introduction to op to electronics jpmakris i

8/4/2019 Introduction to Op to Electronics JPMakris I

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Technological Educational Institute of Crete

Department of Electronics

SOCRATES-ERASMUS INTENSIVE PROGRAM ON

OPTOELECTRONICS, LASERS & APPLICATIONS

Summer School OLA, Crete 2007

Introduction to OptoelectronicsIntroduction to Optoelectronics

(Elem ents of Sem icondu ctors Theor y)(Elem ents of Sem icondu ctors Theor y)

John P. Makris

Lab of Measurements & InstrumentationLab of Measurements & Instrumentation

Dept of ElectronicsDept of Electronics


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Un-bound Energy

States (Conduction

Band)

Energy Band ofBound States

(Valence Band)

Un-bound Energy

States (Conduction

Band)


(Valence Band)


(Valence Band)

Un-bound EnergyStates (Conduction

Band)

Large Gap

(Forbidden States)

No Gap

Small Gap

SemiconductorsMetalsInsulators

Energy Band Representation of Conductivitynergy Band Representation of ConductivityTechnological Educational Institute of Crete






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Energy Distribution FunctionEnergy Distribution Function

Free carriers, electrons and holes, are essential for the operation of activesemiconductor devices. They are introduced in a semiconductor by the process ofdoping.

The number of carriers at any energy level will then depend

on the number of available states at that energy and the energydistribution of the carriers.

Two important functions determine carrier distribution in asemiconductor: a) the energy distribution function and

b) the density of states function.

Remember Paulis exclusion principle for the occupancy ofstates by carriers in a semiconductor, either in the conductionor valence band, or in the impurity levels. Carriers are alsoindistinguishable.

SEMICONDUCTOR STATISTICSSEMICONDUCTOR STATISTICS







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The probability of occupation of an energy level

can never exceed unity, or not more than oneelectron can occupy the same quantum state.

If (Boltzmann approximation) innondegenerate semiconductors:

Energy Distribution FunctionEnergy Distribution Function

The distribution that appropriately describes the occupation of states in asemiconductor is the Fermi-Dirac distribution:







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ThreeThree--Dimensional Density of States FunctionDimensional Density of States Function

The density of states function N(E)dE gives the number of available quantum states

in energy interval between E and E+dE. Consider a cubic region of the crystal with

dimensions L along the three perpendicular directions and impose the condition that

the electron wavefunctions become zero at the boundaries of the cube defined by

values of x, y, and z equal to 0 and L. The boundary conditions are satisfied by a

wavefunction of the form:

and the boundary conditions lead to: ki L = 2 ni, where i=x, y, z

and niare integers. Therefore, each allowed value of kwith

coordinates kx, ky, and kzoccupies a volume (2/L)3 in k-space,

thus, the density of allowed points in k-space is V/(2)3, whereV=L3 is the crystal volume. The volume in k-space defined by

vectors k and k + dk is 4k2dk. he total number of states with

k-values between kand k+dkis: taking

into account the two possible spins.







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ThreeThree--Dimensional Density of States FunctionDimensional Density of States Function

For electrons in the conduction band of a semiconductor:

By combining the above equations, we obtain:

and for unit volume of the crystal:

where Mc is the number of equivalent minima in

the conduction band.

A similar equation holds for the density of states

in the valence band.







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Density of Carriers in Intrinsic and Extrinsic SemiconductorsDensity of Carriers in Intrinsic and Extrinsic Semiconductors

At 0K or very low temperatures, the valence band is filled with electrons and theconduction band is empty. Under such conditions there can be no electrical

conduction. For the latter to occur, a covalent bond has to be broken, for which theminimum energy required is the bandgap energy Eg. This energy can be provided byheat, electric field, optical excitation etc. In an intrinsic semiconductor, without anydopant atoms, the breaking of a covalent bond creates an electron-hole pair, andunder an electric field, the two carriers move in opposite directions to give rise to

two-carrier transport. In an intrinsic semiconductor, therefore,under thermal equilibrium conditions:where ni is the intrinsic carrier concentration. Usually, insemiconductors with Eg~1eV, ni is too small for any practical use.Therefore, doping is used to increase n, or p, and such semicon-

ductors are called extrinsic semiconductors. Under thermalequilibrium conditions, for the conduction and valence bands, thethree-dimensional density of states functions can be written:







