破壞塊材反轉對稱在CdTe/HgTe/CdTe結構的量子自旋霍爾物理的影響

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EFFECT OF BREAKING BULK-INVERSION

SYMMETRY ON THE QUANTUM SPIN HALL

PHYSICS OF CdTe/HgTe/CdTe STRUCTURES

研 究

生： 林儀玹

指導教授： 朱仲夏 教授

霍

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EFFECT OF BREAKING BULK-INVERSION

SYMMETRY ON THE QUANTUM SPIN HALL

PHYSICS OF CdTe/HgTe/CdTe STRUCTURES

研 究

生 ： 林儀玹

_{Student: Yi-Shiuan Lin}

指 導 教 授 ： 朱仲夏 教授

Advisor: Prof. Chon-Saar Chu

國 立 交 通 大 學

電 子 物 理 學 系

碩 士 論 文

A Thesis

Submitted to Department of ElectroPhysics

College of Science

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

in

ElectroPhysics

April 2013

Hsinchu, Taiwan, Republic of China

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影

影響

響

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研 究

生 ： 林儀玹

_{指 導 教 授 ： 朱仲夏 教授}

國立交通大學

電子物理研究所

摘

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Student: Yi-Shiuan Lin

Advisor: Prof. Chon-Saar Chu

Department of Electrophysics

National Chiao-Tung University

Abstract

In this thesis, we consider the effects of the Dresselhaus spin orbital interaction (DSOI), a form of breaking inversion symmetry, on the topological physics of a well known topo-logical structure, namely, the CdTe/HgTe/CdTe quantum well. Related system such as a quantum channel formed out of the above quantum well is also considered. Instead of using model 2D Hamiltonian, and quasi-1D Hamiltonian, for the quantum well and quantum channel, respectively, we have calculated these Hamiltonians from the 3D Kane model. This includes our introduction of the DSOI at the 3D Kane model. Our finding is that the DSOI does not destroy the topological physics (the edge states) but has caused intricate effect on the edge states. This is most evident in the case of a quantum channel, when the edge state, under the effect of the DSOI, is caused to exhibit an edge-switching behavior. Furthermore, in the presence of a potential barrier, the transmission exhibits Fano-type characteristics that can be fine-tuned by the barrier height. There is also a feature of forcing the edge-switching to occur in the immediate vicinity of the transmis-sion region. A detail Fano-type process analysis has been performed to confirm the above finding.

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Abstract in Chinese i

Abstract in English ii

Acknowledgement iii

1 Introduction 1

2 From basic band formulation to effective 2D Hamiltonian 4

3 Edge-state at an open boundary 6

3.1 Edge-state branch . . . 7 3.2 Edge-state branch in the presence of DSOI . . . 10 3.3 Analytic analysis for k=0 edge state . . . 13 4 Topological origin of the edge states 16 4.1 Chern number consideration . . . 17 4.2 Winding number consideration . . . 19 4.3 Spin chern number consideration . . . 20 5 Edge-state branch in a quantum bar 24 5.1 Effective Hamiltonian . . . 25 5.2 Wave function for bulk like and edge like states . . . 29 6 Quantum transport in a quantum bar 34 6.1 The transport property of the effective 1D system . . . 35 6.2 Probing the fano-physics in the transport property . . . 42

6.3 The transport property in the presence of DSOI . . . 49

7 Summary 62 7.1 Conclusion . . . 62

7.2 Future work . . . 65

A Detail derivation: From basic band formulation to effective 2D Hamil-tonian 66 A.1 3D 8 × 8 Hamiltonian . . . 67

A.2 3D 6 × 6 Hamiltonian . . . 69

A.3 2D Hamiltonian . . . 71

A.4 2D Hamiltonian with Dresselhaus spin-orbital interaction (DSOI) . . . 75

A.5 2D band structure: Isotropy and gap-closing . . . 81

A.6 effective 4 × 4 Hamiltonian . . . 89 B Basis definition of H2D

10×10 92

C Basis definition of ˜H2D

10×10 98

D The time reversal operator for Hef f 105

E The eigenvalue and eigenvector of the special 4 × 4 Hamiltonian 107

F Basis definition of HW. 109

3.1 It shows the structure of a open boundary system. The wave function is zero when y ≤ 0 and y = ∞. . . 7 3.2 It shows the eigenenergy of edge state for Hef f with d = 7nm. The black

line is the bulk band. The red solid line is spin-up edge band and the red dash line is spin-down edge band. The blue circle is numerical result for spin-up edge band and blue plus sign is numerical result for spin-down edge band. . . 9 3.3 It shows the eigenenergy of edge state for Hef f with d = 7nm. The black

line is the bulk bands with DSOI. The red lines are edge bands without DSOI. The blue circle is numerical result for edge band with DSOI. . . 12 4.1 It shows the topological phase diagram with Iz. The x axis is δ/A and

the y axis is M/B. The red line is M B − B2_δ2_/A2 _{= 0. The black region}

is undefined. The blue region is topological trivial. The green region is topological non-trivial. . . 23 4.2 It shows the topological phase diagram with Jz. The x axis is δ/A and

the y axis is M/B. The red line is M B − B2_δ2_/A2 _{= 0. The black region}

is undefined. The blue region is topological trivial. The green region is topological non-trivial. . . 23 5.1 It shows the structure of a quantum bar system. We set the origin of y-axis

at the middle of the bar. The wave function is zero when y ≤ −W/2 and y ≥ W/2. . . 25 5.2 It shows the band structure of HW for W = 300nm with δ0 = 0. . . 27

5.3 It shows the band structure of ˜HW for W = 300nm. The blue solid line is

µ = 1 subbands. The red dash line is µ = −1 subbands. . . 28 5.4 It shows the edge states with certain kx and spin in absence of DSOI. . . . 30

5.5 It shows the edge states with certain kx in the presence of DSOI. . . 31

5.6 It shows the column vector part of the two edge states of HW with W =

300nm. The solid line is the µ = +1 state with DSOI and the dash line is the state without DSOI. Those two state are the upper edge bands and their k value is 0.01nm−1. The blue line is |E1, +i component. The red line is |H1, +i component. The black line is |E1, −i component. The green line is |H1, −i component. . . 32 5.7 It shows the column vector part of the two bulk states of HW with W =

300nm. The solid line is the µ = +1 state with DSOI and the dash line is the state without DSOI. Those two state are the lowest conduction sub-bands and their k value is 0.01nm−1. The blue line is |E1, +i component. The red line is |H1, +i component. The black line is |E1, −i component. The green line is |H1, −i component. . . 32 5.8 It shows the density of edge states Φedge₊ with W = 300nm, ∆0₀ = −1 and

E = 6.4meV . (Note: the dimension of y-axis is different to x-axis) . . . 33 5.9 It shows the spin polarizationf edge states Φedge₊ with W = 300nm, ∆0₀ = −1

and E = 6.4meV . (Note: the dimension of y-axis is different to x-axis) . . 33 6.1 It is the structure of a quantum channel system we want to study. An

uniform potential U0 is imposed at the red area between x = 0 and x = L. 36

6.2 This figure shows the definition of the t, r, t0, r0, tI _{and r}I_{. The t, t}0

and tI _{are the transmission coefficients and the others are the reflection}

coefficients. The t and r are at the right boundary, x = L, and the t0 and r0 are at the left boundary, x = 0. . . 38 6.3 It is the transmission from FN versus the incident energy for HW without

DSOI. The width W of the quantum channel is 300nm. The length L of the potential V (x) is 100nm and the value of potential energy U0 is 10meV . 39

6.4 It is the transmission from FN versus U0 for HW without DSOI. The width

W of the quantum channel is 300nm and the incident energy is 5meV . The blue circle is for L = 100nm. The red dash line is for L = 300nm. The black solid line is for L = 500nm. . . 40 6.5 It is the transmission from FN versus U0 for HW and HWedge. The width W

of the quantum channel is 300nm and the incident energy is 5meV . The length L of the potential area is 500nm. The red dash line is for HW. The

blue solid line is for H_Wedge. . . 41 6.6 It is the transmission from FN versus U0 for HW without DSOI. The length

of the potential L is 500nm and the incident energy is 5meV . The blue solid line is for W = 2500nm. The red dash line is for W = 300nm. The black dot line is for W = 350nm. The green circle is for W = 400nm. . . 41 6.7 It is the probability density from FN of the scattering state with E =

