Nonparabolicity and exciton effects in two-photon absorption in zincblende semiconductors

0038-1098/81/320937-04502 00/0

Sohd State Communications, Vol 39, pp 937-940 Pergamon Press Ltd 1981 Printed m Great Britain

NONPARABOLICITY AND EXCITON EFFECTS IN TWO-PHOTON ABSORPTION IN ZINCBLENDE SEMICONDUCTORS M H Wefler Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U S A (Recetved lOMarch 1981 by J Tauc) The two-photon absorption coefficient ts calculated from perturbation theory for direct-gap zmcblende semiconductors using exact nonparabohc energies and matrLx elements The enhancement due to exclton effects ~s included The results are compared to experiment and to other theories including the tunnehng or Keldysh theory

THERE HAS BEEN much interest m the study of twophoton absorption in semiconductors, mltmlly because of the new reformation gained over one-photon absorption and because of the practical consequences such as reducible absorption, and later because of the large discrepancies among the results of experiments and of two theoretical methods, the perturbation and tunnehng theories This was most recently discussed by Valdyanathan et al [1 ] In this work we show that perturbation theory using the simple non-parabolic models suggested by Pldgeon et al [2, 3], when used correctly including exact matrix elements and exclton effects [4], prowdes the best theory of the two-photon absorption coefficient for a variety of direct-gap zmcblende semiconductors We present, for the first Ume, the frequency dependence of the exoton enhancement for a range of values of the exoton binding energy We also show, from the frequency dependence, that the corrected tunnehng theory of Keldysh [5] and Bychkov and Dykhne [6], is less accurate than perturbation theory in the nonparabohc region above the absorption edge The two-photon absorption coefficient K2 = a(2~[I is gwen in second-order perturbation theory, in the notation of [21 and [3], by

+

x cose sine e'~,

(3)

where 7/= [E~ + 8k2P2/3] 1/2, 8 and ~ are the Euler angles of k with respect to the optical polarization, and the energies Egh and ffgl are gwen m equation (8) of [2] Equation (3) would reduce for k ~ 0 to equation (6) of [2], except for the relatwe sign of the two terms m the square brackets The second term is described m [2] as due to a "/-/6 --* UI, U3 --* U6 transition" This change from the order U~ -+ Ut, Ut -> Uf symbohzed m equation (2), Is apparently an attempt to use the exclusion principle for the occupied state t = 6 Nguyen et al [8] showed that this Is not correct unless one also includes an extra (--) sign due to the reversal of the Fermi operators, the net effect Is that one should ignore the exclusion principle for virtual intermediate states The incorrect s]gns m [2] leads to spurious cancellations such as discussed for the hght-hole transltons m [3] We have re-calculated K2 using equations (1) and (2) with the correct signs, and also using exact, nonparabohc matrix elements instead of those for k --* 0 used by Pldgeonetal [2, 3] We find

K2 = (4ne4P/l~n~cZE~)f(hcop/Eg), K2 = (2e4/n~cZrn4w~) ~. ~ ISf,12~(Ef -- E, -- 2hwp) dak, '

where the nonparabonc result fnP(a) Is (with A >>Eg)

(1)

where the sum over mtermedmte states t ~s

fnp, A ~, Eg(ol) --

Sf, ~- ~. (f[~'plt)(tl$'plt) t

St --E~--h6op

(4)

(2)

We first follow the model of Pldgeon etal [2, 3], but instead using the Kane wave functions [7] for A >>Eg For example, for the heavy-hole transition 3(1,*) -* l(c~) (m the notaaon of [2] and [7]) equation (2) gwes

(2a -- 1)3/= / 4(3°t)s/= 3(X3 t OtX-- 1)2 +

3¢~+-

90¢:

(5)

Similarly for A >>Eg we obtain (2ol- 1)3/2 f 12(2a)1/2

937

+ (2or + 1) a/2(80t4 + 4Or=+ 3) / 12txs )

(6)

