Transcript 投影片 1 - National Cheng Kung University

Elementary Excitations
Elementary Excitations ~ long-lived states near the ground state.
( many body effects included )
Quasi-particles: band-electron, polaron, … (Fermions)
Collective excitations: phonon, magnon, plasmon, polariton, … (Bosons)
In between:
Cooper pair, exciton, ….
14. Plasmons Polaritons and Polarons
Dielectric Function of the Electron Gas
Definitions of the Dielectric Function
Plasma Optics
Dispersion Relation for Electromagnetic Waves
Transverse Optical Modes in a Plasma
Transparency of Alkali Metals in the Ultraviolet
Longitudinal Plasma Oscillations
Plasmons
Electrostatic Screening
Screened Coulomb Potential
Pseudopotential Component
Mott Metal-Insulator Transition
Screening and Phonons in Metals
Polaritons
LST Relation
Electron-Electron Interaction
Fermi Liquid
Electron-Phonon interaction: Polarons
Peierls Instability of Linear Metals
Dielectric Function of the Electron Gas
ε(ω,0) : plasmon
ε(0,K) : screening
Definitions of the Dielectric Function:
 D    E  4 ext
D  E  4 P   E
 E  4  4  ext  ind 
Fourier components (ω dependence understood ):




 E    E  K  ei K r   4    K  ei K r 
K

K

D K     K  E  K  →
→

 K  
D  ext
 ext  4 ext
2
→ i K  E  K   4   K 




 D      K  E  K  ei K r   4  ext  K  ei K r 
K

K

i   K  K  E  K   4 ext  K 
ext  K 
 K 
 1  ind
 K 
 K 
E  
   4 
2
→
ext  K  ext  K 

  K 
 K 
 K 
Plasma Optics ( K  0 )
d 2x
m 2  e E
dt
E  ei  t
ne2
P  ne x  
E
2
m

→

 2 m x  e E
 ,0     
Plasma frequency ωp is defined by   p   0
4 ne2
 
m
 p2
    1  2

Adding ion core constant contribution:
2
4 ne2    1   p 
  
        
2 
2


m


4 ne2
 
  m
2
p
  p   0
e
E
m 2
P  
D  
 1  4
E  
E  
4 ne2
    1 
m 2
2
p
x
Dispersion Relation for Electromagnetic Waves
In a non-magnetic isotropic medium:
Plane wave solution:
2 D
2 2

c
 E
2
t
E, D  ei K  x i  t  ei K x//  i  t
→
 , K  2  c2 K 2
Assuming ω real :
ε real & ε > 0 → K real : trans. EM
wave with vph = c / ε1/2
ε real & ε < 0 → K imaginary : damped
wave with depth 1 / K.
ε complex
→ K complex : damped
wave with depth 1 / Re(K).
ε=
spontaneous oscillation.
ε=0
longitudinal plasma wave.
  p2 
        1  2 
  
Transverse Optical Modes in a Plasma
  p2 
        1  2 
  
 , K  2  c2 K 2
→
 2         2   p2   c 2 K 2
  p2 
        1  2 
  
  p
→ E, B  e
→ K2<0
K x
incident wave totally reflected
  p
→
c2 K 2
  
  
2
2
p
transverse wave in plasma
4 ne2
 
m
2
p
p 
2 c
p
Transparency of Alkali Metals in the Ultraviolet
Metal reflects visble light but is transparent to ultraviolet light.
InSb with n = 41018 cm–3
Longitudinal Plasma Oscillations
Condition for longitudinal mode:
  L   0
Reason: For a longitudinal wave,
there is a depolarization field
E  4 P
→
D  E  4 P  0   E
K = 0 plasma wave in thin film
e gas:
 p2
  L   1  2  0
L
→
 L  p
ωp = low freq cut-off of transverse EM wave.
Motion of a unit volume of e gas of concentration n :
d 2u
nm 2  neE  4 n2 e2 u
dt
→
i p t
u  u0 e
For small K,
p 
where
E  4 P   4  neu 
4 ne2
m

