Landau levels based on matrix mechanics Yuki Izaki

( )

2021 3

2021

Landau levels based on matrix mechanics Yuki Izaki

Abstract

In principle, the energy levels under a magnetic field can be precisely calculated based on the k p theory, as was shown by Luttinger and Kohn. In practice, however, it is too complex to calculate the energy levels of electrons in solids because of the noncommuta- tivity of the kinematical momentum operatorπ which does not allow us to use the simple diagonalization technique. Even today, the energy levels under the field for multiband systems can be obtained perturbatively using the Lowdin partitioning. The theory using the Lowdin partitioning can give a good agreement with the experiment on Bi. However, for PbTe, there is a considerable discripancy between the theory and experiment. In this study, we introduce a new nonperturbative technique (π-matrix method) to calculate exactly the energy levels under the magnetic field, considering the noncommutativity of π. By using this technique, we can rigorously calculate the energy of electrons in solids under the magnetic field. We apply this method to PbTe, a typical Dirac electron system, and obtained the spin-splitting parameter that agree with the experiment. This result can rectify the existing discrepancy between the experimental and theoretical results.

We also apply this method to Cd₃As₂, a typical Dirac semimetal. We show that the en- ergy levels of Cd₃As₂ exhibits anomalous properties that the previous studies have never encountered

(π=p−eA)

π π-matrix

PbTe

Cd₃As₂

1 3

2 7

2.1 . . . 7

2.2 . . . 9

2.3 . . . 10

2.4 k·p . . . 12

2.4.1 k·p . . . 12

2.4.2 Luttinger-Kohn . . . 14

2.5 . . . 18

3 23 3.1 . . . 23

3.2 . . . 27

3.3 . . . 28

3.3.1 . . . 29

3.3.2 . . . 30

4 k·p 32 4.1 . . . 32

4.1.1 . . . 32

4.1.2 . . . 36

4.2 k·p . . . 37

4.3 L¨owdin partitioning . . . 37

4.3.1 2 . . . 40

4.3.2 . . . 45

4.4 . . . 47

4.4.1 . . . 47

4.4.2 k·p . . . 48

4.5 Fan-diagram plot . . . 50

4.5.1 . . . 50

4.5.2 . . . 51

4.5.3 . . . 52

5 53 5.1 π-matrix . . . 53

5.2 π-matrix - - . . . 54

5.3 π-matrix - - . . . 56

5.4 π-matrix - k·p - . . . 56

6 59 6.1 PbTe . . . 60

6.1.1 Lent . . . 60

6.1.2 Lach-hab . . . 66

6.1.3 PbTe . . . 67

6.2 Bi . . . 69

6.2.1 Trigonal . . . 70

6.2.2 Bisectrix . . . 71

6.2.3 Binary . . . 72

6.2.4 Bi . . . 73

7 Cd3As2 77 7.1 . . . 78

7.2 . . . 78

7.2.1 Z.Wang . . . 79

7.2.2 Jeon . . . 81

7.2.3 2 . . . 83

7.3 . . . 84

7.3.1 . . . 86

7.3.2 . . . 90

8 93

94 95

1

1.0.1

Bohr-Sommerfeld [1–5]

g

k·p [6]

k·p ¯hk π

π

2

4×4 2 (2×2 )

! "

# $

%

&

' (

) *

!

!"#

$%&"'()*

+,-.

/!"#

012 34

!"#

5*.

5*. 5*.

6734 89.

:;<=>?@ABC

1.0.1

[7–9]

L¨owdin partitining

g [10]

( ) (

)

Bi

PbTe [10–14]

Bi₂Se₃ [15–17]

1

[14] Bi

500 Bi

g [10] PbTe

60% [14, 18]

2 2

[19–22] Bi g

0

[14]

Wien2k VASP

(π-matrix) [23]

π

π (π-matrix

, 1.0.2) PbTe Bi

Cd3As2

MZC fan-diagram plot x

! "

# $

%

&

' (

) *

!

+!"#

,-. /0

!"#

1*2 1*2 1*2

3456789:;<=

678>?

π@ABCDEF

GHIJKLMNN 1*28OPQRSTU +!"#8

VWXY*Z[J$%\]U^

1.0.2

M_fan

8 2

k·p 3

Cd₃As₂

4 k·p

5

π-matrix 6

PbTe

Bi π-matrix

7 π-matrix Cd3As2

2

[24–26]

2.1

p→π=p+eA (2.1.1)

π p

[pi, pj] = 0 (2.1.2)

A B

B =∇×A (2.1.3)

[πi,πj] = [pi+eAi, pj+eAj]

= [pi, pj] +e²[Ai, Aj] +e(Aipj+piAj −Ajpi−pjAi)

3 p=i¯h∇ ψ

piAjψ=−i¯h∇(Ajψ) = (pjAi−Aipj)ψ

[π_i,π_j] =e[A_ip_j+ (p_iA_j−A_jp_i)−A_jp_i−(p_jA_i−A_ip_j)]

=e(piAj −pjAi) =i¯he

!∂Aj

∂xi −∂Ai

∂xj

"

=−i¯he$ijkBk =π×π (2.1.4)

$_ijk π

H= π²

2m (2.1.5)

B z

A= (0, Bx,0)

H = 1 2m

#p²_x+ (p_y+eBx)²$

(2.1.6)

H y ψ y

p_y ¯hk_y

H= 1

2mp²_x+ e²B² 2m

!¯hk_y eB +x

"2

= 1

2mp²_x+ mω_c²

2 (x+X)² (2.1.7)

ωc ≡eB/m X ≡¯hky/(eB) (2.1.7) ψn,ky

ψn,ky(x, y) = 1

%Ly

e^ikyyφn(x+X) (2.1.8)

Ly y Φn(x)

H_n(x) φ_n(x) =

! 1

2ⁿn!√ πlB

"¹₂

e^−12(x/lB)2H_n

!x lB

"

(2.1.9) lB =%

¯

h/(eB) φ

π_±= (πx±πy)/√ 2

π+φn(x) =√

n+ 1φn+1

π₋φn(x) =√ nφn−1

E En=

! n+1

2

"

¯

hωc (2.1.10)

n= (0,1,2,· · ·)

¯

hω_c (2.1.10)

X X

X 0≤X < Lx

0≤ky <(eB/¯h)Lx y

ky 2π/Ly ξ(B)

ξ(B) = Ly

2π eB

¯

h Lx ≡ eBS

h (2.1.11)

S

z

A = (0, Bx,0) (2.1.10)

En =

! n+ 1

2

"

¯

hωc+¯h²k_z²

2m (2.1.12)

E = ¯h²k²/(2m)

m^∗ Ec

E =Ec+ ¯h²k² 2m^∗ [25]

En=Ec+

! n+1

2

"

¯

hω^∗_c +¯h²k_z²

2m^∗_c (2.1.13)

ω^∗_c =eB/m^∗_c

2.2

1/B

- -

EF λ n <λ

! n+ 1

2

"

¯

hωc< EF −¯h²k²_z

2m_e ≡E_F^# (2.2.1)

N

N = (λ+ 1)ξ (2.2.2)

ξ &

λ+¹₂'

¯

hωc =EF λ

(2.1.11) ξ N

λ=n^# λ=n^#−1 N

! λ+1

2

"

= E_F^#

¯ hωc

(2.2.3)

λ 1/B 1/B

1/B

G= 1

En−EF +iδ (2.2.4)

δ

1

x+iδ = P

x −iπδ(x)

→δ(x) =−1 πIm

! 1 x+iδ

"

P δ(x) D

D=(

k

δ(Ek−EF) (2.2.5)

En

D=−ξ π

(

n=0

) Im

! 1

En−EF +iδ

"

dkz

2.2.1 1/B

1/2¯hωc

E_F

2.3

-

-

*

pdq= (n+γ) 2π¯h (2.3.1)

1/B! T⁻¹"

2.2.1 1/B

γ 1/2 1/2 [1,27]

¯

hk→p+eA

*

pdq=

*

(¯hk−eA)·dq

=

*

−e(ρ×B+A)·dρ (2.3.2)

¯

hk˙ =−ev×B r˙=v = 1

¯ h

∂E

∂k ρ

¯

hk=−eρ×B (2.3.2)

*

ρ×B·dρ=−B·

*

ρ×dρ=−2πρ²B

=−2Φ

Φ (2.3.2)

*

A·dρ= )

(∇×A)dσ

= )

Bdσ =Φ

σ (2.3.2)

*

p·dq=eΦ (2.3.3)

(2.3.1) (2.3.3)

Φ= (n+γ)2π¯h

e (2.3.4)

σ S ¯hk˙ =−er˙×B

kˆz

kˆzׯhk˙ = ˆkz·B·(−er)˙ −kˆz(−er)˙ B

=−eB#

˙ r−kˆz

+kˆz·r˙,$

=eBρ˙ (2.3.5)

(2.3.5)

ρ(t)−ρ(0) =− ¯h

eBkˆz×[k(t)−k(0)] (2.3.6)

ρ k ¯h/eB π/2

σ S

σ= ¯h²e²B²S (2.3.7)

(2.3.3)

Φ=B ¯h²

e²B²S= 2π

e (n+γ)

→S = 2πeB

¯

h² (n+γ) (2.3.8)

2.4 k · p

k·p m^∗

k·p

k·p

J.M.Luttinger W.Kohn [6] k·p

2.4.1 k · p

k·p k0

[24] k·p V (x)

Hψ= -p²

2m +V (r) .

ψ=Eψ (2.4.1)

ψ

ψ=e^ik·ru_n,k(r) (2.4.2)

n k 2.4.1

2.4.2 .