−0.2386meV , W = 300nm, L = 100 and U0 = 10meV . . . 44

6.8 It is the transmission of Dip. 1 1 with W = 300nm, L = 100 and U0 =

10meV . The red circle is MS contributed by propagating mode only. The black dot line is FN. The blue soild line is the Fano profile form, equation (6.24). . . 44 6.9 It is the transmission from FN versus the incident energy for HW without

DSOI. The band parameters of HW is from Markus K¨onig’s work.[7] The

width W of the quantum channel is 200nm. The length L of the potential V (x) is 100nm and the value of U0 is 10meV . . . 45

6.10 It is the transmission of Dip. 1 1 with W = 200nm, L = 100 and U0 =

10meV for Markus K¨onig’s band parameters.[7] The black dash line is FN. The blue circle is MS contributed by propagating mode only. The red solid line is the Fano-profile form (equation (6.17)). . . 45 6.11 It is shows the values of |1 − rbbrbb0 − 2rebr0be|, |1 − rbbr0bb| and |2rebr0be| near

the resonant energy. The blue sold line is |1 − rbbrbb0 − 2rebr0be|. The red

6.12 It is the transmission from FN versus U0 for HW and ˜HW. The width W

of the quantum channel is 300nm and the incident energy is 5meV . The blue solid line is for L = 300nm. The red dash line is for L = 500nm. The black dot solid line is for L = 300nm. The green circle is for L = 500nm. The soild line and dash line is for ˜HW and The dot line and circle is for HW. 51

6.13 It shows edge band gap for HW and ˜HW. The black dash line is for HW.

The solid lines are for ˜HW. . . 51

6.14 It is the transmission from FN versus the incident energy for ˜HW. The

width W of the quantum channel is 300nm. The length L of the potential V (x) is 100nm and the value of U0 is 10meV . . . 53

6.15 It shows the transmission at the Dip. 1 1 from FN and MS versus the value of δ of Hef f. The unit of the x-axis is our δ result at d = 7nm. The width

W of the quantum channel is 300nm. The length L of the potential V (x) is 100nm and the value of U0 is 10meV . The dash line is the FN. The solid

line is the MS with the propagating modes only. . . 53 6.16 It shows the value of the Fano factor of the Dip. 1 1. The minus δplot

range shows the result of h0W₋ because the h0W₋ is equal to the h0W₊ with minus C value. The Fano factor derive from the equation (6.24). The other parameters are the same as Fig. 6.15. . . 54 6.17 It shows the value of a of the Dip. 1 1. The other parameters are the same

as Fig. 6.15. . . 54 6.18 It shows the density of Φ++ from FN at the Dip. 1 1 (The incident energy

is 0.0375meV .). The other parameters are the same as Fig. 6.15. . . 55 6.19 It shows the density of Φ−+ from FN at the Dip. 1 1 (The incident energy

is 0.0375meV .). The other parameters are the same as Fig. 6.15. . . 55 6.20 It shows the transmission at the Dip. 1 1 from FN and MS versus the value

of U0. The width W of the quantum channel is 300nm and the length L

of the potential V (x) is 100nm. The circle is the FN. The solid line is the MS with the propagating modes and the longest decay mode only. . . 56

6.21 It shows the value of Fano factor of the Dip. 1 1. The Zµp is derived from

equation (6.31). The circle is for µ = −1.The plus sign is for µ = 1. The other parameters are the same as Fig. 6.20. . . 57 6.22 It shows the value of a of the Dip. 1 1. The other parameters are the same

as Fig. 6.20. (At the potential range near 6.7meV , we can’t obtain a.) . . 57 6.23 It shows the density of the Φedge_E,+(x, y). The potentila energy U0 is 10meV

and the incident energy is 0.0375meV (Dip. 1 1). The width W of the quantum channel is 300nm and the length L of the potential V (x) is 100nm. The incident state is localized at the edge y = −150nm. The reflection and transmission states are localized at the edge y = 150nm. . . 60 6.24 It shows the spin polarization density of the Φedge_E,+(x, y). The parameters

of the system are as the same as Fig 6.23. . . 60 6.25 It shows the density of the Φedge_E,+(x, y). The potentila energy U0 is 15meV

and the incident energy is −2.808meV (Dip. 4 1). The width W of the quantum channel is 300nm and the length L of the potential V (x) is 100nm. The incident state is localized at the edge y = −150nm. The reflection and transmission states are localized at the edge y = 150nm. . . 61 6.26 It shows the spin polarization density of the Φedge_E,+(x, y). The parameters

of the system are as the same as Fig 6.25. . . 61 A.1 It shows the CdTe/HgTe/CdTe quantum well structure. We set z = 0 at

the middle of HgTe region in the following calculation. . . 72 A.2 This picture shows the eigenenergies of the subbands versus the well

thick-ness d. The blue line is E1. The red line is H1. The red dash line is H2. The red dot line is H3. The green line is L1. . . 73 A.3 This figure shows the eigenenergy of H3D

10×10 at Γ point with DSOI. E1, H1

and H3 are coupled by Ckz and blue lines show the energy at Γ point for

them. L1 and H2 are coupled by Ckz and denote red lines. . . 77

A.4 It shows the eigenenergy of ˜S and ˜A subbands. The blue line is the eigenen-ergy of ˜S subband versus d. The red line is the eigenenergy of ˜A subband versus d. . . 79

A.5 The blue solid line is the eigenenergy of H3D

10×10 at Γ point versus d. The

red dash line is the eigenenergy of ˜H3D

10×10 at Γ point versus d. . . 79

A.6 This figure shows the energy dispersion of H2D

ef f with d = 6.58nm. The

black vertical indicates the kc. . . 85

A.7 This figure shows the energy dispersion of ˜H2D

ef f with d = 6.56nm. The

black vertical indicates the kc. . . 85

A.8 This figure shows energy structure of conduction and heavy hole ( ˜S1 and ˜

S2) bands. The ˜E1 is the energy of the 1st highest band. The ˜E2 is the energy of the 2nd highest band. The ˜E3 is the energy of the 3rd highest band. The ˜E4 is the energy of the 4th highest band. . . 86 A.9 This figure shows min( ˜E2 − ˜E3). The blue solid line is for the H2D

ef f. The

red dash line is for the ˜H2D

ef f. . . 86

A.10 This figure shows the energy difference of H2D

10×10versus φ(angle of k) at

d = 6.58nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1. 87 A.11 This figure shows the energy difference of H2D

10×10versus φ(angle of k) at

d = 7nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1. . 87 A.12 This figure shows the energy difference of ˜H2D

10×10versus φ(angle of k) at

d = 6.56nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1. 88 A.13 This figure shows the energy difference of ˜H2D

10×10versus φ(angle of k) at

d = 7nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1. . 88 A.14 It shows the eigenenergy of ˜H2D

10×10and H10×102D with the well thickness 7nm.

The solid line is the eigenenergy of H2D

10×10. The dash line is the eigenenergy

of ˜H2D

10×10. . . 89

A.15 It shows the eigenenergy of conduction and heavy hole bands from two effective Hamiltonian. The blue solid line is from H2D

10×10. The red dash

B.1 This picture show the solution of δE_{. The solid line is the real part. The}

dash line is the imaginary part. The right vertical line is EHg

v (= 0eV ). The

left vertical line is EHg

c (= −0.303eV ). When E1E is between EvHg and EcHg,

all δE _{are real. Otherwise two of δ}E _{will be pure imaginary. Here we only}

show two of roots. The other root are minus times of the roots showed here. 95 B.2 This picture show the solution of γE_{. The solid line is the real part. The}

dash line is the imaginary part. The left vertical line is ECd

v (= −0.57eV ).

