半絕緣砷化镓與砷離子佈植砷化镓天線兆赫茲輻射實驗與模擬分析

國

立

交

通

大

學

光電工程研究所

博

士

論

文

半絕緣砷化镓與砷離子佈植砷化镓

天線之兆赫茲輻射實驗與模擬分析

A study of generation of terahertz radiations from semi-insulating

and arsenic-ion-implanted GaAs photoconductive antennas

研 究 生：周 榮 華

半絕緣砷化镓與砷離子佈植砷化镓

天線之兆赫茲輻射實驗與模擬分析

A study of generation of terahertz radiations from semi-insulating

and arsenic-ion-implanted GaAs photoconductive antennas

研 究 生：周 榮 華 Student：Yi-Chao Wang

指導教授：潘犀靈 教授

Advisor：Prof. Ci-Ling Pan

國 立 交 通 大 學

光 電 工 程 研 究 所

博 士 論 文

A Thesis

Submitted to Department of Photonics & Institute of Electro-Optical Engineering

College of Electrical Engineering National Chiao Tung University In partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Electro-Optical Engineering

July 2008

Hsinchu, Taiwan, Republic of China

半絕緣砷化镓與砷離子佈植砷化镓

天線之兆赫茲輻射實驗與模擬分析

研 究 生 : 周 榮 華

指導教授 : 潘 犀 靈 教授

國立交通大學

光電工程研究所

中文摘要

在實驗上，我們從不同觀點分析這些測量到的兆赫輻射脈衝波形，包括尖 峰峰值，尖峰寬度，負與正峰值的比，以及頻寬。我們發現來自中型孔徑天線中， 測量到的兆赫脈衝的特性顯著地仰賴孔徑大小和光導材料。我們可以從我們的理 論模擬推論，測量到的孔徑相依的行為源自於較大孔徑的天線在其陽極的附近有 一個因陷阱而提高的偏壓電場是較強的，因此引起較大的空間－電荷電場屛避的 效應，以及較大的頻寬。對於大孔徑天線的實驗結果來說，我們的砷離子佈植砷 化鎵天線比半絕緣砷化鎵天線展現了更大的頻寬和更好的發射效率。由已知的實 驗觀察可得知，砷離子佈植砷化鎵天線的光學吸收係數在某些條件下，可以是大 於半絕緣砷化鎵天線的光學吸收係數，我們將此假設代入理論模型中，由模擬結 果我們成功地驗證了砷離子佈植砷化鎵天線的優越特性確實能歸因於在離子佈 植層中比起半絕緣砷化鎵有較大的吸收係數。對於兩者類型的材料，我們觀察到 輻射出來的尖峰兆赫振幅顯示對於泵光辉度有一個異常的相關性，這相關性偏離 了縮放比例規則所給的預測。分析那理論上的和模擬結果，我們推論出這種行為 起因於帶填充和雙光子的吸收效果的發生。此外，在特定的泵光辉度之下，尖峰 兆赫振幅對於偏壓電場的相關性，偏離了從被縮放比例規則預測的線性關係。

A study of generation terahertz radiations from

semi-insulating and arsenic-ion-implanted GaAs

photoconductive antennas

Student : Rone-Hwa Chou Advisor : Prof. Ci-Ling Pan

Department of Photonics and Institute of Electro-Optical Engineering

National Chiao Tung University

Abstract

The work presented here is focused on the experimental analysis and numerical

explanation of the terahertz radiation pulses from biased photoconductive antennas with

different gap sizes. The antenna with gap sizes of 1.5 cm is divided into large-aperture

antennas. The antennas with gap sizes ranging from 20 mμ to 0.5 mm are divided into

mid-size-gap antennas. The photoconductive materials fabricated on the large-aperture

antennas are arsenic-ion-implanted GaAs (GaAs:As+_{) and semi-insulating GaAs}

(SI-GaAs). The photoconductive material fabricated on the mid-size-gap antennas is

Theoretically, we adopt a set of nonlinear partial differential equations including

an electromagnetic wave, drift-diffusion, and Poisson equations to interpret the measured

data. This model considers the space-charge field and near-field screening effects within

the THz antenna. The former plays an important role in the mid-size-gap antennas,

whereas the latter is dominant in the large-aperture antennas. Experimentally, we analyze

these measured terahertz radiation waveforms from different point of views, including the

peak terahertz amplitude, the peak width, the ratio of negative peak to positive peak, and

bandwidth. We find that the characteristics of measured terahertz pulses from the

mid-size-gap antennas depend markedly on both the gap size and photoconductive

material. We deduce from our simulation that the gap-dependent behavior stems from

the fact that an antenna with larger gap has a stronger trap-enhanced bias field near the anode edge and thereby induces larger space-charge field screening effect and bandwidth.

For the large-aperture antennas, our single-energy arsenic-ion-implanted GaAs

antenna exhibits larger bandwidth and better emission efficiency in comparison with semi-insulating GaAs antenna. Our simulation verifies that the superior characteristics for the latter can be partly attributed to larger optical absorption in the ion-implanted layer. For both types of materials,

w

the prediction given by the scaling rule. Analyzing the theoretical and simulation results, we infer that this behavior arises from band filling and two-photon absorption effects. Besides, at specific pump fluence, the dependence of peak terahertz amplitude

on bias field is also distinct from the usual linear relationship predicted by the scaling rule.

Acknowledgements

I am deeply indebted to my advisor, Prof. Ci-Ling Pan, for giving me the opportunity to work in this area, for his precious advice, continuous encouragement and full support during my research.

I am also very grateful to Dr. Tze-An Liu for his assistance and valuable discussion in my research. I hardly finish my research without his kindly help.

I would like to thank my wife, Anne Wu, most heartily for her patience, encouragement, and understanding during the course of this work.

Contents

Chinese abstract……….…………..……….…i

English abstract …..……….…………..……….………...….iii

Acknowledgements...vi Contents.………..……….…………..………..…..…vii Tables captions….……….…………..……….……….………..xi Figures captions……...….……….…………..….….……….xii Chapter 1. Introduction...1 1.1 Background...1

