Terahertz Imaging and Nonlinear Spectroscopy of Semiconductors Using Plasmonic Devices

AN ABSTRACT OF THE DISSERTATION OF

Zachary James Thompson for the degree of Doctor of Philosophy in Physics presented on August 19, 2015.

Title: Terahertz Imaging and Nonlinear Spectroscopy of Semiconductors using Plasmonic Devices

Abstract approved: Yun-Shik Lee

In this dissertation, a series of studies in the field of terahertz (THz) science are presented, specifically using nonlinear THz spectroscopy. We exploit huge field enhancement and subwavelength confinement in plasmonic structures. There are three distinct projects which will be discussed: nonlinear THz spectroscopy using plasmonic induced transparency (PIT), THz-triggered insulator-metal transition (IMT) in nanoantenna patterned vanadium dioxide (VO2) films, and fabrication of sub-diffraction limit imaging bulls-eye structures. We used PIT structures to observe the high-field carrier dynamics in semiconductors, specifically in intrinsic, high resistivity silicon (high-ρ Si) and intrinsic gallium arsenide (GaAs). The PIT structures rely on the coupling of a ”bright mode” in a central half-wave dipole antenna to the ”dark mode” of the adjacent split-ring resonators. We employed these structures because of their sensitivity to carrier dynamics due to the sharp resonance of the ”dark mode.” We observed the response of the PIT oscillation to both low and high THz fields in the presence of an optical pump. Increasing the optical pump power, and therefore the number of carriers, resulted in the damping of the oscillation. With increasing THz field strength, we observed a field induced transparency from the intervalley scattering of the excited carriers and demonstrated THz control of the PIT oscillation. By changing the delay time between the THz and optical pulses, we demonstrated pulse shaping of the PIT waveforms. We demonstrated the THz-triggered insulator-metal transition (IMT) in nanoantenna patterned vanadium dioxide (VO2) films. Vanadium dioxide is a promising material for electronic and photonic applications due to its IMT transition lying near room temperature. We observed that the phase transition is activated on the sub-cycle time scale where strong THz fields drive the electron distribution far from equilibrium. We also observed a lowering the transition temperature of the IMT phase transition for both heating and cooling cycles in nanoslot antenna VO2 films with increasing THz fields and also a narrowing in the width of the observed hysteresis. Using the Fresnel thin-film coefficients, Drude model, and the resistivity in semiconductors we found the activation energy in the insulating phase and show that it can be lowered with THz fields. We employed THz time domain spectroscopy to extract the frequency dependence and to observe the transiently induced IMT from the strong THz fields. We attempted to fabricate sub-diffraction-limit imaging bulls-eye structures in the Oregon State University cleanroom. During the course of the project, recipes for two different types of photoresists, SU-8 2100 and SU-8 5, were developed. We observed lack of adhesion of the metal (Al) layer for the metal-dielectric interface. Lastly the removal of metal for the apertures posed additional problems. While this project did not ultimately succeed, we present an explanation of the issues associated with their fabrication and the steps necessary to complete fabrication. c Copyright by Zachary James Thompson August 19, 2015 All Rights Reserved Terahertz Imaging and Nonlinear Spectroscopy of Semiconductors using Plasmonic Devices

Zachary James Thompson

A DISSERTATION

submitted to

Oregon State University

in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Presented August 19, 2015 Commencement June 2016 Doctor of Philosophy dissertation of Zachary James Thompson presented on August 19, 2015.

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.

Zachary James Thompson, Author ACKNOWLEDGEMENTS

I would like to begin by thanking my adviser, Dr. Yun-Shik Lee for his inﬁnite patience and understanding during my time at Oregon State. I also would like to thank my group members Michael Paul, Andrew Stickel, Byounghwak Lee, and Ali Mousavian for their help inside and outside of the lab, I could not have done this without you guys. I especially thank Morgan Brown and Rick Presley for their invaluable guidance during the course of my fabrication projects and acting as sounding boards for any insane ideas I could conjure. I want to thank my parents, Rick and Sharon Thompson, for their endless words of encouragement and help during this process. I have the best parents in the world. Most importantly, I want to thank my wonderful wife Mikayla. She has kept me sane and dealt with any and all irrational reactions I’ve had to the smallest stimuli. Without her, I would be completely lost. For the rest of the people who have helped over the years (including my sister Sara AKA Fat Kid), I express my undying gratitude with the Del ’n Bones below. It is intended to represent the resilience and dedication required for the path I have chosen and to pay homage to those who are no longer with us. TABLE OF CONTENTS Page 1 Overview 1 1.1 History...... 1 1.2 Sources...... 2 1.3 Detection...... 3 1.4 Applications...... 4

2 Electromagnetic Waves in Nonlinear Media6 2.1 Linear Media and Wave Equation...... 6 2.2 Thin-Film Fresnel Formula...... 10 2.3 Harmonic Oscillator...... 14 2.4 Nonlinear Media...... 16

3 Terahertz Generation and Detection 20 3.1 Generation via Optical Rectiﬁcation...... 20 3.1.1 Phase Matching...... 20 3.1.2 Zinc Telluride...... 22 3.1.3 Lithium Niobate...... 23 3.2 Terahertz Detection...... 26 3.2.1 Bolometer...... 27 3.2.2 Pyroeletric Detectors...... 28 3.2.3 Michelson Interferometry...... 28 3.2.4 Electro-Optic Sampling...... 29

4 Surface Plasmons and Surface Plasmon Polaritons 33 4.1 Theory of Surface Plasmons...... 33 4.2 Drude-Sommerfeld Model...... 33 4.3 Surface Plasmon Polaritons at Interfaces...... 34 4.3.1 Dispersion Relation...... 35 4.3.2 SPP Excitation...... 39 4.4 Excitation Methods...... 40 4.4.1 Otto Method...... 40 4.4.2 Kretschmann Method...... 41 4.4.3 Spatial Periodicity...... 41 4.5 Applications to Terahertz Science...... 42

5 Non-Linear Terahertz Spectroscopy using Plasmon Induced Transparency 43 5.1 Introduction...... 43 5.2 Plasmonic Induced Transparency...... 44 TABLE OF CONTENTS (Continued) Page 5.2.1 Coupling of Two Resonators...... 44 5.3 Coupling of Linear Antenna and Split-Ring Resonator...... 47 5.3.1 Fabrication of PIT Structure...... 52 5.4 Experimental Considerations...... 56 5.5 Initial Testing and PIT Observation...... 56 5.5.1 Resonant Cases in GaAs...... 57 5.5.2 Resonant Cases in Si...... 60 5.5.3 Off Resonant Cases in GaAs...... 64 5.5.4 Off Resonant Cases in Si...... 67 5.6 THz Field Effects of GaAs and Si PIT Structures...... 70 5.7 Optical Excitation in the Wake of a PIT Resonance...... 71 5.7.1 GaAs...... 72 5.7.2 Si...... 74 5.8 THz Control of PIT Resonance...... 75 5.9 Pulse Shaping of PIT Waveforms...... 77 5.10 THz pump-Optical Pump Experiments...... 78 5.11 Summary...... 80

6 Terahertz Field Induced Metal-Insulator Transition in Vanadium Dioxide 81 6.1 Introduction...... 81 6.2 Mott Insulators...... 81 6.3 Sample Structure...... 85 6.4 Experimental Considerations...... 86 6.5 Terahertz Field Induced Absorption...... 87 6.6 Hysteresis and Activation Energy...... 88 6.6.1 Resistivity Derivation...... 89 6.6.2 Resistivity and Activation Energy...... 92 6.6.3 Hysteresis Width...... 96 6.7 Transient Phase Transition...... 98 6.8 Summary...... 101

7 Sub-Diﬀraction Limit Nonlinear Imaging with Plasmonic Devices 102 7.1 Introduction...... 102 7.2 Background...... 102 7.3 Fabrication...... 104 7.4 Future Work...... 106 TABLE OF CONTENTS (Continued) Page 8 Conclusion 108

Bibliography 110

Appendix 124 A PIT Fabrication Recipe...... 125 B Bullseye Fabrication Recipe...... 127 LIST OF FIGURES Figure Page 1.1 The electromagnetic spectrum with the THz gap highlighted in blue.....1

2.1 Cartoon of the transmission...... 10 2.2 A cartoon of the harmonic oscillator model...... 14

2.3 Example of an harmonic and an anharmonic potential. The associated motion of a charge carrier is plotted on the left...... 16

3.1 Example of generation using optical rectification in a phase-matched medium. Where the light is propagating left to right, Eopt is the optical electric field, POR is the polarization of the material, and ET Hz is the THz electric field.. 21 3.2 A cartoon of the walk-off length. Figure 3.35 from Ref. [1]...... 22 3.3 Index of refraction plot. Note that the optical group index and THz phase index are matched at λopt ≈ 810 nm and νTHz ≈ 1.7 THz. Figure 3.37 from Ref. [1]...... 23

3.4 Cartoon of our tilted pulse front setup for LiNbO3. Note that the diﬀraction grating tilts the optical pulse front (red). The THz output pulse front is shown in orange...... 25 3.5 Diﬀraction angle plot...... 26 3.6 Schematic of a compound bolometer. Figure 4.25 from Ref. [1]...... 27

3.7 Schematic of a pyroelectric detector. Figure 4.28 from Ref. [1]...... 28 3.8 A cartoon of our Michleson interferometer using a bolometer...... 29 3.9 A cartoon of our EO sampling setup...... 31

4.1 Interface diagram...... 35

4.2 SPP dispersion curve...... 39 4.3 Otto conﬁguration...... 40 4.4 Kretschmann conﬁguration...... 41

4.5 Grating coupler, one example of a periodic array used to elicit a SPP resonance. 42

5.1 A) Parallel SRRs. B) Anti-parallel SRRs...... 44 LIST OF FIGURES (Continued) Figure Page 5.2 SRRs with 90o rotation. The ﬁgure illustrates the higher energy resonant case (A) where the magnetic dipoles are parallel and the lower energy resonant case where the magnetic dipoles are anti-parallel (B) and the associated line splitting (C)...... 45

5.3 The geometry for Az...... 49 5.4 The square loop cartoon...... 50

5.5 PIT resonance cartoon illustrating the coupling between the dipole antenna and adjacent SRRs...... 51 5.6 Here is an example unit cell of the PIT structure. Dimensions can be found in Table 5.3.1 (below)...... 53 5.7 Optical microscope pictures of the GaAs sample. The pictures on the left are the positive arrays. The pictures on the right are the negative arrays. The top row is the 0.6 THz arrays. The middle row is the 0.9 THz arrays. The bottom row is the 1.2 THz arrays...... 54 5.8 Optical microscope pictures of the Si sample. The pictures on the left are the positive arrays. The pictures on the right are the negative arrays. The top row is the 0.6 THz arrays. The middle row is the 0.9 THz arrays. The bottom row is the 1.2 THz arrays. The psychedelic blue color is due to the improper functioning of the white-balance on the microscope...... 55

5.9 The GaAs PIT 0.6 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 57

5.10 The GaAs PIT 0.6 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 57 5.11 The GaAs PIT 0.9 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 58 5.12 The GaAs PIT 0.9 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 58 5.13 The GaAs PIT 1.2 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 59 LIST OF FIGURES (Continued) Figure Page 5.14 The GaAs PIT 1.2 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 59 5.15 The Si PIT 0.6 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 60 5.16 The Si PIT 0.6 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 60 5.17 The Si PIT 0.9 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 61 5.18 The Si PIT 0.9 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 61 5.19 The Si PIT 1.2 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 62 5.20 The Si PIT 1.2 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 62 5.21 The GaAs PIT 0.6 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 64

5.22 The GaAs PIT 0.6 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 64

5.23 The GaAs PIT 0.9 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 65

5.24 The GaAs PIT 0.9 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 65 LIST OF FIGURES (Continued) Figure Page 5.25 The GaAs PIT 1.2 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 66 5.26 The GaAs PIT 1.2 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 66 5.27 The Si PIT 0.6 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 67 5.28 The Si PIT 0.6 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 67 5.29 The Si PIT 0.9 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 68 5.30 The Si PIT 0.9 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 68 5.31 The Si PIT 1.2 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz field of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.... 69 5.32 The Si PIT 1.2 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz field of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right...... 69 5.33 The GaAs PIT 0.9 THz negative array at 90 degrees. The incident THz field strength was modulated...... 71 5.34 The Si PIT 0.9 THz negative array at 90 degrees. The incident THz field strength was modulated...... 71 5.35 Cartoon of the optical pump setup. A small hole is drilled through a focusing parabolic mirror through which the optical pump passes and overlaps the THz focus...... 72 5.36 The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the THz excitation. The legend shows the carrier concentration. 73 5.37 The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration...... 74 LIST OF FIGURES (Continued) Figure Page 5.38 The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration...... 74 5.39 The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration...... 75 5.40 The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. Increasing the incident THz field greatly modulates the transmission...... 76 5.41 The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. Increasing the incident THz field slightly modulates the transmission. 77

5.42 The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation after the PIT excitation at the time listed in the legend. By delaying the optical pulse we were able to shape the transmitted THz waveform...... 78

5.43 The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation after the PIT excitation at the time listed in the legend. By delaying the optical pulse we were able to shape the transmitted THz waveform...... 78

5.44 The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the PIT excitation. The high optical pump power damps the PIT oscillation...... 79

5.45 The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the PIT excitation. The high optical pump power damps the PIT oscillation...... 79

6.1 A. Orbital diagram for Vanadium when T is below Tc (unstrained) and above Tc (strained). B. Cartoon of the dimerization of valence electrons in neighboring Vanadium atoms...... 82 6.2 Cartoon of the nanoslot antennas...... 85

6.3 Plot of the THz ﬁeld dependence for the nanoslot and bare VO2 (inset)... 87

6.4 Plot of the hysteresis curves for the nanoslot and bare VO2 (inset)...... 89 6.5 Plot of the normalized sheet conductivity comparing the bare and nanoslot VO2 data...... 93 6.6 Plot of the normalized sheet conductivity for the fit comparing the bare and nanoslot VO2 data...... 94 6.7 Plot of the resistivity as a function of temperature. The activation energy fits for temperatures between 35o and 55o C are over-plotted...... 95 LIST OF FIGURES (Continued) Figure Page 6.8 Plot of the hysteresis width as a function of incident electric field. The inset plot show the transition temperature for the nanoslot sample for increasing and decreasing temperature. The black lines are linear fits for the data.... 97 6.9 Waveforms for an incident THz field of 150 kV/cm at 45o C, 65o C, and 67o C. 98

6.10 Plot of the waveforms for incident THz ﬁelds of 150 kV/cm, 300 kV/cm, 630 kV/cm, and 850 kV/cm at 45o (a) and at 65o C (b). The power transmission spectra for each waveform is inset...... 99 6.11 Plot of the waveforms for 150 kV/cm at 67o C (temperature driven transition) and 850 kV/cm at 65o C (ﬁeld driven transition) to help illustrate the lowering of the transition temperature. The yellow shaded area is the waveform for 150 kV/cm at 65o C...... 100

7.1 Example bullseye structure...... 103 7.2 Example of a 0.5 THz bullseye structure (left) and 1.0 THz bullseye structure (right). There are etching pits around the convex edges of the structures from the lack of S1818 adhesion and the resulting etching of the metal layer.... 106 Terahertz Imaging and Nonlinear Spectroscopy of Semiconductors using Plasmonic Devices

1 Overview

Terahertz radiation falls between the infrared (IR) and microwave regions of the electromagnetic spectrum, colloquially know as the ”THz gap” (Fig 1.1). In nature, the THz generally corresponds to molecular rotational and vibrational modes. The photon energy at 1 THz is 4 meV which corresponds to thermal energy at a temperature of 48 K. The energy of this radiation is too low to excite atomic transitions which makes it an extremely useful tool to characterize materials since it acts as a nondestructive, non-contacting probe.

Figure 1.1: The electromagnetic spectrum with the THz gap highlighted in blue.

1.1 History

Terahertz science initially rose out of thermal detection. Room temperature corresponds to photon energies of 6 THz. Pioneering work was done by Rubens [2] in isolating this frequency. Planck recognized his work in 1922 by writing the following: ”Without the intervention of Rubens the formulation of the radiation law, and consequently the formulation of quantum theory, would have taken place in a totally different manner, and perhaps even not at all in Germany.” The next significant step in THz science was in 1965 when difference 2 frequency generation was used to create monochromatic 3 THz light. [3] Finally single-cycle THz sources were realized in 1990 [4] which lead to growth in the THz field. Now the field has become extremely accessible in recent years due to the advent of table-top sources and detectors.