The right vertical line is ECd

c (= 1.036eV ). In the energy range EvCd <

EE

1 < EcCd, all γE are real. Here we only show two of roots. The other root

are minus times of the roots showed here. . . 95 C.1 This picture show the solution of ˜δS_{. The solid line is the real part. The}

dash line is the imaginary part. The right vertical line is EHg

v (= 0eV ).

The left vertical line is EHg

c (= −0.303eV ). When ˜ES is between EvHg and

ECd

c (= 1.036eV ), two of ˜δE are pure imaginary. The other ˜δSs are real.

Here we only show three of roots. The other root are minus times of the the roots showed here. . . 102 C.2 This picture show the solution of ˜γS_{. The solid line is the real part. The}

dash line is the imaginary part. The left vertical line is ECd

v (= −0.57eV ).

The right vertical line is ECd

c (= 1.036eV ). In the energy range EvCd <

EE

1 < EcCd, all ˜γS are real. Here we only show three of roots. The other

Chapter 1

Introduction

The quantum spin Hall effect (QSHE) has been theoretically predicted and experimentally observed in a HgCdTe/HgTe/HgCdTe quantum well system.[1-2] The band structure of the HgCdTe is normal and the band structure of the HgTe is inverted. Changing the width d of the HgTe layer will change the two dimension (2D) electronic structure of the quantum well between normal and inverted region. Specifically, the eigenenergy of the conduction and heavy hole bands at the Γ points exhibit an energy crossing at a critical well width dc (dc = 6.3nm[1]). The 2D band structure is topologically non-trivial in the

d > dc region.

In the topologically non-trivial region, the system supports edge states which is helical, namely, the spin polarization is connected with its propagation direction. Furthermore, time reversal symmetry requires that edge states at the same sample edge exist in a pair: consisting of opposite propagation directions. This is the so called quantum spin Hall effect. These edge states are robust against weak non-magnetic impurities or potential profile.[3] As such, the edge states are expected to have high (almost perfect) transmission through a potential barrier. Even though the gapless spectra of the edge states, in a quantum channel, could open up a gap due to edge state wave function overlapping[4], the back scattering of the edge states remain small as long as the energy of interest is far away from the small energy gap. Here the quantum channel is formed out of the quantum well. Recently, transport characteristics through such a quantum channel in the presence of a potential barrier was studied.[5] They speculate that that transport dip structures

speculation.

On the other hand, the fact that bulk inversion asymmetry (BIA) exists, where the Dresselhaus spin orbital interaction (DSOI) comes into play, has not been seriously considered.[6] An earlier study along this line claimed that BIA effects has no significant bearing on the electronic property of the system.[7] However, the DSOI term do cause mixing in the conduction and heavy hole bands, removing the energy crossing at the Γ point.[8] Would this level anti-crossing destroy the QSHE completely? Or, if the QSHE is to survive, how would the electronic states adjust to maintain the topological nature of the system? We feel that this issue has far from being fully explored.

Still there are a number of recent works that have included DSOI into their effective 2D Hamiltonian for the exploration of the quantum transport of the system.[9-11] Disorder in a quantum channel is found to back scatter the edge states if the impurity is strong enough and the channel is narrow enough, or if bulk states exist at the Fermi energy.[9] Furthermore, back scattering of the edge states can occur in a quantum channel that has abrupt change in the channel width even though there is no impurity.[10] On the other hand, the DSOI is found to provide a scheme monitoring the transmission or reflection of an incoming edge state by the monitering og the Fermi energy.[11] This is due to the interplay of the DSOI and the finite channel width effect, causing the spin of the edge state to precess and the simultaneous switching of its location from one sample edge to the other. This work has not included bulk states of the channel into their consideration. A major focus of our work is to study inticate processes made possible by the presence of both bulk and edge states in a quantum channel.

In this thesis, we perform a careful and detail calculation, both numerically and semi-analytically, on the effects of the DSOI on the effective 2D Hamiltonian in a CdTe/HgTe/CdTe quantum well system, the electronic states in the corresponding semi-infinite system or quantum channel, and the transport characteristic through a potential barrier in the quantum channel. Furthermore, we have discussed in depth on the topological nature (in terms of topological number) of the edge states; and we have explicitly demonstrated the Fano-type mechanism that leads to the transport spectra in the transmission. Our finding is that the DSOI provides us a handle to tune the Fano-spectra. We also demonstrate that by fine tuning the potential barrier in the vicinity of a Fano-spectra, we can switch

the transmitted edge state from one edge to the other of the quantum channel. This scheme of invoking the Fano-physics is more sensitive and less restrictive upon the struc-ture requirement than that proposed in ref[11]. This noval feastruc-ture could find application in furture spintronic devices.

This thesis contains 7 chapters, including this Introduction chapter. In chapter 2, we show our result of the the effective 2D Hamiltonian for the quantum well with DSOI which we have derived from a 3D Kane model. The detail of the derivation is presented in Appendix A. In chapter 3, we solve for the edge states in the semi-infinite 2D system is formed out by the quantum well. Our methods of approach include both semi-analytical and numerical methods. In chapter 4, we explore the topological origin of the edge state in sight of topological quantities such as Chern number, winding number and the spin Chern number. In chapter 5, we derive Hamiltonian for a quantum channel that is formed out of the quantem well system. The electronic states for the quantum channel is obtained and discussed. In chapter 6, we consider the transport through the quantum channel that consists of the a potnetial barrier. Both full numerical and semi-analytical approaches are used to obtain and to analyze the transprot characteristics. Dip structures in the transmission are found and are shown to originate from Fano physics. This Fano-type process is made possible by the coexistance of edge state and bulk like resonance state in the barrier region. Finally, we present a summery of our work (a majonity of it is in chapters 5 and 6), and a suggestion to possible future work.

From basic band formulation to

effective 2D Hamiltonian

In this chapter, we show the effective 2D Hamiltonian of the CdTe/HgTe/CdTe quantum well system with DSOI. And, we also show how DSOI removes the energy-crossing of the conduction and beavy hole bands in the quantum well.

The lattice structures of HgTe and CdTe are zinc blende structure so they do not have center of inversion.[6] It makes those materials be BIA. This symmetry broken makes a spin orbital interaction well known as DSOI. Kane Hamiltonian describes the 3D bulk electronic structure of the HgTe and CdTe (with DSOI) so we use it to start our analysis on the quantum well. How the DSOI comes in the 3D Kane Hamiltonian and the detail derivation of the effective 2D Hamiltonian are presented in Appendix A.

In the resent study[1], the non-trivial edge bands is observed in the energy gap between the conduction and heavy hole bands of the quantum well. We do the derivation presented in Appendix A to obtain the effective 2D 4 × 4 Hamiltonian which describes the bands structure at the energy range of the conduction and heavy hole bands. In the set of basis (|E1, +i, |H1, +i, |E1, −i, |H1, −i), the effective Hamiltonian Hef f is of the form:

Hef f = Dk2+         −M + Bk2 _Ak + 0 δ Ak− M − Bk2 δ 0 0 δ −M + Bk2 _Ak − δ 0 Ak+ M − Bk2         . (2.1)

HAMILTONIAN

Table 2.1: The band structure parameter of Hef f

M (meV ) B(meV /nm2) D(meV /nm2) A(meV /nm) δ(meV )

ours 4.83 753 578 376 1.68

S.C. Zhang[1] 6.86 169 5.14 346

Markus K¨onig[7] 10 686 512 364 1.6

a_{This band parameter is for d = 7nm.}

b_{In ref[1], they didn’t include the DSOI so there are no δ terms.}

The value of the band parameters in Hef f is showed at Table 2.1. The from of the

effective Hamiltonian is as the same as the other’s result.[7] (By applying an unitary transformation, we can obtain the same effective Hamiltonian as their result.) Our value of parameters A, B, D and δ are similar to theirs result. Only the value of M is much different. The difference is because they replace the CdTe layers by the HgCdTe layers. The band parameters of the bulk material aren’t equal so that we have a different set of band parameters. The topological properties of the two set of band parameters are the same. We will show it in chpater 3.

