Transcript 投影片 1 - National Cheng Kung University

15. Optical Processes and Excitons
Optical Reflectance
Kramers-Kronig Relations
Example: Conductivity of Collisionless Electron Gas
Electronic Interband Transitions
Excitons
Frenkel Excitons
Alkali Halides
Molecular Crystals
Weakly Bound (Mott-Wannier) Excitons
Exciton Condensation into Electron-Hole Drops (Ehd)
Raman Effect in Crystals
Electron Spectroscopy with X-Rays
Energy Loss of Fast Particles in a Solid
Optical Processes
Raman scattering:
Brillouin scattering for acoustic phonons.
Polariton scattering for optical phonons.
+ phonon emission (Stokes process)
– phonon absorption (anti-Stokes)
2-phonon creation
k γ << G for γ in IR to UV regions.
→ Only ε(ω) = ε(ω,0) need be considered.
Theoretically, all responses of solid to EM fields are
known if ε(ω,K) is known.
XPS
ε is not directly measurable.
Some measurable quantities: R, n, K, …
Optical Reflectance
Consider the reflection of light at normal incidence on a single crystal.
r   
Reflectivity coefficient
E  refl 
     e i  
E  inc 
Let n(ω) be the refractive index and K(ω) be the extinction coefficient.
→
r   
n  iK  1
n  iK  1
see Prob.3
    n    iK    N  
         i   
Let
Complex refractive index
     n 2  K 2
→
     2nK
E   inc   E 0 exp i k r  t  

E  trans   exp i  n  i K  kr  t 
Reflectance
R
E  refl 
2
E  incl 
2
 r
2
  exp   K k r  exp i  n kr  t 
 2
(easily measured)
θ is difficult to measure but can be calculated via the Kramer-Kronig relation.
Kramers-Kronig Relations
Re α(ω)
 KKR → Im α(ω)
α = linear response
 d2
d
F
2




x

 2
j
j  j
dt
Mj
 dt

Equation of motion:
(driven damped uncoupled oscillators)
d i t
f t   
e
f
 2


x  t    dt    t  t  F t  
t
Linear response:
 
    
j
F
Mj
x   x j  
→
j
fj
  i j  
2
x     F
→

 i j   2j  x j 
2
2
j

j
j
f j  2j   2  i j 

2
j

j
f   dt ei t f  t 

Fourier transform:
x   xj
   
2 2
2
F
M j   2  i j   2j 
fj 
j
Let α be the dielectric polarizability χ so that P = χ E.
P   p j   nex j
j
j
2
2
→  d   d   2  p  ne E
j
j
j
2
 dt
dt

m
→
ne 2
fj 
m
1
Mj
Conditions on α for satisfying the Kronig-Kramer relation:
• All poles of α(ω) are in the lower complex ω plane.
• C d ω α /ω = 0 if C = infinite semicircle in the upper-half complex ω plane.
It suffices to have α → 0 as |ω | → .
• α(ω) is even and α(ω) is odd w.r.t. real ω.
Example: Conductivity of Collisionless Electron Gas
For a free e-gas with no collisions (ωj = 0 ):
    
1
m   i  
 0

 
1 1


i






m  
s    s  1 
 s   1
P  ds 2
 P  ds 2
m 2
 
s   2 m 
s  2
1

4 nex
4 P

    1 
 4 ne2  
E
E
    
D
t
Treating the e-gas as a pure metal: c  H  4 E 
Treating the e-gas as a pure dielectric:

c H 
D
t
n e2
     i     i n e     
m
i 








 
1
m 2

  
m
KKR
E
t
i   E     4   i  E  
2
    
4 ne2
→      1 
m 2
Consider the Ampere-Maxwell eq. c  H  4 J 
Fourier components:
     