Hψ= -p²

2m +V (r) .

e^ik·run,k(r)

= 1 2m

&

−i¯h∇²'

e^ik·run,k(r) +V (r)e^ik·run,k(r)

=Ee^ik·run,k(r) (2.4.3)

2.4.3 1

2m(−i¯h)²/

−k²e^ik·run,k(r) + 2ike^ik·r∇un,k(r) +e^ik·r∇²un,k(r)0

=e^ik·r -¯h²k²

2m + ¯h

mk·p+ p² 2m

.

u_n,k(r)

(2.4.4) (2.4.3) e^ik·r

1p² 2m + ¯h

mk·p+ ¯h²k²

2m +V (r) 2

u_nk(r) =Eu_nk(r) (2.4.5) (2.4.1)

1 H+ ¯h

mk·p+¯h²k² 2m

2

u_nk(r)

H^#= _m^¯hk·p+ ^¯h_2m^2k2

k·p H^# k

n

$n(k) =$n(0) + ¯h²k² 2m + ¯h

2m (

j$=n

|(n0|k·p|j0)|²

$n(0)−$j(0) (2.4.6) m^∗

m^∗= -i

¯ h

d²$n(k) dk²

.−1

(2.4.7) (2.4.6) (2.4.7)

m

m^∗ = 1 + 2 m

(

j$=n

|(n0|p|j0)|²

$_n(0)−$_j(0) (2.4.8)

2.4.2 Luttinger-Kohn

k·p k·p

k·p

Luttinger Kohn

[6] k·p

H0 U

H0 ψnk $n(k)

n k

H₀ψ_nk =$_n(k)ψ_n,k (2.4.9)

ψ

(H₀+U)ψ=$ψ (2.4.10)

ψnk=e^ik·runk (2.4.11)

unk ψnk

Luttinger-Kohn

χnk =e^ik·run0 (2.4.12)

un0 χ ψnk

Luttinger-Kohn χnk

f(r) ψnk

f(r) =(

n

)

dkgn(k)ψnk

=(

n

)

dkgn(k)e^ik·runk (2.4.13)

unk=(

n^!

bnn^!(k)un^!0 (2.4.14)

(2.4.14) (2.4.13)

f(r) =(

n

(

n^!

)

dkg_n(k)b_nn!e^ik·ru_n!0 (2.4.15) n n^#

f(r) =(

n

(

n^!

)

dkgn^!(k)bn^!ne^ik·run0

=(

n

)

dk˜gn(k)e^ik·rχnk (2.4.16)

˜

g_n(k) =(

n^!

g_n!(k)b_n!n(k) (2.4.17)

unk Luttinger-

Kohn χnk

χnk

(ψnk,ψn^!,k^!)= )

ψ^∗_nkψn^!k^!dr

= )

eⁱ

!k−k^!"_·r

u^∗_nkun^!kdr

=δ(k−k^#)δ_nn! (2.4.18)

χnk

(χ_nk,χ_n!,k^!)= )

eⁱ

!k−k^!"_·r

u^∗_n0u_n!0dr

(2.4.19) un0

u^∗_n0un0=(

m

Bⁿⁿ_m^!e^−iKm·r (2.4.20)

B_m^nn! Km (2.4.20)

(2.4.19)

(χnk,χn^!k^!)= )

eⁱ(^k−k^!)·ru^∗_n0un^!0dr

= )

eⁱ(^k−k^!−Km)·r(

m

Bⁿⁿ_m^!dr

= (2π)³(

m

B_m^nn!δ(k−k^!−Km) (2.4.21)

k, k^! k−k^!−K_m = 0

Km m=0

(2.4.21)

(χnk,χn^!k^!)= (2π)³B₀^nn!δ(k−k^!) (2.4.22) (2.4.20)

B_m^nn! = 1 Ω

)

e^iKm·ru^∗_n0un^!0dr (2.4.23)

m= 0 (2.4.18)

B₀^nn! = 1 Ω

)

u^∗_n0un^!0dr

= 1

(2π)³δnn^! (2.4.24)

(2.4.24) (2.4.22)

(χnk,χn^!k^!)=δnn^!δ(k−k^#) (2.4.25)

Luttinger-Kohn χnk

(2.4.10)

ψ=(

n^!

)

dk^#An^!(k^#)χn^!k^! (2.4.26) (2.4.10)

(H0+U)(

n^!

)

dk^#An^!(k^#)χn^!k^! =$(

n^!

)

dk^#An^!(k^#)χn^!k^! (2.4.27) χnk

(

n^!

)

dkχnk(H0+U)χn^!k^!An^!(k^#) =$(

n^!

)

dkAn^!(k)χnkχn^!k^!

(2.4.28)

χnk,χn^!k^! (nk|H|nk) (2.4.28)

(

n^!

)

dk(nk|(H0+U)|n^#k^#)An^!(k^#) =$An(k) (2.4.29) H0

(nk|H0|n^#k^#)= )

e^−ik·ru^∗_n0H0e^ik!·run^!0 (2.4.30)

H₀=p²/2m+V(x) H0e^ik!·rψ= p²

2m

+e^ik!·rψ,

+V (x)+

e^ik!·rψ,

(2.4.31)

(2.4.31) 1 p=−i∇

−1 2m

+∇²e^ik!·rψ,

=− 1 2m

+∇e^ik!·r∇ψ+ik^#∇e^ik!·rψ,

= −1 2m

+

ik^#e^ik!·r∇ψ+e^ik!·r∇²ψ−k^#2e^ik·r+ike^ik!·r∇ψ,

=e^ik!·r

!k^!·p m + p²

2m + k^#2 2m

"

ψ (2.4.32)

(2.4.31)

H0e^ik!·r=e^ik!·r

!k^#·p m + p²

2m + k^#2 2m

"

+V (x)e^ik!·r (2.4.33) (2.4.30)

(nk|H0|n^#k^#)= )

e^−ik·ru^∗_n0H0e^ik!·run^!0

= )

eⁱ(^k!−k)·r

u^∗_n0

!k^#·p m + p²

2m + k^#2

2m +V (x)

"

un^!0dr

= )

eⁱ(^k!−k)·ru^∗_n0

!

H0+ k^#·p m + k^#2

2m

"

un^!0dr

= )

eⁱ(^k!−k)·r

u^∗_n0

!

$n^!+ k^#·p m + k^#2

2m

"

un^!0dr (2.4.34) (2.4.25)

)

eⁱ(^k!−k)·ru^∗_n0u_n!0dr=δ(k−k^#)δ_nn!

=δ(k^#−k)(2π)³ Ω

)

u^∗_n0un^!0dr (2.4.35) (2.4.34)

(nk|H0|n^#k)=δ(k^#−k)(2π)³ Ω

) u^∗_n0

!

$n^!+ k^#·p 2m + k^#2

2m

"

un^!0dr

=δ(k^#−k) -!

$n^!+ k² 2m

"

δnn^!+kαp^α_nn!

m .

(2.4.36)

p^α_nn! (¯h= 1 )

p^α_nn! = (2π)³ Ω

) u^∗_n0

!1 i∇α

"

un^!0dr (2.4.37)

p^α_nn! p^α_nn!(k) =m∂$(k)/∂kα

p^α_nn = 0, p^α_nn! =p_n^α!n= (p^α_nn!)^∗ (2.4.38)

(2.4.29) U (2.4.20) (nk|U|n^#k)=

)

eⁱ(^k!−k)·r

U u^∗_n0un^!0dr

=(

m

)

eⁱ(^k!−k)·rBⁿⁿ_m^!e^−iKm·rU dr

= (2π)³(

m

B_m^nn!U(k−k^#+Km) (2.4.39)

U(k) U

U(k)≡ 1 (2π)³

)

dre^−ik·rU(r) (2.4.40)

Luttinger Kohn U

U k k^# Km

m= 0 m*= 0

(2.4.39) m= 0

(2.4.39)

(nk|U|n^#k)= (2π)³B₀^nn!U(k−k^#)

=δnn^!U(k−k^#) (2.4.41) (2.4.36) (2.4.41) (2.4.29)

!

$n+ k² 2m

"

An(k) + (

n^!$=n

kαp^α_nn!

m An^!(k) +

)

dk^#U(k−k^#)An(k^#) =$An(k) (2.4.42) Luttinger-Kohn

(2.1.1) ¯hk → π

(2.4.11)

k k π

(2.4.12) k

¯ hk → π

[ (2.4.14)] Luttinger-Kohn

Luttinger-Kohn k·p

4 k·p

2.5

m V (r) [28]

i¯h ∂

∂tΨ= [α·p+βm+V(r)]Ψ Ψ=

! φ χ

"

(2.5.1) α,β

αi=

! 0 σi

σi 0

"

,β =

! I2 0 0 −I2

"

(2.5.2)

σx =

! 0 1 1 0

"

,σy=

! 0 −i i 0

"

,σz =

! 1 0 0 −1

"

(2.5.3)

σ (2.5.1)

-! 0 σ σ 0

"

·p+

! I2 0 0 −I₂

"

m+V (r) . - φ

χ .

= ˆE - φ

χ .

⇒σ·pχ+mφ+V (r)φ=+

Eˆ^#+m, φ σ·pφ−mχ+V (r)χ=+

Eˆ^#+m, χ

⇒σ·pχ+V (r)φ−Eˆ^#φ= 0

σ·pφ−2mχ+V (r)χ−Eˆ^#χ= 0

⇒+

Eˆ^#−V (r),

φ−σ·pχ= 0 (2.5.4)

+Eˆ^#+ 2m−V (r),

χ−σ·pφ= 0 (2.5.5)

Eˆ = ˆE^#+m (2.5.6)

(2.5.5) m,V (r) Eˆ^#-m 2mχ=σ·pφ

⇒χ= σ·p

2m φ (2.5.7)

p/m=v

χ∝vφ (2.5.8)

χ φ v

(2.5.5)

χ=+

Eˆ^#+ 2m−V (r),₋1

σ·pφ (2.5.9)

(2.5.6)

+Eˆ^#−V(r),

φ−σ·p+

Eˆ^#+ 2m−V (r),₋1

σ·pφ

⇒Eˆ^#φ= 1

2m(σ·p) 3

1 +Eˆ^#−V (r) 2m

4₋1

(σ·p)φ+V (r)φ (2.5.10) +Eˆ^#−V (r),

/2m 3

1 +Eˆ^#−V (r) 2m

4₋1

/1−Eˆ^#−V (r)