1.2 Overview of terahertz radiation………..2

1.2.1 Terahertz radiation………..2

1.2.2 Sources of THz radiation…..………..3

1.2.3 Applications of THz radiation………..………..6

1.3 Motivation………...8

1.4 Contributions from this thesis………..………...9

1.5 Organization of this thesis………....………..10

Chapter 2. Photoconductive antennas ...15

2.2 Photoconductive materials...18

2.2.1 Semi-Insulating GaAs...19

2.2.2 Low-temperature grown GaAs...20

2.2.3 Arsenic-ion-implanted GaAs...23

Chapter 3. Modeling of THz radiation from photoconductive antennas...29

3.1 Current surge model………...30

3.2 Drude-Lorentz model………..………...31

3.3 Drift-Poisson model………...33

3.3.1 Poisson's equation………..……….……...34

3.3.2 Carrier continuity equations………..….……...34

3.4 Fullwave model………..….………...35

3.5 Numerical algorithm for Poisson's equation...36

3.6 Method of lines...40

3.7 Boundary conditions...44

3.8 Programming methodologies...50

Chapter 4. Large-aperture antennas...55

4.1 Near-THz field screening effect...56

4.2 Experimental Methods...62

4.4 Results and Discussions...66

4.4.1 Pump fluence dependence...66

4.4.1.1 Waveforms and spectra...66

4.4.1.2 Peak width, peak shift, and bandwidth...68

4.4.1.3 Peak THz amplitude...70

4.4.2 Bias dependence...73

4.5 Conclusions...75

Chapter 5. Mid-size-gap antennas...88

5.1 Space-charge bias field screening effect...89

5.2 Theoretical Methodology...90

5.3 Experimental methods...96

5.4 Results and discussions...98

5.4.1 Trap-enhanced bias field...98

5.4.2 Evolutions of parameters in antennas...99

5.4.3 THz radiation waveforms and spectra...101

5.4.4 Dependence of the negative peak on gap size...104

5.4.5 Bias field and pump fluence dependencies...106

5.5 Conclusions...110

Curriculum vitae...125 Appendix A: Mathematica codes for the simulation of large-aperture antennas...126 Appendix B: Mathematica codes for the simulation of mid-size-gap antennas...129

Table Captions

Table 4.1 Parameters used in THz radiation simulations for SI-GaAs and GaAs:As+

Figure Captions

Figure 1:

Fig. 1.1: THz radiation spectrum………...

...

...

Figure 2:

Fig. 2.1: Schematic view of (a) the photoconductive dipole antenna, (b) the bow-tie antenna, and (c) the photoconductive strip line…...28 Fig. 2.2: Schematic of an Auston antenna, used to generate single-cycle bursts of THz

radiation. A sub-picosecond optical pulse excites the semiconducting substrate in a region between two biased electrodes. This radiation is coupled into free space using a hemispherical substrate lens..…...28

Figure 4:

Fig. 4.1: Schematic of a large-aperture photoconductive antenna

with a voltage _b

..

...

b

V

Fig. 4.2: Experimental setup for a large-aperture photoconductive antenna with a voltage V

..

...

Fig. 4.3: Transient normalized photoreflectance changes for SI-GaAs (solid blue circle) and GaAs:As+ _{(open red circle) antennas.}

_...

Fig. 4.4: Measured THz waveforms Er for (a) SI-GaAs, and (b) GaAs:As+ antennas

as a function of time delay t at various pump fluences F. The bias field applied to the antennas was kept at 0.6 kV/cm. (c) Fourier-transformed amplitude spectrum iE of the waveforms in (a), and (d) r r of the waveforms in (b)

………...

i

E

Fig. 4.5: (a) Measured THz waveforms Er , (b) corresponding Fourier-transformed

amplitude spectrum iEr, (c) simulated Er and (d) iEr for SI-GaAs (blue full line) and GaAs:As+ _{(red full line) antennas at pump fluence}

. Both E Eir

2 58 J / cm

F = μ r and are normalized to their peak

amplitude

………...

Fig. 4.6: (a) Measured peak width dt (full line) and peak shift tp (dashed line) obtained

from Fig. 2(a) and 2(b), (b) simulated dt (full line) and tp (dashed line), and

(c) measured peak frequency fp (full line) and bandwidth df (dashed line)

obtained from Fig. 2 (c) and 2(d) as a function of pump fluence F for both SI-GaAs (blue) and GaAs:As+ _{(red) antennas}

_………...

Fig. 4.7: (a) Measured (dashed-marks), (b) fitting (dashed line), and simulated (full line) THz peak amplitude Ermax versus pump fluence F for both SI-GaAs

+

(blue) and GaAs:As (red) antennas. The green dash-cross and full lines are the ratio ρ ( ≡ (Ermax(GaAs:As+) － Ermax(SI-GaAs))/Ermax(SI-GaAs)) for

measured and simulated case. The bias field E was kept at 0.6 kV/cmb

….

GaAs:As+ _{(red dashed-triangle) antennas, and the relative emission}

efficiency ρ (green dashed-cross) as a function of bias field Eb at pump

fluence _{F =58 J / cmμ} 2………...

….

Figure 5:

'

x

K

Fig. 5.1: Geometry for calculating the far-THz field Er(t). is the point on the

antenna’s surface,

x

K

r

G

'

x

x

r

'

x

x

…….

ˆz

Fig. 5.2: Schematic representation of DC-biased PC antennas excited by femtosecond laser pulses. The transmission line length L and width W are equal to 10 mm and 0.1 mm, respectively. The illuminated regions are indicated by a red circular spot. The left and right diagrams illustrate a uniform illumination for gap size G = 0.02 or 0.1 mm, and an asymmetric illumination for G = 0.2 or 0.5 mm, respectively………...114

Fig. 5.3: Simulated bias field E (x) as a function of position x at depth b z=0.1 mμ

for multi-GaAs:As+ _{antennas with gap sizes G of 0.02 (blue), 0.1 (green),}

0.2 (red), and 0.5 mm (black)……….…..115 Fig. 5.4: Simulation results. (a) electron concentration , (b) negative

logarithmic function of the net-charge concentration,

( , )

n x t

log[ ( , )]x t

− Δ , (c)

space-charge field , and (d) near-field THz radiation E(x,t) as a function of the time delays t and gap position x at pump depth

( , ) s

E x t

0.1 m

z= μ

for multi-GaAs:As+ _{antenna with gap size of 0.5 mm. The values of the}

large and small peaks in (b) correspond to net-charge concentrations of and _{4.0 10 (#/ cm )×} 13 −3 , respectively..………...….115 15

1.5 10×

Fig. 5.5: Simulated temporal evolutions of (a) Δ( )t and (b) Eb(t) at x = 0.01 mm and

0.1 m

z= μ for gap sizes G of 0.02 (blue), 0.1 (green), 0.2 (red), and 0.5 mm (black). Inset shows a magnified view at t~ 1.72 ps………...……..116 Fig. 5.6: (a) Simulated and (c) measured THz radiation waveforms Er as a function of

time delays t for gap sizes G of 0.02 (blue), 0.1 (green), 0.2 (red), and 0.5 mm (black). (b) and (d): The corresponding Fourier-transformed amplitudes spectra. The pump fluence is ………...……...………117 r

E _{70 /cmμJ} 2

Fig. 5.7: The negative values min r

E

− (circle marks) of the negative peak of THz

versus gap size G for simulation (blue) and measurement (red). The blue and red solid curves are the fits to the simulated and measured min

r

E

− . To

compare with the measured Δf , the values of the simulated Δf have been shifted downward by 0.5 THz………...…118 Fig. 5.8: (a) The negative values min

r

E

− (circle marks) of the negative peak of THz

waveform, and frequency bandwidth Δf (squares and dashed curves) versus gap size G for simulation (blue) and measurement (red). The blue and red solid curves are the fits to the simulated and measured min

r

E

− . To

compare with the measured Δf , the values of the simulated Δf have been shifted downward by 0.5 THz. Measured results: (b) Bandwidth Δf

