hw2-3

Phys 446 Condensed Matter Physics Homework 2 and 3 Due to March 16th 2012, Friday 9:40 in class 1) Surface plasmons: Consider a semi-infinite plasma on the positive side of the plane  z=0.  z=0. A solution of Laplace’s equation ∇ 2ϕ  = 0 in the plasma is ϕ i (

, y ) = A cos(kx )e

that in the vacuum

− kz

, whence E zi = kA cos( kx )e − kz ; E xi = kA sin( kx )e − kz . (a) Show

ϕ 0 ( x , z)

for  z < 0 satisfies the boundary condition = A cos( kx)ekz for  z 

that the tangential component of  E  be continuous at the boundary; that is, find E  find E  xo. (b) Note that Di = ε (ω ) Ei ; 



D0 = E0 . Show   that the boundary condition that the 



normal component of  D be continuous at the boundary requires that ε (ω ) = − 1 , 

2

whence from the relation ε (ω ) = 1 −

ω  p

2

ω 

for the frequency

ω  s

we have the Stern-Ferrell result: ω s2 =

1 2

2

ω p

of a surface plasma oscillation.

2) Plasmon mode of a sphere: 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 ne 2 / 3m . Because all electrons participate in the oscillation, such an 



excitation is called a collective excitation or collective mode of the electron gas. 3) Plasma frequency and electrical conductivity: An organic conductor has recently 15 1  been found by optical studies to have ω p = 1.80 x 10 s‐ for the plasma frequency, 15 and τ  = 2.83 x 10‐ s for the electron relaxation time at room temperature. (a) Calculate the electrical conductivity from these data. The carrier mass is not known and is not needed here. Take ε (∞ ) = 1 . Convert the results to units (Ω cm)‐1. (b) From the crystal and chemical structure, the conduction electron concentration is 4.7 x 1021cm‐3. Calculate the electron effective mass m*. 4) Gap plasmons and the van der Waals interaction: Consider two semi-infinite media with plane surfaces z surfaces z = 0 and d . The dielectric function of the identical media is ε( ω ). Show that for surface plasmons symmetrical with respect to the gap the 2 2 2 frequency must satisfy ε( ω ) = -tanh (Kd/2), (Kd/2), where K  where K  = k  x + k  y . The electric  potential will have the form φ

= f(z) exp (ik  x x + ik  y y – iωt)

Look for non-retarded so1utions - that is, solutions of the Laplace equation rather than of the wave equation. The sum of the zero-point energy of all gap modes is the nonretarded part of the van der Waals attraction between the two specimens-see N. G. van Kampen, B. R. A. Nijboer, and K. Schram, Physics Letters 26A, 307 (1968)..

5) Reflection at normal incidence: Consider an electromagnetic wave in vacuum, with field components of the form E y (inc) = Bz (inc) = Aei ( kz −ω t )  .

Let the wave be incident upon a medium of dielectric constant ε and permeability  μ = 1 that fills the half-space x > 0. Show that the reflectivity coefficient r( ω ) as defined  by E(refl) = r( ω ) E (inc) is given by n + iK − 1 r (ω ) = n + iK + 1 1/2 where n + iK = ε , with n and K real. Show further that the reflectance is

 R(ω ) =

(n − 1) 2 + K 2 (n + 1) 2 + K 2

6) Conductivity sum rule and superconductivity: We write the electrical conductivity as σ(ω)

= σ1(ω) + iσ2(ω) where σ1, σ2 are real. (a) Show by a Kramers-Kronig relation that

lim ωσ 2 (ω ) =

ω →∞

2 π 

∞

∫ σ (  s)ds . 1

0

This result is used in the theory of superconductivity. If at very high frequencies (such as xray frequencies) σ2(ω) is identical for the superconducting and normal states, then we must have ∞

∫

∞  s σ1

(ω )d ω =

0

∫σ

n 1

(ω ) d ω . 

0

But at frequencies 0 < ω < ω g , within the superconducting energy gap the real part of the conductivity of a superconductor vanishes, so that in this region the integral on the left-hand side is lower by

≈ σ 1nω  . There must be an additional contribution to

 s

σ 1

to balance this

deficiency.  s

n

 s

(b) Show that if  σ 1 (ω < ω g ) < σ 1 (ω < ω g ), as is observed experimentally, then σ 1 (ω) can

have a delta function contribution at  s

σ 2 (ω )

ω

= 0, and from the delta function there is a contribution

≈ σ 1nω g  / ω .  The delta function corresponds to infinite conductivity at zero frequency.

(c) By elementary consideration of the classical motion of conduction electrons at very high frequencies, show that ∞

∫ σ (ω )d ω  = 1

0

π ne

2

2m

result found by Ferrell and Glover.

7) Davydov splitting of exciton lines: The Frenkel exciton band is doubled when there are two atoms A, B in a primitive cell. Extend the theory of eqs. (25) to (29) of Kittel, chapter 11 about Frenkel excitons to a linear crystal AB.AB.AB.AB. with transfer integrals T 1 between AB and T 2 between B.A. Find an equation for both bands as a function of the wavevector. The splitting between the bands at k = 0 is called the Davydov splitting.