10.3 ShockleY-Read-Hall Recombination Processes 499

which can be integrated with the integral:

f dx - _ "!"IOg(A+ BX) (A+Bx)x- A x ·

(10.2.18)

Therefore the expression integrates out to

_ 1 {lo [ no + Po + Snit)] -10 [no + Po + sn(O)]} = -Kt no + Po g Snit) g Sn(O) ,

I {[no + Po + Sn(t)]Sn(O) } K( )

og = no + Po t. (10 .2.19) [no + Po + Sn(O)]Sn(t)

Solving for an(t) yields

Snit) = Sn(O)(no + Po) . eK(no+Po)t[no + Po + Sn(O)] - Sn(O)

(10.2.20)

Checking the bounds of the above result as t goes to zero, we find that the excess concentration becomes simply Sn(O). As t approaches infinity, the excess carrier concentration approaches zero; the electron concentration relaxes back to its equilibrium concentration no through the action of the recombination.

The expression for the excess electron concentration can be simplified if the excess concentration is assumed to be small with respect to the sum of the equilibrium electron and hole concentrations. This implies that

Sn(O) « (no + Po). (10.2.21)

Under this assumption the excess concentration as a function of time becomes

Snit) = Sn(O)e-K(no+Po)t. (10.2.22)

Therefore the excess electron concentration is found to decay exponentially at some characteristic time determined by the product of the recombination rate constant K and the sum of the equilibrium concentrations no and Po. Generally, the characteristic time constant of the decay is called the carrier lifetime for band-to-band recombination and is denoted as T. The expression for the excess electron concentration becomes then

Snit) = Sn(O)e- f . (10.2.23)

10.3 Shockley-Read-Hall ReCombination Processes

Let us now consider band-to-bound generation-recombination mechanisms quantitatively. It is helpful to first make some symbolic definitions:

Rcn is the recombination rate of electrons into traps, en is the trap capture cross section (considered a constant), En is the trap emission cross section (considered a constant),

500 Generation and Recombination Processes in Semiconductors

Ren is the electron emission rate from traps, Nt is the total number of trap states present, f(Et) is the probability that a trap of energy Et is occupied, 1 - f(Et) is the probability that a trap state is empty.

The trap capture rate is proportional to the number of empty traps, which is given as the product of the total number of traps and the probability that a trap is unoccupied:

(10.3.1)

The trap capture rate must also depend on the concentration of free carriers available n, since no capture events can occur if there are no electrons available to be trapped. The trap emission rate depends on the number of filled traps,

(10.3.2)

as well as on the number of vacancies present in the conduction band. Under non degenerate conditions (defined as the condition in which the use of Maxwell­Boltzmann statistics is applicable in determining the electron carrier concentra­tion in the conduction band) the conduction band is virtually empty and the Fermi-Dirac filling factor for electrons in the conduction band can be safely neglected in determining the trap emission rate.

The trap emission cross section En is assumed to be constant, a function of only the material identity. An expression for Elt can be obtained as follows. In equilibrium the trap emission and capture rates must be equal. Therefore

(10.3.3)

The recombination or capture rate is directly proportional to the product of the number of empty traps and the concentration of electrons in the conduction band, as discussed above. Rcn is given then as

[Rat.., = CnN t [1 -Mt)J!z~ (10.3.4)

where Cit is the capture cross section. Similarly, the emission rate is proportional to the number of filled traps and can be written as

i ~~ = En N t!.J3t ) , J (10.3.5)

where the concentration of vacancies in the conduction band has been neglected. Equating the two rates and solving for En yields

I E - C [1 - fo(E t )]

n - no It_ fo(Et

) ~ (10.3.6)

--------------.......... 10.3 . Shockley-Read- Hall Recombination Processes 501

where no is defined as the equilibrium electron carrier concentration and fo(E t )

is the equilibrium distribution function. The equilibrium distribution function is simply the Fermi- Dirac function,

1 fo(Ed = --""-Et -;E~r .

l + ekBT

Therefore 1 - fo(Ed is simply

Et - f r

ekBT Et r:r •

l+e kfiT

With these substitutions, the emission cross section becomes

Et - E/ \

, En = noCne kBT .

(10.3.7)

(10.3.8)

(10.3 .9)

We can simplify expression (10.3.9) by noting that we can obtain the equilibrium electron concentration no easily by integrating the product of the equilibrium distribution function by the density of states by using the approach in Section 5.8. The equilibrium electron concentration is given by Eq. (5 .8.26):

-IE,-Er i

no = Nce~, (10.3.10)

where Nc is defined as the effective conduction-band density of states. Nc is defined by Eq. (5.8.26) as

3

_ 2(2n:m*kB T) 2 Nc - h2 ' (10.3.11)

where m* is the electron effective mass and only one equivalent conduction-band minimum is assumed present. The concentration no can be rewritten in terms of the intrinsic concentration nj as

(Et - Ej )

no = nj e ----rnr , (10.3.12)

where Ej is called the intrinsic level and is equal to the Fermi level in pure, intrin­sic material. The expression for En can be simplified further if it is assumed that the trap energy is aligned with the Fermi level, Et = E f. With this assumption, the expression for the emission cross section is

(10.3.13)

where nl is defined as the equilibrium electron concentration corresponding to the special case of the trap energy's being coincident with the Fermi level. The emission cross section is found then to be equal to the product of the capture cross section and the equilibrium electron concentration.