(square marks) and negative values min r

E

− (point marks) of the negative

peak of THz waveform as a function of gap size G for multi-GaAs:As+ _(blue)

and SI-GaAs (red) antennas...………...….120 Fig. 5.9: (a) Bias field Eb and (b) pump fluence F dependencies of peak THz

amplitude Ermax for multi-GaAs:As+ (blue) and SI-GaAs (red) antennas with

gap size G of 0.02 (square marks), 0.5 (triangle marks), and 1 mm (point marks). The pump fluence F is fixed at _{70 J/cm}μ 2 _{in (a), and the nominal} bias field Eb is kept at 3.5 kV/cm in (b). The dashed lines are the theoretical

Chapter 1

Introduction

1.1 Background

Although terahertz (THz) radiation generated from biased photoconductive (PC) antennas have been widely used in many disciplines, we can found that some experimental phenomena are not able to be explained by proper physical mechanisms. For example, the scaling rule which gives the pump fluence dependence of peak THz amplitude indicates that the peak THz amplitude should saturate at high fluence. However, some experiments show anomalous results that the peak THz amplitude decreases gradually at high fluence. In addition, some fundamental principles involving with carrier dynamics within PC antennas are still unclear by the researchers. For example, it is found experimentally that the space-charge bias field screening effect is dominant in a small-gap photoconductive antenna, whereas plays minor role in a large-aperture photoconductive antenna. We also found that the THz pulse width calculated from some theoretical models is narrower than the measured width. Up to now, theoretical models have provided no interpretations for this inconsistence between theory and experiment.

Over past ten years, many groups have attempted to design an THz radiation emitter with maximum emission efficiency. To reach this purpose, researchers employ different PC materials or metallic patterns to fabricate on the antennas, and also vary the optical pump conditions in order to observe their different performance. Nevertheless, these experimental studies still lack corresponding theoretical works.

1.2 Overview of terahertz radiation

1.2.1 Terahertz radiation

We usually classify the electromagnetic radiation in the frequency range from to as terahertz (THz) radiation where is a largely unexplored frontier area for research in science and engineering in part since reliable sources of high quality THz radiation have been scarce. From the electromagnetic spectrum of Fig. 1.1, we can see that the THz frequency range lies above the high-frequency range of electronics, but below the range of the traditional fields of optics. The fact that the THz frequency range lies in the transition region between electronics and photonics has led to creative ways in source development. Many groups have made great efforts to develop THz radiation sources in both continuous-wave and pulsed way. In particular, pulsed sources used in THz time domain spectroscopy (THz-TDS) as shown in Fig. 1.2 has generated a great deal of interest, and sparked a rapid growth in

12

0.2 10× 12

the field of THz science and technology.

1.2.2 Sources

of THz radiation

Free-Space Electron Laser (FEL) [1-2], and backward wave oscillators (BWO) [3] are electron beam sources that generate relatively high power signals at the terahertz frequency range. All of these devices work based on the interaction of a high-energy electron beam with a strong magnetic field inside resonant cavities or waveguides. Due to this interaction, energy transfer occurs from the electron beam to an electromagnetic wave. Conventional lasers rely on the inversion of an atomic or molecular transition. Thus the wavelength at which they operate is determined by the active medium they use. The FEL eliminates the atomic “middle-man”, and does not rely on specific transitions. Potentially FEL’s offer three main characteristics those are often hard to get with conventional lasers, namely wide tunability, high power and high efficiency. They do this by using a relativistic beam of free electrons that interact with a periodic structure, typically in the form of a static magnetic field. This structure exerts a Lorentz force on the moving electrons, forcing them to oscillate. The basic idea is to cause all the electrons to have approximately the same phase, thereby producing constructive interference (stimulated emission). A key feature of these lasers is that the emitted radiation is a function of the electron energy and we can

The FEL is therefore a widely tunable system and can be tuned to emit THz radiation. However, the main disadvantages of these electron beam devices are bulky and need extremely high fields as well as high current densities.

By injecting a CO2 pump laser light into a cavity filled with a gas that lases,

one also can obtain THz radiation [4-5]. The lasing frequency is fixed dictated by the filling gas. Tunable sources have been developed based on mixing a tunable microwave source with these gas lasers [6-7]. Power levels of 1 - 20 mW are common for 20 - 100 W laser pump power depending on the chosen line, with one of the strongest being that of methanol at 2522.78 GHz. A miniaturized gas laser has been reported with 75 × 30 × 10 cm dimensions and 20 kg weight, which generates 30 mW power at 2.5 THz [8].

One approach to generate THz radiation is the quantum-cascade laser (QCL), first proposed by R. F. Kazarinov and R. A. Suris back in 1971 [9]. For the THz generation the QCL employs the intraband transitions in a biased doped semiconductor heterostructure. The first operating device was demonstrated however only in 1994 by J. Faist and coworkers [10], and despite the impressive progress in the QCL development achieved ever since, these devices are not yet able to operate at a room temperature.

Electro-Optic Sampling (FSEOS). This technique is based on an emission by optical rectification [11], [12]. Optical rectification is a process first observed in the 1960’s that describes how a pulse at optical frequencies can be downshifted by degenerate difference frequency generation inside a non-linear crystal. This effect arises from the second order susceptibilityχ of a crystal. The susceptibility, χ=P/ε₀E measures the degree of polarization, P, caused when an electric field, E, is applied to a dielectric material (ε₀ is the permittivity of a vacuum). The non-linearity between the

polarization, P, and the magnitude of the electric field, E, may be encompassed by

writing P as a power series in

(

)

0 1 2 3 ....

P=ε χ E+χ E +χ E + . Higher order terms such as χ₂ , denote the non-linear response of the materials and are important for the

high electric fields found in laser pulses. In a material subjected simultaneously to waves of frequencies ω₁ and ω₂ , the P will contain a term of the form,

1 2

cosωtcosω t =_⎡⎣cos

(

)

(

)

Ultrafast pulses with a temporal width of approximately 70 fs comprise a large number of difference frequency waves and have a frequency bandwidth in excess of 10 THz. Using an ultrafast visible pulse to excite a crystal that has large second-order susceptibility, such as zinc telluride, produces a time varying polarization of the electron cloud inside the crystal. We can think of the oscillating electron cloud with

and

electric polarization, P, vibrating at the various frequencies (say ω₁ ω₂) that correspond to those that make up the incident pulse of visible light. The electron cloud then re-radiates at THz frequencies, ω_THz =ω ω₁+ ₂, as a result of the beats that form between the various frequency components. The pulse of THz electromagnetic radiation contains a broad range of frequencies, from zero up to the bandwidth of the visible radiation. The power of the THz pulse is derived entirely from the incident laser pulse, so FSEOS has typically lower power.

1.2.3 Applications of THz radiation

Along with the progress of THz radiation sources, recently researchers have become increasingly interested in exploiting THz radiation for numerous technological applications including spectroscopy, sensing, range finding, tomography and microscopy, and imaging [14]. In these applications, THz spectroscopy [15-16] is valuable for distinguishing molecules and studying intermolecular interactions, while tomography relates to imaging through the successive layers of a material. Microscopy with THz radiation is realized by working in the near-field THz radiation pattern, thus overcoming the far-field wavelength limitation on resolution. Imaging remains the most fervently researched and eagerly awaited application of THz radiation. There exists a further uncountable number of applications for THz radiation

and even a brief description of these would be ambitious. We have therefore opted to give a bullet-point review of some of the more interesting applications in biology, chemistry, gas detection, medicine and elsewhere before giving a more comprehensive account of THz radiation imaging.

