quantum cascade lasers relies directly on controlled engineering of ultrafast processes. Population inversion between subbands can be achieved by optimizing the electron tunneling and relaxation dynamics, and the photonic properties of quantum well and superlattice heterostructures. Numerous ultrafast THz and mid-IR studies of intersubband relaxation and dephasing have been reported. Intersubband relaxation and carrier thermalization E

CB (n = 2) (n = 1)

Intersubband

Interband

z (a)

VB k||

Absorption (norm)

E 1

0 20

25

30

35

Frequency (THz) (b)

Figure 4.9 Intersubband absorption. (a) Left: spatial variation of the conduction (CB) and valence band (VB) edges in the heterostructure growth direction, and energy extrema of the lowest subbands (horizontal lines). Right: dispersion along the two-dimensional layers and optical transitions. (b) Typical intersubband absorption line, measured from an n-type doped GaAs multiple quantum well structure.

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have been observed in both n-type and p-type heterostructures, in which spectral changes around the intersubband line are sensitive to transient modifications of the carrier distribution function f(E).69–75 Near-IR photoexcitation of a coherent superposition of several bands also leads to THz emission, as a measure of intersubband or intervalence band oscillations.76–78 This yields information on the decay of intersubband coherence in undoped structures, equivalent to the overall intersubband line shape. Coherent emission was also detected from intersubband plasmons in parabolic quantum wells79 and from Bloch oscillations in superlattices that arise from different physics in high THz fields.80 Separation of homogeneous and inhomogeneous components of intersubband line shapes necessitates nonlinear experiments, realized through mid-IR femtosecond four-wave mixing experiments.81,82 Such timeresolved THz and mid-IR studies provide insight into energy and phase relaxation times and into microscopic carrier–carrier and carrier–lattice interactions of interest for fundamental physics and device design.83 Combined with high field strengths, coherent THz pulses offer new ways to measure and control coherent polarizations between quantized levels in quantum wells or other nanomaterials. Intense THz fields can lead to the generation of intersubband sidebands on the near-IR resonances and to undressing of collective depolarization effects.84,85 Rabi oscillations occur at the highest field strengths. The experiments of Luo et al., discussed in the following section, have provided a vivid example of direct field-resolved measurements in this regime.86 The experiment is illustrated in Figure 4.10(a). An incident coherent THz pulse couples to the (n = 1) to (n = 2) intersubband transition (νIS ≈ 24 THz) in the mid-IR spectral range of an n-type modulation-doped GaAs multiple quantum well structure. The electron density is 5 × 1010 cm–2 in each quantum well. A prism geometry enables strong coupling to the Probe

MIR

PS ZnTe

Sample Electric Field (kV/cm)

(a) 5

Incident

20 0

0 Re-emitted –5 0.0

0.2

0.4

0.6 0.8 Time (ps) (b)

–20 1.0

1.2

0.2

0.4

0.6 Time (ps) (c)

0.8

1.0

Figure 4.10 Rabi oscillations from intersubband transitions in GaAs multiple quantum wells. (a) Experimental scheme. (b) and (c) Measured field transients incident (thin line) and reemitted (thick line) from the sample, for two different THz field amplitudes of 5 kV/cm and 30 kV/cm. (Reprinted with permission. Copyright 2004 by the American Physical Society.)

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intersubband dipole moment oriented perpendicular to the quantum well layers. The incident and reemitted light fields are characterized in the time domain via electrooptic sampling using a 10-µm thick ZnTe crystal. Figures 4.10(b) and (c) show the incident field and the reemitted field for different field strengths. The incident field is determined by use of an inactive undoped sample. The emitted field in turn is obtained by subtracting the incident field from the total field transmitted through a doped sample. In the weak field limit shown in Figure 4.10(b), the reemitted THz field is 180° out of phase from the incident field. This corresponds to absorption of light at the intersubband transition; the incident field creates a coherent intersubband polarization, from which a coherent field is emitted (free induction decay) that partially cancels the incident THz pulse. At higher field strengths, Figure 4.10(c), the reemitted field changes drastically; initially it is out of phase, then vanishes and reappears as inphase with the driving field. Constructive interference with the reemitted field during the second half of the driving pulse corresponds to gain and thus to amplification of the incident field. This is a direct manifestation of Rabi oscillations between the (n = 1) and (n = 2) subbands. A detailed analysis of the data reveals deviations between the simplified two-level Maxwell-Bloch formalism and the measured field traces.86 This can be explained by a Coulomb-mediated collective coupling between different quantum well layers in the nanostructure.86–88 Intersubband coherent control via high-field THz pulses enables new quantum optical experiments in solids, with novel applications (e.g., in photonics and quantum information science). Rabi oscillations were also studied on hydrogenic 1s-2p transitions of donor impurities in GaAs,89 which serve as a model two-level system for such applications. Gain without inversion from intersubband transitions was recently experimentally observed.90 This also motivates coherent population transfer experiments to provide maximum control over THz excitations in nanostructures.91–93

4.3 Superconductors 4.3.1 Overview Far-IR spectroscopy has played an important role in the characterization of superconductors since the BCS phonon-mediated pairing mechanism was proposed and experimentally verified.94 Mattis and Bardeen described the electrodynamics of superconductors and the opening of a spectroscopic gap 2∆0 for quasiparticle excitations.95 In weak-coupling BCS theory, the gap value is given by 2∆0 = 3.5kBTC, with TC being the transition temperature. The electrodynamics was experimentally confirmed by Glover and Tinkham using far-IR techniques.10,96 Nonequilibrium superconductivity also has a long history starting with tunneling experiments.97 The formation of Cooper pairs—also called “quasiparticle recombination”—is a fundamental quantum process in a superconductor arising from the pairing of two unbound quasiparticles that are thermally or otherwise excited out of the condensate’s correlated ground state. It was soon realized that pair breaking by excess phonons complicated matters, as first described by the rate equations of Rothwarf and Taylor.98 Experimental and theoretical progress continued throughout the 1970s and 1980s.99–102 Ultrafast optical spectroscopy of metals blossomed when it was realized that the electron phonon coupling constant λ could be determined from such

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experiments.103–105 For example, this technique was used to measure λ for several conventional BCS superconductors.106 With the discovery of high-TC superconductors in the copper oxides (cuprates), it was realized that all-optical pump–probe spectroscopy could be used for nonequilibrium studies below TC.107,108 A limitation in the interpretation of such studies is that they probe the dynamics at visible or near-IR wavelengths, far above the superconducting gap and other intrinsic low-energy excitations. THz-TDS is a useful experimental tool to study superconductors.106,109 With optical photoexcitation, it provides what, to date, is probably the most insightful experimental approach to interrogate nonequilibrium dynamics in superconductors. Such experiments have the potential to measure quasiparticle interactions that remain hidden in linear spectroscopic measurements. In this section, we first describe linear IR spectra of superconductors using THz-TDS, with an emphasis on MgB2 and YBa2Cu3O7. This highlights commonalities and differences between conventional and high-TC superconductors in their THz electrodynamics. Armed with this knowledge, we proceed in the subsequent sections to time-resolved THz studies of nonequilibrium dynamics in the BCS superconductor MgB2, and in the high-TC cuprates YBCO and Bi-2212.

4.3.2 Far-IR Spectroscopy of Superconductors There is a vast literature on the electrodynamics of superconductors.1,10 Given these excellent reviews and space limitations we restrict our description to studies of superconductors using THz-TDS. The advantage of THz probes is that they significantly penetrate inside the bulk of the material to provide a contactless probe of low-energy excitations such as the superconducting energy gap and the collective superfluid condensate and quasiparticle response.110 Early time domain THz experiments on superconductors concentrated on studying the equilibrium conductivity of conventional metals such as NbN that exhibit an isotropic s-wave superconducting gap,13 and of high-TC cuprates109,111–113 with an anisotropic d-wave gap and more complex physics. In the following, we illustrate the THz response of a superconductor with a time domain THz study of MgB2.114 As discovered in 2001, this material turns superconducting below TC = 39 K, an unexpectedly high transition temperature for a conventional metallic compound.115 The THz experiments described below cover the 2 to 11-meV spectral range, with THz pulses generated and coherently detected using a 250-kHz Ti:sapphire amplifier. Experimental results are shown in Figure 4.11, as obtained from a 100-nm thick MgB2 film. The electric field of THz pulses transmitted through the film is shown in Figure 4.11(a). As the material transitions from the normal (dashed) into the superconducting state (solid line), the THz field increases in amplitude and shifts in phase. This field reshaping arises from changes to the complex conductivity σ(ω). The response is obtained via Fourier transformation of time domain THz signals and by evaluation of standard thin film transmission formulas.116 A useful way to analyze the electrodynamic properties of superconductors is to plot the ratio of the conductivity to its normal state value σN(ω). This is shown in Figures 4.11(b) and (c) for the real and imaginary parts, respectively. A strong depletion in the real part σ1(ω)/σN(ω) is evident in Figure 4.11(b) as the temperature is decreased below TC. An absorption onset occurs at ≈5 meV, as evident in the data,

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EO Signal

3 0 –3 –1

0 1 Time Delay (ps) (a)

2

1 1

0

5 10 Photon Energy (meV)

5 10 Photon Energy (meV)

(b)

(c)

σ2(ω)/σN(ω)

σ1(ω)/σN(ω)

2

0

Figure 4.11 Terahertz conductivity of a 100-nm thick film of MgB2. (a) Time-resolved THz fields measured in transmission at T = 6 K (solid line) and 40 K (dashed line). (b) and (c) Real part σ1(ω) and imaginary part σ2(ω) of conductivity normalized to the normal state (40 K) value σN (ω). Data are shown for T = 6 K (dots), 24 K (squares), and 30 K (triangles). Mattis-Bardeen calculations for an isotropic s-wave gap of magnitude 2∆0 = 5 meV, TC = 30 K, and T = 6 K are shown as thick solid lines. The dashed line in (b) also shows the same calculation for T = 24 K.

particularly at the lowest temperature, T = 6 K. This is the signature of the superconducting gap. The imaginary part σ2(ω)/σN(ω) in Figure 4.11(c) shows the buildup of a component in the superconducting state that strongly increases with decreasing frequency. The latter, 1/ω-like response is the hallmark of a superconductor; it arises from the purely inductive motion of the superfluid in the electromagnetic THz field. The results were compared with calculations with Mattis-Bardeen theory for BCS superconductors with an isotropic s-wave gap,95 shown as lines in Figures 4.11(b) and (c). This theory is valid in the so-called “dirty limit,” which occurs when the superconducting gap 2∆0 is much smaller than the normal state Drude width 1/τ. In this case, absorption sets in for frequencies above the superconducting gap 2∆, in which elastic scattering with rate 1/τ enables momentum conservation during THz absorption. The Mattis-Bardeen theory faithfully represents both real and imaginary parts of the experimental data. The gap in the real part reflects that, at the lowest temperatures, Cooper pair breaking is only possible for photon energies above 2∆0. At higher temperatures, the gap value decreases, and thermally excited quasiparticles exhibit an additional Drude-like response which results in the conductivity σ1(ω) even below

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the gap. The temperature dependence of the superfluid response in σ2, which vanishes at TC = 30.5 K, is equally well reproduced.114 However, this overall agreement is possible only when an artificially small gap value of 2∆0 = 5 meV is assumed in the calculations. This value is almost a factor of two smaller than the ratio expected from the known TC in weak-coupling BCS theory, which predicts 2∆0 = 3.5 k B TC ≈ 9 meV. This persistently small gap size is a fundamental property of MgB2, which is also seen in numerous other experiments. In contrast to surface sensitive measurements, however, the THz probe penetrates through the bulk of the film and thus confirms the general presence of the small gap. The small gap value is explained by the existence of two superconducting gaps, of which the smaller one dominates the optical conductivity.117–119 First-principle band structure calculations confirm this novel physics and indicate that the dominant hole carriers in Boron p orbitals are split into two distinct sets of bands with quasi–two-dimensional and three-dimensional character. The coupling between these bands leads to MgB2’s novel superconducting state with two gaps but a single TC. In contrast to conventional narrow-gap superconductors such as MgB2, the THz electrodynamics of high-TC cuprates is quite different.1,106 Most notably, the superconducting gap has a d-wave symmetry and peaks around 30 THz, such that the typical BCS gap structure in σ1(ω) (Figure 4.11) is not observed in the 1- to 3-THz range. However, as with the BCS superconductors, a strong 1/ω dependence from the inductive condensate response is quite prominent in σ2(ω) below TC. A model that describes the electrodynamics of superconductors at frequencies far below the gap is the two-fluid model. It is useful to approximate the responses of high-TC cuprates in the range up to ≈3 THz. The two fluids in this model refer to thermally excited quasiparticles and the superconducting condensate. With decreasing temperature, more quasiparticles join the condensate. Thus the quasiparticle density nN(T) and the condensate density nSC (T) are temperature dependent. The sum of quasiparticle and condensate densities is constant with temperature: nN(T) + nSC (T) = n where n is the normal state carrier density above TC. This can also be written as X N(T) + XSC (T) = 1 where X N(T) ≡ nN(T)/n is the quasiparticle fraction and XSC (T) ≡ nSC (T)/n is the condensate fraction. The conductivity for the two-fluid model is given as 1 nN (T )e 2 n (T )e 2 + SC 1 / τ(T ) - iw m m

i   pd(w ) + w 

(4.7a)

ε 0 w 2p 1 i  X N (T ) + pd(w ) +  XSC (T ).  2 1 / τ(T ) - iw m 0 l L (0)  w

(4.7b)

s(w ) =

=

The first term is just the Drude response with a temperature-dependent carrier density and scattering time. The second term describes the condensate response with the first term in brackets being a δ-function at zero frequency that describes the infinite DC conductivity of the superconducting condensate. The second term in brackets is a purely imaginary inductive response (obtained by letting τ → ∞ in the Drude model). We can thus see that, on condensation of quasiparticles into the superfluid, spectral weight is transferred from the Drude peak to the zero frequency

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Terahertz Spectroscopy: Principles and Applications

superconducting peak. In Equation 4.7b, the condensate portion is rewritten in terms of the London penetration depth l 2L ( 0 ) =

m m 0 ne 2

(4.8)

which describes how far a DC magnetic field penetrates into the superconductor at zero temperature. These equations highlight an important aspect about the electrodynamics of superconductors; from a simultaneous measurement of σ1(ω) and σ2(ω), the quasiparticle and condensate fractions can be quantitatively determined. The temperature dependence of these fractions then can provide essential information about the superconducting state, to distinguish, for example, isotropic s-wave from anisotropic d-wave gap symmetries.106 Figure 4.12 shows a calculation using the two-fluid model for various values of X N and XSC (keeping X N + XSC = 1). For the calculations, τ = 150 fs, n = 1019 cm-3, and m* = m 0 giving σDC = 104 Ω-1cm-1 for X N = 1. The real part σ1(ω) is shown as a solid line, the total σ2(ω) is plotted as a dashed line, and the Drude contribution to σ2(ω) is plotted as a dotted line. A pure Drude response results for XSC = 0, see Figure 4.12(a). With XSC = 0.2, a clear 1/ω dependence is evident, although the relative Drude contribution to σ2(ω) is still quite large but becomes much smaller at XSC = 0.4. Finally, for XSC = 1 as shown in Figure 4.12(d), the response at THz frequencies entirely arises from the superconducting condensate. 2

σ/104 (Ω cm)−1

1 (a) Xn = 1, Xsc = 0

1

0.5

0.5 0

0

1

2

3

2 σ/104(Ω cm)−1

(b) Xn = 0.8, Xsc = 0.2

1.5

0

1

2

3

6 (c) Xn = 0.6, Xsc = 0.4

1.5

(d) Xn = 0, Xsc = 1 4

1

2

0.5 0

0

0

1

2

Frequency (THz)

3

0

0

1

2

3

Frequency (THz)

Figure 4.12 Model calculation using the two-fluid model for different values of the quasiparticle and condensate carrier fractions.

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141

4

0.4 0

2 0

0.4

0.8

0.4

60 K 0.4

0.3

τ = 175 fs

1.2

1.6

0

95 K

0 0.2

2

0.6

τ = 51 fs 1 0.4 1.6 Freq (THz)

1

1.4

Frequency (THz)

Frequency (THz)

(a)

(b)

0.8

Scattering Time (fs)

Carrier Fractions

0.4

0.1 0.8

Xn

0.6 0.4 Xsc

0.2 20

0.2

1 1.6 Freq (THz)

1

0

0.8

σ1(ω)/104

6

σ1(ω)/104

σ2(ω)/104 (Ω cm)–1

Time-Resolved Terahertz Studies of Carrier Dynamics

40

60

80

1.8

350 250 150 50 0

0

50

100

150

200

Temp (K)

Temp (K)

(c)

(d)

250

Figure 4.13 THz-TDS measurements on YBa2Cu3O7 with two-fluid model fits.

Of course, this idealized situation is not practically realized, though as Figure 4.13 reveals, it provides a good starting point. Specifically, Figure 4.13 shows the results of THz-TDS measurements on a nearly optimally doped film of YBa2Cu3O7 (TC = 89 K) epitaxially grown on <100> MgO using pulsed laser deposition. Figure 4.13(a) shows σ2(ω) at T = 60 K, whereas Figure 4.13(b) shows σ2(ω) at 95 K. The insets show the corresponding real part σ1(ω). The thick solid lines are the experimental data and the dashed lines are fits using the two-fluid model. In Figure 4.13(a), the Drude contribution to σ2(ω) is plotted as a thin solid line with the condensate fraction plotted as a thin dotted line. The two-fluid model provides a reasonable fit to the data. In Figure 4.13(c), X N(T) and XSC (T) are shown as extracted from fits to the experimental data. The relation X N(T) + XSC (T) = 1 is seen to be accurate to within 10%. At the lowest temperature measured, X N(T) = 0.1, which is still a substantial quasiparticle fraction. In Figure 4.13(d), τ is plotted as a function of temperature. Below TC, there is a dramatic increase in the scattering time until ~50 K, where it saturates at 400 fs because of impurity scattering. This increase in τ results in a substantial narrowing of the Drude peak (see inset to Figure 4.13(a)). However, in even the best films, τ is still much shorter when compared with the highest quality single crystals where τ is so long that the Drude peak lies at microwaves frequencies.120

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Time domain THz spectroscopy was also used to study the conductivity in another important cuprate, Bi2Sr2CaCu2O8+d (Bi‑2212). In this material, about 30% of the spectral weight resides in an additional component that follows the temperature dependence of the superfluid but exhibits a finite scattering width 1/τ.121 This has been associated with static fluctuations of the superfluid due to real-space electronic inhomogeneity in Bi-2212 (that can be described as a disordered condensate response). Furthermore, THz-TDS experiments can be carried out on underdoped cuprates, which are known to exhibit a puzzling “pseudogap” phenomenon that appears in numerous spectroscopies and persists to temperatures well above TC. Among various explanations, the pseudogap has been ascribed to a transient phase coherence from short-lived remnants of the superfluid above TC. Here, THz-TDS experiments on underdoped Bi-2212 films have revealed that while such coherence indeed persists above TC, it cannot be the main source of the pseudogap because it vanishes well below the pseudogap temperature.122 Despite the subtleties of the cuprate properties, the two-fluid model (Equation 4.7 and Figure 4.12) is still a reasonable starting point for understanding their THz response well below the gap, and accordingly for insight into the nonequilibrium dynamics. Anticipating the dynamic experiments to be described later, the model suggests how we can fruitfully apply optical pump THz probe spectroscopy to superconductors. A short near-IR pulse breaks Cooper pairs, thereby reducing the condensate fraction and increasing the quasiparticle fraction. Subsequently, the excess photoexcited quasiparticles will recombine to the condensate. This process can be monitored in exquisite detail using THz pulses, allowing for the simultaneous measurement of the dynamical evolution of quasiparticle and condensate densities (i.e., of their time-dependent changes ∆nN(t) and ∆nSC (t)).

4.3.3 Quasiparticle Dynamics in Conventional Superconductors The basic properties of MgB2 and the electrodynamic response as measured using THz-TDS have been discussed previously. In the following, we describe the results of measurements using optical pump THz probe spectroscopy on 100-nm films of MgB2 deposited on sapphire (TC = 34 K).123,124 In these experiments, 100-fs pulses with a center wavelength of 800 nm were employed to photoexcite the sample at fluences sufficiently low (0.1–5 µJ/cm2) such that the condensate was perturbed without driving the sample above TC. In Figure 4.14(a), the imaginary conductivity is plotted as a function of frequency at various delays with respect to the photoexcitation pulse. The initial temperature is 7 K and the excitation fluence is moderately high at 3 µJ/cm2. At ∆t = –5 ps, the 1/ω condensate response dominates. At 3 ps, the condensate response is dramatically reduced and even more so at 10 ps. By 300 ps, the condensate has still not fully recovered. Interestingly, it takes some time (≈10 ps) for the pair-breaking (i.e., reduction in condensate density) to finish. This is shown more clearly in Figure 4.14(b), where the dynamics of σ1(ω) and σ2(ω) at 0.8 THz are plotted as a function of time. As the condensate response decreases, a corresponding increase in the quasiparticle density is observed in σ1(ω). This is followed by a much longer recovery time as quasiparticles rejoin the condensate. The condensate recovery time is plotted in Figure 4.14(c) and shows a pronounced temperature dependence peaking near TC.

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σ2(ω)/104 (Ω cm)–1

6 σ1

4 4

–5 ps

3

300 ps 2

σ2

2

3 ps 10 ps

1

1.0 1.5 Frequency (THz)

0

10 100 200 Time (ps)

(a)

(b) 5

τ(ns)

∆σ (a.u.)

0.8

0.4

4 1.51.2 1 0.6

1

2

0.4

0.1 0

30 10 20 Temperature (K)

1

10 Time (ps)

50

(d)

(c)

Figure 4.14 Optical pump THz probe spectroscopy of MgB2. (a) Dynamics of the imaginary conductivity related to the condensate density at T = 7 K. (b) Conductivity dynamics at 0.8 THz, for T = 7 K. (c) Condensate recovery time. (d) Initial change in the conductivity related to pair-breaking dynamics at various fluences (in µJ/cm2).

Finally, in Figure 4.14(d), the pair breaking dynamics are plotted for various fluences. With increasing fluence, the pair breaking time decreases. The observed pair breaking and pair recovery dynamics in MgB2 can be understood using the phenomenological Rothwarf-Taylor model.98 This model consists of two coupled rate equations describing the temporal evolution of the density of excess quasiparticles and phonons injected into a superconductor. The Rothwarf-Taylor equations are written as

dn = βN - Rn 2 - 2 RnnT dt dN 1 ( N - N0 ) = [ Rn 2 - βN ] . τP dt 2

(4.9a) (4.9b)

Here, n is the excess quasiparticle density, nT is the thermal quasiparticle density, N is the excess density of phonons with energies greater than 2∆, R is the bare

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quasiparticle recombination rate, β is the rate of pair breaking by (>2∆) phonons, and τP describes the relaxation time of the phonons either by anharmonic decay to phonons with energies <2∆ or through phonon escape from the sample (e.g., into the substrate). As Equation 4.9a reveals, in the limit of small nT, the recombination of quasiparticles is a bimolecular process (hence the n2 term) requiring, as expected, two quasiparticles to form one Cooper pair. However, the direct determination of R is made difficult as it can be masked by excess 2∆ phonons breaking additional Cooper pairs (e.g., the βN term). Thus, in many measurements, the phonon decay τp is measured and has been termed the phonon bottleneck. The phonon bottleneck is clearly observed in the condensate recovery dynamics of MgB2 and Pb superconductors.125 In MgB2 the recovery time is largely independent of fluence (Figure 4.14(c)) in contrast to bimolecular recombination kinetics. In MgB2, Pb, and likely all conventional superconductors, the condensate recovery is governed by the decay of acoustic phonons which, as described previously, could be due either to anharmonic decay or phonon escape from the probe volume. In MgB2, no change in τP was observed as a function of film thickness, indicating that phonon escape is not important and the condensate recovery is therefore determined by anharmonic decay processes. This assignment of the recovery is further supported by the observed temperature dependence near TC showing that τp ∝ 1/∆(T). This 1/∆(T) dependence occurs because the superconducting gap decreases in magnitude with increasing temperature. This results, for the same photoexcitation fluence, in a larger density of phonons with energies in excess of 2∆ yielding an increased condensate recovery time. The early time pair breaking dynamics in Figure 4.14(d) are somewhat surprising, because after excitation with a 100-fs pulse, the reduction in condensate density continues for ≈10 ps. This is in contrast with Pb, in which the pair breaking dynamics are complete in ≈1 ps.126 An understanding of this delayed pair breaking dynamics can be obtained from the Rothwarf-Taylor equations. In the limit of small nT and neglecting the term with τp (which is reasonable since the condensate recovery timescale is much longer than the initial pair-breaking dynamics), it can be shown that n(t ) =

 β 1 1 1 1 - +  R 4  2 τ τ 1 - K exp ( -tβ / τ ) 

 τ  4 Rn0 + 1 - 1  2 β  K= ;   τ 4 Rn0 + 1 + 1 2  β 

1 = τ

(4.10a)

1 2R + ( n0 + 2 N 0 ) 4 β

(4.10b)

where n 0 and N0 are, respectively, the initial excess quasiparticle and phonon densities and the time scale of the initial dynamics is given by τ.123 Of particular importance is the dimensionless parameter K, which describes three distinct regimes for the pair breaking dynamics, as manifested in the last term of Equation 4.10(a). For K = 0, the quasiparticles and phonons are in quasi-equilibrium at an elevated temperature.

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145

For this case the initial pair-breaking dynamics would show a step-like dependence at t = 0. The regime 0 < K ≤ 1 corresponds to the situation when the density of excess quasiparticles is larger than the quasi-equilibrium (K = 0) situation, whereas –1 ≤ K < 0 corresponds to the situation in which the excess density of >2∆ phonons is greater than for K = 0. The dynamic signature for this regime is a pronounced rise time and appears to explain the pair breaking dynamics in MgB2 in Figure 4.14(d). That is, a significant fraction of the energy in the photoexcited quasiparticle distribution is initially transferred to high-energy phonons that subsequently break additional Cooper pairs before condensate recovery. This contrasts with a scenario in which the photoexcited quasiparticles reach quasi-equilibrium through direct pair breaking in an avalanche-like process not involving phonons. Fits to the rise time dynamics at various fluences (solid lines in Figure 4.14(d)) using Equations 4.10a and 4.10b permitted a determination of the phonon pair breaking and bare quasiparticle recombination rates,123 yielding β–1 = 15 ± 2 ps and R = 100 ± 30 unit cell/ps, respectively.

4.3.4 Quasiparticle Dynamics in High-TC Superconductors Knowledge of quasiparticle interactions is particularly relevant for the high-TC cuprate superconductors, where the mechanism for charge pairing at temperatures up to ≈100 K remains unresolved.127 These materials show a complex phase diagram as a function of chemical doping; with increased density of hole carriers in the Cu2O planes they transition from an insulating antiferromagnetic state, via a non-Fermi liquid state that shows superconductivity at relatively small TC (“underdoped”) to the state with highest TC (“optimally doped”). As the doping is further increased, the materials show Fermi liquid properties and a progressively smaller TC until they are no longer superconducting. The quest to understand the complex quasiparticle properties and interactions in cuprates has inspired numerous experiments that employ visible or near-IR fs pulses to observe optical reflectivity changes at E ≈ 1–2 eV probe energy.108,128–130 However, it should be understood that this energy scale far exceeds the fundamental excitations of a superconductor, such as the THz gap and collective quasiparticle and Cooper pair response discussed previously. Ultrafast THz studies of superconductors, in contrast, can provide a direct measure of quasiparticle and Cooper pair dynamics, but have remained scarce. In the following section, we discuss these first THz studies of high-TC superconductor ultrafast dynamics. From a simple extrapolation of the dynamics on conventional superconductors one might expect similar dynamics on the high-TC cuprates. For example, near TC it was shown that for conventional superconductors the condensate recovery time is proportional to 1/∆(T). This suggests that the condensate recovery in the cuprates might follow a similar trend, albeit with a shorter lifetime given the larger gaps in these materials. Although such behavior is observed and there are similarities between these two classes of superconductors, there are also important differences in the nonequilibrium dynamics that likely originate from their vastly different microscopic properties. The major microscopic differences are that in the cuprates (1) superconductivity is obtained through chemical doping of Mott-Hubbard insulators, (2) superconductivity is thought to occur in two-dimensional CuO2 planes, (3) the superconducting order parameter has d-wave symmetry resulting in nodes in the superconducting gap (along kX = kY), and (4) there is the distinct possibility

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that the boson mediating pairing is not a phonon but some other excitation such as antiferromagnetic spin fluctuations. The hope is that optical pump THz probe and time-resolved mid-IR studies can provide insight into the properties of the cuprates through, for example, extracting the quasiparticle recombination rate or unmasking other interactions that are hidden in various time-integrated spectroscopies. We will briefly describe the first optical pump THz probe experiments that were performed on near-optimally doped and underdoped samples of yttrium– barium–copper oxide.131,132 The measurements were made on the same epitaxial films in Figure 4.13 and others grown under similar conditions including YBa2Cu3O7-d with TC = 89 K (50 K) for δ = 0 (0.5) and Y0.7Pr0.3Ba2Cu3O7 with TC = 0.5. The Pr compounds provide an alternate approach to achieve underdoping which is easier than controlling the oxygen stoichiometry. Figure 4.15(a) shows σ2(ω) as a function of frequency at various delays after photoexcitation for YBa2Cu3O7 at 60 K, whereas Figure 4.15(b) shows the dynamics for Y0.7Pr0.3Ba2Cu3O7 at 25 K. In both cases, there is a rapid decrease in the condensate fraction followed by a fast picosecond recovery. The condensate recovery is nearly complete by 10 ps, in dramatic contrast to conventional superconductors. The extracted lifetimes of the conductivity dynamics, and thus the condensate recovery time, are shown in Figure 4.15(c) for all of the samples investigated. In the

σim/104 (Ω cm)–1

YBa2Cu3O7-δ 6

Unpumped

0

60 K

2.5

Unpumped

2

10 ps

4 2

Y0.7Pr0.3Ba2Cu3O7 25 K

10 ps

1.5 1

1 ps 0.5

1

1 ps

0.5

2 ps 1.5

2

0

0.4

0.8

2 ps 1.2

1.6

Frequency (THz)

Frequency (THz)

(a)

(b)

5 τσ(ps)

4 3 95 K

2 1

0

10

20

30

40 50 T(K)

60

70

80

90

(c)

Figure 4.15 Condensate dynamics on (a) YBa2Cu3O7 and (b) Y0.7Pr0.3Ba2Cu3O7. (c) Summary of lifetimes measured on near-optimally doped and underdoped films—the lines are to guide the eye. Triangles: YBa2Cu3O6.5, diamonds: Y0.7Pr0.3Ba2Cu3O7. Closed and open circles: two different YBa2Cu3O7 films taken at different fluences.

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1.3 ps

0 10 ps 35 ps 50 ps 4

6 8 10 4 6 8 10 Photon Energy (meV)

0

–2000

(c)

(d)

1

0 –2 –4

0.1

1/∆σ2 (10–3 Ω cm)

1000

(b) t = –1.2 ps

–∆σ2(1000 Ω–1 cm–1)

(a)

∆σ2(ω)(Ω–1 cm–1)

∆σ1 (ω)(Ω–1 cm–1)

optimally doped films (closed and open circles), the condensate recovery is approximately 1.7 ps and increases near TC, consistent with a decrease in the superconducting gap. Above TC at 95 K, the lifetime has decreased to 2 ps and is likely a measure of electron–phonon equilibration in the normal state. Furthermore, the lifetime is independent of fluence indicative of the absence of bimolecular kinetics although this was not pursued in detail in this work.131,132 In contrast, for the underdoped films (triangles and diamonds in Figure 4.15(c)), the lifetime was constant at ≈3 ps even above TC. The constant lifetime as a function of temperature is suggestive of a pseudogap. These results revealed that it is possible to sensitively probe the condensate dynamics in the cuprates using optical pump THz probe spectroscopy. A comprehensive study of the transient picosecond THz conductivity changes after ultrafast optical excitation in a different superconductor, Bi2Sr2CaCu2O8+d (Bi‑2212), has recently been reported.133 The pump-induced change ∆σ(ω) of the CuO2 plane THz conductivity was measured in 62-nm thick optimally doped Bi-2212 films with a transition temperature TC ≈ 88 K. As in YBCO, in the superconducting state a 1/ω component dominates the imaginary part of conductivity σ2(ω) at low frequencies, which provides a direct measure of the condensate density. Figures 4.16(a) and (b) show the conductivity change observed directly after photoexcitation. The imaginary part displays a strong decrease, ∆σ2 < 0, because of the depletion of the condensate via interactions with the photoexcited carriers. This superfluid spectral weight is transferred to the quasiparticles, as shown by the broad increase of σ1. The photo-induced conductivity changes recover on a ps timescale. The changes are well reproduced with the two-fluid model (solid lines in Figure 4.16(a) and (b)) where the total spectral weight is conserved, or (e2/m*) ∆nSC = −(e2/m*) ∆nN.

–6 0 20 40 60 0 20 40 60 Time Delay t (ps)

Figure 4.16 Dynamics of Cooper pair formation observed via optical pump THz probe spectroscopy of the high-TC superconductor Bi-2212. The results are for T = 6 K lattice temperature and 0.7-µJ/cm2 fluence of the near-infrared excitation pulses. (a) and (b) Induced changes of the real and imaginary parts of the THz conductivity (circles) for different pump– probe delays t. Thick lines: two-fluid model. (c) Conductivity dynamics around 5.5-meV probe energy (symbols) plotted on a logarithmic scale. (d) The same data as in panel (c), but plotted as 1/∆σ2. The linear decay on this reciprocal scale reveals a bimolecular kinetics of condensate recovery. Solid lines are guides to the eye. (Reprinted with permission. Copyright 2005 by the American Physical Society.)

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A central result of the Bi-2212 experiments relates to the temporal shape of the relaxation dynamics. Figure 4.16(c) displays the dynamics of the superfluid density in ∆σ2. Clearly, the decay is nonexponential, which is seen by comparison with the single exponential model curve (solid line). Figure 4.16(d) shows the same data on a “reciprocal” scale (i.e., it is plotted 1/∆σ2 versus linear delay time). The decay in Figure 4.16(d) shows a strikingly simple linear time dependence

1 1 = + r ⋅ t. Ds 2 (t ) Ds 2 ( 0 )

(4.11)

This linear time dependence plotted on a reciprocal scale is a benchmark for bimolecular kinetics. By taking the time derivative of Equation 4.11, and taking into account the proportionality ∆σ2 ∝ ∆nSC ∝ ∆nN (confirmed by the two-fluid fits at each time delay) we obtain directly d/dt ∆nN = − R (∆nN)2. A natural explanation of this bimolecular kinetics is the pairwise interaction of quasiparticles as they recombine into Cooper pairs. Further evidence for the bimolecular recombination picture is obtained from the very strong density and temperature dependence, which are fully explained by the bimolecular rate equation, as discussed elsewhere.133 It has been conjectured that the origin of the fast bimolecular decays in Bi-2212 may be linked to quasiparticle recombination from the antinodes of the anisotropic d-wave superconducting gap through emission of short-lived high-energy phonons or spin waves.133 Unlike YBCO, scanning probe microscopies have shown that Bi-2212 exhibits a nanoscale electronic inhomogeneity,134 which may further relax momentum restrictions for quasiparticle scattering. This may also explain the differences in the dynamics between YBCO and Bi-2212. Although these studies on cuprates probed changes in the collective charge motion of quasiparticles and Cooper pairs, they did not probe the superconducting gap. For high-TC superconductors, signatures of this gap occur at higher frequencies around 10 to 50 THz (≈40 to 200 meV). They are of fundamental significance. Among the longest-standing puzzles of the cuprates is the pseudogap, a depression in the electronic density of states.17,135 It mimics the superconducting gap in energy but appears already at temperatures T* much higher than TC. It is widely believed that understanding the correlations that lead to the emergence of this pseudogap will also help clarify the mechanism of superconductivity in the cuprates. Because the pseudogap and superconducting gap are similar in energy, it is difficult to separate them in time-integrated linear spectra. Next we discuss a first ultrafast study that probes changes in the gap region, enabling separation of components along the temporal degree of freedom.136 Optimally doped and underdoped thin films of YBCO (with TC = 88 and 68 K, respectively) were excited at 1.6 eV with a 2-MHz repetition rate cavity-dumped Ti:sapphire laser. Terahertz probe pulses spanning the ultrabroadband range from ≈14 to 43 THz with 150-fs pulse duration were generated in a novel GaSe-based technique25 and were employed to detect transient reflectivity changes around the YBCO gap. As shown in Figure 4.17(a), optimally doped YBCO at low temperatures exhibits a subpicosecond reflectivity increase around 22 THz probe frequency (90 meV), followed by a decay on an approximate 5-ps timescale. The amplitude ∆R/R of this

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T= 14 K

2

–2

1 71 K

0

94 K 0

2 4 Time Delay (ps) (a)

6

20

300 fs

160 fs

5 ps

5 ps

30 ps

30 ps

30 (b)

20 40 Frequency (THz)

(d)

1 ∆R/R0 (×10–3)

0

30

–4

40

–6

(c) (e)

T = 100 K

1

Slow

0 50 K –1

Fast

14 K 0

2 4 Time Delay (ps) (d)

∆R/R0 (×10–3)

2

Amplitude (norm.)

∆R/R0 (×10–3)

3

0 6

0

50

100 Temperature (K) (e)

150

Figure 4.17 (a) Transient change of the reflectivity at 21.8 THz in optimally doped YBCO after near-infrared excitation, for different lattice temperatures T as indicated. (b) and (c) Spectra of the reflectivity change ∆R/R0 at different pump–probe time delays for (b) T = 14 K and (c) T = 94 K. (d) Dynamics in underdoped YBCO probed at 35 THz, for different temperatures. (e) Amplitudes of the fast and slow components (symbols) and the sum total (solid line) of the dynamics in underdoped YBCO.

change slowly decreases as the sample temperature is raised, whereas above TC, it changes into a very fast relaxation dynamics with opposite sign. This difference between the behavior below and above TC is further corroborated by the spectral response shown in Figures 4.17(b) and (c). Below TC, the induced reflectivity change in Figure 4.17(b) shows a strong oscillatory structure. It happens to follow exactly the spectral changes in the stationary reflectivity as the sample temperature is lowered from above TC into the superconducting state.137,138 Hence, below TC the experiment directly shows the subpicosecond reduction of the superconducting gap followed by its reformation on a 5-ps time scale, in agreement with the time scales of condensate depletion and reformation observed in the 1- to 3-THz spectral range.131,133 In contrast, the much faster dynamics and featureless spectra above TC (Figure 4.17(c)) can be attributed to the cooling of a hot carrier distribution analogous to dynamics in metals.139 The mid-IR dynamics in underdoped YBCO show a markedly different dynamics. As seen in Figure 4.17(d) for 35-THz probe frequency, the reflectivity change

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Terahertz Spectroscopy: Principles and Applications

below TC = 68 K cannot be explained by a single decaying component. Rather, it is described by two components: a “fast” reflectivity increase and decay, and a slower ≈5-ps constituent that follows the superconducting gap dynamics of optimally doped YBCO. A more detailed analysis across all probe frequencies showed that both components show a similar (but not identical) spectral shape that corresponds to changes of a conductivity gap around 100 meV. It also provides the temperature dependence of the amplitudes (Figure 4.17(e)). The “slow” component vanishes at TC, as in the optimally doped sample—it is a signature of the superconducting gap dynamics because of breakup and reformation of Cooper pairs. In contrast, the fast ≈700-fs component persists up to the pseudogap temperature T* > TC (T* ≈ 160 K in this sample), and results from ultrafast excitation of a second kind of correlated carrier that lies at the origin of the pseudogap phenomenon. The latter may be due to preformed pairs or antiferromagnetic correlations.135 Thus the femtosecond study provides a separation of different components of the ≈30 THz conductivity gap, which remain hidden in linear spectra. The temperature dependence of the transient signal amplitudes also closely follows the strength of an important, 41-meV antiferromagnetic resonance in cuprates, which supports theories in which spin fluctuations couple strongly to the carriers.136 The experiments discussed show the detailed insight into the dynamics of superconductors obtained by employing THz pulses as probes of the fundamental charge excitations. They motivate further work on a larger range of materials to elucidate the differences or similarities in the pseudogap correlations or quasiparticle recombination kinetics. Moreover, almost all previous time-resolved studies of superconductors have employed photoexcitation in the 1- to 2-eV range. This leads to a cascade of scattering processes as quasiparticles relax to the low-energy states, resulting in a fairly indirect and uncontrolled way to break Cooper pairs. We can envision that in the future resonant excitation of superconductors with intense THz pulses at the gap energy 2∆0 will be possible, leading to direct control of (and consequently even more detailed insight into) the nonequilibrium state of superconductors.

4.4 Half-Metallic Metals: Manganites and Pyrochlores 4.4.1 Overview Manganite perovskites such as R1-xDxMnO3 (where R = La, Nd and D = Ca, Sr) and related compounds are similar to the cuprates in that their parent materials (e.g., LaMnO3) are Mott-Hubbard insulators. As a function of doping, fascinating states occur ranging from charge-ordered insulator to ferromagnetic metal phases. The most studied materials of this class during the past decade are the optimally doped manganites around x ≈ 0.3, which includes La0.7Ca0.3MnO3 and La0.7Sr0.3MnO3. Here, “optimal” refers to the maximum ferromagnetic Curie temperature TC. Above TC, these materials are paramagnetic semiconductors, whereas below TC, they transition into ferromagnetic metals. In the vicinity of TC, an applied magnetic field yields a dramatic decrease in the resistivity. This large negative magnetoresistance has resulted in these materials being termed colossal magnetoresistive (CMR) manganites. They are also often called half-metallic manganites referring to the complete spin polarization that occurs below TC. Several excellent reviews of CMR materials

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151

are available.4,5,7,9,140 Recent research has focused on the structural and electronic properties away from optimal doping, with the nature and origin of intrinsic electronic inhomogeneities receiving considerable attention.5 Although the intense interest in manganites during the past decade141 derives from advances in cuprate superconductivity, these materials have been investigated since their discovery in the 1950s. This initial work led to fundamental theoretical insights related to double exchange and experimental explorations such as neutron scattering studies of their detailed magnetic structure.142–146 In the manganites, the structural units of interest are the MnO6 octahedra. The octahedral symmetry breaks the five-fold degeneracy of the Mn d-orbitals resulting in localized t2g spins (S = 3/2) and higher lying eg spins. It is in this eg‑derived band that transport occurs. Additionally, strong Hund’s rule coupling leads to slaving of the eg to the t2g spins, an essential ingredient to the so-called double exchange. In the double exchange model, electron transport between adjacent Mn ions via an intervening oxygen orbital is enhanced if the core spins on adjacent sites are parallel. However, it is now realized that the double exchange model alone cannot account for CMR behavior because the magnitude of the resistivity in the paramagnetic phase is larger than spin scattering predictions.147,148 Coherent lattice effects such as the Jahn-Teller distortion of the MnO6 octahedron also affect the charge transport. The simple picture is that in the paramagnetic phase lattice distortions can follow the carriers from site to site because disorder (of the t2g spins) reduces the carrier kinetic energy, resulting in polaronic transport. Ferromagnetic ordering increases the carrier kinetic energy. Hopping from site to site becomes too rapid for the lattice distortions to follow the carriers, resulting in a metallic state (though polaronic signatures persist below TC). Although the ultimate origin of CMR in the manganites has not been established, it is clear that the spins and polaronic behavior are basic ingredients to consider. In the quest to more fully understand the origin of CMR, many other materials have been synthesized and characterized.149 A particularly important class of materials in this regard are the pyrochlores such as Tl2Mn2O7, which also display a pronounced negative magnetoresistance.150,151 In Tl2Mn2O7, the transition temperature is TC = 120 K. Pyrochlores at first appear quite similar to perovskite manganites: both exhibit CMR, and MnO6 octahedrons are equally important structural subunits. However, experimental and theoretical studies have highlighted important differences. In particular, in pyrochlores double exchange and Jahn-Teller effects are negligible because of the low carrier density and the absence of the Jahn-Teller Mn3+ ion. In addition, separate subsystems are responsible for ferromagnetic ordering (Mn 3d electrons, t2g with S = 3/2 as in manganites) and conduction properties (Tl 6s electrons). The ferromagnetic ordering of the Mn4+ sublattice likely occurs through frustrated superexchange between Mn on adjacent sites. Below TC, indirect exchange between the Mn and Tl orbitals causes a band of primarily Tl 6s character to shift below the Fermi level. It thus appears that the microscopic origin of CMR in pyrochlores and manganites is quite different, though the possibility remains that CMR in both materials derives from mesoscopic effects related to intrinsic electronic inhomogeneities. Below, we briefly discuss time-integrated infrared spectra of La0.7Ca0.3MnO3 and Tl2Mn2O7, followed by a description of time-resolved studies.152–155 These THz studies provide insight into their fundamental

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properties and highlight important differences between these half-metallic correlated materials.

4.4.2 Optical Conductivity and Spectral Weight Transfer

σreal/103 (Ω cm)–1

For the cuprates, we discussed how, on entering the superconducting state, spectral weight transfers from the Drude-like response to the condensate response represented as a Dirac delta function at zero frequency. In the manganites, spectral weight transfer also occurs as a function of temperature.155 However, for these materials the spectral weight transfer occurs from higher energies (~1 eV) to a coherent Drude-like response with the crossover from paramagnetic semiconductor to ferromagnetic metal. This spectral weight transfer is intimately related to the CMR effect in DC conductivity measurements and involves the spin and lattice degrees of freedom. Figure 4.18 shows the temperature dependence of the optical conductivity from 0.2 to 5 eV as a function of temperature for La0.7Ca0.3MnO3 (TC ~ 260 K).155 This clearly shows spectral weight transfer with decreasing temperature and the development of a Drude peak centered at zero frequency. A detailed discussion of the various features of the optical conductivity has been presented.155 The broad feature at ~1.0 eV has been interpreted in terms of the photon-induced hopping of Jahn-Teller polarons. With decreasing temperature, it is this Jahn-Teller peak that partially evolves into a metallic response below TC. This is just as discussed previously, in that with increasing spin polarization the increased carrier kinetic energy is sufficient to overcome polaronic trapping. Further details regarding the higher energy peaks can be found elsewhere.155 In the pyrochlore Tl2Mn2O7 (TC = 120 K), there is also a strong transfer of spectral weight from higher to lower energies with decreasing temperature.153 However, in this material there is no high energy peak ascribable to polarons and the spectral weight transfer likely comes from higher energy (>2 eV) interband transitions consistent with the crossing of a single Tl 6s band as the ferromagnetic half-metallic state develops. Figure 4.19 shows the temperature dependence of the reflectivity for Tl2Mn2O7. At 295 and 160 K, the carrier density is very low such that the phonons

Mn3+

1.5

O eg

1.0

t2g

Mn4+ t

eg

10 K 100 K 125 K 150 K 200 K 225 K 250 K 300 K

t2g

0.5 La0.7Ca0.3MnO3 1

2

3

4

5

Photon Energy (eV)

Figure 4.18 Optical conductivity in La0.7Ca0.3MnO3 as a function of photon energy at various temperatures. The inset gives a schematic depiction of double exchange. (Reprinted with permission. Copyright 1998 by the American Physical Society.)

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Time-Resolved Terahertz Studies of Carrier Dynamics

0.8 0.6 0.4 0.2

40 K

0.8

160 K

0.6 0.4 0.2

0.8 0.6 0.4 0.2

110 K

0.8

295 K

0.6 0.4 0.2

0.01

0.1

1

0.01 Energy (eV)

0.1

1

Figure 4.19 Infrared reflectivity in Tl2Mn2O7 at various temperatures.

are clearly observable. At 110 K, the development of a Drude peak occurs and it is fully developed at 40 K (though the phonons are still not fully screened). Comparing the 40 K data with Figure 4.1(b), we can see that the response is entirely consistent with Drude-like behavior including the dramatic decrease in the reflectivity at ωs which dips and then rises back up to a value consistent with ε∞. This behavior is quite different from that of the manganites, which are much more spectrally congested because of the distinct polaronic response. In Tl2Mn2O7, the carrier density below TC increases as the square of the magnetization to a maximum value of n ≈ 1019 cm-3, about two orders of magnitude smaller than in manganites.

4.4.3 Dynamic Spectral Weight Transfer in Manganites Figure 4.20 shows the results of THz-TDS and optical pump THz probe measurements on a 100-nm thick film of La0.7Ca0.3MnO3 grown on LaAlO3.152,156 In Figure 4.20(a), the real conductivity is plotted as a function of frequency at various temperatures. These conductivity measurements are in the regime ωτ << 1 as indicated by the flat frequency response. A large increase in the conductivity results with decreasing temperature, up to a value of ≈6 × 103 Ω-1cm-1 at T = 4 K, which is similar to poor metals. Figure 4.20(b) shows the temperature dependence at 0.7 THz. The solid line is a fit with σ(T) = σ0 exp(M(T)/M0), where M(T) ∝ (1-T/TC)b is the magnetization and β = 0.33. The results are in good agreement with DC resistivity measurements. The spin ordering thus clearly plays a role in determining the magnitude of the conductivity. However, as we will see from the ultrafast measurements, thermally disordered phonons (i.e., not polarons) are also quite important. Figures 4.20(c) and (d) show the dynamic changes in conductivity following photoexcitation with 100-fs 1.55-eV pulses at T = 15 and 180 K, respectively.152,156 A two-component conductivity dynamics is observed after the creation of a nonthermal electron distribution. Relaxation of this hot quasiparticle distribution cools through phonon emission. This corresponds to the fast resolution-limited component in the data. The longer time scale is in turn related to the spin-lattice thermalization, in which spins eventually come into thermal equilibrium with the phonons. Between 15 and 180 K, there is a strong change in the relative amplitudes of these two

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Terahertz Spectroscopy: Principles and Applications 6

La0.7Ca0.3MnO3

σr/103 (Ω cm)–1

5

4K

50 K

4

70 K

3

90 K

2

130 K 170 K 200 K 230 K

1 0

6

15 K

5 4 3

1.0

0.5

0

Frequency (THz) (a)

40 Time (ps) (c)

80

10 σr/103 (Ω cm)–1

1.0

180 K

0.9

1 La0.7Ca0.3MnO3 Tc = 250 K

0.1

0

100

0.8 200

0

50

Temperature (K)

Time (ps)

(b)

(d)

100

Figure 4.20 (a) and (b) Temperature-dependent real part of conductivity in La0.7Ca0.3MnO3 measured using THz-TDS. (c) and (d) Two-component decrease in the conductivity after photoexcitation.

components: the fast electron-phonon related decrease in the conductivity and that related to spin-lattice thermalization. At low temperature, the conductivity decrease is dominated by electron–phonon relaxation, whereas closer to TC, the spin-lattice thermalization component becomes more dominant. It is possible, using the timeintegrated and time-dependent data in Figure 4.20 (and more data at other temperatures), to extrapolate the conductivity in the TS -TL plane by expanding ln(σ(TS, TL)) in a power series and performing a least squares fit. TS is the spin temperature and TL is the lattice temperature. This fitting procedure has been accomplished for La0.7Ca0.3MnO3 to third order in TS and TL . The results are shown in Figure 4.21 as contours of constant ln(σ) in the TS -TL plane.152,156 Conventional measurements do not deviate from equilibrium as indicated by the white diagonal line in the figure. However, the optical pump THz probe experiment starting from a point on the equilibrium line allows for access to the portion of the TS -TL plane below the diagonal because the excited electrons preferentially couple to the phonons during the initial 2 ps. This optically induced change in the phonon temperature is shown by the solid black arrow. The magnitude of this change is given by z/CL where z is

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Time-Resolved Terahertz Studies of Carrier Dynamics 1

: Spin-lattice equilibration slope = Cphonon/Cspin

: Initial Excitation α Elaser/Cphonon

Tspin/Tc

0.8

0.6

0.4 Eq

0.2

0

u

ili

a br

tio

n

Li

: ne

=

in

T sp

8.5

2

8 n

no

ho

Tp

7.5 7 6.5

1

6 5.5 5

0

0.2

0.4 0.6 Tphonon/Tc

0.8

1

Figure 4.21 Plot of the natural log of the conductivity in the phonon spin temperature plane. The white diagonal line denotes the equilibrium line (TS = TL) of conventional timeintegrated measurements. The white lines are contours of constant conductivity.

the absorbed laser energy and CL is the lattice specific heat. The sample then returns to the equilibrium line as shown by the dashed arrows with the slope CL/CS (where CS is the specific heat due to the spins). Depending on the initial temperature, z, and CL , S, the observed conductivity decrease can depend predominantly on changes in the spin temperature, the phonon temperature, or both. When the contours of constant conductivity are nearly perpendicular to the TL axis, the change in conductivity is dominated by the phonons. This is the case at low temperatures as shown in the Figure 4.21 (label 1). However, at higher temperatures near TC, the lines of conductivity are now nearly parallel to the TL axis, and thus the changes in conductivity are mostly related to changes in the spin temperature (label 2). In short, changes in conductivity at low temperatures are dominated by thermally disordered phonons, while spin fluctuations and the associated polaronic correlations dominate at higher temperatures.152,156

4.4.4 Carrier Stabilization in Tl2Mn2O7 through Spatial Inhomogeneity For the manganites, the dynamics result from energy transfer from the excited quasiparticle distribution to the lattice and spin degrees of freedom. Furthermore, the metallic-like carrier densities of approximately 1021 cm–3 means that, for moderate fluences, it is not possible to excite a large fractional change in the carrier density. The pyrochlore Tl2Mn2O7, however, has a much lower carrier density, making it possible to photoexcite a large fractional change in the carrier density. In fact, optical pumping at 1.55 eV photoexcites electrons from O 2p orbitals to the Tl 6s-derived band. Furthermore this occurs (below TC) preferentially in the minority spin band

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Terahertz Spectroscopy: Principles and Applications

∆R/R × 100

(i.e., the carriers in the Tl 6s band have their spins oppositely oriented to the Mn t2g spins). Tl2Mn2O7 cannot be grown as a thin film and the synthesis of single crystals is accomplished using high pressure techniques.150 Thus optical conductivity (Figure 4.19) and time-resolved measurements are performed in reflection. To track the changes in the photoexcited carrier density, dynamic shifting of the plasma edge was measured at different wavelengths. In the following, results are presented for measurements at 5-µm wavelength. Figure 4.22(a) shows the photo-induced changes ∆R/R in the reflectivity as a function of time at various temperatures. At all temperatures, the traces show a rapid change in reflectivity at zero delay, followed by a fast relaxation with a timescale of a few picoseconds and a slower component with a 10- to 100-ps timescale. The negative ∆R/R is consistent with a photo-induced increase in the carrier density that causes a shift of the plasma edge to higher energies. The relatively fast ≈10-ps initial relaxation reduces the photoexcited carrier density. For T << TC, nearly all of the carriers follow this recombination pathway. However, near and above TC, a large offset appears in ∆R/R indicating the establishment of a long-lived carrier population.