1.2 Sources

We will briefly discuss several types of THz sources, including free-electron sources, THz lasers, photocurrent sources, and frequency conversion systems. We will begin with free-electron lasers (FEL). [5] These fall under free-electron sources. In this scheme, a population of electrons is excited using a short optical pulse. These excited electrons are then accelerated to relativistic speeds and passed through a magnetic array. This causes the electrons to oscillate in a sinusoidal pattern, thereby radiating narrowband THz radiation. Another example of a free electron source is a backward wave oscillator (BWO). [6] These work by projecting a beam of electrons into a counter propagating, slowly oscillating electromagnetic field. This causes a compression of the electron beam and the oscillation of the beam which emits and amplifies the THz radiation. A prime example of a THz laser is a quantum cascade lasers (QCL), where a material is engineered to have step potentials via periodic stacks of semiconducting materials. [7] An injected carrier will tunnel through the series of potential barriers, emitting a THz photon each time it tunnels. Photo-conductive (PC) antennas work in the following manner. [8] A semiconductor, generally gallium arsenide (GaAs), has two parallel strip-line antennas held at some potential difference. An optical pulse is used to excite carriers in the gap between the two antennas. The resulting motion of the electrons and holes gives rise to a time dependent current, emitting THz radiation. We will briefly discuss our method of generation, a frequency conversion system which 3 relies on optical rectification (OR). A short optical pulse of frequency ω is passed through a nonlinear crystal. The time-dependent polarization in the material is related to the intensity envelope of the optical pulse of frequency ω, which gives rise to a short, single-cycle THz pulse.

1.3 Detection

Detection of THz radiation falls into two categories: incoherent and coherent detection. In the former category, the detectors generally utilize thermal effects in some capacity. The latter uses methods very similar to the generation methods from the previous section and are used to acquire spectral information. Incoherent THz detectors are used for power measurements. Initially bolometers were used to detect thermal (THz) radiation by measuring the change in resistance across a small thermal mass when the mass is heated. [9] One of the issues of bolometers is they generally require liquid helium temperatures to detect THz radiation. Another method of thermal detection are golay cells. [10] Golay cells rely on the expansion of a small volume of gas to deform a flexible mirror and modulate a signal from a LED onto a photodetector. While golay cells are very sensitive detectors, they are generally large in size, reducing their utility. The rise and availability of micromachining has led to a reduction in their size. Lastly, we have pyroelectric detectors. [11] These rely on the heating of a crystal to change the instantaneous polarization. Pyroelectric detectors and bolometers will be discussed further in Sec. 3.2. Coherent THz detectors are used to extract frequency dependence from a transmitted signal. Photo-conductive switches work in much the same manner as PC antenna. First, an optical pulse is used to excite carriers between two strip-line antennas when there is no voltage bias between them. Then the THz pulse will hit the same spot as the optical did, inducing a current, which is measured. Electro-optic sampling uses a nonlinear optical crystal 4 in a similar fashion to optical rectification. The THz pulse is incident on the nonlinear optical crystal which induces a birefringence in the crystal. This physically means that the index of refraction in crystal changes depending on the incident polarization. This birefringence will rotate the polarization of the reference optical beam as it passes through the crystal. This rotation is proportional to the THz electric field. Electro-optic sampling is our primary method of extracting spectral data and will be discussed in-depth in Sec. 3.2.

1.4 Applications

The advent of table-top THz sources has led to an increase in THz research. Due to this increase in accessibility, there have been recent developments in THz science which have yielded promising applications. THz radiation is a useful tool for characterizing materials. High-speed wireless communication has been demonstrated and is being investigated by DARPA for secure communications. It also has great promise for security detection and imaging purposes. THz radiation can be used to characterize organic and semiconductor materials and devices. A prime example of research which has been conducted by our group for this purpose focused on graphene. Graphene has a huge response to THz radiation which allows for characterization of this novel, single layer material. THz radiation has been employed as a nondestructive, non-contacting probe. Due to graphene’s Drude-like response, the sheet conductivity can be extracted from the transmission. [12]. However it is noteworthy that when strong THz fields are applied there is an induced transparency. [13,14] The increase in demand for data transfer has driven wireless communication to THz frequencies. The lower bandwidths of the currently used GHz frequencies have created a push towards higher frequencies. [15] Recently transfer rates of up to 2.5 Gb/s have been observed at 0.625 THz, which illustrate the utility of this band. [16]Although the high transfer rates are attractive, they can only used for relatively short distance, on the order of meters, 5 due to power constraints. [17,18] Terahertz waves are an excellent tool for security detection and imaging. [19, 20] The THz regime is very attractive for security applications due to its non-ionizing nature from its low photon energy (4 meV). Material responses at THz frequencies correspond to molecular rotations. This leads to the ability to differentiate materials based on their spectral response. Owing to the general transparency of dielectrics in the THz regime, it allows for material detection inside packaging and differentiation of materials with similar optical properties due to their very different THz spectral responses. This allows for drug [21] and explosive [22] detection through non-destructive THz spectroscopy. Security imaging can utilize the THz response of materials. [23,24] Polar liquids (water), metals, plastics, and semiconductors all exhibit different responses to THz radiation. Polar liquids are highly absorptive which leads to the ability to differentiate between hydrated and dehydrated substances. Metals reflect nearly all incident THz radiation, leading to the easy detection of concealed weapons. Plastics have low absorption and low refractive index, leading to high transmission. Semiconductors have a high THz refractive index and low absorption. 6

2 Electromagnetic Waves in Nonlinear Media

2.1 Linear Media and Wave Equation

In order to understand the interaction of THz radiation and matter, our derivation must start at the very root of electricity and magnetism. This stems from the fact that the THz pulse used in our lab is generated via nonlinear optical processes (speciﬁcally optical recti- ﬁcation), which means we have to look at the polarization of materials. We will commence by writing Maxwell’s equations in matter. [25]

∂B ∇ × E = − (2.1) ∂t ∂D ∇ × H = J + (2.2) f ∂t

∇ · D = ρf (2.3)

∇ · B = 0 (2.4)

For linear media, we can write the displacement field (D-field) and auxiliary field (H- field) in the following manner:

D = 0E + P = E (2.5) 1 1 H = B − M = B (2.6) µ0 µ

Substituting these into Eq. 2.1 and Eq. 2.2, taking the curl, we ﬁnd the generalized electromagnetic wave equations. 7

∂2E ∂ ∂P ∇ × ∇ × E + µ = −µ J + + ∇ × M (2.7) 0 0 ∂t2 0 ∂t f ∂t

∂2H ∂P ∂2M ∇ × ∇ × H + µ = ∇ × J + ∇ × − µ (2.8) 0 0 ∂t2 f ∂t 0 0 ∂t2

Using the fact that ∇ × ∇ × C = ∇ (∇ · C) − ∇2C and inserting the rest of Maxwell’s equations (Eq. 2.3 and Eq. 2.4), we arrive at the following:

∂2E 1 ∂ ∂P ∇2E − µ = ∇ρ + µ J + + ∇ × M (2.9) 0 0 ∂t2 f 0 ∂t f ∂t

∂2H ∂P ∂2M ∇2H − µ = −∇ × J − ∇ × + µ (2.10) 0 0 ∂t2 f ∂t 0 0 ∂t2

We can simplify these by using Ohm’s law, [26] assuming linear correlation between the

free volumetric current density (Jf ) and E via the electrical conductivity σ, and neglecting

charge ﬂuctuations (∇ρf = 0). We will only proceed with the electric half of this derivation since Maxwell’s equations are cyclic, therefore knowing the electric half determines the

magnetic half. Lastly we can assume the material is non-magnetically permeable (µ = µ0) because the electric response dominates the interaction. Repercussions of this are that M = 0.

∂2E ∂E ∂2P ∇2E = µ + µ σ + µ (2.11) 0 0 ∂t2 0 ∂t 0 ∂t2

(1) If we use P = 0χe E = ( − 0) E we arrive at Eq. 2.12. This speciﬁc form will be particularly useful when dealing with conductors.

∂2E ∂E ∇2E = µ + µ σ (2.12) 0 ∂t2 0 ∂t

However, a majority of materials being studies in this dissertation are generally dielectric 8

and insulating, we then ignore the conductivity term and rewrite the wave equation in the following form:

∂2E n2 ∂2E ∇2E = µ = (2.13) 0 ∂t2 c2 ∂t2

n2 1 q √ It should also be noted that µ0 = 2 = 2 , where n = = R is the index of c v 0 q refraction, c = 1 is the speed of light in vacuum, and v is the speed of the wave. The 0µ0 general solutions of this diﬀerential equation are linearly polarized, monochromatic plane- waves with wave vector k and angular frequency ω.

i(k·x−ωt) i(k·x−ωt) E (x, t) = E0e H (x, t) = H0e (2.14)

Using these solutions with Maxwell’s equations we can relate E and H using k and ω. Namely, the divergence yields that k · E = k · H = 0, meaning that the associated ﬁelds of the wave are perpendicular to the direction of propagation (transverse). Taking the curl

yields that k × E = ωµ0H. Lastly, we use Eq. 2.12 with Eq. 2.14 to obtain the dispersion relation.

2 2 k = µ0ω (2.15)

This formula relates the electric and magnetic properties at a given angular frequency to the propagation and dispersion of that wave in the medium. For our non-magnetic medium, we can relate the free-space wavelength (λ0) to the wave-vector in the following manner:

2πn ω k = = n (2.16) λ0 c

Now if the media of interest is a good conductor, the dispersion relation looks signiﬁcantly diﬀerent. The solutions to the wave equation in Eq. 2.14 can be inserted into Eq. 2.12, but 9

we must note that σ ω in good conductors, which allows us to ignore the ﬁrst term of Eq. 2.12.

2 k ≈ iσµ0ω (2.17)

Another point of note is that k2 is purely imaginary, this means that each of the constituent components of k have the same magnitude.

rσµ ω Re |k| = Im |k| = 0 (2.18) 2

When using this in Eq. 2.14, it is easy to see that a wave propagating into a metal will exponentially decay from the imaginary portion of k. This yields an important result, the skin depth.

r 2 δs = (2.19) σµ0ω

The skin depth corresponds to the length at which the magnitude of the electric ﬁeld decays to e−1. For comparison the skin depth for metals at THz frequencies is on the order of 0.1 µm, which is much smaller than the free space wavelength, 300 µm. In a lossy dielectric, we can equate the dispersion relations for a good conductor (Eq. 2.17) and a dielectric (Eq. 2.15) to ﬁnd the refraction conductivity relation.

2 iσ R = n = (2.20) ω0

Lastly, we will use the time-averaged Poynting vector (Eq. 2.21) to ﬁnd the radiation intensity. We do this because our bolometer measures the transmitted power of our THz beam (T , Sec. 2.2).

1 1 hSi = E × H∗ = v |E |2 kˆ (2.21) 2 2 0 10

The magnitude of the Eq. 2.21 yields the radiation intensity, typically measured in W/m2.

1 I = |hSi| = v |E |2 (2.22) 2 0

Lastly, we will relate the Poynting vector magnitude and the skin depth to ﬁnd the penetration depth. For the purposes of this dissertation, we make use of the penetration depth, as it describes where the intensity decays to e−1. Therefore it relates the |E|2, the imaginary portion of the index of refraction κ, and the absorption coeﬃcient α.

r δs c 1 1 δp = = = = (2.23) 2 2κω α 2σµ0ω

2.2 Thin-Film Fresnel Formula

Figure 2.1: Cartoon of the transmission.

Thin film samples can be said to exhibit Drude-like behavior if their spectral response is flat; meaning the spectral range of interest is far from resonance. This section specifically

pertains to our VO2 samples (speciﬁcally Sec. 6.6.2). Having a Drude-like response allows us to extract the sheet conductivity from the transmission data using the thin-ﬁlm Fresnel 11

formula. In this derivation we assume that we are examining an isotropic and homogeneous thin-ﬁlm at normal incidence. We further assume that there is no destructive interference

in the the thin ﬁlm. This can be done since the VO2 ﬁlm thickness is much less than the

λ wavelength (n2d 10 ). The sapphire substrate, on the other hand, is optically thick at 300 µm with n ≈ 3.1, which also allows us to assume there is no destructive interference in the substrate. We also ignore the absorption in the sapphire [27] and assume the pulses are well temporally separated.

ni − nj 2ni rij = tij = (2.24) ni + nj ni + nj

Now using the transmission and reflection coefficients from 2.24, we can start looking at the transmission through the thin film sample. We can view our transmission in the following way since we are dealing with a pulsed laser system: each transmitted pulse will contribute to the total transmission t, where t(n) corresponds to the nth transmitted pulse. During each trip through the material phase is acquired. The nth pulse exiting the material will have a

total phase of φn = (2n − 1) φ.

∞ X t = t(n) (2.25) n=1

iφd 3iφd 5iφd = t12t23e + t12r23r21t23e + t12r23r21r23r21t23e + .... (2.26) ∞ n iφd X 2iφd = t12t23e r23r21e (2.27) n=0

Since each element in the sum is less than one and monotonically decreasing, we can use the sum of a geometric series to calculate the quantity to which the series converges.

iφd t12t23e t = 2iφ (2.28) 1 + r23r12e d 12

Now we can simplify using Eq. 2.24 and applying the thin ﬁlm conditions: only a small amount of phase is acquired when traversing the sample (φ 1 ,eiφ ≈ 1 + iφ) and the ﬁlm thickness is much less than that of the wavelength (d λ).

4n1n2 t = −iφ iφ (2.29) (n1 + n2)(n2 + n3) e d + (n1 − n2)(n2 − n3) e d t13 (n1 + n3) = (2.30) n1n3 n1 + n3 − n2 1 + 2 iφd n2

n1n3 n1n3 Further, for a metallic thin ﬁlm we can simplify using: 2 1 and (n3 − n1) n2 n2

n2. This is a direct result of the index of the thin ﬁlm being much larger than either the index of air or the substrate. We must not forget to also utilize the refraction conductivity

d 2 relation (Eq. 2.20) and the simpliﬁcation it allows us to make, in2φ2 = 2πi λ n2 = −Z0σd. 1 Where Z0 = = 376.7Ω, the vacuum impedance, and the conductivity, σ, can be re-written c0

in terms of the sheet conductivity, σs = σd. In the end we have the following result:

t (n + n ) t = 13 1 3 (2.31) n1 + n3 + Z0σs

However, we must also account for the reﬂections from inside the substrate. The solution for this is found by applying the sum of a geometric series as we did above in Eq. 2.28 and simplifying.

2iφd 2 4iφd r = r32 + t32r21t23e + t32r21r23t23e + ... (2.32) ∞ n 2iφd X 2iφd = r32 + t32r21t23e r21r23e (2.33) n=0 2iφd t32r21t23e = r32 + 2iφ (2.34) 1 + r21r32e d 2iφd r32 + r21e = 2iφ (2.35) 1 + r21r32e d 13

When inserting the constituent reflection and transmission coefficients and applying the thin-film approximation, this becomes slightly messier than the transmission.

(n − n )(n + n ) + (n + n )(n − n ) (1 + 2iφ ) r = 3 2 2 1 3 2 2 1 d (2.36) (n3 + n2)(n2 + n1) + (n3 − n2)(n2 − n1) (1 + 2iφd) n − n − Z σ = 3 1 0 s (2.37) n3 + n1 + Z0σs

Now that the most tedious part, in terms of derivations1, is behind us, we can press on to transmission through the substrate with and without the thin-film. We are applying this line of logic to bolometer (power/intensity) measurements, but the Fresnel equations are written in terms of the electric field, therefore we must take the norm-square of each term to find the power transmission.

2 2 twith = tt34 + tr34rt34 + tr34r t34 + ... (2.38)

2 2 twithout = t13t34 + t13r34r31t34 + t13r34r31t34 + ... (2.39)

2 2 2 2 2 2 2 4 4 2 Twith = t t34 + t r34r t34 + t r34r t34 + ... (2.40) 2 2 t t34 = 2 2 (2.41) 1 − r34r 2 2 2 2 2 2 2 2 4 2 Twithout = t13t34 + t13r34r31t34 + t13r34r31t34 + ... (2.42) 2 2 t13t34 = 2 2 (2.43) 1 − r34r31

Taking the ratio of Twith and Twithout allows us to determine the properties of the thin-ﬁlm, where R is the relative power transmission.

