TERAHERTZ TIME-DOMAIN SPECTROSCOPY OF LOW-
DIMENSIONAL MATERIALS AND PHOTONIC STRUCTURES
by
CHEN XIA
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Thesis advisor: Dr. Jie Shan
Department of Physics
Case Western Reserve University
January, 2013
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis/dissertation of
Chen Xia ______
Doctor of Philosophy candidate for the ______degree *.
Jie Shan (signed)______(chair of the committee)
Kenneth D. Singer ______
Xuan Gao ______
Lei Zhu ______
______
______
(date) ____08/10/2012______
*We also certify that written approval has been obtained for any proprietary material contained therein. TABLE OF CONTENTS
LIST OF FIGURES ...... 5
ABSTRACT ...... 13
CHAPTER 1 ...... 15
1.1 DEVELOPMENT OF THZ SOURCES ...... 17
1.2 THZ SOURCES BASED ON ULTRAFAST LASERS ...... 18
1.3 NOVEL THZ SOURCES ...... 23
1.4 THZ SPECTROMETER ...... 24
1.5 OUTLINE OF THIS THESIS ...... 30
CHAPTER 2 ...... 32
2.1 INTRODUCTION ...... 32
2.2 FILMS FABRICATION AND CHARACTERIZATION METHODS ...... 34
2.2.1 Preparation of BaTiO3/PMMA nanocomposite films ...... 34
2.2.2 Fabrication of THz photonic crystals ...... 35
2.2.3 Scanning electron microscopy ...... 35
2.2.4 Terahertz Transmission Spectroscopy ...... 36
2
2.3 TRANSFER MATRIX METHOD...... 37
2.3.1 Matrix form in wave propagation ...... 38
2.3.2 Wave propagation in layered system ...... 43
2.3.3 Transfer matrix of photonic crystal ...... 47
2.4 LOCAL FIELD AND EFFECT-MEDIUM THEORY ...... 51
2.4.1 Homogenous system ...... 51
2.4.2 Two-phase system ...... 53
2.4.3 Layered structure ...... 56
2.5 STUDY OF ALL POLYMERIC THZ PHOTONIC CRYSTAL ...... 58
2.5.1 Polymer films incorporated with nanoparticles ...... 58
2.5.2 THz properties of polymer films ...... 61
2.5.3 Polymeric THz photonic crystal ...... 66
2.6 CONCLUSION ...... 69
CHAPTER 3 ...... 70
3.1 INTRODUCTION ...... 70
3.2 EXPERIMENTAL SETUP ...... 73
3.3 THZ EMISSION FROM GRAPHITE SURFACES ...... 74
3.4 FLUENCE DEPENDENCE OF THZ EMISSION FROM GRAPHITE BASAL PLANES ...... 77
3.5 CORRELATION DYNAMICS OF THZ EMISSION FROM GRAPHITE SURFACES ...... 83
3.6 POLARIZATION RESOLVED THZ EMISSION FROM BASAL AND EDGE PLANES OF GRAPHITE ...... 87
3.7 CONCLUSION ...... 92
3
CHAPTER 4 ...... 93
4.1 INTRODUCTION ...... 93
4.2 EXPERIMENT ...... 95
4.3 EXPERIMENTAL RESULTS ...... 99
4.3.1 Static Properties ...... 99
4.3.2 Photoconductive properties...... 101
4.4 CONDUCTION IN AMORPHOUS SEMICONDUCTORS ...... 106
4.4.1 Drude-smith model ...... 107
4.4.2 Quantum mechanical tunneling (QMT) in one-dimensional (1D) systems ...... 114
4.4.3 Resonant Photon Absorption ...... 122
4.5 CONCLUSION ...... 128
APPENDIX: ...... 130
0 ELECTROMAGNETIC MODEL FOR THZ EMISSION FROM SURFACES ..... 130
0.1 THE SNELL’S LAW IN NONLINEAR OPTICS ...... 132
0.2 EMISSION FROM NONLINEAR POLARIZATION AT THE SURFACES OR INTERFACES ..... 134
0.2.1 THZ RADIATION FROM ISOTROPIC POLARIZATION SHEET ...... 138
0.2.2 THZ RADIATION FROM ANISOTROPIC SURFACES ...... 142
0.3 FORMATION OF POLARIZATION SHEET ON GRAPHITE SURFACES ...... 145
REFERENCE…………………………………………………………………………… 147
4
List of Figures
1. Figure 1.1 THz spectrometer based on a Ti:Sapphire femtosecond laser. The THz
radiation is generated in a ZnTe emitter through optic rectification and detected by
another ZnTe crystal via electro-optic sampling…………………………………………………….24
2. Figure 1.2 Schematics of a THz emitter (left) and detector (right) based on (110)
ZnTe. In the generation process, both the pump and the emitted THz fields are
polarized in the {y’,z’}-plane. In the detection process, the applied THz field along
the y’ axis induces birefringence. The principle axes are along {y”, z”}, rotated by
45° from {y’, z’}...... 25
3. Figure 1.3 Electro-optic sampling of THz electric field...... 27
4. Figure 2.1 A plane wave E1 with wave vector k1 and frequency ω is incident from
medium 1 with permittivity ε1 and permeability m1 to medium 2 with permittivity
ε 2 and permeability m2 . The reflection and transmission happen at interface of two
dielectric media, and the reflected field and transmitted field are observed at
' direction θ1 and θ2 with wave vectors k1 and k 2 , respectively...... 39
5. Figure 2.2 Demonstration of p-polarized plane waves propagation through plane
interface between medium i and j inside multilayered system. Ei and E j are fields
approaching the interface from medium i (left) and medium j (right), and plane
' ' waves Ei and E j leaving the interface are observed at reflection directions of θi
and θ j , respectively. All the electric fields are polarized in the plane of incidence, 5
while all the magnetic fields are perpendicular to the plane to satisfy the vector
rules determined by Maxwell equations...... 42
6. Figure 2.3 Multilayered structure is fabricated between substrates n0 and nN +1 . The lth
layer with dielectric function nl has thickness of dl . The electric field propagating in layer
is sum of right-travelling wave Al and left-travelling wave Bl ...... 44
7. Figure 2.4 Demonstration of 1D photonic crystal comprising of alternatively stacked
layers of index of refraction n1 and n2 . The physical period of crystal is d with
neighboring layers to be d1 and d2 in thickness. The amplitude of fields inside nth
unit cell are denoted as an , bn , cn , en , respectively...... 48
8. Figure 2.5 Schematic diagram of two-phase systems that could be applied by a) Maxwell
Garnett, b) Bruggeman, and c) layered structure effective medium theory ...... 58
9. Figure 2.6 Scanning electron microscopy (SEM) images of fractured surfaces of
PMMA/BaTiO3 nanocomposites. (a) 6% v/v, particle size 100 nm ; (b) 19% v/v, 100
nm; (c) 3% v/v, 200 nm; (d) 18% v/v, 200 nm...... 59
10. Figure 2.7 Scanning electron microscopy images of fractured surfaces of PMMA
comprising BaTiO3 nanoparticles: (a) 6% v/v, particle size 100 nm; (b) 3% v/v,
particle size 200 nm. The images show occasional zones of aggregated BaTiO3 ...... 60
11. Figure 2.8 Real (n’) and imaginary (n“) parts of the THz refractive index of PMMA/BaTiO3
nanocomposite films containing a) 0%, b) 3%, and c) 18% v/v BaTiO3 (particle size: 200 nm).
Solid lines represent effective medium theory calculations ...... 63
6
12. Figure 2.9 Real (n’) and imaginary (n“) parts of the refractive index of PMMA/
BaTiO3 nanocomposite films containing a) 0%, b) 6%, and c) 19% v/v BaTiO3 (particle
size:100 nm). Solid lines represent effective medium theory calculation...... 64
13. Figure 2.10 Real part (n’) of the refractive index of nanocomposite a)
PMMA/BaTiO3, b) poly(styrene)/BaTiO3 films at 0.8 THz as a function of BaTiO3
content. The solid line represents an effective medium theory calculation ...... 65
14. Figure 2.11 Optical properties of Bragg mirrors consisting of 6 alternating double
layers of PMMA and a 18 % v/v PMMA/ BaTiO3 nanocomposite between two quartz
substrates. (a) THz electric-field waveform of the signal (dashed, off-set from zero
electric field) and reference (solid) in the time domain. (b) Transmittance in the
frequency domain ...... 68
15. Figure 3.1 Setup of THz Emission spectroscopy ...... 74
16. Figure 3.2 Picosecond THz emission from basal surface of HOPG, and its spectrum ...... 75
17. Figure 3.3 The rotational dependence of THz emission signal. The emission signal
was almost kept at same level while the HOPG sample rotated along its c-axis ...... 76
18. Figure 3.4 The THz emission signal at different linear polarization status of optic
pump. The polarization direction angle of each status was with respect to p-
polarization ...... 76
19. Figure 3.5 Measured peak of the THz radiated electromagnetic field as function of
optical fluence with the saturation modle fitting in solid line ...... 78
7
20. Figure 3.6 The demonstration of graphite surface under optical excitation. The
surface field and radiated field both contribute to the surface current ...... 79
21. Figure 3.7 Schematic illumination of double pump experimental setup ...... 84
22. Figure 3.8 The influence on THz emission from the present of earlier excitation. The
solid line and dotted line represent the THz radiation with and without the earlier
excitation...... 85
23. Figure 3.9 The radiated THz signal from later pulse T2 ()τ as function of delay time
between two pump pulses (in circles). The RMS of THz signal was chosen to
represent its strength, and its dependence on the delay time was fitted with double
exponential function in thick-dashed line. The unaffected radiated signal T20 was
also plotted as reference in triangles and aligned with a thin-dashed base line...... 87
24. Figure 3.10 A schematic illumination of emission measurements in both basal plane
and edge plane configuration (top view)...... 88
25. Figure 3.11 The normalized spectrum of THz radiation from edge plane in
configuration ep1. The blue lines indicate s-polarized and red lines p-polarized
components in signal. The solid and dashed lines correspond to those two
independent measurements E0 and Eπ ...... 90
26. Figure 3.12 The normalized spectrum of THz radiation from edge plane in configuration
ep2. Only p-components observed in both measurements. The solid and dashed lines
correspond to those two independent measurements E0 and Eπ ...... 90
8
27. Figure 3.13 The decomposition of dipole moments contribution in THz radiation in
configuration ep2. The red line and the blue line are radiations from defect-induced
nonlinearity along the c-axis and photo-Dember effect normal to surface plane,
respectively...... 92
28. Figure 4.1 Chemical structure of alkoxy-substituted zinc phthalocyanine derivatives
(ZnPc-OC8)...... 96
29. Figure 4.2 DSC trace for the crystal to liquid crystal phase transition...... 96
30. Figure 4.3 Microscope Images of a 4 mm ZnPc thin film above and below the phase
transition temperature. The images were taken under cross-polarized light...... 97
31. Figure 4.4 Absorption spectrum of ZnPc thin film covering the so-called Q band
ranging from 600nm to ~ 800nm. The two intense π-π* transitions within Q band
was lost due to absorption saturation and/or disordered nature of thin films...... 97
32. Figure 4.5 Setup of optic pump-THz probe spectroscopy...... 98
33. Figure 4.6 The transmitted THz electric-field waveform with (solid line) and without
(dotted line) ZnPc thin films in the beam path...... 101
34. Figure 4.7 Frequency dependence of the refractive index (real part, red; imaginary
part, blue) of ZnPc thin films ...... 101
35. Figure 4.8 THz electric field transmitted through an unexcited sample (solid) and its
change (dashed), 20 ps after the ultrafast photoexcitation...... 102
36. Figure 4.9 The complex photoconductivity of ZnPc at delay time 20ps...... 102
9
37. Figure 4.10 Dynamics of photoconductivity in ZnPc thin films. A series of delay
times (as indicated) have been chosen for the spectral study...... 102
38. Figure 4.11 Complex photoconductivity spectrum at different delay times after
photo excitation in ZnPc thin films ...... 103
39. Figure 4.12 Normalized complex photo conductivity by the pump fluence at delay
time 20ps for differing excitation fluences...... 104
40. Figure 4.13 Experimental dynamics (red) and their double exponential fits (black)
for two different excitation fluences ...... 104
41. Figure 4.14 Effective photoconductivities near delay t=0ps under heating (in red)
and cooling (in blue) process...... 105
42. Figure 4.15 Left: Photoconductivity dynamics at different sample temperatures,
and their double exponential fits. Right: time constants obtained from the fit. .... 106
43. Figure 4.16 Prediction of the Drude Smith model of Eq. (4.11) for the real (left) and
imaginary (right) part of the conductivity for parameter c = 0, -0.5 and -1 ...... 110
44. Figure 4.17 Comparison of an experimental photoconductivity spectrum (dashed
lines with grey error bars) with the Drude Smith (solid lines) model for ZnPc thin
films 20 ps after photoexcitation at 60 J/m2 at room temperature. Top: real part
and bottom: imaginary part of the photoconductivity...... 110
45. Figure 4.18 The average scattering lifetime of ZnPc thin film at different delay time
after photoexcitation of 60 J/m2 ...... 112
10
46. Figure 4.19 The effective d.c photoconductivity of ZnPc thin film at different delay
time after photoexcitation of 60 J/m2 ...... 112
47. Figure 4.20 The estimated carrier density in ZnPc thin films from the Drude Smith
fitting parameters at different delay time after photoexcitation of 60 J/m2 ...... 113
48. Figure 4.21 The persistence of velocity in scattering- the constant c for ZnPc thin
films at different delay time after photoexcitation of 60 J/m2 ...... 114
49. Figure 4.22 Comparison of experimental photoconductivity spectrum (solid lines) of
ZnPc films to the QMT model (dotted lines). The photoconductivity was measured
at pump-probe delay time 20 ps, pump fluence 50J/m2 and sample at room
temperature. The real part of the conductivity is in red, and the imaginary part is in
blue...... 120
50. Figure 4.23 Comparison of experimental photoconductivity spectrum (solid lines) of
ZnPc thin films with the resonant absorption model (dotted line). Photoconductivity
in ZnPc films was measured at pump-probe delay time 15 ps, pump fluence 30 J/m2
at 320 K. The real part of conductivity is in red color, and the imaginary part is in
blue...... 126
51. Figure 4.24 Temperature dependence of carriers’ delocalization length in ZnPc thin
films...... 127
52. Figure 4.25 Temperature dependence of the carriers’ overlap integral in ZnPc thin
films...... 127
11
53. Figure 0.1 Scheme of the three-region pictures in the nonlinear surface generation
phenomenological model. An ultrafast laser excitation is incident at angle of , and
its propagation direction is illuminated in red color with wave vector superscripted훉
with and subscripted i (incidence). The polarization sheet induced is laid on the
interface훚 and recorded as x-y plane. The THz emissions, generated from media
interface, are sharply peaked in both “reflection” and “transmission” directions, and
are denoted with superscript Ω, and subscripts r (reflection) and t (transmission),
respectively ...... 131
54. Figure 0.2 The diagram of interface between mediem 1 and mediem 2. The pillbox
and contour covering the polarization cross the interfaceε to assisst theε deviation of
boudary conditions ...... 136
55. Figure 0.3 The geometry of surface THz generation between mediem ε1 and
mediem ε2 ...... 138
56. Figure 0.4 The relative relation between k , D, E , and H ( B ) ...... 143
12
Terahertz Time-Domain Spectroscopy of
Low-Dimensional Materials and Photonic Structures
Abstract
by
CHEN XIA
Terahertz (THz), the frequency region that bridges the gap between the radio
and optical frequencies in the electromagnetic spectrum, is of both scientific and technological importance. Its broad applications, however, have been limited by the availability of bright sources and sensitive detectors. The invention of modelocked femtosecond lasers has made possible the generation and detection of electromagnetic
transients on the picosecond to sub-picosecond time scale, which correspond to
broadband coherent radiation up to a few or 10’s THz. This time-domain technique has
had a significant impact on spectroscopy, imaging and sensing.
While detectors and emitters have been the main interest in the development of
THz technology, other components such as modulators, reflectors and filters are also of
importance. In Chapter 2 we apply THz transmission spectroscopy to characterize
polymeric materials and nano-composites. We demonstrate the design and fabrication
of THz mirrors based on one-dimensional photonic crystals consisting thermoplastic
13
polymers and BaTiO3 nanoparticles. Stop bands have been achieved in the THz region, and the results agree well with the prediction of the transfer matrix method.
The main focus of this thesis is to study charge transport and carrier dynamics in low dimensional materials using THz time-domain spectroscopy. In contrast to conventional transport measurements, where electrical contacts are required, the THz method based on freely propagating electromagnetic transients can probe the electronic transport in a contactless fashion. In Chapter 3 we investigate highly oriented pyrolytic graphite (HOPG) using the THz emission spectroscopy. The direction, spectrum, polarization and dynamics of the emitted THz radiation from graphite upon ultrafast photoexcitation reveal the emission mechanism as well as the out-of-plane transport properties of graphite.
In Chapter 4 metal phthalocyanines have been investigated by the optical pump/THz probe technique. Disc-like phthalocyanine molecules self-assemble into columns and form quasi-1D conduction channels for efficient charge transport. With appropriate side groups the material can have both crystalline and liquid crystalline phases. The photoconductivity spectrum, its dependence on the pump-probe delay time and sample temperature suggest the mechanism of as well as the influence of crystallinity on the photoconductivity in metal phthalocyanines.
14
Chapter 1
Introduction: Terahertz Time-Domain Spectroscopy
Terahertz (THz) science and technology, in particular, the time-domain THz spectroscopy (TDTS) based on mode-locked femtosecond lasers has attracted considerable attention in both scientific and technological communities in the past two decades [1] [2]. THz spectroscopy commonly refers to the spectroscopic techniques employed in the spectral region ranging from 0.1 to 10 THz, or equivalently, from about
3 cm-1 to 300 cm-1. Although this spectral range is in between microwaves and optical waves, the techniques widely used in either microwave electronics or photonics can not be readily applied in the THz region. This fact along with its potential for applications in various fields has made THz an active research area.
THz spectroscopy has first been applied to study the rotational and low-lying vibrational resonances of simple molecules [3] [4], since these modes fall into the THz spectral range. Later on THz spectroscopy has been extended to the characterization of numerous other systems. They include the phonons and/or free carrier responses in traditional semiconductors [5] [6] and dielectrics [7], surface plasmons in structured metals [8] [9], polarons and polaritons in functionalized organic materials [10], excitons in quantum confined systems such as quantum dots [11] [12], quantum wells [13] and heterojunctions [14] [15], and collective modes in correlated electron systems such as 15
colossal-magnetoresistance magnites [16] and superconductors [17]. In addition, THz radiation is non-ionizing because of its low photon energy and thus provides a convenient noncontact method for medical diagnostics [18], materials imaging [19], and homeland security detection [20].
Applications of THz, however, have historically been limited due to the lack of bright sources and sensitive detectors. The advances in ultrafast laser technology have made possible the generation and detection of coherent THz radiation with high detection sensitivities. In this approach, the THz electric field, rather than its intensity, is measured directly, which ensures information carried in both its amplitude and phase to be captured simultaneously. The THz electric-field waveform transmitted through a sample of interest can be directly related to the complex response of the material without the need of the Kramers-Kronig relation, advantageous for spectroscopic analysis [21] [22]. In addition, the electromagnetic transients of THz radiation resulted from the fs laser based techniques have a duration of picoseconds or sub-picoseconds.