The experiments for THz time-domain spectroscopy (THz-TDS) use coherent pulses of electromagnetic radiation to obtain information about the frequency range between 0.1 and 5 THz. The electromagnetic pulses have sub-picosecond width and field strength on the order of 10-100 V/cm, depending on antenna geometry. In THz-TDS experiments, the picosecond pulses of THz radiation are used to probe different materials. The radiation has several distinct advantages over other forms of spectroscopy: many materials are transparent to THz; THz radiation is safe for biological tissues because it is non-ionizing unlike X-rays, and images formed with THz radiation can have relatively good resolution (less than 1 mm). Also, many interesting materials have unique spectral fingerprints in the THz range, which means that THz radiation can be used to identify them. The emerging field of TDS typically relies on broadband short-pulse THz sources employing photoconductive (PC) antennas [17-18], since the broad bandwidths correspond to short coherence lengths, which are required for high resolution imaging or tomography and sensing. This broadband technique is also widely used to obtain linear spectroscopic information,

such as complex dielectric constants or the conductivity of materials to fully understand the behavior with very broad spectral coverage. As implied in the title of the dissertation, we shall be dealing with ultrafast carrier dynamics in GaAs-based photoconductive THz antennas to simulate the emitted THz transients using numerical simulation. These will be detailed in the ensuing chapters of the dissertation.

1.3 Motivation

Experimentally, the photoconductive antenna based on LT-GaAs lacks of the reproducibility, and thus we want to search a replacement of low temperature grown GaAs (LT-GaAs) without losing the THz emission efficiency. One of the replacement is the arsenic-ion-implanted GaAs (GaAs:As+_{). We want to verify that whether such}

ion-implanted material are able to replace LT-GaAs or not by investigating its characteristics of terahertz radiation. Understanding the detail mechanisms in a photoconductive antenna can help us promote the device efficiency in order to reach an optimum terahertz radiation operation. We expect to adopt a rigorous theoretical model to explain the observed phenomena in experiments. After the validity of the model is verified, we want to vary some parameters of the model to predict its possible results and also give a suggestion on how to fabricate THz antennas with high emission efficiency.

1.4 Contributions of this thesis

This thesis makes some contributions to the knowledge of the generation of THz radiation. The main contributions to THz radiation can be listed as the following seven points:

1. We systematically investigate the characteristics of THz radiation from large-aperture and mid-size-gap biased photoconductive antennas.

2. We use large-aperture and mid-size-gap biased photoconductive antennas with two different materials to compare their distinct characteristics of THz radiation.

3. For mid-size-gap biased photoconductive antennas, we observe their THz radiation under different gap size and two different materials.

4. For large-aperture photoconductive antennas, we construct a rigorous theoretical model to interpret several measured features including THz waveforms, THz pulse width, pump fluence dependence, and anomalous saturation behavior.

5. We construct another rigorous theoretical model to explain the gap-dependent THz waveforms from mid-size-gap photoconductive

antennas, and also demonstrate the carrier and field dynamic behavior which can give us a good insight of fundamental THz radiation principle. 6. We clearly deduce each physical mechanism responsible for the measured

result for large-aperture and mid-size-gap photoconductive antennas.

7. We point out the way to further improve the interpretation and prediction of numerical simulation in future works.

1.5 Organization of this thesis

Chapter 1 gives the fundamental principle involving with THz radiation, and reviews some terahertz radiation sources and their operation principles. The motivation and our contribution are also contained in this chapter.

In the Chapter 2 we describe photoconductive antennas with different types of designs and three kinds of materials.

In the Chapter 3 we review four theoretical models involving with carrier and field dynamics. The numerical algorithm is also provided for each model. We will also suggest several softwares to use in numerical simulation.

In the Chapter 4 we conduct a comparison between the characteristics of terahertz radiation pulses generated using biased semi-insulating and

arsenic-ion-implanted GaAs photoconductive antennas with 1.5-cm aperture size under various pump fluences and bias fields. We use fullwave model to explain the distinct terahertz radiation characteristics between both types of materials.

In the Chapter 5 we investigate the gap-dependent THz pulses from mid-size-gap multi-energy arsenic-ion-implanted GaAs photoconductive antennas in terahertz time-domain spectroscopy experiments. We demonstrate the carrier and field dynamics within the antennas. An explanation for the gap-dependent phenomenon is provided in this Chapter.

Chapter 6 summarizes the results obtained during this work, and provides an outlook on future prospects of the numerical simulation of THz radiation mechanism.

References

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[9] R. F. Kazarinov and R. A. Suris, Fiz. Tekh. Poluprov. 5, 797 (1971) [Sov. Phys. Semicond. 5, 707 (1971)]

[10] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).

[12] A. Bonvalet, Appl. Phys. Lett. 67, 2907 (1995).

[13] O. S. Heavens and R. W. Ditchburn - Insight into Optics (1987)

[14] D. R. Grischkowsky, IEEE J. Sel. Top. Quantum Electron. 6, 1122 (2000).

[15] M. Nagel, C. Meyer, H.-M. Heiliger, T. Dekorsy, H. Kurz, R. Hey, and K. Ploog, Appl. Phys. Lett. 72, 1018 (1998).

[16] B. B. Hu and M. C. Nuss, Opt. Lett. 20, 1716 (1995).

[17] G. Zhao, R. N. Schouten, N. van der Valk, W. T. Wenckebach, and P. C. M. Planken, Rev. Sci. Instrum.73, 1715 (2002).

[18] A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, App. Phys. Lett. 86, 121114 (2005).

Fig. 1.1: THz radiation spectrum.*

Chapter 2

Photoconductive antennas

Besides of FEL, QCL, and FSEOS, THz radiation also can be generated by employing femtosecond laser pulses to excite biased photoconductive (PC) antennas. These devices were first proposed by Auston et al. [1] and subsequently studied by many groups [2-4]. Typical PC antennas have two parallel metallic contacts deposited on the surface of a semiconductor as shown in Fig. 2.1. THz pulses are typically produced by conduction between two electrodes patterned on a semi-insulating GaAs (SI-GaAs), low temperature gallium arsenide (LT-GaAs), or other ion-implanted materials substrate. To generate THz radiation, a femtosecond laser pulse (often a titanium sapphire laser (Ti-Sapphire laser)) has to be focused onto the gap of the photoconductive antennas with a DC bias across the gap. Absorption of the laser pulse whose center frequency exceeds the bandgap of the semiconductor substrate generates free carriers there between the electrodes. The mechanism underlying the THz radiation is to quickly create photoexcited carriers between the externally biased contacts using an ultrafast laser pulse. The charge carriers are then quickly accelerated to the opposite electrodes by the bias field and lead to the formation of an electrical dipole due to the difference of mobility between electrons and holes. This acceleration

of the charge carriers by the external electric field generates an electromagnetic transient. That is, the bias–laser pulse combination allows these charge carriers to rapidly jump the gap, and the resulting fast temporal change in a transient current can generate a pulse of electromagnetic radiation in the THz frequency range. The performance of THz generation depends on the temporal shape of photo-induced current from charge acceleration. The far-field radiation E t from the current _r

