### 國立臺灣大學理學院應用物理研究所 博士論文

### Graduate Institute of Applied Physics College of Science

### National Taiwan University Doctoral Dissertation

### 彎曲結構中的非線性自旋軌道耦合效應 The Nonlinear Spin-Orbit Coupling Effects

### in Curved Structures

### 張鑑源

### Jian-Yuan Chang

### 指導教授：張慶瑞 教授 Advisor：Prof. Ching-Ray Chang

### 中華民國 102 年 6 月

### June 2013

## 謝辭

### 很感謝王景貴爺爺的養育與栽培，

### 也感謝張慶瑞院長的指導與協助。

## 摘要

對於 Rashba 自旋軌道耦合與 Dresselhaus 自旋軌道耦合在一個具有任意形狀的 曲面中，其精確的哈密頓函數被嚴謹地推導獲得。我們發現兩個正交的主曲率可 以控制電子的自旋傳輸，而且在曲面正交方向的局限位勢的漸近行為則是可以忽 略的。另外我們也發現高階的動量項在大曲率的曲面中發揮了重要的作用。曲面 中的線性自旋軌道耦合只誘導產生額外的虛位勢項，然而曲面中的非線性自旋軌 道耦合則會誘導產生額外的虛動能項、虛動量項以及虛位勢項。由於額外的曲率 誘導項以及關聯虛磁場的作用，曲面中的自旋傳輸是不相同於在平面中的。我們 也明確地推導獲得在柱面或球面中的自旋軌道耦合的哈密頓函數，而且在奈米圓 環中的自旋進動以及關聯本徵態也被詳細地分析研究。因此我們推論曲率會顯著 影響彎曲結構中的自旋軌道耦合與自旋傳輸。

## Abstract

The exact Hamiltonians for Rashba and Dresselhaus spin-orbit couplings on a curved surface with an arbitrary shape are rigorously derived. Two orthogonal principal curvatures dominate the electronic spin transport, and the asymptotic behavior of the normal confined potential on a curved surface is insignificant. For a curved surface with a large curvature, the higher order momentum terms play an important role in controlling spin transport. The linear spin-orbit coupling on a curved surface only induces the extra pseudo-potential term, and the cubic spin-orbit coupling on a curved surface can induce the extra pseudo-kinetic, pseudo-momentum, and pseudo-potential terms. Because of the extra curvature-induced terms and the associated pseudo-magnetic fields, spin transport on a curved surface is very different from that on a flat surface. The spin-orbit Hamiltonians on a cylindrical or spherical surface are explicitly derived here, and the spin precession and the associated eigenstates on a nanoring are analyzed in detail. We can conclude that the curvature has a significant influence on the spin-orbit coupling and spin transport in curved structures.

## Table of Contents

口試委員會審定書 i

謝辭 ii

摘要 iii

Abstract iv

Table of Contents v

List of Figures vi

Chapter 1 Introduction 1

Chapter 2 Hamiltonian of Spin-Orbit Coupling for Electron System on a Curved Surface 6 Chapter 3 Hamiltonian Formalism on the Nanotube and the Nanobubble 16 Chapter 4 Hamiltonian of Spin-Orbit Coupling for Hole System on a Curved Surface 30 Chapter 5 Hamiltonian Formalism on the Nanoring 36

Chapter 6 Summary and Discussion 50

Appendix A Asymptotic Behavior for the Confined Potential of an Ultrathin Film 54 Appendix B Definitions of Basic Mathematics on a Curved Surface 56 Appendix C Tensor Transformation of a Hamiltonian without SOC 58 Appendix D Tensor Transformation of Linear Rashba Spin-Orbit Coupling 60 Appendix E Tensor Transformation of Cubic Dresselhaus Spin-Orbit Coupling 62 Appendix F Tensor Transformation in a Cylindrical Nanotubular System 65 Appendix G Tensor Transformation in a Spherical Nanobubble System 68 Appendix H Tensor Transformation of Cubic Rashba Spin-Orbit Coupling 71 Appendix I Tensor Transformation in a Nanoring System 74

Bibliography 78

## List of Figures

Fig. 1. The parametric equation and unit normal vector of the curved surface S 7 Fig. 2. Electrons are confined to a delta function potential well 8 Fig. 3. Electronic band structures of the different toric rings 24 Fig. 4. Various spinors on the different toric rings 25

Fig. 5. Various spinors on the Rashba nanoring 47

Fig. 6. Various spinors on the Rashba nanoring 47

## Chapter 1

## INTRODUCTION

Most current electronic devices are based on charge transport, but the emerging Spin-

tronics technology is manipulating the intrinsic spin of electrons and its associated mag-

netic moment.^{1,}^{2} It is well-known that spin is a quantized vector and that the spin com-

ponents are described by the special unitary group of degree 2, denoted by SU(2). The

spin degrees of freedom, including magnitude and direction, provide a large possibility

for the design of next generation devices.^{1,}^{2} Thus, for example, highly dense recording

devices or quantum computing devices can be constructed if we take advantage of these

large spin degrees of freedom.^{3,}^{4} The appearance of spin current is the start point of Spin-

tronics. When the electron spin is aligned with a certain direction, the electron flow can

give rise to a spin-polarized current. The spin filter is another important device in Spin-

tronics.^{5,}^{6} Only the predetermined orientation of the spin-polarized current is allowed to

pass through the spin filter. The filtering of spin current can be applied to information

processing. In short, the appearance and filtering of spin current are crucial developments

in the field of Spintronics.

In recent years, the manipulation of spin transport is achieved using a real external

magnetic field or the effective magnetic field due to the spin-orbit interaction, and a large

amount of research has been devoted to the spin properties in solid-state systems.^{7–9} The

two interactions usually used to control spin are the external magnetic field and the spin-

orbit coupling (SOC).^{7–9}For controlling spin, the external magnetic field can be tuned by

a driven current while the SOC in solid-state systems is easily controlled by voltage gates.

The two kinds of SOCs usually taken into account in the low-dimensional transport, such

as in a two-dimensional electron gas (2DEG), are the Rashba spin-orbit interaction (RSI)

and the Dresselhaus spin-orbit interaction (DSI). Different curved structures can lead to

the different behavior of the SOC, and thus the physical effects of curved materials are

extensively researched.^{10–19} For instance, a fullerene is composed entirely of carbon, and

is similar in structure to stacked graphene sheets,^{20} in the form of a hollow cylinder or a

hollow sphere. The cylindrical fullerene is called a carbon nanotube or a buckytube, and

the spherical fullerene is called a buckyball.

