準粒子凝聚態在二維系統與自旋有序之應用

國立臺灣大學理學院物理學研究所 博士論文

Graduate Institute of Physics College of Science National Taiwan University

Doctoral Dissertation

準粒子凝聚態在二維系統與自旋有序之應用

Quasi-particle Condensation in Two-Dimensional System and Its Application in Spin Ordering

研究生：陳志宇 Chih-Yu Chen

指導教授：胡崇德 博士 Supervisor: C.D. Hu, Ph.D.

中華民國 108 年 7 月

July, 2019

誌謝

首先感謝我的指導教授胡崇德老師，給我充實的博士生涯，以及口試委員張 慶瑞老師，林育中老師，程思誠教授，陳繩義教授的指教。

再來感謝我的家人，父母，妻子怡菱，女兒玉紓，謝謝你們容忍一個30 幾歲

還不賺大錢的任性兒子老公爸爸，大力相挺讓我無後顧之憂完成學位。還有各位 關心我的親戚們。

三感謝我的各位好朋友們，長洪武術的師母，小圈圈練功團的師兄弟，聆聽

唱片的好朋友們與前輩，3，感謝你們讓生活充滿豐富內涵，當喜怒哀樂的出口。

最後感謝我的三位導師陳清河，林主惟與黃義忠先生，你們影響是一生的。

中文摘要

準粒子凝聚態在凝態物理的應用廣泛，在玻色-愛因斯坦凝聚，超導體與超流

體中扮演重要角色。本論文主要分為兩部分，第二至第六章討論二維半導體中激

子（exciton）的凝聚態，研究顯示一種新的混合態波函數為二維半導體激子凝聚

態的基態，並提供可能的實驗量測方式。

第七至第十章研究磁振子（magnon）的凝聚態，組織現有的 Schwinger-boson

平均場理論，應用於氧化銅材料，以及討論動量非零之玻色-愛因斯坦凝聚態之物

理意義與氧化銅中commensurate-incommensurate 相變生成之可能之微觀機制。

關鍵字：激子、二維半導體、玻色-愛因斯坦凝聚、氧化銅、磁振子、自旋、孤立

子

ABSTRACT

In this thesis, we study the aspects of quasiparticle condensate phenomena. The Bose-Einstein condensation of quasiparticle plays an important role in many areas such as the superconductivity, superfluidity, magnons, polaritons, and of course, one of the main topic of this thesis-exciton. The exciton condensation of two-dimensional (2D) semiconductors is reports in Ch. 2-6. We start from an effective Hamiltonian of 2D semiconductors and show an interesting mixed state of exciton condensate.

The bosonization of electrons can also be a useful mathematical tool to study quantum spin systems. In Ch. 7-10, we extend the Schwinger boson mean field theory (SBMFT) method of ferromagnetic and antiferromagnetic systems. The condensation of Schwinger bosons can describe the ordering phase of spins. We study the commensurate-incommensurate phase transition of CuO as an example.

Key words: exciton, 2D semiconductor, Bose-Einstein condensation, CuO, magnon, commensurate-incommensurate phase transition, soliton

CONTENTS

口試委員會審定書... #

誌謝... i

中文摘要... ii

ABSTRACT ... iii

CONTENTS ... iv

LIST OF FIGURES ... vi

LIST OF TABLES ... viii

CONTENTS 1 Exordium 5

2 Introduction of Exciton Condensate 9

3 Effective Hamiltonian of 2D Semiconductors 11

3.1 The effective Hamiltonian without external field... 11

3.2 The effective Hamiltonian with external field…... 13

4 Coulomb Interaction Revisit 15

4.1 Spin selection rule and 2D Coulomb potential ... 15

4.2 The form factors of cases with and without external field ………... 17

5 The Gap Equation and Its Solution 21

5.1 Gap equation ... 21

5.2.1 Numerical solutions... 23

5.2.2 Approximate form of solutions ... 25

6 Proposed Experiment 29

6.1 Luminal properties of exciton condensation ... 30

6.2 Midgap states of exciton condensation ... 31

7 Introduction of Magnon Condensate in CuO 35

8 Formulation of SBMFT 37

8.1 Spin rotation ... 37

8.2 Schwinger boson mean field theory ... 38

8.3 Free energy and SB Equations... 42

9 Application of SBMFT to CuO 45

9.1 The information of CuO... 45

9.2 Finite momentum BEC of magnon ... 48

9.3 The spin correlation function... 49

10 The c-ic Phase Transition of CuO 53

11 Conclusion 57

LIST OF FIGURES

Fig. 3.1:(a) Two-fold degeneracy in valence and conduction bands without external field. (b)Minus mode (bands 1, green) and plus mode (bands 2, blue) with external field R = 1.2vF ………13 Figure 5.1: For ky = 0.2, the iteration processes (red curve → black curve) of (a)

symmetric and (b) anti-symmetric solution of ∆(kx) without eternal field……….24 Figure 5.2: The contours of symmetric (a) symmetric, (b) anti-symmetric, and (c) mixed solutions without external field for germanene. ……….25 Figure 5.3: The contours of (a) symmetric, (b) anti-symmetric, and (c) mixed solutions with external field R = 1.2vF for germanene. ………...…..25 Figure5.4: The diagram of ∆ at (kx,ky)=(0.2,0) for different semiconductor band gap 2λ………..………...26 Figure 5.5: The green curve is plotted with Eq. (5.9) and blue curve is plotted with Eq.