( )

distribution J (t) (s ∝ v, v is the drift velocity of the carriers) on the propagating axis

of the radiation can be expressed by [5]

( )

( )

where A is the area of the emitter carrying the current distribution and a is acceleration of the carriers. Since the THz radiation in the far field is proportional to the time derivative of the current density, the emitted THz transient depends crucially on the carrier drift velocity. As a result, the THz radiation field has a field shape proportional to the acceleration of the photo-excited carriers. Very short THz pulses (typically ~2 ps) are produced due to the rapid rise of the photo-induced current in the gap and in short lifetime materials such as LT-GaAs. This current may persist for only a few hundred femtoseconds or up to several nanoseconds, depending on the material of which the substrate is composed. This photoconductive antenna puts out a train of pulses, whose repetition frequency is the same as that of the femtosecond pump laser.

Pulse widths are on the order of 100 fs, with average powers below a few hundred nW and a frequency spread of > 500 GHz. The pulse bandwidth is typically centered at about 1 to 2 THz. The details of the spectrum can vary significantly, however, depending on the design of the antenna and pump-laser power, pulse width and shape, and configuration.

2.1 Antenna types

The features of the pump pulses will significantly affect the generated THz radiation. Besides, the characteristics of THz radiation also depend on the designs of photoconductive antennas. Several designs of antennas includes the dipoles, bow tie, and coplanar strip line as shown in Fig. 2.1, or interdigitated structures, logarithmic spiral antennas, and more sophisticated designs derived from microwave theory. The most commonly used are the Hertzian dipole antenna which has been reported to have an emission spectral distribution as high as 2 or 3 THz. The bow-tie antenna possesses a wide frequency bandwidth, and is also used as a photoconductive antenna. For the bow-tie antenna, increased radiation power was observed, although its emission spectrum distributed at frequencies lower than dipole antennas. A unique photoconductive source is the biased coplanar strip line, the metal and semiconductor interface of which was excited with femtosecond laser pulses. The biased strip line

showed a very wideband emission spectrum (< 5 THz).

In 1997, M. Tani et. al. report a quantitative and systematic comparison of the emission properties of the photoconductive antennas [6]. In their experiments, they investigated the characteristics of THz radiation from several photoconductive antennas with three different designs (the three dipoles, bow tie, and coplanar strip line) and two different materials (semi-insulating GaAs (SI-GaAs) and low-temperature grown GaAs (LT-GaAs)). They found that the radiation spectra showed no significant difference for both materials under the same design. In addition, the pump-power dependencies of the radiation power showed saturation for higher pump intensities, which was more serious in SI-GaAs-based antennas than in LT-GaAs-based antennas.

2.2 Photoconductive materials

One of the essential device factors determining the efficiency of THz antennas is the photoconductive materials fabricated on the antenna. To generate high THz power, the photoconductive materials should have large carrier mobility. In addition, a high resistivity is desired for the materials so that the PC antennas can endure high bias field and thus obtain high THz radiation power. To fulfill these conditions, some approaches like doped and ion-implantation are applied to the PC materials. In the

following, we introduce three materials which are commonly fabricated on PC antennas.

2.2.1 Semi-insulating GaAs

One typical material fabricated on biased PC antennas is known as semi-insulating GaAs (SI-GaAs) [7]. By a variety of techniques one may produce SI-GaAs [8-10] with carrier lifetimeτ of several picoseconds as a whole. The c

electrostatic property of SI-GaAs exhibits greatly reduced parasitic capacitance suitable for fast devices, and allows for integration and the implementation of monolithic microwave integrated circuits. SI-GaAs provides semiconductor qualities including thermal stability during epitaxial growth or anneal of ion-implanted active layer, absence of undesirable substrate active layer interface effects, no degradation of active layer properties by outdiffusion of impurities from substrate during thermal processing, and lowest possible density of crystalline defects. Undoped GaAs can be made semi-insulating by the addition of either oxygen or chromium to the melt. For our study, the SI-GaAs grown by the liquid encapsulated Czochralski method. Through a compensation of shallow acceptors by the intrinsic type of deep donor defect, EL2, the resistivity of SI-GaAs antenna is able to reach a value as high as the order of ₁₀17_Ωicm.This high resistivity is about six orders of magnitude greater than

that of silicon and provides excellent isolation and substrate insulation. The resistivity of the semiconductor can be controlled by counter doping with a deep-level impurity that has a conductivity type opposite to that of the impurities introduced during growth.

Hurd et al. [11] found that if pump light is incident upon SI-GaAs, the deep traps within SI-GaAs lead to the photo-induced space charge at the electrode, and therefore produce a space-charge field that screen the bias field and alter the transient response of a PC antenna. The influence of space-charge field depends on the ratio of the deep trap and shallow concentrations.

A particularly appealing phenomenon is provided by

when

a DC external bias is applied to the electrodes of a SI-GaAs antenna: an enhanced bias field will form near the edge of the anode leading to an enhanced THz radiation when femtosecond laser pump pulses are incident upon the edge of the anode [12].

2.2.2 Low-temperature grown GaAs

One usually makes use of molecular beam epitaxy method to grow GaAs thin films on certain device. If the growth temperature during the film deposition is reduced to about 200 - 300 under arsenic overpressure an incorporation of excess group V-atoms (As) occurs. Post annealing at temperatures above 600o _{leads to the}

o

C

nucleation of the excess arsenic in crystalline As precipitates of several nanometer size [13]. These As clusters act as burried Schottky barriers, form localized mid-gap states and are responsible for trapping of electrons and holes [14]. Other types of trapping centers are point defects, which are incorporated during the growth process. They are double donors, which consist of neutral and positively charged arsenic antisites. Energetically the antisite states are localized close to the center of the band gap [15]. The charged antisites as well as the precipitates are believed to influence the electrical properties of the material and the carrier relaxation after optical excitation [16]. Due to the fast carrier trapping the carrier lifetime of such low-temperature grown GaAs (LT-GaAs) is only about 250 ~ 500 fs [17] and depends critically on the growth temperature.

The trapping states may consist of Urbach tail states, which are located energetically close to the conduction band edge and the mid-gap states, mentioned above. The Urbach tail states are caused by disorder in the material, while the mid-gap states formed by the arsenic precipitates and the defects. In LT-GaAs the mid-gap state concentration can be as high as 20 3

10 cm− after growth [18], but

decreases while the subsequent annealing to values of about to depending on growth and annealing temperature [19].