2 2 2 Twith t (1 − r34r31) R = = 2 2 2 (2.44) Twithout t13 (1 − r34r ) 1Nothing is as tedious as TDS. 14

Lastly we can solve for the sheet conductivity of the thin-ﬁlm by substituting in for the constituent parts of Eq. 2.44 and simplifying.

r 4 2 2 2 2 ! 1 2 2 Rn3 + 2n3n4 (2n1 + n4 (2 − R)) + n4 (4n1 + 4n1n4 + Rn4) σs = − n3 + 2n1n4 + n4 − 2n4Z0 R (2.45)

2.3 Harmonic Oscillator

Figure 2.2: A cartoon of the harmonic oscillator model.

The simplest way to describe nonlinear media is to start with the harmonic oscillator, specifically a damped-driven oscillator. We can view the ionic core as being immovable and tethered to the electron with via a spring. In this picture(Fig 2.2), an incident electric field is driving electrons, which oscillate in their respective potential wells. However, they will not oscillate indefinitely due scattering mechanisms which we condense into a damping term. For a Drude-Lorentz oscillator, we have the following equation of motion:

d2x dx q + Γ + ω2x = − E (t) (2.46) dt2 dt 0 m∗

where the charge carrier in question has effective mass m∗ and charge q. If we assume that the incident electric field is a monochromatic plane wave, we can easily find the solutions 15

using the following ansatz.

−iωt x = x0e xˆ (2.47)

Where the displacement from equilibrium x0 is given by:

q |E0| x0 = − ∗ 2 2 (2.48) m ω0 − ω − iΓω

Using this we can relate the displacement to the dipole moment (p = qx) and ﬁnally the macroscopic polarizability per unit volume of the material (P = Np, where N is the carrier density).

−iωt P = qNx = χe (ω) E0e xˆ (2.49)

We next need to use the displacement vector D in its two forms:

D = 0E + P = 0RE (2.50)

Now we can solve for the relative permittivity ( ). We can simplify by inserting the 0 2 Nq2 plasma frequency ω = ∗ and assume the medium is isotropic. This allows the relative p 0m permittivity to take the form of Eq. 2.51.

2 |P| ωp R (ω) = 1 + = 1 + 2 2 (2.51) 0 |E| ω0 − ω − iΓω

2 This is of particular usefulness since n = R, which allow us to rewrite the dispersion relation.

ω k (ω) = p (ω) (2.52) R c 16

2.4 Nonlinear Media

Figure 2.3: Example of an harmonic and an anharmonic potential. The associated motion of a charge carrier is plotted on the left.

The harmonic oscillator used in the previous section featured a perfectly harmonic potential, meaning that the potential energy of the charge carriers in the well is symmetric about its rest position. In order to see second order nonlinear eﬀects, a noncentrosymmetric crystal must be used. This translates into the potential inside the crystal having some

1 2 2 anharmonicity, which in the simplest approximation takes the potential from 2 mω0x to 1 2 2 1 3 2 mω0x + 3 mαx . This non-linear term corresponds to multi-photon processes. We will incorporate this into the Drude-Lorentz model by adding the following term:

d2x dx q + Γ + ω2x + αx2 = E (t) (2.53) dt2 dt 0 m

We will now solve Eq. 2.53 by using a perturbative expansion and collecting terms, in this case, powers of λ. This is possible because the the series should converge since each successive term is much smaller than the previous, explicitly x(1) x(2) ... x(n). The λ term is present for book keeping purposes since this is a recursive solution and every term in this expansion depends on the preceding terms. It is worth solving this equation of motion for the general case since we will need terms for optical rectiﬁcation and electro-optic sampling (Pockels Eﬀect). 17

∞ X x (t) = λnx(n) (t) (2.54) n=1 The ﬁrst order term, which does not include the nonlinear term, has the following equation of motion:

d2x(1) dx(1) q + Γ + ω2x(1) = E (t) (2.55) dt2 dt 0 m

We assume that there are two plane-waves in our derivation. We ﬁnd the solution has the form of Eq. 2.57. This will allow us to explore all the solutions for two photon processes.

(1) (1) −iω1t (1) −iω2t x (t) = x (ω1) e + x (ω2) e + c.c. (2.56)

−iω1t −iω2t q E0e q E0e = ∗ 2 2 + ∗ 2 2 + c.c. (2.57) m ω0 − ω1 − iΓω1 m ω0 − ω2 − iΓω2

The second order term lacks the electric ﬁeld term because the material is only being

driven at frequencies ω1 and ω2 and the resulting perturbation should only be due to the material’s response at those frequencies. This manifests itself in second order term by the the nonlinear term, x(1)2 is of order λ2, due to its dependence upon the ﬁrst order terms.

2 (2) (2) d x dx 2 + Γ + ω2x(2) = −α x(1) (2.58) dt2 dt 0

Now we must be slightly more careful with the second order term in order to elicit the desired result. There will be several terms due to the interaction of the incident waves, specifically x(1)2 should have four distinct types of terms: second harmonic generation (SHG, Eq. 2.59), sum frequency generation (SFG, Eq. 2.60), difference frequency generation (DFG, Eq. 2.61), and optical rectification (OR, Eq. 2.62). 18

2 2 (1) −2iω1t (1) −2iω2t SHG = x (ω1) e + x (ω2) e + c.c. (2.59)

(1) (1) −i(ω1+ω2)t SFG = 2x (ω1) x (ω2) e + c.c. (2.60)

(1) ∗(1) −i(ω1−ω2)t DFG = 2x (ω1) x (ω2) e + c.c. (2.61)

(1) 2 (2) 2 OR = 2 x (ω1) + 2 x (ω2) (2.62)

Of these, OR is of particular interest because it governs how we generate our THz pulse. The fact that the OR term has zero frequency requires explanation of how this can generate THz. This explanation requires an alternative view of what is happening in the material, which complements the view of the harmonic oscillator. This view relies on the second order polarization of the material and we must recall the initial explanation of linear media and the relationships between D, E, and P. For nonlinear media, the D-ﬁeld can be expressed as having both a linear and nonlinear component.

(1) (NL) (1) (NL) D = 0E + P + P = D + P (2.63)

Which leads to the macroscopic polarization of the material to be written perturbatively, in the same fashion as the solution to the harmonic oscillator.

P = P(1) + P(2) + P(3) + ... (2.64)

(1) (2) 2 (3) 3 P = 0 χ E + χ E + χ E + ... (2.65)

It is also important to note that P for any order of this expansion can be can be written in the following manner:

(n) (n) n P = 0χ E (2.66) 19

We need to rewrite Eq. 2.62 as it is of particular importance to this dissertation. We can relate the second order polarization to the second order correction to the harmonic oscillator in a similar way to Eq. 2.49.

2 (2) (2) 2αe N 2 P0 = −Nex0 = h i |E0| (2.67) 2 2 2 2 2 2 2 m ω0 (ω0 − ω ) + Γ ω

(2) (2) 2 P0 = 20χ (0, ω, −ω) |E0| (2.68)

In general all of these processes are written as tensors which depend on the two input and one output frequencies. They are commonly written in the format below and will be used in Sec. 3.1.2 and Sec. 3.1.3 to express the polarization for THz generation. Each process has its own associated tensor owing to the fact that the material should respond diﬀerently when mixing diﬀerent frequencies.

  E2  x         E2   y  Px d11 d12 d13 d14 d15 d16        2       Ez   P  = 20  d d d d d d    (2.69)  y   21 22 23 24 25 26         2EyEz    Pz d31 d32 d33 d34 d35 d36    2E E   z x    2ExEy 20

3 Terahertz Generation and Detection

3.1 Generation via Optical Rectiﬁcation

3.1.1 Phase Matching

A point which has been glossed over is the issue of the propagation of the pump beam, which drives the nonlinear response, and the generated beam. Before we dive into this too

deeply, we must discuss two quantities: the phase velocity (vp) and the group velocity (vg). The phase velocity pertains to the speed at which a given frequency can propagate in a medium. It has the following form:

ω c v = = (3.1) p k n (ω)

The group velocity represents the speed at which a packet of photons can travel in a medium. It can be written in terms of the phase velocity, which yields to following form:

∂ω c v = = (3.2) g ∂k ∂n n (ω) + ω ∂ω In our system, the use of a second order eﬀect dictates that the group velocity of the

2 optical pump beam, the speed at which the intensity proﬁle |E0| propagates, needs to match the phase velocity of the generated THz beam in order for there to be no destructive interference. This is most easily explained with a simple cartoon. 21

Figure 3.1: Example of generation using optical rectification in a phase-matched medium. Where the light is propagating left to right, Eopt is the optical electric field, POR is the polarization of the material, and ET Hz is the THz electric field.

As we can see if there is any ”walk-off” between the beams they will destructively interfere because the polarization of the material will be in the opposite direction of the displacement of the THz pulse. This is quantified by the walk-off length, the distance at which the optical pulse leads the THz pulse by the optical pulse duration τp.

cτp lw = (3.3) (nT − nO)

A useful counterpart to the walk-oﬀ length is the coherence length. This corresponds to the eﬀective interaction length; the distance where the optical pulse will either lead or lag

π behind the THz pulse by a phase factor of 2 . The coherence length depends on the following

quantities: ngr, the generating pulse group index; nT , the THz pulse index; and νT Hz, the 22

Figure 3.2: A cartoon of the walk-oﬀ length. Figure 3.35 from Ref. [1].

THz frequency.

c lc = (3.4) 2νT Hz |ngr − nT |

3.1.2 Zinc Telluride

Generating THz radiation in Zinc Telluride (ZnTe) is relatively easy due to how well phase-matched THz and optical pulses are. For example, we use a transform limited gaussian pulse with a central wavelength of 800 nm to generate our THz pulse, which has 1 THz bandwidth and a central frequency 1 THz. Using the optical and THz indices, nO = 3.2 and

nT Hz = 3.3, this leads to a coherence length of 1.5 mm. Recalling the form for the polarization tensors (Eq. 2.69), the tensor describing the nonlinear response for optical rectiﬁcation for ZnTe, a 43¯ m zinc-blende crystal, [28] can be written as follows:

  0 0 0 1 0 0   (2)   χ = d14  0 0 0 0 1 0  (3.5) OR     0 0 0 0 0 1

Even though this material is remarkably well phase-matched, it does have its drawbacks. 23

Figure 3.3: Index of refraction plot. Note that the optical group index and THz phase index are matched at λopt ≈ 810 nm and νTHz ≈ 1.7 THz. Figure 3.37 from Ref. [1].

Our setup can not produce the extremely high electric ﬁelds required for nonlinear THz

kV spectroscopy (ET Hz > 100 cm ) using ZnTe. However, high ﬁeld THz generation has been demonstrated. [29] Instead we must employ a tilted-pulse-front geometry to generate THz in LiNbO3 (see Sec. 3.1.3) to achieve the required electric ﬁelds.

3.1.3 Lithium Niobate

Lithium Niobate (LiNbO3) is an extremely important nonlinear material. It has many desirable qualities such as high optical transparency over a broad spectral range, it has strong optical nonlinearity, [30] piezoelectricity, [31] and ferroelectricity. [32] This 3m symmetry group crystal [33] is used for THz generation because the electro-optic coefficient we exploit for optical rectification is much larger than the coefficient used in ZnTe, we use d33 = 27 pm pm V ,[34] in comparison to d14 = 4 V in ZnTe. [35] The tensor which describes optical rectification can be found below (Eq. 3.6). 24

  0 0 0 0 d15 −d22   (2)   χ =  −d d 0 d 0 0  (3.6) OR  22 22 15    d15 d15 d33 0 0 0

While THz generation in LiNbO3 is much more efficient than ZnTe, it is also more complicated. The complication arises from the discrepancy between the optical and THz indices; the THz index is 5.2 [36] while the optical index is 2.3, [37] leading to a coherence length of approximately 52 µm. The LiNbO3 crystal must also be MgO doped in order to increase optical damage resistance and suppress the photoreactive effect. [38] In this setup, a diffraction grating is used to tilt the incoming optical pulse front to

the Cherenkov angle (θc in Eq. 3.7). [39] As the pulse travels through the crystal the THz

o pulse is produced at the Cherenkov angle, θc ≈ 63 using Eq. 3.7. This generation scheme is analogous to the sonic-boom of a jet, where the sound waves produced by the engine(s) travel slower than the aircraft itself and create a conical shock front as the sound and jet propagate. In our case, the generated THz pulse is strung out along the shock wave front as the optical pulse travels though, building, until it exits.

−1 nopt θc = cos (3.7) nT Hz

Before going further, we must discuss the choice of diffraction grating and lenses. Firstly, the optical beam diverges after it reflects off the grating and lenses must be used to capture the tilted beam and focus it on to a LiNbO3 crystal. A dual lens setup is preferable to a single lens setup since the second lens helps to restore the spherical aberration minimizes distortion of the pulse front. Also, long focal length lenses are used to minimize wavefront distortion at the cost of optics table space.

In order to further optimize experimental arrangement, the cut angle of the LiNbO3 has to be taken into account. This angle, γ, should be the same as θc to allow the THz pulse to exit the crystal at normal incidence; it must also be matched to the tilt angle of the optical 25

Figure 3.4: Cartoon of our tilted pulse front setup for LiNbO3. Note that the diffraction grating tilts the optical pulse front (red). The THz output pulse front is shown in orange. pulse front. This directly affects the choice of lenses (namely their magnification factor for the optical pulse front β1 and the grating image β2), the groove density of the grating p, and the chosen diffraction angle θd.[40]

λoptp tan γ = gr (3.8) noptβ1cos θd

ph tan θ = tan γ = noptβ2θd (3.9)

However, since our system is not an ideal setup, there has to be trade-offs in the fit of the tilt parameters. If we plot β1 = β2, we can find the optimal efficiency for a given set of lenses at a given diffraction grating groove density. The diffraction angle must stay close to θc, but not too close due to setup geometry constraints; there has to be ample room to place a lens or mirror to redirect the diverging beam without clipping the incoming beam (see Fig. 3.4). 26

Figure 3.5: Diﬀraction angle plot.

Using the above plot, we chose a groove density of 1800 cm−1. This groove density was chosen because an appropriate horizontal magniﬁcation ratio, roughly 0.6, can be achieved using commercially available lenses. It also maximizes the grating eﬃciency due to the

o diﬀraction angle being very close to the Littrow angle (θd − θlitt < 10 ).

3.2 Terahertz Detection

Terahertz detection is very diﬃcult, primarily because the photon energy is 4.1 meV. This is obviously well below interband transitions for semiconductors and is generally associated with rotations and vibrations of molecules. It is also readily absorbed by water, [41]

so most measurements must be done under a N2 purge to overcome this, especially if we want to extract the frequency dependent transmission. We will discuss two types of detectors: incoherent power transmission detectors (bolometers and pyroelectric detectors) and spectrometers (Michelson interferometry and EO Sampling) 27

3.2.1 Bolometer

Our primary detector is a liquid helium (L-He) cooled Si bolometer. Bolometers are an extremely old but robust style of detector. Invented by Samuel Langley, bolometers were initially used for infrared astronomical measurements and long-distance bovine thermal detection. [42] They are prime THz radiation detectors because the radiation corresponds to a 48 K change in temperature (~ω = kBT ).

Figure 3.6: Schematic of a compound bolometer. Figure 4.25 from Ref. [1].

Bolometers measure incident electromagnetic radiation by absorbing radiation and heating a material which has a temperature dependent resistance. This material is situated as a resistors in a circuit. When THz radiation is incident upon the bolometer, the resistance changes, thereby changing the voltage. The signal (voltage) from our bolometer is proportional to the intensity of the incident THz radiation. Due to sensitivity constraints, the bolometer needs to be small in size. This poses major problems since the diﬀraction limit is

λ half the wavelength ( 2 )[43] and the wavelength of THz light is on the order of 300 µm. To circumvent this, the bolometer is placed in a multimode waveguide to collimate the incident radiation, ensuring that all the THz radiation is absorbed. 28

3.2.2 Pyroeletric Detectors

While bolometers are extremely sensitive, the use of L-He make them cost prohibitive. Pyroelectric detectors on the other hand do not require such a cooling scheme, they do however rely on the heating of a material. The incident radiation changes the polarization of the material instead of changing the resistance of the material. The material in question must be a polar crystal, meaning that it must have a permanent electric dipole moment. The dipole moment is sensitive to changing temperature, so any incident THz radiation on the material will change the instantaneous polarization.