(5.10). Intersection of two curves is the solution of (∆0,∆1). (a) (kx, ky) = (0.35, 0.2) (b) (kx, ky) = (0.8, 0.6)…… ……….26 Figure 5.6: Numerical mixed state solution of ∆ as ky = 0.2 (purple) and ky = 0.6 (black) for

germanene. ………..27 Figure 6.1: sNp structure consisting of a normal slab in the yz plane having a thickness d between semiconductors with s-wave and p-wave. ……….31 Figure 6.2: The midgap state energy versus kx for (a) sNp and (b) sNs

structures. ………....33 Figure 9.1: Illustration of the magnetic unit cell and the positions of copper atoms…..46 Figure 9.2: The determination spin directions……….48 Figure 9.3: The dispersion relation of ωβ,k at T = J3/4 and k3 = 0.118. The magnitude

Figure 10.1: The numerical solution of commensurate-incommensurate Tc-ic ≈

0.337J3. ………55

LIST OF TABLES

Table 1.1: Copy from Table 1 of Kohn and Sherrington’s review article [4]. The properties of two types of single bosons. ……… 7 Table 1.2: Copy from Table 2 of Kohn and Sherrington’s review article [4]. The properties of two types of condensate states. ……….. 7 Table 9.1: This table gives the coupling constants between two copper atoms. ...47

Quasi-particle Condensation in

Two-Dimensional System and Its Application in Spin Ordering

Chih-Yu Chen Supervisor: C.D. Hu

July 2, 2019

2

Contents

1 Exordium 5

2 Introduction of Exciton Condensate 9

3 Effective Hamiltonian of 2D Semiconductors 11

3.1 The effective Hamiltonian without external field . . . 11

3.2 The effective Hamiltonian with external field . . . 13

4 Coulomb Interaction Revisit 15 4.1 Spin selection rule and 2D Coulomb potential . . . 15

4.2 The form factors of cases with and without external field . . . 17

5 The Gap Equation and Its Solution 21 5.1 Gap equation . . . 21

5.2 Solution of gap equation . . . 22

5.2.1 Numerical solutions . . . 23

5.2.2 Approximate form of solutions . . . 25

6 Proposed Experiment 29 6.1 Luminal properties of exciton condensation . . . 30

6.2 Midgap states of exciton condensation . . . 31

7 Introduction of Magnon Condensate in CuO 35 8 Formulation of SBMFT 37 8.1 Spin rotation . . . 37

8.2 Schwinger boson mean field theory . . . 38

8.3 Free Energy and SB Equations . . . 42

9 Application of SBMFT to CuO 45 9.1 The information of CuO . . . 45

9.2 Finite momentum BEC of magnon . . . 48

4 CONTENTS 9.3 The spin correlation function . . . 49

10 The c-ic Phase Transition of CuO 53

11 Conclusion 57

Chapter 1 Exordium

Cooper pairs and Bose-Einstein condensation (BEC) are two very profound quantum states. They have common properties and important differences.

The former is the origin of the celebrated superconductivity and BEC is the cause of fascinating superfluidity. On the other hand, the fact that Cooper pairs are formed by fermions and BEC are composed of bosons engenders far-reaching different physical consequences.

Let’s start with superconductors [1]. The physical properties of supercon- ductivity can be described microscopically by BCS theory. In BCS theory, a key idea is electron–electron will form a pair (Cooper pair) in supercon- ductors. More precisely, two electrons near Fermi surface can form pairs via electron-phonon interaction as temperature is lower than some critical value Tc.

Bose-Einstein condensation (BEC) of bosons is a state of matter which a large fraction of bosons occupy the lowest quantum state. This phenomena is first predicted by S.N. Bose in 1924 and later extended by A. Einstein in 1924 to 1925. The experimental discovery of BEC in dilute gas at low tempera- ture in 1995 by W. Ketterle, E. Cornell and C. Wieman [2, 3]. In condensate state, microscopic quantum phenomena, particularly wave-function interfer- ence, become apparent macroscopically. We can say the condensation of bosons is "conventional" since the statistical and spin nature of boson allows the condensation phase. On the other hand, fermions cannot occupy the ground sate with a great amount because the Pauli exclusion principle.

In general, the wave-function of BEC centers at zero momentum since intuitively zero momentum is the lowest energy state. The finite momentum BEC of strongly interacting Bose system is discussed by Yukalov [5]. This phenomena has also been discussed in the past decade in magnon system, see for example, [6, 7]. That is, the condensate state occurs at some finite k = k₀. Its properties of breaking U (1) symmetry of Helium III is discussed

6 CHAPTER 1. EXORDIUM in the work of Bunkov and Volovik [8]. In the article of Kohn and Sherrington [4], they classified bosons in two types. Quoting results are shown in Table 1.1 and Table 1.2.

Apart from the conventional BEC of bosons, the quasi-particles as bosons have more complex and interesting contents, especially for the type 2 con- densates. It allows possibility of the finite momentum Bose-Einstein con- densation which is usually to be zero. We keep Table 1.1 of single boson for comprehensiveness however we will focus on the condensation of quasi- particles summarized in Table 1.2. One needs to bear in mind that some systems may display both type 1 and type 2 mixed situation. For exam- ple the crystalline superconductor may be viewed as type 1 bosons (Cooper pair), in a condensate of type 2 bosons (the normal host crystal).

Looking back to the magnon system [6, 7]. One more question rises: Are magnons type 2 bosons? In the work of J. Hick et. al., they derived an effective boson Hamiltonian to describing the lowest magnon band of YIG, see Appendix of ref. [7]. They used method developed by Holstein and Pri- makoff in 1940 which can be found in original paper or other literature [9].

Such boson derived from spins leads no superfluidity but its condensation of- fers a view point in dealing magnetic ordering system. The Schwinger boson mean field theory (SBMFT), a method of dealing with electron spins, also applies similar idea as that which we are going to use of Holstein-Primakoff bosonization.

The main purpose of this thesis is to present condensation of type 2 bosons on the platform of real physical systems. The exciton condensation of two- dimensional (2D) semiconductors is reported in PART II. We start from an effective Hamiltonian of 2D semiconductors and show an interesting mixed state of exciton condensate. In PART III, we extend the SBMFT of ferro- magnetic and antiferromagnetic systems. The bosonization of electron spins can also be a useful mathematical tool to study quantum spin systems. BEC of Schwinger bosons can describe the ordering phase of spins. We also study the commensurate-incommensurate phase transition of CuO as an example.

