In this chapter, we show that the energy band structures excluding SOI effect in external periodic potential in 2DEG. There are massless Dirac points at the corners of Brilloin zone (K1 points) as graphene system.
And we also do another work for inspecting the numerical results. The wave function be expanded from Γ point and compare the numerical energy dispersion with the results which the wave function be expanded from K1. The result for comparison is exactly the same.
Chapter 3
Energy band structure with SOI effect
In this chapter, we consider the effect of spin-orbit coupling on the energy band structure, we have discussed in Chapter 2. The spin-orbit is arisen from the in-plane gradient of the periodic potential.
3.1 Muffin-tin potential lattice in the presence of SOI
The Hamlitonian H for a 2D MTP system with spin-orbital interaction can be expressed by.
H = p2
2m∗ + V (r) + HSO. (3.1)
The spin-orbit interaction term, in vacuum can be desired by
HSO = eλ
~σ ·(p × E) = −eλ
~ σ ·(p × ∇U) = λ
~σ ·(p × ∇V ) = −λ
~σ ·(∇V × p) , (3.2)
where in-plane electric field E = −∇U (U: electric potential); V (Electric potential en-ergy)= - eU (e > 0 ) ; spin-orbit coupling constant λ=120˚A2 (for InAs)
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
The wave function includes both spin-up and spin-down component in column vector form:
The Hamlitonian of SOI term operates on the wave function leading to:
HSOΨ (r) = −iλ show-ing that spin-up and spin-down are decoupled because HSO depends only on σz. Due to Eq. (3.1), and Schr¨odinger equation HΨ (r) = EΨ (r), and the orthogonal term of plane-wave form eik·rei(K1+Gn)·rcan be a substrate the m’th component to form a matrix equation. For getting the simple numerical formulation, we take off eik·rei(K1+Gn)·r and obtain:
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
This equation shows that the spin-up cn↑part is decoupled with spin-down cn↓(the element only exist on diagonal term). The numerical result is shown on the subsection 3.4.
The Fig. 3.5 shows the lowest two bands with wave vector near K1. We can see that the original Dirac point opens up a gap in the presence of SOI and the numerical result shows Ecn↑ = Ecn↓.
3.1.1 The Analytical result in the presence of SOI by perturba-tion method
The subsection will show the analytical calculation in our system in the presence of SOI . The Schr¨odinger equation for a 2DEG in the presence of SOI, using the Eq. (3.1) and Eq. (3.2) (where we defined HSO = hSOσz, because of ∇V × p is along z direction) can be written in the following form
HΨκ(r) =
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
The wave function Ψκ(r)κ = Ψ (r)k+K1 ( k is very close to K1, all at the equivalent K points (see Fig. 2.4 ) can be approximately expressed as a linear superposition of three plane-wave states, which is linear in k.
The Eq. (3.6) has a spin-orbit term, as show bellow
hSOΨs=±1(r)
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT on both side can be ignore for the matrix form.
hSO(κ) At the K point, the H0 has a doubly degenerate energy -W. Using the correspond eigen-states, |c1i and |c2i given by, Eq. (2.11), Eq. (2.12), we obtain the 2 × 2 subspace repre-sentation of H0, H1 and hSO.
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
where λ is spin-orbit coupling constant. We ignore the energy shift term ˜H0 and obtain:
The Eq. (3.15) shows the lowest two energy bands at K point opens up a gap (2√
3AλW ), the s=1 (spin-up)and s=-1 (spin-down) the energy dispersion is the same ( which is as same as numerical result, see subsection 3.4).
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
−0.1 −0.05 0 0.05 0.1
53.5 54 54.5 55 55.5 56 56.5 57 57.5 58
kx(K
1)
E(meV)
numerical analytical
1.1209(meV)
Figure 3.1: The lowest two bands which wave vector is near K1 (ky=0, −0.1K1 < kx <
0.1K1). The red line: the numerical result for three K point with SOI; blue line: the analytic result for 2 × 2 matrix with SOI, λ=120˚A2(InAs); m∗ = 0.023me; U0 = 165meV;
a=40nm; d=0.663a.
The Fig. 3.5 shows the energy of analytic energy band (restrict in subspace) is higher than numerical energy band (3 × 3 matrix) except the k ≈ 0 (close to K1). Because of the numerical energy band consider the 2W (higher energy), leading the energy higher than analytic energy band (only consider -W).
