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
by
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 Media
6
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 .
3.1.1 Phase Matching . . . . . . .
3.1.2 Zinc Telluride . . . . . . . .
3.1.3 Lithium Niobate . . . . . . .
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20
20
22
23
3.2 Terahertz Detection . . . . . . . .
3.2.1 Bolometer . . . . . . . .
3.2.2 Pyroeletric Detectors . .
3.2.3 Michelson Interferometry
3.2.4 Electro-Optic Sampling .
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26
27
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29
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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 . . . . . . . . . . . . . . . . . . . . .
4.3.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 SPP Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
35
39
4.4 Excitation Methods . . . . . .
4.4.1 Otto Method . . . .
4.4.2 Kretschmann Method
4.4.3 Spatial Periodicity .
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40
40
41
41
4.5 Applications to Terahertz Science . . . . . . . . . . . . . . . . . . . . . . . . .
42
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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 . . . . . . . . . . . . . .
5.3.1 Fabrication of PIT Structure . . . . . . . . . . . . . . . . . . . . . . .
47
52
5.4 Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.5 Initial Testing and PIT Observation .
5.5.1 Resonant Cases in GaAs . .
5.5.2 Resonant Cases in Si . . . .
5.5.3 Off Resonant Cases in GaAs
5.5.4 Off Resonant Cases in Si . .
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56
57
60
64
67
5.6 THz Field Effects of GaAs and Si PIT Structures . . . . . . . . . . . . . . . .
70
5.7 Optical Excitation in the Wake of a PIT Resonance . . . . . . . . . . . . . . .
5.7.1 GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.2 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
72
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
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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 . . . . .
6.6.1 Resistivity Derivation . . . . . . .
6.6.2 Resistivity and Activation Energy
6.6.3 Hysteresis Width . . . . . . . . .
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88
89
92
96
6.7 Transient Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
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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
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
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. . . . . . . . . . . . . . . . . . . . .
3.1
3.4
44
LIST OF FIGURES (Continued)
Figure
Page
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
Here is an example unit cell of the PIT structure. Dimensions can be found
in Table 5.3.1 (below). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
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
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
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
5.2
5.6
5.7
5.8
5.9
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
Plot of the normalized sheet conductivity for the fit comparing the bare and
nanoslot VO2 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
Plot of the resistivity as a function of temperature. The activation energy fits
for temperatures between 35o and 55o C are over-plotted. . . . . . . . . . . .
95
6.6
6.7
LIST OF FIGURES (Continued)
Figure
6.8
6.9
Page
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
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 rectification), which means we have to look at the polarization of materials. We will commence
by writing Maxwell’s equations in matter. [25]
∇×E=−
∂B
∂t
∇ × H = Jf +
∂D
∂t
(2.1)
(2.2)
∇ · D = ρf
(2.3)
∇·B=0
