Semiconductor Junctions
The following energy diagrams are for the potential energy U= e V of an electron in the
electrostatic potential V. The Fermi level EF is given for low temperatures.
A. Semiconductor – Semiconductor:
pn – Junction
When two pieces of n- and p-type semiconductor are put into contact, electrons flow across
the boundary to reach lower energy levels. Neutral dopants become ionized and form a
dipole that gives rise to a potential step. The flow stops when the two Fermi levels coincide.
Separate n- and p-type semiconductors:
n-type
After forming the pn - junction:
p-type
CB
CB
EF
Ionized
Donors
CB
EF
EF
VB
Ionized
Acceptors
VB
VB
Electrons flow downhill from n to p
until the Fermi levels EF line up.
B. Semiconductor 1 – Semiconductor 2:
Two separate semiconductors:
n-type
Depletion:
EF mid-gap
Heterojunction
After forming a junction:
intrinsic
Ionized
Donors
Electrons
EF
Band Offset
Energy U= eV
V = Electrostatic Potential
1
Modulation Doping: Obtain high carrier density without scattering of electrons by dopants.
Dopants are on one side of the junction, electrons on the other. Achieves record mobilities
of >106 cm2/Vs in GaAlAs/GaAs. Used for creating a two-dimensional electron gas.
Semiconductor – Metal: Schottky Barrier
Separate semiconductor and metal:
After forming a junction:
CB
Schottky
Barrier
(n-type)
EF
CB
EF
EF
VB

Electrons trapped
in interface states
VB
+
Depletion width ~ 1/ND+
C. Junction with Bias Voltage, Dynamic Equilibrium
Reverse Bias:
Forward Bias:

I
+
e eV/kT
V
1
Generation (Drift) Current:
Minority carriers are generated
thermally. They drift down the
barrier. I is independent of V.
+

I  1 + e eV/kT
Opposing currents
matched at V = 0
Recombination (Diffusion) Current:
Majority carriers from dopants diffuse
against the barrier (EgeV). I depends
exponentially on V.
2
Rectifying Diode
Reverse Bias
Zero Bias
Forward Bias


+
+
Photodiode
Solar Cell
Light Emitting Diode (LED)

+

Photon creates e h+ pair
+
Apply Reverse Bias Voltage
Produces Voltage
+

e h+ pair recombines into a photon
Apply Forward Bias Voltage
Calculating Carrier Densities and Band Diagrams
A. Homogeneous Semiconductor, Static Equilibrium
Two variables: nh = Hole Density, ne = Electron Density
Two equations:
1. Charge Neutrality:
nh  ne + ND+  NA = 0
ND+ = Ionized Donor Density
NA = Ionized Acceptor Density
Determines the position of the Fermi level in the gap.
2. Mass Action:
nh  ne = const. = [nint(T)]2
nint = Intrinsic Carrier Density
Chemistry analog: H+ + OH  H2O
h+ + e
 photon

Recombination rate 
nh  ne

Generation rate  exp(Eg /kBT)
(e recombines with h+ )
(thermal photon creates e h+ pair)
In equilibrium the rates for the forward and back reaction are equal.
Driving up nh reduces ne , because the extra holes take away electrons by recombination.
3
B. Inhomogeneous Semiconductor (Junction), Static Equilibrium
Charge neutrality is gone. The space charge induced by the flow of carriers across a
junction decays over a Debye screening length ~ 1/n . It is an analog to the ThomasFermi screening length in metals and the London penetration depth in superconductors.
One needs to solve two equations simultaneously (= self-consistently) by iteration:
1. Poisson Equation:
(z)  V(z)
2. Charge Density ρ:
V(z)  (z)
 = Total Charge Density
V = Electrostatic Potential
f(E) = Fermi-Dirac function
D(E) = Density of States
1.
d2V/dz2 = (z) / 0
Poisson Equation
2.
(z) = + e  n+(z)  e  n(z)
n+, n = hole, electron density
n(z) =  f(E)  D[EeV(z)] dE
D = Density of Electron States
(CB + Acceptors)
D+ = Density of Hole States
(VB + Donors)
n+(z) =  [1f(E)]  D+[E+eV(z)] dE
C. Junctions with Bias Voltage, Dynamic Equilibrium
The Fermi level is not flat any more, because it is shifted by the voltage across the junction.
n
p
In the depletion region there are different Fermi levels for electrons and holes (EF ,EF ).
p
EF
+
4