0 –1 –2 –3 –4 0

0 50 K

–1

–1

–3 0 –1 –2 –3 –4

180 K

–2 –3

110 K

–2

0 100

300

500

295 K

0 100

300

500

Time (ps)

PI Carrier Density/1018 (cm–3)

(a) 12 10 8

1

6

2

4

50 ps 500 ps

2 0

3

0

50

100

150

200

250

300

Temperature (K) (b)

Figure 4.22 (a) Photo-induced reflectivity changes at 5 mm wavelength in Tl2Mn2O7 for various temperatures. (b) Experimentally determined photo-induced carrier densities at 50 and 500 ps.

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Time-Resolved Terahertz Studies of Carrier Dynamics (b) T

(a) T << Tc

Tc

λ ξ

t2g

t2g

O 2p hole Tl 6s electron

Figure 4.23 Left: Photoexcited spin singlets for T << TC. The small circle is a domain with spin-down t2g states and embedded in the larger area which is a domain with spin-up t2g states. The e-h pairs recombine quickly in this temperature range. Right: Electron hole pairs after photoexcitation as T approaches TC from below. In this temperature range, holes within λ of a boundary migrate to domains of opposite t2g spin orientation (dotted arrows), effectively preventing electron-hole recombination.

To obtain a more quantitative understanding of the charge and spin dynamics in Tl2Mn2O7, the photo-induced change in carrier density (∆n) was extracted using the time-integrated optical conductivity and the reflectivity dynamics. The calculated photo-induced carrier density at 50 and 500 ps is plotted as a function of temperature in Figure 4.22(b). In region 1 (T < 90 K), the photo-induced carrier density at both time delays is nearly equal, increasing as T approaches TC. As T increases past TC, ∆n(t = 500 ps) remains nearly constant in region 2 (90 < T < 170 K) and then rapidly decreases in region 3 (T > 170 K), completely disappearing by 295 K. In the following paragraph, we focus on region 1. Further details regarding region 2 and 3 can be found in the original publication.154 Energy considerations and dipole selection rules dictate that the pump excitation creates e‑h spin singlets in the minority spin manifold. At the lowest temperatures the e-h singlets are created in a homogeneous background of t2g spins (Figure 4.23). Under these conditions, there is no energetic or dynamic constraint hindering recombination. That is, e-h wavefunction overlap in a spin singlet configuration occurs with a high probability, leading to recombination and a fast recovery of the carrier density (15 ps) as the data in Figure 4.22(a) shows. However, with increasing temperature a long-lived carrier density develops. This indicates that a competing pathway develops that stabilizes a fraction of the photoexcited carriers against recombination. A route for stabilizing a fraction of the initially photoexcited carriers is through the development of spatial fluctuations with a characteristic length scale given by the magnetic correlation length χ(T) of the Mn t2g spins responsible for ferromagnetic ordering. That is, small regions of opposite t2g spin develop in an otherwise homogeneous background. This is schematically depicted in Figure 4.23(a), in which, for T < TC the density of t2g (↓) domains is quite low, whereas it strongly increases upon approaching TC as shown in Figure 4.23(b). Of course, χ is also temperature- dependent, diverging at TC as for any ferromagnet, but χ ≈ 1 for T < 90 K in region 1,

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yielding a correlation length of order 0.1 nm.157 In this simple picture, we emphasize that the density of “defects” of opposite t2g orientation captures the essential physics of carrier stabilization. The spatial inhomogeneity that develops has important implications for photoexcited e-h pairs. From the bandstructure,154 it is energetically favorable for O 2p holes to cross a t2g (↑)–t2g(↓) boundary, whereas there is an energetic barrier for the Tl 6s electrons. Thus the spatial inhomogeneity in the t2g spin manifold leads to spatial segregation of the photoexcited Tl 6s electrons and O 2p holes, which in turn inhibits recombination. However, this process competes with recombination, and spatial segregation can only occur for holes that are initially created within a distance λ of a boundary as depicted in Figure 4.23. The net result is that with increasing temperature the number of “defects” increases, leading to an increase in the Tl 6s carrier density because more O 2p holes are trapped. Thus, in region 1, the amplitude of the fast relaxation component is due to recombination while that of the slow component is due to carrier stabilization. Therefore the photo-induced carrier density at t = 500 ps should be proportional to the ratio of slow to fast amplitudes in the ∆R/R traces. This in turn scales with the ratio of the volume occupied by t2g(↑) spins to that occupied by t2g(↓) spins, including a factor γ = (λ + χ)3. Here, γ is the effective volume where the probability of trapping an O 2p hole is greater than a recombination event. This results in an expression for the excess Tl 6s electron density ∆n given by Dn = C0

γ (1 - M ) 2 ξ 3 - γ (1 - M )

(4.12)

where the magnetization M is normalized to its saturation value and C0 is a proportionality constant. The solid line in Figure 4.22(b) shows ∆n plotted as a function of temperature for T < 90 K, revealing good agreement with the data. This analysis yields λ ~ 0.79 unit cells, indicating that the photoexcited holes can only travel a very short distance before recombining. In summary, spatial inhomogeneity that develops in the t2g spins (responsible for a decrease in M) stabilizes a fraction of the photoexcited carriers.154 These results are consistent with CMR in Tl2Mn2O7 arising from carrier localization from strong spin fluctuations without invoking phonons, which is in dramatic contrast to the results for the manganites where phonons (whether thermally disordered or of a polaronic nature) were crucial. These results also suggest that Tl2Mn2O7 may be a particularly simple example where the transport properties are determined by intrinsic nanoscale inhomogeneity occurring near a second-order phase transition.

4.5 Summary and Outlook We hope this chapter has revealed the potential of time-resolved THz techniques in the study of many-body correlations in bulk and nanostructured semiconductors, superconductors, and strongly correlated transition metal oxides. In semiconductors, we have reviewed experiments that study the free carrier response and buildup of

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Coulomb correlations, distinct signatures of intraexcitonic transitions that test the analogy between e-h gases and atomic systems, and the ultrafast polarization kinetics of intersubband transitions. Time domain THz spectroscopy provides for a direct determination of the real and imaginary parts of conductivity, which in superconductors directly reveals the quasiparticle and superfluid densities and the excitation gap 2∆ of the correlated charge pairs. Time resolving the conductivity dynamics, in turn, gives the most direct insight into the temporal breakup and recombination kinetics of quasiparticles in both conventional and high-TC superconductors. Time-resolved THz spectroscopy of manganites and pyrochlores gives important insight into dynamic spectral weight transfer, quasiparticle interactions with lattice and spin degrees of freedom, and stabilization processes of carriers through spatial inhomogeneities. Of course, these results barely scratch the surface of what is possible and within the context of dynamic studies on nanoscale and correlated materials there is still a great deal to be investigated. Intraexcitonic spectroscopy represents a rising field to investigate low-temperature carrier phases in semiconductors, where THz probes of optically dark states are of increasing interest in the search for excitonic BoseEinstein condensates.38,158 There are also a number of differences in the dynamics of YBCO and Bi-2212 presented in Section 4.3. It is not yet clear if these differences are intrinsic to the properties of these materials or are related to experimental parameters such as the incident fluence. It would be interesting to probe the dynamics for a larger range of materials and over a broader doping range. This holds for the manganites as well where, to date, dynamic optical pump THz probe experiments have only been performed on optimally doped samples (e.g., those that display halfmetallicity). However, as function of doping, the manganites display an enormous number of states including charge ordering, orbital ordering, and various magnetically ordered states. It will be important to understand the dynamics in each of these distinct phases especially as research on these materials focuses more and more on phenomena associated with phase coexistence.5,140 For example, recent mid-IR studies on Nd0.5Sr0.5MnO3, which transitions from paramagnetic semiconductor to ferromagnetic metal to charge-ordered insulator with decreasing temperature suggest that the dynamic polaron response provides a way to probe phase coexistence.159 There are also a host of other materials in which probing the mid- or far-IR dynamics can be expected to provide considerable insight. For example, several visible and x-ray experiments have studied the dynamics of VO2, giving insight into the insulator–metal transition and the cause–effect relationships between lattice and electronic degrees of freedom.160,161 Recent THz experiments complement these previous studies.162 This motivates, more generally, the study of photo-induced phase transitions both in materials where the phase transition is thermally accessible (as in VO2) and those where photoexcitation leads to a metastable state which is not thermodynamically accessible as observed previously.163 Other examples include f-electron intermetallics, in which hybridization between localized f moments and the conductions electrons results in a THz hybridization gap.164 All optical pump probe studies have been used to probe the quasiparticle dynamics as a function of temperature and show a strong dependence on the opening of a hybridization gap,165 but, to date, there have been no experiments which directly probe the dynamics at the gap edge as reported for superconductors.136 Finally, we mention the area of multiferroics,

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which is becoming an increasingly active research area. In multiferroics such as LuMnO3 or Ba0.5Sr1.5Zn2Fe12O22, there are two coexisting order parameters with the most common being ferroelectricity and antiferromagnetism. This leads to magnetoelectric coupling where the magnetic order can be modified with an electric field or vice versa.166–171 There is a great deal to be learned about these materials using both time-integrated and time-resolved optical spectroscopy. For example, using THz spectroscopy, electromagnons have recently been reported.172 This suggests that optical pump THz probe experiments may provide insight into the dynamics and microscopic origin of magnetoelectric coupling in these materials. In addition to probing the dynamics of other correlated-electron materials, experimental and theoretical advances are also expected to provide new opportunities. One area that seems to be very exciting is to use THz pulses with high peak electric fields as described in Section 4.2. Fields on the order of 1 MV/cm will enable coherent control of low-energy transitions in excitons and various nanoscale materials, and may lead to interesting nonlinear dynamics in correlated electron materials such as superconductors, manganites, and multiferroics. The majority of optical pump THz probe studies to date have used near-IR or visible pulses for photoexcitation. It would be of enormous value to have greater spectral agility for the photoexcitation. This is challenging in the case of optical pump THz probe spectroscopy because the pump spot size needs to be quite large (typically several millimeters) for uniform excitation of the THz probe beam, whereas the THz probe ideally must remain sensitive requiring fairly high repetition rates. Finally, we mention the need for a more concerted theoretical effort in this area. For semi-conductors, sophisticated models have been developed that provide insight into the fundamental quantum kinetics of carrier interactions.173 Likewise, it would be of considerable interest if such time domain theories were extended to encompass the physics of strongly correlated electron materials. While our contemporary understanding is naturally limited, we believe that the many facets of time-resolved THz techniques hold promise of unprecedented insight into the basic properties of complex materials.

Acknowledgments We thank our many collaborators on infrared and THz spectroscopy studies reviewed here, in particular D. S. Chemla, T. Elsaesser, M. Woerner, M. A. Carnahan, A. J. Taylor, S. A. Trugman, J. Demsar, R. P. Prasankumar, B. A. Schmid, R. Huber, J. F. Ryan, and many more. Preparation of this review was in part supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-05CH11231 and by the Los Alamos National Laboratory LDRD program. We would also like to thank the authors of various articles for the permission to reproduce their figures in the preceding text.

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5

Time-Resolved Terahertz Studies of Conductivity Processes in Novel Electronic Materials Jie Shan

Case Western Reserve University

Susan L. Dexheimer

Washington State University

Contents 5.1 Introduction................................................................................................... 172 5.2 Charge Transport in Photo-Excited Insulators: Polarons in Single-Crystal Sapphire............................................................................ 173 5.3 Charge Transport in Disordered Electronic Materials: Dispersive Transport in Amorphous Semiconductors and Semiconducting Organic Polymers.......................................................................................... 176 5.3.1 Amorphous Semiconductors............................................................. 177 5.3.2 Semiconducting Organic Polymers.................................................. 182 5.4 Polaron Formation and Dynamics in Molecular Electronic Materials......... 184 5.4.1 Dynamics of Polaron Formation: Quasi-One-Dimensional Systems............................................................................................. 185 5.4.2 Carrier Transport in Pentacene......................................................... 187 5.5 Charge Transport in Nanoscale Materials: Nanocrystalline Semiconductors and Quantum Dots............................................................. 188 5.5.1 Conductivity and Dielectric Screening in Nanoporous TiO2........... 189 5.5.2 Excitons in Semiconductor Quantum Dots...................................... 190 5.6 Extending into Mid-Infrared Spectral Regime: Carrier Dynamics in Graphite...................................................................... 193 5.7 Summary....................................................................................................... 196 Acknowledgments................................................................................................... 196 References............................................................................................................... 196

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5.1 Introduction Terahertz (THz) spectroscopy based on femtosecond laser techniques1–12 has emerged as a powerful probe of charge transport and carrier dynamics. The technique makes use of ultrashort pulses of propagating electromagnetic radiation to measure conductivity in the THz spectral regime. (A frequency of 1 THz = 1012 Hz, and corresponds to an energy of 4.2 meV = 33 cm-1 = 48 K and a wavelength of 300 mm.) In particular, when combined with a time-synchronized femtosecond excitation pulse, THz spectroscopy is suitable for the investigation of electronic charge transport under nonequilibrium conditions.13–17 These attributes permit THz spectroscopy to circumvent many of the constraints of conventional transport measurement techniques. Conventional electrical characterization techniques do not generally allow access to material response at frequencies as high as those of carrier-scattering processes, information that is of great value in examining the fundamentals of charge transport mechanisms, and are not suited for probing very rapid changes in the material response, so that they typically cannot explore the interesting and important regime of nonequilibrium electronic excitation. In addition, in conventional measurements of charge transport, one must have samples amenable to good electrical contact, an issue that can be a severe constraint, especially in studies of nanoscale materials. The THz probing technique, used in conjunction with optical pump excitation, can overcome all of these limitations. In this chapter, we describe the application of this approach to studies of the a number of aspects of fundamental properties of charge transport in novel electronic materials and highlight new insights into the properties and dynamics of charge carriers that have resulted from the distinctive measurement capabilities of time-resolved THz spectroscopy. We focus in particular on carrier dynamics made accessible by recent ultrafast time-resolved techniques; measurements of steady-state properties in this important spectral region using both recently developed time-domain THz and well-established far-infrared/submillimeter techniques have been applied in a wide range of studies of materials, as reviewed extensively elsewhere.1–12 In Section 5.2, we discuss time-resolved THz studies of charge transport in insulators. Unlike in metals and semiconductors, because of difficulties in doping or thermally exciting the system, charge carriers are often present only under nonequilibrium conditions. Insulators, therefore, provide an ideal laboratory to demonstrate the unique capabilities of time-resolved THz spectroscopy. In addition, the subpicosecond timescale of the probe electric field transient implies that we have access to frequencies up to the THz range, frequencies comparable to charge carrier scattering rates in typical crystalline materials or transfer rates between neighboring sites in disordered systems. In Section 5.3, we present examples of how THz spectroscopy can yield important information about charge transport in disordered electronic materials, focusing in particular on dispersive transport in amorphous semiconductors and photoconducting polymers. The short-pulse nature of the THz probe also implies that we have access to the dynamics of rapidly evolving systems. In Section 5.4, we describe time-resolved THz studies of fast photocarrier dynamics in molecular electronic materials, including studies of the dynamics of polaron formation in quasi-one-dimensional systems and of photoconductivity in molecular crystals. Since the THz measurements are made with traveling electromagnetic waves,

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no electrical contacts are needed. In Section 5.5, we describe how one can take advantage of this property to perform measurements of nanoscale materials, systems for which formation of electrical contacts may be difficult and, further, may significantly modify the properties of the material. Finally in Section 5.6, we indicate how the technique of THz spectroscopy can be extended from the far-infrared to the mid-infrared regime and can be used to obtain both amplitude and phase information about material response in this spectral regime after photoexcitation. These ultrabroadband THz sources can also provide pulse durations well below 100 fs, permitting higher time resolution for probing ultrafast processes. Studies of the impact of strongly coupled optical phonons on the electronic charge transport properties in graphite provide an example of this approach. The aim of this chapter is to provide an overview of the diverse applications of time-resolved THz spectroscopy for probing the properties of electronic materials.13–79 Given the extensive literature that has emerged over the past several years based on this powerful experimental methodology, we do not attempt to be comprehensive in our review, but rather to select representative studies that highlight the capabilities of the method and the range of its application. We focus in particular on novel electronic materials in which time-resolved THz spectroscopy has provided new insight into conductivity mechanisms. Time-resolved THz spectroscopy is also especially well-suited for studies of carrier dynamics in conventional semiconductors, superconductors, and strongly correlated electronic materials, as discussed in detail in the chapter by Kaindl and Averitt in this book. The instrumentation and experimental techniques for the time-resolved THz measurements are discussed in detail in the first part of this volume.

5.2 Charge Transport in Photo-Excited Insulators: Polarons in Single-Crystal Sapphire The nature of charge transport in normally insulating solids or liquids is a subject of considerable technologic and fundamental importance. From the technological standpoint, understanding of electronic transport in insulators is crucial for important processes such as electrical and optical breakdown, as well as for their use as radiation detectors. From the basic physics viewpoint, there is inherent interest in being able to examine charge transport in insulators, just as has been done comprehensively in metals and semiconductors. Of particular significance is the role of polarons in ionic insulators. Because of the strong interaction between conduction electrons and phonons in such ionic materials, electrons are surrounded by a cloud of virtual phonons, forming quasiparticles called polarons. Polaronic transport in various regimes, depending on the electron–phonon interaction strength, has been a subject of much theoretical attention. However, as mentioned previously, unlike in metals and semiconductors, charge carriers typically exist in insulators only under nonequilibrium conditions and are not accessible to conventional electrical characterization techniques. Conventional approaches are further impeded by the difficulty of making electrical contacts to insulators and the short lifetime of the carriers prior to trapping or recombination. THz spectroscopy, combined with ultrafast optical excitation, opens up new possibilities to investigate charge transport in insulators.

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Recently, model systems including a variety of insulating liquids53,54 and solids such as single-crystal sapphire (Al2O3)74 and rutile (TiO2)34 have been studied. We use the example of single-crystal sapphire74 to illustrate the approach. In the measurements, conduction electrons were injected in sapphire (band gap Eg = 8.7 eV) through two-photon absorption of the third harmonic of a Ti:sapphire laser. The electric-field waveforms E(t) of THz pulses transmitted through the unexcited sapphire crystal and the pump-induced change in the THz waveform, ∆E(t), were recorded. A delay of several picoseconds after pump excitation was chosen to ensure that any hot carrier effects had abated, but that the carrier density had not decreased significantly. Figure 5.1(a) displays representative waveforms of E(t) and ∆E(t) measured for sapphire at room temperature. To extract the pump-induced complex dielectric function ∆ε(ν) or complex conductivity σ(n), as a function of the frequency n, we take the Fourier transforms of the measured THz waveforms. In the limit of a weak perturbation to the THz response, as is appropriate for the current experimental conditions, a linear relation between the pump-induced response and the ratio of the Fourier transform of the waveforms can be found -1

 2 πnl ε( n) - ε ext ( n)  ∆ E ( n) ∆ε( n) = 2 ε( n) i ,  × c / ε ( n ) ε ( n ) + ε ( n ) E ( n) extt  

σ( n) = -2 πi nε 0 ∆ε( n). Here E(n) and ∆E(n) denote the Fourier transform of the corresponding time-domain quantities, E(t) and ∆E(t), ε0 is permittivity of free space, and ε(n), and εext(n) are the

0.6

E(t)

0 –100 1

σ´ and σ´´(1/Ωm)

Electric Field (a.u.)

100

∆E(t)

0 –1 0

1

2 Time (ps) (a)

3

σ´

0.3

0.0

σ´´

0.5

1.0 Frequency (THz) (b)

1.5

Figure 5.1 (a) Top panel: the transmitted THz electric-field waveform E(t) in single-crystal sapphire (Al2O3) at room temperature; bottom panel: pump-induced change in the THz waveform ∆E(t) (dots) and a fit to the Drude model (solid curve). The pump fluence was 3 J/m2 and the pump-probe delay was 5 ps. (b) Frequency dependence of the real (σ′) and the imaginary part (σ″) of the pump-induced complex conductivity (dots) of single-crystal sapphire at room temperature, extracted from the waveforms in (a). The solid curves represent a fit to the Drude model. (From Shan, J. et al., Phys. Rev. Lett. 90, 247401, 2003. With permission.)

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complex dielectric functions of the unperturbed sample and its surrounding material (e.g., air, sample cell windows), respectively. This expression includes propagation through the excited sample of thickness l, as well as interfacial transmission losses at the boundary of the sample. The influence of multiple reflections is slight and has been omitted for simplicity. The frequency-dependent complex conductivity, σ(n) = σ′(n) + iσ″(n), corresponding to the waveforms of Figure 5.1(a) is shown in Figure 5.1(b). It is dominated by the electron conductivity because of the much smaller value of the electron mass (and, correspondingly, the larger value of the electron mobility) than that of the hole. The experimental results (dots) are described satisfactorily by the simple Drude model for conductivity, σ( n) = ε 0 ω 2p /( γ 0 - 2 πi n), as shown by the solid lines. Here, γ0 denotes the electron scattering rate, and ωp = {ne2/(ε0 m*)}1/2 the plasma frequency, which is related to the density of conduction electrons (n), the elementary charge (e), and the electron effective mass (m*). The corresponding fit to the time domain data is shown in Figure 5.1(a) as a solid line. From the fitting procedure, a room-temperature scattering rate of γ0 = (95 fs) –1 and a plasma frequency of ωp = 2π × 0.15 THz were obtained. The fact that the frequency-dependent conductivity is adequately described by the Drude model indicates that conduction electrons form large polarons in sapphire, quasiparticles that remain mobile but exhibit an enhanced effective mass m* as they travel through the solid because of the electron-phonon interaction. A schematic representation of large polaron transport in ionic crystals is shown in Figure 5.2(a). This behavior contrasts with charge transport in nanostructures that exhibit strong quantum confinement effects and in disordered systems, as discussed in subsequent sections of this chapter. The nature of the charge transport can be further explored by examining the dependence of the conductivity on the sample temperature. With decreasing sample temperature, the conductivity retains its Drude form, but exhibits a sharply reduced scattering rate. As shown in Figure 5.3, the scattering rate drops from γ0 = (95 fs) –1 at room temperature to γ0 = (5 ps)–1 at 40 K, a decrease by a factor of approximately 50. An examination of the temperature dependence of γ0 reveals an activated behavior at high temperatures (T > 200 K) and a weaker variation at low temperatures (T < 200 K). These features are characteristic of LO phonon and acoustic phonon scattering, respectively. A comparison of the experimental findings with large polaron theories revealed important parameters characterizing electron–phonon interactions, as well as distinctions among the various theoretical models. The solid line in Figure 5.3 corresponds to a fit to the polaron model of Low and Pines80 with an effective electron mass of m* = 0.30 m 0 (the electron mass), and the dashed line represents only the acoustic phonon contribution to the total scattering rate. One particularly interesting result of this study is that the electron mobility in high-purity sapphire can be as high as 600 cm2/V⋅s at room temperature and that the mobility increases sharply with decreasing temperature. At 40 K, the electron mobility exceeds 10,000 cm2/V⋅s, a value normally associated with high-mobility semiconductors. Thus, at low temperatures, the complete freezing out of the LO phonons and the reduced interaction with acoustic phonons can lead to high carrier mobility in insulators having sufficiently low defect densities.

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(a) 100

E

50

P

0 –50

R

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(b)

(c)

Figure 5.2 Schematics of charge transport in three representative systems. (a) Polaronic charge transport in ionic crystals in the limit of large polarons. Because of the interaction between conduction electrons and phonons, charge carriers are surrounded by a cloud of virtual phonons. For intermediate coupling, the charge carriers form large polarons, quasiparticles that remain mobile but exhibit an enhanced effective mass m* as they travel through the solid. (b) Polarizability of an exciton strongly quantum confined in a QD of radius R to an externally applied electric field E. In the absence of the field (left), the ground-state exciton possesses a spherically symmetric electron and hole distribution centered inside the dot (only the electron distribution shown), which result in a zero net dipole moment. The externally applied electric field perturbs the exciton wave function and the charge spatial distribution (right), which leads to a net dipole moment P in the QD. (From Dakovski, G.L. et al., J. Phys. Chem. C, 10.1021/ jp069026o, 2007. With permission.) (c) Hopping transport in a disordered system: charge carriers hop between localized electronic states centered at different locations.

5.3 Charge Transport in Disordered Electronic Materials: Dispersive Transport in Amorphous Semiconductors and Semiconducting Organic Polymers THz spectroscopy has proved to be an important tool for examining charge transport properties in disordered systems, which include media as diverse as glasses, amorphous semiconductors, organic polymers, and biological materials. The intense

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Scattering Rate (THz)

25 LO Phonons – hω – s kT e

20 15 10 Acoustic phonons

5 0

T 3/2 0

100

200 300 Temperature (K)

400

Figure 5.3 Temperature dependence of the scattering rate in high-purity sapphire (squares). The solid line is a fit to a transport model that includes LO phonon and acoustic phonon scattering. In particular, the LO phonon scattering is described by the polaron model of Pines and Low. The dashed line shows the acoustic phonon contribution. At the temperatures above 200 K, denoted by the dotted line, LO phonon scattering dominates. (From Shan, J. et al., Phys. Rev. Lett. 90, 247401, 2003. With permission.)

interest in such disordered systems can be attributed both to the technological relevance of many noncrystalline materials and to the novel fundamental phenomena that arise in materials lacking translational symmetry. Mechanisms for charge transport in disordered systems can be very different from the conventional transport processes in crystals. Typically, the disorder results in localization of low-energy carriers, substantially reducing the effective carrier mobility in these materials. Multiple trapping of carriers into localized states with a distribution of binding energies, as well as hopping transport between localized sites at a distribution of distances, can result in dispersive transport, in which photogenerated carriers exhibit an effective time-dependent mobility. The processes associated with the localized carriers not only determine the conductivity of the materials, but also influence diffusion-mediated carrier recombination. These processes reflect the fundamental physics of disordered materials and are also of critical importance for optoelectronic and photovoltaic applications. The sections that follow review studies of two types of disordered electronic materials for which timeresolved THz spectroscopy has provided new insight into conductivity processes. The first section reviews studies of photoconductivity in amorphous semiconductors, focusing in particular on hydrogenated amorphous silicon and silicon–germanium alloys, in which the conductivity is dominated by multiple trapping processes in a distribution of band tail states. The subsequent section reviews studies of photoexcited carriers in conjugated conducting polymers, in which carrier transport is dominated by structural disorder associated with a distribution of effective conjugation lengths.

5.3.1 Amorphous Semiconductors The loss of long-range crystalline order in a semiconductor lattice has a profound impact on the nature and dynamics of the electronic excitations. Amorphous silicon

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Energy

Extended States

Mobility Edge Band Tail States Eg Deep Traps

Density of States

Figure 5.4 Schematic density of electronic states in an amorphous semiconductor, showing the extended transport states in the valence and conduction bands, localized band tail states at the edges of the valence and conduction bands, and mid-gap deep defect states.

is well established as a model system for studying the physics of disorder in semiconductors and has important technological applications in thin film transistors and photovoltaic devices. Amorphous silicon–germanium alloys are also important for solar energy applications, because variation of the germanium content allows the bandgap to be tuned into the near-infrared spectral range. The schematic density of states for appropriate for amorphous silicon and related materials is shown in Figure 5.4. An important reference energy is the demarcation between the extended electronic states, which have transport properties analogous to those of extended band states in crystalline semiconductors, and the distribution of localized, low-mobility band tail states that result from the disorder.81 In nonhydrogenated amorphous silicon, dangling bonds result in a high density of mid-gap defect states that act as deep traps, resulting in poor electronic properties. In devicegrade materials, hydrogenation largely passivates the dangling bonds, so that the conductivity is dominated by carriers in the energy bands and in the bandtails, which extend over a distribution of energies on the order of a few tens of meV below the band edge. Time-resolved THz measurements of photoexcited carrier processes in nonhydrogenated a-Si and radiation-damaged silicon on sapphire, both of which are materials that have a high density of deep defect states, were carried out by Lui and Hegmann.82 As expected, the time-resolved response in these materials was found

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to be dominated by rapid, irreversible trapping into deep defect states, which was manifested as a simple exponential signal decay with a time constant ranging from 0.58 ps to 5.5 ps, correlated with the expected defect density, with minimal dependence on fluence or temperature. In what follows, we will focus on time-resolved studies of photoconductivity carried out by Nampoothiri and Dexheimer63,64,66 that are relevant to device-grade a-Si:H and a-SixGe1-x:H, and demonstrate dispersive transport behavior characteristic of carrier dynamics dominated by bandtail states. The fast dynamics of photoexcited carriers in hydrogenated amorphous silicon (a-Si:H) and related materials have been previously investigated in an extensive series of studies using picosecond and femtosecond pump–probe techniques in the visible and near-infrared spectral ranges.83–91 In these experiments, carriers are optically generated in the extended states, and the resulting induced absorbance is probed during the time evolution of the photoexcited carrier distribution. In general, the response observed at visible and near-infrared probe wavelengths is highly nonexponential, with a strong dependence on the initial photoexcited carrier density. The interpretation of these measurements has generated considerable controversy— different models used to interpret the time-resolved response result in drastically different timescales for the fundamental localization processes, in which carriers initially excited into extended states trap into lower energy band tail states, and therefore have different implications for the nature of the observed bimolecular recombination and its relation to carrier diffusion properties. A critical limitation in this time-resolved work has been the apparent lack of sensitivity of the measured time-resolved optical response to the carrier state; modeling of the time-resolved optical response in terms of bandtail trapping requires the assumption of a difference in optical absorbance between photoexcited carriers in extended states and in localized (bandtail) states, and evidence for these differences has been unclear in the visible and infrared wavelength ranges accessible in earlier work. In contrast, measurements in the THz spectral range prove to be extremely sensitive to carrier localization, allowing clear observation of the carrier trapping dynamics that give rise to the unusual electronic properties characteristic of amorphous semiconductors. Representative measurements on a thin film of a-SixGe1-x:H (x = 0.5) prepared by plasma-enhanced chemical vapor deposition onto a sapphire substrate are presented in Figure 5.5, which displays the time-resolved induced absorbance probed in the near-infrared and THz spectral regions after photoexcitation of carriers into the extended electronic states using optical pulses 35 fs in duration centered at 800 nm generated by an amplified Ti:sapphire laser system.63 The near-infrared probe measurements are carried out with probe pulses at 800 nm. For the THz probe measurements, single-cycle THz pulses are synchronously generated by optical rectification in ZnTe and are detected by electrooptic sampling. In the time-resolved THz measurements presented in Figure 5.5, the induced absorbance signal is detected at the peak of the THz electric field waveform as a function of time delay between the pump and THz probe pulses. This type of measurement reflects the change in absorption averaged over the frequency spectrum of the THz pulse, as long as there is no significant change in the index of refraction or strong dispersion of the response,23 which has been verified to be the case in more detailed frequency-dependent measurements

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0.1

–ΔT/T

–ΔT/T

1.0

0.5

0.0

0

4 8 Delay (ps) (a)

12

0.01

0.001

1

Delay (ps)

10

(b)

Figure 5.5 Time-resolved THz and near-infrared (800 nm probe) response in a-SixGe1-x:H (x = 0.5) after photoexcitation into the extended states with pulses 35 fs in duration centered at 800 nm. (a) THz (black lines) and near-infrared (gray lines) measurements at initial carrier densities of approximately 4 × 1019 cm-3 (dashed lines) and 1 × 1019 cm-3 (solid lines) are presented on a linear scale, normalized to the peak signal amplitude for comparison of the dependence of the dynamics on initial carrier density. (b) Log–log plots together with smooth lines representing the result of a fit of both the THz and near-infrared responses to a model that reflects both an effective time-dependent mobility and diffusion-limited bimolecular recombination.

in which the full THz waveform is measured at a series of constant pump–probe delays. Comparison of the data traces presented in Figure 5.5 demonstrates the significant differences in the carrier response probed in the two spectral regions. The THz measurements clearly show a fast relaxation response on a picosecond time scale, and only a very slight dependence of the overall decay transient on initial carrier density is observed. Measurements on the same sample of the induced absorbance detected at a probe wavelength of 800 nm, carried out at the same initial carrier densities, show a time dependence and a fluence dependence that are dramatically different from those observed at THz probe wavelengths, and that are strikingly similar to those observed in previous subpicosecond pump–probe measurements at visible and near-infrared probe wavelengths on a-Si:H and related materials. The observed decay of the near-infrared induced absorbance is highly nonexponential and is strongly dependent on the initial carrier density. The carrier response is presented in the log–log plot in Figure 5.5(b), along with fits to a self-consistent model based on an effective time-dependent mobility along with diffusion-limited bimolecular recombination. This model includes contributions from time-dependent populations of free (high-mobility) carriers and trapped (low-mobility) carriers. The effects of carrier localization are based on multiple trapping into low-mobility band tail states with an exponential distribution of binding energies, a mechanism that is consistent with established models for carrier transport in amorphous semiconductors that were developed to model time-offlight transient photoconductivity measurements carried out at low temperature on

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σ1 (Ω cm)–1

3.0

2.5

2.0

1.5 0.5

1.0 1.5 Frequency (THz)

2.0

Figure 5.6 Spectral dependence of the real part of the photoconductivity of a-SixGe1-x: H (x = 0.5) at 4-ps delay. The response includes a component of ~ω2 reflecting contributions from carriers in localized states. (From Nampoothiri, A. V. V. and Dexheimer, S. L. Materials Research Society Proc. 808, 97–102, 2004. With permission.)

substantially longer time scales.92 In essence, the effective time-dependent mobility reflects the fraction of time that the carriers are in the high-mobility extended (transport) states, which decreases with time as the carrier distribution undergoes multiple trapping and thermally activated detrapping processes in the distribution of bandtail states. A simple physical model for this process93 predicts an effective mobility that decreases as a power law in time ~ta-1, which is in good agreement with the experimental observations. The observed THz response largely reflects the free carrier population, and the decay transient includes the effect of population loss from bimolecular recombination in addition to the time-dependent mobility. The frequency dependence of the time-resolved THz response also reflects the carrier localization processes inherent to amorphous semiconductors. Figure 5.6 presents σ(ω) for an a-SixGe1-x:H (x = 0.5) thin film on a sapphire substrate at a time delay of 4 ps after photoexcitation.64 The measured conductivity increases with frequency in the THz range giving a strong deviation from a Drude frequency dependence, in contrast to steady-state conductivity measurements in doped crystalline silicon,94 which show only subtle changes from the Drude law in this same frequency range. The frequency dependence fits to a simple expression σ(ω) = σ(0) + bωs that includes a constant component, together with a power law component. The component that is effectively constant over the observed frequency range is consistent with the carrier population in the extended states in the ωτ << 1 limit, with a short scattering time τ owing to the disorder. The observed frequency dependence of the power law component, which fits to an exponent s of approximately 2, is consistent with theoretical models for the low-frequency ac conductivity associated with localized states in disordered media based on the Kubo-Greenwood formalism,95

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reflecting a significant degree of localization of much of the photoexcited carrier distribution on a picosecond time scale.

5.3.2 Semiconducting Organic Polymers Another aspect of conductivity in disordered systems is illustrated by time-resolved THz studies of hopping transport in conjugated conducting polymers, in which carrier transport is dominated by structural disorder associated with a distribution of effective conjugation lengths. Hopping of a charge carrier between localized electronic states centered at different locations shown schematically in Figure 5.2(c) is a common mode of the charge transport in disordered systems,96 and the frequency dependence of the conductivity is one of the key experimental signatures of the hopping transport processes. The conductivity at a given probe frequency n provides a measure for the number of hops per unit time within a time window ~ 1/n. In the dc or low-frequency limit, the carrier mobility is limited by the slowest process while charge carriers drift under an external dc or low-frequency field over macroscopic distances. The measured transport properties depend, to a large extent, on the macroscopic morphology of the sample. On the other hand, conductivity probed in the GHz-THz frequency region reflects transport processes occurring on the time scale of 1 to 1,000 ps, which correspond to hops between close neighbors in disordered systems. Further, the use of high frequencies ensures that the probed movement of charges is within relatively ordered domains. The carrier mobility at high frequencies is therefore mainly determined by the intrinsic properties of the material of study. THz spectroscopy, capable of probing conductivity with ultrashort electromagnetic pulses containing frequencies extending from tens of GHz to a few THz, thus provides an ideal complement to the conventional dc or low-frequency measurements. In addition, because of its contact-free nature and its ability to be combined with ultrafast photoexcitation, the THz approach also permits us to probe the photoconductivity of isolated structures in a disordered system like isolated polymer chains in a polymer network with ps time resolution. One such example is shown below in a time-resolved THz study by Hendry et al.33 in the conjugated polymer MEH-PPV {poly(2-methoxy-5-(2’-ethyl-hexyloxy)p-phenylene-vinylene)} containing 50% PCBM {1-(3-methoxycarbonyl)-propyl-1phenyl-(6,6)C61}. The PCBM serves as an exciton-dissociating electron scavenger. The photo-induced response of the MEH-PPV/PCBM blend was probed at a delay of approximately 10 ps after photoexcitation. On this time scale, the hot carrier effects are negligible and largely only the photoinjected holes remain to contribute to the conductivity. The frequency dependence of the photo-induced hole conductivity is displayed in Figure 5.7. It shows a behavior quite different from a Drude-like response: the imaginary part is negative; and both the real and imaginary parts of the conductivity increase with frequency. The experimental frequency dependence of the photo-induced conductivity can be described by a hopping model with symmetric nearest-neighbor interactions. The time evolution of the mean-square displacement, ∆2(t), of holes moving along isolated MEH-PPV chains can be calculated within this model and is shown in Figure 5.8(a). The numerical data of Grozema et al.97 were obtained based on the tight-binding approximation combined with static torsional disorder (deviation from

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σ(ω, τ) (S/m)

500

0

–500 –1000

0.4

0.6 0.8 1.0 Frequency (THz)

1.2

Figure 5.7 Frequency dependence of the real (σ′, filled squares) and the imaginary part (σ″, empty squares) of the pump-induced complex conductivity of an MEH-PPV/PCBM blended film measured τ = 10 ps after photoexcitation. The lines represent the simulated hole conductivity based on a tight-binding model and a hole density of 8⋅1021 m–3. (From Hendry, E. et al., Phys. Rev. Lett. 92, 196601, 2004. With permission.)

planar alignment of the chain) in the limit of infinitely long chains without chemical defects. Two characteristic stages of the transport process can be identified: at the early stage, the mean-square displacement increases rapidly with time and exhibits nonlinear temporal behavior; whereas after about 10 ps, it becomes diffusive with an essentially linear time dependence. These two stages correspond to the initial fast delocalization of the hole and its subsequent arrival at sites with small hopping probabilities. The intramolecular hole mobility (Figure 5.8(b)) can then be evaluated by 1000 Mobility (cm2V–S–1)

21500

∆2(t)/λ2

21000 20500 20000 19500

0

10 20

30

40

Time (ps) (a)

50 60

70

100 10 1 0.1 0.01 0.001 108

1010 1011 109 Frequency (Hz)

1012

(b)

Figure 5.8 (a) Dimensionless mean-square displacement of holes as a function of time for PPV calculated from a model based on the tight-binding approximation. The straight line indicates the linear increase of the displacement with time in the second stage. (b) Frequency dependence of the intramolecular mobility of holes along PPV chains, obtained from the mean-square displacement data of (a) using the Kubo formula. (Adapted from Grozema, F.C. et al., J. Phys. Chem. B 106, 7791–7795, 2002. With permission.)

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use of the Kubo formalism, essentially a Laplace transform, from the time evolution of the mean-square displacement. At frequencies lower than a few gigahertz, the hole mobility is dominated by the long-term diffusion and is largely independent of frequency. From a few GHz to THz, however, the mobility, arising primarily from the initial nonlinear temporal behavior of the carrier displacement, increases rapidly with frequency. Values as high as approximately 100 cm2/Vs are predicted for the hole mobility at THz frequencies in PPV. With an appropriate value for the carrier density, the model is seen to reproduce the experimental results for the frequency dependence of conductivity (lines in Figure 5.7). Thus the time-resolved THz measurement, as described in the example above, offers unique data on charge transport along isolated polymer chains. Such information is very useful for gaining insight into the relation between the molecular structures and their conducting properties. The THz measurement presents a guideline for design of polymeric materials with desired transport properties. Although to date a relatively small number of studies of time-resolved THz photoconductivity in polymers have been performed,18,33,35,37 the approach has excellent potential for application to a large collection of polymeric materials and, more generally, to other disordered systems. The method permits access of high-frequency transport properties that often reveal the underlying local material properties in spatially inhomogeneous material systems. It should be noted that the notion of spatially inhomogeneous systems is correlated with, but not identical to, the notion of disorder. It applies, for instance, to crystalline nanoparticles in an ensemble. As discussed in the beginning of this section, the dc or low-frequency conductivity of such systems is limited by the least conductive part of the system, and is often determined by the macroscopic morphology of the sample. The use of high frequencies in THz spectroscopy ensures that primarily the intrinsic properties of the material are probed.

5.4 Polaron Formation and Dynamics in Molecular Electronic Materials One of the most distinctive applications of THz spectroscopy is to measure the complex conductivity in a system undergoing rapid change after photoexcitation. The optical pump THz probe technique has been exploited to examine a variety of important basic dynamic processes, including carrier generation, cooling, trapping, and recombination in both bulk and nanoscale materials. The method has also been demonstrated to be extremely powerful in probing the formation dynamics of quasiparticles as the result of many-body interactions of charge carriers among themselves, as reviewed in the discussion of dynamics in semiconductors and strongly correlated electronic materials in chapter 4, and by the interaction of carriers with lattice or molecular vibrations, examples of which are discussed in the following section. In comparison with the well-developed optical pump optical probe technique, THz probing offers a direct and quantitative measure of the charge transport properties, while sharing nearly all of the excellent time resolution of optical probing. On the other hand, in comparison with conventional transport measurements, time-resolved THz spectroscopy allows us to probe highly nonequilibrium systems that are not otherwise accessible.

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In what follows, we will discuss time-resolved THz studies, with emphasis on the dynamic properties, in two classes of molecular electronic materials. First, we review studies of polaron formation dynamics in quasi-one-dimensional mixedvalence metal–halide complexes, materials that serve as model systems for the physics of reduced dimensionality. This is followed by a discussion of polaron transport in organic molecular crystals, focusing in particular on studies of pentacene, a material of current technological interest for which many key scientific and technological questions are still unanswered. These studies show how time-resolved THz experiments can shed light on important issues regarding the nature of charge carriers and transport in molecular materials.

5.4.1 Dynamics of Polaron Formation: Quasi-One-Dimensional Systems The localization of an electronic excitation via interaction with a deformable lattice is an important fundamental process in a wide range of condensed matter systems, and has a dramatic impact on the transport properties of electronic materials. THz spectroscopy has proved invaluable for investigating the impact of polaron states on transport properties, as demonstrated in the example of electronic excitations in sapphire reviewed in Section 5.2. Here, we review time-resolved THz studies of the dynamics of polaron formation, in which a carrier initially excited into a delocalized state interacts with the lattice, resulting in the formation of a localized polaron state in which the carrier is stabilized in a localized lattice distortion. The experiments are carried out on a class of quasi-one-dimensional materials, mixed-valence metalhalide linear chain (MX) complexes, that have proved to be excellent model systems for investigating a number of phenomena inherent to low-dimensional systems.98,99 Quasi-one-dimensional materials are ideal systems for studying the localization process, since their reduced dimensionality can lead to strong electron–phonon interactions, and the linear structure of the materials simplifies the dynamical configuration space, in that the dominant motion associated with the structural rearrangement is expected to occur along the linear axis. In the case of an ideal one-dimensional lattice, the transition from the extended free electronic state to the localized polaron state is theoretically predicted to be a barrierless process, giving rise to extremely rapid dynamics for the photo-induced structural rearrangement.100,101 Previous timeresolved optical experiments on MX materials using vibrational wave packet techniques have resolved the analogous process of the formation of self-trapped excitons (sometimes also referred to as exciton-polarons), and have demonstrated that the exciton localization process takes place on the time scale of a single vibrational period102–104; similar dynamics are expected for formation of polarons from free carrier states. The schematic structure along the chain axis of a platinum-halide MX complex, consisting of covalently bonded extended linear chain of alternating Pt ions and halide (X) ions, is shown in Figure 5.9. The Pt ions also have transverse ligands (for example, ethylenediamine [C2H8N2], or [en]) that provide a nearly square planar geometry about the Pt ions. These materials form crystals with parallel linear chains separated by counterions (for example, ClO4 –). The chains experience minimal interchain interaction, so that the physical properties are dominated by the onedimensional chain structure. The structure is characterized by a periodic bond length

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Terahertz Spectroscopy: Principles and Applications … ‒‒‒‒‒ X–– Pt+(3+δ) – X– ‒‒‒‒‒ Pt+(3–δ) ‒‒‒‒‒ X–– Pt+(3+δ) –X– ‒‒‒‒‒ Pt+(3–δ) ‒‒‒‒‒ …

Figure 5.9 Schematic structure showing two unit cells of a quasi-one-dimensional halidebridged mixed-valence platinum linear chain (MX) system, neglecting the transverse ligands and counterions. The chain structure has a Peierls distortion corresponding to a periodic variation in Pt–halide bond length and a commensurate charge density wave corresponding to a periodic variation in Pt ion valence, with the effective Pt ion charge denoted by 3 ± δ.

distortion (or Peierls distortion) and a commensurate periodic charge disproportionation (or mixed-valence character), giving a charge density wave ground state. The MX complexes have a strong optical intervalence charge transfer transition that effectively transfers charge between inequivalent metal ions.105 Excitation into the lower energy tail of the absorption results in the formation of charge-transfer excitons, whereas excitation at energies high in the band can generate free carriers, either by direct excitation into the electron-hole continuum or by rapid dissociation of highly excited excitons. Optical pump THz probe measurements on oriented crystals of the {[Pt(en)2][Pt(en)2I2].(ClO4)4} complex are presented in Figure 5.10.65 In these experiments, optical pulses 35 fs in duration centered at 800 nm generated by an amplified Ti:sapphire laser system are used to excite the PtI(en) complex well above its band edge. Single-cycle THz probe pulses are synchronously generated by optical rectification in ZnTe and are detected by electrooptic sampling. In these measurements, the differential transmittance signal detected at the peak of the THz electric field waveform corresponds to an induced absorbance, since there is negligible pump-induced change in the index of refraction. The measurements presented in Figure 5.10 show a biphasic induced absorbance response, with a large amplitude component that decays rapidly with a time 0.015

0.003

∆E/E~t–α α = 0.5±0.08 –∆E/E

–∆E/E

0.010

0.005

0.001

0.000 –2

–1

0

1 2 Delay (ps) (a)

3

4

0.0002

3

10

Delay (ps)

100

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Figure 5.10 Time-resolved THz response of [Pt(en)2][Pt(en)2I2].(ClO4)4 after excitation with pulses 35 fs in duration centered at 800 nm. (a) Short-time response: measurements (gray), with a fit (black) to an exponential decay including a convolution with the measured system response, giving a time constant of 400 fs for the initial decay of the induced absorbance. (b) Long-time response: log–log plot, with a fit showing the decay of the signal amplitude ~t-1/2, reflecting diffusion-limited recombination of mobile carriers constrained to a one-dimensional geometry.

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constant of approximately 400 fs, followed a long-lived, smaller amplitude component that decays slowly on a timescale extending beyond 100 ps. The initial fast decay dynamics do not show a significant dependence on initial excitation density. The large amplitude THz response immediately after photoexcitation reflects a population of high-mobility free carriers, and its rapid decay is consistent with the reduction of mobility on polaron formation. The observed 400-fs timescale for this process is comparable to the characteristic vibrational period for LO phonons along the PtI(en) chain axis, consistent with the theoretically predicted barrierless localization dynamics. The longer time signal decays slowly as a power law ~t -1/2, consistent with diffusion-limited recombination of the polaron population with the characteristic time dependence for diffusive motion constrained to one dimension. This result is indicative of hopping transport, characteristic of localized polaron states, in contrast to the band-like transport described in Section 5.2 for large polarons in sapphire.

5.4.2 Carrier Transport in Pentacene

1

2

–5.0 ps 1.1 ps 2.6 ps 19.6 ps

1

10 K 50 K 100 K 0 150 K 0 2 200 K –1 1 Probe Delay Time (ps) 240 K

10 –ΔE/E (%)

THz Electric Field (a.u.)

Semiconducting organic molecular crystals are of interest for a number of potential applications. The photophysics of polyacene materials, in which planar conjugated organic molecules stack to form crystals, have been extensively studied, though many issues regarding carrier generation and transport in the materials remain unresolved. Recently, pentacene materials, including crystals, thin films, and functionalized derivatives have been investigated using time-resolved THz spectroscopy.32,67–69,76 Data from a study by Hegmann et al.32 on functionalized pentacene (FPc) are displayed in Figure 5.11. Figure 5.11(a) presents examples of THz pulses transmitted through an FPc film at 10 K, 5 ps before the pump excitation, and 1.1, 2.6, and 19.6 ps

0

1

m = –1

–1 0

5

10 Time (ps) (a)

15

20

0.1

m = –0.5

1

10 Probe Delay Time (ps) (b)

Figure 5.11 (a) THz pulses transmitted through a functionalized pentacene crystal at 10 K at various pump–probe delays. A 100-fs pump pulse centered at 800 nm excites the sample at time τ = 0, and the pump–probe delay time is the arrival time of the positive peak of the THz probe pulse. (b) Normalized differential transmission in percent of the peak of the THz probe pulse as a function of the pump–probe delay and sample temperature in a log–log plot. Lines labeled m = -1 and m = - 0.5 are guides to the eye for power-law decay of τ-1 and τ-0.5, respectively. The inset shows the temporal evolution in a linear plot for the first 2 ps after photoexcitation. (Adapted from Hegmann, F.A. et al., Phys. Rev. Lett. 89, 227403, 2002. With permission.)

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after the pump excitation, respectively. The transmitted amplitude of the THz pulse is reduced by the presence of photoexcited carriers in the sample, but no significant reshaping of the waveform is observed. This indicates that the complex conductivity arising from the photoexcited carriers is purely real, with no significant frequency dependence within the bandwidth of the THz pulse. In this limit, measurements of the dynamics can be simplified by recording the pump-induced change in the peak THz field while varying the pump –probe delay time τ. The observed frequency response is attributed to a Drude-like conductivity in the regime of very high scattering rate (2πn/γ0 << 1).32 The dependence of the differential transmission of the peak of the THz electric field waveform, ∆E/E, on the pump–probe delay for FPc is displayed in Figure 5.11(b) in a log-log plot for several sample temperatures. The early response is also presented in a linear plot in the inset. The data exhibit a sharp rise (~ 0.5 ps), followed by a fast decay (~ 2 ps). Although these early processes are limited by the time resolution of the optical pump THz probe measurement, the fast rise of the photoconductivity suggests that a population of mobile carriers is formed in FPc over time scales much shorter than 0.5 ps after photoexcitation. The longer time response is dominated by a power-law decay ~ τ -β, a temporal behavior that has been attributed to dispersive transport characteristic of disordered media.106 An intriguing result from the time-resolved THz studies is that the photoconductivity (i.e., the overall signal amplitude) increases with decreasing the sample temperature in Pc and FPc,32,76 in contrast to the expected temperature dependence for thermally activated hopping transport. Hegmann et al.67 interpret their observations in FPc in terms of the molecular polaron limit, with transport involving tunneling between isoenergetic sites.107 The authors suggest that these observations reflect the sensitivity of the time-resolved THz response to the intrinsic material properties, making it possible to observe band-like transport over short length scales, even though dc transport in these materials is dominated by carrier trapping, most likely at interfaces between small ordered domains.