Recent experimental and theoretical studies both show that the SOCs on a curved sur-

face, or the geometric influence of the band structure, can induce the significant pseudo-

10,12–14,18,19

In a previous work, the perturbation theory was applied to the curved materials.^{10} The

work reported the appearance of the band gap and of the spin-splitting due to the SOC, and

indicated the importance of the curvature to the SOC effects in graphenes, fullerenes, nan-

otubes, and nanotube caps. Recently measured asymmetric splittings of the valence and

conduction bands in a quantum dot also indicated that the geometric curvature enhances

the strength of the SOC.^{11} Moreover, the pseudo–magnetic fields were reported recently

due to the exceptional flexibility and strength of graphene membranes.^{12–14} An enor-

mous pseudo–magnetic field greater than 300 Tesla was observed in the Landau levels of

graphene nanobubbles,^{13} and a possible local microstress induced by a pseudo-magnetic

field was possibly also detected by the nanoscale Aharonov-Bohm interferences.^{14} A re-

cent theoretical study had also independently reported that the geometric effect and the

geometric phase can be induced by a curved surface,^{15} on which both Rashba and Dres-

selhaus interactions are expressed as the non-Abelian spin-orbit gauges.^{21,}^{22} An effective

magnetic field appears due to the SOC on a curved surface, and it is no less important

than the contribution of the SOC. Besides, a recent analytic study for the Hamiltonian of

a corrugated single-layer graphene combined with the atomic SOC of carbon indicated a

significant difference between the flat and corrugated graphene films.^{16} A similar analytic

study had also shown that the curvature can enhance the effective strength of the SOC in a

carbon nanotube.^{17} The effective Rashba Hamiltonian for a curved quantum wire with an

18

with a position-dependent curvature was also reported recently.^{19}

From the point of view of both fundamental and applied research, dynamics on curved

structures is of considerable significance to the study of solid-state systems. Recently, a

number of interesting experimental results on low-dimensional curved materials or corru-

gated graphene films promote a clear theoretical understanding of curved structures.^{10–19}

However, because of the complicated interplay between the nonlinear momentum and the

curved surface, only the linear momentum of the SOC had been studied in detail.^{10–19} In

other words, in most previous studies, a curved surface had been assumed to be a nearly

flat surface, and the cubic momentum of the SOC was seldom discussed. Nevertheless,

in a large curvature system, the full understanding of spin transport and of the associ-

ated physical properties can only come from a complete study including the higher order

momentum terms. Recent studies of carbon nanotubes with a very small radius and of

ripples on a corrugated graphene also indicated that the local curvature must not be over-

looked.^{10–12,}^{16–19} Thus, a more general analysis of a Hamiltonian with SOC on a curved

surface is necessary to understand the effect of the curvature on charge and spin transport.

The aim of my doctoral thesis is to understand the influence of the Rashba or Dres-

selhaus Hamiltonian with cubic momenta on a curved surface. Spin transport on curved

surfaces with different curvatures is also analyzed in detail. The thesis is organized as

follows: In Sec. IIwe consider the case of the nearly-free electron system and construct

describe the detailed procedures for obtaining the Hamiltonian with cubic Dresselhaus

SOC on a curved surface. In Sec. IIIwe explicitly derive the effective Hamiltonians of

the two geometric structures, namely the surfaces of a hollow cylinder and of a hollow

sphere. The band structures and the associated spinors on curved surfaces are analyzed,

and the influence of the nonlinear momenta on both cylindrical and spherical surfaces is

explicitly stated. The difference between the flat and curved surfaces can be understood

from the extra curvature-induced terms in the Hamiltonian. The special cases of the car-

bon nanotube and the graphene nanobubble are discussed. In Sec. IV we consider the

two-dimensional hole gas (2DHG) with cubic SOC and exactly derive the cubic Rashba

or Dresselhaus Hamiltonian on a curved surface. We also discuss the curvature-induced

geometric effect and the effective magnetic field. In Sec.Vwe explicitly calculate the ef-

fective Hamiltonians with Rashba or Dresselhaus SOC on a nanoring. The various spinors

on the nanorings are analyzed in the quantized energy levels, and the influence of the non-

linear momenta on a nanoring is also discussed. Finally, a summary and discussion are

given in Sec. VI.

## Chapter 2

## HAMILTONIAN OF SPIN-ORBIT

## COUPLING ON A CURVED

## SURFACE

For the motion of electrons confined to the small neighborhood of the curved surface

S shown in Fig. 1, the position of an electron in curvilinear coordinates q^{1}, q^{2}, q^{3} is

written as

*

R q^{1}, q^{2}, q^{3} =^{*}r q^{1}, q^{2} + q^{3}N qˆ ^{1}, q^{2} , (2.1)

where^{*}r q^{1}, q^{2} is the parametric equation of the curved surface S, and ˆN q^{1}, q^{2} is the

unit normal vector at q^{1}, q^{2}.

Fig. 1. The parametric equation and unit normal vector of the curved surface S.

The Hamiltonian with SOC in curvilinear coordinates is expressed as H = H_{KE}+

H_{SOC}+ V q^{3}, where H_{KE} is the kinetic term, H_{SOC} describes the SOC in solids, and

the surface potential which confines electrons to the small neighborhood of the curved

surface S is

V q^{3} =

0

V

f or
q^{3}

≤ ε, otherwise,

(2.2)

where ε is the thickness of the curved surface S. The surface potential is similar to a

delta function potential well shown in Fig. 2, and the wavefunction of the bound state is

introduced in AppendixA.

Using the curvilinear coordinates and the metric tensor introduced in AppendixBto

express the kinetic term, we obtain

Fig. 2. Electrons are confined to a delta function potential well.

H_{KE} = 1
2mp^{i}p_{i}

= − ¯h^{2}
2m

√1 g

∂

∂ q^{i}

√gg^{i j} ∂

∂ q^{j}

,

(2.3)

where the index values are 1, 2, and 3, g is the determinant of the matrix formed by the

metric components g^{i j}, m is the effective electron mass in solids, and the momentum com-

ponents are defined as p^{i}= −i¯hg^{i j ∂}

∂ q^{j}. We note that the use of the kinetic term signifies
a single-band effective mass approximation, in other words, the influence of the lattice

potential is incorporated into the effective electron mass m, and the momentum operator

p^{i}yields the lattice momentum.

For brevity and clarity, we first consider a Hamiltonian without SOC. Using the math-

ematical definitions given in AppendixB, we perform a similar derivation to that given by

Costa.^{23} By this means we can derive an effective eigen equation; its derivation is given

in AppendixC. From the effective eigen equation in Eq. (C2), the effective Hamiltonian

without SOC in curvilinear coordinates can be written as

H˜_{KP}= − ¯h^{2}
2m

√1

˜ g

∂

∂ q^{m}

pgg˜ ^{mn} ∂

∂ q^{n}

− ¯h^{2}
8m

h

Tr(α_{m}^{n})^{2}− 4Det (α_{m}^{n})i
,

(2.4)

where the indices m and n are 1 and 2, the reduced metric tensor ˜g= g q^{1}, q^{2}, 0, and α_{m}^{n}

are the Weingarten curvature matrix elements for a curved surface.^{24} Next to the ordinary

kinetic term, the second term in Eq. (4) is the geometric potential induced by a curved

surface. We note that it depends only on the surface curvature. In this paper we will use

the geometric potentials for all manner of the curvature-induced potentials. We also note

that the eigenvalues of the Weingarten curvature matrix depend on the selected coordinate

system, and they are usually the orthogonal principal curvatures κ1 and κ2 of the local

curved surface. Tr (α_{m}^{n}), proportional to the mean curvature, is defined to be the trace of

the Weingarten curvature matrix, and Det (α_{m}^{n}), called the Gaussian curvature, is defined

to be the determinant of the Weingarten curvature matrix.

