A planar approach to the lattice dynamics of polar semiconductor superlattices

486

Surface Science 221(1989) 486-512 North-Holland, Amsterdam

A PLANAR APPROACH TO THE LATTICE DYNAMICS OF POLAR SEMICONDUCTOR SUPERLATIKES L. MIGLIO

and L. COLOMBO Dipartimento

di F&a

* “A. VoIta” e Unitd GNSM-CZSM

Received 5 April 1989; accepted for publication

di Pavia, via Bassi 6, I-27100 Pavia, Ita@

24 April 1989

We present a planar approach to the calculation of the dynamical matrix for polar semiconductor superlattices: in-plane and inter-plane Coulombic force constants are derived analytically. This technique obtains a fast convergence of the lattice summations and allows for a helpful comparison between the analytic expressions of single slab force constants and superlattice ones. The origin and the dispersive behavior of macroscopic interface modes are discussed in comparison to Fuchs and Kliewer surface branches. A numerical application to the (GaAs)s(AkAs), system is made in the framework of the bond-charge-m~el dynamics: the sensitivity of the microscopic interface vibrations to the interface force constants parametrization is pointed out by a few examples.

1. Introduction The role of long range Coulombic interactions in determining the zone center anisotropy of some optical branches for GaAs/AlAs superlattices has been recently pointed out in a few publications [l-4]. In particular, refs. [1,2] perform the calculation of the Coulomb dynamical matrix in the framework of a three-dimensional Ewald transformation for the superlattice unit cells, which display a symmetry breaking between the growth direction and the plane perpendicular to it. Thus, the analogy to the dynamical anisotropy of naturaZ &axial systems (like layered transition metal dihahdes, for example [5]) can be evidenced, especially if we consider super-periodic structures made out of * Present address: Institute Romand de Recherche Numerique (IRRMA), PHB-Ecublens, CH-1015 Lausanne, Switzerland.

en Physique

0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

des Materiaux

L. Mglio, L Colombo / Lattice dynamics of polar semiconductor lattices

48-l

very thin atomic layers. However, only very recently some investigations [3,4] have considered the two-dimensional characteristics of these systems and the relations between macroscopic optical modes originated from surfaces (or interfaces, as in our case), i.e. Fuchs and Kliewer [6] modes, and the most anisotropic branches of GaAs/AlAs superlattices. In our opinion a comparison between the dynamics of uncoupled slabs and the one of superlattices is particularly helpful in understanding the two-dimensional features of superperiodic systems, which can be considered in terms of multi-interface materials. The most convenient approach to this problem is a planar point of view, that is the evaluation of in-plane and inter-plane force constants for the atomic planes normal to the growth direction. Here we show how to calculate the exact Coulombic force constants between any two sublattices of the superlattice, by implementing and then summing up the individual inter-plane contributions, already obtained for single slabs [7]. Our technique displays two advantages: The first one is merely practical, i.e. it provides a faster way to calculate the dynamical matrix of superlattices, even for a superperiodicity as small as in the case of (GaAs),(AlAs),. The other one is that the analytical expression for the planar force constants of the superlattice can be easily compared to the one for the slab case, and the similarities (and differences) can be pointed out straigthforwardly. Moreover, our numerical calculation for the GaAs/AlAs system supply a comparison between the adiabatic bondcharge mode [S] (BCM) results and existing calculations performed by rigid ions [2,3] and shell [l] models. The analysis of the dispersion relations along symmetry directions perpendicular to the growth axis is quite interesting, since the microscopic interface modes are strongly dependent on the particular parametrization adopted for interface interactions. In particular, we show that, even within the same dynamical model, slight changes in the choice of the interface force constants may produce large variations in the dispersion of several branches.

2. Coulombic force constants by three-dimensional

Ewald transformation

As shown in several textbooks [9-111 the elements of the Coulombic force constant matrix for any crystal structure can be obtained by means of the Ewald transformation as:

Q,B(q,KKt)=zKz.$e2 y-T.fi(qlKKt) a

[

+SK,~Z~‘~ZK~~T,p(q K”

=

0 1 KK’)

, I

(1)

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor lattices

488

where z, is the effective charge of the K th ion, u, is the cell volume, q, is the ath Cartesian component of the wavevector q and T,,(q 1KK’) can be splitted into a summation over the reciprocal space vector 7:

_

(~+4)a(~+dP

c

I~+!71*

7#0

exp[-ir*(r(K’)-r(K))]

exp[-1r+q12/4P],

(2) and a summation over real space lattice vectors I: Z$

P3/2H,8(@

I r(l’~‘)

-

r(f~)

I)

exp[iq*(r(f’ic’)

- r(h))].

(3)

Here P is a convergence parameter, roughly as big as the reciprocal of the n.n. distance, r(k) is the position of the Kth ion in the Ith cell and Hap(x) contains the complementary error function erfc(x):

H,,(x)

=

I,rfc(x)+-JL($+Z)exp[-x2]] XaXB x2 x3

[

-a,,

]

-$erfc(x)+-$($)

exp[-x2]].

