_ Solid State Communications, Vol. 45, No. 2, pp. 195-198, 1983. Printed in Great Britain.

0038-1098/83/020195-04503.00/0 Pergamon Press Ltd.

ELECTRON-HOLE CORRELATIONS IN Si UNDER UNIAXIAL STRESS

Tapash Chakraborty* and C.E. Campbell*

Institut ftir Theoretische Physik, der Universit~t zu K61n, Ziilpicher Strasse 77, 5000 K61n 41, Germany

(Received 16 August 1982 by B. Miihlschlegel)

Using the many-body variational approach as input, we have evaluated the lifetime of the electron-hole drop in Si under uniaxial stress. Comparing our results with the recent experimental values, we notice that the elec- t ron-hole correlations obtained in our approach has the correct density variation. The theoretical lifetime obtained in the Auger recombination process compares reasonably well with the measured lifetime.

RECENT EXPERIMENTAL studies of the decay kinetics of electron-hole drops (EHD) [ 1-4] in uniaxially deformed Si, provide unique grounds for study of the intrinsic recombination of the EHD. Another important aspect of these experiments is the estimate of density variation of the enhance factor geh(O), which is the probability of finding an electron at a hole. Of late, such experiments have been performed in Ge by Chou and Wong [5]. The situation in Ge is however, quite complicated, since in this case, both radiative and nonradiative recombination processes are important. In fact, the EHD in Ge is about 25% radiatively efficient at zero stress [6]. On the other hand, for Si at liquid He temperatures, the radiative efficiency is estimated [7] to be 5.10 -4. Therefore, nonradiative recombination is the most dominant process for EHD in Si.

For a stress along the [ 1 1 1 ] crystallographic direction Wagner et al. [4] have measured the lifetime of the EHD as a function of stress, expressed in terms of the associated valence band splitting. The most probable decay mechanism in this case is assumed to be the Auger recombination of the EHD. The lifetime is then given by [4, 5 ],

T -1 : (Cegeeh(O) -1- Chgehh(O))/9 2 ( 1 )

where p is the density of electrons or holes, geeh(O) and gehh(O) are the three-particle enhancement factors for the eeh and ehh Auger processes respectively, Ce and ch are the associated Auger coefficients. Employing the Kirk- wood superposition approximation, ge~h ~ geeg2eh and gehh ~ ghhg2h, the lifetime can be expressed in terms of the two-body correlation functions only [5],

r- i = (Cegee(O) + Chghh(O)) g2eh(O ) p2. (2)

*Present address: School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455, U.S.A.

195

To a good approximation, ce and ca can be considered to be insensitive to stress variation [4]. In what follows, we will use the experimental values [8] for highly doped samples, Ce = 2.3 x 10 -31 c m 6 s -1 and Ch = 0.78 X 10 -al cm 6 s -1.

Many-body variational approach for binary systems [9] has been quite successful in the study of the ground- state properties of the EHD [10] in Ge and liquid metallic hydrogen [11-13]. The optimal pair-correlation functions gag(r), ~t,/3 = e or h obtained in this method are in excellent agreement with those obtained in the perturbative methods [11, 12]. In our present work, we assume for simplicity that the bands are isotropic with optically average mass # = 0.123 for electrons and holes. In fact, recent numerical results by Vashishta and Kalia [14] indicate that the exchange-correlation energy of the electron-hole liquid is independent of the different band characteristics of semiconductors. In that case, one expects a similar effect for the optimal pair-correlation functions. The theoretical details are given in [9-11 ]. The resulting gate(0) are then used in equation (2) to obtain the lifetime of the EHD.

Figure 1 depicts the EHD lifetime as a function of the charge carrier density. The experimental curve is from Wagner et al. [4]. Also shown in Fig. 1 : a simple approximation, T -1 ~ p2 (dot-dashed line), the present result (solid line) and the curve obtained by using the me = mh results of Vashishta et al. [ 15] (VBS). Follow- ing [4], all the curves are normalized to a value of r = 135ns at zero stress (p = 3.5 × 10 TM cm-3). Compar- ing the theoretical results of VBS and ours with the experimental result, we find that a correct density variation of r has been obtained by us. This in turn, reflects a correct density variation O f g e h ( 0 ) , obtained in our optimization method. As far as the absolute values are concerned, our result shows a glaring over- esimate of the experimental lifetime throughout the density range.

196 ELECTRON-HOLE CORRELATIONS IN Si UNDER UNIAXIAL STRESS

. I , , ,

\ \,\ 400 ~o ,

"\\o \ \'\ \~, \ '\ o \ \ '\

300 \ \ i \\ , ~ \ Present w o r k ' o-\ \ \ ~ \ \ \ \ \'\ \ Y \ \ " , \ . ~\,,,

200 Experimental----"°x x\, ~ \ ,

VBS

100 ,,; a i i

1 2 3 EHD Density p[101~cn~ 3]

Fig. l. Droplet lifetime r as a function o f charge carrier density p.

II , . ,

400

300

c Q - r

hi I-'

200

100

k ~'kk periment

Theory ~x~o \ ~ o~.... ~ ~ _

:! I I I

1 2 3 /- EHD Density p[1018cr f i3]

Fig. 2. Same as Fig. 1. The theoretical results are here normalized to the high-stress value of 385 ns (see text).