Figure 3.7: Schematic of a pyroelectric detector. Figure 4.28 from Ref. [1].

The pyroelectric material is generally placed between electrodes with a blackened ab- sorber on top of the structure. The polarization of the material points from one electrode to the other, which induces a surface charge density in the electrode. When THz radiation is absorbed in the top layer, it changes the temperature and therefore the polarization of the material. This in turn changes the surface charge density of the electrode and creates a current, which is proportional to the incident THz intensity.

3.2.3 Michelson Interferometry

Michelson interferometry is a well established method for using an incoherent power detector to measure the power spectrum. The interferometer operates by splitting the trans- 29 mitted THz radiation using a beam splitter, in our case a high-ρ Si wafer, and delaying one of the legs by using a delay stage before recombining the signal and measuring. The delay stage will systematically be moved to walk the two beams over each other. The Fourier transform of the resulting interference pattern, the interferogram, is the power spectrum.

Figure 3.8: A cartoon of our Michleson interferometer using a bolometer.

While useful, Michelson interferometry lacks the ability to examine the phase information from the sample.

3.2.4 Electro-Optic Sampling

Electro-optic (EO) sampling is an extremely powerful tool. It allows us to map-out our THz electric ﬁeld as a function of time, which gives us the ability to extract the phase information. This technique is commonly known as THz time-domain spectroscopy (THz TDS).

3.2.4.1 Pockels Eﬀect

To describe EO sampling, we must return to our description of nonlinear media (Sec. 2.4) and revisit the second-order processes. We can re-tool the second order polarization 30 by assuming we have input frequencies of ω (optical) and 0 (THz radiation) to ﬁnd the polarization at frequency ω in the material.

(2) X (2) P (ω) = 2 0χijk (ω, ω, 0) Ej (ω) Ek (0) (3.10) jk

(2) P (2) We will now use the field-induced susceptibility tensor, χij (ω) = 2 k χijk (ω, ω, 0) Ek (0), which describes how the THz electric field modulates the susceptibility of the material, which yields the following simplified form.

(2) X (2) P (ω) = 2 0χij (ω) Ej (ω) (3.11) j

This is known as the Pockels effect. [44] Generally it refers to an induced birefringence in a material via an applied DC electric field. It is commonly used in active Q-switched lasers to modulate the quality factor of the cavity. For our purposes, the THz field acts as the applied DC field since it is oscillating roughly 103 times more slowly than the optical. It is also important to note that if the media in question is lossless, the Pockels effect should have the same magnitude electro-optic coefficient as optical rectification. Luckily this is the case ZnTe in our frequency range (0 to 2 THz) since we are sufficiently far from the transverse optical phonon resonance at 5.3 THz. This makes the coefficient r41 = d14.

3.2.4.2 Signal Extraction

Extracting the information contained in the signal is relatively simple. The linearly

λ polarized optical pulse passes though the ZnTe crystal, a 4 -plate, a Wollaston prism and ﬁnally onto a balanced photodiode. Assuming there is no THz ﬁeld incident on the ZnTe crystal, the linear polarization of the optical pulse should not be altered as it transverses the ZnTe crystal. The linear polarization

λ to changes to circular when it passes through the 4 -plate. The Wollaston prism then splits 31 the circularly polarized beam into its two constituent linear polarizations (with associated intensities Ix, Iy) and projects them onto the balanced photodiode. Since the polarization components are of equal magnitude, there should be no net signal.

Figure 3.9: A cartoon of our EO sampling setup.

When the optical pulse is temporally swept across the THz pulse, in a pump-probe type setup, there will be phase retardation caused by the THz radiation via the Pockels eﬀect depending on where the THz radiation and optical overlap. It is given by the following formula [45]:

ωL ωL ∆φ = (n − n ) = n3 r E (3.12) y x c c O 41 T Hz

While there is some phase retardation, it is generally small and a small angle approximation can be made.

I I I = 0 (1 − sin∆φ) ≈ 0 (1 − ∆φ) (3.13) x 2 2 I I I = 0 (1 + sin∆φ) ≈ 0 (1 + ∆φ) (3.14) y 2 2 32

Now the diﬀerence in the intensities of the two polarizations, Is, caused by the presence of the THz ﬁeld is measured by the balanced photodiode.

ωL I = I − I = I ∆φ = I n3 r E (3.15) s y x 0 0 c O 41 T Hz

The difference in the signals is proportional to the incident THz field, which allows map out the electric field as a function of time. In order to map out the entire waveform, the optical beam must be walked across the THz pulse using a computer controlled delay stage. Each step in the delay stage changes the overlap and allows for a different piece of the THz field to be sampled. 33

4 Surface Plasmons and Surface Plasmon Polaritons

In 1968 A. Otto [46] and R. H. Ritchie [47] created the field of plasmonics. Their initial results are the humble beginnings of a now burgeoning field, where physics exploits resonance properties of metal-dielectric interfaces. Plasmonics are used to circumvent the diffraction limit and are employed in localized fluorescence microscopy, [48–52] near field imaging using subwavelength apertures, [53, 54] and also electric field enhancement. [55, 56] In order to better understand this unusual behavior we must start by building the theory ”from the ground up” and then proceed to the practical applications of plasmonic devices.

4.1 Theory of Surface Plasmons

Our brick and mortar construction starts small but eventually we will derive the dispersion relation and generation conditions. Initially we need to start with electromagnetic theory. The ﬁrst piece of this derivation starts with Eq. 2.50 and arrives at the equation for relative permittivity (Eq. 2.51).

4.2 Drude-Sommerfeld Model

In order to derive and explain the working of surface plasmons, we must begin by con- sidering the free electron gas in a conductor. This is relevant because we are dealing with a metal, which can be approximated to an ideal conductor. Starting from the equation of motion for a free electron interacting with an incoming electric ﬁeld of amplitude E0 and frequency ω (the Durde-Sommerfeld model Ref. [57]), we will solve for the relative permittivity. 34

∂2x ∂x m + m Γ = eE e−iωt (4.1) e ∂t2 e ∂t 0

We should note that because this equation is for a free electron, there is no restoring force and that for visible frequencies interband transitions need to be included for a complete model. [58] It is easy to see the damping term, Γ, can be thought of as a collision frequency through which the incoming electric ﬁeld’s eﬀects are damped via scattering in the metal.

−iωt The solution to this diﬀerential equation comes from the ansatz x = x0e xˆ. Inserting this in to the equation and diﬀerentiating yields the following result, Eq 4.2.

m ω2x + im ωΓx E = − e 0 e 0 (4.2) 0 e

Next we insert the ansatz with this result in the equation for relative permittivity and ﬁnally arrive at the destination Eq. 4.3.

ω2 Γω2 = 0 + i00 = 1 − p + i p (4.3) R ω2 + Γ2 ω (ω2 + Γ2)

Now that we have derived the dielectric function, it is important to note a few things. First, this formula will accurately describe the region of interest (the THz regime) because is suﬃciently far from the plasma frequency. Secondly, operating at THz frequencies is far from the aforementioned interband transitions.

4.3 Surface Plasmon Polaritons at Interfaces

Before continuing, the diﬀerences between a surface plasmon and a surface plasmon polariton (SPP) needs to be addressed. A surface plasmon is a transverse magnetic (TM) wave at the metal-dielectric interface resulting from the charge oscillations induced by the electric ﬁeld from the incident light. The surface plasmon polariton is the special case when the surface plasmon resonance and the evanescent wave of the incoming light couple in the 35 metal. After coupling, this quasi-particle propagates through the metal along the interface.

4.3.1 Dispersion Relation

Moving on from Drude-Sommerfeld model and our dielectric function, we shall press on toward deriving the dispersion relation for a surface plasmon polariton (SPP) and the generation requirements for SPP. It is worth noting that the solutions we want are eigenmodes of the system, which implies that we need only to satisfy certain criteria to achieve excitation. Beginning at the wave-equation for a metal-dielectric interface and using the following plane- wave solutions, we can extract the eigenmodes of the system.

1 ∂2D ω2 ∇ × ∇ × E = − 2 = 2 RE (4.4) 0 ∂t c

ω2 ∇ × ∇ × E − E = 0 (4.5) c2 R

Figure 4.1: Interface diagram.

Our system consists of two semi-inﬁnite media: z < 0 is our dielectric media, generally 36

the sample which is being tested, and z > 0 is a metal of our choosing. It is important to note that the incoming pulse is traveling through a different dielectric before it reaches the metal-dielectric interface shown in Fig. 4.1. Each material has its own unique dielectric function, however for the purposes of this paper we shall assume the first media has a constant, real dielectric function and the metal is completely Drude-like. Assuming that we have p-polarized light that is confined to the x − z plane and that j = 1 (j = 2) is the dielectric (metal), we can use an electric field of the following form1:

  Ej,x     ikxx−iωt ikj,zz Ej =  0  e e (4.6)     Ej,z

Now inserting this electric ﬁeld in to the wave equation above and assuming both space to be source free (∇ · D = 0), we can extract the relationship between the total wave-vector,

2π k = λ , its constituent components, and the relative permittivity.

2 2 2 R,jk = kx + kj,z (4.7)

Further extension of the source-free condition lead to the following relation between the electric ﬁelds resulting from the divergence:

kxEj,x + kj,zEj,z = 0 (4.8)

Utilizing our boundary conditions (E1k = E2k, D1⊥ = D2⊥), we are able to acquire more conditions which our wave must satisfy. Namely:

1 Note that k1,x = k2,x = kx due to boundary conditions. 37

E1,x − E2,x = 0 (4.9)

R,1E1,z − R,2E2,z = 0 (4.10)

Now solving eq. 4.8 for Ej,x and inserting the results in to eq. 4.9, we now have two equations relating the z-components of the electric ﬁeld.

R,1E1,z − R,2E2,z = 0 k2,zE2,z − k1,zE1,z = 0 (4.11)

The above equations can now be re-arranged, solving for the Ej,z components and substituting, so that only the permittivities and kj,z components remain.

R,1k2,z − R,2k1,z = 0 (4.12)

2 Finally we can arrive at the dispersion relation by solving eq. 4.7 and eq. 4.12 for kj,z and doing substitution.

2 2 R,1R,2 2 R,1R,2 ω kx = k = 2 (4.13) R,1 + R,2 R,1 + R,2 c

2 2 2 To ﬁnd kj,z in terms of k , we can do a similar trick by solving eq. 4.7 for kx and setting them equal, which gives us:

2 2 R,j 2 kj,z = k (4.14) R,1 + R,2

Now that we have a dispersion relation, the dielectric functions of the materials are now a

point of interest. For the bound solution, we require that the normal components of k (kj,z) 38

are decaying, meaning that in the z-direction we have an evanescent wave. Since we want a

wave that will propagate along the interface, we need kx to have a real component. Later we will take in to account that the propagating wave in the metal is decaying and therefore has an imaginary component, but let’s not get ahead of ourselves. From here we can conclude

R,1R,2 that in order to have a real kx, the quantity has to have a real component. This R,1+R,2

means R,1R,2 and R,1 +R,2 have the same sign because we need their square root to have a real component. The dielectric functions for metals generally includes a large, negative real part (at THz frequencies, this is on the order of −104 [1]) and for dielectrics it is usually constant and positive (air R ≈ 1 and SU-8 R ≈ 2.9 [59]), so this condition is satisﬁed. Next we must consider SPP propagation at the metal-dielectric interface. The metal’s

0 00 dielectric function R,2 = 2 + i2 is described by the Drude-Sommerfeld model which we derived above and as stated, our dielectric material is assumed to only have a real dielectric constant (R,1 = 1). From this launch point it is easy to see that kx is going to be complex,

0 00 0 which can be succinctly written as kx = kx + ikx, where kx determines the wavelength of the

00 SPP (λSPP ) and kx determines the propagation length. Now we can see that kx is:

s r 0 00 R,1R,2 ω 1 (2 + i2) ω kx = = 0 00 (4.15) R,1 + R,2 c 1 + 2 + i2 c

Normally at this point in the derivation the following approximation would be made:

0 00 |2| >> |2|. This approximation is true for noble metals in the visible (for silver at 633nm

Drude = −18.2+0.5i). We, however, are more concerned with aluminum at THz frequencies.

0 00 4 When using the Drude model |2| << |2| for aluminum at 1THz (Drude = −3.246 × 10 + 6.424 × 105i [1]). Using this, we can see that Eq. 4.15 simpliﬁes to Eq. 4.17.

√ 0 1 |2| ω kx = (4.16) |1 + 2| c 3/2 00 00 1 2 ω kx = (4.17) 2 |2| |1 + 2| c 39

4.3.2 SPP Excitation

Before delving in to excitation, we should look at the relation between the wavelength of a SPP and the free space wavelength. Using the numbers for aluminum, it is shown that √ 2π λ 1|2| √ that λSPP = 0 ≈ , where ≈ 1 = n1. However interesting this result is, there is kx n1 |1+2| still a good distance to cover. An extremely integral point we have glossed over is that in order to generate a SPP, the SPP has to fulﬁll energy and momentum conservation.

Figure 4.2: SPP dispersion curve

Using the dispersion curve for SPP the resonance condition is found by plotting (Fig.

ckx √ 4.2) ω = n , where n = R the index of refraction of the material the pulse is propagating through before it interacts with the metal (for SU-8 at 1 THz n ≈ 1.7 − 1.8 [59]). The deviation from the light line tells us an important piece of information about SPP: as the frequency increases, losses increase at an alarming rate. This exempliﬁes why the THz regime is prime territory for the use of plasmonic devices, the frequency is a factor of 103 lower than that for optical light. However, this graph also shows that excitation requires an increase of wave vector over its free space value. [58] 40

4.4 Excitation Methods

Several techniques exist which serve to increase the wave vector and satisfy this excitation requirement. The three methods in common use are the Otto conﬁguration, the Kretschmann conﬁguration, and lastly using a periodic array.

4.4.1 Otto Method

Starting out we have the Otto configuration, created by Andreas Otto. [46] This method uses a prism which is slowly translated closer to a metal interface. The mechanism for generating the surface plasmon is coupling the evanescent wave from the prism/dielectric interface with the dielectric/metal interface as they are brought closer together. There is an obvious limit to the distance between the prism and the metal because the electric field of the evanescent wave is exponentially decaying as a function of distance, showing that the two interfaces must be extremely close. This poses logistical problems for experimental setups. As with all SPP, an incident laser is needed for excitation. In this setup, the angle of the laser dictates the excitation condition, meaning if the incidence angle of the laser is changed, a minimum in the reflectivity will be observed when the wave vector resonance condition is satisfied.

Figure 4.3: Otto conﬁguration 41

4.4.2 Kretschmann Method

The next method for discussion is the Kretschmann method which was developed in 1971 by Erwin Kretschmann. [60] This method does not rely on a small gap between the prism and the metal, instead the metal layer is directly deposited on the prism. This negates issues arising from controlling the air gap making excitation for SPPs easier and is more commonly used. [61] However the uniformity of the metal layer is critical to the operation. If the layer is too thin, damping effects from the glass will effectively kill propagation of the SPP. A thick metal layer is also problematic because the metal will absorb the beam and no SPP will be generated. As was mentioned with the Otto configuration, the Kretschmann method utilizes a minimum in the reflectivity to denote the excitation of a surface plasmon.

Figure 4.4: Kretschmann conﬁguration

4.4.3 Spatial Periodicity

Lastly SPP can be excited by using a periodic array to couple to the incoming light. [62,63] In this method, a periodic array is added to increase the wave vector, as seen in Fig. 4.5.