(4)

The form of eq. (1) is suitably chosen to display the physical meaning of the Coulombic contributions. In fact, the first term originates from the anisotropic macroscopic field: for q + 0 along a definite direction the longitudinal contribution is positive, while the transverse one is zero. The second term represents the Lorentz field and the third one is the self-field, i.e. the field produced at the ion site due to its own displacement. For perfect crystals displaying at least tetrahedral symmetry [9-111 &&=OIKK’)=fC?,,j,

that the self-field vanishes by cell neutrality, i.e. E,Z, = 0. By approaching the l? point along the qx direction, for instance, it is easily found that expression (1) leads to:

so

for the xx component, and to for the yy, zz components. This anisotropy of the force constant

matrix leads to the splitting between

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor

lattices

489

longitudinal optic (LO) and transverse optic (TO) vibrations in ionic crystals. However, by approaching r along a direction normal to qx the same frequencies are obtained (with the exchange between one TO mode and the LO mode), due to the rotational invariance of Ta8(q= 0 1UK’). In the case of one superlattice, the shape of the unit cell (extended in the growth direction) displays the symmetry breaking occurring in the crystal. As a consequence a new set of 1 and r vectors is obtained and the summations in expressions (2) and (3) no longer provide an isotropic T,,(q = 0 1KK’) term. In particular, by numerically computing this term for the case of (GaAs),(AlAs), superlattice, it is found that its geometric form, as well as the one of the short range contribution, is the following:

&(q=OIKK’)=

T,&=OIKK’)=

,

K,

K'=CUtiOn

K=

cation

(Union)

(union),

~'=union

(5)

(cation),

(6)

where the z-axis lies in the growth direction. This symmetry breaking between z and x, y components is, in turn, responsible for the dispersive anisotropy of the optical branches at zone center in semiconductor superlattices, since by rotating the dynamical matrix the TO-LO splittings no longer match for different directions. However, this straightforward explanation, already mentioned in ref. [3], is somehow misleading. In fact, expressions (2) and (3) are evaluated just on the grounds of geometric information, so that one can use them also for the case of a fictious superlattice made out of one homogeneous material with the same atomic positions of the true superlattice. Yet, in this situation no mismatch is present for the optical branches approaching I? from different directions. Actually, the physical origin of the anisotropy for the real superlattice may be better understood by considering the different dipole moments occurring in GaAs and AlAs slabs, as due to the change of masses (and, in turn, displacements) and, if any, effective charges. In particular, the electric field at site K due to the (qj) phonon branch

reflects the symmetry breaking occurring in the supercell through the summation over dipole moments Z,~ez+(q 1~‘j) of both slabs. Since this symmetry breaking is generated only in the superperiodicity direction, the planar point of view is more suitable then the three-dimensional approach for investigating the Coulombic field effects in real superlattices.

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor lattices

490

3. Interplanar approach to the Coulombic force constants of a superlattice First we outline how to calculate the Coulombic contribution between one atom on site (1~) and the sublattice (I’K’), which lies on the generic (k, h, j) plane. Here we follow the same pathway of ref. [7], except for a correction which will be treated in the following. The starting point is the electric field at any point x, due to an array of dipoles P(/;K’) exp[i2?TK*r(l’~‘)] at the positions r(l’K’), generated by an optical phonon which propagates along the K direction on the (k, h, j) plane:

JePb) = c

&9w)&

r;n B

F

exp[i2rK*R(LI;ic’)]

(8)

.

Ir(LI;K’)-x~

Here the capital letters, like K, label vectors belonging to the (k, h, j) plane and 1= (L, 14) with (1;~‘) labelling different atomic planes of the (k, h, j) family. After a few manipulations exploiting two-dimensional periodicity on (h, k, j) plane, explained in the following, we shall evaluate the double derivative at the x = r(L = 0, Ig~) position and compare eq. (8) to the expression of the dipolar field in terms of the interplanar force constants. In details, by using the integral representation of the reciprocal function

,r(l,Kf) _x,

=

f=/opexp[ --IWO -xI~P~] dp

and by separating the vectors components normal ( -1) to the (h, from the parallel ones (capital case), we transform eq. (8) to:

xexp[i2nK*X]T

-+exp[ ?T

+i2&(R(LI;K’)

-X)]

- IR(LZ$c’)

k,

j)

plane

-X12p2

dp.

The summation over L in square brackets is a periodic function two-dimensional lattice and can be expanded in a Fourier series:

(IO) of the

~;exp[-pt(Ll;r’)-X~2p”+i2nK*(R(LI;.’)-x)]

=$$$exp[-~2~G+K~2/p2-i2*G*(R(Ol;n’)-X)], a where s,

is the area of the planar

unit cell and G is a vector

(II) of the

L. Miglio, L. Colombo / Luttice dynamics of polar semiconductor

lattices

491

two-dimensional reciprocal lattice. We consider now two possible situations: (A) the (&K) site does not lie on the same plane of (&K’), i.e. 1rl (Ij~) ‘&K’)l #O. (B) 1 rL (lg~) - rl (13~‘) 1 = 0 and the form of the Coulombic series strictly resembles a two-dimensional Ewald problem. In the first case the summation over reciprocal lattice vectors in the right-hand side of eq. (11) rapidly converges for any value of p and a straightforward substitution into eq. (10) is possible. By evaluating the integral over dp we obtain: E,d’p(~) = C Cpa(‘;K’)~lexp[i2~~*X] I I a Ba b B xc c

IG~Xlexp[-2nlG+KIlrl((;K’)-xll

-i2mG*(R(l;K’)

-XI].

02)

The double derivative at point x = (L = 0 I, K) is easily performed by choosing the z-axis along the (k hj) direction and taking into account that:

&(Q(+XJ)=S~, Sg= n

ss=

-I

fOrQ(f;K’)>XI,

U=Z,

+1

forrl(l;K’)
a=z,

for (Y= x, y.

ss = 0

03)

We finally obtain 04)

where Q&K I 13~l;~‘) is, by definition, the force constants matrix which takes the following expression: Q,p(K

%Ki;K’)

W+K)a(G+Kb IG+KI

+6

(IzBr s

IG+KI

xexp[-2rlG+KIIrl(1;K’)--rl(Ig~)I-i2nG*(X(I;K’)-X(I,~))]. (15) Turning now to the case (B), i.e. I rl ( 13~) - rA (lj~‘) I = 0, we operate in the same way as in the Ewald procedure. In fact the integration over dp is

492

L. Miglio, L. Colombo / Lattice dynamics ofpolar semiconductor

lattices

splitted into two parts: the first one (0 < p < P) contains the summation on the two-dimensional direct lattice (eq. (ll), left-hand side), the other one (P < p < co)displays the summation on reciprocal lattice vectors (eq. (12) right-hand side):

x

X

26

?zexp[-n21G+K12/p2-i2?rG*(R(OZ+‘)-X)]

-$ xexp[ r

- (R(LZ,K’)

dp

- X)2p2 + i2rK*(R(LZ,K’)

-x>]

dp.

L (16)

We separate the second term, which displays the summation over direct lattice vectors, re-writing it as follows: C CP~(Z,K’) LKI

exp[i2?rK*R(L3K’)]

axa

/3

Xerfc(lr(L.Z,K’)

P

L ax,

) r(

L&K’)

-

X/P

-xlP)

and Z&(x) is given by eq. (4). When the value x = r(L = 0 I, K) is taken into the right-hand side of eq. (17), special care should be taken for the divergent contribution (L = 0 I, K’ = K) in the direct space summation. For this term the replacement

is obtained by subtracting the self-interaction of the dipole (L = 0 I, K) to the left-hand side of eq. (17), before taking this limiting value for x [7]. For what concerns the first term in eq. (16), it is easier to perform the derivatives and to set x i = rI (Zj~) before integrating over dp. For the sake of clarity we outline separately in-plane and normal to plane cases. The purely

L. Miglio, L. Colombo / Lattice dynamics

in-plane

derivatives

are straightforward

- 5 5 $&(/+‘)4r2(G

x2h -1

493

and we obtain

+ K),(G+

K)P

exP[i2rK*K(O+)]

pexp[-a2,G+K12/p2]Ldp

3,

0

P2

= - 5 5 ~pB(I,~‘)4n2(G

x

of polar semiconductor lattices

+ K),(G

erfc(nIG+KI/P),

s ,,:,,

+ K),q

exp[i2?TK*K(OI+c)]

a=x,

y,

p=x,

y.

(19)

a

For mixed in-plane/normal &exP[

-(Q

to plane

double

derivatives

we have:

-Xz)‘P’]

(I+‘)

=2(43K’)-X,)P2eXP[-(5(4K’)-X,)2P2], a = 4x7

Y),

P = X,

Y(Z).

(20)

By setting x I = rl (13~) this term vanishes and so does the whole integral. The last case, double derivative along z, is particularly important since it originates a non-vanishing term which was missing in ref. [7]: $exp[

- XJ’P’]

- (Q W)

L = &{exp[

-(Q

(&KI)

- XJ2P2])

X2p2(~l(~3K’)-X,)-2p2eX+(~l(~,K’)-X,)2p2],

(Y,p=Z,

(21) so

that, by setting

5 5

$JQ(~~K’)

x2J;; -$ sa

=

x I = I-~ (13~) and substituting exp[i2?rK*R(O1,K)

p-2exp[-Ir2(G+K12/p2]

- i2aG*(K(O/,K’)

we have:

- R(O~,K))]

dp

0

$ $ $P&r’) x46 S,

into the integral,

exP[i2aG*(K(k’)

fierfc(7rIG+KI/P)-

-K(k))]

7~I G+ K I

,IG~K,exp[-~21G+K12/P2]

. 1 (22)

494

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor

lattices

Finally we write down the total expression for the dipolar field by summing up the direct term in (17) to a compacted form of (19) and (22). Here again we obtain the force constants matrix for in-plane contributions as: Q,p(K =

t bKhK’)

P3~ffa&+(~~3K’)

-

@jK)

I)

eXp[i27&+(LZ,K’)

- t’(0z3K))]

L

Xexp[i2rG*(R(OZ,K’) sexp[

- R(oZ3K))] -i2rGS(R(0Z3K’)

- R(O/,K))]

VT1G + K 1

and particular attention should be payed to the replacement Has(x = 0) --, H$. Let us turn now to the calculation of the superlattice force constants: to obtain the lattice we have to sum the interplanar force constants of eq. (15) over an infinite number of planes, pertaining to different superlattice cells in the growth direction: +a, c

Qas(K

(OZ,K,

nd,Z;~‘)

exp[i2nq,(nd,

+ rl (Z;K’) - rl (Z,K))].

(24)

n= --m

Here d, is the superlattice periodicity, n is an integer number and q3 ranges in the small (folded) side of the Brillouin zone, along the z-direction. The extended cell contains several atoms (12 in our (GaAs),(AlAs), system) labelled by the index pair (Zg~). We choose now a definite sign for sg in the expression of the interplane contribution inside the cell (n = 0 term) by calculating equation (24) for two ions having a positive distance along the z-axis, i.e. r 1 ( Z;K’) > r , ( Z3~). Therefore, the summation over n for the c@ = xz, zx, yz, zy components is split into negative terms (sa = - 1 for - 00 < n < - 1) and positive ones (ss = + 1 for 0 < n < cc). All the other components do not change sign whatever n we consider. In detail we obtain: Q&L = $4

q3 113~3 6~‘) exp[i2w3(rl

(I$‘)

exp[-2nIG+Kj(rl(z;K’)-rI(~3~))]

- rL (z~K))]

L, Miglio, L. Colon&o / Lattice dynamics of polar semiconductor

x

lattices

495

+fexp[-(2nIG+KI-i2aqg)nd,] n=O

x +fexp[-(2?r(G+KI+i2aq,)nd,]

(25)

n=l

where

+~,,(G+K),+L(G+K)B

exp[-i2?rG*(R(~;K')-R(~3~))].

