Terahertz Induced Non-Linear Electron Dynamics in Nanoantenna Coated Semiconductors at the Sub-Picosecond Timescale

AN ABSTRACT OF THE DISSERTATION OF

Andrew Stickel for the degree of Doctor of Philosophy in Physics presented on August 10, 2016.

Title: Terahertz Induced Non-linear Electron Dynamics in Nanoantenna Coated Semiconductors at the Sub-picosecond Timescale

Abstract approved: Yun-Shik Lee

This dissertation is an exploration of the material response to Terahertz (THz) radiation. Speciﬁcally we will explore the ultrafast electron dynamics in the non- perturbative regime in semiconductors that have been patterned with nanoantenna arrays using broadband, high intensity, THz radiation. Three main semiconductor materials will be studied in this work. The ﬁrst is VO2 which undergoes a phase transition from an insulator, when it is below 67◦ C, to a metal, when it is above 67◦ C. The second and third materials are Si and GaAs which are two of the most commonly used semiconductors.

We study the insulator to metal transition (IMT) of VO2 and its response to high field THz radiation. The near room temperature IMT for VO2 makes it a very promising material for electrical and photonic applications. We demonstrate that with high field THz the IMT transition can be triggered. This transition is induced on a sub-cycle timescale. We also demonstrate a THz field dependent reduction in the transition temperature for the IMT when transitioning from both below Tc to above as well as from above Tc to below. This transition is not equal for the above and below cases and leads to a narrowing of the hysteresis curve of the IMT. The thin film Fresnel coefficients, along with a phenomenological model developed for the nanoantenna patterned VO2, are also used to calculate the sheet conductivity of the VO2 sample. We show, using this sheet conductivity and its relation to the band gap, that the bang gap in the insulating phase has a strong dependence on the incident THz radiation with larger fields reducing the band gap from 1.2 eV at low incident THz fields to 0.32 eV at high incident THz fields.

The ultrafast, non-equilibrium, electron dynamics of GaAs and Si were also explored. GaAs and Si are the two most prevalent semiconductors in use today and with the decrease in size and increase in clock speeds of transistors a deep understanding of the ultrafast high field electron dynamics of these materials is of vital importance. Using THz time domain spectroscopy we investigate the transition rates of electrons excited by a 800 nm optical pump. We show that the optically induced transition for GaAs happens on a shorter timescale than that of Si. We also investigate the THz transmission dependence on the indecent THz field. We show that intense THz fields enhance the transmission through the sample. The increase in transmission is due to intervalley scattering which increases the effective mass of the electrons resulting in a decrease in the conductivity the sample. c Copyright by Andrew Stickel August 10, 2016 All Rights Reserved Terahertz Induced Non-linear Electron Dynamics in Nanoantenna Coated Semiconductors at the Sub-picosecond Timescale

Andrew Stickel

A DISSERTATION

submitted to

Oregon State University

in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Presented August 10, 2016 Commencement June 2017 Doctor of Philosophy dissertation of Andrew Stickel presented on August 10, 2016.

APPROVED:

Major Professor, representing Physics

Chair of the Department of Physics

Dean of the Graduate School

I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request.

Andrew Stickel, Author ACKNOWLEDGEMENTS

First and foremost I would like to thank Professor Yun-Shik Lee. For Advising me throughout the my 6 years of working for him. I would also like to thank Dr. Zack Thompson for working with me, teaching how to tighten a bolt, and not punching me in the face as much as he had the right to. I would also like to thank Dr. Byounghwak Lee and Ali Mousavian for their input and support during my work. All of my friends and faculty in the physics department that have listened to me whine...I mean helped me out. Speciﬁcally I would like to thank Dr. KC Walsh for being a mentor, friend, and all around amazing help who gave me important insights into grad school and helped me out immensely. I would like to thank my teachers Mr. Day, Mr. Morales, and Dr. Deal for encouraging my passion for science and guiding me when I needed it. Also I need to thank Dr. Ritter and Ms. Wise without who this 137 page document would be a dribbling pile of non-sense. Finally, and most importantly, I would like to thank my parents Bryan Stickel and Carol Ernst. From the beginning of my life they have done nothing but support my dreams and put up with my 19,762 questions. They kept me from killing myself, even when those cool metal disks on the oven seemed REALLY fun to play with. Mom, Dad, I love you both. TABLE OF CONTENTS Page 1 Terahertz Radiation and Applications1

1.1 History...... 2

1.2 Sources...... 2

1.3 Detection...... 4

1.4 Applications...... 6

2 Theoretical Foundations for Terahertz Spectroscopy of Semiconductors8

2.1 Maxwell’s Equations and Electromagnetic Radiation...... 8 2.1.1 Electromagnetic Plane Waves...... 10 2.1.2 Electromagnetic Waves at a Boundary...... 16

2.2 Non-linear Optics...... 23 2.2.1 Second Order Non-linear Eﬀects...... 24 2.2.2 The χ(2) tensor...... 26 2.2.3 Phase Matching...... 28

2.3 Ultrafast Optics...... 30 2.3.1 Mode Locking...... 31 2.3.2 Pulse Width and Dispersion...... 34

2.4 Band Theory of Semiconductors...... 36 2.4.1 The Simple 1-D Chain...... 37 2.4.2 Beyond the 1-D Chain...... 39

2.5 THz Field Enhancement by Nanoantennas...... 41 2.5.1 Nanoantenna Enhanced Electric Fields for Long Wavelength Radiation...... 42 TABLE OF CONTENTS (Continued) Page 2.5.2 Nanoantenna Arrays...... 44

3 Terahertz Generation and Detection 46

3.1 THz Generation...... 46 3.1.1 ZnTe and Optical Rectiﬁcation...... 46

3.1.2 LiNbO3 THz Generation...... 48

3.2 THz Detection...... 51 3.2.1 Pyroelectric Detectors...... 52 3.2.2 The Bolometer...... 53

3.3 Experimental Setup and Procedures...... 55 3.3.1 The Laser System...... 55 3.3.2 Layout of Experimental Setup...... 56 3.3.3 Experimental Designs...... 58

4 Vanadium Dioxide Field Induced Transition 65

4.1 Mott Insulators...... 65 4.1.1 The Hubbard model...... 65 4.1.2 Conductivity and the band gap...... 68 4.1.3 Mott Criterion...... 71

4.2 Insulator to Metal transition in VO2 ...... 72

4.2.1 The structural phase transition of VO2 and its consequences on conductivity...... 72

4.3 The Sample...... 74

4.4 THz Dependent Transmission...... 76 TABLE OF CONTENTS (Continued) Page 4.5 Frequency Response...... 80

4.6 THz dependent Hysteresis Narrowing...... 84

4.7 THz Dependent Sheet Conductivity...... 89

4.8 Time Dependent Transmission of the THz Beam and Real Time Fluc- tuations of the Conductivity...... 94

4.9 Conclusion...... 98

5 High Field Electron Dynamics at Sub-picosecond Timescale in GaAs and Si 99

5.1 Intervalley Scattering Induced by Strong THz Fields...... 99 5.1.1 Intervalley Scattering in GaAs...... 101 5.1.2 Intervalley Scattering in Si...... 104

5.2 Nanoantenna Patterned Samples...... 104

5.3 Optical Free Carrier Driven Conductivity...... 106

5.4 Temporal Evolution of Optically Excited Free Carriers...... 112

5.5 THz Induced Transparency Via Intervalley Scattering...... 120 5.5.1 Time Domain Spectroscopy Investigation of THz Intensity Dependent Transmission...... 121 5.5.2 Power Transmission Detection of THz Induced Transparency 127

5.6 Induced Sheet Conductivity...... 133

5.7 Conclusion...... 135

6 Conclusion 137 TABLE OF CONTENTS (Continued) Page Appendix 139

A Derivation of Index of Refraction Vs Wavelength...... 140 LIST OF FIGURES Figure Page 1.1 The electromagnetic spectrum with the THz region highlighted in blue...... 1

2.2 A schematic of all of the internal reﬂections inside a bulk, lossy, dielectric...... 20

2.3 A schematic of all of the internal reﬂections inside a bulk, lossy, dielectric coated with a thin ﬁlm...... 21

2.4 A plot a cavity where N=200 modes have a) random phase (incoherent) b) uniform phase (coherent)...... 32

2.6 A cartoon plot of a nanoantenna array (left) with an SEM images of an actual nanoantenna (right)...... 44

3.1 Three time steps in the generation of the THz pulse via optical rectiﬁcation...... 48

3.2 Cartoon setup of the LiNbO3 based THz generation. The dark lines within the optical pulse represent the the pulse front...... 50

3.3 The basic circuit diagram for the pyroelectric detector made by New England Photoconductor (NEP)...... 53

3.4 The schematic of a basic bolometer [36]...... 54

3.5 The laser setup used in the lab...... 56

3.6 A schematic of the experimental set up used in the lab...... 58

3.7 A schematic of the experimental for Power Dependent Transmission. 59

3.8 A schematic of the experimental set up for Michaelson interferometry. 60 LIST OF FIGURES (Continued) Figure Page 3.9 A schematic of the experimental set up for Time Domain Spectroscopy. 62

3.10 A cartoon ﬁgure of how the TDS detection works...... 63

4.1 A plot of density of states for diﬀerent values of α [127]. T0 is the energy of the top of the valance band and I is the band gap..... 68

4.2 A) Energy plot for the unstrained (M1 state) and strained (R state) V 3d states. B) A representation of the V 3d|| states superimposed of a section of the VO2 crystal lattice...... 73

4.3 A cartoon picture of the VO2 sample. The top layer of gold and sapphire substrate are pictured sandwiching the 100 nm thick VO2 thin ﬁlm. An incoming THz electric ﬁeld is pictured to depict the which side of the sample was exposed to THz...... 74

4.4 An image of the mounted VO2 with the two peltier heaters on either side, the thermistor attached near the center, and the thermal paste and kapton tape attaching it to the mount...... 75

4.5 THz dependent relative transmission though NS coated VO2 at temperatures approaching the transition temperature...... 77

4.6 THz dependent relative transmission though bare VO2 at temperatures above and below the transition temperature...... 79

4.7 The transmitted THz spectra from bare coated VO2 for varying temperatures at diﬀerent incident THz powers...... 81

4.8 The transmitted THz spectra from NS coated VO2 for varying temperatures at diﬀerent incident THz powers...... 82

◦ 4.9 The transmitted THz spectra from NS coated VO2 at 64 C for two diﬀerent incident THz powers...... 83 LIST OF FIGURES (Continued) Figure Page

4.10 The temperature dependent transmission through bare VO2 demonstrating the hysteretic behavior of the phase transition...... 84

4.11 The temperature dependent transmission through NS VO2 demonstrating the hysteretic behavior of the phase transition...... 86

4.12 The temperature dependent transmission, for sample temperature going from 50 degrees up to 80 degrees C, through NS VO2 demonstrating the hysteretic behavior of the phase transition...... 87

4.13 Plot of the narrowing of the hysteresis width as it varies with increasing THz...... 88

4.14 The normalized temperature dependent theoretical sheet conductivity for high incident THz on bare VO2 compared with low incident THz on NS coated VO2...... 90

4.15 The semilog plot of the relative transmission vs both the calculated sheet conductivity based off the phenomenological model and the regression done with the bare sheet conductivity and the fitted sheet conductivity based on the thin film equations for the NS coated VO2. 92

4.16 The resistivity of the samples as a function of temperature for the NS coated VO2...... 93

4.17 Plots of the THz waveform transmitted through the bare VO2 sample. This demonstrates the eﬀects of only varying the temperature on the transmitted waveform...... 95

4.18 The time dependent transmitted THz wave through NS coated VO2 at a) 45 C, b) 62 C, c) 65 C, and d) 67 C on the increasing temperature side of the hysteresis curve...... 96

4.19 The transmitted THz ﬁeld through NS coated VO2 comparing low THz at high temperature to High THz at low temperature...... 97 LIST OF FIGURES (Continued) Figure Page 5.2 A diagram of the intervalley scattering for GaAs as designed by Su et. al. [152]...... 102

5.3 A cartoon of the Nanoslot coated samples...... 105

5.4 THz waveform transmitted through the nanoantenna coated GaAs with τpp = −0.5ps and incident THz ﬁeld of 321 kV/cm...... 107

5.5 THz waveform transmitted through bare GaAs with τpp = −0.5ps and incident THz ﬁeld of 321 kV/cm...... 108

5.6 A plot of THz waveform transmitted through the nanoantenna coated Si sample at diﬀerent incident pump powers. τpp = -3 ps and ET Hz = 320 kV/cm...... 109

5.7 A plot of THz waveform transmitted through bare Si sample at diﬀerent incident pump powers. τpp = -1 ps and ET Hz = 334 kV/cm 110

5.8 The Fourier transform of the transmitted THz waveform when τpp = -3 ps...... 111

5.9 The transmitted THz waveform through nanoantenna coated GaAs −3 at diﬀerent values of τpp for ET Hz = 99 kV/cm and Ne = 9.6e15 cm 113

5.10 The transmitted THz waveform through nanoantenna coated GaAs at diﬀerent values of τpp for ET Hz = 348 kV/cm and Ne = 9.6e15 −3 cm . Not the THz wavefrom for τpp = −2, −3 ps has been magni- ﬁed by a factor of 10...... 114

5.11 The diﬀerential transmission of the THz ﬁeld through nanoantenna coated GaAs vs τpp ...... 115

5.12 The diﬀerential transmission of the THz ﬁeld through nanoantenna coated GaAs vs τpp ...... 116 LIST OF FIGURES (Continued) Figure Page 5.13 The transmitted electric through the nanoantenna coated Si at dif- −3 ferent values of τpp for ET Hz = 340 kV/cm and Ne = 8.9e14cm .. 117

5.14 The transmitted electric through bare Si at diﬀerent values of τpp −3 for ET Hz = 340 kV/cm and Ne = 8.9e14cm ...... 118

5.15 The diﬀerential transmission of the THz ﬁeld through nanoantenna coated Si vs τpp ...... 119

5.16 The diﬀerential transmission of the THz ﬁeld through nanoantenna coated Si vs τpp ...... 120

5.17 THz Waveform transmitted through nanoantenna coated GaAs at −3 diﬀerent incident THz powers with Ne = 1.58e16cm and τpp is a) 1.5 ps, b) 0 ps, and c) -1.5 ps...... 122

5.18 THz Waveform transmitted through bare GaAs at diﬀerent incident −3 −3 THz powers when τpp = −0.5 ps and Ne = a) 0 cm , b) 1.9e15 cm , c) 6.4e16 cm−3, d) 1.3 e16 cm−3, e) 1.8e16 cm−3...... 124

5.19 THz Waveform transmitted through nanoantenna coated Si at ET Hz = 33 kV/cm and 619 kV/cm. There is no optical pump for this experiment...... 125

5.20 THz Waveform transmitted through nanoantenna coated Si at ET Hz = −3 33 kV/cm and 619 kV/cm. τpp = −2 ps and Ne = 1.38cm ..... 125

5.21 THz Waveform transmitted through bare Si at diﬀerent incident THz powers...... 126

5.22 Total relative power transmission through nanoantenna coated GaAs −3 −3 −3 for Ne = 6.4e15 cm , 1.6e16 cm , 2.5e16 cm and τpp = a) 3 ps, b) 1.5 ps, c) 0.5 ps, d) 0 ps, e) -0.5 ps, f) -1.5 ps, g) -3 ps, and h) -10 ps...... 128 LIST OF FIGURES (Continued) Figure Page

5.23 Total relative power transmission through bare GaAs for τpp = −3 ps and -1 ps...... 129

5.24 Total relative power transmission through bare GaAs at low incident pump powers such that the percent change in transmission due to excited carries is less that 25 % for τpp = a) -10 ps, b) -3 ps, and c) -1 ps...... 130

5.25 Total relative power transmission through nanoantenna coated Si −3 −3 −3 for Ne = 3.6e14 cm , 8.9e14 cm , 1.4e15 cm and τpp = a) 3 ps, b) 1.5 ps, c) 0.5 ps, d) 0 ps, e) -0.5 ps, f) -1.5 ps, g) -3 ps, and h) -10 ps...... 131

5.26 Total relative power transmission through bare Si for τpp = a) -3 ps, b) -1 ps, c) 1 ps)...... 132

5.27 The conductivity of GaAs vs incident THz ﬁelds at τpp = −10 ps and free carrier densities of 12.9, 31.8, and 51.1 ·1015 cm−3 ..... 134

5.28 The conductivity of Si vs incident THz ﬁelds at τpp = −10 ps and free carrier densities of 7.2, 28.8, and 28.7 ·1015 cm−3 ...... 135 1 Terahertz Radiation and Applications

Terahertz(THz) radiation is a band of the electromagnetic spectrum that ranges in frequency from 0.1·1012 Hz to 10·1012 Hz. It ﬁlls the gap between the far infrared and microwaves. Radiation of 1 THz has a wavelength of 300 µm which corresponds to an energy of 4.2 meV or 49 Kelvin. Due to this low energy THz radiation is an excellent tool for exploring the electron dynamics in many material systems. Its energy is well bellow the band gap of virtually all semiconductors, which allows for extensive analysis of intra-band electron dynamics without worrying about making excitations between the valence and conduction band. Its long wavelength also allows for it to pass through dielectric materials without much interaction.

Figure 1.1: The electromagnetic spectrum with the THz region highlighted in blue 2

1.1 History

THz radiation has been known about for over one hundred year. Long wavelength sources have been produced as far back as the turn of the 20th century [1]. In 1911 Ribens and Bayer showed that a mercury arc lamps with a quarts shell were able to reproduce long wave radiation reliably [2]. This opened the door to many new experiments that were unavailable before. However, it is only since the 1980s that coherent THz radiation was able to be reliably produced [3]. Using a free electron laser Elias, Hu, and Ramian were able to produce radiation with wavelengths between 390 and 1000 µm; the ﬁrst time a stable coherent source of THz had been produced. Over the years other sources of THz radiation have been produced such as synchrotron radiation [4], air plasmas [5], and optical rectiﬁcation [6].

1.2 Sources

As stated in section 1.1 historically there have been very few sources of THz and before the 1980s there were no sources that were either high powered or coherent. Since the 1980s a number of useful sources have been developed such as the free electron laser, synchrotron radiation, air plasma, photocurrent sources, and materials with non-centrosymmetric symmetry.

The first modern source of THz radiation that was developed was the free electron laser. The basic principle of a free electron laser is that a beam of free 3 electrons is accelerated down a path. The path is lined with a strong magnetic field that alternates direction every few centimeters. This has the effect of causing the electrons to oscillate or “wiggle” up and down. This oscillation causes the electrons to radiate at

2 eBwλw 1 + 2πm c λ = λ 0 (1.1) l w 2γ2

where λl is the wavelength of the emitted radiation, λw is the distance for the applied magnetic ﬁeld to make one complete cycle, Bw is the average magnetic induction, and γ is the energy of the electron divided by its rest mass energy [7]. The radiation produced from this setup is incoherent. However, if a laser with a wavelength equal to λl is co-propagate with the electron beam then the electrons begin to emit in phase with the laser. It turns out that an an external laser source is not needed. The electron beam can be placed in a resonant cavity, which will produce its own coherent beam, allowing for the free electron laser to be tuned to virtually any wavelength.