The motion of an electron is characterized by the wave number k in solids. However,

¯hk is not the real momentum of an electron in solids, but a lattice momentum. The in-

teraction of electrons moving in solids will be modified according to the solid structures.

It is known that the dispersion is spin-degenerate if the Hamiltonian is invariant under

time reversal and space inversion.^{25–28} Thus the symmetry breaking can lead to a disper-

sion with the spin-splitting. In solids, the interfacial or crystal structure may break the

space inversion symmetry, and it is the origin of the SOC. The Rashba SOC is induced

by a structure inversion asymmetry (SIA) of the interfacial potential, and its spin-splitting

is linearly proportional to k.^{25,}^{26} The Dresselhaus SOC is induced by a bulk inversion

asymmetry (BIA) of the crystal potential, and its spin-splitting is proportional to k^{3}.^{27,}^{28}

For the curvilinear coordinates, the effective Hamiltonian with SOC can be derived by

the manipulation of the tensor transformations:

p^{i}= ∂ q^{i}

∂ ¯q^{t}p¯^{t}, σ^{i}= ∂ q^{i}

∂ ¯q^{t}σ¯^{t},
S_{i j}= ∂ ¯q^{k}

∂ q^{i}

∂ ¯q^{t}

∂ q^{j}

S¯_{kt}, and

S_{ii j j}= ∂ ¯q^{k}

∂ q^{i}

∂ ¯q^{k}

∂ q^{i}

∂ ¯q^{t}

∂ q^{j}

∂ ¯q^{t}

∂ q^{j}
S¯_{kktt},

(2.5)

where ¯qand q stand for the old and new coordinates, respectively, σ^{i} are the Pauli ma-

trices, the components of the Rashba tensor are defined as S_{i j}, the components of the

Dresselhaus tensor are defined as S_{ii j j}, and the index values are 1, 2, and 3.

Therefore, we can also apply the same eigenfunction transformation to the linear

Rashba SOC. The detailed derivation is shown in AppendixD. For the Rashba SOC with

linear momenta, the general form is

H_{LRS}=S_{i j}σ^{i}p^{j}

= − i¯hSi jσ^{i}g^{ju} ∂

u,

(2.6)

where i 6= j. From the effective eigen equation in Eq. (D2), the effective Hamiltonian of

the linear Rashba SOC in curvilinear coordinates can be written as

H˜_{LRS}= − i¯hSimσ^{i}g^{mu} ∂

∂ q^{u}
+1

2i¯hS_{i3}σ^{i}Tr(α_{m}^{n}) ,

(2.7)

where the indices m and n are 1 and 2, and other index values are 1, 2, and 3. There

is the k-dependence of the Rashba SOC in the first term. The second term in Eq. (7)

is the curvature-induced geometric potential from the linear Rashba SOC; its strength is

proportional to the mean curvature.

The behavior of spin electrons on a curved surface is similar to that on a flat sur-

face with an extra background potential. From Eqs. (4) and (7), we see that two ex-

tra curvature-dependent geometric potentials are induced from the kinetic term and the

Rashba SOC. One is a spin-independent geometric potential, and the other is a spin-

dependent geometric potential. In the case of a system with a constant curvature, the

spin-independent geometric potential only induces a constant energy shift, but the spin-

dependent geometric potential induces a large difference in the behavior of spin electrons,

particularly in a large curvature system. However, for a system with a spatial dependence

of the curvature, the two curvature-induced geometric potentials both lead to the compli-

cated spin precession in a U-shape channel.^{19}

Similarly, the eigenfunction transformation of the cubic Dresselhaus SOC can be de-

rived. The derivation is very tedious for its nonlinear dependence on the momentum, and

we list the detailed derivation shown in AppendixE for brevity and clarity of the main

text. For the Dresselhaus SOC with cubic momenta, the general form is

H_{CDS}=S_{ii j j}σ^{i}p^{i}p^{j}p^{j}

= (−i¯h)^{3}S_{ii j j}σ^{i}g^{iu} ∂

∂ q^{u}

g^{jv} ∂

∂ q^{v}

g^{jw} ∂

∂ q^{w}

,

(2.8)

where i 6= j. From the effective eigen equation in Eq. (E2), the effective Hamiltonian of

the cubic Dresselhaus SOC in curvilinear coordinates can be written as

H˜_{CDS}=i¯h^{3}S_{mmnn}σ^{m}g^{mu} ∂

∂ q^{u}

g^{nv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

−1

2i¯h^{3}S_{33nn}σ^{3}Tr(α_{m}^{n}) g^{nv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

+1

4i¯h^{3}S_{mm33}σ^{m}
h

3Tr (α_{m}^{n})^{2}− 4Det (α_{m}^{n})
i

g^{mu} ∂

∂ q^{u},

(2.9)

where the indices m and n are 1 and 2, and other index values are 1, 2, and 3. There is

the k^{3}-dependence of the Dresselhaus SOC in the first term. The second and third terms

in Eq. (9) are the curvature-induced geometric effects from the cubic Dresselhaus SOC.

It is worth mentioning that the momentum along the confined normal direction will

induce a geometric effect in the kinetic term as q^{3} becomes infinitesimal for ultrathin

films. Once the spin is coupled with the momentum in the confined normal direction,

the geometric effect on the SOC will be induced by the surface curvature. Therefore,

on a curved surface, the k^{2}-dependence of the kinetic term along the confined normal

direction will contribute a geometric potential with a quadratic dependence of the curva-

ture, and its strength will depend only on the surface curvature. For the linear Rashba

SOC on a curved surface, the k-linear term in the unconfined direction can lead to a k-

dependent spin-splitting. The k-linear term in the confined normal direction contributes

a geometric potential with a linear dependence of the curvature. Its influence is simi-

lar to an extra pseudo-potential term, and its strength depends on the surface curvature

and the Rashba coupling strength. However, for the cubic Dresselhaus SOC on a curved

surface, the effective Hamiltonian becomes very complicated because the momentum in

the confined normal direction is also coupled with the momentum in the unconfined di-

rection. Hence, on a curved surface, the geometric potential with a linear dependence

of the curvature will be coupled with the k^{2}-dependent term in the unconfined direction.

