Download Review of Semiconductors

Review of Semiconductors
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Review of Semiconductors
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Origin of Band Gaps and Band Diagrams
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Direct and Indirect Band Gaps
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Carrier Concentration
– The Fermi-Dirac Distribution
– Density of States
– Carrier Population in a Band
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Intrinsic Material
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Doping of Semiconductors
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Generation and Recombination
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Carrier Transport
– In Electric Field
– Due to Diffusion
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Energy of Electrons
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In ‘free space’ electrons can take on any energy and form a
continuum
The electrons each have a momentum associated with its energy
which means the mass of the electron is related to the energy also
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more generally
Call this the effective mass
Becomes important later
As does this
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Energy of Electrons
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In real world electrons are almost never free
In atoms there is a Coulombic attraction between the protons (+ve)
in the nucleus and the electrons (-ve)
V=
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q is electronic charge (constant) and
r is distance to nucleus, ε 0 is free space
permittivity
When we apply quantum mechanics and solve the Schrödinger
equation we get a series of possible values for the energy (orbitals)
p
s
d
Energy spectrum
“Classical” view of orbitals
Actual
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Band Gaps
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When atoms are put together (e.g. a crystal) the splitting of the
single energy levels form bands of allowed and forbidden energies
Outermost forbidden gap between non-conducting and conducting
bands is referred to as the band gap of the material
Size of band gap determines whether material is a conductor (0
band gap), semiconductor ( <4 eV) or insulator (> 4eV)
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Metal, Semiconductor, Insulator?
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Classification depends on band gap and the number of electrons in
outer most band (conduction)
In a metal the bands can overlap or be partially filled so electrons
available for conduction is high
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Energy Dispersion Curve
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When Schrödinger equation is solved for a crystal get a complex
series of allowed energy states according to k the crystal momentum
Electrons can only occupy energy states on the E-k curves all other
energy states are forbidden
Band gap is minimum difference in energy between two outermost
bands
Using symmetry energy states can be
folded into reduced zone
Near maxima and minima, curves are
parabolic – approximate as ‘free’
Each ‘band’ has it’s own curvature and
hence effective mass
Effective mass varies as a function of k
Actual
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Simplification
Real Crystals
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In reality the E-k curves for crystals are complex with bands varying
in three dimensions
Means that band gap can vary in different directions of the crystal
Conduction between bands is more complex than the picture given
in the simplified scheme – must be aware of this
Top of Valence Band and bottom of Conduction Band don’t always
align – this has massive impact on properties of crystal
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Direct Band Gap
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Direct alignment of Conduction Band minima and Valence Band
maxima
Two requirements for transition between bands 1. The energy
supplied is greater than band gap and 2. the momentum is
conserved
In direct band gap 2. is always
satisfied near the zone centre
and so only need energy
Generally means absorption of light is
greater
Materials such as GaAs, InP, InAs are
examples of direct semiconductors –
used in optoelectronics
Transition sees an electron move from
VB to CB leaves behind a ‘hole’ – refer
to electrons and holes as carriers
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Indirect Band Gap
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CB minima and VB maxima do not align
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Transition requires the addition or subtraction of momentum in order
to satisfy conservation condition
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Require interaction with a third
particle with momentum – phonon
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Three particle transition less likely
hence lower light absorption
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Examples include Si, Ge as well
as III-V materials such as AlAs
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Aren’t solar cells made of Si ?
What gives?