0 2πn The addition of this periodic lattice allows us to now write kx = kx + a , where a is the 2πn periodicity of the lattice and a is the reciprocal lattice vector. It is also important to note this excitation method applies to both stereo (3 dimensional) and planar (2 dimensional) structures. This means that corrugations in the surface, a grating coupler (Fig. 4.5), can be used to elicit the SPP resonance in the same manner as a periodic 42

Figure 4.5: Grating coupler, one example of a periodic array used to elicit a SPP resonance.

array of planar dipole antennas. [64]

4.5 Applications to Terahertz Science

Deriving how SPPs are generated is all well and good, but now we must apply this knowledge to THz science. The THz regime falls into the region where the SPP dispersion relation roughly overlaps the light-line (ω = ck). This means that the Otto and Kretschmann method will not produce SPPs and the wave-vector must be lengthened by other means. Therefore, periodic arrays must be used in order to generate SPPs at THz frequencies. [65] Using a SPP resonance is the best bet to best Bethe and the diﬀraction limit. According to Bethe [66] subwavelength aperture should exhibit magnetic dipole radiation, meaning the

a 4 transmission should scale as λ ,[67] where a is the diameter of the aperture. This is for a single aperture, if there is an array of periodic apertures the SPP resonance will enhance transmission. [68] These structures also lead to charge confinement, giving rise to an electric field enhancement in the near-field. These structures are also on the order of or smaller than the wavelength, leading to confinement of the light, circumventing the diffraction limit. [69–71] For our purposes, we use plasmonic structures to enhance our incident electric field and probe the carrier dynamics. 43

5 Non-Linear Terahertz Spectroscopy using Plasmon Induced

Transparency

5.1 Introduction

Metamaterials, or engineered materials, have a very promising future. Speciﬁcally we will be discussing plasmonic induced transparency (PIT) [72] which is elicited by using coupled resonators. These engineered materials have been employed using various types of coupled resonators, [73–75] even at optical frequencies. [76] They have been shown to have a tunable transparency frequency with the proper material and resonator selection [77] or even a broadband PIT. [78] They also have the ability to slow light due to the phase retardation in their reaction. [79,80] These structures also have high Q-factors [81] and can be employed in dielectric waveguides. [82] Materials which exhibit a PIT can be useful tools for examining material properties since the induced transparency is extremely sensitive to the local environment. Small changes in the carrier concentration of the material or change in the area immediately surrounding the PIT will alter its resonance frequency. In our experiment we employed PIT structures to observe the carrier dynamics of intrinsic high-resistivity silicon (high-ρ Si) and gallium arsenide (GaAs). [83] The materials chosen for this study were selected due to their prevalence in the semiconductor industry and promise as photonic materials. [84, 85] With device sizes shrinking, the high-ﬁeld, high frequency dynamics of materials are or paramount interest. GaAs has great promise as an ultra-high speed electronic material and there is great interest in its high-frequency carrier dynamics. [86–89] While silicon is the premiere electrical material and fundamental research into its carrier dynamics in the far from equilibrium regime are necessary for further func- tionalization. 44

5.2 Plasmonic Induced Transparency

5.2.1 Coupling of Two Resonators

Our plasmonic structures are used to elicit an induced transparency via coupled oscillators. More speciﬁcally, they rely on coupling the bright mode (super-radiant) in one structure to the dark mode (sub-radiant) in a secondary structure, which can lead to a plasmonic induced transparency [90–92]. These resonators are in a conﬁguration where there is no electric dipole interaction due to the two resonant polarizations directions being perpendicular. The magnetic dipoles, however, will either be parallel or anti-parallel. It is useful to think of these as a magnetic ”molecule” where the induced transparency arises due to the susceptibility of the system. To illustrate the coupling, let us consider two adjacent split ring-resonators (SRRs) in the following orientations (Fig. 5.1 and Fig. 5.2): 0o rotation (parallel), 180o degree rotation (anti-parallel), and 90o degree rotation.

Figure 5.1: A) Parallel SRRs. B) Anti-parallel SRRs.

If the SRRs have the same orientation (0o degree rotation, Fig. 5.1 A), meaning their electric dipoles and magnetic dipoles are aligned, there will be transverse coupling between the electric dipoles and lateral coupling between the magnetic dipoles. This leads to resonant absorption above their individual resonant frequency due to the magnetic dipoles being 45

parallel. In comparison, when the relative angle between the SRRs is 180o (Fig. 5.1 B) the energy associated with the resonant absorption will be lower due to the magnetic dipoles being anti-parallel.

Figure 5.2: SRRs with 90o rotation. The ﬁgure illustrates the higher energy resonant case (A) where the magnetic dipoles are parallel and the lower energy resonant case where the magnetic dipoles are anti-parallel (B) and the associated line splitting (C).

The case where the relative angle between the SRRs is 90o (Fig. 5.2) is most important to us. In this conﬁguration the excited SRR will drive sub-radiant SRR via inductive coupling,

π albeit with a 2 phase delay. There are two cases to consider in this geometry: times where the magnetic dipoles are parallel and where they are anti-parallel. Both of these arise from the delay in the response of the sub-radiant resonator. Now that we have established the coupling mechanism, we must explore how the PIT arises. Our magnetic ”molecule” is comprised of two ”atoms”, one with a super-radiant state |ai =a ˜ (ω) eiωt which strongly couples to the incident light and one with a sub-radiant state |bi = ˜b (ω) eiωt which can only weakly couple to the incident light. However, both of these

”atoms” should have a resonance at the same frequency ω0. These can be modeled as two linearly coupled Lorentz oscillators when the excitation frequencies are close to resonance

(expressed by the detuning δ = ω − ω0 ω0). 46

   −1   ˜ a˜ δ + iγa κ gE0   = −     (5.1)  ˜      b κ δ + iγb 0

In Eq. 5.1 we solve for the equation of motion for each ”atom”, where κ represents coupling between atoms and g is a parameter, based on the geometry, which indicates how strongly the super-radiant structure couples to the incident radiation. We can take the inverse of the above matrix and solve the coupled diﬀerential equation to ﬁnd the amplitude of motion for the super-radiant state.

˜ −gE0 (δ + iγb) a˜ = 2 (5.2) (δ + iγa)(δ + iγb) − κ

The amplitudea ˜ can be used to ﬁnd the polarizability and therefore the susceptibility ˜ (qNa˜ = 0χE0). Given that our detuning is small δ γa, γb, κ, we can ignore terms with orders δ greater δ2 and simplify the susceptibility into the following form:

−gqN δ (γ2 − κ2) iγ χ = b + b (5.3) 2 2 2 0 (γaγb + κ ) γaγb + κ

It is important that we note that as the detuning approaches zero, the real term approaches zero. The imaginary term is extremely small because it is inversely proportional to coupling strength. Since the coupling is very strong due to the proximity of the two resonators, there is very little absorption. The phase index can be approximated using the susceptibility near zero detuning can be approximated in the following manner:

χ n = p1 + χ ≈ 1 + (5.4) ph 2

This shows the phase index goes to one as χ goes to zero. Now the group index can be found using the denominator of Eq. 3.2, explicitly:

∂n n = n + ω ph (5.5) gr ph ∂ω 47

∂nph ∂nph −2 We should note that ∂ω |ω=ω0 ≈ ∂δ |δ=0, which is proportional to κ . This can increase the group index if the coupling is small, leading to slow light at the cost of absorption. However, since we assumed strong coupling, this quantity goes to zero and the group index approaches one as well. This is the cause the induced transparency. The susceptibility also depends upon the carrier concentration. Any changes to the carrier concentration will not only serve to damp out the oscillation, but also change the resonance frequency due to the susceptibility changing.

5.3 Coupling of Linear Antenna and Split-Ring Resonator

In order to engineer a metamaterial to have a PIT, the resonance of the individual components contained in the metamaterial must be determined. In the following sections we will derive the resonance for the antenna and the split ring resonators.

λres The dipole antenna should be resonant when L = 2 . However because these antennas λ are patterned on semiconductors the resonant wavelength will not be simply 2 , it should be shorter. The ends of the antenna have an increase in the reactance which not only gives rise to the plasmonic resonance (surface charge wave in the metal), but also increases the eﬀective antenna length. [93] The substrate on which the metal is patterned interferes with the resonance and eﬀectively lengthens the antenna leading to the following formula [94]:

λ res = L + 2δ (5.6) 2neﬀ

However, the delta term can be ignored and is on the order of the lateral antenna dimen- sion [93,95], leading us to our ﬁnal form:

λres = 2neﬀL (5.7)

The split ring-resonator (SRR) can be modeled as a LC circuit. In order to ﬁnd the 48

resonant frequency (ω0), we must calculate the capacitance and inductance of the SRR. The capacitor portion is at the gap of the SRR and can be modeled as a parallel plate capacitor with only air in the gap ( ≈ 0). [96]

A C = 0 (5.8) d wt C = 0 (5.9) d

Now ﬁnding the self-inductance is more complicated (Lself). First the ﬂux through the SRR must be calculated.

Φ = LselfI (5.10) Z = B · dS (5.11) I = A · dl (5.12)

While using the magnetic field to find Lself is relatively straight forward, it is easier to use the vector potential (A) to compute Φ in the geometry of a SRR. [97] So to find A, assume there is a current flowing down a metallic wire of length L which can be solved in the following way:

Z +L 0 µ0I 2 dz Az = 1 (5.13) 4π −L 0 2 2 2 2 (z − z ) + r  1  L h L 2 2i 2 µ I − z − 2 + z − 2 + r 0   = ln 1 (5.14)  h i 2  4π L L 2 2 − z + 2 + z + 2 + r 49

Figure 5.3: The geometry for Az.

1 We can further rearrange the formula and ﬁnd Az in its ﬁnal form .

" ! !# µ I −z + L z + L A = 0 sinh−1 2 + sinh−1 2 (5.15) z 4π r r

Now that we have Az and assuming that the SRR is merely square loop, we can easily find the flux by multiplying the flux due to one of the side-lengths by 4. The bounds of the line integral are for the loop on the inside of the square coil. Lastly, we can divide by the current and find Lself.

H A · dl L = (5.16) self I −w+ L 4 Z 2 = Azdz (5.17) I L w− 2 −w+ L " L ! L !# µ Z 2 −z + z + = 0 sinh−1 2 + sinh−1 2 dz (5.18) π L r r w− 2 2µ √ L − w q = 0 −w sinh−11 + w 2 + (L − w) sinh−1 − w2 + (L − w)2 (5.19) π w √ 1Note: sinh−1z = ln z + z2 + 1 and −sinh−1 − z = sinh−1z 50

Figure 5.4: The square loop cartoon.

Now lastly, we ﬁnd the resonant frequency given by the following:

1 ω0 = √ (5.20) LC v u πd = cu (5.21) u √ q t −1 −1 L−w 2 2 2wt −w sinh 1 + w 2 + (L − w) sinh w − w + (L − w)

The positive antenna arrays’ fundamental resonance manifests itself as transmission peak with strong absorption dips on either side when the incident THz polarization is parallel to the central antenna (0 degree). The absorption dips are the result of the magnetic dipoles for the central dipole antenna and the SRRs being either parallel (low frequency) and anti- parallel (high frequency). The peak in the center is the PIT, which should have the same value for the transmission as the bare substrate. The coupling between the antenna and the SRRs can be explained in the following manner: The coupling between the antenna and the SRR depends on the phase diﬀerence between the capacitive and inductive currents in the SRR from the antenna. [98] The capacitive and inductive surface current produced by the antenna in the SRRs need to have a phase 51

Figure 5.5: PIT resonance cartoon illustrating the coupling between the dipole antenna and adjacent SRRs. diﬀerence of 0 to constructively interfere and enhance coupling. The capacitive current in the SRR arises from the interaction from the edge of the SRR closest to the antenna and the antenna itself. From FE = qE, it is easy to see that the induced current in either of the SRRs should be toward or away from the antenna, depending on the antenna’s polarization

π at that time. The induced capacitive current lags the electric field from the wire by a 2 phase difference. The inductive current caused by the antenna’s current must also be taken into account. We can see that the inductively induced current will be in-phase with the magnetic field due to Maxwell’s equations (specifically Eq. 2.2). The magnetic and electric

π ﬁeld from the wire are 2 out of phase, meaning the induced currents have either a 0 or π phase diﬀerence. The inductive current in the SRRs will be in-phase with the capacitive current if they are placed in line with the base of the antenna. This oscillation creates a capacitance across the gap of the SRRs and excites the narrowband dark mode. 52

The dark mode will then couple back to the antenna and cause it to continue to radiate at its resonant frequency. The dark mode in the SRR can be viewed as storing energy for the oscillation that would have otherwise been radiatively lost in the broadband response of the dipole antenna. The negative antenna arrays’ fundamental resonance manifests itself as having perfect absorption at the PIT frequency when the incident THz polarization is perpendicular to the central antenna (90 degrees). The simplest way to explain this is using Babinet’s principle of complementary screens. This means that instead of having a maximum of transmission at the PIT, the negative structures will have a minimum. The absorption dips for the magnetic dipole coupling will now be transmission peaks. We primarily studied the negative arrays because their response will be clearer due to the structure only transmitting frequencies near these magnetic dipole peaks and not having a background transmission from the bare areas of substrate (compared to the positive arrays).

5.3.1 Fabrication of PIT Structure

The fabrication process for these structures can be found in the appendix. In order to fabricate the antenna arrays a photo-lithography mask was made using the Heidleberg 66FS DWL Mask Writer. On the mask there are three pairs of structures with resonance frequencies and dimensions listed in the table below. Each pair consists of a positive and negative structure. A positive structure has metal for the antenna and SRR and the space between them is bare substrate. The negative structure has the antenna and SRR etched in the metal. 53

Figure 5.6: Here is an example unit cell of the PIT structure. Dimensions can be found in Table 5.3.1 (below).

f (THz) L(µm) w (µm) g (µm) s (µm) l (µm) δy (µm) Px (µm) Py (µm) 0.6 96 6 6 7.5 33 31.5 120 135

0.9 64 4 4 5 22 21 80 90

1.2 48 3 3 3.75 16.5 15.75 60 67.5 The fabrication process for these structures was straight forward. Approximately 1µm of S1818 photoresist was spin-coated and used to pattern the negative of the structures directly on the substrate. The sample with S1818 was placed behind the mask in the photo-aligner and then exposed to UV light. This exposure allows the exposed S1818 to be removed when placing the sample in developer. After developing the photoresist, 500 nm of Al was deposited using the Polaron thermal evaporator. The sample was then placed in an acetone bath to remove the underlying layer of photoresist. Lastly the sample was sonicated to remove excess metal which was not attached to the substrate surface. 54

Figure 5.7: Optical microscope pictures of the GaAs sample. The pictures on the left are the positive arrays. The pictures on the right are the negative arrays. The top row is the 0.6 THz arrays. The middle row is the 0.9 THz arrays. The bottom row is the 1.2 THz arrays. 55

Figure 5.8: Optical microscope pictures of the Si sample. The pictures on the left are the positive arrays. The pictures on the right are the negative arrays. The top row is the 0.6 THz arrays. The middle row is the 0.9 THz arrays. The bottom row is the 1.2 THz arrays. The psychedelic blue color is due to the improper functioning of the white-balance on the microscope. 56

5.4 Experimental Considerations

The laser used for both THz generation and optical excitation is a Ti:Sapphire femtosecond laser system with a Gaussian proﬁle pulse, 1µJ pulse energy, central wavelength

λ0 = 800 nm, bandwidth δλ = 10 nm, and 1 kHz repetition rate. The THz pulse (central

frequency f0 = 1 THz, bandwidth δf = 1 THz) is generated using tilted pulse front geometry

o in LiNbO3. The optical and THz pulses were then overlapped at the focus of a 90 oﬀ axis parabolic mirror, where the optical beam has a beam waist about a factor of 5 larger than the THz beam waist. The THz TDS waveforms were acquired using electro-optic sampling in a 1 mm ZnTe crystal.

5.5 Initial Testing and PIT Observation

Upon testing the structures, we found that a thicker substrate was needed for a full characterization. The first round of samples were fabricated on 300 µm thick high-ρ Si wafers, which correlates into a 6.84 ps delay between the first transmitted pulse and the first internally reflected pulse. This was done to test our capability to fabricate these structures. The second round employed a 3000 µm thick high-ρ Si wafers which allowed for a 68.4 ps delay. However, the TDS waveforms showed a trailing pulse approximately 23 ps behind the first transmitted pulse, limiting the temporal resolution. This limitation was not of consequence because the ratio of the amplitude of the electric field at 22 ps to peak amplitude is approximately 10−3. The GaAs sample was 1 mm thick, which provided an adequate temporal window (24 ps). In our initial characterization we observed the PIT samples resonating at their the fundamental, but not the second harmonic. This is expected due to the fact that the charge distribution must have opposite signs at opposite ends of the antenna, meaning only odd modes should be excited in the dipole antenna. [95] We did, however, observe the second harmonic from the magnetic resonance due coupling between the SRRs in adjacent unit 57 cells. [99–102]

5.5.1 Resonant Cases in GaAs

We observe the PIT peak (dip) in the positive (negative) structures. It is also important to note the absorption (emission) of the second harmonic in the SRRs.

Figure 5.9: The GaAs PIT 0.6 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.10: The GaAs PIT 0.6 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 58

We observe the PIT peak (dip) in the positive (negative) structures. We lack the ability to observe the absorption (emission) of the second harmonic in the SRRs.

