343 - Lindhard Theory of Screening

The TF-theory of screening subsumed a semiclassical approximation to the Schrödinger equation, and requires a slowly-varying potential (a point charge’s potential was shown to satisfy this requirement). The effect of this was that the electron energy levels were only shifted by the potential directly: ε (k ) =

2 2

− eφ (r ) .

ℏ k 

2m

The Lindhard theory uses a perturbative expansion ψ (k ) = ψ k = ψ k0 + ψ k1  rather than the semiclassical   2 2

approximation ak F ≡ φ (k F r ) / φ ′( k F r ) << 1 → ε (k ) ≈  ρ (r ) = −e →  ρ

ind

∑

k

0 k

fk ψ +ψ

1 k

2

= −e

∑f

ℏ k 

2m

1

− eφ (r ) . We   have , −1

k

(V +

k

∑ ε (k ) − ε ( k + q) V (r)) ≡  ρ

∫ ∑ ε (k ) − ε (k + q) f

k

(eφ (r))]e − i⋅q •r d 3r = − e 2

kq

Juxtaposing (1.1) with  ρ

ind 

0

ind 

+ ρ 

q

e − i⋅q•r + e i⋅q•r

(q) = F[ ρ ind ( r )] =ɺ − e [

e − i⋅q•r + ei⋅q•r

 f k − 1 q − f k + 1 q

∫ ℏ (k • q ) / m 2

2

2

(q) ≡ χ (q )φ (q) , we have the susceptibility. Using  f k ∓ 1 q = f k ± 2

see what happens at k • q << k F 2 and at k • q ~ k F 2 , T  = 0 (in which we use  x =

q 2 k F 

and

φ ( q)

2

ℏ ⋅k • q ∂ f k 2m ∂ µ 

d ε  dk

=

2

ℏ k 

2m

d 3k

(1.1)

; 3 4π 

+O

2

 

(q ) , we

),

2 2 3  ℏ (k • q) −1 ℏ (k • q) ∂ fk ∂f k d k   d 3k 2 2 2 ∂n0 ≈ −e ∫ ( ) [ fk + 2 − fk ] = −e ∫ 2 = −e =  χ TF ;  3 3  f k − q − f k + q d 3 k   2 2 ∂  µ 8 π ∂ µ 8 π ∂ µ  m m 2 (1.2)  χ (q) = −e ∫ 2 3  ∞ +1 2 2 ℏ (k • q) / m 4π  T = 0 ( ) ( ) Θ ε − ε − Θ ε − ε   1 1− x 1− x k −q /2 k +q /2 k dk ⋅ 2π d χ  F F  2 2 mk  = −e 2 F 2 ( + ln ); 2 3  = −e ∫ ∫  χ  / 4 π π  2 4 1 ℏ ℏ + k q m x x  0 −1 1 2

1 2

Thus, the Lindhard dielectric constant contains the physics of the semiclassical approximation. Freidel oscillations: the Lindhard dielectric constant also contains more physics: φ (r >> a ) = r −3 cos( 2k F r ) . 

We talk about these in chapter 26. Appendix – evaluation of the integral  I (k F  ) =

1

1 V

∑

k′

'

k F +1

1 k − k′

2

≈

∫∫k 0 −1

1 2

2

+ k ′ − 2kk ′x

k F  / k ′

2

2π k dxdk 8π

3

V

See SSP 02 - 352 - pr 05 - computing lindhard response function.

=

1 4π 2

∫ 0

k / k ′ +1

2

k  1 − x 1+ x ln ln + 1) (1.3) dk  = F 2 ( 2 k ′ k / k ′ −1 4π  2x 1− x

1 k