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Local mode sideband IR absorption and second harmonic intensity calculation for various isolated impurities in GaAs
I Phys. Chem. Solids, 1977, Vol. 38, pp. 1137-1144.
Pergamon Press
Printed in Great Britain
LOCAL MODE SIDEBAND IR ABSORPTION AND SECOND HARMONIC INTENSITY CALCULATION FOR VARIOUS ISOLATED IMPURITIES IN GaAs
Istituto
L. BELLOMONTE di Fisica and Gruppo Nazionale Struttura della Materia University of Palermo, Palermo 90123, Italy (Received 3 August
1916; accepted in revised form II March 1977)
Abstract-We have calculated the intensity and shape of the local mode sidebands and the second harmonic intensity for various isolated impurities in GaAs. To calculate the absorption we have considered the second order dipole moment in the harmonic approximation. The absorption depends on the values of the impurity charge and polarizability, on the corresponding quantities for the nearest neighbours of each impurity and on a charge migration factor. The latter accounts for charge variations during the motion. Although these quantities vary from impurity to impurity, the results are presented in a general form valid also for other impurities. The point group symmetry properties are used in order to give a physical meaning to the results. The effects on the dynamical properties from the mass defect and the local changes in the force constant are also taken into account. 1.INTRODUCTION
In a previous paper[l], to be referred to as I, we have calculated the local mode frequency and the infrared absorption of various isolated impurities in GaAs. In dealing with this topic we have used symmetry considerations in order to break up the problem according to the irreducible representations of the crystal point group under which the model is invariant. To calculate the local mode frequency we have taken into account the effects from the mass defect and local changes in the force constants, whereas in calculating the absorption intensity we have considered the contributions to the dipole moment from the motion of the impurity and its nearest neighbours and from charge migration effects. In this work we are discussing the origin of the second order spectrum and calculating the frequency dependent intensity and shape of the local mode upper sidebands which correspond to the simultaneous excitation of a local mode with T,-symmetry and any perturbed band phonon with A,-symmetry. We consider this contribution to the second order spectrum as the most relevant one and work it out in the framework of the second order dipole moment in the harmonic approximation, as discussed in the following section. According to our model the intensity depends upon five free parameters which correspond to the impurity charge and polarizability e, and ai, respectively, to the corresponding quantities for each nearest neighbour II of the impurity: e. and a., and to a charge migration factor K. The physical properties and the numerical values of these quantities were discussed in I, and no further assumptions are here required. The spectra were calculated for various impurity states. They fall roughly in the frequency range between 500 and 8OOcm-’ in which the experimental information is scarce or missing. Consequently we are able to compare our results with experiment only for 9idp, 9ias and “Bqs in GaAs (see Ref. [2]). The agreement is in this case good. 2.THEMODEL The second order spectrum in crystals with isolated substitutional or interstitial impurites usually consists of
sidebands to the local- or gap-modes, local- or gap-mode second harmonic, and two-band phonon absorptions. The experimental and theoretical aspects of these types of absorption have been exploited by several authors mainly in the case of U-centres in alkali halides or alkaline earth fluorides[3-121. According to their results the mechanisms which account for the sidebands in these crystals are of different types such as anharmonicity, second order dipole moment or a combination of both. Usually the anharmonicity is considered as predominant with respect to the second order dipole moment. Sidebands or second harmonic absorption have also been detected in boron doped, lithium compensated silicon[l3,14], and in III-V compounds[4,15]. There is no full theoretical investigation of these spectra except for a calculation of the combination tones in boron doped silicon in terms of the second order dipole moment [ 161.These different conclusions are justified by the behaviour of the semiconductors with respect to the alkali halides. Intrinsic semiconductors exhibit strong two-phonon absorption[l7] interpreted in terms of the second order dipole moment[ll] in the harmonic approximation. Analogously the experimental data regarding one- and two-phonon absorption in doped semiconductors lead us to the conclusion that the anharmonicity plays a minor role with respect to the second order dipole moment. In fact the various effects caused by anharmonicity such as the temperature dependence of width and position of the local mode resonance and the shit of the second harmonic frequency with respect to twice the local mode frequency are quite appreciable in the spectra of U-centres and negligible in doped semiconductors. This point is supported by comparing the experimental data reported in Ref. [12] with those collected in Ref. [4], pp. % and 156. In our model we consider two effects which account for the origin of the second order dipole moment. The hrst one is related to the charge migration with displacement. Let us consider the contributions from the motion of the charges to the a-component of the dipole moment:
1137
(1)
L. BELLOMONTE
1138
which has T, symmetry at first order, and e. stands for the net charge inside a polyhedron surrounding the impurity or another ion at site n. A uniform compression of the tetrahedron surrounding the impurity, i.e. a displacement of its nearest neighbours according to the A, irreducible representation of the point group, will cause some amount of charge migration so that the charges inside the polyhedra are now different from those in the undistorted position. Consequently when the impurity and its neighbours move according to a T, mode the dipole moment is affected by this charge migration effect, i.e. the two modes give a contribution to the second order dipole moment which is particularly appreciable when the charge migration factor is relevant. This occurs for almost all impurities. Secondly we must consider that the impurity has a charge different of that of the ion being replaced. Therefore it induces a net electrostatic polarization on each of its neighbours. This polarization gives rise to an electric field whose potential at the impurity site has a third order component proportional to xyz. When the impurity moves it experiences a field quadratic in the displacement which induces a dipole moment proportional to the field. This effect is appreciable when the impurity charge and polarizability are different from those of the removed ion. This occurs mainly for donors and acceptors. A quantitative discussion of these effects is given in Section 4. We herewith remark that upon collecting all bilinear terms and introducing the symmetry coordinates p(IJ which are combinations of the ionic displacements, the expression for the second order dipole moment is of the type cL”’= Cririciip(ri)p(ri)
(2)
and accounts for the various types of absorption discussed at the beginning of this section. In (2) the c’s depend on the charges and polarizabilities and on the charge migration factor K, whereas the p’s depend on the dynamical properties. The contributions to the local mode sidebands arise from products of the type p(T,)p(I,) where Ii stands for either A, or E or T,. Among these terms we have selected out those in which the second coordinate belongs to A,, i.e. p(A,) for two reasons: (a) the A, band modes are perturbed only by force constant changes, consequently the calculation of the perturbed frequencies is simpler than in other cases such as those in which the band phonons are also perturbed by the effects of the impurity mass defect, as it occurs for the T, band modes, and (b) these terms give rise to the strongest absorption peaks. The combination of a local mode with a perturbed band mode with E or I’, symmetry gives rise to sidebands which have a different spectral distribution and weaker peaks, as discussed in Section 6. The explicit expressions for the A, modes are worked out in the following sections. The calculation of the local mode second harmonic depends on the terms in (2) in which both coordinates belong to T, and will be considered as a by-product in Section 5.2. The present approach is different from other models
which consider the second order dipole moment in the calculation of the sidebands in alkali halides[5-111. The basic differences lie in the physical approach to solve the dynamical problem and to calculate the dipole moment. In fact we carry out the analysis of the perturbed modes starting from the Lagrangian of the perturbed crystal, instead of using Green’s function approach. In calculating the dipole moment we take into account the microscopic quantities which characterize the behaviour of the impurity and of the ions in its neighbourhood. They directly affect the absorption and their values change from impurity to impurity. According to the results reported in Refs. [S-11] we find. that products of coordinates belonging to different irreducible representations give rise to different peak locations and intensity distribution. However our conclusions are different since the crystal structure is different and the model is different too. 3. THE DYNAMICAL EFFECTS
We discussed in I the manner of deducing the Lagrangian of the perturbed crystal. The expression is reported below for convenience L= Lo+AT-AU = Lo + :m&? 4
-
ei - e,
C{
-e,-r;" If
(3) + 9” + P) 1 (et- e?)a. t(ai- a,)e.* 2
A
1
where X, Y and Z are the components of displacement of the impurity at the site k (k’ being the site of its nearest neighbours), and the subscripts i, r and n refer to the impurity, to the removed ion, and to an impurity nearest neighbour n, respectively. When AU is expanded in terms of the components of displacement x., y. and z. of the ion n and the bilinear terms are collected together, we obtain upon introduction of the symmetry coordinates pi the following expression
>
.? tAa,,p,‘t8;AaipiZ
(4)
in which the first term has T, symmetry and allows us to calculate the effects of the force constant shifts on the local mode frequency as discussed in I. The second term has Al-symmetry and pI is defined as follows: 1 PI=-(x,+x*-x3-x,ty,-y2ty3-y,+z,-zz2-zj 2V3 •t zd.
