Halbleiterphysik - Eine Zusammenfassung

pn
nn +
pn
pn
pn
pn

E = hf = ¯hω
pq = E
c
E = pqc
= hf
c
= h
λ
pq . . . c . . .
¯hω0 = ¯hωsc + Ee
¯h −→ k 0 = ¯h −→ k sc + −→ p e
Ee −→ p e
−→ k
¯h −→ k
−→ k
= 2π
λ
= h 2π h
· =
2π λ λ
¯hω ¯hk

sin cos
• ω(k)
Ψ (x, t) = A. sin (x, t) = A. sin (k.x − k.v.t)
•
•
•
vg = ∂ω(k)
∂k
Ψ(x, t) = exp [i (k.x − ω.t)]
E = ¯hω
p = ¯hk
E = p2
2m
i¯h ∂
∂t Ψ(x, t) = EΨ(x, t)
− ¯h2 ∂
2m
2
∂x2 Ψ(x, t) = p2
⇒ −
2mΨ(x, t)
¯h2 ∂
2m
2
∂
Ψ(x, t) = i¯h Ψ(x, t)
∂x2 ∂t

− ¯h2 ∂
2m
2
Ψ(x, t)
∂x2
+ V (x, t).Ψ(x, t)
△ = ∂2
∂x
= i¯h ∂
Ψ(x, t)
∂t
∂y
∂z 2
∂2 ∂2
+ + 2 2
− ¯h2
2m △ Ψ(−→ x , t) + V ( −→ x , t).Ψ( −→ x , t) = i¯h ∂
∂t Ψ(−→ x , t)
+∞
−∞
Ψ ∗ ( −→ x , t)Ψ( −→ x , t)dV = 1
Ψ(x, t) = 1
√
2π
g(k) · e i(kx−ω(k)t) dk
g(k)
∆x · ∆px ≥ ¯h
2
Ψ( −→ x , t) = e −iωt · ϕ( −→ x )
− ¯h2
2m △ ϕ(−→ x ) + V ( −→ x ).ϕ( −→ x ) = E · ϕ( −→ x )

ϕ(x) = 0
ϕ(x) = A. sin(kx)
E(k) = ¯h2 k2
2m
ϕ(x = 0) = ϕ(x = a) a
k = nπ
. . . (n = 1, 2, 3, . . .)
a
A
ϕn(x) =
2
a sin
nπx
a
En = ¯h2 π 2
2ma 2 n2
L
ϕl,j,n(x, y, z) =
3
2 2
sin
L
lπx
sin
L
jπx
sin
L
nπx
L
En = ¯h2 π2 2mL2 2 2 2
l + j + n
V
V
ϕI(x) = A1e ikx + A2e −ikx
ϕII(x) = B1e κx + B2e −κx
ϕIII(x) = C1e ikx + C2e −ikx
i II
k κ
Ai Bi Ci
ϕI(x = 0) = ϕII(x = 0)
ϕII(x = d) = ϕIII(x = d)
∂ϕI
∂x
∂ϕII
∂x
∂ϕII
(x = 0) = (x = 0)
∂x
∂ϕIII
(x = d) = (x = d)
∂x

⇒ C2 = 0
⇒ A1 = 1
ϕ(x) = A. exp(−ikx)
Ψr(x, t) = A. exp(−ikx).exp(−iωt)
R = Ψ ∗ r(x, t).Ψr(x, t)
ϕ(x) = A. exp(+ikx)
Ψt(x, t) = A. exp(+ikx).exp(−iωt)
R = Ψ ∗ t (x, t).Ψt(x, t)
1 = 10 −10 m
V (r) = − e2
4πε0r
− ¯h2
e2
∆Ψ(r) − Ψ(r) = EΨ(r)
2m 4πε0r
∆ r θ ϕ
Ψ(r, θ, ϕ) = R(r).Θ(θ).ϕ(ϕ)
r θ ϕ n m l
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . (n − 1)
m = −l, −l + 1, . . . , 0, . . . l − 1, l

n
En = − e2
4πε0aB
aB
aB =
· 1
n 2
2
4πε0¯h
= 0.053nm
m0e2
4 m0e
EB = − = −13.6eV
2(4π¯hε0) 2
±¯h/2
s = ± 1
2
¯hω = Em − En
1
¯hω = ER 12 − 1
n2
1
¯hω = ER 2
n = 2, 3, 4, . . .
2 − 1
n2
¯hω = ER
n = 3, 4, 5, . . .
n = 4, 5, 6, . . .
1
3 2 − 1
n 2