The k^{2}-dependent spin-splitting will appear, and it is similar to that induced by an extra

pseudo-kinetic term. Similarly, the geometric potential with a quadratic dependence of

the curvature will also be coupled with the k-dependent term in the unconfined direction.

The k-dependent spin-splitting will also arise, and it is similar to that induced by an ex-

tra pseudo-momentum term. The strengths of the pseudo-kinetic and pseudo-momentum

terms depend on the surface curvature and the Dresselhaus coupling strength. Obviously,

the k-cubic term in the unconfined directions can lead to a k^{3}-dependent spin-splitting.

Nevertheless, there will be no pseudo-potential term in this case because the momenta in

From Eqs. (4) and (7), we obtain a general form of the Hamiltonian which includes the

effect of the confinement of the kinetic term and of the linear Rashba SOC in curvilinear

coordinates:

H˜_{KLR}= − ¯h^{2}
2m

√1

˜ g

∂

∂ q^{m}

pgg˜ ^{mn} ∂

∂ q^{n}

− ¯h^{2}
8m

h

Tr(α_{m}^{n})^{2}− 4Det (α_{m}^{n})i

− i¯hSimσ^{i}g^{mu} ∂

∂ q^{u}
+1

2i¯hSi3σ^{i}Tr(α_{m}^{n}) ,

(2.10)

where the indices m and n are 1 and 2, and other index values are 1, 2, and 3.

From Eqs. (4) and (9), we also obtain a general form of the Hamiltonian which in-

cludes the effect of the confinement of the kinetic term and of the cubic Dresselhaus SOC

in curvilinear coordinates:

H˜_{KCD}= − ¯h^{2}
2m

√1

˜ g

∂

∂ q^{m}

pgg˜ ^{mn} ∂

∂ q^{n}

− ¯h^{2}
8m

h

Tr(α_{m}^{n})^{2}− 4Det (α_{m}^{n})i

+ i¯h^{3}S_{mmnn}σ^{m}g^{mu} ∂

∂ q^{u}

g^{nv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

−1

2i¯h^{3}S_{33nn}σ^{3}Tr(α_{m}^{n}) g^{nv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

+1

4i¯h^{3}S_{mm33}σ^{m}
h

3Tr (α_{m}^{n})^{2}− 4Det (α_{m}^{n})i
g^{mu} ∂

∂ q^{u},

(2.11)

where the indices m and n are 1 and 2, and other index values are 1, 2, and 3.

When electrons are confined in a rather small region, they exhibit the wave properties,

and the confinement yields large electron momenta. Due to the quantum confinement, the

effective Hamiltonian of a constrained particle is different from that of a classical particle.

Costa shows that the confined Hamiltonian should include a geometric potential due to

the contribution of the kinetic term in the confined normal direction.^{23} The geometric

potential induced by the contribution of the linear Rashba SOC in the confined normal

direction was also studied.^{18} From Eqs. (10) and (11), it is obvious that the extra spin-

dependent effect from the SOC on a curved surface is equivalent to an effective magnetic

field acting on a flat surface.^{15} Its amplitude and direction relate to the geometric struc-

ture and the strengths of the Rashba and Dresselhaus SOCs. For the linear Rashba SOC

on a curved surface, there exist two different origins of effective magnetic fields. The

first effective magnetic field originates from the k-dependence of the Rashba SOC in the

unconfined direction, and the second effective magnetic field originates from the pseudo-

potential term. However, for the cubic Dresselhaus SOC on a curved surface, there are

three separate origins of effective magnetic fields. The first effective magnetic field is due

to the k^{3}-dependence of the Dresselhaus SOC in the unconfined directions, the second

effective magnetic field is due to the pseudo-kinetic term, and the third effective magnetic

field is due to the pseudo-momentum term.

## Chapter 3

## HAMILTONIAN FORMALISM ON

## THE NANOTUBE AND THE

## NANOBUBBLE

Recently, a cylindrical nanotube with a controllable radius can be fabricated by a

method known as the self-rolling strained semiconductor layers.^{29} After this advance,

the study of the nanotube system has increasingly attracted researchers’ attention. For

instance, the experimental measurement of magnetotransport in a tubular 2DEG system

has been carried out.^{30,}^{31} Also, a spherical nanobubble with a controllable curvature can

be formed on a very elastic graphene film.^{32} The electronic properties of the graphene

nanobubble are strongly modified by the strain and the surface curvature.^{13} Nevertheless,

very few theoretical studies consider the curvature-induced terms in the Hamiltonian,

and thus the pseudo-potential, pseudo-kinetic, and pseudo-momentum terms derived in

Sec.IIwere hitherto neglected.^{33,}^{34} We give the exact derivations of 2D nanotubular and

nanobubble systems below, and we also discuss the associated curvature-induced physical

phenomena.

For a [111]-grown quantum well in Cartesian coordinates, the x, y, and z axes corre-

spond to [100], [010], and [001] crystallographic directions, respectively, and the Rashba

SOC^{25,}^{26} is expressed as

H_{LRS}=α

¯h (σ^{x}p^{y}− σ^{y}p^{x}) +α

¯h (σ^{y}p^{z}− σ^{z}p^{y})
+α

¯h (σ^{z}p^{x}− σ^{x}p^{z}) ,

(3.1)

where α is the Rashba coupling strength whose unit is meV·nm.

For a [100]-grown quantum well in Cartesian coordinates, the x, y, and z axes corre-

spond to [100], [010], and [001] crystallographic directions, respectively, and the Dres-

selhaus SOC^{27,}^{28} is expressed as

H_{CDS}=β

¯h^{3}σ^{x}p^{x}

p^{y 2}− p^{z 2}
+ β

¯h^{3}σ^{y}p^{y}

p^{z 2}− p^{x 2}

+ β

¯h^{3}σ^{z}p^{z}

p^{x 2}− p^{y 2}
,

(3.2)

where β is the Dresselhaus coupling strength whose unit is meV·nm^{3}.

In such conditions, the components of the Rashba tensor Si j in a Cartesian coordinate

system are

S_{xy}= S_{yz}= S_{zx}= α

¯h and
S_{zy}= S_{yx}= S_{xz}= −α

¯h,

(3.3)

and the components of the Dresselhaus tensor S_{ii j j} in a Cartesian coordinate system are

S_{xxyy}= S_{yyzz}= S_{zzxx}= β

¯h^{3} and
S_{zzyy}= S_{yyxx}= S_{xxzz}= −β

¯h^{3}.