17 3

5 10 cm× −

18 3

5 10 cm× −

therefore the slow recombination of the trapped carriers acts as a bottleneck for the depopulation of the bands via the point defects. High epitaxial GaAs crystals, grown under standard conditions (not LT-GaAs), exhibit a typical electron-hole recombination time of about 100 ~ 500 ps [18]. Therefore it is obvious that localized states in the energy gap due to defects or impurities (trapping states) can act as recombination centers for non-radiative processes, which change the recombination time substantially. If the density of trapped states is sufficiently high, carriers can tunnel through these centers to valence band. Because the trapping of carriers due to these localized states occurs very fast, the recombination time is reduced to about 10 ps. This effect is important for the development of fast PC antennas with sub-picosecond response times, which can be used for THz pulse emission [20]. With a bandwidth extending up to 5 THz and a good signal-to-noise ratio, such THz pulses are a promising tool for far-infrared spectroscopy of solids, liquids, and gases. However due to the introduced defects and impurities the statistical probability of scattering is increased and the mobility of the carriers is therefore reduced, what causes an increase in resistivity. This leads on the opposite side to a decrease in the sensitivity of the device to external electrical fields.

Compared to the SI-GaAs antenna, the LT-GaAs-based antenna has a larger bandwidth of THz radiation owing to its shorter carrier lifetime (< 1 ps), but its

disadvantage lies in the lack of reproducible property.

2.2.3 Arsenic-ion-implanted GaAs

Implanting ions into semi-insulating materials will significantly change their original resistivity and temperature stability, as well as optical properties, such as τ c or absorption coefficient α₀ . The value of τ for As-implanted [21-22] or c LT-GaAs antenna [23] is as short as 1 ps or less; hence, it influences THz radiation characteristics, including: pulse shape, pulse duration, and bandwidth. Concerning the quantity α₀, its value is affected in the presence of impurity, bias field, or ion implantation. Nolte et al. pointed out that LT-GaAs exhibits an excess absorption relative to GaAs, depending on the growth conditions [24]. On the other hand, Lin et

al. [25] found that the band edge of absorption coefficient +

0

α of GaAs:As increases from to for different dosages. Therefore, one can deduce that the absorption coefficient of semi-insulating materials tends to be altered after ion implantation.

3

6.2 10× 4

-2.2 10 cm× 1

In practical experiments, one can choose ions with different energy and dose for implanting the antenna’s surface. The ion-implanted depth is determined by the ion energy, and can be calculated using the Stopping Range of Ions in Matter (SRIM)

software [26]. For As ion energy of 200 keV and 2 MeV, the implanted depths are about 0.1 mμ and 1 mμ , respectively.

+

In addition to As , many groups have investigated THz devices based on GaAs implanted with H+_{, N}+_{, and O}+_{. Salem et al. studied strip-line emitters based on}

GaAs:H, GaAs:N, GaAs:O, and also GaAs:As [27-28]. Recently, Winnerl et al. [29] studied THz emitters implanted with dual energy implants of N+ _{and As ions of} +

References

[1] D. H. Auston, K. P. Cheung, and P. R. Smith, Appl. Phys. Lett. 45, 284 (1984). [2] D. H. Auston, Appl. Phys. Lett. 26, 101 (1975).

[3] G. Mourou, C. V. Stancampiano, A. Antonetti, and A. Orszag, Appl. Phys. Lett. 39, 295 (1981).

[4] X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).

[5] J. T. Darrow, X.-C. Zhang, D. H. Auston, and J. D. Morse, IEEE J. Quantum Electron. 28, 1607 (1992).

[6] M. Tani, S. Matsuura, K. Sakai, and S. Nakashima, Appl. Opts. 36, 7853 (1997). [7] M. S. Markram-Ebied, “Nature of EL2: The Main Native Midgap Electron Trap

in VPE and Bulk GaAs,” in semi-insulating III-V Materials, D. Look, Editor, Shiva Publishing Ltd., England (1984).

[8] N. G. Ainslie, S. E. Blum, and J. F. Woods, J. Appl. Phys. 33, 2391 (1961). [9] G. R. Cronin and R. W. Haisty, J. Electrochem. Soc., 111, 874 (1964).

[10] J. B. Mullin, R. J. Heritage, C. H. Holliday, and B. W. Straughan, J. Cryst. Growth, 3-4, 281 (1968).

[11] C. M. Hurd and W. R. McKinnon, J. Appl. Phys. 75, 596 (1994). [12] S. E. Ralph and D. Grischkowsky, Appl. Phys. Lett. 59, 1972 (1991).

[13] M. Luysberg, H. Sohn, A. Prasad, P. Specht, Z. Lilienthal-Weber, E. R. Weber, J. Gebauer, R. Krause-Rehberg, J. Appl. Phys. 83, 561 (1998)

[14] U. Siegner, R. Fluck, G. Zhang, U. Keller, Appl. Phys. Lett. 69, 2566 (1996) [15] G. D. Witt, Mater. Sci. Eng. B 22, 9 (1993)

[16] G. Segschneider, F. Jacob, T. Löffer, H. G. Roskos, S. Tautz, P. Kiesel, G. Döhler, Phys. Rev. B 65, (2002)

[17] M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanemoto, Y. Abe, N. Tsukada, Jpn. J. Appl. Phys. 1 33, 4807 (1994).

[18] A. Othonos, J. Appl. Phys. 83, 1789 (1998)

[19] D. C. Look, D. C. Walters, G. D. Robinson, J. R. Sizelove, M. G. Mier, C. E. Stutz, J. Appl. Phys. 74, 306 (1993)

[20] P. R. Smith, D. H. Auston, M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988)

[21] M. Tani, K. Sakai, H. Abe, S. Nakashima, H. Harima, M. Hangyo, Y. Tokuda, K. Kanamoto, Y. Abe, and N. Tsukada, Jpn. J. Appl. Phys. 33, 4807 (1994).

[22] S. Gupta, J. F. Whitaker, and G. A. Mourou, IEEE J. Quantum Electron. 28, 2 (1992).

464

tal-Weber, Z. [23] Wang, H.-H. Grenier, P. Whitaker, J. F. Fujioka, H. Jasinski, J. Lilien

[24] D. D. Nolte, W. Walukiewicz, and E. E. Haller, Phys. Rev. Lett. 59, 501 (1987).

[26] J er and J. P. Biersack, available online at http://www.srim.org [25]

G.-R, Lin, C,-C

. F. Ziegl .

is, M. Chicoine, and F.

hler, J. Selected Topics in Quantum Electron. [27] B. Salem, D. Morris, Y. Salissous, V. Aimez, S. Charlebo

Schiettekatte, J. Vac. Sci. Technol. A 24, 774 (2006).

[28] B. Salem, D. Morris, V. Aimez, J. Beerens, J. Beauvais, and D. Houde, J. Phys. Condens. Matter 17, 7327 (2005).

