circular dichroism of two-photon absorption in zinc-blende semiconductors

Februari 1994

~TUCAL ELSEVIER

Optical Matenals 3 (1994) 53—60

Linear/circular dichroism of two-photon absorption in zinc-blende semiconductors D.C. Hutchings Department ofElectronics and ElectricalEngineering, University ofGlasgow, Glasgow G12 8QQ, UK

B.S. Wherrett Department ofPhysics, Heriot- Watt University, Edinburgh EHI4 4AS, UK Received 19 May 1993; revised manuscript received 20 August 1993

Abstract The spectral dependence of linear/circular dichroism in two-photon absorption is calculated for several zinc-blende semiconductors, using the four-band Kane bandsiructure model. It is demonstrated that the incremental dichroism is highly sensitive to the details ofthe chosen band model, and particularly that the split-offvalence band can have a major influence on its value.

1. Introduction

—

across the fundamental band gap in semiconductors, mainly concerned with the determination of flL(w) for a single, plane-polarised beam. The more successful of these treatments (compared to reliable experimental data) are based on a second-order perturbative method, using either a three-band (conduction, heavy-hole and light-hole) or a four-band (conduction, heavy-hole, light-hole and split-off) isotropic model forthe semiconductor bandstructure [4—7].Few studies consider alternative two-photon absorption configurations, although these can potentially provide additional information because different nonlinear susceptibility tensor elements can be accessed. In particular the linear-circular dichroism could prove a useful parameter because it is less sensitive to the experimental conditions than absolute fi measurements have proven to be. Existing two-photon absorption calculations for

There have been various theoretical treatments of the two-photon absorption associated with excita-

circularly polarised light have eitheremployed a twoband model [8,9] or determined the linear/circular dichroism solely at the two-photon band edge

Interest in semiconductor components as nonlinear elements in optical communication and information processing systems has renewed concern over the role of two-photon absorption. Not only can the presence of strong two-photon absorption be a signal for the onset of damage, but also the mechanism provides a fundamental limitation to the achievement of all-optical switching at photon energies below the band edge [1]. It has also been recognised recently that the nonlinear refraction at such photon energies can in turn be related, through a nonlinear Kramers— Kronig relation [2,3], to the two-photon absorption spectrum itself. We address here one aspect of twophoton absorption, namely its polarisation dependence, as exhibited by the linear/circular dichroism the ratio of the coefficients for linearly and circularly polarised radiation (flL//3c)

tion

0925-3467/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD10925-3467(93)E0030-F

54

D.C. Hutchings, B.S. Wherrett / Optical Materials 3 (1994) 53—60

(2hw=Eg) [10]. The principal aim of this paper is to provide a comparison between the two-photon absorption coefficients obtained with circularly and linearly polarised light, for an isotropic Kane bandstructure model that includes the effect of nonparabolicity and the use of exact electronic wavefunctions. In particular the frequency dependence of the linear/circular dichroism is investigated and bandstructure-dependent features are highlighted by examining individual contributions to the third-order susceptibility tensor elements. A more complete treatment oftwo-photon absorption would include the effects of other bands, pre-

dominantly the next higher conduction band. In particular, mixing of these bands into the valence and conduction band produces anisotropy due to momentum matrix-element changes and effective-mass warping. Dykman and Rubo [111 calculate the effect of valence band warping alone on the anisotropy of two-photon absorption in InSb and GaAs, including a calculation of linear/circular dichroism. However, since the dichroism is not specifically due to the anisotropic nature of the crystal, it is of interest first to investigate it using the simpler isotropic band structure model. On comparison with the results of ref. [11] it is noted that linear/circular dichroism in twophoton absorption shows little dependence on that anisotropy which originates from valence-band warping alone,

2. Theory The first-order susceptibility for cubic materials has only diagonal non-zero tensor elements, all of which are equal. Hence the linear optic properties show no linear/circular dichroism or anisotropic effects. Dichroism arises in higher order nonlinear optical processes, however. For the zinc-blende structure (cubic symmetry class 43m), out of the 81 third-order susceptibility tensor elements 21 are nonzero and only

x~~( —w,

w, w)

=x~~( —w, w, w)

(1)

.

The two-photon absorption coefficient is directly related to the imaginary part of the third-order susceptibility tensor. For zinc-blende materials, this relation can be expressed as [1 31, /3(w) = 3w 2 2 ~0n~c ~ 2+ ~

x

(

~.