(3.4)

For a 2D nanotubular system with SOC based on Eqs. (12) and (13), we will express

the Hamiltonian of the nanotubular system in cylindrical coordinates (ρ, φ , z). Using the

tensor transformations given in AppendixFand Eq. (10), we obtain a Hamiltonian with

linear Rashba SOC in a cylindrical coordinate system:

H˜_{KLR}= − ¯h^{2}
2m

1
ρ^{2}

∂^{2}

∂ φ^{2}+ ∂^{2}

∂ z^{2}

− ¯h^{2}
2m

1 2ρ

2

− iα [cos φ σ^{x}+ sin φ σ^{y}− (sin φ + cos φ ) σ^{z}]1
ρ

∂

∂ φ

− iα (−σ^{x}+ σ^{y}) ∂

∂ z+1

2iα [sin φ σ^{x}− cos φ σ^{y}
+ (cos φ − sin φ ) σ^{z}] 1

ρ

.

(3.5)

The first two terms in Eq. (16) are the kinetic terms and the geometric effect due to the

contribution of the kinetic term for cylindrical geometry. The third and fourth terms are

the Rashba SOC along the unconfined φ - and z-directions, respectively. It is interesting

to point out that the linear derivatives with respect to φ and z become anisotropic in

cylindrical symmetry. The effective strength of the Rashba SOC along z-direction remains

as it is on a flat surface; the effective strength of the Rashba SOC along φ -direction

increases inversely with the radius. As for the remaining term, the linear momentum in the

confined ρ-direction contributes a geometric potential with an inverse linear dependence

on the radius (1/ρ).

In order to compare our results with the previously studied SOC in a half-ring chan-

nel,^{19}we let z → 0, and Eq. (16) reduces to a Hamiltonian with Rashba SOC in a 1D toric

ring of the nanotubular system. If we assume that the momentum along z-direction tends

to zero in the nanotubular system, the simplified Hamiltonian must remain Hermitian.

Hence, its form must be

H˜_{KLR}(z → 0) = − ¯h^{2}
2m

1
ρ^{2}

∂^{2}

∂ φ^{2}− ¯h^{2}
2m

1 2ρ

2

− iα [cos φ σ^{x}+ sin φ σ^{y}

− (sin φ + cos φ ) σ^{z}]1
ρ

∂

∂ φ +1

2iα [sin φ σ^{x}

− cos φ σ^{y}+ (cos φ − sin φ ) σ^{z}] 1
ρ

.

(3.6)

Similarly, from the tensor transformations given in AppendixFand Eq. (11), a Hamil-

tonian with cubic Dresselhaus SOC in a cylindrical coordinate system will be

H˜_{KCD}= − ¯h^{2}
2m

1
ρ^{2}

∂^{2}

∂ φ^{2}+ ∂^{2}

∂ z^{2}

− ¯h^{2}
2m

1 2ρ

2

+ iβ cos 2φ

(− sin φ σ^{x}+ cos φ σ^{y})1
ρ

∂

∂ φ

∂^{2}

∂ z^{2}

−σ^{z} ∂

∂ z

1
ρ^{2}

∂^{2}

∂ φ^{2}

−1

2iβ cos 2φ (cos φ σ^{x}
+ sin φ σ^{y}) 1

ρ

1
ρ^{2}

∂^{2}

∂ φ^{2}− ∂^{2}

∂ z^{2}

+1

4iβ cos 2φ

× 3
ρ^{2}

(sin φ σ^{x}− cos φ σ^{y})1
ρ

∂

∂ φ + σ^{z} ∂

∂ z

.

(3.7)

The first two terms in Eq. (18) are the kinetic terms and the geometric effect due to the

contribution of the kinetic term for cylindrical geometry. The cubic momenta in the third

term are related to the Dresselhaus SOC in both unconfined φ - and z-directions. The linear

momentum in the confined ρ-direction contributes a geometric potential with an inverse

linear dependence on the radius (1/ρ), and the geometric potential will be coupled with

the square momentum in the unconfined φ - or z-direction in the fourth term. Also, the

square momentum in the confined ρ-direction contributes a geometric potential with an

inverse quadratic dependence on the radius (3/ρ^{2}), and the geometric potential will be

coupled with the linear momentum in the unconfined φ - or z-direction in the last term.

For a 2D nanobubble system with SOC based on Eqs. (12) and (13), we will express

the Hamiltonian of the nanobubble system in spherical coordinates (ρ, θ , φ ). Similar to

the previous case, using the tensor transformations given in AppendixGand Eq. (10), we

obtain a Hamiltonian with linear Rashba SOC in a spherical coordinate system:

H˜_{KLR} = − ¯h^{2}
2m

1

ρ^{2}sin θ

∂

∂ θ

sin θ ∂

∂ θ

+ 1

ρ^{2}sin^{2}θ

∂^{2}

∂ φ^{2}

− iα [(sin θ + cos θ sin φ ) σ^{x}− (sin θ + cos θ cos φ ) σ^{y}

+ cos θ (cos φ − sin φ ) σ^{z}]1
ρ

∂

∂ θ − iα [sin θ cos φ σ^{x}
+ sin θ sin φ σ^{y}− sin θ (sin φ + cos φ ) σ^{z}] 1

ρ sin^{2}θ

∂

∂ φ +1

2iα [(sin θ sin φ − cos θ ) σ^{x}+ (cos θ − sin θ cos φ ) σ^{y}
+ sin θ (cos φ − sin φ ) σ^{z}] 2

ρ

.

(3.8)

The first term in Eq. (19) only includes the kinetic terms, and there is no geometric

effect in the kinetic term for spherical geometry. The second and third terms are the

Rashba SOC along the unconfined θ - and φ -directions, respectively. We also point out

that the linear derivatives with respect to θ and φ are anisotropic in spherical symmetry.

The effective strength of the Rashba SOC along θ - or φ -direction increases inversely with

the radius; the effective strength of the Rashba SOC along φ -direction is also a function of

θ . As for the remaining term, the linear momentum in the confined ρ -direction contributes a geometric potential with an inverse linear dependence on the radius (2/ρ).

For a comparison of our results and the SOC in a half-ring channel,^{19} we let θ →

π /2, and Eq. (19) reduces to a Hamiltonian with Rashba SOC in a 1D toric ring of the

nanobubble system. Assume further that the momentum along θ -direction tends to zero in

the nanobubble system, and thus the influence of the principal curvature along θ -direction

becomes insignificant. Hence, in order to maintain the Hermitian property, the simplified

Hamiltonian must be

H˜_{KLR}
θ → π

2

= − ¯h^{2}
2m

1
ρ^{2}

∂^{2}

∂ φ^{2}− iα [cos φ σ^{x}+ sin φ σ^{y}

− (sin φ + cos φ ) σ^{z}]1
ρ

∂

∂ φ +1

2iα [sin φ σ^{x}

− cos φ σ^{y}+ (cos φ − sin φ ) σ^{z}] 1
ρ

.