5.5 Charge Transport in Nanoscale Materials: Nanocrystalline Semiconductors and Quantum Dots Another attractive attribute of THz spectroscopy is its ability to probe conductivity using freely propagating electromagnetic radiation. Probing charge transport in a non-contact measurement opens up many possibilities for investigations of nanoscale materials and structures. For such systems, connecting wires for traditional transport measurements may be very difficult, although interesting and important in its own right. The technique of THz spectroscopy provides a powerful complement to the conventional approaches of probing charge transport in nanoscale systems, a central topic in nanoscience and nanotechnology. The method has been exploited in several recent investigations of charge transport in quantum wells31,44,60,61 isolated nanoparticles,24,29,38,78 and assemblies of nanostructures,25,28,36,70,71,77,79 as well as in studies of carrier cooling, trapping, and recombination dynamics in nanostructures.25,26,28,38,62,70 Following, we discuss semiconductor and oxide nanoparticles as a model system

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to illustrate how THz spectroscopy can be employed to reveal the drastically differing nature of the electronic charge transport in two limiting size regimes of nanoparticles. The measurements are very similar conceptually to those described in Section 5.2 for the study of photoexcited insulators. The electric field waveform E(t) of THz pulses transmitted through the unexcited ensemble of nanoparticles and the pump-induced change in the THz waveform, ∆E(t), a few picoseconds after photoexcitation are recorded. A modest reduction in the THz signal is often observed with increasing pump–probe delays resulting from carrier trapping at the nanoparticle interfaces or at interior defects. Assuming that the rate of change of the THz response of the system is sufficiently slow, the induced complex conductivity or susceptibility of the nanocomposites can be extracted, as described in Section 5.2, within an approximation of quasi-steady-state material response through Fourier transform of the recorded waveform of E(t) and ∆E(t).

5.5.1 Conductivity and Dielectric Screening in Nanoporous TiO2 We first examine charge transport in relatively large nanoparticles, for instance, in Degussa P25 titanium dioxide particles. These nanoporous samples of TiO2 particles of about 25-nm diameter consist of a mixture of anatase and rutile type TiO2. Understanding charge transport in this material is of technological importance because dye-sensitized porous networks of TiO2 nanoparticles are essential components of the Graetzel solar cell, an efficient implementation of a dye-sensitized photovoltaic device. Photoconductivity in TiO2 Degussa P25 titanium dioxide particles has been investigated using THz spectroscopy first by Turner et al.77 and later by Hendry et al.36 The frequency dependence of the complex conductivity in a nanoporous TiO2 film was found to differ strongly from that in single-crystal rutile. We use the results of Hendry et al.36 to illustrate the differences. The conductivity for single-crystal rutile is displayed in Figure 5.12(a) and for nanoporous TiO2 films in Figure 5.12(b). One immediate observation is that the conductivity in a nanoporous TiO2 film under the same excitation conditions is more than two orders of magnitude smaller than in a bulk rutile crystal. In addition, the spectral dependence of the conductivity in the nanoporous TiO2 films can no longer be described by the Drude model, although this model satisfactorily reproduces the photo-induced conductivity in bulk rutile (solid lines in Figure 5.12(a)). In particular, the real part of the conductivity in nanoporous TiO2 films increases with frequency; and the imaginary part can even take negative values over the investigated frequency range, whereas the imaginary part of the Drude conductivity is always positive. Data for three representative pump fluences from 0.4 to 1.1 × 1018 photons/m2 are shown in Figure 5.12. Although the normalized photo-induced conductivity in bulk rutile is essentially independent of excitation fluence, the normalized conductivity in nanoporous TiO2 exhibits a strong density dependence. These large differences can be attributed to the local field effects arising from the large dielectric contrast (~100) between that of the TiO2 nanoparticles and the surrounding air. Indeed, a combination of the Drude model for conductivity in individual nanoparticles and Maxwell-Garnett effective medium theory108 for spherical TiO2 particles embedded in air reproduces the experimental spectral dependence

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σ (ω, T = 30 K)/N0 (×10–24 Sm2)

σ (ω, T = 30 K)/N0 (×10–22 Sm2)

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σbulk

10 8 6 4 2 0

0.2

0.4

0.6

0.8

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4

σnano

2 0 –2

ω(THz)

0.6 ω(THz)

(a)

(b)

0.2

0.4

0.8

1.0

Figure 5.12 (a) Spectral dependence of the real part (open symbols) and the imaginary part (filled symbols) of the normalized photoconductivity of single-crystal rutile (TiO2) at 30 K for excitation fluences of 0.40⋅1018 (triangles), 0.82⋅1018 (circles), and 1.06⋅1018 photons/m2 (squares). The lines represent a fit to the Drude model. (b) Normalized photoconductivity in nanoporous TiO2 for the same excitation fluences. The lines represent fits to the MaxwellGarnett (MG) effective medium theory. (From Hendry, E. et al., Nano. Lett. 6, 755, 2006. With permission.)

of the complex conductivity (lines in Figure 5.12(b)) reasonably well. In particular, Maxwell-Garnett effective medium theory with a large filling factor predicts that the spectral shape depends on the plasma frequency (carrier density), just as observed experimentally. Indeed, by scaling the plasma frequency with laser fluence, the main features of the spectral dependence of the complex conductivity of nanoporous TiO2 can be reproduced. Thus we can conclude that the conductivity of a composite of TiO2 nanoparticles of diameter 25 nm is dominated by the Drude conductivity of the individual nanoparticles. In principle, there could also be additional contributions from interparticle charge transport, though in this case of relatively large nanoparticles, the contribution of such transport appears to be slight.

5.5.2 Excitons in Semiconductor Quantum Dots Now let us turn to the case of small nanoparticles. The THz response in semiconductor quantum dots of CdSe, PbSe, InP, InGaAs, and other materials has been studied by several groups.22,24–26,28,29,36,38,62,71,77,78 Following we use the results of Wang et al.78 to show the distinct transport properties of carriers in isolated quantum dots (QDs) in the regime of strong quantum confinement. The pump-induced sheet susceptibility in a dilute suspension of CdSe QDs of 2-nm radius after photoexcitation of single electron hole pairs in the QDs is shown in Figure 5.13. The imaginary part of the sheet susceptibility as a function of frequency n is seen to remain essentially unchanged on photoexcitation, whereas the change in the real part is appreciable, but largely frequency-independent over the investigated frequency range (dashed lines in Figure 5.13). In the language of the sheet complex conductivity, ss(n), the response corresponds to a purely imaginary conductivity, because the conductivity is related

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∆χ´S and ∆χ´´S (10–6 mm)

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∆χ´S

2.0 1.5 1.0 0.5

∆χ´´S

0 0.6

0.7

0.8

0.9

1.0

1.1

Frequency (THz)

Figure 5.13 Spectral dependence of the change in the real (χs′) and imaginary part (χs″) of the photo-induced sheet susceptibility of a dilute suspension of CdSe QDs at room temperature (gray lines). The dashed lines represent a model corresponding to a frequency- independent and purely real induced susceptibility. (From Wang, F. et al. Nat. Mat. 5, 861, 2006. With permission.)

to the susceptibility by σs(n) = –2πinε0χs(n). This response is entirely different from what is observed in photoexcited bulk crystals (sapphire, Figure 5.1(b); rutile, Figure 5.12(a)). For the latter case, a Drude-like THz response, characteristic of free charges, is observed. Nor can the measured spectral dependence of the CdSe QDs be accounted for by local field effects in an effective medium theory for a dilute suspension. For CdSe QDs of radii smaller than the exciton Bohr radius (aB ~ 5 nm), the quantum confinement effects are strong and the carriers occupy discrete energy levels separated by at least tens of meV.109 This situation invalidates the picture of perturbed bulk transport with charge carriers moving in a continuous band of states. In the strong confinement regime, the excitations are more like those of a large atom than those of a small piece of bulk material. The pump-induced susceptibility can therefore be interpreted as the polarizability of photogenerated excitons confined within QDs. Such an interpretation is confirmed by the observation that the response to a THz electric field is given by a real and spectrally flat susceptibility ∆χs. These two features both follow from the existence of well-separated electronic states that are probed by THz radiation with photon energies (of ~4 meV) lying significantly below the electron and hole transitions. As shown schematically in Figure 5.2(b), in the absence of the field (left), the ground-state exciton, which is strongly confined in a QD of radius R, may be represented by spherically symmetric electron and hole distributions centered inside the dot (only the electron distribution shown). This configuration has no net dipole moment. The externally applied electric field perturbs the exciton wave function and leads to a net dipole moment P in the QD (right).

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Experiment

3

α(×104 3)

Model R4 scaling

2

1

0 1.0

1.5

2.0

2.5

Nanoparticle Radius R (nm)

Figure 5.14 Polarizability of quantum-confined excitons in photoexcited CdSe QDs as a function of the QD radius R. Experimental data (symbols) and theoretical predictions based on a multiband effective-mass model (dotted line). The error bars represent the uncertainties associated with the reproducibility of the measurements. For comparison, the solid line shows a simple R4 scaling. (From Wang, F. et al. Nat. Mat. 5, 861, 2006. With permission.)

In addition, such a distortion of the charge distribution follows in phase with the oscillation of the THz electromagnetic radiation, corresponding to a real and spectrally flat susceptibility ∆χs observed experimentally (Figure 5.13(b)). The exciton polarizability of CdSe QDs, defined as α = P/E, was extracted from the pump-induced susceptibility ∆χs using an effective medium theory. It is displayed in Figure 5.14 as a function of the QD radius. Here E is the externally applied electric field and P the dipole moment of the exciton that one would measure if the unexcited QD were embedded in a host with a matched dielectric constant. Of particular interest is the numerical value of the polarizability of the electron hole pair confined to the semiconductor QDs. The polarizability, as shown in Figure 5.14, can easily exceed 10,000 Å3. This is a remarkably large value compared with the polarizability of an electron in an atom or molecule, which is typically approximately 1–10 Å3. The result reflects the highly delocalized nature of the excitation, extending over the entire volume of the QD, as shown in Figure 5.2(b). The exciton polarizability is found to increase approximately as the fourth power of the QD radius (R4, dotted line). Such dependence can be understood qualitatively from elementary quantum mechanics by considering the following expression for the linear polarizability: 2  a ~ er /∆E, where er is the transition dipole moment and ∆E is a typical energylevel spacing of the electron. For an electron confined within a region of characteristic size R, the transition dipole moment is of the order of eR, whereas the energy scale is h 2 / m* R 2, where h is Planck’s constant and m* the electron effective mass. One thus obtains a polarizability of α ~ R 4 / aB, where aB denotes the Bohr radius. Calculations (solid line) within the framework of standard perturbation theory and

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effective mass model are shown to be compatible with the experimental magnitude of the size-dependent polarizability.

5.6 Extending into Mid-Infrared Spectral Regime: Carrier Dynamics in Graphite Following the pioneering work of Auston, Grischkowsky, and their coworkers,1,110 researchers have optimized ultrafast photoconductive switches in the past two decades to permit generation and field-resolved detection of electromagnetic transients up to approximately 5 THz. Such a bandwidth, although impressive, actually reflects the finite response time of photoconductive materials rather than the ideal bandwidth that could be obtained from current state-of-the-art mode-locked laser pulses. For instance, a 10-fs, transform-limited optical pulse (with a bandwidth of ~50 THz), in principle, should permit generation and detection of electromagnetic transients up to approximately 50 THz. This increased spectral range covers many interesting vibrational and electronic resonances in solids and molecular systems. Although some progress has been made lately to extend the bandwidth of THz systems based on photoconductive materials from the far-infrared (<10 THz) to the mid-infrared region,111–113 it is more convenient to employ nonresonant secondorder nonlinear optical materials114 as transducers to optimize the spectral range. Zhang,115,116 Leitenstorfer,117–119 and their coworkers have demonstrated a detection bandwidth larger than 30 THz based on free-space electrooptic sampling in thin inorganic semiconductor crystals such as ZnTe, GaP, and GaSe. Cao et al.120 have done similarly with poled polymers. Huber and coworkers117 have generated THz pulses through phase-matched optical rectification in GaSe crystals with high conversion efficiency and central frequency readily tunable from the far-infrared up to 40 THz. By combining a thin GaSe emitter (90 µm) and ZnTe detector (10 µm) they have also been able to obtain bandwidth-limited THz pulses shorter than 50 fs based on a 10-fs laser.117 Such THz electromagnetic pulses with their exceptionally short pulse duration and ultrabroad spectral tunability are interesting in themselves for enhanced spectroscopic measurements. The combination of these probe pulses with optical excitation pulses of just a few-cycle duration provides further new possibilities in ultrafast dynamics of material excitations in the mid-infrared spectral regime with femtosecond time resolution. This powerful new methodology has already been exploited to study the ultrafast carrier dynamics in graphite,51 carbon nanotubes,70 gases,52 and bulk semiconductors,40–45,55,56 as well as dynamics of vibrational modes in molecules.16,17 We will illustrate this approach with the studies of the ultrafast carrier dynamics and energy relaxation after photoexcitation in graphite.51 Because of its close relationship to carbon nanotubes and the recent discovery of unusual electronic properties in its single-layer form of graphene, graphite has lately been the focus of much research interest. Optical phonons, strongly interacting with electrons, are believed to significantly influence the transport and energy relaxation processes in graphite. By employing the ultrabroadband THz electromagnetic transients that capture signatures of photoexcited carriers in graphite, Kampfrath and

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Terahertz Spectroscopy: Principles and Applications

τ = 0.5 ps

–100

–50 –0.1

0 t(ps) (a)

0.1

–150

lm ∆ετ Re ∆ετ 15 20 Frequency (THz)

25

(b)

Figure 5.15 (a) THz waveform E 0(t) transmitted through unexcited graphite and the pumpinduced changes ∆E(t) measured at 0.5 ps after photoexcitation. (b) Real (circles) and imaginary (dots) part of the pump-induced change in the in-plane dielectric function of graphite at 0.5 ps after photoexcitation calculated from the THz waveforms of (a). Solid lines are fit to a model that includes the contribution of both the direct and indirect optical transitions as described in the text. (Adapted from Kampfrath, T. et al., Phys. Rev. Lett. 95, 187402, 2005. With permission.)

coworkers51 have been able to resolve the role of strongly coupled optical phonons in the ultrafast dynamics of the electronic charge transport properties. In their experiment, a 10-fs optical excitation pulse was used to transfer electrons from the valence band to the conduction band in a graphite film. From time-resolved photoemission measurements121 these photoexcited carriers are known to thermalize within approximately 0.5 ps and can then be described by a Fermi-Dirac distribution with an electronic temperature Te. The transport properties of the thermalized carriers were probed by THz electromagnetic transients. A typical electric field waveform of the THz pulses transmitted through an unexcited graphite film is shown in Figure 5.15(a) (dots). The probe pulses contain frequencies ranging from 10 to 26 THz, which, as will be shown, cover the electron scattering rates γ0 in graphite, and additionally correspond to THz photons with energy ~kBTe. The pump-induced change in the THz electric field waveform is displayed in Figure 5.15(a) as circles. The corresponding pump-induced complex dielectric function can be extracted from the THz waveforms and is shown in Figure 5.15(b) (symbols). This corresponds to the graphite in-plane dielectric function induced by photoexcitation and has an interesting spectral feature: the imaginary part of the dielectric function (dots) is positive for frequencies n < 15 THz, and becomes negative for n > 15 THz, corresponding to pump-induced absorption and bleaching, respectively. The experimental observations can be explained using the simplified graphite band structure sketched in Figure 5.16: A Bloch electron can absorb a probe THz photon either by a direct optical transition, which conserves the electron wave vector, or by an indirect optical transition, such as electron scattering with phonons, lattice defects, or other electrons, which does not conserve the electron wave vector. The photoexcited carriers block some of the originally possible direct optical transitions

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Conduction Band

∆k

Pump 0

DOT

K

k

K´

K

K ×

K´

Γ

∆k Valence Band (a)

K´

K (b)

Figure 5.16 (a) Sketch of the in-plane band structure of graphite close to the K or K′ point where the valence and conduction bands slightly overlap. Arrows indicate possible direct and indirect optical transitions induced by the probe pulse. (b) Brillouin zone perpendicular to the c axis. The Fermi surface is located around the K and K′ points. Arrows mark possible scattering events of electrons and correspond to wave vector changes ∆k that are confined to the vicinity of the Γ and K points. (From Kampfrath, T. et al., Phys. Rev. Lett. 95, 187402, 2005. With permission.)

in a range of kBTe around the Fermi energy and the absorption of THz photons with energy of approximately kBTe decreases. On the other hand, the elevated electronic temperature enhances carrier scattering and results in stronger indirect optical transitions. Therefore the frequency dependence of the imaginary part of the photoinduced dielectric function reflects the interplay between the direct and indirect optical transitions in graphite. More detailed information about the charge transport and energy relaxation can be extracted from a comparison of the experimental time dependence of the dielectric response with a model that includes contributions from both direct and indirect optical transitions. The indirect transitions were described by a Drude response with the carrier scattering rate and plasma frequency as parameters. The direct optical transitions were described by two-dimensional band-to-band transitions with the electronic temperature Te as a free parameter. This model (solid lines, Figure 5.15(b)) is seen to fit the experimental data well. Figure 5.17 displays the inferred electronic temperature Te and the Drude scattering rate γ0 as a function of the pump–probe delay time. The results indicate that immediately after photoexcitation, the electronic temperature is elevated by about 80 K from the room temperature and the electron scattering rate is increased by almost 4 THz from 10 THz. These changes decay away on a time scale of a few picoseconds. On the other hand, at the given pump fluence, the electrons in the graphite film are estimated to reach temperatures as high as 1,200 K immediately after photoexcitation. This suggests that the majority of the initially deposited optical excitation energy is transferred from the photoexcited electrons to a few strongly coupled phonons within the first 0.5 ps. These phonons cool by energy transfer to cold lattice modes on a longer timescale and explain the observed decay of Te over

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Te(K)

exp ﬁt 350

12

Cold SCOPs 10

300 0

1

2 3 τ(ps) (a)

4

5

0

1

2 3 τ(ps)

4

5

(b)

Figure 5.17 Temporal evolution of the electronic temperature Te (a) and Drude scattering rate g (b) extracted from a comparison of the experimental photo-induced dielectric function to a model described in the text. In (a), the dashed line indicates the hypothetical dependence of Te on the pump–probe delay if only cold strongly coupled optical phonons (SCOPs) were involved. The solid line marks an exponential decay with a time constant of 7 ps. (From Kampfrath, T. et al., Phys. Rev. Lett. 95, 187402, 2005. With permission.)

a few ps. The hot optical phonons also substantially contribute to the significant increase of the Drude scattering rate observed in Figure 5.17(b). Therefore the strongly coupled phonons are shown to significantly influence the ultrafast dynamics of both the electronic energy and carrier relaxation in graphite.

5.7 Summary In this chapter, we have reviewed applications of time-resolved THz spectroscopy to studies of carrier dynamics in novel electronic materials with a wide range of transport mechanisms. These studies demonstrate the distinctive measurement capabilities of time-resolved THz spectroscopy.

Acknowledgments The authors would like to thank their collaborators in their time-resolved THz experiments, especially T.F. Heinz (J.S.) and A.V.V. Nampoothiri (S.L.D.), and their colleagues for permission to reproduce figures from their work. Work was supported in part by the National Science Foundation under grants DMR-0349201 (J.S.), DMR0305403, and DMR-0706407 (S.LD.).

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Response 6 Optical of Semiconductor Nanostructures in Terahertz Fields Generated by Electrostatic FreeElectron Lasers Sam G. Carter

University of Colorado, Boulder

John Cerne*

State University of New York at Buffalo

Mark S. Sherwin

University of California, Santa Barbara

Contents 6.1

Introduction..................................................................................................206 6.1.1 Overview of Scientific Results..........................................................207 6.1.2 Semiconductor Quantum Wells........................................................ 212 6.1.2.1 Interband (NIR) Properties................................................. 213 6.1.2.2 Intraband (Terahertz) Properties......................................... 214 6.1.3 Experimental Techniques................................................................. 214 6.1.3.1 In-Plane Terahertz Electric Fields....................................... 214 6.1.3.2 Growth-Direction Terahertz Electric Fields....................... 216 6.1.4 Electrostatic Free-Electron Lasers.................................................... 216

These two authors contributed equally to the writing of this chapter.

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Internal Dynamics of Excitons and Effects of Terahertz Radiation on Excitonic Photoluminescence................................................................. 217 6.2.1 Introduction....................................................................................... 217 6.2.2 Internal Dynamics of Magnetoexcitons Measured by Optically Detected Terahertz Resonance Spectroscopy................................... 218 6.2.2.1 Experimental Results.......................................................... 219 6.2.3 Nonresonant PL Quenching Mechanism.......................................... 227 6.2.3.1 Experiment.......................................................................... 227 6.2.3.2 Results................................................................................. 227 6.2.3.3 Drude Analysis of Carrier Heating..................................... 230 6.2.3.4 Discussion........................................................................... 230 6.3 Near-Infrared-Terahertz Mixing.................................................................. 233 6.3.1 Introduction....................................................................................... 233 6.3.2 NIR-Terahertz Mixing with In-Plane Terahertz Polarization: Sideband Generation and Nonlinear Spectroscopy of Magnetoexcitons........................................................................... 234 6.3.2.1 Experimental Setup............................................................. 234 6.3.2.2 Results................................................................................. 234 6.3.2.3 Discussion........................................................................... 238 6.3.3 NIR-Terahertz Mixing with Out-of-Plane Terahertz Polarization: Excitonic and Electronic Intersubband Transitions.... 239 6.3.3.1 Undoped Square QWs.........................................................240 6.3.3.2 Undoped Asymmetric Coupled QWs.................................. 245 6.3.3.3 Doped Asymmetric Coupled QWs......................................248 6.3.4 Conclusion......................................................................................... 251 6.4 Modifying Exciton States with Terahertz Radiation................................... 251 6.4.1 Background: Static Electric Field Effects......................................... 252 6.4.2 Dynamic Franz-Keldysh Effect........................................................ 252 6.4.3 AC Stark Effect: Autler–Townes Splitting........................................ 256 6.4.4 Conclusions....................................................................................... 263 6.5 Conclusion....................................................................................................264 References............................................................................................................... 265

6.1 Introduction Semiconductor heterostructures are the foundations for revolutionary advances in solid-state optical devices of the last few decades. The family of semiconductors that has had the largest impact is based on elements from groups III (Ga, In, Al) and V (In, As, Sb, N) in the periodic table. Of these so-called III–V semiconductors, the gallium arsenide (GaAs) aluminum gallium arsenide (AlxGa1-xAs) system is the favorite of optical physicists. GaAs can be grown very cleanly by molecular beam epitaxy, and has a direct bandgap near 300 THz (1.5 eV) at low temperatures. This photon energy is extremely convenient for state-of-the-art detectors and sources of near-infrared (NIR) radiation. Furthermore, because the lattice constants of GaAs and AlAs are nearly identical, AlxGa1-xAs can be grown epitaxially

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and strain-free on GaAs for an arbitrary x. Thousands of scientific papers and dozens of monographs have been written describing fascinating physical effects observed in GaAs/Al xGa1-xAs heterostructures. Although the band gap of GaAs is near 300 THz, there are many fundamental energy scales in GaAs/AlxGa1-xAs heterostructures that are between 0.25 and 5 THz (1 and 20 meV). The bound electron-hole complexes called excitons have a binding energy near 2.5 THz (10 meV) in a 10-nm wide GaAs quantum well clad on both sides by Al0.3Ga0.7As. In a high-quality sample, the homogeneous line width of the exciton can be 0.25 THz (1 meV) or less. In a quantum well (QW), the three- dimensional conduction band breaks up into two-dimensional subbands. The spacing between these subbands can be engineered to be at terahertz frequencies. At sufficiently high intensities, energy scales associated with the interaction between the terahertz radiation and the semiconductor heterostructure can rise into the terahertz frequency range. For example, the average kinetic energy of a free electron in GaAs responding to an oscillating terahertz field (the ponderomotive energy) can become comparable to the photon energy. In another example, the product of the terahertz electric field amplitude and the transition dipole matrix element (Rabi frequency) can exceed the line width of an excitonic transition, or even an intersubband transition. These cases cannot be treated with time-dependent perturbation theory, and new quantum physics is explored. Through pure serendipity, the Physics Department at the University of California, Santa Barbara (UCSB) built the world’s first far-infrared free-electron lasers (FELs) at the same time that members of the UCSB Materials Department built world-leading facilities for fabricating III–V semiconductor heterostructures and processing them into devices. (In the early 1980s, the region of the electromagnetic spectrum now usually referred to as “terahertz” was generally called “far-infrared” or “submillimeter.”). This chapter describes a series of experiments, carried out over 15 years, which take advantage of the interband optical properties of III–V semiconductor heterostructures to report on the remarkable effects of terahertz excitation with UCSB’s electrostatic FELs.

6.1.1 Overview of Scientific Results In this chapter, experimental measurements of the following phenomena are described and compared with theory. • Internal dynamics of excitons: • Optically detected terahertz resonances: Transitions between the internal hydrogenic states of quantum-confined excitons in GaAs QWs are observed by optically detected terahertz resonance spectroscopy.1–4 This technique was previously used by others to study impurity-bound electrons as well as cyclotron resonance.5 The terahertz source for other experiments is a molecular gas laser, which is not continuously tunable but has adequate power on many of its lines. Photoluminescence (PL) generated by light tuned above the band gap is monitored while the sample is excited by terahertz radiation and a magnetic field is swept between 0 and 9 T. When the magnetic field tunes the transition between, for example, the 1s → 2p-like state of the exciton

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through the terahertz excitation frequency, a change in the PL intensity is observed.2,3 The observed transition energies for the prominent 1s → 2p+ line are in good agreement with previously published theoretical calculations.6 The mechanisms whereby terahertz radiation can change PL are also investigated. These mechanisms include heating of free carriers, which in turn heat luminescing excitons,1 and a twostep process whereby excitons excited to the nonradiative 2p+ state are thermally excited to a nearby radiative exciton from which they decay.3 See Chapter 4 for recent, time-resolved terahertz absorption measurements of internal transitions of excitons. • Serendipitous discovery of terahertz-optical mixing or sideband generation: When terahertz radiation with frequency f THz is resonant with internal transitions of magnetoexcitons and NIR radiation with frequency f NIR is resonant with their interband transitions, an unexpected phenomenon is observed. In addition to the expected PL, much brighter and narrower NIR emission lines are present.7,8 Closer examination found that they are always at f NIR ± 2f THz. The intensities of these lines, dubbed “sidebands,” depends linearly on NIR power and quadratically on terahertz power. The power dependence confirms the origin of the sidebands as a resonant four-wave mixing process in which two terahertz photons combine with one NIR photon to generate a second NIR photon at the sideband frequency. This novel four-wave mixing process is used as a nonlinear spectroscopic tool to map out additional internal transitions of magnetoexcitons. • Sideband generation from intersubband transitions: It was striking to discover sidebands at f NIR ± nf THz with even integer values of n, but no odd values like the lowest-order sidebands at n = ±1. The absence of the odd sidebands is consistent with symmetry arguments: with the terahertz electric field polarized in the plane of the QW, as it was for all of the experiments discussed so far, the excitonic system is symmetric under inversion. • Symmetry breaking to generate sidebands at fNIR ± fTHz : Sideband generation is a form of all-optical NIR wavelength conversion—NIR light at one wavelength is converted to another via a nonlinear optical process. Clearly, such conversion would require less terahertz power if the power of the sideband increased linearly rather than quadratically with terahertz power. Such a linear dependence on terahertz power is expected in a process involving three rather than four-wave mixing. Three-wave mixing can only occur in systems in which spatial inversion symmetry is broken. Inversion symmetry is broken by performing sideband generation experiments in a sample containing GaAs QWs of slightly different widths quantum mechanically coupled by a thin tunnel barrier.9 For this symmetry breaking to result in the generation of sidebands at n = ±1, it is necessary to couple terahertz radiation into the sample with its electric field polarized in the growth direction, rather than in the plane of the QW. This is a significant technical challenge, and various solutions, along with

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their relative merits, are described in the chapter. Once the coupling challenge was resolved, sidebands at n = ±1 were indeed observed when f NIR was resonant with an excitonic interband transition.9 • Enhancing sideband conversion efficiency: Terahertz sideband generation is of potential interest for optical communications, because it enables information carried on one NIR wavelength to be transferred to a different wavelength without intermediate conversion to electronic information. A sample was designed and grown with the goal of greatly improving conversion efficiency over that observed in initial experiments. Conversion efficiencies (sideband power/incident NIR power) in excess of 2 × 10 –3 are observed at low temperatures, with the NIR radiation traversing only 4 microns of material.10 Sideband generation is observed up to room temperature in this experiment, albeit with much lower efficiency. • Voltage-Controlled Sideband Generation: Terahertz sideband generation becomes all the more interesting, both scientifically and technologically, if one can control it with an applied voltage. In a QW structure, an electric field applied in the growth direction changes the spacing between subbands as well as the energies of excitons via the DC quantum-confined Stark effect. Several devices were fabricated in which different kinds of QWs were grown between two gate electrodes that were transparent to both NIR and terahertz radiation.11–14 The NIR and terahertz resonances in the intervening QWs could be tuned with voltages on the order of 1 V. For fixed NIR and terahertz frequencies, the sideband generation efficiency could be effectively turned off or on, and resonances could be observed when the DC electric field swept through a nonlinear resonance. A relatively complex set of resonances is mapped out for a sample containing asymmetric tunnel-coupled QWs, which have both electronic and hole intersubband transitions in the terahertz frequency range.11 • Sideband Generation from a Three-Subband System: Sideband generation is studied in detail in devices containing a particularly simple active structure, a set of 15-nm GaAs square QWs between two electrical gates.13 The width of the square wells was chosen to make the two lowest heavy hole subbands separated by 2.5 THz (10 meV). The separation between the electronic subbands was greater than ~10 THz (~40 meV), much greater than the applied terahertz radiation. In this structure, sideband generation is particularly simple to analyze, because, only three subbands are coupled by terahertz and NIR radiation—a single electron subband and two hole subbands. As one would expect, with 0 DC electric field across the square well, only n = ±2 sidebands are observed because a square well is symmetric under spatial inversion. As the DC electric field is increased, n = ±1 sidebands become allowed, and grow roughly quadratically in strength. The resonant behavior of the sidebands is examined as a function of both the NIR and THz frequencies, and the results are consistent with

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theoretical analysis based on calculation of the second-order nonlinear susceptibility of this excitonic system. • Sideband generation in doped versus undoped quantum wells: With the exception of the work described in this section, all work discussed in this chapter was performed on undoped QWs. The terahertz radiation has caused transitions between excitonic subbands. When a QW is doped, the terahertz radiation can now directly excite electronic intersubband transitions. Sidebands are excited in two samples containing identical GaAs QWs of slightly different widths separated by a thin tunnel barrier.14 In one sample, the QWs were undoped, in the other they were doped with electrons. The maximum sideband conversion efficiency in the doped sample is found to be approximately the same as that of the undoped sample. However, the mechanisms and details for sideband generation are very different. In the undoped sample, sideband generation proceeds via many excitonic intersubband transitions, some of which involved transitions between hole subbands. In the doped sample driven near the electronic intersubband transition, sideband generation occurs only when the terahertz radiation is resonant with the single available electronic intersubband transition. Sidebands involved in transitions between hole subbands are insignificant. • Nonperturbative and strong-field physics: • Sideband generation in strong terahertz fields: A standard perturbative treatment of sideband generation predicts that the intensity of the nth sideband is proportional to the nth power of the terahertz intensity. In this perturbative limit, the oscillating terahertz field is assumed to be sufficiently small to leave the QW system near a state of thermal equilibrium. Experiments on sideband generation in undoped, asymmetric tunnel-coupled QWs are performed in which the terahertz electric field reached roughly 10 kV/cm.12 In these experiments, the n = 1 sideband intensity exhibits a nonmonotonic dependence on terahertz field strength. At several terahertz frequencies, the sideband intensity first increases with terahertz field strength, as expected, but then reaches a maximum and decreases as the terahertz field is increased further. This behavior is modeled using a nonperturbative method in which the Schrödinger equation in a strong terahertz field is solved exactly using Floquet theory. The complexity of the coupled QW system makes detailed comparison of theory and experiment difficult. • The dynamic Franz-Keldysh effect: When a static electric field is applied to a bulk undoped semiconductor or in the plane of a QW, absorption is observed below the bandgap from tunneling of carriers into the gap, and small oscillations in the absorption occur above the gap. This is known as the Franz-Keldysh effect.15,16 The dynamic Franz-Keldysh effect occurs when a strong oscillating electric field is applied. The most interesting regime occurs when the ponderomotive

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energy (the average kinetic energy of an electron in an oscillating electric field) is comparable to or greater than the photon energy and the frequency is sufficiently high that the electron moves ballistically over a time scale that is long compared with the period of the oscillating field. At terahertz frequencies, the phenomenon is enriched by excitonic effects. The NIR absorption of a sample containing In0.2Ga0.8As QWs is measured in the presence of strong terahertz fields polarized in the plane of the QW.17 Changes in the NIR absorption near the band edge are observed. These are compared with a theory in which nonequilibrium Green functions are used to determine the interband susceptibility in the presence of a terahertz driving field. The calculated exciton positions are in good qualitative agreement with experimental data. • The AC Stark effect or Autler–Townes splitting of excitons: When a static electric field is applied along the growth direction of a square QW, an excitonic red shift occurs whose magnitude is quadratic in the applied electric field. This is the DC quantum-confined Stark effect (QCSE).18,19 This effect is used in experiments described previously to tune excitonic energy levels with an applied voltage. This effect is also used to make QW modulators for optical communications, which can modulate light at tens of gigabits/s. Because the linewidth of an exciton is typically much greater than 10 GHz (the coherence time of the state is much less than the driving period), optical modulators operate in an adiabatic regime that can be modeled by simply using the DC QCSE formalism and, at the end of the calculation, inserting an oscillating electric field. This final section reports on the NIR absorption of InGaAs/AlGaAs square QWs in the presence of oscillating electric fields polarized in the growth direction and ranging in frequency from 1.5 to nearly 4 THz (6–16 meV).20 As in the square well in which sidebands were studied previously,13 the terahertz and NIR fields couple together only three subbands—two hole subbands, separated by nearly 3.5 THz (14 meV), and a single electron subband. The excitation is highly non-adiabatic because the frequency is significantly greater than the ~0.5 THz (~2 meV) exciton line width. In the absence of terahertz excitation, the NIR absorption spectrum consists of a single peak associated with the lowest energy exciton in the system. As the strength of the terahertz field is increased, the interband absorption spectrum changes in a striking fashion. When the terahertz frequency is resonant with the transition between the two lowest hole subbands, the single absorption peak splits into two peaks with nearly equal amplitude and a separation that increases with increasing terahertz intensity. When the terahertz frequency is higher (lower) than this intersubband resonance, a second smaller absorption peak is seen at frequencies higher (lower) than the main exitonic resonance. The experimental results are compared with a theory that takes into account all of the quantized electron and hole states, the Coulomb interaction between electrons and holes, and the effects of strong terahertz fields without resorting to perturbation theory.

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6.1.2 Semiconductor Quantum Wells An introduction to the optical properties of semiconductors is essential to understanding this chapter. Because most of the semiconductor structures considered in this chapter are GaAs QWs, we will begin with the band structure of GaAs. The band structure consists of the allowed energies of electronic states as a function of the wavevector k. In GaAs, there is a direct gap between the lowest empty band, the conduction band, and the highest filled band, the valence band, at the Γ point, where k = 0. These bands are shown in Figure 6.1(a). Only the properties of these two bands near k = 0 will be considered here. The Γ valley of the conduction band is

e2

Heavy Hole Valence Band

Interband Transition

Interband Transition

Egap

Energy

e1

Intersubband

Conduction Band

hh1 lh1 hh2

Light hole Valence band

–15 –10

–5 0 5 k(105 cm–1)

10

15 –15 –10

–5 In-plane

(a)

0 k(105

5

10

15

cm–1)

(c)

Cond. Band e2

65 THz

e1

470 THz

368 THz

Val. Band Al0.3Ga0.7As

hh1 lh1

39 THz

GaAs

Al0.3Ga0.7As

(b)

Figure 6.1 (a) Simplified diagram of the band structure of gallium arsenide (GaAs) near the fundamental gap. (b) Diagram of a GaAs/AlGaAs QW. (c) Simplified diagram of the band structure of a GaAs QW.

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spherically symmetric and parabolic, leading to an effective mass of 0.067m 0. The valence band is more complicated, with two degenerate bands near k = 0, both of which are anisotropic. Because the bands have different curvature, they are called the heavy-hole and light-hole valence bands. In intrinsic GaAs, the valence band is largely full, and the conduction band is largely empty. The thermal distribution of electrons yields very few free carriers even at room temperature because of the significant band gap. The conductivity can be increased by doping GaAs with impurities. To n-dope GaAs with electrons in the conduction band, typically silicon impurities are added. They occupy gallium sites, donating one extra electron which can ionize and provide free carriers. In p-doping, an impurity such as beryllium occupies gallium sites to accept electrons from the valence band. This induces “holes” in the valence band, which also act as carriers. NIR light can excite optical transitions from the valence band to the conduction band, generating electrons in the conduction band and holes in the valence band. With the advent of epitaxial growth techniques, such as molecular beam epitaxy, very thin layers of different materials with similar lattice constants can be grown on top of each other with high quality interfaces. AlAs and GaAs have essentially identical lattice constants, so layers of AlxGa1-xAs of any thickness or fraction x can be grown without strain on a GaAs substrate. Growth in this material system is very mature. The properties of AlxGa1-xAs are very similar to those of GaAs up to x ≈ 0.45, where the gap becomes indirect. The direct band gap can therefore be varied between about 368 THz and 532 THz (1.52 eV and 2.2 eV) at low temperature. This allows the growth of GaAs layers between AlxGa1-xAs barriers, forming potential wells in one dimension in which both conduction band and valence band carriers are confined. When the thickness of the GaAs layer is comparable to the carrier de Broglie wavelength, quantum confinement becomes significant. A GaAs QW is depicted in Figure 6.1(b), illustrating the band diagram. Quantum confinement in the growth direction leads to energy quantization for motion along the confinement direction, but carriers are still free to move in the QW plane. This gives rise to a number of subbands in the valence and conduction bands, with minima determined by the confinement energy in the growth direction. A simple illustration of these subbands is displayed in Figure 6.1(c). 6.1.2.1 Interband (NIR) Properties Interband transitions take place between a valence subband and a conduction subband, with the strength of the transition determined by the overlap of the growth direction wavefunctions. For symmetric QWs, transitions from hh1 (the first heavy hole subband) to e1 (the first electronic/conduction subband) and lh1 (the first light hole subband) to e1 are strong, whereas transitions from hh1 to e2 and hh2 to e1 are not allowed. The interband absorption is strongly affected by the creation of excitons: bound electron-hole pairs. When an electron is excited from the valence band to the conduction band, there is a Coulomb interaction between the electron and the hole left behind in the valence band. The bound states are similar to hydrogenic states but with a binding energy of ~2 THz (~8 meV) and a Bohr radius of approximately 10 nm because of a smaller reduced mass and a larger permittivity. These hydrogenic wavefunctions are also confined to two dimensions in QWs, with in-plane

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wave functions labeled 1s, 2s, 2p, and so on. Interband transitions are only allowed to the s states because of a vanishing electron-hole overlap for states with nonzero angular momentum. The interband absorption spectrum is dominated by the 1s exciton, with the 2s and higher states often blending into the free electron-hole continuum. Excitons can be formed from various conduction and valence subbands. We will label a heavy-hole exciton eαhhβXnl, where α is the conduction subband, β is the heavy hole valence subband, and n,l are the in-plane hydrogenic quantum numbers. Light-hole excitons are labeled eαlhβXnl. For example, the lowest exciton state is e1hh1X1s. The interband optical properties are characterized by PL and NIR transmission measurements. PL is obtained by exciting the sample with a laser tuned above the bandgap. The optically excited carriers thermalize and recombine, emitting light in all directions. Thus low-temperature PL emission is dominated by the lowest interband transition, with emission at higher energies determined by the carrier temperature. Transmission is obtained by measuring the fraction of transmitted light that passes through the sample, either with a tunable continuous wave laser or with a broadband source. Transmission directly probes the density of optically allowed interband transitions. A transmitted (or reflected) probe laser can also contain additional frequency components from terahertz optical mixing. 6.1.2.2 Intraband (Terahertz) Properties In GaAs QWs many intraband transitions are in the terahertz regime. Terahertz electric fields polarized in the growth direction can couple to intersubband transitions within the valence band or conduction band. The frequency of these intersubband transitions varies from mid-infrared to the terahertz regime, depending on the QW structure and the band. Clearly, there must be free carriers in the well for intersubband absorption to occur. This is typically accomplished through doping, but most experiments in this chapter use interband excitation to generate excitons. A growthdirection terahertz electric field can then excite excitonic intersubband transitions, such as e1hh1X1s → e2hh1X1s or e1hh1X1s → e1hh2X1s. These transitions are tuned by the application of a static electric field. Transitions between different in-plane states can be achieved with in-plane terahertz electric fields. These in-plane transitions naturally occur in the terahertz regime because the exciton binding energy is approximately 2 THz (~8 meV). Transitions such as 1s → 2p+, 1s → 2p–, and 1s → continuum can occur, and the resonant frequencies can be tuned by applying magnetic fields. In high magnetic fields, the 1s → 2p+ resonance is similar to the electron cyclotron resonance, and the 1s → 2p– resonance is similar to the hole cyclotron resonance. All of these terahertz resonances can cause direct terahertz absorption. However, it is generally much more sensitive to detect them by observing the effects of terahertz absorption on PL, as is done in this chapter.

6.1.3 Experimental Techniques 6.1.3.1 In-Plane Terahertz Electric Fields The experiments described in this chapter are performed in two different geometries, depending on the polarization of the terahertz electric field. For in-plane

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polarizations, the visible/NIR probe and the terahertz beams are roughly collinear, illuminating the sample at near normal incidence. Therefore, the electric fields of the probe, PL, and terahertz radiation are in the QW plane. The continuous wave visible/NIR probe beam consists of either an argon ion laser operating at 583 THz (2.41 eV) or a tunable titanium (Ti):sapphire laser operating in the 270–430 THz (1.1–1.8 eV) range. The visible/NIR laser excites electron-hole pairs in an undoped semiconductor sample while terahertz radiation passes through the sample. The effect of the terahertz beam is probed by either collecting the reflected or transmitted NIR radiation/PL. The reflection configuration is shown in the inset of Figure 6.2(a). The reflected radiation and PL are collected either by the same optic fiber which delivered the excitation radiation1 or by separate collection optic fibers.2,7 In NIR transmission experiments the transmitted light is either collected with an optical fiber8 or using free space optics.17

I = 0.05 kWcm–2

1.00

–2

PL Ratio

I = 0.5 kWcm

B

CR 0.60

(a) f = 20 cm–1

Ar+ Laser

PL

Sample

FIR

0.95 PL Ratio

CR X1

0.85

1s – 2p+

(b) f = 103 cm–1

PL Ratio

1.00 X1 0.80

1s – 2p+

(c) f = 130 cm–1 0

2

4

6

8

Magnetic Field (T)

Figure 6.2 The ratio of the photoluminescence amplitude with and without far-infrared (FIR) irradiation as function of magnetic field for Sample 1 at three FIR frequencies. The inset in (a) shows the experimental setup. (Reprinted with permission from Cerne J. et al., Phys. Rev. Lett., 77, 1131, 1996. Copyright 1996, American Physical Society.)

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6.1.3.2 Growth-Direction Terahertz Electric Fields For experiments with growth-direction terahertz electric fields, the NIR probe illuminates the sample at near-normal incidence, but the terahertz beam is coupled into the edge of the sample with its polarization in the growth direction. The sample acts as a dielectric waveguide for the terahertz radiation, but the boundary conditions yield a poor growth-direction terahertz field at the surface (where the QWs are located). Various methods are used to improve the terahertz coupling. In several cases, sapphire is pressed or glued onto the sample surface to form an extended dielectric waveguide with the QWs at the center.10,12,13 In typical intersubband absorption measurements, a metal coating on the sample surface provides a much better boundary condition, strengthening the growth-direction terahertz field and eliminating in-plane fields. The metal layer, however, typically prevents a NIR probe from entering the sample. In one terahertz optical experiment, a transparent conductor (indium–tin–oxide) was pressed onto the sample surface to try to eliminate this problem.14 Perhaps the most successful terahertz coupling method is to use a metallic coating on the sample surface and couple the NIR probe in through the sample substrate.20 This requires QW absorption below the band gap of the substrate, which can be achieved by using InGaAs QWs. The NIR source is either a tunable continuous wave (cw) Ti:sapphire laser for terahertz optical mixing experiments or a NIR light-emitting diode for probing transmission. Both sources are focused onto the sample, and the reflected or transmitted light is collected with free-space optics.

6.1.4 Electrostatic Free-Electron Lasers The UCSB FELs remain unique among terahertz sources and FELs. The FELs are powered by an electrostatic accelerator, much like a van de Graaf accelerator, with an electron beam energy tunable from 2 to 6 MeV. A thermionic electron gun, similar in concept to a triode vacuum tube, periodically fires an electron beam around an evacuated beam line, through one of two wigglers that can be chosen depending on the desired frequency, and back up into a collector for energy and charge recovery. Terahertz radiation travels through the electron beam within the wigglers in an electromagnetic cavity defined by metal end mirrors connected by an overmoded parallel plate waveguide. The terahertz emission is continuously tunable from 0.12 to 4.8 THz (0.5 to 20 meV). Within the selected wiggler, tuning is achieved by varying the energy of the electron beam. The peak power is typically at the kilowatt level. The pulses are typically a few microseconds long, and have a typical repetition rate of 1 Hz. The duration of the FEL pulses is much longer than any relaxation time in the semiconductor samples studied in this chapter. The response of the samples is thus in the steady-state limit. The few microsecond duration of the terahertz pulses is compatible with well-developed NIR technology and electronics. For example, a FEL-synchronized pulse of tunable, monochromatic NIR radiation can easily be generated by sending the output of a cw Ti:sapphire laser through an appropriately-triggered acousto-optic modulator. Standard photomultiplier tubes can be used for singlechannel detection with submicrosecond time resolution. Multichannel detection

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with submicrosecond time resolution is accomplished with a gated, intensified CCD camera. The few-microsecond pulses generated by the UCSB FELs have spectral band widths (pulse-to-pulse and within a single pulse) less than one part in 103. This enables most experiments to be carried out with the beam traveling through the ambient atmosphere in one of the many spectral “windows” in which the atmosphere is transparent over laboratory length scales. Rather than continuously tuning the FEL frequency, most experiments are performed at a discrete set of frequencies and the resonance frequencies of the sample are continuously tuned by adjusting either a magnetic or electric field. Most other FELs with user facilities (for example, FELiX in the Netherlands, and, more recently, FELBE in Germany) are based on higher energy radiofrequency linear accelerators. The output of FELiX extends in frequency from 1.2 THz to 100 THz (5–410 meV). FELiX naturally produces a train of “micropulses” with duration measured in picoseconds; these micropulses are contained within a “macropulse” with a duration of a few microseconds. The peak powers available at FELiX are in the MW range, much higher than is available at the UCSB FELs, and hence very promising for nonlinear and non-perturbative experiments. Femtosecond lasers have been synchronized with the output pulses of FELiX and other FELs based on radiofrequency linear accelerators. Thus terahertz optical measurements can and have been performed at FELs like FELiX. However, the synchronization does present an additional challenge, and the frequency resolution of an experiment like sideband generation will be Fourier transform limited by the duration of the FEL micropulse.

6.2 Internal Dynamics of Excitons and Effects of Terahertz Radiation on Excitonic Photoluminescence 6.2.1 Introduction In the simplest approximation, an exciton can be thought of as a hydrogenic complex composed of an electron and a hole. Using interband absorption or PL spectroscopy, it is possible to deduce many of the properties of excitons in QWs. One can, for example, measure a sharp peak in PL or absorption from the e1hh1 exciton in its 1s state, and also, in high-quality samples, observe a nearby peak from an exciton in its 2s state. The 1s and 2s excitons have significant electron-hole overlap, and hence are radiative. However, there exist many nonradiative exciton states with nonzero angular momentum. These states have no signature in single photon interband absorption or PL. However, after a 1s exciton is created via interband excitation, an electromagnetic wave with frequency resonant with, for example, the 1s-2p transition can be absorbed, promoting the exciton from the 1s to the 2p state. In GaAs, these internal transitions of excitons lie in the terahertz frequency range. Just as it is necessary to perform optical spectroscopy to unravel the internal electronic structures of atoms, so is it necessary to perform terahertz spectroscopy to elucidate the internal electronic structures of excitons.

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Terahertz Spectroscopy: Principles and Applications

In the last chapter, Kaindl has described beautiful and relatively recent experiments which directly measure the terahertz absorption spectrum of GaAs QWs as a function of time after they have been photoexcited with interband radiation. Excitonic absorption features form surprisingly quickly after photoexcitation. This section describes earlier, quasi-steady-state measurements of the internal electronic structures of cold excitons. These measurements use a different technique that requires lower optical power but more terahertz power (although milliwatt power levels provided by molecular gas lasers are sufficient). The presence of terahertz absorption is deduced by observing terahertz-induced changes in the excitonic PL. It is possible to optically detect terahertz resonances of excitons by monitoring the PL at a fixed terahertz frequency and relatively low intensity as the internal transitions of excitons are swept through the terahertz frequency by continuously varying an applied magnetic field. In the first part of the chapter, the optically detected terahertz resonances of magnetoexcitons are measured. Great care is taken to ensure that the resonances are associated with free excitons, and not with neutral donors (which also have excitations at terahertz frequencies). The observed resonances compare favorably with theoretical calculations published before the experiment. The rest of the section explores the resonant and nonresonant mechanisms whereby terahertz radiation alters PL.

6.2.2 Internal Dynamics of Magnetoexcitons Measured by Optically Detected Terahertz Resonance Spectroscopy As can be seen in the inset of Figure 6.2(a), an Ar+ laser is used to excite electron-hole pairs in an undoped semiconductor sample while terahertz radiation passes through the sample. Sample 1 consists of fifty 100 Å-wide GaAs QWs between 150 Å-thick Al0.3Ga0.7As barriers, grown by molecular beam epitaxy on a semi-insulating (100) GaAs substrate. The magnetic field is oriented along the propagation directions of the terahertz and visible radiation. The Ar+ all-line laser excitation intensity of 10 W/cm2 creates an exciton density of approximately 3 × 109 cm–2 per well.21 Two techniques are used to measure the PL: a single channel technique using a photomultiplier tube and a multichannel technique using a gated intensified CCD camera. For conventional optically detected terahertz resonance (ODTR) using a photomultiplier tube, the monochromator is set to detect the peak of the PL from the e1hh1X1s. A boxcar integrator gates the PL signal during and after the terahertz pulse. The unquenched, 9 K PL immediately after the terahertz pulse is used to normalize all the data. The magnetic field, B, is varied to sweep excitonic resonances through the fixed energy of the terahertz radiation. At each B, the wavelength of the monochromator is adjusted to track the energy of the PL peak, which shows the expected diamagnetic shift. The second technique using an ICCD camera allows PL spectra over a wide energy range (not just one amplitude) to be captured at each B. This powerful new technique is referred to as multichannel optically detected terahertz resonance (MODTR). A separate MODTR run is made with the terahertz radiation blocked to serve as a reference.

