Ab initio investigations of spin-dependent quantum transport

(1)

HAL Id: tel-02459090

https://tel.archives-ouvertes.fr/tel-02459090

Submitted on 29 Jan 2020

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Ab initio investigations of spin-dependent quantum

transport

Jiaqi Zhou

To cite this version:

Jiaqi Zhou. Ab initio investigations of spin-dependent quantum transport. Micro and nanotechnolo-gies/Microelectronics. Université Paris Saclay (COmUE); Beihang university (Pékin), 2019. English. �NNT : 2019SACLS508�. �tel-02459090�

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(!/e)S/cm

10−5_e2_/h

(9)

∆ ∆ µ1 µ2 z k_|| Γ ∆1 k_|| = (0.526, 0.900)

(10)

E = EF E = EF−0.3 E = EF E = EF − 0.3 k kz σxyz E = EF E = EF − 0.3 Γ 1 2 1 3 √ 2 3 E = EF E = EF+0.84 E = EF E = EF+ 0.84 k kz σxyz E = EF E = EF + 0.84 Γ 1 2 1 3 √ 2 3 1−x x 1−x x x x x x 1−x x 1−x x x E = EF kz σx yz E = EF k_|| kz = 0

(11)

Γ E = EF − 0.02 Γ E = EF−0.15 E = EF−0.36 ′

(12)

σz xy k E = EF E = EF − 0.09 k kz σzxy E = EF E = EF−0.09 Γ ±∞ p d

(13)

(14)

(15)

(16)

(17)

(18)

(19)

• ∆1 (!/e)S/cm (!/e)S/cm (!/e)S/cm E = EF+0, 84 1−x x 1−x x x (!/e)S/cm

(20)

•

(!/e)S/cm

(!/e)S/cm E = EF − 2.0

σz

(21)

σz

xy (!/e)S/cm

(22)

(23)

(24)

(25)

= − _{× 100% =} − _{× 100%.}

= ↑ + ↓ ↑ ↓

(26)

∆ ∆

(27)

↑ _k || = 0 k_|| = 0 ∆1 spd ∆ ∆1 ∆1 ∆1 ∆5 ∆′2 ∆2 ∆1 ∆′₂ ∆2 ∆5 ∆1 ∆1 ↑

(28)

300◦_C

(29)

425◦_C

400◦C

420◦C

(30)

(31)

(32)

Js

Jc

(33)

σzxy(ω) = ! ! BZ d3_k (2π)3 " n fnk ×" m̸=n 2 Im#⟨nk|ˆjx|mk⟩ ⟨mk |−eˆvy| nk⟩ $ (ϵnk− ϵmk)2− (!ω + iη)2 σz xy x y z ϵn ϵm n m fnk ˆjxz = 12{ˆsz, ˆvx} ˆ sz = !₂ˆσz ˆvy = 1_!∂H(k)_∂k_y H(k) ω η • •

(34)

dm dt =−γm × BM+ αm× dm dt + γ Ms T m = M/Ms Ms γ α m BM m T = τFLm× ζ + τDLm× (m × ζ), ζ τFL (m, ζ) τDL (m, ζ)

(35)

(36)

(37)

(38)

(39)

(T + V + V + T + V ) χ(RNI, rNe) = Eχ(RNI, rNe), RNI rNe χ(RNI, rNe) E T V V T V e2 ₌_{! = 2m} e= 1 e ! me T = ₋ " µ=1,NI 1 2Mµ∇ 2 µ, V = " µ̸=ν ZµZν |Rµ− Rν| , T =₋ " i=1,Ne ∇2i, V = " i̸=j 1 |ri− rj| , V =− " µ=1,NI " i=1,Ne Zµ |Rµ− ri| , Mµ Z

(40)

χ(Rµ, ri) = Φ(Rµ)ΨR(ri), ΨR(ri) (T + V + V ) ΨR(ri) = ERΨR(ri). R V Ψ( ) Ψ( ) = ψ1( 1)ψ2( 2) . . . ψN( N), ψi( i) Ψ( ) HΨ( ) = EΨ( )