→
k k
E(k)
Ψ = ΨH(r1) − ΨH(r2) Ψ = ΨH(r1) + ΨH(r2)
E(k)
→
→
→
T = 0

→
→
d
Θ
2d · sin(Θ) = n · λ
2π
kl = lπ
a
l = ±1, 2, 3, . . .
exp i −→ k · −→
x
uk ( −→ x )
Ψk ( −→ x ) = uk ( −→ x ) · e i−→ k −→ x

a0
V (x + a0) = V (x)
Ψk(x) = uk(x) · e ikx u(x) = u(x + a + b)
k
β 2 − α 2
2αβ
· sinh (βb) · sin (αa) + cosh (βb) · cos (αa) = cos (k (a + b))
α 2 = 2m
¯h 2 E β2 = 2m
¯h 2 (V0 − E)
+1 −1 ⇒
+1 −1
P sin(αa)
αa
+ cos(αa) = cos(ka)
P = ma
¯h 2 · V0 · b
P → 0
P → ∞
E(k) E(k) a, P =
const. k α E(k)
k = nπ
a cos
k = nπ
a
− π
π
a ≤ k ≤ + a
− 2π
a
π ≤ k ≤ − a
π
2π
+ a ≤ k ≤ + a
E(k)
→
k = ± 2π
a

E(k)
→
k q
K
ω(q) = 2
M ·
qa
sin
2
ωac(q) = ω0
√2 · 1 − 1 − γ2 2 aq
sin
2
ωopt(q) =
ω0
√2 · 1 + 1 − γ2
2 aq
sin
2
ω0 =
(C1 + C2) (M1 + M2)
γ
M1M2
2 = 16 C1C2
2 ·
(C1 + C2) M1M2
(M1 + M2) 2
q

A + B −
En = n + 1
¯hω
2
¯hω
T
n =
exp
1
¯hω
kBT
− 1
[uvw]
(hkl)
•

•
•
•
Γ
X
L
Γ
EG

EC, EV
ELeitungsband = EC + ¯h2 k 2
2m∗ n
EV alenzband = EV + ¯h2 k2 2m ∗ p
ELeitungsband = EC + ¯h2 k2 1
2m∗ +
n1
¯h2 k2 2
2m∗ +
n2
¯h2 k2 3
2m∗ n3
k = 0
k = 0
IV − IV III − V
m ∗ = ¯h 2 ·
2 d E
dk2 −1

1 1
=
m∗ ¯h 2
⎡
⎢
⎣
∂ 2 E
∂k2 x
∂ 2 E
∂ky∂kx
∂ 2 E
∂kz∂kx
∂ 2 E
∂kx∂ky
∂ 2 E
∂k 2 y
∂ 2 E
∂kz∂ky
∂ 2 E
∂kx∂kz
∂ 2 E
∂ky∂kz
∂ 2 E
∂k2 z
⎤
⎥
⎦
k

→
→
→
ni
n p
p = n = ni(T )
n p
EC

→
D + + e → D ∗
→
→ →
A ∗ → A − + h
p
n
→
p · n = n 2 i
p p = NA n = n 2 i /NA
n n = ND p = n 2 i /ND
E1 E2 n1
n2
n2
n1
= exp − E2
kT
exp − E1
= exp −
kT
E2
− E1
kT
1
f(E) =
E−EF
1+exp kT
f(E) = exp − E−EF
kT
→
EF
EF

Z(E)
E
dN(dkx, dky, dkz) = 2 ·
k
=
k
= 2 · dkx · dky · dkz
2π
L
3
= 2V · 4π · k2 dk
(2π) 3
E = ¯h2 k 2
2m ∗ ⇒ k2 = 2m∗
¯h 2 · E
dE = ¯h2
k · dk
m∗ dN(E) = 2V
(2π) 3
4π
¯h 3 (2m∗ ) 3 1
2 E 2 · dE
Z(E) · dE = dN(E)
V
Z(E) = 1
(2π) 3
4π
¯h 3 (2m∗ ) 3 1
2 E 2
∞
n = f(E)Z(E)dE
p =
EV
∞
EC
(1 − f(E))Z(E)dE
< 1018cm−3
∞
n = f(E)Z(E)dE ≈ NC · exp − EC
− EF
p =
EV
∞
EC
(1 − f(E))Z(E)dE ≈ NV · exp
NC = 2
¯h 3 (2πm∗ nkT ) 3
2 NV = 2
¯h 3
NC NV
kT
− EF − EV
kT
2πm ∗ pkT
3
2