(5)
This term accounts for the perturbation effects from the force constant changes on the Al-modes and will be discussed in detail below. The last term in (4) collects the contributions to AU from the other symmetry modes which are not of interest here since we are not dealing with them. We therefore approximate AU as follows: 4 AU =;Aa
+Aa~pi’
(6)
1139
Local mode sideband IR absorption and second harmonic intensity calculation Table I. Values for various impurity states of: AaA defined by eqn (7). the intensity and frequency of the main peak, the frequency of the lower peak, the experimental and calculated second harmonic integrated intensity
Impurity units
Aa, (IO’ dynlcm)
Intensity main peak
Frequency main peak
Frequency lower peak
(10-20 cm’)
(cm-‘)
(cm-‘)
%oa
-1.721 -0.467 -0.467 -1.221 -1.221 -1.221
0.106 0.0217 0.0205 0.0067 0.0066 0.0064
680.1 738.2 715.2 529.2 524.2 519.7
724.1 782.2 760.4 573.4 568.4 563.5
-0.0581 0.764 0.764 1.257 1.257 1.05
0.0018 0.0148 0.0140 0.0032 0.0039 0.0565 0.0534 0.034
560.2 582.2 572.2 811.2 790.0 627.8 617.8 583.8
605.6 626.2 616.2 780.2 759.2 597.2 587.4 552.8
‘OBGm
“BG. “Mg,. *‘Mg,. Z6M!Z0. *‘AI,. Ti& ‘QS,_ ‘2C*, ‘C,
“Si,. Y&,
1.05
“P *r
0.345
with
-
226 161 I41 47.1 44.1 41.4
17.5 6 8.13
21.6 lo-2 10-* 12.4 11.1 1.83 1.73 10.3
with
6 = COS
AaA = 1.60x 10”{(ei- e,)e,, - 0.35[(ef - e,2)a. + (ai - a,)e.2]}
$ e=( $) Q(7)exp(2riqd
where the summation is over all different stars of wave vectors q, and the coefficients ime are suitable combinations of the exponential terms which appear in the wave
expansion.
The
complex
conjugate
(c.c.1
arises from the fact that we have adopted the definitions: ea(kly)=eZ(kl:q);
Q(y!=Q*(yq).
(hqJ
sin &as;)
sin (&raqJ
+ i sin (haqJ cos (hraqy) cos (hq,)
(7)
in which the charges and polarizabilities are measured in units of e and k, respectively. Using the values of the e’s and the a’s given in I, it is possible to calculate AaA whose numerical values are given in Table 1. We now expand the displacements which appear in (5) in plane waves [19,20] and combine them together in order to construct symmetry combinations of the coordinates Q(q/j) which we shall call normal coordinates and label them by Q(r,]q/j). They are bases for the irreducible representations of the point group Td. This procedure will transform a S, as follows:
Plane
Sec. harm. integr. inten. exptl talc. (10e2’ cm) (1OF cm)
and n, [ are obtained by ciclic permutations from 5. In the following expressions the dependence of the polarization vectors on k, q and j will be omitted. We introduce these definitions of the 0’s in (3) and deduce the equations of motion for (J
where a labels any perturbed frequency belonging to A,. By means of this relation it is straightforward to eliminate p, using eqn (8). The final result is the following expression which yields for the relation the A, perturbed frequencies and is formally similar to eqn (36) of Ref. [21] which is valid for mass defects only.
In order to determine the behaviour of x,(0.3 in the limit of large n, we introduce the squared frequency distribution E(d) so that the number of modes of the unperturbed crystal with frequency o$ between o* and o2 t do’ is @/n)E(o*) d(J), and consider the function x,(z) of the complex variable z
Xl(Z)=
In particular for p, we have:
l61& + qe, + 5ez12E(W2) doZ I
Z-d
(11)
AS z approaches the real axis we can express the real ~~=-~(~)“*~{Oi[4(5e,(k’Iq)+qe,(k’I~) part of this integral as the principal value which we call f,(z).
+[ez(k’ly))
a(A,lq)+c.c.]
(8)
So
that near the real axis we have xl(o.3 = xdz) = f,(z) 7 7&(z)
(12)
L. BELLOMONTE
1140
with
&(z) = 1615ex + qe,f kl*E(z).
(13)
To calculate E(d) we have used the model proposed in Ref. [22]. In the first two figures we plot f,(z) and $,(z) for impurities at the Ga site. The in-band behaviour of these quantities will be discussed in Section 6, we here remark that inspection of Fig. 1 and of the values of Aaa given in Table 1 show that there are no local modes for any impurities, since Aaa is below a critical value, which for GaAs is about 3 x lo4 dynjcm. Analogously T, local modes occur when the mass defect is greater than a certain value. On the other hand in-band resonances could be possible depending upon the value and sign of Aa,+ Equation (8) leaves the normalization unfixed, it is convenient to introduce real normalized coordinated R(a) so that the Lagrangian takes the form L = Z.(&l)
- o,2R2(o)).
The calculation of the second order dipole moment is similar. The motion of the impurity and it’s nearest neighbours gives a relevant contribution to the x-component of the dipole moment A
u) = e,X + 2e,p,
whereas the electrostatic polarization gives the following contribution
and the charge migration effect can be approximated by the simple formula Se = Ke&:.
Sei = -ZKep,
where C(w,‘) is the normalization constant defined as follows
(19)
which for the impurity and for a nearest neigbbour n, becomes, upon expansion of fin to the first order:
The @s are related to the R’S by
(14)
(17)
(2W
when these expressions are introduced back in (17) and (18), and the second order terms are collected together, expression (2) is obtained. We select from (2) the terms which lead to the simultaneous excitation of a T, mode and an A, mode and write them down explicitly (for the x-component of F(~‘):
=Px (2)
and pI written in terms of the R’s becomes:
f {0,(X -
;p.J + D*p7}p,
(21)
with
(224 These expressions will be used below to calculate the sideband absorption. 4.TEEDYPOLEMOMENT
GW
In I we have outlined the approximations for deducing the main contributions to the first order dipole moment.
Expression (21) can be rewritten using the expansion of
-I 6L Fig.
1. Plot
of f&o’)
defined by eqn
(‘12)for impuritiesat Ga-site.
1141
Local mode sideband IR absorption and second harmonic intensity calculation
the symmetry coordinates belonging to T, in terms of the normal coordinates R,(s) of the perturbed lattice given in I, eqn (24) where s labels any Ti perturbed mode and the expansion of p1 in terms of the coordinates R(a) given by (16). We have then (2) = _
PX
e-
simple poles at each z = 02, (b) it has no other poles, since the apparent poles at z = w’, cancel out. It therefore follows that; ;
1
pSB.~(s, a)~~(s)~(a). a (mtmt~)“*
(23)
In calculating the local mode sidebands we drop out the summation over s and retain only the term which corresponds to the local mode resonance whose label is L and the frequency oL. Consequently (23) becomes:
p'(a)
0 z.-
K(z)=
z-
(29)
too*’
Since n is very large the poles of K(z) are very dense on the real axis between 0 and a~,.~.In the limit n + 00,K(Z) becomes an analytical function in this range. As one approaches the real axis the imaginary part tends to the limit lim Zm(K(z)) = +nZ,p’(a)&z
- 2,).
(30)
1
C(o~*)R~(o,)~D,orX(o) - +X&L)) PX(2)= - kIa (mkm*y’* + GX,(~~)I. 2a~mw (24)
The expression for the cross-section becomes then uzbr + 0.1 =
where p(a) = Cb.‘lxdo.3.
(25)
5.1 Sideband intensity Standard perturbation theory allows us to obtain the expression for the two-phonon cross-section, which in the present case is: az(or + WI ) =
(ek)
mtmkq//4a2
x [D,~J~(w~) -
(26) where R denotes the thermal average and B is the local field correction discussed in I. The actual calculation of a2 is complicated by the fact that p(a) has poles in the allowed range. To overcome this difficulty we construct the following meromorphic functions of the complex variable z which serve to cancel out the apparent poles at z = 05. (274
1615eX t qe, t le,r (2 - al;)*
0n
WJ
W& + w, 2+ kl*)* z-oqj
K(z) = H(z) - (XIW/F(Z).
-“’
(16l.k
OL
>
(ti(WL) + W(o.) + 1).
+ 4,&~)1’C*~0~~)
lim Zm(K(z)) (31)
+ v,
+ 5eZ12)*Hz)
fdz)(2mhh - fdz)) - huh’
].
(32)
A similar expression, though more complicated, holds for the simultaneous excitation of a local mode and a T, band mode. The complication arises from the fact that one has to redefine the expressions (27) by taking into account the fact that the Z’,modes are also perturbed by the impurity mass defect.
5.2 Second harmonic The second harmonic intensity calculation is a very simple extension of the formalism developed before. The expression for the dipole moment (21) has to be replaced by an expression which contains products of two coordinates both belonging to T,.
(27b) Px (*)= 0,(X -
&4)(Y - fpJt aw
- :&s
+ ( Y - :p&71 +D,P+s (33)
WC)
with
(274
We notice that
holding for each of the values z = 0.’ which are the roots of the equation F(z) = 0 which by (10) gives the perturbed frequencies belonging to Al. This also means that -C’(o.‘) is the residue of l/F(z) at the value z = 0.‘. The function K(z) has the following properties: (a) it has IPCSVol.38.No.I&X
-
(f,(z) - m~~/AaJ*+ d,b*(z)
ix4hLN + 02x7bd12p’(4
F(z) = XI(Z)- mdAa*
iLk+0.
(
with
+ R(o,)
+ l)C(WA
H(z) = _! c
-4x.&N
+ I,
p2/3*he2
C(z) =
x Mx.40~)
2@*he2
mkmt~c~6a2
A = : [-S(eia. - e.ai)/R3 - 32a&ei - e.)/R6 t Ke/3]
D,=
- e.ai)/R3 t 10ap.(ei - en)/R6t Ke/3] (34b)
D, = -3[5(eia. - e,,a$R’ +68aian(ei - eJR6 + llKe/3] (34c)
1142
L. BELLOMONTE
0
572
1144
17, 6
228
a
w, cm-’
Fig. 2. Plot of $,(o’) defined by eqn (13) for impurities at Ga-site.
and p(L, a) used in (24) has to be replaced by p(L, L) = c2(oL2)[a~&.)
- tx4hd t 2D,cyx(OL)- :xNL))x7(mL) + 4X72(OL)l. (35)
The expression for the integrated cross-section is then,
It is worth to observe that the two expressions for the cross-section depend on the dipole moment and on the dynamical properties in a’ separate manner. In fact the terms which depend on the dipole moment are collected in the D’s, whereas those which depend on the dynamics are collected in the x’s and in Im(K(z)). Consequently one can reformulate or change part of the assumptions made in calculating the intensity without affecting the general aspect of (31) and (36).