THz synchrotron radiation is produced in a similar fashion, at least in the sense that it is depends on the acceleration of electrons to produce radiation. Unlike free electron lasers a synchrotron use orbital electron motion rather than a linear path with a wiggler. A synchrotron uses an electron storage ring, where electrons are kept in a closed loop, and moved in a circle via an external magnetic ﬁeld.

The third source of THz radiation is laser generated air plasma. A high intensity 4 laser can be used to produce an air plasma. The resulting freed electrons can be accelerated either via a pondermotive force, due to a secondary applied laser [8], or by taking advantage of the oscillations produced by the ionization process itself [9].

Another important source of THz radiation takes advantage of photoconductive switches. The essence of this method of THz generation is a high power pulsed laser incident on a a semiconductor wafer, such as GaAs or InP, which has electrodes attached [11]. A bias is applied to the semiconductor synced to the laser pulse which, in conjunction with the incident laser pulse, can produce sub-picosecond THz pulses.

The final method that will be discussed is THz generation via optical rectification. Optical rectification is a second order, non-linear optical effect which mixes an incoming photon with a static electric field. Two types of crystals that are employed in the use of THz generation via optical rectification are ZnTe and

LiNbO3 [12]. As this is the primary method of THz generation which this work employs, a more in-depth explanation can be found in section 2.2

1.3 Detection

Due to it’s low energy THz radiation is extremely diﬃcult to detect using standard photo current detectors, which rely on interband transitions. There are a number of other THz detection devices. Devices such as Golay cells, pyroelectric detectors, and bolometers have been developed as possible ways to detect THz radiation. 5

These types of detectors are often refereed to as power detectors, or incoherent detectors, as they only register the total power of the THz beam. Techniques such as electro-optic sampling allow for detection of the THz waveform, not just total integrated power. THz Time domain spectroscopy will be discussed in section 3.3.3.

The most accurate integrated power detector is the bolometer. The primary components of a bolometer is a thermally reactive resistive element that is attached to a material with well known thermal characteristics such as GaAs/AlGaAs [10] or Si. The base material is coupled to a low temperature bath to help reduce noise. A bias is applied to the resistive element and the resulting current is measured. When photons enter the detector their energy is deposited onto the base material and is transfered into thermal energy. This increase in temperature causes the resistive element to change its resistance, and thus the current. This is an extremely accurate system [13], but it usually requires liquid helium temperatures to run eﬃciently.

One alternative is a Golay cell. This functions based on the principles of the ideal gas law. A chamber that is filled with gas is separated from an optical element by a thin, flexible membrane. The thermal energy of the incoming radiation increases the temperature of the gas which causes an increase in pressure. This causes the membrane to shift, an effect which can be measured with high accu- racy [14]. Thus, if the change in the membrane is known the incoming energy can be calculated based on the particular details of the gas cell. 6

The ﬁnal incoherent detector is the pyroelectric detector. Like the other detectors it operates on the conversion of photon energy into thermal energy. The main operating element of a pyroelectric detector is a crystal that undergoes spontaneous polarization. This spontaneous polarization and the dielectric constant are dependent on the temperature of the crystal. Electrical plates, and an open circuit setup are employed so that when the crystal is heated, and it’s properties altered, a measurable current is produced [15].

1.4 Applications

While THz radiation is diﬃcult to produce and detect, its properties lend themselves to many useful applications. As THz radiation is extremely sensitive to the conductivity of a material [16] it is a useful tool for exploring the electrical properties of a material without having to attach electrodes to make physical contact with the sample. Thus THz radiation allows for non-contact non-destructive analysis of many interesting and novel materials and devices.

THz also has applications in medical imagining [17–20]. The energy of one THz photon is on the order of 4 meV. The energy splitting in many biological systems and materials is also on the order of 4 meV.

THz radiation is also very useful for security purposes. Over the last 15-20 years there has been a large increase in the need for security in high traﬃc areas such as airports, train stations, and ports. This has required the development 7 of devices such as the infamous x-ray scanner which is used to take a full body image of a person to look for any weapons or hidden items. However, as X-rays are ionizing radiation they are very hazardous to one’s health. THz radiation oﬀers an alternative to X-rays, and will result in little to no damage to those being scanned [22,23]. THz radiation can also be employed in the detection of chemicals such as explosives or drugs [24,25]. 8

2 Theoretical Foundations for Terahertz Spectroscopy of

Semiconductors

The usefulness of THz radiation was laid out in Sec 1.4. In order to properly utilize THz radiation a through understanding of a number of concepts, such as electromagnetic radiation and ultrafast optics, must be understood.

2.1 Maxwell’s Equations and Electromagnetic Radiation

All electromagnetic radiation consists of coupled oscillations of electric and magnetic fields. Therefor, before we can discuss any radiation a complete understanding of Electric and Magnetic fields must be developed. All Electric and Magnetic fields are governed by six main equations [26] 9

~ ~ ∇·D = ρf (2.1) ∂B~ ∇~ × E~ = − (2.2) ∂t ~ ~ ∇·B = 0 (2.3) ∂D~ ∇~ × H~ = J~ + (2.4) f ∂t ~ ~ ~ ~ D = 0 (1 + χE) E ≡ 0rE = E (2.5) ~ ~ ~ ~ B = µ0 (1 + χM ) H ≡ µ0µrH = µH (2.6)

where 0 and µ0 are the vacuum permittivity and permeability respectively, and χE and χM , in general, are tensors that are deﬁned as

~ ~ P = 0χEE (2.7) ~ ~ M = µ0χM H (2.8)

Here, P~ and M~ are the microscopic polarization and magnetization of a material. For this work we will treat χE and χM as scaler operators as the materials that are being dealt with behave as such. 10

2.1.1 Electromagnetic Plane Waves

Applying the curl operator to equations 2.2 and 2.4, distributing the curl on the ~ ~ ~ ~ ~ ~ 2 ~ RHS, and applying the vector equality ∇ × ∇ × Q = ∇· ∇·Q − ∇ Q

! ∂∇~ × B~ ∇~ ∇~ E~ − ∇2E~ = − (2.9) · · ∂t ∂∇~ × D~ ∇~ ∇~ H~ − ∇2H~ = ∇~ × J~ + (2.10) · · f ∂t

By applying equations 2.1 and 2.5 to 2.9 and equations 2.2 and 2.6 to 2.10 yields

! ρ ∂ ∂E~ ∇~ f − ∇2E~ = − µJ~ + µ (2.11) · ∂t f ∂t ∂ ∂B~ ∇~ (0) − ∇2B~ = µ∇~ × J~ − µ (2.12) · f ∂t ∂t

Rearranging we get the equations 11

∂2E~ 1 ∂J~ ∇2E~ − µ = ∇~ ρ + µ f (2.13) ∂t2 f ∂t ∂2B~ ∇2B~ − µ = −µ∇~ × J~ (2.14) ∂t2 f

At this point we have something that looks like the inhomogeneous wave equation. However, equations 2.13 and 2.14 can be simplified further if we make the assumption that the charges are uniformly distributed in space, ie that ∇ρf = 0. Also, if we assume a linear relationship between the electric field and the current density, also known as Ohm’s law [27], then the equations can be simplified further to be

∂2E~ ∂E~ ∇2E~ − µ = µσ (2.15) ∂t2 ∂t ∂2B~ ∂B~ ∇2B~ − µ = −µσ∇~ × E~ = µσ (2.16) ∂t2 ∂t

∂E~ ∂2E~ For non-matalic materials it is fair to assume that σ ∂t ∂t2 as the conductivity of non-metalic materials is extremely low compared to the how rapidly the electric field is changing in time. The same can be said for the magnetic field. This means the first temporal derivative of the fields can be set to zero which yeilds the final equations 12

∂2E~ ∇2E~ − µ = 0 (2.17) ∂t2 ∂2B~ ∇2B~ − µ = 0 (2.18) ∂t2

This is the exact form of the wave equation and has a number of important

2 ~ 1 ∂2Q~ results. One is that the standard wave equation is ∇ Q − v2 ∂t2 = 0 which implies 1 that µ = v2 . Since there can be no polarization or magnetization in a vacuum, c 1 √ and v = , then c = √ and n = µrr. n µ00

Also, the general solution to the wave equation is known so these equations can be easily solved. It is not common to solve for the electric and magnetic ﬁelds themselves. We can return to equations 2.2 and 2.3 and derive a solution directly ~ ~ ~ from these equations. Since ∇·∇ × Q = 0 equation 2.3 is automatically satisﬁed if we assume

B~ = ∇~ × A~ (2.19)

where A~ is a generic vector ﬁeld called the vector potential. Similarly, as ∇~ × ∇θ = 0 equation 2.2 is automatically solved if we let

˙ E~ = −∇φ − A~ (2.20) 13

Here φ is generic scaler function called the scaler potential. Equations 2.19 and 2.20 along with equations 2.5 and 2.6 can be plugged into equation 2.4 to produce

! ∂ ∂A~ ∇~ × ∇~ × A~ = µJ~ + µ −∇φ − (2.21) f ∂t ∂t

simplifying and rearranging give the equation

∂2A~ ∂φ µ − ∇~ 2A~ = µσE~ + ∇ −µ + ∇~ · A~ (2.22) ∂t2 ∂t

Again, if we make the assumption that the material is non-conductive then, for a reasonably sized electric field, µσE~ ≈ 0. The main reason for switch from the actual fields found in equations 2.17 and 2.18 to the potentials is that the potentials are gauge invariant. This allows us to produce new potentials without altering the ∂φ ~ associated fields. The gauge we will employ is the Lorentz gauge, µ ∂t − ∇A = 0. Applying these to equation 2.22 gives

∂2A~ µ − ∇~ 2A~ = 0 (2.23) ∂t2

which is another form of the wave equation and all of the same insights still hold. Weather we use equations 2.17 and 2.18 or equation 2.23 it is clear the solutions are simple sinusoidal functions. Focusing on equations 2.17 and 2.18 gives the solution of polarized plane waves of the form 14

~ ~ i(~k·~x−ωt) E (~x,t) = E0e (2.24)

~ ~ i(~k·~x−ωt) B (~x,t) = B0e (2.25)

~ ~ ~ ~ plugging equations 2.24 and 2.25 into Maxwell’s equations yields k·E0 = k·B0 = ~ ~ ~ 0 as well as k × E0αB0. This proves that the electric ﬁeld and the magnetic ﬁeld are perpendicular to each other and both are perpendicular to the wave vector ~k. Also, plugging equation 2.24 into equation 2.17 gives k2 − µω2 = 0 which shows the dispersion relationship for electromagnetic radiation to be

c ω = ~k (2.26) n which leads to c E~ = B~ (2.27) n

So far a non-conductive dielectric media has been assumed. However, if the material is metallic, that is to say that it is highly conductive, then the approximation that lead to equations 2.17 and 2.18 are no longer valid. For a good conductor it is fair to say that σ ω which implies that the second time derivative of the electric and magnetic ﬁelds can be ignored in equation 2.15 and 2.16 instead of the ﬁrst order. This leads to the dispersion relation 15

k2 ≈ iµσω (2.28)

Any complex number can be expressed as Q = |Q|eiφ where |Q| is the magni-

−1 Im(Q) 2 π tude of the complex number and φ = tan Re(Q) . For k the value of φ is 2 . Thus

r √ i π µσω k = µσωe 4 = (1 + i) (2.29) 2

Thus the real and imaginary parts of k are equal. Applying this result to the plane wave solution gives an oscillatory portion related to the real part of k and an exponentially decaying portion related to the imaginary portion. For a lossy

nω dielectric media k is expressed as c . As both ω and c are real, n must be a complex number which means

r krc σ nr = ni = = (2.30) ω 2ω0 16

2.1.2 Electromagnetic Waves at a Boundary

2.1.2.1 Bulk Dielectric Boundary Conditions

Now that we have shown that plane waves are a solution to Maxwell’s equations we can begin to apply these solutions to different situations. One of the most important case studies is what happens at a boundary interface between different materials. In terms of optics, the major difference between materials is the index √ of refraction which we showed can be expressed, in general, as n = rµr. Recall that µr = 1 + χM . As the magnetic susceptibility is very small compared to 1 [28] √ µr ≈ 1 and thus n ≈ r.

To understand how Electric and magnetic ﬁelds behave at the interface between two materials the standard pill box and loop can be applied, as in ﬁgure 2.1a. Assuming a normal incidence wave and taking equations 2.2 and 2.4 and integrating over the area of the loop, as the edges approach the interface, yields

−Ein + Eref + Etrans = 0 (2.31)

µ1Bin + µ1Bref − µ2Btrans = Kf (2.32) 17

(a) A cartoon of an a) pill box and b) (b) The vector picture of the incoming wave, the loop applied at the interface. reﬂected wave, and transmitted wave.

Assuming µ1 ≈ µ2 and no free charge, i.e. a non-magnetic dielectric, and plugging Equation 2.27 into 2.32 yields

n1Ein + n1Eref − n2Etrans = 0 (2.33)

By combining equations 2.31 and 2.33 yields the transmission and reﬂection coeﬃcients

Etrans 2n1 t12 ≡ = (2.34) Ein n1 + n2 Eref n2 − n1 r12 ≡ = (2.35) Ein n1 + n2 18

It is important to note that t+r = 1. This process can also be done for oblique incidence waves and leads to effects such as the Brewster angle. The transmission and reflection coefficients above are just for the electric fields. For most laboratory settings, it is the intensity of the electromagnetic wave that is measured. The intensity of the electromagnetic wave is the magnitude of the Poynting vector [29]. The Poynting vector is defined as

S~ = E~ × H~ (2.36)

using equations 2.6, 2.27, that the electric ﬁeld is perpendicular to the magnetic ﬁeld, c2 = 1 , and the time average of a sinusoidal function is 1 leads to 0µ0 2

D E 1 2 hIi = S~ = cn E~ (2.37) 2 0

which gives the power transmission and reﬂection coeﬃcients as

S n T ≡ trans = 2 |t|2 (2.38) Sin n1 S R ≡ ref = |r|2 (2.39) Sin

Again, we note that T +R = 1. For the power transmission and reflection coefficients there is a physical significance. The transmission plus reflection coefficients 19 must equal 1 in order for there to be energy conservation.

2.1.2.2 Thin Film Transmission and Reﬂection

So far bulk material has been assumed when working with transmission and reflection coefficients, but interesting things happen when you assumes a thin film. In general, when measuring transmission through any material, the transmission detected will be more complicated than just applying the transmission coefficient. Consider a material of thickness L. Each interaction with an interface will have a reflected and transmitted element. This implies that there will be a Fabry-Perot effect. The total transmission can be calculated based on the this. Each round trip the light makes inside the material requires two reflections and by the time the light leaves the material there will have been a total of two transmissions. Thus a general form for the transmission of the electric field, after n round trips in the material, can be calculated

(n) i2φn iφ t = t12 r12r21e t21e (2.40)

2πnL where φ = λ To get the total transmission through the material, one must add all of the individual transmitted rays 20

∞ X (n) X i2φn iφ iφ X i2φn ttot = t = t12 r12r21e t21e = t12t21e r12r21e (2.41) n=0

Figure 2.2: A schematic of all of the internal reﬂections inside a bulk, lossy, dielectric.

As t12, t21, r12, and r21 < 1 the inﬁnite sum is merely a geometric series which converges to.

2 iφ |t12| e ttot = (2.42) 2 i2φ 1 − |r12| e

This reasoning can also be expanded to thin ﬁlms. A ﬁlm is said to be thin if the wavelength of the light is much less than the apparent size of the material, 21

λ that is if nf d 10 . As most of the ﬁlms dealt with in this work are approximately 100 nm and the wavelength for 1 THz is approximately 300 µm the thin ﬁlm approximation is valid.

Figure 2.3: A schematic of all of the internal reﬂections inside a bulk, lossy, dielectric coated with a thin ﬁlm.

When a thin film is applied, the total transmission stays mostly the same, as that of the bulk sample. However, for the thin film the second material is now the thin film and the bulk is the third.

(n) i2φn iφ t = t1f r1f rf3e tf3e (2.43)

which changes the total transmission to 22

iφ t1f tf3e ttot = i2φ (2.44) 1 + r1f rf3e

substituting equations 2.34 and 2.35 gives

2n1 2nf iφ n +n n +n e 4n n t = 1 f 3 f = 1 f tot n1−nf n3−nf −iφ iφ 1 − ei2φ e (n1 + nf )(n3 + nf ) + (n1 − nf )(n3 − nf ) e n1+nf n3+nf (2.45)

iφ ikx Equating the plain wave solution gives e = e |x=d and recalling that λ d implies that kd 1 and that e±iφ can be approximated as 1 ± iφ gives eequation 2.45

t13 (n1 + n3) ttot = (2.46) n1n3 n1 + n3 − iφnf 1 + 2 nf

For our materials, the thin ﬁlm can be considered a good conductor which

n1n3 nω means that nf n1, n3 [30] and 2 1. Also we can equate φ to kd = d = nf c n2πf n2π c d = λ d. Using the relationship between the complex index of refraction of a 1 metal and its conductivity found in equation 2.30 and deﬁning Z0 = = 376.7Ω c0 we get

t13 (n1 + n3) t13 (n1 + n3) ttot = = (2.47) n1 + n3 + Z0σd n1 + n3 + Z0σs 23

This is the thin film transmission coefficient where we have defined the sheet conductivity σs as σd. This is an incredibly useful tool for THz spectroscopy and thin film analysis as it allows for the computation of the sheet conductivity of a thin film only knowing the index of atmosphere and substrate and measuring the total transmission through the sample. This means that the conductivity, a purely material property, can be measured without any contact or destruction of the sample. A similar calculation can be done for the total reflection of the of the thin film. However, as no reflection measurements were taken that exercise will be left to the reader.

2.2 Non-linear Optics

In section 2.1, equation 2.5 stated that the microscopic polarization vector was linearly proportional to the applied electric field. This is actually just an approximation for low strength electric fields. It is a good approximation for most applications [31]. However, when the applied electric field gets large enough, such as with a high fluence laser, then the polarization needs to be changed to

~ X ~ (n) X (n) ~ n P = P = 0 χ E (2.48)

where χ(n) is a rank n tensor that translates the applied electric field vector into the polarization vector. As each higher order term requires an increasingly 24 large electric field to have an appreciable effect one would expect that the χ(2) term would be the next dominate term. However, if the material in question has a centrosymmetric symmetry, then it can only have a symmetric response to an electric field. Thus, by symmetry, this type of material can only have odd terms in the Taylor expansion [32]. Thus, for any material with centrosymmetric symmetry the next highest order term will be χ(3). The χ(3) term gives rise to effects such as four wave mixing and self focusing [33].

There are two common ways to make use of the second order term, the term that the rest of this work will focus on. One is to have a material that has a noncentrosymmetric symmetry such as ZnTe. The other is to take advantage of an interface. Even if a material has centrosymmetric symmetry, at an interface there is no way for the response of an electron to be symmetric [34,35].