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Density of Carriers in Intrinsic and Extrinsic SemiconductorsDensity of Carriers in Intrinsic and Extrinsic Semiconductors

The density of electrons and holes in the conduction and valence bands are given,

respectively, by:

which lead to:

NC and NV are the effective density of states in the conductionand valence respectively. The quantity F1/2() is called the Fermiintegral. For example, for electrons:

where: and







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In the case of a lightly doped or non-degenerate semiconductor, where the Fermi

level is within the forbidden energy gap and , the Boltzmann

approximation is valid and the electrons density becomes:

The Fermi integral in the above equation is of the form of a gamma function and is

equal to (3/2) whose numerical value is . Therefore, it follows that:

Note that in this case


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For a non-degenerate semiconductor the law of mass

actionholds:

where Eg= EC- EV is the bandgap energy and ni is

the intrinsic carrier concentration:

Since the bandgap energy does not depend on the impurity

concentration in a non-degenerate semiconductor and since ni,

is independent of Fermi energy level, which is affected by the

doping level, it follows that this equation is equally valid for

intrinsic and extrinsic semiconductors.







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The position of the Fermi energy level in an intrinsic semiconductor, EFi, (intrinsic

Fermi level) is given by:

Two important points: (i) the Fermi energy has a temperature dependence;

(ii) it has been assumed that the value of the effective mass

is constant in the respective bands, which is not strictly true, but reasonable

assumption.







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In real semiconductors, species are present. Thus, there is some unintentional

impurity concentration of donors and acceptors (in addition to the intentional ones)

present. As a result, in an n-type semiconductor, the net electron concentration inthe conduction band is given by . Similarly, in a p-type:

A semiconductor in which the number of free carriers produced by one type of

dopant is reduced by the presence of the other type of dopant is said

to be compensated. The compensation ratio (ideally desirable

0.1)is defined:

Compensation in SemiconductorsCompensation in Semiconductors







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Consider a non-degenerately doped n-type semiconductor with a small density of

compensating acceptors also present. Electrical neutrality: .

For the donor and acceptor energy level, the Fermi functions are:

Schockley diagram

Numerical solution provides accurate value of EF.






Compensation in SemiconductorsCompensation in Semiconductors


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At 300K, KBT is 26 meV and shallow donors and acceptors, e.g. in GaAs, 4-30 meV,

therefore completely ionized: , . Thus charge neutrality and law

of mass action give for n0:

Similarly, from the quadratic equation with p as a variable:

where it is assumed that for the n-type semiconductor ND>NA.

The free-carrier concentration in a semiconductor as a function

of temperature is very important for many device applications.

This parameter can be obtained from above Eqs.At very high temperatures (known as the intrinsic region), where

the intrinsic generation of electron-hole pairs is a dominant

process:







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In the intermediate temperature range, where ni


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Conduction Processes in SemiconductorsConduction Processes in Semiconductors

In order for a semiconductor material to conduct, electrons and holes must be in

motion in their respective partially filled bands. The carrier motion must have a net

direction, and for this an external force (stimulus) is needed (otherwise the carriershave a scattering limited thermal velocity, which is not directional). Moving electrons

and holes collide with other carriers, impurity centers, and phonons.

An externally applied electric field can move carriers in a

band in the direction of the electric field (drift)

Electrons and holes, like neutral particles, can alsoacquire directional motion due to a concentrationgradient (diffusion)







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Electrical conduction by electrons and holes in their respective bands is similar toconduction in metals by free electrons. Therefore:

The equation of electron subject to an electric field in the x-direction and itssolution:

Then, for the current density:

The steady-state values of velocity and current:

where mobility is defined as the mean drift velocity per unit field.It follows:

The total current density due to drift of electrons and holes andthe conductivity are given by:






Conduction Processes in SemiconductorsConduction Processes in Semiconductors


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Diffusion arises from a non-uniform density of carriers (electrons and holes). In the

event of the absence of any other processes, the carriers will diffuse from region

of high density to a region of low density. The process is identical for neutralcharged particles.