938

TWO-PHOTON ABSORPTION IN ZINCBLENDE SEMICONDUCTORS 25

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Fig 1 Two-photon absorption coefficient K2 vs cz = Theoretical curves (a) including non-parabohclty and exciton effect for A >>Eg, (b) neglecting exciton effect, (c) theory of Pldgeon et al [2,3], (d) parabolic theory, (e) tunneling theory Expermaental points scaled to InSb at 300 K as described In the text circles from [3] for InSb at 300K with X = 10 6/~m (1) and 0 6Mm (2), and at 77K for X = 10 6/am (3), and for (Hg, Cd)Te with k = 10 6tan at 300K (4) and 150K (5), square from [4] for InSb at 300 K with X = 10 6/ma

ho~p/Eg, for InSb at 300 K

The result for K2 for the case of lnSb at 300 K is plotted in Fig l (curve b) using equation (5), and compared to the result of [2] for A >>Eg (curve c) Our result has a completely different frequency dependence, larger as a = boap/E~~ 0 5 without cancellations, but smaller as a + 1 because of the exact matrix elements The result for ~ ~ Eg from equation (6) is larger than that for A >>Eg, by a factor 1 35 as a ~ 12and 1 47 as a ~ 1, not smaller as in [2] The experimental results of [3] are given as dots m Fig 1, where the data are scaled to their equivalent for InSb at 300 K by the factor [E~n~/(O 18 eV)3(3 9) 2] The points numbered 4 and 5, for (Hg, Cd)Te at 300 K and 150 K respectively, were rescaled from the experimental values of 10 and 30 cm MW -1 using Eg = 0 18 and 0 13 eV, respectively, with np = 3 5 The corrected theory given here agrees less well with the data than did the incorrect theory We also have re-calculated K2 for the parabolic case starting with k ~ 0 wavefunctlons and energies as m [4] and find f p , A ~, Eg(ot) =

(2~--1)a/2 ( + 2 9 ~ ) ~V/~'~ 4 12

(7)

This IS plotted (curve d) In Fig 1 This approximation overestimates K2 (The opposite was stated m [2] ) The above result differs shghtly from that of Lee and Fan [4] We deduce that they neglected the energy A in the energy denominator for the spht off-band intermediate

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Fig 2 Theoretical exclton enhancement factor gex for the two-photon absorption K2, for the values shown of the ratio e of the exciton binding energy Eb to the energy gap Eg states in equation (2), resulting in incorrect expressions for their coefficients Iv, by, , fv Before discussing exclton effects, we consider the other major method for calculating multlphoton absorption, the tunnehng theory of Keldysh [5] This theory was shown by Bychkov and Dykhne [6] to be incorrect for even-photon absorption, but they gave no expression for this case for nhoJ >Eg From their equation (33), with some corrections, the lowest-order non-zero term gives In the multIphoton hmIt for n = 2

ft'a>Zg(a)

= const

x ot-l(I)'(o~,

[2(2a--

1)/or]l/2), (8)

where @' is a function related to the Dawson integral qb'(ot,y) = n~-~(2a)V2e- ~

~:eX=sln~(rrotI:~/2)dx

(9)

@' reduces to (2ct -- 1) 3/2 as ct -~ ½, reproducing the "allowed-forbidden" perturbation theory [and equations (5)-(7)], as was remarked In [1] The constant prefactor in equation (8) becomes 16(4 + 29x/~/12)/X/6 when we Include the effects of the complex valence band The result of this corrected tunneling calculation is plotted (curve e) in Fig 1 It clearly overestimates K2 even more than the parabolic perturbation theory, even though a nonparabohc model was used This is because the energy difference Egv was m fact expanded to order k 2 during the calculation [5, 6], and this was done in an exponential factor Thus flus theory IS In effect a parabohc theory So far we have neglected the correction due to exciton effects Lee and Fan [4] showed that these enhance K 2 for trans~tlons from valence band v, by a factor [9]

gVex(a)

[ :

(1 + X ~

"trXvexp(rrXv)](Iv.o, J ~ - x n l ~ - ' ~ J~ Eb: :]' "1

(10>

Vol 39, No. 8

TWO-PHOTON ABSORPTION IN ZINCBLENDE SEMICONDUCTORS

939

Table 1 The two-photon absorption K2 m cm MW-1 from the nonparabohc theory for A ~ 0 [Equation (6)] and wzth exctton enhancement [Equanon (10)] compared with some experimental results, for several zmcblende semwonductors GaAs

InP

ZnTe

CdTe

ZnSe

h~p/Eg

0 82 0 003

0 87 0 004

0 52 0 004

0 78 0 003

0 69 0 008

Equation (6) x Equation (10) Experiment

0 0 0 0 0

0 030 0 044 0 21 b

0 0 0 0

0 0 0 0

0 0092 0 017 0 04 d

a Reference [ 10] b Reference [4]

