LINEAR AND NONLINEAR ELECTRO-OPTICS IN A SEMICONDUCTOR
byHaipeng Zhang
B.A., Electrical Engineering, South China University of Technology, 2006
M.S., Electrical Engineering, University of Colorado, 2010

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirement for the degree ofDoctor of Philosophy
Department of Electrical, Computer, and Energy Engineering
2012This thesis entitled:Linear and Nonlinear Electro-optics In a Semiconductor
written by Haipeng Zhang
has been approved for the Department of Electrical, Computer, and Energy Engineering(Prof. Steven T. Cundiff)(Prof. Rafael Piestun)

Date

The final copy of this thesis has been examined by the signatories, and weFind that both the content and the form meet acceptable presentation standards of scholarly workin the above mentioned discipline.

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Zhang, Haipeng (Ph.D., Electrical Engineering)Linear and Nonlinear Electro-optics In a Semiconductor
Thesis directed by Professor Steven T. Cundiff

Numerous investigative efforts have been made to study the properties ofsemiconductor materials, which are the foundation of modern electronics. In thisdissertation, we study semiconductor electro-optics in both linear and nonlinear
regimes. Three main topics are presented. First, we describe transverse
electroreflectance (ER) and electroabsorption (EA) experiments using a rapidly
oscillating radio-frequency (RF) bias and electrodes that are insulated from thesemiconductor sample. This technique produces an effectively uniform transverse
electric field in a metal-semiconductor-metal structure. Then a pulsed terahertz
emitter that uses the RF biasing technique is described. The effectively uniform
electric field generated by the RF technique allows excitation of a large laser spot,
lowering the photo-excited carrier density for a given pulse energy and increasing
the efficiency of terahertz generation. The last part focuses on quantum interference
control in a semiconductor. Measurement of ballistic current injection and coherent
control of the photo-excited carrier density through the interference of one- and twophoton absorption in the presence of a static electric field, are described. Theseexperiments and results provide better understanding of electro-optical properties
in semiconductor materials.

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Acknowledgements

First I am deeply grateful to my adviser, Professor Steven Cundiff, forproviding tremendous support and encouragement through my graduate career. I
thank him for giving me such a great opportunity to explore exciting science in his
wonderful labs and for offering thoughtful and patient guidance.
I am very thankful to Professor Rafael Piestun, as well as Professor Daniel
Dessau, Professor Juliet Gopinath, and Professor Robert McLeod, on my thesis
committee, for their support and advices.
I would like to thank all the group members of the Cundiff lab, past andpresent, for their precious friendship and enormous help. Particularly I thank Jared
Wahlstrand, for guidance early in my career as a graduate student, for teaching me
valuable knowledge and techniques in lab, and for his proof-reading help andencouragement with this thesis. I’m also grateful to SooBong Choi, Xingcan Dai andAlan Bristow, for helpful discussion and guidance.
I also owe a great debt of gratitude to the JILA shop staff. I would like toespecially thank David Alchenberger for his support in clean-room facilities andsample preparation, as well as Terry Brown and Paul Beckingham, for their help inbuilding custom circuits and electronic devices.
Finally, I express my gratitude to my parents and my beautiful fiancéeBoya, for their love and support.

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Contents

Chapter

1. Introduction ............................................................................................................... 1
1.1 Electroreflectance and electroabsorption based on Franz-Keldysh effect ....... 1
1.2 Photoconductive terahertz generation............................................................... 3
1.3 Coherent control of current injection and carrier density ................................ 5
1.4 Thesis organization ............................................................................................ 7

2. Radio frequency (RF) bias and detection technique ................................................. 9
2.1 Background and motivations ............................................................................. 9
2.1.1 Introduction to trap-enhanced electric field ........................................... 11
2.1.2 Transverse sample geometry with insulated electrodes ........................ 15
2.1.3 Photo-injected carrier screening of the applied electric field ................. 17
2.2 Radio frequency bias generation and synchronization with laser pulses ...... 19
2.3 Frequency mixing lock-in detection ................................................................. 22
2.4 Summary........................................................................................................... 28

3. Electroreflectance and electroabsorption based on Franz-Keldysh effect ............. 29
3.1 Introduction ...................................................................................................... 30
3.1.1 Airy function theory for Franz-Keldysh effect ........................................ 31
3.2 DC bias electroreflectance experiment and results ........................................ 37
3.3 Radio-frequency bias electroreflectance .......................................................... 42

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3.4 Bias, polarization, and optical power dependence the RF bias
electroreflectance .................................................................................................... 48
3.4.1 Bias dependence ....................................................................................... 48
3.4.2 Optical polarization dependence ............................................................. 50
3.4.3 Optical power dependence ....................................................................... 54
3.5 Electroabsorption experiment and results ...................................................... 58
3.6 Summary........................................................................................................... 67

4. Terahertz generation with RF technique ............................................................... 68
4.1 Terahertz generation from photoconductive antenna .................................... 69
4.2 Electro-optic sampling detection system ......................................................... 74
4.3 Carrier screening and THz power saturation ................................................. 77
4.4 RF bias THz generation ................................................................................... 80
4.4.1 Experimental layout ................................................................................ 80
4.4.2 Optical power dependence and bias dependence .................................... 82
4.4.3 Spatial dependence of the THz field........................................................ 87
4.5 THz generation and electroreflectance in LT-GaAs........................................ 88
4.6 Summary........................................................................................................... 95

5. Quantum interference and electric field induced coherent control ....................... 97
5.1 Quantum interference control of ballistic current in a semiconductor .......... 98
5.2 QUIC population control in noncentrosymmetric medium .......................... 103
5.3 Electric field induced population control ....................................................... 104
5.4 Experimental layout – all optical measurement ........................................... 106
5.5 All-optical measurement results and discussion .......................................... 109
5.6 All-electrical measurement of field-induced population control .................. 114
5.7 QUIC of photocurrent injection in Er-doped GaAs ....................................... 122

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5.8 Summary......................................................................................................... 127

6. Conclusion .............................................................................................................. 128

Bibliography ............................................................................................................... 131

Appendix
A. RF bias and detection setup details ..................................................................... 139
A.1 The RF setup .................................................................................................. 139
A.2 DDS software control ..................................................................................... 142
B. Photolithography for metal electrodes ................................................................. 145
C. Thin GaAs sample preparation ............................................................................ 147
C.1 Mechanical lapping ........................................................................................ 147
C.2 Wet etching (gloves and safety glasses required) ......................................... 148

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FiguresFigureFigure 2.1: Measured THz signal in SI-GaAs with electrodes on the surface as afunction of illumination position for an electrode bias of 40 V. Details of thismeasurement will be described in Chapter 4…………………………………………….13
Figure 2.2: Schematic of carrier-injection in a biased metal-semiconductor-metal(M-S-M) structure. Electrons accumulate near the cathode…………………………...14
Figure 2.3: SI-GaAs sample with insulated electrodes. (a) Structure used in theelectroreflectance and THz experiments. (b) Structure used in the coherent control
experiment……………………………………………………………………………………..16
Figure 2.4: The applied bias field (black) and the actual field inside the SI-GaAs(blue) due to carrier screening……………………………………………………………...18
Figure 2.5: RF bias generation apparatus. The modulation frequency f is derived
from the laser repetition rate frep using direct digital synthesis (DDS). An LC
circuit enhances the voltage across the electrodes. BP: band pass filter, PD:
photodetector, AMP: RF amplifier, PS: power splitter, and Mon.: monitor for tuning
the circuit………………………………………………………………………………………20
Figure 2.6: Frequency-mixing lock-in detection scheme. BP: band pass filter, AMP:
RF amplifier, PS: power splitter, LP: low pass filter, FD: frequency doubler (only forelectro-reflectance case), PD: photodetector……………………………………………...23
Figure 2.7: Bias field with frequency f synchronized with the laser pulses. Red: laser
pulses. Black: RF bias. Blue: effective modulation of the bias………………………...24

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Figure 3.1. Change in absorption coefficient due to the electric field in bulk GaAs
predicted by the Airy function theory. Yellow: EB = 10 kV/cm. Red: EB = 5 kV/cm.
Blue: EB = 2 kV/cm…………………………………………………………………………...34
Figure 3.2. Electric field induced change in reflectivity in bulk GaAs predicted by
the Airy function theory. Yellow: EB = 10 kV/cm. Red: EB = 5 kV/cm. Blue: EB = 2
kV/cm…………………………………………………………………………………………...37
Figure 3.3. Diagram of the DC bias ER experimental apparatus. The electrodes are
in direct contact with sample. ND: neutral density filter………………………………38
Figure 3.4. Two-dimensional plot of the ER spectrum with DC bias vs. laser spot
position. Lines show the approximate position of the electrode edges……………….40
Figure 3.5. Extracted electric field vs. the position of the laser spot for DC bias
ER……………………………………………………………………………………………….42
Figure 3.6 Diagram of the RF bias ER experimental apparatus. The sample is in thetransverse ER configuration with insulated electrodes. ND: neutral density
filter……………………………………………………………………………………………..43
Figure 3.7. Two-dimensional plot of the ER spectrum with RF bias vs. laser spot
position. Lines show the approximate position of the electrode edges……………….44
Figure 3.8. Extracted electric field vs. the position of the laser spot. The line is a fit
to the analytical theory of Ref. [39]………………………………………………………..45
Figure 3.9 ER spectra for GaAs with the probe beam centered between theelectrodes, for the optical field parallel to the bias field. The dashed line is an ER
spectrum using a sample with electrodes on the surface of the sample and a DC
bias, and the solid line is an ER spectrum using a sample with electrodes insulated
from the surface of the sample and an RF bias………………………………………….47
Figure 3.10. The evolution of the ER spectrum of GaAs with increasing bias
field……………………………………………………………………………………………...48

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Figure 3.11. Fit of theoretical curve to experimental ER data near the fundamental
band edge of GaAs…………………………………………………………………………….50
Figure 3.12. Polarization dependence. (a) ER spectrum in linear scale for GaAs withbeam centered between electrodes, for the optical field perpendicular (TE, black)and parallel (TM, red) to the RF bias field. (b) Absolute value of ER spectrum inlogarithmic scale, to highlight the positions of the Franz-Keldysh oscillations…….53
Figure 3.13 Two dimensional plots of the ER spectrum vs. laser spot position. (a) 50
nW optical power; (b) 1 µW; (c) 3 µW; (d) 5 µW; (e) 10 µW; (f) 15 µW…………...……55
Figure 3.14. Extracted electric field vs. laser spot position from the data in Fig. 3.13.
(a) 50 nW optical power; (b) 1 µW; (c) 3 µW; (d) 5 µW; (e) 10 µW; (f) 15 µW………...57
Figure 3.15 Transverse electroabsorption (EA) setup. The EA signal is measured asa function of incident photon energy………………………………………………………59
Figure 3.16. Representative electroabsorption spectra for TM (transverse magnetic)
and TE (transverse electric) polarization…………………………………………………60
Figure 3.17. Evolution of the fundamental EA spectrum of GaAs with increasing
electric field……………………………………………………………………………………61
Figure 3.18. Spatial map of the EA signal and electric field in a GaAs sample with a
120 µm gap between electrodes. (a) EA spectra as a function of spot position at 75 V
bias. (b) Electric field versus position near anode edge, found by analysis of the EA
lineshape……………………………………………………………………………………….63
Figure 3.19. The shifting of the EA spectrum near the fundamental bandgap of
GaAs with decreasing temperature………………………………………………………..65
Figure 3.20 Two dimensional plot of EA spectrum vs. laser spot position, under low
temperature. (a) 200 K (b) 150 K; (c) 100 K………………………………………………66
Figure 4.1: Schematic of a GaAs based photoconductive switch for THz generation.
EB is the applied static electric field, j(t) is the time dependent current density…70

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Figure 4.2: Schematic of a typical electro-optic (EO) sampling system for detectionof ultrashort THz pulses…………………………………………………………………….75
Figure 4.3: EO sampling with three relative delays between the THz pulse and theprobe pulse, showing how the signal on the balanced detector changes depending on
the sign of the THz field overlapping with the probe pulse……………………………77
Figure 4.4: Radio-frequency biased THz generation apparatus, showing both theemitter and the EO sampling system……………………………………………………..82
Figure 4.5: THz radiation signal measured using a photoconductive emitter with aradio frequency (RF) bias. (a) Typical THz trace. (b) THz spectrum…………………84
Figure 4.6: Dependence of peak THz field on optical power for various excitationgeometries. Filled circles: large spot, RF technique; empty circles: small spot, RF
technique; filled triangles: large spot, DC technique; empty triangles: small spot,
DC technique…………………………………………………………………………………..85
Figure 4.7: Dependence of the peak THz field on the bias voltage using the RFtechnique……………………………………………………………………………………….86
Figure 4.8: Dependence of the THz peak field on the position of the laser spot for a
focus of 8 µm; the shaded region corresponds to the gold electrodes. Short dash, RFinsulated high optical power (150 mW); long dash, RF-insulated low optical power(15 mW); solid line, dc direct low optical power (15 mW)…………………………88
Figure 4.9: Experimental apparatus for simultaneous ER and THz measurement in
LT-GaAs……………………………………………………………………………………….90
Figure 4.10: Two-dimensional plot of the ER spectrum of LT-GaAs sample with 250
V DC bias vs. laser spot position. The average excitation power is 15 mW. The redline shows the approximate position of the anode edge……………………………….91
Figure 4.11: Electroreflectance signal from LT-GaAs. (a) Under 150 V bias. (b)Under 200 V bias……………………………………………………………………………..92
Figure 4.12: THz electric field measured using a LT-GaAs emitter under DC bias.93

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Figure 4.13: THz electric field (red) and ER signal (blue) versus laser spot
position…………………………………………………………………………………………94
Figure 4.14: THz field as a function of the applied DC bias for an LT-GaAs based
emitter………………………………………………………………………………………….95
Figure 5.1. Schematic of asymmetric carrier distribution in k-space under two-color
excitation……………………………………………………………………………………..100
Figure 5.2. Schematic of the experimental apparatus, including the two-color
interferometer and radio frequency biasing scheme. OPO: optical parametricoscillator, BBO: β-barium borate crystal, BP: bandpass filter, DDS: direct digitalsynthesizer……………………………………………………………………………………107
Figure 5.3. GaAs epilayer attached to a transparent sapphire disk by epoxy withinsulated electrodes…………………………………………………………………………108
Figure 5.4. Modulation of the transmitted probe at the dither frequency as thephase parameter φ 21 = φ 2ω − 2φω is slowly swept. The signal for Edc ║ <001> is shownfor positive (solid black), negative (dashed, red) and zero (blue, dotted)
bias…………………………………………………………………………………………….110
Figure 5.5. Amplitude of the signal as a function of bias voltage for Edc ║ <001> andEdc ║ <011>. The lines are linear fits……………………………………………………112
Figure 5.6. Polarization dependence of the signal for Edc ║ <110>: E2ω, Eω ║ Edc(blue); E2ω ⊥ Edc, Eω ║ Edc (red); Eω ⊥ Edc, E2ω ║ Edc (green); and E2ω, Eω ⊥ Edc(black)…………………………………………………………………………………………113
Figure 5.7. Electrode structure. Multiple gaps are shown; the electrode structure
provides gaps ranging from 6 up to 150 µm…………………………………………….115
Figure 5.8. Bias dependence of field induced QUIC population control, measured by
the electrical detection. Black line shows non-monotonic dependence of the signal
magnitude. Dashed red line shows the phase of the signal which flips by nearly π
when the magnitude dips near zero……………………………………………………...118

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Figure 5.9. Bias dependence of field-induced QUIC signal on three different laser
spot positions for 10 µm electrode spacing………………………………………………119
Figure 5.10. Map of the field-induced QUIC signal magnitude as a function of bias
and the position of the laser spot for 150 µm electrode spacing. The edges of theelectrodes are at 40 and 190 µm…………………………………………………………..120
Figure 5.11. Bias dependence of QUIC signal in a sample with 150 µm electrodespacing, for two laser spots spaced by 12 µm: nearer the electrode (solid) and farther
away (dashed). The amplitude (black) and phase (red) of the QUIC signal are
shown………………………………………………………………………………………….121
Figure 5.12. Pump-probe results. The points are the measured reflectivity as afunction of time delay, and the solid lines are fits of the data (after the coherent
artifact when the pump and probe pulses overlap in time) to A1 exp(−t/τ1) + A2
exp(−t/τ2) + B. (a) Results for GaAs:Er and LT-GaAs. (b) Results for SI-GaAs…...124
Figure A.1: Schematic configuration of RF bias and detection setup. BP: band passfilter, PD: photodetector, PS: powersplitter, AMP: RF amplifier, FD: frequency
doubler (only for ER experiment), LP: low pass filter…………………………………140
Figure A.2: Clock and amplitude control………………………………………………..143
Figure A.3: Mode and frequency control…………………………………………………144
Figure C.1: The lapping tool……………………………………………………………….148
Figure C.2: The wet etching tool………………………………………………………….149

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CHAPTER 1

Introduction
The research presented in this thesis goes under the general heading of‘linear and nonlinear optical properties of a semiconductor.’ This field, which has
been studied extensively during the past fifty years, encompasses the study of lightmatter interaction in a semiconductor. The subfield concerns the optical propertiesof a semiconductor in the presence of a strong electric field that changes slowly
compared with the optical duration. The set of experiments described in this thesis
focused on three main topics: field induced change in reflectivity or absorption ingallium arsenide (GaAs), photoconductive generation of electromagnetic radiationin terahertz (THz) frequency range, and nonlinear optical coherent control ofphotocurrent and carrier density.

1.1 Electroreflectance and electroabsorption based on Franz-Keldysh
effectElectroreflectance (ER) and electroabsorption (EA) based on the FranzKeldysh effect are widely used characterization techniques for semiconductormaterials. A modulated electric field induces a resonance in the differentialreflectance or absorption spectrum for photon energies near the band structure
critical point of a semiconductor. To give a rough physical picture, when the electricfield is applied, the electron- and hole-wavefunctions turn into Airy functions rather
than plane waves. The interference between electron- and hole-wavefunctions leads

2
to oscillations in the reflectance or absorption spectrum. This effect is named after
the German physicist Walter Franz and Russian physicist Leonid Keldysh, whodeveloped theories of it nearly simultaneously [1,2].
There have been two periods of intense interest in the Franz-Keldysh effect.
The first was around 1966–1974 during the development of modulation
spectroscopy (ER and EA), and the second occurred in the late 1980’s whenmodulation spectroscopy and analysis of Franz-Keldysh oscillations was applied tothe study of heterostructures. Practically, ER and EA are used to find the band
structure critical points and/or determine the effective mass of a semiconductor. EA
modulators based on the Franz-Keldysh effect are also widely used in today’s opticalcommunication systems. In previous studies, a variety of techniques have beendeveloped for ER and EA experiments. Among these techniques, one of the mostcommonly used was Schottky barrier ER [3,4] introduced by Aspnes. The bias is
applied between a doped semiconductor substrate and a transparent electrode on
the surface. Later the Schottky barrier technique was replaced by experiments
using heterostructures to produce a more uniform electric field. Another widelyused technique is called electrolyte ER, which uses the interface between a doped
semiconductor and a liquid electrolyte [5]. People also developed a photoreflectancetechnique which relies on the screening of the surface field of a semiconductor byphotocarriers [6]; this technique requires no electrical contacts. Another important
technique is transverse ER or EA with a bias voltage modulated symmetrically
about ground in order to have a uniform electric field, introduced by Rehn andKyser [7], which has the advantage of enabling studies of the polarization
dependence of the optical field with respect to the static electric field.
A transverse biasing scheme is required to do interesting nonlinear
experiments, e.g., coherent control of carrier population [8]. However, the electricfield distribution in the transverse M-S-M (metal-semiconductor-metal) structure is3
not uniform as expected. Since ER and EA are perfect tools for characterizing thefield uniformity, in this thesis research, we studied both ER and EA in thetransverse geometry. The ER and EA results from a GaAs sample with transverse
electrodes and DC bias reveal a highly nonuniform electric field distribution due tocarrier-trapping defects [9]. We developed a novel technique that uses a rapidly
oscillating bias with a frequency near half the laser repetition rate and a sample
with insulated electrodes. The technique produces an effectively uniform electricfield under relatively high illumination (up to 160mW average optical power), and isvery useful in ultrafast experiments that require a uniform transverse electric field,
examples will be discussed in Chapter 4 and 5. Because it allows the study ofmaterials with relatively high conductivity, the technique might find use inmodulation spectroscopy as well.