The second order Polarization term is written as

~ (2) (2) ~ 2 P = 20χ E (2.49)

2.2.1 Second Order Non-linear Eﬀects

Despite its simple appearance equation 2.49 has some very important consequences. If we assume that the applied electric ﬁeld is made of two co-propagating waves then we can write the electric ﬁeld as 25

~ ~ iω1t ~ iω2t E = E1e + E2e + c.c (2.50)

2 ~ (2) (2) ~ iω1t ~ iω2t and applying that to equation 2.49 gives P = 0χ E1e + E2e + c.c . Expanding the electric ﬁeld term gives

2 i2ω1t 2 i2ω2t i(ω1+ω2)t ∗ i(ω1−ω2)t 2 2 E1 e + E2 e + 2E1E2e + 2E1E2 e + c.c. + 2 |E1| + |E2| (2.51)

These terms can be group according to the frequency of the resulting wave.

2 i2ω1t 2 i2ω2t E1 e ,E2 e , c.c. are the second harmonic generation (SHG)terms.

i(ω1+ω2)t 2E1E2e , c.c. are the sum frequency generation(SFG) terms.

∗ i(ω1−ω2)t 2E1E2 e , c.c. are the diﬀerence frequency generation (DFG) terms.

2 2 |E1| , |E2| are the optical rectiﬁcation(OR) terms.

The second order Polarization term can be written as a sum of the diﬀerent frequency terms

~ (2) X ~ (2) (2) ~ ~ P = P (ωn) = 20χ (ω1, ω2; ωn) E1(ω1)E2(ω2) (2.52)

The polarization is often discussed in terms of this frequency dependent form, 26 not the full polarization, as often only one particular type of eﬀect is being used, such as OR or DFG.

2.2.2 The χ(2) tensor

A compact way of writing equation 2.49 is

(2) X (2) Pi = 2 χijkE1jE2k (2.53) j,k

This makes it clear that χ(2) is a rank 3 tensor. In general it can be written as a matrix. It can also be simpliﬁed into what is called the ”d-matrix“. This is done by taking advantage of the symmetries in equation 2.53. The d matrix is deﬁned as dil = χijk.

l= 1 2 3 4 5 6

j,k= x,x y,y z,z y,z;z,y x,z;z,x x,y;y,x

This means that the 18 elements in the d-matrix can represent the 27 elements that are in χ(2). Using the d-matrix, equation 2.53 can be written as the matrix equation 27

  E2  x         E2  (2)  y  Px d11 d12 d13 d14 d15 d16        2       Ez  P (2) = d d d d d d    (2.54)  y   21 22 23 24 25 26       2EyEz  (2)   Pz d31 d32 d33 d34 d35 d36   2E E   x z   2ExEy

While this looks complicated, the d-matrix can often be simpliﬁed for speciﬁc materials. By making symmetry arguments based on the crystal lattice structure many of the elements in the d-matrix can be set to 0 and others can be shown to be equal. For example, in LiNbO3, which has the space group 3m, the d-matrix is [36]

  0 0 0 0 d15 −d22     dLiNbO3 = −d d 0 d 0 0  (2.55)  22 22 15    d15 d15 d33 0 0 0

This d-matrix for LiNbO3 is actually very important for THz generation and we will return to it later. 28

2.2.3 Phase Matching

So far, it has been assumed that the radiation produced from these non-linear processes is in phase with the radiation that generates it and no interference between the two occurs. While it is possible for the incident beam and the resulting beam to maintain constructive interference over the distance of the crystal, the fact that the index of refractions depends on the wavelength of the light means that the assumption is not universally true. If we return to equation 2.17 and use the relation that D~ = E~ + 1 P~ then we get the equation 0

2 ~ 2 ~ 2 ~ 1 ∂ E 1 ∂ P ∇ E − 2 2 = 2 2 (2.56) c ∂t 0c ∂t

Assume a 1-D plane wave for the generated wave and deﬁne it as E3(x, t) =

i(k3x−ω3t) i(k1x−ω1t) A3e with the two incident waves being deﬁned as E1(x, t) = A1e

(1) (2) (2) ±i(k2x−ω2t) ∼ and E2(x, t) = A2e . Using the fact that P3 = P3 + P3 where P3 =

(2) χ E1E2 gives

2 2 2 2 (2) ∂ E3 n3 ∂ E3 1 ∂ χ E1E2 2 − 2 2 = 2 2 (2.57) ∂x c ∂t 0c ∂t

Substituting the functional form for Ei into the above and assuming that Ai is only spatially dependent yields the equation 29

2 2 2 ∂ A3 ∂A3 n ω + i2k − k2A + 3 3 A ei(k3x−ω3t) = ∂x2 3 ∂x 3 3 c2 3 (ω ± ω )2 χ(2)A A 1 2 1 2 i[(k1±k2)x−(ω1±ω2)t] − 2 e (2.58) 0c

2 2 2 n3ω3 As k3 = c2 and ω3 = ω1 ± ω2, depending if the phenomenon is sum or diﬀerence frequency generation, and assuming a slowly varying envelope for A3, ie

2 that ∂ A3 i2k ∂A3 , equation 2.58 simpliﬁes to ∂x2 3 ∂x

∂A i (ω )2 χ(2)A A 3 3 1 2 i(k1±k2−k3)x = 2 e (2.59) ∂x 2k30c

It is useful to deﬁne a new quantity ∆k ≡ k1 ± k2 − k3. If ∆k = 0 then

A3 monotonically increases as a function of x, but if ∆k 6= 0 then A3 must have some spacial oscillations which means there will be points of constructive and destructive interference. Thus, for the best results for harmonic generation of

ωini 1 radiation ∆k = 0. Using ki = c ∆k becomes c (ω1n1 ± ω2n2 − ω3n3). As we know that the index of refraction of a material, in principle, is wavelength dependent (see appendixA) the only way to satisfy ∆ k = 0 and have ω3 = ω1 ±ω2 is if

n1 = n2 = n3 (2.60) 30

This is the phase matching criteria for non-linear wave mixing with electromagnetic radiation. In order to produce any useful radiation from harmonic mixing the index of refraction at the wavelength of the generated light must be equal to the wavelength of the source(s). This condition is diﬃcult to meet for many materials. However there are a number of techniques used to obtain phase matching. One is take advantage of a material that is birefringent. Depending on the diﬀerence in the indexes it is possible to excite a material with a given polarization that will allow for the the index of the incoming wave to match the index of the wave being produced.

2.3 Ultrafast Optics

Ultrafast optics and spectroscopy have opened the door to a huge number of exciting and important ﬁelds [37–40]. Many chemical and electrical interactions happen in less than one nanosecond. Thus, in order to learn about the temporal dynamics of these systems we need to use tools that shorter characteristic times. Out of this necessity ultrafast optics was born. Ultrafast optics is the technique which use a series of coherent pulses that are incredibly short, instead of a contentious wave (CW) laser.

While there are many methods for producing ultrafast laser pulses two will be discussed here. The first is Q-switching. Q-switching works by placing some gain medium inside a resonant chamber that, some how, has a loss mechanism that can 31 be turn on and off. This can be some sort of saturable absorber, a rotating mirror, or laser modulator such as a pockel cell. Initially the loss mechanism is turned on and the gain medium is pump externally to produce a a population inversion. Once the population inversion N(t), is sufficiently high the loss mechanism is turned off. This creates a cavity for the laser to resonate in, causing a massive build up of photons n(t). However, as the photons are produced by the excitation of the population inversion, the number of photons produced per second will drop dramatically due to the lose of the population inversion N(t). This interaction between the population inversion of the gain medium and the photons is what creates the ultra short pulse.

2.3.1 Mode Locking

Mode locking is the primary method that is used to generate short pulses in this work. ML works based on cavity modes and broad band laser generation. The most common gain medium employed for ML is Ti:sapphire, which is titanium doped with sapphire(Al2O3). The primary benefit of this gain medium is its range of wavelengths it can generate [41]. It has been shown that it can produce light at wavelengths between 660 and 986 nm. For a laser cavity that is 1 m there are over 106 different modes that can be supported in this range of wavelengths. The total electric field can be represented as 32

N X i(ωnt+φn) Ene (2.61) n=1

Normally, having this number of modes would be of little use. This is due to the random phase that each mode has, φn. However, if one aligns these phases so they are no longer random then the diﬀerent modes would interfere constructively. A simulation of a cavity with only 200 modes is displayed in ﬁgure 2.4.

Figure 2.4: A plot a cavity where N=200 modes have a) random phase (incoherent) b) uniform phase (coherent).

There are number of ways to induce mode locking. One is called active mode 33 locking. This is where an element modulates the loss of the beam. The element is made to periodically switch back and forth form disrupting the beam to not. This produces an electric ﬁeld of

E(t) = E0 cos(iωnt + φn) (1 − α(1 − cos (Ωt)) ⇒

E(t) = E0 (cos(iωnt)(1 − α) + α cos ((ω ± Ω)t + φn))) (2.62)

2c Thus, if Ω = ωn+1 − ωn = L , where L is the length of the cavity, then this would force all modes to have one uniform phase.

Another technique is Kerr lens mode locking. This takes advantage of the third order non-linear effect where the index of refraction is intensity dependent n = n0 +In1. A nonlinear material is placed next to a hard aperture and when the laser pulse strikes the nonlinear material, any portion of the beam that is mode locked will have a higher intensity, and thus experience a higher index, causing the ML section to be focused down more than the CW wave. The aperture can be designed so that the focused light will make it through but the diffuse light is blocked. Thus the random phases are filtered out and a uniform phase is built up. 34

2.3.2 Pulse Width and Dispersion

While ultrashort pulses are extremely useful they can be difficult to work with, mainly it is very hard to keep a characteristic temporal profile with an ultrafast pulse. The most common type of ultrafast pulse is the Gaussian pulse. It’s electric field can be represented as

Γt2+i(kx−ωt) E(t) = E0e (2.63)

Here Γ is a complex number that describes both pulse duration and the chirp of the pulse. A chirped pulse is a pulsed laser eﬀect where a laser’s frequency can change with time. Ignoring the real part of Γ for a second, the imaginary part

2 of the exponent is φ = ω0t + Γit . As angular frequency is deﬁned as the rate of

dφ change of the angle, we ﬁnd that ω(t) = dt = ω0 + 2Γit. Thus the frequency of the laser increases or decreases with time, depending on the sign of Γi. This is called a chirp.

Lets consider the complex nature of Γ now. The spectral response can be calculated using a Fourier transform.

Z ∞ r 2 2 π −(ω−ω0) −Γt +iω0t iωt E(x = 0, ω) = E0e e dt = E0 e 4Γ (2.64) −∞ Γ

Thus the spectra of a Gaussian pulse is also a Gaussian, centered on the fre- 35

1 quency of the light pulse. The bandwidth of the laser pulse is directly related to Γ .

1 Γr−iΓi Recalling that Γ is a complex number Γ = Γr + iΓi it is better to write as 2 2 . Γ Γr+Γi Thus the intensity spectra I(ω) = |E(ω)|2 has a spectral bandwidth proportional

Γr to 2 2 . Γr+iΓi

Knowing the electric ﬁeld in the frequency regime will also shed light on how a Gaussian pulse propagates through a dispersive media. If we know what the spectrum is at x = 0 we can propagate the electric ﬁeld forward by E(x, ω) =

ikx nω E(x = 0, ω)e . As k = c if n is dependent on the frequency of the light then so must k. If we expand k around the central frequency we get

X 1 d(n)k 1 k(ω) = | (ω − ω )n ≈ k(ω ) + k0(ω )(ω − ω ) + k00(ω )(ω − ω )2 n! dω(n) ω=ω0 0 0 0 0 2 0 0 (2.65)

Using this expansion with the Fourier transform from the frequency regime back to the time domain gives

Z ∞ r 2 π −(ω−ω0) 0 1 00 2 i(k(ω0)+k (ω0)(ω−ω0)+ k (ω0)(ω−ω0) )x E(x, t) = E0 e 4Γ e 2 dω (2.66) −∞ Γ

0 letting ω = ω −ω0, combining terms, and completing the square gives the ﬁnal result 36

00 Γ −2iΓ k x 2 − 0 0 (t−k0x) k(ω0) 1+(2Γ k00x) iω0 t− E(x, t)αe 0 e ω0 (2.67)

Making an analogy to equation 2.63 we ﬁnd that phase velocity of the pulse is

ω 1 k , the group velocity of the pulse is k0 , and the pulse width Γ is now dependent on the distance traveled in the dispersive media

Γ − 2iΓ k00x Γ(x) = 0 0 (2.68) 00 2 1 + (2Γ0k x)

This means that the further a pulse travels into a dispersive media then the smaller Γ will be. This means that the pulse will be stretched out temporally. This is can be a major problem if an experiment requires extremely short pulses.

Also note that even if Γ0 is a real number the pulse will still have an imaginary component to the quadratic portion of its exponent after it passes through the media. This means that the dispersive media will chirp any Gaussian pulse.

2.4 Band Theory of Semiconductors

Virtually no atoms is in total isolation. While it is possible to do work with single atoms [42–44] it is not common. The vast majority of systems studied can be classiﬁed as some sort of bulk material. A material is considered bulk if the dimensions of the object are much greater than many of the characteristic lengths, such as the coherence length. This means that one needs to ﬁnd a way to 37 characterize systems that are more than just single atoms.

2.4.1 The Simple 1-D Chain

Lets ﬁrst start out with the most simple case; a series of hydrogen atoms arranged in a 1-D ring with N atoms. The ﬁrst thing that must be done is to form the ˆ Hamiltonian of this system. Let Hs(x) be the the single atom Hamiltonian. As the system is periodic we can write the continuous variable x as xn = a · n where a is the inter-atomic spacing and n is the nth atom. Thus the total Hamiltonian is

N ˆ X ˆ H(x)tot = Hs(x − xn) (2.69) n=0

The ﬁrst guess at the solution would be the solutions to the single hydrogen atom. We know these won’t be exact anymore due to the interactions between atoms, but they can give us a good start. Instead of just trying one of the hydrogen wave functions we will try a sum of all of them. This method is called the Linear Combination of Atomic Orbitals (LCAO)

X X Ψ(x) = bm(xn)ψ(x − xn)m = φm(xn) (2.70) n,m n,m

where ψ(x − xn)m is the m-th solution to the hydrogen atom centered at the nth lattice site. bm(xn) is the coeﬃcient for the m-th orbital at the nth lattice 38

site. φm(xn) is some theoretical m-th full solution to the Hamiltonian at location xn. This can be simplified by using Bloch’s theorem which states that the wave function from different lattice sites can only differ by a phase factor [45]. Thus, by normalizing Ψ(x) and applying Bloch’s therm to equation 2.70 turns into

r 1 X Ψ(x) = eik(xn)ψ(x − x ) (2.71) N n m n,m

First we can determine something about k by applying the periodic boundary conditions of the 1-D chain. Thus φm(0) = φm(xN ) which implies that bm(0) =

ikxN bm(o)e which is only true if

q · 2π k = (2.72) aN

Where q = 1, 2, 3,...N-1. An expression for the energy of the system can be calculated by taking the expectation value of the Hamiltonian. For a ﬁrst order approximation we will only consider the interaction of a single atomic orbital

X ˆ ik(xn−xn0 ) ˆ hΨ| H |Ψi ⇒ e hψ(x − xn0 )| H |ψ(x − xn)i = n,n0

X ik(xn−x 0 ) e n E hψ(x − xn0 )| ψ(x − xn)i (2.73) n,n0 39

The inner product on the RHS turns into a Kronecker delta δn,n0 . The LHS is slightly more complicated. The nearest neighbor approximation is needed to ˆ simplify the problem. Be letting hψ(x − xn±1)| H |ψ(x − xn)i = β, hψ(x − xn)| ˆ H |ψ(x − xn)i = α, and grouping like terms in the sums we get the relation

q2π E = eikaβ + α + e−ikaβ = α + 2β sin (ka) = α + 2β sin (2.74) N

This is the 1-D dispersion relation for a chain of hydrogen atoms only considering a single state.

2.4.2 Beyond the 1-D Chain

In reality 3-D solids are much more complex. Not only are they shifting from a 1-D to a 3-D case but all of the atomic levels must be taken into account as well. More so, as only hydrogen is hydrogen there are no analytical forms for the atomic orbitals for other atoms. Thus other tools are required to compute the dispersion relations of diﬀerent systems. Before the advent of modern computers a number of tools and approximations were developed to deal with the calculations of band structures [46–49]. Once computers had developed into what we think of them as now, a whole new world of calculations opened up [50–54]. Theories that before that had just been diﬃcult at best to produce results with, such as 40 the Hartree-Fock method, became algorithms a computer could execute with ease. This allowed for a far more detailed analysis of band structures than was available with pen and paper methods.

When all of the terms neglected in the 1-D chain approximation are taken into account, a complete picture of the dispersion relationship for a material can be formed. This is often refereed to as the band structure of the material and can be seen in ﬁgures 2.5a and 2.5b. This is signiﬁcant as many important properties of a material can be understood using the band structure, such as how energetic a photon needs to be to excite free electrons in a system and how good a material is at absorbing said light, ie direct vs indirect band gaps.

(a) A plot of electron energy vs momen- (b) A plot of electron energy vs momentum for GaAs [147] tum for Si [147]

However, these calculations can be misleading. As discussed in section 4.1 in some of these materials the approximations discussed in section 2.4.1 are invalid. When things such as electron electron correlations are neglected, which is often a 41 valid approximation to make, the results are sometimes incomplete, as is the case with Mott Insulators [55,56].

2.5 THz Field Enhancement by Nanoantennas

Under standard conditions light cannot pass through an aperture that is less than the size of its wavelength [57]. It was shown that the transmission through a small aperture goes as

r 4 T α (2.75) λ

Where λ is the wavelength of the light and r is size of the aperture. From this,

r it is clear that as λ drops below 1, the transmission through the aperture drops abruptly. However, in 1998 it was discovered that subwavelength apertures allow high transmission and enhance the electric field in the near zone, if a resonate condition is satisfied [58]. In recent years sub-wavelength apertures have gained importance [59–67]. In this work the subwavelength aperture that will be employed is the nanoantenna array pictured in figure 2.6 42

2.5.1 Nanoantenna Enhanced Electric Fields for Long Wavelength

Radiation

Antenna theory and applications have a wide verity of uses [70–72]. A recent application is for ﬁeld enhancement of free space, long wavelength radiation. The design implemented in this work is a 2D array of slots cut out of a 100 nm think layer of gold that is deposited on top of the sample. Via Babinet’s principle these slots act very smiler to linear antenna of the same dimensions and can be considered bound charge oscillators (BCOs) [73].

