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