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6.2.2.1 Experimental Results

(a)

1.0

X1

1.1

Ar+ Intensity (W/cm2)

104

690 1.0

(b) 0.4 1.2 2.0

0.4 0.2

X1

CR

10 0.6

FIR Intensity (kWcm–2)

PL Ratio

0.8

f = 84 cm–1; IFIR = 1 kW/cm2

1s–2p+

PL Ratio

20

CR

1.2

1s–2p+

6.2.2.1.1 Optically Detected Terahertz Resonances Although s-orbital excitons (e.g., e1hh1X1s and e1hh1X2s) are accessible to interband measurements such as PL and photoluminescence excitation, terahertz radiation is required to directly excite many higher states (e.g., e1hh1X2p). In this section, we will focus on e1hh1X, so the electron and hole subband index will usually be dropped in the notation. Figure 6.2 plots the ratio of the PL amplitudes with and without terahertz irradiation as a function of B. A series of resonances is observed. We focus on two dominant resonances that are observed in all the samples that we studied. We assign these to terahertz-induced electron cyclotron resonance (a and b) and 1s → 2p+ (as discussed in the following section) excitonic transitions (b and c). Note the terahertz-induced enhancement of the e1hh1X1s PL amplitude above 6 T in Figure 6.2(c). The weaker resonances such as X1 are probably excitonic features as well (e.g., 1s → 3p+), but are not observed in all samples. The terahertz resonances are preserved when a Ti: sapphire laser excites carriers only into the QW, below the barrier band gap. The ODTR features become narrower at lower terahertz and visible intensities. In Figure 6.3, the PL ratio as a function of B is shown for several visible (Figure 6.3(a))

f = 103 cm–1 IAr+ = 110 W/cm2

700 0

2

4

6

8

10

Magnetic Field (T)

Figure 6.3 The photoluminescence ratio as a function of magnetic field is shown for several visible (a) and far-infrared (b) intensities in Sample 1. The traces are not offset. (Reprinted with permission from Cerne, J. et al., Phys. Rev. Lett., 77, 1131, 1996. Copyright 1996, American Physical Society.)

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Terahertz Spectroscopy: Principles and Applications

and terahertz (Figure 6.3(b)) intensities. The traces are not offset. Both the cyclotron resonance and excitonic transitions are visible. At the lowest terahertz and visible intensities (top trace in Figure 6.3(a)) the PL is enhanced at all magnetic fields except at 1.9 T, where there is a sharp feature less than 0.3 T (0.12 THz or 0.5 meV or 4 cm–1) wide. This feature is assigned to the 1s → 2p+ free-exciton resonance and broadens dramatically as the visible excitation intensity is increased. The baseline also falls below 1 and cyclotron resonance becomes observable at the higher visible excitation intensities. The dependence on visible excitation intensity is strongest at low terahertz intensity. Figure 6.3(b) shows the dependence on terahertz intensity for a fixed visible excitation intensity. Increasing the terahertz intensity broadens the absorption features and lowers the PL ratio baseline. This means that higher terahertz intensities produce significant off-resonance PL quenching at all magnetic fields and visible intensities. The solid symbols in Figure 6.4 show the energies of the clearest minima in the PL ratio as a function of B. The open symbols represent the 1s-2s energy spacing as deduced from PL measurements. The deepest minima are associated with the filled circles. The dashed lines above and below the solid line represent calculations of the transition frequency of donors in the center or near the edge of the same QW. Clearly, these do not agree with the measurement. The solid black line is a calculation of the 1s-2p+ transition energy (using the low-field, hydrogenic notation22) as a function of B in a QW with identical specification to the one on which measurements are reported.6 The line, which was calculated 10 years before the experiment was done, agrees extremely well with the solid circles, supporting their identification with the 1s-2p+ transition of the exciton. The solid diamonds are identified as the electronic cyclotron resonance. These are linear with B and suggest an electronic effective mass of 0.073 m0, which is in agreement with theory23 and experiment24 for a 100-Å GaAs QW. The slope for the 1s-2p+ exciton transition is roughly the same as that for CR, also consistent with its identification as a 1s → 2p+ -like transition. Weaker, higher energy transitions (solid triangles) have a clearly larger slope, indicating that they are transitions from the 1s ground state to even higher energy states. The inset of Figure 6.4 shows the magnetic field at which the 1s → 2p+ -like transition occurs in four QWs (70, 100, 140, and 150 Å) at two terahertz frequencies (3.1 and 3.9 THz or 12.8 and 16.1 meV or 103 and 130 cm–1). The 1s → 2p+ transition in the top trace of Figure 6.3(a) is roughly four times narrower than the 1s e1hh1X1s PL line. This may be explained by the fact that the 1s → 2p+ transition energy depends only weakly on well width and hence is less sensitive to fluctuations in the well: ∂E1s → 2p+/∂LZ = (∂E1s → 2p+/∂B)(∂B/∂LZ) = 0.007 THz/Å (0.03 meV/Å), where the derivatives on the right-hand side are calculated from experimental data of Figure 6.4 and its inset. For comparison, the interband electron heavy hole (e1-hh1) spacing is much more sensitive to the well width, varying like 0.25 THz/Å (1 meV/Å).25 6.2.2.1.2 Multichannel Optically Detected Resonant and Nonresonant PL Modulation by Terahertz Radiation The previous section demonstrates that the e1hh1X1s PL amplitude is a powerful sensor with which ODTR can illuminate terahertz excitations. However, the

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150

100

8 6 4 2 60

0

Data 103 cm–1 Data 130 cm–1 Theory 103 cm–1 Theory 130 cm–1

CR

50

BR (T)

FIR Frequency (cm–1)

Exciton Center donor Edge donor 2p– –2s Exciton

0

2

4

100 140 Well Width () 6

8

10

Magnetic Field (T)

Figure 6.4 The solid symbols show the energy of the various far-infrared resonances as a function of magnetic field in Sample 1. The excitonic transitions such as the 2s → 2p + (solid circles) and higher energy transitions (solid triangles) can be seen in addition to the free-electron cyclotron resonance (solid diamonds). The thick solid line and thin dotted line represent calculations using excitonic theory,6 whereas the dashed lines are calculated using donor theory.18 The thin solid line is free-electron cyclotron resonance with an effective mass of 0.073 m0. The 1s → 2s energy spacing of the heavy hole exciton (large empty circles) was deduced from interband PL excitation measurements. The inset shows the magnetic field at which the 1s → 2p + exciton transition occurs at two FIR frequencies in four QWs. (Reprinted with permission from Cerne, J. et al., Phys. Rev. Lett., 17, 1131, 1996. Copyright 1996, American Physical Society.)

mechanism behind ODTR is not explained. Also, despite its success, this technique provides limited information. By measuring a PL spectrum over a wide energy range at each B (instead of only the e1hh1X1s PL amplitude), the carrier dynamics over a broad energy range are revealed. MODTR provides a much more complete view of terahertz dynamics by resolving the carrier distribution over a broad energy range. Using MODTR will enable an understanding of some of the mechanisms that underlie ODTR. We also observe a very rich set of phenomena, only some of which are understood. Figure 6.5 shows the difference of the PL spectra from sample 1 with and without terahertz irradiation plotted as a function of B. The terahertz frequency is fixed at 3.1 THz (12.8 meV or 103 cm–1) and the intensity is approximately 50 kW/cm2.

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Terahertz Spectroscopy: Principles and Applications

3

3

F

PL Diﬀerence (arb. units)

2

E

D´

2

1

1

B

3 2 1

0

0

0 –1

–1 –2

G

D

E´

–1 C A

8

M

ag

6

ne

tic

Fie l

4

d ( 3.5 2 T)

1.58

1.56 ) (eV rgy

–2

–2

1.55

1.57 Ene

Figure 6.5 (Color figure follows p. 204.) PL spectra from Sample 1 without terahertz irradiation subtracted from photoluminescence spectra with terahertz irradiation at 3.1 THz (12.8 meV or 103 cm–1) as a function of B. The terahertz intensity is approximately 50 kW/ cm2. The sample temperature is 10 K. (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 66, 5301, 2002. Copyright 2002, American Physical Society.)

The sample temperature is 10 K. Stronger PL quenching is represented by more negative values (shown in blue in color figure) of the PL difference. The false color image of this surface plot is projected on the bottom of the graph. There are strong qualitative differences in the PL modulation at different magnetic fields. At B = 0T, the difference spectrum shows the quenching of the heavy hole 1s (H1s) PL peak amplitude (Figure 6.5(a)), the enhancement of high energy tail of the H1s PL (B), and the enhancement of the light hole 1s (L1s) PL amplitude (C). These effects were previously observed and studied in detail.1 As B increases, the H1s PL amplitude is resonantly quenched at the H1s → H2p + (mz = ± 1) transition at 3.5 T (D and D′), whereas the L1s PL reaches a maximum (E and E′). The terahertz-induced repopulation from H1s to L1s states is clearly seen here at 3.5 T. At 8 T, further resonant quenching is observed (F) which is assigned to the H2p – (mz = –1) transition. Note that at higher than 3.5 T, quenching is reduced and the dominant effect of

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(a)

H3p–

H2p+ L1s

L1s H2p+

L1s H2p+

H1s

H1s

PL ETHz

L1s H1s

(b) 2p––2s

8000

1s – 3p

7000 6000 5000 4000

H2p+

2

900 800 700 600

LX

500

HX 0

1000

4 6 Magnetic Field (T)

8

LX PL Amplitude (arb. units)

ETHz 10

0 HX PL Amplitude (arb. units)

H3p+

1s – 2p+

Energy – EH1s (meV)

terahertz radiation is to enhance and blue shift the H1s PL peak (F). This can seen in the depth (height) of the PL difference surface plot higher than and lower than 3.5 T and is also shown by the color difference between these two regions in the projected image plot. A ridge that begins at 382 THz (1.58 eV) at 0 T and appears to follow the quadratic magnetic field dependence of the H1s PL peak also is indicated (G). This ridge is consistently higher than the H1s state by one terahertz photon energy throughout the magnetic field range. Note that this ridge appears to cross a more strongly curved second ridge at point E′, which is where the L1s is resonantly enhanced. Neither ridge is consistent with magnetic field dependence of PL from the L1s. Also note that the line where the PL difference sheet crosses zero (yellow line in the projected image plot) is nearly independent of magnetic field and does not follow the B2 dependence of H1s PL peak energy. The information contained in Figure 6.5 is more accessible when cross-sections through this surface plot are examined. The resonant enhancement of the L1s PL peak at the expense of the H1s PL amplitude is shown in Figure 6.6(b), where the

400 10

Figure 6.6 Energies relative to the H1s state of higher energy excitonic states are schematically shown in (a). Solid lines indicate energy levels (e.g., H2p+) that are accessible from the H1s state via radiative transitions. When B = B1s–2p+, the terahertz photon energy matches the H1s → H2p+ energy separation. Note that for B = 3.5 T, the H2p+ state is above the L1s state. Insets in (a) are simple representations of the thermal and photothermal carrier distributions induced by terahertz irradiation. The L1s and H1s photoluminescence (PL) amplitudes are plotted as a function of B under terahertz illumination at 10 K (b). These data are obtained from cross-sectional slices that follow the L1s and H1s PL energies in Figure 6.5. (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 66, 5301, 2002. Copyright 2002, American Physical Society.)

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Terahertz Spectroscopy: Principles and Applications

H1s and L1s PL amplitudes are plotted against B. Note that this figure shows the PL amplitudes, not the normalized PL ratios (i.e., PL with divided by PL without terahertz irradiation, as shown in most ODTR spectra), which adds a baseline that depends on B. The H1s → H3p+ feature at 2.1 T splits into two dips (H1s → H3p+ and H1s → H3p–) for B higher than 2.5 T.2,4,26 The L1s PL enhancement is asymmetric in B, with a fast decay above 3.5 T, and a more gradual decrease below 3.5 T. The H1s amplitude shows similar asymmetry and reflects the conventional ODTR results.2,4 Figure 6.6(a) sketches the energy difference between the H1s and higher energy exciton states (based on data published previously2). The diagrams in Figure 6.6(a) sketch the photothermal carrier distributions at different magnetic fields, and will be discussed in greater detail later in this section. Absorption of terahertz radiation can heat carriers, resulting in a quasithermal distribution with a temperature higher than that of the lattice.1 It is thus useful to compare changes in PL induced by terahertz radiation to those induced by changing the temperature of the sample. Figure 6.7 plots the L1s to H1s PL amplitude ratio from Sample 1 as a function of inverse lattice temperature, without (a) and with (b) terahertz irradiation of approximately 100 W/cm2. These data form Arrhenius plots where straight lines indicate thermally activated population of the L1s from H1s states. The slopes of these lines determine the activation energies, which are shown in the legends of Figure 6.7. Note the dramatic decrease in the activation energy when the sample is illuminated by terahertz radiation at 3.5 T. We propose a qualitative explanation of terahertz-induced changes in PL which involves three processes. (1) Absorption: terahertz radiation is absorbed via internal transitions of excitons,2,4,26–28 or by Drude-like free carrier excitations of ionized electrons and holes.1 (2) Quasithermalization: each species of excitons (for example, H1s, H1p+, or L1s), is associated with a band of states with different center-of-mass momenta. The power transferred into the electron-hole system indirectly heats each band of excitons to a temperature warmer than that of the lattice. (3) Luminescence: the luminescence under terahertz irradiation only reflects the population of states that can participate in radiative interband transitions. The radiative states are those with in-plane center of mass momentum KCOM near zero. No states are radiative for higher orbital excitors. We now discuss the experimental data as a function of increasing magnetic field with a fixed terahertz frequency of 3.1 THz (12.8 meV or 103 cm–1). The case B = 0 T has been studied extensively1 and will be discussed in greater detail later in the next section. In this nonresonant case, the mechanism for absorption was shown to be Drude-like heating of ionized electrons and holes. Between 0 T and about 3.5 T, the qualitative features of Figures 6.5 and 6.6 are similar to those at B = 0 T, with a quenching of the H1s luminescence and enhancement of L1s. In this regime, there are many transitions between internal states which have energies below that of the 3.1 THz (12.8 meV or 103 cm–1) pump, and which can be excited by it. One of the most striking features of the data is the resonant transfer of PL from H1s to L1s which occurs near 3.45 T (E and E′ in Figures 6.5 and Figure 6.6). Our explanation is a resonant photo/thermal mechanism, shown schematically in the central cartoon in Figure 6.6(a). Terahertz radiation promotes excitons from H1s to

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Optical Response of Semiconductor Nanostructures in Terahertz Fields 100

(a)

225

0.00 T: EA = 16.4 ± 0.4 meV

LX/HX PL Ratio

3.45 T: EA = 9.35 ± 0.73 meV 8.00 T: EA = 10.5 ± 0.5 meV

10–1

10–2 No THz 100

(b)

3.45 T: EA = 0.19 ± 0.20 meV

LX/HX PL Ratio

4.60 T: EA = 7.82 ± 0.59 meV 8.00 T: EA = 13.0 ± 0.6 meV

10–1

10–2 ωTHz = 103 cm–1 1

2

3

100/Temperature (K–1)

4

Figure 6.7 Arrhenius plot of the e1lh1X1s/e1hh1X1s photoluminescence (PL) amplitude ratio without (a) and with (b) terahertz irradiation. The terahertz intensity is approximately 100 kW/cm2. The straight lines represent activated behavior with an activation energy that is determined by the slope of the line. Activation energies that are obtained from exponential fits are shown in the legends. The e1hh1X2p+ - e1lh1X1s energy spacing measured from PL is approximately 3.4 THz (14 meV) for all B. The e1hh1X2p+ - e1lh1X1s separation is approximately 0.25 THz (1 meV) at 3.45 T. (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 66, 5301, 2002. Copyright 2002, American Physical Society.)

H2p+. The dark H2p+ state is only about 0.25 THz (1 meV) below the radiative L2s state according to PL and ODTR data. Single-photon radiative recombination is forbidden for the 2p+ state, and non-radiative recombination is also weak. Thermal fluctuations, associated with the quasitemperature of the H2p+ excitons, are sufficient to then populate L1s exciton states to a much greater degree than they could be populated via thermal excitation directly from H1s. According to photothermal ionization experiments on shallow donors in GaAs,29 the 2p+ state allows extremely efficient ionization. Previous research29 found that the probability of ionization from the 2p state is much larger than expected from the energy separating this state and the continuum; the ionization probability at 4.2 K is essentially unity. Above approximately 3 T, the H2p+ is actually higher in energy than the HX continuum. L1s

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Terahertz Spectroscopy: Principles and Applications

excitons decay, emitting the observed luminescence. The photothermal hypothesis is supported by the observation that, in Figure 6.7(b), the ratio of L1s to H1s luminescence is nearly independent of temperature when terahertz radiation illuminates the sample at B = 3.45 T. This demonstrates an effective activation energy which is negligible. In contrast, without terahertz irradiation, the L1s/H1s at 3.45 T increases strongly with temperature, consistent with an activation energy on the order of 2.5 THz (10 meV). Above 3.45 T, the changes in PL induced by terahertz irradiation are qualitatively different from those induced at lower magnetic fields. At high magnetic fields in Figure 6.6(a), the terahertz photon energy is smaller than the lowest dominant exciton transition (H1s → H2p+). The high magnetic field photothermal distribution is shown schematically in the cartoon on the right side of the inset in Figure 6.6(a). In this regime, lower energy exciton transitions (e.g., H2p – → H2s) can occur, and the hot carriers can impact with and heat cold excitons, which produces a PL-quenching signal at 8 T in Figure 6.5. The heating is not efficient and high terahertz and visible intensities are required to produce the H2p – → H2s feature. There are several reasons for this reduced efficiency. First, at low temperature, most of the carriers are bound in 1s (ground state) excitons, so the population in excited states (e.g., 2p –, which has an energy that has a characteristic temperature of 80 K above the 1s state), especially at low terahertz and visible intensities, is significantly lower than the 1s population. Higher intensities increase the exciton temperature, and hence enhance the population of 2p – excitons that can participate in the H2p – → H2s transition. Second, PL amplitudes are not as sensitive to temperature changes at higher B. Finally, most hot heavy hole excitons (HXs) (with center of mass momentum KCOM) will eventually cool to contribute to the main HX PL line within the radiative cone (KCOM ≈ 0). Because radiative recombination for the hot HX (KCOM ≠ 0) is suppressed because of momentum conservation, and because nonradiative recombination is weak, most of the hotter HXs have nowhere to go but back to the KCOM = 0 state where they can efficiently recombine radiatively. The closest escape for HXs is the L1s which is over 3.4 THz (14 meV, 163 K) higher in energy at B = 0 (see Figure 6.6(a)). This activated behavior can be seen in Figure 6.7, where the activation energy to populate the L1s state is approximately 2.4 THz (10 meV) without (a) and with (b) terahertz irradiation at 8 T. As a result, heating may increase the H1s lifetime as it journeys through KCOM space, but will not significantly change the quantum efficiency of the H1s PL nor the steady-state ODTR signal. This model does not explain the terahertz-induced enhancement and blue shift of the H1s PL amplitude. One expects an energy shift in the PL close to terahertz resonances because of the AC Stark effect.17,30 The calculated shift changes sign depending on whether the terahertz pump frequency is higher or lower than the transition frequency. One would expect an intensity-dependent blue shift (red shift) in the exciton absorption spectrum when the driving photon energy is higher (lower) than the unperturbed transition energy. The data, on the other hand, show a shift that increases monotonically with B. The apparent lack of a Stark shift may be due to the fact that excitonic transition energies are swept across the energy of the photons in the strong terahertz radiation field. As a result the difference between the transition frequency and the pump

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227

frequency (detuning frequency) is not fixed, but is swept continuously from positive to negative values, and hence no single laser-induced energy shift is observed. This may lead to a broadening of the absorption due to the transition energy shifting as the detuning energy is swept through zero. Furthermore, because the terahertz radiation is both the pump and probe, it is more difficult to isolate terahertz-induced shifts in the exciton transition energy. MODTR has been used to study the terahertz dynamics of magnetoexcitons in GaAs QWs. A simple qualitative model can be used to understand the PL modulation due to terahertz radiation. On the other hand, this work has generated several new puzzles which still require exploration. The MODTR baseline and the higher energy ridge (labeled G in Figure 6.5) that follows the H1s peak are still not clearly understood. The higher energy ridges (G and E′ in Figure 6.5) appear to indicate mixing of excitonic and photonic states. Significant energy shifts of over 0.7 THz (3 meV) are observed in the H1s PL close to H2s– → H2s, which may be related to an AC Stark effect. The information gained from studying excitons in QW will provide an important foundation for research of excitons in even lower dimensional systems such as quantum wires and dots.

6.2.3 Nonresonant PL Quenching Mechanism In this section, we investigate in more detail the quenching of PL by intense terahertz radiation in the absence of a magnetic field. The dominant effect of the terahertz radiation is to heat carriers without significantly heating the lattice. The carrier heating can be described by a Drude free-carrier model, despite the clear presence of excitons. It is suggested that terahertz radiation heats free carriers, which are not contributing to luminescence, and that these hot carriers heat the luminescing excitons. 6.2.3.1 Experiment The sample and experimental setup was identical to the technique using the photomultiplier tube detector that was described in the previous section. In this case, no magnetic field was applied and the temperature, terahertz intensity, and terahertz frequency dependence were explored. 6.2.3.2 Results Figure 6.8 illustrates the effects of lattice temperature and terahertz radiation on the PL spectra. The solid lines in Figure 6.8(a) and Figure 6.8(b) show PL spectra in the absence of terahertz radiation for Sample 1 multiple quantum-well (consisting of fifty 10-nm wide GaAs wells) at 8.4 and 96 K, respectively. At 8.4 K, the PL spectrum consists of a single narrow line (full width at half max <0.5 THz [2 meV]) at 374.1 THz (1.547 eV), which we assign to the heavy hole free exciton transition e1hh1X1s from the lowest electronic state e1 to the highest heavy-hole state hh1. At 96 K, the e1hh1X1s transition has shifted to lower energy because the bandgap narrows as lattice temperature increases, the e1hh1X1s peak is smaller and broader, and a second peak appears at a higher energy corresponding to the excitonic transition e1lh1X1s between e1 and the thermally populated light hole level lh1.

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228 0.16 0.12 0.08

0.50

No FIR

0.25

FIR Power: 300 kW/cm2 FIR Frequency: 66 cm–1

0.04 0.00

PL Intensity (arb. units)

(a) Lattice Temperature: 8.4 K

0.00

(b) Lattice Temperature: 96 K

0.12 0.08

No FIR

PL Without FEL (arb. units)

PL with FEL (arb. units)

Terahertz Spectroscopy: Principles and Applications

150  100  150  ×50 Al0.3Ga0.3As Al0.3Ga0.7As GaAs Z

0.04 0.00 1.525

1.535

1.545

1.555

1.565

Energy (eV)

Figure 6.8 Excitonic photoluminescence (PL) spectra from Sample 1 (100 Å wide Al0.3Ga 0.7As/GaAs multiple QWs) (a) at 8.4 K with FIR (circles) and without far-infrared (solid line) irradiation, and (b) at 96 K with no FIR irradiation. Note the energy shift of the PL in (b) because of the narrower band gap at the higher lattice temperature. (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 51, 5253, 1995. Copyright 1995, American Physical Society.)

The circles in Figure 6.8(a) show the PL spectrum at 8.4 K in the presence of terahertz radiation. The shape and the amplitude of this spectrum closely match the spectrum in Figure 6.8(b). However, the energy of the e1hh1X1s transition is unchanged from its unquenched 8.4 K value, indicating no change in the bandgap energy. The absence of a significant blue shift indicates that the luminescence remains primarily excitonic. From these data, we deduce that the primary effect of the terahertz irradiation is to heat luminescing excitons without heating the lattice. At the highest terahertz intensity, the lattice temperature remains below 15 K. Note that there are subtle differences between the 8.4 K terahertz-irradiated spectrum and the 96 K spectrum. In particular, the PL linewidth is broader at 96 K, perhaps resulting from increased phonon scattering. A comparison of the effects of higher lattice temperature and increasing terahertz radiation intensity can be seen in Figure 6.9. The open triangles show that the peak intensity of the e1hh1X1s PL decreases monotonically with increasing lattice temperature or terahertz intensity. An exciton thermal dissociation model is used to fit the data in Figure 6.9(a) (see Appendix A in reference 1). The solid circles show that the peak intensity of the light hole free exciton (e1lh1X1s) PL initially increases as the lattice temperature or terahertz intensity is increased. This enhancement arises from the thermal excitation of e1hh1X1s into the higher energy e1lh1X1s.

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1.0

(a) PL Amplitudes vs Lattice Temperature

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Heavy Hole

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Light Hole

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50

100

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Light Hole PL Amplitude (arb. units)

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Optical Response of Semiconductor Nanostructures in Terahertz Fields

1.0

(b) PL Amplitudes vs FIR Intensity

0.8 0.6 0.4

Heavy Hole

0.02

Light Hole

0.2 0.0

0.04

FIR Frequency: 20.5 cm–1 Lattice Temperature: 8.8 K

1.0

2.0

3.0

4.0

0.00

Light Hole PL Amplitude (arb. units)

Heavy Hole PL Amplitude (arb. units)

Lattice Temperature (K)

(FIR Intensity (kW/cm2))1/3

Figure 6.9 Heavy-hole (triangles) and light-hole (disks) exciton photoluminescence amplitudes from sample 1 versus (a) lattice temperature and (b) the cube root of the far-infrared (FIR) intensity at a constant lattice temperature of 8.8 K. The fit in (a) arises from an exciton thermal dissociation model which is discussed in Appendix A in reference 1. The FIR frequency is 0.615 THz (2.54 meV or 20.5 cm-l). (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 51, 5253, 1995. Copyright 1995, American Physical Society.)

At approximately 100 K, which corresponds approximately to the exciton ionization energy and the energy separating light and heavy holes (see Figure 6.9(a)), the e1lh1X1s PL intensity also begins to decrease as electron-hole correlations weaken and nonradiative traps strengthen at higher temperatures. The same effect can be seen in Figure 6.9(b) above a terahertz intensity of 16 kW/cm2 and at a fixed lattice temperature of 8.8 K. In the absence of terahertz radiation, the carrier temperature is assumed to be the same as the lattice temperature. This assumption is reasonable for the 36 to 100K temperature range used in our calculations, but begins to fail at lower temperatures31 where the photoexcited carriers tend to have a higher temperature than the lattice. In the presence of terahertz radiation (Figure 6.9(b)), the carrier temperature is deduced by finding a lattice temperature Figure 6.9(a)) at which the e1hh1X1s PL amplitude is the same. The e1lh1X1s PL amplitude or the ratio of the e1lh1X1s to e1hh1X1s

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amplitudes also could be used and yields temperatures within approximately ±20%. The success of such a comparison relies on the assumption that the PL amplitudes depend only on carrier temperature. 6.2.3.3 Drude Analysis of Carrier Heating The terahertz radiation heats photogenerated electrons and holes. We assume that the electrons and holes are at the same temperature. For the purposes of modeling, the electrons and holes can then be treated as an ensemble of particles with an absorption cross-section σ abs and an energy relaxation rate 1/τ energy equal to the average of the electron and hole properties. We also assume that the electrons and holes share the same momentum relaxation time τmom. The terahertz pulses we use are much longer than the carriers’ relaxation time, and thus may be treated as a steady-state excitation. Assuming that the power absorbed is equal to the power lost per carrier, the average energy relaxation time and absorption cross-section are related by Pabs = I THz σ abs = Plost =

kB (TC - TL ) k (T - TL ) . → σ abs τ energy = B C τ energy I THz

(6.1)

ITHz, σ abs , kB, TC, TL , and τ energy are the terahertz intensity, average absorption crosssection, Boltzmann constant, carrier temperature, lattice temperature, and average energy relaxation time respectively. Figure 6.10 plots σ abs τ energy (as given in Equation 6.1) as a function of terahertz frequency. The fits using a simple Drude model are surprisingly good and allow energy and momentum relaxation times to be calculated for different carrier temperatures (see inset of Figure 6.10). At the high frequencies used, the data are close to the 1/ω2 Drude absorption regime,32 in which it is difficult to extract τ energy and τmom separately in the Drude model. This can be seen in the nearly linear behavior when σ abs τ energy is plotted on a logarithmic scale in Figure 6.10 and results in a relaxation time uncertainty of roughly a factor of four at 101 K and a factor of two at 36 K. The uncertainties obtained from the fits in Figure 6.10 were roughly ±20% and ±15% for σ abs τ energy and τmom, respectively, which significantly underestimate the overall uncertainty. However, it is clear that σ abs τ energy decreases in magnitude as temperature is increased. This result is a strong indication of the decrease of τ energy as temperature is increased. The decrease of the energy relaxation time with temperature confirms that carriers can relax faster at higher temperatures as more carriers have sufficient energy to emit fast-relaxation LO phonons.33–35 The energy relaxation time exhibited a power law dependence on carrier temperature, with a fitted exponent of –1.97 ± 0.04 as can be seen in the inset of Figure 6.10 in which the relaxation times extracted from the Drude fits include both electrons and holes. 6.2.3.4 Discussion 6.2.3.4.1 Free-Carrier Terahertz Absorption in an Excitonic System The PL remains excitonic in nature for all the terahertz frequencies and up to the highest terahertz intensities. At the very highest intensities of 700 kW/cm2, the

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36 K 64 K

Time (ps)

49 K

10.0

231

α Tc–2 Energy Relaxation

1.0

σabs τenergy (cm2 s)

Momentum Relaxation

10–25

10–26

1012

0.1

89 K 101 K

0

80 120 40 Carrier Temperature (K)

ω–2

1013

1014

FIR Frequency (rad/s)

Figure 6.10 Drude fit to the absorption cross section at different temperatures. Energy and momentum relaxation times extracted from the Drude fit are shown in the inset. (Reprinted with permission from Cerne, J. et al., Phys. Rev. B, 51, 5253, 1995. Copyright 1995, American Physical Society.)

terahertz electric fields are 12 kV/cm, corresponding to a potential drop of 2.9 THz (12 meV) across an excitonic radius of 100 Å, which is the similar to the exciton binding energy. In these measurements, the terahertz intensities are typically on the order of 100 kW/cm2 (see Figure 6.9) or less, producing terahertz electric fields smaller than the static electric field binding the exciton together, and apparently are insufficient to field ionize the excitons. Similarly, heating excitons may diminish their population via thermal ionization, but will not defeat their dominance in PL from undoped samples even at room temperature.36 One might have expected that when the terahertz photon energy matches the exciton binding energy (or the 1s – 2p transition energy), nonthermal ionization (excitation), and therefore resonant PL quenching would occur. The terahertz frequency is scanned in increments of 0.024 THz (0.1 meV) from 1.7 to 2.7 THz (7 to 11 meV) at an intensity of approximately 1 kW/cm2, but no resonant PL quenching is observed. This may not be surprising since the terahertz power varies strongly (±50% over a 0.03 THz [0.12 meV or 1 cm–1) frequency range and up to a factor of six on wider ranges) during a typical single scan. Furthermore, it is difficult to normalize the PL signal with respect to terahertz power since the signal is not related to the terahertz power in a simple, well-understood manner. This energy range encompasses most of the calculated and measured excitation energies for the e1hh1X1s found in recent literature. A second possible terahertz resonance would involve the heavy to light hole transition induced by 2.7 THz (11 meV) terahertz radiation. This transition is allowed due to strong valence band mixing).37,38 This resonance was explored but not observed.

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Given that the PL remains excitonic during the terahertz pulses and that the carrier temperatures suggest that most of the carriers are still bound as excitons, it is surprising that the terahertz-induced carrier heating could be accurately fitted by a simple Drude model. We believe that the resolution of this mystery lies in the fact that although excitons dominate PL, free carriers dominate terahertz absorption. If one assumes that the relatively high mobility of the free carriers allows them to dominate in absorbing terahertz radiation, it is reasonable to suggest that the terahertz radiation is heating the free carriers, which in turn heat the luminescing excitons. The steady-state picture involves a hot bath of free carriers, injected by the visible excitation radiation and released by exciton dissociation, absorbing terahertz radiation and heating the remaining excitons to their temperature. Excitonic PL is reduced as more excitons thermally ionize at higher temperatures and hot free electrons and free holes are less likely to bind as excitons. Microwave experiments in similar systems have shown that free carriers dominate microwave absorption, and that these hot free carriers collide with excitons.39 As a result, the average exciton temperature increases and the exciton distribution shifts to higher energy, which is reflected by the enhancement of PL on the higher energy side of the H1s PL peak and decrease in PL on the lower energy side of the H1s PL peak. If the excitons are at the same temperature as the free-carrier bath, then the excitonic PL should reflect the free-carrier temperature and the Drude analysis should produce the correct relaxation times for the free carriers that are being heated by the terahertz radiation. 6.2.3.4.2 Relaxation Times Numerous experiments have measured electronic relaxation rates in GaAs quantum heterostructures, and although most of the samples and experimental techniques are similar, the results vary significantly. Our experiment obtained energy relaxation times ranging from 11 ps at 40 K to 3 ps at 80 K (see Table 1 in reference 1). These times are shorter than those measured in other experiments,34,35,40–42 and the TC-2 temperature dependence is much weaker. Except for one study,42 most experiments show a two order of magnitude decrease in energy relaxation time as temperature is raised from 40 to 80 K. Although the temperature dependence of the energy relaxation time does not agree with other experiments, it does explain the power dependence of the carrier heating found in reference 1. Equation 1.1 suggests that carrier temperature Tc is related to the terahertz intensity ITHz as follows:

TC ∝ I THz τ energy ∝ I THzTC-2 → TC3 ∝ I THz .

(6.2)

This result is consistent with Figure 6.5 of reference 1, which shows that the carrier temperature grows as the one-third power of terahertz intensity. Differences between the relaxation times measured here and those in other experiments are discussed in greater detail elsewhere.1 6.2.3.4.3 Conclusion Excitonic PL from Al xGa1-x As QWs at low temperature clearly indicates that free-carrier heating is the dominant effect of terahertz radiation. At maximum

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233

intensities of 700 kW/cm 2 or 12 kV/cm, the terahertz radiation raises the carrier temperature in Sample 1 to higher than 100 K, and lowers the inelastic relaxation time to 1 ps. The average free-carrier energy relaxation time is inversely proportional to the carrier temperature squared. Although excitons dominate the luminescence, free carriers play a much stronger role in terahertz absorption at B = 0. The fact that no excitonic terahertz resonances were observed using steady-state PL may be more indicative of the experimental limitations rather than the weakness of terahertz coupling to excitons. Despite the qualitative nature of this experiment, the Drude model could be applied to the terahertz intensity and frequency dependence of the carrier heating to obtain energy and momentum relaxation times. This experiment not only resolves the dominance of carrier heating in quenching PL in QWs by intense terahertz radiation, but also underscores the need to include this effect when probing the many other exciting properties of these structures.

6.3 Near-Infrared-Terahertz Mixing 6.3.1 Introduction Because terahertz radiation can couple very strongly to transitions in semiconductor QWs, intense terahertz radiation can lead to many interesting nonlinear effects in QWs including harmonic generation,43,44 saturation of intersubband transitions,45,46 Rabi oscillations,47 AC Stark effect,17,30 resonant/nonresonant ionization of excitons,2,4 and quenching of PL.1,21 In many cases, the energy associated with the terahertz electric field coupling to a dipole transition is comparable to both the transition and terahertz photon energies. The nonlinear interaction of terahertz and NIR radiation not only provides rich new physics, but also shows a strong potential for applications in a frequency regime which is underexploited technologically. For example, terahertz–NIR mixing may have applications in modulating light at high frequencies, which is of central importance for optical communication. Further applications could use NIR probes for fast, coherent detection of terahertz radiation.48 This technique also could serve as a novel, nonlinear probe of terahertz transitions in semiconductors. As is demonstrated in this section, NIR–terahertz mixing can dominate the optical emission of quantum heterostructures, so a greater understanding of this effect is crucial to a number of experimental and design scenarios. Nondegenerate three- and four-wave mixing in semiconductors has been studied extensively in the NIR domain and in recent years these experiments have been extended to terahertz frequencies. Most of these experiments involve generating weak terahertz radiation by difference mixing two NIR lasers.49–51 Other experiments have mixed nondegenerate terahertz photons.52 In this section, we discuss the first experiments to mix intense terahertz radiation with NIR radiation to produce NIR sidebands. This section is divided in two parts according to the linear polarization direction of the terahertz radiation, which determines whether the terahertz radiation can couple to intersubband transitions in the QW.

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6.3.2 NIR-Terahertz Mixing with In-Plane Terahertz Polarization: Sideband Generation and Nonlinear Spectroscopy of Magnetoexcitons Low intensity NIR radiation, which is used to excite electron-hole pairs across the energy gap, can mix with high intensity terahertz radiation that is polarized in the plane of the QW to produce intense and narrow sidebands on either side of the NIR probe radiation. The frequency of the sideband radiation fsideband is:

fsideband = fNIR + nfTHz

(6.3)

where f NIR and f THz are the NIR and terahertz frequencies respectively and n = ±2. The sideband intensity is doubly resonant in both the NIR and terahertz, with resonant enhancement when f NIR is tuned to interband absorption maxima and when f THz is tuned to intraexcitonic absorption. Because the electric field of the terahertz radiation is linearly polarized in the plane of the QW, it does not couple to intersubband transitions of the QW. 6.3.2.1 Experimental Setup Two samples are discussed in this section. Sample 1 is measured using the terahertz PL setup (see inset of Figure 6.2). The main difference in the case of sideband measurements is that instead of an Ar+ laser, a continuous wave Ti:sapphire laser (tunable NIR probe with frequencies of 270 to 440 THz [1.1–1.8 eV]) is used to create electron-hole pairs in the sample. The second sample (Sample 2) consists of 25 periods of undoped GaAs/Al0.3Ga0.7As multiple QWs (10 nm/15 nm) on 30 periods of GaAs/Al0.69Ga0.31As (4 nm/40 nm) etch stop layers grown on a semi-insulating GaAs substrate. The substrate was etched to allow NIR transmission measurements to be made. Unlike the measurements on Sample 1 in this and previous sections where NIR radiation and PL are collected in reflection, the measurements on Sample 2 collect the transmitted NIR radiation and PL. 6.3.2.2 Results Figure 6.11 is representative of the observed NIR sidebands collected in reflection at 9 K from Sample 1. The thin lines represent the NIR spectrum, while terahertz radiation illuminates the sample, and the thick lines were obtained in the absence of terahertz irradiation. Figure 6.11(a) shows both the up-converted (n = +2) and downconverted (n = –2) sidebands for a fixed NIR frequency of 386.1 THz (1.597 eV or 12880 cm–1) and terahertz frequency of 3.45 THz (14.3 meV or 115 cm–1). Note the greater intensity of the down-converted sideband at 379.5 THz (1.569 eV or 12658 cm-1) compared to the upconverted sideband at 393.3 THz (1.626 eV or 13118 cm–1). The broad background on the down-converted sideband is enhancement of the energy tail of e1lh1X1s PL due to carrier heating.1 On Sample 2, the sidebands can be seen more clearly, and n = ±4 sidebands appears. Figure 6.12 shows sideband spectra at B = 10 T and f THz = 3.45 THz (14.3 meV or 115 cm–1). In Figure 6.12(a), NIR radiation resonantly creates excitons

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0.2

NIR Intensity (arb. units)

ωHH ωNIR ωHH

0.1

+2ωTHZ

‒2ωTHZ With THz 0.0

Without THz

12500

12900

12700

Frequency

13100

(cm–1)

2

1

NIR Intensity (arb. units)

NIR Intensity (arb. units)

(a)

‒2ωTHZ ωHH

0 12500

12600 Frequency (b)

(cm–1)

0.08

+2ωTHZ

0.00 12972

12980

12976 –1)

Frequency (cm (c)

Figure 6.11 Sidebands and photoluminescense (PL) observed at two terahertz frequencies and three NIR frequencies at 8.5 T. (a) f THz = 3.45 THz (14.3 meV or 115 cm-1); f NIR = 386.4 THz (1.598 eV or 12,888 cm–1); fsidebard = 379.5 THz (1.569 eV or 12,658 cm–1) and 393.3 THz (1.626 eV or 13118 cm–1). The small, broad PL feature at 379.2 THz (1.568 eV or 12,650 cm–1) is a result of PL enhancement because of carrier heating. (b) f THz = 3.1 THz (12.8 meV or 103 cm–1); f NIR = 382.3 THz (1.581 eV or 12753 cm–1); fsideband = 376.1 THz (1.556 eV or 12,547 cm–1). The high-energy PL tail (376.5 to 377.7 THz or 1.557 to 1.562 eV or 12,560 to 12,600 cm–1) is slightly enhanced because of carrier heating. (c) f THz = 3.45 THz (14.3 meV or 115 cm–1); f NIR = 382.1 THz (1.580 eV or 12746 cm–1); fsideband = 389.0 THz (1.609 eV or 12,976 cm–1). PL baseline is zero because the sideband is far above the PL emission peak. (Reprinted with permission from Cerne, J. et al., Appl. Phys. Lett., 70, 3543, 1997. Copyright 1997, American Institute of Physics.)

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Terahertz Spectroscopy: Principles and Applications ωTHz = 115 cm–1, B = 10 T 4

(a) –2ω Sideband –2ωTHz

Intensity (a.u.)

2 × 400

0 12500

4

2

(b)

12600

12700

12800

+2ω Sideband +2ωTHz

+4ω Sideband +4ωTHz × 400

0 12600

12800

× 4000 13000

Energy (cm–1)

Figure 6.12 Typical sideband generation spectra at B = 10 T. f THz = 3.45 THz (14.3 meV or 115 cm–1) for both figures. (a) Downconversion: f NIR = f 2s = 382.1 THz (1.580 eV or 12745 cm–1) and fsideband = f NIR = 375.2 THz (1.552 eV or 12,515 cm–1). (b) Upconversion: f NIR = f1s = 375.2 THz (1.552 eV or 12,515 cm–1) and fsideband = f NIR = 2f THz = 382.1 THz (1.580 eV or 12,745 cm–1), and f NIR + 4f THz = 389.0 THz (1.609 eV or 12975 cm–1). (Reprinted with permission from Kono, J. et al., Phys. Rev. Lett., 79, 1758, 1997. Copyright 1997, American Physical Society.)

in the e1hh1X2s state at 382.1 THz (1.580 eV or 12745 cm–1). A narrow (<0.06 THz [0.25 meV or 2 cm–1]) downconverted sideband that is exactly two terahertz photon energies lower appears at 375.2 THz (1.552 eV or 12,515 cm–1) in the presence of terahertz radiation. The n = +2 and n = +4 upconverted sidebands are shown in Figure 6.12(b), where the NIR radiation resonantly creates excitons in the e1hh1X1s state at 375.7 THz (1.554 eV or 12,531 cm–1). Note that the sideband intensities are typically 1000 or more times smaller than the NIR laser intensity. The n = +4 sideband appeared only at the highest terahertz intensity of approximately 10 kW/cm2. Figure 6.13 shows the dependence of the n = +2 sideband intensity on B at constant f THz = 3.45 THz (14.3 meV or 115 cm–1) and with f NIR always tuned to the e1hh1X1s interband transition as B is changed. Three resonances are observed in this nonlinear optically detected terahertz resonance measurement. Resonance (a) occurs at 4.5 T and corresponds to f THz = f 2p+ – f1s, the 1s → 2p+ transition. This resonance is also observed in the reflection measurements on Sample 1.7 Resonance (b) occurs at 9.5 T and corresponds to 2f THz = f 2s – f1s, a 1s → 2s transition involving two terahertz photons. This transition is forbidden in linear ODTR. Resonance

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+2ω Sideband Intensity (a.u.)

(c)

ωNIR = ω1s ωTHz = 115 cm–1

8

237

6

(b)

4 (a) 2

0

2

4

6

8

10

12

Magnetic Field (T)

Figure 6.13 The magnetic field dependence of the n = +2 sideband intensity for f THz = 3.45 THz (14.3 meV or 115 cm–1) and f NIR = f1s at all magnetic fields. The data demonstrate nonlinear optically detected terahertz resonances. Pronounced resonances occur when (a) f THz = f 2p+ - f1s, (b) 2f THz = f 2s- - f1s, and (c) f THz = f 2p - - f1s. Note that the two-photon resonance (b) is forbidden in linear optically detected terahertz resonance. (Reprinted with permission from Kono, J. et al., Phys. Rev. Lett., 79, 1758, 1997. Copyright 1997, American Physical Society.)

(c) occurs at 11.5 T and corresponds to f THz = f 2p– – f1s, the 1s → 2p– transition. These resonances are in excellent agreement with linear ODTR experiments2,4 and theory6 on magnetoexcitons in QWs. As with measurements in reflection, the n = ±2 sideband emission is maximized when either f NIR, fsideband, or both f NIR and fsideband are at excitonic interband resonances such as f1s or f 2s. The strongest sidebands occur in the doubly resonant condition when f NIR = f1s and fsideband = f 2s. The intensity of the n = +2 sideband depends quadratically on the terahertz intensity for a constant NIR power. At a constant terahertz power, the n = +2 sideband intensity increases linearly with the NIR power up to 50 mW, but saturates at higher NIR power. Therefore, at low NIR power, the n = +2 sideband emission can be described as a third-order nonlinear process involving one NIR photon and two terahertz photons. The same power dependence is found for the n = –2 sideband, but due to its small magnitude, the power dependence of the n = +4 sideband could not be determined. The n = +2 sideband emission depends critically on the polarization of the terahertz radiation. The n = +2 sideband intensity is maximized when the terahertz

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radiation is linearly polarized with an axis parallel to the linear polarization of the NIR excitation laser. The n = +2 sideband intensity drops by 70% to reach a minimum when the terahertz radiation is circularly polarized. This suggests that both left (σ –, electron-cyclotron-resonance inactive) and right (σ+, electron- cyclotron-resonance active) circularly polarized terahertz photons, which add to produce a linear polarization, are simultaneously required to produce the n = +2 sideband. 6.3.2.3 Discussion The n = ±2 sideband generation can be described using a third-order nonlinear sus( 3) ceptibility χijkl which mixes the electric fields Ej and Ek of two terahertz photons with the electric field El of a NIR photon to produce a resulting polarization Pi as follows: Pi ( fNIR ± 2 fTHz ) =

∑

jkl

( 3) χijkl E j ( ± fTHz ) Ek ( ± fTHz ) El ( ± fNIR ).

(6.4)

Although the sideband generation resonates with transitions in the material, no real excitations are created and the process is parametric, with the initial and final states being the same. Assuming that the NIR photon creates excitons and the terahertz radiation excites internal transitions in these   excitons, standard perturbation theory with the exciton–photon interaction - p ⋅ E , where p is the exciton polarization operator, ( 3) leads to the following expression for χijkl for the n = ±2 sideband: 3) = χ(ijkl

- N well h3

∑(f αβg

〈 0 | pi | g 〉〈| p j | β〉〈β | pk | α 〉〈α | pl | 0 〉 NIR

+ 2 fTHz - fg - i Γ/2 p)( fNIR + fTHz - fβ - i Γ//2 p)( fNIR - fα + i Γ/2 p)

+ ( nonresonant terms ) . (6.5) Here Nwell is the well layer density; α, β, and g run over the exciton states; fα = (E α - E 0)/h where E α (E 0) is the energy of the exciton state α (vacuum state 0 ); and Γ is the phenomenologic damping factor. The expression for the n = –2 sideband is obtained by replacing f THZ with -f THZ in Equation 6.5. Using Γ = 0.25 THz (1 meV) in a numerical evaluation of Equation 6.5 leads to an extremely large χ(3) ranging from 10 –6 to 10 –3 esu for the nonresonant and resonant cases, respectively. Equation 6.5 shows that χ(3) is resonantly enhanced when (1) f NIR = f ns (where n is the principal quantum number for the excitonic state and s represents the angular momentum quantum number) (2) fNIR + fTHz = fn′pm ( m = ±1) , or (3) fNIR + 2 fTHz = fn′′s . These resonances agree with experimental observations and can be visualized in Figure 6.14, where the dashed lines represent virtual states that only exist while laser fields are present in the material, whereas the solid lines represent real magnetoexcitonic levels such as 1s, 2s and 2p±. When one of the virtual levels coincides with a real level, χ(3) is resonantly enhanced. In each case of Figure 6.14, two of the three conditions (1–3) are met. In (a), f NIR = f1s and f NIR + f THZ = f 2p-; in (b), f NIR = f1s and f NIR + 2f THZ = f 2s; and in (c), (f NIR + f1s) and (f NIR + f THZ) - f 2p-. The detuning

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Optical Response of Semiconductor Nanostructures in Terahertz Fields

|2s> σ– |2p+> σ+ |1s>

ωTHz ωTHz

|2s> |2p> |1s>

239

|2s> σ+ σ–

∆

|2p> |1s>

σ+ σ–

ωNIR |0>

(a)

|0>

(b)

|0>

(c)

Figure 6.14 Diagrammatic representation of the resonant processes responsible for the observed three resonances in Figure 6.13. The dashed lines represent virtual levels, whereas the solid lines represent real magnetoexcitonic levels. Resonances occur when real levels coincide with virtual levels. The shaded area represents the magnitude of the detuning ∆. (Reprinted with permission from Kono, J. et al., Phys. Rev. Lett., 79, 1758, 1997. Copyright 1997, American Physical Society.)

in (a) is greater than 3 THz (12.4 meV or 100 cm–1), which results in the weak sideband generation observed for peak (a) in Figure 6.13. On the other hand, the detunings of the remaining unsatisfied condition in (b) and (c) are D = (f NIR + f THZ) - f 2p- and D = (f NIR + 2f THZ) - f 2s, respectively. These detunings are less than 0.3 THz (1.2 meV or 10 cm-1), which is consistent with the strong sideband emission that is observed for peaks (b) and (c) in Figure 6.13. Equation 6.5 and Figure 6.14 also explain the observed dependence of sideband generation on the polarization of the terahertz radiation. This is especially clear in cases (b) and (c) in Figure 6.14, in which the adjacent real excitonic states (that are connected by terahertz photons) have opposite angular momentum. As a result, two opposite helicity photons are required to optimize the two-part process. For example, in case (b), a σ – photon induces the 1s → 2p- transition nearly resonantly, but a σ+ photon is required to induce the 2p- → 2s transition in the second part of the process. Similarly, both a σ – and a σ + photon are required for case (c). Therefore, linearly polarized terahertz radiation optimizes sideband generation since it consists of equal parts of σ – and a σ+ polarizations.

6.3.3 NIR-Terahertz Mixing with Out-of-Plane Terahertz Polarization: Excitonic and Electronic Intersubband Transitions In the previous section, only even sidebands (fsideband = f NIR + nf THZ, where n = ±2, 4, …) are observed because of spatial inversion symmetry. The generation of n = ±1 sidebands is particularly desirable for application to wavelength conversion because these sidebands are linear in terahertz fields and therefore much stronger in the perturbative regime. The resonant structure of n = ±1 sidebands is also easier to interpret because it can be described by a χ(2) process. Bulk inversion asymmetry in GaAs has given rise to n = ±1 sidebands in thick, bulk GaAs samples,53 but this process has not been observed in thin heterostructures without a structural asymmetry. Breaking the structural inversion symmetry can be achieved by the growth of asymmetric

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2ω Sideband Intensity (5 nW/div)

Emission (a.u.)

1ω Sideband Intensity (10 nW/div)

Detuning (THz) 0 1 2 3 4 5 6 7 8 ×10–4

1.557 1.567 1.577 1.587 Spectrometer Energy (eV)

0

0.2 0.4 0.6 0.8 Incident 2.5 THz Power (normalized)

1

Figure 6.15 Terahertz power dependence of the n = +1 (1ω, circles) and n = +2 (2ω, triangles) sideband intensity at f THz = 2.5 THz (10.3 meV). Solid lines are linear and quadratic fits. Inset: Typical sideband emission spectrum. Full terahertz power is ≈2 kW at an intensity of ≈2 MW/cm2. The NIR intensity was ≈250 W/cm2. (Reprinted with permission from Phillips, C. et al., Appl. Phys. Lett., 75, 2728, 1999. Copyright 1999, American Institute of Physics.)