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Electron Population in Bands
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Electrons are in constant random (Brownian) motion and are in
thermal equilibrium with each other and the crystal lattice
At absolute zero (0 K) the electrons all occupy the lowest possible
energies with no excess energy
We cannot know the precise energy of a particular electron but we
can know the average energy (given by the temperature) which
should remain the same
Since the electrons have a temperature there will be empty lower
energy states and occupied higher energy states. The electrons in
the higher energy states will relax down to the lower energy states
with the excess energy given off to other electrons which can then
occupy the higher energy states
– This is a dynamic equilibrium, on average it doesn’t change but
individual electrons do change their states
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Free Electrons
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We are interested only in electrons that are able to participate in
conduction or are able to change their energy
When a semiconductor (or insulator) is at 0 K the valence band will be
completely full and the conduction band will be completely empty
For conduction, electrons must be able to move to another physical
location and gain energy
– Electrons in a full band cannot participate in
conduction
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When an electron receives enough energy to
cross the band gap it requires an empty state
in the conduction band to be available – also
leaves behind an empty state in the valence
band
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
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The ‘holes’ left behind in the valence band can be thought of as
particles themselves – in fact it is a lot easier to do so
Holes conduct just as much as electrons do, so we are interested in
not just the electron population but also the corresponding hole
population
Holes have their own properties like effective mass that are very
different to electrons
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Carrier Concentration
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Need to know carrier concentration as well as allowed energy states
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Require the following information:
– number of states available for the carriers, referred to as the density of
states
– the probability a carrier will be in that state, this is given by the
distribution function
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Distribution function depends on what type of particle we are looking
at, there are two broad types:
– Bosons, where the particles can all fill the same energy level. Important
examples include photons and phonons.
– Fermions, where two particles can NEVER occupy the same energy
state. Important examples include electrons and holes.
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Fermi distribution
• Probability distribution function tells us the probability that a particle
occupies a given energy state.
• To find this we need to determine the number of possible arrangements
for the particles where the number and the total energy remains a
constant.
• Mathematically this involves counting up the different arrangements
using probability theory
• Result is that lower energy states are most
probable to be occupied whilst higher energies
are least likely
• Remember that only one particle can occupy an
energy state at one time
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Fermi Distribution
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Fermi-Dirac distribution is result:
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Note that it only takes into account the number
of carriers and the energy of the system –
doesn’t know about allowed or forbidden states
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Depends on temperature of system. At 0K the
lowest available arrangement is for all low
energy states to be filled – hence FD is square
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As temperature increases the probability a
higher energy state is occupied increases
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Fermi Distribution
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Fermi-Dirac distribution is result:
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FD distribution is symmetric – when a carrier is
placed in a higher energy state it is removed
from a lower energy state
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As temperature increases the FD distribution is
“smeared” out
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Since it is symmetric the energy for which the
probability of occupation is half doesn’t change
– this is called the Fermi energy or level
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Fermi Level
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Fermi level (energy) EF is defined by:
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Assuming that the number of carriers does not
change with temperature EF remains the same
for all temperatures
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EF relates to the number of carriers in the
system – when at 0K the Fermi level is the
highest energy of carriers in the system since
all states below are occupied it gives us
information on the number of carriers
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Fermi Level
• FD probability for a hole is fh = 1-f(E)e since it is the
probability that a state is not occupied by an electron
• Fermi level is interpreted as the average energy of the
free carriers in the system
– In equilibrium the average energy must stay the same by
definition so EF must be constant
• Also tells us the filling level of
electrons (and holes) in a system
and so therefore is an indicator of
the carrier concentration
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Approximating the FD Dist.
• FD distribution is not very ‘nice’ to work with as a rule
• Can use an approximation to the FD distribution whne
the energy is away from the Fermi level, called the
Boltzmann distribution:
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Nice and easy to use, in general can be used without
too much worry
• Problem when semiconductor is degenerate:
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Density of States
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The density of states is the number of allowed energy states per unit
volume per unit energy
Want the TOTAL number of energy states, don’t really care about
their momentum
Find two things: E-k relationship and the number of k states per
volume
1 dimensional
3 dimensional
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Density of States
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Remember that DOS gives the maximum number of states that can be
occupied not the actual number
Near the bottom of the conduction band (top of valence band) can
approximate by a parabola, this is not true far away from these regions,
in fact real DOS goes to zero at high energies
• DOS has large effect on properties
like the absorption coefficient since
it determines how many carriers can
be excited across the band gap
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Carrier Concentration
• Find the carrier concentration simply by multiplying the
number of available states by the probability of the state
being occupied
• Note the position of the Fermi level
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Number of Carriers
• Mathematically we have for electrons
• And similarly for holes we have
• BURN THESE INTO YOUR BRAIN!!
