effective 4 × 4 Hamiltonian

The band structure of two route of effective 10 × 10 Hamiltonian is similar to the other when d is not near d_c. We can use any one effective Hamiltonian to describe the system at d = 7nm(See Fig. A.14). In the following analysis, we use the effective Hamiltonian H_10×10^2D to study the topological nature of the system.

Figure A.14: It shows the eigenenergy of ˜H_10×10^2D and H_10×10^2D with the well thickness 7nm.

The solid line is the eigenenergy of H_10×10^2D . The dash line is the eigenenergy of ˜H_10×10^2D .

In the current work[4], the edge state is discovered in the energy gap between the conduction (E1) and heavy hole (H1) bands, so we focus upon the energy range near E1 and H1 subbands. We reduce the effective Hamiltonian H_10×10^2D to an effective 4 × 4 Hamiltonian H_{ef f} with Lo ¨wdin perturbation theory up to 2nd order. In the set of basis (|E1, +i, |H1, +i, |E1, −i, |H1, −i) , H_{ef f} is of the form:

H_{ef f} = H⁽⁰⁾+ H⁽¹⁾+ H⁽²⁾. (A.26)

H⁽⁰⁾ is the eigenenergy of E1 and H1 subbands. H⁽¹⁾ is of the form:

H⁽¹⁾ = form those terms is much smaller than the other terms like C and γ so we drop those terms. Furthermore the M^C terms just correct the value of M₁^E and M₁^H and we can make those terms zero by an unitary transformation. Dropping those terms doesn’t affect the physical picture of H_{ef f}. The a and a₁ are also dropped because C is much smaller than P and we also can make a or a₁ zero in the similar way of M^C. Here we only keep the P² terms in H⁽²⁾ because P is much larger than the others terms.

We redefine the parameters kept in H_{ef f}.

Hef f = Dk²+

EFFECTIVE 2D HAMILTONIAN

Table A.2: The band structure parameter of Hef f

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

ours 4.83 753 578 376 1.68

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

Markus K¨onig[7] 10 686 512 364 1.6

a

This band parameter is for d = 7nm.

b

In ref[1], they didn’t include DSOI.

Where the definition of E₁(z), E₂(z), L₁(z), L₂(z) and H_j(z) are in Appendix B. The value of parameters is showed at Table A.2. Our parameter A, B, D and δ are similar to other’s result. Only the value of M is much different to the other group. The physics of two set of band parameters is the same (See chapter 6.).

We show the band structure from H_{ef f} and H_10×10^2D at Fig. A.15. Those two Hamil-tonian gives the similar energy dispersion so we can use H_{ef f} to study the topological physics of CdTe/HgTe/CdTe quantum well system.

Figure A.15: It shows the eigenenergy of conduction and heavy hole bands from two effective Hamiltonian. The blue solid line is from H_10×10^2D . The red dash line is from H_{ef f}.

Basis definition of H _10×10 ^2D

Both |Hi, +i and |Hi, −i subbands are the solution of the same differential equation. We define ∓H_i(z) as the space part of |Hi, ±i.

hHi, +| zi =

0 0 −H_i(z) 0 0 0

 . hHi, −| zi =

0 0 0 0 0 H_i(z)

 .

Where Hi(z) is the eigenfunction with eigenenergy E_i^H of Schr¨odinger equation:

[E_v(z) + ∂_zT_H(z) ∂_z] H_i(z) = E_i^HH_i(z) . (B.1)

The wave function has specific symmetry at z-direction. We define H_i(z) with odd i as even function and H_i(z) with even i as odd function. For the even function, H_i(z)’s form is

H_i(z) =











b^H_i e^βiHz a^H_i cos(α_i^Hz) b^H_i e^−βiHz

for

z < −d/2

−d/2 < z < d/2 d/2 < z

.

For the odd function, H_i(z)’s form is

H_i(z) =











b^H_i e^βiHz a^H_i sin(α_i^Hz)

−b^H_i e^−βiHz for

z < −d/2

−d/2 < z < d/2 d/2 < z

.