1.2 Photoconductive terahertz generation

The use of ultrashort laser pulses for the generation and field-resolved
detection of electromagnetic pulses in the terahertz (THz) frequency range has
gained widespread popularity in recent years due to the wide range of applicationsin science and technology [10,11,12,13]. This technique has provided powerful tools
for a variety of applications, including THz time-domain spectroscopy (THz-TDS),
high speed device characterization, THz imaging [12,14] and ranging [15].
Electromagnetic radiation in the THz range has several distinct advantages: theTHz radiation can penetrate a wide range of non-conducting materials, it is safe forbiological tissues because it is non-ionizing (i.e. much safer than X-ray), and images
formed with THz radiation can have relatively good resolution compared withmicrowaves. Furthermore, many materials have unique spectral fingerprints in theTHz frequency range, which makes THz radiation especially useful for medical

4
imaging and security scanning. The special frequency range of the THz radiation
also allows for measurements on materials that are difficult to probe. For example,
by using reflections from buried interfaces, defects in electronic devices can be foundand imaged precisely.
There are two commonly used techniques for generation of THz radiation
based on using ultrashort laser pulses. One is optical rectification, which based on
the fact that when short optical pulses pass through a nonlinear crystal (such as
ZnTe or GaP) they will induce electrical polarization proportional to the intensity,and the transient polarization is the source of THz radiation [16,17,18]. The othertechnique, photoconductive THz generation, usually uses a metal-semiconductormetal (M-S-M) structure that is biased with an electric, and sometimes also amagnetic, field. The laser pulses illuminate the semiconductor and give rise to a
current transient that leads to electromagnetic radiation in THz frequency range[19,20,21]. Of the two methods, optical rectification has the advantage of simplicity
compared to the photoconductive emitter. However, under the same illumination
power, the peak THz field generated from optical rectification is usually smaller
than that from the photoconductive emitter.
For the second project in this thesis (discussed in Chapter 4), we used the
RF biased M-S-M structure as a THz photoconductive emitter. In the past few
years, the saturation of THz power from a photoconductive emitter with increasingillumination power has been studied [22,23]. The key limitation of the opticalpower-to-THz conversion efficiency is ultrafast screening of the applied electric field
by photo-injected carriers [23]. In our experiment, we adopted the RF bias systemand the insulated electrode sample that were used to generate an uniform
transverse electric field in our previous ER experiment [24]. The uniform fieldobtained from the RF technique allows us to improve the power saturation

5
threshold by using a large excitation spot, and thus promises better THz generation
efficiency.

1.3 Coherent control of current injection and carrier density

Photocurrent can be generated in a semiconductor not only by driving
optically injected carriers with an external DC bias, but also by direct opticalinjection. A carrier population can be optically injected into an undoped andunbiased bulk semiconductor through two different pathways connecting the same
initial and final states (i.e. through one- and two-photon absorption), and thequantum interference between the two pathways gives rise to an asymmetric
distribution of photocarriers in k-space, which leads to a ballistic charge current.
This effect, predicted [25,26] and experimentally demonstrated [27,28] in the 1990’s,
is known as quantum interference control (QUIC). These optically driven currents
have proven useful in studying ultrafast current relaxation processes [29,30]. Undercertain conditions this QUIC technique can generate even ballistic spin-polarized
currents [31]. By measuring the QUIC current injection in a semiconductor device,
the carrier-envelope-phase offset frequency of a mode-locked Ti:sapphire laser canbe detected [32,33] and stabilized [34,35], an important application for opticalfrequency metrology.
A two-color excitation scheme was used in the early experiments [27,28] togenerate photocarriers through one- and two-photon absorption in an unbiased
GaAs sample. The quantum interference between the one- and two-photon
absorption processes, which connect the same initial and final states in the valence
and conduction bands, leads to an asymmetric distribution of carriers in k-space. By
controlling the relative phase between the two-color excitation pulses (withfrequency ω and 2ω), one can arrange for constructive interference in one direction

6in k-space and destructive interference in the opposite direction. This leads to aballistic current whose sign and amplitude can be coherently controlled by adjusting
the relative phase of the ω and 2ω pulses [27,28]. Typically low-temperature grown
GaAs (LT-GaAs) is preferred for QUIC current injection due to its subpicosecond
carrier lifetime. Short lifetime prevents internal discharge of the charge built up by
the QUIC current [28], and thus helps improve the signal-to-noise ratio. In thisdissertation, we use a similar two-color excitation scheme to measure the ballistic
current injection in an erbium-doped GaAs (GaAs:Er) which has a short carrier
lifetime comparable with that of LT-GaAs, and show that the QUIC signal size fromthe GaAs:Er is the same order of magnitude as from a low-temperature grown
GaAs.
For coherent control of ballistic current, the ω and 2ω fields only modify the
distribution of photocarriers in k-space, without changing the total number ofcarriers. In contrast, in a process arising from the imaginary part of the secondorder nonlinear susceptibility χ ( 2) , the ω and 2ω fields can also be used to change
the total density of carriers, rather than their k-space distribution, allowing
coherent control of the overall carrier density. By using phase controlled opticalpulses with frequency ω and 2ω and a (111)-oriented bulk GaAs sample, Fraser et
al. showed [36,37] that the photo-injected carrier density can be controlled
coherently through quantum interference between the transition amplitudes
associated with one-photon absorption of 2ω and two-photon absorption of ωconnecting the same initial valence band and final conduction band states.
In this thesis, we demonstrate that a constant electric field can convert thealways present third order nonlinear susceptibility χ (3) to an effective χ ( 2) , whichenables coherent control of carrier population without requiring a non-zero χ ( 2) . We
present experimental results from both an all-optical pump-probe measurement andwith electrical detection.

7
1.4 Thesis organization

This dissertation includes 6 chapters. The next chapter, Chapter 2, gives anintroduction to the radio-frequency (RF) bias and detection technique, which is a
general technique that encompasses most of the experiments of this thesis. First thetransverse M-S-M structure and the non-uniform electric field distribution in thisgeometry is discussed. Then a description of our RF bias technique that uses a
rapidly oscillating bias across electrodes insulated from a GaAs sample is given.
The resulting uniform field distribution is essential for most of the experiments
described in the following chapters.
Chapter 3 is focused on electroreflectance (ER) and electroabsorption (EA)
experiments. First an introduction is made to the Airy function theory based onparabolic band approximation for Franz-Keldysh effect. Then a description of theexperimental setup and results for ER experiments in bulk semi-insulating GaAs
(SI-GaAs) is given, for both DC bias and RF bias techniques. The results indicatethat by using the RF bias technique we can achieve a uniform field distribution in
the transverse M-S-M structure. EA experiments and results from a thin GaAs
epilayer are also discussed.
Chapter 4 discusses terahertz (THz) generation from a photoconductive
emitter. A brief introduction to photoconductive THz generation from bulk GaAs
and electro-optic sampling detection is given first, following by a review of thelimiting factor of the THz generation efficiency: the ultrafast screening of theapplied bias. Then our RF technique for THz generation, which improved thegeneration efficiency by minimizing the carrier screening effect, is discussed indetail. The results are also compared with the conventional DC bias THz emitter. In
addition, results of THz generation from an LT-GaAs sample are also presented.

8
Chapter 5 is concerned with the coherent control of ballistic current and theoverall carrier density. We start with an introduction to quantum interference
control and χ ( 2) -enabled population control, following by a theory which predicts
electric field-induced population control. Then measurements of field-induced
population control, including the all-optical pump-probe scheme, and the electricalmeasurement, are presented. The experimental results of ballistic current injection
via QUIC in an erbium doped GaAs are also included.
Chapter 6 summarizes the thesis and offers a brief discussion of future
work. Several appendices follow, containing additional information about the RFtechnique and sample preparation that are not covered in the main text.

9

CHAPTER 2

Radio frequency (RF) bias and detection technique
To provide a basic understanding of the most important experimental
technique in this thesis, an overview of the RF bias and detection technique ispresented in this chapter. First we start with an overview of the physics of semiinsulating GaAs (SI-GaAs) with a transverse electrode geometry, in particular thehighly non-uniform electric field distribution, the so called ‘trap-enhanced electricfield,’ found when a bias is applied across this structure. This overview is followed
by a description of our sample with an insulating layer (SiO2) between the SI-GaAs
surface and metal electrodes. The purpose of using this special sample structure isto achieve a more uniform field distribution. Furthermore, ultrafast carrier
screening of the applied electric field is discussed. Finally, our RF bias and lock-in
detection system is presented; the rapidly oscillating bias field enables the
production of an effectively constant field in ultrafast optical experiments.

2.1 Background and motivations
The electro-optical properties of semiconductor materials have been studied
for several decades. It has been found that the electric field induced change in theabsorption or reflectivity spectrum of a semiconductor has peaks near band
structure critical points. Thus, by modulating an applied DC field and measuring
the change in the absorption or reflectivity spectrum using phase-sensitive (lock-in)
detection techniques, one can learn about the band structure of materials. These10
techniques are known as electro-absorption or electro-reflectance. A variety oftechniques have been developed to generate and modulate electric fields in asemiconductor.
One of the most commonly used techniques involves using the built-in
electric field in the depletion region of a doped semiconductor (e.g. GaAs) [3,4].
Contact between a transparent metal film and a doped semiconductor can produce a
Schottky barrier, and by applying a bias between the doped semiconductor
substrate and the electrode, one can obtain a strong and uniform electric field in thethin depletion layer. In the 1980s, the Schottky barrier technique was improved inexperiments in which a heterostructure is grown with an intrinsic layer surroundedby thin doped layers [38]; this produces a more uniform electric field. As will beshown in the next three chapters, producing a uniform electric field is essential for
many

ultrafast

experiments.

Typically

electro-reflectanceexperiments

areperformed in this longitudinal geometry, with a constant built-in depletion fieldparallel to the optical wave vector.
Another way to apply bias is known as the transverse electrode geometry[7,39,40,41]; it was first reported by Rehn and Kyser [7]. In this geometry, twowidely spaced (typically a few hundred µm) metal electrodes are both on the sameside of a semiconductor (GaAs) surface, and thus the applied electric field isperpendicular to the light wave vector. Although this scheme is less common thanusing longitudinal electrodes, this transverse geometry offers the following
advantages: (1) it is relatively easy to fabricate, with no need for transparent
electrodes; (2) one can vary the polarization of the optical field with respect to theelectric field direction; (3) it can be used with an undoped semiconductor. There area few disadvantages: (1) high voltage is required because of the relatively large
separation between electrodes; and (2) the field distribution in GaAs with metalelectrodes deposited directly on the surface has been found to be strongly non-

11
uniform [9]. As will be described in this thesis, the ability to generate a uniform
electric field along the optical field enables a variety of different experiments onelectroreflectance (ER), electroabsorption (EA), THz generation, and coherent
control. In the following parts of this chapter, the physics of the field distribution
and our technique for achieving a uniform field in a transverse electrode GaAs
sample will be discussed in detail.

2.1.1 Introduction to trap-enhanced electric field

Enhanced electric field near the anode edge of a metal-semiconductormetal (M-S-M) structure, the so-called ‘trap-enhanced electric field’, was firstreported in a terahertz generation experiment [9]. Photoconductive antennae basedon M-S-M structures have been used as sources of terahertz (THz) radiation foralmost three decades. Physically, THz emission from a photoconductive antennarelies on the light-induced conductivity change in a biased semiconductor substrate.
Creation of photocarriers by an ultrashort optical pulse leads to a current transientin the biased M-S-M structure that gives rise to a single-cycle electric field transientwith a temporal length of less than a picosecond, which corresponds to a broadfrequency spectrum extending into the THz frequency range. THz spectroscopy
allows the study of a variety of applications, including chemical detection.
Enhanced THz radiation near the positive biased electrode edge (the anode)
has been found in previous studies of THz generation using GaAs M-S-M structures[42]. Terahertz radiation 50 times stronger can be obtained from illumination near
the anode than from illumination near the cathode [42]. As shown in Figure 2.1, ourmeasurement of the peak electric field of the THz pulse as a function of illumination
position from a GaAs M-S-M structure (DC biased) shows strong enhancement ofTHz electric field when the laser pulse is focused near the anode edge, which is12consistent with a previous study by Ralph and Grischkowsky [9]. At modest bias
voltages, they interpret the enhanced THz signal as most of the applied potential
drops within a few microns of the anode.
Detailed study and modeling showed that the strong electric fieldenhancement at the anode of the transverse M-S-M structure is due to carriertrapping impurities [9,43]. The origin of the field inhomogeneity can be traced to thefabrication process of SI-GaAs. Typically SI-GaAs substrates are taken from liquid
encapsulated Czochralski grown ingots, and the semi-insulating character comesfrom a careful balance between the residual shallow acceptor impurities and the socalled EL-2 donor defects. The EL-2 is a deep donor found in GaAs with excess As
concentration, and the surplus As atoms are found to occupy Ga host lattice sites,
forming antisites, AsGa. These defects, at a concentration of around 1016 cm-3, havethe ability to fully compensate moderate concentrations of acceptors (around 1015
cm-3) which results from the unintentional doping with carbon. Thus in SI-GaAs,
the charge density is determined by the EL-2 defect and the residual acceptors.
Normally the shallow acceptors are fully ionized, but the deep EL-2 donor can bepartially ionized, and always acts as a strong electron trap [9].

13Figure 2.1: Measured THz signal in SI-GaAs with electrodes on the surface as afunction of illumination position for an electrode bias of 40 V. Details of thismeasurement will be described in Chapter 4.

In the transverse M-S-M geometry used in most THz experiments, metalelectrodes are in direct contact with the SI-GaAs surface; when a bias is applied
electrons are readily injected from the cathode. Injection of free carriers under amoderate bias leads to an accumulation of electrons and a depletion of holes near
the anode edge, which is strongly affected by electron velocity saturation andgeometric effects [9]. A schematic plot for the carrier-injection process in atransverse M-S-M structure is shown in Figure 2.2.

14

Figure 2.2: Schematic of carrier-injection in a biased metal-semiconductor-metal(M-S-M) structure. Electrons accumulate near the cathode.

Although the free carriers are not sufficient to directly change the field
distribution significantly because of their relatively low density, it has been shownthat a small change in free carrier density results in a large change in trap
occupancy. With increasing electron density near the anode, the EL-2 traps whichhave a larger capture cross section for electrons than holes, tend to become almost
fully occupied by electrons and thus become charge neutral. Without compensationfrom the EL-2 traps, the ionized acceptors form a space-charge region which only
depends on the net acceptor concentration. In this space-charge region, which is
known as the trap-enhanced electric field (TEF), the field strength can be enhanced
by factor of a few hundred compared to the rest of the gap between electrodes [9].
In contrast, the cathode experiences only small electron depletion, sinceelectrons can be readily injected from the electrode. Furthermore, the accumulation

15
of holes near the cathode is small due to the fact that hole velocity saturation
requires a stronger field. Thus, a space charge region which can strongly affect field
distribution doesn’t exist near the cathode.

2.1.2 Transverse sample geometry with insulated electrodes

The large trap-enhanced fields just described enable efficient THz
generation with low optical excitation level, but many experiments, e.g., fieldinduced population control and THz generation experiments discussed in Chapter 4
and 5, would benefit from having a uniform electric field in the M-S-M structure.
The highly inhomogeneous TEF near the positive biased electrode in semiinsulating materials limits applications, e.g., field inhomogeneity will wash out theFranz-Keldysh oscillations which will be discussed in Chapter 3. In order to achievea uniform field distribution in transverse electrode geometry, we developed atechnique which does not require electrical injection of carriers into thesemiconductor, preventing the TEF from forming.
There are two types of sample structures we used in our experiments.
Shown in Figure 2.3 (a) is the sample mainly used in electroreflectance and THz
generation experiments (discussed in Chapter 3 and 4, respectively). We first
evaporate a 100 nm thick layer of silicon dioxide on the SI-GaAs surface as an
insulating layer, and then Ti:Au electrodes of roughly 200 nm thickness arepatterned on top of the insulating layer using photolithography. The electrodes are
separated by a 100 µm gap. Figure 2.3 (b) shows the thin sample used in thecoherent control experiment, discussed in Chapter 5. Ti:Au electrodes of roughly
200 nm thickness are evaporated on a transparent sapphire disk first using
photolithography, and then a 1 µm thick SI-GaAs epi-layer is attached to thesapphire disk by transparent wax. The wax also acts as a insulating layer between

16
the metal electrodes and the sample. As discussed in the previous section, when free
carriers are injected from the electrodes, TEF forms from uncompensated acceptors
due to increasing occupancy of deep traps. Thus the deleterious effects of the traps
can be avoided if the insulating layer between the electrodes and the SI-GaAs
surface prevents free electrons from injecting into SI-GaAs.

(a)

(b)

Figure 2.3: SI-GaAs sample with insulated electrodes. (a) Structure used in theelectroreflectance and THz experiments. (b) Structure used in the coherent control
experiment.

17
2.1.3 Photo-injected carrier screening of the applied electric field

In the experiments described in later chapters, the semiconductor isilluminated by a laser. The insulating SiO2 layer prevents free-electron injection
from the electrodes, yet it introduces a different problem – screening of the applied
electric field by photoexcited carriers. Since the metal electrodes are insulated fromthe SI-GaAs surface, photoexcited electron-hole pairs can not be swept out by theelectrodes, and they are confined inside the SI-GaAs substrate; the photoinjected
electrons and holes will be separated by the applied bias field, building up aninternal field which cancels out the external one in a process known as Coulomb
screening.
The screening time is roughly equal to the dielectric relaxation time in
GaAs. The dielectric relaxation time can be derived from the Poisson equation. Fora small disturbance of charge density ∆N(t) in the charge equilibrium of asemiconductor, from the Poisson equation,dE∆N (t )
=−
,
εε 0
dx

(2.1)

where εε 0 is the dielectric permittivity of the medium. The continuity equation
gives∂(∆N(t))∂E.
= ∇ ⋅ J = µN
∂t
∂x

(2.2)

Here, because we are treating the case of quasi neutrality, we have neglected allterms in ∇ ⋅ J from the gradient in the carrier concentration (diffusion), N is thetotal carrier concentration, and µ is the carrier mobility. The rate of change of the
disturbance in carrier density can be found from combining Eq. (2.1) and Eq. (2.2),

18∂ (∆N (t ))eNµ=−∆N (t ) ,
εε 0
∂t

(2.3)

which gives a simple solution∆N (t ) = ∆N (0) ⋅ exp(tτdHere τ d =

).

(2.4)

εε 0
= εε 0 ρ is the dielectric relaxation time and ρ is the resistivity. The
Neµtime period over which screening occurs is roughly the dielectric relaxation time.
Details of the geometry would also enter into a more sophisticated calculation. As
the screening field is directly proportional to the photocarrier density ∆N , after anexternal field is suddenly turned on, the electric field inside the sample decaysroughly exponentially at rate 1/ τd. A schematic plot is shown in Figure 2.4.

Time
Figure 2.4: The applied bias field (black) and the actual field inside the SI-GaAs(blue) due to carrier screening.

19

Neglecting the dark resistivity (for SI-GaAs, this is typically on the order of

108 Ω ⋅ cm ,whichyieldsadielectricrelaxation

time

of

10 −4 s[44]),

ρ = 1 /[e ⋅ (nµ n + pµ p )] is inversely proportional to the density of photoexcited carriers.
Here for GaAs at room temperature, µ n = 8500 cm 2 /V ⋅ s is electron mobility, andµ p = 400 cm 2 /V ⋅s is hole mobility. With increasing photocarrier density, ρdecreases, and the screening time is shorter. Thus the faster one modulates the bias
field, the more light one can use before screening affects the experiment. At
typically used modulation frequencies in the range of kHz (which are suitable forthe highest-performing lock-in amplifiers), an extremely low illumination level isrequired (as well as high dark resistivity) [7,40], lowering the signal-to-noise ratio.
A radio frequency bias enables measurement using stronger illumination, as well as
enabling study of higher conductivity materials [45]. In the following section, we
present the combination of RF bias and a synchronized train of short light pulses,
which helps us to get an effectively uniform electric field (around 30 kV/cm), andmakes experiments under much higher optical power (more than 100 mW on
average) possible.

2.2 Radio frequency bias generation and synchronization with laser pulses

In order to avoid strong photo-excited carrier screening in our insulated
electrode sample, a rapidly oscillating bias with modulation frequency in the radio
frequency (RF) range is needed. A diagram of the RF bias generation apparatus isshown in Figure 2.5. Details of the apparatus, including the RF signal levels andparts used, are described in Appendix A.

20

Figure 2.5: RF bias generation apparatus. The modulation frequency f is derived
from the laser repetition rate frep using direct digital synthesis (DDS). An LC circuit
enhances the voltage across the electrodes. BP: band pass filter, PD: photodetector,
AMP: RF amplifier, PS: power splitter, and Mon.: monitor for tuning the circuit.

It will be shown below that to make detection of the signal in theexperiments easier, it is desirable to oscillate the electric field at a frequency near
the repetition rate of the laser. We use mode-locked Ti:sapphire lasers (for the fieldinduced coherent control case we use an optical parametric oscillator pumped by a
Ti:sapphire laser) with repetition rate frep = 89 MHz (frep = 76 MHz for the pumplaser in the field-induced coherent control experiment). First of all, a high speed
photodetector is used to detect the Ti:sapphire laser harmonics. The output signal of
the photodetector is then sent through a bandpass filter which passes only the third
harmonic of the laser repetition rate (3frep). This 3frep signal is used as a clock for a
direct digital synthesizer circuit (DDS), the output of which is a frequency of either f= frep + 1 kHz (for THz generation and field induced coherent control experiments)

21

or f = ½ frep + 1 kHz (for electro-reflectance experiment). The DDS requires the clock
frequency to be at least three times the output frequency. A detailed analysis will bepresented in the next section to explain the reason for choosing a different
frequency f for electro-reflectance and THz generation experiments. The purpose ofderiving f from frep by using a DDS circuit is to prevent drifting of the modulation
frequency, which is the difference between frep and f, due to fluctuations in frep. For
example, a 100 microns change in the laser cavity caused by mechanical vibrationor thermal expansion can induce a 2.6 kHz drift of the laser repetition rate. Thus, if
an independent frequency synthesizer were used to generate the bias, the reference
frequency would drift by that much, which could lead to artifacts over a long scan.
The DDS circuit synchronizes the signal f to the laser pulses and minimizes thedrift between f and frep.
The DDS output is amplified and sent to the sample, where we use aresonant LC circuit which is driven by a 2 W RF power amplifier to enhance thevoltage across the electrodes, as shown in Fig 2.5. The inductor is the secondary coilof an air-core transformer with L = 1200 nH. The capacitor is a sealed glass variablecapacitor: for electro-reflectance, the LC resonance frequency is 44.5 MHz (close to½ frep of the 89 MHz Ti:sapphire laser) and the capacitance is set to be C = 7 pF; forTHz generation the LC resonance frequency is 89 MHz (close to frep), and C = 2 pF;
and for coherent control the LC resonance frequency is 76 MHz (frep of the 76 MHz
pump laser), and C = 3 pF. The sample contributes a few picofarad of additional
capacitance. The secondary consists of 22 turns of 20 gauge magnet wire wound on
a plastic toroid. The primary is a ¾ turn of the same wire. Another ¾ turn of wire isconnected to an oscilloscope and the voltage across it, Vmon, is used as a monitor fortuning the resonance of the circuit. In order to minimize signal reflection, which
prevents part of the RF signal from being transferred to the LC resonance circuit,the impedance of the transformer is tuned to match the 50 Ω output impedance of

22

the RF amplifier by adjusting the position of the wires along the toroid. This
position affects the reactance (imaginary part of the impedance) of the LC resonanceby changing the parasitic capacitance between the wires.

