Solid-State Electronics Chap. 6 Instructor: Pei-Wen Li Dept. of E. E. NCU 1 Chap 6. Nonequilibrium Excess Carriers in Semiconductor Carrier Generation

Solid-State Electronics Chap. 6

Instructor: Pei-Wen LiDept. of E. E. NCU

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Chap 6. Nonequilibrium Excess Carriers in Semiconductor

Carrier Generation and Recombination Continuity Equation Ambipolar Transport Quasi-Fermi Energy Levels Excess-Carrier Lifertime Surface Effects



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Nonequilibrium

When a voltage is applied or a current exists in a semiconductor device, the semiconductor is operating under nonequilibrium conditions.

Excess electrons/holes in the conduction/valence bands may be generated and recombined in addition to the thermal equilibrium concentrations if an external excitation is applied to the semiconductor.

Examples:

1. A sudden increase in temperature will increase the thermal generation rate of electrons and holes so that their concentration will change with time until new equilibrium reaches.

2. A light illumination on the semiconductor (a flux of photons) can also generate electron-hole pairs, creating a nonequilibrium condition.



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Generation and Recombination

In thermal equilibrium, the electrons are continually being thermal generated from the valence band (hereby holes are generated) to conduction band by the random thermal process.

At the same time, electrons moving randomly through the crystal may come in close proximity to holes and recombine. The rate of generation and recombination of electrons/holes are equal so the net electron and hole concentrations are constant (independent of time).



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Excess Carrier Generation and Recombination

When high-energy photons are incident on a semiconductor, electron-hole pairs are generated (excess electrons/holes) the concentration of electrons in the conduction band and of holes in the valence band increase above their thermal-equilibrium value. n = no +n, p = po+ p where no/po are thermal–equilibrium concentrations, and n/p are the excess electron/hole concentrations. np nopo = ni

2 ( nonequilibrium)

For the direct band-to-band generation, the generation rates (in the unit of #/cm3-sec) of electrons and holes are equal; gn’ = gp’ (may be functions of the space coordinates and time)



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An electron in conduction band may “fall down” into the valence band and leads to the excess electron-hole recombination process.

Since the excess electrons and holes recombine in pairs so the recombination rates for excess electrons and holes are equal, Rn’ = Rp’. (in the unit of #/cm3-sec). n(t) = p(t)

The direct band-to-band recombination is spontaneous, thus the probability of an electron and hole recombination is constant with time.

Rn’ = Rp’ the electron and hole concentration.



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Recombination Process

Band-to-Band: direct thermal recombination. This process is typically radiative, with the excess energy releasedduring the process going into the production of a photon (light)

R-G Center: Induced by certain impurity atoms or crystal defects. Electron and hole are attracted to the R-G center and lead to the annihilation of the electron-hole pair. Or a carrier is first captured at the R-G

site and then makes an annihilating transition to the opposite carrier band. This process is indirect thermal recombination (nonradiativ

e). Thermal energy (heat) is released during the process (lattice vibrations, phonons are produced)



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Recombination via Shallow Levels:—induced by donor or acceptor sites. At RT, if an electron is captured at a donor site,however, it has a high probability of being re-emitted into the conduction band before completing the recombination process. Therefore, the probability of recombination via shallow levels is quite low at RT. It should be noted that the probability of observing shallow-level processes increases with decreasing system temperature.

Recombination involving Excitons:It is possible for an electron and a hole to become boundtogether into a hydrogen-atom-like arrangement which moves as a unit in response to applied forces. This coupled e-h pair is called an “exciton”. The formation of an exciton can be viewed as introducing a temporary level into the bandgap slightly above or below the band edge.



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Recombination involving Excitons: Recombination involving excitons is a very important mechanism at low temperatures and is the major light-producing mechanism in LED’s.

Auger Recombinations:In a Auger process, band-to-band recombinationat a bandgap center occurs simultaneously with the collision between two like carriers. The energy released by the recombination or trappingsubprocess is transferred during the collision to the surviving carrier. Subsequently, this high energetic carrier “thermalizes”-loses energy through collisions with the semiconductor lattice.Auger recombination increases with carrier concentration, becoming very important at high carrier concentration. Therefore, Auger recombination mmust be considered in treating degenerately doped regions (like solar cell, junction lasers, and LED’s)



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Generation Process

Band-to-Band generation:

R-G center generation:

Photoemission from band gap centers:



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Generation Process

Impact-Ionization:An e-h pair is produced as a result of the energy released when a highly energetic carrier collides with the crystal lattice. The generation of carriers through impact ionizationroutinely occurs in the high e-filed regions of devices and is responsible for the avalanche breakdown in pn junctions.



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Momentum Consideration

In a direct semiconductor where the k-values of electrons and holes are all bunched near k = 0, little change is required for the recombination process to proceed. The conservation of both energy and crystal momentum is readily met by the emission of a photon.

In a indirect semiconductor, there isa large change in crystal momentum associated with the recombination process. The emission of a photon will conserve energy but cannot simultaneously conserve momentum. Thus for band-to-band recombination to proceed in an indirect semiconductor a phonon must be emitted coincident with the emission of a photon.