1

(26) In (25) the upper sign has to be used for xx, yy, zz, xy and yx components, the lower sign otherwise. By summing up the geometric series we have:

Q,,(K

q3 lb,

= C&

I&‘)

exp[i2v3(rl

KK’) - 5 (bd)]

exp[2aIG+KI(rl(/;K’)-rl(/,K)-d,)-i2Tq,d,] I?I

(27)

l-exp[-(2nIG+KI+i2nq,)d,]

If we consider now the case 1rL ( lj~‘) rl ( lg~) 1 = 0, we notice that the term n = 0 in summation (24) is just the in-plane contribution of eq. (23) and the remaining terms of the series can be summed up in the same way as for eqs. (25)-(27):

Q,,(K =

q3 l by b’)

Q&K

1b&K’) + CA, G

exp[-(2nIG+K)-i2aq3)d3] l-exp[-(2aIG+KI-i2aq3)d3]

exp[-(2aIG+KI+i2lrq,)d,] * l-exp[-(2nIG+KI+i2nq,)d,]

*

(28)

Finally, the rl (1;~‘) < r 1 ( 13~) situation can be treated in the same way as eqs. (25)-(27) by taking into account the ss = - 1 value for the n = 0 case. However, a more practical way is to take the Hermitean conjugates of the corresponding terms with interchanged positions.

496

L Miglio, L Colombo / Lattice dynamics of polar semiconductor

lattices

The dynamical matrix is easily constructed by taking into account eqs. (27) and (28). One problem still remains since, due to translational invariance [9-111 the diagonal terms contain elements evaluated at (K, q3) = (0, 0) (see eq. (1)): special care should be taken to avoid the singular terms when we consider lim, _ 0 for q3=0 and lim,S,, for K = 0. This is shown in the appendices where we demonstrate that any divergence is set to zero by the cell neutrality condition.

4. Application to the (G~As)&~IAs)~ (001) superlattke In this section we report a numerical application of the formalism that we outlined in section 3 to the (GaAs),(AlAs), (001) system. This latter is particularly suitable as a test case, since comparison is possible to other recent results. In particular, the 16-parameter shell-model calculation by Richter and Strauch [l] and the 11-parameter rigid-ion approach by Ren et al. [2,3] provide a good basis for the discussion of the model-dependent dynamical structures in superlattices. In the present work we make use of the bond-charge-model (BCM) bulk dynamics [8], which was originally formulated to take into account the peculiar charge distribution in tetrahedral covalent crystals (Si, Ge, alpha tin, diamond). The success of this model suggested its extension to partially polar III-V semiconductors, whenever the neutron scattering data made the parameter fitting possible (as in the GaAs case). The important feature of the BCM is the subdivision between cores and bond charges (bc) (sharing the bonds between anions and cations in a 3 : 5 ratio), which reproduces the actual charge distribution in III-V materials. This suitable modelling, in turn, allows for a limited set of disposable parameters (6): cation-anion, cation-bc, and anion-bc short range central repulsion; bc-bc angular interaction (different for anion-centered and cation-centered clusters) and screened Coulombic potential among any charged particle of the system. A limited set of parameters is, in our opinion, a significant advantage in fixing the interface force constant: In fact no experimental determination is at present available and any rule of thumb to guess them is somehow arbitrary. For GaAs a very good agreement to neutron dispersion relations is provided by BCM [8] even if its displacement pattern in selected symmetry points of the Brillouin zone is still a matter of discussion [12]. Unfortunately, for the AlAs case no neutron data are available and the few indirect experimental results out of the zone center [13] are questionable. Therefore, while the parameters for GaAs are taken from the original best-fit evaluation by Rustagi and Weber [8], the values for AlAs are obtained by interpolating the ones for other materials in the same group (see the discussion in ref. [14]). However, it turns out that even by setting the mass for Al in place of that for Ga in the

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor

lattices

497

Table 1 Bulk BCM parameters used in the present calculation (All the force constant parameters (i.e. +‘s, B’s and Z’/c) are in units [ e2/u,], where e is the electron charge and U, the unit cell volume; the zone center frequencies (as calculated by BCM) and the atomic masses are in THz and amu, respectively)

Ion-ion 3+1-i 1 II

Ion-bc f&

bc-ion-bc Bl

Effective Z2/Z

m.,,,

6.16

5.80

2.36

2.21

16.05

15.48

5.36 8.24

5.79 8.54

potentials

B2

mcation

AlAs

potentials

f+;’

cc0 %0(O) +0(O)

