Electric field dependence of the direct recombination rate in InSb

Volume 41A, number 4

PHYSICS LETTERS

9 October 1972

ELECTRIC FIELD DEPENDENCE OF THE DIRECT RECOMBINATION RATE IN InSb H. REUTER and K. HLIBNER Institut fdir Angewandte Physik, UniversitiJt Heidelberg, Heidelberg, Germany Received 26 July 1972 Using the approximation of drifted Maxwellians an analytic expression is derived for the electric field dependence of the direct recombination rate.

In low doped semiconductors with band edges at k = 0 the direct recombination is predominantly produced by k-conserving transitions [ 1]. The rate at which photons per unit volume are spontaneously emitted in the energy interval dhv can be written [2] as

rspon(hV) dhv 4Ne2hv IMI2pred(hV)fu(1-fl)dhv, m2h2c 3

(1)

where fu and fl are the probabilities that the upper and lower states involved in the transitions are occupied, N is the refraction index, Pred(hV) the reduced density of states and IM12 the averaged square of the matrix element connecting the band edges which in the case of InSb might be taken approximately as constant. If we do not take into account the effect of reabsorption of the emitted photons, the total number of recombining electron-hole-pairs will be given as

R = f rspon(hV) dhv 0

A

=

4Ne2/m2h2c 3,

M

=

mcmv/(mc+rnv) .

In thermal equilibrium we have in the case of a nondegenerated semiconductor

fO(p) = exp(_lpl2/2mckT) exp([Fc_ec ]/kT),

(3)

where F c is the Fermi level of the conduction band and e c the energy at the band edge. With

n c = (2/h3)(2nmckT) 3/2 exp ([F c - eel/kT)

(4)

and using the adequate expressions for f o and n v we obtain in the thermal equilibrium situation

e +~kT g [27r(mv+mc) k T] 3/2"

tt

r0 = ½h3A

(5)

Having an electric field in the x-direction we take advantage of the simple approximation of a drifted Maxwellian !

and hence the direct recombination rate is r" =R/(nenv) , where n c and n v are the electron and hole densities. Assuming isotropic parabolic conduction and valence bands we obtain hv = eg + p2/2mc + p2/2mv, where p is the momentum of the electron and hole which recombine with one another eg the band gap, and m c and m v are the effective masses. By substituting of hv by p using Pred(hV)dhv = (2/h 3) dPxdpydpz we obtain: +oo

R = (2/h3)A f -

(eg + tpl2/2M)fc(p)fv(P)dPxdpydpz , ~0

(2)

1

l

08J

~

o6! 0.4 0.2, 0

0

100

200 300 400 Electric Field [V/cm]

500

Fig. 1. Direct recombination rate r" divided by r~ as a function of electric field for InSb at 77 K.

329

PHYSICS LETTERS

Volume 41A, number 4

Je -- exp /--

2mc~

exp t--TrY}'

~6)

where qc = -mcl~cE is the momentum of the drift motion, with/,t c the electron mobility and E the electric field. With the corresponding expression for the holes fv with qv = mvlavE the integration of (2) leads to

r,,(E)=~h3A

x exp(-

eg + }kT +~M(qe/m c +qv/mv)z . . . . . X [27r(mc+mv) kT] 3/2

[11 A. Mooradian and H.Y. Fan, Phys. Rev. 148 (1966) 873. [2] G. Lasher and F. Stern, Phys. Rev. 133 (1964) A553.

(qc-qv)2

2(mc-~mv)k T !

+

(7)

½M(/ac-/'tv)2E2~

~ (lacmc+Uvmv)2E2~

f exp.-

(8)

330

Since ½M(~tc-/av)2E 2 ~ eg the bracket might be taken as unity. The exponent is of the order of the kinetic energy of the drift motion of the electrons and holes divided by the thermal energy and is therefore quite small. Taking into account the field dependence of the electron mobility, eq. (8) gives for lnSb at 77 K the weak dependence of the direct recombination rate on the electric field shown in fig. 1.

References

o1

r"(E) = r0 (1

9 October 1972