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Intrinsic Material
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Refers to a ‘pure’ semiconductor (this will be clearer in a second…)
For an intrinsic semiconductor we must have n = p (think about it)
We denote the Fermi energy in intrinsic material as Ei – this is always
the same, also denote carrier concentration as ni
The intrinsic level will sit roughly halfway in the band gap of the
semiconductor but off a little due to differences in the density of states
in the conduction and valence bands
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Law of Mass Action
• ni depends on the band gap, temperature and effective
masses of carriers
• Law of mass action relates n, p and ni
• Will become very important when we have a situation
where n ≠ p, it ALWAYS holds in Equilibrium
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Doping
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Carrier concentrations in semiconductors can be altered to desired
levels - a process called doping
Add small amount of material with less or more outer shell electrons
The doped semiconductor is still electrically neutral it is the number
of free electrons and holes that has changed
Can find the modified carrier concentrations fairly easily
Terminology
n type – added dopant
has an excess of electrons
p type – added dopant has
paucity of electrons or put
another way has excess of
holes
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Doping
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Doping introduces energy levels into the forbidden gap of the doped
semiconductor
In case of n type doping, if the dopant is at energy level ED the excess
electron can move to the conduction band if:
– The electron has enough thermal energy
– There is an energy state vacant in the conduction
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The remaining dopant atom is now ionized
with positive charge
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Similarly for p type doping, the excess hole
moves to the valence band with an electron
moving from the valence band to the dopant
which is now negatively charged
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Doping
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Fermi level moves depending on type and concentration of doping
– Closer to conduction band for n type
– Closer to valence band for p type
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Can calculate carrier concentrations in similar manner to the intrinsic
case
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Doped Carrier Concentrations
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Take n type as example
Assume full ionization of dopant so we have ND+ ≈ ND and since the
doping concentration is much larger than the intrinsic concentration
we also have: n ≈ ND
This is fine for the electron concentration but what about the holes?
Take the law of mass action to find the carrier concentration
Recall n.p = ni2 and so it is relatively straight forward to estimate the
hole concentration:
ni2
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p≈
<< n
ND
Terminology: in this case we refer to the electrons as the majority
carrier and the holes as the minority carriers. In p type material the
monikers are reversed
Fermi levels: left as exercise
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Generation and Recombination
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Generation refers to any process whereby an electron moves from
the valence band to the conduction band
– This leaves a hole in the valence band, often refer to the process as
electron-hole pair generation
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Recombination refers to any process whereby any electron returns
to the valence band
– The term comes from the electron
and hole coming together again
– Electron has not vanished, it is
now in the valence band again
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For each generation process there is an
inverse recombination process
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Recombination
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In thermal equilibrium the nett generation rate is zero. In order for
thermally induced generation to give a nett rate would require
thermal gradient across material – typically only consider optical
generation
Each recombination process has associated with it a lifetime for that
process typically labelled τ
The presence of defects, level of doping and even whether the band
gap is direct or indirect determines what types of recombination are
present and which is dominant
Reducing recombination processes is what photovoltaics is
ultimately about
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Transport - Drift
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Two important transport mechanisms are drift and diffusion
Electrons are in constant random motion but if subjected to an
electric field the motion of a charged particle in the electric field is
superimposed on the random motion
Nett effect is that the electrons (and holes) drift in the direction
expected from classical electromagnetism.
Electrons and holes go in opposite directions (since charge is
opposite
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Drift
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Mobility is a measure of how a carrier responds to an electric field
Mobility of carriers depends on the mean time between scattering
events
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Current due to an electric field consists of both the flow of electrons
and holes
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Diffusion
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Diffusion occurs whenever there are concentration differences
Also depends on a carriers mobility
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Typically have both drift and diffusion, so can write total current for
electrons and holes
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner
Other Stuff
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Poisson equation
div E = ρ/ε mostly means for us:
dE/dx = _q (p – n + ND – NA)
ε
Continuity (Book-keeping) Equations
dJn
1
_ __
q dx
= R-G
dJp
_1 __
q dx
= -(R-G)
We will see these again…..
ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner