The two-photon transition in indirect-band-gap semiconductors

J. Phys. Chum. Solih.

1972. Vol. 33. pp. 643-649.

Pergamon Presr.

Printed in Great Britain

THE TWO-PHOTON INDIRECT-BAND-GAP JICK Lawrence

Radiation

Laboratory.

(Received

University

24 March

TRANSITION IN SEMICONDUCTORS* H. YEE of California,

Livermore.

197 I : irl revisedform

I2 July

Calif.

94550.

U.S.A.

I97 I )

Abstract -The absorption coefficient of the two-photon excitation process in which phonons of the crystals participate during the transition process is calculated for ionic and covalent indirect-band-gap semiconductors. It is shown that this absorption coefficient can reduce to a form which can readily be used to estimate the absorption coefficient for a number of crystals if we use certain approximations. A numerical example was carried out for the GaP crystal. This calculation shows that both the optical phonon-assisted and the acoustical phonon-assisted two-photon absorption coefficients are within the same order of magnitude.

1. INTRODUCTION

optical transitions in semiconductors have been the subject of investigation for many years. both experimentally and theoretically[l-51. The investigation of this indirect process was carried out extensively in germanium and silicon[2.3]. With the recent advances in laser techniques which lead to the development of high intensity lasers. it is now possible to investigate the two-photon excitation process in which the phonons of the crystal participate in the transition. This two-photon indirect process, in which acoustic phonons at low temperature participate in the transition, was investigated by Ashkinadze[6] who also suspected that the two-photon indirect process may have taken place in their experiment when one of their GaP crystals was excited by a neodymium laser[7]. Because the direct transition between the conduction band and the valence band at k = 0 may also take place by threephoton transitions. their result is not very conclusive. In a covalent semiconductor, only the acoustic phonon participates in the twophoton indirect transition. In an ionic semiINDIRECT

*Work performed under Atomic Energy Commission.

the

auspices

of

the

U.S.

conductor (e.g., GaP, SIC and AISb), the optical phonon is also active and the effect of the optical phonon in the two-photon indirect transition may be also important. The purpose of this paper is to present the theoretical calculation of the two-photon indirect transition by considering both the effect of the optical and acoustical phonons in the twophoton indirect transition. Because of the complexity and the uncertainty of the energy bands of crystals, it is not possible to determine mathematical detail for any particular crystal or to obtain an exact equation for the absorption coefficient. However, if we assume that the only states involved in the transition are those states that are in the vicinity of the top of the valence band and the bottom of the conduction band. and if we use the assumption used by Bardeen et al. [l], the absorption coefficient for the two-photon indirect process reduces to a form which can be derived readily. 2. CALCULATION

Using Frohlich’s model for the interaction of electrons and optical phonons[8], the total Hamiltonian of an electron interacting with photons and phonons in an ionic crystal can be written as follows:

644

JICK

H.

YEE

w - hqv + E I,~- % >

S(-3

(I) where A, = AU. r)+ AU*. r) is the potential of the incident light and

vector

where w, is frequency of the optical phonons. The lowest-order total transition probability for the optical-phonon-assisted two-photon transition can be obtained from the thirdorder perturbation theory of quantum and it can be written in the mechanics, following form:

E [ A,%,- (fiw + hw,)] [E,,.,,, - slw,] qilr,. +c

,&,,A,,,.&#.,,

11.nt [ E A,“h,F - (hw + fiw,)] [ E,,.,/ - hw]

where II and tn are to be summed on all valences and all conduction bands. The notation kj” designates the quantum states of an electron in the band ‘II’ with a crystal momentum e.g. kj” = (II. kj). The first six terms (3) correspond to the absorption of the two photons and a phonon. and the last six terms correspond to the absorption of two photons with the emission of a phonon. The Feynman diagrams (Fig. I) show this process. The dotted lines in Fig. I represent the absorption or emission of phonons (depending on whether the arrows point toward UI away from the solid lines). and the wavy lines represent the absorption of photons. The matrix elements appearing in (3) can be written as follows[9. IO]: ‘kj.’

2 A,?,” = g (a . Akj)(j = i orf‘). I-

Fig.

I.

Feynman

diagrams

of

the

two-photon

indirect

excitation

process

tn

= c

(4)

THE

= .a-,,m,,,

TWO-PHOTON

&

(a.

hkj)(j

TRANSITION

= i orf),

IN

INDIRECT-BAND-GAP

SEMICONDUCTORS

-- eAh m,.c [E,-~~,,',,,,,,.,1

m = u

I,

645

1 (E,/-E,,,-t+,)

(5)

,g.,, = +ifi [tl,-,, 1112 h,“h,“’ Ikf-kil

(6)

(9)

eAh

1 m,.c ( E,/-EL,.-Yiw I

(7) After substituting the matrix elements given in equations (4) through (7) into equation (3) and changing the summation for the final and initial states to integrals. it can be shown that the transition probability W, can be written as follows:

> iiw

T=

[EL,, -E,,,

-Xw][E,,,

-E,,r -fiw]

1

(10)

X(hw)[~,i-~,;-(hw+fiw,)]