This short pulse nature of the radiation allows an ultrafast time resolution in time- resolved studies of the dynamic properties of materials [23] [24].
The goal of this chapter is to briefly introduce the background of THz spectroscopy, the history of the THz generation and detection techniques and recent progresses in the THz emitters and detectors. We will then provide a brief description of the THz spectrometer setup employed in this thesis. Finally, we present the outline of the thesis.
16
1.1 Development of THz sources
The beginning of THz spectroscopy can be traced back to the microwave and millimeter-wave research [25]. One early approach is to extend the mid-IR sources and detectors into the far-IR region. An arc lamp and a bolometer have been used as a THz source and detector, respectively, in one of the earliest THz works [26]. However, the weak emission at the long wavelengths and the incoherent nature of the radiation from the lamp offered a poor signal-to-noise ratio in detection in the presence of strong background blackbody radiation.
Far-infrared lasers (optically pumped Si:Sb or gas lasers [27]) offer much higher brightness but with very limited tunability since tuning the wavelength generally requires the change of the lasing medium. The wavelength tunability has been improved with the invention of far-IR sources based on photomixing of two IR or near-IR lasers [28]
[29], or of a fixed-frequency laser with the output of a tunable microwave generator
[30]. The wide application of these continuous wave (cw) THz sources has, unfortunately, been limited by their complex pump systems.
Free electron lasers based on relativistic electron beams are another family that is of intense academic interest. They are capable of generating both quasi-cw [31] and pulsed [32] THz radiation with a broad spectrum (up to a few THz). Alternative designs of powerful THz sources based on acceleration of charged particles include the synchorotrons [33] and gyrotron based cyclotrons [34]. They are attractive mostly because of their ability to provide intense, pulsed THz radiation with moderate pulse
17
width and considerable tunability. Their popularity is usually restrained by the
requirement of large facilities.
1.2 THz sources based on ultrafast lasers
Recently, the ultrafast electro-magnetic transients induced (and detected) by
ultrafast optical pulses have emerged as a convenient coherent source for THz
spectroscopy. Photoconductive and optical nonlinear emitters are two major types of
THz emitters. They rely on frequency down conversion of the ultrafast optical pulses in
either a photoconductive material or a nonlinear crystal. This fs laser based approach
has several advantages: 1) the setups are compact with peak brightness of the emission
comparable to that of synchrotron sources, 2) the emission is coherent with high
detection sensitivities, and 3) the emission is pulsed, allowing time-resolved studies in
the far-IR regime.
The application of photoconductive THz emitters was pioneered by Auston [35],
Grischkowsky [23], X-C. Zhang [36] and Nuss and their coworkers [37]. It started with
the generation and detection of ultrashort electrical transients through laser-controlled
photoconductive switches coupled with transmission lines. The subsequent
development involved the successful coupling of the electromagnetic transients into
free space using photoconductive antennas by Mourou and coworkers [38]. In these
emitters electron-hole pairs are generated in the photoconductive material (a
semiconductor) upon excitation by an ultrafast laser pulse. In a transverse bias electric
18
field, the electrons and holes are accelerated and a macroscopic current through the semiconductor is created. This transient current then generates electromagnetic transients, which are coupled into the free space by the antenna structure.
The freely propagating THz radiation is guided through the sample of interest, collected and refocused by off-axis parabolic mirrors (and/or silicon lenses) onto a THz
detector. The mechanism for detection is very similar to that of generation. In the
detector the THz electric field now plays the role of the bias field as in the THz emitter.
The photoconductive switch is turned on when a gating optical pulse, time synchronized
with the pulse generating THz radiation, impinges it. The resultant photocurrent is
linearly proportional to the electric field of the THz radiation at the instant when the
optical gating pulse and the THz pulse overlap in the detector. And to obtain the entire
THz electric-field waveform, a sampling technique can be applied by varying the delay
time between the THz radiation and the gating pulse.
Typical substrate materials used for THz generation and detection include
radiation damaged Si on sapphire, semi-insulating gallium arsenide (SI-GaAs) [39] [40]
and low temperature grown gallium arsenide (LT-GaAs). Typically the rise time of the
photocurrent is determined by the femtosecond optical pulse duration, and the decay
time of the photocurrent is controlled by the carrier scattering time and recombination
time. Efforts have been made to keep the latter on the order of ps so that the radiated
field has considerable frequency components in the THz region. The subsequent
development of the photoconductive emitters includes the optimization of the designs
19
of antennas in the 1980s’ [41, 42], the substrates (comparison between the large band gap semiconductors (Si, GaAs) and narrow band gap semiconductors (InAs, InSb) [43,
44]) and the study of different photocurrent surge mechanisms [44] [45] [46].
The development of the amplified modelocked laser systems with high peak
power optical pulses has enabled new THz generation and detection techniques in
1990s. To utilize the high power available from the amplified laser systems and to avoid
damaging the standard photoconductive antennas, large aperture antenna structures
have been introduced. On the one hand, these structures permit us to scale up the
excitation power by using larger beams; on the other hand, the increased separation
between the electrodes requires a high bias voltage to maintain the bias electric field,
which is technically challenging. In addition, the large emitted THz electric field can
screen the bias field to reduce the THz emission efficiency. Several studies have
investigated the efficiency of large-aperture photoconductive antennas as a function of
excitation power [47] [48].
An alternative method to generate THz radiation using the amplified laser pulses
is to rely on optical rectification in nonlinear materials [49] [50] [51]. For non-
centrosymmetric materials, the bulk second-order nonlinear susceptibility χ(2) is nonzero
and a nonlinear polarization at the difference of any two frequencies within the
bandwidth of the ultrafast optical pulse is expected. Since the laser pulse has a duration of ~100 fs and its bandwidth is ~ 10 THz, the emission from the nonlinear polarization belongs to the THz regime.
20
The history of using nonlinear materials to generation THz radiation can be
traced back to the 1970s’. Yang et al. demonstrated the generation of a THz pulse from
a LiNbO3 crystal using a picosecond pulsed laser [52], and later Zhang et al. measured and calculated the optical rectification signal from a y-cut LiTaO3 under normal
incidence [53]. Organic crystals were also of interest as THz sources due to their large bulk second-order nonlinear susceptibilities. It was first reported by Zhang et al. that
THz radiation was generated from organic salt dimethyl amino 4-N-methylstilbazolium
tosylate (DAST) through optical rectification, and the generated THz field was orders of
magnitude larger than that from semiconductors and inorganic crystals [54]. Of all the
candidates, ZnTe is perhaps the most popular crystal studied because of its large
nonlinearities and favorable phase matching properties for the output of a Ti:sapphire
ultrafast laser [55]. The successful generation and detection of broadband THz radiation
with a pair of ZnTe crystals by an ultrafast pulse centered at 800 nm was first
demonstrated by Nahata et al. [56]. In addition to ZnTe, cadmium telluride (CdTe) [57]
and gallium phosphide (GaP) [58] have also been explored for THz gerenation and
detection.
The corresponding detection technique for THz radiation based on nonlinear
optics is the electro-optic effect, also known as the Pockels effect [56] [53]. In this
effect, when a bias electric field is applied on a Pockels cell, the polarization of an optical
beam travelling through the cell can be modified by the field-induced birefringence in
the cell. In the detection of a THz transient, the electric field of the THz radiation serves
21
as the bias field, which induces birefringence in the detector. The change of the
polarization state of a probe optical pulse can thus be used to probe the amplitude and direction of the THz transient electric field directly at the instant of their overlap. By varying the delay time between the THz transient and the femtosecond probe pulse, the entire waveform of the THz electromagnetic transient can be sampled.
As a non-resonant technique, optical rectification in nonlinear crystals has a higher damage threshold than in photoconductive materials. The technique potentially can also generate THz radiation of a large bandwidth. Ideally the only limiting factor of the bandwidth of the emitted radiation is the duration (or bandwidth) of the excitation
pulse. For instance, broadband THz generation up to 50 THz is possible with an optical
excitation pulse of ~ 10 fs [59] [60]. Similarly for detection, the time resolution is in
principle given by the pulse duration.
In practice, the efficiency and bandwidth/time resolution of the generation and
detection process of the THz radiation depends on the phase matching effect of the
nonlinear processes. It has been realized [56] that in the collinear propagation geometry
within the nonlinear medium the phase matching condition requires the phase velocity
of the THz wave vΩΩ= cn/ and the group velocity of the optical pulse vωω= cn/ to be
matched. The coherence length for both optical rectification and electro-optic sampling
can be estimated as l=ππ// ∆= k c Ω nnω −Ω [61].
22
1.3 Novel THz sources
In Section 1.2 we have discussed the application of photoconductive materials
(semiconductors) and nonlinear crystals for THz generation. Recently, many other nonconventional materials have been explored for THz generation. Metals thin films are one of them. Beaurepaire et al. reported that ultrafast demagnetization in ferromagnetic Cr/Ni/Cr films under ultrafast laser excitation results in THz emission [62].
Later, iron (Fe) thin films grown by molecular beam epitaxy was found by Hilton et al. also to emit THz radiation, and both the magnetic nonlinearity and non-magnetic surface nonlinearity were thought to be the origin of the emission [63]. The latter work was extended by Kadlec et al. onto gold (Au) and silver (Ag) thin films [64] [65]. Their research revealed the dependence of the THz radiation on differing parameters including the laser fluence, incident angle, and the film thickness.
Other more complicated systems such as GaSb/AlSb superlattices and metal/semiconductor Schottky barriers have been studied for THz generation by Zhang et al. [66, 67]. The effects associated with the interface and surface properties including surface depletion, Shcottky, p-n junction, strain-induced piezoelectric fields have been investigated. Also, THz generation was detected from High-Tg superconductor BSCCO and YBCO thin films under ultrafast photoexcitation; and the role of the photo-Dember effect in the process was discussed by Murakami et al. [68] .
23
1.4 THz spectrometer
Now we turn to a brief description of the THz TDS setup that was employed in
this thesis. Figure 1.1 is the schematic representation. In this setup, THz radiation is
generated and detected based on nonlinear effects in nonlinear crystals. (For details see
above in Section 1.2). The laser source is an amplified Ti:sapphire laser with ~ 1W
average power, centered at 800 nm, with 1 kHz repetition rate, and 50 fs duration. The
THz emitter is a 1mm ZnTe crystal and the corresponding detector is another 2mm ZnTe
crystal. Both emitter and detector crystals are cut in the <110> orientation.
Figure 1.1 THz spectrometer based on a Ti:Sapphire femtosecond laser. The THz radiation is
generated in a ZnTe emitter through optic rectification and detected by another ZnTe crystal via electro-optic sampling.
24
The details on the THz generation and detection processes in ZnTe crystals can be found in [69]. In brief, since the ZnTe crystal has a zinc-blende lattice structure with space group of 43m , the only nonzero elements of the electro-optic coefficient tensor
are rrr41= 52 = 63 [55] with the nonlinear susceptibility tensor as:
000r14 0 0 = = rrijk il 000 0r14 0 (1.1) 000 0 0 r14
Consider normal incidence of the optical excitation on the (110) plane of a ZnTe
crystal. In the crystallographic coordinate the optical field can be described as
EE= 0 (sinθ / 2,− sin θθ / 2, cos ) , where is the angle between the field
polarization and the z-axis of the crystal. The excitedθ THz nonlinear polarization is found
2 2 2 1/2 to have magnitude P= rE14 0 [sinθθ (1+ 3cos )] / 2 . It reaches its maximum of
max 2 2 ETHz ∝ 1/3 rE41 0 at sinθ = 2 / 3 . In this case, the THz polarization is perpendicular to the optical pump polarization.
Figure 1.2 Schematics of a THz emitter (left) and detector (right) based on (110) ZnTe. In the generation process, both the pump and the emitted THz fields are polarized in the {y’,z’}-plane. In the detection process, the applied THz field along the y’ axis induces birefringence. The principle axes are along {y”, z”}, rotated by 45° from {y’, z’}.
25
The generated THz radiation is collected and guided by two pairs of aluminum
coated off-axis parabolic mirrors. These mirrors are 2” in diameter and 4” in focal length.
They are arranged in the so-called 4f scheme with the sample of interest located in the
“image” plane in the middle.
The THz radiation is then refocused onto the ZnTe detector based on EO
sampling. The electro-optic sampling process in the ZnTe detector, as mentioned above
in Section 1.2, can be described as the result of the THz field induced birefringence. The
new refractive index ellipsoid can be determined based on the nonlinear susceptibility
tensor of Eq. 1.1.
xyz2++ 22 +++= 2 2221rEyz41 xyz rExz41 rExy41 , (1.2) n0
where x, y, z are the crystallographic axes of ZnTe crystal, Ex , Ey , Ez are the
corresponding components of applied THz field, n0 is the index of refraction of the
unbiased crystal, and r41 is the nonzero electro-optic coefficient.
The most sensitive detection of the THz field is achieved when the phase
retardation between two orthogonal components of the probe beam from birefringence
is the largest. This occurs in ZnTe when the THz beam is polarized along the [±1, 1, 0]
direction as well as the optical probe. For this geometry it can be derived that the∓ new
indices of refraction along the principle axes are:
3 n0 n′′ ≈− n rE (1.3) y 02 41 THz
26
3 n0 n′′ ≈+ n rE (1.4) z 02 41 THz
And the new principal axes {y” ,z” } are obtained by rotation of {y’ ,z’ } about the axis [1,
1, 0] by 45°. The phase retardation induced can be expressed as function of the field as:
22ππdd3 θ =(n′′−= n ′′ ) nr E (1.5) λλz y 0 41 THz
where d is the thickness of the ZnTe detector and λ is the wavelength of the gating beam.
The most common scheme to extract the phase retardation of Eq. 1.5, or
equivalently, the THz field is shown in Figure 1.3. The ZnTe crystal, in combination with a
compensator, is placed in between two crossed polarizers (P1 and P2). The wave plate
can provide a static phase shift θ0 as necessary. The transmission coefficient of the probe beam through the setup is given by:
2 T =sin [( ∆+θθ0 ) / 2] (1.6)
Figure 1.3 Electro-optic sampling of THz electric field.
27
There are two commonly used configurations of EO sampling, corresponding to
and π /2 . In the so-called null scheme ( ) external wave plates are typically
not used. The residual birefringence or strain-induced birefringence in the ZnTe crystal
provides a small θ0 . If both the THz field-induced contribution and the static
contribution are small ( ∆θθ,10 ), the transmission coefficient of Eq. 1.6 can be
approximated by the leading order term of its Taylor expansion:
2 T =(2θθ0 ∆ +∆ θ ) / 4 (1.7)
Usually, the static birefringence θ0 can be chosen to be much larger than the field
induced birefringence ∆θ : θθ0 ∆ , and then the second term in (1.7) can be ignored.
We thus obtain a signal proportional to the field-induced birefringence ∆θ , which is a
linear function of the THz field ETHz . Another detection scheme is the quarter-wave
( λ /4) scheme in which the phase shift is set to π /2 . The transmission coefficient is
simply (1+∆θ )/2 for small ∆θ 1. In both schemes the measured transmission is a
linear function of the THz electric field.
The near-null detection scheme has the advantage of the highest modulation
depth in the detection, and hence provides a good means for weak signal detection. The
method has been used for single-shot detection of the THz waveform [70]. However,
the distortion of the THz waveform induced by the quadratic term in (1.7) is not
negligible for strong THz fields. For detection of strong THz fields the quarter-wave scheme should be used. The quarter-wave scheme provides the largest slope in the THz field dependence of the probe signal, thus has the highest sensitivity. However, in the 28
shot noise limit, both schemes provide similar signal-to-noise ratios [71]. In our setup, the compensator is a quarter-wave plate, and the second polarized (P2) is replaced by a
Wollaston prism. The two polarization components are detected by a pair of balanced photodiodes. The difference is then detected by a lockin amplifier in combination with modulation of the THz beam (through modulation of the optical pulse that generates the THz radiation using an optical chopper).
The THz spectrometer described above provides a convenient tool for materials characterization in the far-IR. The measurement of the THz waveforms with and without the sample of interest and their Fourier analysis yield the complex dielectric function, or equivalently, the complex conductivity of the sample. The non-equilibrium phenomena can also be investigated by incorporating into the setup an additional optical pump beam. For processes occurring on a time scale much longer than the THz pulse duration
(~ ps), the material can be treated as in quasi-equilibrium and its transient properties can be obtained by measuring the change to the transmitted THz waveform while fixing the delay time between the excitation pulse and, for instance, the peak of the THz waveform. For more rapid processes, however, it is more appropriate to perform a two- dimensional (2D) scan of both the THz gate pulse and the optical pump pulse so that at any instant of the THz waveform the delay time between the THz and the pump is a constant.
29
1.5 Outline of this thesis
The focus of this thesis is on the application of THz time-domain spectroscopy to
the study of far-IR properties and dynamics of low dimensional materials. Examples
include one-dimensional THz photonic crystals, graphite and model organic
semiconductors phthalocyanines. These three examples are the topics of Chapter 2, 3,
and 4, respectively.
In Chapter 2 we apply THz transmission spectroscopy to characterize polymeric
materials and nano-composites. We demonstrate the design and fabrication of THz
mirrors based on one-dimensional photonic crystals consisting thermoplastic polymers
and BaTiO3 nanoparticles. Stop bands have been achieved in the THz region, and the
results agree well with the prediction of the transfer matrix method. In Chapter 3 we investigate highly oriented pyrolytic graphite (HOPG) using the THz emission spectroscopy. The direction, spectrum, polarization and dynamics of the emitted THz radiation from graphite upon ultrafast photoexcitation have used to study the emission mechanism as well as the out-of-plane transport properties of graphite. Finally, in
Chapter 4 we present a systematic study of charge transport in metal phthalocyanines
as model organic quasi-1 D systems by the optical pump/THz probe technique. Disc-like
phthalocyanine molecules self-assemble into columns and form quasi-1D conduction
channels for efficient charge transport. With appropriate side groups the material can
also exist in both crystalline and liquid crystalline phases. The photoconductivity
spectrum, its dependence on the pump-probe delay time and sample temperature have
30
been studied to explore the mechanism of as well as the influence of crystallinity on the photoconductivity in zinc phthalocyanines.
31
Chapter 2
All polymeric THz mirror
2.1 Introduction
The terahertz (THz) regime (1 THz = 1012 Hz = 300 m = 30 cm-1) is one of the last frequency ranges of the electromagnetic spectrum to be widely technologically exploited. This “delay” has been mainly related to the lack of convenient coherent sources, sensitive detectors, and efficient components to control and manipulate THz radiation. In recent years, however, considerable efforts have been devoted to the development of materials and optical elements for the THz regime [2, 14, 72-77]. This
flurry of activities is motivated by the significant impact that the field promises to have
on a plethora of important applications. These range from sensing, biomedical imaging
[78], and spectroscopy [79], to interconnects, switches, and wireless communications
[72, 74]. While THz emitters and detectors have become more readily available through
these efforts, the fabrication of devices for control and manipulation of the radiation
still poses significant challenges, mainly because of the limited availability of suitable
materials.