The DSOI only adds the δ terms into the effective Hamiltonian. The other terms are independent of the DSOI. The δ terms are constant and couple the conduction (E1) and the heavy hole (H1) band. The value of δ isn’t zero in this system so the crossing of eigenenergy at the Γ point is removed. The energy spectrum becomes anti-crossing.

Edge-state at an open boundary

In this chapter, we calculate the energy dispersion of edge bands in the corresponded semi-infinite system showed in Fig.3.1. The condition of the edge state existence both with DSOI and without DSOI is derived. We also discuss the transmission property of the edge states with DSOI.

The gapless edge state exists in the system described by Hef f (equation (2.1)) with

∆ = M B − (Bδ/A)2 > 0 and |B| > |D|. (In the system without DSOI, the condition is ∆ = M B > 0, A 6= 0 and |B| > |D|.) The gapless property of the edge states is due to time reversal symmetry. For semi-infinity boundary, there are only two edge bands in the system whether we include DSOI or not. Time reversal symmetry makes system have a two level degeneracy at Γ point. This energy degeneracy makes the edge bands be crossing. Therefore, the edge bands are gapless and the system is topologically non-trivial if ∆ > 0. On the other hand, the DSOI only makes some systems that are topologically non-tirival without DSOI become topologically trivial. The DSOI does effect the topological phase of the 2D electronic system.

In the end of this chapter, we derive the form of effective 1D Hamiltonian consists of edge branches by time reversal symmetry and the form of the edge states. The edge band gap is zero and the pseudo-spin is only dependent on the k linear terms. Those properties are due to time reversal symmetry. The form of edge states make the k2 _{terms be zero so}

the edge band aren’t scattered by non-magnetic potential or impurities. Only magnetic impurities or field scatter the edge bands so DSOI doesn’t weak the edge bands.

Figure 3.1: It shows the structure of a open boundary system. The wave function is zero when y ≤ 0 and y = ∞.

3.1

Edge-state branch

In this section, we analytically solve the edge branches without DSOI for semi-infinity boundary. (Here we assume we can tune all parameter in Hef f so that we can see what is

condition that edge state exists.) We consider the effective Hamiltonian Hef f with δ = 0

(equation (2.1)), Hef f =   h+ 0 0 h−  . (3.1) hτ =   −M + D+k2 Akτ Ak−τ M + D−k2  .

Where D± = D ± B. The spin up part is decoupled to the spin down part. We can divide

Hef f into two 2 × 2 Hamiltonian h+ and h−.

The open boundary is along x-direction so the kx is the quantum number of the wave

function. The wave function is of the form: Ψ (x, y) = eikxxX

j

Cjeiky,jyξj. (3.2)

Where ξ is the pesudo-spin of Ψ. To form an edge state, we need at least two different decay kys. When A is zero, the E1 and H1 subband are decoupled. We can’t find two of

and energy E, k2 y is of the form: k_y,ν2 = α + νβ D+D− − k2 x. (3.3) Where α = ED +A₂2 − M B, β = pα2_{− D} +D−E+E− and E± = E ± M . According to

the form of hτ, there are only four kys. Two kys are minus times of the others. To form

an edge state, we need two different kys whose decay directions are the same. So only

when all kys are complex, the edge state exists in this system.

Form the Schr¨odinger eqaution, we have two ways to define the pesudo-spin ξ corre-sponding to ky,ν. ξ_ν† = Nν  E−− α+νβ_D+ Akx+ τ iky,ν∗  _{. (3.4)} (ξ_ν0)†= N_ν0  Akx− τ iky,ν∗  E+− α+νβ_D−  . (3.5)

At the edge, y = 0, the wave function msut be zero. It means that the pseudo-spins of two kys must be parallel. From the definition of ky2, ξ+ = ξ−0 and ξ−= ξ+0 , we obtain

ky,++ ky,− = 2β D+D− 1 ky,+− ky,− = −i τ A 2_k x EB − M D. (3.6) From the equation (3.6) and ξ₋0 = ξ₊0 , we have

β D− 2BE − 2DM + A2
 kx EB − M D + EB − M D A2_D +D−kx  = 0. (3.7) If β is zero, there are only two kinds of ky, so β must be non-zero. For 2BE −2DM +A2 =

0, its energy dispersion is not gapless. Here we focus upon the gapless edge bands so we drop this solution.

We obtain the form of the energy dispersion: E = MD B + ρ 1 B p A2_(B2_{− D}2_)k x. (3.8)

Where ρ = ±1. The kx, A, B, D and M , are real and ky must be well defined (equation

Figure 3.2: It shows the eigenenergy of edge state for Hef f with d = 7nm. The black

line is the bulk band. The red solid line is spin-up edge band and the red dash line is spin-down edge band. The blue circle is numerical result for spin-up edge band and blue plus sign is numerical result for spin-down edge band.

the global energy band gap will be closed. There are at least two of pure real ky for given

energy and kx, so no eigenstate has two different decay kys. But two decay kys is the

condition for existence of edge states. So, only if |B| > |D|, the edge states may exist. From equations (3.6), (3.8) and definition of α and β, we find out that ky is of the

form: ky,ν = −iτ ρ A2 _{+ 2νβ} 2pA2_(B2 _{− D}2₎. (3.9) Let Λ = τ ρ A2 2√A2_(B2_−D2₎ and λ = τ ρ β √

A2_(B2_−D2₎. The wave function is of the form:

Ψ (x, y) = N (kx) eikxxeΛy eλy− e−λy
ξ. (3.10)

ξ† = √1 2B  √ B − D −τ sgn (A)√B + D  . (3.11) The eλy_−e−λy _{part can be sin or sinh function so this part doesn’t make the wave function}

decay. Only the eΛy _{part makes the wave function be an edge state. The wave function}

Ψ must be zero at y = ∞ so ρ must be −τ . (For Ψ (y = −∞) = 0, ρ = τ .)

λ2 _{< Λ}2_{, the wave function will be 0 at y = ∞. (If λ is imaginary, λ}2 _{is also smaller than}

Λ2_{.) Form the definition of Λ and λ, we have}

Λ2− λ2 _< M

B . (3.12)

If M B < 0, there is no edge state solution. So far, we only assume that the value of A is non-zero. It is the difference between ref[4] and our result. Our result shows that if A 6= 0, M B > 0 and |B| > |D|, the edge state exists for this Hamiltonian. (Our result is similar to ref[14].)

For Hef f with d = 7nm, we have M B > 0 so we can find gapless edge bands. In Fig.

3.2, we show the analytical and numerical result of the edge branches. Because the two results are the same, our analtyical result works.

3.2

Edge-state branch in the presence of DSOI

In this section, we numerically calculate the energy dispersion of edge state with DSOI. And we also prove the edge bands are gapless.

We consider the effective Hamiltonian Hef f (equation (2.1)),

Hef f = Dkx2+   h+ δσx δσx h−  . (3.13) hτ =   −M + Bk2 _Ak τ Ak−τ M − Bk2  .

Because the root of ky is difficult to analytically solve, we numerically solve eigenenergy

of edge bands. The open boundary is along x-direction so kx is the quantum number of

the wave function. The wave function is of the form: ˜

Ψ (x, y) = eikxxX

j

˜

Cjeiky,jyξ˜j. (3.14)

The Schr¨odinger equation is

Hef f(kx, ky,j)eikxx+iky,jyξ˜j = ˜Eeikxx+iky,jyξ˜j. (3.15)

Where ˜E is the eigenenergy. To compute the ky for given E and kx, we define H00, H 0 1

and H₂0 as the following forms. H₀0 =   N (kx) δσx δσx N (kx)  .H 0 1 =   −Aσy 0 0 Aσy  .H 0 2 =   D + Bσz 0 0 D + Bσz  .