502 Generation and Recombination Processes in Semiconductors

The recombination rate outside of equilibrium is typically different from the generation rate. For example, if the excess carrier concentration exceeds the equilibrium carrier concentration and all external perturbations are then removed, recombination will act to restore the system to equilibrium. Similarly, if the system is initially constructed such that the carrier concentrations are less than the equilibrium concentrations and then the driving forces are removed, generation events will restore the system back to equilibrium. The net trap cap­ture rate is given by the difference between the trap capture and trap emission rates as

(10.3.14)

Substituting the expressions for Ren and Ren given by Eqs. (10.3.4) and (10.3.5) into Eq. (10.3.14) yields

(10.3.15)

When the expression for En given by Eq. (10.3.13) is used, the net recombination rate becomes

(10.3.16)

where nl is the equilibrium electron carrier concentration and n is the none qui­librium electron carrier concentration. It is important to recognize that the distri­bution functions, f(Et) and 1 - t(EI) used in Eg . (10.3.1fure noneg~ilibriuJll distribution functions. Below we will derive an expression for f(E r).

Similar capture and emission rates can be obtained for holes. The hole capture rate depends on the number of holes and the number of filled traps. A trap emits an electron that recombines with a free hole in the valence band during a hole capture event. A hole emission event depends on the number of empty traps. In a hole emission event, an electron within the valence band is captured by an empty trap, leaving a free hole behind in the valence band. Under nondegenerate conditions, the concentration of electrons in the valence band is sufficiently large that hole emission is not limited by it. The capture and the emission rates can then be written as

Rep = CpNt f(Et)p,

Rep = EpNdl - f(Et)]. (10.3.17)

The hole emission cross section Ep can be found from equilibrium conditions in a manner similar to that for electrons. This yields

(10.3.18)

where PI is the equilibrium hole concentration; it is again assumed that the trap and the Fermi levels are coincident:

- (£/-£- )

PI = nj e -'i'gi'-' . (10.3.19)

10,3 Shockley-Read-Hall Recombination Processes 503

The net hole recombination rate Rp is given by the difference of the hole capture and emission rates. Rp becomes

(10.3.20)

Under steady-state conditions, the electron and hole recombination rates Rn and Rp are equal. Equating the expressions given by Eqs. (10.3.16) and (10.3.20 ) for Rn and Rp yields

Cn Nt\n[l - ((E I )] - nl ((Et )} = CpNdP((Et) - Pl [1 - ((Et )]) ,

CnNtn - CnNtn((Et) - CnNtnl ((Et ) = CpNtp((Et ) - CpNtPI

+ CpNtPI ((E/). (10.3.21)

Combining terms and simplifying yields

(10.3.22)

Solving for the nonequilibrium distribution function ((E t ) yields

(10.3.23)

where nl and PI are the equilibrium electron and hole carrier concentrations defined by Eqs. (10.3.13) and (10.3.19), respectively. The electron and hole concentrations nand pin Eq. (10.3 .23) are the nonequilibrium carrier concen­trations. The equilibrium carrier concentrations nl and PI are related through the law of mass action. The law of mass action states that the product of the equilibrium electron and hole concentrations is a constant. From Eq. (5.8.26) the electron concentration in the conduction band is

(10.3.24)

A similar re lationship can be derived for the hole concentration in the valence band. The equilibrium hole concentration is given as

-tEf -Et»

P = Nve kilT . (10.3.25)

The product of the equilibrium concentrations nand P is then

- IE, - E( i - 1£(-£,'

np = NcNve kilT e kilT - (Ec - Ev l

np = NcNve kilT. (10.3.26)

But (Ec - Ev ) is simply equal to the bandgap energy Eg • Therefore the np product becomes

Eg

np = NcNve-kBT (10.3.27)

504 Generation and Recombination Processes in Semiconductors

The np product is constant at fixed temperature. This constant is equal to the square of the intrinsic carrier concentration nj, defined as the electron and hole concentration in pure material at a fixed temperature. The np product is then

2 np =nj . (10.3.28)

Subsequently, the product of ni and PI must also be equal to nf, since both ni

and P1 are equilibrium concentrations. Therefore

(1 0.3.29)

With this result and the expression given by Eq. (10.3.23) for the nonequilibrium distribution function, the electron and hole trap recombination rates become

(1 0.3 .30)

