Title: Semiconductors
1Semiconductors
 At zero temperature
 semiconductors are insulators
 with completely filled bands.
 At higher temperatures they conduct at due to the
thermal excitation of electrons across a
relatively small band gap.  In a semiconductor the highest energy filled band
is called the valence band and the lowest energy
band called the conduction band.  We will consider states near the top of the
valence band to be holes (particles of charge e)
with free electron like dynamics but effective
mass mh  We will consider states near the bottom of the
conduction band to be electrons with free
electron like dynamics but effective mass me 
2Direct Gap Semiconductors
 Direct gap semiconductors the top of the valence
band and the bottom of the conduction band occur
at the same kvector.
3Indirect Gap Semiconductors
Indirect gap semiconductors the top of the
valence band and the bottom of the conduction
band occur at different kvector.
Germanium (Ge) Indirect bandgap 0.8eV Direct
bandgap, _at_ k 0, is 0.66eV
4Direct Optical absorption
 Direct gap semiconductor sharp onset of
absorption when the photon energy is equal to the
bandgap
Optical absorption in the direct gap
semiconductor InSb at 4K
Photon creates an electronhole pair
5Indirect Optical absorption
 A transition across an indirect band gap requires
a photon to be absorbed and a phonon to be
absorbed or emitted.
6Ge Indirect Optical absorption
 Indirect gap semiconductor no sharp onset of
absorption
0.73
For T 300 K Eg (indirect gap) 0.66 eV and EG1
(direct gap) 0.8 eV For T 77K Eg (indirect
gap) 0.73 eV and EG1 (direct gap) 0.88 eV
Note numerical values for E(k) for Germanium in
figure in Kittel, reproduced earlier.
7Number of electrons in conduction band
Silicon _at_ 300K n 2x1016 m3
Note Units are cm3 Electron density increases
exponentially with temperature
8Density of States
 Assume bottom of conduction band and the top of
valence band parabolic i.e.  conduction band E Ec ?2k2/2me
 valence band E Ev  ?2k2/2mh
 Conduction
 band
 valence
 band
Note. I do not set Ev 0 until later in notes
document
9Chemical potential or Fermi level
 The chemical potential, m , is the energy for
which f ½ .  Fermi energy all energy states are occupied
below EF at T 0.  In discussing semiconductors m is often referred
to as the Fermi level !
10Electrons density in conduction band
Density of states Distribution function Total
number density of occupied states
Check see main notes
Fermi level
11Number of holes and electrons
Exactly same argument for holes in the valence
band gives
Distribution function Total number density of
occupied states
This last result is particularly important
True for both intrinsic and extrinsic
semiconductors
12Intrinsic semiconductors n p
In pure intrinsic semiconductors the electrons
and holes arise only from excitation across the
energy gap. Therefore n p
Chemical potential?
13Hydrogenic Donors Acceptors
 An electron added to an intrinsic semiconductor
at T0 would go into the lowest empty state i.e.
at the bottom of the conduction band.  When one adds a donor atom at T0
 the extra electron is bound to positive
 charge on the donor atom.
 The electron bound to the positive
 Ion is in an energy state ED Eg DE
 where DE is the binding energy.
 An electron which moves on to an
 acceptor atom has energy EA

14Magnitude of binding energy
Similar to a hydrogen atom. Ground state
wavefunction is The Bohr radius, a0
4??r?o?2/mee2 determines the spatial extent of
the wavefunction. Hydrogen atom (?r 1 ) a0
0.53 Å.
 Binding energy of an electron in the ground state
of a hydrogen atom is  Typical Semiconductor 50 Å
 me 0.15 me and ?r 15.
 10 meV.