(3.9)

Then, from the tensor transformations given in AppendixGand Eq. (11), a Hamilto-

nian with cubic Dresselhaus SOC in a spherical coordinate system will be

H˜_{KCD}= − ¯h^{2}
2m

1

ρ^{2}sin θ

∂

∂ θ

sin θ ∂

∂ θ

+ 1

ρ^{2}sin^{2}θ

∂^{2}

∂ φ^{2}

+ iβ cos 2θ cos 2φ

(sin φ σ^{x}− cos φ σ^{y}) 1
ρ sin θ

∂

∂ φ

1
ρ^{2}

∂^{2}

∂ θ^{2}

+ (cos θ cos φ σ^{x}+ cos θ sin φ σ^{y}− sin θ σ^{z})1
ρ

∂

∂ θ

1

ρ^{2}sin^{2}θ

∂^{2}

∂ φ^{2}

+1

2iβ cos 2θ cos 2φ (sin θ cos φ σ^{x}+ sin θ sin φ σ^{y}
+ cos θ σ^{z}) 2

ρ

1

ρ^{2}sin^{2}θ

∂^{2}

∂ φ^{2}− 1
ρ^{2}

∂^{2}

∂ θ^{2}

+1

4iβ cos 2θ cos 2φ 8
ρ^{2}

(− sin φ σ^{x}+ cos φ σ^{y}) 1
ρ sin θ

∂

∂ φ

− (cos θ cos φ σ^{x}+ cos θ sin φ σ^{y}− sin θ σ^{z})1
ρ

∂

∂ θ

.

(3.10)

The first term in Eq. (21) only includes the kinetic terms, and the geometric effect

in the kinetic term does not arise here for spherical geometry. The cubic momenta in the

second term are related to the Dresselhaus SOC in both unconfined θ - and φ -directions.

The linear momentum in the confined ρ-direction contributes a geometric potential with

an inverse linear dependence on the radius (2/ρ), and the geometric potential will be

coupled with the square momentum in the unconfined θ - or φ -direction in the third term.

Also, the square momentum in the confined ρ-direction contributes a geometric potential

with an inverse quadratic dependence on the radius (8/ρ^{2}), and the geometric potential

will be coupled with the linear momentum in the unconfined θ - or φ -direction in the last

term.

We use the nearly-free electron approximation to numerically calculate the electronic

band structures and their associated spinors. In these calculations we take the Rashba cou-

pling strength to be α = 60 meV·nm, the Dresselhaus coupling strength β = 60 meV·nm,

the radius of the curvature ρ = 9 nm, the circumference of the toric rings a = 2πρ, and

the effective electron mass m = 0.04me, where meis the electron rest mass.

The solid line in Fig. 3(a) shows the band structure for the kinetic energy and its geo-

metric potential, and for comparison the dashed line in Fig. 3(a) only refers to the kinetic

energy. For a curved surface with a constant curvature, the geometric potential causes a

constant shift in the kinetic energy. The band structures of the three different toric rings

with SOC are shown in Figs. 3(b), 3(c) and 3(d). Fig. 3(b) shows the band structure given

19

Fig. 3. Electronic band structures of the different toric rings for (a) Kinetic energy and
geometric potential, (b) Rashba or Dresselhaus SOC on a flat surface, (c) Rashba SOC on
a cylindrical surface, (d) Rashba SOC on a spherical surface, (e) Rashba SOC with band
mixing on a cylindrical surface, and (f) Rashba SOC with band mixing on a spherical
surface. The model parameters are α = 60 meV·nm, β = 60 meV·nm, ρ = 9 nm, a = 2πρ,
and m = 0.04m_{e}.

shows that two more states appear from the symmetry breaking of the up- and down-spin

under the SOC. It is interesting to mention that the two dispersions of the Rashba and

Dresselhaus spin-splittings are the same if their coupling strengths are equal. However,

the associated spinors are dramatically different, and they are shown in Figs. 4(a) and

4(b). With the equal coupling strengths, both Rashba and Dresselhaus SOCs provide an

equal k-dependent spin-splitting while the spin precessional phases are different.^{21,}^{22}

When we compare the band structure of the half-ring channel on a flat surface (Fig.

3(b)) with the band structure of the toric ring on a cylindrical surface (Fig. 3(c) from

Fig. 4. The various spinors on the different toric rings for (a) Rashba SOC on a flat surface, (b) Dresselhaus SOC on a flat surface, (c) Rashba SOC on a cylindrical surface, and (d) Rashba SOC on a spherical surface. The model parameters are α = 60 meV·nm, β = 60 meV·nm, ρ = 9 nm, a = 2π ρ , and m = 0.04me.

Eq. (17)), it seems that the band structure for the toric ring is shifted, and a larger k-

dependent spin-splitting is observed. This can be understood from the modification of the

geometric potential and of the k-dependent Rashba SOC. For a comparison between Eqs.

(17) and (20), the difference of the two Hamiltonians on the different toric rings is only

a constant potential change. Therefore, the band structure of the toric ring on a spherical

surface (Fig. 3(d) from Eq. (20)) is just shifted by the geometric potential. The various

spinors on the cylindrical and spherical surfaces are shown in Figs. 4(c) and 4(d) from

Eqs. (17) and (20), respectively, and their difference is also quite significant. Hence, it

is still necessary to take into account the out-of-plane spin component on a cylindrical or

spherical surface. The nonzero principal curvature on a curved surface is significant for

the motion of electrons, and thus spin precession on a curved surface is different from that

on a flat surface.

Furthermore, we consider the Hamiltonian of a corrugated single-layer graphene com-

bined with the atomic SOC of carbon,^{16} the effective SOC for π band states is written as

H_{SOC}=

ζ^{0}τz

*

µ ·

*s e^{iτ}^{z}^{ω}ζ ν (τzs_{y}+ is_{x})

e^{−iτ}^{z}^{ω}ζ ν (τzs_{y}− is_{x}) ζ^{0}τz

*

µ ·

*s

, (3.11)

where^{*}s= (s_{x}, s_{y}, s_{z}) andσ = (σ^{*} _{x}, σ_{y}, σ_{z}) denote the real-spin Pauli vector and the pseudo-

spin Pauli vector, respectively, ζ and ζ^{0}are the material parameters, ν = ν

_{*}
r

andµ =^{*}

*

µ

_{*}
r

are the geometric parameters, and τ_{z} = 1 or −1 denotes the Dirac point. Here,

the angle ω is defined counterclockwise from the y-axis to one carbon-carbon bond. For

ω = 0, the Hamiltonian in Eq. (22) can be separated into two terms, and then we obtain

H_{SOC}= ζ ν [σ_{x}⊗ τ_{z}s_{y}− σ_{y}⊗ s_{x}] + ζ^{0}τ_{z}I⊗^{*}µ ·

*s, (3.12)

where I is a unit matrix, and the symbol ⊗ denotes the direct product. The first term in

Eq. (23) is the off-diagonal part of the Hamiltonian, and such form is the Rashba-type

SOC; it corresponds to the first term of the linear Rashba SOC on a curved surface in Eq.