Where N (kx) = − ˜E − M σz + Aσxkx+ (D + Bσz)kx2 and σi is Pauli matrix. We rewrite

the Schr¨odinger equation in this form:

H₀0ξ + H˜ ₁0(kyξ) = −k˜ yH20(kyξ).˜ (3.16)

Combining equation (3.16) and (kyξ) = k˜ y( ˜ξ), we have

  1 0 0 −H₂0   −1  0 1 H₀0 H₁0  ξ 0 = kyξ0. (3.17) Where ˜ξ0† = ξ˜† k∗_yξ˜† 

. In this form, we can get the value of ky and the corresponded

eigenvector by the numerical diagonalization.

Because ˜ξ is a 4 × 1 column vector, we need at least four decay kys to match the

boundary condition at the edge. However there are only 8 kys for given kx and E. (See

equation (3.17).) The edge state only exists at the energy range that all kys are complex.

We can only seek the edge states at the energy range where no bulk state with given kx.

At energy range where all kys are complex, we can find a wave function ˜Ψ that consists

of four complex kys and satisfies ˜Ψ(y = ∞) = 0. At y = 0, we derive an equation that

determines the eigenenergy of the edge states: X

j

˜

Figure 3.3: It shows the eigenenergy of edge state for Hef f with d = 7nm. The black line

is the bulk bands with DSOI. The red lines are edge bands without DSOI. The blue circle is numerical result for edge band with DSOI.

From the numerical result, we find out that there only two edge bands in the system. The edge bands touch the conduction and heavy hole bulk bands. If the edge band gap is zero, the edge bands are still gapless with DSOI.

Let Ψ0 be one of the edge states with kx = 0. The time reversed state ΘΨ0 is also an

edge state with kx = 0. The Ψ0 and ΘΨ0 are different states. (The form of time reversal

operator is listed in Appendix D.)

hΨ0| ΘΨ0i = Z   −Ψ0 1(x, y) ∗ Ψ03(x, y) ∗ + Ψ02(x, y) ∗ Ψ04(x, y) ∗ +Ψ01(x, y) ∗ Ψ03(x, y) ∗ − Ψ0 2(x, y) ∗ Ψ04(x, y) ∗  dxdy = 0. (3.19) The Hamiltonian is time reversal invariant so both Ψ0 and ΘΨ0 are the eigenstates of the system. At Γ point, there is a two level degeneracy protected by time reversal symmetry. In the global bulk energy gap, we only find two edge bands so those edge bands are gapless.

3.3

Analytic analysis for k=0 edge state

In this section, we first try to obtain the condition that edge bands exist for the effective Hamiltonian with DSOI. In the end of this section, we discuss the transport property of the edge bands with DSOI.

The effective 4 × 4 Hamiltonian Hef f with kx = 0 is of this form(equation (2.1)):

Hef f(0, ky) =   h+(ky) δσx δσx h−(ky)  . (3.20) hτ(ky) = Dky2+   −M + Bk2 y iτ Aky −iτ Aky M − Bky2  .

The eigenenergy of edge state is at the energy range where all ky is complex. Only when

|B| > |D|, all the kys are complex at the energy band gap. For case A = 0, we can

divide the Hef f into two 2 × 2 Hamiltonian. For each 2 × 2 Hamiltonian, we can find

four complex kys and two of different kys have the same decay direction. However, the

pseudo-spins of those two kys aren’t the same. The wave function can’t be zero at y = 0

so there is no edge state solution. Therefore, A must be non-zero and |B| > |D|. We define a new set of basis ((|E, +i − i |E, −i)√

2, (|H1, +i + i |H1, −i)√ 2, (|E, +i − i |E, −i)√2, (|H1, +i + i |H1, −i)√2). In this representation, the effective 4 × 4 Hamiltonian will becomes block diagonal form.

˜ Hef f (0, ky) =   ˜ h+(ky) 0 0 ˜h−(ky)  . (3.21) ˜ hµ(ky) = Dky2+   −M + Bk2 y i (Aky+ µδ) −i (Aky+ µδ) M − Bky2  .

In this representation, there are some kx dependent terms at the off-block diagonal part.

The ˜Hef f is block diagonal form only if kx = 0. The following analysis only works with

From numerical result, we obtain the eigenenergy and the form of the edge states: ˜ E = MD B. (3.22a) ˜ Ψµ = ˜Nµ 

ei˜kµy,1y _{+ e}i˜k µ y,2y ˜_{ξ. (3.22b)} ˜ ξ† = √1 2B  √ B − D −sgn (A)√B + D  . (3.22c) Here we have found a pair of ky and a pair of corresponded pseudo-spins that are parallel

to each other. The wave function, ˜Ψµ, are zero at y = 0. Next step, we try to find the

condition that both kys decay at the same direction, y-direction. (The set of equation

(3.22) also works when the edge states aren’t the real state of the system.) Taking equation (3.22a), (3.22b) and (3.22c) into equation (3.21), we have

√

B2_{− D}2_{M − Bk}µ y

2

= −isgn (A) B Akµ_y − µδ
. (3.23) From equation (3.23), ky is of the form

k_yνs= i     A 2b     + s s ∆ B2 +  νδ A + i     A 2b     2 . (3.24)

Where b =√B2_{− D}2 _{and ν = µsgn (A). The edge state doesn’t exist when one of k} y is

real or the imaginary parts of kys aren’t both positive. For the form of ky, one of ky will

be real only if ∆ = 0. The region at where the ky is real is the boundary between the

regions at where kys have opposite sign of imaginary part. There is no other condition

makes the imaginary part of ky change sign. (If the imaginary part of wave vector can’t

be zero, the sign of the imaginary part is unchanged for whole parameter space.) Let |∆| << 1 and s = −1, we have k_yµ≈ i     A 2b     ∆ 2B2_κ2 − µ δ |A|  1 − ∆ 2B2_κ2  . (3.25) Where κ = q δ2 A2 +  _2bA   2

. We obtain that the edge state exist when ∆ > 0.

According to the form of ∆, the DSOI only makes some systems that are topologically non-tirival without DSOI become topologically trivial. (It doesn’t change the topological

phase of the topological trivial region.) There is still a clear boundary between topolog-ically trivial and non-trivial in the topological phase diagram. The DSOI just shifts the boundary of the topological phase.

Finally, we want to discuss the transmission property of edge bands in the semi-infinite system. We compose an effective 1D Hamiltonian Hedge _{consisting of the edge bands.}

Hedge= 2 X i=0 3 X j=0 hinijσjkix. (3.26)

Where hi and nj are parameter and σ0 is 2 × 2 identity matrix. We set the basis vector

as the edge states with kx = 0. The edge subbands are time reversal pair so the time

reversal operator is of the form:Θ = iσyK. The time reversal property of k vector operator

is Θki

x = (−1)ikixΘ. The 1D Hamiltonian Hedge is time reversal invariant so we have

[Hedge_{, Θ] = 0. Therefore, time reversal symmetry leads the following condition.}

Θhinijσj = (−1)ihinijσjΘ. (3.27)

By the symmetry argument, we derive the general form of Hedge. Hedge = h0+ 3 X j=1 njσj ! h1kx+ h2kx2. (3.28)

The direction of pseudo-spin is independent of kx. The back scattering induced by

pseudo-spin changing is suppressed. Though the pseudo-pseudo-spin doesn’t change, the propagation direction may change by kx. However, the form of the basis vector makes h2 zero whether

we include DSOI or not. The scattering process with edge band only is weak. The edge state is robust against a normal impurity and a small potential. (In contrast, we doesn’t imply that edge state is robust to large potential. A large potential that makes bulk bands and edge bands coupled may scatter the edge state.)

Topological origin of the edge states

In this chapter, we try to obtain the topological number by chern number, winding number and spin chern number method. The chen number in this chapter is the Berry’s phase of a energy band.

In chern number consideration, we can’t tell the topological phase changing. The chern number is always zero for the eigenvectors of the effective Hamiltonian with DSOI. The form of the eigenvectors makes this result so the chern number of the eigenvector we maintained above is zero even if δ = 0. On the other hand, we can obtain the non-trivial topological number by another kind of eigenvectors for the effective Hamiltonian without DSOI. The two kinds of eigenfunction both characterize the system, effective Hamiltonian with δ = 0. The chern number of the different kinds of eigenvectors aren’t the same so the chern number isn’t invariant under guage transformation.