[ 11l] and [ 1001.The factor 161Ze,t qe, t leZ12is negligible for acoustic and optic transverse waves and appreciable for longitudinal waves near the Brillouin zone boundary. Consequently the strong sharp peak in the density of states at o = 262cm-’ does not appear at all in &(z) whereas the peaks due to the longitudinal modes are enhanced although in different manners at the two impurity sites. In Fig. 3 we plot ~~(0,. t 0.) for %&. and in Fig. 4 for **Siar.In Table 1 we give the values of the position and intensity of the main peak and the frequency of the lower peak. These spectra are broad by themselves since we are dealing with band absorptions, line broadening mechanisms such as those discussed in Ref. [21] will cause a further broadening. Instrumental resolution also contributes to broadening the lines. For these reasons and from the fact that absorptions of different types will superimpose, such as those arising from intrinsic second order absorption or from combination tones of two or more local- or gap-modes, it wil.be difficult to observe the details of the spectra calculated here. Nevertheless the gross features should be noticeable as it indeed occurs for the data reported by Thompson and Newman[2] for a sample containing silicon and boron, see their Fig, 2 and Table 2 for a list of the various peaks. Let us start with *‘S&. We expect peaks at 582 and 626 cm-‘. These are close to those reported at 589.5 and 627.8 cm-’ in Ref. [2] whose origin was uncertain. For %, we have calculated peaks at 627 and 597 cm-‘. The first one superimposes with the second peak for **SAG, whereas the latter is close to an experimental peak
6. RESULTS AND DISCUSSION
Typical spectra for impurities at Ga site and at As site have similar gross features, which can be summarized as follows: each spectrum is characterized by a main peak, a lower peak and a much weaker peak. For impurities at Ga site they medially occur at oL t 198cm-‘, oL t 242 cm-‘, and oL t 262 cm-‘, respectively, whereas for impurities at As site they are found at wL+ 229cm-‘, wLt 198cm-‘, and oL t 262 cm-‘. The intensity ratio is medially 20: 6: 1 and the average peak width is approximately 13 cm-’ for impurities at Ga site and twice for impurities at the other site. The positions of the main peaks are shifted by not more than ?5 cm-’ by the value and sign of Aa,,. The positions of the main peaks depend on the density of states and on the behaviour of the trigonometric factor 161& + qe, + kl’. The density of states has a sharp weak peak at w = 198cm-’ associated with a critical point due to the longitudinal acoustic branch at q = (4,i,f)i and a broad weak maximum centered at o = 245 cm-’ associated with critical points at or near the zone edge along the following branches: longitudinal acoustic along [lOO], longitudinal optic along
384
441
498 4
2
w.
555 6
612 8
670
crn~l
order spectrum (local mode sidebands) for **Siat Ga-site.
Fig. 3. Calculated second
w, cm-’
Fig. 4. The same as Fig. 3 for **Si at As-site.
Local mode sideband IR absorptionand second harmonic intensity calcuiation at 5% cm-’ which is due to the ‘“Bo.-Y3.,8 pair giving a strong absorption and to a weaker peak at 601.4cm-’ whose origin was unexplained in Ref. [2]. For “Bo. which is an abundant impurity in the sample used in Ref. [2], we expect peaks at 715 and 760 cm-‘. The first one is close to a peak at 698cm-’ assigned tentatively to the interaction ‘“BG,“Si,-“S&.. The present result provides an alternative assignment which could be unlikely since the frequency separation between the calculated and ex~rimental abso~tion is nearly 20 cm-‘. The peak at 760 cm-’ is close to the experimental one at 762.7 cm-’ which in Ref. [2] was tentatively assigned to interstitial “B. In conclusion from the exam of this limited amount of experimental data we can say that the agreement is fairly good. A further remark concerns a qualitative discussion of other types of in-band absorption not considered here, in particular the second order intrinsic absorption and the combination of a local mode and an E, or Z’,band mode. The intrinsic second order absorption follows mainly the combined density of states. At high frequency a peak is expected around 525 cm-‘. This arises from the simultaneous excitation of two transverse optic vibrations. The frequency of this peak is little affected by the values of the mass defect and force constant changes. The intensity is also quite independent of the impurity concentration. This peak is present in the results reported in Ref. [21. The contribution from the simultaneous excitation of a local mode and a perturbed T, band mode can be worked out from (35) by replacing it by p(L, s) where s is any TI band mode. There is medially no appreciable difTerence between the values of the coefficients 4, 4 and those of A, D.,, A. Consequently the cont~butions to the intensities from these terms, i.e. from the dipole moment are medially the same. On the other hand the trigonometric factors introduce appreciable differences in the intensity and shape. For the T, modes these terms are of different types such as Lhe.(k)eZtk’);
1143
this discrepancy we are to rearrage the values of the five free parameters in such a manner that both the first and second order intensities are fitted.
7.CONCLUSlON In I we have shown how it is possible to calculate the local mode frequencies and integrated intensities for various isolated impurities in GaAs using a relatively simple model. We have here extended the model to the second order absorption in order to calculate the local mode sidebands and second harmonic intensities. The results compare favourably with experiments in those cases in which experimental results are available. Unfortunately few results are available in order to have a general view of the final outcome. There are various ways of improving the present results: first by considering interactions with further neighbours and by taking the anharmonic interactions into account. Concerning the latter we believe that the improvements will be very slight according to the fact that the a~~onicity is small in these crystals as discussed before. The extension of the local changes to second or further neighbours might introduce some improvement mainly in the cases of donors or acceptors. In fact we expect that when the impurity has an ionic radius or charge quite different from the corresponding quantity of the ion being replaced, the effects of the presence of the impurity extend far beyond the nearest neighbours. Finally the introduction of the distortion effects could also improve the agreement with experiment. The model in this case should also take into account the fact that during motion the charges surrounding an ion suffer deformations. The shape of the curves plotted in Figs. 3 and 4 will be unaffected, but the intensity will be modified. Acknowledgements-The author is indebted to W. G. Spitzer for reading a preprint of this work and for useful discussions, and to M. H. L. Pryce who introduced him to the field.
Lle,(k)~;
SheX(ne,(k’) + &(k’)),
etc.
which in general follow the behaviour of the density of states in the optic region and determine an enhancement of the transverse wave peaks. Similar results hold for the El band modes. Consequen~y we are to expect a peak at or. + 262 cm-‘. Since some cancellation occurs between these factors and their vahre is smaller than 161teX+ rje, t geJ” we estimate that the intensity will be at wL+ 262 about one fifth of that of the main peaks calculated previously. Very few experimental results are available about the second harmonic intensity. For Al and P the agreement is good in view of the fact that the intensity was calculated using a set of parameters determined by fitting the first order intensity. For Sii* there is disagreement. The calculated intensity for this impurity is smaller than the experimental one, since there is cancelIation among the various terms which con~ibute to 4. To remove
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1144
L. BELLOMONTE
14. Newman R. C. and Smith R. S., Solid State Comm. 5, 723 (1%8). 15. Spitzer W. G., .r. Phys. C/rem. Solids 28, 33 (1%7). 16. Bellomonte L. and Pryce hf. H. L., Proc. Inr’l Conference on Localized Excitafions in Solids (Edited by R. F. Wallis), P. 203. Plenum Press, New York (t%S). _ 17. Johnson F. A., Proc. Phys. Sot. 73, 265 (1959). 18. Lax M. and Burstein E., Phys. Rev. 97, 39 (1955).
