國
立
交
通
大
學
物理研究所
博
士
論
文
I. 低維半導體中的電子自旋馳豫 和
II. 微生物鞭毛間的流體動力交互作用
I. Spin relaxations in low dimensional semiconductors and
II. Hydrodynamic interactions between microorganism flagella
研 究 生：蔡政展
指導教授：張正宏 教授
I. 低維半導體中的電子自旋馳豫 和
II. 微生物鞭毛間的流體動力交互作用
I. Spin relaxations in low dimensional semiconductors and
II. Hydrodynamic interactions between microorganism flagella
研 究 生：蔡政展 Student：Jengjan Tsai
指導教授：張正宏 Advisor：Cheng-Hung Chang
國 立 交 通 大 學
物 理 研 究 所
博 士 論 文
A ThesisSubmitted to Institute of Physics College of Science National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Doctor
in
Physics
July 2013
Hsinchu, Taiwan, Republic of China
I. 低維半導體中的電子自旋馳豫 和
II. 微生物鞭毛間的流體動力交互作用
學生：蔡政展
指導教授
：張正宏
國立交通大學物理研究所博士班
摘
要
在這篇論文中，我們主要探討兩個主題：I﹒低維半導體中的電子自旋馳
豫 II﹒微生物鞭毛間的流體動力交互作用。這兩個主題分別隸屬於凝態和
軟物質兩個不同的物理領域。
在主題 I 中，我們探究低維半導體 (二維電子氣) 中，基於 Elliott-Yafet
和 D’yakonov-Perel’自旋馳豫機制所導致的電子自旋馳豫現象。我們運用
Ensemble Monte Carlo (EMC) 和 Semiclassical path integral (SPI) 方法來研
究這些問題。藉由運用 EMC 和 SPI 這兩個方法來加以計算，我們發現其結
果與一些理論推演和實驗數據互相一致。並且我們也預測出一些有趣的結
果，這些發現可作為將來實驗設計的指導方針。
在主題 II 中，我們試著揭露出細菌地毯鞭毛運動的集體行為。我們提出
兩 個 簡 單 模 型 Microorganism-flagellum-rotor matrix (MFR
matrix) 和
Microorganism-flagellum-rotor sweep (MFR
sweep) 來摹擬真實世界中複雜萬
分的細菌鞭毛轉動和甩動的行為。我們使用 Blake-Oseen tensor (BOT) 來描
述細菌鞭毛彼此之間和鞭毛與示蹤粒子之間的流體動力交互作用。藉由運
用 MFR
matrix和 MFR
sweep這兩個模型以及 BOT 來研究微生物鞭毛陣列的行
為，我們提出一些合理的見解來解釋最近的一些實驗結果。此外我們也預
測出一些有趣的現象，這些發現可作為將來實驗設計的指導方針。
I. Spin relaxations in low dimensional semiconductors and
II. Hydrodynamic interactions between microorganism flagella
student：Jengjan Tsai
Advisors：Dr. Cheng-Hung Chang
Institute of Physics
National Chiao Tung University
ABSTRACT
In this thesis we explore two main topics: I. Spin relaxations in low dimensional
semiconductors and II. Hydrodynamic interactions between microorganism
flagella. These two topics belong to two different physical fields, condensed
matter and soft matter fields.
In Part I we focus on exploring the Elliott-Yafet and D'yakonov-Perel' spin
relaxation mechanisms inducing electron spin relaxation in low dimensional
semiconductors (two-dimensional electron gas). The main exploration
approaches are Ensemble Monte Carlo (EMC) and Semiclassical path integral
(SPI) methods. By utilizing these two methods, some consistent results between
our study and some theoretical and experimental results are obtained. In addition
our study also predicts some interesting findings which may offer as design
guidance for future experiments setting up.
In Part II we try to reveal the bacterial carpet collective behavior. We propose
two minimal models Microorganism-flagellum-rotor matrix (MFR
matrix) and
Microorganism-flagellum-rotor sweep (MFR
sweep) to mimic the real and
complex bacterium flagella rotation and sweep behavior, respectively. And in
order to properly describe the hydrodynamic interaction between the bacterium
flagella themselves and between the bacterium flagellum and tracer particle, the
hydrodynamic interaction Blake-Oseen tensor (BOT) is employed. By utilizing
these two models and BOT in studying microorganism matrix, we give some
reasonable explanation to account for recent experimental results. Besides, our
study also predicts some interesting findings which may offer as design
guidance for future experiments setting up.
誌
謝
Academic adventure is an interesting and a so long road… I think that cost some
time of my life in this adventure is a worth thing… Thanks to the so many
teachers, classmates, friends, and my family, who care me, support me and take
a walk with me during such a little bit long PhD research career… Thank you so
much!
At present, I finish this PhD program, and shall enter into a new field in my
life… Wish myself can go ahead, go better, and go success in near future~~
Well…, all my friends, thank you so much~~ Good bye~~
Contents
Chinese abstract ...………...…..…… i
English abstract ...………...…..…..…… ii
Acknowledgement ……….………....…… iii
I Spin Relaxations In Low Dimensional Semiconductors 1
1 Spin Relaxation Mechanisms 3
1.1 Introduction ………....…... 3
1.2 Elliott-Yafet mechanism ……….………..….... 5
1.3 D'yakonov-Perel' mechanism ………..……….….… 7
1.4 Supplement: general de¯nition of spin relaxation time …………..….….….. 9
Bibliography 11
2 Methods for Carrier Transport, Carrier Scattering, and Spin Evolution 1 3 2.1 Carrier transport ……….………. 13
2.2 Carrier scattering ………..……….….. 16
2.3 Ensemble Monte Carlo method ……….…….…. 19
2.5 Spin evolution under EY mechanism ……….………. 22
2.6 Spin evolution under DP mechanism ………. 24
Bibliography 26
3 EY Spin Relaxation in Quantum Wells and Narrow Wires 27
3.1 Introduction ……….……..….……….…. 27
3.2 The EMC and SPI methods on experimental samples ………….….………. 28
3.3 The size e®ect on the EY relaxation ……….…….…..…. 32
3.4 The impurity e®ect on the EY relaxation ……….………..……. 35
3.5 Conclusion ………....…… 40
3.6 Supplement: spin relaxation process ………...……… 41
Bibliography 43
4 DP Spin Relaxation in Narrow Wires 45
4.1 Introduction ……….……….… 45
4.2 Relaxation of uniform spin modes ………...……….…..…… 47
4.3 Bessel relaxations in ballistic channels ……….…….. 52
4.4 Relaxation of helix spin modes ………...……….……… 55
4.5 Conclusion ………. 59
Bibliography 61
II Hydrodynamic Interactions Between Microorganism
Flagella 63
5 Flagellum Model under Hydrodynamic Interactions 65
5.2 Microorganism-Flagellum-Rotor Model and Blake-Oseen Tensor Role
………….……….…..………….………. 66
Bibliography 72
6 Tracer Particle Responses on Microorganism Flagellum Matrixes 73
6.1 Introduction ……….………….…… 73
6.2 The collective motions of MFRs ……….……..…….. 74
6.2.1 Two MFRs ………..…………..………. 74
6.2.2 Synchronization state in a uniform MFR matrix ………...…….……. 75
6.2.3 Repellency and freezing states in a checkerboard-like MFR Matrix …….. 78
6.3 Tracer particle responses on an MFR matrix ……….……. 82
6.3.1 Circular motion mode ………...………..….. 83
6.3.2 Linear oscillation mode ………...…………..…… 86
6.3.3 Sharp jumping mode ………...…….……. 90
6.4 Conclusion and outlook ………..……….…… 95
Bibliography 96
7 Hydrodynamic Spreading of Forces from Bacterial Carpet 97
7.1 Introduction ………..……..…….. 97
7.2 Experimental approaches and results ………...………..…… 98
7.3 MFR matrix model explanation ………..………..….….. 106
7.4 Supplement ………....……. 110
Bibliography 113
List of Figures
1.1 A tree plot shows the main concepts and key words which is included in Chapters 1 and 2. ..……….………..…….….…… 5
3.1 The layout of a quasi-2D sample. ……….……….………....…… 29 3.2 Plot of relaxation time versus mobility in EY mechanism. ...…….… 31 3.3 The EY spin relaxation time versus the wire width at ¯ve di®erent temperatures. ………...………..…………...… 33 3.4 The EY spin relaxation time versus four wire widths at 50K. …………....… 34 3.5 The DP spin relaxation time versus eight wire widths. ……….………… 34 3.6 Relaxation time analytical deduction in a ballistic narrow wire in EY mechanism.