[29] S. Winnerl, F. Peter, S. Nitsche, A. Dreyhaupt, B. Zimmermann, M. Wagner, H, Schneider, M. Helm, and Klaus Ko

, (b) the bow-tie Fig. 2.1: Schematic view of (a) the photoconductive dipole antenna

antenna, and (c) the photoconductive strip line.*

*M. Tani, S. Matsuura, K. Sakai, and S. Nakashima, Appl. Opts. 36, 7853 (1997).

tic of an Auston antenna, used to generate subpicosecond optical pulse excites the sem

Fig. 2.2: Schema cycle bursts of THz

radiation. A iconducting substrate

*T. D. Do

in a region between two biased electrodes. This radiation is coupled into free space using a hemispherical substrate lens.*

Chapter 3

Modeling of terahertz radiation from

photoconductive antennas

ission efficiency of THz antennas ill allow greater control and wider applicability of THz radiation. Previously, Taylor

et al.

Understanding the physics critical to the em w

[1-2] used the current-surge model to explain several characteristics of large-aperture PC antennas, including: THz pulse width, saturation, and pump wavelength dependence. In their model, the formation of THz pulses is associated with the carrier dynamics in PC material, and the time-dependent THz radiation is proportional to the time derivative of the surface photocurrent. In 1996, Taylor et al. [3] used the drift-Poisson model to reproduce the bipolar THz radiation waveforms generated from different gap size of PC antennas. In their numerical simulation, they found that the negative peak of THz waveform becomes obvious if the pump fluence increases gradually. They attribute the formation of bipolar waveforms to the space-charge bias field screening effect, depending on the pump fluence and photoexcited carrier concentration.

In the following, we describe four theoretical models, including current-surge, Drude-Lorentz, drift-Poisson, and fullwave model. These models are usually adopted to interpret the characteristics of THz radiation.

3.1 Current surge model

This model assumes that the transient surface conductivity σ_s( )t grows rapidly when an antenna is excited by a sub-picosecond optical pulse [4]:

' (1 R) ( ) ( ') ( ') 'c t t t s opt q t t t I t h τ σ μ ν − − −∞ − =

∫

where R is the optical reflectivity of the illuminated area, μ is the carrier mobility, ( )t hν

( )

opt

I t is the optical pump pulse intensity, is the photon energy, and τ_c is the

lifetime of the excited carriers. Or, for the peak surface conductivity:

1 (1 R) ( ) , s opt q t F h σ μ ν − = (3.1.2)

is the value of the time-dependent mobility at the moment of maximum where μ₁

is the incident optical fluence.

and s

σ Fopt

In reality, the response of the photoconductor is slower than the rise time of the intensity of the excitation laser pulse because it takes up to several picoseconds for the transient mobility μ to reach its quasi equilibrium value. ( )t

Solving the Maxwell equations and taking into account the finite size of the emitter, one obtains for the radiated field E t_r( ):

0 0 ( ) ( ) , ( ) 1 s r b s t E t E t N σ η σ η = − + + (3.1.3)

where E_b is the applied bias electrical field, η₀ =1/ε₀c=376.7Ω is the impedance of free space, and N is the refractive index of the semiconductor. Eq. (3.1.3) contains the near-THz field screening effect, and its derivation will be detailed in Chapter 4. By the Eq. (3.1.3), two important points should be noted:

(1) the radiated THz field should raise linearly with the increase of the

applied bias voltage at least in the area of the Ohm’s law validity. At very high bias fields, the field dependence of the surface conductivity

r

E

s

σ have to be

taken into account,

(2) E_r increases and then saturates as the incident optical fluence grows.

3.2 Drude-Lorentz model

One usually use a simple one-dimensional Drude-Lorentz model to describe the behavior of the photo-excited carriers [5]. In Drude-Lorentz model, the current density is given by

v

f

j= −qn (3.2.1)

where nf is the carrier density and v the velocity averaged over the carrier distribution. The contribution by the holes is ignored for simplicity since their

contribution is very minor due their much larger effective mass. The change in carrier density over time can be described by:

+ ( ) f f c dn n G t dt = −τ (3.2.2)

Here τ_c is the carrier lifetime, and the photo-generation rate decided by the optical pump pulse. The motion of these generated carriers is slowed down by scattering, and their velocities can be described as:

( ) G t * v v( ) + l( r d t q ) E t dt = − τ m (3.2.3)

Where is the effective mass of the carriers, is the local field at the position of the carriers and

l

E

*

m

the momentum relaxation time. If E_l is constant, it follows, r τ / * ( ) v(0) t r+q Er l[1 ] v t e e m τ τ t/τr − − = − qz (3.2.4) When an electron (charge –q) separates from a hole (charge +q), a dipole is formed with magnitude μ = − . The negative sign indicates its direction is pointing toward

the hole and its position z is found by integrating equation (3.2.4). The polarisation density is therefore given by:

2 / * ₀ ( ) r f t[1 t r] . f q En P t qn z e d m τ τ τ − = − =

∫

If we consider space-charge screening effect [6], the local field results from combination of the biased and space-charge

l E b E E , that is: _s , s l b s b P E E E E ηε = + = − (3.2.6) ε s P is the dielectric

constant. The induced polarization changes with time, what can be described by [7]: ( ), s s r dP P j t dt = −τ + (3.2.3)

Combining Eqs. (3.2.1) – (3.23) and taking the time derivative, we obtain a second order differential equation:

2 2 2 v v 1 v + p s r r qP d d dt dt m ω * τ η ηε = − − τ , (3.2.3) 2 2 _/ * p nq m ω = ε

with the plasma frequency . Solving these equations gives the information of the carrier dynamics via the parameter and τ_c τ_r necessary to

describe the experimental results.

The Drude-Lorentz model is only applicable if the approximation of a free electron gas is valid. This is the case for a large number of excited electrons. Here we should note that one drawback of the Drude-Lorentz model is that it lacks the near-THz field screening effect. Another drawback is that it only considers the time-dependent behavior without including spatial factor. Nevertheless, the Drude-Lorentz model is still favorable for the simulation of THz radiation due to its simplicity and fast computation.

3.3 Drift-Poisson model

domain. These equations have been derived from Maxwell’s laws and consist of Poisson’s equation, the continuity equations and the transport equations. Poisson’s equation relates variations in electrostatic potential to local charge densities. The continuity equations describe the way that the electron and hole densities evolve as a result of transport processes, generation processes, and recombination processes. We will introduce the Poisson and continuity equations in the following.

3.3.1 Poisson

'

s equation

Poisson's equation belongs to a kind of elliptic equation, and it is of particular importance in electrostatics and Newtonian gravity. In electrostatics, Poisson's Equation relates the electrostatic potential V to the local space-charge density ρ :

(

ε

ρ

∇ ∇i = − (3.3.1)

ε is the local electric permittivity. where

We can write the electric field E in terms of an electric potential V with the gradient of the potential V.

E= −∇V (3.3.2)

3.3.2 Carrier continuity equations

∂ = − + ∇ ⋅ ∂ 1 n n n n G R t q J , (3.3.3) ∂ = − − ∇ ⋅ ∂ 1 p p p p G R t q J , (3.3.4)

Where n and p are the electron and hole concentration, Jn and Jp are the electron and

hole current densities, Gn and Gp are the photogeneration rates for electrons and holes,

R and Rn p are the recombination rates for electrons and holes, and q is the magnitude

of the charge on an electron.