Where α^H_i = r

Eν^Hg−E_i^H

T_H^Hg and β_i^H = r

E_i^H−E^Cd_ν

T_H^Cd . We get the eigenenergy E_i^H through the boundary condition that is from continuity of wave function and Schr¨odinger equation.

For the even function, the boundary condition at z = −d/2 is

−T_H^Cdβ_i^Hcos α^H_i d 2



+ T_H^Hgα_i^Hsin α^H_i d 2



= 0. (B.2)

For the odd function, the boundary condition at z = −d/2 is

T_H^Cdβ_i^Hsin α^H_i d 2



+ T_H^Hgα^H_i cos α^H_i d 2



= 0. (B.3)

For |E1, ±i and |L1, ±i subbands, P kz terms make the symmetry of the |Γ6, ±1/2i component opposite to the |Γ8, ±1/2i component. We set the |Γ6, ±1/2i component is pure real so P kz terms also make the |Γ8, ±1/2i component pure imaginary.

We defined E¹(z) as the space part of |Γ6, ±1/2i subbands and E²(z) as the space part of |Γ8, ±1/2i subbands.

hE1, +| zi =

E₁(z) 0 0 −E₂(z) 0 0

 . hE1, −| zi =

0 E₁(z) 0 0 −E₂(z) 0

 .

Where E₂(z) is pure imaginary and E₁(z) is real. They satisfy the set of Schr¨odinger equations.

[E_c(z) − ∂_zT_c(z) ∂_z] E₁(z) − r2

3P ∂_zE₂(z) = E₁^EE₁(z) . (B.4a) [Ev(z) + ∂zTL(z) ∂z] E2(z) −

r2

3P ∂zE1(z) = E₁^EE2(z) . (B.4b) Where E₁^E is eigenenergy of E1 subband. We let e^δEz as the solution of HgTe region and e^γEz as the solution of CdTe region. δ^E and γ^E are functions of E₁^E and determined by

Schr¨odinger equation(B.4).

All γ^E are pure real (See Fig. B.2) but the part of δ^E may be imaginary. The functional form of conduction band will dependent on E₁^E. When E₁^E > E_v^Hg, two of δ^E will be

From equation(B.4a), we have Di = −i q3

From the continuity of wave function and Schr¨odinger equation at boundary z = −d/2, we derive the set of equations determining eigenenergy :

A₁cosh δ₁^Ed

Figure B.1: This picture show the solution of δ^E. The solid line is the real part. The dash line is the imaginary part. The right vertical line is E_v^Hg(= 0eV ). The left vertical line is E_c^Hg(= −0.303eV ). When E₁^E is between E_v^Hg and E_c^Hg, all δ^E are real. Otherwise two of δ^E will be pure imaginary. Here we only show two of roots. The other root are minus times of the roots showed here.

Figure B.2: This picture show the solution of γ^E. The solid line is the real part. The dash line is the imaginary part. The left vertical line is E_v^Cd(= −0.57eV ). The right vertical line is E_c^Cd(= 1.036eV ). In the energy range E_v^Cd < E₁^E < E_c^Cd, all γ^E are real. Here we only show two of roots. The other root are minus times of the roots showed here.

and E₂(z) are condition of equation(B.7), the eigenenergy is determined by the set of equations:

2

We can follow the similar procedure to get the eigenenergy of L1 subbands. The form of light hole subband is

hL1, +| zi =

Where L2(z) is pure imaginary and L1(z) is real. The Schr¨odinger equation describes L1

subband is the set of the equations: are of the forms:

L₁(z) = The eigenenergy is determined by the set of equation:

2

Basis definition of ˜ H _10×10 ^2D

For ˜Si subbands, we define  of ˜Si subband is the set of equation listed below.