One-dimensional periodic structures that act as Bragg reflectors and allow the
construction of THz mirrors of a spectral window of interest represent an important
32
type of passive devices and are also a good example for the materials design issues at hand. These multilayered elements feature stop bands (corresponding to high reflectivity) which result from interference effects. Their position, width, and reflectivity are governed by the layer thicknesses, the total number of layers in the structure, and the difference between the refractive indices (RI) of the materials which constitute the two different layer types [80].
One-dimensional photonic crystals with stop bands in the THz regime have been fabricated by periodically layering conventional semiconductors [81, 82], ceramics [83,
84], superconductors [85], polymers and air [86], and polymers and semiconductors
[87]. All-polymer THz mirrors would display significant advantages over these devices, since they potentially combine low absorption and dispersion in a large spectral window in the THz regime [21, 88], good mechanical characteristics, scalability, and cost effectiveness. Unfortunately, however, it has proven difficult to design polymer systems with high refractive index and high transmittance in the THz regime [21]. This, in turn, has limited the achievable refractive index contrast in all-polymer multilayer structures, and there with the optical characteristics, in particular, the reflectivity of such elements.
In this Chapter we explored a simple and potentially extremely versatile method to engineer the THz RI of polymeric materials. The approach relies on conventional melt- processing techniques to incorporate high-RI inorganic nanoparticles into conventional host polymers. It has been previously demonstrated that ferroelectric nanoparticles such as barium titanate (BaTiO3) can be used as fillers to increase the dielectric function
33
of polymeric materials up to GHz frequencies [89]. Several methods have been reported for the incorporation of BaTiO3 nanoparticles into polymer matrices including solid state shear milling [90], surface modification [91], electrodeposition [92], and polymerization of nanoparticle suspensions in monomers [89, 93, 94]. While these methods have allowed effective control over the dispersion of nanoparticles, they are generally intricate and require considerable time and effort. Due to the long wavelength nature of
THz radiation, the optical performance of nanocomposites in this wavelength regime is much less affected by scattering caused by nanoparticle aggregation than, e.g. in the visible regime. We speculated that this might allow one to employ much simpler processing techniques without sacrificing the materials performance. With the example of melt-processed PMMA/BaTiO3 nanocomposite films that were assembled into a photonic crystal with stop bands in the THz regime we demonstrate that this approach indeed provides a simple and effective route to large scale fabrication of high-quality
THz photonic crystals by roll-to-roll or co-extrusion methods.
2.2 Films Fabrication and Characterization Methods
2.2.1 Preparation of BaTiO3/PMMA nanocomposite films
PMMA (VM-100 resin) was purchased from Arkema and used as received.
BaTiO3 nanoparticles were purchased from Inframat Advanced Materials (99.95%, 100 &
200 nm) and used as received. Compounding was carried out using a DACA lab-scale, recycling, counter rotating twin screw extruder at 230oC and 100 RPM. The appropriate
34
amounts of polymer and nanoparticles were fed into the extruder and mixed by
recycling through the extruder for 5 min before extrusion. The volume fraction of the
BaTiO3 nanoparticles in the resulting nanocomposites was determined from the mass
loss of the components using thermogravimetric analysis (TGA) and their known
3 3 densities (1.18g/cm for PMMA and 5.85g/cm for BaTiO3). TGA was performed on a
Mettler Toledo TGA/SDTA851e instrument by heating the samples in air from 25 to
600oC at a rate of 10oC/min. All TGA samples showed a sharp mass loss at 350oC
attributed to decomposition of the PMMA matrix.
2.2.2 Fabrication of THz photonic crystals
One-dimensional photonic crystals were constructed by stacking 6 pairs of alternating films of a PMMA/BaTiO3 nanocomposite (18 % v/v) and PMMA. The
structure was mounted between two quartz substrates of 1 mm thickness. The average
thicknesses of the composite and PMMA films were 151 and 189 mm, respectively. A
small amount of silicon oil was applied at the film interfaces to prevent the formation of
air gaps between the layers and to reduce surface scattering.
2.2.3 Scanning electron microscopy
Scanning electron microscopy (Philips XL30 ESEM) was used to examine the morphology of the BaTiO3/PMMA nanocomposites. All SEM experiments were
conducted on freshly cryo-fractured surfaces. A thin layer of palladium was deposited
35
onto the surface using a Denton Desk II sputter coater to achieve good electrical contact with the grounded electrode. The micrographs were taken at room temperature at an acceleration voltage of 10 kV and a working distance of 7.4 – 8.9 mm in a secondary electron imaging mode.
2.2.4 Terahertz Transmission Spectroscopy
THz transmission spectroscopy was carried out to determine the complex
refractive index of monolithic films and the transmissivity of the photonic crystal in the
spectral range of 0.2 - 1.5 THz. Details of the THz setup have been described elsewhere
[56]. In short, it employed a mode-locked Ti: sapphire laser system (Spectra Physics) that
produces 50 fs short optical pulses with a 1 kHz repetition rate centered at 800 nm.
Electromagnetic transients of THz radiation were generated through optical rectification
in a ZnTe crystal and were detected through electro-optic sampling in a second ZnTe
crystal. All measurements were performed at room temperature in a closed
compartment purged with dry air to avoid water vapor absorption of the THz radiation
along its propagation.
An electric-field waveform of the THz radiation E0(t) was first recorded as the reference waveform with no sample in the path of the THz beam. A second (signal) waveform E(t) was then measured when a sample was inserted into the beam
perpendicularly. The complex refractive index n of the sample of interest at each
angular frequency ω was extracted by solving the equation:
36
E()ω pt t = 12 21 2 (2.1) E0 ()ω 1(− pr21 )
Here E(ω) and E0(ω) are the Fourier transforms of the signal and reference THz electric- field waveforms, respectively, p= exp( inω l c ) is the wave propagation factor through
a film of thickness l, c is the speed of light in vacuum, and r and t are the Fresnel
reflection and transmission coefficients at the air (medium 1) and film (medium 2)
interfaces. For normal incidence these coefficients are:
4n tt = (2.2) 12 21 (1+ n ) 2
1− n r = (2.3) 21 1+ n
Similar procedure was used to measure the transmittance of the photonic
crystals, which was defined through the ratio of the Fourier transforms of the reference
and signal waveforms as:
2 E()ω T ()ω = (2.4) E0 ()ω
Here the reference wave was measured with the quartz substrates and the signal
waveform was recorded with the photonic crystal between two quarts substrates.
2.3 Transfer Matrix Method
When dealing with layered media with complex structure, the present of
multiple reflections in between dielectric interfaces and consequently brought in
37
coupling of electromagnetic fields on each interfaces make it difficult to understand the
propagation of electromagnetic waves, and only a few simple cases could be worked out
accurately. The introduction of transfer matrix method could solve this problem, and
makes the analyses of interaction between electromagnetic waves and system of
complicated dielectric features become possible.
2.3.1 Matrix form in wave propagation
The reflection and transmission of a monochromatic plane wave at a plane
interface between two different dielectric materials are familiar. Let n1 and n2 be the refractive indices of medium 1 and 2 beside the interface, the propagation directions of reflection wave and refraction wave are determined by Snell’s law:
nnn11sinθθir= sin = 2 sin θ t (2.5)
Assume the plane wave is incident in xz-plane, with the interface lying in the yz-
plane, the amplitude of the reflected and transmitted waves would be determined by
the amplitude of incident wave and properties (εi ,mi) of both media by introducing the
boundary conditions at interface x = 0 (Figure 2.1). The general expression of fields
takes the form:
−−' ω (EEeiikr11+<' eex kr),it 0 E = 11 (2.6) −ikr2 itω E2ee,0 x>
38
' where E1 , E1 , and E2 are amplitude of the incident, reflected, and transmitted fields,
' respectively, and k1 , k1 , and k 2 are the wave vectors of each field, with their tangential component to be equaled:
' kkk112zz= = z (2.7)
And the corresponding magnetic field H can be obtained by Maxwell equation for electric fields:
i HE= ∇× (2.8) ωm
Figure 2.1 A plane wave E1 with wave vector k1 and frequency ω is incident from medium 1 with
permittivity ε1 and permeability m1 to medium 2 with permittivity ε 2 and permeability m2 . The reflection and transmission happen at interface of two dielectric media, and the reflected field and transmitted field
' are observed at direction θ1 and θ2 with wave vectors k1 and k 2 , respectively.
39
Since the boundary conditions of electric fields and magnetic fields require the
continuous of their components Ey , Ez , H y , H z at interface x = 0, if we break down the field into two independent components, one perpendicular to the incident xz-plane
(subscripted by s) and the other parallel to the incident xz-plane (subscripted by p), it is convenient to derive the reflection coefficient and transmission coefficient for s- polarized component:
' E1s nn1cosθθ 12− cos 2 rs = = En1s 1cosθθ 12+ n cos 2 (2.9) E2s 2n11 cosθ ts = = En1s 1cosθθ 12+ n cos 2
and for p-polarized component:
E ' nncosθθ− cos r =1p = 1 22 1 p θθ+ En1p 1cos 22 n cos 1 (2.10) E 2n cosθ t =2 p = 11 p θθ+ En1p 1cos 22 n cos 1
And those are the well-known Fresnel’s coefficients.
Fields propagating through multiple interfaces inside layered media, although
more complicated, can be analyzed in the similar way as in the simple single-interface
case discussed above. For simplicity, assume the whole layers are isotropic and
homogenous, the general expression of fields on the interface between medium i and j
is:
−−' ω iikrii' kr it (EEiie+< ee), x 0 = E −−' (2.11) iikrjj+>' kr itω (EEjje ee), x 0
40
' ' where Ei and E j are the incident fields from medium i and medium j, and Ei and E j
' ' are fields in their reflection direction, respectively, and k1 , k1 , k 2 and k 2 are the wave vectors of fields, with k ' the mirror image of k ' with respect to yz-plane.
Take p-polarized-field case for example, all fields are parallel to the incident xz-
plane and the boundary conditions yield the continuity of component Ez and H y at interface x = 0 (Figure 2.2):
'' (EEip+=+ ip )cosθθ i ( EE jp jp )cos j ε (2.12) εi ''j (EEip−= ip )( EEjp − jp ) mmij
41
Figure 2.2 Demonstration of p-polarized plane waves propagation through plane interface between
medium i and j inside multilayered system. Ei and E j are fields approaching the interface from medium i
' ' (left) and medium j (right), and plane waves Ei and E j leaving the interface are observed at reflection
directions of θi and θ j , respectively. All the electric fields are polarized in the plane of incidence, while all the magnetic fields are perpendicular to the plane to satisfy the vector rules determined by Maxwell equations.
Those equations can be rewritten in the form of matrix:
EEip jp Dipp()''= D ( j ) (2.13) EEip jp with
cosθθ cos = Dp (2.14) εm− εm 42
The matrix Dp is called the dynamical matrix of p-polarized field in the medium, and θ ,
ε , and m are parameters of the incident angle, the permittivity and the permeability of plane waves in each medium.
Similarly, the matrix form of s-polarized field E propagating through interface can be established as:
Eis E js Diss()' = D ( j )' (2.15) Eis E js
with the dynamical matrix Ds to be
11 = Ds (2.16) εmcos θ− εm cos θ
2.3.2 Wave propagation in layered system
It is apparent that the matrix form of fields’ relation beside interface is independent of the interactions of them with other interfaces. This method has overcome the difficulty of fields coupling inside layered structure initiated by multiple reflections between interfaces successfully, and therefore, was chosen to solve the plane wave propagation problems in layered system systematically [80].
For system with large number of homogenous and isotropic layers (Figure 2.3), the dielectric function is written as:
43
n00, xx< n, x<< xx 1 01 nx( )= ...... (2.17) n, x<< xx N NN−1 nNN+1, xx<
where nl (lN=1,2,..., ) is the refractive index of lth layer, xl is the position of interface
between the lth layer and (l + 1)th layer, n0 and nN +1 are the refractive indices (RI) of
substrates beside the multilayer structure.
Figure 2.3 Multilayered structure is fabricated between substrates n0 and nN +1 . The lth layer with
dielectric function nl has thickness of dl . The electric field propagating in layer is sum of right-
travelling wave Al and left-travelling wave Bl
44
The thickness of each layer dl can be written as function of interface
coordinates xl :
d110= xx −
d2= xx 21 − ... (2.18)
dxxNNN−−−1= 21 −
dxNN=−1 − x N
The monochromatic electric field at frequency ω inside is superposition of right- travelling wave and left-travelling wave in each layer, and has the general form of:
−− − Aeik00xx() x x+< Be ik 00 () x x , x x 00 0 −−iklx() x x l iklx () x− x l E() x= Ael+ Be l, xll−1 << x x (2.19) −− − ik(Nx++ 1) () x x N ik(Nx 1) () x x N xx< AeNN++11+ Be , N
where Al and Bl represent the amplitude of right-travelling and left-travelling waves at
− interface xx= l of lth layer, AN +1 and BN +1 are the magnitude of right-travelling and left-
+ travelling waves at interface xx= N , respectively, and klx is the x-component of wave
vector kl in lth layer:
knlx= lωθcos l / c (2.20)
and the choice of the sign in the phase is depending on the travelling direction of waves.
For most natural material, the permeability is always close to unit ( ml ≈1), so the
dynamical matrix becomes:
cosθθ cos Dl = (2.21) nnll−
for p-polarized wave, and 45
11 Dl = (2.22) nnllcosθθ− cos for s polarized wave, respectively. And as shown in last section, the amplitude of waves is linked by dynamical matrix
A1 −1 A0 = DD10 (2.23) B1 B0 and matrix
AAll−1 −1 = PDll D l−1 ,l= 1,2,..., N , (2.24) BBll −1
−1 where Dl is inverse matrix of Dl , and matrix Pl is the propagation matrix taking into
+ − account the field propagation from interface xx= l−1 to interface xx= l in lth layer:
φ ei l = Pl −iφ (2.25) e l
with φl= kd lx l the propagation phase in lth layer .
Now, incident fields throughout the multilayered structure can be related:
N AAN +10 −1 −−11 = DN+10()∏ DPD ll l D (2.26) BBN +10 l=1
Where A0 and B0 are the amplitudes of incident fields travelling to the right and left at
− surface xx= 0 (from substrate n0 ), and AN +1 and BN +1 are the corresponding amplitudes
+ at surface xx= N +1 (from substrate nN +1 ), of multilayered structure.
46
2.3.3 Transfer matrix of photonic crystal
One dimensional (1D) photonic crystal is a special class of multilayered structure with periodically varying dielectric function. In the following section, the transfer matrix method is employed to analyze the performance of a simplest 1D photonic crystal,
consist of alternating stacked two different layers n1 and n2 .
With the same geometry as previous discussion, the periodic function of RI could be expressed as:
n, ( ndxndd− 1) << ( − 1) − nx()= 21 (2.27) n11, ( n− 1) d − d << x nd
where thickness d1 and d21= dd − are the layer thicknesses of materials n1 and n2 , respectively, in the stack and d is the period of photonic crystal (Figure 2.4).
47
Figure 2.4 Demonstration of 1D photonic crystal comprising of alternatively stacked layers of index of
refraction n1 and n2 . The physical period of crystal is d with neighboring layers to be d1 and d2 in
thickness. The amplitude of fields inside nth unit cell are denoted as an , bn , cn , en , respectively.
Following the discussion of (2.19), the propagation of monochromatic wave in nth unit cell of 1D photonic crystal can be written as:
−−ik11xx() x nd ik () x− nd aenn+ be , nd− d1 << x nd Ex()= − −+ −+ (2.28) ik21xx() x nd d ik 21 () x nd d − << − cenn+ ee , (n 1) d x nd d1
where an and bn are the amplitude of right-travelling and left-travelling waves with x- component of the wave vectors
kn11x = ωθcos 1 / c (2.29)
48
in medium n1 , and cn and en are the amplitude of right-travelling and left-travelling
waves with x-component of the wave vectors
kn22x = ωθcos 2 / c (2.30)
in medium n2 .
In the same means as in (2.24), the amplitude of fields an , bn , cn , en are associated as:
acnn−1 = PD11 D 2 , (2.31) benn
and
cann−1 −1 = PD22 D 1 , (2.32) ebnn −1
where those dynamical matrixes and propagation matrixes are expressed with layer
properties of medium n1 and n2 , respectively:
cosθθii cos for p-polarized nnii− Dii = ,= 1, 2 (2.33) 11 for s-polarized nniiiicosθθ− cos
and
eikix d i = = Pii −ik d , 1, 2 (2.34) e ix i
Substituting (2.33), (2.34) into (2.31) and (2.32), and relating (2.32) with (2.31) to
eliminate the column vector of fields in medium n2 , we obtain the matrix equation:
49
aann −1 = T (2.35) bbnn −1
The matrix elements of T for p-wave are:
22 − i nk nk =ik11x d −+21xx 1 2 T11 e[cos kd22xx ( 22)sinkd22 ] 2 nk1 2xx nk 21 22 − i nk nk T=−− eik11x d [( 21xx 1 2)sinkd ] 12 2 nk22 nk 2x 2 1 2xx 21 (2.36) i nk22 nk = ik11x d 21xx− 1 2 T21 e [( 22)sinkd2x 2 ] 2 nk1 2xx nk 21 i nk2 n2k =ik11x d ++21x 12x T22 e[cos kd2x 2 ( 2 2 )sinkd22x ] 2 nk12x nk21x
And slightly different from p-wave, the s-wave case has the matrix elements:
−ik11x d i kk21xx T11= e[cos kd22xx −+ ( )sinkd22 ] 2 kk12xx
− i kk T=−− eik11x d [ (21xx )sinkd ] 12 2 kk 2x 2 12xx (2.37)
ik11x d i kk21xx T21 = e [ (− )sinkd2x 2 ] 2 kk12xx
ik11x d i kk21xx T22= e[cos kd22xx ++ ( )sinkd22 ] 2 kk12xx
As a result, all the column vector of fields in medium n1 inside the photonic crystal structure can be related:
aan n 0 = T (2.38) bbn 0
And the rest fields appeared in column vector of medium n2 can be derived with assistance of (2.32). Those features provide systematical way to field analyzing of photonic crystal, especially by means of computer simulation.
50
Finally, it is worthwhile to point out that the photonic crystal prepared in the lab is usually considered surrounded by the air. The air coatings and they induced two vacuum/materials interface at the end of 1D crystal structure need to be taken into account in simulation, and this will modified the result.
2.4 Local field and effect-medium theory
2.4.1 Homogenous system
Conventionally, the macroscopic response of medium (say susceptibility χω())
to external applied field E()ω is determined by dielectric function of this medium in microscopic scale (say molecule’s polarizability αω() ). To the most naïve extent, the
physically macroscopic and microscopic properties could be related by density of
microscopic components (say molecules) N :
χω()= N α0 () ω (2.39)
This is valid for dilute system where the molecules are far away from each other, and
interaction between molecules is negligible.
