## 國 立 交 通 大 學

## 光 電 工 程 研 究 所

## 碩 士 論 文

### 氮化銦之兆赫輻射與二次諧波及其載子

### 動力學

**THz Radiation, Second Harmonic **

**Generation, and Carrier Dynamics from **

**InN surface **

### 研 究 生：余科京

### 指導教授：安惠榮 教授

### 氮化銦之兆赫波與二次諧波及其載子動力學

**THz Radiation, Second Harmonic Generation, and **

**Carrier Dynamics from InN surface**

### 研 究 生：余科京 Student：Ke-Jing Yu

### 指導教授：安惠榮 教授 Advisor：Prof. Hyeyoung Ahn

### 國立交通大學

### 光電工程研究所

### 碩士論文

### A Thesis

### Submitted to Department of Photonics and

### Institute of Electro-Optical Engineering

### College of Electrical Engineering

### National Chiao Tung University

### In partial Fulfillment of the Requirements

### for the Degree of

### Master of Science

### in

### Electro-Optical Engineering

### July 2009

### Hsinchu, Taiwan, Republic of China

### 氮化銦之兆赫輻射與二次諧波及其載子動力學之研究

學生：余科京 指導教授：安惠榮教授 國立交通大學光電工程研究所 摘 要 我們利用兆赫波輻射光譜研究氮化銦的輻射機制，其包含瞬時光電流效應和 光整流效應。我們計算出 wurtzite 結構晶體的二階非線性光學之塊體和表面電場 感應的電偶響應，進而透過計算結果，兆赫波的入射角、方向角、極化角度相關 趨勢得到良好的擬合結果。在低功率激發下，氮化銦的輻射機制主要由瞬時光電 流所領導; 隨著激發強度提高, 非線性趨勢更加明顯。經由比對兆赫波的方向 角、極化角之趨勢，我們推測出瞬時光電流和光整流對於氮化銦的兆赫輻射影響。 二次諧波與光整流同屬於二階非線性光學響應，我們利用先前計算結果成功 驗證非對稱晶體的二次諧波主要由塊體電偶響應支配其他表面的響應。實驗上所 得到的擬合係數與理論計算相符。 利用時間解析的光激發探測技術，我們研究氮化銦薄膜和其奈米柱的瞬時載 子特性，並發現在奈米柱中的載子有較短的生命期，推測是因奈米柱的結構會產 生較多的缺陷而使得載子有較快的捕捉時間。我們觀察到薄膜的生命期與載子濃 度之間的反比趨勢，但因由缺陷濃度會影響載子的生命期，各樣品的缺陷濃度不 一致使此反比趨勢些微誤差。**THz Radiation, Second Harmonic Generation, and Carrier Dynamics **

**from InN surface **

Student：Ke-Jing Yu Advisors：Prof. Hyeyoung Ahn

Department of Photonics and Institute of Electro-Optic Engineering, College of Electrical Engineering National Chiao Tung University

**Abstract **

THz emission mechanisms for InN are investigated by using THz emission spectroscopy (TES). The second-order optical response of bulk and surface electric field for wurtzite crystal could successfully explain the incident angle-, azimuthal angle-, and pump polarization- dependence of emitted THz field. According to the azimuthal angle- and pump polarization- dependence of TES, we deduced the interplay of transient current and optical rectification in THz emission for InN.

Second harmonic generation (SHG) is the second-order optical response similar to optical rectification. From the calculated results we successfully demonstrated that bulk electric-dipole contribution dominates any surface contribution in SHF for noncentrosymmetric crystal. The best fitting parameters to azimuthal angle dependent SHG experiment are consistent with the calculated results based on bulk electric-dipole contribution.

We also investigated the transient carrier dynamics for InN epilayer and nanorods via time-resolved optical pump-probe technique. The time evolution of transient reflectivity changes of InN is the interplay of photoinduced absorption and band-filling. The pump fluence independent lifetime of InN epilayer is regarded as defect-related recombination time. We also observe an inverse relationship between carrier lifetime and carrier concentration.

**Acknowledgement **

### 在碩士班兩年，雖然研究過程辛苦但也使我成長許多，體驗到研

### 究的迷人之處，學習到正確的態度，感謝我的指導教授安惠榮老師無

### 私的教導，指引我研究方向及指導正確的研究觀念。感謝王怡超學長

### 在實驗系統架設的幫忙及雷射的維護，陳晉偉學長在二次諧波方面的

### 指導。平時與洪誌彰學長的研究討論讓我收穫很多，學長廣泛地學習

### 以及做事的高效率值得學習，還要感謝潘悉靈老師實驗室的同學陪伴

### 我兩年的修課及實驗，呂助增老師實驗室親切地交流分享二次諧波的

### 研究，還有實驗室的伙伴陪我度過最重要的碩二。最後，要感謝家人

### 的支持及協助讓我能無後顧之憂地完成碩士研究。

**Contents **

**Abstract(C)**

### ……….

### i

**Abstract(E)**

### ………

### ii

**Acknowledgement**

### ………...

### iii

**Contents**

### ………...

### iv

**1 Introduction **

### 1-1 Motivation………... 1

### 1-2 Organization of this Thesis………...

### 3

### Reference……….………….

### 4

**2 **

**Indium Ntride **

### 2-1 The material properties of InN in optical and THz spectral range 5

### 2-2 The optoelectronic applications for II-N semiconductors in

### nonpolar orientation ………..…...

### 7

### References………...…….. 9

**3 **

**THz emission and Second-harmonic-generation **

**from a-plane InN **

### 3-1 THz emission mechanism………..……...

### 10

### 3-1.1 Transient Current……….………

### 10

### Drift Current via Surface Depletion Field……….

### 12

### 3-1.2 Instantaneous Nonlinear Polarization

### (Optical Rectification)………..

### 15

### Bulk Seocnd-Order Optical Response……….……….

### 16

### Surface Second-Order Optical Response……….…….

### 22

### 3-2 Experimental

### Setups...

### 33

### 3-2.1

### THz

### emission

### Measurement

### system...

### 33

### 3-2.2 Reflection Second Harmonic Generation (RSHG) Setups..

### 36

### 3-3 Results and Discussions……….………...

### 37

### 3-3.1 Reflection SH Generation from InN……….……..

### 37

### 3-3.2

### Azimuthal

### Angle

### Dependence of THz Emission from InN

### 43

### 3-3.3 Pump Polarization Dependence of THz Emission from

### InN……….………...

### 49

### 3-3.4 The comparison between SH generation and THz

### emission from InN……….………...

### 53

### References……….……… 55

**4 **

**Carrier Dynamics of InN **

### 4-1 Background………...

### 58

### 4-1.1 Transient Photo-Reflection……….……….

### 58

### 4-1.2 Scattering Processes in Ultrafast Regime…………..……..

### 60

### 4-1.3 Effects of Carrier Generation………... 63

### 4-1.3.2 Band-Gap Renormalization

### 64

### 4-2 Experimental

### Setups……….………

### 66

### 4-3 Results and Discussions……….………...

### 67

### 4-3.1 Fluence Dependent Carrier Dynamic in InN film and

### nanorod……….……….

### 67

### 4-3.2 Concentration Dependent Carrier Lifetime

### 72

### References……….

### 73

**5 **

**Conclusions and Future Work **

** 5.1 Conclusions………... 75 **

**Chap 1. Introduction **

1-1. Motivation

Terahertz (THz) radiation lies in the frequency gap between the infrared and microwaves, typically is referred to as the frequencies from 100 GHz to 10 THz. 1

THz is equivalent to 33.33 cm- 1(wave numbers), 4.1 meV in photon energy, or 300

μm in wavelength. Two main directions of THz development are THz imaging in bio-
and medical fields and THz spectroscopy which has long been inaccessible due to the
difficulty in generation of strong, coherent THz waves. Despite several pioneering
breakthroughs, searching for the methods to generate reliable coherent THz radiation
is still a critical issue in THz science community. One of the simple and efficient
methods is to excite the semiconductor surface with sub-picosecond laser pulses. Up
*to now, p-type InAs is known to be the strongest THz emitter under no bias. Recently, *
due to the small intrinsic bandgap (0.7 eV) and quite large gap (2.8 eV) of the next
valley, InN has been inspired potential application in THz region such as efficient
THz emitter/receiver. The main THz emission mechanism of InN is photo-Dember
effect like InAs; however, THz radiation magnitude from InN films is typically one

order less than that from InAs [1-1, 1-2*]. Recently, Ahn et al. has reported THz *

emission enhancement from vertically aligned InN nanorod arrays [1-3] and nonpolar

*orientation (a-plane) InN film [*1-4], which is at least ten times stronger in amplitude
*than that from polar (c-plane) InN film. To investigate the limitation of THz emission *
from InN, fully understanding of the emission mechanism involving transient current
and instantaneous nonlinear polarization is quite important. In general, the main THz

*emission mechanism of c- and a-plane InN is regarded as photo-Dember effect [*1-3]

at low excitation. In addition to transient current, the influence of optical rectification

*(OR) from a-plane InN becomes significant at high excitation density [*1-4]. We

measurement from InN to reveal the important information of optical rectification contribution. In addition, we also investigated bulk and surface-electric-field (SEF) induced second-order optical response of wurtzite crystal. The calculated results based on the second-order optical response successfully duplicated experimentally observed angular dependence. The interplay of transient current and optical rectification is also investigated in this work.