………...….……… 36
3.7 The polarization versus time in three wires of di®erent widths. ………….… 38
4.1 Relaxation time comparison between numerical and experimental results. .… 47 4.2 Plot of spin polarization versus time in DP mechanism. ....………..……..….. 49 4.3 Spin polarization in a 1D channel. ………. 50 4.4 Relaxation time analytical deduction in a ballistic narrow wire in DP mechanism.
………...…….……… 53
4.5 Long-lived spin eigenmodes. ………..……….. 56 4.6 Relaxation rate versus channel width for long-lived spin eigenmodes. …….... 58
5.1 The layout of an MFR matrix system. ………...………..… 68
6.1 The phase diagram of 2 MFRs. ………...……… 75 6.2 Plot of rotation rate versus the thrust azimuthal angle for the synchronization state. ………..……… 77 6.3 Plot of rotation rates versus ¢ for the repellency and freezing states by utilizing completed BOT. ………..………. 80 6.4 Plot of rotation rates versus ¢ for the repellency and freezing states by utilizing approximated BOT. …………..………..…. 81 6.5 Plot of tracer particle rotation radius versus thrust azimuthal angle for
the circular motion mode. ………..………….……….… 85 6.6 Plot of tracer particle maximal displacement versus thrust azimuthal angle
for the linear oscillation mode. ………..…………..……… 89 6.7 Plot of the tracer particle response trajectories. ……….…… 92
7.1 Some swimming behaviors comparison between bacteria VIO5 and MB136. . 101 7.2 The schematic plot of the experiment setup and the forces measured at various heights above bacterial carpets. ………..……….. 103 7.3 The schematic plot of the distortion of the °ow ¯elds. ……….………... 104 7.4 The extracted values are plotted versus rotational rate measured under the dense bacterial carpet conditions. .………..……… 106 7.5 Plot of drag velocity (or drag force) versus tracer particle located height H.
………..………..…………..……… 108 7.6 Plot of rotation rate enhancement index versus number of MFRs. ..…...… 109 7.7 At Na+concentration 50 mM, °agellum spends
» 80 percent time in CW motions. ………..…. 111 7.8 The MSD converges to normal Brownian motions as measured height
7.9 Calibration of laser power and strength of random noises for precise determination of e®ective trapping force. ……….……… 112 7.10 There is no detectable force can be measured in the lateral directions. ..… 112
Part I
Spin Relaxations In Low
Dimensional Semiconductors
Chapter 1
Spin Relaxation Mechanisms
1.1
Introduction
The achievement of modern science and technology is largely based on the blooming development of electronics, which mainly make use of the charge prop-erty of carriers in solid state physics. However, charge is not the only intrinsic carrier property one can take advantage of from carriers. In addition to it, spin is a not yet widely utilized degree of freedom which may enrich device functions. Sophisticated manipulation on the carrier charge has revolutionized our daily life over a century. The possibility of utilizing carrier spin creates another hope for a new generation of spin-based devices. This expectation has attracted numerous studies in the past decades and boosted our understanding on spin systems in solid state physics. Researches along this direction have formed the field of spin electronics, or simply spintronics.
The expectation for a spin technology era is not a pure mirage. In fact, some spin-based devices have been realized and even in use commercially. A prominent example is the read head and the memory-storage cell of the giant-magnetoresistive (GMR) multilayer structure, composed of alternating ferromagnetic and nonmag-netic metal layers, developed around 1988 [1,2]. Since the magnetoresistance of the device depends on the relative orientation of the magnetizations in the magnetic layers, it can be used to sense changes in magnetic fields. After GMR, the stud-ies on spintronics devices have split into two branches. While one of them tends to perfect the existing GMR-based technology, the other seeks more radically for new mechanisms for tailoring desired spin transport properties from, for instance, semiconductors. The idea of spin field-effect-transistor proposed by Datta and Das for semiconductors served as a milestone example, which stimulates extensive theoretical and numerical studies in spintronics [3]. Despite of those promising
progresses, the route towards a spin era of technology is not obstacle free. From the aspect of basic physics, spin flipping energy can be readily affected by ther-mal fluctuations. How to retain a spin information unaffected against therther-mal fluctuations and extract a spin information from the noisy environment are tricky problems. Without a solution to these problems, spintronics is only a laboratory toy at low temperature environments and cannot be coupled into daily used elec-tric circuits functioning at the room temperature. Furthermore, from the aspect of material science, band unmatching on the metal-semiconductor interface and the fast spin relaxation rate in semiconductors are other notorious problems for device designs. As a whole, problems in constructing spin-based semiconductor devices can be summarized into three key issues: spin injection, detection, and manipulation [4]. During spin manipulation, the life time of a polarized spin state is of special concern, which leads to numerous investigations on various spin relax-ation mechanisms. The importance of life time of spin state can be reflected from the recent interest on the persistent spin mode experimentally created and mea-sured by IBM scientists in 2012 [5]. These spin helices of synchronized electrons persisting for more than a nanosecond is longer than the duration of a modern processor clock cycle, which is regarded as a hope for spin information processing. In fact, the existence of a similar kind of long-lived spin mode along a quasi one-dimensional system has been analytically proved and numerically confirmed in our previous study [6]. Doubtlessly, spin relaxation is a central problem in spintronics. It is the main ingredient in Part I of this thesis.
The samples studied in this thesis are focused on low-dimensional systems, such as quasi-two dimensional (quasi-2D) quantum wells (QWs) and quantum wires (QWires), as well as two dimensional electron gas (2DEG), built of III-V semiconductor heterostructures. Spin evolutions in these systems are strongly affected by the Elliott-Yafet and the D’yakonov-Perel’ spin relaxation mechanisms. To analyze these mechanisms, we employee the Ensemble Monte Carlo Method for carrier dynamics and develop the Semiclassical Path Integral Method for spin dynamics. Our theoretical results agree very well with the existing experimental data, unravel some experimental puzzling questions, and predict several sample behaviors beyond experimentally accessible parameters regimes.
