Download Lecture 19

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10PM-2:00 PM)
Lecture 19, room 304 Nielsen
Chapter 10: Plasmons, Polaritons, and polarons
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
Chapter 10 Plasmons, Polaritons, and Polarons
The dielectric constant ε of electrostatics is defined in terms of the
electric field E and the polarization P, the dipole moment density:
G G
G
G
D = E + 4π P = ε E; ε is the relative permittivity.
Suppose a positively charged particle is placed at a fixed position
inside the electron gas. It will then attract electrons and create a
surplus negative charge around it, thus reducing its field.
If ρ ext is the particle's charge density without screening, the full
charge density should be ρ = ρ ext + ρind .
G
G
D is related to the external applied charge density ρext and E is
related to the total charge density ρ = ρ ext + ρind .
G
G
divD = divε E = 4πρ ext
G
divE = 4π ( ρ ext + ρind ).
Dai/PHYS 342/555 Spring 2006
Chapter 10-2
G
Define ε ( K ) such that:
G G
G G G
G
D( K ) = ε ( K ) E ( K ); ε ( K ) is the relative permittivity.
Then we have
G G
G
G iKG ⋅rG
iK ⋅r
divE = div∑ E ( K )e = 4π ∑ ρ ( K )e .
G G
G
G
G iKG ⋅rG
iK ⋅r
divD = div∑ ε ( K ) E ( K )e = 4π ∑ ρext ( K )e
G
ε ( K ) = ρext ( K ) / ρ ( K ) = 1 − ρind ( K ) / ρ ( K )
The electrostatice potentials satisfy the Poisson equation
∇ 2ϕext = −4πρext ; and ∇ 2ϕ = −4πρ .
G
G
G
ϕext ( K ) ρext ( K )
G =
G = ε ( K ).
ϕ (K )
ρ (K )
Dai/PHYS 342/555 Spring 2006
Chapter 10-3
Plasma optics
The equation of motion of a free electron in an electric field:
d 2x
m 2 = −eE. if x and E have the time dependence e − iωt , then
dt
−ω 2 mx = −eE ; x = eE / mω 2 .
The dipole moment of one electron is − ex = −e 2 E / mω 2 , and
the polarization defined as the dipole moment per unit volume is
P = − nex = − ne 2 E / mω 2 , the dielectric function is then
ωp
4π ne 2
D(ω )
P(ω )
2
1
,
4
/ m.
≡ 1 + 4π
= 1−
≡
−
=
ε (ω ) ≡
ω
π
ne
p
2
2
E (ω )
E (ω )
mω
ω
If positive ion core background has a dielectric constant ε (∞),
2
ε (ω ) = ε (∞) − 4π ne 2 / mω 2 = ε (∞)[1 − ω p2 / ω 2 ].
Dai/PHYS 342/555 Spring 2006
Chapter 10-4
Dai/PHYS 342/555 Spring 2006
Chapter 10-5
Dispersion relation for electromagnetic waves
Maxwell's equation for a nonmagnetic isotropic medium,
G
2
G
∂ D
2 2
= c ∇ E.
2
∂t
G G
G
G
G G
iK ⋅r − iωt
If E ∝ e e
and D = ε (ω , K ) E; then
G 2
ε (ω , K )ω = c 2 K 2 ;
Dai/PHYS 342/555 Spring 2006
Chapter 10-6
G
If ε is real and >0: K is real and a transverse electromagnetic wave
propagates with the phase velocity c / ε 1/ 2 .
G
If ε is real and <0: K is imaginary and the wave is damped with a
G
characteristic length 1/|K | .
G
If ε is complex. For ω real, K is complex and the waves are damped
in space.
ε = ∞. This means the system has a finite response in the absence of
an applied force; thus the poles of ε (ω , K ) define the frequencies of
the free oscillations of the medium.
ε = 0. Longitudinally polarized waves are possible only at the
zero of ε .
Dai/PHYS 342/555 Spring 2006
Chapter 10-7
Dispersion relation for electromagnetic waves
The dispersion relationship
ε (ω )ω 2 = ε (∞)[ω 2 − ω p2 ] = c 2 K 2 .
For ω < ω p we have K 2 < 0, so that K is imaginary. Waves incident
on the medium in the frequency range 0 < ω < ω p do not propagate,
but will be totally reflected.
An electron gas is transparent when ω > ω p , the dispersion relation
is then
ω 2 = ω p2 + c 2 K 2 / ε (∞); this describes transverse electromagnetic
waves in a plasma.
Dai/PHYS 342/555 Spring 2006
Chapter 10-8
Dai/PHYS 342/555 Spring 2006
Chapter 10-9
We consider the plane interface z = 0 between metal 1 at Z > 0 and
metal 2 at Z < 0. Metal 1 has bulk plasmon frequency ω p1 ; metal
2 has ω p 2 . The dielectric constants in both metals are those of free
electron gases. Show that surface plasmons associated with the
interface have the frequency
⎡1 2
⎤
2
+
ω
ω
(
p1
p 2 )⎥
2
⎣
⎦
ω= ⎢
1/ 2
.