GaAs potential

charge (Coulomb

potential) 0.187 10.88 8.88 8.01 69.72 14.92

0.180 8.16 11.91 11.07 26.98 74.92

bulk dynamics of GaAs, the dispersion curves for AlAs are nearly the same as the ones by the interpolation procedure. However, this choice, by far the most common one in superlattice calculations [l-3], drastically rules out the effect of interface force constant in the superlattice dynamics. As a consequence, we performed our calculations with two sets of parameters: (1) a simple mass substitution Ga-Al, retaining the GaAs force constants on both sides (so that comparison is possible with refs. [l-3] and (2) different force constants for GaAs and AlAs with arithmetic average between the two for any interaction crossing the interface. Table 1 displays all the values that we used for both calculations of the (GaAs),(AlAs), superlattice, while its unit cell and Brillouin zone are shown in fig. 1. The planar approach has been adopted for the short-range force constants too: In fact we evaluated both in-plane and interplane contributions by back-Fourier transforming, along the (001) direction, the short range dynamical matrix for bulk GaAs and AlAs. This is a very rapidly converging technique, which we have recently outlined in the calculation of slab force constants for the Si(ll1) surface dynamics [15]. In figs. 2 and 3, we report the dispersion curves (solid lines) for the superlattice phonons, along the borders of the Brillouin zone in the (001) plane

498

L. Miglio, L. Colombo / Lattice dynamics ofpolar semiconductor

lattices

-v -x

‘V

Fig. 1. Unit cell and Brillouin

zone for the (GaAs),(AlAs),

(001)

superlattice.

(K # 0, q3 = 0), by (1) and (2) parametrizations, respectively. Shaded areas represent the projection onto the (001) plane of GaAs and AlAs bulk bands and show the frequency regions where bulk-originated extended or confined modes are expected. This picture, which we formerly introduced for some preliminar calculation of the GaAs/Ge dynamics [16], allows us to share straightforward the solutions among extended, confined and interface modes [17], depending on the frequency position of the branch (bulk GaAs and bulk AlAs, bulk GaAs or bulk AlAs, and gap regions, respectively). The effects of the bulk dynamical model on the superlattice dispersions can be seen by comparison of fig. 2 with the results of refs. [1,3]. In particular, the optical bulk bands of AlAs are nearly dispersionless and well separated in frequency between LO and TO components for the rigid-ion approach [3], while a very similar behaviour is found between shell-model [l] and bondcharge-model calculations, where no gap is present. As a consequence, the most important difference among the three superlattice dynamics is related to the interface mode between LO and TO bands (Fuchs and Kliewer mode for AlAs at small K) that is clearly visible in ref. [3], whereas it merges into the bulk bands for shell model and bond-charge model approaches. Generally

L Mglio,

z

z

L. Colombo / Lattice dynamics of polar semiconductor

FREQUENCY

( CM -’

)

z

z

=:

=:

z

0

lattices

0

500

L. Miglio, L. Colombo / Lottice dynamics of polar semiconductor

FREQUENCY 0 lz :

( CM-’ 2 PI

lattices

) z

8

0

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor e-v0

8.0’

r

r

lattices

501

2

:rz M

Fig. 4. Dispersion (TM) directions,

‘1

relations for parametrization scheme (1) (see text), along (001) (TZ) and (100) plus an angular plot for small q vector ranging from (001) (0 = 0) to (100) (0 = 90) directions (TT).

speaking, we note that even for the lower part of the frequency spectrum a better agreement is achieved between our model and shell model then to the rigid-ion model. Another interesting comparison can be made between fig. 2 and fig. 3: we see that in the first case only a few interface modes fall in the gaps at zone borders (XM direction), whereas in the second case a general softening of the optical bands is present (probably related to the averaged dielectric constant) and many more branches fall in the gaps. Furthermore, the latter situation displays several splittings of symmetry-degenerate frequencies at X and M points. These effects are produced by the mirror symmetry breaking in the force constants for the As atoms lying on the interface plane (actually, no explicit equilibrium conditions have been imposed there). We do not know how much physics is inside such pathologic behaviour, still it is important to point out the sensitivity of several superlattice branches to the choice of interface force constants. Figs. 4 and 5 represent the dispersion relations for parametrization (1) and (2), respectively, along (001) (l?Z) and (100) (f+M) directions, plus an angular plot for small q vector ranging from (001) (B = 0) to (100) (0 = 90) directions (lT). We note that a sizeable anisotropic behaviour between in-plane and normal-to-plane directions is found for two

L. Miglio, L Colombo / Lattice dynamics of polar semiconductor lattices

502

150

100

50

0

M

I-

r

%

Fig. 5. The same as fig. 4 for parametrization scheme (2) (see text).

optical branches in the AlAs region and two other ones in the GaAs band. Fig. 4, in particular, qualitatively agrees with similar plots of refs. [1,3] and fig. 5 displays the same qualitative information, except for the softening of the optical bands and the degeneracy removal that we already mentioned. We do not discuss here the features of the (FZ) dispersion since they are well known from linear-chain investigation [17]. Still, it is worthy to note that along this symmetry direction the sharp classification of the branches between transverse and longitudinal do not allow for any anticrossing behaviour that we see quite frequently for in-plane dispersions. Finally, we note that anisotropic confined branches along the lT direction turn to interface localized displacement patterns as they leave the r point along FM or TX. This numerical result was already mentioned in ref. [3], still the physical origin of the relationship between macroscopic interface modes and anisotropic branches deserves some more discussion and will be treated in the next section.

