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研究生:簡偉恩
研究生(外文):Wayne Jan
論文名稱:雜質金屬的電聲子作用
論文名稱(外文):Electron-Phonon Interaction in Impure Metals
指導教授:吳玉書
指導教授(外文):George Yu-Shu Wu
學位類別:博士
校院名稱:國立清華大學
系所名稱:物理學系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:88
中文關鍵詞:電聲子作用電聲子散射率雜質金屬電子平均自由徑
外文關鍵詞:electron-phonon interactionelectron-phonon scattering ratesimpure metalselectron mean free path
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電聲子(e-ph)在純質金屬(clean metals)系統中的散射率是正比於溫度(temperature)的三次方。
至於在雜質金屬(impure metals)系統中,電聲子的交互作用至今還處於一個被研究的題材。這其中主要的爭議是在於當無序系統在高掺雜極限(dirty limit)下,即在熱聲子的波向量(thermal phonon wave vector)乘上電子的平均自由徑(electron mean free path)遠小於一的情況下,到底是減弱了或者加強了電聲子的交戶作用。這裡的熱聲子的波向量等於兩倍的圓週率(Pi)乘上玻茲曼常數(Boltzmann constant)再乘上溫度再除於浦朗克常數(Planck constant)再除以聲速。在非常低溫時或者是系統無序程度是足夠強的可以滿足此極限。從Shcmid或者在後來經由Reizer 和Sergeyev的理論計算,在高掺雜極限下,電聲子的散射率正比於溫度的四次方,這和純質金屬系統中(滿足熱聲子的波向量乘上電子的平均自由徑遠大於一的條件下)的電聲子散射率來比是減弱的。而在高掺雜極限下,實驗上所量測到的,雖然電聲子的散射率正比於溫度四次方的關係曾有人宣稱觀察到,但大部份的實驗發現到在高掺雜極限下電聲子的散射率是正比於溫度的平方,這對於純質極限(clean-limit)的結果來說是加強了電聲子的散射率。由於理論上的預測和實驗上所觀察到的差異,無序系統對電聲子作用所造成的效應,長期以來一直被拿來當作研究的題材。
由於Schmid 的計算後來經由Reizer 和 Sergeyev用不同的理論方法計算也是得到相同的結果,所以理論學家相信在計算上應不會有什麼問題,因此對於散射率對於溫度二次方的倚變的解釋便傾向於修正在模型上的假設。例如 Beltz 等人便提出強聲子阻尼(strong phonon damping)可修正電聲子的散射率,其修正的結果會和實驗上一致。Sergeev 也提出了靜態隨機的位能(static random potentials),也就是說有些雜質不隨著晶格的振動,因此也可以獲得增強性的電聲子散射。
本論文中我們將研究電聲子在無序金屬系統的散射率。我們將證明若所有的雜質是替代型的(substitutional)那麼就會得到Reizer-Sergeyev的結果,即電聲子的散射率正比於溫度的四次方,這個結果即使在我們考慮了晶格結構的分離性(discreteness)時也成立。然而當我們若允許雜質可以是隨機的位移偏離(random positional shift),我們將會得到電聲子的散射率正比於溫的的二次方這個結果。我們的理論較Beltz 及 Sergeev 等人的學說更符合一般的物理狀況。
In this thesis, we calculate the electron-phonon scattering rate in polycrystalline metals, e.g., Ti1-xAlx, in the limit of dilute impurity concentration. We consider the additional contribution due to the Umklapp process of impurity scattering, which has been neglected in all previous nearly-free-electron calculations but is important for the present problem. We find that, as a result of including the Umklapp process, the scattering rate in the dirty limit (i.e. thermal phonon wave vector times electron mean free path <<1) is enhanced by the disorder due to substitutional impurities in the presence of random lattice shift of crystallites. Specifically, we obtain the scattering rate directly proportion to temperature squared divided by electron mean free path in agreement with previous experiments both in order of magnitude and in functional dependence. This work satisfactorily explains the long-standing discrepancy between theories and experiments regarding the effect of disorder on electron-phonon scattering, for the case of polycrystalline metals with dilute impurity concentration.
We also study the electron-phonon scattering rate in impure metals in the case of single crystals doped with impurities. We show that, if all impurities are substitutional, the previous Reizer-Sergeyev result, the electron-phonon scattering rate is directly proportion to temperature to the fourth power, holds even when discreteness of the lattice structure is taken into account. However, the result is modified when we also allow for random positional shift of impurities, in which case the result, the electron-phonon scattering rate is directly proportion to temperature squared, is obtained.
摘要
誌謝
Contents i
Figure and Table Contents iii
Abstract iv
Publication List v
Chapter 1 Introduction
1.1 Introduction 1
1.2 Outline of the Thesis 4
Chapter 2 Theoretical Background
2.1 Electron-Phonon Interaction 5
2.2 Non-equilibrium Green’s Function 7
2.3 Contour Ordering and Three More Non-equilibrium
Green''s Functions 10
2.4 The Keldysh Formalism 14
2.5 Kinetic Equation 23
2.6 Electron-Impurity Scattering 25
Chapter 3 Electron-Phonon Interaction in Impure
Polycrystalline Metals
3.1 Introduction 28
3.2 Theoretical Model 29
3.3 Conclusion 44
Chapter 4 Electron-Phonon Scattering Rates in Impure Metals
4.1 Introduction 45
4.2 Theoretical Model 47
4.3 Conclusion 65
References 66
Appendix A Non-equilibrium Green’s Function for the System
of Non- interacting Fermions 69
Appendix B Free-Phonon Green’s Functions 75
Appendix C Derivation of the Collision Integral 79
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[17] We define a geometric symmetry to be a symmetry present in the system when impurities and host atoms both are regarded as simple geometric objects undistinguishable from one another. This is somewhat different from the usual notion of symmetry in physics.
[18] See, for example, A. A. Abrikosov, L. P. Gor’kov, I. Ye. Dzyaloshinski, Quantum Field Theoretical Methods in Statistical Physics (Prentice-Hall, 1963), Sec. 39.
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[24] We have adopted the convention where a momentum is associated with an impurity line. Let the impurity momentum be k-G, we then associate i(k-G)*eqλ*Vimp(k-G)/ε(k-G, ) with the e-m.i. vertex, and weigh each vertex with the factor C(G;p,p-k), where p-k = incoming electron momentum and p = outgoing electron momentum, and sum the result over G.
[25] Similar to the case of e-m.i. vertex, the e-i vertices are to be weighted by C(G;p+k,p) and C*(G´;p+k,p) and summed over G and G´.
[26] We write up(r)=ΣG d(G;p)exp(iG∙r), and note that d(G;p) satisfies the Hamiltonian matrix equation
[(p+G)^2/2m-ξp] d(G;p)+ ΣG V(G-G´)d(G´;p)=0,
where V(G-G´) is the Fourier transform of V(r), the crystal potential. Using the property V(r) = V(-r), V(G-G´) is real. Therefore, the solution d(G;p) to the Hamiltonian equation can be chosen to be real. Using such up and up+k to evaluate C(G;p+k,p)≡∫exp(-iG·r) up+k* up dr, one sees that C(G;p+k,p) is real.
[27] This can be seen by carrying out the q expansion to higher order.
[28] W. Jan, G. Y. Wu, and H.-S. Wei, Phys. Rev. B 64 165101.
[29] Sums of the typeΣRi exp[-i(G´-G´) ·Ri] represent interference among various scattering events. The modification with G1 = G2 and G3 = G4 means that the random positional shift of impurities bears certain effect on the interference.
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