Dai/PHYS 342/555 Spring 2006
Chapter 10-10
Longitudinal plasma oscillations
The zero of the dielectric function determine the frequencies of the
longitudinal modes of oscillation. ε (ωL )=0
ε (ωL ) = 1 − ω p2 / ωL2 = 0. When ωL = ω p , there is a free longitudinal
oscillation mode of an electron gas at the plasma frequency as the
low-frequency cutoff of transverse electron magnetic waves.
The equation of motion is
d 2u
nm 2 = − neE = −4π n 2 e 2u,
dt
1/ 2
2
⎛
⎞
4
π
d u
ne
2
+ ω p u = 0; ω p = ⎜
⎟ .
2
dt
⎝ m ⎠
2
Dai/PHYS 342/555 Spring 2006
Chapter 10-11
The displacement establishes a surface charge density − neu on
the upper surface of the slab and +neu on the lower surface. An
electric field E = 4π neu is produced inside the slab.
Dai/PHYS 342/555 Spring 2006
Chapter 10-12
Dai/PHYS 342/555 Spring 2006
Chapter 10-13
Plasmons
A plasma oscillation in a metal is a collective longitudinal excitation
of the conduction gas. A plasmon is a quantum of a plasma oscillations.
Dai/PHYS 342/555 Spring 2006
Chapter 10-14
Electrostatic screening
The electric field of a positive charge embedded in an electron gas
falls off with increasing r faster than 1/ r because of the screen effect.
G
The static screening can be described by ε (0, K ). For a uniform gas
of electrons of charge concentration − n0 e superimposed on a
background of positive charge of concentration n0 e.
G
+
ρ ( x) = n0 e + ρext ( K ) sin Kx
G
ρext ( K ) sin Kx is the electrostatic field applied to the electron gas.
For positive charge, ϕ = ϕ ext ( K ) sin Kx; ρ = ρ ext ( K ) sin Kx.
K 2ϕ ext ( K ) = 4πρ ext ( K ).
The amplitude of the total electrostatic potential ϕ ( K ) = ϕ ext ( K ) + ϕ ind ( K ).
Total charge density variation ρ ( K ) = ρ ext ( K ) + ρind ( K )
K 2ϕ ( K ) = 4πρ ( K ).
Dai/PHYS 342/555 Spring 2006
Chapter 10-15
Thomas-Fermi approximation: a local internal chemical potential can be
defined as a function of the electron concentration at that point. In a
region of no electrostatic contribution to the chemical potential, we have
2/3
2
2/3
=
=
N
⎛
⎞
0
2
2
μ = εF =
3π n0 ) at zero temperature.
(
⎜ 3π
⎟ =
V⎠
2m ⎝
2m
In the region where the electrostatic potential is ϕ ( x), we have
2
2
2
/
3
2/3
=2
=
2
2
μ = ε F ( x) − eϕ ( x) ≅
3π n ) − eϕ ( x) ≅
3π n0 )
(
(
2m
2m
Dai/PHYS 342/555 Spring 2006
Chapter 10-16
At q k F , by a Taylor expansion of ε F , we have
dε F
[ n( x) − n0 ] ≅ eϕ ( x)
dn0
n( x) − n0 ≅
3 eϕ ( x)
n0
or ρind ( K ) = −(3n0 e 2 / 2ε F )ϕ ( K ).
εF
2
ρind ( K ) = −(6π n0 e 2 / ε F K 2 ) ρ ( K ),
ε (0, K ) = 1 − ρind ( K ) / ρ ( K ) = 1 + ks2 / K 2 ;
Where k s2 ≡ 6π n0 e 2 / ε F
Dai/PHYS 342/555 Spring 2006
Chapter 10-17
Screened Coulomb potential
The Poisson equation for the unscreened coulomb potential is
∇ 2ϕ 0 = −4π qδ (r ),
The unscreen potential is ϕ 0 = q / r.
The screen potential is ϕ (r ) = (q / r ) exp(− k s r ).
Dai/PHYS 342/555 Spring 2006
Chapter 10-18
Mott Metal-Insulator Transition
A crystal composed of one hydrogen atom per primitive cell
should be a metal, following free electron theory. A crystal with
one hydrogen molecule per primitive cell is different.
Question: Is a lattice of hydrogen atoms at zero temperature a
metal or an insulator?
Assume metallic state where a conduction electron sees a screened
coulomb interaction from each proton:
U (r ) = −(e 2 / r ) exp(− k s r ),
where ks2 = 3.939n01/ 3 / a0 . At large a0 a bound state may condense
about the protons to form an insulator.
Dai/PHYS 342/555 Spring 2006
Chapter 10-19
Dai/PHYS 342/555 Spring 2006
Chapter 10-20
Electron-electron interaction
The effects of electron-electron interactions are usually described
within the framework of the Landau theory of a Fermi liquid.