2.3 Frequency mixing lock-in detection

In laser spectroscopy experiments, one often periodically modulates someparameter of the experiment and uses phase-sensitive detection to increase
sensitivity and reduce background noise. For example, this can be done by using amechanical chopper in the laser beam or by modulating the bias across the sample,
and using a lock-in amplifier to measure the signal at the modulation frequency. In
our system, we modulate the bias voltage across the sample sinusoidally at thefrequency f, which is near the laser repetition rate frep. One could in principle use alock-in amplifier capable of detecting in the RF frequency range, but they are not as
sensitive and resistant to pickup noise as digital lock-in amplifiers in the audio
frequency range. Thus we use a heterodyne technique to shift the effective
modulation frequency to the audio frequency range, where photodetector circuits
and lock-in amplifiers have the best performance. A diagram of the frequencymixing lock-in detection scheme is shown in Figure 2.6.

23

Figure 2.6: Frequency-mixing lock-in detection scheme. BP: band pass filter, AMP:
RF amplifier, PS: power splitter, LP: low pass filter, FD: frequency doubler (only forelectro-reflectance case), PD: photodetector.

The output of the high speed photodetector used to provide the clock for the
DDS is split by a RF power splitter and then filtered. As discussed previously, partof the output, which selects 3frep, is used as the clock frequency of the DDS. The
other part is filtered to select the fundamental harmonic of frep; this is mixed withpart of the DDS output, which in some cases was doubled, to generate a reference
frequency, fref = 2f – frep ≈ 1 kHz (for electro-reflectance), or fref = f – frep ≈ 1 kHz (forTHz generation). It is proven below that the signal from a lock-in amplifier24

referenced to 2f – frep (or f – frep) is the same amplitude as that from a RF lock-in
amplifier referenced to 2f (or f). With the advantages of not requiring mechanicalchopping and a continuous, wide range of electronically selectable reference
frequencies, this detection scheme combined with the RF bias technique could be
useful in ultrafast experiments where detecting a signal modulated by a uniform
electric field is required. On the other hand, if one only needs a constant field, theRF bias frequency can be set to be equal to the laser repetition rate. This is used in
our field induced coherent control experiment which will be discussed in Chapter 5.
Note that one must carefully adjust the phase between the laser pulse train and thebias waveform in order to achieve the optimum effective DC field strength. A
diagram of the pulse train and bias field for this configuration is shown in Fig 2.7.

Figure 2.7: Bias field with frequency f synchronized with the laser pulses. Red: laser
pulses. Black: RF bias. Blue: effective modulation of the bias.

For the case of electro-reflectance based on the Franz-Keldysh effect, at agiven photon energy h ω the differential reflectivity of the sample ∆R(EB, h ω) =

25

R(EB, h ω) – R(0, h ω) is highly nonlinear in the electric field EB from the bias
(details will be given in the next chapter), but neglecting sign-dependent FranzKeldysh effects that can occur in a noncentrosymmetric crystal under certain
conditions [40], ∆R is even in the electric field, totally independent of its sign. Thus,a Taylor expansion of ∆R can be written as∆ R ( E B , h ω ) = R 2 ( h ω ) E B + R 4 ( h ω ) E B + R 6 ( h ω ) E B + .....
2

4

6

(2.5)

The modulated electric field is of the form EB(t) = E0 sin(2πft). This expansion leads,
for the first term in Eq. (2.5), to∆R(EB , hω ) ∝ R2 (hω )E02 sin 2 (2π ft) =R2 E02 R2 E02−cos(2π ⋅ 2 ft) .
2
2

(2.6)

Thus the RF bias results in a DC change in the reflectivity and a modulated
reflectivity signal at frequency 2f. Higher order terms in Eq. (2.5) result in
harmonics of 2f.
The intensity of the laser beam on the photodetector IPD is proportional toR[EB(t), h ω]⋅I(t), where I(t) is the intensity of the incident laser light. For a modelocked laser with a repetition rate of frep, I(t) can be written as∞

I (t ) = ∑ I 0 ⋅ exp[
n = −∞− (t −

n
f rep

τ2

)2
]

(2.7)

Where τ is a laser pulse width parameter. Thus the light intensity on thephotodetector is26

I PD ∝ ∆R[EB (t), hω ]⋅ I(t)

n 2 
−(t −
)
 ∞
frep 
2
4
6= (R2 (hω )EB + R4 (hω )EB + R6 (hω )EB +...)⋅  ∑ I 0 ⋅ exp[]τ2
 n=−∞




(2.8)

Taking a Fourier transform to the frequency domain and only considering thefundamental harmonic at 2f for the sake of simplicity, the Fourier transform of I(t)is a comb with lines centered at n⋅frep.
∞ ~
 − (mf rep ) 2 δ (ω ' − 2πmf rep )
 ∑ I 0 exp
v2
I PD (ω ) ∝ ∫ m =0× [δ (ω − ω ' ) − 0.5δ (ω − ω ' − 4πf ) − 0.5δ (ω − ω ' + 4πf )] ⋅ exp(−iωt )dt∞ ~∞ ~ − (mf rep ) 2 − (mf rep ) 2δ (ω − 2πmf rep ) − 0.5∑ I 0 exp(
= exp(−iωt ) ⋅ {∑ I 0 exp)δ (ω + 4πf − 2πmf rep )
2
2


v
v
m =0
m
=
0


2
∞ ~
(2.9)
 − (mf rep ) δ (ω − 4πf − 2πmf rep )}− 0.5 ⋅ ∑ I 0 exp


v2
m =0



Here ν is the laser spectral width. By the convolution theorem, the Fouriertransform of IPD(t) has a peak at 2f – frep. Thus if we choose f to be close to ½frep, for
2f slightly different from frep, the pulse train and bias waveform slip with respect toone another, leading to a reflectance signal at the difference frequency. Harmonics
arising from terms Rm(EB)m in Eq. (2.4) lead to peaks at m⋅(2f – frep). In principle,
one could construct the constant-field spectrum using these harmonics, but in theexperiments described in this thesis only the fundamental harmonic is used. The
frequency 2f – frep is set in the audio frequency range as a reference for the lock-in
amplifier.
For THz generation using an RF bias, the signal is approximately
proportional to the bias field, because the THz field is detected through electro-optic
sampling, as described in Chapter 4. We again use the heterodyne technique to27shift the effective modulation, this time choosing f to be close to frep. In this case, fora given time delay between the pump and electro-optical sampling pulses, the THz
field strength is

ETHz (t ) ∝ E B (t ) ⋅ I (t )(2.10)

where EB(t) is the modulated electric field, and I(t) is the intensity of the pump laser
pulse train. Because of the difference between the laser repetition rate and the bias
modulation frequency, the THz field is modulated in the time domain
2

 
 
n

 − t −  ∞  frep  
ETHz ∝ E0 sin(2π ft)  ∑ I 0 exp τ2
 n=−∞









(2.11)As before, take the Fourier transform to the frequency domain and use theconvolution theorem:∞ ~ −(mfrep )2 
ETHz (ω ) ∝ ∑ I 0 exp δ (ω − 2π mfrep )]∗[δ (ω − 2 π f ) − δ (ω + 2π f )]
2
 v

m=0

(2.12)∞ ~ −(mfrep )2  −(mfrep )2 = ∑ I 0 exp δ[ω − 2π (mfrep + f )] − ∑ I 0 exp δ[ω + 2π ( f − mfrep )]
2
2
 v

 v

m=0
m=0
∞

~

When we use the fundamental harmonic (m = 1), the second term on the right handside of Eq. (2.12) isδ [ω + 2π ⋅ ( f − m ⋅ f rep )] = δ [ω + 2π ⋅ ( f − f rep )](2.13)

28

This shows that ETHz(ω) has a peak at f – frep. In this case we set f to be close to frep,
and the reference frequency for lock-in detection is fref = f – frep.
For the field-induced coherent control experiment, the RF bias technique isused without requiring this heterodyne scheme for detecting the signal. Details ofthat experiment will be described in Chapter 5.

2.4 Summary

In this chapter, our experimental techniques for achieving a relatively
uniform field in a SI-GaAs sample with a transverse electrode geometry wereintroduced. The uniformity of the field was proven by spatially resolved
electroreflectance, as will be described in the next chapter. We utilize an insulating
layer in order to minimize the trap-enhanced electric field by preventing carrierinjection from metal electrodes. Combined with a fast oscillating bias in the radio
frequency range, which helps us minimize photo-injected carrier screening, we canachieve an effectively uniform field distribution in the transverse geometry. We also
developed a heterodyne detection technique which can guarantee a good signal-tonoise ratio and a widely adjustable reference frequency without the requirement ofa mechanical chopper.
This RF bias and frequency-mixing technique was applied to experiments
such as electro-reflectance, THz generation, and field induced coherent control.
Each of these experiments is discussed in the following chapters.

29

CHAPTER 3

Electroreflectance and electroabsorption based on Franz-Keldysh effect
In this chapter, we mainly focus on electroreflectance (ER) andelectroabsorption (EA) spectroscopy based on the Franz-Keldysh effect. First we
start with a brief introduction to bulk gallium arsenide (GaAs) and the theory of theFranz-Keldysh effect. Then results are discussed from a typical electroreflectancesetup that uses a low frequency bias that is modulated in the kHz range and a pairof electrodes in direct contact with the semi-insulating GaAs (SI-GaAs) sample. Oneof the goals of the ER/EA experiments was to get very nice and clean experimental
data with more observable Franz-Keldysh oscillations that could be compared to asophisticated theory being developed in collaboration with John Sipe of the
University of Toronto. However, it turned out to be much harder than we thought.
We calculated the electric field distribution between electrodes based on a parabolic
band approximation model, and found a strong trap-enhanced electric field near theanode edge as discussed in the previous chapter. The field nonuniformity washes
out the Franz-Keldysh oscillations, which is the reason why we developed the RFtechnique, because it overcomes the trap-enhanced fields. We then present
experimental results with RF bias ER, which uses a rapidly oscillating bias that issynchronized to the laser pulses. The bias generated by our RF technique is applied
to SI-GaAs with electrodes electrically isolated from the sample surface, as
described in Chapter 2. The measured electric field distribution proves that a
uniform electric field can be achieved successfully by using our RF technique. Moredetailed studies of the bias voltage, optical power and polarization dependence of30

ER signal are also described. ER under high optical power is also discussed. In thelast part, electroabsorption results from a 1 µm thick GaAs epilayer are also
discussed in detail.

3.1 Introduction

Gallium arsenide (GaAs), one of the most commonly used semiconductormaterials in today’s manufacture of electro-optic devices and microwave frequency
circuits, is a direct bandgap III/V compound semiconductor with zinc-blende
structure. In GaAs, near the Γ point of the Brillouin zone where k = 0, there is one
conduction band and three valence band, all of them have parabolic dispersions.
The conduction band is an electron band which has a positive curvature withrespect to k, and the three valence bands are hole states which have a negative
curvature. Two of the valence bands, the heavy hole (hh) and light hole (lh) bands,
are degenerate at k = 0. The heavy hole band has a smaller curvature compareswith the light hole band. The third valence band is split off to lower energy by the
spin-orbit coupling, and is known as the split-off band. Bandgap energy Eg isdefined as the energy difference between the minimum of the conduction band andthe maximum of the valence band, at room temperature Eg = 1.42 eV.
In case of linear absorption where the incident photon energy hω is close to
the bandgap energy of GaAs, the absorption coefficient α is directly proportional to
the joint electron-hole density of states [6]:for hω < E g , g (hω ) = 0 ,1  2µ 
for hω ≥ E g , g (hω ) =
2π 2  h 2 

(3.1)
3/ 2

(hω − E ) .
1/ 2

g

(3.2)

31

µ is the reduced electron-hole mass. From Eq. (3.1) and Eq.(3.2), the linear
absorption coefficient α as a function of incident photon energy hω is:for hω < E g , α (hω ) = 0 ,(3.3)

for hω ≥ E g , α (hω ) ∝ (hω − E g ) .
1/ 2

(3.4)

As a result, there is no absorption if hω < E g (only consider linear absorption case),
and if hω ≥ E g , the absorption increases as a function of (hω − E g )

1/ 2

.

3.1.1 Airy function theory for Franz-Keldysh effect

The Franz-Keldysh effect is the field-induced modulation of the opticalproperties in a semiconductor in the presence of a static electric field [1,2,46,47].
The coherent acceleration of photoexcited carriers driven by the electric field gives
rise to so-called Franz-Keldysh oscillations in the reflection or absorption spectrum
for light with photon energy above the bandgap of the semiconductor. The fieldassisted absorption through a tunneling produces an exponential tail in thereflectivity or absorption spectrum for light with photon energy below bandgap. In
practice, electroreflectance (ER) and electroabsorption (EA) based on the FranzKeldysh effect are used to study band structure critical points because the ER andEA signals are always enhanced at bandgap critical points of semiconductormaterials [48]. The Franz-Keldysh oscillations can also give useful information
about effective masses in new materials and heterostructures [49].
The Franz-Keldysh effect was first predicted independently by WalterFranz and Leonid Keldysh in 1958 [1,2]. Callaway and Tharmalingam made early
contributions as well [50,51]. The theory was developed much more extensively andpractically by David Aspnes starting in 1966, when he extended the theory to32

critical points of various dimension and symmetry [46]. In another work, heextended the theory to both real and imaginary parts of the dielectric function andshowed how to include broadening phenomenologically by adding an imaginary
argument to the Airy functions [47]. Later Aspnes and Rowe took the low-field limit
and showed that under certain conditions, the response is proportional to the thirdderivative of the optical constants [48]. This development put ER on solid
theoretical footing as a technique for finding semiconductor critical points.
Here we only consider the simple case of a direct transition in the vicinityof a normal (M0, which means the valence band curves downward and the
conduction

band

curvesupward)absorption

edge

in

a

directbandgap

semiconductor. This case describes all of our experimental results in the following
sections. When a constant electric field, EB, is applied to a semiconductor, theexperimentally measureable change in absorption coefficient is∆α ( E B ) = α ( E B ) − lim α ( E B ) .
E B →0(3.5)

Under the parabolic band approximation with positive effective mass at the normal
M0 edge, neglecting the Coulomb interaction between electrons and holes, theabsorption coefficient under a constant electric field is [46]α ( E B ) = (N 2 / π )Rθ 1 / 2 {Ai ′ 2 (η ) − ηAi 2 (η )},(3.6)

where η = ( E g − hω ) / hθ , R is a constant, and θ is the electro-optic frequency

θ = [e 2 E B2 / 2hµ ] ,
1/ 3

(3.7)

33

here µ = mcmv/(mc + mv) is the reduced mass between the conduction band andvalence bands, which have effective masses mc and mv [47]. Ai is the Airy function,
which solves [46]d 2 F ( x)= xF ( x ) .
dx 2

(3.8)

There are actually two solutions for F(x), Ai(x) and Bi(x), which are the solutions
regular and irregular at infinity, respectively. The integral representations of thesefunctions are

Ai ( x ) =1 ∞cos( 13 s 3 + xs ) ds ,∫0NBi( x) =1 ∞exp(− 13 s 3 + xs) − sin( 13 s 3 + xs) ds .
N ∫0

[

(3.9)

]

N is the normalization constant of the Airy function and usually taken as π .

(3.10)

34

Figure 3.1. Change in absorption coefficient due to the electric field in bulk GaAs
predicted by the Airy function theory. Yellow: EB = 10 kV/cm. Red: EB = 5 kV/cm.
Blue: EB = 2 kV/cm.

From Eq. (3.5), the experimentally measurable change in absorption
coefficient can be obtained using the asymptotic forms of Ai(x) [46] for x → ∞ ,

{(

)[

]

}

∆α ( E B ) = Rθ 1 / 2 N 2 / π Ai ′ 2 (η ) − ηAi 2 (η ) − − η H (−η ) . (3.11)

where H(x) is the unit step function, equal to one for positive x, and zero for
negative x. The expression for ∆α in Eq. (3.11) is based on the approximation thatthe index of refraction n is constant as the photon energy is tuned across the M0edge, which is a good approximation for the fundamental edge in GaAs. For higher
absorption thresholds the variation of n with field must be taken into account. Thechange in absorption coefficient as predicted by Eq. (3.11) is plotted as a function ofphoton energy in Figure 3.1.

35

In a semiconductor, absorption is related to the dielectric constant throughthe Kramers-Kronig relations, thus the dielectric constant is also affected by the DC
field. In case of electroreflectance, the normalized change in reflectivity under aconstant electric field is [49]∆R / R = a(ε 1 , ε 2 )δε 1 + b(ε 1 , ε 2 )δε 2 ,(3.12)

where ε 1 and ε 2 are the real and imaginary parts of the complex dielectric constantε = ε 1 + iε 2 , and a and b are the Seraphin coefficients, related to the unperturbed
dielectric function. Analogous to the absorption coefficient, the complex dielectric
function under a constant electric field isε ( E − E g , E B ) = ε ( E − E g , 0) + ∆ ε ( E − E g , E B )(3.13)

where ∆ε is the change of ε due to the electric field.
In the experiments, we modulate the DC field and measure the change inreflectivity spectrum of a laser beam. It will be useful to compare the data to asimple theory. Near the fundamental bandgap of a bulk semiconductor, b ≈ 0 , andassuming electromodulation from flatband, where the field-induced change indielectric function is equal to the dielectric change due to the perturbation in thefield modulation experiment, we have
∆R / R ≅ a Re[ ∆ε ( E − E g , E B )] .

(3.14)

Near the fundamental bandgap of a III-V semiconductor, the change of thedielectric constant is [49]36

 Eg − E  ,∆ε ( E − E g , E B ) = ∑ α i G
h
θ
i
i



(3.15)

where i = hh, lh, for heavy-hole and light-hole contributions, θ is the electro-optic
frequency, and2

2e 2 h 2 e ⋅ Pc ,v  2 µ i  3 / 2
1/ 2
αi =
 2  (hθ i ) ,
2 2
m E
 h 

(3.16)

with µ the reduced mass, and Pc ,v the momentum matrix element between
conduction and (heavy hole or light hole) valence band. The ‘electro-optic’ function
G (ξ ) has an Airy function form [52]G (ξ ) = π [ Ai ′(ξ ) Bi ′(ξ ) − ξAi(ξ ) Bi(ξ )] + ξ H (ξ ) + i{π [ Ai ′ 2 (ξ ) − ξAi 2 (ξ )] − − ξ H (−ξ )} , (3.17)

where Ai and Bi are given by Eq. (3.9) and Eq. (3.10). The effect of broadening canbe included by replacing E − E g with E − E g + iΓ in Eq. (3.6) and Eq. (3.14), where Γ
is a broadening parameter. The change in reflectivity as predicted by Eq. (3.14) isplotted as a function of photon energy in Figure 3.2.

37

Figure 3.2. Electric field induced change in reflectivity in bulk GaAs predicted by
the Airy function theory. Yellow: EB = 10 kV/cm. Red: EB = 5 kV/cm. Blue: EB = 2
kV/cm.

3.2 DC bias electroreflectance experiment and results

Electroreflectance (ER) based on the Franz-Keldysh effect is a widely usedcharacterization technique for semiconductor materials. A modulated electric field
produces a resonant differential reflectance spectrum at band structure criticalpoints [48]. In our experiment we start with a transverse electrode sample structure
and DC bias. The metal electrodes are in direct contact with the semi-insulating
GaAs sample, and from the experimental results we observe a highly nonuniform
electric field distribution in the gap between electrodes due to strong trap-enhanced
electric field effects.