Choe at al. show that for a slot, cut out of a ﬁlm of metal, of dimensions a by b, where a b, when light polarized perpendicular to side b is incident in the slot the component of the normalized pointing vector pointing into the slot is

32Re [W ] SN = s (2.76) z 2 2 π [Wa + Ws] where Ws/a corresponds to εsample/air and is equal to

ab Z ∞ Z ∞ ε k2 − k2 W = s/a 0 x s/a 2 q 8π −∞ −∞ 2 2 2 k0 εs/ak0 − kx − ky bk π + ak π − ak 2 sinc2 y sinc x + sinc x dk dk (2.77) 2 2 2 x y 43

If ba< λ this can be simpliﬁed to

3/2 32ε ab 2 2 2 2 s/a ib λ π b λ Ws/a ≈ + εs/a − ln εs/a − + 2γ − 3 (2.78) 3πλ2 λ 4a2 λ2 4a2 where γ ≈ .5777 is the Euler-Gamma constant. it can be shown that resonance will occur when Im[Wa + Ws] = 0 [74]. This leads to the resonance condition for the nanoslot which is

p 2 λres = 2(n + 1)a (2.79)

For Si, which has an index of nSi = 3.418, and a = 60 µm, λres ≈ 300µm which is right at 1 THz. When radiation of the proper wavelength is applied to these systems it has been shown to generate large ﬁeld enhancements [74]. 44

2.5.2 Nanoantenna Arrays

Figure 2.6: A cartoon plot of a nanoantenna array (left) with an SEM images of an actual nanoantenna (right)

A nanoantenna array is a periodic array of slits etched out of a metal ﬁlm that has been layered on top of a substrate. The primary characteristics are the hight (h) and width (w) of the nanoslot(NS), as well as its the slit spacing in the x and y

(dx and dy) directions. Each slot acts as an antenna.

A number of important quantities have been derived for the nanoantena array such as the charge distribution between slots. The surface charge distribution, σ, for a setup like ﬁgure 2.6 is

r ε0E0 λ iωt − iπ σ(x, t) = √ e e 4 (2.80) 2π x

Where E0 is the applied electric ﬁeld, ω is the angular frequency of the light, and x is the distance from the slit edge. It is interesting to note the phase factor at the 45 end of equation 2.80. This implies that the emitted radiation will be out of phase with the incident light.

Possibly the most important quantity that can be calculated is the ﬁeld enhancement γ of the incident radiation [76]

α(ω) Etrans γ = = Einc (2.81) β hw dxdy

Using this formula, it is possible to calculate the strength of the electric ﬁeld inside the gap of the antenna. 46

3 Terahertz Generation and Detection

With the essential theory laid out in Chapter2 it is important to discuss THz radiation itself, how its is generated, how it is detected, and how THz spectroscopy is implemented in a lab setting.

3.1 THz Generation

A wide range of setups and techniques have been used to generate THz radiation such as organic molecules [77], organic crystals such as 4-Dimethylamino-N-methyl- 4-stilbazolium Tosylate (DAST) [84–86], semiconductors such as GaAs [78–80], photocurrent and antenna systems [11, 81], and photoionization of gases [82, 83]. The main method that will be discussed in this section is THz generation via optical rectiﬁcation in non-centrosymmetric systems, mainly ZnTe in section 3.1.1 and LiNbO3 in section 3.1.2.

3.1.1 ZnTe and Optical Rectiﬁcation

ZnTe is an inorganic crystal structure with a cubic lattice space group of 43m¯ [28]. It has a lattice spacing of 0.61020 nm. One of its most important properties it’s 47 index of refraction as it allows for simple a means of THz generation via optical rectiﬁcation [87–90].

Optical rectification, as discussed in section 2.2.2, is a second order non-linear effect. As such, it is based on the d-matrix shown in equation 2.54. Due to the high symmetry of the 43m¯ spacegroup the d-matrix for ZnTe can be simplified to [91]

  0 0 0 d14 0 0     0 0 0 0 d 0  (3.1)  14    0 0 0 0 0 d14

pm The d14 element for ZnTe is 4.2 V [99]. ZnTe, with the laser system described kV in section 3.3.1, will produce a maximum ﬁeld of approximately 100 cm when focused to a beam waist of 400 µm.

THz generation via optical rectification may seem odd at first glance, using a static field to create a time dependent field. However, it is important to remember that the optical excitation laser is not a continuous wave. As it has a finite pulse duration there is a formation of the static electric field, and then a decay. Thus, even though the produced field itself has no frequency component, the temporal limits of the laser pulse drive a time dependent polarization in the ZnTe crystal and this creates the THz field. This is depicted in figure 3.1 48

Figure 3.1: Three time steps in the generation of the THz pulse via optical recti- ﬁcation

3.1.2 LiNbO3 THz Generation

ZnTe is a very useful tool for generating THz as it is very easy to phase match with due to the closeness of the index of refraction for each wavelength [92, 93]. However, the intensity of the THz radiation emitted from ZnTe is fairly low. In order to produce high ﬁeld THz radiation a diﬀerent non-linear crystal is needed. 49

LiNbO3 is an excellent crystal for generating high ﬁeld THz radiation and has a number of useful properties [94–98]. LiNbO3 is a 3m crystal conﬁguration [91] which gives it a d-matrix of [36]

  0 0 0 0 d15 −d22     −d d 0 d 0 0  (3.2)  22 22 15    d15 d15 d33 0 0 0

pm It has a d33 value of 27 V [98]. This makes it highly eﬀerent at generating

THz radiation [100–103]. However, a major downside for LiNbO3 is the fact that the index of refraction for 800 nm is vastly diﬀerent than for 300 µm. This makes standard phase matching extremely diﬃcult. A solution to this issue is to tilt the pulse front of the optical beam and take advantage of the cherenkov radiation created by the velocity mismatch caused by the 800 nm pulse, which can be thought of as a particle, travailing faster than the local speed of light for the THz [104,105].

In order to take advantage of this, the Pulse front of the optical excitation beam must be tilted to the same angle as the cherenkov cone. The set up for THz generation via optical rectiﬁcation in a LiNbO3 crystal uses a blazed grating to tilt the pulse front. A pair of lens (either spherical or cylindrical will work, each having its advantage and disadvantage) are employed to focus the tilted pulse. The ﬁrst

ﬁrst lens, with focal lengths f1, is set f1 distance away from the diﬀraction grating.

The second lens, with focal length f2, is set f1 + f2 away from the ﬁrst lens, creating a telescope. The optical pulse, which now has been magniﬁed by f1 , irradiates the f2 50

LiNbO3 crystal and produces THz parallel to the tilted pulse front. This setup is picture in ﬁgure 3.2

Figure 3.2: Cartoon setup of the LiNbO3 based THz generation. The dark lines within the optical pulse represent the the pulse front.

The angle of the pulse front tilt can can be mathematically expressed as vph,opt cos (γ) = vgr,T Hz. The Pulse front tilt angle, γ is equal to [106]

mλ p tan (γ) = 0 (3.3) ngrβ1cos (θd)

Here m is the diﬀraction order, λ0 is the central wavelength of the optical

(excitation) pulse, p is density of lines in the grating, ngr is the group index of refraction, β1 is the horizontal magnification of the optical pulse, and θd is the diffraction angle. The last variable, the diffraction angle, is the most important for the setup as it is the actual angle we will need to use when aligning the LiNbO3.

In order for the highest eﬃciencies of the LiNbO3 the pulse tilt γ needs to be equal to the tilt of the grating image inside LiNbO3 crystal. To ensure this, the tilt angle θ is used and is deﬁned as 51

tan (θ) = nphβ2 tan (θd) (3.4)

here n is the refractive index of LiNbO3 for the phase velocity of the optical pulse and β2 is equal to the horizontal magniﬁcation of the lens inside the LiNbO3. Using these equations the set up can be optimized to generate THz radiation in

MV excess of 1 cm . One byproduct of this process is even though the generating pulse is on the order of 100 fs, the THz pulse is on the order of 1 ps.

3.2 THz Detection

As section 1.3 discusses, THz radiation is particularly diﬃcult to detect, mainly due to the low energy of their photons; approximately 4.1 meV. However, over the last century and a half there are a large number of diﬀerent types of directors that have been developed. All of these detectors are based around converting the photon energy into thermal energy.

A number of different detectors were developed to detect the ”new” long wave radiation. Long wave radiation ranges from the far infrared to the THz range, ie about 15 µ m to about 1 mm [107]. One detector developed for this type of range of wave lengths is the Pyrgeometer. A Pygeometer works using a hemispherical filter which blocks all radiation below 3.5 µm. A thermophile is placed inside and coated with a black paint which is responsive between 3 to 50 µm. The system 52 is connected to a circuit system such that when the thermophile is irradiated it will alter the voltage in the system [108,109]. Another type of detector is called a Golay detector. A Golay detector works based off of the thermal expansion of a gas against a flexible membrane with an attached mirror [110–112]. As the THz radiation heats the gas it expands, causing the flexible film to flex and altering the aliment of the attached mirror.

These detectors, and others, are good ways to detect THz radiation. However, the detectors that will be focused on in this work will be the Pyroelectric detector and the Bolometer. Both are integrating power detectors.

3.2.1 Pyroelectric Detectors

Pyroelectric detectors work oﬀ the principle that certain types of materials will generate a voltage when heated or cooled [113–115]. Crystals can also exhibit this type of property [116]. By taking advantage of this it is possible to build a THz detector based oﬀ of this material. By connecting a pyroelectric crystal to a circuit it is possible to measure the change in voltage of the crystal due to the thermal heating caused by the incident THz radiation. 53

Figure 3.3: The basic circuit diagram for the pyroelectric detector made by New England Photoconductor (NEP).

3.2.2 The Bolometer

While the pyroelectric detector is convenient in that it functions at room temperature, the primary integrating power detector used in this work is the bolometer. The bolometer has a responsivity that is a factor of approximately 103 times as large as the pyroelectric, making it a much more sensitive device. Originally used as a long range thermal bovine detector [117] bolometers are used to detect THz radiation.

A bolometer uses the conversion of electromagnetic energy into thermal energy to detect incoming THz radiation. However, it uses change in resistance to detect THz radiation. A bolometer uses a thermal absorber, such as a large piece of Si, which is coupled to a cold bath such as a reservoir of liq nitrogen or helium. A ther- 54 mistor is attached to the thermal absorber to measure the changes in temperature caused by the incoming THz radiation. The reason for the cold bath is to reduce thermal noise and fluctuations. As THz radiation is on the order of 4 meV and the thermal energy of a room temperature object, 300 K, is approximately 40 meV any signal that would be generated by the incoming THz radiation would almost certainly be lost within the noise of the thermal fluctuations. Thus by lowering the temperature of the system to 4.1 k with a liquid helium bath, the thermal energy can be lowered to around 0.35 meV. Even liquid nitrogen temperatures would be insufficient as the thermal energy would be approximately 6.5 meV. This is on the order of the energy of the THz photons and is thus too large.

Figure 3.4: The schematic of a basic bolometer [36]. 55

3.3 Experimental Setup and Procedures

With the methods for producing and detecting THz radiation established it is important now to discuss the physical setup of the system. The laser system is a four stage set-up ending with a 800 nm laser pulse with a temporal width of 130 fs and a pulse energy of 1 mJ. The beam is split into a pump beam and a THz beam. The THz beam is split again into a TDS probe beam and a ﬁnal THz beam. The

THz beam is sent through the LiNbO3 setup described in section 3.1.2. The THz beam is focused onto the sample and the transmitted beam is collected by various detection rigs, depending on the experiment.

3.3.1 The Laser System

The laser system consists of four main parts. The first is a 10 W diode laser centered at 532 nm called the Verdi. The laser from the Verdi is sent into the second stage; the Mira. The Mira uses a Ti:Sapphire crystal as a gain medium and produces the 100 fs pulses via mode locking; as discussed in section 2.3.1. The output of the Mira is centered at 800 nm and has a bandwidth of 10-12 nm. This is sent to the final stage; the Legend. The Legend takes the seed pulse from the Mira and, by using another external pump, amplifies the pulses to 1 mJ of energy per pulse. The output pulsed laser has a repetition rate of 1 kHz and each pulse is 130 fs in duration giving a duty cycle of approximately 10−12. This extremely small duty cycle is vitally important to the experiments that will be discussed in 56 section 3.3.3.

Figure 3.5: The laser setup used in the lab.

3.3.2 Layout of Experimental Setup

The output of the laser system in figure 3.5 passes through a 95/5 beam splitter. The 5 % is used as an optical excitation beam with a variable attenuator to control the fluence of the pump beam. The other 95% passes through a second 95/5 beam splitter. The 5 % from the second beam splitter is taken off to used as the THz Time Domain Spectroscopy (THz-TDS) probe beam, which will be discussed in more depth in chapter 3.3.3.3. What is now roughly 90 % of the original laser passes into the setup described in section 3.1.2 to generate the THz radiation. The 57

THz is then focused down to minimum beam waist of approximately 450 µm. The sample is inserted into a 3 axis control stage and moved to the location of the beam waist minimum.

The optical pump line is then aligned to the center of the THz beam so the two are spatially overlapped and co-propagating. Then a wafer of GaAs is used, along with a delay stage set into the THz path, to ﬁnd the temporal overlap of the optical pump beam and the THz beam. After the sample the THz pulse is once again focused down onto a THz detector. For the experiments in this work three main detection arrangements were implemented; total integrated power transmission, michaelson interferometry, and time domain spectroscopy. 58

Figure 3.6: A schematic of the experimental set up used in the lab.

3.3.3 Experimental Designs

The setup illustrated in ﬁgure 3.6 can be used to implement a number of diﬀerent experiments; everything from simple transmitted power detection to wave form capture using THz time domain spectroscopy. 59

3.3.3.1 Power Dependent Transmission

Power dependent transmission is the most straight forward experiment to implement. Either a pyroelectric detector or a Bolometer can be used for these experiments. However, the Bolometer is much more accurate and reliable so it was used for all main detection schemes. To do power dependent transmission measurements the Bolometer is simply placed at the second focus of the THz beam, after the sample, and is optimized to ensure that all light is being captured.

Figure 3.7: A schematic of the experimental for Power Dependent Transmission.

3.3.3.2 Michaelson Interferometry

The second type of experiment is Michaelson interferometry. Smiler to power dependent transmission, Michaelson interferometry use the Bolometer. However, in this set up a Si wafer is used as a 50/50 beam splitter for the THz beam transmitted through the sample. One of of the beam travels along a fixed path while the other is reflected off a mirror that is on a translational stage. The two 60

THz split THz beams are recombined and sent into the bolometer. Using the translational stage the relative path length between the two legs can be altered, this creating an interferogram.

The beneﬁt of this set up is it allows the analysis of the spectral response of the sample; something straight power measurements won’t. However, as THz is readily absorbed by ambient water vapor the area around the THz generation and sample must be purged of all air. This is done by placing an acrylic purge box around the area and ﬁlling it with nitrogen gas.

Figure 3.8: A schematic of the experimental set up for Michaelson interferometry.

3.3.3.3 THz Time Domain Spectroscopy

The final experiment is THz time domain spectroscopy (THz-TDS). Unlike Michael- son interferometry or power dependent transmission THz-TDS does not use the bolometer. THz-TDS takes advantage of the pockels effect, a second order nonlinear effect [118]. As discussed in 2.2 the second order polarization of a material 61 can be described as.

~ (2) X (2) ~ ~ Pi = 20 χijk (ω1, ω2; ωn) E1j(ω1)E2k(ω2) (3.5) jk

As the temporal width of the THz pulse is much longer than that of the 800 nm pulse and the frequency is much lower, a factor of 103, the THz pulse can be treated as a static electric ﬁeld. This can be simpliﬁed to

~ (2) X (2) ~ Pi = 20 χij (ω1, ω2; ωn) E1j(ω1) (3.6) jk where (2) X (2) ~ χij = χijk (ω1, 0; ωn) E2k(0) (3.7) k This shows that the polarization of the material is dependent on applied static field. Thus our THz field, which is quasi static, is able to induce a birefringence in a non-linear crystal, such as GaAs. The set up for TDS can be seen in figure 3.9 62

Figure 3.9: A schematic of the experimental set up for Time Domain Spectroscopy.

Focusing on the detection part of the TDS set up there are three main optical elements; the detection crystal where the birefringence is induced, a quarter wave plate, and a wollaston prism. The wollaston prism separates the two polarizations of the TDS probe beam which are able to be picked up by the ﬁnal element, a balanced photo diode. The system is aligned such that when there is no THz beam the intensity on each photo diode, Ix and Iy, are equal and the output signal is zero. However, as the THz induces a birefringence the circularly polarized optical pump beam becomes elliptically polarized and Ix is no longer equal to Iy and a signal is produced. 63

Figure 3.10: A cartoon ﬁgure of how the TDS detection works.

The time delay between the TDS probe and the THz can be varied allowing for the entire TDS wave form to be mapped out. However, in order to truly understand the output it is necessary to connect how the THz radiation alters the birefringence and thus how the output of the photo diode relates to the THz waveform. As the GaAs can be considered a lossless medium the pockels effect uses the same d-matrix element as optical rectification which for ZnTe is the d14 element. It can be shown that the change in difference in the phases for the x and y polarizations is [36]

Lω Lω ∆φ = (n − n ) = n3 r E (3.8) x y c c O 14 T Hz 64

where nO is the index of refraction of the optical (800 nm) beam. The important thing to note is that the change in angle is linearly dependent on the applied THz

ﬁeld. With the change in phase it is possible to determine the change in Ix/y

I I = I (1 − (+)sin (∆φ)) = 0 (1 − (+)sin (∆φ)) (3.9) x(y) 0,x(y) 2

if the change in the phase is small sin(∆φ) ≈ ∆φ and the diﬀerence in in the intensities is

2Lω I − I ≈ 2∆φ = n3 r E (3.10) y x c O 14 T Hz thus the response from the balanced photo diode will be linearly proportional to the applied THz ﬁeld. This lets us easily detect and map out the transmitted THz waveform. This is a very powerful tool as it allows for not only the characterization of the frequency response but, unlike Michaelson interferometer, TDS preserves the phase information of the THz pulse, as well as the frequency response, leading to a wealth of information lost in other experiments. 65

4 Vanadium Dioxide Field Induced Transition

Vanadium dioxide, VO2, is an incredibly interesting material and has been widely studied for many years [119–122]. VO2 is an insulator at room temperature. How- ever, above 340 K, or 67◦ C, the crystal structure changes from a monoclinic to a rutile structure [123]. This transition has the eﬀect of changing the the VO2 from being an insulator to a conductor [124]. This insulator metal transition(IMT) is caused by changes in the crystal structure which alters the electron correlations which, in turn, changes whether it is energetically favorable to have a free electron gas [125]. This type of transition is a called a Mott transition and forms a class of materials called Mott insulators.

4.1 Mott Insulators

4.1.1 The Hubbard model

The theoretical foundation of Mott Insulators is based on the work of the Hubbard model [126, 127].One of his main contributions was a new approximation for the Hamilton of a many particle systems; the Hubbard model. This model, which accounts for electron-electron(e-e) interactions, is expressed as 66

ˆ X † X † † ˆ ˆ Hhub = −t ai,σaj,σ + U ai,↓ai,↓ai,↑ai,↑ = H0 + Vee (4.1) ,σ i

(†) < i, j > denotes the nearest neighbor approximation, aq,σ is the lowering(raising) operator for the electron at the qth lattice site with spin σ. U and t are deﬁned as

1 X k2 t ≡ t = eik·a (4.2) N 2m k ZZ U = ψ∗ (r)ψ (r)V (r − r‘)ψ∗ (r‘)ψ (r‘)drdr‘ (4.3) Ri Ri ee Ri Ri

N is the number of lattice sites, a is the lattice vector from lattice point Ri to

‘ Rj, and Vee r − r is the electron-electron interaction potential [128]. The ﬁrst term of this model is the standard hopping term for the tight binding approximation in second quantized form with t representing the energy it takes for one electron to transition to a neighboring site. This term is very important for materials like VO2. As t depends on the overlap between adjacent lattice sites it will vary as the spacing between lattice sites is altered.

