半導体物性 - osaka universitykoun/lecs/semicond/...半導体物性...
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半導体物性多電子理論からのアプローチ
2. Many-electron theory大学院講義「半導体物性」
半導体はバンド理論(一電子描写)が最も成功した分野である。それにも関わらずなぜ多電子理論が必要か?
2-1 なぜ全エネルギーか?
1–2
• 全電子からの見方
• 一電子からの見方バンド計算
DFT計算
1つ1つの電子エネルギーは分かる
相互作用をしている多粒子系では限界
1つ1つのエネルギーを足しても全エネルギーにならない
相互作用をしている多粒子系の実験は、全エネルギーの差
結合エネルギー電子分光、光分光も全エネルギーの差
2. Many-electron theory
In the two-electron picture, a single-electron energy is not well defined.
1-3
One-electron model Two-electron model
Ground state
Excited state
2. Many-electron theory
全エネルギーは全てを決める
1–4
物質の結合
その結合エネルギーをどう計算するか?
電荷密度ρ(r)が分かれば求まる
(イオン,金属,共有性,ファン・デル・ワールス結合)
ρ(r) Ε
2. Many-electron theory
状態の変化
1–5
A B
Q=∆H反応熱
反応の方向性
>0
<0
exothermic
endothermic
∆G=∆H–T∆S
>0
<0
inhibit
proceed
=0 equilibrium
化学反応安定構造不純物濃度キャリア放出光学的遷移
2. Many-electron theory
2-2 密度汎関数理論
1–7
実際に解く方程式
N電子系の全エネルギー
コーン・シャム(KS)方程式
電子密度:ρ(r)
ρ(r)の汎関数
一電子近似
ハートリー・フォック(HF)方程式
c.f. 波動関数理論
n(r) = ϕi (r)i∑ 2
− !2
2m∇2 +Vion (r)+VH(r)+VXC(r)
⎡
⎣⎢
⎤
⎦⎥ϕi (r) = ε iϕi (r)
Etot=E[ρ(r)]
− !2
2m∇2 +Vion (r)+VH(r)− 3α
34π
ρ(r)⎧⎨⎩
⎫⎬⎭
1/3⎡
⎣⎢
⎤
⎦⎥ϕi (r) = ε iϕi (r)
2. Many-electron theory
Total energy
Etot[ρ] = T + Uion[ρ] + UH[ρ] + Uxc[ρ]
kinetic energyelectron-ioninteraction
electron-electroninteraction
Ψ −12m
∇ j2 Ψ
j∑
r (r)Vion (r)dr∫
Vion (r) = −Ze2
| r − R |R∑
UH[r ] = r (r)VH(r)dr∫Uxc[r ] = r (r)Vxc (r)dr∫
VH(r) = e dr ' r (r ')r − r '∫
approximate Uxc
(LDA)
1–8
2. Many-electron theory
1–9
Immediate applications of Etot
Eb[A-B] = E[A] + E[B] – E[AB]
Ecoh[A(sol)] = E[A(gas)] – E[A(sol)]
Eform[AmBn] = E[AmBn] – (mE[A]+ nE[B])
Binding energy
Cohesive energy
Formation energy
2. Many-electron theory
1–10
Cohesive and formation energies
2. Many-electron theory
1–11
Etot Ry/cell eV/atom
E(B.C.) eV/atom
B12C3 -102.2094 -92.709 -92.709
alternate -102.3036 -92.795 -92.795 B12 -68.0843 -77.195 -92.686 diamond -22.7329 -154.650
B (atom) -5.1709 -70.354 -85.535 C (atom) -10.7498 -146.260 Ecoh(B) 6.841 Ecoh (C) 8.390 Ecoh (B12C3) 7.260 Eform(B12C3) 0.023
alternate 0.109
E(B.C.) = 154E(B) + E(C)[ ]
Exp.
5.777.37
0.146
Formation energy of boron carbide
D. M. Bylander, L. KleinmanPRB 42 1394 (1990)PRB 42 1316 (1990)
mixed gas
mixed solid
2. Many-electron theory
1–12
Boron Carbide
rh
in
rh
ci
1
2
cc
c
ci
x y
z
3
4
c
0.023 eV 0.109 eV∆H = 0.146 eV (exp.)