The spin relaxation, or spin de-coherence, of electrons and holes are frequently observed phenomenon in solid materials. There exist many relaxation mecha-nisms responsible for this phenomenon. The most prominent ones are the Elliott-Yafet(EY) mechanism [7, 8, 9, 10], the D’yakonov-Perel’(DP) mechanism [11, 12, 13,14,15], the Bir-Aronov-Pikus (BAP) mechanism, and the hyperfine-interaction
Carrier Charge + Spin Metal Semiconductor Others Spin injection Spin manipulation Spin detection Others System Approach
Quasi-2D QWs and Quasi-2D QWires, and 2DEG
Analytical approach
+
Numerical approach- Ensemble Monte Carlo Method &
Semiclassical Path Integral Method
Figure 1.1: A tree plot shows the main concepts and key words which is included in Chapters 1 and 2. They highlight the necessary theoretical foundation and our study approaches in the spin relaxation issue which explored in Part I.
mechanisms. In this chapter we briefly sketch the EY and DP mechanisms.
1.2
Elliott-Yafet mechanism
An electron in a crystal is usually modeled as a particle moving in a perfect lattice described by a periodic lattice potential. The Schr¨odinger equation of the
electron is ·
p2
2m + V (r)
¸
Ψ = EΨ, (1.1)
where m, p and r are the free electron mass, momentum operator and position vector. Here, V (r) is the lattice potential, which follows both point and trans-lational symmetries of the lattice. The wave function Ψ is the well-known Block function
ukeik·r, (1.2)
where uk obeys the above both symmetries of the lattice. In order to introduce
the spin-orbit interaction into (2.1), we start with the four-component Dirac equa-tion and reduce it to two components in the usual way [16]. After taking some approximations, it yields the Schr¨odinger equation
· p2 2m + V (r) + ~ 4m2c2(∇V (r) × p) · σ ¸ Ψ = EΨ, (1.3)
where c and σ are the speed of light and Pauli matrices and the third term is the spin-orbit coupling. The corresponding Hamiltonian operator and the wave func-tions of the electron still have the point and translational symmetries of the lattice. Since the spin-orbit coupling is present in this Hamiltonian, the eigenfunctions Ψ will be linear combinations of different spin functions,
[ak|Sz+i + bk|Sz−i] eik·r, (1.4)
where akand bk are two functions with the same symmetry as V (r) and uk and the
spin states |Sz±i, which have the angular momentum ±12~ along the z direction.
The system we are interested in contains two kinds of symmetries: the inversion symmetry of space and the time-reversal symmetry. Combining the former, which changes wave vector from k into −k, with the latter, which flips the spin states, it yields another set of eigenfunctions with the same k and energy,
[a−k∗|Sz−i − b−k∗|Sz+i] eik·r. (1.5)
The expressions (2.4) and (2.5) indicate that if the spin-orbit interaction (the third term in (2.3)) is present, the eigenfunctions of the Schr¨odinger equation (2.3) are generally no longer a pure spin state. It makes sense to call the eigenfunctions (2.4) and (2.5) the spin-up (+) and spin-down (−) states, respectively, since typically
|ak| and |a−k| ≈ 1 and |bk| and |b−k| ¿ 1.
In the EY mechanism, the spin can be flipped only when its carrier collides with impurities. To discuss this process, let us consider an extra interaction Hint
for scattering, which can be caused by the impurities, heavy holes, phonons, piezo-acoustic modes, or boundaries, etc. [7, 8, 9]. If Hint scatters the electron from k to
k0 without changing its spin, its matrix element is Z
ak0∗Hintakei(k−k 0
)·rdr. (1.6)
If Hint can cause spin flip, the matrix element must be replaced by
Z
(a−k0Hintbk− b
−k0Hintak)ei(k−k
0
)·rdr. (1.7)
The transition rate is proportional to the square of the matrix elements of an interaction Hint between the electron and the impurity or the lattice as represented
in Appendix and the reference paper therein [17]. For a collision process without spin change, the transition rate is related to the electron momentum relaxation
time τp by 1 τp ∝ ¯ ¯ ¯ ¯ Z (ak0∗Hintak)ei(k−k 0 )·rdr ¯ ¯ ¯ ¯ 2 . (1.8)
If this process contains spin flip, the rate is 1 2T1 ∝ ¯ ¯ ¯ ¯ Z ¡ a−k0Hintbk− b −k0Hintak ¢ ei(k−k0)·rdr ¯ ¯ ¯ ¯ 2 , (1.9)
where T1 is the spin relaxation time, which is also referred to the longitudinal time
or spin-lattice time [18].
1.3
D’yakonov-Perel’ mechanism
In the discussion of EY mechanism in Sec. 2.1, the combined effect of inversion symmetry of space and time reversal symmetry yields a twofold degeneracy of single-particle energies
E+(k) = E−(k), (1.10)
where for convenience (+) and (−) denote the two states (2.4) and (2.5). If the spatial inversion symmetry is lifted, the spin-orbit interaction shall lead to a spin splitting of the electron state even at zero magnetic field, B = 0. The spin splitting can be caused by the bulk inversion asymmetry (BIA) of the underlying crystal structure. Examples include the zinc blende structure of III-V (such as GaAs and InSb) and II-VI (such as ZnSe and HgCdTe) compounds without center of inversion. These materials are different from Si and Ge, which have a diamond structure. Furthermore, the spin splitting can be caused by the structure inversion asymmetry (SIA) of the confined potential V (r). This potential may contain a built-in or an external potential, as well as the effective potential from the position-dependent band edges. To the lowest order of the wave vector k, the BIA induced spin splitting is caused by the so-called Dresselhause term, whereas the SIA induced one is generated by the so-called Rashba term. The spin splitting of higher orders of k can be described by, for instance, the 8 × 8 or the 14 × 14 extended Kane model.
For BIA, examples can be found in the conduction bands Γ-point of [001] grown GaAs/AlAs and alike (type-I) quasi-2D quantum wells or two dimensional electron gas (2DEG) systems. In these systems, the Hamiltonian matrix Hkk of
the Dresselhause term is Hkk= Ec+ ˜e + γ 2˜kz(kx2− ky2) γ2 h ˜ k2 z(kx+ iky) − ikxky(kx− iky) i γ 2 h ˜k2 z(kx− iky) + ikxky(kx+ iky) i Ec+ ˜e −γ2˜kz(k2x− k2y) , (1.11) where Ec is the energy at the bottom of the conduction band, ˜e = ~2/2m∗(kx2+
k2
y + ˜kz2) is the energy operator, kx and ky are the electron momenta in x and
y directions, m∗ is the Γ-point conduction-band effective electron mass, γ is the
spin-splitting parameter, and ˜kz is the operator id/dz [19, 20]. This Hamiltonian
can be reduced as HBIA = β h σxkx ¡ k2y − hk2zi¢+ σyky ³ hkz2i − kx2 ´i , (1.12)
with a material-specific coefficient β, where σx and σy are two components of the
Pauli matrices.
For SIA, examples can also be found in the conduction bands Γ-point of [001] grown GaAs/AlAs and alike (type-I) quasi-2D quantum wells or two dimensional electron gas (2DEG) systems. In these systems, the Hamiltonian matrix Hkk of
the Rashba term is
Hkk = α (σ × k) · ν, (1.13)
where α is a pre-factor which depends on the constituting materials and on the geometry of the quasi-2D or 2DEG systems, σ are the Pauli matrices, ~k is the electron momentum, and ν is a unit vector perpendicular to the 2D plane. If we assume that ν is in the z direction, then this Hamiltonian becomes
HSIA = α (σxky− σykx) . (1.14)
A comparison shows that the energy degeneracy of spin orbit interaction or the quasi-spin up and down states can be lifted in different ways in BIA and SIA. The former can be achieved by removing spatial inversion symmetry or time reversal symmetry, while the latter can be accomplished by applying an external magnetic field. The expressions (1.12) and (1.14) have a general form
HSOI =
1
2~σ · Ω(k), (1.15)
where SOI denotes the spin orbit interaction, σ are the Pauli matrices, B(k) is a k-dependent effective magnetic field around which electron spins precess
with the Larmar frequency Ω(k) = (e/m∗)B(k), with the effective electron mass
m∗ [12,13,15]. This expression gives a clear picture why the effective magnetic field
B(k) causes the electron spin relaxation. That is, a collision event will change the electron momentum ~k and subsequently the electron spin precession axis. There-fore, the randomized precession axis will help smearing the electron spin coherence. If the momentum relaxation time is longer than the spin precession period, e.g., under dilute impurities, the electron spins shall precess freely and lose their coher-ence between two collision events. In contrast, if the momentum relaxation time is shorter than the spin precession period, e.g., under dense impurities, the electron spins will not precess much before the carrier changes its momentum or the carrier spin changes its precession axis. It shall lead to the dynamical narrowing which helps preserving spin coherence.
Owing to different underlying mechanisms, the EY and DP spin relaxation times have opposite impurity density dependence. Under the EY mechanism, spin flip is a temporally discrete event and can only happen at a electron-impurity collision. Thus, more frequent collision events will cause faster spin relaxation. Under the DP mechanism, each single spin precesses between two collisions. Less collision events will lead to longer precession and faster spin decoherence. Accord-ingly, there exists a trend between the momentum relaxation time τp and the spin
relaxation times τEY
s and τsDP of EY and DP mechanisms, respectively
τsEY ∝ τp and τsDP ∝
1
τp
. (1.16)
1.4
Supplement: general definition of spin relaxation time
To study spin relaxations, let us consider a system consisting of N+ electrons
with spin state |Sz+i and N− electrons with spin state |Sz−i. The total electron
number is N = N++N−and the net magnetization at any instance can be defined
as
D = N+− N−. (1.17)
If at equilibrium the net magnetization is Deq, one expects that the evolution of
D will follow the equation
dD dt =
Deq− D
with the relaxation time T . Let N∓→± be the number of spins which flip per
second from |Sz∓i to |Sz±i and W∓→± be the transition rate of electrons from
|Sz∓i to |Sz±i. The evolution of D is then given by
dD dt = Deq− D T = 2 (N−→+ − N+→−) 1 = 2 (W−→+− W+→−) . (1.19)
These transition rates W∓→± are proportional to the square of the matrix elements
of an interaction Hint between the electron and the impurity or the lattice that
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Chapter 2
Methods for Carrier Transport, Carrier
Scattering, and Spin Evolution
Using full quantum mechanical approach to study the carrier transport, carrier scattering, and spin evolution of many-body systems in solids is a formidable task. To overcome this complexity, we introduce the Ensemble Monte Carlo (EMC) method and Semiclassical Path Integral to tackle these dynamical problems.
2.1
Carrier transport
When we consider the carrier transport, e.g., electron transport, in a semi-conductor crystal, it is essentially an extremely complicated many-body problem. However, we can focus on the motion of an electron and approximate the effective influence of atomic nuclei and other electrons on the studied electron by a poten-tial V (r). Then the original many-body problem can be reduced to the problem of a single electron [1,2]. Under this reduction, V (r) is still periodic with the same periodicity as that of the crystal lattice. The electronic state under such V (r) can be obtained by solving the Schr¨odinger equation
· p2
2m + V (r)
¸
Ψ(r) = EΨ(r), (2.1)
where m is the free electron mass, Ψ(r) is the eigenfunction to be determined, and
E is the energy eigenvalue. The Bloch theorem tells us that the solutions for a
perfectly periodic potential have the form
where uk,n(r) is periodic with the same periodicity of V (r), k is the wave vector
of electron and n is the index of bands. Besides, the energy eigenvalue Ek,n is
periodic with the periodicity of the reciprocal lattice.
The relation between Ek,n and k, that is the energy band structure, can be
expressed in one period of the reciprocal lattice because of the periodicity of Ek,n.
Conventionally, the first Brillouin zone, which is a period centered about at the origin of the k-space, is used to show the energy band structure. This structure is usually depicted along some significant crystallographic orientations, such as Λ, ∆, and Σ directions. The energy band structure reveals an energy region where electronic states can not be found. This forbidden energy region is termed the energy gap and electronic states are permitted above and below this gap. While the bands above the gap are the conduction bands, those below it are the valence bands. The energy separation between the minimum of the lowest conduction band
and the maximum of the highest valence band is the band gap energy Eg. The
band model offers the information about the energy levels of the band extremes and the relations between the electron energy Ek and the electron wave vector k,
described by various band parameters.
The structures near the conduction band minima and the valence band maxima are important, because carriers located near the band edges are responsible for the transport property. The conduction band near the minimum is frequently approximated by a quadratic function of k. If the band minimum is located at
|k| = 0, Ek can be expressed as Ek= ~2k2 2m∗, (2.3) where k2 = (k2 x+ k2y+ kz2) and 1 m∗ ≡ 1 ~2 ∂2E k ∂k2 (2.4)
is the inverse of the effective mass. The Ek relation given by (2.3) shows that
the electrons in a crystal behave just like electrons moving in a free space, except for a change in the mass. Here ~k plays the role of momentum, which is termed the crystal momentum. Ek represents the electron kinetic energy measured from
the conduction band minimum. Such simple model is rather widely used for simplifying the calculation of carrier transport.