EF EC EF
EV
ND >> NC
ni
n 2
i = n · p = NCNV · exp − EC
− EF
kT
EG
n n > ni > p
p n < ni < p
> ni
n · p = n 2 i
ρ = e (p − n + ND − NA) = 0
n ND > NA
n = ND
− NA (ND − NA)
+
2
2
+ n
4
2 i
p = − ND
− NA (ND − NA)
+
2
2
+ n
4
2 i
ni ND NA
ND
n =
n 2 i
NA − ND
p = NA − ND

→
→
hf >
EC −EV G = Gth +GL
n · p > n 2 i
αn αp
τL
−→ −→ ∗
F = e · E = m d−→ v
dt + m∗−→ v
τ
d −→ v
dt = 0 ⇒ m∗ · −→ v
τ
= e · −→ E

−→ j σ
−→
j = n · e ·
−→
v
σ = e2 · n · τ
m∗ −→
j =
−→
σ · E
µ
µ =
e · τ
m∗ σ = e · n · µ
−→ J n = e · n · −→ v n
−→ J p = e · n · −→ v p
−→ J = −→ J n + −→ J p = e (nµn + pµp) −→ E
σ = e (nµn + pµp)
→
∂f e
−
∂t ¯h E · ∇kf
df
+ v · ∇rf =
dt
f ∇k ∇r
•
•
coll

vn = −µnE
vp = +µpE
1
µ = 1
µi
i
n →
−→ F = ±e · −→ v × −→ B
n p
→
UH = 1 IB
en Z
Jp =
dp
−eDp
dx
Jn =
dn
+eDn
dx
UT
Dn = UT µn
Dp = UT µp
UT = kBT
e

Jp =
dp
epµpE − eDp
dx
Jn =
dn
enµnE + eDn
dx
∂p
∂t
1 Jp(x) − Jp(x + ∆x)
= + G − R
e ∆x
∂p 1 ∂Jp
+
∂t e ∂x
∂n 1 ∂Jn
−
∂t e ∂x
dE
dx
= G − R
= G − R
= ρ
ε
Lp
exp
x = 0
τd
τd = ε
σ
τL >> τd
→

¯hω = ∆E
•
• k
¯hk →
E(k)

¯hω
Zj(ω) = 1
2π 2
(2mr) 3/2
¯h 2
¯hω − EG
fabs = fv(E1) [1 − fc(E2)]
fem = fc(E2) [1 − fv(E1)]
fabs − fem = fv(E1) − fc(E2)
σ
α(ω) = σ(ω) · Zj(ω) · (fabs − fem)
(fabs − fem) > 0 ⇒
⇒
EF c − EF v > EG
rsp(ω) = σsp(ω) · Zj(ω) · fc(E2)[1 − fv(E1)]

m∗
Eex = EG − En

pn
nn +
dn
UT
Jn = eµnnE + eDn = 0 ⇒ E =
dx n
· dn
dx
ϕ − ϕ1 = −
x
x1
Edx = UT
pn
x
x1
1 dn
dx = UT
n dx
n
n1
1
n dn = UT ln n
n1
n = n2
p = n1
UD = ϕ2 − ϕ1 = UT · ln NAND
n 2 i
n
p U
UD
UD − U
pn →
UD UD + |U|
x = −dp x = +dn
Qp + Qn = −eNAdpA + eNDdnA = 0

NAdp = NDdn
dn dp Emax
1
=
N0
1
+
NA
1
ND
2εUD
d = dn + dp =
eN0
pn
p
n
p
n
Is
I =
U
Is · exp
− 1
UT
Is = Js · A = en 2
Dp
i A
LpND
+ Dn
LnNA
•
•
• →
p n lp ln
Lp Ln
n 2 i
n 2 i ∝ exp
− EG
kT

pn
U ∗ B ≈ εE2 max
2eN0
•
•
→
•
→
→
pn
•
•
•
pn
→
U = 0
U > 0
n p →
p
n
→

A B A B
→

npn n p
n pnp n p
BC n p
p n
BC
IC = eDn
∂n eDnnp
∂x A = A
W
UBE np IC IC
UCB UCB
C B → → IC UCB
B = IC
IB
•
• BC →
•
•
•
•
BE BC UCE IB
•

IC = Is
exp
UBE
UT
−
UBC
UT
n n
UDS > 0
p
UGD = UGS − UDS
UGS ≤ 0
|UGD| > |UGS|
D
S → D S
UDS UGS
UDS
UDSS

µ ∝ T −2
•
•
•
•
•
•
•
•
•