The second term represents the electron-electron interaction that is normally neglected in standard tight binding calculations. Here U represents the repulsive energy two electrons have while occupying the same atomic state. Note that the two electrons must have opposite spins, as per the Pauli exclusion principle. 67

ˆ ˆ Without the Vee term in the Hhub any site that has a spin up(down) electron, but no spin down(up) electron, would allow for uninhibited hopping of a spin down(up) electron at that state. The term added by Hubbard incorporates the fact that two electrons repel, thus making two electrons on the same site less energetically favorable. This means that systems were a spin down(up) electron was theoretically able to freely move through a state with a spin up(down) electron, the e-e repulsion makes that transition possibly unfavorable. This means that electrons are no longer able to transition around the lattice as freely as without ˆ the Vee term, thus decreasing the conductivity of the system.

Whether the e-e interactions is a dominant eﬀect, or simply a small pertur- bation, depends strongly on the ratio of U and t. At this point it is useful to introduce the deﬁnition

t α ≡ (4.4) U

As α decreases it becomes less and less energetically favorable to hop between sites. At some critical value of α it will no longer be favorable to hop to a site that has an electron with an opposed spin. This critical α manifests as the formation of a band gap. The density of states of a model system is plotted below for decreasing values of α. 68

Figure 4.1: A plot of density of states for diﬀerent values of α [127]. T0 is the energy of the top of the valance band and I is the band gap.

4.1.2 Conductivity and the band gap

As Figure 4.1 implies there is a strong correlation with the conductivity of a material and its band gap. A simple approximation can be derived using a square lattice model of a crystal [130] and the fact that the conductivity of a semiconductor can be shown to be

σ = e (neµe + nhµh) (4.5)

For a 3D lattice it can be shown that the total number of states, N, in k-space 69

2 1 4 πk3 2 2 is N = 8 3 and the energy of the system is E = ¯h k . This combined with π 3 2m∗ ( a ) e/h 1 dN 1 dN dE −1 the deﬁnition of the density of states D(E) = a3 dE = a3 dk dk gives the 3D density of states for the conduction band, as

√ π 2m∗3 p D(E) = E − Eg (4.6) (πh¯)3

The reason for the energy term being E-Eg is that the density of states must go to zero at the bottom of the conduction band. The density of states, with the

−(E−Ef ) 1 fermi-dirac distribution f(E) ∼= e kT , gives the number of electrons in a given energy range. This will let us calculate the number of free electrons in the conduction band

Z ∞ p ∗3 E Z ∞ 3 E −E π 2me f −E ∗ f g kT p kT 2 kT ne = D(E)f(E)dE = 3 e E − Ege dE = N0me e Eg (πh¯) Eg (4.7) q 2 (πkT )3 N0 = (4.8) 2 (πh¯)3

A smiler calculation can be done for the holes with E ⇒ -E in D(E), as holes have negative energy, and the probability of there being a hole is one minus the probability of an electron. Thus nh can be calculated with

1 near 0 K and E > Ef 70

0 p ∗3 0 Z π 2m −Ef Z √ E ∗ 3 −Ef h kT kT 2 kT nh = D(−E)(1 − f(E))dE = 3 e −Ee dE = N0mh e −∞ (πh¯) −∞ (4.9)

Before we can use these results we must determine what Ef is. To do this we will use the fact that no extra electrons or holes could be created, ne = nh. This

∗ 3 Ef −Eg ∗ 3 −Ef 2 kT 2 kT means that N0me e = N0mh e . Solving for Ef gives

∗ Eg 3 mh Ef = + kT ln ∗ (4.10) 2 4 me

plugging this back into equation 4.7 and 4.9 yields ne = nh ≡ n and

3 −Eg ∗ ∗ 4 2kT n = N0 (memh) e (4.11)

These results, combined with equation 4.5 show

−E 1 ∗ ∗ 3 g σ = = eN (m m ) 4 (µ + µ ) e 2kT (4.12) ρ 0 e h e h

This means that there is a direct relationship between the band gap of a semi-

Eg conductor and its conductivity, and thus the resistivity, when 2 kT. This result is critically important for the Hubbard model. As Figure 4.1 shows, the band gap of a Mott insulator is directly related to α as deﬁned in equation 4.4. Therefor, by 71 varying α we can alter the band gap, and thus the conductivity. In a more practical sense, by measuring the resistivity of the sample we are able to determine the value of the band gap.

4.1.3 Mott Criterion

It is important to be able to quantify at what point a Mott insulator will make the transition between its metallic state and its insulating one. It is clear that the transition depends on t and U in the Hubbard model. These quantities are virtually impossible to measure directly. However, they do correspond to physical properties. As U is the interaction between electrons on a lattice site it should depend on how many electrons there are, per site, as well as the average distance they are away from each other. It can be shown that the IMT transition will occur when [120]

1 aH N 3 ' 0.2 (4.13)

Here aH is the eﬀective Bohr radius and N is the electron density. This formula gives a way to calculate the necessary electron density an insulator, or semiconductor, must have in order to behave like a metal. This criterion can be put to the test by altering the electron density and seeing how the system is eﬀected. One method is by injecting free carriers into an insulator [129]. It was discovered 72

that by injecting free carriers into VO2, the material could be switched from a conducting phase to an insulating phase.

4.2 Insulator to Metal transition in VO2

4.2.1 The structural phase transition of VO2 and its consequences on conductivity

As stated in 4.1.1 the hopping energy t depends on the overlap between adjacent sites. VO2 is know to undergo a structural phase transition at approximately 340 K, or 67 C. This transition takes the material from the high temperature rutile(R) structure to a low temperature monoclinic (M1) structure [131]. In the M1 phase the V-V pairs dimerise. While this has the eﬀect of bringing the V atom closer to its partner, it also moves farther away from the rest of the atoms in the lattice. This has the overall eﬀect of lowering t and thus lowering α. As α is decreased past some critical value the VO2 transitions from being a metal in the R phase to an insulator in the M1 phase.

The other consequence of the dimerization of the V atoms is it’s eﬀect on the degeneracy of the 3d orbitals. In the high temperature R phase the V atoms are equally spaced. This symmetry leads to a degeneracy in the t2g states. The t2g states are the x2-y2, yz, and xz d states. The energy of these states is lower than the other two d states, 3z2-r2 and xy. The 3z2-r2 and xy states are degenerate. 73

When the VO2 undergoes its phase transition to the M1 state, and the V atoms dimerizes, some of the symmetry of the system is broken. This broken symmetry

2 2 breaks the degeneracy of the the t2g states into a lower energy x -y d state, which is often called the d|| state, and the higher energy xz and yz d states which form degenerate π bonds. The reason for the shift up in energy of the xz and yz states is due to its coupling with the O 2p states. In the transition to the M1 phase the hybridization strength between the low energy O 2p states and the V 3d xz and yz states is increased which causes the V 3d xz and yz to increase in energy. On the other hand the hybridization of the V 3d|| states with the O 2p states changes very little over the transition [132]. This all can be seen in ﬁgure 4.2.

Figure 4.2: A) Energy plot for the unstrained (M1 state) and strained (R state) V 3d states. B) A representation of the V 3d|| states superimposed of a section of the VO2 crystal lattice. 74

4.3 The Sample

The samples of VO2 that we worked with had a 300 µm thick substrate of sapphire.

Then a 100 nm thick layer of VO2 is deposited on the sapphire. For samples without nanoantennas, refereed to as bare samples, the preparation is complete. For NS coated samples a thin layer of gold ( 100 nm) is deposited on top of the VO2. Then, using e-beam lithography, a 2 mm by 2 mm array of 200 nm wide (x-direction) by 60 µm tall (y-direction) slots are cut with a periodicity of 60 µm in the x-direction and 70 µm in the y-direction.

Figure 4.3: A cartoon picture of the VO2 sample. The top layer of gold and sapphire substrate are pictured sandwiching the 100 nm thick VO2 thin ﬁlm. An incoming THz electric ﬁeld is pictured to depict the which side of the sample was exposed to THz

The samples were mounted using thermal paste and Kapton tape to two peltier 75 heaters that were run in parallel via a control unit. There is a gap left between the two peltier heaters to allow for the THz transmission. A thermistor was placed near the excitation location to measure the current temperature of the samples during experimentation.

Figure 4.4: An image of the mounted VO2 with the two peltier heaters on either side, the thermistor attached near the center, and the thermal paste and kapton tape attaching it to the mount.

The dual conﬁguration of the peltier heaters allowed for much more even heating and assured that even quick temperature changes did not create temperature 76 gradients, and false reading across the samples. The location of the thermistor was chosen to ensure the most accurate reading at the location that the THz was exciting.

4.4 THz Dependent Transmission

As disused in chapters2 and3 THz radiation is extremely sensitive to changes in the conductivity of materials. Thus THz makes for an excellent tool for study- ing the MIT in VO2. However, due to our ability to produce high ﬁeld THz

kV in excess of 100 cm a new realm of exploration can be opened. The ﬁrst phenomenon that needed testing was the THz transmission dependence on incident THz electric ﬁeld strengths.

The two main samples, NS coated and bare VO2, were exposed to varying intensities of THz radiation. Figure 4.5 shows the THz dependent transmission through the NS sample at various temperatures. 77

Figure 4.5: THz dependent relative transmission though NS coated VO2 at temperatures approaching the transition temperature.

The first thing of note is that at all temperatures there is a decrease in transmission with increasing transmission. This means that as the incident THz field increases the conductivity of the sample must also be increasing. Due to the low energy of a THz photon, 4 meV, it is highly unlikely that the THz field is directly creating free carriers. This means the THz field is inducing a change in conductivity via some other means. As it is well known that the conductivity of VO2 is highly dependent on the atomic arrangement of the Vanadium and Oxygen atoms the change in conductivity would indicate that the THz electric field is altering 78 the structure crystal.

The results of figure 4.5 are even more surprising as the increase in conductivity is not just seen when the temperature of the sample is held near the phase transition. The dependence of the conductivity on the incident THz field is seen at temperatures as low as 45 degrees C. This just speaks to how strongly the THz field is enhanced by the nanoslot.

This is supported by examining the THz dependence on the transmission for the bare sample. Figure 4.6 shows only a weak dependence on the THz ﬁeld. 79

Figure 4.6: THz dependent relative transmission though bare VO2 at temperatures above and below the transition temperature.

While there is some dependence of the transmission on the incident THz field, for the bare sample, it is greatly reduced when compared to the NS coated sample, for a given incident THz field. This is due to the local enhancement of the THz field by the nanoantenna array. The gold layer has no other effects on the sample than to enhance the local THz field. 80

4.5 Frequency Response

The frequency response of the samples is also of interest. Michaelson interferometry was initially used to test the frequency response of the sample, see section 3.3.3.2 for a description of the setup. Scans were conducted at two different different THz powers, for different sample temperatures. For bare samples the frequency results, shown in figure 4.7, were uninteresting. It shows the expected drop in overall transmitted THz power. However, there is no shift or alteration of the spectral response. 81

Figure 4.7: The transmitted THz spectra from bare coated VO2 for varying temperatures at diﬀerent incident THz powers.

However, the response for the nanoslot coated samples was very interesting. Figure 4.8 shows that as the temperature of the sample is increased there is a shift in the central frequency of the transmitted THz pulse. 82

Figure 4.8: The transmitted THz spectra from NS coated VO2 for varying temperatures at diﬀerent incident THz powers.

This is not necessarily an unexpected results. As the resonate frequency of nanoslot array depends on the relative permittivity of both the metal and the substrate [64,133], which is the VO2 layer for these samples. As the VO2 undergoes its phase transition and switch from an insulator to a metal its relative permittivity changes. This will cause a shift in the resonate frequency of the system, which is what is seen. This is very important as this implies that the frequency shift can be used as a means to determine if the VO2 has undergone a phase transition. If we 83 look at just 64 C case and compare the two different THz fields, we can see that the higher THz field has undergone a shift in central frequency which the lower THz field has not.

◦ Figure 4.9: The transmitted THz spectra from NS coated VO2 at 64 C for two diﬀerent incident THz powers.

This shift, along with the large decrease in transmission, indicate that the larger THz ﬁeld is actually driving the phase transition and the switch to the metallic state in the VO2. 84

4.6 THz dependent Hysteresis Narrowing

VO2 has long been known to have hysteretic behavior for its heating and cooling cycles [134–136]. This can be detected via THz radiation using the power transmission setup described in section 3.3.3.1, applying a single incident THz field and slowly varying the temperature of the sample. This is then repeated for different incident THz fields. For the bare samples, there is little, if any effect on the hysteresis curves from the incident THz field.

Figure 4.10: The temperature dependent transmission through bare VO2 demonstrating the hysteretic behavior of the phase transition.

The NS coated samples show a strong dependence on the incident THz field. Figure 4.11 shows two very important things. The first is that as the incident THz 85 power increases the sample begins to undergo the phase transition much earlier when compared to the lower incident THz field. This serves to further support the conclusion that the THz field itself is helping to drive the phase transition.

The other important thing to note is that with higher incident THz fields the phase transition has an earlier onset when the transition is from below the critical temperature to above the critical temperature. However, when going from above to below the critical temperature there is not much change between different THz fields. This means that as the incident THz field is increased the width of the hysteresis narrows. 86

Figure 4.11: The temperature dependent transmission through NS VO2 demonstrating the hysteretic behavior of the phase transition.

While the whole hysteresis is crucial for determining things like the width of the hysteresis loop, Figure 4.12 only shows the half of the hysteresis where temperature is increasing. As this plot is much less busy it is helpful at showing how strong of an eﬀect the diﬀerent incident THz powers have on the hysteresis loop. 87

Figure 4.12: The temperature dependent transmission, for sample temperature going from 50 degrees up to 80 degrees C, through NS VO2 demonstrating the hysteretic behavior of the phase transition.

The width of each hysteresis loop vs the incident THz field can be measured and plotted, as seen in figure 4.13. The width of the loop is defined as the difference of the critical temperature, ie, the temperature when the transmission is exactly half of its maximum value, for the increasing temperature side of the loop vs the decreasing temperature side. This illustrates the dependence the hysteresis curves have on the incident THz field. 88

Figure 4.13: Plot of the narrowing of the hysteresis width as it varies with increasing THz.

The reason for the narrowing of the hysteresis curve has to do with the energy associated with the phase transition [137–139]. For the heating phase the energy it takes to transitions from the monoclick to rutile can be shown to be the energy of the strain plus the shear energy

1 1 U = U + U = E2 + ηGγ2 (4.14) tot,up strain shear 2 2

Where E is Young’s modulus, is the extensional strain, η is the domain shape parameter, G is the shear modulus, and γ is shear strain. However, when the temperature is decreasing the transition from rutile to monoclick is extremely 89 rapid. This means that the shear strain isn’t a factor in the energy of dislocation so for decreasing temperatures the energy is

1 U = U = E2 (4.15) tot,down strain 2

We can ﬁnd the width of the hysteresis cycle with this. The diﬀerence in the transition temperatures is

∆U U − U ηGγ2 ∆T = = tot,up tot,down = (4.16) ∆S ∆S 2∆S

This tells us exactly what the width of the hysteresis curve depends on. From the results of the power dependence and the frequency response we know the THz ﬁeld is having an eﬀect on the physical structure of the crystal itself. Thus we can conclude that the THz is weakening the bonds between the atoms which is thus lowering the shear modulus, G. By lowering G we reduce ∆U and thus lower ∆T .

4.7 THz Dependent Sheet Conductivity

So far, all of the discussion has been about the changes in the transmission of the

THz through the VO2. However, the transmission is not a fundamental property of the sample. The change in transmission is really just a byproduct of the change in the conductivity of the material. As the thickness of the VO2 layer is much 90

much smaller than the wavelength of our light, 100 nm vs 300 µm, the VO2 can be treated as a thin ﬁlm, as discussed in section 2.1.2.2. However, applying equation 2.47 to the NS coated samples is diﬃcult at best.

In order to quantify the sheet conductivity for the NS samples, a different approach is needed. The first thing that needs to be discussed is that the NS coating itself is not having any direct effect on the samples. This can be done by overlaying the sheet conductivities for a low incident THz field on NS coated samples with high incident field THz on bare samples.

Figure 4.14: The normalized temperature dependent theoretical sheet conductivity for high incident THz on bare VO2 compared with low incident THz on NS coated VO2.

As ﬁgure 4.14 shows, for comparable local THz ﬁelds the conductivities will be the same for any sample. That means that if certain material characteristics 91 are determined for the bare sample, they can also be applied to the NS case. A phenomenological model was produced that compared the power transmission through the sample and the sheet conductivity

T log samp = ae−bx + cx + d (4.17) Tair

where a, b, c, and d are perimeters that can be fit, and are assumed to be essentially material properties, and x is the normalized sheet conductivity, x=σsZ0. As the sheet conductivity for the bare sample can be calculated reliably the fitting for the bare sample, based on data from figure 4.14 results in the fallowing fitting parameters

a = 1.698 b = 2.2594 c = -0.2229 d = -1.6941

Using these parameters, the sheet conductivity for the NS coated samples can be computed. 92

Figure 4.15: The semilog plot of the relative transmission vs both the calculated sheet conductivity based off the phenomenological model and the regression done with the bare sheet conductivity and the fitted sheet conductivity based on the thin film equations for the NS coated VO2.

figure 4.15 shows the overlay of the calculated NS values vs the conductivity based on the fitting perimeters demonstrating the validity of this approach. With this result we are able to calculate things such as the resistivity of the NS sample at different temperatures and incident THz fields. This can be seen below in figure 4.16 93

Figure 4.16: The resistivity of the samples as a function of temperature for the NS coated VO2.

As discussed in section 4.1.2 the resistivity of the sample is directly related to the band gap of the material. By simplifying equation 4.12 to

Eg/2kbT ρ = ρ0e (4.18)

This can be fitted to the different resistances in figure 4.16 to determine the band gap of the VO2 at varying incident THz powers. 94

kV ET Hz cm Eg (eV) 90 1.2

300 0.44

560 0.34

790 0.32

This shows that as the THz ﬁeld is increased the band gap decreases. This indicates that the THz ﬁeld is indeed driving the phase transition and increasing the value for α.

4.8 Time Dependent Transmission of the THz Beam and Real Time

Fluctuations of the Conductivity

The ﬁnal phenomenon to investigate is the real time response of the material to the THz waveform. Using integrated detectors like the bolometer and pyroelectric removes all information about the instantaneous interaction of the THz beam with the sample. THz Time Domain Spectroscopy can be used to explore these real time dynamics.

We can ﬁrst investigate the temperature dependence of the samples to give a baseline as to what the phase transition will appear as. The bare samples are the best to do these measurements with as they do not have nearly the dependence on the incident THz ﬁeld as the NS coated samples. 95

Figure 4.17: Plots of the THz waveform transmitted through the bare VO2 sample. This demonstrates the eﬀects of only varying the temperature on the transmitted waveform.