(7). The second term in Eq. (23) is the diagonal part of the Hamiltonian, and it is the

geometric potential induced by the surface curvature; it corresponds to the second term

of the linear Rashba SOC on a curved surface in Eq. (7). The forms of the two Hamilto-

nians in Eqs. (7) and (23) are the same, but with a modification of the Rashba coupling

strength for different materials. Hence, in order to construct the linear Rashba-type SOCs

of most curved materials, we need to modify the Rashba coupling strength α by inserting

the material parameters ξ and ξ^{0} for different materials,^{10,}^{16} i.e., αξ and αξ^{0}. In the in-

stance of a corrugated single-layer graphene with π band states, the material Rashba-type

coupling strength is αξ = aα (ε_{p}− εs) V_{pp}^{π} +V_{pp}^{σ} /12V_{sp}^{σ}^{2}≈ 0.15 meV·nm,^{10,}^{16}and the

curvature-induced Rashba-type coupling strength is αξ^{0}= aαV_{pp}^{π} /2 V_{pp}^{π} −V_{pp}^{σ} ≈ 0.21

meV·nm,^{10,}^{16} where a is the lattice constant, α is regarded as the atomic SOC strength of

p orbitals, ε_{s} and ε_{p}are the atomic energies for s and p orbitals, respectively, and V_{sp}^{σ}, V_{pp}^{σ}

and V_{pp}^{π} represent the coupling strengths for σ and π bands between nearest-neighbor s

and p orbitals or between nearest-neighbor p orbitals. Similarly, for constructing the cubic

Dresselhaus-type SOCs of most curved materials, we need to modify the Dresselhaus cou-

pling strength β by inserting the material parameters ξ and ξ^{0}for different materials, i.e.,

β ξ and β ξ^{0}. Nevertheless, for a curved surface, the linear Rashba SOC only induces the
extra pseudo-potential term, but the cubic Dresselhaus SOC can induce the extra pseudo-

kinetic and pseudo-momentum terms. Therefore, for the cubic Dresselhaus-type SOC,

the second term in Eq. (23) must be modified according to our Eq. (9).

For spin transport in the ballistic regime, we use the nearly-free electron approxi-

plane-wave bases with spinors. Here only the band mixings from an external magnetic

field are discussed, and the other possible band mixing between s and p orbitals is not

analyzed. However, from the second-order k · p perturbation theory, we can obtain the

effective mass of the n-th band, and also obtain the modification of the Rashba coupling

strength, i.e., the material parameters ξ and ξ^{0}. Hence, the band mixing between multi-

bands can be introduced by the material parameters ξ and ξ^{0}. For a corrugated single-

layer graphene with π band states, the influence of the band mixing can be studied using

the second-order k · p perturbation theory with the atomic SOC and the local curvature

as two weak perturbations.^{10,}^{16} The curvature effect breaks the isotropy of the lattice,

and leads to an effective anisotropic coupling between σ and π bands in the momentum

space.^{10} Moreover, it was reported that an external magnetic field also plays an important

role in controlling the band mixing between σ and π bands.^{10} Here we apply an external

magnetic field to a cylindrical or spherical surface, and the resultant spin-splitting with

band mixing due to the external magnetic field appears as shown in Figs. 3(e) and 3(f).

In the absence of an external magnetic field, the spinors will be position-dependent, and

no band gap will arise, as is apparent in Figs. 3(b), 3(c) and 3(d). However, when an ex-

ternal magnetic field is uniformly applied to the entire cylindrical and spherical surfaces,

and the direction of the uniform field is not parallel to the spin directions, the external

magnetic field asymmetrically shifts the energies of different spinors with opposite mo-

and 3(f). Hence, the band-crossing points are obviously broken, and local band gaps are

also observed.

## Chapter 4

## HAMILTONIAN OF SPIN-ORBIT

## COUPLING FOR HOLE SYSTEM ON

## A CURVED SURFACE

The cubic Rashba SOC is due to a structure inversion asymmetry (SIA) of the interfa-

cial potential, and the cubic Dresselhaus SOC is due to a bulk inversion asymmetry (BIA)

of the crystal potential. The spin-splitting of the heavy-hole states is mediated by a cou-

pling to the light-hole states, so that the spin-splitting is proportional to k^{3}.^{35–37} For heavy

holes in the quantum well grown along the [001]-direction, the x, y, and z axes correspond

to [100], [010], and [001] crystallographic directions, respectively, and we have the total

angular momentum with spin projection ±3/2 along the growth direction for heavy holes.

The cubic Rashba SOC in a Cartesian coordinate system is represented as

H_{CRS}= i α

2¯h^{3} σ+p^{3}_{−}− σ−p^{3}_{+} , (4.1)

and the cubic Dresselhaus SOC in a Cartesian coordinate system is represented as

H_{CDS}= β

2¯h^{3}(σ+p_{−}p_{+}p_{−}+ σ−p_{+}p_{−}p_{+}) , (4.2)

where the notations p_{±} = p_{x}± ip_{y} and σ_{±} = σx± iσy denote the ladder operators for

the momentum components and the Pauli matrices, respectively, ¯h is the reduced Planck

constant, α is the cubic Rashba coupling strength whose unit is meV·nm^{3}, and β is the

cubic Dresselhaus coupling strength whose unit is meV·nm^{3}.

To study the cubic Rashba or Dresselhaus SOC in a curved two-dimensional hole

system, we need to analyze the curved system in curvilinear coordinates q^{1}, q^{2}, q^{3}. In

general, the position in curvilinear coordinates is

*

R q^{1}, q^{2}, q^{3} =^{*}r q^{1}, q^{2} +q^{3}N qˆ ^{1}, q^{2},

where ^{*}r q^{1}, q^{2} and ˆN q^{1}, q^{2} are the parametric equation and the unit normal vector

of the curved system. The confined potential of the curved system is similar to a delta

function potential well, and its form is expressed as Eq. (2).

For the Rashba or Dresselhaus SOC with cubic momenta, the general form in curvi-

linear coordinates is written as

H_{CS}=S_{chi j}σ^{c}p^{h}p^{i}p^{j}

= (−i¯h)^{3}S_{chi j}σ^{c}g^{hu} ∂

∂ q^{u}

g^{iv} ∂

∂ q^{v}

g^{jw} ∂

∂ q^{w}

,

(4.3)

where S_{chi j}are the components of the cubic Rashba or Dresselhaus tensor, σ^{c}are the Pauli

matrices, and the index values are 1, 2, and 3. We note that the metric components are

defined as g_{i j} = ^{∂}

*

R

∂ q^{i}·^{∂}

*

R

∂ q^{j}, and the momentum components are defined as p^{i}= −i¯hg^{i j ∂}

∂ q^{j}.
For obtaining the cubic Rashba or Dresselhaus Hamiltonian in curvilinear coordinates,

we can use the tensor transformation rules:

p^{i}= ∂ q^{i}

∂ ¯q^{t}p¯^{t}, σ^{c}=∂ q^{c}

∂ ¯q^{t}σ¯^{t}, and
S_{chi j} =∂ ¯q^{k}

∂ q^{c}

∂ ¯q^{r}

∂ q^{h}

∂ ¯q^{s}

∂ q^{i}

∂ ¯q^{t}

∂ q^{j}
S¯_{krst},

(4.4)

where ¯qand q represent the old and new coordinates, respectively, and the index values

are 1, 2, and 3.