(2.4)
For linear media, we can write the displacement field (D-field) and auxiliary field (Hfield) in the following manner:
D = 0 E + P = E
H=
1
1
B−M= B
µ0
µ
(2.5)
(2.6)
Substituting these into Eq. 2.1 and Eq. 2.2, taking the curl, we find the generalized
electromagnetic wave equations.
7
∂ 2E
∂P
∂
∇ × ∇ × E + 0 µ0 2 = −µ0
Jf +
+∇×M
∂t
∂t
∂t
∇ × ∇ × H + 0 µ0
∂ 2M
∂ 2H
∂P
−
µ
=
∇
×
J
+
∇
×
0 0
f
∂t2
∂t
∂t2
(2.7)
(2.8)
Using the fact that ∇ × ∇ × C = ∇ (∇ · C) − ∇2 C and inserting the rest of Maxwell’s
equations (Eq. 2.3 and Eq. 2.4), we arrive at the following:
1
∂
∂P
∂ 2E
Jf +
+∇×M
∇ E − 0 µ0 2 = ∇ρf + µ0
∂t
∂t
∂t
2
∇2 H − 0 µ0
∂P
∂ 2H
∂ 2M
=
−∇
×
J
−
∇
×
+
µ
f
0 0
∂t2
∂t
∂t2
(2.9)
(2.10)
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.
∇2 E = 0 µ0
∂E
∂ 2P
∂ 2E
+
µ
σ
+
µ
0
0
∂t2
∂t
∂t2
(2.11)
(1)
If we use P = 0 χe E = ( − 0 ) E we arrive at Eq. 2.12. This specific form will be
particularly useful when dealing with conductors.
∇2 E = µ0
∂ 2E
∂E
+
µ
σ
0
∂t2
∂t
(2.12)
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:
∇2 E = µ0
∂ 2E
n2 ∂ 2 E
=
∂t2
c2 ∂t2
(2.13)
q
√
2
= R is the index of
It should also be noted that µ0 = nc2 = v12 , where n =
0
q
refraction, c = 01µ0 is the speed of light in vacuum, and v is the speed of the wave. The
general solutions of this differential equation are linearly polarized, monochromatic planewaves with wave vector k and angular frequency ω.
E (x, t) = E0 ei(k·x−ωt)
H (x, t) = H0 ei(k·x−ωt)
(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 = ωµ0 H. Lastly, we use Eq. 2.12 with Eq. 2.14 to obtain the dispersion
relation.
k 2 = µ0 ω 2
(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:
k=
2πn
ω
=n
λ0
c
(2.16)
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.
k 2 ≈ iσµ0 ω
(2.17)
Another point of note is that k 2 is purely imaginary, this means that each of the constituent components of k have the same magnitude.
r
Re |k| = Im |k| =
σµ0 ω
2
(2.18)
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
δs =
2
σµ0 ω
(2.19)
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.
R = n2 =
iσ
ω0
(2.20)
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 |E0 |2 k̂
2
2
(2.21)
10
The magnitude of the Eq. 2.21 yields the radiation intensity, typically measured in
W/m2 .
1
I = |hSi| = v |E0 |2
2
(2.22)
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.
δs
c
1
δp =
=
= =
2
2κω
α
r
1
2σµ0 ω
(2.23)
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
wavelength (n2 d λ
).
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.
rij =
ni − nj
ni + nj
tij =
2ni
ni + nj
(2.24)
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.
t=
∞
X
t(n)
(2.25)
n=1
= t12 t23 eiφd + t12 r23 r21 t23 e3iφd + t12 r23 r21 r23 r21 t23 e5iφd + ....
= t12 t23 e
iφd
∞
X
r23 r21 e2iφd
n
(2.26)
(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.
t=
t12 t23 eiφd
1 + r23 r12 e2iφd
(2.28)
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).
4n1 n2
(n1 + n2 ) (n2 + n3 ) e−iφd + (n1 − n2 ) (n2 − n3 ) eiφd
t13 (n1 + n3 )
=
n1 n3
n1 + n3 − n2 1 + n2 iφd
t=
(2.29)
(2.30)
2
Further, for a metallic thin film we can simplify using:
n1 n3
n22
1 and
n1 n3
n2
(n3 − n1 ) 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
relation (Eq. 2.20) and the simplification it allows us to make, in2 φ2 = 2πi λd n22 = −Z0 σd.
Where Z0 =
1
c0
= 376.7Ω, the vacuum impedance, and the conductivity, σ, can be re-written
in terms of the sheet conductivity, σs = σd. In the end we have the following result:
t=
t13 (n1 + n3 )
n1 + n3 + Z0 σs
(2.31)
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.
2
r23 t23 e4iφd + ...
r = r32 + t32r21 t23 e2iφd + t32r21
= r32 + t32r21 t23 e
2iφd
∞
X
r21 r23 e2iφd
n
(2.32)
(2.33)
n=0
t32r21 t23 e2iφd
1 + r21 r32 e2iφd
r32 + r21 e2iφd
=
1 + r21 r32 e2iφd
= r32 +
(2.34)
(2.35)
13
When inserting the constituent reflection and transmission coefficients and applying the