19. Born M. and Huang K., Dynamical Theory of Crystal lattices. Pergamon Press, Oxford (1954). 20. Maradudin A. A., Montroll E. W., Weiss G. H. and Ipatova 1. P., Theory of Lattice Dynamics in the Harmonic Approximation. Academic Press. New York (1971). 21. Dawber P. G. and Elliott R. ‘J., Proc. Ro;. Sk. A.273.222 (1%3); Proc. Phys. Sot. 81,453 (1%3). 22. Banerjee R. and Varshimi Y. P., Can. J. Phys. 47,451 (1%9).
Pergamon Press
Printed in Great Britain
LOCAL MODE SIDEBAND IR ABSORPTION AND SECOND HARMONIC INTENSITY CALCULATION FOR VARIOUS ISOLATED IMPURITIES IN GaAs
Istituto
L. BELLOMONTE di Fisica and Gruppo Nazionale Struttura della Materia University of Palermo, Palermo 90123, Italy (Received 3 August
1916; accepted in revised form II March 1977)
Abstract-We have calculated the intensity and shape of the local mode sidebands and the second harmonic intensity for various isolated impurities in GaAs. To calculate the absorption we have considered the second order dipole moment in the harmonic approximation. The absorption depends on the values of the impurity charge and polarizability, on the corresponding quantities for the nearest neighbours of each impurity and on a charge migration factor. The latter accounts for charge variations during the motion. Although these quantities vary from impurity to impurity, the results are presented in a general form valid also for other impurities. The point group symmetry properties are used in order to give a physical meaning to the results. The effects on the dynamical properties from the mass defect and the local changes in the force constant are also taken into account. 1.INTRODUCTION
In a previous paper[l], to be referred to as I, we have calculated the local mode frequency and the infrared absorption of various isolated impurities in GaAs. In dealing with this topic we have used symmetry considerations in order to break up the problem according to the irreducible representations of the crystal point group under which the model is invariant. To calculate the local mode frequency we have taken into account the effects from the mass defect and local changes in the force constants, whereas in calculating the absorption intensity we have considered the contributions to the dipole moment from the motion of the impurity and its nearest neighbours and from charge migration effects. In this work we are discussing the origin of the second order spectrum and calculating the frequency dependent intensity and shape of the local mode upper sidebands which correspond to the simultaneous excitation of a local mode with T,-symmetry and any perturbed band phonon with A,-symmetry. We consider this contribution to the second order spectrum as the most relevant one and work it out in the framework of the second order dipole moment in the harmonic approximation, as discussed in the following section. According to our model the intensity depends upon five free parameters which correspond to the impurity charge and polarizability e, and ai, respectively, to the corresponding quantities for each nearest neighbour II of the impurity: e. and a., and to a charge migration factor K. The physical properties and the numerical values of these quantities were discussed in I, and no further assumptions are here required. The spectra were calculated for various impurity states. They fall roughly in the frequency range between 500 and 8OOcm-’ in which the experimental information is scarce or missing. Consequently we are able to compare our results with experiment only for 9idp, 9ias and “Bqs in GaAs (see Ref. [2]). The agreement is in this case good. 2.THEMODEL The second order spectrum in crystals with isolated substitutional or interstitial impurites usually consists of
sidebands to the local- or gap-modes, local- or gap-mode second harmonic, and two-band phonon absorptions. The experimental and theoretical aspects of these types of absorption have been exploited by several authors mainly in the case of U-centres in alkali halides or alkaline earth fluorides[3-121. According to their results the mechanisms which account for the sidebands in these crystals are of different types such as anharmonicity, second order dipole moment or a combination of both. Usually the anharmonicity is considered as predominant with respect to the second order dipole moment. Sidebands or second harmonic absorption have also been detected in boron doped, lithium compensated silicon[l3,14], and in III-V compounds[4,15]. There is no full theoretical investigation of these spectra except for a calculation of the combination tones in boron doped silicon in terms of the second order dipole moment [ 161.These different conclusions are justified by the behaviour of the semiconductors with respect to the alkali halides. Intrinsic semiconductors exhibit strong two-phonon absorption[l7] interpreted in terms of the second order dipole moment[ll] in the harmonic approximation. Analogously the experimental data regarding one- and two-phonon absorption in doped semiconductors lead us to the conclusion that the anharmonicity plays a minor role with respect to the second order dipole moment. In fact the various effects caused by anharmonicity such as the temperature dependence of width and position of the local mode resonance and the shit of the second harmonic frequency with respect to twice the local mode frequency are quite appreciable in the spectra of U-centres and negligible in doped semiconductors. This point is supported by comparing the experimental data reported in Ref. [12] with those collected in Ref. [4], pp. % and 156. In our model we consider two effects which account for the origin of the second order dipole moment. The hrst one is related to the charge migration with displacement. Let us consider the contributions from the motion of the charges to the a-component of the dipole moment:
1137
(1)
L. BELLOMONTE
1138
which has T, symmetry at first order, and e. stands for the net charge inside a polyhedron surrounding the impurity or another ion at site n. A uniform compression of the tetrahedron surrounding the impurity, i.e. a displacement of its nearest neighbours according to the A, irreducible representation of the point group, will cause some amount of charge migration so that the charges inside the polyhedra are now different from those in the undistorted position. Consequently when the impurity and its neighbours move according to a T, mode the dipole moment is affected by this charge migration effect, i.e. the two modes give a contribution to the second order dipole moment which is particularly appreciable when the charge migration factor is relevant. This occurs for almost all impurities. Secondly we must consider that the impurity has a charge different of that of the ion being replaced. Therefore it induces a net electrostatic polarization on each of its neighbours. This polarization gives rise to an electric field whose potential at the impurity site has a third order component proportional to xyz. When the impurity moves it experiences a field quadratic in the displacement which induces a dipole moment proportional to the field. This effect is appreciable when the impurity charge and polarizability are different from those of the removed ion. This occurs mainly for donors and acceptors. A quantitative discussion of these effects is given in Section 4. We herewith remark that upon collecting all bilinear terms and introducing the symmetry coordinates p(IJ which are combinations of the ionic displacements, the expression for the second order dipole moment is of the type cL”’= Cririciip(ri)p(ri)
