Post on 28-Aug-2020
4. Deep levels
4.Deeplevels
大学院講義「半導体物性」
深い準位の物理
4. Deep levels
深い準位とは
ギャップ中の準位 > kTr.t.
1,伝導特性の不活性化キャリア捕獲
2,様々な発光特性
3,大きな原子位置変位
Jahn-Teller効果
4. Deep levels
GIAのホームページよりhttps://www.gia.edu/gia-about
ダイヤモンドの色と不純物
4.1 Effects of Impurities on Diamond
4. Deep levels
型 I II
Ia Ib IIa IIb
紫外光吸収領域 λ < 340 nm λ < 225 nm
黄色、褐色 ブルー
赤外光吸収領域 2.5 < λ <10 µm 2.5 < λ <6 µm
暗抵抗 (Ω•cm) > 1014 > 1014 10< ρ < 1014
不純物 Nが大量に含まれる N濃度はB濃度より大きい N濃度はB濃度以下
500-2000 ppm 50-500 ppm
Nは2量子複合体を形成している
Nは単独で置換位置に入る
Bが含まれる
作成条件
天然ダイヤモンドのほとんどはこの型
天然のものではめったにないが、高圧合成ダイヤモンドのほとんどはこの型
天然ではあまりない。低圧合成ダイヤモンドの多くはこの型
天然にはほとんど見られないが、合成ではできる
ダイヤモンドの種類
4. Deep levels
Type IBdispersed N atoms
(C-center)
Type IaApairs of N atoms
(A-center)
Type IaA/Bpairs & 4N atoms
(A, B-center)& platelets
Type IaA/B irregular
pairs & 4N atoms, platelets, DL, Void
T>1500
Annealing temperature (°C)
T>2600 T>2500 – 2700
様々な種類のダイヤモンドの作成
4. Deep levels
[111]
(a) (b)
ダイヤモンド中の窒素不純物の配置
IB IA
IaA
4. Deep levels
-3 -2 -1 0 1 2 3 4 5 6
Energy [eV]
0
10
20
30
DO
S [
sta
te/e
V.c
ell
]
Diamond
N
2N
4N:V
ダイヤモンド中の窒素不純物によるギャップ状態
4. Deep levels
4.2 Large atom relaxations
Ga
As Si
(a) (b)
(c) (d)
d 0 DX -
Ga
AsSi
Ga
AsS
d 0
Ga
AsS
DX -
4. Deep levels
縮退した電子軌道が部分的にしか占められていないとき、分子構造の対称性は落ちる。
線形構造以外の分子
H. A. Jahn and E. Teller, Proc. Roy. Soc. (London) A161, 220 (1937); A164, 117 (1938);
4.2.1 Jahn-Teller effects
4. Deep levelsSquare molecule
I II
III
Eσ
Eσ
Eσ'
Eσ'
ψσ'ψσ
degenerate orbitals
deformation
4. Deep levels
Perturbation energy
Selection ruleC4v
C4v E 2C4 C2 2σv 2σd
A1 1 1 1 1 1A2 1 1 1 -1 -1B1 1 -1 1 1 -1B2 1 -1 1 -1 1E 2 0 -2 0 0
If Ea(1) does not vanish, a JT distortion of the b-type occurs for a orbital.
E ⊗ E = A1 + B1 + E
Ea(1) = ϕa ub ϕa
Γa ⊗Γb ⊗Γa ≠ 0
Γb ⊂ Γa ⊗Γa( )
4. Deep levels
CB
VB
V2+ V+ V0 V-
A1
T2
4.2.2 Vacancy
4. Deep levels
QT2,2QT2,3
QT2,1QA1
QE,1 QE,2
Td E 8C3 6σd 6S4 3C2
A1 1 1 1 1 1
A2 1 1 -1 -1 0
E 2 -1 0 0 2
T1 3 0 -1 1 -1
T2 3 0 1 -1 -1
Character table of Td
Coupling of electron and phonon
phononsA1 + E + T2
T2 ⊗ T2 = A1 + E + T2
electrons T2
4. Deep levels
Impurity levels
V2+
V+
V0
V2+V0E(1,2)
E(0,1)
E(0,2)
E0
+
Ev Ecε(+2/0)
+2E
0
+
Ev Ecε(+2/+) ε(+/0)
+2
before afteratom relaxation
V+: unstable
4. Deep levels
Jahn-Teller distorsion
E0(Q) = – kQ221
E1(Q) = – kQ2 + ε1(Q) - µ21
E2(Q) = – kQ2 + ε1(Q) + ε2(Q) - 2µ21
ε1(Q) = εL - VQ
ε2(Q) = ε1(Q) + U
Energy-minimum point
E0(µ) = 0 at Q = 0
E1(µ) = εL–µ–EJT at Q = ––
E2(µ) = 2E1(µ)–η at Q =2 ––kV
kV
η = EJT–UEJT = ––kV2
Ueff = U–2EJT
Numeric estimation
Q = 2u = 0.16 Åk = 12.0 eV/Å ≈ 4 x 1.89 md/ÅV = -3.15 eV/Å
EJT = 0.11 eV
Ueff = 0.13 eV
4. Deep levels
ε(0/+) = E(0) – E(+)
ε(+/2+) = E(+) – E(2+)
V0 → V+
V+ → V2+
Ueff = E(2+)+ E(0) – 2E(+) = ε(0/+) – ε(+/2+)Effective U
2V+ → V0 + V2+
When U<2EJT negative Ueff !