15Number of Electrons in the conduction band
 Consider a semiconductor with ND donor atoms per
m3  ND0 and ND number density of neutral and
ionised donors
At T 0 all electrons in the lowest available
energy states. No electrons are excited from
donor states into conduction band n 0
ND0 ND
16T Room temperature
 kBT gt (EC ED) number of available states in
conduction band gtgt ND . Therefore almost all the
donors will be ionised. n ND ND  Relevant regime for all electronics. Note that
the density of electrons is independent of
temperature.  The chemical potential is well below EC and the
expression obtained for n in an extrinsic
semiconductor can be used to give 
 where
 Silicon Nc 2.6x1024 so for n 1022
T gtgt Room temperature
In this limit number of electrons excited across
the bandgap becomes larger than number of donors.
Behaves like an intrinsic semiconductor.
17ntype semiconductors
ln(n)
ln(m)
18ptype semiconductors
NA
ln(p)
Number of holes in valence band
ln(T)
Eg / 2
Fermi level
ln(m)
EA
ln(T)
19Compensated semiconductor
Compensated semiconductor both donors and
acceptors present. ND donors per m3 and NA
acceptors per m3 For ND gt NA have an ntype
semiconductor with n ND  NA for T 300K For
NA gt ND have an ptype semiconductor with p NA
 ND for T 300K
Conduction Band
Ec Eg
NA electrons fall into acceptor states
20Impurity Bands
 Have considered the impurities as isolated atoms.
Reasonable as doping level normally one donor
per 106 semiconductor atoms. 
 At very high donor concentrations, one has
substantial overlap between the donor or acceptor
wavefunctions.  Above a critical doping level one has an impurity
energy band with a finite conductivity.  Electron density at which this metal insulator
transition occurs?  aB 50 Å lattice constant, a 2.5 Å.
 Need b aB 20a . i.e. one donor per 203 8000
semiconductor atoms
f(r)
aB
b
21Mobility of semiconductors
Both electrons and holes carry current in the
same direction in a semiconductors. Conductivity
s neme pemh me electron mobility, mh
holes mobility In considering scattering of
electrons and holes it is important to consider
mobility as the numbers of carriers varies with
temperature. Conductivity se ne2tp /me
Mobility me se/ne etp /me Mean Free Path
Le tpv so me e Le /vme Le /v The
electron and hole distributions are
nondegenerate and ltEgt kBT ltvgt lt 2E/me gt
T1/2. T gt TD number of phonons increases as T.
L T1 . m T3/2 T ltlt TD ionised
impurity scattering dominates. Similar to
Rutherford scattering scattering cross section
E2 T2. So L T2 and m T3/2
22 Hall Effect in Semiconductors
 ntype semiconductors (ngtgtp) RH 1/ne
 ptype semiconductors (pgtgtn) RH 1/pe.
 The Hall effect is used to obtain the carrier
densities in semiconductors.  In an electric field electrons and holes drift in
opposite directions.  Consider case of n p and me mh.
 Have no Hall field.
 The free carrier Hall coefficient is generally
(Hook and Hall p153)  RH ( p mh2  n me2 )/e(n me p mh )2
23Semiconductor devices Inhomogeneous
semiconductors
 All solidstate electronic and optoelectronic
devices are based on doped semiconductors.  In many devices the doping and hence the carrier
concentrations are nonhomogeneous.  In the following section we will consider the pn
junction which is an important part of many
semiconductor devices and which illustrated a
number of key effects
24The pn semiconductor junction ptype / ntype
semiconductor interface
We will consider the pn interface to be abrupt.
This is a good approximation. ntype ND donor
atoms per m3 ptype NA acceptor atoms per
m3 Consider temperatures 300K Almost all donor
and acceptor atoms are ionised.
25Electron andhole transfer
 Consider bringing into contact ptype and ntype
semiconductors.  ntype semiconductor Chemical potential, m
below bottom of conduction band  ptype semiconductor Chemical potential, m
above top of valence band.  Electrons diffuse from ntype into ptype
filling empty valence states.
26Band Bending
 Electrons diffuse from ntype into ptype
filling empty valence band states.  The ptype becomes negatively charged with
respect to the ntype material.  Electron energy levels in the ptype rise with
respect to the ntype material.  A large electric field is produced close to the
interface.  Dynamic equilibrium results with the chemical
potential (Fermi level) constant throughout the
device.  Note Absence of electrons and hole close to
interface  depletion region
27Electrostatic voltage drop, Df0
 In equilibrium m constant.
 Electrostatic voltage difference, Df0, between n
and p regions.  For x gtgt 0
 for x ltlt 0

 Since Eg Ec Ev
28Depletion region
Depletion region
 Assume the electric field in the region of the
junction removes all the free carriers creating a
depletion region for dpltx lt dn.  The ionised impurities are fixed in the lattice.
So charge density is  r eND per m3 for 0 ltx ltdn
 r eNA per m3 for dp lt x lt 0.
n,p
n,p
Electron and
Electron and
Hole Density
Hole Density
r
r
Net charge
Net charge
density
density
E
dn
dp
0
0
0
x
a
0
Depletion region
The total charge in the depletion region must be
zero as the number of electrons removed from the
right equals the number of holes removed from the
left i.e. NDdn NAdp.
29Electric field E(x)
 Can calculate the electrostatic potential, f(x)
from the Poissons equation  Charge density ?(x) eND for 0 lt x lt dn ?(x)
eNA for dp lt x lt 0  Boundary condition E 0 for x gt dn and x lt
dp  So integration gives
30Electrostatic potential, f(x)
 Integration of E gives the potential ?(x).
 Since F 0 for x lt dp and F DF0 for x lt
dn.  ?(x) is continuous at x 0 so
So since NDdn NAdp
Resulting depletion width is 100nm to 1mm. Self
consistent
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32pn junction with a forward bias
 Forward bias ptype region biased positive with
respect to ntype region.  The voltage is dropped across depletion region
since the free carrier density is low and
therefore the resistivity is high.  Total potential across the depletion layer is ??
??0 V
33Generation Current
 Electronhole pairs created in the depletion
region move apart in the strong electric field. A
generation current, Jgen, in the negative
xdirection results.  Magnitude of the generation current density is
 Jgen A exp(Eg/2kBT) where A is a constant.

p

type semiconductor
n

type semiconductor
p

type semiconductor
n

type semiconductor
34Recombination Current
E
C
Df
e
0
E
C
m
E
V
Df
e
0
E
V
p

type semiconductor
n

type semiconductor
 Electrons with energies greater than e?F0 can
move into the ptype material where they
recombine with holes.  A recombination current, Jrec, in the positive
xdirection results  Jrec B exp(e ?F0/2kBT) where B is
constant.
p

type semiconductor
n

type semiconductor
p

type semiconductor
n

type semiconductor
35CurrentVoltage Characteristic
 At equilibrium, without a bias voltage Jgen
Jrec 0  With external positive voltage V the Jgen is
unchanged, but Jrec becomes 

 Total net current density is


36Applications of pn junctions
 pn junction diodes Excellent diodes, which can
be used for rectification of AC signals.  Light emitting diodes (LEDs) and lasers In
forward bias one has an enhanced recombination
current. For direct band gap semiconductors light
is emitted.  Solar cells If photons with hngtEg are absorbed
in the depletion region of a pn junction one has
an enhanced generation current. The energy of the
photons can be converted to electrical power in
solar cells based on this mechanism.  pnp junction transistors Transistors based on
the properties of pn junctions can also be
produced.  See Hook and Hall p1848.
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