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