E_c(z) + T_c(z)k²_z
S˜₁ⁱ(z) +

Where ε^Hg,c_i = E_c^Hg− ˜E_i^S, ε^Hg,v_i = E_v^Hg− ˜E_i^S, ε^Cd,c_i = E_c^Cd− ˜E_i^S and ε^Cd,v_i = E_v^Cd− ˜E_i^S. For

function at boundary z = −d/2, we derive the set of equations:

The Schr¨odinger equation is continuous at boundary z = −d/2. We have

ε→0lim

C depends on the materials so it is a function of z. By treating Ck_z terms in the equation(C.5b) and (C.5b) as ¹₂{C, kz}, we obtain

The (k_zC) term contribute a δ-function at the boundary. This δ-function givens an addition in the boundary condition. From equation(C.5) and (C.6), we derive the set of

equations: Those two sets of equation above determine the eigenenergy.

We can do the similiar produre to find ˜Ai subbands. We define   of ˜Ai subband is the set of equation listed below.

E_c(z) + T_c(z)k²_z
A˜ⁱ₁(z) +

Figure C.1: This picture show the solution of ˜δ^S. The solid line is the real part. The dash line is the imaginary part. The right vertical line is E_v^Hg(= 0eV ). The left vertical line is E_c^Hg(= −0.303eV ). When ˜E^S is between E_v^Hg and E_c^Cd(= 1.036eV ), two of ˜δ^E are pure imaginary. The other ˜δ^Ss are real. Here we only show three of roots. The other root are minus times of the the roots showed here.

Figure C.2: This picture show the solution of ˜γ^S. The solid line is the real part. The dash line is the imaginary part. The left vertical line is E_v^Cd(= −0.57eV ). The right vertical line is E_c^Cd(= 1.036eV ). In the energy range E_v^Cd < E₁^E < E_c^Cd, all ˜γ^S are real. Here we only show three of roots. The other root are minus times of the roots showed here.

A˜ⁱ₃(z) are function at boundary z = −d/2, we derive the set of equations:

2

From Schr¨odinger equation at boundary z = −d/2, we derive the set of equations: Those two sets of equation above determine the eigenenergy.

Appendix D

The time reversal operator for H _{ef f}

The time reversal operator Θ is of the form:

Θ = U_ΘK. (D.1)

K is the conjagate operator and UΘ is the unitary transformation. The definition of UΘ

is

[UΘ]ij = hi| Θ |ji . (D.2)

Where |ii is the ith basis vector. The time reversal relation of the angular momentum state |j, mi is

Θ |j, mi = e^iπ[j−m]|j, −mi . (D.3) From the definition of basis vectors, we have

|H1, ±i = ∓H₁(z) |Γ8, ±3/2i . (D.4)

|E1, ±i = E₁(z) |Γ6, ±1/2i + E₂(z) |Γ8, ±1/2i . (D.5)

The quantum number j of Γ6 subbands is 1/2, and the quantum number j of Γ8 subbands is 3/2. E₂(z) is pure imaginary and H₁(z) and E₁(z) are real. According to the property

of the basis vector, the time reversal relation of E1 and H1 subbands is

Θ |H1, ±i = ∓ |H1, ∓i . (D.6)

Θ |E, ±i = ± |E, ∓i . (D.7)

In the set of basis vector (|E1, +i, |H1, +i, |E1, −i, |H1, −i) , the time reversal operator Θ is of the form:

Θ =





0 −σ_z σ_z 0



K. (D.8)

Appendix E

The eigenvalue and eigenvector of the special 4 × 4 Hamiltonian

Consider a 4 × 4 Hamiltonian H_4×4:

H_4×4 = Dk²+





h₊ F F^† h−



. (E.1)

Where h_τ and F are

h_τ =





−M + Bk² Ak_τ Ak−τ M − Bk²



.

F =





ak₊ δ + bk² δ + bk² a⁰k−



.

The relation between h₊ and h− is

h− = Vµh+V_µ^†. (E.2)

Where V_µ = µe^−iσzφ, the k is the amplitude of the k vector and the φ is the polar angle of the k vector. Let the eigenvector ϕ of this form:

ϕ^†=

˜

ϕ^† ϕ˜^†V^†



. (E.3)

The Sch¨odinger equation becomes a set of equations:

Dk²+ h₊
˜ϕ + F V_µϕ = E ˜˜ ϕ. (E.4a) Dk²+ V_µh₊V_µ^†
V_µϕ + F˜ ^†ϕ = EV˜ _µϕ.˜ (E.4b)

Because F V_µ is hermitian, the equations (E.4a) and (E.4b) are the same.