QW structures or by using a static electric field, typically in the growth direction. The polarization of the terahertz beam must also be in the growth direction. Sidebands with n = ±1 are first observed in asymmetric coupled QWs (ACQWs), in which the terahertz radiation couples to intersubband transitions.9 Figure 6.15 displays the n = ±1 and n = +2 sidebands that are observed in the ACQWs, including the respective linear and quadratic dependences on terahertz power. Gated devices are also developed, in which the intersubband resonances are tuned by a static electric field.11,12,14 Figure 6.16 displays the sideband signal in an ACQW as a function of the static electric field, illustrating voltage-controlled sideband generation. In a simple square QW, this static field is used to break the inversion symmetry to generate n = ±1 sidebands.13 Several theoretical articles examine sideband generation in this system as well.54–56 We will describe experiments in an undoped square QW first, because it is the simplest system and the resonant behavior can be explained in detail. Next, sideband generation in undoped ACQWs will be discussed, and finally sideband generation in n-doped QWs will be presented. 6.3.3.1 Undoped Square QWs The active region of this sample consists of three undoped 150 Å GaAs QWs separated by 300 Å Al0.3Ga0.7As barriers. These QWs are between two doped 80 Å QWs that are ohmically contacted and used to apply static electric fields to the sample. All of these QWs are grown on a distributed Bragg reflector (DBR), which reflects the incident NIR light along with the generated sidebands (see Figure 6.17 for a schematic of the conduction and valence band profiles). The reflectivity (essentially transmission with the DBR) of the sample at 19 K with no static field (Ebias = 0) is

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Optical Response of Semiconductor Nanostructures in Terahertz Fields 0.8

0.3

n = +2

0

1540

1545

1550

1555

EµHνX

1560

EαHβX

ωNIR + ωTHz

0.4

n = +1

100

ωTHz

0.5

E1HH2X-E2HH1X

x104 Laser Line

ωNIR

0.6

200 NIR emission (nW)

Sideband Conversion (×10–4)

0.7

NIR Energy (meV)

E2HH1X-E1HH2X

0

0.2 0.1 0 –25

–20

–15

–10

–5

0

5

10

15

DC Electric Field (kV/cm)

Figure 6.16 Sideband voltage scan at f NIR = 373.8 THz (1.546 eV) and f THz = 2.0 and THz (8.2 meV). Peak assignments are derived from a excitonic nonlinear susceptibility theory. Inset: n > 0 sideband spectrum at 0 V gate bias, f NIR = 372.9 (1.542 eV) and f THz = 2.0 THz (8.2 meV). (Reprinted with permission from Su, M.Y. et al., Appl. Phys. Lett., 81, 1564, 2002. Copyright 2002, American Institute of Physics.)

displayed in Figure 6.18(a). The spectrum is relatively simple with strong absorption at the heavy-hole and light-hole excitons, e1hh1X1s and e1lh1X1s. These excitons are formed from an electron in the first conduction subband and a hole in the first heavy-hole or light-hole valence subbands, respectively. An exciton with a hole in the second heavy-hole valence band, e1hh2X1s, is about ~2.4 THz (~10 meV) above e1hh1X1s, but interband excitation of the e1hh2X1s is not optically allowed for Ebias = 0. Excitons formed from higher electron or light-hole subbands are at energies sufficiently far away that they will be ignored. We consider only the 1s in-plane state of these excitons as they dominate the absorption strength. Front Gate

Back Gate

DBR Fermi Energy

Active Area

Figure 6.17 Schematic diagram of the conduction band structure of the studied sample. (Reprinted with permission from Ciulin, V. et al., Phy Rev. B, 70, 115312, 2004. Copyright 2004, American Institute of Physics.)

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–21

31

0

1.0

–2

hh2X Energy (meV)

Normalized Reflectivity

Electric Field (kV/cm) –10 0 10 21

lh1X

0.5

–4

–6

0.0

hh1X 1.53

1.54

1.55

–8

–2

–1

0

Energy (eV)

Bias (V)

(a)

(b)

1

2

3

Figure 6.18 (a) High-resolution normalized reflectivity spectra at 19 K and zero bias. The arrow indicates the spectral position of the e1hh2X1s transition that does not appear for unbiased structures. (b) Energy change of the e1hh1X1s reflectivity line as a function of bias and estimated electric field at 19 K, solid squares. The dashed line shows the result of calculations. (Reprinted with permission from Ciulin, V. et al., Phys. Rev. B, 70, 115312, 2004. Copyright 2004, American Physical Society.)

A static electric field red shifts the e1hh1X1s and e1lh1X1s resonances with some broadening according to the quantum-confined stark effect (QCSE).18,19 The energy of the e1hh1X1s absorption is plotted in Figure 6.18(b) as a function of the static field, along with the calculated dependence. The red shift is due to the electron and hole wavefunctions moving away from each other to lower energy at opposite sides of the QW. Due to leakage current and other effects, the tuning is poor for Vbias < -1.5 V, and the integrated PL was not constant for | Vbias | > 2 V. The experiments are typically performed at Vbias = + 1V (Vbias = +10 kV/cm) to avoid these effects. By breaking the spatial inversion symmetry of the QW, e1hh2X1s becomes optically allowed, with an energy that is relatively constant with Ebias. Figure 6.19(a) shows the experimental geometry for sideband generation. The terahertz beam is edge-coupled into the sample, which acts as a dielectric waveguide. Sapphire is pressed onto the sample surface to extend the waveguide, placing the QWs at the center of the waveguide where the terahertz field is stronger. The NIR beam from a continuous wave Ti:sapphire laser passes through the sapphire, interacts with the terahertz-driven QWs, and is reflected back out by the DBR, along with any sidebands. The sample is held at 19 K or 25 K for these experiments. A typical

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E = ETHz +Edc Sapphire NIR Laser ENIR

Thz FEL Active Area Sample

Sideband Conversion Eﬃciency × 104

1.2

243

Laser × 10–4

1.0

n=1

n = –1 0.8 0.6 0.4 0.2

ETHz

n=2

n = –2 0.0 (a)

1.528

1.532

1.536

Energy (eV) (b)

Figure 6.19 (a) Schematic drawing of the experimental configuration. (b) Sideband spectra at 25 K for f THz = 0.65 THz (2.7 meV) with a bias Vbias = 1 V and a terahertz laser intensity of approximately 0.1 MW cm–2. (Reprinted with permission from Ciulin, V. et al., Phys. Rev.B, 70, 115312, 2004. Copyright 2004, American Physical Society.)

sideband spectrum at f THz = 0.65 THz (2.7 meV) is displayed in Figure 6.19(b). The n = ±1 sidebands are much stronger than the n = ±2 sidebands and only appear when a static electric field is applied to the sample. The resonant behavior of sideband generation is examined by scanning the NIR laser energy while simultaneously scanning the detection energy to follow a particular sideband. Figure 6.20(a) displays n = –1 sideband resonance spectra for a series of terahertz frequencies. There is a lower energy peak (A) when the incident NIR energy is resonant with e1hh1X1s, which is particularly strong at the two lower terahertz frequencies. A higher energy peak (B) occurs when the sideband energy is resonant with e1hh1X1s, so that the two resonances are always separated by the f THz (see inset of Figure 6.20). The level diagrams in Figure 6.20(b) display these two resonances. There is an overall decrease in the amplitude of these resonances as the terahertz frequency increases, although there is an increase in the B resonance at f THz = 2.5 THz (10.4 meV), corresponding to the e1hh1X1s - e1hh2X1s transition. The double resonance condition is fulfilled at this terahertz frequency, so that f NIR is resonant with e1hh2X1s and fsideband is resonant with e1hh1X1s. The sideband resonant behavior can be well understood by a simple χ(2) model, with three states: the ground state (vacuum—no excitons | 0〉), e1hh1X1s ( |1〉), and e1h1X1s ( | 2〉). The lh1X exciton is not included as sidebands associated with this exciton are quite weak. Taking an expression for χ(2) from previous work57 and

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244

(a) 3

hh1X

EnergyB–EnergyA

B

A 2

10

Low-energy Peak A

0

0 5 10 THz Photon energy (meV)

High-energy Peak B

5.5 meV

1

|2> |1>

8.2 meV 10.4 meV

0 hh1X

|0> (d)

2

3

|χ(2)|2 (arb. units)

|0>

2.7 meV

(c)

(b)

|2> |1>

5

2

2.7 meV 5.5 meV

1

8.2 meV

Max|χ(2)|2 (arb. units)

n = –1 Sideband Intensity (arb. units)

Terahertz Spectroscopy: Principles and Applications

1

10.4 meV 0 1.530

1.535

1.540

1.545

NIR Laser Energy (eV)

0

0

5

10

THZ Photon Energy (meV)

Figure 6.20 (a) Sideband resonance spectra, showing the n = –1 sideband intensity as a function of the NIR laser energy. The measurements were taken at 25 K and Vbias = 1 V (Edc = 10 kV/cm) for f THz = 0.65 (triangles), 1.3 (circles), 2.0 (diamonds), and 2.5 (squares) THz (2.7, 5.5, 8.2, and 10.4 meV) and terahertz laser intensities of approximately 25, 30, 40, and 55 kW cm–2, respectively.26 The spectra have been offset for clarity. For each curve the offset is shown by a small line on the right side of the panel. Inset: energy difference between the two resonances labeled A and B as a function of ω THz. The line shows the predicted oneto-one-correspondence. (b) Schematic level model of the different transitions. (c) | χ( 2 ) |2 calculated using Eq. 6.6 Ciulin et al.13 for the f THz and Edc used experimentally. For the numerical evaluation of the plotted curves, we used g = 0.25 THz (1 meV) and a GaAs band gap of 368 THz (1.52 eV). (d) Calculated maximum of | χ( 2 ) |2 (f NIR) as a function of f THz. (Reprinted with permission from Ciulin, V. et al., Phys. Rev. B, 70, 115312, 2004. Copyright 2004, American Physical Society.)

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245

applying it to this three level system gives χ(2 ) ( fNIR + fTHz , fNIR , fTHz ) ∝

∑(f αβ

α0

z m 0x α m αβ mβx 0

- fNIR - fTHz - i g α 0 2 p) ( fβ 0 - fNIR - i g β 0 2 p)

(6.6)

x z where mβ0 is the interband dipole matrix element, m αβ is the intersubband dipole matrix element (for α =/ b), and ga0 is the interband dephasing rate. For n = +1 (n = –1) sidebands, f THz is positive (negative). The index β represents the exciton intermediate state created by NIR radiation, and α represents the state into which the β state is excited by terahertz radiation. The product of dipole moments in the numerator is zero without an applied electric field. For α = 1, b = 2 there is a double resonance when f NIR is resonant with e1hh2X1s and fsideband is resonant with e1hh1X1s (n = –1 sideband). This is an excitonic intersubband resonance, as it occurs when f THz is equal to the e1hh1X1s – e1hh2X1s splitting. This double resonance corresponds to the B resonance in Figure 6.20(a) at f THz = 2.5THz (10.4 meV). For α = 2, b = 1, there is also a double resonance, which occurs for n = +1 sidebands. The α = b terms can only have a double resonance for small terahertz frequencies (2pf THZ < g), with f NIR and fsideband resonant with a single exciton z state. The dipole moment m αα represents the dipole moment between the electron and hole, which is non-zero with inversion asymmetry. The α = b = 1 term is much larger than the α = b = 2 term because of the stronger interband matrix element of e1hh1X1s compared with that of e1hh2X1s. The α = b = 1 term represents a quasistatic modulation of the exciton absorption, and accounts for the strong sideband generation at low terahertz frequencies. Figure 6.20(c) plots | χ( 2 ) |2 from this model as a function of f NIR for a series of terahertz frequencies matching the experimental data. The two calculated resonances appear at the same energies as in experiment, with the amplitudes following the same trend. The maximum of | χ( 2 ) |2 (f NIR) is plotted in Figure 6.20(d) as a function of f THz, displaying the strong low-frequency, quasistatic response, and the excitonic intersubband resonance near f THz = 2.4 THz (10 meV). Although other undoped QW structures (i.e., ACQWs) may have much more complicated interband absorption spectra than this simple system, the same low-frequency response and excitonic intersubband resonances are expected.

6.3.3.2 Undoped Asymmetric Coupled QWs There are several advantages of using ACQWs for sideband generation. First of all, inversion asymmetry is built into the structure without the application of static electric fields. Second, using coupled wells allows the intersubband spacing to be controlled by the tunnel barrier width and also by applied electric fields. The disadvantage of ACQWs is the complexity of the energy levels. The typical ACQW structure is inset in Figure 6.21. The GaAs QWs are of width 100 Å and 120 Å, with a 25- Å Al0.2Ga0.8As tunnel barrier. In this system, the energy splitting between the two lowest conduction subbands is designed to be ~2.5 THz (~10 meV), whereas there are multiple closely spaced valence subbands.

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Terahertz Spectroscopy: Principles and Applications 3.5 NIR Probe

2.5

Sapphire 30 Al% 0

THz Field

Active Spectrometer Region

2.0

×50 120  100 

1.0 n = –1

0.5 0.0

n = +1

1.5 NIR Laser

Sideband Conversion (×103)

3.0

n = +2 n = +3

n = –2 ×10

–0.5 1.525

×10 1.535

÷100

×10 1.545

×10

1.555

NIR Energy (eV)

Figure 6.21 Transmitted sideband spectrum taken at 20 K with a FEL frequency of 1.5 THz (6.4 meV). The transmitted beam at f NIR is divided by 100 and the n = +3, +2, –1, –2 sidebands are multiplied by 10 for clarity. The terahertz power was approximately 1 kW and the NIR power was 0.2 mW. The experimental geometry and the conduction band diagram of a sample well are inset in the figure. (Reprinted with permission from Carter, S.G. et al., Appl. Phys. Lett., 84, 840, 2004. Copyright 2004, American Institute of Physics.)

The sideband spectrum in Figure 6.21 is from a sample with 50 periods of ACQW, designed to maximize the n = ±1 sideband conversion efficiency for application to wavelength conversion.10 The substrate of this sample was etched away, and the epitaxial layer was glued between two pieces of sapphire. This placed the active region at the center of a dielectric waveguide. The sidebands are quite strong at low temperatures, with an n = +1 conversion efficiency of approximately 0.2% (ratio of sideband power to incident NIR laser power). This conversion efficiency is comparable to those in other wavelength converters,58,59 and represents a very high nonlinearity considering the approximately 4-µm thick active region. Sidebands are also observed at sample temperatures up to room temperature with reduced conversion efficiency. The resonant behavior in this ACQW is also explored.11 An applied static field is used to tune the excitonic intersubband transitions into resonance with the terahertz electric field. This sample consisted of 10 ACQWs, with gate QWs and a DBR, as in the square QW structure displayed in Figure 6.17. The linear optical properties are characterized by measuring the PL as a function of the gate voltage. This PL is plotted as a colormap in Figure 6.22, along with the calculated exciton energies formed from two conduction subbands and several valence subbands. There are

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247

1565

Energy (meV)

1560 1555 1550 1545 1540 1535 1530 –20

–15

–10

–5

0

5

10

DC Electric Field (kV/cm)

Figure 6.22 (Color figure follows p. 204.) Experimental photoluminescence spectra as a function of the DC electric field. Theoretical e1hhnX and e2hhnX energies are indicated by circles and squares, respectively, where n represents the heavy hole subband. Observed excitonic intersubband resonances at 2.0 and 2.5 THz (see Figure 6.23 and Figure 6 of Reference 11) are indicated by the dotted and solid arrows, respectively. (Reprinted with permission from Su, M.Y. et al., Appl. Phys. Lett., 81, 1564, 2002. Copyright 2002, American Institute of Physics.)

several avoided crossings in the spectrum, which occur when the subband energies in the two QWs approach each other. The calculated exciton energies are qualitatively similar to the measured energies, although the complexity of this spectrum makes it difficult to interpret. The vertical arrows represent excitonic intersubband transitions that are expected to generate sidebands at f THz = 2.0 or 2.5 THz (8.3 or 10.3 meV). Figure 6.23 displays a color map of the n = +1 sideband signal at f THz = 2.0 THz (8.3 meV) as a function of the static electric field and f NIR. As f NIR is scanned, the detection energy is set to always detect the n = +1 sideband signal. There are several resonances, each labeled by the excitonic intersubband transition calculated to give rise to the resonance. This experiment demonstrates that sideband generation can be controlled by a voltage, which also controls the terahertz resonances. In many of these previously discussed sideband experiments, the terahertz intensity is low enough that the system can be modeled perturbatively with χ(2) or χ(3). This perturbative regime applies when the dipole oscillation energy, or Rabi energy µE, is less than  times the dephasing rate g of the system, where µ is the intersubband dipole moment and E is the terahertz field amplitude. When mE ≥ g , nonperturbative behavior, such as Rabi oscillations can occur.47 Sideband generation in ACQWs was explored in this nonperturbative regime and in the strong-field regime in which µE is greater than the terahertz photon energy.12 These experiments were performed at ~20 K on the same gated ACQW sample described in the

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Terahertz Spectroscopy: Principles and Applications 1

1560

0.9

1550

0.8

E1HH2X-E2HH3X

1545

1540

E1HH2X-E2HH1X

E2HH1X-E1HH2X

0.7 0.6 0.5 0.4 0.3 0.2

1535

1530 –20

Sideband Conversion (×10–4)

NIR Energy (meV)

1555

E1HH1X-E2HH1X –15

–10

–5

0

0.1 5

10

0

DC Electric Field (kV/cm)

Figure 6.23 (Color figure follows p. 204.) Sideband excitation voltage scan at f THz = 2.0 THz (8.2 meV). (Reprinted with permission from Su, M.Y. et al., Appl. Phys. Lett., 81, 1564, 2002. Copyright 2002, American Institute of Physics.)

previous paragraph, near the e1hh2X1s - e2hh1X1s excitonic intersubband resonance. Figure 6.24 plots the n = +1 sideband conversion as a function of the terahertz electric field amplitude for three terahertz frequencies. The observed non-monotonic behavior, in particular at f THz = 1.5 THz (6.2 meV), clearly indicates that the effect of the terahertz beam is nonperturbative. This behavior can occur as the exciton absorption lines are shifted or even split due to the AC Stark effect, as observed in Section 6.4. These shifts can bring the exciton levels out of resonance, thereby decreasing the sideband generation. The sideband generation is modeled by a threelevel system in the strong-field regime (mE ≥ hf THZ) using Floquet theory. Although these experimental observations clearly demonstrate a nonperturbative effect of the terahertz field on the exciton system, the physical cause is better illustrated by directly measuring the exciton absorption. 6.3.3.3 Doped Asymmetric Coupled QWs There are several differences between sideband generation in undoped and doped QWs. First of all, in doped QWs excitonic effects are screened by the electron gas, leading to much broader interband resonances. Second, the lowest energy interband transitions cannot occur due to Pauli blocking. One might expect sideband generation to be much weaker in doped QWs for these reasons. However, in doped QWs, the terahertz electric field can drive a real intersubband polarization without optical

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Optical Response of Semiconductor Nanostructures in Terahertz Fields 2.4

1.5 THz

2.0 THz

249

2.5 THz

Sideband Conversion (10–5)

2 1.6 1.2 0.8 0.4 0

0

2 4 6 8 THz Electric Field Strength (~kV/cm)

10

Figure 6.24 Resonant terahertz field strength dependence of sideband generation at 1.5 THz, 2.0 THz, and 2.5 THz (6.2 meV, 8.2 meV, and 10.4 meV). The absolute terahertz electric-field scale is accurate only to within a factor of 2. (Reprinted with permission from Su, M.Y. et al., Phys. Rev. B, 67, 125307, 2003. Copyright 2003, American Physical Society.)

excitation, which allows different nonlinear processes that give strong terahertz– NIR mixing. Sideband experiments are performed in a sample essentially the same as the previously discussed gated ACQW sample,11,12 but each ACQW was modulation doped with an estimated electron density of 1.2 × 1011 cm–2 per well.14 The splitting between the lowest conduction subbands is estimated at ~1.5 THz (~6.2 meV) for zero bias, but intersubband absorption is due to collective excitations of the electron gas (intersubband plasmons),60 which occur at a higher frequency. The intersubband absorption is measured with Fourier transform infrared spectroscopy, giving a resonance frequency that varied between 2.4 THz (10 meV) and 2.0 THz (8 meV) with gate voltage. Figure 6.25 displays a sideband resonance map of the n = +1 sideband signal at f THz = 2.0 THz (8.2 meV) as a function of the static electric field and f NIR for this doped sample and an undoped reference sample. The peak sideband conversion efficiencies for these two samples are roughly equal, but the resonant behavior is quite different. The undoped sample shows multiple excitonic intersubband resonances, similar to those in previous work.11 The lowest energy resonance simply consists of an electron transition, while the higher energy resonances involve hole transitions. In the doped sample, only the lower resonance involving electron transitions appears significant. This strikingly different resonant behavior implies that terahertz-NIR mixing in n-doped QWs is primarily sensitive to electronic intersubband transitions. Also, the sideband resonance for the doped sample occurs at a NIR energy that is below the NIR absorption (not shown), where Pauli blocking prevents interband transitions.

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Terahertz Spectroscopy: Principles and Applications

(a)

Involve Hole Transitions

NIR Energy (eV)

1.55

×105

(b)

6

1.54

4

e2hh1 X e1hh1 X

e2 e1 2

Ground

1.53

hh1 –10

–5

0

5

–10

–5

0

5

0

DC Electric Field (kV/cm)

Figure 6.25 The n = +1 sideband resonance maps of (a) the undoped and (b) doped samples at f THz = 2.0 THz (8.2 meV), taken by varying f NIR and Ebias, while always measuring the n = +1 sideband. The sideband conversion for each point is represented on a grey scale, displayed on the right. The maps were taken at ~12 K with a terahertz intensity of ~18 kW/cm2 and NIR intensity of ~25 W/cm2. (Reprinted with permission from Carter, S.G. et al., Phys. Rev. B, 72, 155309, 2003. Copyright 2005, American Physical Society.)

The resonant behavior can be partially understood by using a simple χ(2) model with three subbands (two conduction and one heavy-hole) and non-interacting particles. The expression for χ(2) is similar to Equation 6.6), but with an integral over the in-plane wavevector k, and an additional term proportional to ∞

m m m x 1α

z αβ

x β1

rββ ( k ) - rαα ( k ) . α1 ( k ) - fsideban nd - i g α1/2 p][ fαβ ( k ) - fTHz - i g αβ /2 p]

∑ ∫ kdk [ f

α ,β = 2 ,3 0

(6.7)

The indices 1, 2, 3 represent the heavy-hole valence subband, and the first and second conduction subbands, respectively. f m1(k) is the interband transition frequency at a given k and conduction subband α. rbb(k) is the population in subband β at wavevector k. This additional term requires an intersubband population difference and has both an interband and intersubband resonance in the denominator. This term dominates near the intersubband transition. Figure 6.26(a) displays the maximum n = +1 sideband signal as a function of terahertz frequency, along with the maximum of | χ( 2 ) |2 . There is a clear resonance at ~2 THz (~8 meV), where the intersubband plasmon energy has been independently measured, as well as a significant low-frequency response. These results demonstrate the sensitivity of terahertz-NIR mixing to collective excitations of the electron gas, which is a fundamentally different nonlinear process than that for undoped QWs.

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Optical Response of Semiconductor Nanostructures in Terahertz Fields

+1ω Sideband Conversion (×104)

1.4

(a)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

Percent Change

251

(b)

–5 –10 –15 –20 –25

0

5 10 THz Frequency (meV)

15

Figure 6.26 (a) Peak values of the n = +1 sideband resonance spectra solid dots for the doped sample and the peak values of the χ(2) model curves solid line plotted versus f THz. (b) Percent change in the integrated PL of the doped sample with the terahertz field on at an intensity of ~40 kW/cm2. (Reprinted with permission from Carter, S.G. et al., Phys. Rev. B, 72, 155309, 2005. Copyright 2005, American Physical Society.)

6.3.4 Conclusion Terahertz-NIR mixing has been demonstrated as a useful spectroscopic technique for optically detecting terahertz resonances and has also shown potential for all-optical wavelength conversion. This technique is particularly sensitive to the symmetry of the QW system. In-plane terahertz fields will only generate even (n = ±2, ±4, …) sidebands from inversion symmetry in the QW plane. On the other hand, growthdirection terahertz fields can generate odd sidebands for asymmetric QW structures. The use of static electric fields to alter the symmetry and tune the resonances has proven particularly useful. Terahertz–NIR mixing has also revealed interesting nonperturbative behavior as illustrated previously,12 in which the terahertz field significantly changes the interband optical properties. This subject will be further studied in Section 6.4 by directly probing the interband absorption.

6.4 Modifying Exciton States with Terahertz Radiation The previous section on terahertz–NIR mixing generally treated the effects of the terahertz field on the interband absorption perturbatively. In this section, we discuss significant, nonperturbative changes in the interband absorption as the QW exciton states are “dressed” by the terahertz field. The changes in exciton absorption occur as the coupling energy between the terahertz field and the exciton exceeds the

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Terahertz Spectroscopy: Principles and Applications

interband absorption line width and even approaches the terahertz photon energy. A unique feature of these effects is that there is negligible absorption of the terahertz radiation as there are no free carriers in the QWs without optical excitation. First, we will discuss static electric field effects as background, and then we will show the two dominant terahertz field effects in QWs: the dynamic Franz-Keldysh effect and the AC Stark effect.

6.4.1 Background: Static Electric Field Effects Before discussing how the optical properties of QWs are altered by an intense terahertz field, it is helpful to review the effects of static electric fields. The effect of a static electric field is qualitatively different when applied in the growth direction versus in the QW plane. In plane electric fields accelerate free carriers, while growth direction electric fields “tilt” the potential well. In-plane fields lead to the Franz-Keldysh effect, which can also be observed in bulk semiconductors since the carriers are free to move in the direction of the field. The Franz-Keldysh effect results in absorption below the bandgap from tunneling of carriers into the gap and oscillations in the absorption above the bandgap from spatial oscillations in the electron probability density.16 Electric fields in the growth direction concentrate electron and hole wave functions on opposite sides of the QW, lowering the energy. A red shift and some broadening of the exciton absorption are observed according to the QCSE.18,19 This effect was illustrated in Section 6.331 for a 150Å GaAs QW. If the electric field is modulated at low frequencies, the optical absorption adiabatically follows the field according to the static effect. This quasistatic limit is the case for QW modulators that use the QCSE to modulate NIR light at frequencies up to approximately 100 GHz. As the frequency becomes larger than the exciton dephasing rate, which is usually in the terahertz regime, the effect takes on a new dynamic quality. Thus, for in-plane terahertz fields, the dynamic Franz-Keldysh effect (DFKE) is observed, and for growth direction terahertz fields, the AC QCSE is observed.

6.4.2 Dynamic Franz-Keldysh Effect As in the static Franz-Keldysh effect, the dynamic version occurs when the electric field is polarized in a direction in which carriers are free to move. The field therefore accelerates the carriers, increasing their kinetic energy. This increase in kinetic energy is given by the ponderomotive energy, EKE =

e2 E 2 , 4 mω 2

(6.8)

where e and m are the carriers’ charge and mass, respectively, and the driving electric field is given by Ecos (ωt). In the static limit this energy diverges since the carriers will continue to accelerate indefinitely. Clearly, the momentum scattering rate will limit this acceleration. Consequently, drive frequencies much less than this scattering rate will give an effect essentially the same as the static Franz-Keldysh effect.

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γ << 1

253

γ~1

MPA

ω

DFKE

Not Well Known Well Known

FKE

γ >> 1

E

Figure 6.27 A schematic map of electrooptic phase space as a function of amplitude and frequency of the perturbing field, showing DFKE regime “between” MPA and FKE. (Reprinted with permission from Nordstrom, K.B. et al. Phys. Rev. Lett., 81 457, 1998. Copyright 1998, American Physical Society.)

The momentum scattering rate, and perhaps other decoherence processes, define the boundary between the quasistatic and dynamic effects, which often occurs in the terahertz frequency range. The different regimes of the driven carriers can also be described by defining a dimensionless parameter, g, as the ratio of the ponderomotive energy to the photon energy of the driving field. g=

EKE e2 E 2 = . ω 4 mω 3

(6.9)

Figure 6.27 maps the different regimes, determined by g, as a function of E and ω. In the quasi-static regime of the Franz-Keldysh effect, the low drive frequencies yield g (>>) 1. For high frequencies and moderate fields, g (<<) 1. In this regime, multiphoton effects are observed,62 but the ponderomotive energy is small. At frequencies greater than decoherence rates and for g (~) 1, the ponderomotive energy is significant, and the effects cannot be considered quasistatic. This dynamic regime has been the subject of much theoretical attention,63 but few experimental results on this subject have been published. The theoretical effect of the DFKE on the interband absorption of a two-dimensional system is displayed in Figure 6.28(a). There is significant belowgap absorption, as in the static effect, and a blue shift in the band gap equal to the EKE. This shift occurs because the energy of carriers in the band is increased by the ponderomotive energy. The DFKE has been clearly observed in room temperature bulk GaAs with an intense (~109 W/cm2) MIR pump, demonstrating strong belowgap absorption and a blue shift in the band-gap.64 The DFKE occurs fairly naturally at terahertz frequencies, because the frequencies are low enough for the ponderomotive energy to be significant with moderate fields (~1–10 kV/cm) and the frequencies are higher than decoherence rates. In this section, we will discuss the observation of the DFKE in InGaAs QWs driven by terahertz frequency electric fields, in which the interband absorption is dominated by excitonic effects.17

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(a)

α

EKE

E = Egap E (b)

(c) |2'> |2>

ωTHz

ω12 ∆

ω*12 |1> |1'>

THz NI

R

Figure 6.28 (a) DFKE in an ideal two-dimensional system. In a strong terahertz field (dashed curve; solid curve is zero field), the “edge” shifts to higher frequency and subgap absorption increases. (b) The AC Stark effect: a strong field applied at frequency f ∼ f12 . causes f12 to shift. For f < (>) f12, the transition shifts to f12* > (<) f12. D = (f12 -f12*)/2 is the shift of |1〉 with respect to distant energy levels. (c) Schematic of experiment. (Reprinted with permission from Nordstrom, K.B. et al., Phys. Rev. Lett., 81, 457, 1998. Copyright 1998, American Physical Society.)

The excitonic DFKE is observed in a sample with 20 periods of 80 Å In0.2Ga0.8As QWs separated by 150-Å GaAs barriers on a GaAs substrate. The experiments are performed in He vapor at approximately 7 K. The indium in the QWs lowers the exciton energy to below the bandgap of the GaAs substrate, allowing transmission measurements without substrate removal. The transmission of the sample is probed with a continuous wave Ti:sapphire laser, tuned across the lowest heavy-hole exciton resonance, and detected by a silicon photodiode. The NIR transmission spectra without the terahertz field are displayed in Figure 6.29, labeled “I = 0.” The exciton absorption is strongly inhomogeneously broadened with a full width at half max of 1.5 THz (6 meV). When only discussing the DFKE, it is sufficient to only consider the 1s inplane state of the hh exciton since the DFKE is a non-resonant phenomenon. However, an in-plane terahertz field will also couple the 1s state to the 2p state, leading to the AC Stark effect, which will be discussed in the next section. The relevant states in this system therefore consist of the ground state (no excitons) and the 1s (optically allowed) and 2p (optically forbidden) states of the lowest heavy-hole exciton. The energy separation between the 1s and 2p states is estimated at 2 THz (8 meV) from

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1.39

I = 12 I=0 1.40 Energy (eV) (a)

1.41

Transmission (arb. units)

Transmission (arb. units)

Optical Response of Semiconductor Nanostructures in Terahertz Fields

1.39

255

I=7 I=0 1.40 Energy (eV)

1.41

(b)

Figure 6.29 Experimental transmission of MQW near e1hh1X1s exciton with (a) f THz = 0.60 THz (2.5 meV) at ITHz = 0, 1, 2, 4, 12 (arbitrary units). (b) f THz = 4.1 THz (14 meV) at ITHz = 0, 1, 2, 4, 7 (arbitrary units). Arrows connect calculated centers of experimental peaks and point in the direction of increasing ITHz. (Reprinted with permission from Nordstrom, K.B. et al., Phys. Rev. Lett., 81, 457, 1998. Copyright 1998, American Physical Society.)

magnetotransmission measurements.8 The light-hole exciton is shifted 12 THz (50 meV) from the heavy-hole exciton by strain, and higher exciton states (from higher energy electron and hole subbands) should be sufficiently far away in energy to be ignored. The terahertz beam is focused down to the same spot as the NIR probe but with a larger diameter (0.5 to 2.5 mm depending on wavelength), as illustrated in Figure 6.28(c). The effect of this in-plane terahertz field on the NIR transmission is displayed in Figure 6.29 for (a) f THz = 0.6 THz (2.5 meV) and for (b) f THz = 3.4 THz (14 meV). Below the 1s → 2p transition, at f THz = 0.6 THz (2.5 meV), there is a red shift for low terahertz intensities followed by a blue shift as the intensity increases. Above the 1s → 2p transition, at f THz = 3.4 THz (14 meV), there is a monotonic increase in the absorption energy with increasing terahertz intensity. The exciton absorption is broadened and decreased in both cases, presumably due to exciton ionization. The position of the exciton absorption is quantified by taking the average of the energies contained in the exciton line, weighted by the absorption strength. This “center of mass” is plotted in Figure 6.30 as a function of g (see Equation 6.9), which is proportional to the terahertz intensity, for (a) f THz = 0.6 THz (2.5 meV) and for (b) f THz = 2.5 THz (10.5 meV). The absolute values of g are estimated to within a factor of two or three. Again, there is a red shift that reverses to a blue shift below the transition in Figure 6.30(a), and there is a monotonic blue shift above the transition in Figure 6.30(b). The qualitatively different behavior below and above the 1s → 2p transition is explained by competition between the AC Stark effect and the DFKE. The schematic in Figure 6.28(b), with 1s →| 1〉 and 2 p →| 2 〉 , illustrates the AC Stark effect. For drive frequencies below the original transition energy, the level separation is increased and the 1s state is red shifted. For frequencies above the original transition energy, the level separation is decreased and the 1s state is blue shifted. The DFKE is expected to blue shift the exciton energy by EKE , regardless of the frequency. Thus for frequencies below the 1s → 2p transition, the AC Stark effect and DFKE act in

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Exciton Shift (meV)

Exciton Shift (meV)

256

0.4

0

–0.4

0

0.4

γ

0.8

1.2 0.8 0.4 0

0

(a)

0.08 γ

0.16

(b)

Figure 6.30 Center of mass of measured (squares) and calculated (triangles) exciton transmission peak versus γ for (a) f THz = 0.60 THz (2.5 meV) f12. (Reprinted with permission from Nordstrom, K.B. et al., Phys. Rev. Lett., 81, 457, 1998. Copyright 1998, American Physical Society.)

opposite directions. At low intensities, the AC Stark effect dominates, leading to a red shift in the 1s exciton, but at higher intensities the AC Stark effect saturates, leading to a blueshift from the DFKE. For frequencies above the 1s → 2p transition, the two effects act in concert to give a blue shift. For comparison, theoretical calculations are performed that use nonequilibrium Green’s functions to determine the interband susceptibility in the presence of a driving field. The electron-hole interaction is included by using the Bethe-Salpeter equation.65 The calculated exciton positions are plotted in Figure 6.30, giving good qualitative agreement with experimental data. These experiments present the first experimental demonstration of the DFKE in which the effect of the driving field is no longer quasistatic and the ponderomotive energy is significant compared with the driving field photon energy. The shifts observed in these experiments are relatively small compared to the inhomogeneously broadened line width (less than 10%), but the shifts are still clearly visible and reveal the interplay between the DFKE and the AC Stark effect. The DFKE occurs as free carriers are driven back and forth in the driving field, whereas the AC Stark effect occurs as the driving field couples different excitonic states. When excitonic effects are unimportant, as in a previous work,64 the DFKE is observed by itself. When the driving field is polarized in a direction in which the carriers are quantum-confined, the DFKE is no longer relevant, and the AC Stark effect is the only effect of importance. This will be discussed next.

6.4.3 AC Stark Effect: Autler–Townes Splitting The AC Stark effect is closely related to Rabi oscillations as both are due to a nearresonant “pump” field coupling together two states. Much of the important physics of these phenomena can be explained by considering a simple two-level system with eigenstates labeled a and b, having energies hfa and hf b and a dipole moment m. These states are coupled by the pump electric field of amplitude E and frequency f, with the interaction energy defined by hW = 2mE, where W is the Rabi frequency. When

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257

the Rabi frequency W is much less than the dephasing rate g between a and b, the effect of the driving field can be treated perturbatively, and the original eigenstates still represent the system well. For W ≥ g the effect of the field is nonperturbative, and the states are “dressed” by the field. The resulting wave function can still be written as a linear combination of the original eigenstates with time-dependent coefficients, Ca (t) and Cb (t).    ψ (r , t ) = Ca (t )ua (r )e -2 pifat + Cb (t )ub (r )e -2 pifbt . (6.10)   The original eigenstates are ua (r ) and ub (r ) Solving for the time-dependent coefficients in the presence of the driving field and taking the rotating wave approximation (RWA), we obtain

Ca (t ) = e +2 piWDt/2 ( A+ e -2 piW′t/2 + A- e -2 piW′t/2 ),

D + W′   D - W′ Cb (t ) = e -2 piDt/2  A+ e -2 piW′t/2 + A- e +2 piW′t/2  .   W W

(6.11)

The detuning ∆ is defined as f – f ba, where f ba = f b – fa, and the generalized Rabi 2

frequency W′ is W + D 2 . The coefficients A+ and A– are determined by initial conditions (see reference 66 for more details on this derivation). In the time domain, the probability of finding the system in state a or b oscillates at the generalized Rabi frequency. In the frequency domain, the wave functions have additional frequency components: -( D ± W′ )/2 for state a and +( D ± W′ )/2 for state b. On resonance, the two frequency components of state a, fa - ( D + W′ )/2 and fa - ( D - W′ )/2 have equal amplitudes. Off resonance (D > W), the component with the smaller frequency shift dominates state a. When this driven system is probed by a weaker beam, there can be a splitting or shift in the linear absorption because of these additional frequency components. The original states of the system are “dressed” by the strong coupling of the states in the presence of the driving field. In semiconductors, AC Stark shifts in interband absorption have been observed due to an interband pump, typically detuned below the band gap.67,68 Resonant pumping of interband and intersubband transitions has resulted in Autler-Townes splitting of the linear absorption.69,70 Autler-Townes splitting occurs when the strongly coupled states are probed by absorption to or from a remote third state. This splitting was first observed in a molecular system driven by a strong radio frequency field and probed with a microwave field.71 Autler-Townes splitting is often the precursor of electromagnetically induced transparency, in which a pump beam causes an otherwise absorbing resonance to become transparent.70,72 These AC Stark effects in semiconductors have primarily been induced using visible to mid-infrared pumps, in which there was significant absorption of the pump. Changes in the interband QW absorption due to the AC Stark effect at terahertz frequencies has been given a great deal of theoretical attention. For in-plane fields, the coupling between s and p exciton states has been studied,73,74 and for growth direction fields, intersubband coupling has been studied.75–77 Experimentally, the

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previously discussed measurements of the DFKE with in-plane terahertz fields show some evidence of the AC Stark effect, but the effects are small and competed with the DFKE.17 In this section, we will focus on experiments in which a growth direction terahertz field couples two heavy-hole exciton states together, and clearly manifests an Autler-Townes splitting in the interband absorption.20 These experiments are performed at low temperature (~10 K) on a sample with ten periods of undoped In0.06Ga0.94As QWs (each 143 Å) with Al0.3Ga0.7As barriers (300 Å). A 6% indium concentration is used to bring the QW exciton absorption below the bandgap of the semi-insulating GaAs substrate, allowing transmission of the NIR probe through the substrate. A 100-nm layer of aluminum is deposited on the epitaxial layer to improve the boundary conditions for the terahertz field. The aluminum layer eliminates in-plane terahertz fields and significantly enhances the growth direction field. The NIR probe consists of incoherent light from a 353-THz (1.46 eV) LED focused onto the sample backside. As illustrated in Figure 6.31 (left), the probe is transmitted through the substrate, interacts with the QWs, reflects off the aluminum layer, and is transmitted back through the QWs and substrate. The reflectivity of the sample (essentially double-pass transmission) without terahertz excitation is plotted as the lowest spectrum in Figure 6.31 (left). The strong absorption is due to the e1hh1X1s. The exciton e1hh1X1s is expected to be 3.34 THz (13.8 meV) above e1hh1X1s from calculations, but interband transitions to this state are optically forbidden. The light-hole exciton, e1lh1X1s, is shifted to much higher energies due to strain, and other higher exciton states (e.g., e2hh1X1s, e2hh2X1s) are expected to be sufficiently high in energy that the terahertz field only weakly couples them to e1hh1X1s. The states considered in this experiment are the ground state (no excitons), e1hh1X1s, and e1hh2X1s. These states will be referred to as the ground state, hh1X, and hh2X for simplicity in the remainder of this section. The different in-plane states (2s, 2p) are not considered as they have weaker oscillator strengths and 1s to 2p transitions will only occur for in-plane terahertz fields. The terahertz field couples hh1X and hh2X, and the NIR beam probes the transition from the ground state to e1hh1X. Schematics of these energy levels and fields are displayed to the right of the measured spectra in Figure 6.31 (left), with the terahertz frequency below, on, and above the e1hh1X – hh2X resonance. The terahertz beam is focused onto the edge of the sample (see top portion of Figure 6.31, left), with the maximum intensity inside the sample estimated at ~1 MW/cm2 (~15 kV/cm electric field amplitude). The effect of the terahertz field on the reflectivity is displayed in Figure 6.31 (left) for terahertz frequencies below, on, and above the hh1X – hh2X resonance. Below resonance in Figure 6.31A (left), at f THz = 2.52 THz (10.4 meV), the absorption line red shifts with increasing terahertz intensity, with a weaker absorption line appearing above the undriven exciton energy at the highest intensities. On resonance in Figure 6.31B (left), at f THz = 3.42 THz (14.1 meV), there is a symmetric splitting of the absorption that increases with intensity. Above resonance in Figure 6.31C (left), at f THz = 3.90 THz (16.1 meV), a weaker absorption line appears below the undriven exciton energy. These observations are clear manifestations of an Autler-Townes splitting because of terahertz-induced coupling of the hh1X and hh2X states. The two absorption lines observed represent the

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259

Al In Al e1

Substrate

THz

1 0.8

h2X h1X 0

B

665 365 210 0

0.6

h2x (dark)

h1x

0.4 1.6

0

C

320 220 1.2 140 1 0 0.8 1.4

0.6

h2x (dark)

h1x

0.4 351

353

355

NIR Frequency (THz)

357

Reflectivity (theory)

1.2

h2x (dark)

h1x

h2X

1.4

930 710 460 175 0

h1X

Reflectivity

1.6

A

h1

h1X h2X

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

QWs h2 Aluminum NIR Frequency (eV) 1.455 1.465 1.475

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 1.6 1.4 1.2 1 0.8

NIR Frequency (eV) 1.455 1.465 1.475 A

910 680 455 230 0

B

910 680 455 230 0

C

350 235 115 0

0.6 0

0.4 351

353

355

357

NIR Frequency (THz)

Figure 6.31 (Left). Reflectivity spectra for a series of terahertz intensities at f THz = (a) 2.52 THz (10.4 meV), (b) 3.42 THz (14.1 meV), and (c) 3.90 THz (16.1 meV). The spectra are offset and labeled according to the terahertz intensities (in kW/cm2). Level diagrams illustrating the detuning from the expected h1X-h2X resonance are shown to the right of each graph. Schematics of the experimental geometry and the band diagram of a QW along with the relevant subband energies are displayed above. The doublesided arrow represents the lowest excitation of the system, the h1X exciton. The AlGaAs barriers are labeled “Al” and the InGaAs layer is labeled “In.” (Right). Calculated reflectivity spectra for a series of terahertz intensities at f THz = (A) 2.52 THz (10.4 meV), (B) 3.42 THz (14.1 meV), and (C) 3.90 THz (16.1 meV). The spectra are offset and labeled according to the terahertz intensity (in kW/cm2). The absorption strength and energy position of the spectrum for zero terahertz field were set to best fit the measured reflectivity. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

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1.462

353.4

1.461

353.2 353.0

1.460

352.8

1.459

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

(b)

3 2 1

0

NIR frequency (eV)

(a)

353.6

200 400 Intensity (kW/cm2)

600

Splitting (meV)

Splitting (THz)

NIR Frequency (THz)

terahertz dressed states, each consisting of a linear combination of hh1X and hh2X. This Autler-Townes splitting can be observed at temperatures of up to 78 K. Calculated reflectivity spectra are displayed in Figure 6.31 (right) for the same terahertz frequencies and comparable terahertz intensities. These calculations are similar to the model discussed previously in this section in that they calculate how the terahertz field couples the states of the QW. The calculations are for an infinitely deep QW, and they include all of the quantized electron and hole states, the Coulomb interaction, and the effects of the terahertz field without resorting to perturbation theory.77 The model is equivalent to solving the semiconductor Bloch equations in the low-density limit.78 The agreement between the experimental and calculated spectra is quite good, although there are some differences. First, the observed splitting is more symmetric at f THz = 3.42 THz (14.1 meV) (Figure 6.31B, left) than in the calculations (Figure 6.31B, right), indicating the actual hh1X – hh2X resonance is closer to 3.42 THz (14.1 meV) than the calculated value of 3.34 THz (13.8 meV). More significantly, the experimental spectra show a red shift in both dressed states with increasing terahertz intensity that is not seen in calculations. This red shift is likely due to heating of the sample due to the intense terahertz field. The red shift can be seen more clearly in Figure 6.32(a), which plots the energies of the two absorption lines as a function of terahertz intensity. These energies are obtained by fitting the spectra to two Lorentzians. On resonance, the two dressed state energies are expected to

0

Figure 6.32 (a) Positions of the absorption lines and (b) the splitting as a function of terahertz power at f THz = 3.42 THz (14.1 meV). Error bars represent the uncertainty from the fits. The small squares in (b) give the calculated splitting, and the curve fits these points to a square root function, expected from theory. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

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symmetrically shift up and down in energy relative to the undriven exciton line, but terahertz-induced heating red shifts both of these lines as the intensity increases. The splitting between the absorption lines should be independent of this heating effect and is plotted in Figure 6.32(b), along with the calculated splitting. The splitting on resonance is expected to be proportional to the Rabi frequency, 2µE, as found in the simple Rabi model. The splitting should therefore be proportional to the square root of the terahertz intensity. Over the limited range of terahertz intensities in which the two absorption lines can be resolved, the measured splitting appears linear but is also not far from the expected square root behavior. Further measurements must be performed to better characterize the dependence of the splitting on terahertz intensity. Terahertz-induced changes in the QW reflectivity are measured over a wide range of terahertz frequencies from 1.53 to 3.90 THz (6.3 to 16.1 meV). The absorption spectra at several terahertz frequencies, with roughly equal intensities, are shown in Figure 6.33, along with the fits to two Lorentzians. The absorption spectra are obtained by correcting for the linear decrease in the reflectivity with energy (presumably due to the tail of absorption by the GaAs substrate) and taking the negative natural logarithm. Figure 6.34 displays the fitted absorption line positions as a function of terahertz frequency at a relatively low intensity of ~300 kW/cm2. This was the maximum intensity for f THz = 3.75 and 3.90 THz (15.5–16.1 meV). These positions represent the energies of the two dressed states, with the dark (light) circles representing strong (weak) NIR absorption. There is a clear anticrossing of the dressed states at the hh1X – hh2X resonance, as expected from theory. On resonance, the oscillator strength is equally shared between the dressed states because of the strong coupling of the two exciton states. Off resonance, the dressed state near the undriven hh1X energy (the dashed line in Figure 6.34) dominates the absorption strength, while the weaker dressed state approaches the hh2X energy with decreasing terahertz frequency. Clearly, in the zero-frequency limit, the electric field will produce an absorption line at hh2X since the field breaks the inversion symmetry of the QW, making hh2X optically allowed. The measured absorption positions follow the calculated dressed states quite well. These calculated positions from the model equivalent to the SBE are quite similar to positions obtained from the Rabi model: fdressed = fe1hh1X - ( D ± W 2 + D 2 ) / 2 . On resonance, the splitting is determined by the Rabi frequency, whereas far off resonance, the splitting is determined by the detuning. The deviations from calculations at 1.53 and 1.98 THz (6.3 and 8.2 meV) may be due to a higher terahertz intensity than estimated. The lack of a blue shift above resonance is probably due to heating. These experiments provide the first clear evidence of the terahertz AC Stark effect in the interband absorption spectrum, with a clear manifestation of AutlerTownes splitting. Although the AC Stark effect has previously been observed in semiconductors with MIR and optical pumps, this experiment at terahertz frequencies is unique for several reasons. First of all, there is negligible absorption of the terahertz pump because there are no free carriers in the QWs without optical excitation. This reduces heating of the sample, which tends to broaden the exciton line width, and likely makes the observation of this effect possible with a quasi-continuous wave pump. Second, the Rabi frequency, which is given by the splitting on resonance, is a significant fraction of the pump frequency because of the large intersubband dipole

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Terahertz Spectroscopy: Principles and Applications NIR Frequency (eV) 0.8 0.6

1.455

1.460

1.465

1.470

A

fTHz = 3.90 THz

B

fTHz = 3.75 THz

0.4 0.2 0.0 0.6 0.4 0.2

Absorbance

0.0 0.6

C

fTHz = 3.42 THz

0.4 0.2 0.0 0.6

D

fTHz = 3.09 THz

0.4 0.2 0.0 0.6

E

fTHz = 2.82 THz

0.4 0.2 0.0

351

352

353

354

355

356

NIR Frequency (THz)

Figure 6.33 Absorption spectra and fits for f THz = 3.90 THz (16.1 meV) (A) to 2.82 THz (11.7 meV) (E). The thick black lines represent the absorption obtained from the reflectivity spectra by correcting for the downward slope due to continuum absorption and by taking the negative natural logarithm. The gray curves are fits to two Lorentzians; the other two curves are the individual Lorentzians. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

moment and the small pump frequency. On resonance at the highest terahertz intensity, the ratio Ω/f is ~0.2, and below resonance at 1.5 THz (6.2 meV), the ratio is close to 0.5. This system is therefore approaching the strong field regime, in which Ω/f ≥ 1, where the RWA breaks down and terahertz replicas can appear in the spectra. These large Rabi frequencies allow the observation of terahertz dressed states over a wide range of detunings that exceed half of the resonance frequency. Finally, the terahertz-driven QW is essentially a QW modulator driven at terahertz frequencies. As previously stated, growth-direction electric fields in QWs are currently used to modulate NIR light through the QCSE at frequencies up to approximately 50 GHz. At these frequencies which are lower than the carrier dephasing time, the QW absorption adiabatically follows the field. At terahertz frequencies,

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Drive Frequency (meV) 5

10

15

20 1.468 1.466 1.464

354

1.462 1.460

353

1.458

NIR Frequency (eV)

NIR Frequency (THz)

355

1.456

352 1.5

2.5

3.5

4.5

Drive Frequency (THz)

Figure 6.34 Measured absorption line positions (large circles) versus terahertz frequency for a terahertz intensity of approximately 300 kW/cm2. The absorption strength for each point is represented on a grayscale; darker circles indicate stronger absorption. The standard error of the positions from fitting is indicated by error bars except where smaller than the circle diameter. The data points and solid line represent the calculated absorption positions at a terahertz intensity of 375 kW/cm2. Near the resonance at f THz = 3.4 THz (14.1 meV), the calculated spectra were fit to two Lorentzians to determine absorption positions; away from resonance, the positions were taken at the minimum of the reflectivity for each peak. The horizontal dashed line marks the undriven exciton line position. (Reprinted with permission from Carter, S.G. et al., Science, 310, 651, 2005.)

quantum coherence between exciton states of the QW is preserved, leading to terahertz dressed states. This quantum coherence will be important to consider as the frequency of optoelectronic devices, such as QW modulators, continues to increase. One possible application of quantum coherence in a QW modulator is that two optical beams resonant with different dressed states of the QW interact very strongly, with sensitivity to the phase between the two beams. This effect allows for the intermodulation of two beams at arbitrarily low intensities.79

6.4.4 Conclusions The effect of a terahertz driving field on the interband absorption of QWs depends qualitatively on the polarization of the terahertz radiation. In-plane terahertz fields drive carriers back and forth, leading to the DFKE, which blue shifts excitons according the ponderomotive energy, EKE. The terahertz field also couples to inplane exciton transitions, leading to the AC Stark effect, which competes with the DFKE. Growth-direction terahertz fields couple to intersubband transitions in QWs, leading to a quantum-confined AC Stark effect. This effect is essentially the result of Rabi oscillations between two states, which dress the states. Perhaps the most dramatic evidence of these dressed states is the observation of an Autler-Townes splitting of the exciton absorption.

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6.5 Conclusion This chapter has described a number of experimental methods and results which arise from studying the optical properties of semiconductor heterostructures under illumination by essentially monochromatic, quasi-continuous wave radiation at frequencies ranging from 0.3 to 4 THz (1.2 to 16 meV). For some of the experiments, such as optically detected terahertz resonance, only modest terahertz power is necessary because this is essentially a linear spectroscopy tool. Others, such as the experiments in which terahertz radiation modifies the optical absorption (and hence the band structure) of semiconductor QWs, require terahertz electric fields on the order of 10 kV/cm, or intensities of several hundred kW/cm2. During the time that this chapter’s experiments were performed, terahertz source technology has made tremendous advances. From the higher frequency end, terahertz quantum cascade lasers have worked their way down to lower than 2 THz (8 meV) with average emitted powers exceeding 10 mW. From the lower frequency end, reliable sources based on multiplied microwave oscillators can now be purchased with powers ranging from 50 mW at 0.25 THz (1 meV) to approximately 10 µW at 1 THz (4 meV). Both of these advances, which are sure to continue, bode well for transitioning at least some of the experimental methods described in this chapter to experimental tabletops that include the terahertz source, and, perhaps, into useful devices. For example, the intensities inside a terahertz quantum cascade laser cavity can exceed 10 kW/cm2, enabling highly nonlinear experiments on structures embedded into the laser cavity. Sideband generation in a mid-infrared quantum cascade laser has already been demonstrated.80

kBΔT/IFIR (cm2/s)

10–21

6.3 meV

10–22

10–23

10–24 10

ω–2 Carrier Temperature = 15 K Carrier Temperature = 30 K Carrier Temperature = 50 K 100 FIR Frequency (cm–1)

Figure 6.35 kB∆T/ITHz versus far-infrared (FIR) frequency for the quantum dot sample at carrier temperatures of 15, 30, and 50 K. The lattice temperature is 7 K. Carrier temperature is determined from the PL lineshape, and the terahertz intensity required to heat carriers to 15, 30, and 50 K is measured at each terahertz frequency. For all three carrier temperatures, the heating efficiency reaches a maximum for FIR energy of 1.5 THz (6.3 meV). (From Cerne J. et al., Phys. Rev. Lett., 77, 1131, 1996. Copyright 1996, American Physical Society.)