Second harmonic generation (SHG) is considered as second-order optical response similar to optical rectification. SHG has been identified as a tool for studying bulk and surface symmetry in materials. Since InN is not centrosymmetric, one can expect that bulk dipole contribution dominates any surface term. Reflection SHG from InN film has significant six-fold rotation symmetry about surface normal which is well described by our calculation of bulk electric dipole contribution.

Many direct bandgap semiconductors generate THz radiation when they are optically excited by femtosecond pulses; however, their performance as THz emitters varies dramatically. In general, THz emission strength depends on the electron and hole mobility. In the quasi-classical approach to electron transport in semiconductors, the mobility parameter is used to describe the scattering events taking place in the material. Through the time evolution of carrier dynamics and THz emission, we found that carrier scattering and non-radiative recombination would occur when THz emission takes place. It means that these mechanisms may influence THz radiation. The pump fluence dependence of carrier dynamic of InN film and nanorods was studied by optical pump-probe spectroscopy. Additionally, concentration dependent lifetime of InN film is observed.

1-2. Organization of this Thesis

In this work we calculated bulk and surface-electric-field induced second-order
optical polarization for wurtzite structure crystal. Basing on the calculation results, we
explored the azimuthal angle and pump polarization dependence of THz radiation
*from c- and a-plane InN film surface. The azimuthal angle dependence of reflection *
*SHG from both c- and a-InN was examined and well described by our calculations. *
Besides, we investigated the carrier dynamic of InN film and nanorods.

In chapter 1, our motivation in this work is briefly introduced. In chapter 2, we introduce the material properties including growth condition and optical properties of InN in optical and THz range. In chapter 3, we studied reflection SHG and THz radiation. We present the analysis theories and origins of the second-order optical responses of wurtzite crystal. Then, experimental arrangements are presented, which contain ultrafast laser source, electro-optic sampling based THz emission spectroscopy (TES-EOS), and reflection SHG system. Finally, we present and discuss the azimuthal and polarization rotation angle-dependent results. In chapter 4, we focus on carrier dynamics in InN film and nanorods. In the beginning, we introduce scattering process and effect of carrier generation which are relevant to the investigation in this chapter. After that, an optical pump-probe spectroscopy used in exploring ultrafast dynamic is described. Eventually, we discuss pump fluence dependent carrier dynamics and concentration dependent carrier lifetime. Finally, our conclusion and future work are shown in chapter 5.

**References **

[1-1] R. Ascazubi ,I. Wilke, K. Denniston, H. Lu, and W. J. Schaff, “Terahertz
**emission by InN,” Appl. Phys. Lett. 84 4810 (2004) **

[1-2] G.. D. Chern, E D. Readinger, H. Shen, M. Wraback, C. S. Gallinat, G.
Koblmuller, and J. S. Speck, “Excitation wavelength dependence of terahertz
**emission from InN and InAs,” Appl. Phys. Lett. 89 141115 (2006) **

[1-3] H. Ahn, Y. P. Ku, Y. C. Wang, C. H. Chuang, S. Gwo, and C. L. Pan,

“Terahertz emission from vertically aligned InN nanorod arrays,” Appl. Phys.
**Lett. 91 132108 (2007) **

[1-4] H. Ahn, Y. P. Ku, C. H. Chuang, C. L. Pan , H. W. Lin, Y. L. Hong, and S.
Gwo, “Intense terahertz emission from a-plane InN surface,” Appl. Phys.
**Lett. 92 102103 (2008) **

**Chap 2. Indium Nitride **

**2-1. The material properties of InN in optical and THz spectral range **

*In this work, the a-plane InN epitaxial film (~1.2 μm) was grown by *

plasma-assisted molecular beam epitaxy (PA-MBE) on*γ-plane{1 102} sapphire wafer, *

*while the c-plane (000 1) InN epitaxial film (~2.5 μm) was grown on Si(111) using a *
*double-buffer layer technique. The back side of γ-plane sapphire wafer was coated *
with a Ti layer for efficient and uniform heating during the PA-MBE growth. The
*growth direction of the InN film was determined using a 2θ-ω x- ray diffraction scan. *
*Near-infrared photoluminescence was detected from the as-grown a- and c-plane InN *
*films at room temperature. Unintentionally doped n-type carrier concentrations of *
7.0×1018 and 3.1×1018 cm-3* and electron mobilities (μ) of 298 and 1036 cm*2/Vs were
*determined by room-temperature Hall effect measurements for a- and c-plane InN *
film, respectively. InN has wurtzite structure with bulk symmetry, 6mm as shown in

Fig. 2-1*. The atom arrangement in the surface of a- and c-plane InN is plotted in *Fig.
2-2.

The dispersion of optical properties is one of the most important material properties. The optical constants of the InN layers has been determined by spectroscopic Ellipsometry operating at the spectral range between 0.6 and 4.2 eV

[2-1] and by terahertz-time domain spectroscopy (THz-TDS) operating at THz range

[2-2]. Recently, the frequency-dependent terahertz conductivity of InN epilayer has been investigated by THz-TDS [2-2] and reveals Drude-like behavior.

The InN nanorods (750 nm) were grown on Si3N4/Si(111) at sample

temperature of 330oC (LT-NR). The N/In flux ratios was ~2.6 for LT-NR and was

adjusted at different growth temperature to ensure that the growth proceeded in the columnar mode. The morphologies and size distribution of InN nanorods were analyzed using field-emission scanning electron microscopy. The SEM image of the

hexagonal- shaped LT-NR exhibits nanorods with a uniform diameter of ~130 nm. The LT-NR exhibits an average aspect ratio (height/diameter) of ~6 and an aerial density of ~5×109 cm-2. The frequency dependent refractive index and conductivity

are determined by THz-TDS [2-2] at THz frequency range. The frequency

dependence is well-described by Drude-Smith model which modify the backward scattering events of carriers.

Figure 4-3. The diagram of wurtzite structure of InN.

**2-2. The optoelectronic applications for III-N semiconductors in nonpolar orientation **
III-Nitride group semiconductors have a wurtzite structure which posses a
strong built-in polarization field including spontaneous polarization and piezoelectric

polarization results from (In+, Ga+, Al+…)-N- arrangement. From the lattice

*orientation, it is divided into polar plane (c-plane) and nonpolar plane (a- and *

*m-plane). These inhomogeneous properties of lattice influence the distribution of *

photo-current and electric field and even the optoelectronic properties. Hence, the effect of polarization field has been investigated and is an important issue in application for III-N group semiconductor. We discuss the consequences of the very large electrostatic fields that exist within thin layers. The positive and negative charges in material would be separated along the direction of polarization field, and they accumulated at the two sides of polarization field or at the discontinuous region of the field. The carrier accumulation due to polarization field results in the band bending and varies the band-gap and the relative potential of electrons and holes. The carrier accumulation directly affects the radiative recombination rate and leads to

*redshift in emission. P. Waltereit et al [*2-3] has proposed a way to remove the

electrostatic field via nonpolar growth. In the nonpolar orientation, the material will
not carry spontaneous polarization components. The band profile of material grown in
nonpolar direction would thus be unaffected by electrostatic fields. It means that the
nonpolar material exhibits higher transition energy and electron-hole overlap of
*essentially unity. P. Waltereit et al represented fast radiative recombination time and *
*blueshift in luminance emission for nonpolar plane (m-plane) multiple quantum wells. *
At the room temperature, this fast radiative lifetime for nonpolar samples results in
high quantum efficiency.

Figure 2-3. The effect of electrostatic field on emission properties of GaN/(Al,Ga)N
multiple quantum wells. The blueshift in emission and fast radiative
*lifetime of c-plane with respect to the electric-field-free m-plane are the *
consequences of the intrinsic polarization field. [2-3]

In addition, terahertz emission efficiency has been improved by using the
*nonpolar crystal orientation. The dramatic terahertz emission enhancement of a-plane *
InN [see Fig. 2-4*] has been observed by Ahn et al [*2-4]. In the nonpolar orientation,
the intrinsic polarization field parallel to surface can induce a photo-current along the
sample surface. It is well-known that the radiation of in-plane current is more efficient
than that of current normal to surface due to the limited surface emission cone.
Therefore, photo-carrier induced terahertz radiation can be enhanced via nonpolar
growth.

*Figure 2-4. THz emission amplitude from a- and c-plane InN film. [*2-4] Inset shows
*the comparison between n-type InAs and a-plane InN. *

**Reference **

[2-1] H. Ahn, C. H. Shen, C. L. Wu, and S. Gwo, “Spectroscopic Ellipsometry
study of Wurtzite InN Epitaxial Films on Si(111) with Varied Carrier
**Concentrations,” Appl. Phys. Lett. 86 201905 (2005) **

[2-2] H. Ahn, Y. P. Ku, Y. C. Wang, C. H. Chuang, S. Gwo, and C. L. Pan,
“Terahertz spectroscopic study of vertically aligned InN nanorods,” Appl.
**Phys. Lett. 91 163105 (2007) **

[2-3] P. Waltereit, O. Brandt, A. Trampert, H.T. Grahn, J. Menniger, M. Ramsteiner,
M. Reiche, and K. H. Ploog, “Nitride semiconductors free of electrostatic
**fields for efficient white light-emitting diodes,” nature 406 865 (2000) **
[2-4] H. Ahn, Y. P. Ku, Y. C. Wang, C. H. Chuang, S. Gwo, and C. L. Pan,

“Terahertz emission from vertically aligned InN nanorod arrays,” Appl. Phys.
**Lett. 91 132108 (2007) **

**Chap 3. THz emission and Second-harmonic-generation from a-plane InN **
**3-1. THz emission mechanism **

The emission of THz radiation from semiconductor surfaces is very complicate due to multiple competing mechanisms leading to the radiation. Contributions from photo-carrier acceleration in the depletion field, photocarrier diffusion, and optical rectification have all been reported [3-1, 3-2, 3-3]. The relative magnitude of radiation resulting from the various processes is strongly dependent on excitation fluence. To investigate the limitation of THz generation, full recognizing of all different mechanisms leading to emission is quite important.

3-1-1. Transient Current

By means of above band-gap (resonant) nature of the excitation process, in a conventional transport picture one assumes that electron-hole pairs (EHP) are created by vertical transitions in real space within the penetration depth of the incident optical field. Subsequently, EHPs are accelerated by the surface depletion/accumulation field and diffuse to semiconductor inner due to gradient concentration, which leads to drift current and diffusion current, respectively. The photo-generated free carriers are expected to flow in surface normal direction and can be the source of THz wave in Maxwell equation. The direction of transient current leads to azimuthal angle independent optical property of THz field. Under steady pump power fluence, the pump polarization issue can be neglected if we ignore Fresnel coefficient for Tp and

Ts. The transient current induced THz field is proportional to photoexcited carrier

*density n and the mobility μ for electron and hole. *

To study the photoexcited THz radiation from semiconductors, we qualitatively
**describe the electric field E in dipole approximation when the carrier’s velocity is **
much less than the velocity of light. We calculate the electric field of an arbitrary

motion point charge, using Lienard-Wiechert potentials [3-4]
0
1
( , )
4 ( )
*qc*
*t*
*rc*
πε
=
− ⋅
**V r**
**r v** , ( , )*t* =*c*2 ( , )*t*
**v**
**A r** **V r**

**and the expression of correspond electric field E **
*t*
∂
= −∇ −
∂
**A**
**E** **V**

Here we skip the detailed gradient calculation process and just quote the finally
results [3-4]
2 2
3
0
( , ) [( ) ( )]
4 ( )
*q* *r*
*t* *c* *v*
πε
= − + × ×
⋅
**E r** **u r** **u v**
**r u**
where **u**≡*c***r**−**v**

Due to the carrier’s velocity is much less than the velocity of light (**u**≅*c***r**), we

further modify the equation of moving electron induced electric field

3 2 3

( / 4*e* πε*r* ) ( / 4*e* πε*c r* )[ ( )]

= + × ×

**E** **r** **r** **r v (3-1) **

*where r is the distance from the carrier to observation position, *ε is the dielectric
* constant, c is the velocity of light, and v is the acceleration of the carrier. The first *
term of Eq. (3-1) corresponds to Cloumb field; the magnitude decrease as

*2*

_{r and can }*be ignore at far field. The second term describes the radiation field which is p-wave.*

*radiation term in Eq. (3-1) can be rewritten as*

**Substituting carrier velocity v into current density J = ev, the photo-carrier induced**2 _{ˆ}

(sin / 4 *c r Ja*)

θ = θ πε θ

**E** (3-2)

where θ is the angle between carrier accelerated direction and observed direction.

Eq. (3-2) indicates that radiation field induced by accelerated carriers is proportional to time derivative of photo-carrier generated transient current density and the

**Figure 3-1 The charge accelerated in a direction with an angle**θ with respect to
observation.

Drift Current via Surface Depletion Field

In wide bandgap semiconductors such as GaAs (Eg = 1.43 eV) and InP (Eg = 1.34 eV), due to the discontinuous structure in the air/semiconductor interface, there exists a surface state in the forbidden gap between valance and conduction bands. Due to the band bending to continue Fermi-level on air/semiconductor interface, a depletion region forms, where a surface built-in field exists. The field direction is normal to surface, and strength is a function of Schottky barrier potential and dopant concentration, which is shown in Eq (3-3).

( ) ( )
*d*
*eN*
*E x* *W x*
ε
= − (3-3)
where ε * is material permittivity, N is dopant concentration, and W* is the depletion

*depth which can be written as a function of potential barrier V *
2
[ *kT*]
*W* *V*
*eN* *e*
ε
= − (3-4)
Therefore, surface depletion field induced total transient current is the sum of drift
current for electrons and holes,

( )

*n* *p* *n* *p* *d*

*J* =*J* +*J* = *en*μ +*ep*μ *E*

For large bandgap semiconductors, the direction of the band bending-induced electric
field depends on the dopant types so that the directions of drift current are opposite
*for n- and p-type semiconductors. After excitation, the photoexcited carriers in surface *

depletion field can be accelerated by an appropriate electric field and the resultant
transient electric dipole can lead to the generation of terahertz pulses and the
*polarization of THz radiation is p-polarized for both p- and n-type semiconductors *
and THz field strength is proportional to the time derivative of the transient current as
the description in Eq. (3-2), that is,

THz *J t*( )
*E*
*t*
∂
∝
∂ (3-5)

Meanwhile, for narrow band gap semiconductors which possess a narrow depletion region near the surface and the so-called the photo-Dember effect dominates the THz emission from them.

Figure 3-2 Photo drift current via surface depletion field (a) for typical n-type semiconductor (b) for typical p-type semiconductor.

Diffusion Current via Photo-Dember Effect

For the narrow bandgap semiconductors, such as InAs and InSb, the band bending is not significant and for photoexcitation with near IR light, the excess energy of the narrow semiconductor is very large. In addition, the optical absorption length is typically of the order of a few hundred nm. Photoexcitation increases the photoexcited carriers near the surface and then the carriers diffuse into semiconductor. Due to the different diffusion velocity between electrons and holes, the spatial charge separation

can be formed and induces the build-in electric field (or photo-Dember field). The photo-Dember current is the sum of the diffusion current for electrons and holes,

diffusion *n* *p* ~ *e* *h*
*n* *p*
*J* *J* *J* *eD* *eD*
*x* *x*
∂Δ ∂Δ
= + − +
∂ ∂ (3-6)

*where n*Δ and Δ*p* are the densities of photoexcited electrons and holes, *D and _{e}*

*h*

*D indicate diffusion coefficient of electrons and holes, respectively. The diffusion *

*coefficient D is defined by the Einstein relation, D k T*= * _{B}* μ/

*e*, where μ is the mobility of electrons or holes which can be obtained by Hall measurement. Hence, THz radiation is proportional to the difference in mobility and temperature between the electrons and holes, and the gradient carrier density. Since the mobility of electrons is always greater than that of holes, the photo-Dember current induced THz fields for different dopant semiconductors (n-type or p-type) have the same irradiative direction, which is outward and normal to surface as shown in Fig. 3-3. Therefore, the photo-Dember effect is dominant for THz emission from the narrow bandgap

semiconductors with the large mobility ratio of electrons and holes (μ μ* _{e}*/

*) and*

_{h}richer excess energy of carriers (*T ) _{e}* [3-5]. Finally, because of small effective mass
and high mobility, the photo-current has fast rise- and decay time character which is
benefit to obtaining a broadband THz spectrum.

Figure 3-3 Photo diffusion current from different diffusion velocity of electrons and holes.

**3-1.2. Instantaneous Nonlinear Polarization (Optical Rectification) **

As ultrafast laser excited semiconductor via resonant (above band-gap) or
nonresonant (below band-gap) excitation process, because of spatial separation of
**final electron and hole states, this process leads to an instantaneous polarization P**0

whose second time derivative 2 2

0*/ t*

∂ **P** ∂ determines the radiated signal. In general,

linear polarization takes place in dielectric material, while nonlinear polarization

occurs when nonlinear coefficient _{χ}(2)_{E}_{+}_{χ}(3)* _{E}*2

_{+}

_{...}

_{is comparable to linear }

coefficient _{χ at high excitation density.}(1) _{ }

In addition to transient current induced by free carriers, the instantaneous
polarization from bound charge also contribute THz emission. Free carriers dominate
THz radiation at low excitation, while under high-density excitation nonlinear effect
become significant and the screening effect of surface electric field occurs via the
high density carriers, which results in the decrease of the transient current. According
*to the excitation density dependence of the temporal wave forms by M. Nakajima et *

*al *[3-6], the dominant radiation mechanism changes from the drift current, for low
density excitation, to the diffusion current and the optical rectification, for
high-density excitation. Except for transient current, in this section we focus on the
nonlinear optical effect on the THz radiation. Optical rectification can be regards as a
second order optical response which includes second harmonic generation, sum
frequency generation, and differential frequency generation. For an anisotropic
medium, the instantaneous nonlinear polarization in dipole approximation can be
expressed by
(2) (2)
3 1 2 3 1 2 1 2
( ) ( ; , ) ( ) ( )
*i* ω ω ω= + =χ*ijk* −ω ω ω *j* ω *k* ω
**P** **E** **E** (3-7)

THz emission occurs via the differential frequency generation of two identical input

optical pulses (that is, Ω = − ). Thus, the influence elements of nonlinear ω ω

polarization are crystal symmetric characters, the polarization of optical pump and crystal orientation. The effective second-order susceptibility (2)eff

*ijk*

χ can have bulk-

( *bulk*

*ijk*(2)

χ ) and surface-electric-field induced ( (2)SEF

*ijk*

χ ) contributions. For the surface-

electric-field contribution, we take only the term of the first-order in the expansion and then the effective nonlinear susceptibility becomes.

(2)eff (2)_{( ;} _{, ) 3} (3)_{( ;} _{, ,0)} surf

*ijk* *ijk* *ijkl* *Ez*

χ =χ Ω −ω ω + χ Ω −ω ω (3-8)

where (2)bulk

*ijk*

χ is the intrinsic bulk second-order susceptibility tensor which is

determined by the symmetry properties of the bulk crystal and we refer _{3} (3) surf

*ijklEz*

χ to

surface-electric-field (SEF) induced effective surface susceptibility. In the following, we will discuss the bulk- and SEF induced second-order optical responses separately. It should be noted here that the calculation that follows is sufficient to describe the process of second harmonic generation, as the only difference is that the polarization

of the lattice occurs at the sum frequency ( 2ω ω ω= + ), whereas the far-infrared

generation occurs at the difference frequency (Ω = − ). ω ω

Then, by substituting the bulk- (SEF-) susceptibility into nonlinear polarization and applying transformation matrix crystal (surface) coordinate to lab coordinate, we could get bulk- (SEF-) second-order optical response (ex. THz field and SHG).

Bulk Second-Order Optical Response

Bulk optical polarization can contribute to Terahertz emission and SHG from semiconductor surfaces in reflection geometry. The second-order susceptibility tensor is determined by according to crystal bulk symmetry, and then the intrinsic bulk

nonlinear polarization contributed to THz radiation becomes
bulk_{( )} (2)bulk_{(} _{;} _{, ) (} _{) ( )}

*i* Ω =χ*ijk* −Ω −ω ω *j* −ω *k* ω

**P** **E** **E** (3-9)

The index notations are determined by the crystal coordinate corresponding to crystal orientation. It should be noted that except the radiated frequency there is the same angularly form for second-order polarization such as SHG and THz. Generally, for crystal without inverse symmetry, it is well known that bulk electro-dipole contribution to SHG dominates over any contribution from surface term. Since InN has a wurtzite structure, (2)bulk

*ijk*

χ has symmetry properties of the 6mm bulk crystal.

The nonvanishing (2)bulk

*ijk*

χ elements in 6mm symmetry can be expressed as

15 24 33

d =d = ⋅a d and d_{31} =d_{32} = ⋅b d_{33}, where a = b = -0.5 in the bound-charge

theory [3-7].

*Here we use d coefficients to simply the expression of electric susceptibility *

tensor and their relations are (2) (2)

15 *xxz* *xzx*
*d* =χ =χ , (2) (2)
24 *yyz* *yzy*
*d* =χ =χ , (2)
31 *zxx*
*d* =χ ,
(2)
32 *zyy*
*d* =χ , and (2)
33 *zzz*

*d* =χ . Therefore, the bulk susceptibility tensor for 6mm

symmetry [3-8] can be written as

15
( )bulk
15
31 31 33
0 0 0 0 0
0 0 0 0 0
0 0 0
*ijk*
*d*
*d*
*d* *d* *d*
χ 2
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
(3-10)

*The bulk susceptibility tensors for a- and c-plane are the same since both of *
*them have the same bulk symmetry, 6mm. For a-plane InN, defining the beam *
coordinate system as ( , , )*s*ˆ κˆ *z*ˆ , where *ˆz is normal to the crystal surface, s-axis is on *
*the crystal surface and parallel to an s -polarized pump beam , and ˆ*κ = × . The *ˆz s*ˆ
crystal orientation is selected to be ˆx=[1120], ˆy=[1100], and ˆz [000 1]= .

second-order polarization in the lab coordinate system as a function of azimuthal

angle

### φ

*, pump polarization θ, and incident angle by applying the transformation*

matrix *As l*→ and *Ac s*→ which represent the coordinate transformation from surface
to lab and from crystal to surface, respectively.

(2) (2)

lab *s l* *c s* crystal

*P* =*A A P*_{→} _{→} (3-11)
The optical rectification induced second-order radiation fields are

(2) (2)
eff
*s* *s* *s*
*E* = Ω*A L* ⋅*P* , (2) (2) (2)
eff
*p* *p* *s z* *c k*
*E* =*A* Ω*L* ⋅⎡_{⎣}*F P* −*F P* ⎤_{⎦}. (3-12)

The coefficients *F , _{s}*

*F ,*

_{c}*A ,*

_{s}*A*, and

_{p}*L in Eq. (3-12) are defined as the*

_{eff}

projection of the reflected SHG light on the spatial coordinates inside the medium, the
corresponding amplitude, and effective phase-matching length, respectively. Their
expressions are
sin /
*s*
*F* = Θ *N*, 2
1
*c* *s*
*F* = −*F* , 2
cos
*p*
*p*
*T*
*A* = π
Θ,
2
cos
*s*
*s*
*T*
*A* = π
Θ

where Θ denotes the incident angle of reflection SHG light, and *T and _{p}*

*T are the*

_{s}*Azimuthal angle dependence of bulk contribution for wurtzite crystal *

The generated second-order radiation field for *s- or p- polarized harmonic light *

due to the bulk anisotropic source as a function of the azimuthal angle become

bulk , (bulk) , (bulk) , (bulk)

, , eff a0 + (cos ) (sin )

*s p* *s p* *m* *s p* *m*

*s p* *s p* *m* *m*

*E* =*A* Ω*L* ⎡_{⎣} *b* φ +*c* φ ⎤_{⎦} (3-13)

the subscript notation indicates *s- or p-polarized bulk electric field. The corresponding *

coefficients are listed in Table I, and all the distinct elements not shown in the table
are zero. The coefficients *f , _{s}*

