Semiconductors

1
Semiconductors

• 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

2
Direct Gap Semiconductors

• Direct gap semiconductors the top of the valence
  band and the bottom of the conduction band occur
  at the same k-vector.

3
Indirect Gap Semiconductors
Indirect gap semiconductors the top of the
valence band and the bottom of the conduction
band occur at different k-vector.
Germanium (Ge) Indirect bandgap 0.8eV Direct
bandgap, _at_ k 0, is 0.66eV
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Direct 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 electron-hole pair
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Indirect Optical absorption

• A transition across an indirect band gap requires
  a photon to be absorbed and a phonon to be
  absorbed or emitted.

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Ge 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.
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Number of electrons in conduction band
Silicon _at_ 300K n 2x1016 m-3
Note Units are cm-3 Electron density increases
exponentially with temperature
8
Density 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
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Chemical 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 !

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Electrons density in conduction band
Density of states Distribution function Total
number density of occupied states
Check see main notes
Fermi level
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Number 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
12
Intrinsic semiconductors n p
In pure intrinsic semiconductors the electrons
and holes arise only from excitation across the
energy gap. Therefore n p
Chemical potential?
13
Hydrogenic 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

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Magnitude 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.

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Number 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
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T 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.
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n-type semiconductors
ln(n)
ln(m)
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p-type semiconductors
NA
ln(p)
Number of holes in valence band
ln(T)
Eg / 2
Fermi level
ln(m)
EA
ln(T)
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Compensated semiconductor
Compensated semiconductor both donors and
acceptors present. ND donors per m3 and NA
acceptors per m3 For ND gt NA have an n-type
semiconductor with n ND - NA for T 300K For
NA gt ND have an p-type semiconductor with p NA
- ND for T 300K
Conduction Band
Ec Eg
NA electrons fall into acceptor states
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Impurity 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
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Mobility 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
non-degenerate and ltEgt kBT ltvgt lt 2E/me gt
T1/2. T gt TD number of phonons increases as T.
L T-1 . m T-3/2 T ltlt TD ionised
impurity scattering dominates. Similar to
Rutherford scattering scattering cross section
E-2 T-2. So L T2 and m T3/2
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Hall Effect in Semiconductors

• n-type semiconductors (ngtgtp) RH -1/ne
• p-type 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

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Semiconductor devices Inhomogeneous
semiconductors

• All solid-state electronic and opto-electronic
  devices are based on doped semiconductors.
• In many devices the doping and hence the carrier
  concentrations are non-homogeneous.
• In the following section we will consider the p-n
  junction which is an important part of many
  semiconductor devices and which illustrated a
  number of key effects

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The p-n semiconductor junction p-type / n-type
semiconductor interface
We will consider the p-n interface to be abrupt.
This is a good approximation. n-type ND donor
atoms per m3 p-type NA acceptor atoms per
m3 Consider temperatures 300K Almost all donor
and acceptor atoms are ionised.
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Electron andhole transfer

• Consider bringing into contact p-type and n-type
  semiconductors.
• n-type semiconductor Chemical potential, m
  below bottom of conduction band
• p-type semiconductor Chemical potential, m
  above top of valence band.
• Electrons diffuse from n-type into p-type
  filling empty valence states.

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Band Bending

• Electrons diffuse from n-type into p-type
  filling empty valence band states.
• The p-type becomes negatively charged with
  respect to the n-type material.
• Electron energy levels in the p-type rise with
  respect to the n-type 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

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Electrostatic 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  

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Depletion 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 m-3 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.

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Electric 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

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Electrostatic 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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p-n junction with a forward bias

• Forward bias p-type region biased positive with
  respect to n-type 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

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Generation Current

• Electron-hole pairs created in the depletion
  region move apart in the strong electric field. A
  generation current, Jgen, in the negative
  x-direction 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
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Recombination 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 p-type material where they
  recombine with holes.
• A recombination current, Jrec, in the positive
  x-direction results
•   Jrec B exp(-e ?F0/2kBT) where B is
  constant.

p
-
type semiconductor
n
-
type semiconductor
p
-
type semiconductor
n
-
type semiconductor
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Current-Voltage 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
•  

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Applications of p-n junctions

• p-n 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 p-n 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.
• p-n-p junction transistors Transistors based on
  the properties of p-n junctions can also be
  produced.
• See Hook and Hall p184-8.

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