2m (2.5.11)

pV (r) =V (r)p−ie¯h∇φ (2.5.12) (σ·∇V (r)) (σ·p) = (∇V (r))·p+iσ·[(∇V (r))×p] (2.5.13) (2.5.9)

Eˆ^#φ= 1

2m(σ·p) (σ·p)φ− 1 2m

Eˆ^#

2m (σ·p) (σ·p)φ + 1

2m 1

2m(σ·p) (V (r)) (σ·p)φ+V (r)φ

= p² 2m

3

1−Eˆ^#−V (r)

2mc² +V (r) 4

φ+ 1

4m² (−i¯h) (σ·∇V (r)) (σ·p)φ

= p² 2m

3

1−Eˆ^#−V (r)

2m +V (r) 4

φ+ ¯h²

4m² (∇V(r)) (∇V (r)) + ¯h

4m²[∇V (r)×pψ1] (2.5.14)

Eˆ^#−V (r) p²/2m m

+Eˆ^#−V (r),

p² p⁴/2m

∇φ·∇= dV (r) dr

∂

∂r (2.5.15)

∇φ= 1 r

dV (r)

dr r (2.5.16)

(2.5.14) Eˆ^#φ=

!p²

2m +V (r)− p⁴

8m³− ¯h² 4m²

dV (r) dr

∂

∂r + 1 2m²

1 r

dV (r) dr S·L

"

φ(2.5.17)

S L

S = 1

2¯hσ, L=r×p (2.5.18)

(2.5.17)

W L Γ X W K

E n er gy (e V)

2.5.1 Wien2k [29] PbTe

HSOC= 1 2m²

1 r

dV (r) dr S·L

!

= ¯h

4m²σ·∇V (r)×p

"

(2.5.19)

2.5.1 PbTe PbTe

-

En

En,+, En,−

En,σ =

! n+ 1

2

"

¯ hωc+σ

2gµBB (2.5.20)

Ez =En,+−En,− =gµBB (2.5.21)

g g µB µB =e¯h/2m

Ec=En,±−En−1,± = ¯hωc (2.5.22) MZC

MZC = EZ

EC

= gµBB

¯ hωc

= g

2 (2.5.23)

(2.5.23) g g

[13, 15–18, 30–33]

3

2005 Kane-Mele [34]

Z2

Z2

2007 Fu Bi

[35]

[36]

Cd3As2

3.1

[37]

[38–43]

[44]

R

H(R)|φn(R))=En(R)|φn(R)) (3.1.1)

|φn(R)) n En En

R

H[R(t)]|φn[R(t)])=i¯h∂

∂t|φn[R(t)])=En(R)|φn[R(t)]) (3.1.2)

R n

|φ_n[R(t)])=e^iθ(t)|φ_n[R(t)]) (3.1.3)

θ(t) (3.1.1) (3.1.3)

¯ he^iθ(t)

!

−∂θ(t)

∂t |φn[R(t)])+i∂

∂t|φn[R(t)])

"

=En[R(t)]e^iθ(t)|φn[R(t)]) (3.1.4) (3.1.4) (φn[R(t)]|

∂θ(t)

∂t =i(φn[R(t)]| ∂

∂t|φn[R(t)]) − 1

¯

hEn[R(t)]

θ

θ(t) =−1

¯ h

) t 0

En[R(t^#)]dt^# +i

) t

0 (φn[R(t^#)]| ∂

∂t^#|φn[R(t^#)])dt^# (3.1.5)

|φn[R(t)])

|φn[R(t)])=e^iΦB[R(t)]e^−i#0tdt!En[^R(^t!)]^/¯h|φn[R(t)]) (3.1.6) ΦB[R(t)] =i

) t 0

dt^#(φn[R(t^#)]| ∂

∂t^# |φn[R(t^#)]) (3.1.7) (3.1.6)

(3.1.6) R

ΦB R

|φn[R(t)])=|φn[R(t= 0)])

(3.1.4)

−¯he^iθ(t=0)∂θ(t)

∂t |φ_n[R(t)])=E_n[R(t)]e^iθ(t)|φ_n[R(t= 0)]) (3.1.6)

|φn[R(t)])=e^−i#0tdt!En[^R(^t!)]^/¯h|φn[R(t)]) (3.1.8)

ΦB = 0 (3.1.6) R

ΦB

θ_n(t) |φ_n(R))

|φ^#_n(R))=e^iθn(t)|φn(R)) (3.1.9)

ΦB ΦB

|φ^#_n(R)) Φ^#_B

Φ^#_B(t) =i ) t

0

dt^#(φ^#_n(R(t^#))| ∂

∂t^#|φ^#_n(R(t^#)))

=γ(t)− ) t

0

dt^#(φn(R(t^#))|∂θ_n(R)

∂t^# |φn(R(t^#)))

=γ(t)− ) t

0

dt^#∂θn(R)

∂t^#

=γ(t) +θn(R(0))−θn(R(t)) (3.1.10)