In winding number consideration, we can obtain the condition of the topological phase changing that we obtained in chapter 3. But it only works for 2 × 2 Hamiltonian and we must drop the small wave vector region of the 2D k space to obtain this result.

In spin chern number consideration, we can obtain the topological non-trivial condi-tion which we obtained at chapter 3. However, the phase diagram is dependent on the generator that generates the eigenvectors to calculate the spin chern number. It means that we need a proper way to obtain the generator.

4.1

Chern number consideration

The definition of Berry’s curvature is of the form[18]:

* Bj * R= ∇* R× D jR* ∇*_R   j * RE=D∇* Rj * R ×   ∇*_Rj * RE. (4.1) This is the Berry’s curvature of state j represented by vector of parametersR. Substitut-* ing k vector and eigenvectors of the system with DSOI forR and state j, we obtain the* Berry’s curvature of the system with DSOI. The chern number is equal to integrate the Berry’s curvature for whole k space.

The effective 4 × 4 Hamiltonian Hef f (equation (2.1)) is

Hef f = Dkx2+   h+ δσx δσx h−  . (4.2) hτ =   −M + Bk2 _Ak τ Ak−τ M − Bk2  .

The eigenenergy Eρµ and eigenvector of Hef f are (See Appendix E.)

Eρµ = Dk2+ ρ q (M − Bk2₎2_{+ (Ak + µδ)}2_{, (4.3)} ϕ†_ρµ = √1 2  αρµ₁ (k) e−iφ αρµ₂ (k) µαρµ₁ (k) µα₂ρµ(k) e−iφ  , (4.4) where α₁ρµ(k) = Nρµ(Ak + µδ), αρµ2 (k) = Nρµ(k) (Eρµ− Dk2 + M − Bk2) and Nρµ(k) =

h

(αρµ₁ )2+ (αρµ₁ )2i

−1/2

.

The Berry’s curvature of the ϕρµ is always zero even if δ = 0. It is because the spin

up and spin down parts of the eigenvector are equal weighting. The Berry’s curvatures from the spin up and spin down parts for a given k are totally canceled.

Actually, the Berry’s curvature is only zero for non-zero k and diverges at Γ point. It is because there is a two level degeneracy protected by time reversal symmetry. From an another formula of Berry’s curvature, this degeneracy makes the Berry’s curvature diverge at Γ point.

*

"

X hm| ∇H |ni × hn| ∇H |mi #

Where Ei is the eigenenergy of the wave function.

Obtaining the chern number in this case is similar to calculate the total electric flux on a plane from an electric charge that is on that plane. The electric flux is zero except of the location of the charge and is undefined at the location of the charge. If we want to calculate the flux from the charge, we need to shift that charge slightly out of the plane. It is similar to add some terms that remove the time reversal protected degeneracy at Γ point in this case.

We have tried to compute the chern number by adding some term breaking time reversal symmetry. Taking those terms very small, we can obtain the chern number that decribes the original system. However, the chern number is only dependent on the terms we adding. We can’t calculate the chern number in this way so we can’t tell the topological phase change by the chern number consideration.

On the other hand, we have two set of eigenvectors which are the eigenatates of the effective Hamiltonian without DSOI. We have another set of the eigenvectors ˜ϕρµthat are

this form: ˜ ϕρ+ * k= 1 p2ε (ε + ρ [M − Bk2_])   χρ+ 0  . (4.6a) ˜ ϕρ− * k= 1 p2ε (ε + ρ [M − Bk2_])   0 χρ−  . (4.6b) Where ε = q A2_k2_{+ (M − Bk}2₎2_{, χ}† ρµ = 

Ake−iµφ ρε + M − Bk2  _{and µ indicates}

the spin. The Berry’s curvature of ˜ϕρµ is not zero for all k. The corresponded chern

number is of ther form:

Cρµ =

ρµ

2 [sgn (M ) + sgn (B)] . (4.7) But, the chern number of ϕρµ is always zero. This form of wave function also describes the

right state, even if DSOI disappears, because the Hamiltonian is invariant under gauge transformation. We obtain two very different chern numbers from two different set of eigenvectors that are all eigenfunctions of the same effective Hamiltonian. It shows the chern number depends on the wave function form we choosing. The chern number isn’t invariant under gauge transformation so it may not be a good topological number. (The

Chern number for the Chern class is the summation of chern number for all occupied bands.[15] The Chern number is invariant under gauge transformation.)

4.2

Winding number consideration

Consider a 2 × 2 Hamiltonian h as this form: h = g0 * k+X i gi * kσi. (4.8)

The chern number is also determined by the winding number of*g * k  .[15] C = 1 4π Z dkdφX lmn εlmn(ˆg)l(∂kg)ˆ m(∂φg)ˆ _n. (4.9)

If we have a 2 × 2 Hamiltonian characterizing the system, we can derive the chern number from the winding number of this Hamiltonian.

From the form of the eigenvector of Hef f, we have the unitary transformation V that

can transform the Hef f (equation (2.1)) to the block diagonal form.

˜ Hef f = V Hef fV†=   ˜ h+ 0 0 h˜−  . (4.10) ˜ hµ = Dk2+   −M + Bk2 _{(Ak + µδ) e}iφ (Ak + µδ) e−iφ M − Bk2  . (4.11)

The ˜hµ describes the same eigenenergy and eigenvector as Hef f so we can obtain the

topological number by ˜hµ.

For ˜hµ,

*

gµ*kis of the form:

*

The corresponded chern number Cµ is Cµ= − 1 2  M − Bk2 Eµ kf ki . (4.13) Where Eµ= q (Ak + µδ)2+ (M − Bk2₎2_.

If we integral whole k space, the chern number is always not an integer. Cµ = 1 2  B |B| + M √ M2_{+ δ}2  . (4.14)

For µ = −sgn (δA), we can find the special k_c0 = |δ/A| where _∂k∂ M −Bk_E 2

µ    k=k0 c = 0. If we only integral between k_c0 and k = ∞, the chern number is zero for ∆ = M B − [Bδ/A]2
< 0 and the chern number is ±1 for ∆ > 0. It is the same result that we obtained in the Chapter 3.

It seems that we can exactly find the topological number by this method. However we may need to choose the range of integration. Here we must drop the contribution of the region k < δ/A. The Hef f works for small k so the winding number mainly depends on

the the region where Hef f can’t characterize. In addition, we can only calculate the chern

number for a 2 × 2 Hamiltonian. The winding number can’t be a topological number for general systems.

4.3

Spin chern number consideration

The Berry’s curvature in presence of DSOI is zero for k 6= 0 and diverge at Γ point. We can’t defined a proper topological number from the chern number of the eigenvector of Hef f (equation (2.1)). To obtain the topological number, we need a way to redefined the

eigenvector. The new set of vector must be smooth and unique at the whole k space. In recent years, a new definition of topological number that is called spin chern number has been established.[16][17] The spin chern number is the chern number of the vector ϕ0.

Where ϕ0 is the linear superposition of the valence bands of the system. ϕ0_i = valence band X j C_j0iϕj. (4.15)

ϕ0_i is the eigenvector of a spin operator S with eigenvalue m0_i. If the direct bulk energy gap and spin gap ∆m = m1− m2 are both non-zero, we have a well defined spin chern

number.

Consider a spin operator Iz[17] is of the form:

Iz =   I2×2 0 0 −I2×2  . (4.16)

For Hef f without DSOI, we can distinguish the spin up and the spin down by Iz.

There-fore, we use Iz to calculate the spin chern number of Hef f with DSOI.

To derive ϕ0, we project I_z0 onto the eigenstates of Hef f and drop the conduction bands.