(41)

E =⟨Ψ|H|Ψ⟩ =" i ⟨ψi|Hi| ψi⟩ + 1 2 " i,j ⟨ψiψj|Hij| ψiψj⟩ , Hi Hij ⎡ ⎣−∇2_{+ V} _{( ) +} " i′_(̸=i) ! d ′|ψi′( ′)| 2 | ′_{− |} ⎤ ⎦ ψi( ) = Eiψi( ). −∇2 V ( ) V Ψ(r1, r2, . . . , rN) = 1 √ N ! ) ) ) ) ) ) ) ) ) ) ) ) ) ψ1(r1) ψ2(r1) · · · ψN(r1) ψ1(r2) ψ2(r2) · · · ψN(r2) ψ1(rN) ψ2(rN) · · · ψN(rN) ) ) ) ) ) ) ) ) ) ) ) ) ) .

(42)

E = ⟨ψi(ri)|H|ψi(ri)⟩ ψ ⎡ ⎣−∇2_{+ V} _{( ) +} " i′(̸=i) ! d ′|ψi′( ′₎_|2 | ′_{− |} ⎤ ⎦ ψi( )− " i′(̸=i),|| ! d ′ψ ∗ i′( ′)ψi( ′) | ′ _{− |} ψi′( ) = Eiψi( ). * i′₍_̸=i),|| + d ′ ψ∗i′( ′)ψi( ′) |′_{− |} ψi′( ) || • V (r)

(43)

ρ(r) • E [ρ( )] = T + V + V + T + V = T [ρ( )] + E [ρ( )] + V ( ) + E , T [ρ( )] E [ρ( )] V ( ) E = T + V V ( ) E F [ρ( )] F [ρ( )] = T [ρ( )] + E [ρ( )], T [ρ( )] E [ρ( )]

(44)

ρ( ) T [ρ( )] E [ρ( )] , −∇2_{+ V} KS[ρ( )] -ψi( ) = Eiψi( ). ∇2 VKS VKS[ρ( )] = V ( ) + V [ρ( )] + V [ρ( )] = V ( ) + ! d ′ ρ( ′₎ | − ′_|+ δE [ρ( )] δρ( ) . ρ( ) = N " i=1 f (Ei, µ)|ψi( )|2. f (Ei, µ) µ VKS[ρ(r)] ρ(r) V [ρ(r)]

(45)

VKS[ρ(r)]

ψi(r)

Ei ψi(r)

ρ(r) ψi(r)

(46)

ρ(r) ρ(r) ρ(r) ρ′_(r) E [ρ( )] E [ρ( )] E [ρ( )] E [ρ( )] = T [ρ( )]_{− T [ρ( )] + E [ρ( )] − E} [ρ( )], T [ρ( )] T [ρ( )] E [ρ( )] E [ρ( )] E [ρ(r)] = ! drρ(r)ε [ρ(r)],

(47)

ε [ρ(r)] V [ρ( )] = δE [ρ( )] δρ( ) ≈ d dρ( ){ρ( )ε [ρ( )]} = ε [ρ( )] + ρ( ) dε [ρ( )] dρ( ) . ε [ρ( )] ε [ρ( )] ε [ρ( )]

(48)

µ1 µ2

(49)

|ψ⟩ H|ψ⟩ = E |ψ⟩ . H |ψ⟩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ H1 τ1 0 τ₁† Hd τ2† 0 τ2 H2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |ψ1⟩ |ψd⟩ |ψ2⟩ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠= E ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |ψ1⟩ |ψd⟩ |ψ2⟩ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, H1 |ψ1⟩ H2 |ψ2⟩ Hd |ψd⟩ τ1,2 G(E) (E_{− H)G(E) = I,}

(50)

|v⟩ |ψ⟩ = −G(E)|v⟩. G G† G g1 g2 |ψ2⟩ = g2(E)τ2|ψd⟩ , g2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ E_{− H}1 −τ1 0 −τ1† E− Hd −τ2† 0 −τ2 E− H2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ G1 G1d G12 Gd1 Gd Gd2 G21 G2d G2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ I 0 0 0 I 0 0 0 I ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. G1d = g1τ1Gd, G2d = g2τ2Gd, Gd = (E− Hd− Σ1− Σ2)−1, Σ1 = τ1†g1τ1 Σ2 = τ2†g2τ2 |v⟩ |ψR_⟩

(51)

|ψA_⟩ (E_{− H)|ψ⟩ = −|v⟩,} ) )ψR4 ))ψA4 |ψR⟩ = −G|v⟩, |ψA⟩ = −G†|v⟩. (E_{− H)(|ψ}R⟩ − |ψA⟩) = (E − H) 5 G_{− G}†6_{|v⟩ = (I − I)|v⟩ = 0.} A = i5G_{− G}†6. |ψ⟩ = A |v⟩ |v⟩ G = 1 E + iδ− H = " k |k⟩⟨k| E + iδ − ϵk , iδ |k⟩ H ϵk A =" k |k⟩⟨k| 2δ (E− ϵk)2+ δ2 δ→0 = 2π" k δ(E_{− ϵ}k)|k⟩⟨k|. A

(52)

|ψ1,n⟩ n a1 a1 |ψ1,n⟩ + |ψR⟩ |ψ1,n⟩ |ψ1,n⟩ τ1|ψ1,n⟩ = (E − H1)|ψ1,n⟩ |ψR_⟩ _|ψ 1,n⟩ (H1+ τ1+Hd+ τ1†+ τ2†+ H2+ τ2)(|ψ1,n⟩ + ) )ψR4_{) = E(} |ψ1,n⟩ + ) )ψR4_), (H − E)))ψR4₌_−τ† 1|ψ1,n⟩. |ψR_⟩ −τ1†|ψ1,n⟩ ) )ψR4 _{= Gτ}† 1|ψ1,n⟩ . |ψd⟩ |ψd⟩ = Gdτ1†|ψ1,n⟩ .

(53)

|ψ2⟩ |ψ2⟩ = g2τ2|ψd⟩ = g2τ2Gdτ1†|ψ1,n⟩ . |ψ1⟩ = (1 + g1τ1Gdτ1†)|ψ1,n⟩ . ρ =" n f (En, µ)|ψn⟩ ⟨ψn| , f (En, µ1) µ1 |ψd,n⟩ = Gdτ1†|ψ1,n⟩ . ρd,1 |ψ1,n⟩ ρd,1 = ! ∞ −∞ dE" n f (E, µ1)δ(E− En)|ψd,n⟩ ⟨ψd,n| = ! _∞ −∞ dEf (E, µ1)Gdτ1† a1 2πτ1G † d = 1 2π ! ∞ −∞ dEf (E, µ1)GdΓ1G†d.

(54)

Γ1 = τ1†a1τ1 = i(Σ1− Σ†1) ρ = 1 2π ! _∞ −∞ dE " i=1,2 f (E, µi)GdΓiG†_d. 0 = ∂ * i|ψi|2 ∂t = i7#_⟨ψ1|τ1|ψd⟩ − ⟨ψd|τ1†|ψ1⟩ $ +#_⟨ψ2|τ2|ψd⟩ − ⟨ψd|τ2†|ψ2⟩ $8 . * i|ψi|2 i |i⟩ ⟨i| j −e ij =−i(⟨ψj|τj| ψd⟩ − ⟨ψd|τj†|ψj⟩), j = 1, 2. i 1 2 =−i(⟨ψ2|τ2| ψd⟩ − ⟨ψd|τ2†|ψ2⟩). i 1 2 =⟨ψ1,n|τ1Gd†Γ2Gdτ1†|ψ1,n⟩,

(55)

Γ2 = τ2†(g2†− g2)τ2 n I 1 2 = ! ∞ E=−∞ dEf (E, µ1) " n δ (E− En)⟨ψ1,n|τ1Gd†Γ2Gdτ1†|ψ1,n⟩ = 1 2π ! ∞ E=−∞ dEf (E, µ1) Tr(G†_dΓ2GdΓ1) I = I 1 2− I 2 1 = 1 π ! ∞ −∞

dE[f (E, µ1)− f(E, µ2)] Tr(G†_dΓ2GdΓ1).

ρ(r) VKS[ρ(r)] H[ρ(r)] Σ1 Σ2 ρ′_(r) _ρ′_(r) ρ(r) ρ(r) ρ(r) ρ′_(r) ρ(r)

(56)

(57)

(58)

(59)

z

x y

(60)

k 31_{× 31 × 1} Gσ = e2 h " k_|| Tσ(k||, EF),

(61)

σ Tσ(k||, EF) σ k_|| = (kx, ky) e h 20_{× 20 × 1 k} k 300_{× 300 × 1} k 10× 10 × 10 σ_yzx =−e 2 ! 1 V N3 k " k Ωx_yz(k), k × × k

(62)

z

(63)

(64)

∆1 ↑ ↑ ↑ ↑ Γ ∆1 s pz dz2 ∆1 ∆1

(65)

k_||

∆1

k_|| = (0, 0)

(66)

Γ ∆1 ↓ ↓ ↓ _{= 0.8991} k_|| = (0.526, 0.900)

(67)

k_|| = (0.526, 0.900)

(68)

k_|| = (0.526, 0.900)

(69)

− ₌ ↑ ₊ ↓ ₌ ↑ ₊ ↓ ↑ ↑ ↓ ↓ ↓ ↓ −412 (!/e)S/cm −841 (!/e)S/cm E = EF−0.3 (!/e)S/cm E = EF + 3.82

(70)

σ_yzx =₋e 2 ! 1 V N3 k " k Ωx_yz(k), k Ωx_yz(k) =" n fnkΩxn,yz(k), Ωx_n,yz(k) = !2 " m̸=n −2 Im[⟨nk|12{ˆσx, ˆvy}|mk⟩⟨mk|ˆvz|nk⟩] (ϵnk − ϵmk)2 . Ωx yz(k)

(71)

E = EF E = EF − 0.3 E = EF E = EF − 0.3 k Ωx n,yz(k) = ⎧ ⎪ ⎨ ⎪ ⎩ sgn(x) lg|x|, |x| > 10, x 10, |x| ! 10.

(72)

E = EF − 0.3 −841 (!/e)S/cm k E = EF Γ Γ E = EF − 0.3 Γ Γ Γ kz σyzx E = EF E = EF − 0.3 Γ 1 2 1 3 √ 2 3 kz

(73)

Γ E = EF E = EF − 0.3 E = EF−0.3 (!/e)S/cm E = EF + 0.84 (!/e)S/cm −1713(!/e)S/cm E = EF − 4.42 −1577 (!/e)S/cm E = EF − 3.74

(74)

E = EF E = EF + 0.84 E = EF E = EF + 0.84 k E = EF + 0.84

(75)

E = EF + 0.84 (!/e)S/cm k E = EF E = EF+ 0.84 kz σyzx E = EF E = EF + 0.84 Γ 1₂ 1₃ √₃2 E = EF E = EF + 0.84

(76)

(!/e)S/cm E = EF (!/e)S/cm E = EF + 0.84 (!/e)S/cm β − (!/e)S/cm ± β ± (!/e)S/cm β β − − − − − EF EF − 0.4 EF + 1.24 EF + 3.82 EF + 0.84 ± ±

(77)

1−x x 1−x x x x

x x

(!/e)S/cm E = EF + 0.84

EF

(78)

1−x x

1−x x x

(79)

E = EF

kz σyzx

(80)

Γ

(81)

− (!/e)S/cm (!/e)S/cm (!/e)S/cm

E = EF + 0.84

1−x x

1−x x x

(82)

(83)

(84)

(85)

k 1_{× 21 × 21} k 1× 150 × 150 k 6_×16×16 k _× _× 10−5_e2_/h ↑ ↓ ↑ ↓ ↓ _{= 1.11}_{× 10}−3 _e2_/h ↓ _{= 9}_{× 10}−5 _e2_/h

(86)

k_|| kz = 0

(87)

↑ _{= 0.16} _Γ ↑ ↑ ↓ ↑ ↑ ₌ ↓ _{= 0.57} _Γ ↑ ₌ ↓ _{= 1.11}_×10−3_e2_/h ↓ ↓ _{= 1.82}_{× 10}−3 _e2_/h ↓ _{= 1.19}_×10−4 Γ ↑ ₌ ↓ _{= 0.57} _Γ Γ ↓ _{= 0.57} _1.19 _{× 10}−4 E = EF − 0.02

(88)

Γ

E = EF − 0.02

(89)

E = EF − 0.02 E = EF − 0.02 ↑ ₌ ↓ _{= 0.91} Γ E = EF − 0.15 E = EF − 0.36 ↑ _{= 0.95} _{E = E} F − 0.15 ↓ = 0.94 E = EF−0.36 ↓ _{= 1.19}_{× 10}−4

(90)

x E = EF − 0.15 E = EF− 0.36 E = EF − 0.15 ↑ _{= 0.95} _Γ E = EF−0.36 ′ k_|| = (0.283, 0.283) ′

(91)

↑ _{= 9.5}_{× 10}−3 ′ ′ ′ ↓ _{= 3.8}_{× 10}−8 Iσ = e h !

Tσ(E)[f (E− µ1)− f(E − µ2)]dE,

f µ1 µ2

(92)

(93)

k_||

(94)

(95)

− ! E = EF − 2.0

(96)

(97)

x y kx ky a b c a b c k 12_{× 10 × 6} k × ×

(98)

σ_αβγ α β γ α β α β C4v σz xy −σzyx σyzx −σzyx σzxy −σyxz σz yx −361 (!/e)S/cm σzxy (!/e)S/cm (!/e)S/cm σx yz σzyx σxzy σyzx σzxy σyxz −18 45 ₋₈₈ 286 ₋₁₇₆ ₋₃₆₁ _−0.72 −44 −44 −61 103 ₋₂₀₄ ₋₁₅ _−0.54

(99)

σzxy −204 (!/e)S/cm = max α,β,γ(σ γ αβ(EF))/σxx α,β,γ(σ_αβγ (EF)) σxx × × −0.72 −0.54 σz yx σyzx σxyz −390 −178 (!/e)S/cm (!/e)S/cm _−0.5 σz yx σz yx −528 (!/e)S/cm E = EF−0.094 −400 (!/e)S/cm E = EF+0.47 σzxy σyzx (!/e)S/cm E = EF − 0.048

(100)

σz xy

−204 (!/e)S/cm E = EF (!/e)S/cm E = EF − 0.09

Ωz n,xy(k)

(101)

σz xy k E = EF E = EF− 0.09 E = EF Γ ₋ (!/e)S/cm k

(102)

k kz σzxy E = EF E = EF − 0.09 Γ Γ −0.09 Γ Γ Γ k kz E = EF E = EF − 0.09 Γ E = EF

(103)

Γ

E = EF − 0.09 Γ

(104)

−34 (!/e)S/cm (!/e)S/cm E = EF− σz yx −361 (!/e)S/cm σz xy −204 (!/e)S/cm

(105)

(106)

x

(107)

±∞

2.866_×√2 = 4.053

(108)

1× 21 × 1 k

(109)

(110)

= ) ) ) ) ↑₋ ↓ ↑₊ ↓ ) ) ) )× 100%, ↑ ↓ p d p p d

(111)

p d

(112)

(E, V )

−V/2 ≤

(113)

(114)

(115)

∆1

(116)

(!/e)S/cm

σz yx

(117)

(118)

(119)

(120)

(121)

/ /

(122)

(123)

(124)

(125)

(126)

(127)

(128)

(129)

(130)

(131)

(132)

(133)

(134)

(135)

VSe2

2

(136)

(137)

WTe2

(138)

Mn3X X = Ge

(139)

(140)

(141)

(142)

(143)

(144)

(145)

(146)