+

~

~

4)

.

(2)

Here n 0 is the linear refractive index and ê is a unit vector in the polarisation direction (e.g. for propagation along the z-axis, linear polarised light along the x-axis corresponds to ê=x and circularly polarised light to é= (~± ij5)/~J~). For linear polarisation, Ié é I = 1 and for circular polarisation I êê I = 0. ~ is used to represent Im~~5( —w, w, w) etc. The twophoton absorption anisotropy parameter i is given by 7= X~xxx—

~

~

~

(3)

X~xxx

In this paper the band structure described by Kane [14,15] is employed; this consists of four doubly-degenerate bands (one conduction and three valence) as shown in fig. 1, obtained by including kp and spinorbit interaction terms. Although the initial basis functions have cubic symmetry, this band structure model is isotropic; the band energies are independent of direction in k-space. To include anisotropy the influence of higher bands must be incorporated 3) element equal the summedium of the three off-diagonal [13,16]. isFor an to isotropic the diagonal~( elements (a=0). Hence our model will produce just two, independent tensor elementsX~~~~ andxic~~~~. The ratio of the two-photon absorption coefficients is then given from eq. (2) by /

3L/pc ~ (4) It is also useful to express an incremental dichroism,

(3)

four of these are independent [12], ~ ~ and Furthermore, by restricting our analtion frequency is present), intrinsiconly permutation ysis to the degenerate regime (where one radiasymmetry reduces the number of independent elements to three because: ~

~

in the isotropic limit, as 1~_p C\ ~ //3 fl1~ ,,,,

the point being that ~

proves to be the tensor ele-

D.C. Hutchings, B.S. Wherrett / Optical Materials 3(1994)53—60

ton energies is equal to the energy difference between a valence band (occupied) and a conduction band (unoccupied). Thus in eq. (6), /3 refers to a conduction band state; also w2=w3=w (and w1 = —w). This reduces the number of ST permutations from 24 to four (with identical denominators); these are shown diagrammatically in fig. 2. Note that as this is a resonant situation, a damping term F should be ineluded in the resonant denominator. However, above the band edge itself F may be taken to be vanishingly small, allowing the substitution:

E

~Jcb

Eg k \~i~

~

I\

Qpg_2W 1

lh

Fig. I. Band structure for a zinc-blende semiconductor near the centre ofthe Brillouin zone as given by Kane. In the calculations performed in this paper only the set of four doubly degenerate (in spin) bands are considered: conduction band (cb), heavyhole valence band (hh), light-hole valence band (lh) and splitoff valence band (so).

ment most sensitive to details of the bandstructure. From a density-matrix treatment, the third-order susceptibility is in general [12]: xW(wi,w 2,w3) e4 1 3!h3~ 0m~(w1+w2+w3)w1w2w3 XST g,a,p,y ~

(~?agwIc02w3) (ê~Pga)(êji’ap)

(~kPpy)(~1’P~g)

55

=

__

Qflg2wiF

—4

—

2w

Q~g 2w)

+i7t~5(

—

(7)

.

Two-photon absorption is manifested by the delta function term in eq. (7), while the real part gives rise to an associated nonlinear refraction [3]. On evaluating eq. (6) for two-photon absorption, some simplification is possible by factorising the terms: 4 2ire 3m~w4 XXXXX~ 30h 2

~ (~p~)(~p~) I >< v,c XXYXY=

(6)

X

Q,~—w 4 3 2ire 3m~w4 0h 2 ç-~~ (j•p.) 2(Q~~—w) + (fi~p~) (~~iv) I I

where m 0 is the free electron mass, ê, is the unit vector in the direction ofthe ith polarisation and Pafi and hQpa are the momentum matrix element and energy difference respectively, taken between the states a and /3. Here ST denotes that the expression which follows it is to be summed over all 24 permutations of the pairs (1, w1 + w2+ w3), (j, w1), (k, w2) and (I, w3) that can be generated by the diagrammatic technique of Ward [17]. Ideally the sum should be performed over all electron states, a, /3, y and over all occupied electron states g. In practice these summations are taken over a limited number of states which are deemed to be dominant. Two-photon absorption is a resonant effect which occurs in semiconductors when the sum of two pho-

X cl (Q~ 2w), —

I

k

0)

I

~

I 0)

~~js~)/~”

~

(0

I -0)

Ic

I 0)

0)

~

J

-0)

k

I

~/~//1

~

Ic

0)

0)

I

0)

-(0

~

Fig. 2. Diagrammatic representation ofthe four transition schemes which contribute to two-photon absorption [171.