4 2 ne2
    
  
m
D
E
 4 E 
t
t
→
→
    i

   1
4 
pole at ω = 0
Electronic Interband Transitions
R & Iabs seemingly featureless.
R
Selection rule
   c k    v k 
allows transitions  k  B.Z.
dR/dλ
→ Not much info can be obtained from them?
Saving graces:
Modulation spectroscopy: dnR/dxn,
where x = λ, E, T, P, σ, …
Critical points where
k  c  k    v  k    0
provide sharp features in dnR/dxn which
can be easily calculated by pseudopotential method (accuracy 0.1eV)
Electroreflectance: d3R/dE3
Excitons
Non-defect optical features below EG → e-h pairs (excitons).
Mott-Wannier exciton
Frenkel exciton
Properties:
• Can be found in all non-metals.
• For indirect band gap materials, excitons near direct gaps may be unstable.
• All excitons are ultimately unstable against recombination.
• Exciton complexes (e.g., biexcitons) are possible.
Exciton can be formed if e & h have
the same vg , i.e. at any critical points
k  c  k    v  k    0
3 ways to measure Eex :
• Optical absorption.
• Recombination luminescence.
• Photo-ionization of excitons
(high conc of excitons required).
GaAs at 21K
I = I0 exp(–α x)
Eex = 3.4meV
Frenkel Excitons
Frenkel exciton: e,h excited states of same atom; moves by hopping.
E.g., inert gas crystals.
Lowest atomic transition of Kr = 9.99eV.
In crystal it’s 10.17eV.
Eg = 11.7eV → Eex = 1.5eV
Kr at 20K
The translational states of Frenkel excitons are Bloch functions.
Consider a linear crystal of N non-interacting atoms. Ground state of crystal is
 g  u1u2
uN 1uN
uj = ground state of jth atom.
If only 1 atom, say j , is excited:  j  u1u2
u j 1 v j u j 1
uN 1uN
In the presence of interaction, φj is no longer an eigenstate.
For the case of nearest neighbor interaction T :
H  j   j  T  j 1   j 1 
Consider the ansatz
j = 1, …, N
 k   e ik j a  j
j
H k   e ik j a H  j   e i k j a  j  T  j 1   j 1  
j
j
  e i k j a   T  ei k a  e  i k a    j    2T cos ka  k
j
 ψk is an eigenstate with eigenvalue
Ek    2T cos ka
Periodic B.C. → k 
2 s
Na
s
N
,
2
,
N
1
2
(N-fold degenerated)
Alkali Halides
The negative halogens have lower excitation levels
→ (Frenkel) excitons are localized around them.
Pure AH crystals are transparent (Eg ~ 10 eV)
→ strong excitonic absorption in the UV range.
Prominent doublet structure for NaBr ( iso-electronic with Kr )
Splitting caused by spin-orbit coupling.
Molecular Crystals
Molecular binding >> van der Waal binding → Frenkel excitons
Excitations of molecules become excitons in crystal ( with energy shifts ).
Davydov splitting introduces more structure in crystal (Prob 7).
Weakly Bound (Mott-Wannier) Excitons
Bound states of e-h pair interacting via Coulomb potential
 e4
En  Eg  2 2 2
2  n
1
where


e2
U 
r
1
1

me mh
are
n = 1, 2, 3, …
For Cu2O, agreement with experiment
is excellent except for n = 1 transition.
Empirical shift for data fit gives
  cm 1   17,508 
800
n2
With ε = 10, this gives μ = 0.7 m.
Cu2O at 77K
absorption peaks
Eg = 2.17eV = 17,508 cm–1
Exciton Condensation into Electron-Hole Drops (EHD)
Ge:
 E
g
e VB 
  e  CB    h VB 
 ~1ns

 exciton
 ~8s


For sufficiently high exciton conc. ( e.g., 1013 cm−3 at 2K ), an EHD is formed.
→ τ ~ 40 µs ( ~ 600 µs in strained Ge )
Within EHD, excitons dissolve into metallic degenerate gas of e & h.
Ge at 3.04K
FE @ 714 meV : Doppler broadened.
EHD @ 709 meV : Fermi gas n = 21017 cm−3.
EHD obs. by e-h recomb. lumin.
Unstrained Si
Raman Effect in Crystals
1st order Raman effect
(1 phonon )
    
    
k  k  K
k  k  K
Cause: strain-dependence of electronic polarizability α.
Let
  0  1 u  2 u2 
u = phonon amplitude
u t   u0 cos  t
E t   E0 cos  t
Induced dipole: p t   1 u t  E t   1 u0 E0 cos  t cos  t
 1 u0 E0 cos     t  cos     t 
App.C:
I     
I     
Anti-Stokes
nK  1 u nK
2
nK  1 u nK
2
 nK  1
 nK
→
Stokes
nK
I    
 e

I    
nK  1
 / kB T
T 0

0
1st order Raman
λinc = 5145A
K0
GaP at 20K.
ωLO = 404 cm−1. ωTO = 366 cm−1 .
1st order: Largest doublet.
2nd order: the rest.
Si
Electron Spectroscopy with X-Rays
XPS = X-ray Photoemission Spectroscopy
UPS = Ultra-violet Photoemission Spectroscopy
Monochromatic radiation on sample :
KE of photoelectrons analyzed.
→ DOS of VB (resolution ~ 10meV)
Only e up to ~ 50A below surface can escape.
4d
Excitations from deeper levels are
often accompanied by plasmons.
E.g., for Si, 2p pk ~99.2eV is
replicated at 117eV (1 plasmon)
and at 134.7eV (2 plasmons).
5s
 ωp  18eV.
Ag: εF = 0
Energy Loss of Fast Particles in a Solid
Energy loss of charged particles measures Im( 1/ε ).
Power dissipation density by dielectric loss:
EM wave:
P

1
4
1
 E2
4
 cos  t    cos  t    sin  t 
Isotropic medium:
1
4
1
D
E
4
t
Re  Ee i t  Re  i  Ee i  t 
Particle of charge e & velocity v :
P  , k  
P
D  r, t   

1
 E 2 
8
e
rvt
D, k    , k  E , k 
 D  , k   i  t 
i  t
Re 
e

 Re  i D , k  e


,
k

 

1

 D 2  , k 
4
 1 

1 



  sin  t    cos  t    sin  t 
  

 


1
P  , k  
 D 2  , k 
4
 1 

1 



  sin  t    cos  t    sin  t 
  

 


1
   , k
1
P  , k  
 D2 , k  Im    1  D 2  , k   2 
8
   8

 1 
Energy loss function =  Im 



,
k




1
 1 
2

 D  , k   
8
 
 1   k0 v 
2 e2
P    
Im 
 ln
 v        