*f ,*

_{c}*F ,*

_{s}*F ,*

_{c}*t ,*

_{s}*t*,

_{p}*A ,*

_{s}*A*, and

_{p}*L in Table I are*

_{eff}

as defined as the projection of fundamental light and reflected SHG on lab
coordinated inside medium, the transmission coefficients, the corresponding
amplitude, and the effective phase-matching length, respectively. Due to the
azimuthal angle dependent experiment of the nonlinear OR induced electric field are
selected as (2)
*pp*
*E , * (2)
*ps*
*E , * (2)
*sp*
*E , and * (2)
*ss*

*E* by polarizer [notated as Ein,out; for example,

E*ps *represent *s-polarized electric field irradiated by p-polarized fundamental light]. *

We re-write the expressions of optical rectification induced electric field in Table II. It
is noted that *φ is defined as the angle between the ˆk axis and the crystal *[1100]

Table I. General expressions of bulk contribution to second-order response for *a- and *
*c-plane *

*Coefficients of bulk second order response from a-plane *
bulk
*p*
*E* 1(bulk) eff 15 31 33 15
(bulk)
3 eff 15 31 33
(bulk) 2 2 2 2 2 2
1 eff 15 33 15 15 31 31
{ [2 (2 2 2 )] (2 )
{ [2 (4 2 2 )]}
{ (2 )} { (2 ) [ (2 ) ]
*p*
*p* *s p* *s* *p* *c* *c* *s* *s*
*p*
*p* *s p* *s* *p* *c* *c*
*p*
*p* *s* *s* *c* *p* *p* *s* *c s* *c* *c* *s*
*b* *A* *L* *t t E E F* *f* *d* *d* *d* *F* *f d*
*b* *A* *L* *t t E E F* *f* *d* *d* *d*
*c* *A L* *t E F* *d* *d* *t E F* *f f d* *F f* *d* *d* *f d*
⎡ ⎤
= Ω _{⎣} − + − _{⎦}
⎡ ⎤
= Ω _{⎣} + − _{⎦}
= Ω − + − + +
(bulk) 2 2 2 2 2
3 eff 15 33 31 15 31 33
}
( 2 ) { (2 )}
*p*
*p* *s* *s* *p* *p* *c*
*c* *A L* *t E* *d* *d* *d* *t E f* *d* *d* *d*
⎡ ⎤
⎣ ⎦
⎡ ⎤
= Ω _{⎣} − + − + + − _{⎦}
bulk
*s*
*E* (bulk) 2 2 2 2 2 2
1 eff 15 31 15 33 31
(bulk) 2 2 2 2 2
3 eff 15 31 33 33 15 31
(bulk)
1 eff 15 31 33
2(bulk)
3 ef
( 2 ) ( (2 ) )
(2 ) ( 2 )
( 4 )( )
*s*
*s* *s* *s* *p* *p* *c* *s*
*s*
*s* *s* *s* *p* *p c*
*s*
*s* *s p* *s* *p* *c*
*s*
*b* *A L* *t E* *d* *d* *t E* *f* *d* *d* *f d*
*b* *A L* *t E* *d* *d* *d* *t E f d* *d* *d*
*c* *A L* *t t E E* *f* *d* *d* *d*
*c* *A L*
⎡ ⎡ ⎤⎤
= Ω _{⎣} − − + _{⎣} − − _{⎦}_{⎦}
⎡ ⎤
= Ω _{⎣} + − + − − _{⎦}
⎡ ⎤
= Ω _{⎣} − + − _{⎦}
= Ω _{f}⎣⎡*t t E Es p* *s* *p*(4 )(2*fc* *d*_{15}+*d*_{31}−*d*_{33})⎤⎦

Coefficients of bulk second order response from *c-plane *

bulk
*p*
*E* (bulk) 2 2 2 2 2 2
0 eff 15 31 33 31
a*p* = { 2 + }+ { }
*p* *p* *p* *c c s* *s c* *s s* *s* *s* *s*
*A* Ω*L* ⎡_{⎣}*t E* −*F f f d* +*F f d* *F f d* *t E F d* ⎤_{⎦}
bulk
*s*
*E* (bulk)
0 eff 15
a*s* = (2 )
*s* *s p* *s* *p s*
*A L*Ω ⎡_{⎣}*t t E E f* *d* ⎤_{⎦}

Table II. Bulk contribution for different polarization combination from *a- and c-plane. *

Coefficients of bulk second order response from *a-plane *

bulk
*pp*
*E*
(bulk) 2 2 2 2
1 eff 15 15 31 31
(bulk) 2 2 2
3 eff 15 31 33
{ (2 ) [ (2 ) ]}
{ (2 )}
*pp*
*p* *p* *p* *s* *c s* *c* *c* *s*
*pp*
*p* *p* *p* *c*
*c* *A L* *t E F* *f f d* *F f* *d* *d* *f d*
*c* *A L* *t E f* *d* *d* *d*
= Ω ⋅ − + +
= Ω ⋅ + −
bulk
*sp*
*E*
(bulk) 2 2
1 eff 15 33
(bulk) 2 2
3 eff 15 33 31
{ (2 )}
( 2 )
*sp*
*p* *s* *s* *c*
*sp*
*p* *s* *s*
*c* *A L* *t E F* *d* *d*
*c* *A L* *t E* *d* *d* *d*
= Ω ⋅ −
= Ω ⋅ − + −
bulk
*ps*
*E*
(bulk) 2 2 2 2
1 eff 15 33 31
(bulk) 2 2 2
3 eff 33 15 31
( (2 ) )
( 2 )
*ps*
*s* *p* *p* *c* *s*
*ps*
*s* *p* *p c*
*b* *A L* *t E* *f* *d* *d* *f d*
*b* *A L t E f d* *d* *d*
⎡ ⎤
= Ω ⋅ _{⎣} − − _{⎦}
= Ω − −
bulk
*ss*
*E* 1 (bulk) eff 2 2 15 31
(bulk) 2 2
3 eff 15 31 33
( 2 )
(2 )
*ss*
*s* *s* *s*
*ss*
*s* *s* *s*
*b* *A L* *t E* *d* *d*
*b* *A L* *t E* *d* *d* *d*
= Ω ⋅ − −
= Ω ⋅ + −

Coefficients of bulk second order response from *c-plane *

bulk
*pp*
*E* (bulk) 2 2 2 2
0 eff 15 31 33
a*pp* = { 2 + }
*p* *p* *p* *c c s* *s c* *s s*
*A L*Ω ⋅*t E* −*F f f d* +*F f d* *F f d*
bulk
*sp*
*E* (bulk) 2 2
0 eff 31
a*sp* = { }
*p* *s* *s* *s*
*A L*Ω ⋅*t E F d*
bulk
*ps*
*E* all coefficients 0=
bulk
*ss*
*E* all coefficients 0=

For wurtzite structure, the bulk contribution of second-order response for

*a-plane has one-fold and three-fold symmetry; however, that for c-plane is azimuthal *

angle independent regardless of *s- or p-polarized electric field, as it can be seen in *

Table I and Table II.There are interesting results that bulk electric dipole for *a-plane *

orientation ONLY contributes to azimuthal angle dependence, while that for *c-plane *

ONLY contributes azimuthal angle independence.

It should be noted that due to the complicate coefficients it is difficult to organize them to a simple expression of cosθ, cos(3 )θ , sinθ and sin(3 )θ . Hence,

in Table I and Table II, we simply describe electric field by a linear combination of
cosθ, _{cos}3_{θ , sin}θ_{, and } _{sin}3_{θ . }

Surface-Electric-Field-Induced Second-Order Optical Response

In non-centrosymmetric crystal, the intrinsic surface contribution to optical response is negligibly small compared to the bulk contribution. However, when DC electric field is sufficiently large, the electric-field induced effective second-order optical polarization should be considered. In this session, the surface-electric-field (SEF) induced second-order polarization will be described. As we did in bulk contribution, we start from determining SEF induced second-order susceptibility tensor.