38

Figure 3.3. Diagram of the DC bias ER experimental apparatus. The electrodes are
in direct contact with sample. ND: neutral density filter

The experimental apparatus for DC bias ER is shown in Figure 3.3. For alight source we use a mode-locked Ti:sapphire laser with a spectrum centered at 820
nm, with full width, half maximum 40 nm and repetition rate frep ≈ 89 MHz. The
pulse energy is 5.5 nJ with about 70 fs duration (we use a laser as our light source
for all of our ER and EA experiments because it is required in experiments using
the radio-frequency bias). The laser output is passed through a Spectral Products
Model CM 110 monochromator, with a 600 line/mm grating blazed at 500 nm and a
resolution of 1.5 nm. After the monochromator, the spectrally resolved light is sent
through a neutral density filter and then focused on the sample using a lens with a
focal length of 75 mm. The average power at the sample is approximately 45 nW.
The FWHM beam waist at the focus on the sample is measured to be 26 µm using
the knife edge formed by the Au-GaAs interface (there is a large difference inreflectivity between GaAs and Au). Back reflected light from the sample is captured
by a beam splitter and measured with a custom high gain photodetector. The

39

sample is a (100)-oriented SI-GaAs wafer with two Au electrodes patterned directlyon the GaAs surface, separated by a 100 µm gap. The sample position is controlledwith respect to the focusing lens using a stepper motor. A square-wave voltage,
which is generated by a high-voltage square wave generator, is applied across thegap between electrodes so as to produce a periodic modulating field. We modulate
the bias between 0 V and 410 V at 1 kHz. The modulation frequency is produced bya Stanford Research System DS-345 function generator. The ER signal is measured
by a lock-in amplifier at this modulation frequency.
By moving the sample position with respect to the laser focal spot with amotorized translation stage, we measure the ER signal distribution in the gap
between electrodes. A map of the ER spectrum versus spot position is shown in
Figure 3.4, where the x axis corresponds to incident photon energy, and the y axis
corresponds to focal spot position. Here the polarization of the laser light is parallelto the applied electric field. The edges of two Au electrodes, located around 15 µm
and 115 µm with respect to the y axis, are indicated by the red lines. The field
induced variation of reflected light intensity, ∆R, is normalized by the total reflected
intensity R, which is directly proportional to the average output of thephotodetector. From the ER signal distribution we find that the ER signal isstrongly enhanced near the positively charged electrode edge, which is around 115µm in Figure 3.4. As the electrodes are in direct contact with the SI-GaAs sample,

free-carrier injection from electrode will lead to increased free-electron occupancy of
the EL-2 defect near the anode edge, giving rise to the trap-enhanced electric field(TEF) in this region. The ER signal enhancement near the anode edge is mainly
related to the strong TEF [9].

40

Figure 3.4. Two-dimensional plot of the ER spectrum with DC bias vs. laser spot
position. Lines show the approximate position of the electrode edges.

We estimate the electric field distribution in our sample by using theexperimental data combined with the Airy function theory. In GaAs at the photon
energies used here, the ER line shape is given to a good approximation by theelectro-optic function G(x) [47]. For a uniform electric field within the laser spot, thetheoretical reflectivity as a function of laser energy ћω predicted in Eq. (3.14) can be
written in a simplified form:∆R (hω , E B ) ∝hω − E g + i Γ
1G[].
θ (EB )hθ ( E B )(3.18)

41

Here Γ is the broadening parameter, EB is the bias field, and Eg is the band gap
energy (for GaAs at room temperature, Eg = 1.42 eV). The electro-optic frequency θ
is defined in Eq. (3.7).
Using µ = 0.036 me (only light holes contribute for optical polarization
parallel to bias field [53]), one can derive from Eq. (3.18) a relation between theelectric field strength and the energy of the first dip above the band gap in the ER
spectrum. We choose the position of the first dip above gap because it is not strongly
affected by either field inhomogeneity or broadening. The electro-optic energy isrelated to the difference between the energy of the dip Edip and the band gap energy
Eg by: Edip = Eg + •θ(EB), where  is a real number which can be found analyticallyfrom Eq. (3.18) (here we got  = 1052). Once  is found, the actual electric fieldstrength can be calculated through Eq. (3.7).
We estimated the actual electric field distribution inside this direct contact
sample with DC bias from the ER data. The estimated electric field versus position
between electrodes is shown in Figure 3.5. The field strength near the positively
biased electrode edge is almost 100 times stronger than other part of the gap, whichis comparable to other results on the trap-enhanced field effect [9]. The highly
inhomogeneous TEF also affects the shape of the Franz-Keldysh oscillations
strongly by making the oscillations above bandgap diminish quickly with increasing
photon energy. The field nonuniformity within the laser spot acts like a hugebroadening parameter.

42

Figure 3.5. Extracted electric field vs. the position of the laser spot for DC bias ER.

3.3 Radio-frequency bias electroreflectanceIn this section, we present our ER experiments using a transverse electrodesample structure with RF bias and heterodyne detection technique as discussed in
Chapter 2; we discuss the main results of our electroreflectance experiments withRF technique, and compare the results with the DC biased direct contact electrodesample.
A diagram of the experimental apparatus is shown in Figure 3.6. The
optical part of the setup is identical to that of the DC bias experiment in section 3.2.
An RF bias of frequency f = ½ frep + 1 kHz ≈ 44.5 MHz is applied across theelectrodes on the sample. The bias voltage is around 400 V (measured by themonitor of the LC circuit, as shown in Figure 2.5). The lock-in amplifier measures atfref = 2f – frep.

43

We studied a semi-insulating (100)-oriented GaAs wafer. The sample
includes a 100 nm SiO2 layer between Au electrodes and GaAs as an insulating
layer. The detailed sample geometry was described in Chapter 2. Position of thesample with respect to the laser focal spot is controlled using a motorizedtranslation stage.

Figure 3.6 Diagram of the RF bias ER experimental apparatus. The sample is in thetransverse ER configuration with insulated electrodes. ND: neutral density filter

The main purpose of our RF technique is to achieve a more uniform electricfield distribution in a transverse electrode geometry. By scanning the sample under
the laser spot with the stepper motor stage and recording the ER spectrum as afunction of spot position, we can map the electric field between two Au electrodes. A
plot of the ER signal versus position is shown in Figure 3.7. The polarization of thelaser light is also parallel to the applied electric field.

44

Figure 3.7. Two-dimensional plot of the ER spectrum with RF bias vs. laser spot
position. Lines show the approximate position of the electrode edges.

The edges of the two Au electrodes are located around 60 µm and 160 µm
with respect to y axis, as indicated by the red lines in Figure. 3.7. The field induced
variation of reflected light intensity ∆R is normalized by the total amount ofreflected intensity R, which as in the DC bias ER experiment, is recorded directlyfrom the output of the photodetector by an Agilent digital multimeter. We can see
the ER signal ∆R/R is now distributed fairly uniformly across the gap, and from thisdata we can estimate the actual electric field in the sample between electrodes
using the simplified Airy function approximation based on parabolic-bandassumption, as introduced previously.
The estimated electric field from our RF ER data is shown in Figure 3.8.
We find, as expected, that the field is weakest (but most uniform) in the center of

45

the two electrodes. The field near the electrode edges is higher than in the center
due to geometric effects. The field distribution in Figure 3.8 is far more uniform
compared with the one shown in Figure 3.5 – the difference between the electrodeedge and the center of the gap is only around a factor of 2 in Figure 3.8, whereas in
Figure 3.5 the difference is close to a factor of 100. This is proof of the uniform
electric field achievable by using the RF technique.

Figure 3.8. Extracted electric field vs. the position of the laser spot. The line is a fit
to the analytical theory of Ref. [39].

The calculated electric field distribution from our data was fit to an
analytical theory from the appendix of Ref. [39]. This theory, which assumes anidealized two-dimensional potential distribution with simple boundary conditions
for a typical transverse electrode geometry, can describe the bias field distribution

46

in sample structures such as the one studied here quite accurately, at least in theabsence of trap-enhanced fields. The field on the surface of the gap between twoelectrodes is:

E=

πV
4dK (1 − m)

[1 − m ⋅ cosh 2 (πV2d)] −1 / 2 .

(3.19)

Where K is the complete elliptic integral of the second kind, d is the sample
πsthickness, V is the bias voltage, and m = cosh − 2 ( ) (where s is the distance between
4delectrodes). The theoretical result for our sample (of 300 µm thickness, neglecting
the thin insulating layer) is shown as a solid line in Figure 3.8. In the center of thegap, the ER spectrum is consistent with a peak electric field of 26.5 kV/cm. The best
fit of the field distribution yields a peak bias of 430 V. With electrodes in directcontact with the sample and a similar bias of 410 V, the electric field extracted fromthe ER spectrum with the laser spot in the center of the gap is only 5.5 kV/cm.
A direct comparison of single ER traces using both DC and RF bias isshown in Figure 3.9. The dashed line shows an ER trace using direct contact sample
with a DC bias of 410 V modulated a 1 kHz, with the laser spot focused midway
between the electrodes. The solid line shows an ER trace using a sample withelectrodes insulated from the surface and RF bias, also focused at the same position.
Both traces are taken with optical field parallel to the bias field.

47

Figure 3.9. ER spectra for GaAs with the probe beam centered between theelectrodes, for the optical field parallel to the bias field. The dashed line is an ER
spectrum using a sample with electrodes on the surface of the sample and a DC
bias, and the solid line is an ER spectrum using a sample with electrodes insulated
from the surface of the sample and an RF bias.

The spacing between Franz-Keldysh oscillations is related to the electricfield. From Figure 3.9 we can see that even if a bias with same amplitude is applied,
for the sample with direct contact electrodes the effective electric field in the centerof the gap is much smaller compared with the insulated electrode sample, which has
a more uniform field distribution. This is an important advantage of the RF bias
technique.

48

3.4 Bias, polarization, and optical power dependence the RF bias
electroreflectanceMore results from the RF bias electroreflectance, including the ER signal
dependence on the applied bias, the optical polarization of incident light, and theoptical power, will be discussed in this section.

3.4.1 Bias dependence

We also studied the dependence of ER signal on applied RF bias. The
magnitude of the RF bias can be controlled by adjusting the output amplitude of the
DDS (direct digital synthesizer). In this way, one can change the bias amplitude
itself without inducing any phase shift between the RF bias and the Ti:sapphirelaser pulse train (if one uses RF attenuators to adjust the amplitude, a phase shift
can be induced). The change of the ER signal with increasing electric field is shownin Figure 3.10.

Figure 3.10. The evolution of the ER spectrum of GaAs with increasing bias field.

49

In Figure 3.10, we can see the entire oscillation period changes withincreasing electric field in the manner that is predicted by the parabolic band
theory: the oscillation peaks and dips above band gap shift toward higher energy.
Physically, this is due to the fact that photo-carriers are accelerated more quickly bystronger electric fields, thus gaining higher energy.
By finding the best fit of the ER spectrum between the theory prediction
and the experimental data, one can also measure the broadening energy Γ, which isrelated to the decay processes of free carriers such as collisions, trap recombination,phonon emission, etc [39]. Figure 3.11 shows a ‘best fit’ to the ER spectrum at afield of 13.3 kV/cm. The ‘best fit’ occurs for Γ = 4.5 meV. We attribute the poor fit at
low energies to the fact that the semiconductor is transparent at those energies.
This results in experimental artifacts due to the reflection of light from the backsurface of the sample, which has been seen in previous experiments [82].

50

Figure 3.11. Fit of theoretical curve to experimental ER data near the fundamental
band edge of GaAs.

3.4.2 Optical polarization dependence

The polarization dependence of electro-absorption (EA) spectra has beenstudied both theoretically and experimentally in the past as a way of studying thesymmetry of band structure critical points [41,53]. The origin of the polarization
dependence of the Franz-Keldysh absorption spectrum near the band gap in GaAs isrelated to how the heavy hole (hh) – conduction band (CB) and light hole (lh) –conduction band transitions contribute differently to the bulk absorption.
Consider incident light with variable polarization on bulk GaAs in thepresence of a DC field. Here transverse electric (TE) is defined as the electric field of
the incident light pointing perpendicular to the field vector of the applied electricfield, and transverse magnetic (TM) refers to when the electric field of the incident
light is parallel to the field vector of the applied electric field. Then the bulk

51electroabsorption of GaAs can be decomposed into its hh-CB and lh-CB
contributions, writing for TE and TM electroabsorption, respectively [53],13α TE (ω , E B ) = P 2α hh (ω , E B ) + P 2α lh (ω , E B )α TM (ω , E B ) =4 2
P α lh (ω , E B )3(3.20)(3.21)

Where P is the momentum matrix element, and α lh (ω ) and α hh (ω ) differ from eachother by the combined density of states in the perpendicular direction with respect
to the field and the square of the envelope function overlaps along the field
direction. As shown in Eq. (3.20) and Eq. (3.21), the TM polarized light onlyinteracts with the lh band, whereas the absorption of the TE polarized light isaffected by both hh and lh bands. Due to the fact that the lh-CB combined density ofstates is larger than the hh-CB combined density of states, and the decay of the
below band gap tunneling tail for hh is faster than lh, α lh (ω ) is larger than α hh (ω ) ,and the ratio between them grows with increasing photon energy. As a result, theabsorption of TM polarized light which consists of only lh-CB contributions, isstronger than the absorption of TE polarized light, which consists of mainly hh-CB
contributions. This explains the polarization dependence of the below gap
exponential tail of ER, which was studied previously. The above gap dependence iswhat we are more interested in, and it is simply related to the fact that the effective
masses are different.
Ruff et al. studied the EA spectrum of a low temperature grown GaAs (LTGaAs) epi-layer, they show that with photon energy above the band gap, the FranzKeldysh oscillations for TM polarized light show longer oscillation periods comparedwith TE polarized light, and below band gap, the TM absorption is stronger than TE

52[41]. The results are in good agreement with theory calculations using both k•pmodel and 2D diagonal approximation.
Although the best available theories calculate only the EA spectrum, theER is related through the Kramers-Kronig relations [47], and thus the samepolarization dependence can be expected, with a somewhat different line shape near
the band edge. Taking advantage of our transverse M-S-M sample geometry, we canstudy the polarization dependence of the ER signal at the fundamental band gap of
GaAs. A half-wave plate in front of our SI-GaAs sample rotates the polarization of
the incoming laser beam with respect to the direction of the applied electric field inthe sample. The polarization dependent ER signal plotted with both linear and log
scales is shown in Figure 3.12.

53

(a)

(b)

Figure 3.12. Polarization dependence. (a) ER spectrum in linear scale for GaAs withbeam centered between electrodes, for the optical field perpendicular (TE, black)and parallel (TM, red) to the RF bias field. (b) Absolute value of ER spectrum inlogarithmic scale, to highlight the positions of the Franz-Keldysh oscillations.

Figure 3.12 (a) shows the ER spectrum of SI-GaAs in linear scale for two
different polarization configurations with the laser spot midway between theelectrodes. And the absolute value of the field induced reflection change in54logarithmic scale is shown in Figure 3.12 (b). The strong dips in the spectrum are
due to the zero crossings of the reflectivity spectrum. In both linear and logarithmic
scales, a clear difference in the differential reflectivity between TE (black) and TM(red) polarized light is evident. Our ER results match previous EA results andtheoretical calculations in that, for energies above the band gap, the oscillation
period is longer for TM incident polarization than for TE, as expected because of thedifference between the effective masses of the light and heavy holes.

3.4.3 Optical power dependence

We also studied the optical power dependence of the ER signal with RFbias. A continuously adjustable neutral density filter in front of the SI-GaAs sample
is used to control the incoming beam power precisely. In order to study the influenceof photocarrier generation on the electric field distribution inside the sample, we
also performed spatial scans by scanning the sample under the laser spot andmeasuring the ER spectrum with a variety of optical powers. The two-dimensionalplots of the ER spectrum versus laser spot position are shown in Figure 3.13, withaverage optical power at the sample from 50 nW up to 15 µW. The two red lines ineach figure indicate the edges of the two electrodes.

55

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.13 Two dimensional plots of the ER spectrum vs. laser spot position. (a) 50
nW optical power; (b) 1 µW; (c) 3 µW; (d) 5 µW; (e) 10 µW; (f) 15 µW.

56

From Figure 3.13, with increasing optical power the rate of photocarrier
generation clearly affects the ER signal distribution in the gap between electrodes.
Under relatively low optical power (less than 5 µW), the normalized ER signal ∆R/R
distributes uniformly cross the gap. When the optical power reaches 5 µW, the ∆R/R
signal starts decreasing, and remains roughly constant under higher optical power.
The calculated electric fields versus the position of laser spot under different opticalpower corresponding to spatial scans in Figure 3.13 are shown in Figure 3.14.

57

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.14. Extracted electric field vs. laser spot position from the data in Fig. 3.13.
(a) 50 nW optical power; (b) 1 µW; (c) 3 µW; (d) 5 µW; (e) 10 µW; (f) 15 µW.

58

In Figure 3.14, the electric field distribution under different optical powers
is estimated using the mapping method that was discussed in section 3.2. The
electric field near the edge of each electrode increases constantly with increasing
optical power. Interestingly, the electric field around the center of the gap between
electrodes remains constant with average optical power below 10 µW, and withhigher optical power (above 10 µW) the field in the center drops to almost zero. The
optical power dependence of the electric field distribution can be explained partiallyby the ‘optical biasing effect’ which is induced by optical injection of free carriers [9].
As described in Chapter 2, our sample has a SiO2 insulating layer between As
electrodes and SI-GaAs surface, this SiO2 layer prevents carrier injection fromelectrodes. But under strong illumination, large amount of optical injected free
carriers can also be captured by traps in SI-GaAs, and thus have a strong effect on
the potential distribution. Other effects, like high field dynamics of free carriers,
can also be involved. More theoretical work, e.g., calculation of Franz-Keldysh effectthat take into account Coulomb interactions between electrons and holes understrong optical illumination, would be needed to fully understand the optical power
dependence of the ER signal.

3.5 Electroabsorption experiment and results

Although the sample preparation can be a bit more complicated,
experiments based on the electric field induced change in absorption spectrum canprovide a better signal-to-noise ratio than electroreflectance. Also, the experimentdiscussed in Chapter 5, about the field-induced population control, had to be done intransmission in a thin sample. So we also performed electroabsorption (EA)
experiments. The experimental apparatus for EA is very similar to the ER setup.
We use the same Ti:sapphire laser as the light source. To provide spectral

59

resolution and achieve a tight focus at the sample, we used a 4f pulse shaper with a
movable slit at the focal plane to spectrally resolve the incident light. The homemade pulse shaper is used as a monochromator in this experiment. The spectral
resolution of the pulse shaper is around 0.3 nm. We focus the spectrally resolved
laser beam after the pulse shaper onto a thin GaAs sample, and simply collect thetransmitted light, then focus it again onto a silicon photodiode for detection. Figure
3.15 is a diagram of the experimental apparatus.

Figure 3.15 Transverse electroabsorption (EA) setup. The EA signal is measured asa function of incident photon energy.

The sample we studied is an undoped GaAs epilayer of thickness 1 µm,
grown by molecular beam epitaxy with an AlAs etch stop. The GaAs sample isattached to a sapphire disk first using transparent wax, and the substrate is then60

removed by mechanical thinning and chemical etching. More details about the thinsample can be found in Appendix C. After thinning and etching, Ti:Au electrodes ofroughly 200 nm thickness are evaporated on the GaAs surface, separated by 120µm. The gap between electrodes is made by a shadow mask using a thin magnet

wire. A DC bias generated by the high-voltage square wave generator is applied tothe sample. The function generator produces a modulation frequency which is set to
be a reference signal for the high-voltage pulser and the lock-in amplifier. The
sample position with respect to the laser focal spot is also controlled by a stepper
motor stage. In Figure 3.16, representative EA experimental spectra for two
orthogonal optical polarizations of incident beam are plotted. For energies above theband gap, the EA oscillation period is slightly longer for TM polarized light than for
TE. The results are consist with previous studies [41,53].

Figure 3.16. Representative electroabsorption spectra for TM (transverse magnetic)
and TE (transverse electric) polarization.

61

Multiple EA traces taken under different bias field strength are shown in
Figure 3.17. The incident light is TM polarized. Just as in the ER case, the main
dips at the bandgap critical point are fixed, and the oscillation peaks above bandgap
shift toward higher energy with increasing electric field. As discussed in Section 3.1,
in the parabolic band approximation, with electric fields sufficiently large that
Coulomb effects can be neglected, the EA lineshape takes the form of the electrooptic function F ( x ) = [ Ai ' ( x )] 2 − xAi 2 ( x ) − − x H ( − Re[ x]) . For a uniform electric fieldwithin the laser spot, the theoretical differential absorption as a function of laser
energy ћω predicted in Eq. (3.11) can be written in a simplified form:∆α (hω , E B ) ∝hω − E g + i Γ
1F().
θ (EB )hθ ( E B )(3.22)

Figure 3.17. Evolution of the fundamental EA spectrum of GaAs with increasing
electric field.

62

A spatial scan is shown in Figure 3.18 (a) with an average optical power of
30 nW. The EA signal is measured as a function of incident photon energy and the
focal spot position on the sample. Here we modulate the bias between 0 V and 75 V
at 50 Hz, the modulated EA signal is measured by the lock-in amplifier, and the DC
signal on the photodiode is also recorded by a digital multimeter. We divide themodulated signal by the DC signal to give ∆T/T. Then the modulated absorption∆α = − log( ∆T / T + 1) / L , where L is the thickness of the sample. Since here we have athin GaAs layer with Au electrodes in direct contact, in Figure 3.18 (a), most of the
EA signal is concentrated near the positively biased electrode edge (around 150 µm),
due to the trap-enhanced electric field effect [9].

(a)

63

(b)

Figure 3.18. Spatial map of the EA signal and electric field in a GaAs sample with a
120 µm gap between electrodes. (a) EA spectra as a function of spot position at 75 V
bias. (b) Electric field versus position near anode edge, found by analysis of the EA
lineshape.

As with ER, by fitting our EA data to the theoretical EA lineshape based on
the parabolic band approximation, the electric field distribution between electrodes
was estimated. Similar to the ER case, rather than fit the entire lineshape, which is
distorted by the nonuniformity of the field, we instead find the position of the firstpeak above the band gap. This first peak above gap is not strongly affected by either
field inhomogeneity or broadening. The electric field strength is related to thedifference between the peak position Epeak and the band gap energy by

E B = [1052( E peak − E g )]3 / 2 . Using the above relation we calculated the field
distribution using the EA spectrum. The extracted electric field near the positively
biased electrode edge as a function of position is shown in Figure 3.18 (b). The
result, with strong trap-enhanced fields near anode edge, is very similar to the ER
experiment as expected.

