拓撲絕緣體(Topological insulators)的發現為架構二維電子系統開闢了一條新的路徑。這些二維電子系統駐留在三維拓撲絕緣體的表面，而且已知它們對於非磁性的干擾有很強的抵抗性。在實驗上已知生長三維拓撲絕緣體的表面並非平坦，而是由類似粗糙的梯田所組成。在這裡，我們探討表面粗糙程度(surface roughness)對三維拓撲絕緣體的傳遞性質之影響。表面粗糙程度引起的「自旋連接」(spin-connection)修正了狄拉克費米子(Dirac fermions)。透過考慮表面粗糙程度的過渡階段，我們構造了由粗糙程度引起的渦流(vortices)結構之有效哈密頓量。通過有效哈密頓量(effective Hamiltonian)，我們可以找到散射截面(scattering cross section)和相應的電導率(conductivity)。我們的分析指出即使在粗超的階段，狄拉克費米子的電導率仍然是類似於平坦的表面。粗糙程度引起的狄拉克費米子在三維拓撲絕緣體的表面傳遞之特性是溫和的變化。

The discovery of topological insulators (TIs) has pointed out a new route to construct two dimensional electronic systems. These two dimensional electronic systems reside on surfaces of 3D TIs and are known to be robust against non-magnetic perturbations. Experimentally, however, it is known that surfaces of grown 3D TIs are not flat and are composed of rough terraces. Here we investigate effects of surface roughness on transport properties of three dimensional topological insulators. It is shown that surface roughness induces spin connection correction to Dirac fermions. By considering surface roughness in the roughnening phase, we construct effective Hamiltonian due to vortices configuration of roughness. Through the effective Hamiltonian, we find the scattering cross section and the corresponding conductivity. Our analyses indicate that even in the rough phase, the conductivity of Dirac fermions still resemble that of flat surfaces. The roughness induces mild change on transport properties of Dirac fermions on surfaces of 3D TIs.

1 Introduction 1
2 Methods 5
2.1 Review of previous work . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Basic formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The eective Hamiltonian in a Dirac form . . . . . . . . . . . . . . 14
2.4 To nd dierential cross section for scattering by vortices . . . . . . 18
2.5 Scattering cross section, lifetime, and conductivity . . . . . . . . . . 22
3 Numerical results and discussion 28
4 Conclusion 34
Bibliography 34

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