3 vF2 2
K   p 1 
k 
 10  2
p




See J.M.Ziman, “Principles of the
Theory of Solids”, 2nd ed., §5.7.
Plasmons
Plasmon = quantum of plasma wave. (boson)
Creation of plasmon in metal thin film
by inelastic e scattering. Einc  10 keV
spectrometer for plasmon
Al
Mg
Einc = 2 keV
Harmonics of 10.3 eV (surface)
& 15.3 eV (volume) losses
Harmonics of 7.1eV (surface)
& 10.6 eV (volume) losses
Prob. 1
Electrostatic Screening
Ref: G.D.Mahan, “Many-Particle
Physics, 2nd ed., § § 5.4-5
Static screening ε(0,K).
Jellium model: uniform e gas of charge density – n0 e
with uniform background charge + n0 e .
Let background charge be disturbed (e.g. by impurities):
Gauss’ law:
 D  4  ext
Fourier components:
i K  D K   4 ext  K 
Longitudinal component:
EL  
Isotropic medium:
Linear screening:
 E  4  
→
   x  n0 e  ext  x
  x  ext  x  ind  x
i K  E  K   4  ind  K   ext  K  
4
4
ext  K 
EL  K  
 ind  K   ext  K 
iK
iK 
4
  K   2  ind  K   ext  K  
K
DL  K  
  K   lim
ext  0
DL  K 
EL  K 
ext  K 
0 
ind  K   ext  K 
 lim
ext
ext  K 
 K  
ind  K   ext  K 
Let T = 0.
For φ = ̶ 0    
0
F
2
2m
3 2n0 
2/3
   F  x   e  x  
For φ  0
2
3 2 n  x  
2m
2/3
 e  x    F0
K << kF
d
 F  x    F0   n  x   n0  F
dn
d F
dn

n  n0
d
 n  x   n0  F
dn
→
n  n0
2 F0

3 n0
→
 e  x 
n  n0
3 e  x 
n  x   n0  n0
2
F
2
6 n0 e2   K 
3 e  K 
ind  K   e  n  K   n0    n0

F
K2
2
F
ind  K 
k S2
 1 2
  0, K   1 
K
 K 
k 
2
S
6 n0 e2
F
k 
2
S
6 n0 e2
F
D   
3n
2
1/3
43 
  n0 
a0   
→
 F a0
2/3

3

n


0
e2 2
a0 
2
me
2
= Bohr radius
kS2  4 e2 D  F 
1 / kS = Thomas-Fermi screening length
For Cu with n0 = 8.5  1022 cm–3 , kS = 0.55A.
kS2
  0, K   1  2
K
→
kS2
  0, K  0   2
K
Limit not the same
 p2
 ,0   1  2

→
 p2
   0,0    2

Full theory for ε(ω, K) was due to Lindhard. (see Mahan § 5.5.B)
Screened Coulomb Potential
Consider a point charge q in an e gas.
20  4 q r 
In the absence of the e-gas
f r   
Fourier transform:

3
f K  e
→
q
r
( unscreened potential of q )
 r   
i K r
d 3K
 2 
3
e i K r
4 q
K2
0  K   4 q
 K  
 K  K 2
 K 
kS2
 K   1 2
K
Thomas-Fermi screening:
4 q
 2 
0  K  
→
By definition:
d 3K
0 
2
 K  
→

4 q
K 2  kS2
K2
q
K2
e i K r  ei K r
i K r cos
 r  
dK  d cos   d 2 2 e
  dK 2 2
3 
K

k
0
K  kS
iK r
 2  0 1
S
0

1

q
K ei K r
2q
K sin Kr

Im  dK 2 2

dK
2
2

 r 
K  kS
r 0
K  kS

K
1 1
1 




K 2  kS2 2  K  ikS K  ikS 
q
K ei K r
  r   Im  dK 2 2
 r 
K  kS

q
Im  i e kS r 
r
( contour closing in upper complex K plane )
q
  r   e k
r
S
r
Note: T-F screening is valid only for K << kF .
Friedel oscillations set in for large K.
Calculations based on the screened potential for the residual
resistivities of substitutional alloys of Cu with metallic elements
of different valencies agreed well with experiment.
Pseudopotential Component
In a metal of valency z and ion density n0 :
4 q
 K   2 2
K  kS
k 
2
S
→
6 zn0e2
F
4 zn0e2
U  0  zen0  0  
kS2
→
2
U 0    F
3
Mott Metal-Insulator Transition
Independent e model → c-H with 1 atom per primitive cell is a metal.
c-H2 with 1 molecule per primitive cell is an insulator.
Under extreme pressure (equal inter- & intra- molecule H dist.), c-H2 becomes metallic.
With e-e interaction considered, c-H at T = 0 may either be metallic or insulating
depending on the lattice constant a.
Mott: Critical lattice constant separating the metallic & insulating state is
aC  4.5a0  4.5
2
me2
 2.25 A
Each e sees a screened potential from each proton:
Ref: N.F.Mott, E.A.Davis, “Electronic Processes
in Non-Crystalline Materials”, 2nd ed., §4.2.
e 2  kS r
U r    e
r
1/3
n01/3
43 
2
kS   n0   3.939
a0   
a0
n0 = e density
For very large kS , there is no bound state → solid is metallic.
Bound state 1st appears when kS < 1.19 / a0 .
→
n01/3 1.42
3.939
 2
a0
a0
For a s.c. lattice,
n0 = 1/a3 .
→
aC 
3.939
a0  2.78a0
1.42
Mott (metal-insulator) transition can be caused by changes in
composition, pressure, strain, Ba , …
Metallic phase: independent e (band model).
Insulator phase: strong e-e correlation (Hubbard model).
Doping of semiconductor can also
induce a metal-insulator transition.
nC  3.74 1018 cm3
rD  32 108 cm
6
→ aC  1.44 10 cm
Assuming P arranged in s.c.lattice:
1
nC  3  0.33 1018 cm3
aC
P in Si
nC
Screening and Phonons in Metals
4 ne2
  , K     , K  
M2
kS2 4 ne 2
 1 2 
K
M2
K,   0
kS2 4 ne2