64

Due to field inhomogeneity and broadening, we find that the Franz-Keldysh
oscillations above the bandgap were mostly washed out in our room temperature EA
and ER experiments. Hoping to see more oscillations we also studied EA in asample at low temperature. A cryostat which has AR coated windows on both sides
was used to cool the GaAs sample with liquid helium. EA traces under different
temperatures are shown in Figure 3.19, with incident beam power around 20 nW,
and a 75 V bias modulated at 50 Hz. With temperature decreasing, crystal latticespacing shrinks. Thus the bandgap energy of GaAs increases and the EA traces
show up at higher energy. A sharp exciton (caused by the attraction between thephotoexcited electron and hole) feature appears when the temperature is around
100 K, and this makes it harder to understand the current data since the simple
theory does not account for the electron-hole interaction. This indicates that astronger electric field is possibly required to separate the Franz-Keldysh oscillations
from the excitonic effect [83].

65

Figure 3.19. The shifting of the EA spectrum near the fundamental bandgap of
GaAs with decreasing temperature.

66

We also performed two dimensional spatial scans under low temperature bymoving the cryostat with the sample inside and recording the EA spectrum as afunction of laser focal spot position. The plots of EA signal versus position at threedifferent temperatures are shown in Figure 3.20. The red lines indicate the edge ofelectrodes on GaAs surface. The positive biased electrode edges are around 20 µm,

(a)

(b)

(c)

Figure 3.20 Two dimensional plot of EA spectrum vs. laser spot position, under low
temperature. (a) 200 K (b) 150 Κ; (c) 100 K

67with EA signal concentrated near these regions, probably due to the electric field
inhomogeneity caused by the trap-enhanced field effect. However, at 200 K and 150
K, EA signal also shows up near the cathode edge which can not be only explainedby the trap-enhanced field, but disappears at 100 K. We do not have a good
explanation for the low temperature EA signal distribution at this point, and wehaven’t seen more oscillations as expected. Furthermore, the ice on the window of
the cryostat and other issues like the thermal expansion of the wax layer make ourEA signal not perfectly repeatable. The EA experiment under low temperature stillneeds further technical improvement. Ideally, the RF bias technique could be
adapted to a low temperature experiment so that a uniform field could be achieved.
This might require somehow mounting an LC circuit on the sample holder inside
the cryostat.

3.6 Summary

In this chapter, our experimental results of transverse electroreflectanceand electroabsorption were discussed. By using a radio frequency bias combinedwith a sample whose Au electrodes are insulated from the GaAs surface, we
successfully minimized the trap-enhanced field effect and achieved a uniform
electric field distribution, as measured by ER under relatively low optical power. We
compared our ER results using the RF technique side by side with the conventionaltransverse ER using DC bias, and also presented further studies such as bias
dependence, optical power and polarization dependence. Furthermore, we showed
our DC bias EA result from a 1 µm thick GaAs epilayer. The EA results are similar
to the ER, but with a better signal-to-noise ratio. The low temperature EA data is
not fully understood at this point. Further study and improvement of theexperimental apparatus is required.

68

CHAPTER 4

Terahertz generation with RF technique
An important ultrafast optical application of biased M-S-M structures is thegeneration of broadband THz pulses. The application of the RF bias technique tothis problem is the subject of this chapter.
We start with a brief introduction about photoconductive terahertz (THz)generation, and the electro-optic sampling detection system we use to measure THz
emission. Then the key limiting factor of photoconductive THz generation efficiency,
ultrafast screening of the applied electric field, is discussed. Finally we show
experimental results using the RF bias technique, and compare our data with DCbias results. As discussed in previous chapters, the RF bias technique and sample
with insulated electrodes allow application of a strong, effectively constant field
without carrier injection from the electrodes, and thus the trap-enhanced electricfield does not form. The resulting uniform electric field distribution allows
excitation with a large laser spot, lowering the photo-injected carrier density for agiven pulse energy and increasing the pulse energy at which ultrafast screening of
the field during the laser pulse occurs. Furthermore, the insulated electrodes
sample shows less susceptibility to damage. Our RF technique shows the potential
for further improvement if a more powerful RF power amplifier is available. For
comparison, previous results of THz generation using low temperature grown GaAs
(LT-GaAs) are discussed.

69

4.1 Terahertz generation from photoconductive antenna

Ultrafast broadband terahertz radiation, which refers to electromagnetic
waves at frequencies between 0.3 and 10 THz, has been exploited for a variety of
applications including imaging, sensing, and spectroscopy [42]. One of the mostwidely used sources for THz radiation [54] is a biased photoconductive switch: asemiconductor, such as GaAs, with metal electrodes on the surface, which is excitedby ultrafast optical pulses from a mode-locked laser. Although higher peak THz
fields can be achieved using amplified laser systems that operate at kilohertz
repetition rates, for spectroscopy applications sources employing oscillators withrepetition rates near 100 MHz are attractive for their better noise performance,
relative simplicity and lower cost. The transverse electrode sample geometry which
we used to study electroreflectance can also work as a photoconductive dipole
antenna for THz generation.
The GaAs based photoconductive THz emitter relies on the absorption ofultrashort pump pulses with a photon energy above the bandgap in order togenerate electron-hole pairs. In undoped GaAs, the creation of photocarriers by
ultrashort optical pulses gives rise to a sudden, many orders of magnitude, change
in the electrical resistance. Because the GaAs is biased with a static electric field,
the change in its resistance will lead to a current that has a strong timedependence, radiating electromagnetic waves in the THz frequency range. A
diagram of a typical THz photoconductive switch (sometimes called an antenna)
with a DC bias applied to its electrodes is shown in Figure 4.1. It is comprised of a
GaAs substrate with metal electrodes deposited on the surface. A laser beam is
focused to a spot within the gap between the electrodes to create free carriers in the
GaAs substrate. A DC bias is applied to the electrodes to create a constant electricfield in the gap.

70

x
z

y

Figure 4.1: Schematic of a GaAs based photoconductive switch for THz generation.
EB is the applied static electric field, j(t) is the time dependent current density.

The laser beam creates a time-dependent photocarrier population in the
GaAs substrate. Neglecting the background carrier density, the rate of change ofthe carrier density n is
dn(t )n(t )=−+ G (t , z ) ,
τc
dt

(4.1)

where τ c is the carrier lifetime, z is the propagation distance of the laser pulses, andG(t,z) is the optically injected carrier generation rate, at the surface of the GaAs
substrate, it is [54]G (t , z = 0) = G (t ) = α ⋅I (t )
,
hω

(4.2)

71

here α is the absorption coefficient of GaAs and ћω is the incident photon energy.
Assuming a Gaussian pulse with pulse duration τ p , the laser intensity is −ln 2 ⋅ t 2 .
I(t) = I 0 exp 2 τp 

(4.3)

Here assume α is constant, and one absorbed photon generates one electron-hole
pair. Since the carrier lifetime τ c (~10 ns in SI-GaAs, a few 100’s of fs for LT-GaAs)
is much longer than the pulse duration, the approximate solution of Eq. (4.1) isn(t ) = e

−

t

t

τc

t

τc

t

∫ G (t ) ⋅ e dt = ∫ G(t )dt − N ⋅ e
−∞

0

−

0

t

τc

.

(4.4)

The first term refers to the change in carrier density within the laser pulse
duration, and the second term refers to the carrier density change after the laser
pulse. N0 is the bulk carrier density without illumination.
Before excitation there are almost no free carriers present, so even with astrong applied bias field EB, the current is essentially zero. After excitation, thepulse generates electron-hole pairs and the conductivity σ of the GaAs is increased
dramatically,σ (t ) = n(t ) ⋅ e ⋅ µ ,(4.5)

where by definition µ = Vd / E B ( Vd : drift velocity) is the usual mobility of free
carriers. The time dependent current density isj (t ) = σ (t ) ⋅ E (t ) = σ (t ) ⋅ [ E B + ETHz (t )] .

(4.6)

72

This current is driven by the total field E(t) in the semiconductor [54], which
consists of the applied electric field EB plus the generated THz field ETHz(t).
Only nonmagnetic material is considered here, thus B = µ•H everywhere.
According to Maxwell’s equations,∇ × E (t ) = − µ∂ H (t ),∂t∇ ⋅ H (t ) = j (t ) +

(4.7)

∂
[ε r ε 0 E (t )] .
∂t

(4.8)

Inside the semiconductor, assume D = εrε0E by neglecting dispersion and nonlinear
effects. From Eq. (4.7) and Eq. (4.8), one can derive [54]E (t ) = −B (t ) = −µ ⋅c⋅ j (t ) ,
1+ εr
µ

1+ εr

(4.9)

⋅ j (t ) ,(4.10)

where c is the speed of light in vacuum. To obtain the THz field transient, use Eq.
(4.9) in Eq. (4.6),

j (t ) = σ (t ) ⋅ [ E B −µ ⋅c⋅ j (t )] ,1+ εr

(4.11)and solving for j(t) givesj (t ) =σ (t ) ⋅ (1 + ε r )1 + ε r + σ (t ) ⋅ µ ⋅ c⋅ EB .

(4.12)

73

Inserting the expression for j(t) into Eq. (4.9) gives

E (t ) = E B + ETHz (t ) = −µ ⋅ c ⋅ σ (t )⋅ EB .
1 + ε r + µ ⋅ c ⋅ σ (t )(4.13)The THz near field is roughly Gaussian. Taking into account the carrier lifetimeand mobility, the THz far field is a single cycle pulse of a few hundred femtoseconds
duration.
Summarizing the processes described above, THz generation from asemiconductor based photoconductive antenna with external bias can be thought ofas occurring in a few steps. First of all, an optical pulse creates carriers in asemiconductor with static bias field. The photoinjected electrons and holes areaccelerated in the DC field, moving in opposite directions and creating a current
transient. The time dependent current leads to a transient THz electric field which
is roughly the time derivative of the current. In the case of strong illumination,
photocarriers of very high density can be created, and carrier separation by the bias
field can build up an opposing field to screen the applied DC field. It will be shown
later that this screening process limits the strength of the THz field that can beachieved using this technique. These processes all happen within less than apicosecond. On the much longer time scale of approximately 10 ns between the laser
pulses, the photocarriers can either recombine or travel to the electrodes, and theTHz emitter system goes back to its initial state, ready to repeat these processes
again.

74

4.2 Electro-optic sampling detection system

In time domain THz spectroscopy, the amplitude and phase of a THz pulse
are measured as a function of time after it transmits through a sample. This allows
the full characterization of the THz response of the sample [54]. This technique
works because one can easily measure the THz electric field. Two types of timegated THz pulse detection schemes have been widely used. One is based on asemiconductor photoconductive antenna, and the detection mechanism, which basedon the fact that a current transient can be detected when photo-injected carriers ina photoconductive antenna is driven by the THz field, is very similar to themechanism used to generate THz pulses discussed in previous section. The otherone is based on the electro-optic effect, in which an electric field applied to anonlinear crystal leads to a change in its refractive index. We use the second
scheme, which is usually called electro-optic (EO) sampling, in our experiments for
detecting free space THz radiation.
Figure 4.2 shows a schematic diagram of an EO sampling system. Here alinearly polarized beam from the same Ti:sapphire laser that is the light source forgenerating THz radiation is used as a probe beam for EO sampling. The THz
radiation generated from the photoconductive antenna and the probe beam are
carefully aligned to propagate in the same direction through the nonlinear crystal (a1 mm thick (110) oriented ZnTe crystal is used in the experiments). The THz field
can induce birefringence in the ZnTe crystal, which induces a small ellipticity in theprobe polarization that is proportional to the THz field. Following the ZnTe crystal
there is a λ/4 waveplate and a Wollaston prism. Two orthogonally polarizedcomponents of the probe beam after the Wollaston prism are focused separately
onto two photodiodes of a balanced photodetector. The time delay between the THz
pulse and the optical probe pulse is controlled by a retroreflector mounted on a

75translation stage driven by a stepper motor in the probe beam path (not shown in
the figure).

Figure 4.2: Schematic of a typical electro-optic (EO) sampling system for detectionof ultrashort THz pulses.

When the THz field and the probe pulse do not temporally overlap in the
ZnTe crystal, as shown in Figure 4.3 (a), there is no field induced birefringence, andthe probe beam polarization remains unchanged. After the λ/4 waveplate, the probe
beam will be circularly polarized, and two equal polarization components of theprobe will be separated by the Wollaston prism, and thus the output of the balanced
detector is zero [55]. The amount of ellipticity is linear in the THz field, and thus sois the voltage on the balanced detector. By scanning the delay between the THz
pulse and probe pulse, the THz electric field can be mapped out as a function of
time.

76

As shown schematically in Figure 4.3 (b) and (c), by changing the timedelay, the positive or negative part of the THz field overlaps the probe pulse in –ydirection in the ZnTe crystal, the THz electric field can induce birefringence in the
crystal based on the second order nonlinear effect, and this leads to a polarization
rotation of the probe pulse. The field induced change in refractive index of ZnTe is∆n = r41 n3 ETHz, where n is the linear refractive index and r41 is a component of theelectro-optic tensor. The phase shift of the probe pulse is [56]∆ϕ THz (ω ) =ωcn 3r41ETHz (ω )L ,(4.15)

where L is the thickness of the ZnTe crystal. With a phase change ∆φTHz (ω ) in theprobe polarization, after the λ/4 waveplate, the field component of the probe pulse iny direction has a π/2 phase shift, which makes the probe polarization slightly
elliptical. After the Wollaston prism, the two orthogonally polarized components
have slightly different intensities which can be detected. The intensity difference on
the balanced photodetector is [57]∞S = I 1 − I 2 = ε 0 c Re ∫ a 0 (ω ) ∆φTHz (ω )dω ,2(4.16)

0where c is the speed of light, ε 0 is the dielectric constant, and a 0 (ω ) is theamplitude of the probe optical field, a 0 (ω ) = A probe (ω ) exp[ik (ω ) L] . We see that theoutput signal from the balanced detector is proportional to the actual THz electricfield and the length of the crystal. In practice, dispersion in the crystal results insome distortion in the measured THz pulse shape, particularly for thick crystals or
extremely short THz pulses. A complete description of EO sampling in thefrequency domain is given in Reference [57].

77

(a)

y

(b)

y

(c)

y

x

x

x

Figure 4.3: EO sampling with three relative delays between the THz pulse and theprobe pulse, showing how the signal on the balanced detector changes depending on
the sign of the THz field overlapping with the probe pulse.

4.3 Carrier screening and THz power saturation

Generation of THz radiation from a biased photoconductive antenna isuseful because of its good noise performance and relative simplicity at low cost.

78

However, the optical-to-THz conversion efficiency is very small (typically ~0.002%)[58]. A number of different emitter designs and biasing techniques have been
explored to improve the conversion efficiency of the photoconductive THz antenna.
These include so-called “semilarge” emitters using widely spaced electrodes, highvoltage bias, and high optical power [59]; emitters using lower bias, tight focusing,and strong trap enhanced fields [9,58]; and emitters with interdigitated, closely
spaced electrodes, with half of the gaps blocked [60]. Because they do not require
high voltage or special electrode structures, emitters based on trap-enhanced fields
are attractively simple. The key limitation is ultrafast screening of the applied field
by photoinjected carriers [23,43,58]. This screening depends strongly on the laser
excitation spot size, becoming less important for a larger spot.
As discussed previously, in a biased photoconductive antenna theacceleration of photocarriers induced by the bias electric field gives rise to the THz
emission. At the same time, the carriers moving in the bias field can partially
screen out the bias [23]. The screening of the bias field can be classified into two
main categories: direct Coulomb screening from the separation of the photocarriers,
and radiation screening from the generated THz radiation itself [61,62]. Both of thetwo screening effects are related to the spatial dynamics of the carriers, such as
their velocity and acceleration in the applied electric field. Consider a point charge
moving with velocity v and acceleration a in a bias field, the electric field it
generates is [63]:



q 
1
(1 − v 2 / c 2 )( Rˆ − v / c) Rˆ × [( Rˆ − v / c) × a ] 
E=
{
+
} ,

4πε  1 − Rˆ ⋅ v / c 3
R2
c2 R


 tr

(4.17)

where R is the position vector directed from charge q to the point of observation, c isthe speed of light and ε is the permittivity of the medium. The position vector,

79

velocity and acceleration are evaluated at the retarded time tr = t – R/c. When thedrift velocity of the photocarriers is much less than the speed of light, the electricfield in Eq. (4.16) can be split into two parts: the contribution of the radiation field(Er) and Coulomb field (Ec) [63,64]

E ≅ Ec + Er =q  Rˆ q  Rˆ × ( Rˆ × a )  2 +
4πε  R  t 4πεc 2 Rtr

(4.18)r

Both the Coulomb field, which is induced by charge separation, and theradiation field, which is from the acceleration of charges, are in the opposite
direction to the applied bias field, and thus contribute to the reduction of theeffective carrier-driving electric field in a photoconductive antenna. When thescreening field approaches the external bias field within the time scale of the opticalpulse, the THz emission is suppressed. In a typical experiment, when the excitation
spot size is fixed, with increasing optical power, the increase in carrier density leads
to a rapid decrease in the distance between carriers, and an increase in the numberof carriers that contribute to the Coulomb screening, thus the Coulomb screening
field increases with increasing optical power. Further more, under higher carrier
density, the radiation screening also gets stronger due to the carrier numbers which
contributes to the radiation screening also increase. As a result, the THz radiation
saturates quickly under high optical excitation power. For a fixed opticalpower/carrier density, the Coulomb screening is dominant when the excitation spot
size is no larger than 100 µm FWHM (this value probably depends somewhat on theTHz wavelength), and when the spot size is larger than 100 µm, radiation screening
dominates [23]. On the other hand, under a given optical power, one can decrease
the carrier density by using a large excitation spot. A bigger spot leads to anincreasing distance between carriers and a decreasing number of carriers that
contribute to the Coulomb and radiation screening field [23,43,58].

80

Due to their strong dependence on carrier density, both Coulomb andradiation screening field decrease with increasing excitation spot size. As a result,
one can think about improving the optical power-to-THz conversion efficiency byminimizing the screening field with larger excitation spot. However, simply
increasing the excitation spot size is not the best option, since the electric field in a
typical M-S-M structure THz emitter is strongly enhanced only within a few
microns near the anode edge due to the trap-enhanced field (TEF) [9]. If a large spot
size means only a small portion of the excitation spot can overlap with the narrow
TEF region, the THz generation efficiency will be strongly compromised. Thus inprevious studies, an elliptical focusing technique was adopted to obtain effective
THz generation using a maximally broadened excitation aperture in the TEF region[23,58].

4.4 RF bias THz generation

Instead of using an elliptical laser spot [23,58] or a complicated electrodestructure that lowers the effective active area of the device [60], here we use the RFbiasing technique, which results in a uniform electric field across the sample.
Unlike the trap-enhanced field device, using a large laser excitation spot with an
RF bias does not lower the average field within the laser spot. The large spotincreases the saturation threshold of optical excitation, and there are a number ofother advantages as well.

4.4.1 Experimental layout

We use a simple electrode structure that is insulated from the sample, as
described in Chapter 2, and rapidly oscillate the bias at a frequency near the laser

81

repetition rate to prevent the screening of the field by space charge during the time
between pulses. Although an oscillating bias has been used at lower repetition rates
corresponding to amplified laser systems [65], as far as we know it had not beenpreviously scaled up to the repetition rates of Ti:sapphire or fiber oscillators.
Figure 4.4 shows a schematic of the THz generation and detection
apparatus. The light source is the same mode-locked Ti:sapphire laser we used in
our ER and EA experiments, with a spectrum centered at 830 nm, a FWHM of 50
nm, and a repetition rate frep = 89 MHz. The pulse energy is 5.5 nJ with about 70 fs
duration. The laser beam is divided by a beam splitter, one output of which is used
to pump the photoconductive emitter. The other output is used as the probe pulse inan EO sampling system using a 1-mm-thick ZnTe (110) crystal. The emitter is abulk semi-insulating (100)-oriented GaAs wafer, with a 100-nm-thick layer of siliconoxide on the surface as an insulating layer, and Au electrodes separated by 100 µm
on top of the insulating layer. More details about the sample structure werediscussed in Chapter 2. The THz radiation from the photoconductive antennapasses through the back of the sample with a silicon hemisphere lens attached forbetter output coupling and THz collection efficiency, and is collected and focused
into the EO sampling system by a pair of parabolic reflectors. The probe beam is
directed into the ZnTe crystal using a mirror mounted on a long post. This blocks asmall portion of the THz beam.

82

Figure 4.4: Radio-frequency biased THz generation apparatus, showing both theemitter and the EO sampling system.

A radio-frequency bias is applied across the electrodes with a frequency fnear the laser repetition rate. The THz signal is modulated at the difference
between the RF bias frequency and the laser repetition rate, and that difference canbe set to any frequency. We use a modulation frequency in the kHz frequency range,
where the lock-in amplifier has the best performance. More information about the
RF bias and heterodyne detection system can be found in Chapter 2 and Appendix
A.

4.4.2 Optical power dependence and bias dependence

As demonstrated in the previous chapter, because the electrodes are
insulated from the GaAs sample, current injection and the resulting trap-enhanced
fields do not form [24]. Although the bias field strength we have been able togenerate (~10 kV/cm at 89 MHz and ~30 kV/cm at 45 MHz as measured byelectroreflectance [24]) using the RF bias technique is not as strong as the localized

83

trap-enhanced fields seen in samples with DC bias (~200 kV/cm) [9], the area overwhich the electric field extends is considerably larger, essentially the entire region
between the electrodes.