(3.1.10) ΦB

Φ^#_B

R(t) = R(0) ΦB = Φ^#_B

t=0 t=T

C ΦB[C]

Φ_B(T) [C] =i ) T

0

dt^#(φ_n(R(t^#))| ∂

∂t^#|φ_n(R(t^#)))

=i

*

C

dR·(φn(R)| ∂

∂R|φn(R))

≡

*

C

dR·An(R) (3.1.11)

An(R)≡i(φn(R)| ∂

∂R|φn(R)) (3.1.12)

C

(3.1.11)

Φ_B(T) [C] =

*

C

dR·A_n(R)

= )

S

dS ∇R×An(R)

= )

S

dS·Bn(R) (3.1.13)

Bn(R)≡ ∇R×An(R) =i 5∂φn

∂R 66 66×

66 66∂φn

∂R 7

(3.1.14) An

Bn B(R)

Bn,i(R) =i$ijk

5∂φn

∂Rj

66 66∂φn

∂Rk

7

(3.1.15)

$ijk (3.1.15)

Bn,i(R) =i$ijk

(

m

5∂φn

∂Rj

66 66φm

75 φm

66 66∂φn

∂Rk

7

(3.1.16) m

[1]m=n

(φn|φn)= 1 Rk

∂

∂Rk (φn|φn)= 5∂φn

∂Rk

66 66φn

7 +

5 φn

66 66∂φn

∂Rk

7

= 0 (3.1.17)

5∂φ_n

∂Rk

66 66φ_n

7

=− 5

φ_n 66 66

∂φ_n

∂Rk

7

(3.1.18)

5∂φn

∂R_j 66 66φn

7

=− 5

φn

66 66

∂φn

∂R_j 7

(3.1.19) (3.1.16)

Bn,i(R) =i$ijk

5∂φn

∂R_j 66 66φn

75 φn

66 66

∂φn

∂R_k 7

=i$ijk

5 φn

66 66

∂φn

∂Rj

75∂φn

∂Rk

66 66φn

7

= 0 (3.1.20)

[2]m*=m

H|φn)=En|φn) Rk

∂H

∂Rk|φn)+H 66 66∂φn

∂Rk

7

= ∂En

∂Rk |φn)+En

66 66∂φn

∂Rk

7

(3.1.21) (φm| n*=m,Em*=En

5 φm

66 66∂φn

∂Rk

7

= 1

En−Em

5 φm

66 66∂H

∂Rk

66 66φn

7

(3.1.22)

5∂φn

∂Rj

66 66φm

7

= 1

En−Em

5 φn

66 66

∂H

∂Rj

66 66φm

7

(3.1.23) (3.1.16)

Bn,i(R) =i$ijk

(

m$=n

(φn|∂R^∂Hj|φm) (φm|∂R^∂H_k|φn)

(En−Em)² (3.1.24)

(3.1.24) B_n(R)

n (3.1.24)

3.2

- [1, 45]

∆σ_xx∝cos -

2π

!F

B +γ±δ

".

(3.2.1)

F γ -

Sn = 2πeB

¯

h (n+γ) (3.2.2)

γ ΦB γ = ¹₂−^Φ_2π^B [2] δ 3

1/8 0

M

[1]

F

B =n+1 2 ±1

2M (3.2.3)

3.2.1 PbTe Sn M [14]

δ (3.2.3)

(3.2.1)

2π

!F B + 1

2− ΦB

2π

"

= 2π

! n+1

2 ±1 2M+1

2 −ΦB

2π

"

= 2π(n+ 1)±1

2M −ΦB

2π

(3.2.2) (3.2.1) cos

0

±1

2M−ΦB

2π = 0→φB =πM (3.2.4)

0 1

3.2.1

Pb1−xSnxTe PbTe

SnTe

MZC = 1 [14]

3.3

Armitage [46] 1928

[28]

[47]

Herring [48]

Adler Bell [49, 50]

2011 Wang

[51]

2011 X.Wang

[52]

3.3.1

[1]

[2]

k ≡ −k TRIM(time-reversal

invariant momenta) k=k₀

k =−k0

[3]

[53]

I4/mcm c

C2v C2v

Cd3As2 I41/acd C4

Cd3As2

[53]

3.3.2

-

S = 2πeB

¯

h (n+γ±δ)

δ 0 1/8 [1, 42]

- γ

γ = ¹₂− ^Φ2π^B π

0 fan-diagram plot

[54, 55] Cd₃As₂

3.3.1 fan-diagram plot

π [ ΦB/(2π) =0.56,0.58] [56]

C.M.Wang

[22]

δ π 0

3.3.1 Cd3As2 [56]

4

k · p

. k·p

π

[57–62] Bi 2

[63] Bi

PbTe g

L¨owdin

partitioning g

[10] Bi PbTe g [10, 14]

L¨owdin partitioning k·p

4.1

4.1.1

Bi Cohen Blount k·p

[11]

Bi L Cohen-Blount

H=







∆ 0 ¯hk·t ¯hk·u 0 ∆ −¯hk·u^∗ ¯hk·t^∗

¯

hk·t^∗ −¯hk·u −∆ 0

¯

hk·u^∗ ¯hk·t 0 −∆





 (4.1.1)

∆ t, u

vi,j

t≡v1,3=v4,2

u≡v1,4=−v3,2

Re(t) = 0 [12]

H=∆β+i¯hk·

> ₃ (

µ=1

W(µ)βαµ

?