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Many interesting theoretical predictions have been made on the response of undoped76,79 and doped81–83 QWs in strong terahertz fields, and these are worthy of pursuit. However, perhaps the biggest challenges and opportunities for optically detected terahertz science on semiconductor heterostructures lies in the investigation of lower dimensional structures, like quantum wires and dots. Some preliminary experimental results are shown in Figure 6.35. Here, terahertz-induced changes in PL from a sample containing quantum dots are investigated. The dependence on frequency is strikingly different from the f –2 observed generically for the same experiment in QWs (see Figure 6.10). Terahertz radiation is absorbed resonantly with a peak near 1.5 THz (6.3 meV).84 This likely indicates some sort of intraband transition within the quantum dots. The heterogeneity of most quantum dots presents new challenges, perhaps necessitating optically detected terahertz resonance spectroscopy under an optical microscope so that single quantum dots can be studied. The experiments described in this chapter are only the beginning. The future will continue to involve free-electron lasers, but will also be broadened to laboratories with ever-improving tabletop terahertz sources.

References 1. Cerne, J., et al., Quenching of excitonic quantum-well photoluminescence by intense far-infrared radiation: free-carrier heating. Physical Review B, 1995. 51(8): 5253–62. 2. Cerne, J., et al., Terahertz dynamics of excitons in GaAs/AlGaAs quantum wells. Physical Review Letters, 1996. 77(6): 1131–4. 3. Cerne, J., et al., Photothermal transitions of magnetoexcitons in GaAs/AlxGa1-xAs quantum wells—art. no. 205301. Physical Review B, 2002. 66(20): 5301. 4. Salib, M.S., et al., Observation of internal transitions of confined excitons in GaAs/ AlGaAs quantum wells. Physical Review Letters, 1996. 77(6): 1135–8. 5. Wright, M.G., et al., Far-infrared optically detected cyclotron resonance observation of quantum effects in GaAs. Semicond. Sci. Technol., 1990. 5: 438. 6. Bauer, G.E.W. and T. Ando, Exciton mixing in quantum wells. Phys. Rev. B, 1988. 38: 6015. 7. Cerne, J., et al., Near-infrared sideband generation induced by intense far-infrared radiation in GaAs quantum wells. Applied Physics Letters, 1997. 70(26): 3543–5. 8. Kono, J., et al., Resonant terahertz optical sideband generation from confined magnetoexcitons. Physical Review Letters, 1997. 79(9): 1758–61. 9. Phillips, C., et al., Generation of first-order terahertz optical sidebands in asymmetric coupled quantum wells. Applied Physics Letters, 1999. 75(18): 2728–30. 10. Carter, S.G., et al., Terahertz electro-optic wavelength conversion in GaAs quantum wells: Improved efficiency and room-temperature operation. Applied Physics Letters, 2004. 84(6): 840–842. 11. Su, M.Y., et al., Voltage-controlled wavelength conversion by terahertz electrooptic modulation in double quantum wells. Applied Physics Letters, 2002. 81(9): 1564–1566. 12. Su, M.Y., et al., Strong-field terahertz optical mixing in excitons. Physical Review B, 2003. 67: 125307. 13. Ciulin, V., et al., Terahertz optical mixing in biased GaAs single quantum wells. Physical Review B, 2004. 70(11): 115312. 14. Carter, S.G., et al., Terahertz-optical mixing in undoped and doped GaAs quantum wells: from excitonic to electronic intersubband transitions. Phys. Rev. B, 2005. 72: 155309.

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15. Franz, W., ErnfluB eines elektrischen Feldes auf eine optische Absorptionskante. Z. Naturforsch., 1958. 13: 484. 16. Keldysh, L.V., Influence of a strong electric field on the optical characteristics of non conducting crystals. Sov. Phys. JETP, 1958. 34: 788. 17. Nordstrom, K.B., et al., Excitonic dynamical Franz-Keldysh effect. Physical Review Letters, 1998. 81(2): 457–60. 18. Mendez, E.E., et al., Effect of an electric field on the luminescence of GaAs quantum wells. Phys. Rev B, 1982. 26(12): 7101–4. 19. Miller, D.A.B., et al., Band-edge electroabsorption in quantum well structures: the quantum-confined stark effect. Physical Review Letters, 1984. 53(22): 2173–6. 20. Carter, S.G., et al., Quantum coherence in an optical modulator. Science, 2005. 310: 651. 21. Quinlan, S.M., et al., Photoluminescence from Al/sub x/Ga/sub 1-x/As/GaAs quantum wells quenched by intense far-infrared radiation. Physical Review B, 1992. 45(16): 9428–31. 22. Greene, R.L. and P. Lane, Far-infrared absorption by shallow donors in multiple-well GaAs-Ga1-xAlxAs heterostructures. Phys. Rev. B, 1986. 34: 8639. 23. Ekenberg, U., Nonparabolicity affects in a quantum well: Sublevel shift parallel mass, and Landau levels. Phys. Rev. B, 1989. 40: 7714. 24. Michels, J.G., et al., An optically detected cyclotron resonance study of bulk GaAs. Semiconductor Science and Technology, 1994. 9(2): 198–206. 25. Bimberg, D., F. Heinrichsdorff, and R.K. Bauer, Binary aluminum arsenide/gallium arsenide versus ternary gallium aluminum arsenide/gallium arsenide interfaces: a dramatic difference of perfection. Vac. Sci. Technol., 1992. B10: 1793. 26. Nickel, H.A., et al., Internal transitions of confined neutral magnetoexcitons in GaAs/ Alx GA1-xAs quantum wells. Phys. Rev. B, 2000. 62: 2773. 27. Nickel, H.A., et al., Internal transitions of neutral and charged magnetoexcitons in GaAs/AlgaAs quantum wells. Phys. Status Solidi B, 1998. 210: 341. 28. Herold, G.S., et al., Physica E, 1998. 2: 39. 29. Stillman, G.E., C.M. Wolfe, and D.M. Korn. 1972. Warsaw, Poland. 30. Birnir, B., et al., Nonperturbative resonances in periodically driven quantum wells. Physical Review B, 1993. 47(11): 6795–8. 31. Schnabel, R.F., et al., Influence of exciton localization on recombination line shapes: InxGa1-xAs/GaAs quantum wells as a model. Phys. Rev. B, 1992. 46: 9873. 32. Perkowitz, S., J. Far infrared free-carrier absorption in n-type gallium arsenide. Phys. Chem. Solids, 1971. 32: 2267. 33. Hirakawa, K., et al., Blackbody radiation from hot two-dimensional electrons in AlxGa1-xAs/GaAs heterojunctions. Phys. Rev. B, 1993. 47: 16651. 34. Hirakawa, K. and H. Sakaki, Energy relaxation of two dimensional electrons and the deformation potential constant in selectively doped alluminum gallium arsenide/ gallium arsenide heterojunctions. Appl. Phys. Lett., 1986. 49: 889. 35. Asmar, N.G., et al., Energy relaxation at THz frequencies in Al/sub x/Ga/sub 1-x/As heterostructures. Semiconductor Science and Technology, 1994. 9(5S): 828–30. 36. Christen, J. and D. Bimberg, Line shapes of intersubband and excitonic recombination in quantum wells: Influence of final-state interaction, statistical broadening, and momentum conservation. Phys. Rev B, 1990. 42: 7216. 37. Chang, Y.C. and R.B. James, Saturation of interrsubband transitions in p-type semiconductor quantum. Phys. Rev. B, 1989. 39: 12672. 38. Planken, P.C.M., et al., Terahertz emission in single quantum wells after coherent optical excitation of light hole and heavy hole excitons. Physical Review Letters, 1992. 69(26): 3800–3.

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39. Ashkinadze, B.M., et al., Microwave modulation of exciton luminescence in GaAs/ AlxGa1-xAs quantum wells. Phys. Rev. B, 1993. 47: 10613. 40. Shah, J., et al., Energy-loss rates for hot electrons and holes in GaAs quantum wells. Phys. Rev. Lett., 1985. 54: 2045. 41. Tsubaki, K., A. Sugimura, and K. Kumabe, Warm electron system in the n-AlGaAs/ GaAs two-dimensional electron gas. Appl. Phys. Lett., 1985. 46: 764. 42. Yang, C.H., et al., Hot-electron relaxation in GaAs quantum wells. Phys. Rev. Lett., 1985. 55: 2359. 43. Heyman, J.N., et al., Temperature and intensity dependence of intersubband relaxation rates from photovoltage and absorption. Physical Review Letters, 1995. 74(14): 2682–5. 44. Markelz, A.G., et al., Subcubic power dependence of third-harmonic generation for in-plane, far-infrared excitation of InAs quantum wells. Semiconductor Science and Technology, 1994. 9(5S): 634–7. 45. Craig, K., et al., Far-infrared saturation spectroscopy of a single square well. Semiconductor Science and Technology, 1994. 9(5S): 627–9. 46. Craig, K., et al., Undressing a collective intersubband excitation in a quantum well. Physical Review Letters, 1996. 76(13): 2382–5. 47. Cole, B.E., et al., Coherent manipulation of semiconductor quantum bits with terahertz radiation. Nature, 2001. 410: 60–63. 48. Wu, Q. and X.C. Zhang, Free-space electro-optic sampling of terahertz beams. Applied Physics Letters, 1995. 67(24): 3523–5. 49. Sirtori, C., et al., Far-infrared generation by doubly resonant different frequency mixing in a coupled quantum well two-dimensional electron gas system. Appl. Phys. Lett., 1994. 65: 445. 50. Chui, H.C., et al., Tunable mid-infrared generation by difference frequency mixing of diode laser wavelengths in intersubband InGaAs/AlAs quantum wells. Appl. Phys. Lett., 1995. 66: 265. 51. Paiella, R. and K.J. Vahala, Four-wave mixing and generation of terahertz radiation in an alternating-strain coupled quantum well. IEEE J. of Quant. Electr., 1996. 32: 721. 52. Hart, R.M., G.A. Rodriquez, and A.J. Sievers, Optics Lett., 1991. 16: 1511. 53. Zudov, M.A., et al., Time-resolved, nonperturbative, and off-resonance generation of optical terahertz sidebands from bulk GaAs. Phys. Rev B, 2001. 64: 121204(R). 54. Shimizu, A., M. Kuwata-Gonokami, and H. Sakaki, Enhanced second-order optical nonlinearity using inter- and intra-band transitions in low-dimensional semiconductors. Appl. Phys. Lett., 1992. 61(4): 399–401. 55. Maslov, A.V. and D.S. Citrin, Optical absorption and sideband generation in quantum wells driven by a terahertz electric field. Physical Review B (Condensed Matter), 2000. 62(24): 16686–91. 56. Maslov, A.V. and D.S. Citrin, Enhanced optical/THz frequency mixing in a biased quantum well. Solid State Communications, 2001. 120(2–3): 123–7. 57. Boyd, R.W., Nonlinear optics. 1992: Academic Press, New York. 58. Masanovic, M.L., et al., Monolithically integrated Mach-Zehnder interferometer wavelength converter and widely-tunable laser in InP. IEEE Photonics Technology Letters, 2003. 15: 1117. 59. Hsu, A. and S.L. Chuang, Wavelength conversion by dual-pump four-wave mixing in an integrated laser modulator. IEEE Photonics Technology Letters, 2003. 15: 1120. 60. Helm, M., The basic physics of intersubband transitions. Semicond. Semimetals, 2000. 62: 1.

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61. Weman, H., et al., Impact ionization of free and bound excitons in AlGaAs/GaAs quantum wells. Semicond. Sci. Technol., 1992. 7: B517–9. 62. Nathan, V., A.H. Guenther, and S.S. Mitra, Review of multiphoton absorption in crystalline solids. Journal of the Optical Society of America B-Optical Physics, 1985. 2(2): 294–316. 63. Yacoby, Y., High-frequency Franz-Keldysh effect. Phys. Rev., 1968. 169(3): 610. 64. Srivastava, A., et al., Laser-induced above-band-gap transparency in GaAs. Physical Review Letters, 2004. 93(15): 157401. 65. Haug, H. and S. Schmitt-Rink, Electron theory of the optical properties of laserexcited semiconductors Prog. Quantum Electron., 1984. 9(1): 3–100. 66. Boyd, R.W., Nonlinear optics. 1992, Boston: Academic Press. xiii, 439. 67. Mysyrowicz, A., et al., “Dressed excitons” in a multiple-quantum-well structure: evidence for an optical stark effect with femtosecond response time. Physical Review Letters, 1986. 56(25): 2748–51. 68. Frolich, D., et al., Optical quantum-confined stark effect in GaAs quantum wells. Physical Review Letters, 1987. 59(15): 1748–1751. 69. Shimano, R. and M. Kuwata-Gonokami, Observation of Autler-Townes splitting of biexcitons in CuCl. Physical Review Letters, 1994. 72(4): 530–33. 70. Serapiglia, G.B., et al., Laser-induced quantum coherence in a semiconductor quantum well. Physical Review Letters, 2000. 84(5): 1019–22. 71. Autler, S.H. and C.H. Townes, Stark effect in rapidly varying fields. Phys. Rev., 1955. 100: 703–22. 72. Harris, S.E., Electromagnetically induced transparency. Phys. Today, 1997. 50: p. 36. 73. Johnsen, K. and A.-P. Jauho, Quasienergy spectroscopy of excitons. Physical Review Letters, 1999. 83(6): 1207–10. 74. Dent, C.J., B.N. Murdin, and I. Galbraith, Phase and intensity dependence of the dynamical Franz-Keldysh effect. Phys. Rev B, 2003. 67: 165312. 75. Liu, A. and C.Z. Ning, Exciton absorption in semiconductor quantum wells driven by a strong intersubband pump field. J. Opt. Soc. Am. B, 2000. 17(3): 433–9. 76. Maslov, A.V. and D.S. Citrin, Optical absorption of THz-field-driven and dc-biased quantum wells. Physical Review B, 2001. 64(15): 155309/1–10. 77. Maslov, A.V. and D.S. Citrin, Numerical calculation of the terahertz field-induced changes in the optical absorption in quantum wells. IEEE J. Sel. Top. Quantum Electron., 2002. 8(3): 457. 78. Haug, H. and S.W. Koch, Quantum theory of the optical and electronic properties of semiconductors. 2004, River Edge, NJ: World Scientific. 79. Maslov, A.V. and D.S. Citrin, Mutual transparency of coherent laser beams through a terahertz-field-driven quantum well. Journal of the Optical Society of America B-Optical Physics, 2002. 19(8): 1905–1909. 80. Zervos, C., et al., Coherent near-infrared wavelength conversion in semiconductor quantum cascade lasers. Appl. Phys. Lett., 2006. 89: 183507. 81. Galdrikian, B. and B. Birnir, Period doubling and strange attractors in quantum wells. Physical Review Letters, 1996. 76(18): 3308–11. 82. Batista, A.A., B. Birnir, and M.S. Sherwin, Subharmonic generation in a driven asymmetric quantum well. Phys. Rev B, 2000. 61: 15108. 83. Batista, A.A., et al., Nonlinear dynamics in far-infrared driven quantum-well intersubband transitions. Phys. Rev B, 2002. 66: 195325. 84. Cerne, J., et al. Hot excitons in quantum wells, wires, and dots, Hot carriers in Semiconductors IX, K. Hess, J-P Leburton, and U. Ravaioli, eds., 305 (1995). 85. Xu, X. et al., Coherent optical spectroscopy of a strongly driven quantum dot. Science, 2007. 317: 929–932.

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Section III Applications in Chemistry and Biomedicine

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Terahertz Spectroscopy of Biomolecules Edwin J. Heilweil and David F. Plusquellic National Institute of Standards and Technology

Contents 7.1 7.2 7.3 7.4 7.5

Introduction.................................................................................................. 269 Experimental Procedures............................................................................. 270 Theoretical Spectral Modeling Methods..................................................... 272 Weakly Interacting Organic Model Compounds......................................... 274 Small Biomolecules as Crystalline Solids................................................... 276 7.5.1 Amino Acids..................................................................................... 277 7.5.2 Polypeptides...................................................................................... 278 7.5.3 Nucleic Acid Bases and Other Sugars..............................................280 7.5.4 Other Small Biomolecules................................................................ 282 7.6 Terahertz Studies of Large Biomolecules.................................................... 286 7.6.1 DNAs and RNAs.............................................................................. 286 7.6.2 Proteins............................................................................................. 288 7.6.3 Polysaccharides................................................................................. 292 7.7 Terahertz Studies of Biomolecules in Liquid Water.................................... 293 7.8 Conclusions and Future Investigations........................................................ 294 Acknowledgments................................................................................................... 295 References............................................................................................................... 295

7.1 Introduction A better comprehension of biomolecular function and activity in vivo requires a detailed picture of biopolymer secondary and tertiary structures and their dynamical motions on a range of timescales.1–3 At the cores of all proteins, for example, are the substituent amino acid building blocks that determine their structure and function. In an effort to develop methodologies to directly monitor complex macromolecule dynamics in real time, we4 and others5,6 suggested that an atomic level picture of the concerted motions of polypeptide chains and DNAs may be accessible through accurate measurement of low-frequency vibrational spectra. These vibrations are expected to occur in the terahertz (THz) frequency regime and may be observed using Raman, low-energy neutron7 and infrared absorption spectroscopies or related optical techniques. However, for even naturally occurring proteins and DNAs with >30 kDalton (kDa) molecular weights and large numbers of constituent peptide units or bases, one would expect the density 269

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of overlapping states to be so high in this frequency range that contributing absorption bands would “smear out” and yield essentially structureless spectra. It would clearly be of highest priority to obtain low-frequency vibrationally resolved spectra for biologic systems in aqueous-phase environments because this condition would most closely mimic their natural environment. However, this scenario has not been immediately feasible for most systems using far-infrared absorption spectroscopy because (1) absorption by the amino acids and most other biomolecules is masked by much stronger water absorption in the 1 to 3 THz spectral region8 and (2) spectral broadening arising from the full accessibility to conformational space and the rapid time scale for interconversion in these environments. Despite this limitation, recent studies of biomolecules in aqueous or high relative humidity environments reviewed in this chapter have revealed detailed information about the dynamics of these systems through careful broadband absorption measurements. Also reviewed here is the work performed to determine whether pure solid samples of organic model systems and protein fragments do indeed show sharp spectral features that are uniquely determined from their individual molecular symmetries and structures. Spectral THz absorption data and spectral modeling methods discussed in this review are the first of their kind and demonstrate that these expectations are indeed borne out.

7.2 Experimental Procedures As discussed in detail in our previous publications, solid organic species (e.g., di-substituted benzenes) and lyophilized samples of individual purified amino acids and short polypeptides were used as received from Sigma-Aldrich, Inc. or ICN Biomedicals, Inc. Powders requiring low-temperature storage were maintained at 273 K until use to prevent decomposition and exposure to atmospheric water. Matrix diluted samples for THz absorption measurements were rapidly prepared by first weighing 2 to 10 mg of solid and homogenizing the material in a mortar and pestle to reduce the solid particle size distribution. This procedure ensures particle sizes sufficiently smaller than THz wavelengths to reduce baseline offsets at higher frequencies arising from nonresonant light scattering. Each sample was thoroughly mixed with approximately 100 mg of spectrophotometric grade high density polyethylene powder (Sigma-Aldrich, Inc. with <100 micron particle size or MicroPowders, Inc. with <10 micron particle size) and pressed as a pellet in a 13-mm diameter vacuum die at the lowest possible pressures (approximately 200 psi or 1.4 × 106 Pa) to minimize decomposition from transient heating or high pressure. The 2.5-mm thick pellets have sufficient path length to eliminate etaloning artifacts (approximately 2 cm–1 period) often observed in THz spectra. Sample pellets were mounted in an aluminum sample holder fixed in position with a Teflon securing ring for 298 K measurements. Pellets were also encapsulated in a brass fixture that adapts to the copper cold finger of a vacuum cryostat (Janis Research Company, Inc., Model ST-100) fitted with 3-mm thick high-density polyethylene windows for broadband infrared studies at 4, 10 or 77 K. Infrared absorption spectra in the 0.2 to 22 THz range were obtained with a modified Nicolet Magna 550 Fourier transform infrared spectrometer (FTIR) using

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a silicon-coated broadband beam splitter and deuterium triglyceride sulfide (DTGS) room temperature photovoltaic detector fitted with a high density polyethylene window. The dry-air purged, globar source FTIR has sufficient sensitivity to generate high quality spectra in the 1.2 to 22 THz (40 to 700 cm-1) range for both room temperature pellets and samples placed in the liquid nitrogen or helium cooled cryostat positioned in the sample compartment. All measurements made using the FTIR (after ~1 hour of sample compartment purging to eliminate water vapor interference) were obtained at 4 cm–1 spectral resolution and averaging 64 interferometric scans. In standard fashion, spectra were converted to optical density (OD) units after ratioing raw sample transmission spectra (Tsample) to that of a pressed 100 mg polyethylene blank (TPE) pellet (OD = –log10(Tsample/TPE) obtained under identical acquisition conditions. Absorption band intensities were also found to obey Beer’s Law (OD = εcl where ε is the molar extinction coefficient, c the molar concentration, and l the sample thickness) as the amount of investigated solid varied in several pellet samples of the same thickness and weight. This finding strongly indicates that all observed absorption features directly arise from the sample in the host matrix rather than from pellet mixing inhomogeneity or other optical anomalies. Lower frequency THz time-domain spectroscopy (THz-TDS) was used to collect spectra between 0.4 and 8 THz (10 cm–1 and 225 cm–1) for solid disks and 10 to 90 cm–1 for liquid samples and compared with the FTIR spectra whenever possible. The THz-TDS spectrometer we employ is now a common apparatus and a review of the technique and set-up are reported elsewhere.9 Details and modifications made to our instrument to extend the typical 0.3 to 3 THz spectral bandwidth out to approximately 8 THz are described as follows. Broadband THz pulses were generated from amplified 800-nm fs laser pulses (Ti:sapphire MIRA oscillator and Legend amplifier system, Coherent Lasers, Inc., with ~45 fs pulse duration, 1 kHz repetition rate, and an average power of 300 mW) by difference frequency mixing in a 0.150-mm thick GaP(110) crystal. Four 90° off-axis parabolic aluminum coated mirrors were used to propagate, focus, and collect the THz pulses to interrogate the sample in transmission mode. Transmitted THz pulses were then focused onto a second 0.150-mm GaP(110) detector crystal gated by weak (~200 nJ) 800-nm pulses for electrooptic detection. These weaker gate pulses were generated from the original input 800-nm pulses by reflection from the front surface of a quartz wedge before reaching the generation crystal. Linearly polarized gate pulses, modulated by the instantaneous THz field in the second 0.150-mm thick GaP detector crystal, were subsequently detected by balanced matched silicon detectors, and time-domain data was collected by a lock-in amplifier (Stanford Research System Model 830).9 The analog time domain data were transferred to a computer, digitized, and analyzed by a LabView interface and fast-Fourier transformed to obtain the THz power spectrum. In similar fashion to FTIR measurements described previously, raw sample spectra are corrected by using a PE pellet background spectrum to obtain sample absorption spectra. FTIR and THz-TDS spectra were collected after purging each instrument to obtain a dry air environment and thereby remove gas phase water interference. Analysis for comparison of averaged spectra (512 scans for FTIR and 20 for THz-TDS measurements) to theory yielded an intensity uncertainty from the baseline of less than ±0.005 optical density (OD) units (k = 1; type B analysis).

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High-resolution THz absorption spectra were obtained using an instrument described in detail elsewhere.10 Briefly, THz radiation from 0.06 to 4.3 THz was generated by the photomixing of two near IR laser at the surface of a low-temperature-grown GaAs photomixer.11 The two lasers include a fixed frequency diode laser operating near 850 nm (ΔνFWHM ≈ 0.0001 cm‑1) and a standing-wave Ti:sapphire laser having a resolution of ΔνFWHM ≈ 0.04 cm‑1. The Ti:sapphire beam is combined with the output from a fixed frequency diode laser and focused by an aspherical lens onto the photomixer. A maximum power of ≈1 µW of THz radiation is obtained from 0.06 THz to 3 THz. The focused THz beam passes through the cryogenically cooled sample at 4.2 K and is detected by a liquid helium-cooled silicon composite bolometer. Power detection sensitivity of the bolometer is <1 nW up to 3 THz in a 400 Hz bandpass (NEP of the bolometer is 1 pW/Hz1/2).

7.3 Theoretical Spectral Modeling Methods To better understand the origin and assign the absorption features observed in our experimental spectra to molecular motions, a complementary series of calculations were performed to try to extract a picture of the intramolecular and intermolecular vibrational modes for these samples. Obtaining both accurate vibrational mode frequencies from properly minimized structures and extracting infrared intensities throughout the THz spectral region is crucial when comparing experimental observations to theory. Gaussian 03 Calculations: To model THz spectra of minimally interacting (no hydrogen bonding) model systems such as dicyanobenzene, gas phase density functional geometry optimizations and vibrational frequency calculations for each molecule were performed by the Gaussian software package (G03W Rev B.02)12 with the Becke-3-Lee-Yang-Parr (B3LYP) functional and 6–311++G* basis set under the Windows 32 operating system environment. Initial starting atomic geometries for each molecule were obtained from the NIST chemical structure database,13 and the structures optimized as free space molecules. Positive vibrational frequencies and close agreement of bond lengths and angles with published crystallographic data for the systems examined confirmed that the minimized structures were obtained. CHARMM Force Field Calculations for amino acids: Calculations of the THz vibrational spectrum of l-serine, l-cysteine, l-tryptophan, and dl-tryptophan were accomplished using the CHARMM 22 (Chemistry at HARvard Molecular Mechanics) empirical force field.10,14 The method used is similar to the previous work of Gregurick et al. for the calculation of the vibrational spectra of small peptide15 and sugar molecules.16 For the amino acids, crystallographically determined atomic coordinates were used as the starting structure for the calculations.17–22 To represent the crystal environment of the experimental work, we constructed a pseudo-crystal lattice comprising single-unit cells; the macroscopic crystal was then reproduced by using periodic boundary conditions in all calculations. We first optimized the geometry of the crystals using an adopted Newton-Raphson method until the gradient change was less than 10 –4 kcal/mol. Using the steepest descent method, the structure

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was gently optimized further to ensure an energy minimum, whereby all eigenvalues of the mass-weighted Hessian were positive. The THz vibrational frequencies were obtained from the normal mode eigenvalues of the mass-weighted Hessian and the intensities of each normal mode were calculated from their dipole derivatives. The dipole derivatives were calculated numerically from the displacement of the equilibrium structure along each normal mode direction. Computed spectra were represented by taking the calculated intensity for each absorption peak convoluted with a Gaussian spectral function with a fullwidth at half maximum (FWHM) corresponding to 20 cm–1. All calculated spectra were strictly obtained at the harmonic level and the calculations were performed on a UNIX-based silicon graphics dual processor system taking about 15 to 30 minutes per simulation. DFT Spectral Analyses: Density functional therory (DFT) calculations are performed on the crystal structures of selected amino acids using the freeware program CPMD (Car-Parrinello molecular dynamics), version 3.9.2.23 For l-serine, the calculations used the crystal structure coordinates.17–22 However, to save CPU time, the l-cysteine amino acid calculations used the minimized structure output of CHARMM described previously. For these calculations, a super cell was created with four molecules per cell and then applied periodic boundary conditions to simulate the larger crystal lattice. The one-electron orbitals were expanded in a plane wave basis set with a kinetic energy cutoff of 80 Ry restricted to the gamma point of the Brillouin zone. Medium soft norm-conserving pseudo potentials of Martins-Trouiller were used for all the elements.24 The energy expectation values were computed in reciprocal space using the Kleinman-Bylander transformation.25 To calculate the intensities of each normal mode, we allowed the system to propagate for a total time of 5 ps by applying ab initio molecular dynamics using 600 u for the fictitious mass of the electron. In these cases, a time step of 0.12 fs was used and recorded positions and charges of each atom at every step. The total dipole moment was calculated at each step of the ab initio molecular dynamics using the Berry phase polarizability.26 IR intensities were computed from the autocorrelation of the dipole magnitude followed by taking the Fourier transform.27 Again, each spectral peak was represented as a Gaussian band shape with a FWHM corresponding to 20 cm–1, which closely mimics the observed experimental widths. CPMD calculations were performed on the NIST multinode LINUX cluster using 4 to 8 nodes and took approximately 2 to 3 weeks for the optimization and frequency calculations. In contrast, the molecular dynamics simulations used to extract dipole derivative intensities required several months of computation for each species. DFT calculations for crystalline tripeptides were performed using the DMol3 software package.28 All calculations included all electrons and made use of a double numerical plus polarization atomic orbital basis set (equivalent to a double-ζ). The exchange correlation functional based on the generalized gradient approximation in the parametrization of Perdew and Wang (PW91)29 was selected for its known reliability to accurately model hydrogen bonded systems. The crystal cell parameters of the monoclinic cells were held fixed at the values reported from the x‑ray studies. The harmonic frequencies and normal mode displacements were calculated at the

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gamma point by a two-point numerical differentiation of the forces. The intensity for each normal mode, Qi, was calculated using

∂µ i I i ∝ υ′ υ′′ ∂Q i

2

3N

≅

∑ j

2

e j q i , j - e oj q io, j ≅ ∆Q i

3N

∑ j

(

e oj q i , j - q io, j ∆Q i

)

2

where the sum is overall 3N Cartesian coordinate displacements, q j – q jo, derived from the eigenvectors of the Hessian by dividing by square root of the atomic masses and ΔQi represents the root-mean-square displacement of atoms in mode Qi. The static charges, ejo, were set to the Mulliken atomic charges of the optimized structure.

7.4 Weakly interacting Organic Model Compounds In weakly associated solid-state systems, most intermolecular interactions are dominated by noncovalent, electrostatic, or van der Waals forces. To explore whether THz spectra of weakly interacting planar molecular solids (crystals) and analogous solution phase systems can be modeled using available force fields and the detailed molecular interactions in this spectral region, we measured and calculated the THz spectra of the 1,2-, 1,3-, and 1,4- isomers of dicyanobenzene (DCB) and related dinitrobenzene structures which differ only by the ring locations of the two cyano (or nitro) side groups. Vibrational THz spectroscopy and mode assignment investigations of the DCB isomers were previously conducted,30 but little attention was paid to the lower frequency THz region of the spectrum. Because the DCB homologs are known to have planar, weakly interacting structures in the solid state,30 investigation of the DCB isomers could reveal whether available cost-effective isolated gas phase molecule spectral calculations using DFT adequately agree with experimental THz spectra. In addition, having strongly coupled electronegative cyano groups coordinated to the p-bonded benzene ring affords the possibility that partial atomic charges undergoing vibrational dipolar motions could enhance THz absorption intensities making comparison to theory straightforward. Many of these experimental and theoretical characteristics were confirmed and are briefly reviewed here. To first give the reader a flavor for how many THz modes involve motion of the constituent atoms, the vector representations of the calculated vibrational modes for frequencies below 200 cm–1 of each molecule are shown in Figure 7.1. 1,2-DCB and 1,3-DCB exhibit three IR-active modes in this spectral range but 1,4-DCB has only two IR-active modes due to inversion symmetry. The calculated number of frequencies and intensity profiles match very well with the full experimental spectral observations. Figure 7.2 presents the collection of experimental and modeled vibrational spectra from the gas-phase Gaussian calculations, FTIR and THz-TDS for solutions (in CHCl3) and solids (in PE matrix) measurements for all three isomers at room temperature (~298 K). As one can see, the calculated spectral frequencies and relative intensities agree extremely well for all three isomers. There appears to be no well-defined frequency shift pattern one might apply for the calculated spectra to exactly overlap the observed features. Comparing

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119 cm–1

169 cm–1

197 cm–1

1, 3-DCB

120 cm–1

125 cm–1

198 cm–1

1, 4-DCB

79 cm–1

128 cm–1

Figure 7.1 (Color figure follows p. 204.) Displacement vector representation of the lowest non-zero infrared-active modes of the 1,2- 1,3-, and 1,4-dicyanobenzene isomers obtained from Gaussian 03 vibrational analysis of optimized isolated gas phase structures.

the THz-TDS and FTIR measurements of solid and liquid-phase samples (eliminating intermolecular modes) shows that these systems exhibit two intense IR-active intermolecular (phonon) features for all three species in the spectral region below 100 cm–1. This study yielded two significant observations: (1) The Gaussian software package may be reliably used to interpret low frequency THz spectral region for weakly interacting organic systems and it is able to help distinguish and classify observed vibrational features as intramolecular or intermolecular (phonon) modes. This study and our experience with similar molecular structures (e.g., dinitrobenzenes) also show that DFT with B3LYP level and 6–311G* or higher order basis adequately determines

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50 40

(a)

30 20 10

(b)

(d)

(c)

Absorbance (a.u.)

0 50 40 30

1, 3-DCB (a)

(b)

20 10

(d)

(c)

0 50 40 30 20 10 0

(e)

(e)

1, 2-DCB (a) (b)

(d)

(c) 50

100

150

200

250

300

350

400

450

500

550

600

(e) 650

700

Wavenumber (cm–1)

Figure 7.2 Experimental and simulation spectra of DCB isomers. Spectra of solid phase in PE matrix measured by (a) THz-TDS and (b) FTIR, of solution phase (in chloroform) measured by (c) FTIR and (d) THz-TDS, and (e) DFT simulations. To simplify, solution spectral intensities are multiplied by 10, while the solid THz spectra are scaled by 2. Spectra are vertically offset for clarity.

the solid-state equilibrium structure and internal modes with reasonably good accuracy for systems exhibiting weak intermolecular interactions. (2) To the best of our knowledge, the observed intensities of the lowest two phonon modes of 1,4-DCB are the strongest THz absorption features reported to date. These isomers and their readily identifiable spectral features may be of great interest in future studies using THz spectroscopy to analyze internal and phonon mode properties of solids.31

7.5 Small Biomolecules as Crystalline Solids The THz spectra of biomolecular crystalline solids have been reported for a variety of oligopeptides, simple sugars, pharmaceutical species, and other small biomolecules. In contrast to the larger systems discussed in later sections, a common feature of these studies is the observation of relatively small number of sharp vibrational features attributed to a discrete number of intramolecular or phonon modes of the solid. Observing vibrationally resolved features is primarily a result of two factors. Depending on the space group, typically only one or two conformations of the

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monomer subunits are dominant in the crystal and therefore, the small number of unique atoms per unit cell results in a small number of vibrational modes that are IR active below 3 THz. Second, when employing the proper crystal growth procedures that minimize defects, narrow inhomogeneous line widths (<1 cm–1) often result especially for systems where strong hydrogen bonding networks exist throughout the solid. Moreover, at lower temperatures approaching 4 K, the spectral lines can significantly sharpen and shift depending on the impact of anharmonicity and crystal cage effects. Temperature-dependent studies can give information about the character of the mode as well as the anharmonicity of the potential energy surface. A few examples in each of these areas will be reviewed.

7.5.1 Amino Acids In an effort to reveal the underlying THz spectral contributions from the individual amino acids of a protein, we began a series of experiments to measure the THz IR absorption spectra for many of the 20 naturally occurring solid-state (lyophilized) amino acids. As discussed, we found that these spectra demonstrate a high degree of complex absorption features and structural information in the 1 to 22 THz (33 cm–1 to 700 cm–1) spectral range.32 These results complement a related investigation of shortchain polypeptides also published by our group and discussed later in this chapter. We obtained the solid-state spectra for l-serine and l-cysteine to determine whether an atomic substitution (–OH for serine versus –SH for cysteine) can be identified in the THz spectrum.33 Significant differences were noted, especially in the lower frequency phonon region (<100 cm–1) indicating the crystalline lattice interactions are strongly affected by the simple atomic substitution. THz spectroscopy is also highly sensitive to the enantiomeric crystalline structures of biomolecules. The stereochemistry of many biologic and pharmaceutical species plays a significant role in whether they are functional or even toxic. To this end, an investigation into whether the THz spectra of enantiomeric mixtures of leftand right-handed species can be uniquely identified was undertaken by our group.34 Spectra for l-, d- and dl-tryptophan (1:1 l- and d-tryptophan) as well as various mole percent mixtures were studied using broadband FTIR spectroscopy. Although the spectra for pure d- and l-tryptophan are identical, admixtures from about 5 to 50% mole fractions could readily be identified from changes in absorption intensities of several spectral features. Figure 7.3 shows examples of these spectra as well as a single fluorine-substituted analog which generates a completely different spectrum compared with the native amino acid. Principal component spectral analysis could readily be employed to quickly analyze these spectra and readily extract “contaminant” concentrations, a task that is difficult for any other analytical method. Starting from available x-ray crystal structures for each amino acids species,13 application of inexpensive DFT methods12 did not produce modeled THz spectra that agreed well with the experiment. Apparently the currently available basis sets and parameterized potential energy functions are not adequate to model intermolecular hydrogen bonding in these systems. It was suggested that application of modern ab initio structure, frequency, and intensity calculations will eventually enable assignment of THz spectral features to both internal and external vibrational motions

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6

298 K

F6, DL-Tryptophan

Absorption

5 4

DL-Tryptophan

3 2

D-Tryptophan

1 0 –1

L-Tryptophan 0

100

200

300

400

500

Wavenumber (cm–1)

Figure 7.3 Experimental infrared absorption spectra for the pure d- and l-tryptophan (Trp) enantiomers, their 1:1 mixture (dl-tryptophan) and F6-fluorine-substituted dl-tryptophan amino acid in the solid state. Comparison of the l-Trp and d-Trp stereoisomers shows the spectra are identical, whereas the dl-Trp mixture has a distinctly different spectrum. The F6dl-Trp species also shows different spectral features from the dl-Trp mixture.

of these complex hydrogen-bonded molecular solids. CHARMM and CPMD approaches were found to agree better than simple gas-phase theory and these results were included in our publication.33

7.5.2 Polypeptides Because polypeptides have C- and N-terminations and are therefore sequence- specific, we investigated a series of polypeptides to examine whether their crystalline forms yield qualitatively different THz spectra. As may be seen in Figure 7.4, every peptide that was examined, including reverse sequences, produced a different and readily identifiable spectrum. From these results, it can be concluded that shortchain crystalline polypeptides may be uniquely “fingerprinted” by simply acquiring the THz spectrum in the 50 to 500 cm–1 spectral range. Several mixed polypeptide sequences with up to three amino acids were also studied (e.g., Val-Gly-Gly, glutathione), but sharp spectral features were not present for species containing more than 10 amino acids. This observation may be the result of overlapping spectral density as the molecular weight increases, or the spectral features are swamped by hydrogenbonded water, which is difficult to remove from these complex biomaterials. The sensitivity of the THz region to different β-sheet structures and the impact of cocrystallized water were investigated by our group35 for three different crystalline forms of the tripeptide, NH3+-Ala3-O–, using a continuous wave THz source and THz-TDS methods.36 Depending on the conditions during recrystallization, trialanine is known from x-ray crystal work37 to exist in one of two β-sheet forms, a dehydrated parallel β-sheet (p-Ala3) and a hydrated anti-parallel β-sheet (ap-Ala3-H2O). A third form discovered in this study was the dehydrated form, ap-Ala3. As shown in Figure 7.5, the

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Terahertz Spectroscopy of Biomolecules Peptide Absorption Spectra (300 K)

6 5

Asp-Arg-Val-Tyr-Ile-His-Pro-Phe-His-Leu

Absorbance

4

Glutathione

3

Val-Gly-Gly

2

Ala-Gly-Gly

1 0 –1 0.0

Gly-Ala-Ala 2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

THz

Figure 7.4 Representative THz infrared absorption spectra for several di- and tripeptides showing the variability in spectral content and uniqueness for each structure. For comparison (top), a random sequence 10-mer is also shown and exhibits loss of distinguishing absorptions, most likely from spectral overlap or hydrated water.

THz Spectra

Absorbance

p-Ala3

4.2 K

ap-Ala3-H2O

ap-Ala3

1.0

2.0

3.0

ν/THz

Figure 7.5 THz absorption spectra of the three different crystalline forms of trialanine. Although the conformational forms of the monomers are similar, the parallel and hydrated antiparallel β-sheets have different space groups (P 12 and C 2, respectively, the dehydrated form is unknown but likely C 2). These spectra indicate the sensitivity to the different space groups as well as the impact of the hydrogen bonded (structural) water.

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THz spectra (0.6 cm–1 to 100 cm–1) obtained at 4.2 K for the three forms are qualitatively different. This is in sharp contrast to the mid-IR region where the FTIR spectrum38 of ap-Ala3-H2O is similar to that of the p-Ala3 and nearly identical to the dehydrated form, ap-Ala3. The significance of this result is the dramatic impact the environment and the arrangement of hydrogen bonds have on the THz spectrum because the conformational forms of the monomers are essentially the same. The small size of these systems enabled rigorous comparisons with quantum chemical (density functional theory) and classical models (CHARMM). In sharp contrast to the weakly bound organic systems discussed previously in which gas phase predictions were sufficient, calculations on these hydrogen-bonded crystals required the use of periodic boundary conditions. Results from the DFT calculations using the DMol3 program suite28 are shown together with the experiment in Figure 7.6 for the parallel and hydrated antiparallel sheet forms. Spectral predictions of the parallel sheet with its highly cross-linked hydrogen bonding network are in fair agreement with many of the observed features. However, the weak intersheet binding in the antiparallel sheets is a severe challenge for theory where clearly the current level fails to achieve even qualitative agreement. Furthermore, the predictions from the classical model, CHARMM, in Figure 7.6 indicate that the similarities regarding spectral predictions were accidental because the corresponding nuclear motions at these two levels were vastly different.35 These results suggest the need for improved density functionals that target multibody effects present in hydrogen bonding networks and improved force field models that include polarizability terms for water and three-atom hydrogen bonding terms for periodic solids. Current theoretical studies are under way that (1) use a triple zeta basis set and a fully relaxed crystalline cell geometry at the DFT level and (2) treat anharmonicity of the modes using VSCF theory at the classical level.39

7.5.3 Nucleic Acid Bases and Other Sugars The Jepsen group40 studied the dielectric functions of the four nucleobases, adenine (A), guanine (G), cytosine (C), and thymine (T) and their corresponding nucleosides, dA, dG, dC, and dT (d = dioxyribose) of DNA from 0.5 to 4.0 THz at 10 K and 300 K using THz-TDS. The spectra of A, G, C, and T are dominated by absorption features between 1 THz and 3.5 THz. At 10 K, the broad room temperature features blue shift by 5% and resolve into several narrow peaks. The blue shift with decreasing temperature is attributed to decreasing bond lengths. The spectra above 1.5 THz of dA, dG, dC, and dT are similar to the nucleobases but a second group of resonances from 1 THz to 2 THz have narrow asymmetric line shapes attributed to the sugar groups. Quantum chemical calculations were performed on the gas phase tetramer of thymine using density functional theory (B3LYP/6-31G basis). The absorption spectrum and index of refraction were calculated for the minimum energy geometry (beginning with the x-ray crystal structure) and compared with experiment. The calculations predict four low frequency infrared-active modes arising from in-plane and out-of-plane intermolecular motions of the hydrogen bonded cluster. Although definitive assignments are not possible using this approach, nevertheless, the correct

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Terahertz Spectroscopy of Biomolecules p-Ala3 β-sheet

Absorbance

CHARMM

DMol/PW91

Expt

1.0

ν/THz

2.0

3.0

ap-Ala3-H2O β-sheet

Absorbance

CHARMM

DMol/PW91

Expt

1.0

ν/THz

2.0

3.0

Figure 7.6 Predicted and observed THz spectra of the parallel (top) and hydrated antiparallel (bottom) β-sheet forms of crystalline trialanine from DFT/PW91 (DMol3 software suite) and CHARMM.

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number of features was reproduced. The significance of these results is the observation of the well-resolved and distinct spectral fingerprints of the nucleobases and nucleosides arising from the arrangement of the molecules in the microcrystalline structures. The THz dielectric properties of the three sugars—glucose, fructose, and (protonated and deuterated) sucrose—were studied by the Jepsen group41 from 0.5 THz to 4.0 THz using THz-TDS methods. Crystalline forms were shown to have sharp features (3 cm–1), whereas absorption by the amorphous forms were featureless and increased monotonically with frequency. With increasing temperature, all absorption features of glucose and fructose broadened and red shifted. However, for sucrose, the two lowest frequency resonances of the protonated and deuterated samples (peak 1 at 1.38/1.28 THz for the protonated/deuterated forms and peak 2 at 1.69/1.63 THz) initially blue shifted with increasing temperature and then above a crossover temperature (240 K for peak 1 and 120 K for peak 2), shifted to lower energy. They found the frequency shifts with deuteration Δ ν /ν = –7% and –3.5% are larger than expected based on simple harmonic mass weighting alone (Δ ν /ν ~ 1.2%). Furthermore, they argue the anomalous blue shift is not easily explained based on vibrational anharmonicity because red shifts arising from hydrogen bonding stretch potentials are expected with increasing temperature. They suggest the initial blue shift at low temperatures may be a result of the relatively larger impact of van der Waals forces on the hydrogen bonding potentials compared with the higher temperature limit where anharmonicity presumably dominates.

7.5.4 Other Small Biomolecules The Jepsen group42 also studied the dielectric functions of the biomolecules, benzoic acid, and the monosubstituted analogs, salicylic acid (2-hydroxy-benzoic acid), 3- and 4-hydroxybenzoic acid, and aspirin (acetylsalicylic acid) from 0.5 THz to 4.5 THz at 10 K and 300 K using THz-TDS. This series was studied to elucidate the sensitivity of THz vibrational modes to small changes in the overall structure as well as the hydrogen bonding environment. Despite the similarity of the molecular structures, the absorption spectra shared very little in common. The authors attributed modes below 72 cm–1 to lattice vibrations and higher frequency resonances to dimer modes (torsions, out-of-plane bends and asymmetric stretches of the hydrogen bonds) from previous theoretical work.43 The broadening and red shift of the features with increasing temperature was attributed to mechanical anharmonicity of the potential energy surfaces. Based on fits of the 10 K data to a frequency-dependent complex dielectric function, the center frequencies, line widths, and oscillator strengths (and integrated band intensities) are reported for the three isomers. For the all-trans form, these same parameters are used to model the room temperature data after red shifting all frequencies by 4.5 cm–1 and increasing line widths by a factor of two. Comparison of the absorption features of the three forms enabled some speculation of the types of nuclear motions involved.

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Similar spectroscopic studies of small pharmaceutical species have also been conducted using THz-TDS spectroscopy by the Taday group.44 Investigations of the spectrum below 3 THz were conducted on active drugs carbamazepine, piroxicam, and theophylline in the solid state and showed that polymorphism and hydration strongly affect the THz spectrum. This and earlier work by the same group at Teraview in the United Kingdom demonstrates that THz methods can be successfully applied to the study of drug species and how subtle hydration, environmental, and structural effects alter drug potency and their controlled manufacture in the pharmaceutical industry. The impact of mechanical anharmonicity on THz vibrational features was investigated by our group using THz continuous wave methods by measuring the temperature dependence of the THz absorption spectrum (0.06 to 4.3 THz) of crystalline biotin.45,46 The THz spectra obtained at 4.2 K and 298 K are shown in Figure 7.7. A quantitative model was developed to account for observed changes in line shapes and the redistribution of intensity over this temperature range. Such studies are unique to this region because the fundamental vibrational transition frequencies (1 THz ≈ 33 cm–1) are comparable to the thermal energy available (kT ≈ 200 cm–1) at room temperature. With sufficient spectral resolution (<0.1 cm–1), the enormous change in the level populations that occur over the temperature range from 4.2 K to 298 K may be exploited to recover upper limits on the mechanical anharmonicity of the vibrational modes. The lower part of Figure 7.7 illustrates the impact of temperature on the line shape when (positive) mechanical anharmonicity is introduced at the level of 0.1% for a 1-THz vibration. The model calculations superimposed in Figure 7.7 were obtained as follows. The 4.2 K spectrum was first least-squares fit to Gaussian line shape model containing thirteen lines. The asymmetrically broadened and redshifted features in the room temperature spectrum were then fit using just one adjustable parameter per line, the mechanical anharmonicity. This simple model was shown to explain the intensity redistribution and the asymmetric distortion of the line shape at room temperature. With the further relaxation of the center frequencies in the room temperature fit, the quality of the fit shown in Figure 7.7 was comparable to that obtained at 4.2 K. Because crystal cage dimensions are known to vary by 1 to 3% over even smaller temperature ranges,47–52 such shifts are expected based on current theoretical results where the crystal parameters were fully relaxed. The best-fit Gaussian line widths obtained at 4.2 K are shown in the top panel of Figure 7.8 to increase linearly with frequency. The line shape function and linear increase are both consistent with an inhomogeneous broadening mechanism that likely results from the distribution of crystal lattice defects. Consequently, Lorentzian widths arising from vibrational dephasing (or damping) must be less than these widths enabling conservative lower limits on the vibrational lifetimes of 5 ps to 1 ps. The anharmonicity factors determined for each line are shown in the lower panel of Figure 7.8 as a function of frequency and are all positive, indicating the anharmonicity associated with hydrogen bond stretching is more important than negative contributions arising from torsional motions. Moreover, the observed increase with

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Terahertz Spectroscopy: Principles and Applications Biotin νe, ∆νFWHM, κ(νe) Fit

Absorbance (base 10)

T = 4.2 K

χ2Fit = 1.8

1

0

1.0

Absorbance (base 10)

1

T = 298 K

ν/THz

2.0

3.0 χe, νe Fit χ2Fit = 1.8

0

1.0

ν/THz

2.0

3.0

χ1 – 1×10–3

Figure 7.7 THz absorption spectra of crystalline biotin obtained at 4.2 K (top panel) and 298 K (middle panel) where the linearly increasing baseline has been removed. Superimposed on the spectra are model predictions with residuals. The absorption features in the 4.2 K spectrum were first identified and fit to a Gaussian line shape model. The best fit parameters were held fixed in fits to the 298 K spectrum except for the center frequencies and the anharmonicity factors (χe). An example of the impact of anharmonicity (χe = 0.1%) on the line shape at 1 THz is illustrated at the bottom for the two temperatures.

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Terahertz Spectroscopy of Biomolecules Biotin Gaussian Widths

6 5

∆ν/cm–1

4 3 2 1 0

0

1

ν/THz

2

3

Anharmonicity Factors

1.0

χe/%

0.75

0.50

0.25

0.0

0

1

2

3

ν/THz

Figure 7.8 Trends observed in the THz absorption spectra of biotin. In the top panel, the increasing Gaussian widths with frequency (obtained from the 4.2 K spectrum) are consistent with an inhomogeneous broadening mechanism. In the lower panel, the increasing anharmonicity factors with frequency (obtained from the 298 K spectrum) suggest the vibrational force fields contain increased contributions from hydrogen bonding stretch potentials.

frequency is as expected because the higher frequency modes are likely to have more hydrogen bonding stretch character. The anharmonic constants represent upper limits since contributions from other line broadening factors would only reduce their magnitude. The significance of this study is that it provides quantitative data to access the impact of anharmonicity on THz spectra and to refine vibrational force field models to account for the temperature dependence of absorption features.

