03_RevSemicon

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ELEG620: Solar Electric Systems University of Delaware, ECE Spring 2009 S. Bremner

Review of Semiconductors


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Review of Semiconductors

Origin of Band Gaps and Band Diagrams

Direct and Indirect Band Gaps

Carrier Concentration The Fermi-Dirac Distribution

Density of States

Carrier Population in a Band

Intrinsic Material

Doping of Semiconductors

Generation and Recombination

Carrier Transport In Electric Field

Due to Diffusion


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Energy of Electrons

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

. more generally

Call this the effective mass

Becomes important later

As does this


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Energy of Electrons

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)

When we apply quantum mechanics and solve the Schrdingerequation we get a series of possible values for the energy (orbitals)

V=q is electronic charge (constant) and

ris distance to nucleus, is free space

permittivity0

Energy spectrum Classical view of orbitalsActual

s p

d


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Band Gaps

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 ( 4eV)


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Metal, Semiconductor, Insulator?

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


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Energy Dispersion Curve

When Schrdinger 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 its own curvature andhence effective mass

Effective mass varies as a function of k

Actual Simplification


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Real Crystals

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 dont always

align this has massive impact on properties of crystal


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Direct Band Gap

Direct alignment of Conduction Band minima and Valence Bandmaxima

Two requirements for transition between bands 1. The energy

supplied is greater than band gap and 2. the momentum isconserved

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


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Indirect Band Gap

CB minima and VB maxima do not align

Transition requires the addition or subtraction of momentum in order

to satisfy conservation condition

Require interaction with a third

particle with momentum phonon

Three particle transition less likelyhence lower light absorption

Examples include Si, Ge as well

as III-V materials such as AlAs

Arent solar cells made of Si ?

What gives?


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Electron Population in Bands

Electrons are in constant random (Brownian) motion and are inthermal equilibrium with each other and the crystal lattice

At absolute zero (0 K) the electrons all occupy the lowest possibleenergies with no excess energy

We cannot know the precise energy of a particular electron but wecan know the average energy (given by the temperature) whichshould remain the same

Since the electrons have a temperature there will be empty lowerenergy states and occupied higher energy states. The electrons inthe higher energy states will relax down to the lower energy stateswith the excess energy given off to other electrons which can thenoccupy the higher energy states

This is a dynamic equilibrium, on average it doesnt change butindividual electrons do change their states


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Free Electrons

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

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


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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 holepopulation

Holes have their own properties like effective mass that are very

different to electrons


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Carrier Concentration

Need to know carrier concentration as well as allowed energy states

Require the following information:

number of states available for the carriers, referred to as the density ofstates

the probability a carrier will be in that state, this is given by the

distribution function

Distribution function depends on what type of particle we are lookingat, 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.


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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 arrangementsfor 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 anenergy state at one time


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Fermi Distribution

Fermi-Dirac distribution is result:

Note that it only takes into account the number

of carriers and the energy of the system

doesnt know about allowed or forbidden states

Depends on temperature of system. At 0K the

lowest available arrangement is for all low

energy states to be filled hence FD is square

As temperature increases the probability ahigher energy state is occupied increases


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Fermi Distribution

Fermi-Dirac distribution is result:

FD distribution is symmetric when a carrier is

placed in a higher energy state it is removed

from a lower energy state

As temperature increases the FD distribution is

smeared out

Since it is symmetric the energy for which the

probability of occupation is half doesnt change this is called the Fermi energy or level


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Fermi Level

Fermi level (energy) EF

is defined by:

Assuming that the number of carriers does not

change with temperature EF

remains the same

for all temperatures

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 usinformation on the number of carriers


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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 thefree 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 ofelectrons (and holes) in a system

and so therefore is an indicator of

the carrier concentration


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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 theBoltzmann distribution:

Nice and easy to use, in general can be used without

too much worry

Problem when semiconductor is degenerate:


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Density of States

The density of states is the number of allowed energy states per unit

volume per unit energy

Want the TOTAL number of energy states, dont really care about

their momentum

Find two things: E-k relationship and the number of k states per

volume

1 dimensional 3 dimensional


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Density of States

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 sinceit determines how many carriers can

be excited across the band gap


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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


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Number of Carriers

Mathematically we have for electrons

And similarly for holes we have

BURN THESE INTO YOUR BRAIN!!


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Intrinsic Material

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


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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


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Doping

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

Terminologyn type added dopant

has an excess of electrons

p type added dopant has

paucity of electrons or putanother way has excess of

holes


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Doping

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

The remaining dopant atom is now ionized

with positive charge

Similarly for p type doping, the excess hole

moves to the valence band with an electron

moving from the valence band to the dopantwhich is now negatively charged


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Doping

Fermi level moves depending on type and concentration of doping

Closer to conduction band for n type

Closer to valence band for p type

Can calculate carrier concentrations in similar manner to the intrinsic

case


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Doped Carrier Concentrations

Take n type as example

Assume full ionization of dopant so we haveND+

ND and since thedoping 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:p


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Generation and Recombination

Generation refers to any process whereby an electron moves fromthe valence band to the conduction band This leaves a hole in the valence band, often refer to the process as

electron-hole pair generation

Recombination refers to any process whereby any electron returnsto the valence band The term comes from the electron

and hole coming together again

Electron has not vanished, it isnow in the valence band again

For each generation process there is an

inverse recombination process


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Recombination

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

The presence of defects, level of doping and even whether the band

gap is direct or indirect determines what types of recombination arepresent and which is dominant

Reducing recombination processes is what photovoltaics is

ultimately about


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Transport - Drift

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


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Drift

Mobility is a measure of how a carrier responds to an electric field

Mobility of carriers depends on the mean time between scattering

events

Current due to an electric field consists of both the flow of electrons

and holes


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Diffusion

Diffusion occurs whenever there are concentration differences

Also depends on a carriers mobility

Typically have both drift and diffusion, so can write total current for

electrons and holes


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Other Stuff

Poisson equation

divE = / mostly means for us:

dE/dx = (p n + ND NA)

Continuity (Book-keeping) Equations

= R-G

= -(R-G)

We will see these again..

q

_

1

1

q

q

_

_

dJn

dJp__

__

dx

dx

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