Since electrons in crystal behave just like electrons in free space, except for the change in the mass. This picture suggests that the motion of electrons in
a crystal may be described by the classical equations of motion. The idea is valid when the potential energy felt by the electrons varies slowly compared to the crystal potential so that quantum mechanical effects such as reflection, interference and tunneling can be ignored. Following this concept, the classical motion of an electron can be described by the equation of motion based on its total energy Hamiltonian
H = Ek+ U, (2.5)
where Ek is the kinetic energy and U is the potential energy. For an electron in a
conduction band, one has
H = Ek+ Ec(r), (2.6)
where Ek represent the kinetic energy in terms of the crystal momentum and the
effective mass and Ec(r) is the conduction band minimum. Then the equations of
motion of the system are the Hamiltonian dynamics
dk dt = − 1 ~∇rH (2.7) dr dt = 1 ~∇kH, (2.8)
where ∇r is del operator with respect to position vector r and ∇k is the del
operator with respect to wave vector k. We can easily check that for the quadratic band, the group velocity v = dr/dt simply gives
v = ~k
m∗, (2.9)
which has the similar form of the free electron momentum divided by mass. Due to the advances of modern semiconductor fabrication techniques, we can easily grow compositionally non-uniform heterostructure semiconductors. For in-stance, by placing two compositionally different materials next to each other, a heterojunction is established. A thin two-dimensional conducting layer, termed two dimensional electron gas (2DEG), is formed at the interface of the heterojunc-tion, for example, the interface between GaAs and AlGaAs. Besides, with the use of modern epitaxial growth techniques, the alloy composition can be varied on an atomic scale, so a very sophisticated layer structures consisting of several barriers and wells can be fabricated. For example, a triple alloys compound AlGaAs-GaAs-AlGaAs offers a quantum well in the layer GaAs which the thickness of the well may be about 100˚A or less. Because of the confined electron motion in the well,
the electrons behavior just like a quasi-two dimensional motion. The electrons running at the thin two dimensional conducting layer or the confined well are called the two dimensional electron gas (2DEG). Studying 2DEG is important, since quantum confinements frequently exist in modern heterostructure devices.
Let us assume the aforementioned electron motion is confined in the z di-rection and the electron can move freely in the xy plane. The corresponding three-dimensional Schr¨odinger equation is
− ~2
2m∗∇
2Ψ(r) + E
c(r)Ψ(r) = EΨ(r). (2.10)
The strategy to solve (2.10) is trying to separate the variables. We assume the plane wave solutions in the xy direction since the electrons are free to move on the xy plane. So the total wave function is represented as
Ψ(r) = Cψ(z)eikxxeikyy, (2.11)
where C is the normalization coefficient. Substituting (2.11) into (2.10), we get an equation for ψ(z), − ~2 2m∗ ∂2ψ(z) ∂z2 + Ec(z)ψ(z) = Enψ(z), (2.12) where En= E − Ek (2.13)
is the energy associated with confinement in the z direction and Ek = ~
2
2m∗(kx2+ky2)
is the kinetic energy associated with the motion parallel to xy plane.
2.2
Carrier scattering
Carrier motion in semiconductor crystals is mainly made up of the scattering and drift processes. Here we briefly introduce the theory for scattering process, which plays a role in our study. The scattering theory is based on Fermi’s golden rule, which is derived from the first-order time-dependent perturbation theory. It gives the transition probability per unit time between two eigenstates of the unperturbed Hamiltonian H0caused by the perturbation potential H0(r, t). While
Hamiltonian in solid crystals. At first let us write down the Schr¨odinger equation [H0+ λH0(r, t)] Ψ(r, t) = i~∂Ψ(r, t)
∂t , (2.14)
where λ is a real dimensionless parameter. We assume the equation for the un-perturbed Hamiltonian H0 has been solved as
H0ψk= Ekψk, (2.15)
where Ek is the energy eigenvalue and ψk is the corresponding eigenfunction. The
time evolution of the eigenfunction can be represented as Ψ0
k(r, t) = ψk(r)e
−iEkt
~ . (2.16)
Since the eigenfunctions Ψ0
k(r, t) form a complete and orthonormal set, the solution
of the perturbed problem can be constructed by the linear combinations of Ψ0 k(r, t),
Ψ(r, t) =X
k
ck(t)Ψ0k(r, t), (2.17)
where the coefficient ck(t) describes how the perturbation makes the component at
Ψ0
k(r, t) vary with time. Substituting (2.17) into (2.14) and multiplying both sides
of the arranged equation by ψ∗
k0e −iEk0 t
~ , integrating with respect to r, and using
the orthogonality of ψk, we obtain the following differential equation for ck(t)
i~∂ck0(t) ∂t = λ X k hk0|H0|kic k(t)e i(Ek0 −Ek)t ~ , (2.18)
where hk0|H0|ki is the expectation value defined as
hk0|H0|ki =
Z
Ω
ψk∗0(r)H0ψk(r)dr, (2.19)
with Ω the volume of the crystal.
The expression (2.18) indicates that ck(t) depends on time if λ is not zero.
Since ck(t) is expected to vary slowly with time if the perturbation is weak, it can
be expanded as a power series of λ,
ck(t) = c(0)k + λc (1)
k (t) + λ2c (2)
k (t) + · · · . (2.20)
sides, we have i~∂c (0) k0(t) ∂t = 0 i~∂c(1)k0(t) ∂t = P khk0|H0|kic (0) k e i(Ek0 −Ek)t ~ i~∂c (2) k0(t) ∂t = P khk0|H0|kic (1) k (t)e i(Ek0 −Ek)t ~ ... (2.21)
The first equation of (2.21) shows that the zero-order coefficients c(0)k0 are time
independent. The first-order approximation c(1)k0 (t) of ck0(t) can be evaluated from
the second equation of (2.21). The first-order approximation will be sufficient precise, provided that the interaction is very weak.
For an initial state in a definite unperturbed eigenstate ki, the above results
give ck0(t) = c(0)k i = 1 ck(t) = c(0)k = 0, for k 6= ki (2.22) and subsequently i~∂c (1) k0 (t) ∂t = hk 0|H0|k iick(t)e i(Ek0 −Eki)t ~ . (2.23)
As an application of (2.23), we consider a constant perturbation turned on at t = 0
H0(t) =
(
0, for t<0
H0, for t≥0. (2.24)
Substituting (2.24) into (2.23) and carrying out some integrations, we obtain
c(1)k0 (t) = 1 i~hk 0|H0|k iick(t)e ωt 2 sin( ωt 2 ) ¡ωt 2 ¢ t, (2.25) where ω = (Ek0−Eki)
~ . The probability of finding an electron with the wave vector
k0 at time t is then given by |c(1)
k0 (t)|2. Thus, the transition rate S(ki, k0) from the
state ki to the state k0 is
S(ki, k0) = lim t→∞
|c(1)k0 (t)|2
t . (2.26)
By using the relation limt→∞π1sin
2αx
αx2 = δ(x), the transition rate becomes
S(ki, k0) =
2π
~ |hk
0|H0|k
Integrating S(ki, k0) given by (2.27), with respect to all accessible final states k0,
we obtain the scattering rate,
W (k) = Ω
(2π)3
Z
S(ki, k0)dk. (2.28)
This formula is independent of the dimension of the systems. For quasi 2DEG, we need only to put the wave function given by (2.11) to obtain the corresponding scattering formula.
2.3
Ensemble Monte Carlo method
The Monte Carlo transport calculation is usually referred to the single particle Monte Carlo method or the ensemble Monte Carlo (EMC) method. As discussed above, carrier transport in a semiconductor crystal is a many-body problem with a huge number of mutually interacting carriers. However, if in some parameter regimes the carriers can be approximately treated as an ensemble of independent carriers, the macroscopic behaviors of the system might be approached by the long time behavior of a single particle. It is the principal idea of the single particle Monte Carlo method. This method is a useful for calculating carrier transport, especially in the case of steady-state carrier transport under a static and uniform electric field. However, if the problems of interest are not steady, the long time average has to be replaced by ensemble average, which gives rise to the ensemble Monte Carlo method. This method can be used more widely for many other purposes, such as carrier diffusion, the carrier transport in an inhomogeneous field, the non-stationary behavior of carriers, etc. In the study of spin evolution, we need to use the ensemble Monte Carlo method to monitor the transient process of electron spin. But each member, i.e., single particle, in the ensemble follows the same calculation process as that in the single particle Monte Carlo method.
The ensemble Monte Carlo method is based on the successive and simultaneous calculations of the motions of many carriers during a small time increment ∆t. The method is essentially dynamic and thus is suitable for the analysis of transient carrier motion. A key step to execute the ensemble Monte Carlo calculation is deciding the free flight time, that is the duration between two successive scattering events. This duration depends on the total scattering rate which is the sum of various scattering rates of individual scattering mechanisms. The probability density, P (τ ), of finding an electron traveling for a time τ without being scattered
is expected to follow the relation dP (τ ) dτ = − P (τ ) ³ 1 WT(Ek) ´, (2.29)
where the total scattering rate
WT(Ek) =
N
X
j=1
Wj(Ek) (2.30)
is the sum of the scattering rates of N different scattering mechanisms. Since the scattering rate of each scattering mechanism is a function of electron energy Ek,
the total scattering rate is also a function of Ek.
The solution of (2.29) is P (τ ) = WT(Ek) exp · − Z τ 0 WT(Ek)dt ¸ . (2.31)
To determine the free flight time by P (τ ), we have to evaluate the integral in (2.31). Unfortunately, there is no analytical form for that integral because of the complicated form of general Wj(Ek). A simple strategy to get rid of this problem is
adding a virtual scattering process, called self-scattering, with the scattering rate
W0(Ek) to the original total scattering process, so that the new total scattering
rate Γ becomes a constant [3],
W0(Ek) = Γ − N X j=1 Wj(Ek) or Γ = N X j=0 Wj(Ek). (2.32)
The inclusion of the self-scattering makes no change to the k wave vector of the particle and has an advantage that (2.31) can be recast simply as
P (τ ) = Γ exp−Γτ . (2.33)
The free flight time τ for a carrier scattering process is a random variable following the distribution P (τ ). This distribution gives the mean free flight time τm =
R∞
0 τ P (τ ) dτ = 1/Γ. To see which value Γ should is taken, let us consider a
carrier with Fermi velocity vF, which has the free flight length l = vFτ and the
mean free path lmfp=
R∞
0 lP (τ ) dτ = vF/Γ. That is, once vF and lmfp of a system
are known, Γ can be decided. In numerical simulations, τ can be generated by substituting a uniformly distributed random number x ∈ [0, 1] into the following
the formula
τ = −ln(x)
Γ . (2.34)
Equivalently, one can calculate its mean free path by
l = −lmfpln(x). (2.35)
This classical picture is valid, when the sample size is larger than the de Broglie wave length of the carriers.
2.4
Semiclassical Path Integral formalism
With the knowledge of carrier transport and scattering, now we proceed to the evolution of carrier spin. Since the spin dynamics of the semiconductors discussed below is related to the electron dynamics by the spin-orbit coupling, how the spin evolves is decided by the carrier evolution discussed in the last chapter. While the former is stochastic due to impurity collisions, the latter is deterministic and fully decided by the former. In the following, in combination with the ensemble Monte Carlo method, we apply the Semiclassical Path Integral (SPI) method to the EY and DP spin relaxation mechanisms in a quasi-2DEG system to reveal the intriguing behaviors of carrier spin evolution.
The original semiclassical path integral method was formulated for Rashba systems [4, 5, 6], which has the Hamiltonian
H = H0+ HSOI, (2.36)
where H0 consists of the kinetic and potential energies of an electron in the system
and HSOIrepresents its spin orbit interaction (SOI). Since the energy of spin orbital
coupling in the interested material is usually much smaller than the kinetic and potential energies, the electron trajectory γ can be determined purely by H0. The
spin dynamics of this electron will be described by an evolution operator in the path integral formalism,
Uγ = exp · −i ~ Z γ HSOI(t) dt ¸ . (2.37)
· · · , ξnγ, its trajectory γ will comprise nγ straight segments,
γ = γnγ + · · · + γ2+ γ1. (2.38)
The corresponding spin evolution operator Uγ becomes a product
Uγ = Uγnγ × · · · × Uγ2 × Uγ1 (2.39)
of the individual operators Uγj = exp
£
−i
~HSOItγj
¤
, where tγj is the time the
electron spends to travel through the distance γj. These operators own a time
order property and do not commute with each other. This formalism is originally for Rashba Hamiltonian, but can be easily extended to other systems governed by the DP mechanism, such as the Dresselhaus Hamiltonian, and even to the EY mechanism as we shall show below.
2.5
Spin evolution under EY mechanism
For systems under EY mechanism, the Hamiltonian of the electron can be separated into two parts
H = H0 + Hint. (2.40)
The unperturbed part H0 has the form shown in the expression (1.1). The second
term Hint can be treated as a perturbed Hamiltonian which contains several
inter-actions responsible for electron scattering. These scattering potentials could arise from impurities, heavy holes, phonons, piezo-acoustic modes, and boundaries.
For EY mechanism, the spin may flip when the carrier is scattered. The crucial task is to determine the spin flip probability in each scattering event. Let us consider a scattering described by Hint, which changes the electron momentum
from k to k0. According to Sec. 1.1, if the electron spin does not flip during this scattering, the electron momentum relaxation time τp is related to Hint by
1 τp ∝ ¯ ¯ ¯ ¯ Z a∗ k0Hintakei(k−k 0 )·rdr ¯ ¯ ¯ ¯ 2 . (2.41)
If the spin flips during the scattering, the spin relaxation time T1, often called the
longitudinal time or spin-lattice time, is given by 1 2T1 ∝ ¯ ¯ ¯ ¯ Z
(a−k0Hintbk− b−k0Hintak)ei(k−k 0 )·rdr ¯ ¯ ¯ ¯ 2 , (2.42)
which has around the same proportionality constant as (2.41). When the electron encounters a scattering at event ξ, whether its spin state flips or not will be determined by the stochastic operator [8]
Uξ = Ã 0 1 1 0 !
:= Iflip, flip probability φ
à 1 0 0 1
!
:= Inonflip, nonflip probability 1 − φ
(2.43)
where 0 ≤ φ ≤ 1. To realize this process in a calculation, one can randomly take a number x between 0 and 1 at a scattering. Then the operator
Uξ = Θ(φ − x)Iflip+ Θ(x − φ)Inonflip (2.44)
will decide stochastically whether the spin will flip, where Θ is the Heaviside function. The flip probability φ is related to the flip rate in Eqs. (2.41) and (2.42) by 1 τp 1 2T1 = 1 − φ φ or φ = τp 2T1+ τp . (2.45)
That is, φ can be calculated from T1 and τp, provided they have been measured
from experiments.
In a simulation, if an electron encounters nγ times of scattering at points ξ1,
ξ2, · · · , ξnγ, the corresponding spin evolution operator Uγ will be
Uγ = Uξnγ × · · · × Uξ2 × Uξ1. (2.46)
Since this formula for the EY mechanism resembles (2.39) for the Rashba systems, one can regard (2.46) as a generalization of (2.39). However, Uγ in (2.46) is
as-sumed to flip the spin only at discrete times when scattering events occur, whereas that in (2.39) changes the spin at any time. With the microscopic information on each individual spin from (2.46), any macroscopic average of a crowd of spins can be calculated. Suppose the quasi-2DEG system is on the xy plane. The main quantity of concern is the spin polarization in z direction
Pz(t) = 1
n(t,D)
X
electrons at (t,D)
sz(t), (2.47)
in an observation window D at time t. When the spin of an electron is polarized to z direction, sz is set to 1. Therefore, the maximum value of |Pz(t)| is 1, which
corresponds to all electrons being aligned in z direction.
2.6
Spin evolution under DP mechanism
For systems with Rashba SOI, the Hamiltonian consists of two parts
H = H0+ HR, (2.48)
where H0 represents the sum of the kinetic and potential energies of an electron
with its effective mass in a quasi-2DEG. The second part HR = α (σ × k) · z
represents the Rashba SOI, where α is the spin orbit coupling constant, σ stands for Pauli matrices, ~k denotes the electron momentum, and z is the unit
vec-tor perpendicular to the quasi-2D sample. The Hamiltonian HR will cause spin
precession, when the carrier of the spin moves along a classical trajectory. A char-acteristic length determining this precession is the spin rotation length Lso = ~
2
αm∗,
where m∗ is the effective mass of the particle. In real semiconductor materials, the
energy ratio HR/H0 can be as large as 1/10, such as that in the InSb sample [9].
But even for this ratio, HR is still small compared with H0. In such systems,
the electron dynamics is not affected strongly by its spin dynamics, so that in the leading approximation classical trajectories are determined by H0. Thus, we
can apply the ensemble Monte Carlo method to determine the trajectory and scattering process for each electron in quasi-2DEG system.
For a free electron moving along a straight trajectory γ of length l, the dynam-ics of its spin state is governed by the evolution operator U in the path integral formalism [6] U = exp · −i ~ Z γ HR(t) dt ¸ = exp · −i l Lso b · σ ¸ , (2.49)
where b = z × k/|k|. This operator represents simply the spin rotation. If an electron collides with impurities or boundaries nγtimes, its trajectory γ will consist
of nγ straight segments γ = γnγ+ · · · + γ2+ γ1, The corresponding spin evolution
operator Uγ becomes a product
where the individual operators Uγj = exp · −i lj Lso bj· σ ¸ = 1 cos µ lj Lso ¶ − i(bj · σ) sin µ lj Lso ¶ (2.51)
along different straight segments do not commute with each other. In analogy to the case in of EY mechanism, the microscopic information on each individual spin from (2.50) allows us to calculate any macroscopic average of a crowd of spins, as discussed in (2.46), in analogy to (2.47).
Bibliography
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Simulation (Springer-Verlag Wien New York, 1989).
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(Artech House, Boston, London, 1993).
[3] K. Kurosawa, Journal of the Physical Society of Japan 21, Supplement, 424 (1966).
[4] A. G. Mal’shukov, V. V. Shlyapin and K. A. Chao, Phys. Rev. B 66,
081311(R) (2002).
[5] C.-H. Chang, A. G. Mal’shukov and K. A. Chao, Phys. Lett. A 326, 436
(2004).
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Chapter 3
EY Spin Relaxation in Quantum Wells and
Narrow Wires
In the following, we apply the ensemble Monte Carlo method and the semiclas-sical path integral method to study the spin relaxation of the Elliott-Yafet mecha-nism in low-dimensional systems. In quantum wells, the spin properties calculated by these methods confirmed the experimental results. In two dimensional narrow wires, size and impurity effects on the Elliott-Yafet relaxation were predicted, in-cluding the wire-width-dependent relaxation time, the polarization evolution on the sample boundaries, and the relaxation behavior during the diffusive-ballistic transition. For ballistic narrow wires, we derived an exact relation between the Elliott-Yafet relaxation time and the wire width, which confirmed the above sim-ulations.
3.1
Introduction
Spin relaxation is one of the central issues in the study of spintronics [1, 2, 3]. This phenomenon is ubiquitous in materials with spin polarization and has a long research history dating back to the Elliott-Yafet (EY) relaxation in simple met-als (see [3] and recent papers citing this review). The study in this context is largely motivated by a fundamental interest in material properties. However, pur-suing efficient spin manipulation in devices might further boost the progress in this field. Today, several types of mechanisms responsible for different spin relax-ations have been found [4, 5, 6, 7], and among these, the D’yakonov-Perel’ (DP) and EY mechanisms play an essential role. The former is due to spin precession between the momentum scattering events, while the latter happens ”during” the
momentum scattering events. These mechanisms affect the spin dynamics in var-ious materials. For instance, in zinc-blende semiconductors at low temperatures, the spin relaxation is dominated by the DP mechanism [8, 9, 10, 11, 12, 13]. In In-GaAs/InP multiple-quantum wells at room temperature [14,15] and the Te-doped InSb/Al0.15In0.85Sb at low temperatures [16], the spin lifetime depends mainly on
the EY mechanism [17, 18, 19, 20]. In the past, a large number of experimental and theoretical studies have been devoted to the DP mechanism, either in the 3D bulk or in low-dimensional systems like quantum wells (QWs) and 2D narrow wires [13, 21, 22, 23, 24, 25, 26, 27, 28]. However, comparatively less effort has been put into studying the EY relaxation, especially in low-dimensional systems [29].
In this work, we apply the ensemble Monte Carlo method and the semiclassi-cal path integral method to investigate the EY relaxation in QWs and 2D narrow wires in both diffusive and ballistic regimes. The study gave results in accor-dance with the experimentally measured values in real samples [16]. Based on this consistency, we used these methods to study the impurity and sample size effects on the EY relaxation under broad sample conditions. The main issues were how the relaxation time changed with sample width, how the polarization evolved on the boundary, and how the impurity density variation from diffusive to ballistic regimes affected the EY relaxation. Furthermore, the DP relaxation was calculated under the same sample conditions in order to compare it with the EY results. Finally, an analytical formula was derived for ballistic narrow wires, which confirmed our simulations and revealed exactly how the EY relaxation time varied with the wire width.
This chapter is organized as follows. In Sec. 2, the validity and precision of using the EMC and the SPI methods on the experimental samples are examined and compared with the theoretical results. In Sec. 3 and Sec. 4, the effects of size and impurity, respectively, on the EY relaxation are studied and compared with the DP relaxation. Finally, a summary and discussion are given in Sec. 5, and a supplementary material for spin relaxation process is represented in Sec. 6.