I_z0 =Dϕ−+ * k ϕ−− * kEσx. (4.17)

The eigenvector of I_z0 is of the form: ϕ0_ν = √1 2  sgnhDϕ−+ * k ϕ−− * kEi  ϕ−+ * kE+ ν   ϕ−− * kE. (4.18)

The value of direct spin gap ∆m0 is 2  D ϕ−+ * k ϕ−− * kE . The spin gap is zero if ϕ−+

*

k and ϕ−−

*

k are orthogonal. If the ϕ−+

*

k and ϕ−−

*

k are different spin state of ˆn, they are orthogonal. Therefore, the spin gap is zero if the vector *g+*k(equation (4.12)) is equal to −*g−*k at *k. The two vectors are anti-parallel only when the z component of*gµ*k is zero and k < |δ/A| so the spin chern number is undefined when M B > 0 and M B − B2_δ2_/A2 _{< 0. At the well defined}

region, the spin chern number Cν is

Cν = −

ν

The Fig. 4.1 shows the topological phase diagram. The topological non-trivial region is equal to what we proved in chapter 3.

We have proved the spin chern number can describe the topological property of system. However, there is still a problem because we find out that the topological phase diagram is dependent of the generator.

We use the operator Jz to obtain spin chern number. The Jz is of the form:

Jz = 1 2         1 0 0 0 0 3 0 0 0 0 −1 0 0 0 0 −3         . (4.20)

For Hef f without DSOI, we can also distinguish the spin up and the spin down by Jz.

The basis vector of Hef f is the eigenvector of Jz. Jz is also a reasonable operator for spin

chern number. We numerically calculate the topological phase diagram of Jz showed in

Fig. 4.2.

With different spin operator, the undefined region is not the same. The undefined region of one operator may be the topologically non-trivial region of another. It needs to know how to find the proper spin operator that indicate all topologically non-trivial region.

Figure 4.1: It shows the topological phase diagram with Iz. The x axis is δ/A and the y

axis is M/B. The red line is M B − B2_δ2_/A2 _{= 0. The black region is undefined. The}

blue region is topological trivial. The green region is topological non-trivial.

Figure 4.2: It shows the topological phase diagram with Jz. The x axis is δ/A and the

y axis is M/B. The red line is M B − B2δ2/A2 = 0. The black region is undefined. The blue region is topological trivial. The green region is topological non-trivial.

Edge-state branch in a quantum bar

In section 5.1, we derive the effective 1D Hamiltonian of the width W quantum bar (quantum channel). We find out that the effective 1D Hamiltonian has a symmetry that we call pseudo-parity. Because the operator y, the transverse direction, mixes the states with different pseudo-parity, it only simplifies the system in some cases. In chapter 6, we consider a system without y-dependent potential. Therefore, we use the pseudo-parity to simplifiy the calculations.

In section 5.2, we discuss the property of the edge states and the edge channels. Where the edge channels are the eigenstates of the 1D Hamiltonian with specific wave vector. The edge states are the linear combination of the edge channels. The DSOI terms mix spin so the edge channels isn’t a pure spin state. The edge channels are localized at the two edges so the magnetic impurity near any edges of the sample will effect all of edge channels.

On the other hand, by the edge injection, we can generate an edge state that is a pure spin state and localized at an edge of the sample Where the edge state is linear superposition of the edge channels that have the same energy and propagation direction. The location of the edge state is determined by the relative phase between the edge channels. The finite size effect[4] and DSOI make relative phase vary in x, the longitudinal direction. The location of the edge state will change when it is propagating. The spin polarization of the edge state changes with the location so the spin polarization will automatically precess.

5.1

Effective Hamiltonian

In this section, we derive the effective 1D Hamiltonian of the quantum bar. Then, we define the pseudo-parity operator and prove that it is a symmetry of the effective 1D Hamiltonian.

We consider a quantum bar showed in Fig. 5.1. The boundary is along x-direction. The wave function Φ is zero when y ≥ W /2 or y ≤ −W /2. Where W is the width of the quantum bar. We derive an effective 1D Hamiltonian HW to describe the band structrue

with this boundary condition. We seperate Hef f (equation (2.1)) into two parts, H0 and

H0. H0 contains all ky dependent terms and M . H0 contains the δ and all kx dependent

terms. We use H0 to derive the basis vectors of HW.

Figure 5.1: It shows the structure of a quantum bar system. We set the origin of y-axis at the middle of the bar. The wave function is zero when y ≤ −W/2 and y ≥ W/2.

In the set of basis (|E1, +i, |H1, +i, |E1, −i, |H1, −i) , H0 is of the form:

H0 =   Dk2 y− (M − Bky2)σz− Akyσy 0 0 Dk_y2− (M − Bk2 y)σz+ Akyσy  . (5.1) |E1, +i and |H1, +i subbands are decoupled to |E1, −i and |H1, −i subbands. |E1, ±i and |H1, ±i subbands are coupled by Aky terms and form |S; i, ±i and |A; i, ±i

sub-bands. The E1 component of |S; i, ±i subbands is even function. The E1 component of |A; i, ±i subbands is odd function.

For Hef f with d = 7nm, the value of M B and ∆ (∆ = M B − (Bδ/A)2) is positive.

There are four edge subbands in the system. The higher two subband is |S; i, ±i subbands and the others are |A; i, ±i subbands. In addition, the lowest conduction bulk subbands are |A; i, ±i subbands and the highest valence bulk subbands are |S; i, ±i subbands.

the highest valence bulk subbands are |A; i, ±i subbands.) The basis vectoers are of the form:

hS; i, +| = ( hS; i| 01×2 ), hS; i, −| = ( 01×2 hS; i| σz ),

hA; i, +| = ( hA; i| 01×2 ), hA; i, −| = ( 01×2 hA; i| σz ).

Where |S; ii is the pseudo-spin of |S; i, +i and |A; ii is the pseudo-spin of |A; i, +i. (The detail definition of basis is listed in Appendix F.)

We define the set of basis vector (|S; i, +i, |A; i, +i, |S; i, −i, |A; i, −i). In this set of basis vector, the effective Hamiltonian HW is defined as this form:

[H_W(kx)]_ij = hi| (H0+ H0) |ji = Eiδij + hi| H0|ji . (5.2)

Where |ii is the ith basis vector. The H0 is of the form: H0 = Dk_x2+   Bk2 xσz+ Akxσx δσx δσx Bkx2σz+ Akxσx  . (5.3)

The σz doesn’t couple E1 and H1 component so it does not couple |S; i, ±i and |A; i, ±i

subbands. The σxcouples E1 and H1 component so it only couples |S; i, ±i and |A; i, ±i

subbands. (The same kind of subbands aren’t coupled by the σx.)

The analytical form of HW is

HW (kx) =   E0+ B0k2 x+ A 0_k x δ0 −δ0 _E0 _{+ B}0_k2 x− A0kx  . (5.4) Where E0 contain the eigenenergy of all basis vectors.

E0 =X

i

E_iS|S; ii hS; i| + E_iA|A; ii hA; i|
. (5.5) Where ES

Figure 5.2: It shows the band structure of HW for W = 300nm with δ0 = 0.

subband. The other terms are of the form. B0 = D + BX

ij

(|S; ii hS; i| σz|S; ji hS; j| + |A; ii hA; i| σz|A; ji hA; j|). (5.6)

A0 = AX

ij

hS; i| σx|A; ji (|Sii hA; j| + |A; ji hS; i|). (5.7)

δ0 = −δX

ij

hS; i| iσy|A; ji (|Sii hA; j| − |A; ji hS; i|). (5.8)

The band structure of HW is showed in Fig. 5.2 and Fig. 5.3.

The effective Hamiltonian HW has a symmetry we call parity. The

pseudo-parity operator πp is of the form:

πp =   0 nz nz 0  . (5.9) Where nz =P i

(|S; ii hS; i| − |A; ii hA; i|). We have [E0, nz] = 0, [B0, nz] = 0, nzA0nz =

−A0 _{and n}

zδ0nz = −δ0. According to those equations above, we have [πp, HW] = 0.