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7.6 THz Studies of Large Biomolecules 7.6.1 DNAs and RNAs Some of the earliest work on DNA in the THz region (0.09 to 13.5 THz) was performed by Wittlin and coworkers53 over the temperature range from 5 K to 300 K on highly oriented films of Li-DNA and Na-DNA. Five vibrational modes were identified including the lowest frequency modes at 1.35 THz for Li-DNA and 1.23 THz for Na-DNA. These modes were found to soften with hydration. A simple lattice dynamic model was reasonably successful in explaining the vibrational modes and the impact of hydration on these modes. Two artificial RNA single strands composed of polyadenylic acid (poly-A) and polycytidylic acid (poly-C) were studied by the Jepsen54 and Tominaga6 groups using a THz-TDS imaging method over the frequency range from 0.1 THz to 4.0 THz (ΔνInst ≈ 0.5 cm–1). Their technique was used to record images of spotted arrays on a nonpolar polymer substrate with densities that varied between 2 × 105 g/mole to 7 × 105 g/mol (corresponding to chain lengths of 600 to 2000 A or C units). The thin films were formed by dissolving samples in water and by repeated spotting/evaporation of the droplets. A surface profiler was used to character the sizes of the spots which were 40 to 50 μm thick with diameters from 1 to 1.5 mm. Bulk samples in pressed pellet form were first studied and shown to give unstructured absorption that increased linearly between 0.1 THz and 2.5 THz. Poly-C was found to absorb more strongly than poly-A. The index of refraction of poly-C is 10% larger than poly-A. Analysis on the thin films was performed by taking the difference in integrated absorption between poly-A and poly-C. Differences of 10% for 200-μg samples permitted the easy discrimination between these two forms of RNA. Reliable image contrasts were obtained for spot sizes of 1 mm in diameter and for sample weights of ≈100 μg or more. However, they found little discrimination for sample weights of 20 μg. Furthermore, no sharp spectra features were observed for either bulk samples, freestanding thin films or spots. They noted that etalon artifacts were largest when the sample absorption was lowest because of multiple bounces between the interfaces. This study contradicts Globus’s work55 on RNA polymers discussed below where structured absorption is reported. They argue that continuous unstructured absorption is expected in the amorphous condensed phase because of the increasing density of IR-active modes with frequency and the high sensitivity to local environment where additional damping is expected from random interactions with neighboring molecules. They concluded that the lack of distinct spectral features represents a severe limitation for many biologic applications but noted that the careful determination of absorption coefficients could potentially offer additional information about the biomolecule and its environment. In a series of studies by Globus and coworkers,56–60 THz spectra of single- and double-stranded DNA and RNA samples have been measured using an FTIR spectrometer typically performing within the frequency range from 10 to 25 cm–1 and at a resolution of 0.2 to 2 cm–1. In the first study, spectra of DNA samples in liquid gel phase were compared with solid film spectra. Thin samples were sealed in polyethylene or polycarbonate films having thicknesses from 10 to 100 μm. These optical path lengths are smaller than THz wavelengths, thereby reducing radiation losses

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and minimizing standing wave interference patterns from cell reflections. Polarization properties of absorption were used to characterize samples and many examples were presented to establish the reproducibility of the runs. For liquid gel samples, improved sensitivity from increased fidelity of features (0.3 to 0.5 cm–1) was reported and compared with the corresponding solid phase spectra. The samples were further characterized using ultraviolet absorption spectra. In these THz studies, discrimination between the native (double-stranded) and denatured (single-stranded) forms of DNA was claimed based on changes in four features. In a second study of DNA, the Globus group58 investigated the THz spectra of four different types of samples: randomly oriented DNA, partially oriented DNA, artificial poly ribonucleic acid, and Bacillus subtilis spores. The four sample types were prepared in various manners to achieve a thin film that was either free-standing or applied to a near-transparent substrate. In general, transmission of samples exceeded 90% with structure at the level of 1 to 2%. This apparent structure was found to vary depending upon sample thickness, sample orientation, and water content and interpreted to result from a high density of modes. Additional DNA studies by Globus and coworkers60,61 included single- and doublestranded salmon and herring DNA sodium salts, chicken egg ovalbumin, and Bacillus subtilis spores. Film samples on nearly transparent substrates were prepared from a water gel in varying concentration and air dried while liquid samples were sealed between thin polyethylene films. Sample thickness varied between 1 and 250 μm and attempts were made to align the molecules mechanically. It was again noted the extreme sensitivity of the spectra to sample preparation including drying conditions that particularly affect phase transitions of the DNA liquid crystal gels as well as sample thickness, initial concentration in water, and orientation. From reflection and transmission spectra, the frequency-dependent index of refraction and absorption coefficient were determined using an interference spectrometry technique. Transmission spectra of the Bacillus subtilis samples appear roughly linear at a level of 88 to 96% across the 15 cm–1 range with variations of the order of 1%. Results for liquid samples of double-stranded DNA also show roughly linearly decreasing absorption with frequency from 97 to 92% across the same range with similar 1% variations. Absorption by single-stranded DNA samples60 was generally ~20% greater than for double-stranded DNA. Globus and coworkers58 also examined the THz spectra of known sequences of single-stranded RNA potassium salts and double-stranded RNA sodium salts. Single-strand samples consisted of poly-adenine (A), poly-guanine (G), poly-cytosine (C), and polyuracil (U) and double-strand samples consisted of paired homopolymers C-G and A-U. Films of thickness less than 250 μm were prepared with sample concentrations of 2 to 20% in water. Films were either free standing or applied to near-transparent 8-μm thick polycarbonate substrates. Attempts to align the strands were largely unsuccessful and spectral sensitivity to sample preparation, alignment, and interference effects were examined. The frequency-dependent absorption coefficient was calculated for the four single-stranded homopolymers and for the two double-stranded homopolymers, C-G and A-U. Results of theoretical calculations using JUMNA (junction minimization of nucleic acids) and LIGAND programs are given using the FLEX force field for hydrogen bonding for RNA fragments.

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Terahertz spectroscopy is also sensitive to DNA strand hybridization and has been shown to serve as an analytical tool for DNA sequencing. In several elegant articles by Bolivar and Kurz,61,62 strong THz difference absorption upon hybridization was demonstrated using THz-TDS spectroscopy, indicating that direct measurement of free strands and hybridized pairs can be readily identified.61 A novel wave guide approach was further developed, enabling direct hybridization with femtomole sensitivity (comparable to fluorescence tagging methods).62 Early work in our group with Roitberg and Markelz63 reported the THz absorption spectra of calf thymus DNA, bovine serum albumin (BSA; similar to human serum albumin), and type I collagen that have molecular weights of 12 MDa, 66 kDa, and 360 kDa, respectively. Dry samples in polyethylene powder of varying thickness were investigated to access artificial contributions from sample etalon effects. After scaling for molecular path length, samples, of different thickness all showed Beer’s law behavior. Comparison with undiluted samples showed no measurable difference in the spectra of diluted and undiluted samples, indicating that spectral contributions from the instrument could not be resolved by the 0.7-cm–1 resolution. Spectral differences in the two systems studied ensured that the observed data were indeed probing biomolecular signatures. For all three biomolecules, broadband absorption spectra devoid of any features were observed at room temperature and for relative humidities (r.h.) of <5%. For DNA and BSA, absorbance increased roughly linearly with frequency over the observed range of 0 cm–1 to 60 cm–1, whereas absorbance for collagen increased at a faster rate. Both DNA and BSA were investigated under different conditions of r.h. to study the hydration dependence of the THz response. Pure samples were used to ensure that polyethylene did not hinder sample hydration. Both BSA and DNA exhibited changes in the THz absorbance over the 0 cm–1 to 60 cm–1 range observed. It was noted that both BSA and DNA are expected to have conformations that depend on the degree of hydration. DNA has a predominantly helical A conformation at r.h. between 45% and 92% but a disordered helical A conformation at lower hydration and predominantly helical B conformation at higher hydration. Of the three spectra, the first was taken at <5% r.h., the second at 43% r.h., which is near the transition point from disordered to ordered helical A state, and the third at 70% r.h., where a predominantly ordered helical A conformation is to be expected. Interestingly, the first and last spectra, corresponding to the disordered and ordered helical A conformational states are distinctly different, whereas the second spectrum near the transition point share features from both.

7.6.2 Proteins The conformation of BSA as a function of hydration is not definitively known but is expected to be similar to human serum albumin. The latter has a greater content of β-sheet conformation for r.h. <16% but increasing α-helical content for 30% to 50% r.h. The spectra observed for BSA at <5% r.h. is similar to that of denatured BSA but red shifted while the spectrum observed for BSA at 77% r.h. shows some differences including an increased red shift. These differences may reflect

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the expected changes from β-sheet to α-helix in the conformational composition of the sample. An important aspect of sample preparation is the time for structural equilibration. Several days may be required for structural equilibrium to be reached after conditions of humidity are changed. The samples studied here, however, were exposed for only 90 minutes before the acquisition of spectra. It is therefore not possible to attribute the changes observed in the THz spectra to the conformation changes expected for equilibrium hydration levels with any certainty. In light of our more recent work on dipeptides65 and tripeptides,35 structurally integrated water, non– hydrogen-bonded structural water and superficially adsorbed water can have varying impacts on the THz response. The three different retinal isomers that occur in the photoactive proteins, rhodopsin, bacteriorhodopsin, and isorhodopsin were investigated by the Jepsen group64 between 10 cm–1 and 100 cm–1 at 10 K and room temperature. The three different crystalline forms prepared in polyethylene pellets included the all-trans, 9-cis, and 13-cis isomers that differ only in their conformational structure. The absorption data illustrate the utility of the region for the clear distinction of isomeric samples. Furthermore, the broad absorption features present at room temperature are shown to sharpen and blue shift with decreasing temperature. The Markelz group has gone on to investigate several other large biomolecules, including hen egg white lysozyme, horse heart myoglobin, and bacteriorhodopsin in collaboration with Whitmire, Hillebrecht, and Birge.66 Lysozyme and myoglobin powder were diluted with PE powder and pressed into pellet form. Samples were temperature and humidity controlled and spectra were acquired at 77 K and room temperature. For both substances, broadband absorption without identifiable peaks was observed to increase from 5 cm–1 to 40 cm–1 and then leveled off up to 80 cm–1. At 77 K, absorption was less than at room temperature across the observed spectrum but remained featureless. The THz spectrum of the organic semiconductor α-hexathiophane in crystalline form was acquired using an FTIR instrument confirm the featureless properties of these samples. The experimentally measured spectra were compared to the density of states calculated by CHARMm. These calculations were performed on myoglobin and lysozyme structures available from the protein database without any additional water. The state density for both was roughly 500 modes in the first 80 cm–1 with an average separation of 0.2 and 0.16 cm–1 for lysozyme and myoglobin, respectively. In both cases, the density of states increased roughly linearly with frequency and leveled off near 60 cm–1 in contrast to the experimentally observed maximum absorbance near 40 cm–1. This similarity between the calculated state density and observed THz absorption spectra suggested that all modes are IR-active and oscillator strength is frequency-independent. However, from CHARMm and quantum chemical calculations on small crystalline systems, unphysical results were found for even these simple systems without a high degree of verisimilitude in the model to mimic the surrounding molecular environment. In particular, the optimized geometry was not consistent with the known geometry without the surrounding molecular environment which, in turn, led to unrealistic outcomes regarding dipole–dipole interaction energies. Furthermore, results employing periodic boundary conditions were

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still unsatisfactory. Although the state density found in CHARMm calculations was roughly correct, the positions and intensities of lines could not be reconciled with experimental data. Markelz in collaboration with Whitmire, Hillebrecht, Birge, and others67 reported the first demonstration of THz response to conformational state in some interesting and well-conceived studies of bacteriorhodopsin (BR). BR undergoes a light-activated photocycle consisting of a series of intermediate conformational states labeled K through O which are all identifiable by their ultraviolet/Vis absorption spectra. The end result of the photoisomerization processes and the interactions between the chromophore and its binding pocket enabled by conformational changes is the transport of a proton from the inner to the outer side of the cell membrane, which generates a potential gradient to drive formation of ATP. In this photocycle, the particular conformation state labeled M has the useful property that its 10 msec lifetime can be prolonged indefinitely at 233 K. Hence, under appropriate conditions of humidity, the M state can be photoinduced at <570 nm and then captured at 233 K to permit the THz spectra to be measured and compared with the spectrum of the ground state conformation. This work investigated two forms of BR, the wild type (WT) and the D96N mutant, which involves a single residue change. In the D96N mutant, the 96th aspartic acid residue of the 248-residue chain is replaced by asparagine. This seemingly small change of structure results in a 1,000-fold increase in the photocycle time of the molecule.68 It has been suggested that this increase is due to a loss of conformational flexibility with mutation. Differences in THz response between the WT and mutant BR may therefore suggest a possible correlation with conformational flexibility because motions permitting conformational state transitions may be decomposed into a series of THz vibrational motions. Therefore conformational transitions requiring access to the lowest frequency collective modes may impact the THz response. The THz absorption spectra of the two types of BR were first obtained at 80% r.h. and room temperature for 200-μm thick films on Infrasil quartz substrates. In both cases, absorption over the range 0 cm–1 to 60 cm–1 is broad and increases linearly with frequency but the absorption of the D96N is lower by a factor of 1.8. No difference was observed in spectra taken at 30% r.h., but at <5% r.h., absorption in both species decreased markedly. The change was too large to be explained by massloading effects of water content and may result from intramolecular coupling effects and/or collective modes of interior water clusters. The normal mode frequencies and eigenvectors derived from CHARMm were used to verify that indeed the very lowest frequency modes are responsible for the largest structural fluctuations while higher frequency modes involve only smaller rms fluctuations. Additional calculations performed on both WT and D96N used an approximation in which the dipole moment squared is proportional to its derivative. Absorption was then estimated using an ad hoc method to account for anharmonicity with no explicit reference to temperature where the sum over states was taken using harmonically approximated states that included only n = 0 to n = 1 transitions. Some agreement was found in comparisons with the initial BR spectra taken at room temperature and 80% r.h. However, little difference was found

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between the calculated spectra of the WT and mutant BR in contrast to the large differences observed. In studies comparing illuminated and nonilluminated BR, the THz spectra were first obtained at room temperature for the ground state and then again after cooling to 233 K. For the WT-BR, the broadband featureless absorption decreased markedly with decreasing temperature at the upper end of the 0 cm–1 to 60 cm–1 range. Conversely, absorption decreased in D96N-BR with decreasing temperature at the lower frequencies in the range. The most striking effect occurred, however, on the photoconversion to the M state at 233 K in which the THz absorption increased relative to that at room temperature. In contrast, the absorption strength remained essentially unchanged in D96N-BR after conversion from the ground state to the M state. On illumination with <480-nm light, the THz response of WT-BR reverted to that previously observed for the ground state at 233 K and at room temperature. A change in M state absorbance was suggested to result from either a change in dipole moment or a change in the distribution of normal modes. Because the former differences are nearly identical for both WT and D96N, changes in state density were suggested to explain the THz response in WT. However, it should be noted that the ultraviolet/Vis spectra used to confirm conformation state indicated a lower M state content for illuminated WT-BR at 233 K than for illuminated D96N-BR at 233 K. Therefore the presence of N or P states in illuminated WT-BR might conceivably be responsible for the observed absorption increase. The significance of these results is the demonstration of the sensitivity of the THz response to changes in conformation state and to changes of a single residue in the peptide sequence. These studies also raise unanswered questions relating to the quality of theoretical models. The CHARMm results showed no obvious basis for the lower THz absorption of D96N-BR in the ground state compared with WTBR, nor explained the striking difference in the THz response of WT-BR compared with mutant D96N BR under illumination and conformation change. It is important to note that the CHARMm calculations contained no bound water, whereas more recent studies of crystalline di-65 and tripeptide35 systems have shown that the presence of water can strongly affect the THz spectra. These studies have also suggested that the harmonic approximation may not be adequate to reproduce finer details of the observed THz spectra. The Tominaga group reported the earliest known low frequency THz study of cytochrome C solid in 2002.6 In more recent work with Chen, Knab, and Cerne, Markelz studied cytochrome C (CytC), a string of 104 amino acids having a covalently bounded heme group.68 This electron transfer protein has two forms that are structurally quite similar yet have quite different thermal stability, hydrogen exchange rate, and proteolytic digestion rate. The bound heme group exists in two oxidation states and x-ray measurements, NMR structural measurements and compressibility measurements have indicated that the oxidated state, which is the less stable and more active state, has greater conformational flexibility than the reduced state in spite of their great structural similarity. The THz absorption coefficient of the more flexible, oxidized states is much greater than that of the more stable, less active reduced state. As in the studies of large molecules discussed above, the response is

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broad, essentially featureless, and aptly characterized as “glass-like” over the 10 cm–1 to 80 cm–1 range observed. In fitting the dielectric response, it was assumed that all modes are active and oscillator strength is more or less uniform across the frequency range of interest. Differences in the fits can be due to differences in dipole coupling strength or state density. The predicted dipole moment difference between the oxidized state and the less flexible reduced state is only about 6%. Furthermore, the difference in response is still too large to be explained by increases in state density resulting from the larger water content (and hence system size) of the oxidized state. This outcome is consistent with an increased number of low-frequency modes for the more flexible oxidized state and illustrates the use of THz spectra as a measure of the accessibility of the low frequency modes crucial to conformational flexibility. In the most recent work from the Markelz group,69 the THz response of hen egg white lysozyme was investigated as a function of hydration for r.h. from 3% to 39%. Other workers using NMR, specific heat and microwave spectroscopy measurements have previously identified a transition point near 27% r.h. at which the first hydration shell is filled and bulk water begins to accumulate. Evidence of this transition is also seen in the THz spectra in this work for which very uniform thin films (≈100 μm thick) of lysozyme were prepared on quartz substrates. The lysozyme samples were allowed to reach equilibrium in a humidity-controlled cell at room temperature prior to obtaining spectra. For lysozyme, the absorption coefficient α increases nearly linearly with frequency, and the index of refraction n is nearly flat for all levels of hydration shown. Of particular interest is the sudden increase in the entire absorption curve at the 27% r.h. and the subsequent shifts which are even larger as hydration is increased. A similar effect is seen for index n at the same transition point. The apparent discontinuity is indicative of the transition from occupation of bound water sites to bulk water accumulation. In previous work, attempts were made to model large biomolecules using CHARMm to obtain the normal mode distribution and then with simplifying assumptions about the dipole interactions to estimate the THz absorption spectrum. Such models were not especially successful and, in this work, a new approach was tried. Because the system at room temperature is believed to be composed of multiple conformations, one approach is to consider the conformational lifetime rather than determining the normal mode distribution of each particular conformation. Several such models were investigated but results were not entirely satisfactory. Simple oneor two-relaxation time models gave reasonably good fits, but unfortunately unrealistically short lifetimes as compared with the relaxation times found experimentally by others. On the other hand, good fits can be found by adding a sufficient number of oscillators and relaxation times but no physical model justifies such an approach and so little is gained.

7.6.3 Polysaccharides Investigation of the THz spectroscopy of polysaccharides has also been performed in our group (unpublished). Although only a few studies have been conducted, the

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Figure 7.9 Inelastic neutron spectrum (top) and THz absorption spectrum (bottom) for dispersed cellulose powder in polyethylene matrix. The triple-helix polymer exhibits distinct positive absorption features at 1, 3, 5, 7, 9, and higher frequencies, presumably arising from internal collective motions of the helix.

infrared absorption and inelastic neutron spectra of simple purified cellulose powder have been obtained (Figure 7.9). Cellulose is a hydrogen-bonded triple helix, where each strand is composed of polyglucose units. There are clear absorption and scattering features that occur at 3, 5, 7, 9, and 11 THz as shown in the figure. Although there are no modeling results for this system, it is surmised that the periodicity of these systems probably produces interstrand twisting or stretching motions that show up as harmonics in the THz spectrum. Further studies of modified cellulose (different crystalline forms or substituted species such as nitrocellulose) are clearly warranted, as are periodic boundary CPMD or related solid-state calculations to help identify the origin of these unique polymeric spectral features.

7.7 Terahertz Studies of Biomolecules in Liquid Water Two recent studies published at nearly the same time have addressed the use of THz spectroscopy to probe biomolecules in aqueous environments. For example, the Havenith group70 found that the THz region provides a sensitive probe of soluteinduced changes in water’s hydrogen bonding network near the vicinity of a biomolecule. The simple sugar, lactose, has been investigated using a p-Ge-laser THz source71 over the spectral region from 2.3 THz and 2.9 THz. Highly accurate absorption measurements were performed on bulk water and lactose power and compared with total absorption measured for a series of different concentrations of lactose in water. The challenge of these measurements arises because the absorption coefficient of water is 450 cm–1 at 3 THz and therefore measurements were performed in a cell 150 μm in length. Absorption for solutions was found to be greater than the fractional sum

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(by volume) of the bulk absorptions. This amounted to 65% of the total absorption for 0.7 mol/L solution! Further measurements of volume and modeling of hydration shells indicated the solute-induced perturbations extended more than 5.13 ± 0.12 nm from the surface of lactose. This volume corresponds to 123 water molecules/lactose. Simulations using molecular mechanics (and the CHARMm force field) revealed a slowing down of the hydrogen bond rearrangement dynamics in the hydration spheres. This study demonstrates that THz spectroscopy probes the coherent solvent dynamics on the subpicosecond timescale and gives quantitative measure of the extent the solute influences biological water (i.e., water near the surface of a biomolecule). In a THz study (0.1 to 1.2 THz) of myoglobin (Mb) using THz-TDS, Durbin and Zhang72 reported an increase in the absorption per protein molecule in the presence of “biologic” water. Absorption measurements were made on samples between 3.6 to 90 wt% water73 as confirmed by weight loss on heating and accurate optical pathlength determinations critical to this analysis were determined from comparisons made using 100-μm and 200-μm cuvettes. As with the other large biomolecules described previously, nearly continuous broadband absorption that increased linearly with frequency was observed. However, deviations of the normalized absorption from an idealized noninteracting Mb–water model showed enhanced THz absorption rather than a reduction expected on theoretical grounds from a decrease in the 2.5 Debye moment of bulk water to 0.8 Debye because of hindered motions experienced by biologic water. The increased absorption and hence, polarizability of Mb across the THz region (especially above 90 wt% water), is suggested to result from the enhanced mobility of the side chains along the protein74 and for more than 90 wt% water, increased rotational freedom of Mb that occurs when the average separation (>6 nm) becomes significantly larger than radius of gyration (1.5 nm). The significance of these latter two studies is that through quantitative THz absorption measurements, the solvent–biomolecule interfacial structure and dynamics may be examined directly.

7.8 Conclusions and Future Investigations Application of terahertz spectroscopic methods to the study of important biological species, ranging from simple sugars to complex DNAs has been reviewed. We have tried to give the reader a glimpse at the different types of systems investigated to date, and hope that there is ample evidence that the investigations performed have revealed new and exciting properties of these systems. IR methods have improved a great deal in the last decade, so novel studies of complex systems can now be performed. We apologize for not including references to all studies and investigators involved in this rapidly growing field. Although there are many hurdles to overcome, investigations of complex biomolecular species in the solid state are under way and many new groups are encouraged to become involved in this growing research area. Interpretation of resultant THz spectra absolutely requires theoretical modeling to deduce the origins of spectral absorptions and perform intra- and intermolecular vibrational mode assignments. At present, some approaches are adequate and yield quite good agreement with experimental spectra. These methods are also improving with the availability of more advanced computational methods and sophisticated

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hardware. Feedback between experimental and modified theoretical approaches will continue to advance our ability to interpret and extract structural details of larger molecular systems from THz spectra. This interplay between experimentalists and theorists will hopefully produce better force fields and computational methods to help predict and confirm spectral assignments for high molecular weight biomolecules. Finally, we might suggest that time-resolved investigations of biomolecular dynamics and function in aqueous environments are the ultimate goals for experimentalists. The challenge of probing biomolecule spectra in a highly absorbing media is great, and the influence of hydrogen-bonding on solutes by water of hydration is unknown, but recent studies of species in water give us hope that this goal will one day be achieved.

Acknowledgments We wish to thank all of our NIST colleagues, postdoctoral associates, guest researchers, and students for their participation in the THz spectroscopy work reviewed in this chapter. Without their efforts, research prowess, and attention to detail the results we provided to the scientific community over the last 13 years would not have been possible.

Endnote Certain commercial equipment, instruments, or materials are identified in this article to adequately specify the experimental procedure. In no case does identification imply recommendation or endorsement by NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

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11. K. A. McIntosh, E. R. Brown, K. B. Nichols, O. B. McMahon, W. F. DiNatale, T. M. Lyszczarz, Appl. Phys. Lett., 67 (1995), 3844. 12. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, et al., Gaussian 98 (Revision A.11.3), Gaussian, Inc., Pittsburgh PA, 2001. 13. See: http://webbook.nist.gov/chemistry 14. A. D. MacKerell, N. Banavali, Development and current status of the CHARMM force field for nucleic acids. Biopolymers 56 (2000) 257. 15. R. B. Gerber, C. M. Chaban, S. K. Gregurick, B. Brauer, Vibrational spectroscopy and the development of new force fields for biological molecules. Biopolymers, 68(3) (2003), 370. 1 6. S. K. Gregurick, S. A. Kafafi, J. Carbohydrate Chemistry, 18 (1999) 867. 17. M. M. Harding, H. A. Long, Acta Cryst. B24 (1968) 1096. 18. B. Khawas, Acta Cryst. B27 (1971) 1517. 19. T. J. Kistenmacher, G. A. Rand, R. E. Marsh, Acta Cryst. B30 (1974) 2573. 20. C. H. Görbitz, B. Dalhus, Acta Cryst. C52 (1996) 1756. 21. M. N. Frey, M. S. Lehman, T. F. Koetzle, W. C. Hamilton, Acta Cryst. B29 (1973) 876. 22. K. A. Kerr, J. P. Ashmore, T. F. Koetzle, Acta Cryst. B31 (1975) 2022. 23. R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. 24. N. Trouiller, J. L. Martins, Phys. Rev. B 43 (1991) 1993. 25. L. Kleinman, D. M. Bylander, Phys. Rev. Lett. 4, (1982) 1425. 26. R. Resta, Phys. Rev. Lett., 80 (1998) 800. 27. D. A. McQuarrie, Statistical mechanics. Harper-Collins Publishers: New York, 1976. 28. B. Delley, J. Chem. Phys. 113 (2000) 7756. 29. J. P. Perdew, Y. Wang, Phys. Rev. B, 45 (1992) 13244. 30. J. Higgins, X. Zhou and R. Lin, Spectrochim. Acta A, 53 (1997) 721. 31. J. Melinger, N. Laman, S. Sree Harsha, and D. Grischkowsky, Appl. Phys. Lett., 89 (2006) 251110. 3 2. M. Kutteruf, C. Brown, L. Iwaki, M. Campbell, T. A. Korter, E. J. Heilweil, Chem. Phys. Lett., 375, (2003) 337. 33. T. M. Korter, R. Balu, M. B. Campbell, M. C. Beard, S. K. Gregurick, E. J. Heilweil, Chem. Phys. Lett., 418, (2005) 65–70. 34. R. Balu, S. K. Gregurick, E. J. Heilweil, Determination of enantiomeric composition by terahertz spectroscopy: mixtures of D- and L-tryptophan. In preparation. 35. K. Siegrist, C. R. Bucher, I. Mandelbaum, A. R. Hight Walker, R. Balu, S. K. Gregurick, and D. F. Plusquellic, J. Am. Chem. Soc., 128, (2006) 5764. 36. M. Yamaguchi, K. Yamamoto, M. Tani and M. Hangyo, IEEE conference proceedings on biological and medical applications, ultrafast chemistry and physics, 13th Int. Conf. on Infrared and Millimeter Waves, paper WB3-3, p. 477. 37. A. Hempel, N. Camerman, A. Camerman, Biopolymers, 31, (1991) 187; J. K. Fawcett, N. Camerman, A. Camerman, Acta. Cryst. B. 31 (1975) 658. 38. W. Qian, J. Bandekar, S. Krimm, Biopolymers, 31, (1991) 193. 39. Z. Bihary, R. B. Gerber, J. Chem. Phys. 115, (2001) 2695. 40. B. M. Fischer, M. Walther and P. U. Jepsen, Phys. Med. Biol, 47, (2002) 3807. 41. M. Walther, B. M. Fischer, and P. Uhd Jepsen, Chem. Phys., 288, (2003) 261. 42. M. Walther, P. Plochocka, B. M. Fischer, H. Helm and P. U. Jepsen, Biopolymers (2002) 310. 43. H. R. Zelsmann, Z. Mielke, Chem. Phys. Lett. 186, (1991) 501. 44. J. A. Zeitler, K. Kogermann, J.antanen, T. Rades, P. F. Taday, M. Pepper, J. Aaltonen, C. J. Strachan, Int. J. Pharmaceuticals, 334, 78 (2007). 45. D. F. Plusquellic, T. M. Korter, G. T. Fraser, R. J. Lavrich, E. C. Benck, C. R. Bucher, J. Domench, A. R. Hight Walker, in Terahertz sensing technology. Volume 2: Emerging scientific applications and novel device concepts, World Scientific, 2003 p. 385.

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4 6. T. M. Korter and D. F. Plusquellic, Chem. Phys. Lett. 385, (2004) 45. 47. B. Dalhus, C. H. Gorbitz, Acta. Crys. C, Vol. 52, (1996) 1759. 48. Dawson and Mathieson, Acta Crys. 4, (1951) 475. 49. W.-Q. Wan, Y. Gong, Z.-M. Wang, C.-H. Yan, Chinese J. Struct. Chem. 22, (2003) 539. 50. B. Dalhus, C. H. Gorbitz, Acta. Chem. Scand. 50, (1996) 544. 51. W. Q. Wang,Y. Gong, Z. Liang, F. L. Sun, D. X. Shi, H. J. Gao, X. Lin, P. Jiang, Z. M. Wang, Surf. Sci. 512, (2002) 379. 52. K. Tori, Y. Iitaka, Acta Cryst. B 26, (1970) 1317. 53. A. Wittlin, L. Genzel, F. Kremer, S. Haseler, A. Poglitsch, Phys. Rev. A, 34 (1986) 493. 54. B. M. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, P. U. Jepsen, Optics Express 13, (2005) 5205. 55. T. W. Crowe, T. Khromova, B. Glemont, J. Hesler, J. Phys. D: Appl. Phys. 39, (2006) 3405. 56. T. Globus, D. Wollard, T. W. Crowe, T. Khromova, B. Gelmont and J. Hesler, J. Phys. D 39 (2006) 3405. 57. T. R. Globus, D. L. Woolard, A. C. Samuels, B. L. Gelmont, J. Hesler, T. W. Crowe, M. Bykhovskaia, J. App. Phys., 91 (2002) 6105. 58. T. Globus, M. Bykhovskaia, D. L. Woolard, B. Gelmont, J. Phys. D 36 (2003) 1314. 59. T. Globus, R. Parthasarathy, T. Khromova, D. Woolard, N. Swami, A. J. Gatesman, J. Waldman, Proc. SPIE, 5584 (2004). 60. R. Parthasarathy, T. Globus, T. Khromova, N. Swami, D. Woolard, App. Phys. Lett. 87 (2005) 113901. 6 1. M. Brucherseifer, M. Nagel, P. H. Bolivar, H. Kurz, Appl. Phys. Lett., 77, (2000) 4049. 6 2. M. Nagel, F. Richter, P. H. Bolivar, H. Kurz, Phys. Med. Biol., 48 (2003) 3625. 63. A. G. Markelz, A. Roitberg, E. J. Heilweil, Chem. Phys. Lett. 320 (2000). 64. M. Walther, B. Fischer, M. Schall, H. Helm, P. Uhd Jepsen, Chem. Phys Lett. 332, (2000) 389. 6 5. K. Siegrist, D. F. Plusquellic, CW THz studies of the crystalline dipeptide nanotubes. In preparation. 66. A. Markelz, S. Whitmire, J. Hillebrecht, R. Birge, Phys. Med. Biol. 47 (2002) 3797. 67. S. E. Whitmire, D. Wolpert, A. G. Markelz, J. R. Hillebrecht, J. Galan, R. R. Birge, Biophys. J. 85, (2003) 1269. 68. J.-Y. Chen, J. R. Knab, J. Cerne, A. Markelz, Phys. Rev. E. 72, (2005) 040901. 69. J. Knab, J.-Y. Chen, A. Markelz, Biophys. J. 90, (2006) 2576–2581. 70. U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, PNAS, 103 (2006). 71. E. Bründermann, D. R. Chamberlin, E. E. Haller, Appl. Phys. Lett. 73 (1998) 2757. 72. C. Zhang, S. M. Durbin, J. Phys. Chem. B. 110 (2006) 23607. 73. C. Zhang, E. Tarhan, A. K. Ramdas, A. M. Weiner, S. M. Durbin, J. Phys. Chem. B. 108 (2004) 10077. 74. S. Bone, R. Pethig, J. Mol. Biol. 157 (1982) 157; S. Bone, R. Pethig, J. Mol. Biol. 181 (1985) 323.

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8

Pharmaceutical and Security Applications of Terahertz Spectroscopy J. Axel Zeitler

University of Cambridge

Thomas Rades

University of Otago

Philip F. Taday TeraView Ltd.

Contents 8.1 Introduction................................................................................................... 299 8.2 Pharmaceutical Materials Setting.................................................................300 8.3 Applications of Terahertz Spectroscopy in Pharmaceutics.......................... 303 8.4 Security Applications.................................................................................... 313 8.5 Conclusions................................................................................................... 320 Acknowledgments................................................................................................... 320 References............................................................................................................... 320

8.1 Introduction The terahertz region of the electromagnetic spectrum is typically considered to occupy frequencies between 100 GHz and 4 THz (3 cm–1 to 130 cm–1; 3 mm to 75 µm); this bridges the gap between the microwave and millimeter region of the electromagnetic spectrum and infrared region. This region of the spectrum has been traditionally difficult to access, but recent developments in sources and detectors have seen a rapid commercialization of the technology. These advances have been accompanied by much interest in possible applications which has focused on the areas of pharmaceutical, security, and medical imaging. There are unique properties of terahertz radiation that make it a powerful technique in the pharmaceutical and security market sectors. One of these is that many materials are semitransparent to terahertz radiation; this allows it to penetrate many everyday physical barriers such as typical clothing and packing materials with modest attenuation. As we will show, many chemical substances, pharmaceuticals and 299

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explosive materials exhibit characteristic spectral responses in this frequency range. Because terahertz radiation has low photon energies, it is nonionizing, which is useful in a security application. From a pharmaceutical point of view, the average powers generated by the pulsed technique do not induce chemical or phase changes in the material under investigation. Although they are eventually absorbed by water vapor in the atmosphere, terahertz waves will propagate modest distances through normal atmospheric air, so that terahertz can be considered for standoff detection over distances of several tens of meters. In this chapter, we introduce the application into the pharmaceutical and security areas.

8.2 Pharmaceutical Materials Setting Most pharmaceuticals have to be administered at very low doses, usually in the milligram range, but frequently as little as a few micrograms are required. These small quantities necessitate the formulation of the drug in a pharmaceutical dosage form, sometimes also called a drug delivery system. The field within the pharmaceutical sciences that is concerned with the research and development of pharmaceutical dosage forms is called pharmaceutical formulation science, and the translation of these formulations to industrial products is studied in the field of pharmaceutical technology. The need to develop a dosage form that is convenient to administer, however, is only one reason for research and development work in formulation science and pharmaceutical technology. The three major aspects of importance in formulating and optimizing drug delivery systems are to: 1. Optimize the bioavailability of a drug, Bioavailability means that the drug actually reaches the target site in the body at sufficiently high concentrations. A rationally developed delivery system should help to improve the drug’s activity. The objective in formulation sciences is to develop the right delivery systems for a given drug.1 2. Optimize the stability of the drug and the dosage form—chemical, physical, and microbiologic stability in the dosage form and in the body. A drug that degrades within a few minutes, although active in a laboratory situation, cannot ever be approved by to a patient. Current drug development research in the pharmaceutical industry takes the issue of stability extremely seriously. For a drug to be developed into a viable medicine with a chance to successfully enter the marketplace, it has to be stable. The earlier in the drug discovery and development process stability problems are identified and tackled the faster and less costly is the development.2 3. Develop feasible formulation techniques. A large range of delivery systems that can be formulated in the laboratory have no chance to be produced in larger scale or at acceptable costs. The reality of drug development is that a resulting medicine must be affordable. This requires large-scale production using simple production processes. For any excipient (a pharmacologically

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inactive additive used to convert a drug into a dosage form) the regulatory bodies apply the same very stringent criteria as for the drug itself. The costs associated with the approval of a new excipient for human use are extremely high (in the order of tens to hundreds of millions of dollars). Therefore priority in most drug development work is given to approved excipients and technically feasible delivery systems. In the light of these important aspects of dosage form development it is no surprise that throughout the 20th century and today, the tablet is the dominating dosage form on the market. More than 70% of all prescription drugs are administered as tablets. Tablets have a range of advantages as dosage forms. They allow mass production at low costs, contain the same dose of drug in each unit, allow the formulation of practically all solid drug substances, and allow the control of drug release in the gastrointestinal tract. As a solid dosage form, they provide a high chemical stability of the drug and thus a long shelf life. Finally, administration of the dosage form to the patient is simple and convenient and thus patient compliance is high.3 A wide range of different tablet types have been developed to optimize drug delivery. Various types of tablets can be differentiated either according to their route of application or according to their formulation technology. By far the most often used route of application for tablets is the oral administration (i.e., tablets are administered through the mouth into the gastrointestinal tract). There the drug is released from the tablet and can be taken up into the body through the epithelial cells of the gastrointestinal tract. Technologically, a simple tablet is swallowed as a whole and then disintegrates into granules in the stomach. These granules further disintegrate into individual particles and from these the drug dissolves to be taken up into the body, either by passive diffusion or with the help or influx transporter proteins in the epithelial cells. Some tablets are dissolved before administration (e.g., effervescent tablets), some are administered as chewable tablets. Sandwich tablets consist of two or more layers containing different drugs or different drug formulations, allowing the release of drugs from the various layers at different kinetics. Finally, through the addition of polymers as coating materials (coated tablets) or dispersion in the tablet as a matrix (matrix tablets), drug release can be modified and controlled. For example, the release of a drug can be inhibited in the harsh acidic environment of the stomach, thus protecting the drug against chemical or enzymatic degradation. Other coating polymers lead to a zero-order drug release throughout the whole gastrointestinal tract, thus enabling longer dosing intervals for the patient (for example, once per day instead of three times per day). For coated tablets, the quality, thickness, and thickness distribution of the polymer coat are crucial to tablet performance. Tablets are made by mixing the solid drug with various excipients, such as fillers (as bulking agents), binders (to allow the powder particles to form granules), and disintegrants (to allow the disintegration of the tablet in the gastrointestinal tract). Usually the powder mixture cannot be compressed into a tablet as such, but has to

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be granulated before compression. Granulation is an agglomeration process in which the individual particles (drug and excipient) are made to adhere to form larger particles called granules. Granulation is usually required to prevent particle segregation (to allow for a high dose uniformity), to improve particle flow (to allow for a high production speed), and to improve the compression characteristics of the mixture (to allow the formation of tablets of sufficient mechanical strength to withstand packaging and transport stresses). During the granulation process, the drug may be exposed to granulation fluids (solvents of binder solutions), high compression forces, and elevated temperatures during the agglomeration process or drying of the granules. The granulation step is followed by the compression step in the tableting machine. In a modern rotary tablet press, thousands of tablets can be produced per minute. Successful compression requires the granules to be free flowing into a die between two punches (often glidants are added to the granules to improve the flow properties), to cohere on compression between the punches (often binders are also added to the granules to improve compression properties), and finally the formed tablet must not adhere onto the punches and die when it is ejected from the tableting machine (usually a lubricant is added to the granules to facilitate the ejection step). If the tablet is to be coated, the coating steps follow compression using specialized coating instrumentation.3 From this, it is clear that drugs today are usually administered orally as solid dosage forms. However, they can only act if the drug dissolves reasonably fast and at sufficient concentration in the gastrointestinal tract. It is also clear that, during their formulation to a solid dosage form (usually a tablet), drugs are exposed to solvents, heat, pressure, and a range of excipients. All of these factors may influence the physical and chemical stability of the drug. Indeed, the two most important problems in the development of solid drugs and solid dosage forms are drug solubility and crystalline polymorphism (i.e., the ability of a chemical compounds to crystallize in different crystalline lattices).4 Many drugs have low aqueous solubilities that may prevent their use as active ingredients in drug formulations, because of the resulting low bioavailability of the drug in the body. Increasing the solubility or the dissolution rate of a drug is therefore of highest significance in pharmaceutical formulation science and pharmaceutical technology. Several approaches can be taken to improve the solubility of drugs in the solid state. For example, drugs can be used in a thermodynamically metastable polymorphic form, because these often show a higher solubility and dissolution rate than the stable polymorphic form. It is also possible to convert the crystalline material into an amorphous form. Because this form is in a higher energetic state as the crystalline form, its solubility is usually also much higher. However, using thermodynamically metastable or unstable systems is not without its risks, because the drug may convert back to the stable form during the shelf life of the tablet. Therefore the formulation of amorphous solid solutions of drugs and polymers has been suggested as an alternative to a neat amorphous drug. Another possibility is to use drugs with very small particle sizes, either by subjecting larger particles to specific milling processes (micronization, nanomilling) or controlled precipitation. Although it is possible to produce small particles of a drug in the stable polymorphic form, due

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to the high specific surface area of a micrometer- or nanometer-sized particles, the chemical stability is often reduced and frequently secondary agglomeration occurs. Also the formation of amorphous compound during a milling process often cannot be avoided, leading to further stability problems.5 As stated previously, solubility and dissolution rate are influenced by polymorphism and degree of crystallinity. With the advent of combinatorial chemistry in the drug discovery process, drugs will become larger and contain more functional groups, allowing them to crystallize in even more different polymorphic forms and with even lower aqueous solubility. Thus the problem of poor solubility and crystalline polymorphism will become considerably larger in the future. To allow the rational development of dosage forms and reliable production in the industrial setting it is therefore of critical importance to be able to accurately distinguish and control the solid-state form of the drug present (e.g., in a tablet). Ideally, such determination should be fast, allowing at-line or even in-line measurements to improve efficiency and quality of the formulation and production processes. X-ray diffraction and thermal analytical techniques (such as differential scanning calorimetry) can be used to detect and to some extent to quantify crystalline or amorphous forms of a drug; however, these techniques are slow compared with spectroscopic techniques and cannot easily be used at-line or in-line. On the other hand, the information from near-infrared, mid-infrared, or Raman spectra is predominantly at the molecular level, whereas the properties that need to be monitored predominantly are at the intermolecular and lattice levels (i.e., polymorphic form or crystalline versus amorphous form). There is therefore a need for a spectroscopic technique monitoring the intermolecular level in pharmaceutical formulation and production technology.6–9 The ability to probe vibrational modes originating from the translations and liberations of drug molecules in their crystalline environment is very appealing in the pharmaceutical materials context because it directly relates to the solid-state properties.

8.3 Applications of Terahertz Spectroscopy in Pharmaceutics Although pharmaceutical applications of far-infrared spectroscopy before the advent of pulsed terahertz technology were limited to a few isolated studies,10–12 recent activity in the field has been substantial. A study of the characterization of different polymorphic modifications of ranitidine hydrochloride has set the scene for the use of terahertz technology in a pharmaceutical materials context.13 It was demonstrated that terahertz spectroscopy can be used to detect different polymorphic forms of the drug not only in the pure material but, more importantly, in the final tablet by a simple transmission experiment without the requirement of any sample preparation. A characterization in the terahertz range of different sugars used as excipients in drug formulation highlighted how the spectral features, originating from intermolecular interactions, of the materials investigated depend on the crystalline environment and that amorphous materials do not exhibit any distinct spectral features.14 The applicability of the technology to study the solid-state properties in drugs was further established by measuring the

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12 *

*

Absorbance (decadic)

Absorbance (decadic)

8 6 4

*

8 6 4 2

2 0

*

10

10

20 40 Wavenumber (cm–1) (a)

60

0

20

40

Wavenumber (b)

60

(cm–1)

Figure 8.1 Terahertz transmission spectra through samples of commercially available tablets without further sample preparation. The spectral features of carbamazepine form III are marked with an asterisk. Spectra are offset in baseline for clarity. (a) Tablets containing 200 mg carbamazepine; (b) tablets containing 400 mg carbamazepine.

terahertz spectra of a range of pharmaceutical drugs that are widely used in the field as model compounds for different solid-state properties.15 A range of other materials with potential pharmaceutical use has since been characterized.16–20 Terahertz radiation can propagate through most pharmaceutical dosage forms. Especially at lower frequencies between 10 and 60 cm–1 where the dynamic range of the pulsed terahertz spectrometers is largest, sufficient spectral information can be extracted to identify the active ingredient. Because the measurements are performed in transmission, the volume of the dosage form that is measured is much higher compared with a reflection measurement. For example, the spectral fingerprint of carbamazepine can be extracted from the transmission spectra of several tablets on the market (Figure 8.1). Here, tablets containing 200 mg (Figure 8.1a) and 400 mg (Figure 8.1b) of carbamazepine from different manufacturers were studied. Carbamazepine, a drug used for the treatment of epilepsy, is known to exist in at least four polymorphic modifications. As can be seen from the terahertz spectra, the excipients used by different manufacturers vary, whereas the spectrum of carbamazepine, the active ingredient, is similar in all tablets. The terahertz spectra reveal that all tablets contain the thermodynamically most stable polymorphic modification of carbamazepine form III. Some of the excipients are crystalline and exhibit unique spectral features, whereas the majority of the excipients used to formulate the tablets are amorphous and therefore do not contribute spectral at features at terahertz level. When performing terahertz spectroscopy with samples at very high concentrations

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without prior dilution of the sample material in a transparent matrix, such as polyethylene, it is important to be aware of spectral artifacts because the dynamic range of the instrument can be exceeded due to very strong absorption. In the pulsed terahertz spectrometers based on photoconductive emission and detection, the dynamic range decreases exponentially with increasing frequency.21 The spikes on some of the spectral features in Figure 8.1, especially at higher wavenumbers, are artifacts as the dynamic range is exceeded and are not real spectral features. To avoid saturation effects the sample material has to be diluted. Similar to midinfrared spectroscopy in which the sample material is mixed with KBr, in terahertz spectroscopy the sample is mixed with polyethylene or poly(tetrafluoroethylene) (PTFE), two materials that are almost transparent in the terahertz range. Typically a pellet is between 5 and 13 mm in diameter and contains between 5 and 40 mg of sample material, depending on its absorption coefficient. Depending on the desired spectral resolution, the pellets are preferably thicker than 2 mm to avoid the acquisition of multiple echo reflections in the terahertz waveform at longer delay times, which would lead to oscillations in the frequency domain, so-called etaloning. Figure 8.2 shows the terahertz spectra of three different polymorphic modifications of matazachlor and two different polymorphic forms of acetazolamide. The spectra were acquired after diluting 20 mg of the drug material with 360 mg polyethylene and compressing the mixture into a pellet of 13-mm diameter. In addition to transmission spectroscopy, in which the spectrum of a pharmaceutical tablet or a pellet containing the diluted sample material is recorded, terahertz spectra can also be acquired by attenuated total reflectance (ATR). Here, the sample is pressed tightly onto a crystal of a material with a very high refractive index, such 3.5

2.5 2.0

Form I 3.0 Absorbance (decadic)

Absorbance (decadic)

3.0

1.5 1.0

Form II

0.5 0.0

Form III 20

40

60

Wavenumber (cm–1) (a)

80

2.5 2.0 1.5 1.0 0.5 0.0

Form II Form I 20

40

60

80

Wavenumber (cm–1) (b)

100

Figure 8.2 Terahertz transmission spectra of a 20-mg sample material dispersed in 360 mg polyethylene. (a) Metazachlor; (b) acetazolamide.

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Terahertz Spectroscopy: Principles and Applications Evanescent Wave Sample

ATR Crystal

θ

Reﬂected Terahertz

Incident Terahertz

Figure 8.3 Schematic diagram of an attenuated total reflectance set-up.

1.5

2.0

Absorbance (decadic)

Absorbance (decadic)

as silicon. Terahertz radiation is passed through the crystal at an angle below the critical angle so that total internal reflection occurs at the surface, where the sample material in pressed against the crystal (Figure 8.3). At the interface between the crystal and the sample material, an evanescent terahertz electric field can interact with the sample and the spectrum of the sample can be extracted from the reflected radiation. ATR has the advantage that no sample preparation or dilution is necessary. The sample material is just pressed onto the ATR crystal and the spectrum can be recorded immediately. This set-up has the additional advantage that spectra can be acquired without the need to wait for the purge of the sample chamber with dry nitrogen gas to remove any atmospheric water vapor every time the sample is changed, because the instrument with the ATR unit only needs to be purged before starting the first experiment and then is kept under constant purge for the remainder of the

1.5 1.0 0.5 0.0

Monohydrate Anhydrate 20

40

60

80

Wavenumber (cm–1) (a)

1.0 0.5

Monohydrate Anhydrate

0.0 100

20

40

60

80

Wavenumber (cm–1)

100

(b)

Figure 8.4 (a) Transmission. (b) Attenuated total reflectance spectra of piroxicam normalized to maximum intensity.

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experiments. Figure 8.4 shows an example of spectra acquired from a sample pellet in transmission (Figure 8.4a) and from the same materials by ATR (Figure 8.4b). Although the peak positions for the different spectral features are the same in the transmission spectrum and the ATR spectrum, the relative intensities are different. This is because the penetration depth de of the effervescent electric field into the sample material in ATR is dependent on the wavelength of the radiation.  n 2n cos θ  λ d e ⊥ =  1 22 2  2  n1 n2  π n1 sin 2 θ - n22  n 2n cos θ  2n 2 sin 2 θ - n22 λ d e =  1 22 2  2 12 2 θ - n2 2 2 2 n n n n s in 1 2   1 2 2 π n1 sin θ - n2

(

)

(8.1)

where n1 and n2 are the respective refractive indices of the ATR crystal and the sample material, θ is the angle of the incident terahertz pulse and λ is the wavelength of the radiation. From Equation 8.1 follows that with increasing frequency the penetration depth is decreasing. Over the terahertz range the penetration depth varies over one order of magnitude between about hundreds of micrometers at 2 cm–1 to tens of micrometers at 100 cm–1. This difference in penetration depth causes the overpronunciation of the features at low wave numbers in Figure 8.4b. Further to the applications for the identification of different solid-state modifications, the quantitative ability of TPS (sometimes also referred to as THz-TDS for terahertz time-domain spectroscopy) for the determination of one polymorphic form in the presence of another was investigated.22 The quantification of traces of a different polymorphic modification of a drug is very important for quality control during manufacturing. Phase transformations of part of the product can be induced by the processing steps as outlined previously.5 The polymorphic impurities can then act

0.20

Absorbance (decadic)

Absorbance (decadic)

0.25 0.15 0.10 0.05 0.00

–0.05

44

46

Wavenumber (cm–1) (a)

48

0.20 0.15 0.10 0.05 0.00 –0.05

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Lactose Content per Pellet (wt%) (b)

Figure 8.5 (a) Detail of the absorption feature of lactose α-monohydrate at 46 cm–1 for pellets containing a 0- to 5-mg sample material. (b) Linear fit (dashed line) of the absorbance reading against the amount of lactose α-monohydrate. The thin black lines mark the 95% confidence interval.

