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Semiconductor Junctions
The following energy diagrams are for the potential energy U = q   = e   of an
electron in the electrostatic potential  . The Fermi level EF is given for low temperatures.
When two neutral semiconductor pieces are put into contact, electrons flow across
the boundary to reach lower energy levels. Neutral dopants become ionized and build up an
electrostatic potential (“space charge”). This process stops when the two Fermi levels
coincide.
1a) Semiconductor – Semiconductor:
pn – Junction
Separated p- and n-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.
1b) Semiconductor – Semiconductor:
Two separate semiconductors:
n-type
Depletion:
EF mid-gap
Heterojunction
After forming a junction:
intrinsic
Ionized
Donors
Electrons
EF
Band Offset
Energy U= e
 = Electrostatic Potential
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Modulation Doping: Achieve high electron 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 (see Lecture 35).
2) 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+
3) Metal Oxide Semiconductor (MOS):
For a p-type semiconductor = nMOS (conducting via n-type carriers using inversion)
Flatband
Accumulation
Inversion
Charging of a Capacitor
Metal Ins. Semicond.
EF
EF
EF
VGate: 
Field Effect Transistor
(FET) see also p. 5.
VGate: 
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Calculating Carrier Densities and Band Diagrams
A. Homogeneous Semiconductor, Static Equilibrium
Two variables: nh = Hole Density, ne = Electron Density
Two equations:
1. Charge Neutrality:
ND+ = Ionized Donor Density
NA = Ionized Acceptor Density
nh  ne + ND+  NA = 0
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.
B. Inhomogeneous Semiconductor (Junctions), 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)  (z)
2. Fermi Dirac Distribution:
(z)  (z)
1.
d2/dz2 = (z) / 0
2.
(z) = + e  n+(z)  e  n(z)
n(z)
=  fe(E)  D[E+U(z)] dE
n+(z)
=  [1fe(E)]  D+[E+U(z)] dE
fe(E)
= 1 / {exp[(EEF)/kBT]  1}
 = Total Charge Density
 = Electrostatic Potential
U(z) = q(z) = Potential Energy
q = Charge

D (E) = Density of States for
Conduction Band + Acceptors
D+(E) = Density of States for
Valence Band + Donors
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C. Junction with Bias Voltage V, Dynamic Equilibrium
Reverse Bias:
Forward Bias:

+

I
+
e eV/kT
Two opposing currents,
matched at V=0
V
1
Generation (Drift) Current:
Minority carriers are generated
thermally and drift down the
barrier, independent of V.
I = 1 + e eV/kT
Recombination (Diffusion) Current:
Majority carriers from dopants
diffuse against the barrier (EgeV),
depends exponentially on V.
Rectifying Diode
Reverse Bias
Zero Bias
Forward Bias

+

+
Photodiode
Solar Cell
Light Emitting Diode (LED)

+


Photon creates e h+ pair
Reverse Bias
+
+

eh+ pair recombines into a photon
Forward Bias
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Field Effect Transistor (FET)
nMOS:
Two Cuts:
Off:
VGate  0
Gate
Source
n
Drain
n
Channel
p
On:
VGate > 0
Inversion
This cut:
Source  Channel  Drain
= back-to-back diodes
Other cut:
Gate  Oxide  Channel (p. 2)
CMOS (Complementary Metal Oxide Semiconductor)
Back-to-back nMOS and pMOS with common gate.
Low power consumption, draws current only during switching.
pMOS
in
out
out
in 1
nMOS
in 2
Inverter
NAND
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DRAM (Dynamic Random Access Memory)
Single field effect transistor + storage capacitor.
Needs to be refreshed due to leakage current.
CCD (Charge-Coupled-Device)
Array of MOS capacitors that store
light-induced charge and pass it on
along the array for readout.
Dark
Illuminated
High voltage on pixel 2.
H
Readout by pulsing
pixel 3 to high voltage,
which transfers charge
from pixel 2 to pixel 3.
HEMT (High Electron Mobility Transistor) = MODFET (Modulation-Doped FET)
Modulation doping achieves high mobility in the channel by putting the dopants into an
adjacent layer and let the electrons spill over. Fastest transistors (GaAlAs/GaAs, SiGe/Si).
Quantum Well Laser
A semiconductor with smaller band gap is inserted at the pn junction of a LED (see slide).
Both electrons and holes are trapped in the narrow-gap semiconductor. That enhances the
probability for an electron to meet a hole and recombine into a photon. If the quantum well
is narrow enough (single digit nm), the bands become quantized in the direction
perpendicular to the well, and their density of states gets concentrated. That enhances the
number of electron-hole pairs that are able to contribute to a narrow laser line.
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