Figure 4.17 shows some very important things. Even at early times, such as 3.4 ps, the transmission through the sample at 65 C is higher than for 74 or 76 degrees C. This was expected as the sheet conductivity should have little dependence on the incident THz field. Figure 4.18 below is a series of THz-TDS scans at different sample temperatures and incident THz fields for NS coated VO2. 96

Figure 4.18: The time dependent transmitted THz wave through NS coated VO2 at a) 45 C, b) 62 C, c) 65 C, and d) 67 C on the increasing temperature side of the hysteresis curve.

Even as low as 45 degrees there is a clear difference between the low incident THz and the high incident THz. At 65 C an interesting effect happens. The low power THz waveforms still show all of the signs of passing through the insulator phase of the VO2. However, the high field THz waveforms, while at short time, exhibit similar behavior. At later times they very quickly deviate from low THz waveforms. The most notable phenomenon is the lack of continued oscillation. This indicates that the THz waveforms have encountered a higher conductivity than their low THz field counterparts.

It is of interest to compare the low temperature, high incident ﬁeld THz wave- 97 form to the high temperature, low incident ﬁeld THz waveform. Figure 4.19 shows the waveform for 850 kV/cm with the VO2 at 65 C plotted against the waveform for 150 kV/cm with the VO2 at 67 C. The THz wavefrom for a 150 kV/cm incident

THz ﬁeld with the VO2 at 65 C is plotted in the background for reference.

Figure 4.19: The transmitted THz ﬁeld through NS coated VO2 comparing low THz at high temperature to High THz at low temperature.

This ﬁgure clearly shows that high incident THz ﬁelds dynamically drive the

VO2 samples into the metallic state. 98

4.9 Conclusion

In this chapter we have investigated the causes behind the MIT transition of VO2. We have developed a phenomenological theory for the field induced insulator to metal transition. We have shown that samples of VO2 without any nanoslot res- onators are much less dependent on the strength of the incident THz field. We have also shown that conductivity of the nanoslot coated VO2 samples display a large dependence on the incident THz field, due to the lack of field enhancement. Not only is the conductivity, resistivity, and the band gap dependent on the THz field, but that the changes induced by the THz field are rapid and dynamic. The fact that the THz radiation reduces the atomic bond strength leads to both a drop in the band gap and the sheer modulus with increasing incident THz fields. Finally we demonstrated that these transitions happen on extremely short times scales; with the duration of a single THz pulse. 99

5 High Field Electron Dynamics at Sub-picosecond Timescale in

GaAs and Si

GaAs and Si are the two most prevalent semiconductors in use today [140]. Each has unique properties; the most important being a direct (GaAs) [141] versus indirect (Si) [142] band gap. The wide spread use of these semiconductors makes understanding these systems an integral part of developing new technologies. With the size of transistors and other devices shrinking to the order of nanometers [143] with faster and faster switching rates it is more vital than ever to explore the high ﬁeld, sub-picosecond, electron dynamics of these semiconductors.

5.1 Intervalley Scattering Induced by Strong THz Fields

To understand these electron dynamics it is important to ﬁrst the fundamental properties of electrons in periodic systems. Basic band theory was covered in section 2.4. However, for virtually all real world systems an analytic solution to the band structure is not an option. Thankfully there are a number of techniques that can be used to calculate the band structure of real world materials [144–147]. 100

(a) A plot of electron energy vs momen- (b) A plot of electron energy vs momentum for GaAs [147] tum for Si [147]

Semiconductors are materials that have a gap in their band structure between the valiance band and the conduction band. They are often broken up into two main categories, direct and indirect band gaps. Direct band gap materials, such as GaAs, have no change in momentum between the top of the valiance band and the bottom of the conduction band while indirect band gap materials, such as Si, have a diﬀerence in momentum between the top of the valiance band and the bottom of the conduction band. This is a vital diﬀerence for optical excitations of electrons into the conduction band.

It is much easier to use electromagnetic radiation to excite an electron to transition across the band gap in direct band gap materials due to the high energy to momentum ratio that photons have. For example, the band gap in Si is approximately 1.1 eV. A photon of that energy has a wavelength of about 1160 nm. The momentum of that photon is p =hk ¯ and so the momentum of the photon is 101

1 proportional to λ . The momentum of an electron at the edge of the brillouin zone 1 is a where a is the lattice spacing. For Si the lattice spacing is on th order of .54 nm. Thus the change in the momentum for an electron going from the Γ point to the X point is on the order of 1000 times larger than a photon with the energy to induce to bring the electron from the top of valence band to the bottom of the conduction band. In order to make these indirect transitions the photon must couple with another source of momentum, often a phonon, in order to conserve energy and momentum as the electron transitions to the conduction band.

5.1.1 Intervalley Scattering in GaAs

Fig 5.1a above shows calculated band structures for GaAs (left) and Si (right). The direct band gap for GaAs and the indirect band gap for Si can be seen. Another feature of note, for GaAs, is the L valley. This valley is where hot electrons are most likely to transition to from the Γ valley [148–151]. If an electron inside the Γ valley is given enough energy, it is able to transition to the L valley via coupling with a phonon [150,152], as shown in Figure 5.2 bellow.

In previous work it was demonstrated that the energy imparted to to the electrons by a 1 MV/cm THz field was too small to induce a transition from the valence band to the conduction band [63]. The mean free path for GaAs approximately is 26 nm when the electron density is on the order of 1018cm−3 [153], given a thermal velocity of 4.4·105. This gives an average energy of approximately 300 102 meV imparted by the 1 MV/cm THz field. However, the nanoantenna array will enhance the THz field by a factor of 20. This gives an energy of up to 6 eV, more than enough to induce the excitations seen in [63].

Figure 5.2: A diagram of the intervalley scattering for GaAs as designed by Su et. al. [152].

However, as depicted in figure 5.2, the energy scales are much different if the electron is already excited into the conduction band. In order to induce intervalley scattering the energy of the electron must be approximately equal to the valley it is scattering into. As depicted above, the energy gap between the Γ valley and the X valley is on the order of 300 meV, which means that a THz field of 1 MV/cm could induce scattering into the side valley. 103

When the THz ﬁeld is enhanced by the nanoantenna arrays, it imparts an 20 times larger to the electron. Figure 5.1a shows that there are a large number of side valleys that the electron could scatter to, form the bottom of the conduction band, given that much energy.

Intervalley transition is detectable due to the change in bulk conductivity the system undergoes when electrons transitions from the Γ valley to the L valley. This is mainly due to the change in eﬀective mass. An electron’s eﬀective mass is a measure of how an electron responds to external forces at the bottom, or top, of a band. The energy of any electron can be written as a function of its momentum. This is called the dispersion relation. Taking a Taylor expansion of E(k) out to second order, it can be shown that the mass of the electron, near the an extrema in the band structure can be written as

h¯2 m∗ = (5.1) d2E 2 dk2

As equation 5.1 implies, the eﬀective mass is inversely proportional to the curvature of the band structure at the extrema. Thus, if we examine the band

d2E structure of GaAs we see that dk2 will be much larger at the Γ point than at the ∗ ∗ L point. Thus mΓ < mL. This reduces the conductivity of sample as

ne2τ σ = (5.2) m∗ 104

This decrease in conductivity would cause an increase in transmission for THz radiation.

5.1.2 Intervalley Scattering in Si

Despite the presence of an indirect bandgap in Si, intervalley scattering is not that much more complicated than for GaAs. As shown in ﬁgure 5.1b, there are many diﬀerences in the band structure between GaAs and Si. GaAs has two sidevalleys X and L that are near the energy of the Γ valley. Si only has one such sidevally; the L valley.

5.2 Nanoantenna Patterned Samples

The GaAs and Si samples used in these experiments have a similar nanoantenna structures as the VO2 samples discussed in section 4.3. A 300 µm substrate of either GaAs or Si was coated with a layer of gold, approximately 100 nm thick. Slots were cut in the sample that were 60 µm in the y-direction and 200 nm in the x-direction. A 2 mm by 2 mm region was patterned with these nanoslots such that each slot was separated by 70 µm in y-direction and 60 µm in the x-direction, when measured from the bottom left corner of one nanoslot to bottom left corner of its neighbor. 105

Figure 5.3: A cartoon of the Nanoslot coated samples

Each nanoantenna patterned sample has an accompanying reference sample, or bare sample, but without the nanoslots.

The experiments preformed on these sample is the pump probe style analysis, as discussed in section 3.3. The sample is excited with a 800 nm, 130 fs, optical pump beam and then probed with a THz pulse. This gives a great deal of control and flexibility to the experiments. It means that the fluence of the optical pump beam can be changed. This effectively alters the number of free carries generated.

The pump-probe delay can be altered. For convention the delay τpp = tT Hz - tpump.

This means that at positive τpp the THz pulse is incident on the sample first while a negative τpp has the pump beam incident first. Finally the THz power can be altered. This final parameter is of the most interest as it will allow us a glimpse into the High-field electrons dynamics in the conduction band for GaAs and Si. 106

5.3 Optical Free Carrier Driven Conductivity

As touched on in section 5.1 Semiconductors are a material that intrinsically have very small conductivities due to the energy gap between their valence band and conduction band. In order to investigate the carrier dynamics in GaAs and Si the THz transmission dependence on optical carrier density needs to be established. The optical carrier density is calculated assuming that one photon interacts with one electron. Thus there is linear direct relationship between the intensity of the pump beam, the number of incident photons, and the number of free carriers generated. As long as the total energy of the incident pump beam, Upump, is known, a simple calculation can be made to ﬁnd the density of free carriers Ne generated

Upump N = α ¯hω (5.3) e Aδ

where α = .0107 is a numeric constant that depends on the reflection coefficient R as well as the beam profile, A is the cross sectional area of the beam when it is incident on the sample and δ is the skin depth of the material for a give incident pump wavelength. This says that as the incident pump energy grows the density of free electrons grows as well. This allows for the fine tuning of the optical electron density and allows for the exploration of the effects that different electron densities have on the transmission of the the THz probe beam.

The transmission of THz through a material is directly dependent on the con- 107 ductivity of that material. The free electron density, and the conductivity, depend on the optical pump energy. As the optical pump energy is increased the THz transmission will decrease. Figures 5.4 and 5.5 demonstrates the eﬀect that altering the energy of the incident optical beam has on the transmitted THz waveform through nanoantenna coated GaAs.

Figure 5.4: THz waveform transmitted through the nanoantenna coated GaAs with τpp = −0.5ps and incident THz ﬁeld of 321 kV/cm 108

Figure 5.5: THz waveform transmitted through bare GaAs with τpp = −0.5ps and incident THz ﬁeld of 321 kV/cm

The higher the carrier density the lower the THz transmission, due to the increase in conductivity. An interesting thing to note is for both nanoantenna patterned and bare GaAs, there does seem to be some sort of threshold. The plots

15 −3 15 −3 of Ne = 1.92·10 cm and Ne = 6.45·10 cm are virtually on top of each other, with the exception of time delay less than 1 ps when. In contrast the transmission

16 for Ne = 1.27 · 10 is markedly decreased after the pump beam is incident. In all cases the THz transmission, when comparing between varying pump energies, is nearly identical when the THz is incident before the optical pump beam.

Both nanoantenna and bare cases imply that there is a delay between when the pump beam hits the sample and the full excitation of the electrons. The ﬁnite 109 temporal width of the optical beam does play a role here. As the pump beam is Gaussian in time, not delta like. There is also a characteristic excitation time for the optical electron density. The transition is not instantaneous, and this will be discussed much more in depth in section 5.4.

The eﬀects of increased optical carrier density in nanoantenna coated and bare Si are similar to that of GaAs, due to its indirect band gap and thus a much larger skin depth at 800 nm, the optical carrier density is only on the order of 10 % of the optical carrier density in GaAs. A decrease in THz transmission is still observed which implies an increase in conductivity.

Figure 5.6: A plot of THz waveform transmitted through the nanoantenna coated Si sample at diﬀerent incident pump powers. τpp = -3 ps and ET Hz = 320 kV/cm 110

Figure 5.7: A plot of THz waveform transmitted through bare Si sample at diﬀerent incident pump powers. τpp = -1 ps and ET Hz = 334 kV/cm

The effects of the increasing energy of the pump beam are apparent at t ≥ 0. A clear trend is present in both Si samples. As pump energy is increased there is a decrease in the transmitted THz. However, the change for the nanoantenna coated sample is not nearly as large as for either GaAs sample. It is also interesting to note that there is a smooth transition across all incident probe energies. This is very different when compared to the GaAs case where there seems to be a very well defined transition.

The change in transmitted THz waveform for the nanoantenna coated sample 111 is not large compared to GaAs so it is useful to try another tool to examine the THz transmission dependence on incident pump power. The time dependent THz electric ﬁeld E(t) can be Fourier transformed into E (ω) as seen in ﬁgure below.

Figure 5.8: The Fourier transform of the transmitted THz waveform when τpp = -3 ps

This ﬁgure really demonstrates the diﬀerence in the transmission for increasing pump energies. This also serves to illustrate the change in conductivity as there is a clear shift peak frequency. Section 2.5 discuses the role of the relative permittivity of the substrate on the resonance frequency. Figure 5.8 shows a clear redshift in central frequency, as well as a broadening of the spectrum, with increasing pump energy. This is very indicative of the increase in conductivity we would expect. 112

5.4 Temporal Evolution of Optically Excited Free Carriers

One of the most important applications of ultrafast lasers is the study of short timescale events. Section 5.3 only discussed the eﬀects of increasing the optical free carrier density. How these excited carriers evolve in time is of great interest.

By controlling the time delay, τpp, between the 800 nm pump beam and the THz probe beam1 the temporal dependence can be mapped out.

THz-TDS was initially employed to study the temporal dependence of GaAs and Si as it would allow for even more detail in the evolution of the optical carrier density over time. Both the nanoantenna sample and the bare sample exhibited nearly identical time evolutions as seen in ﬁgures 5.9 and 5.10

1by controlling the path length diﬀerence,see section 3.3.3 113

Figure 5.9: The transmitted THz waveform through nanoantenna coated GaAs at −3 diﬀerent values of τpp for ET Hz = 99 kV/cm and Ne = 9.6e15 cm 114

Figure 5.10: The transmitted THz waveform through nanoantenna coated GaAs −3 at diﬀerent values of τpp for ET Hz = 348 kV/cm and Ne = 9.6e15 cm . Not the THz wavefrom for τpp = −2, −3 ps has been magniﬁed by a factor of 10.

When τpp ¡ 0 the transmission dropped dramatically for both samples, indicating the excitation time was incredibly short. For the nanoantenna coated sample, when

kV the THz incident ﬁeld is 99 cm , there is a clear change in the transmission between

τpp = −1.5 ps -0.5 ps, and the positive times. When τpp = −0.5 ps is of particular interest as it illustrates the dynamics of the excited electron dynamics. When t is less than τpp the waveform lines up will with the positive τpp waveforms. However, when t ≥ 0 there are large deviations from the positive τpp waveforms.

This time dependence is important as it leads to important insights. Figure 5.9 and 5.10 both consistently shows a lag of 0.5 ps between the pump arrival at the sample and the onset in a change of the transmission. This delay is consistent with the data presented in section 5.3. Part of this delay is the ﬁnite temporal 115 width of the pump beam. It is also due to the characteristic excitation time of the sample.

THz-TDS, while very useful, doesn’t tell the whole story. Using a bolometer and power transmission setup will allow for a more complete mapping of the temporal evolution of the excited carriers.

Figure 5.11: The diﬀerential transmission of the THz ﬁeld through nanoantenna coated GaAs vs τpp

By exploring the transmission dependence on τpp a more complete picture of the excitation time for GaAs can been seen. By looking at the time it takes from the THz transmission to go from 0 percent change to a maximum change will give an estimate of the excitation time.

Figure 5.11 also depicts the Transmission dependence on the strength of the incident THz ﬁeld. There is strong positive correlation between the incident ﬁeld strength and the transmission through the sample. This will be discussed much 116 more in-depth in section 5.5.

As we are interested in the time dependence of the excited electron, not the THz induced transparency, we can normalize all of the plots in ﬁgure 5.11 to unity to compare the thermilization time of the electrons.

Figure 5.12: The diﬀerential transmission of the THz ﬁeld through nanoantenna coated GaAs vs τpp

The ﬁrst thing of note, is that once normalized to unity, all of the plots overlap nicely. This indicates that the thermilization time is not THz dependent but is a material property. The an error function can be ﬁt to the plots and the thermilization time calculated. tT herm,GaAs = 1.08 ± 0.04 ps.

Comparing the Si samples to the GaAs shows some interesting, if not unex- 117

pected, results. Figure 5.13 shows transmitted THz waveform the τpp dependence for nanoantenna coated Si.

Figure 5.13: The transmitted electric through the nanoantenna coated Si at dif- −3 ferent values of τpp for ET Hz = 340 kV/cm and Ne = 8.9e14cm .

Unlike GaAs, the Si sample shows a much longer rise time. For GaAs the transition is very sharp, like in ﬁgure 5.10. However, The Si samples show a very gradual decay in the transmission after the pump beam is incident on the sample. This trend is also seen in the bare Si sample. 118

Figure 5.14: The transmitted electric through bare Si at diﬀerent values of τpp for −3 ET Hz = 340 kV/cm and Ne = 8.9e14cm .

Initially it appears as if the THz transmission undergoes a phase shift. This phase shift can be explained as before, as a result of the drift in percent relative humidity. However, unlike before, the transition is much more pronounced. This diﬀerence can be explained by considering the slow excitation time for Si when compared to GaAs.

As for GaAs, a power based transmission detection scheme, using a bolometer, is employed to study the transmission dependence on τpp on a much ﬁner scale. 119

Figure 5.15: The diﬀerential transmission of the THz ﬁeld through nanoantenna coated Si vs τpp

Just as for GaAs, ﬁgure 5.15 can be normalized to unity. 120

Figure 5.16: The diﬀerential transmission of the THz ﬁeld through nanoantenna coated Si vs τpp

Again, the thermilization time is independent of incident THz ﬁelds and ﬁtting to an error function give a thermilization time of tther,Si= 1.795 ± 0.08 ps.

5.5 THz Induced Transparency Via Intervalley Scattering

The true focus of these experiments is the dependence of the THz transmission on the incident THz ﬁeld strength. It has already been shown that GaAs has a strong

MV interaction with incident THz fields in excess of 10 cm [63]. Without an optical excitations it was shown that large THz fields were able to create a number of free 121 carriers in GaAs and this increase its conductivity. However, the interaction of GaAs and Si that have a high free carrier density with large THz fields have not been widely explored.

We will employ two main methods to study the samples under these conditions, the ﬁrst being THz-TDS. The second is a power dependent THz transmission using a bolometer.

5.5.1 Time Domain Spectroscopy Investigation of THz Intensity

Dependent Transmission

For nanoantenna coated GaAs comparing high and low incident ﬁelds yields startling results as illustrated in ﬁgure 5.17 122

Figure 5.17: THz Waveform transmitted through nanoantenna coated GaAs at −3 diﬀerent incident THz powers with Ne = 1.58e16cm and τpp is a) 1.5 ps, b) 0 ps, and c) -1.5 ps.

This plot is one of the most important ﬁgures produced in this work. At

τpp = 1.5ps there is little change between the transmission of the three diﬀerent

THz fields. This is due to the fact that at positive τpp there are no free carriers for the THz to interact with. As τpp becomes more and more negative the THz pulses with a lower incident field strength are attenuated much more than the THz with large incident fields.