56

4 3e 2ire 3m~w4 0h

XXXYY=

x[

[

D.C. Hutchings, B.S. Wherrett / Optical Materials 3(1994)53—60

1*]

(~ •p~~) (~ ~

Q~—w

A >> Eg), and Weiler [5], who included nonparabol-

icity and exact wavefunctions (for the limits A>>Eg and AUZ
~ ~

0—2w)

(8)

.

—0)

Here the summations are to be performed over conduction band states c (empty), valence band states v (filled) and all states i (empty or filled). These will, in general, involve a summation over bands and an integration over k-space. The first two imaginary susceptibilities, X~xxxand ~ can also be derived by a transition-rate approach [7]. The susceptibility term ~ however, cannot be obtained directly from a transition-rate approach although its form is somewhat similar to the other terms. It should be noted that ~ can never be isolated experimentally, always occurring in combination with at least one of or X~’yxy~ Calculation of the susceptibilities in eq. (8) involves the intermediate process of determining the four summations, (~Pci)(I~~)

(j~p~1) ~

w

—

V

~ (i~p~) (i~p~0) Q1~—w Q~—w (9) followed by a summation over conduction and valence states. This can be simplified further because the calculation photon wavevector is negligible compared to the electron wavevector and hence only electron states with the same wavevector are coupled. Hence, in the summation over states there will only be one integration of k-space. The Kane band-structure model has been used with some success under various degrees of approximation in computing two-photon absorption for linearly polarised light. The momentum matrix element pairs in eqs. (8), (9) consists of one “allowed” matrix element, between a conduction and valence band, and one “forbidden” matrix element, between a band and itself or between two valence bands. The latter for-

‘r’

ref. [7], in the present work the

ô(Q~

j

—

As in

‘

~

‘

bidden terms are identically zero at the zone centre. Lee and Fan [4] used this model for parabolic bands with zone-centre 2wavefunctions (i.e. energies and wavefunctions to order cxk). panded to order k Such a calculation was extended by Pidgeon et al. [18], who included nonparabolicity (in the limit

k•p perturbation matrix is diagonalised numerically without resorting to any of the above limited expansions in either energy or wavefunction. The limiting cases used in ref. [5] cannot be employed as most of the semiconductors examined here have bandgaps and split-off energies of comparable magnitudes.

3. Results It has been shown previously that the results of the four-band model compare favourably with discrete wavelength measurements of the degenerate, linearpolarisation two-photon absorption coefficients in both narrow and wide gap materials [7]. Good agreement has also been obtained for the spectral dependence of two-photon absorption in the range 0.5
8 (eV) 0.1 75 0.36 1.42 1.44

A (eV) 0.85 0.38 0.34 0.91

E~(eV) 21.3 21.5 25.7 20.7

ZnTe ZnSe

2.26 2.67

0.92 0.42

19.1 24.2

_____________________________________________

D.C. Hutchings, B.S. Wherrets / Optical Materials 3(1994) 53—60

confirms the isotropic nature of the band structure model, In InSb (fig. 3) the split-off energy is much greater than the fundamental band-gap and so transitions from the split-offband occur at energies beyond the one-photon absorption edge. However, transitions for which the split-offband is the intermediate level are still included in this calculation and make a small contribution to the overall two-photon absorption [7]. For linear polarised light ~ transitions from the light-hole band contribute slightly more than those from the heavy-hole band. This is in contrast to one-photon absorption, where transitions from the heavy-hole band dominate due to the larger densityInSb 1’

4x10

Total

3x10

XXXX

2x105

~

-15

~

lxi 0

0 2x105

I I

I

I

Ih

XXXX ~

/

“

lxi 015

~-

hh

0

I I

XYXY

-is

I

—

I

I

hh

lxi 0 05

Ih

10~

x

~

0

I

ixiO15

I

I

—

—

— —

~ ,‘

0 -lxlO15

.—

~

008

.

0.10

012

0.14

0.16

Fig. 3. Calculated imaginary part ofthe third-order susceptibility as a function of frequency, for InSb. The top graph shows the total contribution to the susceptibility tensor elements w, w), Img~(—w, w, o) and Img~~(—w,w, o). The othergraphs show the separate contributions arising from transitions originating from the heavy-hole and light-hole valence bands.