024 034 23 b 015 e 030 ~

e Reference [11] d Reference [12]

w h e r e E ~ IS the exoton binding energy [10], Xv = [Ebv/(2h~ p Egv)] 1/2, and -

-

Jv = 2Y2v f ~ d t I[( l t i + ' t)/(1--t)] + ~ Yv

x exp -- 2Xv tan -I ~'~v ] | '

(11)

with Yv - [Ebv/(Egv -- N~op)]1/2 Evaluating Jv numerically gives enhancement factors gex(Ol) plotted m Fig 2 for various values of e -- Ebv/Eg b For a ~ 1/2 and e ~ 1 ,gex(a) ~ 2rr [e/(2a -- 1)] a/2 The enhanced K~ for InSb IS plotted (curve a) in Fig 1, using our nonparabohc result for A >>Eg In equation (5), multiplied by gex from equation (10) with Ebv ~-- 0 4 meV In these calculations, we have neglected the effect of higher bands wtuch contribute "allowed-allowed" transmons due to the lack of inversion symmetry Lee and Fan [4] showed that these are extremely weak effects for InSb, InP, GaAs and ZnTe There has been much variation m the reported experimental results (see [1 ]) Most of the results of [3] and [4] for the narrow gap materials as shown m Fig 1 are m reasonable agreement with our calculations For the larger-gap materials which have relatwely larger exclton binding energies we obtain from equation (6) for A ~ Eg the results shown m Table 1 Because these materials have larger exc~ton enhancements gex than InSb and Hgi-xCdxTe, it is not possible simply to scale these data for A ~ Eg to the results shown m Table I In Table 1 our results agree fmrly well with some of the experimental results, but our results are a factor of 2 - 5 smaller than some of the expenmental results As was discussed m [2], erroneous experimental results usually are too large We agree with Valdyanathan et al [1 ] that more accurate experiments are needed In summary, we have obtained expressions for the two-photon absorption including nonparabohclty and

0030 013 034 b 008 ~

034 076 13 e 025 f

e Reference [13] Reference [14]

exclton effects The frequency dependence shows that perturbation theory gives more accurate results than the tunneling theory m the nonparabohc region 2hoop > EK Our expressions, equations (4)-(6), with the exctton enhancement from equations (10) and (11), should give reasonable estimates of K2 for a large variety of zlncblende semiconductors, over a wide range of frequencies and energy gaps The most useful improvement to these results would be re-calculation of the exclton enhancement mcludmg nonparabohctty, rather than mcludlng the small effects of higher bands over large regions of k space as suggested in [1 ]

Acknowledgements - The author ts grateful to Dr D G Seller and Dr A T Futro for helpful comments on this work REFERENCES ! 2 3 4 5 6 7 8

9 10 11

A Valdyanathan, A H Guenther & S S Mltra, Phys Rev B22, 6480 (1980) C R Pldgeon, B S Wherrett, A M Johnston, J Dempsey & A Mlller, Phys Rev Lett 42, 1785 (1979) A M Johnston, C R Pldgeon & J Dempsey, Phys Rev B22, 825 (1980) CC L e e & H Y Fan,Phys Rev B9,3502(1974) L V Keldysh, Zh Eksp Teor Flz 47, 1945a (1964) [Soy Phys JETP20, 1307(1965)] Yu A Bychkov & A M Dykhne, Zh Eksp Teor Fzz 58, 1734 (1970) [Soy Phys JETP 31,928 (1970)] E O Kane, J Phys Chem Sohds 1,249 (1957) V T Nguyen, A R Strnad & Y Yafet, Phys Rev Lett 26, 1170 (1971) G D Mahan,Phys Rev 170,825 (1968) A Baldereschi & N O Lipari, Phys Rev B3,439 (1971) M Bass, EW V a n S t r y l a n d & A F Stewart,Appl Phys Lett 34, 142 (1979)

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TWO-PHOTON ABSORPTION IN ZINCBLENDE SEMICONDUCTORS V V Arsenev, V A Dneproslol, D N Klysho & A N Penln,Zh Eksp Teor Ftz 56,780 (1969) [Soy Phys JETP 29,413 (1969)]

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B Bosacchl, J S Bessey& F C Jam, J Appl Phys 49, 4609 (1978) J H Bechtel&WL Smlth, Phys Rev B13,3515 (1976)