2. Many-electron theory
...
1-13
ε i
kXΓ
ε i
Ii = E(!,ni ,!)− E(!,ni −1,!)
I (1) = E(N )− E(N −1),I (2) = E(N −1)− E(N − 2),
Etot = I (i )i=1
N
∑
2-3 一電子固有値の意味 orbital energy
ionization energy
2. Many-electron theory
If it were
then
1-14
I (i ) = εN+1−i
Etot = ε ii=1
N
∑ − 12
ρ(r)VH (r)dr∫ − ρ(r) Vxc (r)− εxc (r)[ ]dr∫
Actually,
Ii = E(!,ni ,!)− E(!,ni −1,!) = ε i
Etot = ε ii=1
N
∑
全体 ≠ 部分の和2. Many-electron theory
Statistics of impurity levels in a gap in semiconductors
1-15
nd =1
12exp β εd − µ( ){ }+1
1exp β εd − µ( ){ }+1
nd =1+ exp −β εd − µ +U( ){ }
12exp β εd − µ( ){ }+1+ 12 exp −β εd − µ +U( ){ }
FD distribution
U → ∞
U → 0
2. Many-electron theory
Ionization energies and eigenvalues are different things.
1-16
He atom
Relaxation of wave functions by removing an electron.
1s -1.8359
He+
1s -4.0
total energy-5.7234
C. C. Roothaan et al.Rev. Mod. Phys. 32 (1960) 186.
(Ry units)
Ionization energy1.7234
(-1.1404)
(-5.6685)
LDA
2. Many-electron theory
1 11.26
2 24.383
3 47.887
4 64.492
5 392.077
6 489.981
sumi I(i) 1030.08
Ionization potentials of carbon atom
(eV)Carbon
LSD
2p↑ 3.725
↓ 5.903
2s↑ 11.465
↓ 13.838
1s↑ 258.937
↓ 259.813
Etot 1019.501
orbital energy
CRC Handbook of Chemistry and Physics, 67th ed.
1-17
2. Many-electron theory
The relaxation effect of wave function becomes insignificant when N → ∞.
1-18
Koopmans’ theorem
E(!,ni ,!)− E(!,ni −1,!) = ε i
2. Many-electron theory
Koopmans’ theorem
Janak theorem
E(!,nN )− E(!,nN −1) = εN
E(!,ni ,!)− E(!,ni −1,!) ≈ ε i (!,ni − 0.5,!)
1-19
∂E(!ni!)∂ni
= ε i
E(!,ni ,!)− E(!,ni −1,!) = ε i
Significance of eigenvalues
the highest occupied state
HF approach
DFT
transition state
Perdew, et al.
2. Many-electron theory
2-4 Quasi-Particles
Interacting electrons
G(k,ω ) = 1ω − ε k − Σ(k,−iω )
!ε k = ε k +ReΣ(k,−i !ω +δ )
Γ k = − ImΣ(k,−i !ω +δ )Width:
Shift:
2. Many-electron theory
Valence band structure of Ge
circles:ARPESAngle-Resolved Photoelectron Spectra
solid linesCalculation
H. X. Chen et al., Phys. Rev. B42, 7429 (1990)
2. Many-electron theory
Lifetime of electrons
X. J. Zhou et al., Synchrotron Radiat. News 18, 15 (2005)
(0,0) – (π,π) direction
La2-xSrxCuO4
2. Many-electron theory
Quasiparticle bands in GeM. Rohlfing, et al., Phys. Rev. B48, 17791 (1993)
Calculation Experiments A. L. Wachs, et al., PRB 32, 2326 (1985).J. E. Ortega, et al., PRB 47, 2130 (1993).
2. Many-electron theory
Energy gap problem
k
N electrons
EDFT
k
N+1 electrons
EDFT
µ(N)
µ(N+1)Eg
Eg,DFT
∆
Ec = Etot(N +1) − Etot
(N )
Ev = Etot(N ) − Etot
(N−1)
Eg = µ (N+1) − µ (N )
= εN+1(N) − εN
(N)( )+ εN+1(N+1) − εN+1
(N)
= εN+1(N+1) − εN
(N)
Δ
2. Many-electron theory