7

single bosons type 1 type 2

examples

4He atoms, exciton

tightly bounded fermions

nature complexes of even number particle-hole bound of real fermions complexes form of Green’s

functions and density matrices

singular as functions singular as functions

of momentum of momentum

sum variables difference variables

momentum properties carry may or may not carry

mechanical momentum mechanical momentum Table 1.1: Copy from Table 1 of Kohn and Sherrington’s review article [4]. The properties of two types of single bosons

condensed states type 1 type 2

example He II excitonic phase

type of additional order

off-diagonal diagonal

long-range order long-range order

superfluidity yes no

form of Green’s functions and density

matrices

macroscopic singularities macroscopic singularities as functions of momentum as functions of momentum

sum variables difference variables Table 1.2: Copy from Table 2 of Kohn and Sherrington’s review article [4]. The properties of two types of condensate states.

8 CHAPTER 1. EXORDIUM

Chapter 2

Introduction of Exciton Condensate

Electrons in semiconductors can form quasi-particle states. The binding of an electron and a hole by attractive Coulomb interaction gives rise to an exciton. If the density is high enough, the overlap of excitons becomes signif- icant. At low-enough temperature they form condensate and their collective behaviors such as Andreev reflection [10], superradiance [11] and Josephson tunneling [12] are similar to those of the Cooper pairs in superconductors.

The investigation of condensed phase of excitons has made great progress in the past few decades [13]. The theoretical scheme for condensation phase and collective behaviors was constructed by pioneer works of Kozlov et al.

[14, 15]. They pointed out in a crystal with closely-lying bands the Coulomb interaction became important. At low temperature, the density of excitons can be very high. Once condensed, the gap function ∆_k becomes finite and the system enters a BCS-like state. Later, the theory of exciton condensa- tion in bulk germanium and silicon is discussed by the work of M. Combescot and P. Nozières [16]. Their results were in good agreement with experiments in aspects of ground state energy and critical density. The two-dimensional (2D) exciton instability in an InAs-GaSb-based system has been reported by Naveh and Laikhtman [17]. They formulated the gap equation of exci- ton condensation in their case I where electrons and holes are separated and compared it with that of BCS theory. They also discussed how electric field effects the density of condensation. The electron-hole system of InAs-GaSb bi-layers is also discussed recently by Pikulin and Hyart [18]. They study small tunneling and large tunneling between layers in which the systems dis- play s-wave exciton condensation and quantum spin Hall (QSH) insulator property, respectively. This QSH insulator phase exhibits topological non- trivial property with p-wave exciton parameter. Despite these interesting

10 CHAPTER 2. INTRODUCTION OF EXCITON CONDENSATE properties, however, the exciton condensation in single-layer materials is still an issue which needs more investigation.

After the discovery of graphene [19], more and more 2D materials have been synthesized. Silicene (2D silicon) and germanene (2D germanium) [20, 21] will be discussed in this work. These low-bulked honeycomb ma- terials have topological properties related to QSH effect. The existence of QSH effect in 2D system was first proposed by Kane and Mele in graphene [22]. They showed the band gap can be opened by spin-orbit coupling (SOC).

However, the first order SOC is shown to be rather weak in subsequent works [23, 24]. Hence the QSH effect of graphene can only happen at incredibly low temperature. On the other hand, low-bulked 2D semiconductors of heavier elements have much stronger SOC [20, 23]. Furthermore, it was shown that p-wave superconductivity could be stable in a 2D system with Rashba inter- action [25, 26]. These works give us motivation to study the combination of exciton condensation and 2D semiconductors.

In chapter 2-6, we will discuss some interesting physical results owing to SOC. We also take one step further to discuss the system under effect of external electric field. Our study shows that the SOC, intrinsic or extrin- sic, plays an important role in exciton condensation in 2D materials. The latter clearly favors the mixed states of s-wave and p-wave. Chapter 2-6 is organized as follows. Chapter 2 is the introduction of exciton condensation of semiconductors. In Ch.3, we discuss the effective Hamiltonian considering SOC and Rashba interaction. It gives the theoretical basis of this paper. In Ch.4, we discuss the Coulomb interaction in details. The consideration of spin configurations and characters of bands are of great importance. In Ch.5, we derive the gap equation of exciton condensation, which is the central equa- tion of exciton condensaiton. We then show that the exciton condensation should be formed in p-wave-like state due to the mixed state nature of spins.

Experiments are proposed to verify such state in Ch.6.

Chapter 3

Effective Hamiltonian of 2D Semiconductors

In this chapter, we demonstrate the effective Hamiltonian of silicene and germanene. The cases with and without external electric field are presented in part A. and part B, respectively. The band structures and eigenstates are also calculated. Silicene and germanene have honeycomb crystal structure from top view which is similar to graphene. They have a zig-zag geometry from side view. That is, the sublattice A and B are not coplanar but they are still 2D systems. In the work of C.C. Liu et al. [27], they proposed an effective low energy Hamiltonian of 2D semiconductors.