thin-film approximation, this becomes slightly messier than the transmission.
(n3 − n2 ) (n2 + n1 ) + (n3 + n2 ) (n2 − n1 ) (1 + 2iφd )
(n3 + n2 ) (n2 + n1 ) + (n3 − n2 ) (n2 − n1 ) (1 + 2iφd )
n3 − n1 − Z0 σs
=
n3 + n1 + Z0 σs
r=
(2.36)
(2.37)
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 + tr34 rt34 + tr34
r t34 + ...
2 2
twithout = t13 t34 + t13 r34 r31 t34 + t13 r34
r31 t34 + ...
2 2 2
4 4 2
Twith = t2 t234 + t2 r34
r t34 + t2 r34
r t34 + ...
=
t2 t234
2 2
1 − r34
r
2 2 2
2 4 2
Twithout = t213 t234 + t213 r34
r31 t34 + t213 r34
r31 t34 + ...
=
t213 t234
2 2
1 − r34
r31
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
(2.43)
Taking the ratio of Twith and Twithout allows us to determine the properties of the thin-film,
where R is the relative power transmission.
R=
1
Nothing is as tedious as TDS.
2 2
Twith
t2 (1 − r34
r31 )
= 2
2 2
Twithout
t13 (1 − r34 r )
(2.44)
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.
1
σs = −
2n4 Z0
r
n23 + 2n1 n4 + n24 −
Rn43 + 2n23 n4 (2n1 + n4 (2 − R)) + n24 (4n21 + 4n1 n4 + Rn24 )
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:
d2 x
dx
q
+Γ
+ ω02 x = − ∗ E (t)
2
dt
dt
m
(2.46)
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.
x = x0 e−iωt x̂
(2.47)
Where the displacement from equilibrium x0 is given by:
x0 = −
|E0 |
q
2
∗
m ω0 − ω 2 − iΓω
(2.48)
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 = N p, where N is the carrier
density).
P = qN x = χe (ω) E0 e−iωt x̂
(2.49)
We next need to use the displacement vector D in its two forms:
D = 0 E + P = 0 R E
(2.50)
Now we can solve for the relative permittivity ( 0 ). We can simplify by inserting the
plasma frequency ωp2 =
N q2
0 m∗
and assume the medium is isotropic. This allows the relative
permittivity to take the form of Eq. 2.51.
ωp2
|P|
=1+ 2
R (ω) = 1 +
0 |E|
ω0 − ω 2 − iΓω
(2.51)
This is of particular usefulness since n2 = R , which allow us to rewrite the dispersion
relation.
k (ω) =
p
ω
R (ω)
c
(2.52)
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
anharmonicity, which in the simplest approximation takes the potential from 12 mω02 x2 to
1
mω02 x2
2
+ 13 mαx3 . This non-linear term corresponds to multi-photon processes. We will
incorporate this into the Drude-Lorentz model by adding the following term:
dx
q
d2 x
2
2
+
Γ
+
ω
x
+
αx
=
E (t)
0
dt2
dt
m
(2.53)
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 (t) =
∞
X
λn x(n) (t)
(2.54)
n=1
The first order term, which does not include the nonlinear term, has the following equation
of motion:
q
d2 x(1)
dx(1)
+ ω02 x(1) = E (t)
+
Γ
2
dt
dt
m
(2.55)
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.
x(1) (t) = x(1) (ω1 ) e−iω1 t + x(1) (ω2 ) e−iω2 t + c.c.
=
q
q
E0 e−iω1 t
E0 e−iω2 t
+
+ c.c.
m∗ ω02 − ω12 − iΓω1 m∗ ω02 − ω22 − iΓω2
(2.56)
(2.57)
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
2
the nonlinear term, x(1) is of order λ2 , due to its dependence upon the first order terms.
(1) 2
d2 x(2)
dx(2)
2 (2)
+
Γ
+
ω
x
=
−α
x
0
dt2
dt
(2.58)
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,
2
specifically x(1) 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
SHG = x(1) (ω1 ) e−2iω1 t + x(1) (ω2 ) e−2iω2 t + c.c.
(2.59)
SFG = 2x(1) (ω1 ) x(1) (ω2 ) e−i(ω1 +ω2 )t + c.c.
(2.60)
DFG = 2x(1) (ω1 ) x∗(1) (ω2 ) e−i(ω1 −ω2 )t + c.c.
2
2
OR = 2 x(1) (ω1 ) + 2 x(2) (ω2 )
(2.61)
(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.
D = 0 E + P(1) + P(N L) = D(1) + P(N L)
(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)
P = 0 χ(1) E + χ(2) E2 + χ(3) E3 + ...
(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:
P(n) = 0 χ(n) En
(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)
P0
(2)
P0
2αe2 N
(2)
= −N ex0 =
m2 ω02
h
(ω02
−
2
ω2)
+ Γ2 ω 2
i |E0 |2
(2.67)
= 20 χ(2) (0, ω, −ω) |E0 |2
(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.