3.2 The position symmetry for muffin-tin triangular lattice
There are an external muffin-tin triangular potential in the 2DEG. This structure has a symmetry property for rotating 60◦ alone the z-axis. We can interpret the symmetry
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
property from Fig. 3.2, each muffin-tin triangular structure (a), (b), (c) in real-space have the corresponding BZ (a), (b), (c) in k-space. For example, the figure (a) rotate 60◦ to becomes figure (b) in real-space and the K system in (a) change to K0 system in (b) relatively in k-space. Because of the action for rotating 60◦ doesn’t change the structure, the rotating symmetry is tenable.
Figure 3.2: This figure shows the rotating symmetry property for triangular lattice, the original system (a) in real-space correspond to (a) in k-space, then rotate π3 from central point to become (b) in real-space and k-space , and do the same work to become (c) in real-space and k-space, where ˜n is an integer(the blue point note the system which has been rotated ).
For the analytic calculation, the wave function expanded from K10 point(using the K10, K20, K30 be the basis), the method is as same as K system. There are only H1 term different
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
from K system which the k become -k.
H1 = −~υ0k
Therefore, we obtain the subspace (as show in Eq. (3.12) and Eq. (3.13)) representation of H0, H1 and hSO , and count the energy dispersion. The result of energy dispersion in K0 system is as same as K system E = −W ±q¡~v
0
2 k¢2
+ 3 (sAλW )2. This result prove the position symmetry property( show in Fig. 3.2) which is authentic.
3.3 The numerical result compare with single well system in the presence of SOI
Using the same numerical program calculates the case of U0=-300meV. When the extra potential is negative, the MTP resembles many single wells. Such wave under MTP can be illustrated by the overlapping wave functions of the nearest single wells.
HSO = −λ
The disk-shaped potential with step-like profileV (r) = −V0θ(d2 − r) gives rise to SOI term, where d is the diameter of the single well.
HSO = −λV0δ¡d
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
wave function into the Schrodinger equation, the radial differential equation reads
· d
The radial function Rsl(r) has different coefficients for inside and outside the disk, that depend on the index s, given by
Rsl(r) =
The HSO is nonzero only at r=d2, the boundary condition that bring forth the spin de-generacy is given by
rdRl(r)
Finally,the wave function is continuous at boundary, we obtain the coefficient Cls , Els , energy level, and orbital quantum number.
Compared with the energy level of a single well, we can obviously discover the each energy band is almost same level (see Fig. 3.3). There are two energy bands equal to same orbital quantum number(l) when the |l| 6= 0. For the (l)=0 case, only have one energy band because the l=0 did not have opposite quantum number. This work can provide a method to prove the program which is authentic.
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
−300
−250
−200
−150
−100
−50 0
E(meV)
−300
−250
−200
−150
−100
−50 0
E(meV)
Γ M K Γ Γ M K Γ
l=0 l=1 l=2 l=0 l=3 l=1
Figure 3.3: Energy band structure with parameters units typical for InAs are: effective mass m∗=0.023me; a = 40nm; SO coupling constant λ=120˚A2 (a) numerical muffin-tin potential (U0 = −300meV ) in the presence of SOI. Compared with (b) single well (V0=300mev) in 2DEG.
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
3.4 Results for energy band structure in the presence of SOI
In our numerical examples, physic parameters are chosen for InAs in the practical ex-perimental parameters. The Fig. 3.4 shows the energy dispersion with SOI which open
Figure 3.4: Parameters units typical for InAs are: effective mass m∗=0.023me; U0 = 165meV , a = 40nm ; SO coupling constant λ=120˚A2 The blue line is without SOI and the red line is with SOI which spin-up and spin -down are flipping in muffin-tin lattice .
up gaps at K and Γ points (the magnitudes of each gaps are shown in Fig. 5.7), the
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
spin-up states and spin-down states are same energy dispersion but spins opposite site in z-direction (Fig. 3.4). And for the lowest two energy band, the spin-up and spin-down
34.5 35 35.5 36
E(meV)
K
Figure 3.5: The lowest two bands which wave vector is near K1. Red circle: the system with SOI; blue star: the system without SOI, λ=120˚A2(InAs); m∗ = 0.023me; U0 = 165meV; a=40nm; d=0.663a.
states mixing at K1 point without SOI (see Eq. (3.5)).