3.2
The EMC and SPI methods on experimental samples
The spin relaxation caused by the EY mechanism has been explored by some experimental groups [14,15,16]. In [16], a sample is InSb/Al0.15In0.85Sb single QW
grown by MBE on the GaAs substrates. The QW has a well width of 20 nm (corresponding to the height in Figure 3.1) and was uniformly Te-doped (sample number me1831F). The electron density in this sample is 5.7×1011 cm−2 at 77K
W ~200( m) Observation window area~1x200( m 2 ) z y H~20(nm) 2DEG in QW L~200( m) x
Figure 3.1: A quasi-2D sample and an observation window which is a stripe of area 1 × 200 µm2. For a quantum well with large L and W , the stripe is long
and the average spin behavior therein is almost the same as that in the whole well (for cases in Figures 3.2 and 3.3). For a narrow wire with large L but small W , the stripe is short and the average spin behavior inside it is a local spin dynamics along the wire (for the case in Figure 3.7).
and 7.3×1011 cm−2 at 300K. Since the carrier concentration of semiconductor is
proportional to T [30], the concentration for other T in between can be linearly interpolated, as ne(T ) ≈ (0.0072T + 5.15) × 1010 cm−2. The mobility of this
sample was measured by means of the Hall effect and behaves as log10µ(T ) ≈
0.28 × log10T − 0.55 m2V−1s−1 within T = 50 ∼ 300K. For more temperature
dependent factors in spin relaxations, it is referred to [23].
Figure 3.2 shows the product of the spin relaxation time with the temperature,
τsT , versus the carrier mobility µ. Its inset depicts the spin relaxation time versus
the temperature of the sample. In both plots, the triangles are the experimental data measured from the sample me1831F, which is mainly governed by the EY mechanism. The black dots are calculated from the formula [16]
1 τs = CEYη2 µ 1 −m ∗ m ¶2 E1e E2 g kT 1 τp . (3.1)
Therein, m is the free electron mass, m∗ denotes the effective mass in the
con-duction band, Eg represents the band gap, E1e stands for the confinement energy
of the lowest electron subband, τs is the EY mechanism induced spin relaxation
time which is equal to T1 in Eq. (1.9), and η = ∆/(Eg + ∆) with the spin orbit
splitting energy ∆. The momentum relaxation time τp is related to the mobility
µ by τp = µm∗/e and the dimensionless constant CEY is believed to be of the
order of unity. The black dots in Figure 3.2 are calculated from (3.1) by using the
Table 3.1: The simulation protocols. T1 is calculated by (3.1), τp = µm∗/e, vF = (~/m∗) √ 2πne, lmfp= vFτp, and φ is calculated by (2.45). T(K) 50 70 100 120 150 170 200 250 300 T1(ps) 2.54900 1.99790 1.54320 1.35230 1.15060 1.05090 0.93424 0.79486 0.69656 τp(ps) 0.06619 0.07263 0.08014 0.08428 0.08963 0.09278 0.09703 0.10320 0.10852 vF(µm/ps) 1.5381 1.5580 1.5874 1.6067 1.6353 1.6540 1.6817 1.7270 1.7710 lmfp(µm) 0.10180 0.11316 0.12722 0.13541 0.14656 0.15345 0.16318 0.17821 0.19219 φ 0.01282 0.01785 0.02531 0.03022 0.03749 0.04228 0.04937 0.06096 0.07227
E1e ≈ 0.08 eV and CEY ≈ 7.5 [16]. Recall that τs can be affected by various
scattering potentials mentioned in Sec. 1.2. Among others, phonons will become more significant at high T .
Figure 3.2 shows that both the experimental and theoretical studies give the relation τsT ∝ µ for most µ. But two experimental points have an opposite trend
τsT ∝ µ−1 at high µ, which corresponds to the high T regime in the sample
me1831F, as known from the empirical µ(T ) relation mentioned at the beginning of this section. One believes that this opposite trend is because the DP mechanism overrides the EY mechanism in the high µ regime, according to the current under-standing that τsT ∝ µ for EY mechanism and τsT ∝ µ−1 for DP mechanism [16].
The latter is supported by the observation on the sample me1833 (remotely n-doped with Te 20 nm above the well) in [16], which follows the DP mechanism and has the property τsT ∝ µ−1.
Next, the relaxation properties will be calculated by the ensemble Monte Carlo method and the semiclassical path integral method. To compare with above ex-perimental results, the simulations needs to insert the following exex-perimental pa-rameters. First, the spin flip probability φ will be calculated by (2.45), where how
τp = µm∗/e and T1 vary with T is based on the above empirical relation µ(T ) and
the black dots in the inset of Figure 3.2, which are calculated from (3.1). Second,
vF can be derived from vF = ~/m∗
√
2πne with the above empirical electron
den-sity ne(T ). Notice that since ne lies between 5.5 × 1011 and 7.3 × 1011 cm−2, the
corresponding de Broglie wavelength λF =
p
2π/ne ranging from 34 to 30 nm is
larger than the sample hight 20 nm, as shown in Figure 1. Thus, the electrons are confined in the z direction of the sample. Third, the size of the experimental sample was not explicitly mentioned in [16]. However, (3.1) therein is referred to [14, 15], where the sample sizes are about 2 inches (approximately 5 × 104 µm)
in length. Our simulation is performed on a smaller square of 2 × 102 µm in length
for less computational consumption. Both the experimental and simulation sam-ples belong to bulk systems. Since their scales are much larger than the de Broglie
wavelength λF (30 ∼ 34 nm), the electron motion on the xy plan is more
particle-like and the validity of ensemble Monte Carlo method and the semiclassical path integral method are justified. We put 4 × 106 electrons into our 2D sample, which
are initially in the standard initial condition and follow the simulation protocols at 50 ∼ 300K in Table 1. The time course of the polarization Pz(t) is recorded in
the middle of the sample (Figure 3.1).
0.8 1 1.2 1.4 150 200 250 100 200300 1 2 3 Analyticity Simulation Experiment 0.4 R e l a xa t i o n t i m e x t e m p e r a t u r e S T ( p s K ) Electron mobility (m 2 V -1 s -1 ) 50 120 R e l a xa t i o n t i m e S ( p s) Temperature (K)
Figure 3.2: A comparison between the analytical, numerical and experimental relations between τsT and mobility µ, as well as between τs and temperature T
(inset). The large triangle size indicates the experimental error bar.
The observed Pz(t) is an exponential function with a relaxation time τs. During
the temperature variation in Table 3.1, the relations (τsT, µ) and (τs, T ) can be
calculated, which are plotted as red squares in the main plot and the inset of Figure 3.2, respectively. Note that our recent theoretical study and simulation reveal that the Pz(t) of the DP relaxation in a narrow wire will transit from an exponential
function to a Bessel function during the impurity density decline [27]. Such Pz(t)
deviation from an exponential function will not occur in the EY mechanism, as we shall prove in Sec. 4. Thus, here we can characterize Pz(t) properly by the
parameter τs without worrying its deformation.
The red squares in Figure 3.2 calculated by our methods show very close values to the theoretical and experimental results for most µ, with the same relation
τsT ∝ µ. The opposite experimental trend τsT ∝ µ−1 in the high µ regime is
not to see in our simulation. It indirectly supports the previous hypothesis that