F Vµ = µ





ak (δ + bk²)e^iφ (δ + B⁰k²)e^−iφ a⁰k



. (E.5)

Therefore, we can derive the eigenenergy of H_4×4 by the 2 × 2 Hamiltonian H_2×2^µ .

H_2×2^µ = Dk²+





−M + Bk²+ µak (Ak + µ [δ + bk²]) e^iφ (Ak + µ [δ + bk²]) e^−iφ M − Bk²+ µa⁰k



. (E.6)

The eigenenergy is

E_ρµ = Dk²+ µa₊k + ρ q

(M − Bk²+ µa−k)²+ (Ak + µ [δ + bk²])². (E.7)

Where a_ν = (a + νa⁰)/2. We have the form of ˜ϕ so we also have the form of ϕ.

ϕ_ρµ = Nρµ

√2

















[Ak + µ (δ + bk²)] e^iφ [E_ρµ− Dk²+ M − Bk²− µa⁰k]

µ [Ak + µ (δ + bk²)]

µ [E_ρµ− Dk²+ M − Bk²− µa⁰k] e^iφ

















. (E.8)

Appendix F

Basis definition of H _W .

Substituting −i∂_y for k_y, H₀ is of the form:

H₀(W ) =





h₊(0, −∂_y; W ) 0 0 h₋(0, −∂_y; W )



. (F.1)

Where

hτ(0, −i∂y; W ) =





−M − [B + D] ∂_y² τ A∂_y

−τ A∂_y M + [B − D] ∂_y²



. (F.2)

We have h− = σ_zh₊σ_z so we can obtain the form of the eigenstates.

hS; i, +| yi =

S₁ⁱ(y) S₂ⁱ(y) 0 0

 . hA; i, +| yi =

Aⁱ₁(y) Aⁱ₂(y) 0 0

 . hS; i, −| yi =

0 0 S₁ⁱ(y) −S₂ⁱ (y)

 . hA; i, +| yi =

0 0 Aⁱ₁(y) −Aⁱ₂(y)

 .

Where S₁ⁱ(y) and S₂ⁱ(y) are the elements of |Si, +i. Aⁱ₁(y) and Aⁱ₂(y) are the elements of

|Ai, +i. H₀ is real so S₁ⁱ(y), S₂ⁱ(y), Aⁱ₁(y) and Aⁱ₂(y) are all real.

The Schr¨odinger equation of S subbands is the set of equation:

− M + [B + D] ∂_y²
S₁ⁱ(y) + A∂_yS₂ⁱ(y) = E_i^SS₁ⁱ(y) . (F.3a)

Where E_i^S is the eigenenergy. We let e^λy be the solution of S₁ⁱ(y) and S₂ⁱ(y). From equation (F.3), we derive the form of λ:

λ²_ν = α + νβ

B²− D². (F.4)

Where α = −M B + E_i^SD + ^A₂² and β = p

α² + ([E_i^S]²− M²)(B²− D²). The value of A² is larger than 2M (B − D) and M B is positive in our parameter and ref[1]. Therefore when |E_i^S| < M , we can find that all λs are real and the eigenstate in this energy range is edge like state. In the other hand, when |E_i^S| > M , we can find that two λs are pure imaginary and the eigenstate in this energy range is bulk like state.