3.1 The effective Hamiltonian without external field

The obtained Hamiltonian around K in crystal momentum space for low- energy states (φ_1↑, φ_1↓, φ_4↑, φ_4↓) is

H₀^band = (₁− λ^2nd_so )I₄+

 h₁₁ v_Fk₊I₂ v_Fk₋I₂ −h₁₁



, (3.1)

where

h11 = −λsoσz− aλR(kyσx− kxσy), (3.2) I4 and I2 are 4 × 4 and 2 × 2 identity matrices, respectively. σi is the Pauli matrix for spins and a is the lattice constant of 2D semiconductors. Eq. (3.1)

12CHAPTER 3. EFFECTIVE HAMILTONIAN OF 2D SEMICONDUCTORS is in the representation of the low-energy states φ1 and φ4

|φ₁i = u₁₁

p^A_z + u₂₁

s^A + u₃₁

√2(

p^B_x − i p^B_y),

|φ4i = u11

p^B_z − u21

s^B − u₃₁

√2(

p^A_x + i

p^A_y), (3.3) where u₁₁, u₂₁ and u₃₁ are normalized coefficients. One can find details in Ref. [27] and we here simply treat them as coefficients of linear combination of states. Superscripts A and B denote two sublattices of the honeycomb lattice. The complete basis are the direct product of orbitals and spins φ₁₍₄₎ ⊗ |↑ (↓)i. λR is the strength of internal Rashba interaction, λso = λ^1st_so + λ^2nd_so where λ^1st_so and λ^2nd_so are the two kinds of spin-orbit interactions considered in (25) and (30) of Ref. [27]. v_F is the Fermi velocity and the relation to the tight-binding potentials are given by (22) of Ref. [27].

Silicene and germanene have different physical properties from graphene due to the low-buckled geometry. The most important is it gives rise to two directions of SOC. One lies on the honeycomb plane, which appears in graphene as well. The other is perpendicular to the plane in the form of

−it₁µ_ij(~σ × ~d⁰_ij)_z, see (4) of Ref. [27]. The other geometry induced interaction is the intrinsic Rashba interaction aλR, which is much weaker than the SOC, see Eq. (39) of [27]. In this work, we first simplify the notations by letting r ≡ aλ_R and λ ≡ λ_so to avoid unnecessary complications. The degenerate eigenvalues of H₀^band are ±k where

_k = q

(r²+ v_F²)k²+ λ². (3.4) As we can see from Eq.(3.4), 2λ represents the energy gap between valence band and conduction band at momentum K. The band gap E_g is 2λ. With eigenvalues given by Eq. (3.4), we have four eigenvectors







 b₁ b₂ a₁ a₂









= 1

√M_k









A_k B_k 1 0

0 1 −B_k^∗ A^∗_k C_k −A_k 0 1

1 0 −A^∗_k C_k















 φ_1↑

φ_1↓

φ_4↑

φ_4↓









, (3.5)

where

Ak = _k− λ v_Fk−

Bk = irk₊ v_Fk−

Ck = −ir

v_F , (3.6)

and M_k= [(_k− λ)²+ k²(r²+ v_F²)]/(kv_F)² is nothing but the normalization of eigenvectors. b1and b2 are conduction bands since they both have energy +k

whilst a₁ and a₂ are valence bands which are of energy −_k. Thus we have a band structure with two degenerate conduction bands and two degenerate valence bands, see FIG. 3.1(a) where the parameters of germanene are used.

3.2. THE EFFECTIVE HAMILTONIAN WITH EXTERNAL FIELD 13

-1 -0.5 0.5 1 k

-1 -0.5 0.5 1 Ε_k

Eg

(a)

-1 -0.5 0.5 1 k

-0.5 -0.25 0.25 0.5 Εk

band 1 band 2

(b)

Figure 3.1: (a) Two-fold degeneracy in valence and conduction bands without external field. (b)Minus mode (bands 1, green) and plus mode (bands 2, blue) with external field R = 1.2vF.

3.2 The effective Hamiltonian with external field

Now we consider the system in an external electric field E_zz which is per-ˆ pendicular to the 2D xy plane. The induced Rashba interaction becomes R(kyσx− kxσy), where R is the Rashba parameter which is proportional to E_z [28]. In the example given in FIG. 3.1(b), R = 1.2v_F means that we need an electric field of energy about 0.92eV per lattice cite of germanene [27], which is 0.23eV·Å. In practice, we can reach this by adding a gate voltage in the magnitude of order 10V [29]. One can see the change of the Rashba constant due to the gate voltage. Though the effect depends on properties of materials, the order of magnitude should be the same and it is very promis- ing to achieve in laboratory. With the basis (φ_1↑, φ_1↓, φ_4↑, φ_4↓)^T, the effective Hamiltonian at K point is

H₁^band = (₁− λ^2nd_so )I₄+

 h₁₁ v_Fk₊I₂ v_Fk−I₂ h⁰₁₁



, (3.7)

where h₁₁ = −λ_soσ_z− R(k_yσ_x− k_xσ_y), and h⁰₁₁ = λ_soσ_z − R(k_yσ_x− k_xσ_y).

In contrast to the effective Hamiltonian without external field, the lower right hand corner is not −h11. The difference is caused by the formation of Rashba interaction. The Rashba interaction of the case without external

14CHAPTER 3. EFFECTIVE HAMILTONIAN OF 2D SEMICONDUCTORS field is intrinsic, which has opposite directions on A and B sites. The Rashba interaction due to the external field, on the contrary, has the same direction on A and B sites and it can be much stronger than the intrinsic one. The external field breaks the mirror symmetry of internal Rashba interaction.

The eigenvalues of Hamiltonian (3.7) are ±_1k with

_1k=p

λ²+ k²(R − v_F)², (3.8) and ±_2k with

2k=p

λ²+ k²(R + vF)². (3.9) The degeneracy is lifted, see FIG. 3.1(b). The bands with R + v_F and R − v_F are in blue and green, respectively. The band 1, with _1k, has the smaller magnitude. Only at k = 0, the bottoms of two conduction bands and tops of two valence bands touch each other, and the energy gap is still of magnitude 2λ at k = 0. We then solve for the eigenvectors. For energy ±_1k,

b⁰₁ = 1 n₁









−ik(R − v_F)

k+

k (_1k+ λ)

ik−

k (_1k+ λ)

−k(R − v_F)









, a⁰₁ = 1 n₁









i(_1k+ λ) k₊(R − v_F) ik−(R − v_F)

_1k+ λ









. (3.10)

For energy ±_2k, we have

b⁰₂ = 1 n₂









−ik(R + v_F)

k+

k (_2k+ λ)

ik−

k (_2k+ λ) k(R + v_F)









, a⁰₂ = 1 n₂









−i(_2k+ λ)

−k₊(R + v_F) ik−(R + v_F)

_2k+ λ









. (3.11)

Similar to ai and bi before, b⁰_i and a⁰_i correspond to conduction band and va- lence band states, respectively. The normalization factor is n_i = 2pik(_ik+ λ).