As section 2.1.2.2 discussed, the transmission of THz radiation through a ﬁlm 123 is inversely related to the conductivity of that ﬁlm. As concluded in section 5.3 the decrease in transmission is the direct result of the increase in conductivity created by the large scale generation of free carriers produced by the optical pump pulse.

However, the high field THz pulses have a much smaller change in transmission when compared to the low incident field THz pulses. The conclusion is that high field THz must drive the optical excited carriers into states that lower the conductivity of the GaAs, thus allowing for higher transmission of the THz pulse.

As both pulses has the same pump energy and same τpp the high ﬁeld THz must be altering the conductivity of the sample.

Equation 5.2 is the Drude conductivity for a material containing free carrier. As the energy of a THz photon is far too small to induce the transition of a conduction band electron to the valiance band, the high field THz pulse must be altering either the scattering time, which is unlikely, or the effective mass. Looking at equation 5.1 shows that the effective mass is inversely related to the curvature of the band structure, near an extrema.

The conclusion that we came to is that the high THz fields are causing the electrons to undergo intervalley scattering. The side valleys for GaAs, X and L, have a much smaller curvature, when compared to the Γ point, see figure 5.1a, so they will have a larger effective mass and create a lower conductivity.

The next step is to see if this eﬀect extends to the bare GaAs sample. Figure 5.18 test varying THz ﬁelds and the resulting THz transmission. 124

Figure 5.18: THz Waveform transmitted through bare GaAs at diﬀerent incident −3 −3 THz powers when τpp = −0.5 ps and Ne = a) 0 cm , b) 1.9e15 cm , c) 6.4e16 cm−3, d) 1.3 e16 cm−3, e) 1.8e16 cm−3.

As shown, even the bare sample, with it much lower fields, sees the effect of this intervalley transition. The THz induced transparency is reduced, as expected, for the much lower local THz fields.

The Si samples were expected to interact with the high THz ﬁelds much less than the the GaAs samples. This is due to a combination of the larger skin depth, which is a result of the indirect band gap, and the fact that the change in the curvature of the band when scattering to sidevallys is smaller, compared to GaAs as illustrated in ﬁgures 5.1a and 5.1b. 125

Figure 5.19: THz Waveform transmitted through nanoantenna coated Si at ET Hz = 33 kV/cm and 619 kV/cm. There is no optical pump for this experiment.

Figure 5.20: THz Waveform transmitted through nanoantenna coated Si at ET Hz = −3 33 kV/cm and 619 kV/cm. τpp = −2 ps and Ne = 1.38cm . 126

Figure 5.21: THz Waveform transmitted through bare Si at diﬀerent incident THz powers.

As expected the changes are not nearly as large as for GaAs. However, there is still an appricable change in transmission for the nanoantenna coated Si sample.

One important comparison between the GaAs and the Si samples is the rate at which the conductivity changes. For both GaAs samples the THz driven changes in conductivity are nearly instantaneous. When looking at the THz dependence for Si there is a clear onset for the increase in transmission. For Si, at early times in the THz pulse, the low and high THz pulses are nearly overlapping. It is only as the pulse propagates do the transmission for the low and high ﬁeld pulses diverge. 127

In contrast, there is virtually no overlap between low and high THz pulses in the GaAs samples.

5.5.2 Power Transmission Detection of THz Induced Transparency

As discussed above time domain spectroscopy is a vital tool for examining the time dependent effects on the transmitted THz pulses. It allows for the real time mapping of the THz field and gives the ability to explore real time changes in transmitted THz waveform that other detection methods do not. It does not, however, lend itself to concrete analysis. In order to get the total transmitted electric filed the waveform must be Fourier transformed and then integrated along the frequency axis. This leads to a number of numerical errors. Thus an integrated detector, such as a Si-bolometer is required to fully understand the dynamics of these systems. Figures 5.22- 5.26 employ the power dependent transmission detection scheme outlined in section 3.3.3.1.

The THz-TDS results for GaAs indicated a strong dependence on the incident THz ﬁeld. The power dependent transmission supported this. 128

Figure 5.22: Total relative power transmission through nanoantenna coated GaAs −3 −3 −3 for Ne = 6.4e15 cm , 1.6e16 cm , 2.5e16 cm and τpp = a) 3 ps, b) 1.5 ps, c) 0.5 ps, d) 0 ps, e) -0.5 ps, f) -1.5 ps, g) -3 ps, and h) -10 ps. 129

Figure 5.23: Total relative power transmission through bare GaAs for τpp = −3 ps and -1 ps

At positive τpp there is little dependence on THz field strength, which is expected as not many free carries have been excited. As τpp goes more negative, less than 0, there is a clear dependence. The differential transmission, for high pump energies, gets as large as 2.36 or a 236 % increase in transmission over low THz fields.

The bare GaAs had some very interesting behavior, ﬁgure 5.23. At pump energies that were commiserate with those used to study the nanoantenna sample the high pump power had little change in transmission due to increasing THz. This is the opposite trend from the nanoantenna coated GaAs.

However, when the pump energy was severely reduce such that the change in transmission due to the pump beam was commiserate with the nanoantenna tests 130

ﬁgure 5.24 was produced.

Figure 5.24: Total relative power transmission through bare GaAs at low incident pump powers such that the percent change in transmission due to excited carries is less that 25 % for τpp = a) -10 ps, b) -3 ps, and c) -1 ps.

This shows that for pump energies that are low, the bare sample fallows the trend set by the nanoantenna coated GaAs. This is a very interesting discovery as it indicates that at high pump energies, i.e. high concentrations of electrons, there are new and exciting electron dynamics that have not been explored yet in this work. It is currently theorized that this change in behavior is due to a short mean free path coupled with the high electron density, but this work is still ongoing.

The nanoantenna coated Si show behavior that also is agreement with THz- TDS. 131

Figure 5.25: Total relative power transmission through nanoantenna coated Si for −3 −3 −3 Ne = 3.6e14 cm , 8.9e14 cm , 1.4e15 cm and τpp = a) 3 ps, b) 1.5 ps, c) 0.5 ps, d) 0 ps, e) -0.5 ps, f) -1.5 ps, g) -3 ps, and h) -10 ps. 132

Figure 5.26: Total relative power transmission through bare Si for τpp = a) -3 ps, b) -1 ps, c) 1 ps)

The change in transmission was much smaller for nanoantenna coated Si. The diﬀerential transmission only went as high as .31, or a 31 % increase in transmission.

The bare Si was indeed the least interesting, staying in agreement with the time domain spectroscopy results. As ﬁgure 5.26 shows there is little, if any dependence of the transmission on the incident THz Field. The largest increase in transmission was on the order of 10 %. 133

5.6 Induced Sheet Conductivity

As discussed in section 4.7 the THz transmission through a material is important, but it alone is not an indicator of any material property. Using the method described in section 2.1.2.2 the THz transmission is directly related to the conductivity of the sample. The thin ﬁlm sheet conductivity described in equation 2.47 is a useful tool here. Despite the fact that the thickness of the sample itself is of the order of the wavelength of the THz radiation, the skin depth of both samples is much smaller, less than 10 µm in Si and even less in GaAs. This means that the depth of the material which is excited by the pump beam can be approximated as a thin ﬁlm inside the sample itself.

But just as discussed in 4.7 the nanoslots create an issue for applying equation 2.47. Again we turn to a phenomenological ﬁt.

−bx log (Tsamp) = ae + cx + d (5.4)

where x = σsZ0.

Using the bare transmission at τpp = −10 with high incident THz ﬁelds and correlating that to the nanoantenna samples at low incident THz ﬁelds gives coef- 134

ﬁcients of

aGaAs = 0.1604, bGaAs = 1.00, cGaAs = −0.0401, dGaAs = −4.1404 (5.5)

aSi = 1.552, bSi = 1.001, cSi = −0.0104, dSi = −3.7502 (5.6)

Which produces a normalized sheet conductivity, σsZ0, of

Figure 5.27: The conductivity of GaAs vs incident THz ﬁelds at τpp = −10 ps and free carrier densities of 12.9, 31.8, and 51.1 ·1015 cm−3

for GaAs; and for Si 135

Figure 5.28: The conductivity of Si vs incident THz ﬁelds at τpp = −10 ps and free carrier densities of 7.2, 28.8, and 28.7 ·1015 cm−3

Thus, we can see that as the incident THz ﬁeld increases the conductivity of the sample drops signﬁcaltly.

5.7 Conclusion

In this chapter we explored the sub-picosecond dynamics of electrons inside two of the most common semiconductors, GaAs and Si. Their dependence on incident pump energies, pump probe time delay, and incident THz ﬁeld strength was explored using a THz probe. It was found that, as was well known, high energy pump 136 beams generated a large number of free carriers. This had the eﬀect of increasing the conductivity and dropping the THz transmission through the sample.

The Pump-Probe time delay τpp showed expected behavior as well. When it was positive there was little to no change in transmission for the THz probe beam.

As a positive τpp meant the THz probe hit before the 800 nm pump, this result was good. It was shown that the excitation time for GaAs was approximately a factor of two smaller than for Si, with or without Nanoslots.

Finally the conductivity’s dependence on the incident THz field was fascinat- ing and unexpected. Both nanoantenna coated GaAs and nanoantenna coated Si exhibited large drops in conductivity with even low field THz. However, the larger the THz field the stronger the change. The transmission never returned to zero pump-probe levels. This all implied that the changes in conductivity were due to an increase in the effective mass of the electron. This change was induced via scattering of electrons to side valleys where the curvature of the band was significantly less. 137

6 Conclusion

Over the course of my time working for Dr. Lee I have had the chance to work on a number of exciting and difficult projects. Aside from necessary tasks such as experiential design and layout I have had the chance to work with Nanoslot coated VO2 and study the metal to insulator transition. I also had the opportunity to explore the sub-picosecond electron dynamics of some of the most important materials on earth, GaAs an Si. All of this is made possible by the implementation of high field Terahertz radiation generated via optical rectification in LiNbO3 using tilted pulse fronts. This, along with a verity of methods to detect THz radiation, allowed for a wide range of novel experiments.

In my investigation of VO2 we discovered that the metal insulator transition can be triggered using high intensity THz radiation, if the sample is patterned with nanoslots. When the sample is placed at a temperature that is approximately 2 degrees Celsius below the transition temperature a strong THz pulse would cause the sample to transition to a conducting phase. It was also found that the hysteresis loop present in the phase transition would narrow with increasing THz radiation. This was found to be due to the weakening of the atomic bonds by the THz radiation and reducing the Sheer modulus G. Finally it was found, by analyzing the temperature dependent resistivity, that the activation energy, which is directly 138 related to the band gap is signiﬁcantly reduced by the incident THz ﬁeld.

I also investigated the effects of high field THz radiation on Si and GaAs and its effects on their ultra-fast electron dynamics when free carriers were generated by an 800 nm pump pulse. It was shown that for lower energy pump pulses fewer electrons were excited into the conduction band. By changing the delay between the pump and probe, τpp, it was found that the electron excitation time for GaAs was about half the time of that of Si. Finally by using both time domain spectroscopy and using an integrated power detector it was discovered that the THz radiation pushed free electrons that sat in the Γ valley into the X and L valleys for GaAs and pushed electrons from the X valley into the L valley for Si. This caused a dramatic reduction in conductivity for both GaAs and Si, with a much more significant change for samples patterned with nanoslots. For NS coated GaAs there was an increase in transmission as high as 235 % and up to 35% for NS coated Si. 139

APPENDIX 140

A Derivation of Index of Refraction Vs Wavelength

2 eE~ (t) x¨ + Γx ˙ + ω0x = m

~ ±iωt ±iωt since E(t) = E0e assume that x(t) = x0e

±iωt 2 ±iωt thusx ˙ = x0 · (±iωe ) andx ¨(t) = −x0ω e

±iωt eE0 e x = 2 2 m ω0 −ω ±iΓω

Nex Ne2 χ = ~ = 2 2 0E(t) m(ω0 −ω ±iΓω)0 2 2 2 Ne ω0 −ω Γω r = 1 + χ = 1 + 2 ∓ i 2 m0 2 2 2 2 2 2 2 2 (ω0 −ω ) +Γ ω (ω0 −ω ) +Γ ω

1 √ i φ n = (r) 2 = Ze 2

s 2 2 2 2 2 2 Ne ω0 −ω Ne Γω Z = 1 + 2 + 2 m0 2 2 2 2 m0 2 2 2 2 (ω0 −ω ) +Γ ω (ω0 −ω ) +Γ ω −1 Ne2 Γω φ = tan 2 m0 1+ Ne ω2−ω2 m0 ( 0 )

2πc ω = λ

v 2 2 2 ! ! u 2 2πc 2πc 2 ω − 2 u Ne 0 ( λ ) Ne Γ λ Z = t 1 + 2 2 2 + 2 2 2 m0 2 2πc 2 2πc m0 2 2πc 2 2πc ω0 −( λ ) +Γ ( λ ) ω0 −( λ ) +Γ ( λ )

2 Ne Γ 2πc −1 m0 ( λ ) φ = tan 2 2 1+ Ne ω2− 2πc 0 0 ( λ ) 141

Bibliography

[1] M. Kimmit “Restrahlen to T-Rays 100 Years of Terahertz Radiation” Journal of Biological Physics 29, 77 (2003)

[2] H. Rubens and O. von Baeyer “On extremely long waves, emitted by the quartz mercury lamp” Philosophical Magazine Series 6 Volume 21, Issue 125, 689 (1911)

[3] L. Elias, J. Hu, G. Ramian “The UCSB electrostatic accelerator free electron laser: First operation” Nucl. Instr. and Meth. 237 , 203 (1984)

[4] W. Duncan and G. Williams “Infrared Synchrotron Radiation from Elec- tron Storage” Appl. Optics 22, 2914 (1983)

[5] H. Roskos, M. Thomson, M. Kre, and T. Lﬄer “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications” Laser and Photonics Reviews Volume 1, 349 (2007)

[6] Y.S. Lee et. al. “Generation of narrow-band terahertz radiation via optical rectiﬁcation of femtosecond pulses in periodically poled lithium niobate” Appl. Phys. Lett. 76, 2505 (2000) 142

[7] Charles A Brau. “Free-electron lasers.” Science, 239(4844), 1115 (1988)

[8] Z.Sheng, K. Mima, J. Zhang, and H. Sanuki “Emission of Electromag- netic Pulses from Laser Wakeﬁelds through Linear Mode Conversion” PRL 94, 095003 (2005)

[9] V. B. Gildenburg and N. V. Vvedenskii “Optical-to-THz Wave Conver- sion via Excitation of Plasma Oscillations in the Tunneling-Ionization Process” PRL 98, 245002 (2007)

[10] P.K.D.D.P. Pitigala et al “Highly sensitive GaAs/AlGaAs heterojunc- tion bolometer” Sensors and Actuators A 167, 245 (2011)

[11] D. You, R. Jones, and P. Hucksbaum “Generation of high-power sub- single-cycle 500-fs electromagnetic pulses” Opt. Lett. 18, 290 (1993)

[12] A. Stepanov, J. Hebling, and J. Kuhl “Eﬃcient generation of subpicosecond terahertz radiation by phase-matched optical rectiﬁcation using ultrashort laser pulses with tilted pulse fronts” Appl. Phys. Lett. 83, 3000 (2003)

[13] P Richards “Bolometers for infrared and millimeter waves” J. Appl. Phys. 76, 1 (1994)

[14] M. Golay. “The theoretical and practical sensitivity of the pneumatic infra-red detector” Review of Scientiﬁc Instruments, 20(11), 816 (1949) 143

[15] H. Beerman “The pyroelectric detector of infrared radiation” Electron Devices, IEEE Transactions on, 16(6), 554 (1969)

[16] M. Liang, et al. “Terahertz Characterization of Single-Walled Carbon Nanotube and Graphene On-Substrate Thin Films” IEEE TRANSAC- TIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, 2719 (2011)

[17] T. Ouchi et al. “Terahertz Imaging System for Medical Applications and Related High Eﬃciency Terahertz Devices” J Infrared Milli Terahz Waves 35, 118 (2014)

[18] E. Parrott, Y. Sun, E. Pickwell-MacPherson “Terahertz spectroscopy: Its future role in medical diagnoses” Journal of Molecular Structure 1006, 66 (2011)

[19] F. Wahaia et al “Detection of colon cancer by terahertz techniques” Journal of Molecular Structure 1006, 77 (2011)

[20] J. Agbo, R. Gnanasekaran, and D. Leitner “Communication Maps: Ex- ploring Energy Transport through Proteins and Water” Isr. J. Chem. 54, 1065 (2014)

[21] M. van Exter, C. Fattinger, and D. Grischkowsky “Terahertz time- domain spectroscopy of water vapor” OPTICS LETTERS Vol. 14, 1128 (1989) 144

[22] W.D. Duncan et. al “An Optical System for Body Imaging from a Dis- tance Using Near-TeraHertz Frequencies” J Low Temp Phys 151, 777 (2008)

[23] E. Heinz et. al “Passive Submillimeter-wave Stand-oﬀ Video Camera for Security Applications” J Infrared Milli Terahz Waves 31, 1355 (2010)

[24] M. Yin, S. Tang, and M. Tong “The application of terahertz spectroscopy to liquidpetrochemicals detection: A review” Applied Spectroscopy Re- views 51, 379 (2016)

[25] A. Davies, A. Burnett, W. Fan, E. Linﬁeld, and J. Cunningham “Tera- hertz spectroscopyof explosives and drugs” Materialstoday vol. 11 issue 3, pg 18-26 (2008)

[26] J. Maxwell “A Dynamical Theory of the Electromagnetic Field” Phil. Trans. R. Soc. Lond. 1865 155, 459 (1865)

[27] Georg Simon Ohm “Die galvanische Kette, mathematisch bearbeitet” Riemann (1827)

[28] L.Berger et al. “Handbook of Chemistry and Physics” (CRC Press LLC,Oct. 27, 2003) 84rd ed. pg. 4-130

[29] John Jackson “Classical Electrodynamics” John Wiley & Sons, Inc. 1999 3erd ed. pg 259

[30] Eugene Hecht “Optics” Addison Wesley, 2002 4th ed. pg 68 145

[31] David Griﬃths “Introduction to Electrodynamics” Prentice Hall, 1999 3erd ed. pg 330

[32] Robert W. Boyd “Nonlinear Optics” Academic Press, 2008 3erd ed. pg 2

[33] Robert W. Boyd “Nonlinear Optics” Academic Press, 2008 3erd ed. pg 11

[34] Y. Shen “OPTICAL SECOND HARMONIC GENERATION AT IN- TERFACES” Annu. Rev. Phys. Chern. 40, 327 (1989)

[35] P. Guyot-Sionnest, W. Chen, and Y. Shen “General considerations on optical second-harmonic generation from surfaces and interfaces” PRB 33, 8254 (1986)

[36] Y. Lee “Principles of Terahertz Science and Technology” Springer, 2009 pg 81

[37] J. Zheng et al. “Ultrafast Dynamics of Solute-Solvent Complexation Ob- served at Thermal Equilibrium in Real Time” Science 309, 1338 (2006)

[38] N. Tamai and H. Miyasaka “Ultrafast Dynamics of Photochromic Sys- tems” Chem. Rev. 100, 1875 (2000)