(2)
and accounts for the various types of absorption discussed at the beginning of this section. In (2) the c’s depend on the charges and polarizabilities and on the charge migration factor K, whereas the p’s depend on the dynamical properties. The contributions to the local mode sidebands arise from products of the type p(T,)p(I,) where Ii stands for either A, or E or T,. Among these terms we have selected out those in which the second coordinate belongs to A,, i.e. p(A,) for two reasons: (a) the A, band modes are perturbed only by force constant changes, consequently the calculation of the perturbed frequencies is simpler than in other cases such as those in which the band phonons are also perturbed by the effects of the impurity mass defect, as it occurs for the T, band modes, and (b) these terms give rise to the strongest absorption peaks. The combination of a local mode with a perturbed band mode with E or I’, symmetry gives rise to sidebands which have a different spectral distribution and weaker peaks, as discussed in Section 6. The explicit expressions for the A, modes are worked out in the following sections. The calculation of the local mode second harmonic depends on the terms in (2) in which both coordinates belong to T, and will be considered as a by-product in Section 5.2. The present approach is different from other models
which consider the second order dipole moment in the calculation of the sidebands in alkali halides[5-111. The basic differences lie in the physical approach to solve the dynamical problem and to calculate the dipole moment. In fact we carry out the analysis of the perturbed modes starting from the Lagrangian of the perturbed crystal, instead of using Green’s function approach. In calculating the dipole moment we take into account the microscopic quantities which characterize the behaviour of the impurity and of the ions in its neighbourhood. They directly affect the absorption and their values change from impurity to impurity. According to the results reported in Refs. [S-11] we find. that products of coordinates belonging to different irreducible representations give rise to different peak locations and intensity distribution. However our conclusions are different since the crystal structure is different and the model is different too. 3. THE DYNAMICAL EFFECTS
We discussed in I the manner of deducing the Lagrangian of the perturbed crystal. The expression is reported below for convenience L= Lo+AT-AU = Lo + :m&? 4
-
ei - e,
C{
-e,-r;" If
(3) + 9” + P) 1 (et- e?)a. t(ai- a,)e.* 2
A
1
where X, Y and Z are the components of displacement of the impurity at the site k (k’ being the site of its nearest neighbours), and the subscripts i, r and n refer to the impurity, to the removed ion, and to an impurity nearest neighbour n, respectively. When AU is expanded in terms of the components of displacement x., y. and z. of the ion n and the bilinear terms are collected together, we obtain upon introduction of the symmetry coordinates pi the following expression
>
.? tAa,,p,‘t8;AaipiZ
(4)
in which the first term has T, symmetry and allows us to calculate the effects of the force constant shifts on the local mode frequency as discussed in I. The second term has Al-symmetry and pI is defined as follows: 1 PI=-(x,+x*-x3-x,ty,-y2ty3-y,+z,-zz2-zj 2V3 •t zd.
(5)
This term accounts for the perturbation effects from the force constant changes on the Al-modes and will be discussed in detail below. The last term in (4) collects the contributions to AU from the other symmetry modes which are not of interest here since we are not dealing with them. We therefore approximate AU as follows: 4 AU =;Aa
+Aa~pi’
(6)
1139
Local mode sideband IR absorption and second harmonic intensity calculation Table I. Values for various impurity states of: AaA defined by eqn (7). the intensity and frequency of the main peak, the frequency of the lower peak, the experimental and calculated second harmonic integrated intensity
Impurity units
Aa, (IO’ dynlcm)
Intensity main peak
Frequency main peak
Frequency lower peak
(10-20 cm’)
(cm-‘)
(cm-‘)
%oa
-1.721 -0.467 -0.467 -1.221 -1.221 -1.221
0.106 0.0217 0.0205 0.0067 0.0066 0.0064
680.1 738.2 715.2 529.2 524.2 519.7
724.1 782.2 760.4 573.4 568.4 563.5
-0.0581 0.764 0.764 1.257 1.257 1.05
0.0018 0.0148 0.0140 0.0032 0.0039 0.0565 0.0534 0.034
560.2 582.2 572.2 811.2 790.0 627.8 617.8 583.8
605.6 626.2 616.2 780.2 759.2 597.2 587.4 552.8
‘OBGm
“BG. “Mg,. *‘Mg,. Z6M!Z0. *‘AI,. Ti& ‘QS,_ ‘2C*, ‘C,
“Si,. Y&,
1.05
“P *r
0.345
with
-
226 161 I41 47.1 44.1 41.4
17.5 6 8.13
21.6 lo-2 10-* 12.4 11.1 1.83 1.73 10.3
with
6 = COS
AaA = 1.60x 10”{(ei- e,)e,, - 0.35[(ef - e,2)a. + (ai - a,)e.2]}
$ e=( $) Q(7)exp(2riqd
where the summation is over all different stars of wave vectors q, and the coefficients ime are suitable combinations of the exponential terms which appear in the wave
expansion.
The
complex
conjugate
(c.c.1
arises from the fact that we have adopted the definitions: ea(kly)=eZ(kl:q);
Q(y!=Q*(yq).
(hqJ
sin &as;)
sin (&raqJ
+ i sin (haqJ cos (hraqy) cos (hq,)
(7)
in which the charges and polarizabilities are measured in units of e and k, respectively. Using the values of the e’s and the a’s given in I, it is possible to calculate AaA whose numerical values are given in Table 1. We now expand the displacements which appear in (5) in plane waves [19,20] and combine them together in order to construct symmetry combinations of the coordinates Q(q/j) which we shall call normal coordinates and label them by Q(r,]q/j). They are bases for the irreducible representations of the point group Td. This procedure will transform a S, as follows:
Plane
Sec. harm. integr. inten. exptl talc. (10e2’ cm) (1OF cm)
and n, [ are obtained by ciclic permutations from 5. In the following expressions the dependence of the polarization vectors on k, q and j will be omitted. We introduce these definitions of the 0’s in (3) and deduce the equations of motion for (J
where a labels any perturbed frequency belonging to A,. By means of this relation it is straightforward to eliminate p, using eqn (8). The final result is the following expression which yields for the relation the A, perturbed frequencies and is formally similar to eqn (36) of Ref. [21] which is valid for mass defects only.
In order to determine the behaviour of x,(0.3 in the limit of large n, we introduce the squared frequency distribution E(d) so that the number of modes of the unperturbed crystal with frequency o$ between o* and o2 t do’ is @/n)E(o*) d(J), and consider the function x,(z) of the complex variable z
Xl(Z)=
In particular for p, we have:
l61& + qe, + 5ez12E(W2) doZ I
Z-d
(11)
AS z approaches the real axis we can express the real ~~=-~(~)“*~{Oi[4(5e,(k’Iq)+qe,(k’I~) part of this integral as the principal value which we call f,(z).
+[ez(k’ly))
a(A,lq)+c.c.]
(8)
So
that near the real axis we have xl(o.3 = xdz) = f,(z) 7 7&(z)
(12)
L. BELLOMONTE
1140
with
&(z) = 1615ex + qe,f kl*E(z).