Figure 5.11: The GaAs PIT 0.9 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.12: The GaAs PIT 0.9 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 59

We observe the PIT peak (dip) in the positive (negative) structures. We lack the ability to observe the absorption (emission) of the second harmonic in the SRRs.

Figure 5.13: The GaAs PIT 1.2 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.14: The GaAs PIT 1.2 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 60

5.5.2 Resonant Cases in Si

We observe the PIT peak (dip) in the positive (negative) structures. We also observe the absorption (emission) from the second harmonic in the SRRs.

Figure 5.15: The Si PIT 0.6 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.16: The Si PIT 0.6 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right. 61

We observe the PIT peak (dip) in the positive (negative) structures. We lack the ability to observe the absorption (emission) from the second harmonic in the SRRs.

Figure 5.17: The Si PIT 0.9 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.18: The Si PIT 0.9 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right. 62

We observe the PIT peak (dip) in the positive (negative) structures. We lack the ability to observe the absorption (emission) from the second harmonic in the SRRs.

Figure 5.19: The Si PIT 1.2 THz positive array when the THz polarization is parallel to the central antenna (0 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.20: The Si PIT 1.2 THz negative array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right. 63

We observed important features in the oﬀ resonant cases where the THz polarization was aligned with the central antenna (0 degrees) in the negative arrays and also in the positive arrays when the THz polarization was orthogonal to the central antenna (90 degrees). In this geometry we observe the resonance excitation in the SRRs. The positive arrays only exhibited narrowband absorption from the coupling of adjacent SRRs. [103,104] In the negative arrays, however, we observed coupling between the bright mode in the adjacent SRRs and the darkmode in the central antenna. This led to an asymmetric PIT resonance which stems from the unit cell being asymmetric, meaning a second dipole antenna would have to be placed between adjacent SRRs to elicit the PIT. 64

5.5.3 Oﬀ Resonant Cases in GaAs

We observe the SRR peak (dip) in the negative (positive) structures. We also observe the PIT absorption in the negative structures due to coupling between the bright mode in the SRRs and the dark mode in the antenna.

Figure 5.21: The GaAs PIT 0.6 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.22: The GaAs PIT 0.6 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 65

Figure 5.23: The GaAs PIT 0.9 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.24: The GaAs PIT 0.9 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 66

Figure 5.25: The GaAs PIT 1.2 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.26: The GaAs PIT 1.2 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 67

5.5.4 Oﬀ Resonant Cases in Si

Figure 5.27: The Si PIT 0.6 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.28: The Si PIT 0.6 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 68

Figure 5.29: The Si PIT 0.9 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.30: The Si PIT 0.9 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 69

Figure 5.31: The Si PIT 1.2 THz positive array when the THz polarization is perpendicular to the central antenna (90 degree) with an incident THz ﬁeld of 300 kV/cm. The waveform is on the left and relative power spectrum is on the right.

Figure 5.32: The Si PIT 1.2 THz negative array when the THz polarization is parallel to the central antenna (90 degree) with an incident THz ﬁeld of 290 kV/cm. The waveform is on the left and relative power spectrum is on the right. 70

We found the resonance of the PIT to be slightly off of their intended values. The effective index for the PIT negative structures based on the length of the dipole antenna, neff,GaAs ≈

2.60 and neﬀ,GaAs ≈ 2.40, is higher than the background index nbg, given by Eq. 5.22.[105]

In this equation ns is the refractive index of the substrate. It is also noteworthy that the effective index changes as a function of frequency. This possibly stems from neglecting the δ term in Eq. 5.6. While this change is intuitive, it decreases with frequency and the effective index differs from the substrate index by approximately 1.

r 1 + n2 n = s (5.22) bg 2

f0 fres,N90,GaAs neﬀ,GaAs fres,N90,Si neﬀ,Si 0.6 THz 0.58 THz 2.69 0.62 THz 2.52

0.9 THz 0.90 THz 2.60 0.98 THz 2.39

1.2 THz 1.23 THz 2.54 1.33 THz 2.35

5.6 THz Field Eﬀects of GaAs and Si PIT Structures

Previous studies have shown that high THz fields (> 100 kV/cm) have the ability to drive carriers into high momentum states through various processes, such as: intervalley scattering, [106, 107] coherent ballistic transport, [108, 109] and effective mass anisotropy. [110] In our study we observed little change in the transmission in both GaAs and Si PIT samples. This contrasts previous work where interband excitations due to impact ionization have been observed. [111–114] The PIT sample lacks the field enhancement to elicit this effect. 71

Figure 5.33: The GaAs PIT 0.9 THz negative array at 90 degrees. The incident THz ﬁeld strength was modulated.

Figure 5.34: The Si PIT 0.9 THz negative array at 90 degrees. The incident THz ﬁeld strength was modulated.

5.7 Optical Excitation in the Wake of a PIT Resonance

During this study we employed a secondary optical line which we used to perform THz TDS in the presence of an optical pump. It is important to note that most of the carriers come from the optical pump excitation; the concentration of free carriers in intrinsic Si and GaAs are 1.5 × 1010 cm−3 and 1.8 × 106 cm−3 respectively. The optical pump power was modulated using a neutral density ﬁlter. For these experiments we calculated the carrier concentration since it is a much more useful quantity compared to the optical excitation pulse energy by itself.

U (1 − R) N = (5.23) ~ωAoptδ 72

Figure 5.35: Cartoon of the optical pump setup. A small hole is drilled through a focusing parabolic mirror through which the optical pump passes and overlaps the THz focus.

In this formula N is the carrier density in cm−3, U is the pulse energy, 1 − R is fraction of the pulse not reﬂected (we assume all light not reﬂected is absorbed since the ratio of the

sample thickness to penetration depth is a minimum of ≈ 222), ~ω is the average energy

per photon, Aopt is the area of the optical excitation, and δ is the penetration depth at 800 nm (GaAs δ ≈ 0.76 µm [115], Si δ ≈ 13.5 µm [116]). It is also important to note that the recombination time for the electrons in both Si and GaAs (on the order of ns) and the time scales in this experiment diﬀer by three orders of magnitude, therefore recombination can be neglected.

5.7.1 GaAs

The GaAs sample exhibited an extraordinary response to optical excitations. The modulation caused by the optical excitation in GaAs arises the increase in conductivity of the sample from the photoexcited carriers. These excited carriers have high mobility due to their

∗ low effective mass (m ≈ 0.07me), allowing them to damp the resonance via carrier-carrier scattering. This effect can be seen in the figure below (Fig. 5.36). 73

Figure 5.36: The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the THz excitation. The legend shows the carrier concentration.

At high excitation carrier concentrations (Fig. 5.37), there is limit to the damping and oscillation is observed. This oscillation appears to have an onset of N ≈ 8 × 1017. A possible mechanism for the oscillation is plasmon-phonon coupling. Specifically, the plasmon excited by the incident optical pulse couples to the both the transverse optical (TO) and longitudinal optical (LO) phonon. [117] Kuznetsov, et. al. [118] showed that when a DC electric field (100 kV cm ) is applied and an ultrafast pulse (50 fs) excites carriers in GaAs, the induced plasmon couples to the optical phonon modes and an oscillation is observed. The increase in carrier concentration causes the plasmon frequency to approach the frequency of the TO and LO phonons. Near N ≈ 8 × 1017 the plasmon frequency exceeds the TO frequency and increases damping. Another mechanism which increases damping is a decrease in dephasing time of the the plasmon-phonon oscillation with increasing carrier concentration. [119, 120] The dephasing occurs from phonon-phonon interactions due to the crystal anharmonicity. [121] Increasing carrier concentrations also cause the LO phonon to decay more quickly, specifically the LO phonon decay time scales inversely with carrier concentration. [122] 74

Figure 5.37: The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration.

5.7.2 Si

The eﬀects in Si pale in comparison to that of GaAs not only because of the bandgap of Si being indirect, requiring phonons to mediate the interaction with the optical pump, and the carriers having high eﬀective mass. In spite of this, Si exhibited a decrease in transmission with increasing optical pump power. This is caused by the increase in carrier-carrier scattering due to the increase in charge density [123]. Increasing the carrier concentration from 1015 to 1017 reduces the scattering time from 200 fs to 100 fs. This in turn increases the imaginary portion of the susceptibility, damping the PIT oscillation.

Figure 5.38: The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration. 75

The low carrier concentration excitation show the damping of the PIT oscillation at long time scales. With increasing carrier concentration the damping also increased. A noticeable phase shift in the PIT oscillation occurs as well. There is a slight shift in the frequency peaks for the lowest carrier concentration.

Figure 5.39: The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. The legend shows the carrier concentration.

We observed an increase in damping with an increase in carrier concentration at high carrier concentrations as well. This is mostly due to carrier-carrier scattering. In order to have the possibility of observing the plasmon-phonon coupling we saw in GaAs, we would have to increase our pump power by a factor of 50. The phase shift is also present.

5.8 THz Control of PIT Resonance

When THz ﬁeld dependent measurements were taken on the GaAs sample at low optical excitations, we observed a THz induced transparency. The induced transparency is caused by the intervalley scattering of the excited carriers, which appears to have an onset

kV around 100 cm . Intervalley scattering is considered the dominant mechanism for THz induced transparency, where carriers from the conduction band Γ valley are driven to either the L valley or the X valley where they have low mobility and cannot damp the oscillation. The low mobility stems from the excited carriers having high eﬀective masses in these side valleys. [106] 76

The induced transparency is most pronounced in the GaAs sample (Fig. 5.40) where the PIT oscillation at high ﬁelds began to approach the level of transmission without the presence

∗ me of the optical pump. The low eﬀective mass (m ≈ 15 ) of the excited carriers near the conduction band minimum (Γ valley) allows the THz PIT oscillation to accelerate the carriers into high momentum states and drive them into the adjacent side valleys, primarily the L valley. The velocity required for these high momentum states occurs on the subpicosecond timescale (approximately 150 fs)[87], faster than the scattering time (approximately 190 fs). [124] The mobility in both side band is much lower than the zone center due to the

high effective mass. The effective masses are 0.22me and 0.58me in the L and X valleys, respectively. [125] It is also worth noting that the there is a significant amount of band bending at the Γ valley minimum. The effective mass of the excited carriers increases by approximately 15% when the carrier concentration exceeds 1018 [117], however this does not play a significant role. There is a noticeable phase shift in the waveform at low THz fields that disappears as the THz fields are increased.

Figure 5.40: The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. Increasing the incident THz ﬁeld greatly modulates the transmission.

Transmission increases with increasing THz fields in the Si PIT as well. However, the effects are not as great as the GaAs PIT due to the effective mass of the carriers being

∗ me much higher (m ≈ 5 ) at the bottom of the conduction band (X valley) and the change in momentum required to elicit intervalley scattering (into the L valley) is large as well. [126] It is important to note that Si lacks a minimum at the Γ point. The decrease in scattering time 77 due to the increasing carrier concentration is also a contributing factor. What little increase

kV in transmission is observed has an onset above 150 cm . We do see the same disappearance of the phase shift with increasing THz ﬁelds.

Figure 5.41: The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation at 0.667 ps. Increasing the incident THz ﬁeld slightly modulates the transmission.

5.9 Pulse Shaping of PIT Waveforms

The fast decay of the PIT resonance in GaAs allowed for pulse shaping. By changing the delay time between the incident THz and optical pulses we can arbitrarily truncate the resonance with great precision. After the application of the optical pulse, the PIT resonance will damp after approximately 1.5 ps. As stated before, a possible contributing factor to the damping may be the dephasing of the plasmon-phonon interaction. Regardless, the fast damping allows us to control the number of cycles in our emitted pulse, thereby allowing us to change our oscillation from a narrowband resonance to broadband with relative ease. 78

Figure 5.42: The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation after the PIT excitation at the time listed in the legend. By delaying the optical pulse we were able to shape the transmitted THz waveform.

The decay of the oscillation of in the Si PIT is gradual and takes approximately 3 ps to damp. This causes the waveform to have a longer tail than GaAs, but still allows for pulse shaping. The phase shift we saw in the previous sections is still present. Like the GaAs PIT, we were able to change the narrowband resonance into broadband.

Figure 5.43: The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation after the PIT excitation at the time listed in the legend. By delaying the optical pulse we were able to shape the transmitted THz waveform.

5.10 THz pump-Optical Pump Experiments

The GaAs PIT exhibited an increase in transmission with increasing THz fields despite the presence of a strong optical pump. This is possibly due to intervalley scattering. It is 79 note worthy that the increase in transmission occurs at the oscillation peak nearest to the optical excitation. There is a notable phase shift at low THz fields that disappears with the increase in THz fields.

Figure 5.44: The GaAs PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the PIT excitation. The high optical pump power damps the PIT oscillation.

At high carrier concentrations, Si PIT has nearly the same increase in transmission as the GaAs PIT, most likely caused by intervalley scattering. There is also a noticeable phase shift in the oscillation with increasing THz ﬁelds towards the PIT resonance. The spectrum shows the PIT resonance beginning to re-form.

Figure 5.45: The Si PIT 0.9 THz negative array at 90 degrees with an optical excitation 0.667 ps after the PIT excitation. The high optical pump power damps the PIT oscillation. 80

5.11 Summary

We observed the subwavelength dynamics and modulation in the waveforms of PIT structures patterned on semiconductors in the presence of an optical pump. The primary cause of damping in the PIT oscillation is due to the increased conductivity in GaAs and Si, which increase damping with increasing carrier concentration. We observed an increase in transmission with increasing THz ﬁelds, primarily at low carrier concentrations in GaAs. This induced transparency can be attributed to intervalley scattering and demonstrates THz control of the PIT resonance. Using the optical pump to truncate the waveforms, we were able to perform precision pulse shaping in both materials. We also observed that THz pump- optical pump experiments elicit intervalley scattering with increasing THz ﬁelds. This line of experimentation demonstrates the utility of PIT structures for a broad range of purposes. 81

6 Terahertz Field Induced Metal-Insulator Transition in Vanadium

Dioxide

6.1 Introduction

Vanadium dioxide is a superb Mott insulator. It has been studied extensively for the last 50 years. [127–129] Its phenomenally low insulator-metal transition (IMT) temperature

(Tc ≈ 343K [127,130,131]) makes it an ideal material for an optical switch [132] in photonic integrated circuits. [133] Many experiments have been performed to characterize the carrier dynamics of this material and probe the phase transition and it has been debated whether the phase transition is due to electronic correlations or a lattice distortion. Recently it has been shown to involve both of these simultaneously. [134] Other studies have begun to explore the nature of the transition, [135,136] but the underlying microscopic mechanism is not yet fully understood. The transition can be triggered on time scales much faster than the thermalization time in the non-equilibrium regime. [137–141]

We investigated the phase transition of VO2 using nanoslot antennas. [142] Nanoslot antennas are a relatively simple plasmonic device which are easy to design and fabricate.

They have been used in previous studies of VO2 and provide a platform if active photonic devices which exploit field enhancement and subwavelength confinement of light. [143–147] The field enhancement has the ability to elicit nonlinear optical effects and give rise to effective dielectric constant modulation. [148,149]

6.2 Mott Insulators

The conventional band theory of solids centers around several assumptions. It assumes the molecular orbitals of the material can be written as a linear combination of atomic 82

orbitals. A tight-binding approximation is made which means the electron is primarily interacting with its ion core and has a small probability of hopping to its nearest neighboring atom. Lastly, it is assumed that the electrons are non-interacting, independent particles and the repulsion between electrons in the material can be ignored. Mott insulators closely follow this theory, save for the last assumption; their electrons can not be considered independent and the coulomb repulsion between the electrons must be accounted for.

In Mott insulators, speciﬁcally VO2, there are two characteristic ways to elicit the IMT. One method is via the structural phase transition where the material begins to conduct due to the change in potentials from the lattice distortion. The second method is characterized by the change in electronic correlations.

Figure 6.1: A. Orbital diagram for Vanadium when T is below Tc (unstrained) and above Tc (strained). B. Cartoon of the dimerization of valence electrons in neighboring Vanadium atoms.