The force of diffusion acting on electron is given by: where

is the force per unit area acting on the distribution of electrons. The

motion of carriers by diffusion is limited by collisions and scattering.

The velocity due diffusion is given by:

or , where is the diffusion coef-

ficient for electrons and the relevant

diffusion current is expressed:

and similarly for holes:






Diffusion Processes in SemiconductorsDiffusion Processes in Semiconductors


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The diffusion constants for electrons and holes can be written:

from which we get the Einstein relation:

If electric field and concentration gradients are present in a

semiconductor, the total current density for electrons & holesand the total current density are:






Diffusion Processes in SemiconductorsDiffusion Processes in Semiconductors


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Important InsightsImportant Insights

(i) The process of drift is essentially controlled by majority carriers.

(ii) Since the density of minority carriers is small, a concentration gradient of

minority carriers is easily produced, by current injection in a p-n junction or by

intrinsic photoexcitation. Thus, diffusion is essentially controlled by the density

of minority carriers.

(iii) No net current flow at equilibrium. Thus, if a concentration

gradient is somehow induced in the material, a diffusion

current is produced which is exactly balanced by a drift

current due to a built-in electric field that is accommodated

by band-bending that constitutes a potential gradient. For a

p-type material, the built-in field is:







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Bulk Recombination PhenomenaBulk Recombination Phenomena

In a semiconductor, excess minority carriers are generated by intrinsic

photoexcitation or injection across a forward-biased p-n junction (the density of

majority carriers is not usually affected) which after a mean lifetime, generally

recombine with majority carriers. In an n-type semiconductor, the net rate of

recombination (radiative or nonradiative) of holes is given by:

The radiative processes usually involve the absorption or emissionof a photon with energy close to the bandgap (e.g. band-to-band

downward transition of an electron, in which a photon is

emitted). There is a small probability that during such a

downward transition phonons may be emitted, in which case

the recombination becomes a nonradiative process.







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Bulk Recombination PhenomenaBulk Recombination PhenomenaNonradiative recombination is more likely to take place via levels within the bandgap

of the semiconductor. Defects with deep energy levels in the forbidden energy gap

of large bandgap semiconductors act as carrier recombination or trapping centers

and adversely affect device performance. The processes are: (a) electron capture,(b) electron emission, (c) hole capture and (d) hole emission, with corresponding

rates (in cm-3s-1):

where cn(p) and en(p) are the carrier capture and emission rates

with units (cm3s-1) and (s-1), respectively,

at the deep (trap) level and N is the trap

concentration. The parameters se and share the electron and hole capture cross

sections at the trap. Under equilibrium,and with no generation of carriers by any

means (generation rate G = 0):







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If carriers are also generated at a rate G, then the semiconductor is under

nonequilibriurn conditions. In the pair-generation process, an electron is raised from

the valence band to the conduction band, leaving behind a hole. In steady-state

conditions, the rate at which carriers enter a band is equal to the rate at which they

leave the band. Therefore, for an n-type semiconductor:

Under steady-state non-equilibrium conditions:

It can be shown that:

The net rate of recombination through deep-level traps under

steady-state nonequilibrium conditions is given by:







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The capture cross section is a measure of how close to a trap center a carrier has

to come to get captured. Usually, for an electron trap se>>sh and for a hole trap

sh>>se (for normal traps se(h)~10-15-10-13 cm2). For nonradiative recombination (mid-

bandgap) centers se

=sh

=sr

, then:

If this is not true, the recombination rate will decrease. For

example, if ET-EFi increases (e.g. for true trapping centers or

shallow donor and acceptor levels) then se>>sh or sh>>se. Finally, itis important to note that true recombination centers can also act

as generation centers. For low-level injection in an n-type

semiconductor, n>>p and so

So, the lifetime is not a function of the majority carrier density,

n. The rate-limiting step in the recombination process is the

concentration of minority carriers.






introduction to op to electronics jpmakris i

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