(13)
To calculate E(d) we have used the model proposed in Ref. [22]. In the first two figures we plot f,(z) and $,(z) for impurities at the Ga site. The in-band behaviour of these quantities will be discussed in Section 6, we here remark that inspection of Fig. 1 and of the values of Aaa given in Table 1 show that there are no local modes for any impurities, since Aaa is below a critical value, which for GaAs is about 3 x lo4 dynjcm. Analogously T, local modes occur when the mass defect is greater than a certain value. On the other hand in-band resonances could be possible depending upon the value and sign of Aa,+ Equation (8) leaves the normalization unfixed, it is convenient to introduce real normalized coordinated R(a) so that the Lagrangian takes the form L = Z.(&l)
- o,2R2(o)).
The calculation of the second order dipole moment is similar. The motion of the impurity and it’s nearest neighbours gives a relevant contribution to the x-component of the dipole moment A
u) = e,X + 2e,p,
whereas the electrostatic polarization gives the following contribution
and the charge migration effect can be approximated by the simple formula Se = Ke&:.
Sei = -ZKep,
where C(w,‘) is the normalization constant defined as follows
(19)
which for the impurity and for a nearest neigbbour n, becomes, upon expansion of fin to the first order:
The @s are related to the R’S by
(14)
(17)
(2W
when these expressions are introduced back in (17) and (18), and the second order terms are collected together, expression (2) is obtained. We select from (2) the terms which lead to the simultaneous excitation of a T, mode and an A, mode and write them down explicitly (for the x-component of F(~‘):
=Px (2)
and pI written in terms of the R’s becomes:
f {0,(X -
;p.J + D*p7}p,
(21)
with
(224 These expressions will be used below to calculate the sideband absorption. 4.TEEDYPOLEMOMENT
GW
In I we have outlined the approximations for deducing the main contributions to the first order dipole moment.
Expression (21) can be rewritten using the expansion of
-I 6L Fig.
1. Plot
of f&o’)
defined by eqn
(‘12)for impuritiesat Ga-site.
1141
Local mode sideband IR absorption and second harmonic intensity calculation
the symmetry coordinates belonging to T, in terms of the normal coordinates R,(s) of the perturbed lattice given in I, eqn (24) where s labels any Ti perturbed mode and the expansion of p1 in terms of the coordinates R(a) given by (16). We have then (2) = _
PX
e-
simple poles at each z = 02, (b) it has no other poles, since the apparent poles at z = w’, cancel out. It therefore follows that; ;
1
pSB.~(s, a)~~(s)~(a). a (mtmt~)“*
(23)
In calculating the local mode sidebands we drop out the summation over s and retain only the term which corresponds to the local mode resonance whose label is L and the frequency oL. Consequently (23) becomes:
p'(a)
0 z.-
K(z)=
z-
(29)
too*’
Since n is very large the poles of K(z) are very dense on the real axis between 0 and a~,.~.In the limit n + 00,K(Z) becomes an analytical function in this range. As one approaches the real axis the imaginary part tends to the limit lim Zm(K(z)) = +nZ,p’(a)&z
- 2,).
(30)
1
C(o~*)R~(o,)~D,orX(o) - +X&L)) PX(2)= - kIa (mkm*y’* + GX,(~~)I. 2a~mw (24)
The expression for the cross-section becomes then uzbr + 0.1 =
where p(a) = Cb.‘lxdo.3.
(25)
5.1 Sideband intensity Standard perturbation theory allows us to obtain the expression for the two-phonon cross-section, which in the present case is: az(or + WI ) =
(ek)
mtmkq//4a2
x [D,~J~(w~) -
(26) where R denotes the thermal average and B is the local field correction discussed in I. The actual calculation of a2 is complicated by the fact that p(a) has poles in the allowed range. To overcome this difficulty we construct the following meromorphic functions of the complex variable z which serve to cancel out the apparent poles at z = 05. (274
1615eX t qe, t le,r (2 - al;)*
0n
WJ
W& + w, 2+ kl*)* z-oqj
K(z) = H(z) - (XIW/F(Z).
-“’
(16l.k
OL
>
(ti(WL) + W(o.) + 1).
+ 4,&~)1’C*~0~~)
lim Zm(K(z)) (31)
+ v,
+ 5eZ12)*Hz)
fdz)(2mhh - fdz)) - huh’
].
(32)
A similar expression, though more complicated, holds for the simultaneous excitation of a local mode and a T, band mode. The complication arises from the fact that one has to redefine the expressions (27) by taking into account the fact that the Z’,modes are also perturbed by the impurity mass defect.
5.2 Second harmonic The second harmonic intensity calculation is a very simple extension of the formalism developed before. The expression for the dipole moment (21) has to be replaced by an expression which contains products of two coordinates both belonging to T,.
(27b) Px (*)= 0,(X -
&4)(Y - fpJt aw
- :&s
+ ( Y - :p&71 +D,P+s (33)
WC)
with
(274
We notice that
holding for each of the values z = 0.’ which are the roots of the equation F(z) = 0 which by (10) gives the perturbed frequencies belonging to Al. This also means that -C’(o.‘) is the residue of l/F(z) at the value z = 0.‘. The function K(z) has the following properties: (a) it has IPCSVol.38.No.I&X
-
(f,(z) - m~~/AaJ*+ d,b*(z)
ix4hLN + 02x7bd12p’(4
F(z) = XI(Z)- mdAa*
iLk+0.
(
with
+ R(o,)
+ l)C(WA
H(z) = _! c
-4x.&N
+ I,
p2/3*he2
C(z) =
x Mx.40~)
2@*he2
mkmt~c~6a2
A = : [-S(eia. - e.ai)/R3 - 32a&ei - e.)/R6 t Ke/3]
D,=
- e.ai)/R3 t 10ap.(ei - en)/R6t Ke/3] (34b)
D, = -3[5(eia. - e,,a$R’ +68aian(ei - eJR6 + llKe/3] (34c)
1142
L. BELLOMONTE
0
572
1144
17, 6
228
a
w, cm-’
Fig. 2. Plot of $,(o’) defined by eqn (13) for impurities at Ga-site.
and p(L, a) used in (24) has to be replaced by p(L, L) = c2(oL2)[a~&.)
- tx4hd t 2D,cyx(OL)- :xNL))x7(mL) + 4X72(OL)l. (35)
The expression for the integrated cross-section is then,
It is worth to observe that the two expressions for the cross-section depend on the dipole moment and on the dynamical properties in a’ separate manner. In fact the terms which depend on the dipole moment are collected in the D’s, whereas those which depend on the dynamics are collected in the x’s and in Im(K(z)). Consequently one can reformulate or change part of the assumptions made in calculating the intensity without affecting the general aspect of (31) and (36).