In condensed homogenous system where the distances between molecules are
no longer small, the external field induced dipole moment on molecules would cause a
local field E()ω , and makes a difference between the applied field E()ω and the field that really experienced by molecules Eloc ()ω :
Eloc ()ωω= EE () + () ω (2.40)
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If limit our discussion in isotropic medium, where the dielectric properties can be represented within scalar system, the local field arising from polarization of neighboring spherical molecules could be found [95]:
EP()ω= ()/3 ωε0 (2.41)
From microscopic point of view, the dipole moment of given molecule, as microscopic characteristic of molecule, is proportional to electric field acting on the molecule:
loc pE()ω= εαω0 () () ω (2.42) with αω() the molecule polarizability. Considering the fact that the collective effect of bound dipole moment p()ω , by averaging over a macroscopically small, but
1 microscopically large, volume V , is the polarization P()ω : P()ω=∑ pp () ωω = N (), V v the local field Eloc ()ω could be related to polarization P()ω as:
loc PE()ω= ε0 N αω () () ω (2.43)
After inserting (2.43) into (2.41) and put it back to (2.40), the relation between local field Eloc ()ω with external field E()ω is established:
EE()ω= (1 − N αω ()/3)()loc ω (2.44)
On the other hand, the field relation is straightforward by substituting the definition of
macroscopic effective electric susceptibility PE()ω= εχω0 ()() ω into (2.41) and (2.40) in turn, from the macroscopic point of view:
EE()ω= 3loc ()/(()3) ω χω + (2.45) 52
Comparing of (2.44) and (2.45) will lead to the relation between macroscopic and
microscopic dielectric properties of homogenous system is:
Nαω() χω()= (2.46) 1− Nαω ( )/3
A correction factor of β=−>1 / (1N αω ( ) / 3) 1 is found in order to take into
account the local field effect in microscopic scale, and eliminate possible overestimate
of microscopic parameter αω() determined without the present of local field from
(2.39). An alternative and more general used expression of (2.46) is to write the
molecule polarizability αω() in terms of relative dielectric constant εω()/ ε0 = χω ()1 + :
εω( )/ ε − 1 Nαω() 0 = (2.47) εω( )/ ε0 + 2 3
This famous Clausius- Mossotti equation was found by Mossotti (in 1850) and Clausius
(in 1879) independently [95].
2.4.2 Two-phase system
Since the original Clausius- Mossotti equation could only be used to treat single
phase homogenous systems, a modified version was generated later, connected the
macroscopic properties (effective dielectric constant) of multiple-phase system εωeff ()
to each of its constituents. Take system with two different constituents of dielectric
function εω1()and εω2 () for example, the polarization, to the first-order
approximation, could be derived by following equation (2.43):
53
loc PE()ω= ε0 (NN 11 αω () + 2 αω 2 ()) () ω (2.48)
where αω1() and αω2 ()are the polarizability of molecule of two dielectrics, and N1 and
N1 are the density of dipole moment p1()ω and p2 ()ω in system, respectively. In the
same way as (2.47), the relation of molecule polarizabilities αω1(), αω2 ()and mixture’s
effective dielectric constant εωeff () could be derived:
ε( ωε )/− 1 (NNαω ()+ αω ()) eff 0 = 11 2 2 (2.49) εeff ( ωε )/0 + 2 3
If molecules exhibit the same microscopic dielectric behavior in mixture system as in
single-phase system (bulk material), the equation (2.47) could be employed in (2.49), by
substituting each molecule polarizabilities with its bulk dielectric function in single-
phase system separately, and the Clausius- Mossotti equation for two-phase system is
produced:
ε ωε− eff ( )/0 1 εωε10( )/−− 1 εωε 20 ( )/ 1 = ff12+ (2.50) εeff ( ωε )/0+++ 2 εωε 10 ( )/ 2 εωε 20 ( )/ 2
where the fNNNii=/(12 + ) is the volume fraction of molecules, and satisfying
ff12+=1.
Another approximation of Clausius- Mossotti equation for two-phase system is
achieved by Maxwell Garnett [96, 97] when considering nanomaterials with
nanospheres of size much smaller than incident wavelength, but sufficiently large to
possess dielectric features. When fraction of nanospheres is much smaller than host
material ffgh =1 − f g, instead of thinking both nanostructure and matrix materials
54
embedded in the vacuum, the picture of guest nanospheres dispersed in host matrix materials is proposed (Figure 2.5a). And the equation (2.50) would then be simplified by
replacing dielectric constant of vacuum ε 0 with the host medium εωh () in (2.50):
εeff () ω−− εωh () εωg () εωh () = fg (2.51) εeff () ω++ 2 εωh () εωg () 2 εωh ()
within which the effective medium εωeff () is determined by host εωh (), guest εωh ()
and the volume fraction of guest nanospheres fg .
Although Boyd et al. [98] claimed using of volume fraction fg = 0.5 in their theoretical studies as upper bound, an approximation that could be used without the limit brought in by guest- host picture is desired. Bruggeman effective-medium theory
[99] is one of the theories that removed asymmetry of guest and host in two phase system, and satisfy the requirement. In this theory, both of components were assumed to be incorporated in a medium of so-called “effective medium” (Figure 2.5b), and
hence the effective dielectric constant εωeff () was suggested to replace ε 0 in Clausius-
Mossotti equation (2.50). The left-hand side of equation (2.50) was canceled, and right- hand side of equation (2.50) was left comprised of bulk properties of each constituents
εω1()and εω2 (), and their volume fraction in mixture:
εωεω12()−−eff () εωεω ()eff () ff12+=0 (2.52) εωεω12()++ 2eff () εω () 2 εωeff ()
55
More general theory was found by Zeng et al. [100] to describe fillers with more
complicated topological structure by introducing the geometric factor g into
Bruggeman equation (2.52):
εωεω12()−−eff () εωεω()eff () ff12+=0 (2.53) gg[()εωεω12−+eff ()] εeff [ εωεω () −+eff ()] εeff
For spherical components, g =1/3, the equation decays back to (2.52), and for two- dimensional circular components, g =1/2.
2.4.3 Layered structure
Another big type of two-phase system is layered structure (Figure 2.5c). The
system is composed of layered dielectrics εω1()and εω2 (), and their volume fraction were obtained by comparing layer thickness of each materials:
fii= lll/(12 + ) (2.54)
with ff12+=1.
For s-polarized beam with electric field parallel to the layers, the continuity of
electric field in layer plane requires the averaged electric displacement of whole
structure to be:
DDss()ω=+=+f1 1, () ω f2 D 2, s () ω ( ff1 εω 1 () 2 εω 2 ())()Es ω (2.55)
Depending on definition DE()ω= εωeff ()() ω, the effective dielectric function can be
derived:
εeff, s () ω=ff 11 εω () + 2 εω 2 () (2.56) 56
For p-polarized field, in the same way, the continuity of electric displacement normal to the layers yields the average of electric field as:
EEpp()ω=+=+ff1 1, () ω2 E 2, p () ω ( ff1 / εω 1 () 2 / εω 2 ())()Dp ω (2.57)
And combined with the effective dielectric constant definition again, (2.57) will give:
1/εeff, p () ω=ff 11 / εω () + 2 / εω 2 () (2.58)
It is notable from (2.56) and (2.58) that the effective dielectric function is different for the s- and p-polarized waves, introducing birefringence into layered structure. Further investigation in [101] showed that the conclusion of anisotropy could also be applied to nonlinear processes for layered system.
57
(a)
(b)
(c)
Figure 2.5 Schematic diagram of two-phase systems that could be applied by a) Maxwell Garnett, b)
Bruggeman, and c) layered structure effective medium theory
2.5 Study of All Polymeric THz Photonic Crystal
2.5.1 Polymer films incorporated with nanoparticles
To study the feasibility of our approach we employed a model system consisting of PMMA as a common melt-processible matrix polymer and BaTiO3 as a ferroelectric
material that exhibits a high RI in the THz regime [102]. As introduced, PMMA/BaTiO3
blends were prepared by conventional melt-mixing in a counter-rotating twin-screw
extruder and the materials were subsequently processed into ~150 μm thin films by 58
compression molding. The BaTiO3 content was systematically varied between ca. 2 and
20 % v/v and two different particle sizes (100 and 200 nm) were explored. All materials prepared are white-opaque in the visible spectral region, due to scattering of light. SEM images of the nanocomposites containing BaTiO3 nanoparticles of 100 nm and 200 nm diameter at low (~5%) and high (~18%) concentration are shown in Figure 2.6.
(a) (b)
(c) (d)
Figure 2.6 Scanning electron microscopy (SEM) images of fractured surfaces of PMMA/BaTiO3 nanocomposites. (a) 6% v/v, particle size 100 nm ; (b) 19% v/v, 100 nm; (c) 3% v/v, 200 nm; (d) 18% v/v,
200 nm.
The SEM images show that in all cases the nanoparticles are rather well dispersed in the PMMA matrix, with most particles well isolated from one another or in small aggregates of <10 particles The SEM images also show occasional zones (see
59
Figure 2.7, analysis of the available images indicates that these ‘defects’ account for
0.025% of the sample’s volume) with typical dimensions of ~20 x 40μm that are
comprised of densely aggregated BaTiO3. These zones were primarily found near the
surface of the extrudate, and are perhaps the result of wall effects during the extrusion
process.
(a) (b)
Figure 2.7 Scanning electron microscopy images of fractured surfaces of PMMA comprising BaTiO3
nanoparticles: (a) 6% v/v, particle size 100 nm; (b) 3% v/v, particle size 200 nm. The images show
occasional zones of aggregated BaTiO3
It is important to estimate the magnitude of the scattering effects which the
observed BaTiO3 aggregates can exhibit in the THz regime. For the composite to act as an effective medium with small scattering losses, the size of the filler clusters should be much smaller than the wavelength λ ~ 0.3mm (1 THz). In this limit, the interaction
between the radiation and the particles can be treated as Rayleigh scattering. The total
cross section of Rayleigh scattering for a spherical particle of diameter d and RI of nf
embedded in a host of RI nh is given by [95]:
60
21π 56dn 2− σ = ()2 (2.59) 32λ 42n +
where n = nf /nh is the relative RI of the particle. The attenuation coefficient α of the
radiation due to scattering in the limit of dilute medium can thus be estimated from:
d 3 α 4 πη4 (2.60) λ 4
where η is the volume fraction of the particles. For the nanocomposites studied here, it
is assumed that the fillers are randomly distributed in the host polymer and the relative
RI of the filler n>>1, which, as shown below, is appropriate for BaTiO3 (nf ~ 10) in PMMA
(nh ~ 1.5). This estimate suggests that at 1 THz, the attenuation for nanocomposites containing 10% v/v amounts to ~ 10-4 cm-1 and 10-1 cm-1, for particle diameters of 1 and
10 mm, respectively. These values are small in comparison to the absorption coefficient
of many neat polymers [21]. Therefore, scattering of THz radiation in dilute polymer
composites is negligible even with filler particles as large as 10 mm.
2.5.2 THz properties of polymer films
The frequency dependence of the complex RI of these composites in the THz
spectral region was characterized by the THz time-domain spectroscopy. Figure 3.8
summarizes the results for three representative samples comprising 0, 3, and 18% v/v
BaTiO3 (particle size: 200 nm), in the spectral range of 0.2-1.5 THz. The neat PMMA
(Figure 2.8a) has a real part n’ of RI ~ 1.54, which is essentially flat in this spectral
window. Its absorption, reflected in the small imaginary part n” of the RI, is not
61
significant. The absorption coefficient increases rapidly with frequency. At 1 THz it is ~
12 cm-1. These results agree well with data in the literature [21]. Upon incorporating
BaTiO3 nanoparticles into PMMA, both n’ and n’’ increase. n’ of the composite
comprising 18% v/v BaTiO3 is increased to ~2.26, which represents an increase of almost
50% compared to that of neat PMMA. The typical experimental uncertainty (about 3%)
arose primarily from the spatial variation of the samples both in their composition and
thickness. The dependence of the RI on the frequency is relatively weak in the frequency
range of 0.2-1.5 THz. No significant dependence on the size of the particles was observed (by comparing Figure 2.8 and Figure 2.9).
a)
(b)
62
(c)
Figure 2.8 Real (n’) and imaginary (n“) parts of the THz refractive index of PMMA/BaTiO3 nanocomposite films containing a) 0%, b) 3%, and c) 18% v/v BaTiO3 (particle size: 200 nm). Solid lines represent effective medium theory calculations
a)
(b)
63
(c)
Figure 2.9 Real (n’) and imaginary (n“) parts of the refractive index of PMMA/ BaTiO3 nanocomposite films containing a) 0%, b) 6%, and c) 19% v/v BaTiO3 (particle size:100 nm). Solid lines represent effective
medium theory calculation.
The experimental RI data for the nanocomposites were compared with
calculations according to effective medium theories (solid lines, Figure 2.8). Data for the
neat PMMA matrix were taken from Figure 2.8a, while the properties of the BaTiO3
nanoparticles, which defer significantly from that of bulk BaTiO3 crystals, were extracted
from Ref.23. Several effective medium models (see section 3.5) including the
geometrical mean model [103] were found to describe the experimental results
satisfactorily. In Figure 2.8 results from the geometrical mean model are shown, in
which the RI of the composite nc is related to the RI of the filler and the host by:
ηη1− nnnc= ( fh )( ) (2.61)
64
a) (b)
Figure 2.10 Real part (n’) of the refractive index of nanocomposite a) PMMA/BaTiO3, b)
poly(styrene)/BaTiO3 films at 0.8 THz as a function of BaTiO3 content. The solid line represents an
effective medium theory calculation
This model has also been successfully used to describe the properties of polymer/BaTiO3 nanocomposites up to GHz frequencies [89]. The effective medium
model was further tested by studying the dependence of nc on the volume fraction η of
BaTiO3. Figure 2.10 compares experimental data of the real part of nc measured at 0.8
THz and values predicted by the geometrical mean model for different volume fractions
in the range of 0-20%. The experimental data again agree with the effective medium model within the experimental uncertainties. No significant influence of the particle size was observed. To examine the effect of polymer matrix on the dielectric response and to demonstrate broader applicability of the approach, both nanocomposites comprised of PMMA/BaTiO3 and poly(styrene)/BaTiO3 were fabricated. These materials were
found to behave in a similar manner (see Figure 2.10a and b) and their experimentally
determined RIs correlate well with the values predicted by the effective medium theory.
65
Thus, composite polymeric materials with the desired value of the RI can be designed
and engineered using the simple extrusion method. Such materials can potentially be extremely useful for THz device applications.
2.5.3 Polymeric THz photonic crystal
To demonstrate that the melt-processed PMMA/BaTiO3 nanocomposites display properties that are useful for THz devices, we elected to fabricate THz Bragg mirrors, which are important elements for the manipulation of THz radiation. The reflectivity R of such a photonic crystal structure at the stop bands is determined by the ratio of the RI of the two neighbouring layers n1/n2 and the number of the layer pairs N [76]. In
absence of a substrate, the reflectivity at the stop band centre can be expressed by:
n RN= tanh2 ( ln(1 )) (2.62) n2
Thus to achieve, for example, a reflectivity of 95%, only 6 pairs of alternating layers of neat PMMA and a 18% v/v PMMA/BaTiO3 nanocomposite have to be used if losses are
ignored. We chose to create a device with a stop band in the frequency range of 0.2-0.3
THz, which is of particular interest for short-range wireless communications. The quarter-wave stack design for photonic crystals (nidi = λ/4) calls for layer thicknesses di
of ~151 and ~189 mm for the nanocomposite and PMMA, respectively.
The individual films were stacked to form a photonic crystal (Details see section
2.2.2). The element was characterized by THz time-domain spectroscopy (TDS) in
transmission mode. The electric-field waveform of the signal (photonic crystal between 66
two quartz substrates) and reference (two quartz substrates without the photonic crystal) in a time window of 50 ps are shown in Figure 2.11a. The radiation after 10 ps in the reference waveform (solid line) is due to multiple reflections at the substrates and the various optical components of the spectrometer. The pulse was significantly modified upon insertion of the photonic crystal into the beam path (dashed line, Figure
3.11a). The properties of the photonic crystal can be more easily understood in the frequency domain (Figure 2.11b). The transmission of the structure (solid line) was determined from the experimental waveforms shown in Figure 2.11a. A stop band is clearly observed at 0.25 THz with a width of about 0.1 THz. The transmission in the stop band is as low as 5%, a clear indication of a photonic crystal of satisfactory quality. For a more quantitative examination, we simulated the spectral dependence of the transmission of the structure based on the transfer-matrix formalism [76]. Figure 2.11b reveals that the simulation agrees very well with the experimental data, including the position of the stop band, its width and transmittance.
a)
67
(b)
Figure 2.11 Optical properties of Bragg mirrors consisting of 6 alternating double layers of PMMA and a
18 % v/v PMMA/ BaTiO3 nanocomposite between two quartz substrates. (a) THz electric-field waveform of the signal (dashed, off-set from zero electric field) and reference (solid) in the time domain.
(b) Transmittance in the frequency domain
Minor discrepancies between the model and experiment, particularly in the transmission band, are likely due to minor irregularities of the periodic structure and the inhomogeneity (both in composition and thickness) of the composite layers. Figures
2.11b also shows that the performance of this photonic crystal is limited to frequencies
< 0.5 THz. This is primarily due to the intrinsic properties of BaTiO3, which has a large
absorption above 0.5 THz due to soft phonon modes. Alternative filler and matrix materials with low absorption in THz, such as calcite and polystyrene, will be explored next to improve the performance of the all solid THz photonic crystals.
68
2.6 Conclusion
In conclusion, we demonstrated that polymeric materials with engineered RI for applications in the THz regime can readily be fabricated and processed by conventional melt-processing of inorganic nanomaterials and conventional host polymers. Because of the long wavelength nature of the THz radiation, scattering due to aggregation of the inorganic, RI-increasing fillers does not play a major role, and the RI can continuously be tailored through simple compositional variation. The approach appears to be broadly applicable to other fillers and matrix polymers and opens up many new opportunities for applications of layered polymeric materials in controlling and manipulating the technologically relevant THz radiation.
69
Chapter 3
THz Emission from Graphite Surfaces
3.1 Introduction
Our discussion of the THz time-domain spectroscopy so far concerns the use of the THz radiation emitted from a source for spectroscopic studies of samples of interest.
The radiation, however, can also be investigated to learn the properties of the emitter itself. This approach is often called the THz emission spectroscopy.
Noncentrosysmetric crystals such as ZnTe and GaP have been identified as one of the first THz emitters. They rely on optical rectification of the femtosecond ultrafast light pulses based on the bulk second-order nonlinearities [104]. Shortly, emission from centrosymmetric crystals was also reported [105] [106] [107], and was assigned to second-order nonlinearities from the surfaces/interfaces, where the inversion symmetry of the centrosymmetric materials is broken [108]. Later, systematic experimental [109]
[110] and theoretical studies based on phenomenological models [111] [112] have been performed to establish the connection between the THz emission and the surface properties of the emitters.
The second example of the application of the THz emission spectroscopy concerns the surge of photo-excited carriers at/near the emitter surfaces. Upon the
70
excitation by femtosecond optical pulse, a transient current or polarization is formed
and give rise to the THz emission. An external bias field can be employed to drive the carriers as in the photoconductive emitters. In unbiased semiconductors the built-in
depletion/accumulation field formed by band bending at the semiconductor surfaces
can drive photoexcited carriers and produce polarization perpendicular to the surface.
The Transient current can also be formed through carrier diffusion, resulting from
density gradient of photocarriers, in material surfaces/interfaces with unequal electron
and hole mobilties. In the in-plane direction diffusion usually results in zero net current
because of the centrosymmetry. In the direction perpendicular to the surface, a charge
dipole in the vicinity of a semiconductor surface is formed due to the break of symmetry
provided by semifinite nature of the surface. In an isolated sample, where the
macroscopic flow of a current is prohibited, an ultrafast photo-Dember field is
established. THz emission based on this mechanism is known as the photo-Dember
effect.