Derivations based upon the Boltzmann transport theory have shown that the current densities in the continuity equations may be approximated by a drift-diffusion model. In this case the current densities are expressed as:

μ

= − ∇

n n

J q n V (3.3.5)

3.4 Fullwave model

Basic equations for fullwave analysis are three-dimensional Maxwell curl equations, including conductive current components. Although Boltzmann's transport equation for electrons may, in a strict sense, be solved for subpicosecond regions, current-continuity equations based on drift-diffusion for both electrons and holes are used as first-order approximations. These equations are

μ ∂ ∇ × = − ∂ H E t , (3.4.1) ε ∂ ∇ × = + + ∂ n p E H J J t, (3.4.2)

∂ = − + ∇ ⋅ ∂ 1 n n n n G R t q J , (3.4.3) ∂ = − − ∇ ⋅ ∂ 1 p p p p G R t q J , (3.4.4) μ = ∇ + ∇ J_n q _nn V qD_n n (3.4.5) μ = ∇ − ∇ p p p J q p V qD p (3.4.6)

where E and H stand for the electric and magnetic field, n and p are the electron and hole concentrations, respectively, R and G are the recombination and generation rates, respectively, μ is the magnetic permittivity, μ and n p, refer to the electron (hole) mobility and diffusion coefficient.

, n p

D

Sano et al. [8] adopt Eqs. (3.4.1) - (3.4.6) to analyze the propagation of an

electromagnetic wave within a PC dipole antenna with respect to different direction. In 2003, Hughes et al. [9] apply a vector approach to simulate the THz transients generated from a PC dipole antenna.

3.5 Numerical algorithm for Poisson's equation

In the drift-Poisson and fullwave model, Poisson's equation plays an important role in determining the profile of the electrostatic field of a THz antenna. Hence it is a essential project for us to solve Poisson's equation by a proper numerical algorithm. As we know, most of the electrostatic problems that come up in the real world are too hard to solve with formulas, or even by Fourier series. For such problems the most

widely used method is to use a spatial grid. This can be done in 1, 2, or 3 dimensions, but the technique will be illustrated here in 2 dimensions

Suppose that you have an electrostatics problem in infinitely-long geometry so that one of the dimensions (say the y dimension) is irrelevant. Then the equation that must be solved is Poisson's equation in the two dimensions x and z:

2 2 2 2 0 V V x z ε

ρ

= −

subject to boundary conditions on the electrostatics potential V . In THz antennas, x usually stands for the gap direction, z refers to the direction of pump pulse propagation (or penetration depth). To solve this problem on a grid we first choose a rectangular computation region bounded in x by x_min and x_max and in z by and . We then subdivide the x interval into

min

z

max

z N subintervals and the z interval _x

into N_z subintervals to form a grid in the x-z plane. The x-subintervals have length

max min

( ) / _x

x x x N

Δ = − and the z-subintervals have length Δ =z (z_max−z_min)/N_z . Because the ends of the intervals are included the grid is of size

and the (

(N_x+ ×1) (N_z +1)

i

x ; ) position of the grid point labeled by (i, k) is given by z_k

min

i ( 1)

x =x + − Δxi

;

(

2 1, , 1, ( , ) 2 2 2 | i k i k i k i k V V V V x x + − + − ∂ _≈ ∂ Δ , (3.5.3) 2 , 1 , , 1 ( , ) 2 2 2 | i k i k i k i k V V V V z z + − + − ∂ _≈ ∂ Δ , (3.5.4)

where . Substituting Eqs. (3.5.3) and (3.5.4) into Eq. (3.5.1), the grid version of Poisson's equation is approximately given by

, i j ( , )i k V =V x z 1, , 1, , 1 , , 1 , 2 2 0 2 2 i k i k i k i k i k i k i k V V V V V V x z ρ ε + − + − ₊ + − + − Δ Δ = − , (3.5.5)

It is by no means clear that this approximation is useful just by looking at it, but a large number of clever people have discovered over the years that it is useful to rewrite this equation by solving for Vi j, :

1 1, 1, , 1 , 1 , , 2 2 2 2 0 2 2 i k i k i k i k i k i k V V V V V x z x z ρ ε − + − + − ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢⎣ ⎥⎦ + + = + + + Δ Δ Δ Δ , (3.5.6)

The reason that this is a good idea is that it turns out that in this form the equation can be solved by iteration, just as we can solve the equation x = cos x by iteration:

, (3.5.7)

1 n

n cos

x ₊ = x

where n counts successive iterations. If you haven't seen this trick before, try it. Start with 1, then calculate cos(1), then cos(cos(1)), etc.. After doing this a bunch of times you will get the number 0.7391, which solves the equation x = cos x. The same trick works for Eq. (3.5.6): make a guess for across the whole grid, compute the right hand side of Eq. (3.5.6) from this guess and use the equation to get a new set of values for . Rewriting Eq. (3.5.6) in the symbolic form

, i j V , i j V

( )

Rhs

=

this iteration scheme can be written as

( )

1 n

V+

=

Rhs

Amazingly, this actually works and is called successive relaxation. It is, however, painfully slow. But the people who discovered this trick also discovered how to improve it in various ways and have, in fact, developed an entire field of study based on this problem. We will just discuss here the simplest improvement that is useful, but if you would like to see why this really works and be introduced to even better methods. The simple, but effective, improvement we will use is called Successive Over-Relaxation, or SOR for short, and it simply modifies the iteration scheme given above in the following way:

( )

(

)

1 n 1 n

n

V₊ = ×ω Rhs V − −ω V , (3.5.10)

ω ω

where is a number between 1 and 2. Using = 1 is just simple relaxation and using ω = 2 makes the iteration unstable so that V goes to infinity. But between these two extremes there is an optimum value of ω that makes the iteration converge much better than simple relaxation. You can get close to the best value of

ω by experimenting, but here is some rough guidance. For a 20 20× grid use = ω

1.7; for a 30 30× grid use ω = 1.8; for a 40 40× grid use = 1.85; and for a ω grid use ω

50 50× = 1.9.

but what if we want V (x, z) at some arbitrary point, then we have to interpolate the data in two dimensions. There are lots of ways to do this, but the simplest (and usually adequate, but least accurate) way is bilinear interpolation. In this scheme you begin by finding the four grid points that surround the point (x, z) at which you want to find the potential V (x, z). In what follows these points will designated by the symbols 00, 10, 01, and 11 denoting, respectively, the lower left point, the lower right point, the upper right point, and the upper left point.

Generally, the electric field E is useful for the practical calculation. Now suppose we have obtained V (x, z), we have to find E at all the grid points, and then you can use bilinear interpolation on the x and z-components of E just as it was described for V. To find the components of E recall that

i V j V E V x z ∂ ∂ = −∇ = − ∂ ∂ , (3.5.11)

So we just need to know how to numerically compute these partial derivatives using values of V on the grid. This is most easily accomplished by using a simple centered difference approximation: 1, 1, ( , ) | 2 i k i k i k V V V x x + − − ∂ _≈ ∂ Δ |( , )i k i k, 1₂ i k, 1 V V V z z + − − ∂ _≈ ∂ Δ (3.5.12) ;

3.6 Method of lines