Hence, in the tight-binding model, the Hamiltonian with external field can still be diagonalized.

Chapter 4

Coulomb Interaction Revisit

4.1 Spin selection rule and 2D Coulomb poten- tial

In this section, we first discussed the Coulomb interaction in a 2D system and paid special attention to spin configurations. Analogous to the interact- ing fermions in BCS theory, where the interaction leads to Cooper pairs in superconductors, attractive Coulomb interaction between electrons and holes plays a crucial role in forming the exciton condensate state. We consider a comprehensive Hamiltonian of an electronic system

Hi = H_i^band + V (4.1)

where H_i^band is the effective Hamiltonian in Eq. (3.1) or Eq. (3.7) under the representation of basis (φ_1↑, φ_1↓, φ_4↑, φ_4↓)^T. Index i = 0 and i = 1 represent the cases without and with external field, respectively. The Coulomb term is V = 1

2 X

ijmn

X

k,k⁰,q

X

σσ⁰

hk + q, i; k⁰, j|V_q|k⁰− q, m; k, ni c^†_i,k+q,σc^†_j,k0−q,σ⁰c_m,k⁰_,σ⁰c_n,k,σ. (4.2) Operator c’s represent φ₁ or φ₄ with i, j, m and n being either 1 or 4, (see Eq. (3.3)), and σ and σ⁰ are spin indices. The effective two-dimensional Coulomb potential in momentum space [30] is

Vq = 2πe²

|q|(1 + 2πα_2D|q|), (4.3)

where α_2D is the 2D polarizability caused by electron screening of bands.

The magnitude of α can be calculated by Eq. (10) of Ref. [31]

α_2D = e² 2π



− 1

E_g+ x

  

xM

0 . (4.4)

16 CHAPTER 4. COULOMB INTERACTION REVISIT x_M is the energy of the state at the first Brillouin zone boundary. For ger- manene, x_M ≈ 25E_g in the case without external field. On the other hand, x_M is about 6.25E_g for band 1 and much larger (∼50 times) than E_g for band 2 as R = 1.2vF, see FIG. 3.1(b).

Same as the work in Ref. [32], we only consider the Coulomb interac- tion between even numbers of the electrons in valence bands and conduction bands. That is, terms like b^†a^†aa or b^†a^†bb are omitted since they contribute little to exciton pairing. In our 2D system described by H_i^bandthe eigenvectors are spin mixed states and Coulomb interaction doesn’t change spins. Hence, we start with the Coulomb interaction in φ representation then extract out the coefficients of operators b^†a^†ab, b^†a^†ba because the spin configurations are clearer in φ representation. Eq. (4.2) in momentum space is written in the second quantization form

1 2

X

ijmn

X

k,k⁰,q,σσ⁰

V_q^ijmn((1 − δσσ⁰)c^†_i,k+q,σc^†_j,k0−q,σ⁰cm,k⁰,σ⁰cn,k,σ+ δ_σσ⁰c^†_i,k+q,σc^†_j,k0−q,σ⁰c_n,k⁰_,σ⁰c_m,k,σ),

(4.5)

In (4.5), the first term is the direct term and the second contains the exchange term. The possible spin configurations (σσ⁰σ⁰σ) are (↑↑↑↑), (↓↓↓↓), (↑↓↓↑) and (↓↑↑↓). Coulomb interaction in momentum representation for direct terms and exchange terms are proportional to (|k − k⁰| + 2πα2D|k − k⁰|²)⁻¹ and (|k − k⁰+ q| + 2πα_2D|k − k⁰+ q|²)⁻¹, respectively. Since the screening effect is dominant, electrons with all parallel spins give negligible contribu- tion since direct and exchange terms approximately cancel each other. In summery, only spin configurations (↑↓↓↑) and (↓↑↑↓) survive.

Furthermore, the overlap of wavefunction φ₁ and φ₄ is small accord- ing to Eq. (3.3). This is owing to the small overlap of |pzi and |pxi + i |p_yi orbitals and that between wave functions of different sublattices (A and B). The summations such as P

k,k⁰,qV_q¹¹¹⁴φ^†_1,k+q↑φ^†_1,k0−q↓φ1,k⁰↓φ4,k↑ or P

k,k⁰,qV_q¹⁴¹⁴φ^†_1,k+q↑φ^†_4,k0−q↓φ_1,k⁰_↓φ_4,k↑ can be dropped. Thus, with above ar- guments, we simply consider eight sets of integrations. They are (φ^†_1↑, φ^†_1↓, φ_1↓, φ_1↑)^T, (φ^†_1↑, φ^†_4↓, φ_4↓, φ_1↑)^T, (φ^†_4↑, φ^†_1↓, φ_1↓, φ_4↑)^T, and (φ^†_4↑, φ^†_4↓, φ_4↓, φ_4↑)^T and the four others with ↑ and ↓ exchanges.

4.2. THE FORM FACTORS OF CASES WITH AND WITHOUT EXTERNAL FIELD17

4.2 The form factors of cases with and without external field

According to Eq. (3.5), the inverse transformation of operators φ_i,σ with respect to a and b are







 φ1↑

φ_1↓

φ_4↑

φ4↓









= 1

√Dkk









(k− λ)k− 0 irk² vFk²

−irk₋² v_Fk² −(_k− λ)k− 0 v_Fk² −irk₊² 0 −(_k− λ)k₊

0 (k− λ)k+ vFk² irk²















 b1

b₂ a₁ a2







 ,

(4.6) where D_k= (_k− λ)²+ k²(r²+ v_F²). A similar equation can be found for the case with external electric field.

We substitute (4.6) into (4.5) and consider the pairing in the same band.

That is, terms with b^†₁a^†₁a₁b₁ or b^†₂a^†₂a₂b₂ are kept but terms as b^†₁a^†₁a₁b₂ are excluded. This is because the pairing between different bands is much weaker than that between the same band [33]. The effect of multi-band leads to a form factor f (k, k⁰, q). As a result, the second quantized Coulomb interaction is of the form

1 2

X

k,k⁰

h

V_qa^†_k+qa^†_k0−qa_k⁰a_k+ V_qb^†_k+qb^†_k0−qb_k⁰b_k+ 2f (k, k⁰, q)V_qb^†_k+qa^†_k0−qa_k⁰b_ki , (4.7) where the form factor f (k, k⁰, q) will be given below in Eqs. (4.10), (4.12) and (4.13). Form factor is not considered in most of the previous works [32, 35]. However, in the case of 2D semiconductors they are important. It has profound effect in determining the forms of the solutions of the gap equa- tion as shown in next section. We treat f × V as the effective interaction.

In the condensed phase, which is the main purpose of this article, a strong correlation between the k conduction electrons and −k valence holes is ex- pected. The e-h pairings with zero center-of-mass velocity have the lowest possible kinetic energy. The same approximation is also used in calculation of superconductivity. Hence, we can replace k⁰by k for further simplification:

1 2

X

k,k⁰,q

h

V_qa^†_k+qa^†_k0−qa_k⁰a_k+ V_qb^†_k+qb^†_k0−qb_k⁰b_k+ V_qf (k, k⁰, q)b^†_k+qa^†_k0−qa_k⁰b_ki

≈ 1 2

X

k,p

h

V_qa^†_ka^†_pa_ka_p+ V_qb^†_kb^†_pb_kb_p+ 2f (k, p)V_qb^†_ka^†_pa_kb_pi ,

(4.8)

18 CHAPTER 4. COULOMB INTERACTION REVISIT where k − p = q. Thus we have the two-dimensional exciton Hamiltonian (without external electric field)

H₀ =X

k

^v_k(a_1ka^†_1k+a_2ka^†_2k) + ^c_k(b^†_1kb_1k+ b^†_2kb_2k)

+X

k,p

V_k−pf₀(k, p)(b^†_1ka^†_1pa_1kb_1p+ b^†_2ka^†_2pa_2kb_2p), (4.9) where we consider interband interaction only and absorb the intraband inter- action into the band shapes which are obtained by the effective Hamiltonian (3.1). The explicit expression of the form factor f₀ without external field is

f₀(k, p) = (_k− λ)²(_p− λ)²+ k²p²v⁴_F + 2v_F²(_k− λ)(_p− λ)k · p

[(k− λ)²+ k²(r²+ v²_F)][(p− λ)²+ p²(r²+ v²_F)] . (4.10) Notice that in Hamiltonian (4.9), the band labeled 1 and 2 are decoupled so that we can diagonalize the bands separately. They still possess the two-fold degeneracy.

Similarly, in order to discuss the excitons under external field, we express φ’s in terms of a⁰ and b⁰.







 φ_1↑

φ1↓

φ_4↑

φ_4↓









= 1 4













ik(R−vF)n1

1(1+λ)

−in₁

1

ik(R+vF)n2

2(2+λ)

in2

2

k−n1

k1

k−(R−vF)n1

1(1+λ)

k−n2

k2

k−(R+vF)n2

2(2+λ)

−ik+n1

k1

−ik+(R−vF)n1

1(1+λ)

ik+n2

k2

−ik+(R+vF)n2

2(2+λ)

−k(R−v_F)n1

1(1+λ)

n1

1

k(R+vF)n2

2(2+λ)

n2

2



















 b⁰₁ a⁰₁ b⁰₂ a⁰₂







 .

(4.11) Here we dropped the indices of k. By the same approximations, we have the form factor

f₁ = kp 2_1k_1p



(R − v_F)²+

 (R − v_F)⁴

(_1k+ λ)(_1p + λ)+(_1k+ λ)(_1p+ λ) k²p²

  k · p 2



, (4.12) for band 1 and

f₂ = kp 2_2k_2p



(R + v_F)²+

 (R + v_F)⁴

(_2k+ λ)(_2p+ λ) +(_2k+ λ)(_2p+ λ) k²p²

  k · p 2



(4.13) for band 2. We consider only the valence band 1 and conduction band 1 because they have the smaller band gap and larger density of states near Fermi surface. The resulting Hamiltonian with external field is,

H₁ =X

^v_ka⁰_1ka^0†_1k+ ^c_kb^0†_1kb⁰_1k+X

V_k−pf₁(k, p)b^0†_1ka^0†_1pa⁰_1kb⁰_1p. (4.14)

4.2. THE FORM FACTORS OF CASES WITH AND WITHOUT EXTERNAL FIELD19 It is important to point out that the form factor contains a term with angular

dependence. The solutions of the gap equations can be anisotropic because of its presence.

20 CHAPTER 4. COULOMB INTERACTION REVISIT

Chapter 5

The Gap Equation and Its Solution

We derive the gap equation analogous to BCS theory in superconductors.

It turns out that there are three types of solutions distinguished by the symmetric property in k-space. We showed the lowest energy state is the mixture of symmetric and anti-symmetric solutions.

5.1 Gap equation

When electron-hole pairs form in a system, it is natural to consider the commutation relation between pairs. The excitons behave like bosons as the mean distance between two excitons is much larger than the extension of excitons [32]. In the condensed phase, on the other hand, electrons and holes are to be treated separately as fermions and BCS states prevails. The BCS ground state [34] is

|ΨBCSi =Y

k

(uk+ vkc^†_kc^†_−k) |Φi , (5.1) where c^†_kc^†_−k is a Cooper pair operator which create an electron pair in mo- mentum space near Fermi surface.