Surface-electric-field-induced effective susceptibility tensor, (2)SEF

*ijk*

χ is derived

from the third-order bulk electric susceptibility tensor
(2)SEF _{3} (3)bulk_{( ;} _{, ,0)} surf

*ijk* *ijkl* *Ez*

χ = χ Ω −ω ω

It means that a surface electric field ( surf

*z*

*E* ) breaks the bulk symmetry ( (3)bulk

*ijkl*

χ ) and

creates a new effective susceptibility ( (2)SEF

*ijk*

(3)

*ijkl*

χ for wurtzite crystals with 6mm symmetry has only 18 nonzero elements: (3)

*xxxx*
χ ,
(3)
*xxyy*
χ , (3)
*xyyx*
χ , (3)
*xyxy*
χ , (3)
*xzzx*
χ , (3)
*xzxz*
χ , (3)
*xxzz*
χ , (3)
*yyyy*
χ , (3)
xx
*yy*
χ , (3)
xx
*y y*
χ , (3)
xyx
*y*
χ , (3)
zz
*yy*
χ , (3)
zz
*y y*
χ ,
(3)
z z
*y y*
χ , (3)
zzzz
χ , (3) (3)
zzyy zzxx
χ =χ , (3) (3)
zyyz zxxz
χ =χ , (3) (3)
zyzy zxzx
χ =χ .

To properly examine the effective second-order SEF response, we transform the
third-order bulk susceptibility tensor to surface coordinate so that the surface field lies
along the ˆz axis in surface coordinate. Using the transfer matrix, * _{R , we expect }*rot
the form of the effective SEF-induced second-order tensor to be [3-13]

(2)SEF _{3} surf rot rot rot rot (3)crystal

*z* *l* *m* *n* *zo* *lmno*
*lmno*

*E* *R R R R*

αβγ α β γ

χ =

### ∑

χ (3-14)For *a-plane InN, it becomes *

15
(2)SEF
24
31 32 33
0 0 0 0 0
0 0 0 0 0
0 0 0
*ijk*
χ
∂
⎡ ⎤
⎢ ⎥
=_{⎢} ∂ _{⎥}
⎢∂ ∂ ∂ ⎥
⎣ ⎦
(3-15)

where ∂* _{ij}* are constants to represent SEF tensor elements
(3) (3)
15 3

*Esz*[χ

*yyxx*χ

*yxyx*] ∂ = + (3) (3) 24 3

*Esz*[χ

*zzxx*χzxzx] ∂ = + (3) 31 3

*Esz*χ

*xyyx*∂ = ⋅ (3) 32 3

*Esz*χ

*xzzx*∂ = ⋅ (3) 33 3

*Esz*χ

*xxxx*∂ = ⋅

For *c-plane InN, it is *
15
(2)SEF
15
31 31 33
0 0 0 0 0
0 0 0 0 0
0 0 0
*ijk*
χ
∂
⎡ ⎤
⎢ ⎥
=_{⎢} ∂ _{⎥}
⎢∂ ∂ ∂ ⎥
⎣ ⎦
(3-16)
where
(3) (3) (3) (3)
15 3*Esz*[χ*xzxz* χ*xxzz*] or 3*Esz*[χ*yyzz* χ*yzyz*]
∂ = + +
(3) (3)
31 3*Esz* χ*zxxz* or 3*Esz* χ*zyyz*
∂ = ⋅ ⋅
(3)
33 3*Esz* χ*zzzz*
∂ = ⋅

We apply the transfer matrix to the effective SEF-induced susceptibility, and evaluate the nonlinear polarization.

*Azimuthal angle dependence of surface contribution for wurtzite crystal *

Applying the same coordinate axes as for the bulk case, the anisotropic *s- or *

*p-polarized, SEF-induced radiated fields becomes *

surf , (surf) , , (surf)

, , eff + cos (n ) sin (n )

*s p* *s p* *s p*

*s p* *s p* *n* *n*

*E* = *A* Ω*L* ⎡_{⎣}*a* *b* φ +*c* φ ⎤_{⎦} (3-17)

the index notation is same as that for bulk contribution case and the coefficients are

listed in Table III. The SEF-induced optical response for different polarization

combination is listed in Table IV.

Examining the bulk- and SEF-induced second order electric field, it is not
*possible to separate the bulk- and SEF-induced contribution for c-plane InN due to the *
identical susceptibility. However, it becomes possible to distinguish them in *a-plane *

InN since the built-in electric field distorts the crystal symmetric and creates a new SEF-induced susceptibility [3-15].

Table III. General expressions of SEF induced contribution to second-order response
for **a- and c-plane **

Coefficients of SEF induced second order response from *a-plane *

surf
*p*
*E*
[ ]
[ ]
(surf) 2 2 2 2 2 2
eff 31 32 31 32 33 24 15
(surf) 2 2 2 2 2
2 eff 32 31 31 32 15 24
(surf)
2
1/ 2 ( ) [1/ 2 ( ) ] [ ( )]
[1/ 2( )] [ 1/ 2( )] [ ( )]
*p*
*p* *s* *s* *s* *p* *p* *s* *c* *s* *c* *c s*
*p*
*p* *s* *s* *s* *p* *p* *s* *c* *c* *c s*
*p*
*p*
*a* *A* *L* *t E* *F* *t E* *F* *f* *f* *F f f*
*b* *A* *L* *t E F* *t E F f* *F f f*
*c* *A* *L*
⎡ ⎡ ⎤⎤
= Ω _{⎣} ∂ + ∂ + _{⎣} ∂ + ∂ + ∂ − ∂ + ∂ _{⎦}_{⎦}
⎡ ⎡ ⎤⎤
= Ω _{⎣} ∂ − ∂ + _{⎣} ∂ − ∂ − ∂ − ∂ _{⎦}_{⎦}
= Ω eff⎣⎡*t t E E F fs p* *s* *p*[ *s*[ (*c* ∂ − ∂31 32)]−*F fc*[ (*s* ∂ − ∂15 24)]]⎤⎦
surf
*s*
*E*
(surf )
eff 15 24
(surf )
2 eff 24 15
(surf ) 2 2
2 eff 15 24
[ ( )]
[ ( )]
[ ( )]
*s*
*s* *s p* *s* *p* *s*
*s*
*s* *s p* *s* *p* *s*
*s*
*s* *p* *p* *s c*
*a* *A L* *t t E E f*
*b* *A L* *t t E E f*
*c* *A L* *t E f f*
⎡ ⎤
= Ω _{⎣} ∂ + ∂ _{⎦}
⎡ ⎤
= Ω _{⎣} ∂ −∂ _{⎦}
⎡ ⎤
= Ω _{⎣} ∂ −∂ _{⎦}

Coefficients of SEF induced second order response from *c-plane *

surf
*p*
*E* (surface) 2 2 2 2 2 2
15 31 33 31
a*p* = { 2 + ( + )}+ { }
*p* *eff* *p* *p* *c c s* *s* *c* *s* *s* *s* *s*
*A* Ω*L* _{⎣}⎡*t E* −*F f f* ∂ *F f* ∂ *f* ∂ *t E F*∂ ⎤_{⎦}
surf
*s*
*E* (surface)

### {

### }

eff 15 a*s*= (2 )

*s*

*s p*

*s*

*p*

*s*

*A L*Ω ⎡

_{⎣}

*t t E E*

*f*∂ ⎤

_{⎦}

Table IV. SEF induced contribution for different polarization combination from a- and c-plan

Coefficients of SEF induced second order response from a-plane
surf
*pp*
*E* (surf) eff 2 2 2 31 32 2 33 24 15
(surf) 2 2 2
2 eff 31 32 15 24
[1/ 2 ( ) ] [ ( )]
[ 1/ 2( )] [ ( )]
*pp*
*p* *p* *p* *s* *c* *s* *c* *c s*
*pp*
*p* *p* *p* *s* *c* *c* *c s*
*a* *A* *L* *t E* *F* *f* *f* *F f f*
*b* *A* *L* *t E* *F f* *F f f*
⎡ ⎤
= Ω ⋅ _{⎣} ∂ + ∂ + ∂ − ∂ + ∂ _{⎦}
⎡ ⎤
= Ω ⋅ _{⎣} ∂ − ∂ − ∂ − ∂ _{⎦}
surf
*sp*
*E* [ ]
[ ]
(surf) 2 2
eff 31 32
(surf) 2 2
2 eff 32 31
1/ 2 ( )
[1/ 2( )]
*sp*
*p* *s* *s* *s*
*sp*
*p* *s* *s* *s*
*a* *A L* *t E* *F*
*b* *A L* *t E F*
= Ω ⋅ ∂ + ∂
= Ω ⋅ ∂ − ∂
surf
*sp*
*E* (surf ) 2 2
2 eff [ ( 15 24)]
*ps*
*s* *p* *p* *s c*
*c* = Ω*A L* ⋅*t E f f* ∂ − ∂
surf
*ss*
*E* all coefficients 0=

Coefficients of SEF induced second order response from c-plane
surf
*pp*
*E* (surface) 2 2 2 2
15 31 33
a*pp* = { 2 + ( + )}
*p* *eff* *p* *p* *c c s* *s* *c* *s*
*A*Ω*L* ⋅*t E* −*F f f* ∂ *F f* ∂ *f* ∂
surf
*sp*
*E* (surface) 2 2
31
a*sp* = { }
*p* *eff* *s* *s* *s*
*A*Ω*L* ⋅*t E F*∂
surf
*ps*
*E* all coefficients 0=
surf
*ss*
*E* all coefficients 0=

*Pump polarization dependence for wurtzite crystal *

We know the instantaneous second-order polarization is expressed as (2) (2)

*i* *ijk* *j* *k*

*P* =χ *E E* . Therefore, in order to examine the effect of optical rectification, the

dependence on either the effective susceptibility or the electric field driven by the laser pulses should be investigated. The azimuthal angle rotation dependence discussed before shows the contribution of the effective electric susceptibility tensor

(2)eff

*ijk*

χ , in which the bulk and surface OR mechanism in THz emission could be

investigated. On the other hand, we vary the quantity projection of *s-polarized and *

*p-polarized fundamental beam on sample surface via rotating pump polarization at a *

fixed incident angle to identify the effect of electric field on THz emission.