[39] M. Paul et al. “High-ﬁeld Terahertz Response of Graphene” New Journal of Physics 15, 085019 (2013) 146

[40] J. Danielson et al “Interaction of Strong Single-Cycle Terahertz Pulses with Semiconductor Quantum Wells” PRL 99, 237401 (2007)

[41] P. F. Moulton “Spectroscopic and laser characteristics of Ti:Al203” J. Opt. Soc. Am. B 3, 125 (1985)

[42] A. Ashkin et al “Observation of a single-beam gradient force optical trap for dielectric particles” Optics Letters Vol. 11, 288 (1986)

[43] Z. Hu and H. J. Kimble “Observation of a single atom in a magneto- optical trap” Optics Letters Vol. 19, 1888 (1994)

[44] P. Maunz et al “Cavity cooling of a single atom” Nature 428, 50 (2004)

[45] N. Ashcroft and N. Mermin “Solid State Physics” Sanders College 1976 pg 133

[46] F. BELEZNAYT and M. LAWRENCE “The ‘muﬃn-tin’ approximation in the calculation of electronic band structure” J. Phys. C: Solid State Phys. 1 1288 (1968)

[47] J. HUBBARD “The approximate calculation of electronic band structure” Proc. Phys. Soc. 92 921 (1967)

[48] K. Terakura et. al “Band theory of insulating transition-metal monox- ides: Band-structure calculations” PHYSICAL REVIEW B 30, 4734 (1984) 147

[49] A Neckel et al “Results of self-consistent band-structure calculations for ScN, ScO, TiC, TiN, TiO, VC, VN and VO” J. Phys. C: Solid State Phys. 9 579 (1975)

[50] R. Coehoorn et al. “Electronic structure of MoSez, MoS2, and WSez. I. Band-structure calculations and photoelectron spectroscopy” PHYS- ICAL REVIEW B 35, 6195 (1987)

[51] K. Walsh “Hartree-Fock Electronic Structure Calculations for Free Atoms and Immersed Atoms in an Electron Gas” Oregon State Uni- versity (2009)

[52] V. I Anisimov et al “First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method” J. Phys.: Condens. Matter 9 767 (1997)

[53] K. Schwarz *, P. Blaha “Solid state calculations using WIEN2k” Com- putational Materials Science 28, 259 (2003)

[54] M. Fischetti and S. Laux “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge eﬀects” PHYSICAL REVIEW B 38, 9721 (1988)

[55] V. Anisimov, J. Zaanen, and O. Andersen “Band theory and Mott insulators: Hubbard U instead of Stoner I” PHYSICAL REVIEW B 44, 943 (1991) 148

[56] A. Liechtenstein, V. Anisimov, J. Zaanen “Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators” PHYSICAL REVIEW B 52, R5467 (1995)

[57] H. BETHE “Theory of Diﬀraction by Small Holes” Phys. Rev. 66, 163 (1944)

[58] T. Ebbesen “Extraordinary optical transmission through sub-wavelength hole arrays” NATURE VOL 39, 667 (1998)

[59] F. Garcia-Vidal et al. “Light passing through subwavelength apertures” REVIEWS OF MODERN PHYSICS VOLUME 82, 729 (2010)

[60] H. Lezec et al. “Beaming Light from a Subwavelength Aperture” Science vol. 297, 820 (2002)

[61] C. Genet and T. Ebbesen “Light in tiny holes” Nature Vol. 445, 39 (2007)

[62] M. Seo “Active Terahertz Nanoantennas Based on VO2 Phase Transi- tion” Nanoletters 10, 2064 (19)

[63] Y. Jeong et al. “Large enhancement of nonlinear terahertz absorption in intrinsic GaAs by plasmonic nano antennas” Appl. Phys. Lett. 103, 171109 (2013) 149

[64] Z. Thompson et al. “Terahertz-Triggered Phase Transition and Hystere- sis Narrowing in a Nanoantenna Patterned Vanadium Dioxide Film” Nano Lett. 15, 5893 (2015)

[65] F. Garcia-Vidal “Focusing light with a single subwavelength aperture ﬂanked by surface corrugations” APL 83, 4500 (2003)

[66] H. Rigneault “Enhancement of Single-Molecule Fluorescence Detection in Subwavelength Apertures” PRL 95, 117401 (2005)

[67] T. Xu et al. “Subwavelength imaging by metallic slab lens with nanoslits” APL 91, 201501 (2007)

[68] A. Otto “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reﬂection” Zeitschrift fr Physik Volume 216, 398 (1968)

[69] R. H. Ritchie et al. “SURFACE-PLASMON RESONANCE EFFECT IN GRATING DIFFRACTION” PRL 21, 1530 (1968)

[70] R. Chatterjee “Antenna Theory and Practice” Wiley Eastern Limited, New Delhi 1998

[71] M. Ma “Theory and Application of Antenna Arrays” John Wiley & Sons Inc. 1974

[72] R. Elliott “Antenna Theory and Design” Prentice-Hall inc. Englewood Cliﬀs, 1981 150

[73] J. Choe et al. “Slot antenna as a bound charge oscillator” OPTICS EXPRESS 20, 6521 (2012)

[74] J. Kang et al. “Substrate eﬀect on aperture resonances in a thin metal ﬁlm” OPTICS EXPRESS 17, 15652 (2009)

[75] H. Park et al. “Terahertz nanoresonators: Giant ﬁeld enhancement and ultrabroadband performance” APPLIED PHYSICS LETTERS 96, 121106 (2010)

[76] M. Seo et. al “Terahertz ﬁeld enhancement by a metallic nano slit operating beyond the skin-depth limit” NATURE PHOTONICS VOL 3, 152 (2009)

[77] A. Sinyukov et al. “Resonance enhanced THzTHz generation in electro- optic polymers near the absorption maximum” Appl. Phys. Lett. 85, 5827 (2004)

[78] C. Weiss, R. Wallenstein, and R. Beigang “Magnetic-ﬁeld-enhanced generation of terahertz radiation in semiconductor surfaces” Appl. Phys. Lett. 77, 4160 (200)

[79] K.L. Vodopyanov “Optical THz-wave generation with periodically- inverted GaAs” Laser and Photonics Reviews 2, 11 (2008)

[80] X.-C. Zhang, Y. Jin, and X. F. Pvla “Coherent measurement of THz optical rectiﬁcation from electro-optic crystals” Appl. Phys. Lett. 61 2764 (1992) 151

[81] H. Ito et al. “Continuous THz-wave generation using antenna-integrated uni-travelling-carrier photodiodes” Semicond. Sci. Technol. 20, S191 (2005)

[82] M. Kress et al. “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves” Optics Letters 29, 1120 (2004)

[83] K. Y. Kim et al. “Coherent control of terahertz supercontinuum generation in ultrafast lasergas interactions” Nature Photonics 2, 605 (2008)

[84] H. Adachi et. al “High-Quality Organic 4-Dimethylamino-N-methyl- 4-stilbazolium Tosylate (DAST) Crystals for THz Wave Generation” Japanese Journal of Applied Physics, Volume 43, Part 2, Number 8B (2004)

[85] Taniuchi, T, Okada, S,and Nakanishi, H. “Widely-tunable THz-wave generation in 2-20 THz range from DAST crystal by nonlinear diﬀerence frequency mixing” Electronics Letters 40, 1 (2004)

[86] A. Schneidera, I. Biaggiob, and P. Gntera “Optimized generation of THz pulses via optical rectiﬁcation in the organic salt DAST” Optics Communications 224, 337 (2003)

[87] T. Yajima and N. Takeuchi “Far-Infrared Diﬀerence-Frequency Genera- tion by Picosecond Laser Pulses” 1970 Jpn. J. Appl. Phys. 9 1361 (1970) 152

[88] A. Rice et al “Terahertz optical rectiﬁcation from ¡110¿ zinc-blende crystals” Appl. Phys. Lett. 64, 1324 (1994)

[89] A. Nahata et al. “A wideband coherent terahertz spectroscopy system using optical rectiﬁcation and electro-optic sampling” Appl. Phys. Lett. 69, 2321 (1996)

[90] F. Blanchard et al. “Generation of 1.5 J single-cycle terahertz pulses by optical rectiﬁcation from a large aperture ZnTe crystal” Optics Express Vol. 15, Issue 20, pp. 13212 (2007)

[91] Peiponenet al. “Terahertz Spectroscopy and Imaging” (Springer, 2013) 1st ed. pg. 9

[92] G. Gallot et al. “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation” Applied Physics Letters 74, 3450 (1999)

[93] D. T. F. MARPLE “Refractive Index of ZnSe, ZnTe, and CdTe” JOUR- NAL OF APPLIED PHYSICS 35, 539 (1963)

[94] J. Midwinter “Assessment of lithium-meta-niobate for nonlinear optics” Applied Physics Letters 11, 128 (1967)

[95] R. Smith and F. Welsh. “Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate” Journal of applied physics 42, 2219 (1971) 153

[96] S. Abrahams, J. Reddy, and J. Bernstein “Ferroelectric lithium niobate 3 single crystal x-ray diﬀraction study at 24 c” Journal of Physics and Chemistry of Solids 27, 997 (1966)

[97] R. Weis and T. Gaylord “Lithium niobate: summary of physical properties and crystal structure.” Applied Physics A 37, 191 (1985)

[98] T Feurer et al. “Terahertz polaritonics” Annu. Rev. Mater. Res. 37, 317 (2007)

[99] Q. Wu and X. Zhang “Ultrafast electro-optic ﬁeld sensors” Applied physics letters 68, 1604 (1996)

[100] J. Hebling et al. “Velocity matching by pulse front tilting for large-area THz-pulse generation” Optics Express 10, 1161 (2002)

[101] A. Stepanov, J. Hebling, J. Kuhl “THz generation via optical rectiﬁca- tion with ultrashort laser pulse focused to a line” Applied Physics B 81, 23 (2005)

[102] J. Lhuillier at al. “Generation of THz radiation using bulk, periodically and aperiodically poled lithium niobate Part 1: Theory” Applied Physics B 86, 185 (2006)

[103] K. Yeh et al. “Generation of high average power 1 kHz shaped THz pulses via optical rectiﬁcation” Optics Communications 281, 3567 (2008) 154

[104] J. Hebling et al. “Generation of high-power Terahertz pulses by tilted- pulse-front excitation and their application possibilities.” JOSA B 25, B6 (2008)

[105] H. Hirori et al “Single-cycle terahertz pulses with amplitudes exceeding

1 MV/cm generated by optical rectiﬁcation in LiNbO3” Appl. Phys. Lett. 98, 091106 (2011)

[106] J. Fülöp et al. “Design of high-energy terahertz sources based on optical rectification” OPTICS EXPRESS 18, 12311 (2010)

[107] J. Byrnes “Unexploded Ordnance Detection and Mitigation” Springer 2009 1st ed. pg 21-22

[108] B. Albrecht and S. K. Cox “Procedures for Improving Pyrgeometer Performance” Journal of Applied Meteorology 16, 188 (1977)

[109] J. Foot “A New Pyrgeometer” Journal of Atmospheric and Oceanic Technology 3, 363 (1985)

[110] Marcel J. E. Golay “Theoretical Consideration in Heat and Infra-Red Detection, with Particular Reference to the Pneumatic Detector” Rev. Sci. Instrum. 18, 347 (1947)

[111] K. Yamashita, A. Murata, M. Okuyama “Miniaturized infrared sensor using silicon diaphragm based on Golay cell ” Sensors and Actuators A 66, 29 (1998) 155

[112] A. Kaveev, E. Kaveeva, E. Adaev “Pulse Shape Measurement Using Golay Detector” J Infrared Milli Terahz Waves 33, 306 (2012)

[113] Y. Wada and R. Hayakawa “Piezoelectricity and Pyroelectricity of Poly- mers” Japanese Journal of Applied Physics 15, 2041 (1976)

[114] J. Bergman et al. “PYROELECTRICITY AND OPTICAL SEC- OND HARMONIC GENERATION IN POLYVINYLIDENE FLUO- RIDE FILMS” Appl. Phys. Lett. 18, 203 (1971)

[115] M. Broadhurst et al. “Piezoelectricity and pyroelectricity in polyvinylidene ﬂuorideA model” J. Appl. Phys. 49, 4992 (1978)

[116] G. Su et. al “A new pyroelectric crystal l-lysine-doped TGS (LLTGS)” Journal of Crystal Growth 209, 220 (2000)

[117] E. Barr “The infrared pioneersIII. Samuel Pierpont Langley” Infrared Physics 3, 195 (1963)

[118] F. Pockels “On the eﬀect of an electrostatic ﬁeld on the optical behaviour of piezoelectric crystals” Abh. Gott 39, 1 (1894)

[119] G. Anderson, “Studies on Vanadium Dioxide” ACTA Chemica Scandi- navica 10, 623 (1956).

[120] N.F. Mott, “Metal-Insulator Transition” Reviews of Modern Physics 40, 677 (1968). 156

[121] B Fisher, “Moving boundaries and travelling domains during switching of VO, single crystals” J. Phys. C: Solid State Phys, 2072 (1975).

[122] K. Appavoo et. al., “Role of defects in the phase transition of VO2 nanoparticles probed by plasmon resonance spectroscopy” Nanoletters 12, 788 (1975).

[123] Tatsuyuki Kawakubo, “Cryastal Distortion and Electric and Magnetic

Transition in VO2” Journal of the Physical Society of Japan 20, 516 (1965).

[124] A. Zylbersztejn, N.F. Mott, “Metal-insulator transition in vanadium dioxide*” PRB 11, 4383 (1975).

[125] S. M. Woodley “The Mechanism of the Displacive Phase Transition in Vanadium Dioxide” Chemical Physics Letters 453, 167 (2008)

[126] J Hubbard “Electron Correlations in Narrow Energy Bands ” Proceedings of the Royal Society A 276, 238 (1963)

[127] J. Hubbard “The Mechanism of the Displacive Phase Transition in Vana- dium Dioxide III” Proceedings of the Royal Society A 281, 401 (1964)

[128] Alexander Altland and Ben Simons “Condensed Matter Field Theory” (Cambridge University Press, Mar 11, 2010) 2end ed. pg. 60

[129] G Stefanovich, A Pergament and D Stefanovich “Electrical switching and Mott transition in VO2” J. Phys.: Condens. Matter 12, 88378845(2000) 157

[130] Mendel Sachs “Solid State Theory” (Mcgraw-Hill Book Company, 1963) 1st ed. pg. 246

[131] M. Imada, A. Fujumori, Y. Tokura “Metal-insulator transitions” Rev. Mod. Phys., Vol. 70, 1039(1998)

[132] Takashi Mizokawa “Metalinsulator transitions: Orbital Control” Nature Physics, Vol. 9, 612(2013)

[133] J Zhang, J Ou, K MacDonald, and N Zheludev “Optical response of plasmonic relief meta-surfaces” J. Opt. 14, 114002 (2012)

[134] J. De Natale1, P. Hood1 and A. Harker “Formation and characterization

of grain-oriented VO2 thin ﬁlms” J. Appl. Phys. 66, 5844 (1989)

[135] V. Klimov et al. “Hysteresis loop construction for the metal- semiconductor phase transition in vanadium dioxide ﬁlms” Technical Physics 47, 1134 (2002)

[136] R. Lopez et. al. “Enhanced hysteresis in the semiconductor-to-metal

phase transition of VO2 precipitates formed in SiO2 by ion implantation” APPLIED PHYSICS LETTERS 79, 3161 (2001)

[137] A. Roytburd, J. Slutsker “Thermodynamic hysteresis of phase transfor- mation in solids” Physica B 233, 390 (1997)

[138] Yijia Gu et. al. “Thermodynamics of strained vanadium dioxide single crystals” J. Appl. Phys. 2010, 108, 083517 (2010) 158

[139] Xiaofeng Xu et. al. “The extremely narrow hysteresis width of phase

transition in nanocrystalline VO2 thin ﬁlms with the ﬂake grain structures” Applied Surface Science 261, 83 (2012)

[140] Simon M. Sze, Kwok K. Ng “Physics of Semiconductor Devices” John Wiley & Sons Inc., 2007 1st ed. pg. 1

[141] John P. Walter and Marvin L. Cohen “Calculation of the Reﬁectivity, Modulated Reﬁectivity, and Band Structure of GaAs, GaP, ZnSe, and ZnS” Phys. Rev. 183, 763(1969)

[142] Chelikowsky, J.R. & Cohen “Electronic structure of silicon” Physical Review B 10, 5095-5107(1974)

[143] Markoﬀ, John IBM Discloses Working Version of a Much Higher-Capacity Chip, 07/09/2015 Retrieved from: http://www.nytimes.com/2015/07/09/technology/ibm-announces- computer-chips-more-powerful-than-any-in-existence.html

[144] G. B. Bachelet et. al. “Structural-energy calculations based on norm- conserving pseudopotentials and localized Gaussian orbitals” Physical Review B 24, 4745(1981)

[145] D. R. Hamann “Semiconductor Charge Densities with Hard-Core and Soft-Core Pseudopotentials” PRL 42, 662(1978) 159

[146] Giovanni B. Bachelet and Niels E. Christensen “Relativistic and core- relaxation eﬀects on the energy bands of gallium arsenide and germa- nium” Physical Review B 31, 879(1985)

[147] F. Manghi, G. Riegler, and C. M. Bertoni “Band-structure calculation for GaAs and Si beyond the local-density approximation” Physical Re- view B 31, 3680 (1985)

[148] Jagdeep Shaw, et. al. “Determination of Intervalley Scattering Rates in GaAs by Subpicosecond Luminescence Spectroscopy” PRL 59, 2222 (1987)

[149] M.C. Nuss, D.H. Auston, and F. Capasso “Direct Subpicosecond Mea- surement of Carrier Mobility of Photoexcited Electrons in Gallium Ar- senide” PRL 58, (1987)

[150] Joseph Birman, Melvin Lax, and Rodney Loudon “Intervalley-Scattering Selection Rules in III-V Semiconductors” Physical Review 145, 620 (1966)

[151] P.C. Becker, et. al “Femtosecond intervalley scattering in GaAs” APL 53, 2089 (1989)

[152] Su et. al “Terahertz pulse induced intervalley scattering in photoexcited GaAs” Optics Express 17, 9620 (2009) 160

[153] Zoltn Mics et. al. “Density-dependent electron scattering in photoexcited GaAs in strongly diﬀusive regime” Applied Physics Letters 102, 231120 (2013);

[154] D.K. Ferry “First-order optical and intervalley scattering in semiconductors” Phys. Rev. B 14, 1605 (1976)

[155] T. Ichibayashi et al. “Ultrafast relaxation of highly excited hot electrons in Si: Roles of the L - X intervalley scattering” PRB 84, 235210 (2011)

[156] A. Onton “Evidence of Intervalley Scatter of Electrons in the Extrinsic Photoconductivity of n-type Silicon” PRL 22, 288 (1969)

[157] D. Long, J. Myers “Scattering Anisotropies in n-Tpye Silicon” Phys. Rev. 120, 39 (1960)

[158] D. Long “Scattering of Conduction Electrons by Lattice Vibrations in Silicon” Phys. Rev. 120 2024 (1960)