Electrons dans les solides - TD 4

U(x)
U(x)v(x−na)a
x=na U(x)Pnv(x−na)
v(x)
v(x)
U(x)
v(x)
v(x)ε= ¯h2K2/2m m K
v(x)=0 |x| ≥ a/2
ψg(x)
ψg(x) = exp(iKx) + rexp(−iKx), pour(x≤ −a/2),
=texp(iKx), pour(x≥+a/2),
t r

v(x)ψd(x)ψg(−x)
ε ψd(x)
ψd(x) = texp(−iKx), pour(x≤ −a/2),
= exp(−iKx) + rexp(iKx), pour(x≥+a/2),
v(x)
v(x)
ψ(x)
ψ(x)ψg(x)ψd(x)−a/2≤
x≤+a/2
ψ(x)
k
K ψ(x)
x
x=−a/2
cos(ka) = t2−r2
2texp(iKa) + 1 2texp(−iKa)
v(x)
r t K
v(x)
r t
φ1φ2
v(x)W(φ1, φ2)φ0
1(x)φ2(x)−φ1(x)φ0 2(x)

W(φ1, φ2)x
|r|2+|t|2W(ψg, ψ∗
g)x≤ −a/2x≥+a/2
rt∗W(ψg, ψ∗
d)x≤ −a/2
x≥+a/2
r t t =|t|exp(iδ)
cos(ka) = cos(Ka +δ)
|t|
cos(ka)K
|t| ∼ 1|r| ∼ 0δ∼0
∆εgap ∼2πn ¯h2
ma2|r|
n
|t| ∼ 0|r| ∼ 1
|t|