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Low-level injection: the excess carrier concentration is much less than the thermal equilibrium majority carrier concentration, e.g., for a n-type semiconductor, n = p << no.

High-level injection: n no or n >> no

For a p-type material (po >> no) under low-level injection, the excess carrier will decay from the initial excess concentration with time;

where n0 is referred to as the excess minority carrier lifetime (n0 1/p0)

and the recombination rate of excess carriers Rn’ = Rp’=

For a n-type material (no >> po) under low-level injection,

Rn’ = Rp’=

0

)(

n

tn

0/)0()( ntetntn

0/)0()( pntetptp 0

)(

p

tp



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Continuity Equations

Consider a differential volume element in which a 1-D hole flux, Fp+

(# of holes/cm2-sec), is entering this element at x and is leaving at x+dx.

So the net change in hole concentration per unit time is

----continuity equation for holes

Similarly, the continuity equation for electron flux is

ptp

p pg

x

F

t

p

ntn

n ng

x

F

t

n



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Ambipolar Transport

If a pulse of excess electrons and holes are created at a particular point due to an applied E-field, the excess e-s and h+s will tend to drift in opposite directions. However, any separation of e-s and h+s will induce an internal E-field and create a force attracting the e-s and h+s back.

The internal E-field will hold the pulses of excess e -s and h+s together, then the electrons and holes will drift or diffuse together with a single effective mobility or diffusion coefficient. This is so called “ambipolar diffusion” or “ambipolar transport”.

Fig. Show the above situation



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Ambipolar Transport



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Ambipolar Transport



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Quasi-Fermi Levels

At thermal-equilibrium, the electron and hole concentrations are functions of the Fermi level by

Under nonequilibrium conditions, excess carriers are created in a semiconductor, the Fermi energy is strictly no longer defined. We may define a quasi-Fermi level, EFn, for electrons and a quasi-Fermi level, EFp, for holes that apply for nonequilibrium. So that the total electron and hole concentrations are functions of the quasi-Fermi levels.

exp and exp

kT

EEnp

kT

EEnn FFi

ioFiF

io

exp and exp

kT

EEnpp

kT

EEnnn FFi

ioFiFn

io



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Quasi-Fermi Levels

For a n-type semiconductor under thermal equilibrium, the band diagram is

Under low-level injection, excess carriers are created and the quasi-Fermi level for holes (minority), EFp, is significantly different from EF.



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Excess-Carrier Lifetime

An allowed energy state, also called a trap, within the forbidden bandgap may act as a recombination center, capturing both electrons and holes with almost equal probability. (it means that the capture cross sections for electrons and holes are approximately equal)

Acceptor-type trap: – it is negatively charged when it contains an electron and it is neutrall when it does n

ot contain an electron.

Donor-type trap:– it is positively charged when empty and neutral when filled with an electron



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Shockley-Read-Hall Theory of Recombination

Assume that a single recombination center exists at an energy Et within the bandgap. And there are four basic processes that may occur at this single trap.

Process 1: electron from the

conduction band captured by an

initially neutral empty trap. Process 2: electron emission from a

trap into the conduction band. Process 3: capture of a hole from the

valence band by a trap containing an

electron. Process 4: emission of a hole from a

neutral trap into the valence band.



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In Process 1: the electron capture rate (#/cm3-sec):

Rcn = CnNt(1-fF(Et))n

Cn=constant proportional to electron-capture cross sectionNt = total concentration in the conduction band n = electron concentration in the conduction bandfF(Et)= Fermi function at the trap energy

For Process 2: the electron emission rate (#/cm3-sec):

Ren = EnNtfF(Et)

En=constant proportional to electron-capture cross section Cn

In thermal equilibrium, Rcn = Ren, using the Boltzmann approximation for the Fermi function,

In nonequilibrium, excess electrons exist,

n

tccnn C

kT

EENCnE

exp'

)()(1 'tFtFtnencnn EfnEfnNCRRR



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In Process 3 and 4, the net rate at which holes are captured from the valence band is given by

In semiconductor, if the trap density is not too large, the excess electron and hole concentrations are equal and the recombination rates of electrons and holes are equal.

In thermal equilibrium, np = ni2 Rn = Rp = 0

))(1()( 'tFtFtpp EfpEpfNCR

kT

EENp vt

v exp'

RppCnnC

nnpNCCRR

ppCnnC

pCnCEf

pn

itpnpn

pn

pntF

)'()'(

)( and

)'()'(

')(

2



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Surface Effects

Surface states are functionally equivalent to R-G centers localized at the surface of a material. However, the surface states (or interfacial traps) are typically found to be continuously distributed in energy throughout the semiconductor bandgap.



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Surface Recombination Velocity

As the excess concentration at the surface becomes smaller than that in the bulk, excess carriers from the bulk region diffuse toward the surface where they recombine, and the surface recombination velocity increases.

An infinite surface recombination velocity implies that the excess minority carrier concentration and lifetime are zero.



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Homework

6.14 6.17 6.19

Documents

Solid-State Electronics Chap. 6 Instructor: Pei-Wen Li Dept. of E. E. NCU 1 Chap 6. Nonequilibrium Excess Carriers in Semiconductor Carrier Generation