57

of-states near the centre of the Brillouin zone, and occurs because the “forbidden” matrix-element is zero at the zone-centre. However, for circularly p0larised light ~ the contribution from the heavyhole band dominates. This difference between the contributions to the third-order susceptibilities from transition originating from the heavy-hole and light-hole valence bands has its origin in the orientation of the momentum vector matrix-element. In the case of the matrix-element taken between the heavy-hole and conduction bands, the momentum vector matrix-element is perpendicular to the electronic wavevector k. However, in the light-holecase the two vectors are more closely aligned. Another consequence arising from this relative orientation is that the susceptibility term ~ is negative for the heavy-hole band. Note that this does not correspond to two-photon gain as this tensor dcment cannot be related to an interband transition and always occurs in a combination with other (positive) elements. Since the contribution to the same tensor element from the light-hole is positive, there is a partial cancellation in the overall x~ from the two bands. This results in a weak dichroism that is however sensitive to the details of the cancellation. In the cases of GaAs and ZnSe (Figs. 4, 5), A
.

to be positive (/3 > /3~),although this is not necessarily true for the individual band contribulions (in particular the transitions originating from found

58

D.C. Hutchings, B.S. Wherrett / Optical Materials 3 (1994) 53—60

GaAs

ZnSe

::~:: :~ i--~~~TT~ Total ~xxx~

4 1

~

~

~0

‘ih

~XXXX

\

810:

Total

~L:

________

-20

0

/h/:?~

:::~: 9

X~jX~/

xxyy

2x10’

-2x10’9

//

Ih

~

...-

\,./

2x102°

-

xxyy

~

hh

~,/

so

.2x1020~~~

~

Photon Energy (eV) Fig. 4. Calculated imaginary part ofthe third-order susceptibility as a function offrequency, for GaAs. The lower graphs show the separate contributions arising from transitions originating from each of thethree valence bands.

the heavy-hole band). A very noticeable feature in fig. 6 is the cusp, giving a minimum / 3L/flc~1.08 in all of the materials studied at the onset oftransitions from the split-off band. The increasing incremental dichroism for hw>A reflects the small split-off contribution to In the case of the two-band model the linear/circular dichroism can be determined analytically and an approximate result was obtained for narrow-gap semiconductors by simply summing independent contributions from heavy-hole and light-hole valence bands [9]. The results for InSb are similar to ours but deviate both at high photon-energies, where the split-off band is important, and at the two-photon band edge (2hw~E5),where the simpler model predicts a negative incremental dichroism. Our results ~

0

-.~

Photon Energy (eV)

Fig. 5. Calculated imaginary part of the third-order susceptibility as a function of frequency, for ZnSe.

at the two-photon edge agree with the calculations of ref. [10]; the present calculations, however, emphasise the significant dispersion of the dichroism at higher photon energies. Dykman and Rubo [11] have performed a calculation of the contribution of the valence band warping to the anisotropy of two-photon absorption for InSb and GaAs. The coefficients calculated in this reference canbe used to determine linear/circular dichroism. In both materials the predicted dichroism values are similar to ours, even producing the minimum at the split-off threshold for GaAs. It is interesting to note that in this same reference, the authors also calculate the dichroism for InSb using an isotropic band model neglecting the split-off band; the results do not agree well with our present result. In factthe neglect of the split-offband gives a dichroism of less than unity in the same manner as in ref. [9].

D.C. Hutchings, B.S. Wherrett / Optical Materials 3(1994)53—60 1.8

4. Conclusions

-

1~

t

ZoSe GaAs

ZnTe

CdTe .

1 6

CdTe

~-

ZnTe

GaAs ZnSe

1.2

~

~

\

1.0

0.5

59

-~

0.6

/

‘~

,“

.~..

-

~

,~‘

in~~

/

,‘

InAs

/ ~

“~..,.

0.7

,~‘

,“

,~“

,‘

0.8

~--

0.9

1.0

Fig. 6. Calculated two-photon absorption linear/circular dichroism as a function of the ratio of the photon-energy to band-gap for the various Ill—V and Il—VI semiconductors, as indicated. The upper arrows indicate the positions of the thresholds for transitions from the split-off band (2hW=E~+z1).