The THz emission spectroscopy has been successfully employed in a variety of
studies. The technique has been applied to study surface nonlinearities [113], to design
broadband devices [114] and for imaging and sensing applications [115]. It has also been
applied to investigate the dynamics and transport properties of photoexcited carriers in
a variety of model systems ranging from traditional semiconductor materials [116],
inorganic crystal [117], supercondcutors [118], semiconductor heterojunctions [66], to
71
organic materials [54]. Thin metal films [64] [65] including ferromagnetic materials [63]
[62] have also been studied.
In this chapter, we will focus on the study of ultrafast dynamics of photoexcited
carriers on graphite surfaces by the THz emission spectroscopy. The topic is fundamentally important in view of the recent interest in graphitic materials such as carbon nanotubes and graphene. It is also important for applications such as carbon- based high-speed electronics, photodetectors and other optoelectronic devices.
Ultrafast dynamics in graphite has been studied using the time-resolved optical
reflection and transmission spectroscopy [119] [120] [121] and optical pump-THz probe
spectroscopy [122]. Valuable information has been learned about the carrier relaxation
and cooling dynamics in graphite. THz emission was observed from graphite surfaces in
an independent study [123] although the mechanism is still unclear. In this chapter we
describe a systematic study of the THz emission process from graphite surfaces
including the fluence, polarization and crystal orientation dependence. We have also
developed a double pump emission correlation measurement [24] which helps to
elucidate the relaxation dynamics of photo-excited carriers relevant for THz emission.
This mechanism could be important for highly absorbing materials.
The chapter is organized as follows: First of all, the setup of THz emission
spectroscopy is introduced in Sect. 3.2. The part two was a comprehensive study of THz
emission from graphite basal planes covering from its generation mechanism, to the
fluence dependence and correlation dynamics. This will be of the scope from Sect. 3.3 to
72
Sect. 3.5. The last part (Sect. 3.6) is the complementary investigation to the emission mechanism, including the polarization resolved analysis for emission from both edge plane and basal planes of graphite.
3.2 Experimental Setup
In this experiment, we employ the THz emission spectroscopy to investigate highly oriented pyrolytic graphite (HOPG). The graphite samples (SPI Supplies, grade SPI-
1) were 10×10×1mm in size, with lateral grain size of 0.1mm. The experimental setup is modified from the conventional THz time-domain spectrometer as shown in Figure 3.1: an 800 nm ultrafast laser beam was focused onto the surface of graphite. The incident angle is 45 degrees if not otherwise specified. The laser beam was about 1 mm in radius, and the power of laser beam was systematically controlled up to a maximum of 50 J/m2
which is well below the known optical damage threshold of HOPG 1300 J/ m2 [124]. The
reflected THz emission was guided and collected by a pair of parabolic mirrors and focused onto a ZnTe THz detector (More details on the THz detection by electro-optic
sampling in ZnTe and the balance detection scheme refer to Chapter 1). Wire grid polarizers with an extinction ratio of ~ 100:1 were inserted in-between the parabolic mirrors to analyze the polarization state of the THz radiation. A pair of polarizer is necessary in order to calibrate the detection sensitivity of THz spectrometer in different polarization direction.
73
Figure 3.1 Setup of THz Emission spectroscopy
3.3 THz Emission from Graphite Surfaces
An electromagnetic (EM) transient was detected from graphite surfaces (basal
plane) upon the optical excitation by fs pulse. The electric-field waveform is shown in
Figure 3.2. The EM transient has duration of ~ picosecond, and hence contains spectral components in the THz region. The emission is peaked along the reflection direction.
This directionality of the THz emission indicates its coherent nature, and excludes the
mechanism of thermal emission [125]. This observation is compatible with the
independent study of [123].
Within the experimental accuracy we did not observe any discernible
dependence of the emission on the azimuthally rotational angle around the surface 74
normal (Figure 3.3). This can be understood since the sample, with domain sizes much smaller than the pump beam size, can be considered isotropic.
We also found that the emission depends on the direction of the linearly polarized optical pump beam as a cosine function (Figure 3.4). This dependence can be accounted for by the polarization dependence of the linear reflection and scattering of the pump beam at the surface of graphite. This result indicates that the THz emission from the basal plane is independent of the pump polarization.
Figure 3.2 Picosecond THz emission from basal surface of HOPG, and its spectrum
75
Figure 3.3 The rotational dependence of THz emission signal. The emission signal was
almost kept at same level while the HOPG sample rotated along its c-axis
Figure 3.4 The THz emission signal at different linear polarization status of optic pump.
The polarization direction angle of each status was with respect to p-
polarization
Furthermore, we found the emission from the basal plane is predominately p- polarized. It can be attributed to polarization along the surface normal, i.e. the c-axis of
graphite. Polarization in the basal plane can also give rise to the p-polarized THz
emission. But as we discuss below, this component is likely to be negligible.
Graphite, belonging to the symmetry group of 6/mmm, is a centrosymmetric
material and THz emission by bulk second-order nonlinearity is forbidden. The observed
THz emission from graphite surfaces can arise only from the surface second-order
76
nonlinearity or from the injected transient current by optical excitation [113]. In general,
the effect of optical rectification due to the surface nonlinearity is much less important than the current transient involving photo-excited carriers in semiconductors [126].
Similar result is expected for graphite, which presumably also explains the pump
polarization independence observed experimentally. In an unbiased sample, current can
form either due to the surface accumulation/depletion field or/and the photo-Dember
effect. Since the emission was not very sensitive to the surface condition and
environment which can change the doping and surface fields, the photo-Dember effect
is likely the main mechanism for THz emission from graphite basal planes.
3.4 Fluence Dependence of THz Emission from Graphite Basal
Planes
In this section, we study the THz emission as a function of the pump fluence. The
measurements were performed on the basal plane of HOPG, in the power range from 0
2 to 50Jm / . The waveform of the generated THz pulse was found to be largely
unchanged, which corresponds to the fluence independence of the emission spectrum.
Moreover, the THz radiation field depends linearly on the pump fluence at the low
fluence regime, and gradually saturate for fluences > 30Jm / 2 (see Figure 3.5).
77
Figure 3.5 Measured peak of the THz radiated electromagnetic field as function of optical fluence with
the saturation modle fitting in solid line
A microscopic theory has been developed to describe this sublinear behavior induced by surface field on semiconductor surface [127]. As we discussed above, the
main mechanism responsible for the THz emission from graphite basal plane surface is
the photo-Dember effect [46]. To demonstate the obseved sub-linear fluence
dependence of the emitted field, below we describe a similar phenomenological model
for our fluence dependence measurements based on photo-Dember effect (Figure 3.6).
78
Figure 3.6 The demonstration of graphite surface under optical excitation. The surface field and
radiated field both contribute to the surface current
Here, the separation of photoexcited electons and holes due to their different diffusivities sets up an effective dipole near the surface of graphite. The effective current due to the charge diffusion can be described as
Jznl =eD() − D ∂∂ n ˆ d he , (3.1)
where Dh , De are the out-of-plane diffusion constant of holes and electrons, respectively, i.e. along the c-axis of graphite. They are related to the carrier mobility by
Einstein’s relation:
D= () kT em (3.2) he., B he.
Here, T is the electronic temperature, and mhe, is the mobility for either holes or electrons. We have assumed that the system is undoped.
79
In a strongly absorbing material such as graphite for excitation at 800 nm, the initial photo carrier density is an exponential function of layer depth with a characteristic
thickness corresponding to the laser penetration depth d : n= n0 exp( − zd / ) . The initial photocurrent density can be rewritten as:
Jnl = nm − kT/ d d , (3.3) or in terms of surface sheet photocurrent density as:
− jdz, = n sm kT/ d (3.4)
m − m− = mm − where is the mobility difference eh, and ns =∫ ndz is the sheet photocarrier density near the surface.
The associated space-charge field, i.e. the Dember field EtD (), establishes on an ultrafast time scale, which also influenecs charge transport. Then, The net sheet current density is given by
−+ jz()t= n smm kT / d − n sD e E () t (3.5)
+ + where m is the sum of mobilities m= mmeh + . In another hand, the Dember field can be related to the sheet current density given by (0.56). Combining with equation (3.5) we thus obtain the Dember field:
− nts () kT m EtD ()= + (3.6) nss() t+ n′ ed m
where the ns′ is the effective photocurrent density:
2 + εε02zc 1 n′ = j/ Eem = (3.7) s zD tanθ sin θme +
80
In general the carrier mobility is also a function of time on the ultrafast time
scale. The values in Eq. (3.6) can be considered as their averaged values. In the limit of not very strong saturation, i.e., the radiated field approaches its maximum approximately when the surface conductivity reaches its maximum and we evaluate the
peak of radiated THz as:
nmax kT m − max = s ED max + (3.8) nss+ n′ ed m
max where ns is the initial photocarrier density on the surface of graphite, given by:
ω max (1− R ) nFs = (3.9) ω
with F the incident fluence of excitation and the percent reflectance of the
excitation beam at the surface, which was estimated to be 0.5 at 800 nm.
It is clear that the model of (3.8) predicts the fluence dependence observed in
the measurements: at low optical fluences , the radiated field is proportional
to optical excitation, and with the increasing of the optical fluence, the emitted field from graphite surface deviates from its linear dependence. At sufficiently large fluence,
max for which nnss ′ , the corresponding Dember field approaches the saturation value of
− max kT m ED = + (3.10) ed m .
max The critical fluence Fc corresponding to nnss= ′ is given as
n ′ω F = s (3.11) c (1− Rω )
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We use the saturation model of (3.8) to fit the fluence dependence of the peak
of the radiated field (solid line, Figure 3.5). There are two fitting parameters including
the critical fluence Fc and the emitted field after saturation. The critical fluence Fc was found to be 30Jm / 2 , corresponding to an effective sheet photocarrier density of
20 2 nms′ =1.2 × 10 / at the graphite surface according to (3.11). For the geometry of our
experiment with the incident angle of 45° for the excitation and an out-of-plane
dielectric constant of 10 [122] [128] [129], the sum of the electron and hole mobility
along the c-axis of graphite is determined from (3.7) to be:
(3.12)
The second fitting parameter is the saturation field, from which the difference
between the carrier mobilities m − could be estimated. The saturation field was found to
be 4 V/ cm . It was calibrated using an large-aperture GaAs photoconductive switch
[127], for which the bias field was fixed at 200 V/ cm . The saturation field observed
from graphite surfaces was ~ 2% of the value observed from the GaAs photoconductive
switch, which is given by the bias field. According to equation (3.10), the mobility difference is determined as
mm−+/≈× 2 10 −3 (3.13)
or m − ≈⋅0.4cm2 / V s by combing with the result for the sum of carrier mobilities.
Unlike the in-plane mobility, the out-of-plane mobility of carriers in graphite is
not well studied and no direct measurement was found in the literature. The out-of-
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plane thermal conductivity was measured to be about 100 times smaller than the in-
plane thermal conductivity [128]. We assume that the same ratio also applies to the
electrical conductivity based on the Wiedemann-Franz law [129]. If we take a typical in-
plane electron mobility of m ≈⋅10,000 cm2 / V s [130] for graphite, the out-of-plane
2 mobility can be estimated to be mc ~ 100cm / V⋅ s . This value is on the same order of
magnitude as the values extracted based on the simple photo-Dember model from our
experiment.
3.5 Correlation Dynamics of THz Emission from Graphite Surfaces
In this section we investigate the dynamics of photo-excited carriers that is relevant for THz emission from graphite surfaces by using a double-pump correlation technique. The setup is similar to the basic THz emission spectrometer except for the introduction of a Michelson interferometer into the pump arm [131] (Figure 3.7). The
optical excitation pulse is split into two pulses with a controlled delay time. We
modulate the second excitation pulse using an optical chopper and measure the
electric-field waveform of the THz emission using a lock-in amplifier. We compare the results for the case with and without the first pulse.
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Figure 3.7 Schematic illumination of double pump experimental setup
Figure 3.8 shows the THz emission detected without the first excitation pulse
( Et()in dashed line) and with the first excitation pulse ( in solid line). For this 20
example the delay time τ between the pulses is 1 ps. The fluences of the first and
second excitation pulse are, respectively, 40Jm / 2 and 10Jm / 2 . It is clear that the
presence of the first pulse enhances the generation efficiency of THz radiation from the
second pulse, and also narrows slightly the width of the radiated electric-field waveform.
As the delay time between the two pulses increases, the enhancement effect quickly decreases on a picosecond time scale and eventually diminishes for a delay time longer
than 10 ps (see Figure 3.9).
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Figure 3.8 The influence on THz emission from the present of earlier excitation. The solid line and
dotted line represent the THz radiation with and without the earlier excitation.
To qualitatively describe the effect, we first ignore the small changes in the
ττ= 2 = 2 waveforms. We construct parameters T22( )∫ E (, t ) dt and T20 ∫ E20 () t dt , the
square root of the total radiated energy induced by the second excitation pulse with and
without the first pulse. The delay time dependence of the parameters is shown in Figure
3.9. T20 , as expected, is independent of the delay time (dotted line). T2 ()τ , on the other
hand, can be described by a double exponential function (dashed line) with an initial fast
decay component with a time constant of ~ 0.1 ps, followed by a slow process of ~ 5 ps.
The origin of the observed enhancement effect in the THz emission correlation measurements is not well understood. The simple photo-Dember model with a sublinear fluence dependence discussed in the previous section would predict a bleaching effect, i.e. reduced THz emission in the presence of the first excitation pulse.
The simple model does not take into account the significant change in the electronic
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temperature and its effect on diffusion coefficient (or carrier mobility) and the current by diffusion. The model does not consider the ultrafast carrier recombination through
Auger process in the plane of graphene either. On the one hand, the electronic temperature in the presence of the first excitation pulse is higher than that in the absence of the first pulse, which would lead to higher diffusion current since the effect depends on temperature linearly. Which is more, carrier mobilities are expected to be lower at high temperature due to enhanced phonon scattering and the ultrafast carrier recombination is sensitive to the electronic temperature. A systematic study of the effects is warrant for a future study.
The two time scales observed in the experiment could be related to the electron relaxation and cooling process. The fast process possibly corresponds to scattering between electrons and the strongly-coupled-optical-phonons (SCOPs) [121] [120]. The subsequent slow process might be associated with the cooling of SCOP modes via its interactions with other phonon modes [119] [122]. Similar time scales have been observed in optical pump/optical probe measurements in graphene.
We note that optical amplification, however, has been observed in graphitic materials. For instance, an induced negative photoconductivity in the near-infrared region under the strong photoexcitation conditions was observed in monolayer graphene by optical pump-probe measurements [132]. Another example involves amplified emission in the THz region in epitaxial graphene by optic-pump/THz-probe spectroscopy, and was interpreted as stimulated emission of THz [133].
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Figure 3.9 The radiated THz signal from later pulse T2 ()τ as function of delay time between two pump pulses (in circles). The RMS of THz signal was chosen to represent its strength, and its dependence on the delay time was fitted with double exponential function in thick-dashed line. The unaffected
radiated signal T20 was also plotted as reference in triangles and aligned with a thin-dashed base line.
3.6 Polarization Resolved THz Emission from Basal and Edge
Planes of Graphite
To investigate the emission mechanisms, we performed the polarization
resolved measurements from both the basal and edge plane of graphite. The pump
polarization was kept at p-polarized state and the incident angle was fixed to be 45
degrees. The polarization state of the THz emission was determined by wire-grid
polarizers as described in Sect. 3.1. For the basal plane there was only configuration “b”
since the emission is independent of the azimuthal angle (Figure 3.3). We have also
tested two configurations for the edge plane (Figure 3.10), where in configuration “ep1”
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the graphite c-axis is normal to the plane of incidence and in configuration “ep2” the c-
axis is in the plane of incidence. In each configuration, two independent measurements
E0 and Eπ were made with samples rotated by 180 degrees about the edge plane normal
and compared to eliminate any potential artifact induced by sample and/or beam
profile non-symmetry.
Figure 3.10 A schematic illumination of emission measurements in both basal plane and edge plane configuration (top view).
The results are summarized in Table 3.1. Note that the polarization source must be perpendicular to the plane of incidence to obtain s-polarized emission, and in the
plane of incidence for p-polarized emission. Emission from the basal plane “b” is
predominately p-polarized. It can be attributed to polarization along the surface normal,
i.e. the c-axis of graphite. Polarization along the basal plane in the absence of a bias
electric field is not expected, which is verified by the lack of azimuthal angle
dependence of the emission. A more quantitatively analysis is described in section 3.4.
The THz emission from the edge plane is more complex. In configuration “e1” with the c-axis normal to the plane of incidence, both s- and p- polarized emission was
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detected. The s-polarized component is peaked around 1.2 THz and the p-polarized component around 0.6 THz. And the two measurements E0 and Eπ are consistent within the signal-to-noise ratio of the measurements (Figure 3.11). The result from configuration “e2” with the c-axis in the plane of incidence is very different. Only p- polarized emission was observed. Moreover, to our surprise, very different E0 and Eπ both the spectral dependence and amplitude were observed (Figure 3.12).
Config Observed Possible Dipole Possible uration Components Direction Mechanism “b” p c photo- Dember effect normal ( ⊥ ) “ep1” p x defect- s c induced “ep2” p y+c nonlinearity (c-axis) + photo-Dember effect normal ( ⊥ )
Table 3.1 Summary of polarization resolved THz emission from basal plane (“b”) and edge planes(“ep”)
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Figure 3.11 The normalized spectrum of THz radiation from edge plane in configuration ep1. The blue lines indicate s-polarized and red lines p-polarized components in signal. The solid and dashed lines correspond to those two independent measurements E0 and Eπ .
Figure 3.12 The normalized spectrum of THz radiation from edge plane in configuration ep2. Only p- components observed in both measurements. The solid and dashed lines correspond to those two independent measurements E0 and Eπ .
In configuration “e1” the s-polarized emission arises from polarization along the c-axis and the p-polarized, in the graphene plane and into the bulk of the sample. The distinct spectra for s- and p- polarization suggest that the emission presumably arise
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from different origins. Since E0 and Eπ have the same spectrum, it is obvious that the generation mechanisms of each polarization component are independent. The p- component is from the photo-Dember effect within graphene. The mechanism of the s- polarized component is not well understood. Since it originates from polarization along the c-axis, this contribution could potentially arise from structure defects like mismatches between graphene layers and their induced nonlinearity or built-in field.
Such structure defects in HOPG have been identified in the literature [123]. The assign of relatively low frequency components to carrier dynamics in the photo-Dember effect and relatively high frequency components to the defect induced nonlinearity also agrees with the resonant and non-resonant nature, respectively.
An analysis of the configuration “e2” using the above model shows that the p- polarized emission has a mixture of the two contributions: the defect-induced nonlinearity along the c-axis and photo-Dember effect normal to the surface plane of graphene. Under inversion of the sample, the nonlinear polarization undergoes a change of sign and the photo-Dember polarization remains unchanged. Thus, we can construct (E0 - Eπ)/2 and (E0 + Eπ)/2 to represent the emission from each. We observe a low-frequency component peaked around 0.6 THz and a high-frequency component peaked around 1.2 THz as in Figure 3.13.