Vol. 45, N o . 2

Vol. 45, No. 2 ELECTRON-HOLE CORRELATIONS IN Si UNDER UNIAXIAL STRESS

"~1

8

7

l I I

~ 4

E~ 3

Theory

i i |

0 0.5 1 2

Experiment

rs

15

14

12

10~ 'E t~

c o 8% T - -

c -

CL 0~

6

4

2

Fig. 3. Enhancement factor geh(O) and the enhanced density Peh.

197

Given the different density variation ofgeh(0) by VBS and by us, the disagreement with the experimental results might be attributed to some extent to the super- position approximation. Since to date, no numerical values for the true three-particle correlation functions are available, there is no means to decide which way the curve would shift for an exact three-particle enhance- ment. However, the superposition approximation is the simplest approximation, expected to be quite good at small separation. For the remaining disagreement, one could possibly think of some other mechanism, in addition to the Auger decay, which might reduce the EHD lifetime. In particular, for the [100] stress, Wagner et al. proposed a model of free exciton evaporation from the EHD surface. A similar process in this case can reduce the lifetime. Various other nonradiative recom- bination processes in semiconductors have been reviewed by Nimtz [16]. Such studies are, however, beyond the scope of the present paper.

It should however be noted that the present approach (and the VBS), which ignores the different band characteristics of Si, is best suited in the high stress limit. At small or zero stress, multiplicity and anisotropy of the bands might have a profound effect in the enhancement. Keeping this in mind, we have normalized

our theoretical curve to the high stress lifetime of 385 ns reported in [4]. As shown in Fig. 2, the theoretical results are in good agreement with the experimental lifetime in the low and intermediate density region, but expectedly, deviate near the density corresponding to zero-stress. An improved calculation of the enhancement factors, including the different band characteristics, is expected to account for the disagreement near zero stress.

Recently, Gourley and Wolfe [2] reported the experimental results for EHD recombination in Si under a large [100] stress. In this limit of large stress, they estimated the enhancement factor gea(O) from the lifetime T assuming that the dominant process is the Auger decay. Their results for the enhancement factor and the enhanced density Pea = geh(O)P are plotted in Fig. 3, together with our theoretical results (solid lines). The experimental values are normalized to our zero-stress value geh(O) = 2.55. While the theoretical curve for geh(O) increases rapidly with increasing r8 (dimensionless interparticle separation in units of exciton Bohr radius), the experimental esti- mates seem to have reached a constant value beyond r8 -- 1.5. The apparent disagreement between the results in Fig. 3 might be due to the assumption that the

198 ELECTRON-HOLE CORRELATIONS IN Si UNDER UNIAXIAL STRESS Vol. 45, No. 2

Auger decay is the only process to explain the observed lifetime.

Acknowledgements - The work was supported in part by the National Science Foundation under Grant No. DMR-7926447 and by the Alexander von Humboldt- Stiftung. One of us (TC) wishes to thank Giinter Nimtz and K. Heift for informative discussions.

REFERENCES

1. V.D. Kulakovskii, V.B. Timofeev & V.M. Edel- shtein, Soy. Phys. - JETP 47, 193 (1978).

2. P.L. Gourley & J.P. Wolfe, Phys. Rev. Lett. 40, 526 (1978); Phys. Rev. B24, 5970 (1981).

3. A. Forchel, B. Laurich, J. Wagner, W. Schmid & T.L. Reiniecke, Phys. Rev. B25, 2760 (1982).

4. J. Wagner, A. Forchel, W. Schmid & A. Sauer, Solid State Commun. 42,275 (1982).

5. H.-h. Chou & G.K. Wong, Phys. Rev. ~ett. 41, 1677 (1978).

6. K. Betzler, B.G. Zhurkin & A.L. Karuzskii, Solid State Commun. 17,577 (1975).

7. J.D. Cuthbert,Phys. Rev. B1, 1552 (1970). 8. J. Dziewior & W. Schmid, Appl. Phys. Lett. 31,

346 (1977). 9. Tapash Chakraborty, Phys. Rev. B25, 3177

(1982);J. Low Temp. Phys. 48, 151 (1982); Phys. Rev. B (in press).

10. Tapash Chakraborty & P. Pietil~inen, Phys. Rev. Lett. 49, 1043 (1982).

11. Tapash Chakraborty, A. Kallio, L.J. Lantto & P. Pietil~iinen, to be published.

12. Sudip Chakravarty & N.W. Ashcroft, Phys. Rev. BI8, 4588 (1978).

13. C.E. Campbell & J.G. Zabolitzky, to be published.

14. P. Vashishta & R.K. Kalia, Phys. Rev. B25, 6492 (1982).

15. P. Vashishta, P. Bhattacharyya & K.S. Singwi, Phys. Rev. B10, 5108 (1974).

16. Gtinter Nimtz, Physics Reports 63,267 (1980).

NOTE ADDED IN PROOF

The experiment on the absolute measurement of the density dependence of the enhancement factor geh(O) for the electron-hole liquid in Ge has been reported by J.C. Culbertson & J.E. Furneaux, Phys. Rev. Lett. 49, 1528 (1982). The results are in excellent agreement with the theoretical prediction of [10].