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as seeds for the phase transition in the bulk of the dosage form, compromising the quality over the shelf life of the product or even during the production process. New quantitative techniques with fast acquisition speeds, such as terahertz spectroscopy, probing the properties of the crystal form therefore are very useful tools. The sensitivity in the terahertz range is very high compared with other techniques employed for the quantification of the solid-state properties of drugs.23–26 In a quantitative study of polymorphic mixtures, the detection limits for the discrimination of one form from another were found to be in the range of 1 to 5 percent.22 The quantitative performance of terahertz spectroscopy has also been demonstrated for amino acids27 and enantiomers.28 TPS has an excellent sensitivity for the detection of minute amounts of crystalline solid in a transparent amorphous matrix. In Figure 8.5a, the spectral feature of lactose α-monohydrate at 46 cm–1 is highlighted over a range of very low concentrations. The signal response is linear to a very small concentration of lactose (Figure 8.5b). The low-frequency spectral features of sulfathiazole, a very complex polymorphic system with at least five polymorphic forms, of which three are very difficult to distinguish, was characterized by terahertz spectroscopy and low-frequency Raman spectroscopy.29 With both techniques, it was possible to readily distinguish between the different forms. The speed of the spectral acquisition in terahertz pulsed spectroscopy (TPS) allowed to follow the dynamics of the phase transitions between the different forms in situ while the lower melting modifications were heated to their melting temperatures. Although the thermodynamic properties of drugs are usually well characterized, the mechanism and kinetics of the phase transitions in different polymorphic forms often are very complex. Terahertz spectroscopy is a very promising approach to directly study the transformation process and the mechanisms behind such a phase transition.30,31 Yet, to fully exploit the spectral information in the terahertz range, two future developments are essential: first, the development of methodologies to readily assign the spectral features to molecular/crystal properties; and second, the development of instruments with much higher bandwidths than the current generation of pulsed spectrometers, which are limited to frequencies up to 4 THz. Over the last year, several groups have developed different approaches to assign the terahertz spectral features.32–37 It is commonly acknowledged that for meaningful calculations the solid-state environment has to be included into the calculation of the electronic structures. Calculations of the normal modes in isolated molecules are of very limited use for the interpretation of terahertz spectra. However, peak assignments in the terahertz range are still strongly limited by the computation time and costs and cannot be performed on a routine basis. With hydrate forms, another very important class of solid-state modifications in the pharmaceutical field has been studied. Hydration and dehydration are very common processes during the manufacture of a pharmaceutical dosage form—for example, in granulation and drying processes—and have a direct impact on the solid-state properties of many drugs.5,38 Both the hydrate forms and the dehydration processes have been studied by terahertz spectroscopy.39,40 Different hydrate forms can be identified in principle in the same way as different polymorphic forms. However, the study of the dehydration process is not as straightforward compared with the solid-state

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transformations observed in polymorphic modifications. In the case of hydrates, the water molecules that are confined in the crystal structure in the hydrate form will leave the crystal after reaching its dehydration temperature. This is true for all different hydrate forms, even though there are differences in the mechanisms of this process depending on whether the water molecules form channels in the crystal structure or not.41 However, all experiments that have been published to date using terahertz spectroscopy for the analysis of dehydration processes use transmission set-ups in which the hydrate form is either mixed with a transparent, hydrophobic, filling material, or is compacted directly to form a pellet. The dehydration process is strongly influenced by the sample preparation. It was found that the observations by terahertz spectroscopy are different to the reported results obtained from samples in powder form.40 Although the phase transition of the dehydration process is mainly driven by the thermodynamics and kinetics of the solid-state form itself, the step of the hydrate water leaving the crystal structure is governed by the way the sample pellet is prepared. To illustrate this effect, we show the terahertz spectra acquired in situ during a dehydration process under nonisothermal conditions, where the hydrate form is either (1) dispersed in a PTFE matrix without changing its initial needle-like particle morphology, (2) gently ground into small particles and then dispersed into the PTFE matrix, or (3) compressed in a separate layer on top of a PTFE pellet. The PTFE is necessary as a bulking material because the hydrate form itself cannot be used undiluted to form pellets of sufficient mechanical strength. On heating, at around 360 K, a phase transition can be observed in the theorphylline monohydrate pellets independent of the sample preparation (Figure 8.6a–c). From the spectra, it becomes evident that the evaporation of the hydrate water is strongly dependent on the matrix environment and shows how diffusion out of the pellet is possible (Figure 8.6). In the case of the hydrate particles with needle-like morphology, the molecules can diffuse out of the pellet along channels formed by the crystals. The majority of the hydrate water evaporates in one step from the pellet and can be detected by its distinct spectral signature (black dots around 370 K in top plot of Figure 8.6. However, some evaporation is already detected below the temperature of the phase transition in the solid form and after the evaporation “burst,” although the additional concentrations of water vapor are very weak. In the case of the sample with the dispersed small particles of the hydrate, no evaporation is observed below the transition temperature and, rather than a distinct period of high evaporation rate, the spectral signature of the water vapor is only observed at much higher temperatures and with less intensity and in a more continuous fashion (Figure 8.6, middle). For the third pellet with the theophylline monohydrate in the surface layer, weak water vapor signatures can be detected well below the transition temperature. The intensity of the vapor signals remains constant during further heating until about 400 K. At higher temperatures the rotational transitions at 83 cm–1 in the spectrum of water vapor suddenly become very intense (Figure 8.6, bottom), possibly because of a strong interaction as the water molecules evaporate from the surface of the sample pellet. For a conclusive discussion of these observations, further experimental and theoretical work is required. The examples are included into this chapter solely to increase the awareness of such processes when experiments with polycrystalline powder samples in compacted pellets are performed.

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507

(a)

Temperature (K)

485 462 437 414 389 364 338 313 10 20 30 40 50 60 70 80 90 100 110 120 Absorbance (decadic) Wavenumber (cm–1) 0 0.5 1.1 1.6 513

(b)

Temperature (K)

487 464 440 414 389 364 339 313 10 20 30 40 50 60 70 80 90 100 110 120 Absorbance (decadic) Wavenumber (cm–1) 0 0.5 1.1 1.6 518

(c)

Temperature (K)

493 465 440 415 389 364 339 313 10 20 30 40 50 60 70 80 90 100 110 120 Absorbance (decadic) Wavenumber (cm–1) 0 0.5 1.1 1.6

Figure 8.6 Contour plot of terahertz spectra acquired in situ during the dehydration of 15 mg theophylline monohydrate, depending on the sample pellet preparation. Top: Crystals of needle-like morphology dispersed in poly(tetrafluoroethylene) (PTFE). Middle: Gently ground crystals dispersed in PTFE. Bottom: Theophylline monohydrate crystals compressed into a separate layer onto the PTFE pellet.

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The overall process of dehydration can involve the collapse of the crystal structure leading to an amorphous form from which the anhydrate crystallizes again. Such processes will be very difficult to study in a compacted pellet especially if the drug material is embedded in a hydrophobic matrix. The development of different sampling techniques, such as diffuse reflectance, where the sample can be used in its powder form is highly desirable. However, diffuse reflectance experiments will be difficult to perform with pulsed terahertz radiation. Nevertheless, it is possible for some compounds to measure the dehydration kinetics from a hydrate in its compacted form.39 As an example of a process that can be studied by TPS in an actual pharmaceutical formulation—and where the investigation of a compacted sample is highly desirable—we show how the changes in the solid-state properties of a tablet can be followed. A tablet containing 80 mg lactose α-monohydrate as a filler, magnesium stearate as a lubricant, and 200 mg carbamazepine as the active ingredient is prepared and measured directly without further sample preparation by TPS in transmission. For the formulation of the sample tablet, we use the commercially available form III of carbamazepine, which is the thermodynamically stable form used in the tablets on the market. After heating, form III converts into form I at a transition temperature of about 443 K. Both forms have very different spectra in the terahertz range (Figure 8.7a). Lactose α-monohydrate is one of the most widely used excipients in the pharmaceutical industry. It exhibits very good compaction properties and is widely used as a filling material for tablet formulation as its drug–powder mixtures can sometimes be used without a prior granulation step. The monohydrate form is very stable at ambient conditions. Figure 8.7b shows its terahertz spectrum at room temperature. The terahertz transmission of the whole tablet shows the spectral features of carbamazepine form III and lactose α-monohydrate (Figure 8.8). Magnesium stearate is used as a lubricant and, typically, less than 1 mg would be distributed in the tablet. It does not contribute to the spectral features of the tablet. The tablet is transparent enough for terahertz radiation to allow spectral acquisition in transmission without exceeding the dynamic range of the spectrometer.

Absorbance (decadic)

Absorbance (decadic)

2.0 1.5 1.0 0.5 0.0

20

40

60

80

100

Wavenumber (cm–1) (a)

120

0.8 0.6 0.4 0.2 0.0

20

40

60

80

100

Wavenumber (cm–1) (b)

120

Figure 8.7 Terahertz spectrum of (a) lactose α-monohydrate and (b) carbamazepine form III (solid line) and form I (dashed line) monohydrate form at room temperature.

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Absorbance (decadic)

4.0

3

3.5 3.0

2

II

2.5

4

2.0 1.5

I

1.0

1

0.5 0.0

10

20

30

40

50

60

70

Wavenumber (cm–1)

80

Figure 8.8 Terahertz spectrum of the whole tablet. Roman numbers refer to spectral features of the lactose α-monohydrate and Arabic numbers refer to spectral features of carbamazepine form III.

After heating the sample from room temperature using a variable temperature cell at a rate of 5 K/min, a general red shift, broadening, and decrease in absorbance of the spectral features can be observed (Figure 8.9). From about 400 K onward, the spectral features I and II of lactose α-monohydrate start to rapidly decrease in intensity, whereas features 1, 2, and 3 of carbamazepine form III continue their slight intensity decrease and red shift. This decrease of the lactose peaks is associated with the dehydration of the monohydrate to its anhydrous form, which leads to 303 K

Temperature (K)

4

5

6

443 K 423 K 403 K

293 K

I

10 20 Absorbance (decadic) 0

1

2 30

II 40

Wavenumber

(cm–1)

50

3 60

70

1.3 2.5 3.8

Figure 8.9. Contour plot of the terahertz spectra acquired during heating of a carbamazepine tablet.

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a collapse of the crystal structure into a disordered phase. At 423 K the lactose has become fully amorphous, whereas carbamazepine is still in its form III crystal structure. Further heating to 443 K, however, leads to the phase transition from form III to form I, as reported previously.30 The spectral features of carbamazepine form I increase in intensity and shift to the blue while cooling the sample back to room temperature. There is no indication for recrystallization of the amorphous lactose during the cooling. This example highlights two advantages of TPS: first, it is possible to study processes directly in the final dosage form where the conditions can be very different from the situation in a powder mixture because of compaction; and second, transmission through a tablet is usually possible, except for unusually big specimens, allowing the sampling of the volume within the tablet rather than being restricted to spectroscopic information from the surface of the tablet only as would be the case in a reflection experiment. In the disordered or amorphous state, terahertz spectra are featureless compared with the crystalline state.14,15 The absorption coefficient of such materials increases with frequency and the refractive index is significantly higher than in the crystalline phase. The lack of long-range order and symmetry in an amorphous phase leads to the disappearance of the vibrational modes, originating from intermolecular interactions, which can be observed for crystalline samples. After heating amorphous material slowly from the glassy state through its glass transition temperature, slight changes in the absorption coefficient can be observed as the material relaxes. After the glass transition, which can be detected in the terahertz spectra, the crystallization process can be studied.42

8.4 Security Applications Events around the world have shown a requirement for heightened security in public places such as airports, subway, and other mass-transport systems. The long security lines (taking several hours to clear the security gate) at airports are now a daily occurrence for all travelers. This has driven the need for quick, safe, and reliable methods for personal screening. The methods must generate very low levels of false positives, which are very costly in time for the security agents at airport gates. There is no one screening technique that fulfills all the current requirements; in fact, some techniques give rise to health and privacy concerns. We have investigated the terahertz spectra of many of the common explosives and will discuss some of the challenges to be overcome in using the technology in practical security applications. The terahertz absorption spectra that are presented here were acquired using a TPS spectra 1000 transmission system (Teraview, Cambridge, UK). Samples of pure explosives and other materials were diluted to 5% (w/w) with polyethyplene powder. The particle size was kept to below 80 µm to reduce scattering effects. Discs of the mixture were prepared by pressing 420 mg of sample at 2 tons in a 13-mm diameter die. A reference sample of 400 mg of pure polyethylene was prepared. The thickness of the discs was about 4 mm. Sample preparation occurred about 24 hours before measurement to allow the disc to relax to its final thickness. The commercial plastic explosives were recorded without any additional sample preparation. Each rapid-scan

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spectrum, the average of 1,800 co-added scans, took 1 minute to record. For all measurements, the sample compartment of the spectrometer was purged with dry nitrogen at 10 L/minute to minimize absorption by atmospheric water vapor. Blackman-Harris three-term apodization, commonly used in Fourier transform infrared spectroscopy, was applied to the time domain waveform. The absorption spectra and refractive index of each material were reported. The terahertz absorption and refractive index spectra of (2,4,6)-trinitrotoluene, cyclo-1,3,5-trimethylene (RDX), pentaerythritol tetranitrate (PETN), and cyclotetramethylene tetranitramine (HMX) are shown in Figure 8.10. Also shown are the spectra for two commercial plastics explosives, SX-2 and Semtex-H. The commercial plastic explosives are a blend of RDX and PETN together with a plasticizer to allow for easy use of the material. Table 8.1 lists the main peaks observed in the spectra shown in Figure 8.10. RDX may have different impurities depending on the source. The absorption spectrum of PETN has been recently been modeled by Allis and Korter.37 The symmetry group for the crystalline form of PETN is S4. As has been pointed out previously in this chapter, the normal mode calculations of isolated molecules are of limited when use in the solid state; therefore, consideration must also be given to electronic structure of the crystalline state. Figure 8.11 (top) shows the comparison between isolated molecule and the observed spectrum for PETN. The calculated stick spectrum shows (Figure 8.11, top) the eleven lowest frequencies as calculated by density functional theory for the isolated molecule and their descriptions (Table 8.2). There is no agreement between the observed spectrum and this calculated isolated molecule spectrum. Figure 8.11 (middle and bottom) takes into account the boundary conditions that apply in a solid-state crystal. Figure 8.11 (middle) shows the intensity calculation with a Hirshfeld dipole intensity,43 whereas the bottom shows the calculated spectrum with a Mulliken dipole intensity44 giving a much better fit to the observed spectrum. Allis and Korter37 are now able to give some assignments to the observed bands, which are reproduced in Table 8.3. The bands labeled a–e in Figure 8.11 are in effect large-scale crystalline vibrations. In PETN, there are nine possible external modes (three transitional and six rotational modes). Bands labeled f–s have been assigned as isolated molecule vibrations, but shifted to higher frequencies because of packing interactions in the solid state. Cady and Larson have identified two polymorphs of PETN.45 The spectra and assignments reported here and elsewhere37 refer to the stable form I. HMX, on the other hand, has four known polymorphs, given the labels α, β, γ, and δ. The β-form of HMX is the most stable and is the form reported here and elsewhere.34 Most of the features observed in the absorption spectrum of β-HMX can either be assigned low-frequency optical translation or rotational external modes in much the same way as PETN. There are also low-lying combination bands from the isolated molecule spectrum; these are shifted to higher frequency because of solid-state interactions. No other assignments have been completed for any other materials. The commercial plastic explosives are mixtures of PETN and RDX in different portions. The strong RDX peak at 27 cm–1 can clearly be resolved in the Semtex-H and SX-2 shown in Figure 8.10. This low-frequency peak is extremely interesting because it provides identification of RDX-based explosive through many common clothing materials. Figure 8.12 shows the terahertz absorption spectra for a range of

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1.8 n

20

250

10

Refractive Index

1.7

ext.

Extinction/cm–1

Extinction/cm–1

Frequency/THz 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 2.4 300

2.1

200

2.1

100

2.0

50

0 20

30 40 50 Wavenumber/cm–1

60

0

70

n

1.8 10 20 30 40 50 60 70 80 90 100 110 120 (b)

Frequency/THz 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 2.0 200

Frequency/THz 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6 2.0

1.9

1.7 1.6

100

1.5 1.4

50

150

1.6 n

1.3 0

1.1 10 20 30 40 50 60 70 80 90 100 110 120 Wavenumber/cm–1

(d)

Frequency/THz 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3

Frequency/THz 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 50 2.0 1.8

ext. 1.7

40 n

1.6

40 Extinction/cm–1

50

Refractive Index

60 Extinction/cm–1

1.2 10 20 30 40 50 60 70 80 90 100 110 120 Wavenumber/cm–1

(c)

20

1.9 ext.

30

1.8

20 1.7 10

10

n

1.5

1.6

0

0 10 20 30 40 50 60 70 80 90 100 110 Wavenumber/cm–1 (e)

1.5 1.4

50

1.2

30

1.9

1.7

100

1.3

n

ext.

1.8 Extinction/cm–1

150

200

1.8 Refractive Index

Extinction/cm–1

ext.

70

1.9

Wavenumber/cm–1

(a)

0

2.2

150

1.6

10

2.3 ext.

Refractive Index

Frequency/THz 0.9 1.2 1.5

Refractive Index

0.6

Refractive Index

0.3

10 20 30 40 50 60 70 80 90 100 110 Wavenumber/cm–1 (f)

Figure 8.10 Terahertz absorption and refractive spectra of explosives. (a) TNT, (b) βHMX, (c) form I of PETN, (d) RDX, (e) Semtex-H, and (f) SX-2.

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3, 4 8, 9 2

10, 11

5

PETN THz Exp. Mulliken Dipole Intensity (Left) Calculated IR Intensity (Middle) Hirshfeld Dipole Intensity (Right)

6

Intensity

PETN THz Exp. Hirshfeld Dipole Intensity

c, d, e PETN THz Exp. Mulliken Dipole Intensity

l, m, n a, b

f, g h

0

20

40

60

80

i, j

Wavenumber/cm–1

k

o, p q

100

r, s 120

Figure 8.11 Calculated terahertz spectrum of the explosive of form 1 of PETN and comparison with the observed room temperature spectrum (solid black line). The isolated-molecule (top) and solid-state VWN-BP/DNP normal modes (0-120 cm-1) with solid-state intensities provided by the Hirshfield (middle) and Mulliken (bottom) difference-dipole results. (Reprinted with permission from Allis et al. (2006). Copyright 2006 Wiley)

clothing materials. All materials show increased absorption with respect to increasing wave number; this is possible because of increased scattering from the clothing fibers. This highlights that lower terahertz frequencies below 1 THz, may be useful in standoff detection.

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Table 8.1 Terahertz Absorption Band Positions for Explosives Sample

Band Positions (cm–1) Pure Explosive Ingredients

RDX

27, 35, 46, 51, 66, 74, 103

PETN (form 1)

67, 72, 96

b-HMX

59, 82, 95

TNT

53, 73, 127 Commercial Plastic Explosives

Semtex-H

27, 35, 46, 51, 68, 100

SX-22

27, 35, 46, 51, 66, 74, 102

Table 8.2 Lowest Vibrational Frequencies in Isolated Molecules for PETN (form 1) and β-HMX Mode Number

Frequency/cm–1

1

21.7

Description

Reference

Molecule = PETN (form I) ONO2 twisting motions about C2-O1

37

2

22.3

ONO2 twisting motions about C2-O

37

3

37.3

C-ONO2 twisting motions about C1-C2

37

4

37.3

C-ONO2 twisting motions about C1-C2

37

5

43.9

ONO2 twisting motions about C2-O1

37

6

50.0

C-C-O bending motions into center

37

7

53.6

ONO2 twisting motions about C2-O1

37

8

57.0

C-ONO2 twisting motions about C1-C2

37

9

57.0

C-ONO2 twisting motions about C1-C2

37

10

118.3

C framework rotation within fixed N atoms

37

11

118.3

C framework rotation within fixed N atoms

37

Molecule = b-HMX

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1

21.2

Ring twisting and N-NO2 pendulum motion

34

2

44.3

Ring twisting and N-NO2 pendulum motion

34

3

56.3

Symmetric ring twisting motions

34

4

60.6

Vertical NO2 and ring stretches

34

5

62.6

Asymmetric ring twisting motions

34

6

79.9

Symmetric vertical NO2 rotations

34

7

82.4

Asymmetric vertical NO2 rotations

34

8

113.8

Symmetric horizontal NO2 rotations

34

9

119.3

Asymmetric horizontal NO2 rotations

34

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Table 8.3 Assignment of Room Temperature Terahertz Absorption of PETN (form 1) in 20- to 120-cm–1 Region34 Mode Number*

Observed Wave Number cm ~1

Calculated Wave Number/cm–1

Assignment #N

a

39.4

OT

b

39.7

OT

c

60.4

OR1

60.7

OR2

d

67

e

62.2

OR3

73.0

Out of phase (3,4)

g

73.1

Out of phase (3,4)

h

81.4

6+6

f

72

i

85.3

Out of phase (3,4)

j

85.4

Out of phase (3,4)

k

93.2

6–6

l m

96

97.6

In phase (3,4)

97.6

In phase (3,4)

n

97.8

2+2

o

110.6

In phase (8,9)

p

100.8

In phase (8,9)

q

105.0

5+5

r

111.2

Out of phase (8,9)

s

111.4

Out of phase (8,9)

Note: OT, optical translation external mode; OR, optical rotational external mode. Isolated molecule modes from Table 8.2. *See Figure 8.11.

Clothing is one obstacle to standoff detection; another is atmospheric water vapor absorption. Because of the large dipole moment of the water molecule, there are strong rotational lines in the 5 to 120 cm–1 region of the spectrum. Figure 8.13 shows the terahertz absorption spectrum of water vapor; this was measured over a 50-cm path length with a relative humidity of 40%. Shown for comparison is the terahertz absorption spectrum of RDX. There are clearly windows within the water vapor spectrum that allow for detection at distance, maybe over a path length of a few meters, of threat materials but this depends on the relative humidity on the day of measurement. Other challenges beyond the subject of this chapter include the following: are there signatures from confusion materials (i.e., pharmaceuticals) that interfere with the explosive spectrum? Can a catalog of materials be constructed for the terahertz region? To this point, we have considered only transmission spectra of material, this will be impractical for a stand-off system. Therefore, can signals be detected in reflection?

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2.4

3.0

80

Cotton Wool Silk PVC Suede Polyester Nylon Leather

7 6 5 4

70 60 50 40

3

30

2

20

1

10

0

20

40

60

Wavenumber/cm–1

80

100

Attenuation per Layer (dB)

8

Absorbance (decadic)

Frequency/THz 1.2 1.8

0.6

319

0

Figure 8.12 Terahertz absorption spectra of common clothing materials.

0.25

Frequency/THz

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

10

20

30

40

50

60

70

80

90

100

Absorption

0.20

0.15

0.10

0.05

0.00 Wavenumber/cm–1

Figure 8.13 Terahertz absorption spectrum of water vapor, the sharp spectrum, measured over a 50-cm path length and a relative humidity of 40%. For comparison, the absorption spectrum of RDX is shown.

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Some proof-of-concept measurements undertaken by TeraView Limited have shown that it is possible to measure the specular reflection from the surface of a piece of explosive. These reflections are governed by the refractive index media and careful analysis is required. Can diffuse reflection be used? Over what distance can a material be detected? Ten meters would be ideal, but this depends on source power and detector sensitivity. This leads to designing practical systems—can a pulsed or continuous wave system be constructed?

8.5 Conclusions We have introduced the application of terahertz pulsed spectroscopy into two very diverse areas—pharmaceutical and security industries. We have discussed tentative assignments of the spectra obtained by commercial devices. Although the pulsed terahertz techniques are embolic, they show much promise in these two important commercial areas.

Acknowledgments The authors gratefully acknowledge Terry Threlfall, University of Southampton, for providing the samples of metazachlor and acetazolamide and Clare Strachan, University of Helsinki, for the samples of piroxicam and theophylline, Thomas Lo for measuring the spectra of explosives and David Newnham for measuring the spectra of common materials.

References 1. Hoehner, B. and Benet, L. Z., Factors influencing drug absorption and drug availability, in Modern Pharmaceutics Banker, G. S. and Rhodes, C. T. Marcel Dekker, New York, 2002, 93–118. 2. Guillory, J. K. and Poust, R. L., Chemical kinetics and drug stability, in Modern Pharmaceutics, Banker, G. S. and Rhodes, C. T. Marcel Dekker, New York, 2002, 139–166. 3. Kottke, M. J. and Rudnic, E. M., Tablet dosage forms, in Modern Pharmaceutics, Banker, G. S. and Rhodes, C. T. Marcel Dekker, New York, 2002, 287–334. 4. Brittain, H. G., Polymorphism in pharmaceutical solids, Marcel Dekker, New York, 1999, 427. 5. Govindarajan, R. and Suryanarayanan, R., Processing-induced phase transformations and their implications on pharmaceutical product quality, in Polymorphism, Hilfiker, R. Wiley-VCH, Weinheim, Germany, 2006, 333–364. 6. Craig, D. Q. M., Characterization of polymorphic systems using thermal analysis, in Polymorphism, Hilfiker, R. Wiley-VCH, Weinheim, Germany, 2006, 43–80. 7. Lubach, J. W. and Munson, E. J., Solid-state NMR spectroscopy, in Polymorphism, Hilfiker, R. Wiley-VCH, Weinheim, Germany, 2006, 81–94. 8. Chalmers, J. M. and Dent, G., Vibrational spectroscopic methods in pharmaceutical solid-state characterization, in Polymorphism, Hilfiker, R. Wiley-VCH, Weinheim, Germany, 2006, 95–138. 9. Ochsenbein, P. and Schenk, K. J., Crystallography for polymorphs, in Polymorphism, Hilfiker, R. Wiley-VCH, Weinheim, Germany, 2006, 139–166. 10. Klonoff, D. C., Noninvasive blood glucose monitoring, Diabetes Care 20 (3), 433–437, 1997.

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11. Lozano, C. M., Cox, O., Muir, M. M., Morales, J. D., Rodriguez-Caban, J. L., Vivas-Mejia, P. E., and Gonzalez, F. A., Cytotoxic anionic tribromo platinum(II) complexes containing benzothiazole and benzoxazole donors: synthesis, characterization, and structure-activity correlation, Inorganica Chimica Acta 271 (1–2), 137– 144, 1998. 12. Miller, L. M., Smith, G. D., and Carr, G. L., Synchrotron-based biological microspectroscopy: From the mid-infrared through the far-infrared regimes, J. Biol. Phys. 29 (2–3), 219–230, 2003. 13. Taday, P. F., Bradley, I. V., Arnone, D. D., and Pepper, M., Using terahertz pulse spectroscopy to study the crystalline structure of a drug: a case study of the polymorphs of ranitidine hydrochloride, J. Pharm. Sci. 92 (4), 831–838, 2003. 14. Walther, M., Fischer, B. M., and Jepsen, P. U., Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared, Chem. Phys. 288 (2–3), 261–268, 2003. 15. Strachan, C. J., Rades, T., Newnham, D. A., Gordon, K. C., Pepper, M., and Taday, P. F., Using terahertz pulsed spectroscopy to study crystallinity of pharmaceutical materials, Chem. Phys. Lett. 390 (1-3), 20–24, 2004. 16. Walther, M., Plochocka, P., Fischer, B., Helm, H., and Jepsen, P. U., Collective vibrational modes in biological molecules investigated by terahertz time-domain spectroscopy, Biopolymers 67 (4-5), 310–313, 2002. 17. Upadhya, P. C., Shen, Y. C., Davies, A. G., and Linfield, E. H., Terahertz time-domain spectroscopy of glucose and uric acid, J. Biol. Phys. 29 (2-3), 117–121, 2003. 18. Upadhya, P. C., Shen, Y. C., Davies, A. G., and Linfield, E. H., Far-infrared vibrational modes of polycrystalline saccharides, Vib. Spectrosc. 35 (1–2), 139–143, 2004. 19. Fischer, B., Hoffmann, M., Helm, H., Modjesch, G., and Jepsen, P. U., Chemical recognition in terahertz time-domain spectroscopy and imaging, Semicond. Sci. Technol. 20 (7), S246-S253, 2005. 20. Ning, L., Jingling, S., Jinhai, S., laishun, L., Xiaoyu, X., Meihong, L., and Yan, J., Study on the THz spectrum of methamphetamine, Opt. Express 13 (18), 6750–6755, 2005. 21. Jepsen, P. U. and Fischer, B. M., Dynamic range in terahertz time-domain transmission and reflection spectroscopy, Opt. Lett. 30 (1), 29–31, 2005. 22. Strachan, C. J., Taday, P. F., Newnham, D. A., Gordon, K. C., Zeitler, J. A., Pepper, M., and Rades, T., Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity, J. Pharm. Sci. 94 (4), 837–846, 2005. 23. Shah, B., Kakumanu, V. K., and Bansal, A. K., Analytical techniques for quantification of amorphous/crystalline phases in pharmaceutical solids, J. Pharm. Sci. 95 (8), 1641–1665, 2006. 24. Bugay, D. E., Newman, A. W., and Findlay, W. P., Quantitation of cefepime 2HCl dihydrate in cefepime 2HCl monohydrate by diffuse reflectance IR and powder x-ray diffraction techniques, J. Pharm. Biomed. Anal. 15 (1), 49–61, 1996. 25. Byard, S. J., Jackson, S. L., Smail, A., Bauer, M., and Apperley, D. C., Studies on the crystallinity of a pharmaceutical development drug substance, J. Pharm. Sci. 94 (6), 1321–1335, 2005. 26. Mackin, L., Zanon, R., Park, J. M., Foster, K., Opalenik, H., and Demonte, M., Quantification of low levels (<10%) of amorphous content in micronised active batches using dynamic vapour sorption and isothermal microcalorimetry, Int. J. Pharm. 231 (2), 227–236, 2002. 27. Ueno, Y., Rungsawang, R., Tomita, I., and Ajito, K., Quantitative measurements of amino acids by terahertz time-domain transmission spectroscopy, Anal. Chem. 78 (15), 5424–5428, 2006.

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28. Ji, T., Zhao, H. W., Zhang, Z. Y., Ge, M., Wang, W. F., Yu, X. H., and Xu, H. J., Terahertz time-domain spectroscopy of D-, L-, and DL-penicillamines, Acta Phys.Chim. Sin. 22 (9), 1159–1162, 2006. 29. Zeitler, J. A., Newnham, D. A., Taday, P. F., Threlfall, T. L., Lancaster, R. W., Berg, R. W., Strachan, C. J., Pepper, M., Gordon, K. C., and Rades, T., Characterization of temperature induced phase transitions in the five polymorphic forms of sulfathiazole by terahertz pulsed spectroscopy and differential scanning calorimetry, J. Pharm. Sci. 95 (11), 2486–2498, 2006. 30. Zeitler, J. A., Newnham, D. A., Taday, P. F., Strachan, C. J., Pepper, M., Gordon, K. C., and Rades, T., Temperature dependent terahertz pulsed spectroscopy of carbamazepine, Thermochim. Acta 436 (1-2), 71–77, 2005. 31. Upadhya, P., Nguyen, K., Shen, Y., Obradovic, J., Fukushige, K., Griffiths, R., Gladden, L., Davies, A., and Linfield, E., Characterization of crystalline phase transformations in theophylline by time-domain terahertz spectroscopy, Spectr. Lett. 39 (3), 215–224, 2006. 32. Korter, T. M., Balu, R., Campbell, M. B., Beard, M. C., Gregurick, S. K., and Heilweil, E. J., Terahertz spectroscopy of solid serine and cysteine, Chem. Phys. Lett. 418 (1-3), 65–70, 2006. 33. Day, G. M., Zeitler, J. A., Jones, W., Rades, T., and Taday, P. F., Understanding the influence of polymorphism on phonon spectra: lattice dynamics calculations and terahertz spectroscopy of carbamazepine, J. Phys. Chem. B 110 (1), 447–456, 2006. 34. Allis, D. G., Prokhorova, D. A., and Korter, T. M., Solid-state modeling of the terahertz spectrum of the high explosive HMX, J. Phys. Chem. A 110 (5), 1951–1959, 2006. 35. Saito, S., Inerbaev, T. M., Mizuseki, H., Igarashi, N., and Kawazoe, Y., Terahertz vibrational modes of crystalline salicylic acid by numerical model using periodic density functional theory, Jpn. J. Appl. Phys. Part 1—Regul. Pap. Brief Commun. Rev. Pap. 45 (5A), 4170–4175, 2006. 36. Saito, S., Inerbaev, T. M., Mizuseki, H., Igarashi, N., Note, R., and Kawazoe, Y., Terahertz phonon modes of an intermolecular network of hydrogen bonds in an anhydrous beta-D-glucopyranose crystal, Chem. Phys. Lett. 423 (4-6), 439–444, 2006. 37. Allis, D. G. and Korter, T. M., Theoretical analysis of the terahertz spectrum of the high explosive PETN, ChemPhysChem 7 (11), 2398–2408, 2006. 38. Griesser, U. J., The importance of solvates, in Polymorphism, Hilfiker, R. WileyVCH, Weinheim, Germany, 2006, 211–234. 39. Liu, H. B. and Zhang, X. C., Dehydration kinetics of D-glucose monohydrate studied using THz time-domain spectroscopy, Chem. Phys. Lett. 429 (1-3), 229–233, 2006. 40. Zeitler, J. A., Kogermann, K., Rantanen, J., Rades, T., Taday, P. F., Pepper, M., Aaltonen, J., and Strachan, C. J., Drug hydrate systems and dehydration processes studied by terahertz pulsed spectroscopy, Int. J. Pharm. 344(1–2), 78–84, (2007). 41. Morris, K. R., Structural aspects of hydrates and solvates, in Polymorphism in pharmaceutical solids, Brittain, H. G. Marcel Dekker, New York, 1999, 126–79. 42. Zeitler, J. A., Taday, P. F., Pepper, M., and Rades, T., Relaxation and crystallization of amorphous carbamazepine studied by terahertz pulsed spectroscopy, J. Pharm. Sci. 96(10), 2703–2709, 2007. 43. Hirshfeld, F. L., Bonded-atom fragments for describing molecular charge densities Theor. Chim. Acta, 44, 129–138, 1977. 44. Mulliken, R. S., Electronic population analysis on LCAD-MO molecular wave- functions. I. J. Chem. Phys. 23, 1833, 1955.

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45. Cady, H. H. and Larson, A. C., Pentaerythritrol tetranitrate II: Its crystal structure and transformation to PETN I, an algorithm for refinement of crystal structures with poor data, Acta Crystallogr. B 31, 1864–1869, 1975. 46. Zeitler, J.A., Today, P.F., Newnham, D.A., Pepper, M., Gordon, K.C., Rades, T., Terahertz pulsed spectroscopy and imaging in the pharmaceutical setting, Journal of Pharmacy and Pharmacology. 59 209–223 (2007). 47. Zeitler, J.A., Today, P.F., Gordon, K.C., and Rades, T., Solid-State transition mechanism in carbamazepine polymorphs by time-resolved terahertz spectroscopy, Chem. Phys. Chem. 8 (13), doi:10,1002/cphc.200700261, 2007.

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Index A AC Stark effect, 209, 211, 256–263 Adenine, 280 Amino acids, 277–278 Amorphous semiconductors, 177–182 Amorphous silicon–germanium alloys, 178 Amorphous solid solutions of drugs, 302 Applied electric field, 8 Attenuated total reflectance, 305–307 Auger recombination, 132 Autler–Townes splitting of electrons, 211, 256–263

B Bacillus subtilis, 287 Bacteriorhodopsin, 289–291 Bandgap, 23 Benzoic acid, 282 N‑Benzyl‑2‑methyl‑4‑nitroaniline, 53 Bi‑2212, 142, 147 Bioavailability, 300 Biomolecular kinetics, 148 Biomolecules amino acids, 277–278 benzoic acid, 282 crystalline solids, 276–279 description of, 269–270 DNA, 286–288 enantiomeric crystalline structure of, 277 experimental procedures for, 270–272 in liquid water, 293–294 nucleic acid bases, 280–282 pharmaceutical drugs, 282 polypeptides, 278–280 polysaccharides, 292–293 proteins, 288–292 RNA, 286–288 salicylic acid, 282 sugars, 282 summary of, 294–295 weakly interacting organic model compounds, 274–276 Biphasic‑induced absorbance response, 186 Blackbody radiation, 2 Bloch electron, 194 Bloch oscillations, 36 Bohr radius, 191 Boltzmann transport equation, 122

Bose‑Einstein condensation, 132 Bovine serum albumin, 288 Bruggeman effective medium approximation, 95–96 Bulk semiconductors drude response, 122–125, 127 free carrier dynamics in, 122–127 intersubband transitions, 134–136 intraexcitonic spectroscopy, 128–133 overview of, 121–122

C Carbamazepine, 283, 304, 311, 313 Carbon nanotubes, 193 Carrier density, 7–8 Carrier dynamics free, in bulk semiconductors, 122–127 in graphite, 193–196 time‑resolved terahertz spectroscopy studies of, 76 Carrier heating, terahertz‑induced, 230, 232 Carrier localization, 179–181 Carrier transport, in pentacene, 187–188 Cellulose, 293 Charge transport in disordered electronic materials, 176–184 in nanoscale materials, 188–193 in photo‑excited insulators, 173–176 CHARMm Force Field Calculations, 272–273, 289 Clausius‑Mossotti equation, 95–96 Clothing, 318–319 Coherence length, of pulsed near‑infrared laser radiation, 48 Coloumb interactions, 120 Conductivity of half‑metallic metals, 152–153 in nanoporous titanium dioxide, 189–190 Conjugated conducting polymers, 182 Controlled precipitation, 302 Cooper pairs, 136, 147 Crystalline polymorphism, 302 Crystalline semiconductors, 37 Current surge model, 114 Cyclotetramethlene tetranitramine, 314 Cyclo‑1,3,5‑trimethylene, 314 Cytochrome C, 291 Cytosine, 280

325

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D DAST, 42, 53, 64 DC polarization, 45 DC second‑order nonlinear polarization, 45 Deconvolution, 95 Dehydration, 308–309 Demagnetization, 112–113 Destructive interference between terahertz pulses, 32 DFT spectral analyses, 273–274 Dielectric stack, 88, 90 Difference scans, 82 Differential time domain spectroscopy, 31–32 Diffraction-induced pulse distortions, 99 Direct charge transfer, 114 Disordered electronic materials amorphous semiconductors, 177–182 charge transport in, 176–184 description of, 176–177 semiconducting organic polymers, 182–184 DNA, 286–288 Doped asymmetric coupled quantum wells, 248–250 Double exchange model, 151 Drude response, 122–125, 127, 139 Drug delivery systems formulating and optimizing of, 300–301 oral, 302 tablets, 301–302 Drug formulations, 302 Drug solubility, 302 d‑Tryptophan, 278 Dynamic Franz‑Keldysh effect, 210–211, 252–256 Dynamic range, 22 Dynamic spectral weight transfer, in manganites, 153–155

E Electric field, 5, 12, 65–68 Electric polarization, 45 Electron bunch length, 66 Electrooptic crystals, 52–56 Electrooptic detection applications, 65–68 description of, 44–49 experimental results of, 57–65 Erbium‑doped fiber laser, 56 Excitation polarization, 100 Excitons description of, 129–130, 133, 217 internal dynamics of, 218–227

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modifying the states of, with terahertz radiation, 251–263 in semiconductor quantum dots, 190–193 s‑orbital, 219 Explosives screening, 313–318

F Far‑infrared spectroscopy, of superconductors, 136–142, 159 FDTD method, 110 FELiX, 66, 217 Femotosecond laser pulses, 50 Floquet theory, 210 Fourier transform infrared spectroscopy, 13, 15, 94 Free carriers absorption, 13–14 in bulk semiconductors, 122–127 terahertz absorption, 230–232 Free‑electron lasers characteristics of, 216–217 FELiX, 217 locations of, 207 powering of, 216 at University of California, Santa Barbara, 207, 217 Freely propagating subpicosecond terahertz pulses, 44 Fresnel transmission, 17 Fructose, 282 Functionalized pentacene, 187–188

G Gallium arsenide advantages of, 206–207 band gap of, 207 description of, 3, 6–7, 12, 60 exciton formation, 130 growth of, 206 intrinsic, 213 quantum well, 212–213 quasiparticle formation in, 126 Gallium phosphide, 47, 62 Gallium selenide, 47, 51–52 Gaussian 03, 272 Gaussian beam, 25, 29 Gaussian envelope, 4 Glucose, 282 Granulation, 302 Graphite, 193–196 Group velocity mismatch, 60, 63 Growth‑direction terahertz electric fields, 216 Guanine, 280 Guoy phase shift, 99

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327

Index

H Half‑metallic metals optical conductivity of, 152–153 overview of, 150–152 Heavy to light hole transition, 231 Hen egg white lysozyme, 289, 292 High‑TC superconductors, 145–150 Hirshfeld dipole intensity, 314 Hopping transport, in conjugated conducting polymers, 182

I III–V semiconductors, 206 Indium antimonide, 51 Indium arsenide, 51 Induced polarization, 8 In‑plane terahertz electric fields description of, 214–215 near-infrared-terahertz mixing with, 234–239 Input impedance, 91 Intersubband transitions description of, 134–136 sideband generation from, 208–209 Intraexcitonic spectroscopy, 128–133 Intramolecular charge transfer in orienting field, 107–111 Intrinsic gallium arsenides, 213 Inverse Fourier transform, 18 Iron, 56 Isorhodopsin, 289

J Jones matrix formalism, 104–105

K Kleinman‑Bylander transformation, 273 Klienman symmetry, 108–109 Kubo‑Greenwood formalism, 122

L Lactose α‑monohydrate, 307–308, 311–312 Laguerre‑Gauss modes, 30 Laser light modulation, 45 LiNbO3 crystal, 52 Linear dispersion theory, 18 Linear electrooptic effect, 44–49, See also Electrooptic detection Lorentzian widths, 283 Lorentz‑Lorenz equation, 95–96

M Magnesium stearate, 311 Magnetoresistance, 121

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MA1:MMA, 53 Manganites carrier stabilization of, 155–158 double exchange model, 151 dynamic spectral weight transfer in, 153–155 octahedral symmetry, 151 optical conductivity of, 152–153 overview of, 150–152 Mattis‑Bardeen theory, 138–139 Maxwell‑Garnett theory, 95–96 Maxwell’s equations, 3–4, 14, 87, 109 MEH‑PPV, 182 Metals, 56–57 (–)2‑(α‑Methylbenzyl‑amino)‑5‑nitropyridine, 53–54 Mid‑infrared spectral bandwidth, 193–196 Mixed‑valence metal‑halide linear chain complexes, 185 Molecular electronic materials, 184–188 Mott‑Hubbard insulators, 145, 150 Multichannel optically detected terahertz resonance, 218

N Narrow‑gap superconductors, 139 Near‑infrared laser frequency, 48 Near‑infrared‑terahertz mixing description of, 233 with in‑plane terahertz polarization, 234–239 with out‑of‑plane terahertz polarization, 239–251 4‑Nitro‑4’‑methylbenzylidene aniline, 54 Noise, 31 Nonresonant PL quenching mechanism, 227–233 Nucleic acid bases, 280–282 Nyquist frequency, 23

O Optical delay line, 23 Optical pulse, 5 Optical pump probe spectroscopy, 33–35, 129, 146 Optical rectification definition of, 103 response functions for, 113 terahertz emission spectroscopy, 103–107 terahertz‑frequency radiation generation experimental results of, 49–57 principles of, 44–49 semiconductors used to produce, 50–52 Optical systems description of, 24 terahertz beam propagation through, 27 Optically detected terahertz resonance description of, 207–208, 219–220

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Terahertz Spectroscopy: Principles and Applications

future challenges for, 265 magnetoexciton dynamics measured using, 218–227 nonresonant PL quenching mechanism, 227–233 Organic electrooptic crystals, 64–65 Orthoexciton decay, 133

Pump scans, 76, 82 Pyrochlores optical conductivity of, 152–153 overview of, 151–152 Tl2Mn2O, 151–152

P

Quantum wells, semiconductor description of, 212–213 doped asymmetric coupled, 248–250 gallium arsenide, 212–213 intense terahertz field effects on, 252 interband properties, 213–214 intraband properties, 214 nonlinear effects in, from terahertz radiation, 233 undoped asymmetric, 245–248 undoped square, 240–245 Quasiparticle dynamics, in superconductors conventional, 142–145 high‑TC, 145–150 Quasiparticle recombination, 136

Parabolic band approximation, 130 Partial oscillator strength sum rule, 130 Peierls distortion, 186 Pellicle beam splitter, 43 Pentacene, 187–188 Pentaerythritol tetranitrate, 314–18 Pharmaceutical formulation science, 300 Pharmaceutical technology, 300 Pharmaceuticals delivery forms of, 300 description of, 282 drug delivery systems, See Drug delivery systems terahertz spectroscopy applications, 303–313 Phase‑coherent terahertz pulse, 30 Phonon bottleneck, 144 Phonon‑mediated pairing mechanism, 136 Photoconductive antennas terahertz radiation detection using, 10–12 terahertz spectroscopy using, 13–19 Photoconductive switches, 103 Photoconductive terahertz emitters, 10 Photoconductive terahertz generation process of, 6–10 source of, 6–7 theoretical background for, 3–5 Photoconductivity, 75 Photo‑Dember effect, 100 Photoexcitation, 81, 88, 184 Photo‑excited insulators, 173–176 Photoluminescence, 207 Photon energy, 1, 74 Piroxicam, 283 Planck’s constant, 192 Plastic explosives screening, 313–318 Platinum‑halide mixed‑valence complex, 185 Pockels’ effect, 44–49 Polar formation and dynamics, 184–188 Polymorphism, 303 Polypeptides, 278–280 Polysaccharides, 292–293 Poly(tetrafluoroethylene), 305, 309–310 Ponderomotive energy, 253 Proteins, 288–292 Pseudogap, 148 Pulse repetition frequency, 23 Pulsed near‑infrared laser radiation, 48

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Q

R Rabi frequency, 257 Rabi oscillations, 247, 256 Ranitidine hydrochloride, 303 Rayleigh range, 26 Refractive index, 16 Rhodopsin, 289 RMS noise, 31 RNA, 286–288 Rotational transition frequencies, 2 Rothwarf‑Taylor equations, 143–144 Rothwarf‑Taylor model, 143

S Salicylic acid, 282 Sandwich tablets, 301 Sapphire, single‑crystal, 173–173 Schrödinger equation, 210 Security applications description of, 313 screening for explosives, 313–318 Semiconducting organic polymers, 182–184 Semiconductor(s) amorphous, 177–182 bulk drude response, 122–125, 127 free carrier dynamics in, 122–127 intersubband transitions, 134–136 intraexcitonic spectroscopy, 128–133 overview of, 121–122

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329

Index doped, 126 electronic properties of, 122 heterostructures, 206 III–V, 206 nondegenerative three‑ and four‑wave mixing in, 233 optical properties of, 122 quantum wells description of, 212–213 gallium arsenide, 212–213 interband properties, 213–214 intraband properties, 214 nonlinear effects in, from terahertz radiation, 233 second‑order susceptibility, 104 Semiconductor quantum dots, 190–193 Sidebands conversion efficiency of, 209 description of, 239–240 generation of from doped asymmetric coupled quantum wells, 248–250 from intersubband transitions, 208–209 in strong terahertz fields, 210 from three‑subband system, 209–210 from undoped asymmetric quantum wells, 245–248 from undoped square quantum wells, 240–245 voltage‑controlled, 209 Simple tablet, 301 Single‑crystal sapphire, 173–173 S‑orbital excitons, 219 Sparrow criterion, 32 Spectral modeling methods, 272–274 Spectrometer description of, 19, 28 time‑resolved terahertz spectroscopy, 78–81 Spectroscopy, See Terahertz spectroscopy Specularly reflected terahertz pulse, 33 Standoff detection, 318 Stark effect, See AC Stark effect Static electric fields, 252 Sucrose, 282 Sugars, 282 Sulfathiazole, 308 Superconductors Bi‑2212, 142, 147 far‑infrared spectroscopy of, 136–142 high‑TC, quasiparticle dynamics in, 145–150 Mattis‑Bardeen theory, 138–139 narrow‑gap, 139 overview of, 136–137 quasiparticle dynamics in, 142–145

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T Tablets, for drug delivery, 301 Terahertz absorption spectra of, for clothing and water vapor, 319 advances in, 264 beam propagation, 24–30 frequencies of, 299 generation of, 3–10 optical mixing, 208 photon energy conversion of, 1, 74 signal, reflection measurements of, 32–33 Terahertz electric fields description of, 5, 12, 65–68 growth‑direction, 216 in‑plane, 214–215 Terahertz emission spectroscopy demagnetization, 112–113 experiments data analysis, 99–103 emitted waveform, 100–101 excitation polarization, 100 far field, 98–99 focusing optics, 99 near field, 98–99 sample orientation, 100 setup of, 97–98 general formalism, 113–114 intramolecular charge transfer in orienting field, 107–111 materials studied using, 96 optical rectification, 103–107 photoconductive switches, 103 principles of, 75 pulse emission, 97 shift currents, 103–107 time‑resolved terahertz spectroscopy vs., 75, 96 Terahertz emitters bandwidth performances of, 54 photoconductive, 10 types of, 53 Terahertz frequency radiation, See Terahertz radiation Terahertz gap, 1–2 Terahertz phase velocity, 63 Terahertz pulses advantages of, 120–121 delay changes, 20 destructive interference between, 32 errors in, 18 freely propagating subpicosecond, 44 frequency range for, 5 linear dispersion theory, 18 measurement of, 5, 14–15

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Terahertz Spectroscopy: Principles and Applications

optical pulses vs., 5 phase‑coherent, 30 shape of, 36 specularly reflected, 33 transient current element, 9 Terahertz radiation carrier heating, 230 electrooptic detection of applications, 65–68 experimental results, 57–65 gallium phosphide for, 62 organic crystals for, 64–65 zinc selenide for, 62–63 from metals, 56–57 nonlinear optical techniques for detecting, 58 optical rectification for producing experimental results of, 49–57 principles of, 44–49 pharmaceutical applications of, 304 photoconductive antenna detection of, 10–12 properties of, 299–300 semiconductor heterostructure and, interactions between, 207 Terahertz radiation pulse bandwidth of, 46 subpicosecond, 49 Terahertz spectral region description of, 1 optical sources scaled down to, 2 using ultrafast optical pulse generation, 13 Terahertz spectrometer, 19, 28 Terahertz spectroscopy adaptations, 30–37 advantages of, 74 attributes of, 172 Bloch oscillation measurements, 36 conductivity probing using, 188 description of, 74, 172 differential time domain spectroscopy, 31–32 emission, See Terahertz emission spectroscopy experimental considerations, 19–24 extensions of, 30–37 frequency resolution, 23–24 motion carrier influences on, 36 noise effects on, 31 pharmaceutics applications of, 303–313 photoconductive antennas used for, 13–19 principles of, 172 purpose of, 14 semiconductor quantum dots, 190–193 short‑pulse nature of, 172 source–detector considerations, 24, 27 stability of, 20 thin film considerations, 31–32

7525_C009.indd 330

time domain, See Time domain spectroscopy time-resolved, See Time-resolved terahertz spectroscopy vibrational, 274 wavelength, 23 Theophylline, 283, 309 Thermionic electron gun, 216 Thin films, 31–32 Third‑order nonlinear susceptibility, 108–109 Three‑subband system, for sideband generation, 209–210 Thymine, 280 Time domain spectroscopy bulk semiconductors drude response, 122–125, 127 free carrier dynamics in, 122–127 intersubband transitions, 134–136 intraexcitonic spectroscopy, 128–133 overview of, 121–122 crystalline solids detected using, 308 description of, 14, 121 lower frequency, 271 manganites carrier stabilization of, 155–158 double exchange model, 151 dynamic spectral weight transfer in, 153–155 octahedral symmetry, 151 optical conductivity of, 152–153 overview of, 150–152 overview of, 42–44 pyrochlores optical conductivity of, 152–153 overview of, 151–152 Tl2Mn2O, 151–152 schematic diagram of, 43 superconductors Bi‑2212, 142, 147 far‑infrared spectroscopy of, 136–142 high‑TC, quasiparticle dynamics in, 145–150 Mattis‑Bardeen theory, 138–139 narrow‑gap, 139 overview of, 136–137 quasiparticle dynamics in, 142–145 time‑resolved terahertz spectroscopy vs., 74–75 Time‑resolved terahertz spectroscopy amorphous semiconductors, 177–182 carrier dynamics studied using, 76 complex transmission coefficients, 88–91 conductivity calculations, 84–88 deconvolution, 95 definition of, 75 description of, 13, 120–121 dielectric stack, 88, 90

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331

Index difference scans, 82 examples of, 77–78 experiments data collection, 81–83 data workup, 84–96 finite difference time domain simulations of, 88 setup for, 78–81 short pump‑delay times, 92–95 spot size, 83 gallium arsenide studies, 77 history of, 77–78 introduction to, 75–77 materials studied using, 77 photoconductivity calculation of, 84–88 description of, 75 photoexcitation, 81, 88 polarizable excitons, 77 porous media, 95–96 principles of, 75 pump scans, 76, 82 pump‑delay times, 92–95 requirements for, 78 semiconducting organic polymers, 182–184 spectrometer, 78–81 spot size, 83 terahertz emission spectroscopy vs., 75, 96 time domain spectroscopy vs., 74–75 Titanium dioxide, 189–190 Tl2Mn2O carrier stabilization of, 155–158 description of, 151–152 Transfer function, 17

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Transient current element, 9 Transmission spectroscopy, 305–306 Two‑fluid model, 139–140

U Ultrafast optical spectroscopy, 136 Undoped asymmetric quantum wells, 245–248 Undoped square quantum wells, 240–245 University of California, Santa Barbara, 207, 217

V Vibrational frequencies, 2 Vibrational terahertz spectroscopy, 274 Voltage‑controlled sidebands, 209

W Water vapor, 318–319 Weakly interacting organic model compounds, 274–276 Wollaston prism, 44

Y YBCO, 141, 148–149 Ytterbium‑doped fiber laser, 56

Z Zinc selenide, 62–63 Zinc telluride, 47, 51, 55, 57, 60

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3

3

F

PL Diﬀerence (arb. units)

2

E

D´

2

1

1

B

3 2 1

0

0

0 –1

–1 G

–2

E´

D

–1 C A

8

M

ag

6

ne

4 tic Fie 3.5 2 ld (T )

1.58

1.56 ) 1.57 (eV rgy Ene

–2

–2

1.55

Figure 6.5 Figure caption on page 222.

1565

Energy (meV)

1560 1555 1550 1545 1540 1535 1530 –20

–15

–10

–5

0

5

10

DC Electric Field (kV/cm)

Figure 6.22 Figure caption on page 247.

7525_Art_ColorInsert.indd 1

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1

1560

0.9 0.8

1550

E1HH2X-E2HH3X

1545

E1HH2X-E2HH1X

1540

E2HH1X-E1HH2X

Sideband Conversion (×10–4)

NIR Energy (meV)

1555

0.7 0.6 0.5 0.4 0.3 0.2

1535

E1HH1X-E2HH1X

1530 –20

–15

–10

–5

0

0.1 5

10

0

DC Electric Field (kV/cm)

Figure 6.23 Figure caption on page 248. 1, 2-DCB

119 cm–1

169 cm–1

197 cm–1

1, 3-DCB

120 cm–1

125 cm–1

198 cm–1

1, 4-DCB

79 cm–1

128 cm–1

Figure 7.1 Figure caption on page 275.

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