·
HSO, p2
2m∗ + V (r)
¸
=
·
(∂xV ) py− (∂yV ) px, p2
2m∗ + V (r)
¸
6= 0
. (3.22)
Because of Eq. (3.22), the states at K point is a superposition state with the basis is the eigenstate without SOI, leading to open up a gape for first lowest energy band and second
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
lowest energy band (Fig. 3.5).
1 2 3 4 5 6 7 8 9 10 11
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Crange
gap(meV)
Figure 3.6: This figure shows the value of gap for first lowest energy band and second lowest energy band which depend on Crange(show in appendix A).
The Fig. 3.6 shows the magnitude of the gap ( between first lowest energy band and second lowest energy band) would decrease when the Crange ( orbital index) increase.
We have trying other parameters for different a, d, U0, and roughly discuss the results, because for Crange=1 ( shown in appendix E).
3.5 The relationship between time reversal property and our system
The numerical results show the energy dispersion of spin-up and spin-down states are same energy dispersion. We analyze the these results by time reversal symmetry. The wave function and Schr¨odinger equation are given by
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
where the periodic function ucκ,s=1(r) = P
n
Time reversal operator Θ acts on Eq. (3.25), one obtain that
(HN.SO− shSO) e−iκ·r|u−κ,−si χ−s= Eκ,se−iκ·r|u−κ,−si χ−s, (3.26)
CHAPTER 3. ENERGY BAND STRUCTURE WITH SOI EFFECT
The eigenstate of Eq. (3.26) is |u−κ,−si, so the eigenenergy of this system must be E−κ,−s which implicate Eκ,s = E−κ,−s, and because Eκ,−s= E−κ,−s, which comes from the parity operator π acting on Eq. (3.25), we obtain
(HN.SO+ shSO) e−iκ·r|u−κ,si χs = Eκ,se−iκ·r|u−κ,si χs, (3.28)
where the rotating symmetry for triangular lattice (see Fig. 3.2 shows the system in our model with inversion symmetry, the result Eκ,s = Eκ,−s is proven.
3.6 Brief summary
Thus far in this chapter, we show that the energy band structure with SOI effect in external periodic potential in 2DEG. There exist a massless Dirac point at K1 without SOI effect ( as show in chapert2), and we considered the MTP gradient which arise the SOI, the degenerated energy at K point can open up a gap, and we also have an analytic calculation to prove the numerical result is authentic.
Chapter 4
Berry curvature with SOI effect in our system
Berry curvature is as a local gauge potential and gauge field associated with the Berry phase. These concepts were introduced by Michael Berry in a paper published in 1984 [14] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Such phase have come to be know as Berry phases. In this chapter, we will show the Berry curvature with SOI effect in our system.
4.1 Berry phase
In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quan-tum adiabatic theorem applies to a system whose Hamiltonian H(κ) depends on κ that varies with time t. If the n’th eigenvalue εn(κ) remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the eigenstate ¯
¯uκ(0),n®
will remain in an instantaneous eigenstate ¯
¯uκ(t),n®
, up to a phase, throughout the process. The state at time t can be written as
|Ψn(t)i = eiγn(t)e−
i
~
Rt 0
dt0εn(κ(t0))¯
¯uκ(t),n®
, (4.1)
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
where the second exponential term is ”dynamic phase factor” and the first exponential term is the geometric term, with γn being the Berry phase. By plugging into the time-dependent Schr¨odinger equation, we can obtain the solution of γn(t)
γn(t) = i From Stoke’s theorem, we have
γn(C) = i the arbitrary phase attached our expression in Eq. (4.3). To examine this we consider the following gauge transformation |˜uκ,ni = eiξ(κ)|uκ,ni, where the eiξ(κ) is a κ dependent phase factor. We get huκ,n | ∇κuκ,ni = i∇κξ (κ) + huκ,n| ∇κuκ,ni, and in substituting into Eq. (4.3), the additional term ∇κ× ∇κξ (κ) = 0. This step shows that the Berry curvature is independent of arbitrary phase factor which dependent on κ in the wave function. As such, the definition of Berry phase in Eq. (4.3) is uniquely defined.
4.2 Berry curvature analysis
For a closed path C that forms the boundary of a surface S , the closed-path Berry phase can be rewritten using Stokes’ theorem as γn =R
S
dS · Ωn(κ).
From Eq. (4.3), we get:
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
The formulation is as shown below: a complete set P
n0 of the second row is zero.