We apply the eigenfunction transformation to the cubic Rashba or Dresselhaus SOC.

The derivation is quite nontrivial for its nonlinear dependence on the momentum, and it is

shown in AppendixHfor brevity and clarity of the main text. Hence, from the effective

eigen equation in Eq. (H2), the effective Hamiltonian of the cubic Rashba or Dresselhaus

SOC in curvilinear coordinates can be written as

H˜_{CS}=i¯h^{3}S_{clmn}σ^{c}g^{lu} ∂

∂ q^{u}

g^{mv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

−3

2i¯h^{3}S_{clm3}σ^{c}Tr(α_{m}^{n}) g^{lv} ∂

∂ q^{v}

g^{mw} ∂

∂ q^{w}

+3

4i¯h^{3}S_{cl33}σ^{c}
h

3Tr (α_{m}^{n})^{2}− 4Det (α_{m}^{n})i
g^{lu} ∂

∂ q^{u}
+3

8i¯h^{3}S_{c333}σ^{c}
h

12Tr (α_{m}^{n}) Det (α_{m}^{n}) − 5Tr (α_{m}^{n})^{3}i
,

(4.5)

where the indices l, m and n are 1 and 2, and other index values are 1, 2, and 3. We

note that α_{m}^{n} are the Weingarten curvature matrix elements of the curved surface,^{24} and

the associated eigenvalues are usually the orthogonal principal curvatures. Tr (α_{m}^{n}) is the

trace of the Weingarten curvature matrix, and Det (α_{m}^{n}) is the determinant of the Wein-

garten curvature matrix. The first term in Eq. (28) is the k^{3}-dependence of the Rashba or

Dresselhaus SOC, and others are the curvature-induced geometric effects from the cubic

Rashba or Dresselhaus SOC. Those curvature-induced terms correspond to the pseudo-

kinetic, pseudo-momentum, and pseudo-potential terms on a curved surface.

From Eqs. (4) and (28), we obtain a general form of the Hamiltonian which includes

the effect of the confinement of the kinetic term and of the cubic Rashba or Dresselhaus

SOC in curvilinear coordinates:

H˜_{KCS}= − ¯h^{2}
2m

√1

˜ g

∂

∂ q^{m}

pgg˜ ^{mn} ∂

∂ q^{n}

− ¯h^{2}
8m

h

Tr(α_{m}^{n})^{2}− 4Det (α_{m}^{n})
i

+ i¯h^{3}S_{clmn}σ^{c}g^{lu} ∂

∂ q^{u}

g^{mv} ∂

∂ q^{v}

g^{nw} ∂

∂ q^{w}

−3

2i¯h^{3}S_{clm3}σ^{c}Tr(α_{m}^{n}) g^{lv} ∂

∂ q^{v}

g^{mw} ∂

∂ q^{w}

+3

4i¯h^{3}S_{cl33}σ^{c}
h

3Tr (α_{m}^{n})^{2}− 4Det (α_{m}^{n})i
g^{lu} ∂

∂ q^{u}
+3

8i¯h^{3}S_{c333}σ^{c}
h

12Tr (α_{m}^{n}) Det (α_{m}^{n}) − 5Tr (α_{m}^{n})^{3}i
,

(4.6)

where the indices l, m and n are 1 and 2, other index values are 1, 2, and 3, ˜g= g q^{1}, q^{2}, 0

is the reduced metric tensor, and m is the effective heavy-hole mass in solids. The first

term is the ordinary kinetic term and its geometric potential induced by the surface cur-

vature. The other terms in Eq. (29) are just from the cubic Rashba or Dresselhaus Hamil-

tonian on a curved surface.

We emphasize that the momentum along the confined normal direction of the ultra-

thin film will induce a geometric effect in the Hamiltonian. Once the momentum in the

confined normal direction is coupled with the spin, the geometric effect on the SOC will

appear due to the surface curvature. For the cubic Rashba or Dresselhaus SOC on a curved

surface, the effective Hamiltonian becomes very complicated because the momentum in

the confined normal direction is also coupled with the momentum in the unconfined di-

rection. Hence, on a curved surface, the geometric potential with a linear dependence

of the curvature will be coupled with the k^{2}-dependent term in the unconfined direction.

The k^{2}-dependent spin-splitting will appear, and it is similar to that induced by an ex-

tra pseudo-kinetic term. Similarly, the geometric potential with a quadratic dependence

of the curvature will also be coupled with the k-dependent term in the unconfined di-

rection. The k-dependent spin-splitting will also arise, and it is similar to that induced

by an extra pseudo-momentum term. The strengths of the pseudo-kinetic and pseudo-

momentum terms depend on the surface curvature and the cubic Rashba or Dresselhaus

coupling strength. Obviously, the k-cubic term in the unconfined directions can lead to a

k^{3}-dependent spin-splitting. The k-cubic term in the confined normal direction contributes

a geometric potential with a cubic dependence of the curvature. Its influence is similar to

an extra pseudo-potential term, and its strength depends on the surface curvature and the

cubic Rashba or Dresselhaus coupling strength. Therefore, there are four different origins

of effective magnetic fields. The first effective magnetic field is due to the k^{3}-dependence

of the Rashba or Dresselhaus SOC in the unconfined directions, and the other effective

magnetic fields originate, respectively, from the extra pseudo-kinetic, pseudo-momentum,

and pseudo-potential terms.

## Chapter 5

## HAMILTONIAN FORMALISM ON

## THE NANORING

The persistent spin current (PSC) was proposed by many researchers in an equilibrium

system.^{38–40} Rashba first found out a background PSC in an infinite system with Rashba

SOC, and he pointed out that the definition of the conventional spin current need to be

redefined. The PSC is protected by the time reversal symmetry, and it can not be detected

by the method of the conventional charge current or of the spin accumulation. Some recent

results indicated that the PSC on a nanoring with SOC can be detected.^{39,}^{40} Here we

analytically study the curvature-induced quantized energy levels on a nanoring and their

associated spin precessions. The quantized energy levels of the PSC and the transitions

between different energy levels provide a feasible way to detect the existence of the PSC.