The polarization of pump beam was rotated by a half-wave plate, while
maintaining the azimuthal angle and incident angle of sample. As discussed in
azimuthal angle-dependence experiment, we examined the influence of pump
*polarization direction on the emission of p- and s- THz waves. We assumed that the *

pump beam is composite of *p- and s- polarized electric field with a fundamental *

frequency, i.e. pump

*p*

**E** **= E**pumpcos*θ and * **E**pump*s* **= E**pump sin*θ. The linear polarization *

angle θ is defined as the angle between the linear polarized pump beam and a

*p-polarized pump beam. The photo-carrier generated THz field is independent on *

pump polarization; however, surface and bulk nonlinear responses are possible
proportional to _{(} pump 2_{)}

*p*

*E* , _{(} pump 2_{)}

*s*

*E* and _{(} pump pump_{)}

*p* *s*

*E* *E* .

For example, for (100) face of n-InAs, emitted *s-polarized THz signal * THz

*s*

*E* is

proportional to pump pump

*p* *s*

*E* *E* , while THz

*p*

*E* is proportional to a linear combination of

pump 2
(*E _{p}* ) and

_{(}pump 2

_{)}

*s*

*E*, i.e. THz 1 2cos(2 )

*p*

*E*∝ +

*c*

*c*θ and THz 3sin(2 )

*s*

*E*∝

*c*θ [3-12].

Fig. 3-4 shows the *s- and p-polarized THz field amplitude from (100) n-InAs as a *

function of pump polarization rotation. The pronounced two-fold symmetry for *s- and *

*p-polarized THz field was interpreted as the polarization rotation symmetric property. *

The two main contributions to a large offset of *p-polarized THz field compared to *

*s-polarized THz field are transient current and polarization angle independence of OR. *

The 45 degree angle shift between *s- and p-polarized THz fields is attributed to the *

difference between polarization dependence of cos(2*θ) and sin(2θ). *

**-50** **0** **50** **100** **150** **200** **250** **300** **350** **400**
**0**
**1000**
**2000**
**3000**
**4000**
**5000**
p-polarized THz
s-polarized THz
**THz**
** f**
**ield amplit**
**ude (a.u.)**

**Pump polarization angle (deg)**

Figure 3-4 Detected *p- (squares) and s- (circles) polarized THz field as a function of *

the linear pump polarization angle. 0o_{ corresponds to a }_{p-polarized pump }

beam while 90o_{ corresponds to an }_{s-polarized beam. }

Here we calculated the dependence of pump polarization based on the
second-order optical response of bulk and SEF contribution. Since the incident angle
and azimuthal angle were fixed in pump polarization rotating experiment, the
coefficients of *f , _{s}*

*f ,*

_{c}*F ,*

_{s}*F ,*

_{c}*t ,*

_{s}*t*,

_{p}*A ,*

_{s}*L , and*

_{eff}

*A*were chosen to be

_{p}constant during whole process. Substituting the dependence of pump

*p*

**E** and **E**pump* _{s}* on

the pump polarization rotation angle into Table I and Table III, we obtain the

For *a-plane wurtzite crystal, *
*p- THz: * THz _{sin(2 )} _{cos(2 )}
*p* *p* *p* *p*
*E* ∝*a* +*b* θ +*c* θ
*s- THz: * THz _{sin(2 )} _{cos(2 )}
*s* *s* *s* *s*
*E* ∝*a* +*b* θ +*c* θ

For *c-plane wurtzite crystal, *
*p- THz: * THz _{cos(2 )}
*p* *p* *p*
*E* ∝*a* +*c* θ
*s- THz: * THz _{sin(2 )}
*s* *s*
*E* ∝*b* θ

For *a-plane orientation, both p- and s- THz field contain the polarization dependence *

of sin(2*θ) and cos(2θ) and offset signal a. Therefore, the discrepancy of polarization *

dependence for *s- and p- THz is mainly determined by the coefficients of bulk- and *

SEF-induced contribution. The polarization dependence for *c-plane wurtzite crystal is *

similar to (100) face InAs. One can expected that polarization dependent characters
*for c-plane InN will be identical to that for (100) InAs such as large offset for p- THz *
and phase shift between s- and p-polarization.

*Incident angle dependence for wurtzite crystal *

The incident angle dependence of transient current induced THz field has been reported [3-25] and the result is shown in Fig. 3-5. At normal incident, the surge current induced THz field is zero since there is no EM field radiated along direction of current. THz amplitude increases as the incident angle increases and has a maximum at near Brewster angle. At the incident angle larger than Brewster angle, THz amplitude is reduced due to Fresnel loss. However, the incident angle can influence Fresnel coefficient in the calculated second-order optical response. Solid line in Fig. 3-5 is the calculated result according to the concept described above [in

Table I] and the solid dots are the experimental data in Ref. 3-24, showing the good agreement. Meanwhile, the authors of Ref. 3-24 tried to fit their experimental data

based on the 2

eff

*d coefficient and as it can be seen in *Fig. 3-5(b), it only can fit the
data around the small angle and is diverged as incident angle is greater than 50 degree.
Therefore by taking into account the Fresnel coefficient, we could fit the dependence
of THz emission on the full range of incident angle. It is well known that reflection or
*transmission for s- and p-polarized light is quite different each other at large incident *
angle due to Fresnel coefficient. It is the reason why we still can well describe the
THz data even when the incident angle is greater than 50 degree.

**-80** **-60** **-40** **-20** **0** **20** **40** **60** **80**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**

**Normaliz**

**ed bulk contribution **

Data
**Normaliz**

**ed THz output power**

**Incidnet angle (deg.)**

**(a)**

Figure 3-5. Normalized *p-polarized THz output power data (dots) *[3-24] is fitted by
(a) normalized *p-polarized bulk contribution (blue solid curves) (b) *

square of the effective nonlinear coefficient (black solid curves) plotted as a function of incident angle of the pump beam.

**3-2. Experimental Setups **

In this chapter, we will describe the laser system, EO sampling based THz emission system, reflection second harmonic generation system, and optical pump probe system used in this work.

**3-2.1. THz Emission measurement system **

The setup of the Electro-Optic THz system is shown in Fig. 3-8. An amplified

Ti:sapphire laser providing 50fs, 800nm, 2mJ pulsed at repetition rate of 1kHz is used
*to drive this system. The linearly s-polarized incident beam is divided into three *
separated beams by two beam splitters. The transmitted beam from the first beam
splitter is used for optical pump-terahertz probe study and will be discussed later. The
reflected beam from the first beam splitter is divided into the pump beam and the
probe beam by the second beam splitter. Polarization of the pump beam is rotated to
*linearly p-polarized by a half-wave plate and is the used to generate linearly *

*p-polarized THz pulsed in a semiconductor surface emitter such as InAs at the *

incident angle of 70 degrees to the surface normal which is close to the Brewster angle. Any reflected laser beam from the emitter is blocked by a teflon sheet which has a high transmission in the terahertz region. The generated THz radiation is collimated and focused onto the sample by a pair of gold-coated off-axis parabolic mirrors with focal lengths of 3 and 6 inches respectively. The transmitted THz radiation is again collimated and focused onto a 2-mm-thick (110) ZnTe crystal for free space electro-optic sampling by another pair of parabolic mirrors with the same focal lengths with previous pair. A pellicle beam splitter which is transparent to the THz beam and has a reflectivity of 5% for 800nm light is used to make the probe beam collinear with the THz beam in the ZnTe crystal. The linear polarization of the probe beam is perpendicular to the polarization of the THz beam and we adjust the

azimuth angle of the ZnTe crystal to achieve the highest modulation efficiency. The
linear polarization of the probe beam without being modulated by the THz radiation is
converted to circular polarization by a quarter-wave plate; Polarization of the probe
beam modulated by the THz radiation is converted to ellipsoid polarization by a
quarter-wave plate. A Wollaston beam splitter is used to divide the modulated
*ellipsoid polarized probe beam into a linear s-polarized beam and a p-polarized beam. *
A balanced detector with two silicon photodiodes is used to detect the differential
signal between two individual probe beams and the signal is proportional to the THz
electric field. A motor stage within the probe beam path is used to scan the delay time
between the probe pulse and the THz pulse imposing on the ZnTe crystal to obtain the
entire THz time-domain waveform. In order to increase the signal to noise ratio, an
optical chopper and a lock-in amplifier are used. The entire THz beam path is also
located in a closed acrylic box for nitrogen purge. An example of a terahertz pulse
with its corresponding spectrum under humidity of 5% generated by this setup is
shown in Fig. 3-9.

