1. Plasmons, polarons, polaritons
M.C. Chang
Dept of Phys
Plasmons, polarons, polaritons
Dielectric function; EM wave in solids
Plasmon oscillation -- plasmons
Electrostatic screening
Electron-electron interaction
Mott metal-insulator transition
Electron-lattice interaction -- polarons
Photon-phonon interaction -- polaritons
Peierls instability of linear metals
For mobile positive ions

2. Dielectric function: general discussion (Use CGS unit)
real space
(k,)-space
Take the Fourier “shuttle” between 2 spaces:
Question: What is the relation between D(r,t) and E(r,t)?

3. EM wave in solids Maxwell eqs. Phase velocity of light
But what is the conductivity ?

4. AC conductivity for metals (consider uniform EM wave, k=0)
where plasma frequency p(4ne2/m)
 <<1
( ,  can be as large as 100 GHz)
Attenuation of plane wave due to nI
exponential decay

5. EM wave propagates with phase velocity C/ EM wave is damped
 >>1
if positive ion charges can be distorted
EM wave propagates with phase velocity C/
EM wave is damped
What happens near =p?
Shuttle blackout

6. Also show that this leads to ()=0
can have longitudinal EM wave!
Homewrok:
Assume that , then from (a) Gauss’ law, (b) equation of continuity, and (c) Ohm’s law, show that
Also show that this leads to ()=0
On the contrary, if ()0, the EM wave can only be transverse.
When , there exists charge oscillations called plasma oscillation
(our discussion in the last 2 pages involves only the uniform, or k=0, case)
Energy dispersion

7. A simple picture of (uniform) plasma oscillation
For copper, n=81022 /cm3
p=1.61016/s, p=1200A, which is ultraviolet light

8. Experimental observation: plasma oscillation in Al
metal
Plasmon
10.3 eV surface plasmon
15.3 eV bulk plasmon
The quantum of plasma oscillation is called plasmon

9. Dielectric function; EM wave in solids
Plasmon oscillation -- plasmons
Electrostatic screening
Electron-electron interaction
Mott metal-insulator transition
Electron-lattice interaction -- polarons
Photon-phonon interaction -- polaritons
Peierls instability of linear metals

10. Electrostatic screening
Thomas-Fermi theory (1927)
Valid if (r) is very smooth within F
Fermi energy:
Electrochemical potential:

11. Dielectric function but For free electron gas, D(F) = mkF/22,
 (ks/kF)2=(4/)(1/kFa0)  O(1)
Screening of a point charge
Comparison:

12. Why e-e interaction can usually be ignored in metals?
Average e-e separation in a metal is about 2 A
Experiments find e mean free path about A (at 300K)
At 1 K, it can move 10 cm without being scattered! Why?
A collision event k1 k2 k3 k4
calculate the e-e scattering rate using Fermi’s golden rule
The summation is over all possible initial and final states that obey energy and momentum conservation

13.  -1    (E1+E2=E3+E4; k1+k2=k3+k4)
Pauli principle reduces available states for the following reasons:
If the scattering amplitude is roughly of the same order for all k’s, then
 -1    (E1+E2=E3+E4; k1+k2=k3+k4)
k1,k2 k3,k4
A. 2 e’s inside the FS cannot scatter with each other (energy conservation + Pauli principle), one of them must be outside of the FS
B. One e is “shallow” outside, the other is “deep” inside also cannot scatter with each other, since the “deep” e has nowhere to go
C. If |E2| < E1, then E3+E4 > 0 (let EF=0)
But since E1+E2 = E3+E4, 3 and 4 cannot be very far from the FS if 1 is close to the FS.
Let’s fix E1, and study possible initial and final states 1 2 3

14.  -1  V(E2)/3k  V(E3)/3k (# of states for scatterings)
# of initial states = (volume of E2 shell)/3k
# of final states = (volume of E3 shell)/3k
(E4 is uniquely determined)
 -1  V(E2)/3k  V(E3)/3k (# of states for scatterings)
 -1  (4/3k)2 kF2|k1-kF|×kF2|k3-kF|
Total # of states for particle 2 and 3 = [(4/3)kF3/ 3k]2
The fraction of states that “can” participate the scatterings
= (9/kF2) |k1-kF|× |k3-kF|
 (E1/EF)2  (kT/EF)2  10-4 at room temp (1951, V. Wessikopf)
Also, we know that the e-e scattering rate  T2

15. 3 types of insulator: insulator conductor
1. Band insulator (Wilson, 1931)
(due to e-lattice interaction)
2. Mott insulator (1937)
(due to e-e interaction)
insulator
metal
A sharp (quantum) phase transition
Si:P alloy
e-e interaction is not always unimportant!
1977

16. 3. Anderson insulator (1958), due to disorder
1977
Localized states and degree of disorders
localized states
disorder

17. Dielectric function; EM wave in solids
Plasmon oscillation -- plasmons
Electrostatic screening
Electron-electron interaction
Mott metal-insulator transition
Electron-lattice interaction -- polarons
Photon-phonon interaction -- polaritons
Peierls instability of linear metals

18. Electron-lattice interaction -- polarons
rigid ions: band effective mass m*
movable ions: drag and slow down electrons
larger effect in polar crystal such as NaCl
smaller effect in covalent crystal such as GaAs
The composite object of “an electron + deformed lattice” (phonon cloud) is called polaron
# of phonons surrounding the electron:

19. Phonons in metals
Regard the metal as a gas of electrons (mass m) and ions (mass M)
The ion vibrating frequency appears static to the swift electrons
However, to the ions,
The longitudinal charge (ion+electron) oscillation at =0 is interpreted as LA phonons in the Fermi sea
For long wave (both  and k are small), we have
For K, v=1.8105 cm/s (theory), vs 2.2105 cm/s (exp’t)

20. EM wave (=ck) Resonance between photons and TO phonons: polaritons 
(LO phonons do not couple with transverse EM wave, why?)
EM wave (=ck)
Optical mode
Acoustic mode  k
dispersion of T(k) ignored in the active region
ignore the charge cloud distortion of ions
assume there are 2 ions/unit cell
resonance
Dipole moment of a unit cell

21. Consider a single mode of uniform EM wave
Total polarization P=Np/V, N is the number of unit cells
Damping region
()
(0)
()  L T
If the charge cloud distortion of ions is considered,
Static dielectric const.

22. LST relation Experimental results NaCl KBr GaAs L/T [(0)/()]1/2
Dispersion relation
The composite object of photon+TO phonon is called polariton.  k
TO mode
LO mode L T
=C/01/2k
=C/1/2k

23. π- electrons delocalized along the chain
Peierls instability of 1-dim metal chain
sp2 bonding
An example of 1-d metal: 聚乙炔 (poly-acetylene, 1958)
π- electrons delocalized along the chain
Dimerization opens an energy gap 2-3 eV (Peierls instability) and becomes a semiconductor