In the ﬁrst case, speciﬁcally, the material will be conducting if the lattice spacing is small enough to screen of the Coulomb attraction from the nuclei. In VO2 this would be the displacement of the Vanadium in the lattice which leads to the overlap of the valence

∗ band, the Vanadium’s 3dx2−y2 , and π bonding orbitals, consisting of the Oxygen’s 2p orbital with the Vanadium’s 3dyz and 3dzx orbitals. [135, 150] The crystal structure changes from monoclinic in the insulating phase to tetragonal (rutile) in the conducting phase. [151, 152] It is also important to note that the adjacent Vanadium atom’s valence electron prefers to dimerize when the lattice is below the transition temperature (unstrained). [153, 154] This applies to an equilibrium phase transition. 83

The second category is mathematically complicated. The model, most importantly, applies to non-equilibrium conditions. It expounds upon conventional band theory; the interaction of electrons in the material must be taken into account. In order to account for this we must use the Hubbard model Hamiltonian. [155]

X † † X † X H = − tij ciσcjσ + ciσcjσ + Ejcjσcjσ + U ni↑ni↓ (6.1) hiji,σ jσ i

The ﬁrst sum is the hopping probability amongst the nearest neighbors and consists of

† two pieces. First, it contains creation (ciσ) and annihilation (ciσ) operators for atom i with an electron with spin σ. It also includes the transfer integral (orbital overlap tij) deﬁned in Eq. 6.2 (below). The second sum is the on-site energy between the electron at its ion core. The ﬁnal sum is the energy from the coulomb interaction between electrons in a given atom, where U is the average intra-atomic energy (Eq. 6.3) for electrons occupying a given state in a given atom.

Z ∗ 3 tij = ψi ψjd x (6.2)

ke2 U = (6.3) r12

In Mott insulators, the U term dominates. It makes it more favorable for an atom to have a singly occupied state. An electron trying to ”hop” to an adjacent atom would need to be hopping into an unoccupied state otherwise it would have to overcome the coulomb repulsion from another electron already occupying that state. Now it is important to note how the inter-atomic distance d also plays a role. With increasing d the lattice should be insulating, this decreases the hopping integral. Mott postulated that there should be a critical distance d0 where the transition to a metal should occur. This arises because the hopping probability increases with decreasing distance and makes it energetically favorable for the electron to hop to an adjacent atom. There should be an associated activation energy (which will appear 84

in Eq. 6.29) if the spacing is above the critical distance in order to generate carriers. [156]

2EA = I − E (6.4)

In this equation I is the ionization energy and E is the electron affinity of the atom. As the distance decreases this activation energy should decrease as the electron affinity increases due to the existence of the electron-electron interaction term. However, as continuous as this may appear, the transition should be of a discontinuous, first-order nature. This arises from the Mott’s theory which is based on a cubic lattice with a single electron. This implies that there should be an electron-hole bound state in the atom with an associated mutual potential − −e2 , where κ is the background dielectric constant, and binding energy which κr12 must be overcome in order to conduct. This pair formation will occur unless the other electrons in the lattice can screen the electron-hole interaction. Which leads to the following potential energy which utilizes the constant q from the Thomas-Fermi approximation:

−e2 −e2 → exp (−qr) (6.5) r r 4me2 3n 1/3 q2 = 0 (6.6) ~2 π

In Eq. 6.6 n0 is the number of carriers. Using the assumption that no bound states exist in the above potential, there is a ﬁnal ramiﬁcation of this model, the Mott Criterion. [156–159]

∗ This states that as the eﬀective Bohr radius, aH , or the carrier concentration increases, the material will begin to conduct.

∗ 1/3 aH n ≤ 0.25 (6.7)

This ﬁnal equation is of particular interest because it implies that the phase transition can be induced easily in two ways. The transition can be induced by injecting carriers, thereby 85 increasing n. This has been shown experimentally [160] and theoretically. [161] Another way to induce the phase transition is dissociating the electrons from the ion cores.

6.3 Sample Structure

Two types of samples were tested in this study: bare VO2 and nanoslot VO2. Both types of samples have 100 nm thick VO2 ﬁlms and were fabricated by Professor Dai-Sik Kim’s group at Seoul National University. Each sample was created on a 300 µm c-plane sapphire substrate using reactive RF-magnetron sputtering. It is important to note that the VO2 samples are polycrystalline with typical grain sizes around 0.5 µm. The nanoslot samples were created using electron beam lithography and have rectangular slots measuring 200 nm by 60 µm with a periodicity of 70 µm in the length direction and 60 µm in the width direction. This periodicity is done to minimize coupling eﬀects between nearest neighbor antennas. [162] These antennas are resonant at 0.9 THz at room temperature.

Figure 6.2: Cartoon of the nanoslot antennas.

The primary focus of this research was on the nanoslot antenna arrays. The incident THz pule excites the plasmonic resonance by coupling to the periodic structure. The nanoslot antennas on the sample are used to conﬁne charge on the long sides from the incident THz pulse. This excitation causes a surface plasmon resonance due to the periodicity of the 86

antenna array structure. The charge confinement along the edges of the antenna enhances the electric field across the gap of the nanoslot. This can be modeled using diffraction theory as two Sommerfeld half planes being brought together [163] which yields a surface charge density in the following form:

r 0 |E0| λ −iωt − πi σ (x, t) = √ e e ( 4 ) (6.8) 2π x

It is extremely important to find the field enhancement due to the nanoslot array, where the ratio of the peak electric field in the near zone (r λ) and the peak THz electric field gives the field enhancement. Owing to the fact the bolometer only detects the THz radiation in the far field (r λ), there has to be a way to relate the near field to the observed far field in a simple way. This is accomplished by using antenna coverage ratio (β = AAntenna = .0029), AUnit cell

the relative power transmission in the far ﬁeld (R), and the pulse duration ratio (τr, varies from 1.1 and 2.5, Fig. 6.9).

|E | 1 rR α = near = (6.9) |ET Hz| β τr

This formula yields a 20 to 30-fold field enhancement when comparing the incident THz field and near field from the nanoslots. It is also important to note that the near field decays exponentially in the direction of the optical axis (z-axis). This indicates that the

−z/w enhanced ﬁeld is primarily interacting with 100 nm VO2 since Enear ≈ Enear (0) e and

deff ≈ w = 200 nm.[148] The sapphire substrate did not exhibit any THz ﬁeld dependence and is thus ignored.

6.4 Experimental Considerations

The laser used for THz generation is a Ti:Sapphire femtosecond laser system with a

Gaussian proﬁle pulse, 1µJ pulse energy, central wavelength λ0 = 800 nm, bandwidth δλ = 87

10 nm, and 1 kHz repetition rate. The THz pulse (central frequency f0 = 1 THz, bandwidth

δf = 1 THz) is generated using tilted pulse front geometry in LiNbO3. We performed three separate types of experiments in this investigation: power transmission, Michelson interferometry, and THz TDS. We used our L-He cooled Si:Bolometer to acquire the spectrally integrated THz transmission for Michelson interferometry and hysteresis curves. It is important to note that the hysteresis curves had to have the temperature manually cycled while acquiring data. Keeping the change in temperature as a function of time constant proved to be diﬃcult due to VO2 having a highly variable heat capacity and thermal conductivity near the IMT transition temperature. [164] Michelson interferometry and THz TDS allowed us to extract the frequency response of the material. However, only the THz TDS data allows us to view the temporal dynamics of the IMT. The THz TDS waveforms were acquired using electro-optic sampling in a 1 mm ZnTe crystal.

6.5 Terahertz Field Induced Absorption

Figure 6.3: Plot of the THz ﬁeld dependence for the nanoslot and bare VO2 (inset)

We observed a strong nonlinear response to THz ﬁeld in the nanoslot VO2 sample. 88

We tested both the bare and the nanoslot VO2, starting from room temperature, at various temperatures leading up to and into the phase transition. When the temperature of the sample was fixed and the THz field was modulated, we saw a reduction in transmission in the nanoslot sample but little response from the bare sample. As we can see from the inset, for the bare sample at low temperatures (45o C) there is no discernible change and only a slight decrease in the transmission near the transition temperature (71o C). The nanoslots, however, exhibited a large response to increasing THz fields. For example, at 65o C the transmission drops by 75% as the THz field was raised from 100 kV/cm to 800 kV/cm. The reduction in transmission through the nanoslot sample corresponds to an increase in conductivity. At increasing THz fields, the nanoslot VO2 film is driven into the metallic state. This is possibly caused by the THz fields disturbing and breaking down the electronic correlations, transiently eliciting the IMT by collapsing the bandgap between the

∗ π and 3dx2−y2 orbitals. This implies that the IMT is initiated through the interaction of the correlated electrons with the THz ﬁelds.

6.6 Hysteresis and Activation Energy

The VO2 samples exhibited hysteresis when the THz field was fixed and the sample temperature was increased above the IMT and then decreased back to room temperature. This was performed for several THz field strengths on both the bare and the nanoslot samples. The inset plot shows how the bare sample responded. At fields below 350 kV/cm there is no change in the response. However, when the incident field is raised to 680 kV/cm, the onset of the hysteresis during the heating phase slightly lowers in temperature. This agrees well with the field dependent data from the previous section. The nanoslot samples, on the other hand, exhibited a large deformation to the hysteresis. The nanoslot antennas are highly sensitive to any changes in the VO2. Not only does the transition temperature lower significantly, the entire hysteresis curve shifts towards lower temperatures. There is noticeable flattening in 89

Figure 6.4: Plot of the hysteresis curves for the nanoslot and bare VO2 (inset) the hysteresis as well.

6.6.1 Resistivity Derivation

We must derive a few things in order to further describe the material. We ﬁrst need to utilize the conductivity in terms of the transmission (from Sec. 2.2). Using sheet conductivity in terms of the relative power transmission, we can ﬁnd the resistivity which is a much more fundamental property of the material. This will allow us to relate the activation energy discussed in Eq. 6.4 to our data. We will begin by counting the number of carriers in the material. In order to do this we must approximate a Fermi-Dirac distribution for small temperatures (Eq. 6.11) and calculate the density of states per unit volume (Eq. 6.14, for a cubic lattice).

1 f F −D (E) = g (E) (6.10) E−EF 1 + e kT

EF −E ≈ e kT g (E) (6.11) 90

2ρ (E) g (E) = DOS (6.12) V 2 dν = (6.13) V dE 3 2 1 2m 1 = E 2 (6.14) 2π2 ~2

Now the conductivity of the VO2 thin-ﬁlm can be written in the following manner:

σ = e (neµe + nhµh) (6.15)

Using Eq. 6.11 and Eq. 6.14 we ﬁnd the number of electrons as a function of temperature, the Fermi energy, and the bandgap by integrating over all the bands lying above the Fermi energy.

Z ∞ F −D ne = fe dE (6.16) EG 3 2 Z ∞ 1 2me EF 1 −E kT 2 kT = 2 2 e (E − EG) e dE (6.17) 2π ~ EG 3 2 2πmekT EF −EG = 2 e kT (6.18) h2

We will recycle the same two equations to count the holes. However, we will move our zero of integration to the Fermi energy and integrate toward the ground state. 91

Z 0 F −D nh = fh dE (6.19) −∞ Z 0 F −D fe = 1 − ghdE (6.20) −∞ ge 0 Z E−EF = e kT ghdE (6.21) −∞ 3 2 Z 0 1 2mh 1 E−EF 2 kT = 2 2 (−E) e dE (6.22) 2π ~ −∞ 3 2 2πmhkT −EF = 2 e kT (6.23) h2

We assume we have an intrinsic semiconductor, ne = nh, which allows us to ﬁnd the Fermi energy in terms of band gap, eﬀective masses, and temperature by equating Eq 6.18 and Eq. 6.23.

EG 3 mh EF = + kT ln (6.24) 2 4 me

Lastly this is used to ﬁnd the number of carriers n = ne = nh for use in Eq. 6.15.

3 2 2πkT 3 −EG n = 2 (m m ) 4 e 2kT (6.25) h2 h e

This allows us to rewrite the conductivity in the following manner:

3 2 2πkT 3 −EG σ = 2e (µ + µ ) (m m ) 4 e 2kT (6.26) e h h2 h e

It is important to note that the mobility and eﬀective mass have small temperature dependence. Also, a majority of the temperature dependence is contained in the exponential term and we therefore can make the approximation that the term preceding the exponential are constant for our intents and purposes. 92

−EG σ ≈ σ0e 2kT (6.27)

We ﬁnd the temperature dependence of the resistivity in terms of the activation energy by taking the inverse of Eq. 6.27, where half the activation energy is equal to the bandgap energy (EG/2 = EA).

d ρ = (6.28) σs

EA = ρ0e kT (6.29)

6.6.2 Resistivity and Activation Energy

Before starting any further analysis, we must relate the nanoslot VO2 transmission to the bare VO2 transmission. We will do so by making the following argument: the sheet conductivity for the nanoslots corresponds to the sheet conductivity of the bare sample for a given temperature and THz field. This means that if we take the highest THz field for the bare sample and normalize the sheet conductivity and do the same for the lowest THz field from the nanoslot data, we can find the conductivity as a function of transmission for each sample. This can be done because the limits of the conductivity for the nanoslot sample should correspond to the same values as the bare sample. 93

Figure 6.5: Plot of the normalized sheet conductivity comparing the bare and nanoslot VO2 data.

Using this correlation for normalized sheet conductivity as a function of transmission, we can use the phenomenological model in Eq. 6.30 to ﬁnd the sheet conductivity for each nanoslot hysteresis curve. This will allow us to plot the conductivity for any of the nanoslot samples.

x = σsZ0 (6.30)

logR = ae−bx + cx + d (6.31)

After using a nonlinear curve ﬁt in Matlab, the following constants were found for our ﬁt. 94

a = 1.9980 (6.32)

b = 1.8594 (6.33)

c = −0.4229 (6.34)

d = −1.9941 (6.35)

Figure 6.6: Plot of the normalized sheet conductivity for the ﬁt comparing the bare and nanoslot VO2 data.

Taking the fit curve, we are now able to plot the conductivity and therefore resistivity as a function of temperature for each hysteresis curve. These data sets allow us to find the activation energy for a given THz field by fitting Eq. 6.29 to the data. 95

Figure 6.7: Plot of the resistivity as a function of temperature. The activation energy ﬁts for temperatures between 35o and 55o C are over-plotted.

ET Hz (kV/cm) EA (eV) 90 0.60

300 0.22

560 0.17

790 0.16

As we can see, the resistivity flattens with increasing THz fields. This is possibly caused by the THz fields driving the carriers far from equilibrium and lowering the activation energy. 96

6.6.3 Hysteresis Width

We can see from the data that the hysteresis width decreases as we increase the incident THz ﬁeld. This is caused by a softening of the shear strain. During the heating phase there is a structural change (monoclinic to rutile). However, there is also a change in the volume and shear strain during the transition. [165] The diﬀerence in the shearing strains is the

origin of the hysteresis in VO2.[166,167] Speciﬁcally, we can write the the shear strain when increasing the temperature in the following manner:

1 1 U = Eε2 + ηGγ2 (6.36) h 2 2

where E is Young’s modulus, ε is the extensional strain, η is the domain shape parameter, G is the shear modulus, and γ is the shearing strain. The hysteresis arises from the lack of shear energy in the cooling cycle evidenced by Fan et. al. [168] and can be written in the following form:

1 U = Eε2 (6.37) c 2

Now taking the diﬀerence of these yields the change in free energy, which is equal to the strain energy diﬀerence.

(Th − Tc) ∆S = ∆U (6.38) 1 = ηGγ2 (6.39) 2

Solving the above equation for the change in temperature, we arrive at the hysteresis width.

ηGγ2 ∆T = T − T = (6.40) h c 2∆S 97

Our strong THz ﬁelds weaken the molecular bonds, thereby reducing the shear modulus G, giving rise to the narrowing of the hysteresis loop.

Figure 6.8: Plot of the hysteresis width as a function of incident electric ﬁeld. The inset plot show the transition temperature for the nanoslot sample for increasing and decreasing temperature. The black lines are linear ﬁts for the data.

The observed decrease in hysteresis width comes primarily from the increasing temperature portion of the hysteresis curve. This half decreases by four degrees while the decreasing side only decreases by approximately one degree. 98

6.7 Transient Phase Transition

As the VO2 was heated, we observed the dynamics of the THz field induced IMT in the nanoslot sample. This was done at low THz fields (150 kV/cm) to observe the structural IMT in the nanoslot sample. When analyzing the temperature induced IMT it is easy to see that the sample transitioning from an insulator at 45o C, to barely insulating at 65o C and finally a half-metal at 67o C (Fig. 6.9). We argue that A damping of the resonance of the nanoslots was observed as the temperature increases in the inset plot. Specifically the resonance decreases when the sample reaches 65o C due to the increase in the conductivity and disappears when the film becomes half-metal at 67o C due to the film conducting.