S₁ⁱ(y) is even and S₂ⁱ(y) is odd. For edge like state, S₁ⁱ(y) and S₂ⁱ(y) are of the forms:

S₁ⁱ(y) = P+

cosh [λ+y]

cosh [λ₊w] + P−

cosh [λ−y]

cosh [λ−w]. (F.5a)

S₂ⁱ(y) = P₊Q₊sinh [λ₊y]

sinh [λ₊w] + P−Q−

sinh [λ−y]

sinh [λ−w]. (F.5b) Where w = W /2. Taking this form into equation(F.3b), we obtain

Q₊ = _Aλ¹

+ M + E_i^S− _B−D^α−β
tanh [λ₊w] and Q− = _Aλ¹

− M + E_i^S− _B−D^α+β
tanh [λ−w]. Be-cause the value of A is very large, the value of λ is too large to numerically calculate the eigenenergy with the normal form of wave function. We set the wave function in this way[4] to avoid the numerally overflow near the edges. The wave function is zero at the edge. Therefore the equation that determines the eigenenergy is of the form:

Q−= Q₊. (F.6)

For bulk like state, S₁ⁱ(y) and S₂ⁱ(y) are of the forms:

S₁ⁱ(y) = P₊cosh [λ₊y]

cosh [λ₊w] + P−cos [λ−y]. (F.7a) S₂ⁱ(y) = P₊Q₊sinh [λ₊y]

sinh [λ₊w] + P₋Q₋sin [λ₋y] . (F.7b) Where λ−= Im[λ−], Q₊ = _Aλ¹

+ M + E_i^S−_B−D^α−β
tanh [λ₊w] and Q− = _Aλ¹

− M + E_i^S− _B−D^α+β
. For the similar reason of edge like state, the wave function

of λ₊ part must be this form. The wave function is zero at the edge. Therefore the equation that determines the eigenenergy is of the form:

Q−sin [λ−w] = Q₊cos [λ−w] . (F.8)

By the similar way, we can also obtain the eigenenergy of A subbands. The Schr¨odinger equation of A subbands is the set of equation:

− M + [B + D] ∂_y²
Aⁱ₁(y) + A∂yAⁱ₂(y) = E_i^AAⁱ₁(y) . (F.9a) M + [B − D] ∂_y²
Aⁱ₂(y) − A∂yAⁱ₁(y) = E_i^AAⁱ₂(y) . (F.9b)

Where E_i^A is the eigenenergy. Aⁱ₁(y) is odd and Aⁱ₂(y) is even. For the edge like A state, Aⁱ₁(y) and Aⁱ₂(y) are of the form:

Aⁱ₁(y) = P₊Q₊sinh [λ₊y]

sinh [λ₊w]+ P−Q−

sinh [λ−y]

sinh [λ−w]. (F.10a) Aⁱ₂(y) = P₊cosh [λ₊y]

cosh [λ+w] + P−

cosh [λ−y]

cosh [λ−w]. (F.10b)

Taking this form into equation(F.9a), we obtain Q+ = _Aλ¹

+ M − E_i^A−_B+D^α−β
tanh [λ+w]

and Q−= _Aλ¹

− M − E_i^A− _B+D^α+β
tanh [λ−w]. The eigenenergy is determined by the equa-tion.

Q−= Q₊. (F.11)

For bulk like state, Aⁱ₁(y) and Aⁱ₂(y) are of the forms:

Aⁱ₁(y) = P+Q+

sinh [λ₊y]

sinh [λ₊w] + P−Q−sin [λ−y] . (F.12a) Aⁱ₂(y) = P₊cosh [λ₊y]

cosh [λ₊w]+ P−cos [λ−y]. (F.12b) Where λ−= Im[λ−], Q₊ = _Aλ¹

+ M − E_i^A− _B+D^α−β
tanh [λ+w] and Q− = _Aλ¹

− M − E_i^A− _B+D^α+β
. The eigenenergy is determined by the equation.

Q−sin [λ−w] = Q₊cos [λ−w] . (F.13)

The analytical form of γ ˜ _{M S}

Consider a N × N matrix γ. Let vector |i⁰i is eigenvector of γ with eigenvalue g_i.

γ |i⁰i = g_i|i⁰i . (G.1)

The set of vector (|i⁰i) contains N elements and it is linear independent.

The matrix γ is non-hermitian so the vectors |i⁰i are not orthonormal. We need an orthonormal set of vector so we define a orthonormal set of vector 

˜i
.