(4.1.2)

W(1) = Im (u) W(2) = Re (u) W(3) = Im (t) (4.1.2)

W(1) = (γ,0,0) W(2) = (0,γ,0) W(3) = (0,0,γ) (4.1.2)

H =

! ∆ i¯hγk·σ

−i¯hγk·σ −∆

"

(4.1.3) (4.1.3)

HDirac=

! m p·σ p·σ −m

"

[8, 63] E

A, B, C, D det

- A B C D

.

= detAdet/

D−CA⁻¹B0

[64, 65] E det

- ∆−E i¯hγk·σ

−i¯hγk·σ −∆−E .

= det [∆−E] det#

(−∆−E)−(−i¯hγk·σ) (∆−E)⁻¹(i¯hγk·σ)$

= 0 (4.1.4) I ∆ =∆I E = EI k_± = (kx±iky)/√

2,γ^# =

√2γ

(−i¯hγ^#k·σ) (∆−E)⁻¹(i¯hγ^#k·σ) = ¯h²γ^#2

∆−E

! kz k₋ k+ −kz

" !

kz k₋ k+ −kz

"

= ¯h²γ^#2

∆−E

&

k_x²+k²_y+k_z²' I (4.1.4)

−∆²+E²−¯h²γ²&

k_z²+k₋k₊'

= 0

→E =±

@

∆²+ ¯h²γ²(k_z²+k₋k+) (4.1.5) 4.1.1

(4.1.3) ¯hk→π

H =

! ∆ i¯hγπ·σ

−i¯hγπ·σ −∆

"

(4.1.6)

E Hψ=Eψ

H²=

! ∆ i¯hγπ·σ

−i¯hγπ·σ −∆

" !

∆E iγπ·σ

−iγπ·σ −∆

"

=

! ∆²+γ²(π·σ) (π·σ) 0

0 ∆²+γ²(π·σ) (π·σ)

"

(4.1.7)

(π·σ) (π·σ) =

! ¯h²k_z²+ 2π₋π+ 0 0 ¯h²k_z²+ 2π₊π₋

"

2π+π₋ =π_x²+π²_y+i(π×π)_z

= (2n+ 1)e¯hB+e¯hB 2π₋π₊ =π_x²+π²_y−i(π×π)_z

= (2n+ 1)e¯hB−e¯hB

4.1.1 2 ∆= 0.1,γ= 1

∆²+ (2n+ 1±1)e¯hBγ²+ ¯h²k_z²γ²−E²= 0 (4.1.8)

→E =± A

∆²+ 2γ²

! n+ 1

2± 1 2

"

e¯hB+ ¯h²k_z²γ² (4.1.9) j=n+ 1/2±1/2 j = 0

(4.1.8)

Ez =En,+−En,−

= A

∆²+ 2γ²

! n+ 1

2+ 1 2

"

¯

heB+ ¯h²k_z²γ²− A

∆²+ 2γ²

! n+ 1

2− 1 2

"

¯

heB+ ¯h²k_z²γ² (4.1.10) E_C =E_n,+−E_n−1,+

= A

∆²+ 2γ²

! n+ 1

2+ 1 2

"

e¯hB+ ¯h²k_z²γ²− A

∆²+ 2γ²

! n+ 1

2− 1 2

"

e¯hB+ ¯h²k_z²γ² (4.1.11)

EZ Ec MZC

MZC = 1 (4.1.12)

1

4.1.2

Zhu [13] Bi

[66–69]

σ 2g^#µBB

g^# g [13, 33, 70]

E_n,σ^Ex−Dirac= A

∆²+ 2∆¯hω_c

! n+1

2 +σ 2

"

+¯h²kz2

2mz

+σ

2g^#µ_BB (4.1.13)

E_Z^Dirac EZ=E_Z^Dirac+g^#µBB

MZC = EZ

EC

= 1 + g^#µBB

E_n,+^Dirac−E_n^Dirac_−1,+ (4.1.14)

4.2 k · p

k·p π

k·p

k·p

H= p²

2m +V (r) + ¯h

4m²c²σ·∇V (r)×p (4.2.1)

Luttinger-Kohn [6] k·p

[71, 72]

(

n^!

-!

En+ ¯h²k² 2m

"

δnn^!δσσ^!+ ¯hk·v^σσ_nn^!!