5. Single slab versus multislab dynamics A useful comparison between single slab (SS) and multislab (MS), i.e. superlattice systems may be drawn by considering the dispersion branches of

L.. Miglio, L Colombo / Latticedynamics of polar semiconductor lattices

503

the latter for (K # 0, cI~= 0) wavevector. Although all the MS modes are virtually proper eigenfrequencies of an extended-cell bulk crystal, in case of well separated optical bands for the two components (say GaAs and AlAs), it si convenient to share among confined (c), microscopic interface (mi) and macroscopic interface (Mi) modes (see also the former section and ref. [3]). The c modes are referred either to AlAs or to GaAs and are determined by the dispersion of the bulk optical branches for the corresponding material. In particular, the numerical values of these vibrations for a fixed K are generated by the frequencies of their respective branches in the bulk at one set of q3 values, which are integer multiples of the bulk reciprocal vector, divided by the number of effective atomic layers contained in that slab. This is the same rule which is used to obtain the surface-projected bulk modes in the SS, when a slab adapted Brillouin zone approach is used [15,18]. Thus, as long as the effects due to the different boundary conditions between the surface and the interface are neglected, a straightforward comparison may be drawn between the surface-projected plot of an n-layers SS and the corresponding confined modes for a MS. For what concerns the mi vibrations, i.e. the ones originated by the microscopic force constant changes at the interface, a comparison with the surface counterparts is quantitatively meaningless. In fact, the microscopic situation for a GaAs/AlAs interface is completely different with respect to the ones for a GaAs/void, A&s/void cases. Qualitatively speaking, we may expect stiffer frequencies for the interface atoms than the co~~pon~ng vibrations for the loosely bounded surface atoms. This, in turn, is very likely to produce a merging of the mi modes into the c bands for most of the K values. Finally, the case of Mi branches deserves some more explanation before a straightforward comparison with the slab counterparts can be made. The former are produced by the symmetry breaking of the macroscopic dielectric properties of the MS, along the growth direction. Therefore, their frequency position can be predicted either by a dielectric continuum approach or by a lim K _ OOMifor a microscopic, i.e. lattice dynamics, calculation. The Mi vibration for a material/void case, SS situation, is the Fuchs and Kliewer mode f6]: it is well known that a degenerate pair of such sagittal surface phonons is present, between the LO and the TO bulk bands at small K, in any polar material. As K ---, 0 the two branches acquire a bulk character and split increasingly, merging into the LO and the TO branches, separately. Actually, this behaviour is very similar to what happens in the polar plot of fig. 4, where we note that, for GaAs and AlAs separately, the anisotropic pair of modes splits in frequency and merges to the LO and TO region, as ZC vanishes and q3 increases. Physically speaking this is not too strange: the anisotropy of the macroscopic field is responsible both for the appearance of Mi modes and for their angular dispersion at the zone center, in the case of

504

I_. Miglio, L Colombo / Lattice dynamics of polar semiconductor lattices

superlattices. In particular, the absence of macroscopic field for any propagation direction in the K plane of a SS (no infinite planes of dipoles are present, normal to that direction) corresponds to the kl+ 0, q3 # 0 situation for a MS (the stripes of dipoles in the infinite planes acquire a q3 phase factor that destroy the macroscopic field). Therefore, at small K it is correct to compare Mi dispersion curves for a MS with those for a SS, yet as K--j 0 a small q3 # 0 should be considered for the superlattice branches. Quantitatively speaking, it is easy to demonstrate that the anisotropic pair of branches in fig. 4 (both in GaAs and AlAs) converge, for 15:f 0, q3 = 0 (8 = 90 ” ), to the frequency of the Fuchs and Kliewer modes (8.4 and 11.4 THz, respectively). These values are calculated in the appendices by a simple dielectric continuum approach for the GaAs/AlAs interface; it turns out that they are slightly softer than the ones corresponding to the GaAs/void and ALAS/void cases. From the lattice dynamics point of view we have numerically checked that the dispersion along a 1y symmetry direction for the Mi modes with a small q3 # 0 value in the (GaAs),(AlAs), system has the same behaviour as the SS case. Still, better insight is obtained by the comparison between long-wavelength interplanar force constants for a finite system and those for a superperiodic one, i.e. a Xr -+ 0 analysis of eqs. (27) and (28) with respect to (15) and (23). This is explained in the appendices. Here we discuss just the most significative results: lim eq. (15) = eas(K-

K-r0

0

113~li~‘)

xexp[-2~lGllrl(l;K’)-r((IjK)I] Xexp[ -i2rrG*(R(/;fc’)

-R(r3K))].

(29)

L. Miglio, L. CoIombo / Lutiice dynamics of polar semiconductor lattices

rl (,;K’)

- r~ (/SK) d,

xexp[ -i2?rG*(R(/;K’)

505

1 * 2aIKld,

-R(~JK))]

exp[-2alGl(rl(l;K’)-~I(fjK))] l-exp[-2a]GIds] +

-

l-exp[-2nIG]d,]

K ~~~+c eq. (27)=Q,,(K+O, - 7 =-

1.

exp[2nIGI(rl(l;~‘)--TI(~~~)--d3)]

q3+OI13KI;K’)

4n2

$a [

- +$

I K I +$dKJ

+ V,,

+ iL(Q

xexp[i2w3(rl (W) -rl(b))] 1 1 - exp[i2aq,d,]

G#O

T

G,GB

1

+x!$_

m

a

Xexp[i2nq,(rl

(I;K’)

+

-

1

1 1 - exp[i2sq,d,]

L$, rl

I G I +$,(G,)

I

+ iL($)

(IgK))] eXp[ -i2sG*(R(l;K’)

exp[2nIG](rl(1;K’)-rl(lj~)-dj)-i2nq,d,] *

(30)

l-exp[-2n]GIdg+i2~qjd3]

1 -R(lsK))]