One can therefore use a much larger laser spot size,

resulting in a higher saturation threshold for the THz power emitted as a functionof optical pump power. The electroreflectance data presented in [24] were acquired
using very low optical powers in the nanowatt range, whereas to generatesignificant THz power, much larger optical powers in the milliwatt range must be
used. Given the strong dependence of screening on the photoexcited carrier density,
it is not obvious that the uniform field observed in electroreflectance will exist at
high optical powers; however, our results confirm this fact. A typical THz trace,
measured by EO sampling, obtained using the RF technique and the THz power
spectrum, found by taking the discrete Fourier transform of the time domain data,
are shown in Figure 4.5. The oscillations after the main THz pulse are caused bywater absorption lines in the air as well as absorption and dispersion in the ZnTe
crystal.

(a)

84

(b)

Figure 4.5: THz radiation signal measured using a photoconductive emitter with aradio frequency (RF) bias. (a) Typical THz trace. (b) THz spectrum.

The peak electric field can be found using Eq. (4.14) from the EO samplingsignal. We find a peak THz field of 20 V/cm for an input power of 160 mW. Using an
estimated spot size of 50 µm for the THz beam focused at the ZnTe crystal, our THz
power is around 100 µW and the optical power-to-THz efficiency is close to 0.1%,
this result is comparable with other techniques [58]. The maximum THz signal was
found for a laser spot size of ~80 µm FWHM. We measured the dependence of theTHz field on the optical power; the results are shown in Figure 4.6. For a large spot(~80 µm), shown as filled circles in Figure 4.6, we find a steadily increasing THz
field, whereas for a small spot (~8 µm), shown as empty circles in Fig 4.6, we findsaturation behavior. This behavior is consistent with theories of ultrafast screening[23,43,58], because screening is much stronger for a small excitation spot. The peakfluence at which saturation begins is consistent with previous studies [58].

85

Figure 4.6: Dependence of peak THz field on optical power for various excitationgeometries. Filled circles: large spot, RF technique; empty circles: small spot, RF
technique; filled triangles: large spot, DC technique; empty triangles: small spot,
DC technique.

The narrower emission pattern (and thus, better collection efficiency by theTHz optics) for a larger excitation spot size may also explain part of the differencein the signal size between the dc and RF bias techniques. This is an important
advantage, because in a typical spectroscopy apparatus, the THz is collected bysimilar optics and sent through a sample. It is much easier to collimate a THz
source with a narrower emission pattern. We found that Vmon, which is the monitor
for tuning the LC resonant circuit (as shown in Figure 2.5), was lower when thelaser was on the sample than when there was no illumination, a sign that thepresence of free carriers spoils the Q of the LC resonant circuit.
The bias dependence of the THz field strength generated by the RFtechnique, shown in Figure 4.7, shows a roughly linear dependence over theaccessible range of RF powers, as the basic theory presented in Section 4.1 predicts.

86This shows the potential for improved performance if the bias field can be increased,
for instance by using a more powerful RF amplifier.

Figure 4.7: Dependence of the peak THz field on the bias voltage using the RFtechnique.

.
To investigate the role of carrier injection in these structures, we comparedemitters with insulated electrodes to emitters with an identical electrode pattern,
but no insulating layer. With an RF bias, the direct contact samples show a similar
power and bias dependence but are susceptible to failure at the highest opticalpower and largest bias. Examination under a microscope revealed damage where
the laser was focused, the migration of the gold across the gap has shorted out theelectrodes. With a constant bias of 40 V, we find a smaller THz field, which can bepartially explained by the fact that the THz was emitted only from the region of thesample near the electrodes. The samples with a dc bias also showed susceptibility to
failure. The insulated samples showed no signs of damage after long term operation

87

at the highest optical and RF power, which is an important advantage over typicalphotoconductive emitter designs.

4.4.3 Spatial dependence of the THz field

To investigate the field distribution in the presence of strong opticalexcitation, we also examined the dependence of the signal on the position of thelaser spot (of width 8 µm) on the emitter. By scanning the sample under the laser
spot with a stepper motor, the peak THz field as a function of the spot position
between the two Au electrodes was measured. Results are shown in Figure 4.8. We
compared the insulated, RF-biased emitter with an emitter with direct contacts
with a constant bias of 40 V. The RF bias produces a more uniform field
distribution, as measured by the THz emission (which is proportional to the bias
field). The emitters with direct contacts showed the trap-enhanced field effect, with
THz emission peaked near the anode [9]. We found, interestingly, that for RF bias,
the emission was peaked near the electrodes at low optical power, but very uniform
across the sample at high optical power. Neither of these results is consistent withthe distribution measured by electroreflectance at very low optical power [24],which, as discussed in the previous chapter, was consistent with a calculation for a
perfect dielectric with this electrode geometry [39]. The high-field dynamics ofcarriers and the effect of the electrodes on the screening are probably involved; moretheoretical work is needed.

88

Figure 4.8: Dependence of the THz peak field on the position of the laser spot for a
focus of 8 µm; the shaded region corresponds to the gold electrodes. Short dash, RFinsulated high optical power (150 mW); long dash, RF-insulated low optical power(15 mW); solid line, dc direct low optical power (15 mW).

4.5 THz generation and electroreflectance in LT-GaAs

Evidence from electroreflectance (ER) and THz emission has been given for
the trap-enhanced field in a DC biased sample in the previous chapter and this one.
But one may wonder, how well does the field measured by ER correlate with thespatial dependence of the THz emission? We decided to measure ER and the THz
radiation from the same sample simultaneously to answer this question. For thisexperiment, low temperature-grown (LT) GaAs was used because it makes a morereliable THz emitter. The unique properties of low-temperature-grown GaAs (LTGaAs), such as subpicosecond carrier lifetime, large resistance and high carrier

89

mobility, make it one of the most widely used materials in photoconductive THz
emitters [64,66].
Light from the Ti:sapphire laser, the same one we used in our ER and THz
generation experiments using SI-GaAs, is focused onto the LT-GaAs sample by amicroscope objective. The focal spot size was measured to be approximately 5 µm
FWHM. The back-reflected light from the sample is collected by a beamsplitter,narrowed by a monochromator to about 1 nm bandwidth (with a variablewavelength), and then focused on a photodiode which measures ∆R at themodulation frequency of the bias field. Note that, unlike in the ER measurements
described in the previous chapter, here we excite with a broadband pulse andmonochromatize after reflection from the sample. The theory predicts an identical
ER spectrum; one normally narrows the spectrum first so that the optical power at
the sample is minimized, but here a broadband pulse is required to create THz
radiation. On the back side of the sample, the emitted THz radiation is collimatedand focused by the off-axis parabolic mirrors into an electro-optic (EO) samplingsystem. The sample is mounted on a stepper motor in order to examine the laser
spot position dependence of the ER signal and the peak THz field. A schematic plotof the experimental setup is shown in Figure 4.9.

90

Vbias

Figure 4.9: Experimental apparatus for simultaneous ER and THz measurement in
LT-GaAs.

The LT-GaAs sample was grown by Dr. Richard Mirin at NIST. It consistsof a 1.8 µm layer of GaAs grown at 400 o C on a semi-insulating GaAs substratewith no buffer layer. Gold electrodes with 180 µm spacing are evaporated on the LTGaAs surface directly. The gap between two electrodes is made by a very thin wire
which acts as a shadow mask. A bipolar square waveform bias changing from 0 V toa positive value, then back to 0 V, then a negative value, is applied to the sample
with a repetition frequency of 1 kHz. The amplitude of the positive and negativesquare wave can be adjusted anywhere between 0 V and 500 V. This special
waveform is supposed to give a more uniform electric field, according to reference[7]. The EO sampling signal was detected by a lock-in amplifier referenced to 1 kHz,91

while the field induced change in reflectance ∆R was measured using a second lockin amplifier referenced to the second harmonic of 1 kHz.

Figure 4.10: Two-dimensional plot of the ER spectrum of LT-GaAs sample with 250
V DC bias vs. laser spot position. The average excitation power is 15 mW. The redline shows the approximate position of the anode edge.

By modulating the bias field, a change in reflectance spectrum is detectedthat on first glance looks similar to the Franz-Keldysh oscillation. By moving thesample under the focal spot, in Figure 4.10, spatial map of the ER signal shows
strong enhancement near the anode edge due to the trap-enhanced electric field, as
expected for this sample structure with electrodes in direct contact to GaAs.
However, in order to get strong THz emission, a high optical power of 15 mW isapplied onto the sample. Under this strong illumination the field modulated
reflectance signal, which shows a very strong and unusual dependence on both
optical power and bias field, is not consistent with the Franz-Keldysh lineshape in92reflectance, as discussed in the previous chapter. Figure 4.11 (a) and (b) show theER signal under 150 V and 200 V bias. The red curves in both (a) and (b) are the ER
signal with 15 mW incident optical power.

Figure 4.11: Electroreflectance signal from LT-GaAs. (a) Under 150 V bias. (b)Under 200 V bias.

93

The green curves correspond to 1.5 mW and the blue curves correspond to 0.15 mW
incident optical power. All the ER signals are normalized by the voltage from thephotodiode to provide ∆R/R. Whereas at low excitation, the Franz-Keldysh
lineshape was observed in ER in the experiments described in the previous chapter,the data in Fig. 4.11 is clearly inconsistent with the Franz-Keldysh theory. The
unexpected signal might be related to the field modulation of the photoinjected
carrier density, which is much higher in the case of strong illumination. Also, with ahigh photocarrier concentration, a strong carrier screening effect must also be
involved, and this makes the results more difficult to understand.
By adjusting the time delay of the probe pulse, the THz electric field
generated from the LT-GaAs photoconductive antenna can be mapped out by theEO sampling system. In Figure 4.12 the THz electric field is plotted as a function ofprobe time delay. The bias applied across the gap between electrodes is 225 V.

Figure 4.12: THz electric field measured using a LT-GaAs emitter under DC bias.

94

The spatial dependence of the THz peak field on the excitation spot positionis also studied by moving the laser spot cross the gap between electrodes. As we
expected, the THz field is also strongly enhanced near the edge of the anode due tothe trap-enhanced electric field. The peak THz field can be found at the same
position where we find the largest ER signal. The spatial dependence of the peakTHz field and the peak absolute value of the ER signal is shown in Figure 4.13.
Illuminating near the edge of the anode gives rise to a THz radiation about 30 to 50
times stronger than illuminating the center of the gap between electrodes.

Figure 4.13: THz electric field (red) and ER signal (blue) versus laser spot position.

Under a maximum bias of 250 V, a peak THz field of 7.34 V/cm ismeasured. The field strength we found is a reasonable value compared withprevious results [58], considering that the focus is tight on the sample and we did

95

not use a silicon hemisphere lens to capture the highly divergent THz radiation. In
Figure 4.14, the measured THz field is plotted as a function of the amplitude of theapplied bias. The THz field shows a roughly linear dependence on the bias.
However, for the THz emitter with electrodes in direct contact to the LT-GaAs,
increasing the bias beyond 300 V will result in sparking from one electrode to theother, causing irreversible damage.

Figure 4.14: THz field as a function of the applied DC bias for an LT-GaAs based
emitter.

4.6 Summary

In summary, we have developed a new technique for generating THz
radiation effectively through a photoconductive THz emitter. The emitter with its
gold electrodes insulated from the GaAs surface is biased using a rapidly oscillating

96

bias near the laser repetition rate. Its chief advantage over photoconductive
switches utilizing trap-enhanced fields is in spreading the electric field over a wider
area. This provides improvement of the optical power saturation because of thelarger excitation area. Our technique also does not require a specially patternedsample, like interdigitated structures; though using electrodes patterned on thesurface with a thin insulating layer between is optimal. The RF signal is amplified
by a passive LC circuit to produce the high voltage; it does not require active highvoltage electronics, which often produce electrical pickup. One can also modulate
the bias field at the repetition rate, leading to an effectively constant train of THz
pulses—we have tested that scheme using a mechanical chopper to modulate thepump beam (we set the phase of the bias waveform to optimize the THz emission).
This technique can be an attractive alternative for generation of THz
radiation using unamplified mode-locked laser. Currently we are limited to bias
fields of roughly 30 kV/cm at radio frequencies, but the field can be increased
further by optimizing the LC resonant circuit or using a more powerful RF
amplifier. Other improvement like using an antireflection coating as the insulating
layer to minimize the reflection loss can also help increasing the optical power-to
THz conversion efficiency.
We also studied the electroreflectance and THz generation simultaneously
from a DC-biased LT-GaAs sample with Au electrodes patterned directly on top.
The existence of an enhanced electric field near the positively biased electrode isobserved from the spatial dependence of both THz radiation and the ER signal. The
ER signal is not consistent with the lineshape expected for the Franz-Keldysh effectwhen an optical power is used that is sufficient to generate significant THz
radiation. A peak THz electric field of 7.34 V/cm from this LT-GaAs emitter isdetected near the edge of anode, where we also find the strongest ER signal.

97

CHAPTER 5

Quantum interference and electric field induced coherent control
In this chapter, experiments on quantum interference control (QUIC) via
one- and two- photon absorption pathways and coherent control of the carrier
population injection rate in a semiconductor are discussed in detail.
First we start with a brief introduction about quantum interference
injection of ballistic current and carrier population control in GaAs, and then follow
with a description of a theory that predicts that an electric field enables coherent
control of the carrier population. Then, in the experimental part, two measurementsof the field induced coherent control will be discussed. An all-optical measurement
and results are presented first. The results include the bias and optical polarization
dependence of the carrier population in a GaAs epilayer. Then an electricalmeasurement scheme, in which we detect the change in the photocurrent read out
by the same electrodes used to apply the bias to the sample, is discussed. The lastpart describes our measurement of ballistic current injection (without an appliedfield) via QUIC in erbium doped GaAs.

98

5.1 Quantum interference control of ballistic current in a semiconductor

When a semiconductor is illuminated with coherent optical beams at
frequency ω (leads to two-photon absorption if ћω is less than the band gap) and 2ω
(leadstoone-photon

absorption),quantuminterference

betweendifferent

absorption pathways that connect the same initial and final states can cause effects
that depend on the relative phase between the frequency components, enabling
control over the end state of the absorption process. This is analogous to theinterference of two beams in a Young’s double slit experiment. This quantum
interference in semiconductors leads to the injection of a ballistic charge current[25,26,27] or pure spin current [31], where the current direction is controlled by thepolarization of the light and the relative phase between the light components at
frequency ω and 2 ω. By measuring the current injected in a semiconductor device,it has been shown that the phase evolution of the pulse train from a mode-locked
laser can be measured [32,33]. The signal can be used to lock the laser’s carrierenvelope offset frequency [34,35], an important function for optical frequency
metrology using mode-locked lasers.
For one-color excitation in an unbiased semiconductor, either one-photon
absorption of the 2 ω field or two-photon absorption of the ω field can generate a net
carrier density, but neither can generate a net charge current without taking
advantage of some material asymmetry. On the other hand, for excitation with thetwo frequencies simultaneously, the two types of quantum mechanical pathways
that connect the valence band and conduction band are both present, and quantum
interference between these two pathways leads to phase-dependent charge
distributions that are asymmetric in k-space. Since the momentum is proportional
to k, an asymmetric charge distribution is equivalent to a net current. As a result,

99

the ballistic charge current which comes from the asymmetric distribution can becontrolled coherently.
Let us examine quantum interference in absorption in a little more detail.
When both ω and 2 ω fields excite a semiconductor and satisfy ћω < Eg < ћ2ω, thetotal carrier injection rate can be written as the sum of three terms [36]:n& = n& 2ω + n&ω + n& I ,(5.1)

where the first two terms correspond to the population injection rates for one- andtwo-photon absorption acting alone, respectively. The last term is the quantum
interference part of the population injection rate. Each of these total carrier
injection rates is a sum over the Brillouin zone of the rate for a particular
wavevector k, for the interference term:vv& I (k ) ∝ ∑ a cv2ω (k ) * a cvω (k ) + c.c = 0 .
n& I = ∑
n
v
v
k

(5.2)

c ,v , kHere acv2ω and acvω are the one-photon and two-photon transition amplitudes. It canbe shown that, for a simplified model of the bands in an isotropic semiconductor,
acv2ω is independent of k and acvω is proportional to |k|. One hasvv
2
a cv2ω (k ) * a cvω (k ) ∝ k E 2ω Eω e i ( 2φω −φ2ω ) ,(5.3)

which is antisymmetric in k-space. As a result,n& I (k ) ∝ k E 2ω Eω e i ( 2φω −φ2ω ) + c.c.
2

(5.4)

100

is antisymmetric in k-space [26,28]. Therefore, QUIC allows for the optical injectionof a total carrier population n& I (k ) that is distributed asymmetrically in k-space, as
shown in Figure 5.1. The asymmetry results from constructive interference in one
direction (determined by the optical polarization) in k-space, and destructive
interference in the opposite direction. This leads to more carrier injection in one
direction than the other, i.e., more in +k than in –k here in Figure 5.1.

Figure 5.1. Schematic of asymmetric carrier distribution in k-space under two-color
excitation.

Charge current in the semiconductor can be defined as a weighted averageof the carrier population as a function of k. While the interference term sums to zerofor the carrier injection rate, the asymmetric distribution of carriers in k-spaceresults in a ballistic current:
v
v 2
v
v v
J& ∝ ∑v qv (k ) ⋅ a cv2ω (k ) + a cvω (k ) ,c ,v , k

(5.5)

101
v v
where v (k ) is the velocity of carriers in k-space. The net one- and two-photon charge

currents are zero, thus the total charge current is proportional to the interferenceterm only:
v
v
v
v v
J& ∝ ∑v qv (k ) ⋅ a cv2ω (k ) * a cvω (k ) + c.c. .

(5.6)

c ,v , kUsing symmetry arguments for an isotropic material and a parabolic band model,
an equation for the ballistic charge current injection rate has been derived [67]:
v v v
v
v v
q
J& = −iD [ B1 Eω ( Eω ⋅ E 2*ω ) + B2 E 2*ω ( Eω ⋅ Eω* )] + c.c. ,
h

(5.7)

v
v
where Eω and E 2ω are the electric fields of ω and 2ω beam. The constant D is

defined asD=

2 q 3 E P ( 2hω − E g )60π m
h 4ω 4

3/ 2

,

(5.8)

where EP is the Kane energy, Eg is the bandgap energy, q is the electron charge, andm is the electron mass. The coefficients B1 and B2 depend on material parameters(i.e., Ep and Eg) and the effective mass of the various band involved,

B1 = 3.2 − 0.3ε ,(5.9)

B2 = 0.036 + 0.19ε ,(5.10)

where the energy dependence termε=2hω − E ghω.

(5.11)

102

In case of QUIC when ћω < Eg < ћ2ω, ε lies in the range 0 < ε < 1. For bulk GaAs
at room temperature with ћω = 0.8 eV and Eg = 1.42 eV, ε = 0.225 and B1 = 3.13, B2= 0.079 can be found. Assuming two pulses with frequency ω and 2ω propagating in
the z-direction with parallel linear polarizations along x-direction, the electric fields
of incident beams areandvEω = Eω ⋅ e iφω xˆ

(5.12)

v
E 2ω = E 2ω ⋅ e iφ2ω xˆ .

(5.13)

Puting Eq. (5.12) and Eq. (5.13) into Eq. (5.7), the ballistic charge current isq2
J& = 2 DB1 Eω E 2ω sin(2φω − φ 2ω ) xˆ .
h(5.14)As a result, for parallel linear polarizations of the incident beams, a charge current
will be injected along the direction of the incident polarizations. By adjusting therelative phase of the ω and 2ω beams, the magnitude and sign of the current can becontrolled. This effect has been observed previously both electrically [27] and bymeasuring the emitted THz radiation [84]. In the last section of this chapter, ourmeasurement of current injection in Er-doped GaAs will be described.

103

5.2 QUIC population control in noncentrosymmetric medium

In the previous section, ballistic current generation through quantum
interference in an unbiased semiconductor was discussed. In a (100)-oriented
zincblende semiconductor (such as GaAs, the subject of all the experiments in thisthesis) with two-color excitation, the photo-injected carrier-distribution in k-space ismodulated and controlled by QUIC, but without changing the overall carrier
density. For other crystal orientations, such as (111)- and (110)-oriented GaAs , theoverall carrier density can also be controlled coherently by QUIC. This process may
be observed when the optical fields are aligned along certain crystal axes.
In a medium that lacks inversion symmetry in k-space, e.g. (111)-oriented
GaAs, Eq. (5.2), the approximations for the transition amplitudes made earlier
break down. Roughly, the amplitude for one photon absorption has a slight linear
dependence on k, and thus the sum over k of n& I (k ) , is not zero. The interferencecomponent of the total carrier population is [68]:n& I = ∑n& I (k ) = ∑η Ijkl Eωj* Eωk * E 2l ω + c.c.
v
k(5.15)

jkl

where the tensor η I is related to the imaginary part of the second ordersusceptibility χ ( 2) [36]. For the general case of ω and 2ω beam both linearlyv
polarized, using Eω = Eω ⋅ e iφω and likewise for the 2ω component, Eq. (5.15) reduces
to:n& I = η Ijkl (ω ) E 2jω Eωk Eωl ⋅ exp(iφ 2ω − 2iφω ) + c.c. .

(5.16)

From Eq. (5.16), the interference part of the total carrier population n& I depends on
the parameter φ 2ω − 2φω involving the relative phases of the two fields. In a crystal

104with center of inversion symmetry, in the absence of an external field, η Ijkl (ω ) = 0 . If
there is no center of inversion symmetry, then in general η Ijkl (ω ) ≠ 0 for ћ2ω above
the bandgap, and coherent control of population is possible even in the absence ofan external field [36]. We refer to this as the χ ( 2) -enabled population control process
because it is enabled by the second-order nonlinear susceptibility. It has been
experimentally observed in a (111)-oriented GaAs sample using optical [36] andelectrical [69] detection.