The HW and πp have simultaneous eigenvectors so we can obtain a more simple form of

Figure 5.3: It shows the band structure of ˜HW for W = 300nm. The blue solid line is

µ = 1 subbands. The red dash line is µ = −1 subbands.

|S0

; i, µi = (|S; i, +i + µ |S; i, −i)√2 and |A0; i, µi = (|A; i, +i − µ |A; i, −i)√2. |S0_{; i, µi and |A}0_{; i, µi are eigenvectors of π}

p with the eigenvalue µ. In this

represen-tation, the effective Hamiltonian ˜HW is of the form:

˜ HW (kx) =   h0W₊ (kx) 0 0 h0W₋ (kx)  . (5.10)

Where h0W_µ (kx) = ˜E + ˜Bkx2 + ˜Akx + µ˜δ. Here µ also indicates the quantum number of

pseudo-parity. The other terms are of the forms: ˜ E =X i E_iS|S0; ii hS0; i| + E_iA|A0; ii hA0; i|
. (5.11) ˜ B =D + BX ij (|S0; ii hS; i| σz|S; ji hS0; j| + |A0; ii hA; i| σz|A; ji hA0; j|). (5.12) ˜ A =AX ij hS; i| σx|A; ji (|S0; ii hA0; j| + |A0; ji hS0; i|). (5.13) ˜ δ =δX ij

hS; i| iσy|A; ji (|S0; ii hA0; j| − |A0; ji hS0; i|). (5.14)

operater (See Appendix D), we derive the time reversed state of basis vector.

Θ |S0; i, µi = −µ |S0; i, −µi . (5.15a) Θ |A0; i, µi = µ |A0; i, −µi . (5.15b) The time reversal operator in this representantion is of the form:

Θ =   0 n˜z −˜nz 0  K. (5.16) Where ˜nz = P i

(|S0; ii hS0; i| − |A0; ii hA0; i|). The time reversed states of |S0; i, +i and |A0_{; i, +i are |S}0_{; i, −i and |A}0_{; i, −i. The eigenstates of h}0W

+ and h 0W

− are time reversal

pair. It seems that the eignstates of h0W₊ aren’t coupled with the eigenstates of h0W₋ by the normal impurity or potential. But the operator y can couple the basis vectors with different pseudo-parity symmetry. According to the symmetry of the basis vector, we can obtain the form of operator y.

y =   0 y0 y0 0  . (5.17) Where y0 = P ij

hS; i| y |A; ji (|S0_{; ii hA}0_{; j| + |A}0_{; ji hS}0_{; i|). In this set of basis vector,}

the πp is of the form :

πp =   1 0 0 −1  . (5.18)

The operator y doesn’t commute with πpso the potential in terms of odd power of operator

y can mix the states with different pseudo-parity.

5.2

Wave function for bulk like and edge like states

In this section, we discuss property of the edge channels and the edge states in the quantum bar. Where the edge channels are the eigenstates of the effective 1D Hamiltonian. The edge state is the superposition of the edge channels with the same energy and propagation direction.

Figure 5.4: It shows the edge states with certain kx and spin in absence of DSOI.

form that contains only one spin (Fig. 5.4). At the lower edge, y = −W/2, the spin up state is left going and the spin down state is right going. At the upper edge, y = W/2, the spin of the right and left going state is opposite to the edge states at the lower edge. The edge channels aren’t robust against the magnetic impurity. However the magnetic impurity near the upper edge hardly scatters the edge channels at the lower edge. The edge channel only scattered by the impurities at the same edge.

For HW with DSOI, the DSOI and finite size effect couple the two edge channels that

are the eigenstates of HW without DSOI and have the same propagation direction. Those

two edge channels are mixed together to form two edge channelss ψ_µpedge with the same p but different µ (Fig. 5.5 and Fig. 5.6). Where µ is the quantum number of pseudo-parity and p is the propagation direction (p = + is right going.). The edge channel has specific wave vector kedge

µp but it can’t be the form that contains only one spin component. And it

is localized at both two edges so the magnetic impurity near any edges scatters the edge channels. The DSOI terms makes the edge channel become weaker to magnetic impurity. If we inject charge current at an edge of the sample, we predict that we can generate an edge state Φedge

p which is at the edge. The Φedgep is the linear combination of the ψedgeµp

with same p and energy.

Φedge_p (x, y) = √1 2

h

ψ_+pedge(x, y) + ei∆00ψedge

−p (x, y)

i = √1

2e

ikedge_+p xh

f_+pedge(y) + ei∆0(x)f−pedge(y)

i

. (5.19)

Where fedge

µp (y) is the column vector part of ψedgeµp (x, y) and ∆

0_{(x) = ∆}0 0+



k_−pedge− k_+pedgex. The location of the edge state depends on the phase difference ∆0. We choose the guage

Figure 5.5: It shows the edge states with certain kx in the presence of DSOI.

that spin up part of f_µpedge is the same sign such that the edge state is at one of the sample edges if ∆0(x) = 0 or ∆0(x) = π.

In the system without DSOI, the k_µpedge is equal to each other. The ∆0 is constant so the location of the edge state doesn’t change. On the other hand, the DSOI and finite size effect make kedge_µp different. The ∆0 becomes a function of x and the edge state isn’t always localized at one edge. (Fig. 5.8 and Fig. 5.9) When the edge state propagates, the location of the edge state is from a edge of the quantum bar to the other. After propagating a length Lc we call edge-switching length, the edge state will becomes localized at the other

edge. (The value of Lc can be larger than 100um. At that case, the edge state performs

like the edge channel without DSOI.) The spin polarization of the edge state depends on the location and the propagation direction of the edge state. Therefore, the spin polarization of the edge state will automatically changes. (This property is also showed in ref[11].)

For the bulk like states, the DSOI makes the bulk states with certain kx isn’t a pure

spin state too (Fig. 5.7). We also can define a set of bulk state that is a pure spin state initially. The spin of those state also varies like edge state we showed in Fig. 5.9. In contrast, the density of the bulk states are localized at the middle of the ribbon.

Figure 5.6: It shows the column vector part of the two edge states of HW with W = 300nm.

The solid line is the µ = +1 state with DSOI and the dash line is the state without DSOI. Those two state are the upper edge bands and their k value is 0.01nm−1. The blue line is |E1, +i component. The red line is |H1, +i component. The black line is |E1, −i component. The green line is |H1, −i component.

Figure 5.7: It shows the column vector part of the two bulk states of HW with W = 300nm.

The solid line is the µ = +1 state with DSOI and the dash line is the state without DSOI. Those two state are the lowest conduction subbands and their k value is 0.01nm−1. The blue line is |E1, +i component. The red line is |H1, +i component. The black line is |E1, −i component. The green line is |H1, −i component.

Figure 5.8: It shows the density of edge states Φedge₊ with W = 300nm, ∆0₀ = −1 and E = 6.4meV . (Note: the dimension of y-axis is different to x-axis)

Figure 5.9: It shows the spin polarizationf edge states Φedge₊ with W = 300nm, ∆0₀ = −1 and E = 6.4meV . (Note: the dimension of y-axis is different to x-axis)

Quantum transport in a quantum

bar

In this chapter, we fully numerically and semi-analytically calculate the transport of the edge channels and edge states in a quantum bar consisting of a potential barrier. Where the edge channels are the eigenstates of the effective 1D Hamiltonian. The edge state is the superposition of the edge channels with the same energy and propagation direction.

First, we discuss the transmission structure through a square potential in the quantum bar without DSOI. The edge band gap is not zero in a quantum bar because of finite size effect.[4] We show that the edge channels are back scattered by a potential at the energy range near the energy gap. The pseudo-spin of the edge channel at that energy range becomes energy dependent so the reflection is not zero. Though the non-zero edge band gap makes the edge channels scattered by the potential, the edge channel still totally transmits at the energy range not near the energy gap.

On the other hand, we show the transmission dips at the energy range not near the edge band gap is due to the Fano resonance. The edge channel forms the continuum spectrum and the resonance bulk channel forms the discrete state (quasi-bound state). Because the life time of the quasi-bound state are very long (the quasi-bound state is very well), we need to change the representation channel to calculate the Fano factor and life time of quasi-bound state. For the system without DSOI, the Fano factors are unique and the edge channels are totally back scattered at the dip energy.