There is a useful relation for obtaining the numerical formulation:
huκ,n0| (∇κH) |uκ,ni = huκ,n0| (∇kH − H∇κ) |uκ,ni
= huκ,n0| ∇κEκ,n|uκ,ni − huκ,n0| Eκ,n∇κ|uκ,ni
= ∇κEκ,nhuκ,n0 | uκ,ni + Eκ,nhuκ,n0 | ∇κuκ,ni − Eκ,n0huκ,n0 | ∇κuκ,ni
= (Eκ,n− Eκ,n0) huκ,n0 | ∇κuκ,ni ,
(4.7)
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
where the H (κ) comes from H (κ) = U (κ) HU†(κ) = e−iκ·rHeiκ·r. Because of Schr¨odinger equation: H |Ψκ,n(r)i = HU†(κ) |uκ,n(r)i = εn(κ) |Ψκ,n(r)i = εn(κ) U†(κ) |uk,n(r)i.
From Eq. (4.7), we obtain:
huκ,n0 | ∇kuκ,ni = huκ,n0| ∇κH |uκ,ni
Eκ,n− Eκ,n0 , n 6= n0 , (4.8)
and substituted to Eq. (4.5). The numerical calculation of Berry curvature is read as:
Ωn(κ) = iX
n06=n
huκ,n| ∂kxH (κ) |uκ,n0i huκ,n0| ∂kyH (κ) |uκ,ni − (x ↔ y)
[En0(κ) − En(κ)]2 z.ˆ (4.9)
The Eq. (4.9) shows explicitly, that the Berry curvature is due to the restriction to a single band n and to the resulting virtual transitions to other bands n0 6= n, and the numerical result n0 is the effective number for two bands which are the nearest for each higher energy and lower energy (Because of the denominator [En(κ) − En0(κ)]2 in Eq. (4.9)).
For example, the n=1, n0=2, 3 and another case the n=4, n0=2, 3 ( lower energy), 5, 6 ( higher energy).
4.2.1 The analytic result of Berry curvature
The wave function Ψκ,s(r) may be approximately expressed as a linear combination of three plane-wave states.
The term H1 + HSO, when restricted to the sub-Hilbert space spanned by the two vectors (the degenerate eigenvectors of lowest two bands) is represented by a 2 × 2 matrix H˜1+ ˜HSO (shown in chapter 3.1.1 ), the meaning of this step is that we only focus on the Berry curvature of k-space near K point(the Dirac point of the lowest band).
H˜1+ ˜HSO =
−~v20kx −~v20ky+ i√
3sλW A
−~v20ky − i√
3sλW A ~v20kx
, (4.10)
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
Then we use the above equations, the Eq. (4.9) in this case becomes (the analytic result of Berry curvature)
Ωn=±1(κ)
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
4.2.2 The Berry curvature of numerical result compare with the analytic result
This section we will use the Berry curvature analytic formulation Eq. (4.13) to compare with the numerical results which consider three K points( unperturb points and same energy).
Figure 4.1: The inset shows the contour of Berry curvature for the lowest band ( in spin-up case) by considering three K point and we chose (a) ky=0kymax , (b) ky=0.3kymax and (c) ky=0.6kymax (black line) corresponding to the dispersion which is expanded from K1
(kx = ky = 0) in the main panel. The blue line is analytic result n=-1, s=1 (Crange=1);
the red line is numerical result (the lowest band).
From the Fig. 4.1 and Fig. 4.2 we can observe the magnitude of analytic Berry curva-ture at K point confirm to the numerical result. The analytic Berry curvacurva-ture match the
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
Figure 4.2: The inset shows the contour of Berry curvature for the second lowest band (in spin up case) by considering three K point and we chose (a) ky=0kymax , (b) ky=0.3kymax and (c) ky=0.6kymax (black line) corresponding to the dispersion which is expanded from K1 (kx= ky = 0) in the main panel. The blue line is analytic result n=1, s=1 (Crange=1);
the red line is numerical result (the second lowest band).
numerical results except K point.
The effect Berry curvature for first lowest energy band distribute around K point, and when Crange(shown in appendix A) increase the Berry curvature would stable (Crange ≈ 11, are shown in Fig. 4.3).
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
Figure 4.3: The Crangeincrease to 11, the numerical Berry curvature would almost stable.