Let |Φi be the state with full valence band and empty conduction band and replace e-e pair with e-h pair in the ground state [35]. The wave function of exciton condensate is

|Ψ_exi =Y

k

(u^∗_k− v^∗_kb^†_ka_k) |Φi . (5.2) where b^†_kak is the creation operator of e-h pair. It is also instructive to see the form of quasi-particle excitation. For excitons, excitations can be defined

22 CHAPTER 5. THE GAP EQUATION AND ITS SOLUTION as αk = u_ka_k − v_kb_k, and βk = v^∗_ka_k + u^∗_kb_k. Comparing with the BCS theory, we can easily find the analogous Bogoliubov quasi-particle operators γ_k= u_kc_k− v_kc^†_−k, γ−k= u_kc−k+ v_kc^†_k. The coefficient u_kand v_k satisfy the condition |uk|²+ |v_k|² = 1 in both exciton condensation and superconductor cases. Similar to the BCS theory, we minimize the total energy with respect to u_k and v_k and the results are

u_k= 1 2

 1 + ξ_k

E_k

1/2

, v_k = 1 2



1 − ξ_k E_k

1/2

, (5.3)

where

ξ_k= (^c_k− ^v_k)/2, E_k² = ξ_k²+ ∆²_k. (5.4) Using the effective Coulomb potential given by Hamiltonian (4.9), the self consistent gap equation at zero temperature can be determined as

∆_k =X

p

f_i(k, p)V_k−p ∆_p 2q

ξ_p² + |∆_p|²

, (5.5)

where i = 0, 1 are for the case without or with external field.

5.2 Solution of gap equation

We have derived the gap equation in the last subsection. More explicitly, we have an integral equation as

∆(k_x, k_y) = 1 (2π)²

Z

B.Z.

f_i(k, p)V_k−p ∆(p_x, p_y)

2pξ_k²+ ∆²(px, py)dp_xdp_y. (5.6) The domain of integration is over the first Brillouin zone instead of an ar- tificial cutoff introduced in the work of Kozlov and Maksimov [14]. In their work, case B of part 3, they assumed the gap remains a constant value ∆(0) for p < p1, the cutoff. They also argued that ∆(p) decreased rapidly as p > p₁. Determining of p₁ is based on dispersion relation which is a rela- tively simple band structure (p) = p²/2. Let ∆(0) = p²₁/2 and solving for p1 leads to p1 = p2∆(0). There is no reason to set any cut-off momentum here since the dispersion relation here is close to being linear. The gap ∆ does not behave as Kozlov and Masksimov [14] had assumed. It will be even clearer in the solution level which we will see later.

5.2. SOLUTION OF GAP EQUATION 23

5.2.1 Numerical solutions

We solve the integral equation with the following recipe. First we give a specific value k_yi and a trial ∆ and evaluate the integration. This will give a rough curve of ∆(k_x) versus k_x, called ∆₀(k_x). Second we feed the ∆₀(k_x) back into integration and repeat the processes. We can start with trial wave functions with different symmetries in region k_x = [^−π_a ,^π_a], then do the it- eration. Finally the output function will converge to a form with enough accuracy, which is set as 10⁻³ in this work. We apply the same process for the case with external field. The region of k_y = [−^π_a,^π_a] is divided by 33 in- tervals. We did the iteration to get ∆(k_x) for every k_yi. The grids are 33 × 33 in the first Brillouin zone. Momentum k_x and k_y are centered at k = 0 which is the K point of germanene and silicene.

Parameters are given in [27] for germanene. The length is measured for lattice constant a ≈ 4Å. Energy is in unit of Fermi velocity times k, where v_F ≈ 4.57 × 10⁻⁵ cm/s. λ, half of band energy gap, is 46meV and internal Rashba interaction r is 10.7meV for gemanene. By Ref. [31], in the case without external field x_M = 25E_g thus α⁰_2D ≈ _2πE^e2

g(¹₁ − ₂₆¹) × 2 = 48.07Å, where the factor of 2 is the number of degenarate bands, see FIG. 3.1(a). As for the case with external field α_2D¹ ≈ _2πE^e2

g[(¹₁ − _7.25¹ ) + 1] = 46.55Å, where the first term comes form band 1 and the second term comes from band 2, see FIG. 3.1(b). Examples of symmetric and anti-symmetric iterations are shown in FIG. 5.1. The red curve is the initial one and the black curve is the final result.

For a comprehensive consideration, the ground state can be either purely symmetric or anti-symmetric but also the linear combination of them. In or- der to give a complete physical pictures, we give three contour plots of gap functions in the first Brillouin zone in FIG. 5.2. and FIG. 5.3. Apparently, the symmetric solution is nodeless. The antisymmetric one has a node line.

Interestingly, the gap function of the mixed solution almost vanishes in left half plane. It is equally possible that its magnitude is vanishingly small in either upper or lower half plane.