If we replace w, by (-w,$). %&l,F by :2”&L,r . and i/I’.1,1,, by XC-k,, . in (9) and ( IO). we obtain gl- and x2-. respectively. The transition probability W, given in Equation (8) is very difficult to evaluate. Therefore, we will consider only the case when the states that are involved in the absorption process are those that are in the vicinity of the conduction band around k = k, (see Fig. 2). For this case. we can use the assumptions which were used by Bardeen et al. [ 11. That is, we assume that the matrix elements in the g’s are constant (independent of k) and make the following approximation: (1 I)

(E,,r - El,. j 2 Eh,c - E,;

(E~,.-E,,~s+) .2&y

= E,orE,,,+fiws

= x&k,m z 0

n#m

(12) (13)

646

JICK

H.

YEE

EC -EC k kF 0 F

x

nkp(2hw+ fiw,-- E,j3

+ (+

l&-l*+%

Is*-12)

x (nh, + 1)(2hw - hw, - EJ3

(16)

where LE;:

Fig.

2.

Simple

energy

band diagram calculation.

x [+!$

1

-E" 0

kF

used

(L&J.)]“*

in

(Ek,. - E,“v T hq‘)

present

(14)

and

+

1 (Ekp.--c--fiw)

+L

1

1

mc EL -EL,< *fro,) (

=

(efiw,lhT-

(Eke,- E,,+fiw)

(18)

I nF

1

1)

=

n k,-k,

The meaning of the E’s and the k, are given in Fig. 2. After substituting the matrix elements given in (13) through (15) and the equations in (11) and (12) into (8) and using the formula for the two-photon absorption coefficient a0 = (2hwwJl). we obtain the optical phononassisted two-photon absorption process as follows:

Now we consider the participation of the acoustic phonon in the two-photon indirect transition process. For the interaction of the electrons with the acoustic phonon, we use the deformation potential approximation for the interaction Hamiltonian[ 111. H! = ~%nt(fi)~‘* ‘” (2Vp)“Z cr,e+lk‘r - akte --A.r . (19) ( > The total Hamiltonian for the acoustic case can be obtained by replacing the second term in (1) with that given in ( 19). The matrix elements for the interaction of

’

THE

TWO-PHOTON

TRANSITION

IN

INDIRECT-BAND-GAP

the electron with the acoustic phonon can be written as follows [ 1 I]:

SEMICONDUCTORS

+ l&l 2(+)

x Ed,,, (n,+ $pJ

=

Iki-k

)‘I2 (m = c, u)

(20)

1 (El,,-

(&l/L

xE d?U(n k,-k, + p2

(m = c,u).

(21)

Again. if we assume that the states which are involved in the absorption process are those states that are in the vicinity of the top of the valence band and the bottom of the conduction band, we can make the same approximation for the terms (E,,,-Ekf) and (E,,,-- E, & hw,) as that given in ( 1 I) and (12) and make the following approximation for the matrix elements given in (20) and (2 I): %I( k,"k,"

=

kjk.-

=

l&l

@I

(,)‘I2

i -&in, w,l/2

(2vp)l/2

(,)‘I2

(2Vp)“”

(24)

I

where the g’s are:

l(h)“‘i

(2&))'!2

l&l

1)

I&,IPl(n,+

x (2hw - hw, - EJ3

647

(22)

(nkp)112

i

-(&l/2

&,n (Q,

+ 1)‘12. (23)

Using the approximations in ( 11) and (I 2) together with the third-order perturbation theory, it can be shown that the acoustictwo-photon absorption phonon-assisted, process is as follows:

x([k’1’(+) IEd’

+dc

Ekon’ ho,)

I

E

Edc ( E,~,- Ebb<+h) I

-- I

1

+&

1 E - Eke,- hJ) (

&c

+J-

1

k,’

1

1

1

mc ( EG - E,; 2 fi%)

( Ek<- E,,,+ fh) I (26)

where Edc and Ed,, are the deformation pOtential for the conduction and the valence bands respectively, w, is the acoustic phonon frequency, and p is the mass density of the crystal. As an order-of-magnitude calculation, we use GaP as our example and assume that the excitation source is a neodymium laser (hu = 1.17 eV). The other parameters which we use for the calculation are as follows: Ekpc- Ekes= - 3 eV (Ref. [12])

hw = 1.17 eV

Ee-EEkkFC=0.5eV(Ref.[13]) + 1~2+12(9)

I&I’]

p = 4.13 (g/cm3)

nkJ2fiw+hw,-EJ3

v, = 2 X 105 cmlsec + [ Ii?,-1’ (+)

IEd

mc = 0.34 m

Edc = 6 ev (Ref. [14])

JICK

648

ml = 0.75 m

fiw, = 0.05

Ed,, = I2 eV (Ref. [ 151)

eV

](c.O](~.p]v.O)]* tn*

I = 50MW/cm* =--3 E,

4m,

E.. = 2.23 eV.

m

Substituting the parameters given above into equations (16) and (241, we obtain the optical phonon-assisted and acoustic-assisted twophoton absorption coefficients respectively as follows: CX,,(optical phonon case) = 2 X 10e6/cm (Y, (acoustic phonon case) = 1 X IO-‘/cm. 3. DISCUSSION

