Some Calculations
At room temperature kT = 0.0259eV
At room temperature ni for Si = 1.5 x 1010/cm3
Solve this equation for E = EF
1
𝑓 𝐸 =
1 + 𝑒 (𝐸−𝐸𝐹 )/𝑘𝑇
Let 𝑇 → 0𝐾 find f(E<EF) and f(E>EF)
Let T = 300K and EF = 0.5eV plot f(E) for 0 < E < 1
EC
EV
Fermi-Dirac plus Energy Band
More Calculations
At room temperature kT = 0.0259eV
At room temperature ni for Si = 1.5 x 1010/cm3
If Na = 2 x 1015 /cm3 find po and no
The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eV
What is the value of EC – EF for intrinsic Si at T= 300K
The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eV
What is the value of Ei – EF if Na = 2 x 1015 /cm3 at T= 300K
The band gap of Si at room temp is 1.1eV or EC – EV = 1.1eV
What is the value of EF – Ei if Nd = 2 x 1015 /cm3 at T= 300K
Intrinsic Carrier Concentrations
SEMICONDUCTOR
ni
Ge
2.5 x 1013/cm3
Si
1.5 x 1010/cm3
GaAs
2 x 106/cm3
Which element has the largest Eg?
What is the value of pi for each of these elements?
Si with 1015/cm3 donor impurity
Diffusion Processes
n(x)
𝜑𝑛 𝑥0 =
n1
n2
Since the mean free path is a small differential,
we can write:
𝑛1 − 𝑛2
x0
x0 - l
x0 + l
𝑙
(𝑛 − 𝑛2 )
2𝑡 1
𝑛 𝑥 − 𝑛(𝑥 + ∆𝑥)
=
𝑙
∆𝑥
Where x is at the center of segment 1 and ∆𝑥 = 𝑙
In the limit of small ∆𝑥
𝑙2
𝑛 𝑥 − 𝑛 𝑥 + ∆𝑥
𝑙 2 𝑑𝑛(𝑥)
𝜑𝑛 𝑥 =
lim
=
2𝑡 ∆𝑥→0
∆𝑥
2𝑡 𝑑𝑥
𝑙2
2𝑡
≡ 𝐷𝑛 or 𝐷𝑝
Drift and Diffusion Currents
E(x)
Electron drift
Hole drift
n(x)
Electron & Hole
Drift current
Electron diffusion
Hole diffusion
p(x)
Electron Diff current
Hole Diff current
𝐸(𝑥)
𝑉 𝑥 =
−𝑞
E(x) =
𝑑𝑉(𝑥)
𝑑𝑥
𝑑𝑉(𝑥)
𝑑 𝐸𝑖
1 𝑑𝐸𝑖
=−
=
E(x) =
𝑑𝑥
𝑑𝑥 −𝑞
𝑞 𝑑𝑥
The Einstein Relation
At equilibrium no net current flows so any concentration gradient would be
accompanied by an electric field generated internally. Set the hole current equal to 0:
𝐽𝑝 𝑥 = 0 = 𝑞𝜇𝑝 𝑝 𝑥 𝐸 𝑥 − 𝑞𝐷𝑝
𝐷𝑝 1 𝑑𝑝(𝑥)
E(x)=
𝜇𝑝 𝑝(𝑥) 𝑑𝑥
Using for p(x)
qE(x)
E(x) =
𝐷𝑝 1 𝑑𝐸𝑖 𝑑𝐸𝐹
−
𝜇𝑝 𝑘𝑇 𝑑𝑥
𝑑𝑥
Finally:
𝑑𝑝(𝑥)
𝑑𝑥
𝐷𝑝 𝑘𝑇
=
𝜇𝑝
𝑞
𝑝0 = 𝑛𝑖 𝑒
𝐸𝑖 −𝐸𝐹 /𝑘𝑇
0
The equilibrium Fermi Level does not vary with x.
D and mu
Dn
(cm2/s)
Dp
mun
(cm2/V-s)
mup
Ge
100
50
3900
1900
Si
35
12.5
1350
480
GaAs
220
10
8500
400
Message from Previous Analysis
An important result of the balance between drift and diffusion at equilibrium
is that built-in fields accompany gradients in Ei. Such gradients in the bands
at equilibrium (EF constant) can arise when the band gap varies due to
changes in alloy composition. More commonly built-in fields result from
doping gradients. For example a donor distribution Nd(x) causes a gradient in
no(x) which must be balanced by a built-in electric field E(x).
Example: An intrinsic sample is doped with donors from one side such that:
𝑁𝑑 = 𝑁0 𝑒 −𝑎𝑥
Find an expression for E(x) and evaluate when a=1(μm)-1
Sketch band Diagram
Diffusion & Recombination
Jp(x)
Jp (x + Δx)
x
Rate of
=
Hole buildup
x + Δx
Increase in hole conc
In differential volume
Per unit time
-
Recombination
Rate
𝜕𝑝
1 𝐽𝑝 𝑥 − 𝐽𝑝 𝑥 + ∆𝑥
𝛿𝑝
=
−
𝜕𝑡 𝑥→𝑥+∆𝑥 𝑞
∆𝑥
𝜏𝑝
𝜕𝛿𝑝
1 𝜕𝐽𝑝 𝛿𝑝
=−
−
𝜕𝑡
𝑞 𝜕𝑥 𝜏𝑝
𝜕𝛿𝑛
1 𝜕𝐽𝑛 𝛿𝑛
=−
−
𝜕𝑡
𝑞 𝜕𝑥 𝜏𝑛
If current is exclusively Diffusion
𝐽𝑛 𝑑𝑖𝑓𝑓 = 𝑞𝐷𝑛
𝜕𝛿𝑛
𝜕𝑥
𝜕𝛿𝑛
𝜕 2 𝛿𝑛 𝛿𝑛
= 𝐷𝑛
−
𝜕𝑡
𝜕𝑥 2
𝜏𝑛
And the same for holes
And Finally, the steady-state
Determining Diffusion Length
𝜕𝛿𝑛
𝜕 2 𝛿𝑛 𝛿𝑛
= 𝐷𝑛
−
=0
𝜕𝑡
𝜕𝑥 2
𝜏𝑛
𝜕 2 𝛿𝑛
𝛿𝑛
𝛿𝑛
=
=
𝜕𝑥 2
𝐷𝑛 𝜏𝑛 𝐿2
𝐿𝑛 =
𝐷𝑛 𝜏𝑛