It can therefore be concluded that in narrow gap semiconductors, the split-off band is important in the determination of dichroism in two-photon absorption, even though it may be well removed in energy. However, our comparison with [11] does seem to indicate that the valence band warping is not a very significant factor, Although experimental determinations of twophoton absorption dichroism in semiconductors have been limited, the predictions of the four-band model are found to be in excellent agreement. In InSb at low temperatures there was found to be insignificant dichroism at 10.5 ~.tm (at the two-photon band edge) but a dichroism of around 1.1 at 9.5 ~im [21] (1.1 predicted). In wider-gap semiconductors, the dichroism was determined to be 1.25 in ZnSe at 700 nm [22] (1.20 predicted) and 1.19 in ZnTe at 1.06 l.tm [23] (1. 1 7 predicted). Note that this agreement is in spite of the factthat the perturbative models underestimate the absolute values by a factor of threeto-four close to the two-photon band edge [6,7]. This difference for ZnTe has been attributed to Coulombic effects or allowed-allowed transitions, which both become important for 2/lw near the band edge, but apparently has little influence on the dichroism.

In this paper the spectral dependence of the linear/ circular dichroism in several zinc-blende semiconductors has been determined using the isotropic Kane four-band model for the semiconductor bandstructure. To investigate fully the resulting features, the individual band contributions to the imaginary thirdorder susceptibility tensor elements are identified. It is noted that the difference in the two-photon absorption for linear and circular polarised light is proportional to the third-order susceptibility tensor element ~ The calculations in this paper indicate that the contribution to ~ associated with the heavy-holes is negative, which results in a partial cancellation with .

.

.

the light-hole term, and hence this tensor element is very sensitive to the precise model applied. Transilions via the split-off band have a noticeable influence, even in small gap semiconductors and for larger gap materials the influence of the split-offband is very significant due to the above partial cancellation. The effect of split-off band transitions on the linear/circular dichroism is further enhanced because they make little contribution to the susceptibility tensor element ~ which determines the two-photon absorption coefficient for circularly polarised light. This results in a very pronounced feature in the spectral dependence of the dichroism, consisting of a minimum value of/3L/flc~1.08 at the onset of split-off band transitions (2hw=Eg+4), followed by an increase at higher frequencies. Hence we conclude that the spectral dependence of two-photon absorption linear/circular dichroism provides a possible means for probing the split-off band. The calculations presented here are in excellent agreement with the urnited number ofexperimental results that are available.

Acknowledgements D.C. Hutchings gratefully acknowledges support through a Royal Society of Edinburgh/Scottish Offlee Education Department personal research fellowship. We also thank the Nonlinear Optics group at the Centre for Research in Electro-Optics and Lasers (CREOL) for the use of their ZnTe data prior to publication.

60

D.C. Hutchings, B.S. Wherrett /Optical Materials 3(1994)53—60

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[131 M.D. Dvorak, W.A. Schroeder, D.R. Anderson, A.L. Smirl and B.S. Wherrett, IEEE J. Quantum Electron., in press. [14] E.O. Kane, J. Phys. Chem. Solids 1(1957) 249. [151E.O. Kane, in: Lecture Notes in Physics: Narrow Gap Semiconductors Physics and Applications, Vol. 133, ed. W. Zawadzki (Springer-Verlag, New York, 1980). 13. [16] P. Pfeffer and W. Zawadzki, Phys. Rev. B 41(1990) 1561. [17] J.F. Ward, Rev. Mod. Phys. 37 (1965) 1. [18] CR. Pidgeon, B.S. Wherreti, AM. Jonston, J. Dempsey and A. Miller, Phys. Rev. Lett. 42 (1979)1785. [19] B.N. Murdin, C.R. Pidgeon, AK. Kar, D.A. Jaroszynski, J.M. Ortega, R. Prazeres, F. Glotinand D.C. Hutchings, Opt. Mater.Hellwege, 2 (1993)ed., 89. Landolt-Börstein Numerical Data and [201 K-H. Functional Relationships in Science and Technology, Vols. 17a, l7b, Group III (Springer, Berlin, 1982). [21]A.M. Danishevskii, E.L. Ivchenko, S.F. Kochegarov and MI. Stepanova,Sov.Phys.JETPLett. 16(1972)440. [22] J.A. Bolger, PhD Thesis, Heriot-Watt University (1992), unpublished. [23]R.DeSalvo, M. Sheik-Bahae, A.A. Said, D.J. Hagan and E.W. Van Stryland, CREOL, University of Central Florida, Orlando, U.S.A. (private communication).