5.3 Electric field induced population control

The χ ( 2) nonlinearity in general is only nonzero for certain opticalpolarizations, but when a DC electric field is applied, one can get an effective χ ( 2)that enables coherent control of carrier injection without aligning the optical fields
along certain crystal directions. Just as in the case of electric-field induced second
harmonic generation [70], when a DC electric field is applied to a GaAs sample, it
can combine with the always present third-order susceptibility χ (3) to produce aneffective χ ( 2) . This effect enables control of the carrier population through quantum
interference between one- and two-photon absorption, even in a crystal with acenter of inversion symmetry [71].
By extending a theoretical framework for the one-photon Franz-Keldysh
effect to multi-photon processes, Wahlstrand and Sipe calculated QUIC in thepresence of a static electric field. In Eq. (5.18), the tensor η Ijkl (ω ) , which describes
the efficiency of the field-induced population control process as a function of ω, canbe found analytically by assuming the valence band to conduction band velocity
matrix element Vcv is k-independent and assuming parabolic bands separated by adirect bandgap hω g [71]:

105

2


 2  2ω − ω g  − 2ω − ω g Ai 2  − 2ω − ω g  −  Ai '  − 2ω − ω g   Ai  −ΩΩ   Ω  Ω 
e 3 E dc2


zzz
z 2
.
η I (ω ) = 3 4 Vcv +ω2Ω
2h ω Ω









(5.17)In Eq. (5.17), Edc is the applied DC electric field, Ω = (e 2 E dc2 / 2 µh )1/ 3

is the electro-

optic frequency, µ is the reduced mass, and Ai(x) is the Airy function.
The spectrum of the field-induced population injection tensor shows FranzKeldysh oscillations for 2ω > ωg. The oscillations are expected to be strongly damped
at room temperature, just like the case of the one-photon Franz-Keldysh effect [47].
Eq. (5.17) can thus be simplified by using the asymptotic form of the Airy function,
which gives the spectrum in terms of sin 2 ( x) and cos 2 ( x) , and replacing them withthe average value of 0.5,

η Izzz (ω ) =2ω − ω g 
2e 4 µ 1 / 2 z 2 
1
E ,
V
−
cv
 4 2ω − ω dcω
2πh 9 / 2ω 4g

(5.18)

for 2ω > ωg. The tensor η I (ω ) is a real number and the population injection rate
through quantum interference is thus directly proportional to cos(φ 2ω − 2φω ) , whereφ2ω − 2φω is the relative phase between ω and 2ω beams. In this limit of strong
damping, using expressions for the one- and two-photon absorption under theparabolic band approximation, the fractional change in carrier injection rate caused
by quantum interference can be found as
6e 2 E dc Eω2 E 2ω ω (4ω g − 7ω )n& I,=n& 16e 2 Eω4 (2ω − ω g ) 2 + 3E 22ω µhω 4 (2ω − ω g )(5.19)

106

for 2ω > ωg. As a result, in the large-broadening limit, the QUIC induced carrier
injection rate is linear in the applied electric field Edc .

5.4 Experimental layout – all optical measurement

In order to observe field-induced population control experimentally, we
performed an all-optical measurement on (100)-oriented GaAs samples at room
temperature. A diagram of the experimental apparatus is shown in Figure 5.2. The
technique is similar to what was used by Fraser et al. to observe conventional
population control in (111) GaAs [36]. The most important difference is the additionof a bias across the sample. A two-color pump pulse injects carriers through oneand two-photon absorption and quantum interference. Because the absorption
coefficient of the GaAs is affected by the photoexcited carrier density, thetransmission of a probe pulse is also sensitive to the density [72]. We look for achange in transmission that depends on the phase parameter φ 21 = φ 2ω − 2φω . The
light source for the pump beam is an optical parametric oscillator (OPO) using
cesium titanyl arsenate (CTA), with a signal wavelength tuning range from 1400
nm to 1650 nm, and typical output power of 200 mW. The OPO is synchronously
pumped by a mode-locked Ti:sapphire laser with 1.3 W output power at a repetition
rate of frep = 76 MHz. The signal output of the OPO, centered at 1580 nm, is doubled
using a barium borate (BBO) crystal, and the two harmonics are split by a prism
into two paths, the relative delay of which can be adjusted using mirrors mounted
on piezoelectric transducers. The harmonics are recombined using the same prism.
We use the residual pump light, centered at 827 nm, as a probe beam. The pulse
widths of the OPO fundamental, second harmonic, and probe pulses areapproximately 160 fs, 220 fs, and 120 fs, respectively.

107

Figure 5.2. Schematic of the experimental apparatus, including the two-color
interferometer and radio frequency biasing scheme. OPO: optical parametricoscillator, BBO: β-barium borate crystal, BP: bandpass filter, DDS: direct digitalsynthesizer.

The pump and probe beams are focused on a common spot between theelectrodes. The ω beam average power is typically 160 mW and the 2ω beam
average power is 6 mW. Part of the probe beam is split off before the sample andused as a reference for balanced detection. The balance detection helps us cancel thenoise induced by fluctuations in the probe beam. The spot size of the pump beams isroughly 25 µm, and the spot size of the probe beam half that. It is important to havea smaller spot for the probe beam, in order to make sure it only detects the phasesensitive carrier-injection from the two-color pump. To detect only the contribution
from QUIC processes, we dither φ 21 using a mirror mounted on a piezoelectrictransducer. A lock-in amplifier referenced to the dither frequency measures only the

108changes in the probe transmission T due to changes in φ 21 ; the signal is proportional
to dT / dφ 21 . This detection scheme works fine for our experiment, since we’reinterested in observing field-induced QUIC, instead of doing an absolutemeasurement. Using another mirror mounted on a piezoelectric transducer, we
slowly ramp the phase parameter by more than 2π and average the data overapproximately 100 ramp cycles.

Figure 5.3. GaAs epilayer attached to a transparent sapphire disk by epoxy withinsulated electrodes.

We use a 1 µm thick, undoped GaAs epilayer grown by molecular beam
epitaxy. The GaAs sample is attached to the sapphire disk using transparent epoxy,and the substrate is then removed by chemical etching. A 100 nm thick insulating
SiO2 layer is deposited on the surface. More details about the thin sample can befound in Appendix C. Using photolithography, Au electrodes separated by 100 µm
are then patterned onto the sample. A schematic plot of the GaAs sample structureis shown in Figure 5.3. To apply an effective dc electric field Edc, we use a radio

109

frequency (RF) bias synchronized to frep. As mentioned numerous times previously,
this technique results in an effectively uniform electric field in the plane of thesample [24] while avoiding highly nonuniform trap-enhanced fields [9]. We derive a
harmonic of frep from a photodiode and use it as a clock for direct digital synthesis(DDS) of a synchronized, variable phase bias waveform at frep. A resonant LC circuit
passively enhances the voltage across the electrodes. Detailed information about the
RF bias system was given in Chapter 2. The phase of the waveform is optimized sothat the peak of the waveform occurs when the optical pulse arrives at the sample.
The sign of the field may be reversed by changing the phase of the RF signal by πusing the DDS. Linear electroabsorption on the sample studied here, with analysisof the spectra as described in Chapter 3, at the same bias voltage and optical power,
show that we achieve fields of approximately 20 kV/cm. Two samples were studied:
one with the RF field pointing along a <001> crystal direction and the other withthe field along <011>.

5.5 All-optical measurement results and discussion

The primary result of the all-optical measurement of electric field induced
phase-dependent population control is shown in Figure 5.4. When there is no bias
applied (blue dotted line), the transmitted probe power shows no modulation at thedither frequency as the phase parameter φ 21 = φ 2ω − 2φω is swept by the slow ramp
piezo. Only when the uniform electrical field [24] generated by the RF bias is
applied do we clearly observe the change in probe transmittance as a function of thephase parameter φ 21 , with both positive (solid black line) and negative bias (dashed
red line). Here the electric field Edc ║ <001> crystal direction. For light at ω and 2ω
normally incident on the (100) GaAs sample, for which population control enabled

110by the crystal symmetry is forbidden [36], we observe modulation of the transmitted
probe beam only when the external bias is applied.

Figure 5.4. Modulation of the transmitted probe at the dither frequency as thephase parameter φ 21 = φ 2ω − 2φω is slowly swept. The signal for Edc ║ <001> is shownfor positive (solid black), negative (dashed, red) and zero (blue, dotted) bias.

The signal scales as P21ω/ 2 Pω as predicted by the theory, where Pω ( P2ω ) is thepower of the ω (2ω) beam. The maximum observed modulation of the carrier density
was 2%, which is within an order of magnitude of the theoretical prediction (theexpected carrier density modulation from Eq. (5.21) is 0.6%). For a variety ofreasons, it is nearly impossible to extract absolute efficiencies from the experiment.
In an optically thin sample, the fractional change in the pump-probe transmission
would equal the fractional change in the carrier density. Due to the fact that the
thickness of our sample is comparable to the coherence length (~1.2 µm [37])between the ω and 2ω beams, the amplitude of the modulation varies as a functionof depth, complicating the analysis. Also, a cascaded process arising from electric

111

field-induced second harmonic generation (EFISH) [70] followed by opticalinterference causes a change in the rate of carrier injection that depends on therelative phase φ 21 . This is a two-step process in which second harmonic generationof ω generates a field at 2ω which then interferes optically with the light at 2ω. Like
population control, this process leads to a change in the injected carrier density that
depends on φ 21 , and thus would produce an identical signal in the experiment. In
this case, the second harmonic generation would be electric field-induced [70].
However, in a bulk semiconductor like the one studied here, there is ample reason
to expect that the direct process previously described in terms of the multi-photon
Franz-Keldysh effect [71] is dominant. In future experiments, it could be
distinguished from the direct coherent control processes by using a layered
structure [73]. Other complicating factors like multiple reflection and phase walkoff
inside the sample make quantitative comparison of the experimental results andtheoretical calculations challenging. Considering all these experimental issues andthe simplified theoretical assumptions, the agreement between the theory
calculation and the experimental data is reasonable. We have observed additional
evidence of field induced QUIC in experiments on low-temperature grown GaAs andEr-doped GaAs that use electrical [27,69] rather than optical detection. Biasing theelectrodes used to read out the injected photocurrent results in an enhancement in
the signal that is inconsistent with current injection alone.
The magnitude of the bias can be controlled by adjusting the output
amplitude of the DDS. The bias dependence results of the population control signal
are shown in Figure 5.5, where the amplitude of the differential transmission signal
is plotted as a function of the bias voltage for Edc ║ <001> (blue dots) and Edc ║<011> (red squares). The lines are linear fits to the data points. The signal isconsistent with the theoretical prediction in Eq. (5.20) of a linear dependence onbias, for both positive and negative bias (the latter is not shown).

112

Figure 5.5. Amplitude of the signal as a function of bias voltage for Edc ║ <001> andEdc ║ <011>. The lines are linear fits.

Two quarter waveplates were used in the two-color interferometer asshown in Figure 5.2. By double passing the ω and 2ω beam through them, thequarter waveplates act as half waveplates here. By rotating these waveplates, thedependence of the population control signal on the polarization of the optical fields
was also studied. Due to the polarization dependent reflection at the interfaces ofthe prisms in the interferometer, it is unfortunately not feasible to continuously
adjust the polarization. Here we studied four different polarization configurations
as shown in Figure 5.6, with the DC field along <110> crystal axis. The modulationof the probe transmittance is largest for all optical fields parallel to the applied
electric field (blue). When the polarization of ω (green) or 2ω (red) or both of them

113

(black) is rotated by ninety degree, the probe modulation signal falls by more than afactor of five.

Figure 5.6. Polarization dependence of the signal for Edc ║ <110>: E2ω, Eω ║ Edc(blue); E2ω ⊥ Edc, Eω ║ Edc (red); Eω ⊥ Edc, E2ω ║ Edc (green); and E2ω, Eω ⊥ Edc (black).

The results of the polarization dependence are consistent, within the noise,with the theoretical calculations using a 14-band k ⋅ p model [53,70], which predictthat the signal should be smaller by a large factor for Eω ⊥ Edc and zero when E2ω ⊥Edc. The strong polarization dependence of field-induced population control isrelated to the contribution of intraband matrix elements to the two-photon
absorption pathway, which leads to an enhanced efficiency when Eω ║ Edc. Thisenhancement also plays a role in the polarization dependence of the two-photon
Franz-Keldysh effect [85]. As in the one-photon FKE, the oscillation period depends
weakly on the DC field direction because of band warping [74]. The χ ( 2) -enabled
process is predicted to be affected weakly by the DC field, leading to oscillations inη Ijkl (ω ) . In our experiment, polarization impurity causes a small component of the

114

beam to be polarized parallel to the DC field, and we attribute the non-zero signal
for E2ω ⊥ Edc seen in Figure 5.6 to that experimental imperfection, which is aparticular problem for the 2ω beam because of the P21ω/ 2 dependence. The
polarization dependence for the DC field along <100> (not shown) is noisier but
essentially the same as the data for the DC field along <110> in Figure 5.6. We havealso observed a DC field induced modulation of the transmitted 2ω beam that
depends on the relative phase φ 21 , which was observed in the analogous process
enabled by χ ( 2) [36].

5.6 All-electrical measurement of field-induced population control

In addition to the all-optical pump-probe detection, the modulation ofcarrier-density through QUIC can also be measured electrically, which is a morepractical technique for applications. It has been proved that the two-color
population control in a (111)-oriented GaAs can produce an easily measureablechange in the photocurrent across a biased metal-semiconductor-metal structure[69]. Here we present the results of an electrical measurement for field-induced
population control in a (100)-oriented low temperature grown GaAs (LT-GaAs), andcompare the results with the all-optical detection.
In order to prevent the quick relaxation of the QUIC current due to ascreening DC field created by separation of carriers, we use a LT-GaAs sample,
which has a subpicosecond carrier lifetime and a high breakdown threshold [41].
Because of the short lifetime, the carriers get “frozen” in space before the field canpull them back, resulting in a larger signal in QUIC current injection experiments[28]. A pair of Au electrodes is patterned on the surface of the LT-GaAs sample,
providing a series of gaps between the electrodes from 6 µm up to 150 µm, as shownin Figure 5.7. In order to detect the change in the photocurrent read out from the115

electrodes, our Au electrodes are in direct contact with the LT-GaAs (the insulated
electrodes sample could not be used for this experiment).

Au

LT-GaAs

Figure 5.7. Electrode structure. Multiple gaps are shown; the electrode structure
provides gaps ranging from 6 up to 150 µm.

We apply a DC bias across the electrodes and measure the photocurrent
from the same electrodes as a function of the phase between the two-color excitation
beams. Reading out the photocurrent using electrodes has been used previously in
QUIC of ballistic current injection [27,28,33,75,76] and population control [69]experiments. In our case the signal is a combination of two processes: QUIC current
injection [27], which ballistically injects a photocurrent that depends on

φ 21 = φ 2ω − 2φω , and field-induced modulation of the carrier injection rate [71], which
modulates the carrier density and thus the photocurrent through the biased

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structure. The average current injected by QUIC in the absence of a field is < 500
pA. We use a circuit with separate transimpedence amplifiers for low frequency andhigh frequency components to detect the modulated photocurrent. This design keeps
the DC bias across the sample constant while greatly amplifying signals at highfrequencies. The gain of the DC transimpedence amplifier is 1 kΩ and the gain of
the AC transimpedence amplifier is 1 MΩ, with a low frequency cutoff of 50 Hz.
A mode-locked Er-doped fiber laser produces 150 fs pulses at 1550 nm at arepetition rate of 26 MHz. The output is stretched using dispersion compensatingfiber, amplified, and then compressed using a grating compressor. The average
power at the output of the compressor is approximately 80 mW. The light is doubled
and the harmonics split using a prism-based interferometer. The interferometer
setup is very similar to the one shown in Figure 5.2. One major difference is that in
the all-optical experiment, where a mirror mounted on a piezoelectric transducer isused to modulate φ 21 , here an acousto-optic modulator (AOM) provides the phasemodulation. Inside the interferometer, the 775 nm light (the 2ω beam) is frequency
shifted using an AOM by fAOM ≈ 78 MHz, and then recombined with the ω beam.
The frequency shift of the 2 ω beam is equivalent to a linear phase ramp. The phaseparameter φ 21 is typically modulated at approximately 20 kHz. The amplified signal
is measured using a lock-in amplifier referenced to fdet = 3frep − fAOM. The reference
signal is generated using an RF phase detector.
The AOM technique has many advantages over implementations using amoving mirror to dither the phase. The phase modulation ramps constantly over 2π,
and the modulation frequency can be varied over a wide range simply by varying
fAOM, with no complications arising from resonances in the piezoelectric transducer.
The bandwidth limit caused by the capacitance of the electrode structure has beenfound to be one of the limiting factors in carrier-envelope phase detectors [77], and a
two-pulse setup using an AOM could be used to optimize a carrier-envelope

117

detection system with no need for a carrier-envelope phase stabilized source. Also,
due to periodic tilting of the mirror on the piezoelectric transducer as it shakes, a
DC offset has been encountered in previous experiments [78]; this artifact does notarise when an AOM is used to modulate the phase.
The recombined ω and 2ω beams are focused on the sample surface using a
microscope objective. A glass slide is placed in the beam to sample some of the lightthat reflects from the sample, allowing imaging of the sample surface and precise
placement of the laser spot. At the sample, the maximum power available istypically 30 mW at 1550 nm and 400 µW at 775 nm. In all of the data presented
using electrical detection, the optical fields are linearly polarized along the DC field
direction.
By adjusting the bias field applied on a 10 µm gap of the LT-GaAs sample
between two electrodes and detecting the modulated photocurrent as a function of

φ 21 , a bias dependence of the QUIC signal (black) and the phase of the electricallydetected signal (red dashed) were measured; both are shown in Figure 5.8. Unlike
the linear bias dependence measured by the all-optical detection, shown in Figure5.5, the bias dependence measured by the electrical detection is not monotonic. As
the magnitude of the bias is increased from zero, the signal decreases to nearly zero
first, then increases rapidly, for both positive and negative bias. The phase of thesignal reverses at the same time that the signal dips to near zero. The signal isgreatly enhanced at a bias of ±2 V. The maximum fractional modulation of thephotocurrent by QUIC is 5 × 10 −3 , about a quarter as large as the signal observed in
the all-optical experiment [71], where an electric field of approximately 20 kV/cm
was present. This is consistent with the electric field calculated from the voltage
and electrode spacing of 10 µm.

118

Figure 5.8. Bias dependence of field induced QUIC population control, measured by
the electrical detection. Black line shows non-monotonic dependence of the signal
magnitude. Dashed red line shows the phase of the signal which flips by nearly π
when the magnitude dips near zero.

The photocurrent signal includes two components which are roughly π outof phase. One is a bias-independent signal, which mainly comes from ballistic
current injection via QUIC [27]; the other one depends on the magnitude of the bias,
which essentially comes from field-induced population control. Although the fieldinduced population control signal is supposed to be linear in the electric field andthus the bias, the direction of the photocurrent is the same sign as the bias,
resulting in an even symmetry of the bias dependence. From Figure 5.8, the evensymmetry of the bias-dependent photocurrent agrees with our interpretation interms of field-induced population control.
In previous studies of either the χ ( 2) enabled population control [69], or thequantum interference control of ballistic current injection [78], in case of no bias,
the QUIC signal is enhanced when the laser beam is focused near an electrode edge.
When a bias is applied, the field-induced QUIC signal also shows strong dependence
on the laser spot position. Figure 5.9 shows the bias dependence for three different
laser spot positions on a sample with 10 µm electrode spacing. This electrode119

spacing produces the optimal signal-to-noise ratio at zero bias. The spot size in thisdata is approximately 6 µm. When the laser spot is centered between the electrodes(the data in Figure 5.8 and the green curve in Figure 5.9), the bias dependence issymmetric. But when the laser spot is focused on one side (red curve) of thestructure or the other (blue curve), the enhancement of the signal is greater for
either positive or negative bias.

Figure 5.9. Bias dependence of field-induced QUIC signal on three different laser
spot positions for 10 µm electrode spacing.

The connection between the bias and spatial dependence is somewhat
clearer in a sample with wider spaced electrodes. Results are shown for a sample
with electrodes spaced by 150 µm in Figure 5.10. The φ 21 -dependent photocurrent isshown as a function of bias voltage and the position of the laser spot. Regardless ofthe bias, the signal is strongest when the laser spot, which in this case isapproximately 12 µm wide, is focused near the electrode edges. Depending on which

120

electrode the laser spot is near, we observe an enhancement of the signal for
positive or negative bias. For a bias of the opposite sign, the signal magnitude isindependent of bias.

Figure 5.10. Map of the field-induced QUIC signal magnitude as a function of bias
and the position of the laser spot for 150 µm electrode spacing. The edges of theelectrodes are at 40 and 190 µm.

The bias dependence when the laser spot is positioned near an electrode isshown in Figure 5.11. Data for two laser spots are shown, separated by 12 µm. For
the spot closer to the electrode (solid lines), a dip in the amplitude (black) isobserved at -2.5 V, and as the bias is swept through that voltage, the phase (red)changes by approximately 0.63π. The data for the spot farther from the electrode(dashed lines) shows no discernible dip in the amplitude (black), and the phase (red)121changes by approximately π/2. The data for the laser spot focused near the otherelectrode displays the same behavior, but at positive rather than negative bias. Fora wide electrode spacing, the observed phase shift is closer to the value of π/2
predicted by the theory [79] if we identify the bias-independent signal as current
injection and the rest as field-induced population control.