0
K 2 M2
Longitudinal mode:
v
T-F ε for e
kS2 4 ne2
  , K   2 
K
M2
→
4 n e 2
M k S2
phonon

→
4 n e 2  F

2
M 6 ne

2 F
 vF
3M
4 n e 2
K v K
2
M kS
m
3M
LA phonon
Good agreement with experiment for alkali metals.
E.g., for K, vcalc = 1.8105 cm/s , vexp = 2.2  105 cm/s along [100] at 4K.
Plasma:
4 ne2 4 ne2
 ,0  1 

M2 m 2
→
4 ne2
 

2
p
1 1 1
 
 M m
Polaritons
TO phonon + photon → polariton
Maxwell eqs with
J = 0,  =1 :
→
E 
B
0
c t
 B 
2
   E  2 2  E  4 P   0
c t
2
 E  2 2  E  4 P   0
c t
 E
  E    E  E 
2
Plane wave solution:
P due to TO phonon:
Oscillatory solution:
→

 E  4 P 
c t
transverse mode
2
2
2
K E  2  E  4 P   0
c
d2u
q
2


u

E
P=Nqu →
T
2
dt
M
2
d2P
Nq 2
2
 T P 
E
2
dt
M
Nq 2
    P  M E
2
 c 2 K 2   2 4 2 

 E  0
2
2
2  P 
  Nq




 
T

M

2
T
→
 2 2
4 Nq 2  2 2 2 2
2
   c K  T 
   c K T  0
M 

4
 2 2
4 Nq 2  2 2 2 2
2
   c K  T 
   c K T  0
M


4
 0
K=0:
4 Nq 2
  
M
2
2
T
photon
polariton
Nq 2
P
E
2
2
M    T 
→
4 Nq 2
E  4 P  1 
   
M   2  T2 
E
Including e contribution →

4 Nq 2
  0      
M T2
4 Nq 2
        
M   2  T2 
→
T2
           0      2
  T2
T2
  2     T2   0
           0      2

2
  T
 2  T2
 L   0
→
L2     T2   0
LST relation
  2  L2 
         2
2 





T 
    0 for T    L
→
L 
3
T
2
ei K x  e
 K x
EM wave reflected
SrF2
ωL = ωLO
ωT = ωTO
E / / xˆ   E  0
D  E  4 P  0
E / / zˆ   E  0
  0
forbidden gap
GaP
LST Relation
Lydane-Sachs-Teller relation:
2
  0
LO