4.2.3 The relationship between time reversal property and Berry curvature
The numerical results show that the spin-up and spin-down Berry curvatures are opposite sign (see Fig. 4.4 and Fig. 4.5), the curvature satisfies Ωn,s(κ) = −Ωn,−s(−κ). In this subsection, we will derive some symmetry relation of the Berry curvature. The specific symmetry we consider are the inversion symmetry (via parity operator π) and the time reversal symmetry ( via time-reversal operator Θ). Our Hamiltonian H has the property
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
[H, π] = 0 and [H, Θ] = 0. We start from the expression of the Berry curvature
Ωn(κ) = i h∇κuκ,n| × |∇κuκ,ni = ∇κ× An,s(κ) , (4.14)
where An,s(κ) is called Berry connection ( like a vector potential in k-space), the curve of An,s(κ) is Berry curvature. Because of the inversion symmetry ( proven by rotating 60◦ in chapter 3), we insert ππ−1 into Berry connection Eq. (4.14)
An,s(κ) = i huκ,n,s| ππ−1∇κππ−1|uκ,n,si = i hu−κ,n,s| ∇κ|u−κ,n,si = −An,s(−κ) . (4.15)
Here, we point out that π−1∇κπ=∇κbecause κ in ∇κ, or in |∇κuκ,ni is a classical vector, not an operator. Furthermore, π−1eiκ·r|uκ,n,si =e−iκ·r|u−κ,n,si, where π−1r = −rπ−1, and π−1|uκ,n,si=|u−κ,n,si. The symmetry property of the Berry connection we obtain in Eq. (4.15) is for the same spin index but for opposite κ. Corresponding, the symmetry property of Berry curvature is given by
Ωn,s(κ) = ∇κ× An,s(κ) = −∇κ× An,s(−κ) = Ωn,s(−κ) , (4.16)
Eq. (4.16) is resulted form inversion symmetry. The symmetry relation for Ωn,s(κ), due to time-reversal symmetry is derived in following. Denoting |αi = |uκ,n,si,
|βi = i∇κ|uκ,n,si we have
An,s(κ) = i huκ,n,s| ∇κ|uκ,n,si = hα | βi . (4.17)
Corresponding, we denote |˜αi = Θ |αi = |u−κ,n,−si,
¯¯
¯ ˜β E
= Θ |βi = −i∇κ|u−κ,n,−si. We have the identity hα | βi = Dβ˜
¯¯
¯ ˜α E
such that
hα | βi = Dβ˜
¯¯
¯ ˜α E
= i (∇κhu−κ,n,−s|) |u−κ,n,−si = −i hu−κ,n,−s| ∇κ|u−κ,n,−si = An,−s(−κ) ,
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
(4.18)
Between the second and the third steps in Eq. (4.18), we used the relationship (∇κhuκ,n,s|) |uκ,n,si =
− huκ,n,s| ∇κ|uκ,n,si, which is derived as follows
∇κhuκ,n,s | uκ,n,si = 0
⇒ (∇κhuκ,n,s|) |uκ,n,si + huκ,n,s| ∇κ|uκ,n,si = 0
⇒ (∇κhuκ,n,s|) |uκ,n,si = − huκ,n,s| ∇κ|uκ,n,si .
(4.19)
From Eq. (4.18), we obtain the Berry curvature
∇κ× An,s(κ) = ∇κ× An,−s(−κ) = −∇−κ× An,−s(−κ) = −Ωn,−s(−κ) . (4.20)
Eq. (4.20) is resulted from time-reversal symmetry. Inversion and time-reversal symme-tries together give us
Ωn,s(κ) = −Ωn,−s(−κ) = −Ωn,−s(κ) . (4.21)
This symmetry in Eq. (4.20) is clearly demonstrate in our numerical results, presented in Fig. 4.4 and Fig. 4.5. Thus confirming the validity of our numerical calculation.
4.2.4 The numerical result of Berry curvature
This subsection shows the numerical results of Berry curvature for the lowest energy band and the second lowest energy band ( the other figures for high energy band are showed in appendix D).
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
Figure 4.4: The Berry curvature of the lowest energy band for n=1 (a) the contour for spin-up (b) the contour for spin-down (n0=2,3); λ=120˚A2 (InAs); m∗ = 0.023me; U0 = 165meV; a=40nm; d=0.663a ( where kx=ky=0 is Γ point).
