Lecture 18 Lecture 18 Quantum Theory of Extrinsic Semiconductors Suggested reading: 5.1-5.2
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Lecture 18Lecture 18
Quantum Theory of Extrinsic Semiconductors
Suggested reading: 5.1-5.2
5 key semiconductor properties
1. Negative temperature coefficient of resistance conductivity increases with increasing
temperature
2. Photoconductivityresistivity depends on
5. Electroluminescencean applied voltage can resistivity depends on
the light intensityan applied voltage can
produce light
image from crunchgear.com
3. Photovoltaic effects 4. Rectification
image from allwin21.com image from instructables.com
light can produce current & voltage
resistance varies with the direction of current flow
Silicon is the most important semiconductor in today’s electronics & photovoltaics (Images courtesy of IBM and Sunpower)
Modern Si fabrication begins with wafer-pulling
200 mm and 300 mm Si wafers.SOURCE: Courtesy of MEMC, Electronic Materials, Inc. & Texas Instruments
Si is sp3 hybridized and bonds with 4 neighbors
A simplified two-dimensional illustration of a Si atom with four A simplified two dimensional illustration of a Si atom with four hybrid orbitals hyb. Each orbital has one electron.
Electrons in covelent bonds valence band
χ=electron affinity
Left: A simplified two‐dimensional view of a region of the Si crystal showing covalent bonds.
Right: The energy band diagram at absolute zero of temperature.
How do electrons enter the conduction band?
A two-dimensional pictorial view of the Si crystal showing covalent bonds as two lines where each line is a valence electron.
1: Photon excitation across the bandgap Eg
A photon with an energy greater than Eg can excite an electron from the VB to the CB.
1: Photon excitation across the bandgap Eg
A photon with an energy greater than Eg can excite an electron from the VB to the CB.
When a photon breaks a Si-Si bond, a free electron and a hole in the Si-Si bond is created.
2: Thermal excitation across the bandgap Eg
Thermal vibrations of atoms can also break bonds and thereby create electron-hole pairs.
Electrons and holes
A pictorial illustration of a hole in the valence band wandering around the crystal due to the tunneling of electrons from neighboring bonds.
Electron-hole recombination
An electron in the conduction band can “recombine” with a hole in the valence band, often emitting photons (“direct gap” semiconductors such as GaAs & InP)
or heat (“indirect gap” semiconductors such as Si & Ge) in the process.
Conduction in semiconductors
Electron collisions
with thermal vibrationsvibrations
When an electric field is applied, electrons in the CB and holes in the VB can drift and contribute to the conductivity.
Applied field bends the energy bands since the electrostatic PE of the electron is –eV(x) and V(x) decreases in the direction of Ex, whereas PE increases.
Both electrons and holes contribute to conduction
J=envde+epvdh
vde = eEx and vdh = hEx
J=current density
vd = drift velocity of the electrons vdh = drift velocity of the holesvde = drift velocity of the electrons, vdh = drift velocity of the holes
n=electron concentration in CB, p=hole concentration in VB
e = electron drift mobility, h = hole drift mobility, Ex = applied electric fielde d b y, h d b y, x pp d d
Both electrons and holes contribute to conduction
J=envde+epvdh
vde = eEx and vdh = hEx
J=current density
vd = drift velocity of the electrons vdh = drift velocity of the holesvde = drift velocity of the electrons, vdh = drift velocity of the holes
n=electron concentration in CB, p=hole concentration in VB
e = electron drift mobility, h = hole drift mobility, Ex = applied electric fielde d b y, h d b y, x pp d d
Conductivity of a Semiconductor
d l h l h
= ene + eph
= conductivity, e = electronic charge, n = electron concentration in the CB
e = electron drift mobility, p = hole concentration in the VB, h = hole drift mobility
f(E), g(E), and Carrier Concentrations
Electron Concentration in the Conduction Band (CB)
dEEfEgdEn CBE )()(
nE= number of electrons per unit energy per unit volume in the conduction band
Electron Concentration in the Conduction Band (CB)
dEEfEgdEn CBE )()(
nE= number of electrons per unit energy per unit volume in the conduction band
CE
dEnn CE
E dEnn
A E i l f kT b l E f(E) l k lik hAssume EF is at least a few kT below EC, so f(E) looks like the Boltzmann distribution:
f(E) ~ exp[-(E-EF)/kT]( ) p[ ( F) ]
Also, approximate upper limit of integral as infinity
Electron Concentration in the Conduction Band (CB)
kTEENn Fc
c)(exp
n = electron concentration in the CB, Nc = temperature dependent constant, Ec = conduction band edge, EF = Fermi energy, k =
B l T
Boltzmann constant, T = temperature
Looks just like the Boltzmann Distribution Function, f(E)!
n ~ f(E)*g(E)
Electron Concentration in the Conduction Band (CB)
kTEENn Fc
c)(exp
n = electron concentration in the CB, Nc = temperature dependent constant, Ec = conduction band edge, EF = Fermi energy, k =
B l T
Boltzmann constant, T = temperature
Effective Density of States at CB Edge
N 22me
*kT2
3 / 2
Nc = effective density of states at the CB edge, me* = effective mass of the electron in
Nc 2h2
c y g ethe CB, k = Boltzmann constant, T = temperature, h = Planck’s constant
Hole Concentration in the Valence Band (VB)
kTEENp vF
v)(exp
p = hole concentration in the VB, Nv = effective density of states at the VB edge, EF = Fermi energy, Ev = valence band edge, k = Boltzmann constant, T = temperature
Effective Density of States at VB Edge
Nv 22mh
*kTh2
3 / 2
Nv = effective density of states at the VB edge, mh* = effective mass of a hole in the
VB k B lt t t T t t h Pl k’ t t
h
VB, k = Boltzmann constant, T = temperature, h = Planck’s constant
Let’s consider the product of n and p:
kTE
NNnp gvc exp
kTvc
h d 2 h d d h lThe np product is a constant, ni2, that depends on the material
properties Nc, Nv, Eg, and the temperature.
If i i d ( b d i ) p t d t k p t t If n is increased (e.g. by doping), p must decrease to keep np constant.
We call ni the “intrinsic concentration”.
Mass action law applies in thermal equilibriumd i th d k ( ill i ti )and in the dark (no illumination)
Electron-hole recombination & generation
The rate of recombination between electrons and holes is proportional to the concentration of carriers
Rate of recombination R ~ np
The rate of generation will depend on how many electrons are available for excitation at EV (i e NV) how many empty states are available for excitation at EV (i.e., NV), how many empty states are available at EC (i.e., NC), and the probability that the electron will
make the transition:
Rate of generation G ~ NV NC exp[-Eg/kT]
*** Thermal equilibrium: R = G ***
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