1
2
TO
 
Soft mode: ωTO = 0  ε(0) → 
derived for s.c.lattice with
2 atoms in primitive cell
(ferroelectric)
partial propagation
NaCl
λTO = 6110−4 cm , λLO = 3810−4 cm
p :  film: |K| small
Strong Absorption
LiF thin film on Ag
30。 incidence
neutron inelastic scattering
Electron-Electron Interaction
e-e coulomb interaction → e-e scattering, inertial drag, …
Fermi Liquid (Landau):
Quasi-particle description of low-lying excited states.
Q.P. = e + distorted e cloud
m → m*
For alkali metals, m*  1.25 m.
Electron-Electron Collisions
Puzzle: rS ~ 2A but l ~ 104 A for conduction e’s in metal.
Reasons: 1. Exclusion principle. 2. Screening
Forbided
Allowed
 3   4  1   2  0
→
 2  1
Only fraction
ε1 /εF allowed
T = 0: only (ε1 /εF )2  10–10 can scatter
Only fraction
ε1 /εF allowed
2
T 0:
k T 
   B  0
 F 
Numerical calculations with screening → σ0  10–15 cm2 < unscreened Rutherford value
At 300K,
k BT
 102
F
→
  10  0  10 cm
4
19
2
→
lee 
1
 104 cm
n
le ph  103 cm
 e-ph scattering dominates at room T.
At l-He temperature, ρ ~ T2 (e-e scattering dominant)
For In at 2K, l = 30 cm.
Electron-Phonon Interaction: Polarons
Effect of e-ph interaction:
• ρ  T, e.g., for Cu, ρ = 1.55 μΩ-cm at 0C, ρ = 2.28 μΩ-cm at 100C.
• m* > m ( polaron = e + strain field ).
rigid lattice
deformable lattice
Polaron effect is strong in ionic, but weak in covalent, crystals.
e-ph coupling constant
1
deformation energy

2
LO
 number of ph around a slow e.
m*pol from
cyclotron exp.
Theory:
m*pol


2
 1  0.0008  
*
m 

1
2
 1    0.003 
 6

 1 
 1

 m* 1   
 6 
Large polaron: band-like with m*.
Small polaron: moves by ( thermally activated ) hopping.
In polar crystal with degenerate band edge, e or more likely, h , can be
self-trapped by inducing lattice deformation.
Conductivity by Ag+ hopping in MAg4I5 (M = K, Rb, NH4 ) can have σ 105 that of typical ionic crystals
Peierls Instability of Linear Metals
At T = 0, a 1-D e gas is unstable w.r.t. static lattice deformation of G = 2kF .
→ energy gap created at εF . E.g. TaS3 .
Equilibrium deformation Δ is given by
d
 Eelectronic  Eelastic   0
d
For an elastic strain Δ cos 2kF x
1
Eelastic  C  2 cos 2 2k F x
2
1
 C2
4
Let the lattice potential seen by e be
U  2 A cos 2kF x
 k   k  Ck  UGCk G  0
G
2
k2
k 
2m
→
 k   k

 U
U
k 2 k
F
→ U G  2kF   A
 Ck 

0
  k  Ck 2 kF 
for
k  2kF  2kF
 k   k

 U
→
U
k 2 k
F
 Ck

  k  Ck 2 kF

0

for
 k2   k  k  2 k   k  k k  2 k  U 2  0
F
k 

F

1
k  k 2kF 
2





1
k  k 2kF
4

2
K  kF
k  kF  K with
U  A
U 2
2
2
1
1
2
2

k  k 2kF 
k F  K    k F  K  
k F2  K 2 


 2m
2
2m 2 
2
2
1
1
2
2

K  K 2 kF 
 kF  K    K  kF    kF K

2
2m 2
m
→
K 
2
k

2m
2
F
k 2K 2
4
 A2 2
2m 2m
 K 
1
  xF  xK  
2
dK

d
2 2
F
2
2
xF xK  A 
A2 
xF xK  A 2  2
2
2
K2
xK 
m
2
k F2
xF 
m
x
2
K kF
m
dK

d
A2 
2
xF xK  A 
2
d  electronic 2

d

K2
xK 
m
2
2
d K


dK
0 d 

2 A2  k F

 xF
 dK
0
xF xK  A 2  2
0
xF
 dx
A2 
kF
kF
1
x A
2
2
2
k F2
xF 
m
x
2 A2  kF

sinh 1 F
 xF
A
d  elastic d  1 2 
1

 C   C 
d
d  4
2

x
d
1
2 A2  kF
sinh 1 F  0
 Eelectronic  Eelastic   C 
d
2
 xF
A

→
xF
 C xF
 sinh
A
4 A2 kF
→
k
 C 2 kF
 sinh
mA
4 A2 m
2 2
F
K kF
m
dK/  = # of orbitals
per unit length
dx 
2
x
2
xF
kF
dK  dK
kF
m
2
k
 C 2 kF
 sinh
mA
4 A2 m
2
k F2
A
 C 2 kF
m sinh
4 A2 m
2 2
F
For x >> 1,

1
2
2
 x x  x
sinh x e  e
e
  C 2kF 
2 2 kF2
A
exp  

2
m
4
A
m


where
 2e x

1
 4W exp  
 N  0 V



2
k F2
W
 C.B.width
2m
N  0 
2m
 DOS at  F
 2 kF
2 A2
V
 effective e-e interaction
C
c.f. BCS gap