Download Semiconductor Physics

Semiconductor Physics
10p PhD Course
18 Lectures
Nov-Dec 2011 and Jan – Feb 2012
Literature
Semiconductor Physics – K. Seeger
The Physics of Semiconductors – Grundmann
Basic Semiconductors Physics - Hamaguchi
Electronic and Optoelectronic Properties of Semiconductors - Singh
Quantum Well Wires and Dots – Hartmann
Wave Mechanics Applied to Semiconductor Heterostructures - Bastard
Fundamentals of Semiconductor Physics and Devices – Enderlein & Horing
Examination
Homework Problems (6p)
Written Exam (4p)
Additionally
Your own research area.
Background courses (Solid State Physics, SC Physics, Sc Devices)
Course Layout
1. Introduction
2. Crystal and Energy Band structure
3. Semiconductor Statistics
4. Defects and Impurities
5. Optical Properties I : Absorption and Reflection
6. Optical Properties II : Recombinations
7. Carrier Diffusion
8. Scattering Processes
9. Charge Transport
10. Surface Properties
11. Low Dimensional Structures
12. Heterostructures
13. Quantum Wells/Dots
14. Organic Semiconductors
15. Graphene
16. Reserve and Summary
Semiconductor Statistics
Carriers
• Effective masses
• Density of States
Carrier Statistics
• Intrinsic Number Density
• Fermi-Dirac Statistics
• Fermilevel
Dopands
• Donors
• Acceptors
• Statistics
• Compensation
• High Doping
Bandstructure
Real Space
Reciprocal Space
For free electron
For electron in a crystal
Constant Energy Surdaces
Effective Mass
Effective mass are anistropic
Strongly in materials with diamond like structure, such as Si
Weakly for semiconductor with wurtzite structure, such as GaAs
Polaronic effect: In crystals with ionic binding a moving electron polarizes
the Ions.
α Fröhlich Coupling Constant
Nonparabolocity: Energy momentum dispersion is only parabolic for small
k-values. Deviates at higher energies.
Temperature Dependence of Bandgap
Bandgap decreases with increasing
temperature
• Increased electron-phonon interaction
• Expansion of the lattice
Empirical Varshni formula
Physical formula based on BE
phonon model
Density of States
Density of States, DOS, number of possible energy states between E
and E+dE. The density of states for a given band is
Energydispersion where ki can take values ±πn/L (in
the first Brillouin zone) with n < N are number of unit
cells in one dimension
Number of states up to the highest occupied state, EF, which has the
momentum kF
Density of States
Number of states in 3D
The density of states is the derivate
of number of states
Density of states in 3D
Fermi vector
Fermi Energy
Fermi-Dirac Statistics
Electrons, which are spin-half fermions, are statistically described
by the Fermi-Dirac statistical distribution (fermi equation). fe is the
occupancy probability for a electron state at energy E
EF is the fermi level, or more accurate the chemical potential µ.
lim µ(T) = EF T→ 0
fe (E= µ) = ½
The fermi level is a material property. Temperature independent.
Chemical potential is temperature dependent.
In semiconductor physics the term fermi level are often used
instead of chemical potential. fe (E=EF) = ½ at all temperatures.
Fermi-Dirac Statistics
Boltzmann approximation
The fermi-dirac distribution are
often approximated by the
Boltzmann distribution.
Boltzmann approximation not
possible:
• At high temperatures
• Low bandgaps
• Close to bandedges.
The Hole Concept
A hole is a missing electron in an otherwise filled band.
The hole concept is useful to describe the properties at the valence bandtop.
The hole is a quasi particle acting as a positive particle.
The electronic structure of the holes in the valence band are more
complicated than the electrons in the conduction band.
Three different, heavy holes, ligh holes, split-off holes.
Carrier Concentrations
Density of electrons in the conduction band and holes in the valence
band is given by
Carrier Concentrations
With the Boltzmann approximation this can be analytically solved to give
electron density in the conduction band as
And hole density in the valence band as
With
and an effective hole mass mh as
Intrinsic Carrier Concentration
The product of electron and hole density is
Which is independent of the position of the fermi level (as long as
Boltzmann approximation is valid) , and is a material constant.
ni = √np is called the intrinsic carrier concentration.
Intrinsic Conduction
In an ideally pure semiconductor the fermi level is in the middle of the
bandgap and n=p or ni = pi.
Eg [eV]
Si
1.124
GaAs
1.43
GaP
2.26
SiC
3.1
Diamond 5.5
ni [cm-3]
1.0 E10
1.8 E6
2.7 E0
9.3 E-23
4.8 E-63
Defects
Point Defects
Vacancy – The vacancy created by a missing atom A is denoted VA.
Interstitial – An atom occupying an insterstitional site is denoted IA.
