Home Assignment - V (EEL732)Adersh Miglani

Adersh.Miglani@gmail.com16-Aug-2013

Assignment: For Shockley-Read-Hall (SRH) recombinationmodel, derive the proof of the following expression for rate ofrecombination at recombination trap level between forbiddenenergy gap.

R =np− n2i

τp0(n+ nt) + τn0(p+ pt)

whereτp0 = 1

CpNt

τn0 = 1CnNt

Cn is electron capture cross-sectionCp is hole capture cross-sectionNt is number of available trapsnt and pt are net electron and hole concentration at the traplevel Et

Solution: In the ideal semiconductor material, there areno allowed energy states within bandgap. But, in practice,allowed energy states are always present between Ec and Ev

energy levels due to lattice defects in semiconductor crystalor because of undesired impurity atoms. These energy statesare called traps. While determining mean carrier lifetime therecombination at these traps are considered as per SRH theoryof recombination. There are four processes through which freecarriers recombine at these traps.

1) electron capture: An electron falls from conduction bandinto the trap. The rate of electron capture (r1) dependson the density of electrons in the conduction band,density of empty traps and probability that electronelectron falls into the trap.

2) electron emission: An electron jumps from trap to theconduction band. The rate of electron emission (r2)depends on the density of traps occupied by electronsand probability that an electron may jump to conductionband.

3) hole capture: Trap captures a hole from valence band (anelectron jump from the trap to the valence band leavinga hole at the trap). The rate of hole capture is denotedby (r3).

4) hole emission: Trap emits an hole into the conductionband (an electron jump into the trap from valence bandleaving a home in the valence band). The rate of holeemission is denoted by (r4).

The product of electron thermal velocity vth and electroncapture cross-section Cn is volume swept out per unit timeby an electron. The number of total allowed traps is denotedby Nt at an energy level Et within bandgap. And, f(Et) is

the probability that a trap is occupied. So, the rate of captureof electrons during electron capture process is given by thefollowing expression

r1 = n[Nt(1− f(Et))]vthCn

where n is the concentration of electrons in the conductionband and [Nt(1− f(Et))] is density of empty traps at Et.

Rate of electron emission depends on the density of occu-pied traps, Ntf(Et), and en, the probability that an electronmay jump to conduction band.

r2 = [Ntf(Et)]en

At thermal equilibrium rate of electron capture and electronemission should be equal, r1 = r2.

n[Nt(1− f(Et))]vthCn = [Ntf(Et)]en

en = nvthCn

(1− f(Et)

f(Et)

)

en = nvthCn

1− 1

1+exp(

Et−EfkT

)1

1+exp(

Et−EfkT

) = nvthCnexp

(Et − Ef

kT

)

en = ni exp

(Ef − Efi

kT

)vthCnexp

(Et − Ef

kT

)en = nivthCnexp

(Et − Efi

kT

)= nivthCnλn

The rate of hole capture depends on the density of occupiedtraps, Ntf(Et), and concentration of holes in the valence bandalong with volume swept out by a hole per unit time

r3 = p[Ntf(Et)]vthCp

The rate of hole emission density of empty traps at Et andprobability that a hole would be emitted to valence band.

r4 = Nt[1− f(Et)]ep

By considering the thermal equilibrium condition, r3 = r4,expression for ep can be derived as below

ep = vthCpni exp

(Efi − Et

kT

)= nivthCpλp

As per SRH recombination theory, when the thermal equi-librium is disturbed, the concentration of minority carriers(electrons in case of p-type semiconductor) are generated. Dueto the generation of electron-hole pairs, the relative change inthe majority carrier hole concentration is a small fraction. On

the other hand, the relative change in the concentration ofelectrons is appreciable. Under this situation, the net rate ofelectron capture r1 − r2 should be equal to net rate of holecapture by traps at Et which is rate of recombination, R

R = r1 − r2 = r3 − r4

1) First, let us consider r1 − r2

n[Nt(1− f(Et))]vthCn − [Ntf(Et)]en = R

nNtvthCn − nNtvthCnf(Et)−Ntenf(Et) = R

f(Et) =nNtvthCn −R

Nt[nvthCn + en]=nNtvthCn −R

αn

2) Now, let us consider r3 − r4

p[Ntf(Et)]vthCp −Nt[1− f(Et)]ep = R

pNtf(Et)vthCp −Ntep + f(Et)Ntep = R

f(Et) =R+Ntep

Nt[pvthCp + ep]=R+Ntep

αp

3) Now, equate the two expressions for f(Et)

nNtvthCn −R

αn=R+Ntep

αp

nNtvthCnαp −Rαp = Rαn +Ntepαn

R =nNtvthCnαp −Ntepαn

αp + αn

Use expression for αp and αn

R =nNtvthCnNt[pvthCp + ep]−NtepNt[nvthCn + en]

Nt[pvthCp + ep] +Nt[nvthCn + en]

4) Nt term is common in numerator and denominator.

R =nNtvthCn[pvthCp + ep]−Ntep[nvthCn + en]

[pvthCp + ep] + [nvthCn + en]

5) Use expression for ep and en

R =nNtvthCn[pvthCp + nivthCpλp]−Ntep[nvthCn + nivthCnλn]

[pvthCp + nivthCpλp] + [nvthCn + nivthCnλn]

6) vth term is common in denominator and numerator.

R =nNtvthCnCp[p+ niλp]−NtepCn[n+ niλn]

Cp[p+ niλp] + Cn[n+ niλn]

7) Use expression for ep

R =nNtvthCnCp[p+ niλp]−NtnivthCpλpCn[n+ niλn]

Cp[p+ niλp] + Cn[n+ niλn]

8) Divide numerator and denominator by NtvthCnCp.

R =n[p+ niλp]− niλp[n+ niλn]

1NtvthCn

[p+ niλp] +1

NtvthCp[n+ niλn]

9) The terms τn0 = 1NtvthCn

and τp0 = 1NtvthCp

are meanelectron and hole lifetimes at trap level.

R =np+ nniλp − nniλp − n2iλnλp]

τn0[p+ niλp] + τp0[n+ niλn]

R =[np− n2i ]

τn0

[p+ niexp

(Efi−Et

kT

)]+ τp0

[n+ ni

(Et−Efi

kT

)]R =

[np− n2i ]

τn0[p+ pt] + τp0[n+ nt](1)

where pt = niexp(

Efi−Et

kT

)and nt = ni

(Et−Efi

kT

).

10) Expression for pt and nt can be written in terms of Nv

and Nc, respectively, as follows

pt = Nvexp

(−(Et − Ev)

kT

)nt = Ncexp

(−(Ec − Et)

kT

)