These results and analysis on the emission mechanisms are very preliminary. A more systematic study is warranted for a better understanding the THz emission process in graphite.
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Figure 3.13 The decomposition of dipole moments contribution in THz radiation in configuration ep2.
The red line and the blue line are radiations from defect-induced nonlinearity along the c-axis and photo-Dember effect normal to surface plane, respectively.
3.7 Conclusion
In this chapter, the THz emission spectroscopy techniqe had been used to study the surface transport properties of graphite. The photo-Dember effect combined with a local optical rectification induced dipole along c-axis were though to be the dominate mechanism in the genertion of THz radiation from graphite surfaces. The c-axis transient mobility of photocarriers was extracted from excitation fluence dependent measurement with the assistance of a saturation theory proposed by reference 31. The photocarrier dynamics within the basal plane was also studied. A short process of ~0.1 ps and a long process of ~5 ps were found, and they were possibly corresponding to the carrier-SCOP scattering and cooling of strong coupled phonon modes.
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Chapter 4
Probing Photoconductive Properties of Phthalocyanine
with THz Spectroscopy
4.1 Introduction
Semiconducting organic materials have been of scientific interest for many decades [134]. Their mechanical flexibility and programmable electronic and optical properties through chemical modification/functionalization have made these materials attractive for many applications in electronics, optoelectronics and photovoltaic cells
[135]. The compatibility of organic materials with solution based processing has also rendered large area devices based on organic materials cost effective.
In this chapter we investigate phthalocyanines (Pcs) as a model organic semiconductor using the technique of THz time-domain spectroscopy. Pcs are aromatic macrocyclic compounds. They are well known as dyes and have been widely used in photographic copiers and printers [136]. The Pc molecules possess a planar geometry as shown in Figure 4.1. These disk-like molecules self-assemble into columns with significant overlap of the delocalized π-electrons of adjacent molecules, thereby providing quasi-1D channels for efficient charge transport [137]. Furthermore, the Pcs have strong absorption bands in the visible and near-IR region (called Q band). These
93
attractive electronic and optical properties, coupled with their high thermal stability,
have recommended Pcs as an excellent candidate for solar energy conversion
applications [138] [139].
Recent studies have revealed several interesting morphological, optical and
transport properties of Pcs. By controlling the solubility of Pcs in solvents the growth of
extremely long fibers of 10’s and 100’s nm width have been demonstrated [140]. These fibers can further be manipulated by DC electric fields [141]. The transport characteristics of Pc-based field-effect transistors [142] [143] and the photoconductivity
dynamics following a femtosecond excitation of Pcs in solution [144] have been
reported. Furthermore, the charge recombination kinetics between the molecular
columns have been probed by pulse radiolysis techniques [145]. On the theoretical side,
hopping transport in p-doped Pc as a function of doping has been studied [146].
Many basic questions such as the main photoexcitation products, nature of the
electronic charge transport and the influence of the crystalline structure on transport in
Pcs, however, remain unanswered. Here we apply the optical pump-THz probe
spectroscopy to investigate systematically the photoconductivity spectrum up to THz
frequencies as a function of the pump-probe delay time and sample temperature.
The technique of optical pump-THz probe spectroscopy based on femtosecond
lasers has emerged as a powerful contactless probe of the frequency dependent
complex conductivity in the THz range. The spectroscopic information can be utilized to
identify signatures of different contributions to the conductivity. When combined with a
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time-synchronized ultrafast pump pulse, the technique also enables us to monitor the dynamics of photoexcited species with a ps to sub-ps time resolution. The method has been successfully applied to the study of a variety systems including quantum dots
[147], nanotubes [148], liquids [149], insulators [150], as well as organic crystals and polymers [151] [152] [153] [154]. In the latter, the bound charges, phonons, polarons, and other intermolecular and intramolecular interactions have been shown to dominate the THz response.
Below, we will first introduce the experimental details including the sample fabrication and experimental setup in Sect. 4.2. We will then present the experimental results on both the static and photo-induced THz properties in Sect. 4.3. The experimental conductivity spectrum will then be compared to several transport models in Sect. 4.4. Finally we summarize in Sect. 4.5.
4.2 Experiment
4.2.1 Sample Fabrication
In this study we have used zinc 2,3,9,10,16,17,23,24-octakis(octyloxy)-29H,31H- phthalocyanine (ZnPc-OC8) consisting of a disc-shaped zinc phthalocyanine core peripherally substituted with eight n-alkoxy side chains. Its chemical structure is shown in Figure 4.1. The compound was purchased from Aldrich (batch # 04815HC). (The synthesis of this compound has been previously described in [137].) We further purified the as-purchased powder by column chromatography (silica, chloroform/pyridine, 2:1
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by vol.) and repeated recrystalization from butanol. The differential scanning
calorimetry (DSC) traces in Figure 4.2 show a phase transition at temperature of 130 oC
(100 oC) or 400K (370K) on heating (cooling). Such a transition is known in similar phthalocyanine derivatives (metal-free and zinc or copper containing cores) [155]. It
corresponds to the transition from the crystalline (low temperature) to the liquid
crystalline (LC) phase (high temperature).
H17C8O OC8H17
N N N H17C8O OC8H17 N Zn N
H17C8O OC8H17 N N N
H17C8O OC8H17 Figure 4.1 Chemical structure of alkoxy- Figure 4.2 DSC trace for the crystal to substituted zinc phthalocyanine derivatives liquid crystal phase transition. (ZnPc-OC8).
Thin films of ZnPc-OC8 on quart substrates were used for THz measurements.
They were fabricated by placing the ZnPc powder onto a 1 mm-thick quartz plate with
preliminarily deposited cylindrical silica beads of controlled diameter (4 mm or 11 mm)
and covering with a second quartz plate. We then heated the sample above the phase
transition temperature and pressed the top quartz plate, which applied a shear force to
achieve a homogeneous film thickness. Thinner films (<1 mm estimated from the optical
absorption) were drop-cast from chloroform solution onto quartz without silica beads
and the solvent was evaporated at 60 oC for 2 hours. Further procedure included
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annealing of the films at 350 oC and controlled cooling at a rate of 10 K/min. To avoid
moisture, the entire fabrication procedure was performed in a glove box (N2). And all
cells were sealed with epoxy afterwards.
The typical microscopy images of thin film samples are shown in Figure 4.3 under
crossed polarizers. Figure 4.4 corresponds to the absorption spectrum. All
measurements were performed in air if not otherwise specified.
Figure 4.3 Microscope Images of a 4 mm ZnPc thin Figure 4.4 Absorption spectrum of ZnPc thin film film above and below the phase transition covering the so-called Q band ranging from 600nm temperature. The images were taken under cross- to ~ 800nm. The two intense π-π* transitions polarized light. within Q band was lost due to absorption saturation and/or disordered nature of thin films.
4.2.2 Experimental Setup
Optic pump-THz probe spectroscopy was employed to investigate the phthalocyanine materials. Refer to chapter 1 for details of the setup. In brief, it used a
Ti:sapphire amplifiered laser system which delivers optical pulse of 50 fs in duration, peaked at 800 nm and with 1 KHz repetition rate. The THz electromagnetic transient 97
was generated and detected through nonlinear optical rectification and electro-optic sampling in ZnTe crystal, respectively. For the pump-probe scheme, an additional 800 nm optical beam was introduced to induce excitations in the sample (Figure 4.5). The pump-probe delay time was controlled by varying the pump beam path lengths via a mechanical translation stage.
Figure 4.5 Setup of optic pump-THz probe spectroscopy.
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4.3 Experimental Results
4.3.1 Static Properties
We first determine the dielectric properties of unexcited phthalocyanine thin
films. When the THz radiation Ein ()ω propagates through a thin film of thickness l , it
will experience 1) transmission at the sample ( n()ω ) - environment (usually the air
n0 =1 ) interfaces ( t12 for entering and t21 for leaving the sample); 2) propagation
through the slab of thickness l : p= exp[ in (ωω ) l / c ] ; and 3) multiple reflections
between two sides of the slab. Combining these effects the transmitted THz field after
the slab (assuming normal incident) is given by:
pt t ωω= 12 12 EE()in () 2 (4.1) 1(− pr21 )
where tt12 21 and r21 are Fresnel coefficients for transition and reflection at the interafces.
2 For normal incidence we obtain tt12 21 = 4() nωω n0 (() n+ n0 ) and
r21=−+( nn 0 (ωω )) ( nn0 ( )) .
Now we compare the transmitted THz field without the thin film Et0 () and with
the thin film Et() in the beam path. The transmitted THz filed without the sample
E0 ()ω can be derived by directly replacing the sample’s dielectric property n()ω by that
of the air n0 :
E00()ωω= Ein () p = Ein ()exp( ω in 0 ω l /) c (4.2)
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And now we can establish the relationship between the detected THz electric field with
( E()ω ) and without ( E0 ()ω ) the thin film sample:
tt ω= ω12 12 ωω− EE()0 ()2 exp[(()1)/]in l c (4.3) 1(− pr21 )
The corresponding measurements on a (thick) ZnPc-OC8 sample are
demonstrated in Figure 4.6. The THz complex refractive index of the ZnPc-OC8 sample
was then numerically calculated based on equation (4.3) using the iteration method.
The results are shown in Figure 4.3. In this measurement since the sample is relatively
thick the multiple reflection effect has been estimated weak, and hence has been
ignored in the calculation.
The extracted dielectric function (or the complex refractive index) for ZnPc-OC8
films in the THz frequency range is illustrated in Figure 4.7. The refractive index of ZnPc-
OC8 films has very week dispersion; the real part of the refractive index has a value of
n(ω )= 1.17 and the imaginary part is close to zero. This suggests that effects of long-
range interaction and/or collective molecular vibrations (i.e. low energy phonons) in the
frequency range of 1 – 3 THz in ZnPc-OC8 thin film samples are weak.
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Figure 4.6 The transmitted THz electric-field Figure 4.7 Frequency dependence of the waveform with (solid line) and without refractive index (real part, red; imaginary part, (dotted line) ZnPc thin films in the beam blue) of ZnPc thin films path.
4.3.2 Photoconductive properties
The complex photoconductivity ∆σ was determined by comparing the transmitted THz electric field Et() through the unexcited sample and its induced change
∆Et(,τ ) due to photoexcitation (Figure 4.8 for pump-probe delay time = 20 ps). In the
limit of a thin film on a transparent substrate, one can use the simple relation:
∆=∆E()/()ωω E inlc ω / . The photoconductivity spectrum corresponding to the
measurement of Figure 4.8 is shown in Figure 4.9. Here the photoconductivity is
complex with a positive real part and a negative imaginary part. The real and imaginary
parts of the photoconductivity are comparable in magnitude, and both amplitudes
increase monotonically with frequency. These features of Figure 4.9 are drastically
different from the Drude conductivity, a phenomenon often observed as band transport
in crystalline inorganic and organic materials [134] [135]. In the Drude model both the
real and imaginary part of the conductivity are positive with the real conductivity 101
peaked at zero frequency and the imaginary conductivity peaked around the scattering
rate.
Figure 4.8 THz electric field transmitted Figure 4.9 The complex photoconductivity of through an unexcited sample (solid) and its ZnPc at delay time 20ps. change (dashed), 20 ps after the ultrafast photoexcitation.
More results on the photoconductivity dynamics are shown in Figure 4.10 and
Figure 4.11. In Figure 4.10 the pump-induced change at the peak of the THz transient has been used. The photoconductivity was found to be present even at 100’s ps after photoexcitation. The main change in the frequency dependence of the photoconductivity with delay time lies in its amplitude, not in the spectral shape.
Figure 4.10 Dynamics of photoconductivity in ZnPc thin films. A series of delay times
(as indicated) have been chosen for the spectral study.
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Figure 4.11 Complex photoconductivity spectrum at different delay times after photo
excitation in ZnPc thin films
Furthermore, the photoinduced response scales linearly with the excitation
fluence in the fluence range of 20J/m2 ~ 50J/m2 for a fixed pump-probe delay time
(Figure 4.12). Similarly, identical dynamics have been observed for two distinct pump fluences (Figure 4.13), which can be described well by a double exponential function
with a slow decay component on the time scale of ~ 70ps and a fast component on the
time constant of ~ 3ps.
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Figure 4.12 Normalized complex photo Figure 4.13 Experimental dynamics (red) and their conductivity by the pump fluence at delay time double exponential fits (black) for two different
20ps for differing excitation fluences excitation fluences
Finally, the photoconductivity in ZnPc was studied as a function of the sample temperature in the range of 80-450 K (Figure 4.14). We note that at low temperatures
(below 370-400 K) the sample is in a crystalline phase and at high temperatures the sample is in a LC phase with no translational order, but with directional order. In the crystalline phase the photoconductivity increased with increasing temperature. When the temperature passed the phase transition temperature at 370K, the photoconductivity was significantly reduced; this could be due to the suppression of carrier mobility in liquid crystalline phase. This temperature dependence trend of photoconductivity could be reproduced in both heating and cooling procedures, with activation energy of the crystalline phase derived to be ~10 meV.
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Figure 4.14 Effective photoconductivities near delay t=0ps under heating (in
red) and cooling (in blue) process.
We have also followed the photoinduced change in the peak of the THz transient as a function of sample temperature (Figure 4.15). The rise times of all dynamics are comparable for all temperatures, showing the same instantaneous feature of photoinduced response as at room temperature. The decay dynamics is evidently dependent on the sample temperature. Compared to the slow decay at high temperatures (~450K) over the entire 80 ps window shown in Figure 4.15, the dynamics at low temperatures (~80K) consist of two pronounced components: a noticeable initial rapid decay within the first few picoseconds, followed by a flat part over the time window of ~ 80ps. We used the double exponential function to fit the experimental data.
105
The slow decay time constant decreased with increasing sample temperature in the range of 200ps to 50ps, while the fast component increased from sub-picosecond to about 8ps (Figure 4.15 right). The fast decay could correspond to the carrier localization or bimolecular recombination process while the slow decay corresponding to trapping by deep sites.
Figure 4.15 Left: Photoconductivity dynamics at different sample temperatures, and their double exponential fits. Right: time constants obtained from the fit.
4.4 Conduction in amorphous semiconductors
To interpret the experimental results presented in Sect. 4.3, it is desirable to first understand the spectral dependence of the photoconductivity and identify the signature of the photoexcitation products, which contribute to the photoconductivity. Therefore, in this section we compare the experimental photoconductivity spectrum to several
106
available transport models. We note that charge transport at the THz frequencies in poor conductors remains one of the most challenging theoretical problems in condensed matter physics. As we show below, none of the existing models can satisfactorily explain our experimental observations. Our results indicate that a more comprehensive model is required to understand the experiment.
4.4.1 Drude-smith model
As already mentioned above, the Drude model shows the positive real and imaginary part of the conductivity, with the real part peaked at the d.c. frequency and the imaginary part peaked around the scattering rate. However, it has been observed that in some strongly localized systems [156, 157], the maximum of the real conductivity shifts to higher frequencies. To account for such a dependence, N.V Smith extended the
Drude model by applying a simple impulse response approach and a Poisson distribution for the response times [158]. We note that the Drude Smith model is a largely phenomenological model and the physical interpretation of the parameters is debatable.
Here we primarily use this model to parameterize the spectrum.
In the Drude Smith model, a system under an impulse electric field Et()= δ () t at time t = 0 develops current jt(), and the Fourier transform of the current response is determined by the frequency dependent conductivity σω():
jE()ω= σω ()() ω (4.4)
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Since the Fourier transform of a -function is unit, the relation (4.5) implies that the frequency dependence of the conductivityδ is the Fourier transform of the current response jt():
∞ σω( )= j ( t )exp( iω t ) dt (4.5) ∫0
At very short time intervals the externally applied impulse field dominates all interactions, so that the carriers can be treated as free particles, and the velocity
0+ resulted from the impulse is v(0+ )= eE ( t ) dt / m= e / m , giving the current ∫0− j(0)= nev (0) = ne2 / m at time t = 0. If the carriers experience scattering and relax back to its equilibrium state, the current response can be modeled as:
jt( )= j (0)exp( − t /τ ) (4.6) with τ the relaxation time. The widely-used Drude model can be obtained from (4.6) and (4.7):
ne2τ / m σω()= (4.7) 1− iωτ
Note that in the Drude model, the main mechanism for conductivity is attributed only to homogenous scattering. Therefore, if we take into account inhomogeneous scattering and multiple collision events, deviation from the Drude model is expected.
Now assume the scattering events are randomly distributed in time and the probability of a carrier experiencing n scattering events within a time interval t is determined by the Poisson distribution:
ptn( , ,ττ )= ( t / )n exp( − t / τ ) / n ! (4.8) 108
Here τ is the average interval between two scattering events. A modified current function
takes the form:
∞ = −+ττn (4.9) jt( ) j (0)exp( t / )[1∑ n=1 cn ( t / ) / n !]
Here, cn are empirical fitting coefficients which reflect the scatting mechanism. Fourier
transform of (4.10) gives rise to the Drude-Smith model for conductivity:
2 neτ / m ∞ c σω = + n ( ) [1 ∑ = n ] (4.10) 1 −−iiωτ n 1 (1ωτ )
It is obvious that the Drude model corresponds to the first term of the Drude Smith
model. Often it is sufficient to keep only the first two terms to describe the relatively
smooth frequency dependence of the conductivity ( cc1 = ):
ne2τ / m c σω( )= [1 + ] (4.11) 11−−iiωτ ωτ
Of particular interest is the case with a negative c . As clearly shown in Figure 4.16 , the
real part of the conductivity is no longer peaked at the dc frequency – the oscillator strength has shifted to the higher frequencies. For the imaginary part, it is possible to obtain negative values at lower frequencies before it switches its sign. If we limit the discussion to elastic scattering, c can be interpreted as the expectation value of the
scattering angle cosθ , and a negative c value would correspond to back scattering
dominated events.
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Figure 4.16 Prediction of the Drude Smith model of Eq. (4.11) for the real (left) and imaginary (right) part of the conductivity for parameter c = 0, -0.5 and -1
We note that our experimental photoconductivity spectrum shares some of the features of the Drude-Smith model. For instance, for a negative value of c and for frequencies below the (negative) peak of the imaginary part, the model qualitatively agrees with experiment.
Figure 4.17 Comparison of an experimental photoconductivity spectrum (dashed lines with grey error bars) with the Drude Smith (solid lines) model for ZnPc thin films 20 ps after photoexcitation at 60 J/m2 at room temperature. Top: real part and bottom: imaginary part of the photoconductivity.
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As an example we show in Figure 4.17 comparison of an experimental photoconductivity spectrum with the Drude-Smith model. The agreement is excellent.
2 Three fitting parameters have been used: the effective d.c. conductivity στ0 = ne/ m , the average interval between two scattering events τ , and the so called “persistence of velocity” – constant c .
The fitting parameters as a function of pump-probe delay time are summarized below in Figure 4.18 (the average interval between two scattering events τ ), Figure 4.19
2 (effective d.c. conductivity στ0 = ne/ m ), 4.20 (the carrier density n) and 4.21 (constant c) for excitation fluence 60 J/m2.
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Figure 4.18 The average scattering lifetime of ZnPc thin film at different delay time after photoexcitation of 60 J/m2
Figure 4.19 The effective d.c photoconductivity of ZnPc thin film at different delay time after
photoexcitation of 60 J/m2
112
Figure 4.20 The estimated carrier density in ZnPc thin films from the Drude Smith fitting parameters at
different delay time after photoexcitation of 60 J/m2
The average time interval between scattering events is mostly independent of
the pump-probe delay time and is kept at about 400 ~ 800 fs for the entire 200 ps. Such
a long scattering time would correspond to an effective mobility of photo-excited
carriers mτ= e/ m ~ 1032 cm / Vs , comparable to that of electrons in crystalline Si at room
temperature 1400cm2 / Vs [129]. This value seems surprisingly high for organic
semiconductors although Hall mobility on the order of 100’s cm2/Vs has been reported
in single crystal bulk Pc [159] and other organic semiconductor [160].
The dynamics of the effective dc conductivity parameter (Figure 4.19) shows the
instantaneous generation of photoconductivity at the level of 0.1 Ω−−11cm upon photoexcitation, which is followed by two decay components in the first 200ps.
Combining Figure 4.18 and 4.19 we predict the dependence of the carrier density on the
pump-probe delay time. The values of the density are on the order of 1018 cm-3, which
113
would correspond to a phtoexcitation yield of 10-4 -10-5 (ratio of the carrier density and the absorbed photon density). The last fitting parameter -- constant c is seen to be roughly ~ -1 and independent of the pump-probe delay.
Figure 4.21 The persistence of velocity in scattering- the constant c for ZnPc thin films at different
delay time after photoexcitation of 60 J/m2
4.4.2 Quantum mechanical tunneling (QMT) in one-dimensional (1D) systems
There has been an extensive work (both experimental and theoretical) on the behavior of the ac conductivity in amorphous semiconductors. Measurements have been performed in chalcogenide (Group VI) and pnictide glass (Group V) and their alloys
(with Si or Ge). A sublinear frequency dependence of the real conductivity has been universally observed [161, 162]. To explain this result analytical framework has been developed mainly based on the pair approximation with the transition of the carriers
114
between two localized states assisted either by quantum mechanical tunneling or/and
classic resonant mechanism. The theoretical work has been summarized in the reviews
by A. R. Long [163] and S. R. Elliott [164] . Here we will consider the 1D case, which is
considerably simpler from the theoretical standpoint. But most importantly, as we
mentioned in previous sections, the discotic ZnPc molecules self-assemble into columns
through the π-π interactions, forming quasi-1D conduction channels. Therefore, the 1D
transport models are most appropriate for ZnPc thin film systems.
In the pair approximation, it is assumed that the a.c. loss is mainly due to the
transition of carriers between two localized states. The dielectric response of the occupied pair sites to an external field is parallel to the field in the quasi-1D case. If the interactions between the pair sites are ignored, the total response can be obtained by
simply summing over all possible configurations of the pair sites. To evaluate the a.c.
conductivity, the Debye model [165] for the dielectric response is usually applied. The
relaxation of the polarization density of localized sites Pt() to its equilibrium state can
be treated (similarly to that of a dipole) linearly with a time constant τ
dPt() Pt () = − (4.12) dt τ
which gives rise to the exponential time dependence for the polarization
Pt( )= P0 exp( − t /τ ) (4.13)
Next we can use the linear response theory PE()ω= εχω0 ()() ω to relate the
frequency dependence of the dielectric response to the polarization density. And the
115
frequency dependence of the polarization density is calculated from Equation (4.13)
through a Laplace transform. Thus we obtain the frequency dependent dielectric
response:
χχ χω() =00 =(1 + itω ) (4.14) 11−+itωω22 t
And the frequency dependent conductivity is related to the dielectric response as
σω()=−+i ε0 (()1) χω ω.
It is clear that for carrier relaxation in disordered systems such as amorphous
semiconductors, the relaxation characteristics such as the relaxation time τ are not
homogeneous. We assume that the relaxation time is described by a continuous
distribution function n()τ and express the real part of the frequency dependent conductivity as:
∞ ωτ2 σω( )= αnd () τ τ (4.15) 1 ∫0 1+ωτ22
Here subscript 1 is used to denote the real part of a physical quantity and α is the polarizability of a single pair site. If the distribution function of the relaxation time obeys an inverse power law n(ττ )∝ 1/ and the polarizability is a constant, the real
conductivity is expected to depend on frequency linearly:
∞ ω σ() ω ∝∝d()ωτ ω (4.16) 1 ∫0 1+ωτ22
116
In general, the relaxation distribution does not exactly follow the inverse power law, which gives rise to a departure from the linear spectral dependence such the sub-linear dependence often observed in experiments [161, 162].
The frequency dependence of the polarizability of the pair sites αω() under the
application of an external field was first addressed by Pollak and Geballe [166] under the
pair approximation. They assumed that two localized sites are separated by a distance
R and by an energy ∆E . Denote the probability of a carrier occupying site i to be fi
( ff12+=1 under the pair approximation). The rate of change of the occupation
probability is governed by:
f1''=−=− Wf 12 2 Wf 21 1 f 2 (4.17)
where Wij is the transition rate from site i to j . Equation (4.17) can also be rewritten
for ft1() only:
f1 '(=−++ W21 WfW 12 ) 1 12 (4.18)
The application of an external field perturbs the occupation probabilities of the
pair sites. We can use the Boltzmann statistics to evaluate the occupation probabilities
before the application of the external field
exp(−∆E / kT ) ft(= 0) = (4.19) 1 1+ exp( −∆E / kT )
and after the application of the external field
exp(−EeR cosθ / kT )exp(−∆ E / kT ) ft()=∞= (4.20) 1 1+ exp( −EeR cosθ / kT )exp(−∆ E / kT )
117
The results of equation (4.19) and equation (4.20) now can be used as boundary
conditions for the time dependence of the occupation probability ft1():
f11( t )= f (0) + [ f 1 ( ∞− ) f 1 (0)](1 − exp( −Wt )) (4.21)
−1 where the transition rate WW=+=12 W 21 τ . If the applied field is weak so that
eER kT , the dielectric response can be derived using the Laplace transform of
Equation (4.21):
NeRE32 2cos 2θ 1 ωτ j= NeRcosθ f = (4.22) 1 4kTτ cosh2 (∆−E 2 kT ) 1 iωτ
Here N is the linear density of the pairs (dipoles), and θ is the angle between the
applied external field and the dipole moment.
Next we consider the physical mechanisms for carrier relaxation. If quantum
mechanical tunneling (QMT) is the principle carrier transfer mechanism with the
assistance of phonons, the relaxation time τ should be proportional to the overlap of
two neighboring wave functions (with delocalization range of r ):
τ exp(2Rr / ) τ = 0 (4.23) cosh(∆E / 2 kT )
Taking into account the density of pair sites in 1D systems
p() R dR= NdR (4.24)
we obtain an expression for the real conductivity spectrum in 1D amorphous
semiconductors by integrating over all possible angles θ , relaxation times τ , and
energy splitting ∆E (which are all assumed to be randomly distributed):
118
σω= ω θ ∆ 1() Re(()/∫ j Ep () R dRd d ( E ))
42 τ 2 (4.25) πNe ω rmax Rd() ωτ = ∫τ 22 24kT min 1+ωτ
The conductivity is dominated by pair sites with energy spliiting ∆≤E kT . Thus for simplicity we have assumed ∆=E 0 in Equation (4.25) to neglect the term
[cosh21 (∆ 2kT )]− .
Furthermore, we note that the integrand ωτ1+ ω22 τ is sharply peaked at
ωτ =1 in both τ - and R - space. The integral of Equation (4.25) can thus be evaluated
by treating the integrand as a δ -function and using the lower limit τ min = 0 , and the
upper limit τ max = ∞ to obtain the conductivity as [163, 167]
22 σω1()= Ce ω Rω (4.26)
Here Rω is the characteristic tunneling distance determined by equation (4.23):
r 1 Rω = ln( ) (4.27) 2 ωτ 0 and the parameter C= π 34 N r/ 96 kT .
The imaginary part of the conductivity has a different form from that of the real part. It cannot be evaluated by approximating the integrand ωτ1+ ω22 τ as a δ - function. However, Long [163] developed a new technique that integrates the imaginary part of the conductivity by parts and derived the following dependence for the imaginary part of the conductivity in 1D amorphous semiconductors:
'2 3 σω2 ()= −Ce ω Rω (4.28)
119
with the parameter C'= π 24 N/ 72 kT .
Figure 4.22 Comparison of experimental photoconductivity spectrum (solid lines) of ZnPc films to the
QMT model (dotted lines). The photoconductivity was measured at pump-probe delay time 20 ps,
pump fluence 50J/m2 and sample at room temperature. The real part of the conductivity is in red, and the imaginary part is in blue.
Now we will compare our experimental result to the QMT model. The model
accounts for the a.c. conductivity arisen from quantum mechanical tunneling (assisted by phonons). However, in general there could also be a d.c. contribution to the conductivity from conduction through percolation [168]. Since these two mechanisms
are completely independent, we express the total conductivity as the sum of the two
contributions σtotal= σ dc + σω(). A representative example is shown in Figure 4.22.
Fitting the measurement results (solid lines) to the model (dotted lines) yields three
parameters: the characteristic tunneling distance r , the typical tunneling time τ 0 , and
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the d.c. conductivity σ dc . The parameters corresponding to the fit shown in the figure
are:
r= 2 nm
τ 0 =1fs (4.29) −−11 σ dc =0.02 Ω cm
We first compare the characteristic tunneling distance r= 2 nm to the distance
between neighboring ZnPc discs in 1D columns c= 0.4 nm [137]. This result shows that
photoinduced carriers are delocalized in 1D columns over the distance of several
molecules. Such a result seems physically reasonable for an amorphous material [169,
−−11 170]. Next, from the extracted value for the d.c. conductivity σ dc =0.02 Ω cm and the
19 3 experimental value for the photoexcitation density of n0 =4 × 10 / cm , we evaluate the
product of the quantum yield for free carrier generation and the carrier mobility to be
ηm = 0.003cm2 / Vs . Quantum yield for free carrier generation at the level of 0.1% ~ 1%
[171, 172] have been commonly reported for organic thin films. If we use these values,
we obtain the carrier mobility in ZnPc films to be 0.3 ~ 3cm2 / Vs , which lie well within the range of hole mobilities reported in organic thin films [173, 174]. The difficulty of using the QMT model to describe the photoconductivity in ZnPc, however, lies in the
extracted value for the tunneling time τ 0 =1fs . Typically one would expect this value to
be comparable to the inverse of the phonon vibration frequencies since the transition is
assisted by phonons, which are on the order of ν ~ 10−13 s . The tunneling time extracted
from the QMT model is 1-2 orders of magnitude smaller than the vibration period,
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which seems unphysical and suggests that the QMT model is not applicable to the
experimental photoconductivity in ZnPc.
4.4.3 Resonant Photon Absorption
In the previous section we considered the QMT process of carriers between two
localized sites assisted by phonons as the main a.c. loss mechanism in 1D amorphous
semiconductor. An alternative a.c. loss mechanism is resonant absorption of a photon of
energy ω from the applied a.c. electric field. This mechanism was first studied by
Tanaka and Fan[175]. Later Blinowski and Mycielski [176] performed detailed analysis on crystalline semiconductors with impurity bands. And Mott [177] [178] extended the
model to amorphous semiconductors with localized states randomly distributed in both
space and energy.
Let us assume two coupled localized sites separated by R in distance and ∆E in
energy. The probability of resonant absorption of a photon of energy ω from the field
is proportional to I 2 , where I is the overlap integral of the two localized states
I= I00exp( − Rr / ) (4.30)
Here r0 is the delocalization length of the states. Including this coupling interaction the
new energy levels of the states become
± E=( EE12 + )/2 ±∆ E /2 (4.31)
where E1,2 are the energies of the states without interactions and
2 2 1/2 ∆=E[( E12 − E ) + 4 IR ( )] . 122
Resonant absorption can occur when the photon energy associated with the a.c.
+− electric field ε =[ε00 exp(itωω ) +−ε exp( it )] / 2 satisfies ω =EE − ,and the electronic
transition from the lower energy level E − to the upper energy level E + is allowed. This
transition probability is determined by the Fermi’s golden rule:
− 2πω2 −+E m ω We± =ε0 −x +δω( −∆ E )exp( )[1− exp( − )] / ∑ (4.32) 4 kT kT
Here ∑ is the partition function of the localized pair states, and eε0 −+x is the
transition dipole element between the two localized states with energies E − and E + .
The transition matrix element can be easily evaluated as:
2 22 2 eε00−+=x ( eε ⋅x ) IR ( )/ ∆ E (4.33)
And the conductivity of the whole system can be evaluated by integrating over all the
localized pair states (per unit volume):
W NE2 () σω()= 2∑ ± = f dE−+ dE W dx 1 22∫∫∫± εε00 4πω22e N 2 ( E )[1−− exp( /kT )] = f ) (4.34) 32ω −+E − m × dE−+ dE dRR42 I( R )δω ( −∆ E )exp( ) ∫∫∫ kT
It is important to note that this conductivity depends on the photon energy and
temperature exponentially σω1( )∝− [1 exp( −ω /kT )]. At high temperature ω kT ,
[1−− exp(ω /kT )] → 0 , resonant absorption is negligible and the phonon-assistant effects such as QMT are expected to dominate. On the other hand, at low temperature
123
ω kT , [1−− exp(ω /kT )] → 1and the resonant absorption mechanism plays an
important role in the a.c. conductivity.
Below we will focus on the low temperature limit. In this limit, we replace the term [1−− exp(ω /kT )] → 1. At the same time, in this limit also holds true the
condition of
E± − m kT (4.35)
so that the partition function as well as the Gibbs function goes to unit. Equation (4.34)
now can be simplified by making the substitutions ∆=EEE12 − and
− E=( EE12 + −∆ E )/2 (as in (4.31)):
22 2 4π eN () Ef − 42 σω1()= dR d∆ E dE R I( R )(δ ω−∆ E ) (4.36) 32ω ∫∫ ∫
We note that the integration over variable E − automatically satisfy
mω−≤≤ E − m, so that ∫ dE − =ω , and Equation (4.36) can be further simplified as:
4πω22eN 2 () E σω() = f dRIRR2( ) 4 / [(ω )2− 4 IR 2 ( )] 1/2 (4.37) 1 3 ∫
And this 1D integration yields [179]:
22 2 2 4 2 σω1() = πe N ω Rω / 3( kT ) (4.38)
−1 where N= kTN() E f is the total density of states, and RIω = αωln(20 / ) is the
distance between two localized states with the highest absorption rate.
It is interesting to compare the conductivity resulted from resonant absorption
(4.38) with that from phonon-assistant QMT (4.28). The main difference between these
124
two conductivities is the role of ω ↔ kT . In the case of resonant absorption, the a.c.
loss is proportional to the energy difference of the two pair states ω =EE+− − . But in
the phonon-assisted QMT, the energy difference of the two pair states under phonon-
dominated environment is proportional to kT .
Now we consider the correction to the model when the intersite correlation
2 Ee12 = /4πε0 ε Rω becomes important. In the pair approximation at low temperature
such that E12 ω kT , the a. c. conductivity is modified as :
42 3 2 σ1() ω= πe N ω Rω r00/12 εε ( kT ) (4.39)
which is obtained from the result of Equation (4.38) without the intersite correlation
2 effect by replacing the energy loss ω →=Ee12 /4πε0 ε Rω . Combining with the log- frequency dependence of the distance between the localized states with the highest
−1 absorption rate RIω = αωln(20 / ) , Equation (4.39) predicts a sublinear frequency
dependence, which is often observed in experiment. Furthermore, the intermediate
temperature case is also of interest in practice. When E12 kT ω , the a. c.
conductivity is further modified as:
22 2 4 2 σω1() = πe N ω E12 tanh( ω / 2kT ) Rω r0 / 3( kT ) (4.40)
Next we will compare our experiment to the resonant absorption model. Note
that since in the derivation we did not take into account the level broadenings, the
imaginary part of the conductivity vanishes and cannot be directly compared to
experiment. Let us first evaluate the intersite correlation energy in ZnPc thin films:
125
2 E12 = e/ 4πε0 ε Rω ~ 100 meV (4.41)
In this estimate we have used the dielectric constant ε (THz ) ~ 1.2 from the static THz
measurement and correlation distance Rω ~ 10 nm assuming the length is comparable
to the distance of a few molecules. Compared with the room temperature energy
kT~ 26 meV and the probe THz photon energy ω ~4meV , the correlation energy is
the most significant energy scale in the conduction. Therefore, Equation (4.40) should be a better fit. An example of comparison of the experimental photoconductivity spectrum (solid lines) to the resonant absorption model of Equation (4.40) (dotted lines) is shown in Figure 4.23.
Figure 4.23 Comparison of experimental photoconductivity spectrum (solid lines) of ZnPc thin films with
the resonant absorption model (dotted line). Photoconductivity in ZnPc films was measured at pump- probe delay time 15 ps, pump fluence 30 J/m2 at 320 K. The real part of conductivity is in red color, and
the imaginary part is in blue.
There are two fitting parameters in equation (4.40): the delocalization length of
the photoexcited carriers r0, defined as φφ=00exp( −x /r ) for the wave function, and the 126
overlap integral I0 describing the wave function overlap between two pair sites
I= I00exp( − Rr / ). Since the real part of the conductivity is relatively flat within the
entire spectral window (the photon energy is presumably much smaller than the
resonance energy), the stability and accuracy of the fitting procedure become an issue.
For sample temperature ranging from 80K to 370K (before the crystalline to liquid
crystalline phase transition), the fitting seems stable and the fitting parameters seem
physical. We summarize the temperature dependence of the fitting parameters in
Figure 4.24 and Figure 4.25. These measurements were performed at pump-probe delay
time 15 ps and excitation fluence 30 J/m2.
Figure 4.24 Temperature dependence of carriers’ Figure 4.25 Temperature dependence of the delocalization length in ZnPc thin films. carriers’ overlap integral in ZnPc thin films.
For the entire temperature range the carrier delocalization length is ~10 -40 nm,
which corresponds to 10’s of ZnPc molecule thickness. Similar values have been
reported for other organic thin films [169, 170]. The overlap integral, on the other hand,
remains ~ 41± meV for the entire temperature range. Since the overlap integral is determined by the configuration of the neighboring molecules, we do not expect it to vary significantly with sample temperature. Above the phase transition temperature
127
(>370K), the fitting procedure becomes unstable and very large unphysical values are required for the overlap integral to fit the experimental spectrum. Therefore, the resonant absorption model is also not sufficient to describe our observations.
4.5 Conclusion
In this chapter, we employed optical pump-THz probe spectroscopy to investigate the behavior of photoconductivity in organic semiconductors. We chose phthalocyanines as a model system due to their promises in organic electronics and optoelectronics applications. Both complex photoconductivity spectrum and dynamics have been measured in thin films from 80K to 450K, covering a crystal to liquid crystal phase transition. Real photoconductivity has been observed instantaneously after photoexcitation, which then decays on the 100’s ps time scale following a double exponential dependence. The decay times are sensitive to sample temperature and sample crystalline structure.
We have also attempted to understand the photoconductivity spectrum based on several models including the phenomenological Drude-Smith model, the quantum mechanical tunneling theory and the resonant photon absorption theory. However, none of the models has been able to describe the experimental results satisfactorily. A more comprehensive theory is thus required to describe THz frequency dependence of conductivity in amorphous semiconductors. From the experimental standpoint, as a future direction, it is desirable to extend the spectral window of the THz probe by
128
extending the THz bandwidth and by combing it with other time resolved techniques such as optical pump-probe spectroscopy. The high-energy photoexcitation products can thus be independently monitored and separated from the low-energy excitations
(free carriers) in the contributions to the a.c. conductivity.
129
Appendix:
0 Electromagnetic Model for THz Emission from Surfaces
The electromagnetic model for second harmonic generation from surfaces and interfaces has been developed for the surfaces and interfaces of isotropic materials [180]
[181] [182]. Here we extend the model to THz generation process involving surfaces and interfaces of anisotropic media. We also focus on the photo-Dember effect, but the model is also applicable to other emission mechanisms.
In our treatment, the system is divided into three regions: the two bulk materials
1 and 2 and the interfacial zone between them (Figure 0.1). Since the material in the interfacial zone has either its properties or its neighbors significantly different from the bulk material 1 or 2, we assign distinct dielectric response function to it. This approach has been used in earlier treatments of nonlinear optics such as second-harmonic generation at surfaces and interfaces [182].
130
Figure 0.1 Scheme of the three-region pictures in the nonlinear surface generation phenomenological model. An ultrafast laser excitation is incident at angle of , and its propagation direction is illuminated in red color with wave vector superscripted with and subscripted훉 i (incidence). The polarization sheet induced is laid on the interface and recorded as훚 x-y plane. The THz emissions, generated from media interface, are sharply peaked in both “reflection” and “transmission” directions, and are denoted with superscript Ω, and subscripts r (reflection) and t (transmission), respectively
In THz emission only the thin surface layers are involved in the generation process, such as the strongly absorbing region near the surface in the photo-Dember effect, and are considered as interfacial zone. Since the thickness of interface is much smaller than both the bulk material size and the wavelength of the optical excitation, the polarization responsible for radiation in the interface zone will be treated as an infinitely thin polarization sheet
Pnl (,) rt= pr (,) tzδ () (0.1) 131
arising from the excited current transient
Prnl (,)t= Jrnl (,) t dt (0.2) ∫ .
Below we derive the Snell’s law for the determination of the radiation direction and the
boundary conditions from the Maxwell equations for the determination of the emission
strength.
0.1 The Snell’s law in nonlinear optics
We first review the Snell’s law in linear optics. Assume the electromagnetic field
Ei incident onto interface between medium ε1 and medium ε2 at angle θi , the reflected
field Err()θ and the transmitted field Ett()θ could be related to the incident field:
1. The tangential component of wave vector k for all the three fields are the same in
order to satisfy continuity of electric field on interface:
kkisinθθθ ir= sin rt = k sin t (0.3)
2. The basic dispersion relationship between the wave vector and dielectric function
k = nω yields: c
knknknirt///112= = (0.4)
We can easily obtain the familiar Snell’s law in linear optics by combining those two equations:
nnn11sinθθir= sin = 2 sin θ t (0.5)
132
In order to extend Snell’s law to nonlinear optics, a couple of modifications need
to be made. First, to account for absorption in media complex dielectric functions are
introduced. Instead of the trigonometric functions sinθi , sinθt and sinθr we will be
using the ratio of the components of the wave vectors. Secondly, more than one photon
energy (or frequency) will be introduced due to the nonlinear process. For the process
of either sum or difference frequency generation, the tangential components of k wave
vector are still conserved:
ωω12ΩΩ kkix,,±== ix k rx ,, k tx (0.6)
with the subscript x denoting the tangential component of physical variable and
Ω=ωω12 ± corresponding to the frequency of the sum (+) or difference (-) frequency
ω generation. Recalling the basic dispersion relationship kn= ⋅ , each tangential c
component can be written as:
kknω kk=xx ⋅= ⋅ (0.7) x k kc
In optical rectification, the frequencies ωω12~ and frequency Ω=ωω12 − . Combining
(0.6) and (0.7), we obtain the Snell’s law for optical rectification:
ω ΩΩ kkki nnn(ω )ix,=Ω=Ω ()rx ,, ()tx (0.8) iiωi r ΩΩt kkki rt
We note that the nonlinear and linear forms of Snell’s law are not equivalent. It
is apparent that, in dispersionless media such as in vacuum, the prediction of nonlinear
Snell’s law is the same as its linear form for reflection. However, in media with strong
133
dispersion the reflection angle does not have to be identical to the incident angle since the fields involved in nonlinear “reflection” correspond to two different frequencies, hence different refractive indices.
In what follows, we will represent any vector F by its tangential and normal components
k FF= ⋅ t (0.9) t k
k FF= ⋅ n (0.10) n k
0.2 Emission from nonlinear polarization at the surfaces or
interfaces
Once a polarization sheet Pnl (,) rt= pr (,) tzδ () on interface is determined, the radiation from the polarization sheet Prnl (,)t can be derived from the Maxwell
equations with their modified form:
∇⋅B =0 ∇⋅DP =−∇⋅ nl (0.11) ∇×HD −∂ ∂tt =∂ Pnl ∂ ∇×EB +∂ ∂t =0
Here, the D and H are the conventional linear electric displacement and
magnetic field:
l D=ε00 EP += εε E (0.12)
134
B=m00 HM += mm H (0.13)
where ε 0 and ε are vacuum and medium relative permittivity, and m0 and m are vacuum and medium permeability, respectively, in linear optics.
Since the nonlinear polarization sheet source Pnl brings singularity into the problem, the continuity of all the field components cross the interface will be altered.
To this end, we first integrate the Maxwell equations in a source layer of finite thickness and then let the thickness to vanish:
00++ 0+ 0 + B dn=+=() D Pnl dn H dn = E dn =0 (0.14) ∫∫00−−n nn ∫∫0−t 0 − t
where the subscript n denotes the normal component and t denotes the tangential
component of the fields. The results show that all the normal and tangential components of fields are continuous cross the interface as in the absence of the
nonlinear polarization sheet, expect for the normal component of electric displacement
D (and the electric field E ):
00++ 0 + D dn=εε E dn =−=− Pnl dn p (0.15) ∫∫00−−n 0 nn ∫ 0 −n n
The boundary conditions for the fields can be derived by applying Gauss and
Stocks theorem as shown in Figure 0.2:
135
Figure 0.2 The diagram of interface between mediem 1 and mediem 2. The pillbox and contour
covering the polarization cross the interface to assisst theε deviation ofε boudary conditions
The matching condition for magnetic field B can be derived directly from the first
equation in Maxwell equations (0.11) by applying the Stokes divergence theorem to the
pillbox in Figure 0.2:
00++ ∆=BBnn − B n =0 (0.16)
The matching condition for D can be determined by integrating the second equation in (0.11) over the volume of pillbox with surface area S as in Figure 0.2:
∆ ⋅ = ∇⋅ =− ∇⋅nl =− ∇ ⋅ Dnt S∫∫DP dV dV ∫ p dS
or
∆Dnt = −∇ ⋅p (0.17)
ˆ with the operation ∇=∂∂t / t represents the divergence on the interface tangential. And
for H the matching condition would be solved by integration over the contour in Figure
136
0.2 with the length of long arms to be l, considering the finiteness of D in tangential
direction:
∆H ⋅ l =H ⋅ d l =( ∇× H )( ⋅ ntˆ ׈ )dS (0.18) t ∫∫
Then after applying the third equation of the Maxwell equations (0.11), combined with
the continuity equation (0.14), we get:
0+ ∆H =() ∂ Pnl ∂×t nˆ dn =∂ pn ∂× t ˆ (0.19) t ∫0−
The last matching condition of the electric field E may be more complicated
regarding its infinite nature of the polarization sheet beside the boundary. Applying the
curl theorem on last equation in Maxwell (0.11):
∫ El⋅=d 0 (0.20)
Unlike the usual boundary condition, there is a discontinuity of electric field in the
normal direction. Expand the previous equation, we have:
0+ ∆E ⋅= l() Ell+− − E dn (0.21) t∫0− nn
The equation can be rewritten with assistance of discontinuity equation (0.15):
' ∆Et = −∇ tnp / εε0 (0.22)
' by using the relation DEnn= εε0 inside the polarization sheet.
In summary, the boundary conditions with considering the infinitely thin polarization sheet Pnl (,) rt= pr (,) tnδ () on the interface of media can be written as:
137
∆=Bn 0 ∆D = −∇ ⋅p nt (1.23) ∆Hpt =∂/ ∂×t nˆ ' ∆Et = −∇ tnp / εε0
0.2.1 THz radiation from isotropic polarization sheet
We now consider THz generation from surfaces of isotropic materials. For simplicity, the polarization sheet is assumed to be in the x-y plane, and the optical excitation was incident in the x-z plane. The geometry has been demonstrated in Figure
0.3:
Figure 0.3 The geometry of surface THz generation between mediem ε1 and mediem ε2
Since the polarization sheet is excited by optical pump beam:
Er( ,t )=E exp( ik00xz x )exp( −− ik z )exp( iω t ) (0.24) we expect the polarization to have the same spatial variation as that of the optical excitation:
138
pr( ,t )= p ( x , t ) =P exp( ikx x )exp(−Ω i t ) (0.25) where P is the amplitude, and kkxx=00 ⋅Ω/ω = sin θω n ( ) Ω c is the preserved x-
component of the wave vector from the optical excitation with incident angle θ .
We express the radiated THz electric fields in both media as:
Er1( ,t )= E11 eˆ exp( i k 1 ⋅ r )exp( −Ω it ) (0.26)
Er2( ,t )= E22 eˆ exp( i k 2⋅ r )exp( −Ω it ) (0.27)
Ω where the wave vectors are k =ε 1/2 () Ω⋅ in most non-ferromagnetic materials ( m =1). ii c
Since the radiated fields should preserve the spatial variation as the polarization sheet
in the tangential direction, the wave vectors ki on either side of the interface can be
expressed as:
k11=kkxz xzˆ − ˆ (0.28)
k22=kkxz xzˆ + ˆ (0.29)
with the tangential component determined as
22 kiz= kk i − x (0.30)
and the signs of the wave vectors denote to the propagation direction of fields. Since
k can be complex if the dielectric function ε ()Ω is complex. k is usually complex as i i iz
well and the field will decay away from the surface. Since unit vectors of the radiated
field eˆ1 and eˆ 2 satisfy ekˆii⋅=0 , they can be rewritten as epˆ1=ˆ 1 =(k 11zx k )( xˆ + kk 1 ) zˆ ,
epˆ 2==−+ˆ 2(k 22zx k )( xˆ kk 2 ) zˆ for p-polarized field, and eˆi = syˆ = ˆ for s-polarized field.
Here, we have the components of the radiated fields:
139
k k EEeˆ =⋅+=+()1z xzˆ x ˆ EE xzˆ ˆ 11 1 kk 1xz 1 11 (0.31) k2z kx EE22eˆ = 2 ⋅−() xzˆ +ˆ =−EE2xz xˆ + 2 zˆ kk22 for p-polarized field, and
EEeyˆ = ˆ 11 1 (0.32) EE22eyˆ = 2ˆ for s-polarized field. The boundary conditions on interfaces of mediem 1 and mediem
2 can then also be expressed in components: ε
ε +− Bz21zz(0)(0)0=−== Bz +− D21z(0)(0) z=−==− Dz z ikxxP +− HzHzi21yy(0)(0)=−==ΩP x +− (0.33) Hz21xx(0)(0)= − Hz = =−Ω iP y +− Ez21yy(0)(0)0=−== Ez +− ' E21x(0)(0) z=−==− Ex z ikxzP /εε0
Apply the boundary conditions (0.33) to the field components of (0.31) and
(0.32), and also consider the general relation (0.12) and (0.13) between fields
kE×=ω B = ωm0 m H, the radiated fields from the polarization sheet are derived as:
ik12 k Eip, = pˆ i⋅P (0.34) εε0() 21kkzz+ ε 12
22 ik12ik EE1,ss= 2, =⋅⋅ssˆ P= ˆ P (0.35) εε0112()kkzz++εε 0212 () kkzz where subscript p and s denote the different polarization state of the radiated fields, respectively, in medium i(i=1, 2).
140
It is interesting to note the form of radiated fields in the absence of boundaries,
i.e. εεε12= = :
2 ik0 Eip, =pˆ i ⋅P (0.36) 2εε00k z
2 ik0 Eis, =sˆ ⋅P (0.37) 2εε00k z
1/2 where kc0 =ε ⋅Ω() denotes the wave vector in medium ε . They can also be
expressed in a compact form by using vector eˆi :
2 iki eEˆii⋅= eˆi ⋅P (0.38) 2εε0 ik iz
Similarly, we can express the radiated field in a compact form in the presence of
boundaries:
2 iki eEˆii⋅= eˆi ⋅() LP (0.39) 2εε0 ik iz
where L is a 3x3 matrix with only nonzero diagonal elements given by the Fresnel
transmission coefficients:
2k ε 22kkε L =jz i xxˆ ˆ ++iz yyˆ ˆ iz i zzˆˆ (0.40) kkjzεε i+++ iz j kkkkjz iz jzεε i iz j
By comparing (0.38) and (0.39), it is clear that, in the presence of the interfaces,
the radiated fields are modified by the Fresnel transmission coefficients according to L :
the s-polarized field is corrected by component yyˆ ˆ , and p-polarized field by the
components xxˆ ˆ and zzˆˆ . This local-field correction is introduced due to the discontinuity
of the linear optical response εε12≠ on the interface. 141
0.2.2 THz radiation from anisotropic surfaces
The discussion above concerns isotropic systems. For materials such as graphite,
we have to extend our discussion onto anisotropic systems. We will focus on layered materials, which can be described as uniaxial crystals. In this case, both the Snell’s law and Maxwell equations do not require changes. However, the dielectric function will have a tensor form as
DExx ε = ε DEyy (0.41) DEzz ε ⊥
or in components as
DExx= ε
DEyy= ε (0.42)
DEzz= ε ⊥
where the subscript and ⊥ denote the dielectric properties in the basal plane and along the c-axis, respectively. We assume the interface between vacuum and layered material parallel to its basal plane.
Additionally, the boundary conditions, in which the dielectric functions are involved, need to be modified. Boundary condition
' ∆Et = −∇ tnp / εε0 (0.22) will be replaced by
' ∆Et = −∇ tnp / εε⊥ 0 (0.43)
Similarly, the other boundary conditions can be rewritten for this case as:
142
+− Bz21zz(0)(0)0=−== Bz +− D21z(0)(0) z=−==− Dz z ikxxP +− HzHzi21yy(0)(0)=−==ΩP x +− (0.44) Hz21xx(0)(0)= − Hz = =−Ω iP y +− Ez21yy(0)(0)0=−== Ez +− ' E21x(0)(0) z=−==− Ex z ikxzP /ε ⊥
We also note that since the dielectric function is no longer a scalar, the electric field E is no longer perpendicular to the wave vector k (See Figure 0.4). And Equ.
(0.26), (0.27) based on (0.9) and (0.10) are no longer invalid.
Figure 0.4 The relative relation between k , D , E , and H ( B )
Fortunately, the vectors D, H ( B ) and k are perpendicular to each other, and we will use D instead of E in the discussion below and similar derivation as for the isotropic media can be applied here. The radiated electric displacements corresponding to the THz fields in each medium are:
Dr1( ,t )= D11 eˆ exp( i k 1 ⋅ r )exp( −Ω it ) (0.45)
Dr2( ,t )= D22 eˆ exp( i k 2⋅ r )exp( −Ω it ) (0.46)
143
The wave vectors in components are k11=kkxz xzˆ − ˆ and k22=kkxz xzˆ + ˆ with the
magnitude of wave vectors ki :
Ω k =εφ1/2 (,) Ω⋅ (0.47) ii c
where dielectric function εφi (,)Ω is not only determined by THz frequency Ω but also
22 by k 'spropagation direction φ , and kiz= kk i − ix . The unit vectors eˆ1 and eˆ 2 now are not referred to the radiated electric E but to the electric displacement D, and can be
kkiz ix written as epˆii==±+ˆ ( )( xˆ ) zˆ for the p-polarized field and eˆi = syˆ = ˆ for the s- kkii
polarized field, with the constraint ekˆii⋅=0 .
The radiated electric displacement in anisotropic systems can then be solved by
k ×=− combining the anisotropic boundary conditions (0.44) and field condition ω HD
with its comonents defined in (0.45) and (0.46):
ik k ε 1 2 1, Dip, = pˆ i⋅P (0.48) εε0() 2,kk 1zz+ ε 1, 2
2 ik2 DD1,ss== 2, sˆ ⋅ P (0.49) ε 01()kkzz+ 2
where pˆ i is is the modified unit vector for the p-polarized displacement
ε kkiz ix ppˆ ii==±+ε ˆ xˆ zˆ in medium i (i=1,2). And it is obvious that the radiated field is kk00ε ⊥′
strongly dependent on the relative direction between the sheet polarization status
P and the modified unit vector pˆ i or sˆ by their dot product.
144
0.3 Formation of Polarization Sheet on Graphite Surfaces
The surface polarization sheet arises from the photocurrent. For a given
frequency Ω , it can be expressed as:
Prnl (,)t=∫ Jrnl (,) tdt =− Jrnl (,)/ t it Ω=− Jnl (,)rz tˆ / it Ω (0.50)
This thin polarization sheet has no projection along the unit vector syˆ = ˆ , therefore no s-
polarized radiated field is expected. This prediction has been confirmed by our
observation of the dominant p-polarized emission from the graphite basal plane.
By making use of equation (0.48), the electric displacement in both media
(vacuum and graphite) can be written as:
ik11 kxxεε 2 x D1, pz= P (0.51) εε0() 21xzkk+ ε 12 x z ε z′
ik21 kxxεε 2 x D2, pz= − P (0.52) εε0() 21xzkk+ ε 12 x z ε z′
On the vacuum-graphite interface, the in-plane dielectric function of graphite ε 2 in THz
range is usually on the order of 103 [122], which is much larger than the vacuum value of
ε1 =1, so
εε21xzkk 12 x z (0.53)
and we can omit the second term ε12xzk in the denominator of (0.51). In addition, the sheet polarization can be rewritten in the form of a sheet current following equation
(0.50):
145
Pzz=−Ωji/ (0.54)
nl with Jz= jzzδ ()ˆ . If we ignore the dielectric difference between the interfacial region
and bulk graphite, the electric field in either media can be simplified as:
ik1 kx jz E1, pz≈=−P tanθ (0.55) εε0zz′kc 1 εε02z
ik 2 1 j ≈− x = z θθ E2,zp Pz 2 tan sin (0.56) εε0zz′kc 1 ε 2 z εε 02z
Here, kc= Ω tanθ = kk and sinθ = kk are introduced for propagation of optical 1 , xz1 11x
excitation with incident angle of θ from vacuum. It is clear that in the near-field the
radiation is proportionally to the photo-induced current as expected, and the out-of-
plane dielectric function of graphite and incident angle together play an important role
in determining the radiation strength. The z-component of emitted field E2,zp in
graphite was expressed specifically instead of the field E2, p itself for the convenience of the Dember field discussion in the section 3.4.
146
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