Preliminaries: from previous section, you had,
[13.1] → dN = 2deg
g n (r, k , t )
(2π )
3
number of electrons in n th 1 1 = β (ε − µ ) +O d r ⋅ d k = 3 3 +1 band at time t in d r ⋅ d k e 3
3
(1.1)
And, the relaxation-time approximation is encapsulated as, the relaxation-time equilibrium dt 0 d g t t t d t ↔ ≡ ∈ + × [13.3] → ( , , ) P Pr r ( c o l l i s i o n [ , ] ) r k n = × g n ( r, k); (1.2) approximation distribution τ n Also: Liouville’s theorem from appendix H, that, rɺn (t ′) ≡ i1ℏ [rn (t ′), H ] ≡ +∂ p H ∂ (r, p) Jacobian of
→ 1 ′ k ≡ − ∂ t H H [ ( ) , ] n r ∂(r, k ) i
kɺ n (t ′) ≡
=
ℏ
volume element in rp-space is = ↔ 1 canonical coords same as vol. element in rk-space ; (1.3)
Hydrodynamic analogy, and a nd a collision-probability-laden-state-space: Let a probability of collision be associated with every volume-element in phase-space. Now: consider that ∃rn (t ′) ∈ d 3 r ′ × d 3 k ′ and 3
3
ɺn (t ′) ≡ ∃k n (t ′) ∈ d r ′ × d k ′ , the solutions to r
1 iℏ
[rn (t ′), H ] and kɺ n (t ′) ≡
1 iℏ
[k n (t ′), H ] (which are semiclassical)
with the “initial” conditions rn (t ) = r and k n (t ) = k . As indicated, they are in the phase-space-volume-element 6 3 3 ∈ d Vφ ′ ≡ d r ′ × d k ′ . Using (1.1), we compute dN ′ , the number of electrons emerging from the collisions that
happen in the phase-space-coordinate as, 0 relaxation time approximation g n (rn (t ′), k n (t ′), t ′) 3 g n (rn (t ′), k n (t ′), t ′) dt ′ 3 ′⋅ d k′ ≈ dN = dN ′ = d r d 3r ⋅ d 3k 3 3 τ n (rn (t ′), 4π 4π ), k n (t ′)) )) d N (t ) =
d 3r ⋅ d 3 k
t
∫ −∞
4π 3
d 3r ⋅ d 3k × [ Pn (r , k , t ; t ′)] × d t ′ ≡ [ g n (r, k , t )] 3 4 τ n (rn (t ′), k n (t ′)) π
g n0 (rn (t ′), ), k n (t ′), ), t ′)
(1.4)
Thus, we have an expression for the non-equilibrium distribution function, from which we can eliminate all mention of the canonical coordinates, and just consider time-dependence, t
g n (r, k , t ) =
∫ −∞
t
g n0 (rn (t ′), ), k n (t ′), ), t ′) dt ′ dt ′ d ≈ dt ′ = [operator ]; (1.5) Pn (r, k , t ; t ′)dt ′ = g (t ) = g 0 (t ′) P(t ; t ′) ; τ n (rn (t ′), k n (t ′)) τ (t ′) τ (t ′) dt ′ −∞
∫
With the “operator approximation” shown in (1.5), we can integrate by parts and write, t t t 0 dt ′ dP(t , t ′) dg (t ) 0 0 0 = g (t ′) − g (t ) = g (t ′) P(t , t ′) = g (t ′)dt ′ P(t , t ′)dt ′ ′ ′ ′ ( ) τ t d t d t −∞ −∞ −∞
∫
We compute the derivative dg 0 (t ′) dt ′
=
∫
dg 0 ( t ) dt
dg 0 d ε n dkn
∫
(1.6)
for the g 0 (t ) = g 0 (ε n (k n (t ′)) )),T (rn (t ′ ), ), µ (rn (t ′ )) )) ) functional-dependence, ℓ
d ε n dk ℓ dt ′
+
dg 0 dT drn
ℓ
dT dT dr ℓ dt ′
ℓ
+
dg 0 d µ drn
−df
d µ dr ℓ dt ′
d ε
=(
)vℓ ( −eE ℓ − ∇ ℓ µ − (ε − µ )
∇T
); T
(1.7)
Finally: putting (1.7) into (1.6), we finally compute the nonequilibrium distribution function as, t
0
g (t ) = g (t ′) −
− df
∫ ( dε )v (−eE
−∞
ℓ
ℓ
ℓ
− ∇ µ − (ε − µ )
∇T
T
) P(t , t ′) dt ′
Collision probability as an integral-equation: In (1.5), we can compute the collision-probability as,
(1.8)
fraction of e- ∈ band n
dt ′ ′ ′ ′ ′ = × − ∈ + = + − d P t t dt P t t d t 1 Pr(collision [ , ]) ( , )(1 ) ( ) τ (t ′) having no collision ∈[t,t ′]
P(t , t ′) =
Invoking the definition of a derivative
∂ f ( t , t ′ )
∂P (t , t ′) ∂t ′
∂t ′
=
≡ lim ∆t →0
P (t , t ′ )
τ (t ′)
f ( t , t′ +∆ t ) ∆t
(1.9)
, the (1.9) is equivalent to,
↔ P(t , t ′) = exp( −
t
1
∫ τ (t ′′) dt ′′) t ′
(1.10)