 Px 
 d11 d12 d13 d14 d15 d16



 P  = 20  d
 y 
 21 d22 d23 d24 d25 d26



Pz
d31 d32 d33 d34 d35 d36
Ex2





  E2 


y




2
  Ez 




  2Ey Ez 




 2E E 
z x 



2Ex Ey
(2.69)
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:
vp =
c
ω
=
k
n (ω)
(3.1)
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:
vg =
∂ω
c
=
∂n
∂k
n (ω) + ω ∂ω
(3.2)
In our system, the use of a second order effect dictates that the group velocity of the
optical pump beam, the speed at which the intensity profile |E0 |2 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 .
lw =
cτp
(nT − nO )
(3.3)
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
behind the THz pulse by a phase factor of π2 . 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.
lc =
c
2νT Hz |ngr − nT |
(3.4)
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 4̄3m zinc-blende crystal, [28] can be
written as follows:

(2)
χOR

 0 0 0 1 0 0 



= d14 
0
0
0
0
1
0




0 0 0 0 0 1
(3.5)
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
) using ZnTe. However, high field THz generation has been
spectroscopy (ET Hz > 100 kV
cm
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
,
V
[34] in comparison to d14 = 4
pm
V
in ZnTe. [35] The tensor which describes optical
rectification can be found below (Eq. 3.6).
24

(2)
χOR

0
0
0 d15 −d22
 0

=
0
 −d22 d22 0 d15 0

d15 d15 d33 0
0
0





(3.6)
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
pulse is produced at the Cherenkov angle, θc ≈ 63o 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.
θc = cos
−1
nopt
nT Hz
(3.7)
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]
tan γ =
λopt p
gr
nopt β1 cos
θd
tan θ = tan γ = nph
opt β2 θd
(3.8)
(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
diffraction angle being very close to the Littrow angle (θd − θlitt < 10o ).
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 = kB T ).
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
half the wavelength ( λ2 ) [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 absorber 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.
P (2) (ω) = 2
X
(2)
0 χijk (ω, ω, 0) Ej (ω) Ek (0)
(3.10)
jk
(2)
We will now use the field-induced susceptibility tensor, χij (ω) = 2
P
k
(2)
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.
P (2) (ω) = 2
X
(2)
0 χij (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
polarized optical pulse passes though the ZnTe crystal, a
λ
-plate,
4
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
to changes to circular when it passes through the λ4 -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]:
∆φ = (ny − nx )
ωL
ωL 3
=
n r41 ET Hz
c
c O
(3.12)
While there is some phase retardation, it is generally small and a small angle approximation can be made.
I0
(1 − sin∆φ) ≈
2
I0
Iy = (1 + sin∆φ) ≈
2
Ix =
I0
(1 − ∆φ)
2
I0
(1 + ∆φ)
2
(3.13)
(3.14)
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.
Is = Iy − Ix = I0 ∆φ = I0
ωL 3
n r41 ET Hz
c O
(3.15)
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 considering 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
me
∂ 2x
∂x
= eE0 e−iωt
+
m
Γ
e
2
∂t
∂t
(4.1)
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.
The solution to this differential equation comes from the ansatz x = x0 e−iωt x̂. Inserting this
in to the equation and differentiating yields the following result, Eq 4.2.
E0 = −
me ω 2 x0 + ime ωΓx0
e
(4.2)
Next we insert the ansatz with this result in the equation for relative permittivity and
finally arrive at the destination Eq. 4.3.
ωp2
Γωp2
R = + i = 1 − 2
+i
ω + Γ2
ω (ω 2 + Γ2 )
0
00
(4.3)
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 planewave solutions, we can extract the eigenmodes of the system.
∇×∇×E=−
1 ∂ 2D
ω2
=
R E
0 ∂t2
c2
∇×∇×E−
ω2
R E = 0
c2
(4.4)
(4.5)
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