.

cn^!σ^!(k) =Ecnσ(k) (4.2.2)

n 2n×2n

H= ¯h²k² 2m +













E1 0 ¯hk·t1 ¯hk·u1 ¯hk·t2 ¯hk·u2 · · · 0 E1 −¯hk·u^∗₁ ¯hk·t₁^∗ −¯hk·u^∗₂ ¯hk·t^∗₂ · · ·

¯

hk·t^∗₁ −¯hk·u1 E3 0 ¯hk·s1 ¯hk·w1 · · ·

¯hk·u^∗₁ ¯hk·t1 0 E3 −¯hk·w^∗₁ ¯hk·s^∗₁ · · ·

¯

hk·t^∗₂ −¯hk·u2 ¯hk·s^∗₁ −¯hk·w1 E4 0 · · ·

¯

hk·u^∗₂ ¯hk·t2 ¯hk·w^∗₁ ¯hk·s1 0 E4 · · ·

... ... ... ... ... ... . ..











 (4.2.3)

v^↑↑_mn=(ψm↑|vψn↑)=(Cvψm↑|Cψm↑)=(ψn↓|vψm↓)=v^↓↓_nm (4.2.4) v_mn^↑↓ =(ψ_m↑|vψ_n↓)=(ψ_m↑|vCψ_n↑)=(CvCψ_n↑|Cψ_m↑)=−v_nm^↑↓ (4.2.5) C = JK Cψn↑ =ψn↓,(φ|ψ) =(Cψ|Cφ),C² =−1 J

K (4.2.3)

tn =v_0n^↑↑,un =v^↑↓_0n E2n =E2n−1 (4.2.4), (4.2.5)

4.3 L¨owdin partitioning

k·p π

π

L¨owdin partitioning

2×2 L¨owdin partitioning

(4.2.3) [10, 71]

H=H⁽⁰⁾+H⁽¹⁾+H⁽²⁾+· · · (4.3.1)

H⁽⁰⁾mm^! =H⁰mm^! (4.3.2)

H⁽¹⁾mm^! =H^#mm! (4.3.3)

H⁽²⁾mm^! = 1 2

(

l

H^#mlH^#lm!

! 1

Em−El

+ 1

Em^!−El

"

(4.3.4)

H⁽²⁾11 = H13H31

E1−E3

+ H14H41

E1−E3

+· · ·

= (π·t1) (π·t^∗₁) E1−E3

+(π·u1) (π·u^∗₁) E1−E3

+· · ·

=(

n$=1

1

E₁−E_2n−1[(π·tn) (π·t^∗_n) + (π·un) (π·u^∗_n)]

=(

n$=1

-π·αn ·π

2 +(−i)e¯h

2c An·B .

(4.3.5)

αn,i,j = 1 E1−E2n−1

&

tnit^∗_nj +tnjt^∗_ni+uniu^∗_nj+unju^∗_ni'

(4.3.6)

An= 1 E1−E2n−1

(tn×t^∗_n+un×u^∗_n) (4.3.7)

(t·π) (t^∗·π) = π·α·π

2 +(−i)e¯h

2 (t×t^∗)·B (4.3.8)

H⁽²⁾12 =(

n$=1

1 E1−E2n−1

[−(π·tn) (π·un) + (π·un) (π·tn)] (4.3.9)

=(

n$=1

ie¯h E1−E2n−1

(tn×un)·B (4.3.10)

H⁽²⁾21 =(

n$=1

1 E1−E2n−1

[−(π·u^∗_n) (π·t^∗_n) + (π·t^∗_n) (π·u^∗_n)] (4.3.11)

=(

n$=1

−ie¯h

E₁−E_2n−1(t_n^∗ ×u^∗_n)·B (4.3.12)

H⁽²⁾22 =(

n$=1

1 E1−E2n−1

[(π·t^∗_n) (π·tn) + (π·u^∗_n) (π·un)]

=(

n$=1

-π·αn·π

2 −(−i)e¯h

2 An ·B .

(4.3.13)

Heff = π·α·π 2

+(

n$=1

- imµB

E₁−E_2n−1

! −(tn×t^∗_n+un×u^∗_n) 2tn×un

−2t^∗_n×u^∗ tn×t^∗_n+un×u^∗_n

"

·B .

(4.3.14) α

α_ij = δ_ij m +(

n$=1

tnit^∗_nj +tnjt^∗_ni+uniu^∗_nj+unju^∗_ni E1−E2n−1

(4.3.15)

µB =e¯h/2m (4.3.5)

¯

hk → −i¯h∇+eA≡π (4.3.15)

(4.3.14)

¯ hωc

¯

hωc =e¯hB

@

detα(α⁻¹)_ii (4.3.16)

i Ez

E²_z =m²µ²_B



4 66 66 66

(

n$=1

(tn×un)·B E1−E2n−1

66 66 66

2

− B(

n$=1

(tn×t^∗_n+un×u^∗_n)·B E1−E2n−1

C2



=m²µ²_BB·G·B (4.3.17)

Gij = 4



(

n$=1

tn×un

E1−E2n−1





i



(

n$=1

t^∗_n×u^∗_n E1−E2n−1





j

−



(

n$=1

tn×t^∗_n+un×u^∗_n E1−E2n−1





i



(

n$=1

tn×t^∗_n+un×u^∗_n E1−E2n−1





j

(4.3.18)

i E_z =±(g/2)µ_BB g 2m√

G_ii

z MZC

MZC = ∆Ez

¯ hωc

=

A Gzz

αxxαyy−α²_xy (4.3.19)