Figure 5.11. Bias dependence of QUIC signal in a sample with 150 µm electrodespacing, for two laser spots spaced by 12 µm: nearer the electrode (solid) and farther
away (dashed). The amplitude (black) and phase (red) of the QUIC signal are
shown.

We have observed the same general trends when using an OPO and tuningћω from 0.72 eV to 0.83 eV, showing that this effect is not localized to any particular
photon energy. We have also observed the same bias dependence in an Er-doped
GaAs sample [76]. With a widely spaced electrode structure and higher voltage, wehave also observed field-induced QUIC in a semi-insulating (SI) GaAs sample. In
SI-GaAs, no signal could be seen at zero bias. In all experiments, we observe thestrongest signal near the electrode edges.

122

From the electrical measurement, the symmetric distribution of thephotocurrent signal in bias dependence is consistent with a combination of ballistic
current injection and field-induced population control. However, many questions
remain about the interpretation of the data. The phase shift observed
experimentally, which varies depending on the electrode spacing and the laser spot
position, doesn’t agree with the π/2 phase shift predicted by the theory of current
injection and population control [79]. The enhancement of the photocurrent signal
near the electrode edge could be partially explained by the enhancement of electricfield near electrode edge due to a combination of the trap-enhanced field and thephoto-Dember effect (separation of electrons and holes generates a transient electricfield). A better model of the carrier dynamics in the metal-semiconductor-metal
structure is required to fully explain the experimental results.

5.7 QUIC of photocurrent injection in Er-doped GaAs

Quantum interference control of ballistic current injection in bulk
semiconductor has been used to measure and stabilize the phase evolution of thepulses from a mode-locked laser [33,34]. However, the relatively small signal-tonoise ratio limits the application of this technique. In order to gain a larger QUIC
signal, a semiconductor with a short carrier lifetime is preferred, because the short
lifetime prevents internal discharge of the charge built up by the QUIC current
during the time between laser pulses [28]. The most commonly used material for
QUIC of photocurrent is low-temperature grown GaAs (LT-GaAs), which has a
subpicosecond carrier lifetime.
Gallium arsenide with erbium doping or embedded ErAs nanoparticles(GaAs:Er) is another material with an ultrashort carrier lifetime. Since it can begrown using molecular beam epitaxy (MBE) at standard temperatures, the GaAs:Er123

has the advantage of being able to be more easily incorporated in heterostructures,
which is widely used in today’s electronics industry. Also GaAs:Er is expected tohave better crystal structures for the GaAs, and hence better usability. In thissection, we compare the carrier lifetime and the size of the QUIC signal from three
types of samples: an Er-doped GaAs (GaAs:Er), a LT-GaAs, and a semi-insulating
GaAs (SI-GaAs), and show experimental results that prove the QUIC signal from a
GaAs:Er is comparable to that in a LT-GaAs, and much stronger than that in a SIGaAs.
The LT-GaAs and GaAs:Er samples were grown using MBE, by Dr. Richard
Mirin at NIST Boulder. The LT-GaAs sample consists of a 1.8-µm layer of GaAs
grown at 400 o C on a semi-insulating GaAs substrate with no buffer layer. The
GaAs:Er sample consists of 6 µm of Er-doped GaAs grown on top of a 500-nm thickAl0.8Ga0.2As layer on a GaAs substrate. The Er concentration is 6 × 1018 cm-3, as
measured by secondary ion mass spectroscopy (SIMS) on a reference sample. As thepenetration depth of above-bandgap light in GaAs is on the order of 1 µm, the signal
from each sample is essentially entirely due to the LT-grown or Er-doped region.
We performed a standard optical pump-probe experiment to measure thecarrier lifetime of each sample. A mode-locked Ti:sapphire laser with 860 nm
wavelength and a repetition rate of 76 MHz was used as our light source. The pulse
width of the laser was about 150 fs. The average output power of the laser is around
1.3 W, we only use a small portion of it in order to prevent damage to the sample.
Light from the laser was split into pump and probe beams using a beam splitter,with roughly 30 mW average power in each beam. The pump beam was chopped at
1.5 kHz by a mechanical chopper, and that frequency was used as the reference
frequency of the lock-in amplifier. The reflected probe beam was detected with aphotodiode, the pump and probe beams were cross polarized, and a polarizer wasused before the detector to minimize the background from scattered pump light.

124Time delay between the pump and probe beam is controlled by a retroreflector
mounted on a translation stag driven by a stepper motor in the probe beam path.
Changes in the reflected probe beam power were recorded as a function of timedelay between pump and probe beam. Figure 5.12 shows the results of the pumpprobe experiment for each sample.

(a)

(b)

Figure 5.12. Pump-probe results. The points are the measured reflectivity as afunction of time delay, and the solid lines are fits of the data (after the coherent

125

artifact when the pump and probe pulses overlap in time) to A1 exp(−t/τ1) + A2
exp(−t/τ2) + B. (a) Results for GaAs:Er and LT-GaAs. (b) Results for SI-GaAs.

In order to calculate the carrier lifetime, pump-probe data from the threesamples are fit to an exponential decay model. The signal from LT-GaAs could be fit
using a single exponential with a decay constant of 280 fs. The signal from the
GaAs:Er sample required two exponentials for a good fit, with τ1 = 500 fs and τ2 =4.5 ps. We ascribe the carrier lifetime as the longer of the two decay constants. The
measured lifetimes in both LT-GaAs and GaAs:Er are much shorter than in SIGaAs, which was measured to be 354 ps. These results are in reasonable agreementwith previous studies of similar samples [80,81].
We patterned Au electrodes on the surface of the samples using standard
photolithographic techniques. The electrode structure is similar to the one shown in
Figure 5.7, which provides gaps between electrodes of various widths. The GaAs:Er
sample had gaps from 2 µm to 6 µm, while the LT-GaAs sample had gaps from 8 µm
to 26 µm. While ideally we would have had the same gap width to compare theresults in GaAs:Er and LT-GaAs, the signal depends only weakly on the gap width
between 6 µm and 8 µm, so we compare the signal from the 6 µm gap on the
GaAs:Er sample to the signal from the 8 µm gap on the LT-GaAs sample.
QUIC current injection is measured using the prism interferometer setup
which is similar to the one shown in Figure 5.2. Here we only use the interferometerpart to modulate the phase φ 21 and measure QUIC current electrically through atransimpedance amplifier as described in the third paragraph of section 5.6,
without the requirement of a probe beam and external bias.

126

Sample
GaAs:ErLT-GaAs
SI-GaAs
LT-GaAs

Lifetime
4.5 ps
280 fs354 ps
280 fsGap width
6 µm
8 µm
30 µm
30 µm

QUIC signal
0.6 mV
1.6 mV
0.7 mV347 mV

Table 5.1. Summary of data for the samples studied.

The quantum interference signal was optimized by moving the sample withrespect to the focusing lens. This optimizes the focus and the position of the laser
spot with respect to the electrodes. As observed in the previous experiment [78], the
QUIC current is very sensitive to the position of the laser spot. We found the
maximum QUIC signal when the laser spot is focused near electrode edge, thisresult consists with the previous study [78]. The peak-to-peak voltage of the QUIC
signal from GaAs:Er with a 6 µm gap and from LT-GaAs with an 8 µm gap wererecorded, we found that they were roughly comparable. As a control, we measured
the signal from a semi-insulating GaAs sample with a 30 µm gap and found that thesignal was 500 times smaller than from an equivalent LT-GaAs sample. The data issummarized in Table 5.1. The results prove that GaAs:Er is a suitable material
with good signal-to-noise ratio for QUIC of ballistic current injection. The smaller
signal from the GaAs:Er sample compared to the LT-GaAs sample is related to its
slightly longer carrier lifetime, which can be shorten by heavier Er doping.

127

5.8 Summary

In this chapter, we discussed quantum interference control of ballistic
currents carrier population in a semiconductor. We started with a brief introduction
about the theory of both QUIC of ballistic current and coherent control of carrier
population. Then our main experimental results were presented. First we showed
the all-optical measurement of the electric field-induced population control in bulk
GaAs. The linear bias dependence of the population control signal and thepolarization dependent results show the largest signal when the optical fields areparallel to the applied electric field, consistent with theoretical predictions. We alsopresented the experimental scheme and results of the electrical measurement offield-induced population control. The phase shift and the spatial dependence of theelectrically detected signal were discussed. In the last part we showed theexperimental results of ballistic current injection via QUIC in an erbium doped
GaAs sample, and compared the results with that from a low-temperature grown
GaAs. This showed that Er-doped GaAs is suitable substitute for LT-GaAs forsemiconductor-based carrier-envelope phase detection.

128

CHAPTER 6

Conclusion
This thesis contains a detailed description of our radio-frequency (RF) bias
technique and the main results of three different projects related to the linear andnonlinear optical properties of gallium arsenide. The RF bias technique, presented
in Chapter 2, is an essential method that helps us achieve a uniform electric field ina bulk GaAs sample with transverse electrode M-S-M structure. By combining afast oscillating bias in the radio frequency range applied to electrodes insulated
from the GaAs sample with a pulsed light source, a uniform electric field between
the transverse electrodes can be produced. In addition to the RF bias system, aheterodyne detection technique that can guarantee a wide range of reference
frequencies for lock-in detection with good signal-to-noise performance is also
developed.
The first project, discussed in Chapter 3, presents experimental results ofelectroreflectance (ER) using both the conventional DC bias and the RF bias
technique. Spatially resolved ER data proves that by using the RF bias combinedwith the insulated electrode GaAs sample, we successfully minimize the trapenhanced electric field and obtain a uniform transverse electric field. Results ofelectroabsorption experiment from a thin GaAs epilayer with DC bias are also
included in this chapter. Possible future work includes: 1. one could study the odd
Franz-Keldysh effect, which appears for certain crystal orientation and optical field
directions, using a transverse geometry; 2. one could go to low temperatures using

129

the RF bias. The FK oscillations would be stronger; 3. study higher conductivity
samples, for which the RF technique is the only way to get an ER signal.
The second project, presented in Chapter 4, involves THz generation from a
photoconductive emitter. We adopt the RF bias system, which gives a uniform
electric field distribution in the GaAs emitter, to generate THz radiation. The
uniform field in the emitter allows use of a large excitation spot to minimize theultrafast carrier screening. The emitted THz radiation is detected by an electrooptical sampling system, and compared with results from a DC biased emitter. The
RF technique clearly shows improved optical power-to-THz conversion efficiency,
and a great potential for further improvement. In addition, THz generation from a
low-temperature grown GaAs photoconductive emitter with DC bias is also
discussed. Possible future directions include: 1. increase the bias field by narrowing
the electrodes or increasing the bias voltage, to see if the THz field can be further
improved; 2. study the dynamic of the DC field in the semiconductor, understanding
it might help optimize the technique.
The third project, discussed in Chapter 5, is focused on electric fieldinduced quantum interference control of photoexcited carrier population in bulk
GaAs. We show theory predictions and experimental evidence that a constantelectric field enables control of the photocarrier injection rate by interference of oneand two-photon absorption in GaAs. Results from an all-optical measurement andanelectricalmeasurement

schemearepresented.

Furthermore,quantuminterference control of ballistic current injection in an erbium-doped GaAs isdiscussed. Possible future work: 1. study the wavelength dependence of the fieldinduced QUIC and compare with the theory; 2. do more experiments to understandthe spatial dependence of the QUIC signal using electrical detection.

130

Together, these results serve as useful tools for understanding electro-optic
properties and physical processes such as carrier transport in semiconductormaterials.

131

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139

Appendix A

RF bias and detection setup details
In this appendix the experimental details of the radio frequency (RF) bias
and detection are provided.

A.1 The RF setup

Here we present a brief overview of the RF bias components and setup
procedure. A schematic configuration of the RF bias and detection apparatus isshown in Figure A.1.

(1) Start with picking up the laser pulse train signal by a high speed
photodetector PD1 (QQ062A2, designed by JILA electronics shop). The
signal can be monitored by connecting the photodetector output to a RF
spectrum analyzer.
(2) Amplify the output of PD1 using a RF amplifier AMP1 (Mini-Circuits
ZFL-500 HLN or a similar product)(3) Send

PD1

outputtoaRFpowersplitter

PS1(Mini-Circuits

powersplitter ZSC-2-1). The PD1 output is split into two parts, part 1 isused to generate the RF bias across the sample and part 2 is used togenerate a reference frequency for lock-in detection.

140(4) Send part 1 of the pulse train signal through a band pass filter BP1(K&L tunable band pass filter 5BT-250/500-5, 5 MHz bandwidth), which
only selects the third order harmonic 3frep of the laser repetition rate.

Figure A.1: Schematic configuration of RF bias and detection setup. BP: band passfilter, PD: photodetector, PS: powersplitter, AMP: RF amplifier, FD: frequency
doubler (only for ER experiment), LP: low pass filter.

(5) Then use the 3frep signal as the clock signal (CLK input) of the direct
digital synthesizer (DDS), and make sure the 3frep signal has amplitude

141

around 0 dBm by adding RF amplifiers or attenuators. See Section A.2
for DDS software control information.
(6) Split the DDS output into two by another powersplitter PS2, here thefrequency is f ’ = frep + 1 kHz (or ½frep + 1kHz for ER). Send f’ throughRF amplifier AMP2 (Mini-Circuits ZFL-500 HLN) first and adjust theamplitude of f to be around 0 dBm by adding attenuators if necessary.
Then the output of AMP2 is sent into another amplifier AMP3 (MiniCircuits ZHL-1-2W, with 29 dBm gain and 2W maximum output power).
Here make sure the amplitude of AMP2 output doesn’t exceed 0 dBm toavoid potentially damaging the high power amplifier AMP3.
(7) The amplified output signal of AMP3 is used to drive the LC resonancecircuit which amplifies the signal further, producing the high voltage
RF across the sample. The capacitor used here is a sealed glass variablecapacitor (TM90M), and the inductor is the secondary coil of an air-coretransformer consisting of 22 turns of 20 gauge magnet wire wound on a
toroid (Micrometal Inc. T50-0 for 76 MHz and 89 MHz, and T50-17 for
44.5 MHz).
(8) Back to part 2 of the output of PS1, send the signal through anotherband pass filter BP2 (K&L tunable band pass filter 5BT-63/125-5, 5
MHz bandwidth) and select the laser repetition rate frep this time, set
frep to be one of the two inputs (labeled ‘L’ or ‘R’) of a frequency mixer(Mini-Circuits ZAD-1).
(9) From the output of PS2, f ’ = frep + 1 kHz (or ½frep + 1kHz for ER) is then
set to be the other input of the mixer. For ER experiment which uses or½frep + 1kHz for RF bias frequency, double f ’ using a frequency doubler
FD (MiniCircuits FD-2+), for other experiments FD is not required. The
output of the mixer (labeled ‘I’) after a low pass filter LP (MiniCircuits

142

BLP-5) is set as the reference frequency for lock-in detection, here fref =1 kHz (or 2 kHz for ER).

A.2 DDS software control

The DDS we use in our experiments is an AD9854 CMOS 300 MSPS
Quadrature DDS customer evaluation board made by Analog Devices, Inc. The
circuit can be controlled using the AD9854 customer evaluation software which isavailable on the manufacturer’s website.

(1) Connect the DDS circuit board to a computer with the AD9854 software
installed and start the program, first checking the parallel port
connection by selecting ‘Parallel Port’ -> ‘Check Parallel Port
Connection’. Always make sure the selected parallel port is properly
connected to a DDS evaluation board.
(2) Click ‘Clock and Amplitude’ button as shown in Figure A.2. Type in areference clock frequency first (the clock frequency needs to be threetimes the output frequency, i.e., 267 MHz clock frequency for an output
around 89 MHz), hit ‘Enter’ button on the keyboard (every number typed
in here needs to hit ‘Enter’ to make it valid), and choose ‘Send CLK Info
To DUT’. Then select output amplitude to be ‘Output amplitude isalways full scale’ and click ‘Send Amplitude Scaling Info To DUT’. One
can also adjust the RF bias amplitude by choosing ‘Output amplitude isprogrammable’ and enter three hexadecimal numbers in ‘Q-Channel
Multiplier’. The last step is to uncheck the ‘Comparator Power Down’ inthe ‘Power Down’ box, then click ‘Send Control Info To DUT’.

143

Figure A.2: Clock and amplitude control(3) Click ‘Mode and Frequency’ button and enter 44.501 MHz for ER, or
89.001 MHz for THz generation, or 76.000 MHz for coherent control, as
shown in Figure A.3. The phase of DDS output signal can also be
adjusted with respect to the DDS clock signal, by typing in four

144

hexadecimal numbers in ‘Phase Adjust #1’ box. The final step is to click‘Send Mode, Frequency, Phase Info to DUT’ button.
More detailed information about the DDS circuits can be found in AD9854
DDS user’s manual.

Figure A.3: Mode and frequency control.

145

Appendix B

Photolithography for metal electrodes
In this appendix the patterning of metal electrodes by photolithography isdescribed. We use negative photoresist NR7-1500PY.

(10) Clean the sample substrate with acetone and deionized (DI) water.
Bake the sample on a hot plate at 150 0 C for 15 minutes; this helps
remove water from the sample surface.
(11) Place the substrate on the spin coater, put a few drops of the negative
photoresist on the sample surface, and set the spin to be 3000 rpm for
40 seconds.
(12) Prebake the sample with photoresist on the hot plate for 90 seconds at
150 0 C .
(13) Start the photolithography machine (Karl Suss, Mode MJB-3) in thefollowing order: a) turn on the main power supply; b) turn the two
switches on for compressed gas; c) turn on the lamp power supply; d) hit
the ‘power’ button on the control panel.
(14) Put the photomask on the holder of the lithography machine with its
coating side down and push the ‘vacuum mask’ button on the control
panel.
(15) Place the substrate on the sample plate of the photolithographymachine; turn the z knob clockwise first, to lower the sample plate inorder to prevent the sample from crushing against the mask. Align the146position of the sample with respect to the mask by adjusting the two
knobs that control x and y direction of the sample plate.
(16) Pull the lever on the left side of the photolithography machine
clockwise. Adjust the z knob counterclockwise to make contact between
the mask and the sample surface. Slowly and carefully raise the sample
up until some interference fringes can be seen on the sample surface.
This indicates a good contact between the sample and the mask.
(17) Push the lever on the left side of the photolithography machinecounterclockwise until the contact indicator on the control panel is on.
Adjust the exposure time to be 30 seconds, hit the ‘exposure’ button.
(18) After exposure, lower the sample plate using the z knob and take thesample out. Turn off the photolithography machine in reverse order ofthe procedure in (4).
(19) Postbake the sample with the hot plate for 90 seconds at 100 0 C .
(20) Dip the sample in MF-CD-26 developer for 2~5 seconds, then rinse
with DI water and check the pattern under a microscope.
(21) After metal deposition (done by David Alchenberger in the Keck Lab),
soak the sample in acetone for about 15 minutes to lift off the exposedpart, rinse with DI water again, and check the pattern under amicroscope.

147

Appendix C

Thin GaAs sample preparation
In this appendix the preparation of the 1 µm thick GaAs sample isdescribed. The sample consists of a 1 µm thick, undoped GaAs epilayer grown on topof a semi-insulating GaAs substrate by molecular beam epitaxy. The epilayer andthe substrate are separated by a thin AlGaAs stop-etching layer roughly 100 nm
thick.

C.1 Mechanical lapping(22) Attach the polished surface of the sample to a ½ inch transparent
sapphire disk (made by Meller Optics, Inc.) by either clear wax or
optical epoxy (Norland Optical Adhesive 81). The optical epoxy requires
UV cure for one or two hours.
(23) Thin the sample down to a few microns thick by using the lapping tool
(as shown in Figure C.1, it helps thin the sample by a fixed amount by
adjusting the screw on top) and the lapping paper (Precision Surfaces
International, PSI-1612-11 lapping film sheets), which is like sandpaper with a very fine grit.

148

C.2 Wet etching (gloves and safety glasses required)(1) Get 100~200 mL of hydrogen peroxide (H2O2) solution from the fridge inKeck lab.
(2) Get 10~15 mL ammonium hydroxide (NH3:OH) from the flammablecabinet in Keck lab.

Figure C.1: The lapping tool.

(3) Prepare two beakers and rinse them with DI water first. Then fill one ofthem with DI water, and use it to rinse the pH probe and the neutral
solution flask (remove the neutral solution bottle from the flask first) of
the pH meter (Fisher Scientific, Accumet Model 20) in the Keck lab
until the reading of the pH meter is stabilized around 7.
(4) Add 100~200 mL H2O2 solution to the other beaker, put the probe andthe neutral solution flask into the beaker until the reading of the pH
meter is stable.

149(5) Use a pipette to add NH3:OH solution into the H2O2 solution slowly.
Add a few drops each time, shake the beaker slowly and wait until thereading of the pH meter is stable. Keep adding NH3:OH solution until
the pH value of the solution reaches around 8.1.
(6) Store the solution in a Nalgene bottle for transportation (the wet
etching process needs to be performed in a fume hood in a differentroom), and lightly cover the bottle (gas may be produced and needs a
way to escape the bottle to avoid building up pressure).
(7) Clean the probe and the flask using DI water, and reattach the neutral
solution bottle to the flask.

Figure C.2: The wet etching tool.

(8) In a fume hood, pour the etchant solution into the etchant container ofour etching tool (shown in Figure C.2), and put ice around the etchant150

container to keep the temperature low. Fix the sample on the sample
holder, point the syringe needle towards the sample and start etching
until the AlGaAs stop-etching layer is reached.
(9) Rinse the sample, clean the etching tool, label the used etchant ashazardous waste and store in the Satellite Accumulation Area.