I-

(31)

Now, taking into account that the superlattice is super-periodic in the growth direction, we consider the following situation: (i) d, is large with respect to the interatomic distance; (ii) q3 is very small since the Brillouin zone is folded back along that direction and we consider a small part of it. (iii)

506

L. Miglio, L. Colombo / Lattice dynamics of polar semiconductor

1G ( is large with respect

to J(K, q3) 1. Therefore

lattices

we have that:

exp[ - 2s 1G ) d3] + 0, exp[i2aq,(

rl (I&‘)

- rl (I+))]

+ 1

and eq. (31) approaches (29), while eq. (30) still displays a qualitatively different behaviour. In particular, for the in-plane components a term like K,K,/ 1K * I is present, which reminds us the macroscopic field contribution for bulk infinite systems. The physical interpretation of the latter feature is that for q3 = 0 a macroscopic electric field is provided by the in-phase displacement of striped infinite planes of dipoles, yielding a contribution which is equivalent to the one of a bulk crystal with an effective dielectric constant. Therefore, the Fuchs and Kliewer pair for the slab can be compared to the anisotropic branches of the superlattice at small K only if a q3 different from 0 is allowed. Actually, if a projection of the superlattice dispersion relations is performed onto a surface (or interface) plane (summation over q3) these macroscopic modes broaden into a band as K -+ 0: This feature is also obtained by a dielectric continuum approach performed to interpret electron energy loss measurements from superlattice surfaces [19]. In conclusion we think that for what concerns the macroscopic interface modes in polar superlattices (multislabs) a fair amount of information can be obtained by comparing to the corresponding surface modes for single slabs. Unfortunately, the same is not true for the microscopic modes originated at the interface, because the atomic environment is different. Moreover, no fitting procedure as in the case of surface phonons is here possible due to the lack of an experimental probe comparable to the inelastic atom-scattering.

We are very grateful to the Supercomputer Computations Research Institute and to the Center for Material Research and Technology of the Florida State University in Tallahassee for the financial and computational support and for the kind hospitality. We thank Giorgio Benedek (University of Milano) for the careful reading of the manuscript and many helpful comments.

Appendix vibrations

A. Estimate

of the frequency

values

for macroscopic

interface

We consider wavevector values which are sufficiently small to allow for a macroscopic approach (and dispersionless behaviour of wLo and wro), still

L.. Miglio, L Colombo / Lattice dynamics of polar semiconductor

lattices

507

outside the region where the photon-polariton interaction occurs (unretarded field approximation, c + co). From Maxwell equations for an interface between two dielectric media (a and b), we obtain the dispersion relations for the electromagnetic modes [20]: (A.la)

(A.lb)

(A.lc) where the conditions for the appearance fo interface modes require real K and . . imagmary q3, that is: c= + cb < 0,

(A4

and

ca< --e b ,

(A.3a)

or

Zb< -ea.

(A.3b)

In the frequency region where the dielectric constant of one medium is negative, i.e.: CO-E,

c(w) = cm +

l-

b/%o)2

<

0,

(A.4)

therefore, that medium is interface actiue. Now, for the GaAs (a)/AlAs (b) interface, we have o!+.o< oio -Co\o -Cor& and conditions (A.2) and (A.3) are satisfied in two distinct regions [20]: o;o < w < w;,

GaAs interface active,

(A.5a)

&o
AlAs interface active.

(A.5b)

Here wg and 0: are the upper limits of the polariton dispersion curves, i.e. the frequencies of the Fuchs and Kliewer modes for large K. To obtain their values we substitute the explicit expression (A.4), for both a and b media, into eq. (A.la) and set K + 00, i.e. C’(w) = -rb(o). In this way we recover a

508

L Miglio, L Colombo / Lattice dymzmics

ofpolar

semiconductor

lattices

quadratic equation in a*, which provide us with the following solutions [20]: J

I

=

s

2(t;

+c”m) (

e: @aLo)* + (&o)‘) (

+ C((&o)‘+

i[[fBq~(wrLo~2+(wbTo)2)+~~(~~bLu)2+ -4(r;

Woo)*)

two*)“)]”

+fL)(Z”m(WaLo)2(WhTO)2+~~(gbLOf2(Oar0 2 I’* . ))I

i

(A-6)

In table 1 are displayed the actual values of z,, wro, wLo for both GaAs and AlAs: the final results for the interface frequencies are 8.4 and 11.4 THz, respectively. These values are slightly lower than the GaAs/void and AL&,/void cases, i.e. 8.8 and 11.8 THz, respectively. Appendii constants

B. Small wavevector

limit of superlattice

and single

slab force

At small values for K and q3, we obtain the following limits for single slab and superlattice planar force constants: lim eq. (15) = $&(K

x-0

+ 0 113~lj~‘)

xe~[-2,lGrj~,fr;K’)-I,(~3K)/]

Xexp[ -i2?rG*(R(I;K’)

- R(l+))]3

lim eq. (23) = Q+(K -0

1 13~13K’)

K+O

=P3~f&&+(&i,K’)

(B-1)

-‘(ol,K)I)