Ej = 
 0

Ej,z


 ik x−iωt ik z
e x
e j,z


(4.6)
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,
k=
2π
,
λ
its constituent components, and the relative permittivity.
2
R,j k 2 = kx2 + kj,z
(4.7)
Further extension of the source-free condition lead to the following relation between the
electric fields resulting from the divergence:
kx Ej,x + kj,z Ej,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,1 E1,z − R,2 E2,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,1 E1,z − R,2 E2,z = 0
k2,z E2,z − k1,z E1,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,1 k2,z − R,2 k1,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.
kx2
R,1 R,2 2
R,1 R,2 ω 2
=
k =
R,1 + R,2
R,1 + R,2 c2
(4.13)
2
To find kj,z
in terms of k 2 , we can do a similar trick by solving eq. 4.7 for kx2 and setting
them equal, which gives us:
2
=
kj,z
2R,j
k2
R,1 + R,2
(4.14)
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
that in order to have a real kx , the quantity
R,1 R,2
R,1 +R,2
has to have a real component. This
means R,1 R,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
dielectric function R,2 = 02 + i002 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,
which can be succinctly written as kx = kx0 + ikx00 , where kx0 determines the wavelength of the
SPP (λSP P ) and kx00 determines the propagation length. Now we can see that kx is:
r
kx =
R,1 R,2 ω
=
R,1 + R,2 c
s
1 (02 + i002 ) ω
1 + 02 + i002 c
(4.15)
Normally at this point in the derivation the following approximation would be made:
|02 | >> |002 |. 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.
When using the Drude model |02 | << |002 | for aluminum at 1THz (Drude = −3.246 × 104 +
6.424 × 105 i [1]). Using this, we can see that Eq. 4.15 simplifies to Eq. 4.17.
√
kx0
=
1 |2 | ω
|1 + 2 | c
(4.16)
3/2
kx00 =
1 002
ω
2 |2 | |1 + 2 | c
(4.17)
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
√
√
1 |2 |
that λSP P = 2π
≈ nλ1 , where |1 +
≈
1 = n1 . However interesting this result is, there is
k0
|
2
x
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.
√
4.2) ω = cknx , 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.
The addition of this periodic lattice allows us to now write kx0 = kx +
periodicity of the lattice and
2πn
a
2πn
,
a
where a is the
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
4
transmission should scale as λa , [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 functionalization.
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,
albeit with a
π
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 = ã (ω) eiωt which strongly couples to the incident light and one with a sub-radiant state
|bi = b̃ (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 