Negative U
4. Deep levels
dynamic JT effect
縮退したE モードの変位が時間的に変化
QE,1 cosωt+ QE,2 sinωtしかし、ポテンシャルに非調和項があると、回転は止り3つの局所安定状態が生じる。
元の対称性が復元
4. Deep levels
4.3 Calculation of Deep Levels
GaAs1–xPx
Isoelectronic impurities for P
Direct gap Indirect gap
GaAs GaP
Impurity Role ElectronegativityN acceptor 3.0P 1.64Bi donor 1.24
Luminescence centers
LED
Laser
Characteristics of deep levels
Localized WFs Break of k-selection rules
Interactions with remote bands
Green’s Function Method
Formal matters
ρ(E) = − 1πIm Tr(G){ }
Φ = (E − H0 )−1VΦ
(H0 +V )Φ = EΦ
G0 = limη→0 E − H0 + iη[ ]−1
4. Deep levels
H0Φ0 = E0Φ0
Φ = Φ0 +G0VΦ
ρ(E) = 1πddEIm log(detG){ }
or
δρ(r,E) = − 1πddEIm log det I −G0 (r,r,E)V[ ]( ){ }
G0 (r,r ',E) = r G0 r ' = limη→0 r | k k | r ' (E − Ek + iη)k∑ −1
ρ0 (r,E) = r | k 2δ (E − Ek )k∑
G = G0 +G0VG
Dyson equation
δρ(r,E) = 1πddEIm log detG(r,r,E)
detG0 (r,r,E)⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎪
⎫⎬⎭⎪
density-of-states
G = 1E − H
4. Deep levels
Unperturbed spectrum
Resonant state solution
det I −G0V[ ]= 0
G0 = limη→0 E − H0 + iη[ ]−1
Perturbed spectrumRe{G0}
Im{G0}
1/U
ρ0 (E) = − 1πIm Tr(G0 ){ }
Φ = G0VΦ
Φ = Φ0 +G0VΦ
Rmax
Φ = (E − H0 )−1VΦ
EM E-EM
ER EL
E in the band
E outside of the band
V =V11 00 0
⎛
⎝⎜
⎞
⎠⎟
G0 =(G0 )11 (G0 )12(G0 )21 (G0 )22
⎛
⎝⎜
⎞
⎠⎟
G0V =I11 − (G0 )11 0−(G0 )21V11 I22
⎛
⎝⎜
⎞
⎠⎟
det I −G0V( ) = det I11 − (G0 )11[ ]U > 1/Rmax Bound states
Ulocalized states> 0
resonant states< 0
4. Deep levelsPhase shift, δ(E)
sum rule(Conservation of states)
δρ(E)dE = Nimp∫
δρ(E) = 1πdδ (E)dE
δ (E) = − tan−1 Im det 1−G0V( ){ }Re det 1−G0V( ){ }
G0 (E) = R0 (E)+ iI0 (E)
change: (2n+1)π/2singular points
δ (E) = tan−1 I0 (E)U1− R0 (E)U
δ (E) = − tan−1 I0 (E)R0 '(E)(E − ER )
δρ(E) = Γ2π
1
E − ER( )2 + Γ2
4
Γ = 2I0 (ER )R0 '(ER )
Γresonance> 0
antiresonance< 0
δ(E)EM-EM ER EL
δρ(E)
E
-π
-π/2
4. Deep levels
H. P. Hjalmarson, et. al., Phys. Rev. Lett. 44, 810 (1980).
Energies of the A1 symmetry deep impurity levels in various diamond- and zincblende-type semiconductors.
Localized states
E ∝1/V1
1. Hyperbolic relation
2. Asymptotic behavior
E→ Evac
V1→∞as
independent of V1
3. Lower bound
Vmin = 1/ Rmax
4. Deep levels
N impurity in GaP
H. P. Hjalmarson, et al., Phys. Rev. Lett. 44, 810 (1980)
GaAs1–xPx
Direct gap Indirect gap
GaAs GaP
N- + h
Small binding energy (9 meV)
Exciton
N levelShallowDeep levelResonant
Despite this, strong luminescence occurs when N is doped.
(Green light)
Luminescence center
4. Deep levels
4.5 DX center
n-AlxGa1–xAs
Te dope shallow donor
deep donorx>0.22
4. Deep levels
D. V. Lang, et al., Phys. Rev. B 19, 1015 (1979)
4. Deep levels
N. Chand, et al., Phys. Rev. B 30, 4481 (1984)
QConfiguration coordinate
Etot
Eopt
EeEcapE0
DX-
d0 + e
(a) (b)
QConfiguration coordinate
Etot
Eopt
EeEcap
E0
DX-d0 + e
Ga
As Si
(a) (b)
(c) (d)
d 0 DX -
Ga
AsSi
Ga
AsS
d 0
Ga
AsS
DX -
D. J. Chadi and K. J. Chang, Phys. Rev. B 39, 10063 (1989)
4. Deep levels