The Berry curvature distributions imply that the energy difference with others band is the main effect and the wave function term is the minor effect for leading the main contribution of Berry curvature , we can observe these from Eq. (4.9). And the important information in Fig. 4.4 and Fig. 4.5 is that the periodic triangular lattice in our system with inversion symmetry (K = K0 in k-space), the Berry curvature distributions is the same at six corners of BZ.
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
Figure 4.5: The Berry curvature of the second lowest energy band for n=1 (a) the contour for spin-up (b) the contour for spin-down (n0=2,3); λ=120˚A2 (InAs); m∗ = 0.023me; U0 = 165meV; a=40nm; d=0.663a ( where kx=ky=0 is Γ point).
4.3 Brief summary and discussion
This chapter we introduced the Berry curvature which comes from the Berry phase, and also show the numerical results of Berry curvature for each energy bands in first BZ.
Finally, we discuss the correlation between Berry curvature and time reversal property to guarantee the correct numerical results are correct.
CHAPTER 4. BERRY CURVATURE WITH SOI EFFECT IN OUR SYSTEM
However, the another important thing is to compare with the Berry curvature of graphene (see Fig. 4.6). Because of graphene without the inversion symmetry property ( the Berry curvature K 6= K0, see Fig. 4.6 (b)), we can understand simply from Fig. 4.7.
Figure 4.6: The (a) show the energy dispersion of graphene in BZ. (b) The Berry curvature of graphene in BZ. [15]
Figure 4.7: There are without rotating symmetry in graphene, which imply K 6= K0 in k-space (the blue point note the system which has been rotated ).
Chapter 5
Searching for quantum spin Hall effect in our system
The QSH systems are insulating in the bulk, they have an energy gap separating the va-lence and conduction bands, but on the boundary they have gapless edge or surface states that are topologically protected and immune to impurities or geometric perturbations [16], [17],[18], [19].
Therefore, this chapter we will use the topological invariant ( Chern number and Z2
number) to examine the edge state and classify the insulator for open boundary case.
5.1 The Chern number of the energy band
The Chern invariant is rooted in the mathematical theory of fiber bundles ( Nakahara, 1990), but it can be understood physically in terms of the Berry phase ( Berry, 1984) associated with the Bloch wave functions|un(κ)i. Provided there are no accidental de-generacies when k is |un(κ)i transported around a closed loop, acquires a well defined Berry phase given by the line integral of An = i hun| ∇κ|uni. This may be expressed as a surface integral of the Berry flux (Berry curvature) Fn = ∇ × An. The Chern invariant is the total Berry flux in the Brillouin zone, and distinguishes the two states ( bulk and
CHAPTER 5. SEARCHING FOR QUANTUM SPIN HALL EFFECT IN OUR SYSTEM
edge state) is a topological invariant similar to the genus.
The Chern theorem, which states that the integral of the Berry curvature over a close manifold is quantized in unit of 2π .This number is the so-called Chern number.
Cn= 1 2π
Z
BZ
d2κFn. (5.1)
The total Chern number, summed over all occupied bands, Coccupied =PN
n=1
Cnthat is invari-ant even if there are degeneracies between occupied bands, provided the gap separating occupied and empty bands remains finite. A fundamental consequence of the topological classification of gapped band structures is the existence of gapless conducting states at interfaces where the topological invariant changes. Such edge states are well known at the interface between the integer quantum Hall state and vacuum ( Halperin, 1982)and deeply related to the topology of the bulk quantum Hall state.
Figure 5.1: The rotating symmetry property for triangular lattice, the original system in real-space correspond to the BZ in k-space (top right), rotating x axis 180◦ the position in real-space is not change ( the blue point note the system which has been rotated ) and ky becomes −ky in k-space ( bottom right). The (a) shows the pink triangular is the smallest repeated unit cell in BZ.
CHAPTER 5. SEARCHING FOR QUANTUM SPIN HALL EFFECT IN OUR
Figure 5.2: The hexagon black line shows the area of Brillouin zone and the area of triangle dashed line is the basic area for integrating which we mention at below in main panel, and the inset shows the contour of Berry curvature near K1 ( red box) is isotropic
Figure 5.2: The hexagon black line shows the area of Brillouin zone and the area of triangle dashed line is the basic area for integrating which we mention at below in main panel, and the inset shows the contour of Berry curvature near K1 ( red box) is isotropic