Now the question occurs: which solution gives the lowest energy? To an- swer this, we have calculated the energy of each solution. The wave function of exciton condensed state is similar to BCS theory. The form of energy can be deduced from the BCS theory [36] which reads

E =X

k



_k(1 + 2x_k) − (1

4 − x²_k)^1/2∆_k



, (5.7)

24 CHAPTER 5. THE GAP EQUATION AND ITS SOLUTION

-3 -2 -1 1 2 3 k_x 0.1

0.2 0.3 0.4 0.5 D_kx

(a) symmetric solution

-3 -2 -1 1 2 3 k_x

-0.4 -0.2 0.2 0.4 D_kx

(b) anti-symmetric solution

Figure 5.1: For k_y = 0.2, the iteration processes (red curve → black curve) of (a) symmetric and (b) anti-symmetric solution of ∆(k_x) without eternal field.

where x_k = ± ^k

2√

²_k+∆²_k. We insert ∆_k obtained from the integral equation into Eq. (5.7). In the three cases, the anti-symmetric ∆_k state has the high- est energy thus it cannot be the condensation state. Another possible form of solution, ∆0(kx+ iky), which is common in considering p-wave superconduc- tors is also considered. This solution gives energy comparable to that of the antisymmetric state. Thus it cannot be the condensation either. We found the energy of mixed solution is slightly lower than the symmetric one. Defin- ing a parameter η ≡ (E^s− E^m)/E^m, where superscript s and m stand for symmetric and mixed, respectively. η^int≈ 0.006 when only intrinsic Rashba interaction is considered. However, η^ex ≈ 0.03 as an external electrical field R = 1.2v_F is applied. We can see the energy difference is not significant when there is no external field. It is more advantageous to the mixed wave state by applying electric field. In short, the mixed state is the ground state of exciton condensation under external electrical field.

We plot the diagram of ∆ versus band gap (E_g = 2λ) of semiconduc- tor, see FIG. 5.4. The red curve is the case without external field while the black one is the case with external field. The bandwidth of germanene is also pointed out in FIG. 5.4. The regions under the curves are where the conden- sation states are stable. This diagram shows that the exciton condensation state is the ground state of the system thus the e-h pairs can, in principle, survive for a long time. As we estimated before, α_2D⁰ > α¹_2D, thus the 2D Coulomb potential with external field is stronger. This leads to a larger ∆

5.2. SOLUTION OF GAP EQUATION 25

Figure 5.2: The contours of symmetric (a) symmetric, (b) anti-symmetric, and (c) mixed solutions without external field for germanene.

Figure 5.3: The contours of (a) symmetric, (b) anti-symmetric, and (c) mixed solutions with external field R = 1.2vF for germanene.

5.2.2 Approximate form of solutions

To give a clearer picture, we try an approximation solution of ∆(k) ≈ ∆₀(k)+

∆₁(k) cos θ_k where θ_k = tan⁻¹(k_y/k_x). This can show us the amplitudes of the s-wave and p-wave in the mixed state. Then the integral equation in polar coordinate is

∆0(k)+∆1(k) cos θk= 1 π³

Z 2π 0

Z π 0

V (k, θk; p, θp)f1(k, θk; p, θp)K(p, θp)pdpdθp, (5.8) where the integration kernel K(p, θp) = ^{∆0(p)+∆1(p) cos θp}

2√

²_p+(∆0(p)+∆1(p) cos θp)². Coulomb potential V and form factor f₁ can be found in Eq. (4.3) and Eq. (4.12), respectively. Our first equation is given by integrating θ_k over 0 to 2π and solving ∆₀ for a given k. More specifically,

∆0 = 1 π³(2π)

Z 2π 0

Z 2π 0

Z π 0

V (k, θk; p, θp)f1(k, θk; p, θp)K(p, θp)pdpdθpdθk

(5.9)

26 CHAPTER 5. THE GAP EQUATION AND ITS SOLUTION

0.5 1 2 2 Λ

0.5 1 D

Ge

with E HblackL without E HredL

Figure 5.4: The diagram of ∆ at (k_x, k_y) = (0.2, 0) for different semiconductor band gap 2λ.

0.742 0.744 0.746 D0

0.25 0.3 0.35 D1

(a)

0.889 0.89 0.891 0.892 0.893 0.894D0 0.35

0.4 D1

(b)

Figure 5.5: The green curve is plotted with Eq. (5.9) and blue curve is plotted with Eq. (5.10). Intersection of two curves is the solution of (∆₀, ∆₁). (a) (kx, ky) = (0.35, 0.2) (b) (kx, ky) = (0.8, 0.6)

Then we multiply cos θ_k on both side of Eq. (5.8) and perform an integration

1

2πR dθ_k, which leads to

∆₁ = 2 π³(2π)

Z 2π 0

Z 2π 0

Z π 0

V (k, θ_k; p, θ_p)f₁(k, θ_k; p, θ_p)K(p, θ_p) cos θ_kpdpdθ_pdθ_k. (5.10) Now we have two coupled equations for ∆0and ∆1. For a given set of (kx, ky), there will be two curves from Eq. (5.9) and Eq. (5.10). The intersection is the solution. For example in FIG. 5.5(a), (k_x, k_y) = (0.35, 0.2), the numerical results of Eq. (5.9) (green) and Eq. (5.10) (blue) are shown. These two curves intersect at point (∆₀, ∆₁) = (0.744, 0.305). Our approximate gap ∆₀ +

∆₁cos θ_k = 0.744 + 0.305 cos(tan^{−1 0.2}_0.35) ≈ 1.01. This should be compared with the iteration result for (kx, ky) = (0.35, 0.2) of mixed state solution in FIG.5.6 (purple), where the numerical result is ∆ ≈ 1.05, the accuracy is reasonable and the difference comes from the integration boundary [37].

Another example is in FIG. 5.5(b), (kx, ky) = (0.8, 0.6) and solved (∆0, ∆1) =

−1 0.6

5.2. SOLUTION OF GAP EQUATION 27

-0.5 0.4 0.8 1.2 1.6 k_x 0.3

0.6 0.9 1.2 D

Figure 5.6: Numerical mixed state solution of ∆ as k_y = 0.2 (purple) and k_y = 0.6 (black) for germanene.

close to the numerical result ∆ ≈ 1.23, see FIG. 5.6 (black). The amplitude of p-wave type (anti-symmetric) solution if exists is usually much smaller than that of the s-wave (symmetric) solution in superconductors. Here the magnitude of ∆₀ and ∆₁ are comparable in this 2D system with external field. Hence, the mixed state has significant symmetric and antisymmetric parts.

28 CHAPTER 5. THE GAP EQUATION AND ITS SOLUTION