[ 11l] and [ 1001.The factor 161Ze,t qe, t leZ12is negligible for acoustic and optic transverse waves and appreciable for longitudinal waves near the Brillouin zone boundary. Consequently the strong sharp peak in the density of states at o = 262cm-’ does not appear at all in &(z) whereas the peaks due to the longitudinal modes are enhanced although in different manners at the two impurity sites. In Fig. 3 we plot ~~(0,. t 0.) for %&. and in Fig. 4 for **Siar.In Table 1 we give the values of the position and intensity of the main peak and the frequency of the lower peak. These spectra are broad by themselves since we are dealing with band absorptions, line broadening mechanisms such as those discussed in Ref. [21] will cause a further broadening. Instrumental resolution also contributes to broadening the lines. For these reasons and from the fact that absorptions of different types will superimpose, such as those arising from intrinsic second order absorption or from combination tones of two or more local- or gap-modes, it wil.be difficult to observe the details of the spectra calculated here. Nevertheless the gross features should be noticeable as it indeed occurs for the data reported by Thompson and Newman[2] for a sample containing silicon and boron, see their Fig, 2 and Table 2 for a list of the various peaks. Let us start with *‘S&. We expect peaks at 582 and 626 cm-‘. These are close to those reported at 589.5 and 627.8 cm-’ in Ref. [2] whose origin was uncertain. For %, we have calculated peaks at 627 and 597 cm-‘. The first one superimposes with the second peak for **SAG, whereas the latter is close to an experimental peak
6. RESULTS AND DISCUSSION
Typical spectra for impurities at Ga site and at As site have similar gross features, which can be summarized as follows: each spectrum is characterized by a main peak, a lower peak and a much weaker peak. For impurities at Ga site they medially occur at oL t 198cm-‘, oL t 242 cm-‘, and oL t 262 cm-‘, respectively, whereas for impurities at As site they are found at wL+ 229cm-‘, wLt 198cm-‘, and oL t 262 cm-‘. The intensity ratio is medially 20: 6: 1 and the average peak width is approximately 13 cm-’ for impurities at Ga site and twice for impurities at the other site. The positions of the main peaks are shifted by not more than ?5 cm-’ by the value and sign of Aa,,. The positions of the main peaks depend on the density of states and on the behaviour of the trigonometric factor 161& + qe, + kl’. The density of states has a sharp weak peak at w = 198cm-’ associated with a critical point due to the longitudinal acoustic branch at q = (4,i,f)i and a broad weak maximum centered at o = 245 cm-’ associated with critical points at or near the zone edge along the following branches: longitudinal acoustic along [lOO], longitudinal optic along
384
441
498 4
2
w.
555 6
612 8
670
crn~l
order spectrum (local mode sidebands) for **Siat Ga-site.
Fig. 3. Calculated second
w, cm-’
Fig. 4. The same as Fig. 3 for **Si at As-site.
Local mode sideband IR absorptionand second harmonic intensity calcuiation at 5% cm-’ which is due to the ‘“Bo.-Y3.,8 pair giving a strong absorption and to a weaker peak at 601.4cm-’ whose origin was unexplained in Ref. [2]. For “Bo. which is an abundant impurity in the sample used in Ref. [2], we expect peaks at 715 and 760 cm-‘. The first one is close to a peak at 698cm-’ assigned tentatively to the interaction ‘“BG,“Si,-“S&.. The present result provides an alternative assignment which could be unlikely since the frequency separation between the calculated and ex~rimental abso~tion is nearly 20 cm-‘. The peak at 760 cm-’ is close to the experimental one at 762.7 cm-’ which in Ref. [2] was tentatively assigned to interstitial “B. In conclusion from the exam of this limited amount of experimental data we can say that the agreement is fairly good. A further remark concerns a qualitative discussion of other types of in-band absorption not considered here, in particular the second order intrinsic absorption and the combination of a local mode and an E, or Z’,band mode. The intrinsic second order absorption follows mainly the combined density of states. At high frequency a peak is expected around 525 cm-‘. This arises from the simultaneous excitation of two transverse optic vibrations. The frequency of this peak is little affected by the values of the mass defect and force constant changes. The intensity is also quite independent of the impurity concentration. This peak is present in the results reported in Ref. [21. The contribution from the simultaneous excitation of a local mode and a perturbed T, band mode can be worked out from (35) by replacing it by p(L, s) where s is any TI band mode. There is medially no appreciable difTerence between the values of the coefficients 4, 4 and those of A, D.,, A. Consequently the cont~butions to the intensities from these terms, i.e. from the dipole moment are medially the same. On the other hand the trigonometric factors introduce appreciable differences in the intensity and shape. For the T, modes these terms are of different types such as Lhe.(k)eZtk’);
1143
this discrepancy we are to rearrage the values of the five free parameters in such a manner that both the first and second order intensities are fitted.
7.CONCLUSlON In I we have shown how it is possible to calculate the local mode frequencies and integrated intensities for various isolated impurities in GaAs using a relatively simple model. We have here extended the model to the second order absorption in order to calculate the local mode sidebands and second harmonic intensities. The results compare favourably with experiments in those cases in which experimental results are available. Unfortunately few results are available in order to have a general view of the final outcome. There are various ways of improving the present results: first by considering interactions with further neighbours and by taking the anharmonic interactions into account. Concerning the latter we believe that the improvements will be very slight according to the fact that the a~~onicity is small in these crystals as discussed before. The extension of the local changes to second or further neighbours might introduce some improvement mainly in the cases of donors or acceptors. In fact we expect that when the impurity has an ionic radius or charge quite different from the corresponding quantity of the ion being replaced, the effects of the presence of the impurity extend far beyond the nearest neighbours. Finally the introduction of the distortion effects could also improve the agreement with experiment. The model in this case should also take into account the fact that during motion the charges surrounding an ion suffer deformations. The shape of the curves plotted in Figs. 3 and 4 will be unaffected, but the intensity will be modified. Acknowledgements-The author is indebted to W. G. Spitzer for reading a preprint of this work and for useful discussions, and to M. H. L. Pryce who introduced him to the field.
Lle,(k)~;
SheX(ne,(k’) + &(k’)),
etc.
which in general follow the behaviour of the density of states in the optic region and determine an enhancement of the transverse wave peaks. Similar results hold for the El band modes. Consequen~y we are to expect a peak at or. + 262 cm-‘. Since some cancellation occurs between these factors and their vahre is smaller than 161teX+ rje, t geJ” we estimate that the intensity will be at wL+ 262 about one fifth of that of the main peaks calculated previously. Very few experimental results are available about the second harmonic intensity. For Al and P the agreement is good in view of the fact that the intensity was calculated using a set of parameters determined by fitting the first order intensity. For Sii* there is disagreement. The calculated intensity for this impurity is smaller than the experimental one, since there is cancelIation among the various terms which con~ibute to 4. To remove
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