κ
 ã 
 δ + iγa

  = −

b̃
κ
δ + iγb

 g Ẽ0 


0
(5.1)
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.
ã =
−g Ẽ0 (δ + iγb )
(δ + iγa ) (δ + iγb ) − κ2
(5.2)
The amplitude ã can be used to find the polarizability and therefore the susceptibility
(qN ã = 0 χẼ0 ). Given that our detuning is small δ γa , γb , κ, we can ignore terms with
orders δ greater δ 2 and simplify the susceptibility into the following form:
iγb
−gqN δ (γb2 − κ2 )
+
χ=
0
(γa γb + κ2 )2 γa γb + κ2
(5.3)
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:
nph =
p
χ
1+χ≈1+
2
(5.4)
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:
ngr = nph + ω
∂nph
∂ω
(5.5)
47
We should note that
∂nph
|
∂ω ω=ω0
≈
∂nph
| ,
∂δ δ=0
which is proportional to κ−2 . 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.
The dipole antenna should be resonant when L =
λres
.
2
However because these antennas
are patterned on semiconductors the resonant wavelength will not be simply
λ
,
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]:
λres
= L + 2δ
2neff
(5.6)
However, the delta term can be ignored and is on the order of the lateral antenna dimension [93, 95], leading us to our final form:
λres = 2neff L
(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]
0 A
d
0 wt
C=
d
C=
(5.8)
(5.9)
Now finding the self-inductance is more complicated (Lself ). First the flux through the
SRR must be calculated.
Φ = Lself I
Z
= B · dS
I
= A · dl
(5.10)
(5.11)
(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:
µ0 I
Az =
4π
Z
+L
2
dz 0
1
(z − z 0 )2 + r2 2