Figure 6.9: Waveforms for an incident THz ﬁeld of 150 kV/cm at 45o C, 65o C, and 67o C. 99

We can see in Fig. 6.10 that the waveforms have roughly the same transmission for the first cycle, but deviate in the trailing portions of the waveforms. The THz field induced IMT is evidenced by the reduction in the transmission as the THz field was increased. This means the THz field induced IMT is nearly instantaneous and occurs on the picosecond time scale, within less than a half cycle of the THz radiation. The speed of this transition indicates it is not a thermal effect due to the thermalization time being on the order of microseconds. This implies that strong THz fields transiently modulate the electron distribution which induces an IMT on the subpicosecond timescale, without exciting the structural IMT.

Figure 6.10: Plot of the waveforms for incident THz ﬁelds of 150 kV/cm, 300 kV/cm, 630 kV/cm, and 850 kV/cm at 45o (a) and at 65o C (b). The power transmission spectra for each waveform is inset. 100

For clarification, we can compare the nanoslot waveform at 67o C with an incident THz field of 150 kV/cm matches with the nanoslot waveform at 65o C with an incident THz field of 850 kV/cm (Fig. 6.11). This illustrates how the two waveforms are well matched and giving creedence to the THz induced IMT. The transition in the nanoslot sample with an incident field of 150 kV/cm is primarily temperature driven while the 850 kV/cm will have a strong field driven component.

Figure 6.11: Plot of the waveforms for 150 kV/cm at 67o C (temperature driven transition) and 850 kV/cm at 65o C (ﬁeld driven transition) to help illustrate the lowering of the transition temperature. The yellow shaded area is the waveform for 150 kV/cm at 65o C.

Finding and exposing the underlying mechanism for the transient IMT at the microscopic level is both important and challenging. A possible mechanism for the transient IMT could be the Poole-Frenkel eﬀect, however the threshold behavior of the activation energy indicate that more complicated mechanisms may be involved. [169,170] 101

6.8 Summary

This study demonstrates the ability of strong THz pulses to transiently induce the IMT transition in VO2 thin films. The nanoslot antennas allowed the incident THz field to be enhanced by a factor of 20. These high THz fields drives electron distributions far from equilibrium thereby inducing the IMT. The strong THz fields also lowered the transition temperature for both the heating and cooling cycles and reduced the hysteresis width. The decrease in the width of the hysteresis can be attributed to the THz fields weakening the molecular bonds, and therefore the shear modulus along with the hysteresis width. Lastly, we observed the THz transiently triggering the IMT on the sub-cycle timescale in the THz TDS waveforms. This shows the field induced IMT is a non-thermal process and the structural phase transition has little effect on the conductivity at the onset. These results show the utility and ability of nanostructures on VO2 as ultrafast photonic and electronic devices. 102

7 Sub-Diﬀraction Limit Nonlinear Imaging with Plasmonic Devices

7.1 Introduction

Sub-diffraction limit, nonlinear imaging with a plasmonic bullseye structure has been demonstrated. [70] However, the group which created it used different fabrication techniques. The corrugations were directly micro-machined and the metal (Au) used was deposited via sputtering. The associated cost with micro-machining and choice of metal make this method cost prohibitive. Our method was intended to be more cost efficient and produce a greater quantity using photo-resist (specifically SU-8) and a different metal (Al). The bullseye structure we attempted to fabricate were intended to be used in a THz microscope. Before diving too deep into this chapter there are several things to note. This project did not produce viable structures during its 18 months. It did, however, produce a working SU-8 fabrication recipe that is currently used by several departments (EECS, Physics) at OSU. This work saved tens, if not hundreds, of man hours. We will discuss the different attempts made to fabricate these structures, the ultimate reason for failure, and where future work can complete this project.

7.2 Background

The bullseye structure works in the following manner. It uses a grating to couple the SPP resonance to a subwavelength aperture. Specifically, the THz pulse moves through the dielectric, which makes up the structure (corrugations), and then couples to the metal, via the surface plasmon resonance. This occurs on the far side of the structure, inducing the surface plasmon polariton. Next, the SPP couples to the aperture, which allows the SPP to radiate. Since the aperture diameter is very small (d λ), the light travels only a 103 short distance before diverging due to diffraction. However, in that short distance the light can be used for near-field, sub-diffraction limit imaging. If a sample needs to be imaged, the structure can be placed at the beam focus and then the sample can be raster scanned, generating a high-resolution image. The shape of the aperture also factors in. A bowtie aperture, displayed on the right of Fig. 7.1, confines carriers at the tips and enhances the electric field in the near field when the incident polarization is in-line with the points on the bowtie.

Figure 7.1: Example bullseye structure.

f (THz) d (µm) a (µm) h (µm) g (µm) L(µm)

0.5 67 184 17 6 2647

1.0 33 92 9 3 1322

1.2 28 77 7 2.5 1102

These structures are especially useful in the THz because most material exhibit Drude- like behavior, namely graphene. This method of sub diffraction limit imaging would allow intra-grain conductivity to be extracted from the transmission (Eq. 2.45). We should also 104 note that the structure acts as a notch filter and the emitted light is extremely narrow-band, allowing for a researcher to make several structures with different resonance frequencies for probing a materials frequency response. For this reason we were intending to fabricate 9 structures with resonance from 0.5 THz to 2.4 THz. The above table (Fig. 7.2) is for reference on the size of the structure.

7.3 Fabrication

Lithography provides a cost effective method for the fabrication of micron scale devices. To do this, photoresist is used as the fabrication medium. The photoresist can be employed as a barrier medium or the desired structure itself. The two photoresists used in the cleanroom here on campus, S1813 and S1818, have thicknesses on the order of 1 − 2 µm, where the desired structures require thicknesses closer to 10 µm. This requirement led us to use SU- 8 photoresist. It was chosen due to its low cost and ability to produce high-aspect-ratio structures down to the sub-micron scale. [171–174] The structures we attempted to fabricate have two distinct, but equally important steps. The first step is fabricating the structure on the substrate. The second and final step is layering the Al and then removing the material for the aperture. Most of the work was done in the Oregon State University cleanroom located in Owen hall. When we began this project, we had to make two photo-lithography masks using the Heidleberg 66FS DWL Mask Writer. The masks are a negative image of the desired pattern. The photoresist we used, SU-8, is a UV activated polymer. Exposing an area to UV light will cause the SU-8 to crosslink, meaning UV causes the polymers within the photoresist to bond with each other, which then keeps the SU-8 from dissolving when the structure is placed in developer. [171] Initially SU-8 2100 series photoresist was used because it only requires a single layer of photoresist in order to make our structures. The 2100 series photoresist can be layered 105 between 550 µm down to 100 µm and we believed that we could tune the development time to only remove the top 10 µm, the desired height of the corrugations. The kinematic viscosity of this photoresist is 4 × 104 cSt (about a factor of four more viscous than honey, but with roughly the same stickiness), making the use of disposable pipettes insufficient for depositing the photoresist on the substrate. The pipettes could not effectively draw up the SU-8 and also created bubbles in the photoresist, causing uneven distribution during the spincoating process. A syringe was used to deposit SU-8 instead, alleviating the bubble issue and evening out the distribution. After several trials the 2100 series photoresist proved to be ill suited for our purposes due to difficulty in developing to the desired height of the features and the lack of uniformity of the surface due to uneven development. We switched to SU-8 5, which required layering the photoresist because its maximum thickness is 10 µm. This required changing the fabrication recipe because of over-development issues. The bottom layer can not be developed until the final development of the top layer, where the actual bullseye structure corrugations are made. Developing the bottom layer prior to spin-coating and exposing the top layer results in a lack of adhesion between the layers. Once this was learned, we proceeded onto layering the Al layer. We initially attempted to layer the Al in the cleanroom, however the facilities proved to be insufficient for layering the 1µm of Al on a three dimensional structure. The Polaron thermal evaporator can only layer at normal incidence, which did not coat the sides of the structure. In order to circumvent this, the Al deposition was done through Tektronix Inc in Beaverton, OR. Tektronix has an evaporator with a planetary inside which rotates the sample on a 45o angle while depositing the Al. This ensures an even layer over the structure, even the vertical portions. Lastly, the metal needs to be removed in the center for the apertures. A layer of S1818 photoresist is used as a barrier layer to cover the metal while apertures are removed in an acid etch. After attempting to remove the metal in the center for the apertures, we discovered that the metal had not completely adhered to the surface. This adhesion problem has also 106

Figure 7.2: Example of a 0.5 THz bullseye structure (left) and 1.0 THz bullseye structure (right). There are etching pits around the convex edges of the structures from the lack of S1818 adhesion and the resulting etching of the metal layer.

been observed at other universities1. The corners exhibited the poorest adhesion. The acid etch was a poor removal method for the apertures because it etched non-uniformly. Also, the S1818 layer also did not adhere to the corners of the structures causing them to be unintentionally etched. We will discuss how to remedy these issues in the following section.

7.4 Future Work

The metal adhesion issue has a known solution, electroplating. A small amount of metal is deposited on the surface of the SU-8. The sample is then placed in an acid bath with one electrode on the sample and the other on a mass of the metal chosen for deposition. When a large voltage bias is applied the metal starts depositing on the SU-8. The voltage bias will draw metal atoms into the SU-8 causing dendritic growth of the metal into the top layer of the photoresist. This serves to anchor the metal and will alleviate the adhesion issue. The acid over-etching the apertures and the lack of adhesion between the S1818 and the

1Paul McEuen’s group at Cornell University had a similar issue layering metal (Au) on SU-8. 107 metal layer will become the next foreseeable primary issue. A reactive ion etch system is being installed in the cleanroom and is now the most likely candidate to solve the etching issue. Another method utilizes a focused ion beam (FIB) to remove the apertures. This would alleviate the need for the S1818 barrier layer and present a signiﬁcant increase in control of the fabrication. However the top layer of SU-8 will adsorb some of the ions from the FIB bombardment. This could also prove to be a time consuming process as well, due to the volume of the apertures needing to be milled (on the order of µm3). 108

8 Conclusion

During my tenure in Dr. Yun-Shik Lee’s lab I have worked on many challenging experiments. We employed plasmonic induced transparency devices to perform nonlinear THz spectroscopy of semiconductors. We also demonstrated the THz-triggered IMT in VO2. An attempt to fabricate plasmonic devices for sub-diffraction-limit nonlinear imaging was performed as well. Our work using the PIT structure on Si and GaAs yielded several interesting results. These structures use the bright mode in a half-wave dipole antenna coupling to the dark mode in adjacent SRRs to exhibit a very narrow resonance which is extremely sensitive to local carrier dynamics. We employed an optical pump to modulate the PIT oscillation after it had been excited by an incident THz pulse. The optical response of the GaAs PIT sample was much greater than that of the Si PIT due to GaAs having a direct bandgap and lower effective mass carriers. We observed damping of both PIT oscillations with increasing optical pump power (carrier concentration). The excitation in GaAs damped more rapidly than Si with increasing carrier concentration due to the higher mobility of the excited carriers and the ensuing carrier-carrier scattering. At extremely high carrier concentrations plasmon- phonon coupling was observed in the GaAs sample. The damping observed in the Si PIT is primarily due to carrier-carrier scattering. We demonstrated THz control of the PIT resonance. This was done by increasing THz fields to cause the intervalley scattering of the excited carriers and inducing a THz transparency. We also demonstrated pulse shaping of the PIT waveforms by using the optical pump to truncate the oscillation. Lastly we observed a slight induced transparency of the PIT oscillation in the presence of strong optical pump pulses due to intervalley scattering. We demonstrated that strong THz pulses have the ability to transiently induce the IMT 109

transition in VO2 thin films. We employed nanoslot antennas to enhance the incident electric field and drive electron distributions far from equilibrium. We observed a lowering of the transition temperature during the heating and cooling cycles as well as a narrowing of the hysteresis width. The strong THz fields weaken the molecular bonds, reduce the shear modulus, and cause a narrowing of the hysteresis. We also demonstrated that the IMT is induced transiently using THz time domain spectroscopy to show that the transition is triggered on sub-cycle timescales. Lastly we presented our attempt at fabricating plasmonic devices for sub-diffraction- limit nonlinear imaging. This proved fruitful in that we created recipes for two types of SU-8 photoresists, SU-8 2100 and SU-8 5. We observed a lack of adhesion of the metal layer for the metal-dielectric interface, primarily around the convex edges of the structure. We experienced issues during the removal of metal for the apertures due to lack of adhesion between the S1818 barrier layer and the non-uniformity of the acid etch. We presented means to solve these issues and foreseeable future issues. 110

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APPENDIX 125

A PIT Fabrication Recipe

1. Cut or cleave wafer into 1in by 1in (25mm by 25mm) square substrates.

2. Clean substrates using acetone, isopropyl alcohol (IPA), then water.

3. Dry the substrate using compressed air or compressed nitrogen (N2).

4. Place the substrate on a 95oC hotplate to evaporate excess water.

5. Program spin-coater with the desired parameters, then place the substrate on the spin-coater.

6. Center the substrate then apply enough MCC Primer 80/20 to cover the entire substrate.

7. Run the spin-coat program.

8. Next apply enough S1818 photoresist to cover the entire substrate.

9. Verify there are no air bubbles in the photoresist. If there are, rinse with acetone and start over.

10. Run the spin-coat program.

11. Remove the substrate and perform the soft bake.

12. Align the substrate to the mask and expose under UV lamp.

13. Develop the sample for 30 seconds in 5:1 mixture of water to 351 Developer.

14. DIP the sample in water, DO NOT SPRAY.

15. DO NOT use compressed gas, allow the sample to air dry. 126

16. Deposit 1µm of Al.

17. Place in acetone and dissolve the photoresist.

18. Wait 5 minutes and then sonicate for 5 minutes.

19. Rinse with water.

Process MCC Primer 80/20 S1818 or S1813

Thickness NA 1µm

rpm rpm Spin-coat 1 3000rpm, 5000 s , 10s 3000rpm, 5000 s , 10s rpm rpm Spin-coat 2 4000rpm, 5000 s , 20s 4000rpm, 5000 s , 20s Soft Bake NA 2m @ 85oC

Exposure NA 10s

Development NA 30s

Rinse NA Water

Dry NA N2 or Air

Initially, the inverse of this process was used. Meaning, that metal was deposited first, then photoresist is added as a barrier layer for an acid etch. However, over-etching was an issue for both substrates. The GaAs samples had a second problem due to the reactivity of gallium, which led to a search of acid etches in the CRC Handbook of Metal Etchants [175]. We at first used Al due to its cost effectiveness and the material properties [176]. All the etchants for Ni, Cr, Al, Au, and Ag were found to either destroy the photoresist or etch the GaAs. 127

B Bullseye Fabrication Recipe

1. Cut substrate into 1in by 1in (25mm by 25mm) square substrates.

2. Clean substrates using acetone, isopropyl alcohol (IPA), then water.

3. Dry the substrate using compressed air or compressed nitrogen (N2).

4. Place the substrate on a 95oC hotplate to evaporate excess water.

5. Program spin-coater with the desired parameters, then place the substrate on the spin-coater.

6. Center the substrate then apply enough MCC Primer 80/20 to cover the entire substrate.

7. Run the spin-coat program.

8. Next apply enough SU-8 photoresist to cover the entire substrate using a syringe.

9. Verify there are no air bubbles in the photoresist. If there are, rinse with acetone and start over.

10. Run the spin-coat program.

11. Remove the substrate and perform the soft bake.

12. Blanket expose the SU-8 under the UV lamp.

13. Perform post-exposure bake.

14. Repeat 8-13 for second layer.

15. Align the substrate to the mask and expose under UV lamp. 128

16. Develop the sample for 3 minutes 20 seconds.

17. Rinse with IPA and dry with compressed N2.

Process SU-8 5 SU-8 2100

Thickness 15µm 130µm

rpm rpm Spin-coat 1 500rpm, 100 s , 10s 500rpm, 100 s , 10s rpm rpm Spin-coat 2 1000rpm, 300 s , 30s 2000rpm, 300 s , 30s Soft Bake 2m @ 65oC , 9m @ 95oC 15m @ 65oC, 40m @ 95oC

Exposure 9s 14s

Post-Exposure Bake 1m 30s @ 65oC , 2m @ 95oC 5m @ 65oC, 15m @ 95oC

Development 3m 20s 15m

Rinse IPA IPA

Dry N2 or Air N2 or Air