We can simplify the first term in Eqs. (10.3.30) by rewriting it in terms of its least common denominator as

(10.3.31)

With the above simplification, the electron and hole trap recombination rates are

R - R - C N [Cnnn i + Cpnp - Cnnn1 - Cpni PI ] n - p- n t Cn(n+nl)+Cp(P+PI) ,

Cn NtCp (np - nn Rn = Rp = -:----'-'---,----'--

Cn(n + n1) + Cpt P + pd (1 0.3.32)

If it is assumed that the electron and hole trap capture cross sections Cp and Cn

are the same, then the expressions for the trap recombination rates reduce to

CNt(np - nf) h f~). hI Rn = Rp = , 1\ ') (10.3.33)

(n+n1)+(p+ pd (? ~

it "'- P where C is the trap capture cross section. Note that th~ recombination rate depends on the np product. In equilibrium, the np product satisfies the law of mass action and is thus simply equal to nf. As a result, the numerator of Eq. (10.3.33) vanishes and the net electron and hole recombination rates are zero. However, when the np product is greater than nf, there is an excess con­centration of carriers in the semiconductor. The system will then try to restore itself to equilibrium through recombination. The expression for Rn and Rp above

10.3 Shockley-Read-Hall Recombination Processes 505

is positive, implying a net recombination rate. If the np product is less than nf, there is a net depletion of free carriers . The system will then try to restore itself to equilibrium through generation. The expression for Rn and Rp in this case is negative, implying a net generation rate.

The electron and hole recombination rates can be reformulated in terms of an expression involving only the excess minority carrier concentration and a characteristic lifetime. It is common to define electron and hole lifetimes rn and rp as

1 rp = rn == CN

t' (10.3.34)

With these definitions, Rn and Rp become

(10.3.35)

The nonequilibrium electron and hole concentrations nand P can be written in terms of the equilibrium concentrations no and Po and the excess concentrations on and op as

n = no + on, P = Po + op. (10.3.36)

A semiconductor is classified as n type if the equilibrium electron concentration no exceeds the equilibrium hole concentration Po. Conversely, a material is p type if the equilibrium hole concentration is greater than the corresponding electron concentration. In an n-type semiconductor the holes are said to be the minority carriers in the material. If the excess electron concentration is small with respect to the equilibrium electron concentration in an n-type semiconductor, then the excess electron concentration can be neglected with respect to the equilibrium electron concentration. The excess hole concentration, however, may be com­parable with or greater.than the equilibrium hole concentration. The generation rate is given then as

R = (no + on)(po + op) - nr rp(no + on + nl) + rn(po + op + PI)

(10.3 .37)

But nl and PI are defined as the equilibrium electron and hole concentrations, assuming the trap energy lies near the Fermi level, Et ~ EF • If it is further assumed that the trap energy is around midgap, then nl and PI are essentially equal to the intrinsic electron and hole concentrations and can be approximated as being equal to n i . With this approximation and the assumptions that for n-type material,

noop» Poop, noop » onop, (10.3.38)

506 Generation and Recombination Processes in Semiconductors

R can be written as

R = no8p Tp(no + 8n + ni)'

(10.3.39)

wEtere tEte second term in tbe denominator of Eg. (10.3.37) bas been neglected since all its terms are much less than no. We can simplify the expression for R in Eg. (10.3.39) further by neglecting ni and 8n in the denominator with respect to no. R finally can be written as

no8p ap R=-=-, (10.3.40)

noTp Tp

which expresses the recombination rate in terms of the excess minority carrier concentration and its lifetime.

10.4 Impact Ionization Transition Rate

The inverse Auger effect is commonly referred to as impact ionization. During an Auger event, an EHP recombines, while after an impact ionization event an EHP is produced. As discussed in Section 10.1, an impact ionization event occurs when a high-energy carrier makes a collision with the lattice, transferring its excess kinetic energy to a bound electron in the valence band, promoting it into the conduction band, and leaving a hole behind in the valence band. The initiating carrier can be either an electron or a hole. To generate an EHP, the incident carrier must have a kinetic energy at least equal to the bandgap energy. As such, impact ionization is a threshold process with a threshold energy of at least the bandgap energy. Typically, the threshold energy for impact ionization, defined as the minimum energy for which an impact ionization event will occur, is greater than the bandgap energy due to momentum conservation.

We can derive an expression for the impact ionization rate assuming that the interaction arises from a screened Coulomb scattering event. Let us consider an incident high-energy electron that makes a collision with an electron in the valence band, producing two electrons in the conduction band. The momentum of the collision must be conserved, which requires that the k vectors of the carrier species must be conserved. Defining the initial k vector as kl and the k vectors of the final states as k~ and k;, the k-vector conservation requirement gives

(10.4.1)

The electron-electron interaction is essentially a two-body collision. The matrix element of the interaction can be written as

(10.4.2)