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6.OpticalProcessesandElectronDynamics
6. Optical Properties大学院講義「半導体物性」
6. Optical Properties
配置座標上の表現
6. Optical Properties
6.1 Fundamental Optical Spectra
Reflectance measurement
R =!n −1!n +1
2
ε(ω ) = !n2!n = n + iκ
α = 4πκ / λ0
I = I0 exp(−αd)
Complex refractive index
Complex dielectric functionExtinction index
Absorption coefficientWavelength of light in vacuum
6. Optical Properties
Causality relation (Kramers-Krönig relation)
ε r (ω ) = 1+2πP ω 'ε i (ω ')
ω '2−ω 2 dω '0
∞
∫
ε i (ω ) = − 2ωπP ε r (ω ')
ω '2−ω 2 dω '0
∞
∫
H.R. Philipp and H. Ehrenreich, in Semiconductors and Semimetals, 3, 93 (1967)
6. Optical Properties
Optical transition
H = H0 +emcA ⋅p
Electron-radiation interaction
HeRHeR = −er ⋅E
Electric dipole approximation
long-wavelength limit
kn = ukn (r)exp[ik ⋅r]
W = 2π!
c HeR v2δ Ec(kc )− Ev(kv )− !ω( )
q∑
Transition rate
Pcv = c e ⋅p v = ukc(r)*(e ⋅p)ukv(r)dr∫
= 2π!
emω
⎛⎝⎜
⎞⎠⎟2
Pcv2δ Ec(kc )− Ev(kv )− !ω( )
q∑
kc = kv vertical transition
dipole approximation
= q ⋅(∇kukc(r)*)(e ⋅p)ukv(r)dr∫ quadrupole approximation
Joint density of states
J(Ecv ) =14π 3
dSk∇kEcv
∫
6. Optical Properties
Imaginary part of dielectric constant near Van Hove singularities
J(E)∝ E − E0( )1/2E(k) = E(0)+ a1k1
2 + a2k22 + a3k3
2
6. Optical Properties
Comparison between theory and experiment
M. L. Cohen and J. R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, (Springer, Berlin, 1989)
Theory
M. Cardona, L. F. Lastras-Martinez, and E. E. Aspues, Phys. Rev. Lett. 83, 3970 (1999)
Experiment
6. Optical Properties
Theory
C. W. Higginbotham, PhD Thesis, Brown University (1999)
Experiment
6. Optical Properties
Direct transition
M. Cardona, in Solid State Physics, Nuclear Physics and Particle Physics, ed by I. Saavedra (Benjamin, New York, 1968), pp. 737-816
6. Optical Properties
Indirect-transition
G. G. MacFarlane and V. Roberts, Phys. Rev. 97, 1714 (1955); 98, 1865 (1955)
!ω = Ecv ± Ep
kc − kv = ∓Q
W = 2π!
f Hep i i HeR 0Ec − !ωi
∑2
δ Ec(kc )− Ev(kv )− !ω ∓ Ep( )kc ,kv∑
6.2 Absorption Edge Spectra6. Optical Properties
Absorption-emission processes 6. Optical Properties
Absorption/emission spectra of very pure GaAs
Absorption spectrum of rubyin the infrared, visible, and ultraviolet
G. F. Imbush, in Luminescence Spectroscopy, edited by M. D. Lumb (Academic, New York, 1978). D. D. Sell, Phys. Rev. B 6, 3750 (1972); 7, 4568 (1972)
Luminescence from impurities Luminescence from host crystals
e-h recombination
6. Optical Properties
Ψ(r,Q) =ψ r (r,Q)χn (Q) Born-Oppenheimer product
F(ω ) = i P f 2
Condon approximation
Franck-Condon approximation
i P f = kn P lm
= k P l n m
= Pkl (Q)χn (Q)χm (Q −Q0 )
electronic partphonon part
F(ω ) = Pkl2 χn χm
2
6. Optical Properties
6.3 Electron-lattice interactions
Stokes shift6. Optical Properties
G. F. Imbusch, in Luminescence Spectroscopy, edited by M. D. Lumb (Academic, 1978)
Electron-Phonon interaction
Absorption-emission processes 6. Optical Properties
Q
E
e
g
|0> → |n>
phonon mediated transition
6. Optical PropertiesAbsorption coefficient
α (ω )dω = nε4πN!c
Pmni 2
ωmn ρn − ρm( )ω<ωnm<ω+dω∑
F(ω ) Shape function
Pmni = ϕm Pi ϕn
Electron-phonon interaction
l,n Pi m,n ' = dQ∫ χ ln* (Q)χmn ' (Q) drϕl
*(r,Q)Pi∫ ϕm (r,Q)
F(ω ) = l,nµl Pi m,nν
m 2ρl ,nµ
l δ !ω − Em,ν + El ,µ( )mν∑
lµ∑
Pl→ni (Q)
T → 0 El ,m =U0 +Um (Q)
χmn'* (Q)χmn ' (Q ')
n '∑ = δ (Q −Q ')
Fl→m (ω ) = ρln dQ Pmni (Q)
2χ ln (Q)
2δ !ω −U0 −Um (Q)+ El ,n( )∫n∑
Frank-Condon’s principle
completeness
T → high Fl→m (ω ) = dQ Pmni (Q)
2Sl (Q)δ !ω −U0 −Um (Q)+Ul (Q)( )∫
Sl (Q) = ρln χ ln (Q)2
n∑
→exp −Ul (Q) / kT( )dQexp −Ul (Q) / kT( )∫
6. Optical Properties
Line shape
= exp −S( ) m!n!
⎛⎝⎜
⎞⎠⎟ S
n−m Lmn−m S( )⎡⎣ ⎤⎦
2
S = A2
2
εam = m + 12
⎛⎝⎜
⎞⎠⎟ !ω a
εbn = n + 12
⎛⎝⎜
⎞⎠⎟ !ωb + Eab −
12A2!ω
At T = 0 K
At high T
Only m=0 is allowed.
p = n–m
peaked at n ≃ S
Fnm = χbn | χam2
Fn0 =Sn
n!e−S
L0n S( ) = 1
En = (Eab − !ω )+ n!ω = E0 + n!ω
Fp = exp px − S coth x( ) I p Scseh(x)( )x = !ω
2kT
Q
E
e
g
Q
E
a
b
Eaba
(a) 12Mω 2Q2
(b) (a)+ Eab − A!ωMω!
⎛⎝⎜
⎞⎠⎟1/2
Q
6.4 Special Topics: DX center6. Optical Properties
Large-Lattice-Relaxation modelD. V. Lang and R. A. Logan, Phys. Rev. Lett., 59 635 (1977)
6. Optical Properties
6. Optical Properties
D. V. Lang, et al., Phys. Rev. B 19, 1015 (1979)
6. Optical Properties
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
N. Chand, et al., Phys. Rev. B 30, 4481 (1984)H. P. Hjalmarson, et al., Phys. Rev. Lett. 44, 810 (1980)
6. Optical Properties
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