Substitutional – An atom C replacing a host atom A is denoted CA.
Antisite – When a host atom B occupies the site of another host atom A
Intrinsic Defects – Defects involving host atoms.
Extrinsic Defects – Defects including foreign impurities.
Shallow defects
• Energylevels close to the valence or conduction band
• The coulomb part of the ion-core determines the energy level
• The defect wavefunction is given by the Bohr radius
• Actings as dopands
Deep defects
• Energylevel away from the bandedges
• Short range part of the potential determines energylevel
• Wavefunction in the order of the lattice constant
• Normally non-wanted defects
Defects
Energylevels of different impurities in Si
Defects
Energylevels of different impurities in GaAs
Defects
Energylevels of different impurities in Si
Dopands: Donors or Acceptors
Donors :
Impurities close to the conduction band with more valence
electrons than the host atom.
Acceptors : Impurities close to the valence band with less valence
electrons than the host atom
Donors
In Si phosphour (P) and Arsenic (As) are
donors.
They have one valence electron (5) more
than Si (4).
For each impurity 4 valence electron is
required for the crystal bonding.
The remaining electron and the extra
positive charge creates an extra energy
level below the conduction band.
Donors
Donorlevels in
group IV
semiconductors
Donorlevels in
group III-V
semiconductors
Effective Mass Theory
In Effective-Mass Theory (EMT) the energy levels of the donor is
obtained from Bohr’s theory for the hydrogen atom.
The energy level for the electron in hydrogen is scaled with the ratio of
the effective mass to the free electron mass and with the relative
dielectric constant.
For Si the EMT energy level is 6 meV, below
The conduction band edge (hydrogen
continuum)
More detailed calculation using the
Anisotropic effective mass gives 9.05 meV.
Defect Statistics
Concentrations of donors with an electron neutral donor ND0 = N1, and
the concentration of of ionized donors ND+ = N0
The ratio of neutral and ionized donors are given by,
or with degeneracies. Typically gD = 2.
Number if electrons in the conduction band is equal to number of
ionized donors.
Defect Statistics
Carrier density in Si doped with
1015 cm-3 of phosphour
In an inverse temperature scale
the slope of the electron
concentration is the activation
energy equal to the donor energy
Temperature Dependence of Fermi Level
At T=0 the fermilevel is between the highest populated state, the
donor, and the lowest unoccupied state, bottom of the conductionband
At higher temperatures the fermi level moves towards the center of
the bandgap.
Semiconductor Conductivity
Acceptors
The statistics is similar as for electrons on donors. NA number of acceptors
NA0 number of neutral acceptors (with hole), NA- number of charged
acceptors with electron
The degeneracy is more complicated than for donors due to the valence
band structure. In Si typically ga = 6.
Acceptors
An acceptor has less valence electrons
than the host.
In Si Bor (B) and Aluminum (Al) are
acceptors.
They have one valence electron (3) less
than Si (4).
One electron is missing to make the bonds,
which is captured from the valence band,
leaving a hole.
Alternatively, the hole on the acceptor is
excited into the valence band.
Acceptors
Some acceptors have levels
close to the effective mass
value while some are much
higher.
Semiconductors where the
conductivity is by holes are
labelled p-type materials.
Temperature Dependence of Fermi Level
Compensation
When donors and acceptors are both present, some of the impurities will
compensate each other.
Electrons from donors will recombine with holes at the acceptors.
The conductivity type will be determined by the relative concentrations and
activation energies, which determines the fermilevel.
Fermilevel determined from the charge neutrality condition:
n = NC exp((Ef-EC)/kT)
P = NV exp(EV-Ef)/kT)
N0 = ND+
Compensation
We assume Boltzmann statistics
ND > NA material will be n-type
p = 0 and NA- = NA
The charge neutrality simplifies to:
After some algebra:
High Doping – Bandgap Narrowing
At low doping levels the impurity atoms are isolated.
With increasing concentration the distance decreases and the wavefunction
start to overlap. Carriers can move from impurity to impurity.
A periodic arrangement of impurities creates an energy band according to
the Kronig-Penney model.
At high doping the impurity band overlap with the conduction band.
High Doping – Bandgap Narrowing
This effect is known as bandgap
narrowing or the Mott transition
This happens when the distance
between impurities becomes
comparable to the Bohr radius.
Dependent on doping concentration
and doping energy level.
Quasi Fermi Levels
The statistics has sofar only considered semiconductors in thermal
equilibrium.
During non-equilibrium conditions, e.g. external excitation or carrier
injection, the electron and hole densities can take arbitrarily forms.
The fermi level is not constant through the structure.
There will instead be separate quasi-fermi levels for electrons and holes,
respectively.
F½ is the Fermi function, and Fn(r) and Fp(r) the quasi fermi levels.