AND

CONCLUSIONS

The absorption coefficient for the twophoton transition in which the phonons participate in the transition is theoretically calculated. Because of the complexity of the energy band of the crystal. we have made several assumptions to simplify the calculation. It is of interest to mention that the factor (2hw?hw,E,) that appears in (16) and (24) has the same exponent as in the forbidden single-photon indirect transition. e.g.. the matrix elements for the vertical transition at k = 0 or at k = kF are not allowed[l]. The matrix elements for the vertical transition are zero at the two momentum points just mentioned. and for the two-photon indirect forbidden transition, the factor (2hw & hw, E,) will have an exponent of 4. We have neglected the terms that appeared in (3) containing matrix elements that induced inter-band transition by phonons. When the bottom of the lowest conduction band is closed to the edge of the Brillioun Zone, these terms become zero. Hence for semiconductors like Ge, GaP and Si[ 161. this approximation should be a good one. From the numerical calculation which we made on the GaP crystal, it seems that. within the approximation used in this paper,

H. YEE

the effect of the acoustic phonon in assisting the two-photon indirect transition is more important than the optical phonon. At 50-MW laser intensity, the three-photon absorption coefficient for the direct-transition at k = 0 is in the same order of magnitude ( 1Om5/cm) as that of the two-photon indirect transition, and the two absorption processes may be in competition with each other. Since the absorption coefficient for the two-photon indirect transition is linearly proportional to the light intensity and the three-photon absorption coefficient is proportional to the square of the light intensity. a measurement of the absorption coefficient versus excitation intensity may determine which absorption process is dominant. Lt is of interest to point out that phononassisted two-photon absorption coefficients which we just calculated for the GaP crystal is much smaller than that of the absorption coefficient for the direct-two-photon transition around X-= 0 (see Fig. I) when the crystal was excited by a ruby laser (hv = I.8 eV) at the power intensity of 50 MW/cm”. At this power intensity. the direct-twophoton absorption coefficient is about O.l/ cm.[7]. The reason why the direct-twophoton absorption coefficient is much larger than the phonon-assisted two-photon absorption coefficient is because the former is a second-order optical absorption process and the latter is a third-order absorption process. When the GaP crystal is excited by a neodymium laser (hu = 1.17 eV). there is no direct-two-photon absorption around k = 0 because the band gap at k = 0 is more than twice the neodymium laser energy. REFERENCES I. BARDEENJ., BLATT F. J. and HALL L. H.. Proc. of Atlantic City Phorocotldlcciirif~ Conference. p. 146, John Wiley. New York and Chapman & Hall. London ( 1956). 2. MACHARLANE G. G.. MCLEAN T. P.. QUARRINGTON J. E. and ROBERTS V.. Phys. Rec. 108. I377 ( 1957). 3. MACHARLANE G. G.. MCLEAN T. P., QUAR-

THE

TWO-PHOTON

TRANSITION

IN

INDIRECT-BAND-GAP

RINGTON J. E. and ROBERTS V.. Phvs. Rec. 111. 1245 (19.58). 4. MCLEAN T. P.. T11e Absorprim Edge Spectrum of Semicondrcc.lors. in Progress irt Semicotidurrors. p. 53. (Edited by A. F. Gibson, R. E. Burgess and F. A. Kriieer) John Wilev. New York ( 1960). FAN H. f..Rep. Progr.~19. 107 (1956). ASHKINADZE B. M.. BOBRYSHEVA A. I., VITIU E. V.. KOVARSKII V. A.. LELYAKOV A. V.. MOSKALENKO S. A., PYSHKIN S. L. and RADAUTSAN S. I.. Proc. Ninth Intern. Conf. tm Pll.vsics of Sernico/ldrrcrors [in Russian]. Moscow ( 1968). ASHKINADZE B. M., KRETSU I. P.. PYSHKIN S. L. and YAROSHETSKII I. D.. Fiz. Tekh. Poluproc. 2, 15 I I ( 1968) (Societ Phys.-Semicond. 2. 1261 (1969)).

x. 9. IO. I I. 12. 13. 14. 15. 16.

SEMICONDUCTORS

649

FReHLlCH H., Proc. R. Sm. Land. AM. 230 (1937). KARPLUS R. and LUTTINGER J. M.. Phys. Rec. 95. Il54(1954). EHRENREICH H.. J. Phys. Chem. Solids 2. 131 (1957). BARDEEN J. and SHOCKLEY W.. Phys. Rec. 80.72 ( 1950). ZALLEN R. and PAUL i.. Phys. Rec. 134. Al628 (1964). DEAN P. J.. KAMINSKY C. and ZETTERSTROM R. B.. J. oppl. Phys. 38,355 I ( 1967). EPSTEIN A. S.. J. Phys. Chem. So/ids 27. I61 I ( 1966). BALSEV I.,J. plrys. Sot. Jupm 22. IO1 (1966). COHEN M. B. and BERGSTRESSER T. K.. Phys. Reu. 141.789 ( 1966).