˜1 = |1⁰i for m = 1. (G.2a)

| ˜mi = Nm |m⁰i −

m−1

X

n=1

|˜ni h˜n |m⁰i

!

for m > 1. (G.2b)

The set of vector (|i⁰i) is linear independent so the subset of vector (|j⁰i ; j ≤ n⁰) is linear independent and expands a n⁰ dimension vector space V⁰. According to the definition of vector

˜i, we have



˜i = ˜N_i|i⁰i − ˜N_i

i−1

X

j=1

C_ij ˜j

= ˜N_i|i⁰i − ˜N_i

i−1

X

j=1

C_ijN˜_j|j⁰i + ˜N_i

i−1

X

j=1

C_ijN˜_j

j−1

X

k=1

C_jk  

˜kE

=

i

X

k=1

C˜_ki|k⁰i. (G.3)

The subset of vector 

˜j ; j ≤ n⁰
is linear independent and expands a n⁰dimension vector space ˜V . From the equation (G.3), the vector space ˜V is equal to the vector space V⁰.

For i > n⁰, the vector  ˜i

is orthogonal to the all elements of the subset of vec-tor 

˜j ; j ≤ n⁰
. So it is also orthogonal to the all elements of the subset of vector (|j⁰i ; j ≤ n⁰). The matrix γ doesn’t change vector |i⁰i. The vector γ

˜i is of the form.

γ ˜i =

i

X

k=1

g_kC˜_ki|k⁰i. (G.4)

Therefore the matrix element˜j γ

˜i is zero when j > i.

[1] B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, Science 314, 1757 (2006).

[2] Markus K¨onig, Steffen Wiedmann, Christoph Br¨une, Andreas Roth, Hartmut Buh-mann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang, Science 318, 766 (2007)

[3] Hasan, M.Z. and C.L. Kane (2010). ”Colloquium: Topological insulators,” Reviews of Modern Physics. vol. 82, pp. 3045-3067.

[4] Bin Zhou, Hai-Zhou Lu, Rui-Lin Chu, Shun-Qing Shen and Qian Niu, Phys. Rev.

Lett 101, 246807 (2008).

[5] L. B. Zhang, Feng Zhai, and Kai Chang, Phys. Rev B 81, 235323 (2010).

[6] Roland Winkler: Spin-Orbital Coupling Effects in Two-Demensional Electron and Hole systems (Springer, Berlin, Heidelberg, 2003).

[7] Markus K¨onig, Hartmut Buhmann, Laurens W. Molenkamp, Taylor Hughes, Chao-Xing Liu, Xiao-Liang Qi, and Shou-Cheng Zhang , J. Phys. Soc. Jpn. 77, 031007 (2008).

[8] R. Winkler, L.Y. Wang, Y.H. Lin, C.S. Chu, Solid State Communications 152 2096 (2012).

[9] Y. Takagaki, Phys. Rev B 85, 155308 (2012).

[10] Y. Takagaki, J. Phys. Condens. Matter 24 (2012) 435301

[11] Viktor Krueckl and Klaus Richter, Phys. Rev. Lett 107, 086803 (2011).

[12] E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C. R. Becker, G.

Landwehr, H. Buhmann, and L. W. Molenkamp, Phys. Rev B 72, 035321 (2005).

[13] M. Cardona, N. E. Christensen, and G. Fasol, Phys. Rev. B 38, 1806 (1988).

[14] Wen Yu Shan, Hai Zhou Lu and Shun Qing Shen, New Journal of Physics 12 (2010) 043048.

[15] Xiao-Liang Qi, Taylor L. Hughes, Shou-Cheng Zhang, Phys. Rev B 78, 195424 (2008).

[16] Emil Prodan, Phys. Rev B 80, 125327 (2009).

[17] Huichao Li, L. Sheng, D.N. Sheng, and D.Y. Xing, Phys. Rev B 82, 165104 (2010).

[18] J.J. Sakurai: Advanced Quantum Mechanics (Addison-Wesley, Reading, MA, 1967).

[19] Shuichi Murakami, arXiv1006.1188v3.

在文檔中 破壞塊材反轉對稱在CdTe/HgTe/CdTe結構的量子自旋霍爾物理的影響 (頁 103-129)