**0** **2** **4** **6** **8** **10** **12**
**-0.6**
**-0.4**
**-0.2**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
**1.2**
**TH**
**z fie**
**ld(a.u**
**.)**
**Delay time(ps)**
**(a)**
**0.0** **0.5** **1.0** **1.5** **2.0** **2.5**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
**1.2**
**A**
**m**
**plitud**
**e(a**
**.u.)**
**Frequency(THz)**
**(b)**

Figure 3-9 THz time-domain (a) waveform and (b) its corresponding

spectrum generated by the electro-optic THz emission system using InAs as emitter under humidity of 5%

**Emitter **
**ZnTe**
**λ/4 Plate**
**Wollaston **
** Prism ** **Pellicle **
**Balance **
**D t t**
**Motor **
**St**
**λ/2 Plate **
**Chopper **

3-2.3. Reflection Second Harmonic Generation (RSHG) Setups

Fig. 3-10 displays an experimental setup for second harmonic generation from semiconductor surface. Mode-locked Ti:saphire regenerative amplifier is used as pump source filtered by a IR filter and linearly polarized by half wave plate and polarizer. Linearly polarized pump beam is spited to signal arm and reference arm which is used to reduce laser frustration influence. In the reference arm, we choose BBO as NLO crystal to generate 400 nm light as the reference. To efficiently generate SHG, we adjust incident angle to reach phase matching and modify azimuthal angle of BBO to obtain maximum SHG. There is a ND filter in front of BBO and sample to avoid high density damage. Before collecting to Hamamatsu R3896 PMTs, there are bandpass filers and a dichroic mirror to only select 400 nm light. PMTs are connected to multimeters which are monitored by Labview program. The incident angle of focused pump beam is 45 degree and the sample is rotated about its surface normal to examine azimuthal dependent reflection SHG signal. The azimuthal angle dependent results of SHG list in section 3-3.1.

P M T (r ef e ren ce) sam ple NL O cr y s ta l 2w filter dichroic mirror w 2w polarizer lens B.S. Ti:saphire Amplifier ND filter PMT (S i gn a l) IR filter half-wave plate

**3-3. Results and Discussions **

First, we discuss the azimuthal angle dependence of RSHG from *a- and c-plane *

InN surface. Then, the azimuthal angle- and pump polarization- dependent THz field are investigated by EO sampling based THz emission system. Finally we compare THz radiation with SH generation results in summary.

3-3.1. Reflection SH Generation from *a-plane InN *

In the RSHG measurement configuration, since two identical fundamental photons are incident in semiconductor, frequency conversion with the SHG would take place if the momentum and energy are conserved. Since the susceptibility tensor is related to the symmetry of crystal, the effective second-order susceptibility

(2)eff_{( )}

*ijk*

χ φ in lab coordinate is varied as we rotate azimuthal angle about surface

normal. Since susceptibility tensor in lab coordinate also dependent on coordinated transformation, one can expect the function of azimuthal angle dependence is dissimilar for different crystal orientation. Therefore, we are able to distinguish crystal symmetry and orientation via detecting azimuthal angle dependence.

For the measurement, the pump fluence is adjusted to approximately 0.32

mJ/cm2. The SH intensities are measured as the samples are rotated about their

surface normal. The results are shown in Fig. 3-11 for *a-plane InN and in* Fig. 3-12

for *c-plane InN. *

Solid lines are obtained by the bulk electric dipole contribution shown in Table

II and they show the excellent agreement with the experimental data. Generally, for

crystals without inverse symmetry, it is well known that bulk electro-dipole contribution to SHG dominates any contribution from surface term.

beam projection in crystal coordinate are different due to the different orientation. One can predict that samples with different crystal orientation have different azimuthal angle dependence of bulk contribution. The SHG rotational anisotropy result of SHG

*pp*

*I* is named as pp-SHG for the convenience of the following discussion.

As expected, the RSHG data for *a-plane InN show that pp-SHG and ps-SHG are 90 *

degree phase shifted from ss- SHG and sp-SHG, respectively. The RSHG for *a-plane *

InN shows one-fold and three-fold symmetry, while that for *c-plane InN only has *

azimuthal angle independent response. Note that both ss-SHG and ps-SHG intensity is zero and the ss-SHG is most sensitive to the symmetry of the surface structure since its polarization combination contains only anisotropic nonlinear susceptibility tensor

elements [3-17]. Therefore, in the literature [3-18], to discover the surface

configuration of the ZnO thin films, the ss-SHG measurement has been used since the ss-SHG intensity is zero for the bulk of c-plane orientation, as described in Table II.

Figure 3-11. Normalized reflection SHG for different polarization combination from

*a-plane InN surface. *

**0.0**
**0.5**
**1.0**
**pp-SHG**
**0** **60** **120** **180** **240** **300** **36**
**0.0**
**0.5**
**1.0** _{ps-SHG}**0.0**
**0.5**
**1.0**
**ss-SHG**
**0** **60** **120** **180** **240** **300** **360**
**0.0**
**0.5**
**1.0**
**sp-SHG**

Figure 3-12. Normalized reflection SHG for different polarization combination from

*c-plane InN surface. *

**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
** pp-SHG**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
**ps-SHG**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
**ss-SHG**
**0** **60** **120** **180** **240** **300** **360**
**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**
**sp-SHG**

Therefore, the azimuthal angle dependencies can be fitted by the relation shown
in Table II. For the *a-plane InN these are * SHG

*pp*

*I* = [b1sinφ +c1sin3φ ]2, *I*SHG*ps* =

[b2cosφ +c2cos3φ ]2, *Isp*SHG= [b3sinφ +c3sin

3_{φ ]}2_{, and } SHG

*ss*

*I* = [b4cosφ +c4cos3φ ]2. The

coefficients obtained from the best fitting were b1 = 0.11256 ± 0.00878, c1 = -0.19858

± 0.00993, b2 = -0.17288 ± 0.00586, c2 = 0.23627 ± 0.00741, b3 = -2.58619 ± 0.00741,

c3 = 3.33633 ± 0.09824, b4 = 0.10746 ± 0.00622, c4 = -0.1936 ± 0.00691. The

coefficient ratios in Table V, we find good agreement with experiments. In Table V,
*the coefficient a and b are defined as d*15/*d*33 and*d*31/*d*33, respectively.Both *a and b *

used in calculation were -0.5 according to the bond-charge theory [3-7]. The values for the complex refractive index used in the calculations for the optical pump and

*second-harmonic wavelengths were n = 2.424+i0.531 and N=2.259+i 0.561* [3-19],

respectively. The coefficients *f _{s}*,

*f*,

_{c}*F*,

_{s}*F*,

_{c}*t*,

_{s}*t*,

_{p}*A*,

_{s}*L*

_{eff}, and

*A*in Table V are as defined in Ref. 3-10.

_{p}In addition, we measured the reflection second harmonic generation from

*c-plane InN, and fitting the data by using * SHG

*pp*

*I* = a1, *I*SHG*ps* = a2, *Isp*SHG= a3, and

SHG

*ss*

*I* =

a4. The fitting coefficient were a1 = 0.04076 ± 0.00221, a2 = 5.8×10-4 ± 7×10-5, a3 =

0.02589 ± 0.00143, and a4 = 0.00155 ± 2.3×10-4. The coefficient ratios in Table V for