i1 
h
L
L 2
2 2
−
z
−
+
z
−
+
r
2
2
µ0 I 

=
ln 
1 
h
i
4π
2
2
− z + L2 + z + L2 + r2
(5.13)
−L
2
(5.14)
49
Figure 5.3: The geometry for Az .
We can further rearrange the formula and find Az in its final form1 .
"
µ0 I
Az =
sinh−1
4π
−z +
r
L
2
!
+ sinh
−1
z+
r
L
2
!#
(5.15)
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
Lself =
=
4
I
A · dl
I
Z −w+ L
(5.16)
2
Az dz
(5.17)
w− L
2
!
!#
L
L
−z
+
z
+
2
2
sinh−1
+ sinh−1
dz
(5.18)
L
r
r
w− 2
q
√
2µ0
L−w
2
−1
−1
2
=
−w sinh 1 + w 2 + (L − w) sinh
− w + (L − w)
(5.19)
π
w
µ0
=
π
1
Z
−w+ L
2
"
Note: sinh−1 z = ln z +
√
z 2 + 1 and −sinh−1 − z = sinh−1 z
50
Figure 5.4: The square loop cartoon.
Now lastly, we find the resonant frequency given by the following:
1
ω0 = √
LC
v
u
πd
u
= cu
√
t
2wt −w sinh−1 1 + w 2 + (L − w) sinh−1
(5.20)
L−w
w
q
2
− w2 + (L − w)
(5.21)
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 antiparallel (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
at that time. The induced capacitive current lags the electric field from the wire by a
π
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
field from the wire are
π
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
in LiNbO3 . The optical and THz pulses were then overlapped at the focus of a 90o 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
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 5.23: 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
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 5.25: 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
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 5.27: 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
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 5.29: 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
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 5.31: 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
nbg =
1 + n2s
2
(5.22)
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.
N=
U (1 − R)
~ωAopt δ
(5.23)
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 f s) 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 f s to 100 f s. 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
around 100
kV
.
cm
Intervalley scattering is considered the dominant mechanism for THz in-
duced 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
of the optical pump. The low effective mass (m∗ ≈
me
)
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 f s) [87], faster than the scattering time (approximately 190
f s). [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
much higher (m∗ ≈
me
)
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
in transmission is observed has an onset above 150
kV
.
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 pumpoptical 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]
H=−
X
X
X
tij c†iσ cjσ + ciσ c†jσ +
Ej c†jσ cjσ + U
ni↑ ni↓
jσ
hiji,σ
(6.1)
i
The first sum is the hopping probability amongst the nearest neighbors and consists of
two pieces. First, it contains creation (c†iσ ) 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
tij =
ψi∗ ψj d3 x
U=
ke2
r12
(6.2)
(6.3)
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
2
−e
, where κ is the background dielectric constant, and binding energy which
potential − κr
12
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)
r
r
1/3
4me2 3n0
2
q =
~2
π
(6.5)
(6.6)
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, a∗H , or the carrier concentration increases, the
material will begin to conduct.
a∗H n1/3 ≤ 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:
0 |E0 |
σ (x, t) = √
2π
r
λ −iωt −( πi4 )
e
e
x
(6.8)
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
AUnit cell
= .0029),
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).
|Enear |
1
α=
=
|ET Hz |
β
r
R
τr
(6.9)
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
enhanced field is primarily interacting with 100 nm VO2 since Enear ≈ Enear (0) e−z/w and
def f ≈ 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) =
1+e
≈e
E−EF
kT
EF −E
kT
g (E)
g (E)
(6.10)
(6.11)
90
2ρDOS (E)
V
2 dν
=
V dE
3
1
2m 2 1
= 2
E2
2π
~2
g (E) =
(6.12)
(6.13)
(6.14)
Now the conductivity of the VO2 thin-film can be written in the following manner:
σ = e (ne µe + nh µh )
(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
∞
ne =
feF −D dE
(6.16)
EG
3
Z
1
−E
1
2me 2 EF ∞
2 e kT dE
kT
e
(E
−
E
)
= 2
G
2
2π
~
EG
32
EF −EG
2πme kT
=2
e kT
2
h
(6.17)
(6.18)
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
fhF −D dE
−∞
Z 0 feF −D
gh dE
=
1−
ge
−∞
Z 0
E−EF
e kT gh dE
=
nh =
(6.19)
(6.20)
(6.21)
−∞
3 Z
E−EF
1
1
2mh 2 0
2 e kT
= 2
(−E)
dE
2π
~2
−∞
3
2πmh kT 2 −EF
e kT
=2
h2
(6.22)
(6.23)
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
EF =
+ kT ln
2
4
mh
me
(6.24)
Lastly this is used to find the number of carriers n = ne = nh for use in Eq. 6.15.
n=2
2πkT
h2
32
3
−EG
(mh me ) 4 e 2kT
(6.25)
This allows us to rewrite the conductivity in the following manner:
σ = 2e (µe + µh )
2πkT
h2
32
3
−EG
(mh me ) 4 e 2kT
(6.26)
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 ≈ σ0 e 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 ).
ρ=
d
σs
(6.28)
EA
= ρ0 e 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 = σs Z0
logR = ae−bx + cx + d
(6.30)
(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
Uh = Eε2 + ηGγ 2
2
2
(6.36)
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
Uc = Eε2
2
(6.37)
Now taking the difference of these yields the change in free energy, which is equal to the
strain energy difference.
(Th − Tc ) ∆S = ∆U
1
= ηGγ 2
2
(6.38)
(6.39)
Solving the above equation for the change in temperature, we arrive at the hysteresis
width.
∆T = Th − Tc =
ηGγ 2
2∆S
(6.40)
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 Drudelike 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 SU8 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
1
Paul 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 plasmonphonon 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-diffractionlimit 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 95o C 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
Spin-coat 1
3000rpm, 5000 rpm
, 10s
s
3000rpm, 5000 rpm
, 10s
s
Spin-coat 2
, 20s
4000rpm, 5000 rpm
s
, 20s
4000rpm, 5000 rpm
s
Soft Bake
NA
2m @ 85o C
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 95o C 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
Spin-coat 1
, 10s
500rpm, 100 rpm
s
, 10s
500rpm, 100 rpm
s
Spin-coat 2
1000rpm, 300 rpm
, 30s
s
2000rpm, 300 rpm
, 30s
s
Soft Bake
2m @ 65o C , 9m @ 95o C
15m @ 65o C, 40m @ 95o C
Exposure
9s
14s
Post-Exposure Bake
1m 30s @ 65o C , 2m @ 95o C
5m @ 65o C, 15m @ 95o C
Development
3m 20s
15m
Rinse
IPA
IPA
Dry
N2 or Air
N2 or Air