A Fermi gas is a system of noninteracting fermions; the same
system with interactions is a Fermi liquid.
Consider that we gradually turning on the interactin between
electrons, they will have two effects:
1. The energies of each on-electron level will be modified.
Dai/PHYS 342/555 Spring 2006
Chapter 10-21
2. Electron will be scattered in and out of the single electron
level, which are no longer stationary. Whether this scattering
is serious enough to invalidate the independent electron picture
depends on how rapid the rate of scattering is. If the scattering
rate is low, electron-electron relaxation time is much larger
than other relaxation time, then we can ignore it and use the
independent electron theory with modified mass.
In metal, although conduction electrons are crowded together
only 2 A apart, they travel a long distance before colliding with
each other due to
1. Exclusion principle.
2. The screening of the coulomb interaction between two electrons.
Dai/PHYS 342/555 Spring 2006
Chapter 10-22
Dai/PHYS 342/555 Spring 2006
Chapter 10-23
Suppose N electron state consists of a filled Fermi sphere (at
T = 0) plus a single excited electron in a level with ε1 > ε F .
In order for this electron to be scattered, it must interact with
an electron of energy ε 2 , which must be less than ε F . The exclusion
principle requires that these two electrons can only scatter
into unoccupied levels, whose energies ε 3 and ε 4 must be greater
than ε F . Or ε 2 < ε F , ε 3 > ε F , ε 4 > ε F .
In addition, energy conservation requires that
ε1 + ε 2 = ε 3 + ε 4 .
If ε1 is exactly ε F , ε 2 , ε 3 , ε 4 must also be ε F . Thus the allowed wave
vectors occupy a region of K space of zero volume. The life time
of an electron at the Fermi surface at T = 0 is infinite.
Dai/PHYS 342/555 Spring 2006
Chapter 10-24
When ε1 is different from ε F , some phase space becomes available
since the other three energies can now vary within a shell of thickness
of order ε1 − ε F about the Fermi surface, leading to a scattering
rate of order (ε1 − ε F ) 2 .
If the excited electron is superimposed not on a filled Fermi surface,
but on a thermal equilibrium distribution of electrons at nonzero T .
There will be partially occupied levels in a shell of width k BT about
ε F . This provides an additional range of choice of order k BT , and
therefore leads to a scattering rate as ( k BT ) . At temperature T , an
2
electron of energy ε1 near the Fermi surface has a scattering rate
1
τ
= a (ε1 − ε F ) 2 + b ( k BT ) ,
2
where the coefficients a and b are independent of ε1 and T .
Dai/PHYS 342/555 Spring 2006
Chapter 10-25
Thus the electron life time due to electron-electron scattering
can be made as large as one wishes by reducing T and by considering
electrons sufficiently close to the Fermi surface.
Assume that the temperature dependence of τ is taken into account by
a factor 1/ T 2 . We expect from lowest-order pertrubation theory that
τ will depend on the electron-electron interaction through the square
of the Fourier transform of the interaction potential.
2
2
2
2 2
1
2 ⎛ 4π e ⎞
2 ⎛π = ⎞
∝ ( k BT ) ⎜ 2 ⎟ ∝ ( k BT ) ⎜
⎟ ,
τ
⎝ mk F ⎠
⎝ k0 ⎠
Dai/PHYS 342/555 Spring 2006
Chapter 10-26
Fermi liquid theory: quasiparticles
If the independent electron picture is a good approximation, then at least
for levels near the Fermi energy, electron-electron scattering will not
invalidate this picture. If electron-electron interactions are strong,
what happens?
Landau suggested that we can use the independent "quasiparticles"
that obey the exclusion principle. The independent electron picture
is quite likely to be valid if
1. We are only dealing with electrons within k BT of ε F .
2. We are deadling with "quasiparticles"
3. We allow for the effects of interaction on the ε vs K relation.
Dai/PHYS 342/555 Spring 2006
Chapter 10-27
Electron-phonon interaction: polarons
When electron is in a lattice, it may
deform the lattice and induce
strain field. The electron+associated
strain field is called a polaron.
A large polaron moves in a band while
a small polaron spends most of its time
trapped on a single ion.
Dai/PHYS 342/555 Spring 2006
Chapter 10-28
Peierls instability of linear metals
Dai/PHYS 342/555 Spring 2006
Chapter 10-29
The frequency of the uniform plasmon mode of a sphere is
determined by the depolarization field E = −4π P / 3 of a
sphere, where the polarization P = − ner , with r as the average
displacement of the electrons of concentration n. Show from
F = ma that the resonance frequency of the electron gas is
ω02 = 4π ne2 / 3m. Because all electrons participate in the oscillation,
such an excitation is called a collective excitation of the electron
gas.
Dai/PHYS 342/555 Spring 2006
Chapter 10-30