L

-P+

c

exp[-i2sG*(R(013K’)-R(013K))]nIGl

G#O

&erfc(aIGl/P)-

~~~d-~21G12/f’2~])7

(B.2)

L. Miglio, I. Coiombo / Lattice dynamics of polar semiconductor lattices

lim

K-+O,q,-+O

eq. (27) = Q,@(K --, 0, q3 + 0

lKl

113~1;~‘)

i2wq3(r1 -

rl

(/3K))]

I1

- rl ( 13~)

l+WKI(q(+‘)

T

1

+s,,s,,lKI+iss=(K,)+is,,(Kg)

(/;K’)

-

-

2~~~~1k;,(~i~*~~~3(‘K)’ 3

d3)

-

i2aq3d3

2~ 1K 1d, - i21rq3d,

I

1

~+S.,S,,IGl+isB.(C,)+iS.,(Gg) x[l +i2aq3(rl(l$c’) x

1 - exp[ -2a

1G 1d,](l - i2?rq,d,)

exp[2n IG I(rl

lim

exp[ -iZaG*(R(I;K’)-R(13K))]

exp[-2slGl(r,(f;K’)--rl(13K))] [

f

-rl(l+))]

509

(IRK’)

-

rl

(13~)

-

d,)](l

- i2sq3d3)

1 - exp[ -277 1G 1d,](l + i2aq3d3)

I-

(B-3)

eq. (28) = Q,,( K + 0, q3 --, 0 113~13~‘)

R+O,q,+O

- JKJ l-2eIKId,-i2aq3d3

+ h,$,

+i$dL)

+4&$)

r 1-2aIKId3+i2rq3d3

2~ 1K 1d, + i2?rq,d,

Xexp[ -i2rG*(R(I$c’)

WI

2n 1K 1d, - i2nq3d3

1

1

-R(/,K))]

exp[ -2n 1G 1d,](l - i21rq3d3) l-exp[-2~1GId,](l-i2sq,d,) exp[-2?rIGId3](l+i2?rq3d3) * I-exp[-2nIGId3](l+i2rq3d3)

1*

(B-4)

510

L. Miglio, I_. Colombo / Lattice dynamics of polar semiconductor lattices

Now, by setting q3 = 0, we re-write (B.3) and (B.4) and find the expression for small K values: K iyj=, +

eq. (27) = Q,,(K

+ 0, q3 = 0 113~I;~‘)

7

xexp[ -i2rG*(R(I;K’) x

exp[-201GI(r,(I;K’)--Ti(lj~))] i

+

-

-R(/,K))]

l-exp[-281GId3]

ex&WGl(rJW)

-QW

-41

l-exp[-2aIGId3]

lim eq. (28) = Q,,( K + 0, q3 = 0 K+O,q,=O =

eq. (B.2) + F

1’

(B.5)

) ljKI,K’)

a

1 ’ [ 2rlKId,-1+

1

2nIKl4

i ‘-

$$

+4&,IGI

Xexp[ -i2TG*(R(I;tc’)

-

+i&(Ga)

+iL(%>

1

R(~~K))]

1 exp[+2nIGld,]-1

)I

’

1 exp[+2m(GId3]-1

1.

(B.6)

L Miglio, L. Colombo / Lattice dynamics of polar semiconductor

On the contrary, if we set K = 0 before small q3, we obtain:

+ ew[2aIGl(rl(lh’) -

-rl(b)

evaluating

-&)I(1

the limiting

-i2w4)

l-exp[-2aIGldJ](l+i2nq,d,)

[eq.(B.2)forK=O]+O+X

X

[

behaviour

at

(B.7)

1G 1d,](l

- i2aq3d3)

1G 1d,)] (1 + i2aq,dg)

l-exp[-2alGId,](l+i2~q~d~)

1

-R(13~))]

l-exp[-2nIGld,](l-i2sq,d,) exp[ -2n

’

+*,,s,,IGI+iss,(G,)+iS~=(GB)

exp[ -i2aG*(R(l;K’) exp[ -2n

x

1.

511

&-lzb(-&-I)] [

c,c,

lattices

1.

03.8)

In the case of small, in-plane wavevectors, i.e. (B.5) (B.6), we see that a non-analytic behaviour arises for the G = 0 terms, while the single slab expressions (B.l) and (B.2) do approach a finite value. However, a useful comparison can be drawn between single slab limits and superlattice ones, for K -+0 and q3 # 0. In the text it is shown that no divergences appear in this case and that for sufficiently large values of d, the superlattice terms (B.3) and (B.4) approach the single slab format as (B.l) and (B.2). For what concerns the diagonal contributions in the dynamical matrix (see eq. (1)) i.e.

(B.9) the (K = 0,q3 = 0) elements do not provide any divergency is dropped out in the summation over reciprocal vectors.

if the G = 0 term This procedure is

512

L.. Miglio, L. Colombo / Lattice dynamics of polar semiconductor

lattices

allowed by the fact that this term vanishes by cell neutrality condition. In fact, both (B.5) and (B.6) provide non-analytic terms that are equal and independent on l;‘~“, so that a summation over all the effective charges in the supercell may be extracted from the summation (B.9), i.e.

z /“.” it?“I(”

3

)x co+)x +)

and the G = 0 contribution

c ,I

13r

is erased by the cell neutrality

condition

z,,,lT = 0. I,

References WI E. Richter and D. Strauch,

Solid State Commun. 64 (1987) 867; E. Richter, Gitterdynamik in GaAs/AIAs Uberstrukturgittem, Diplomarbeit der Universitat Regensburg (1986). I21 S. Ren, H. Chu and Y.C. Chang, Phys. Rev. B 37 (1988) 10746. 131 S. Ren, H. Chu and Y.C. Chang, Phys. Rev. B 4 (1988) 8899. [41 H. Chu, S. Ren and Y.C. Chang, Phys. Rev. B 37 (1988) 10764. I51 G. Benedek and A. Fray, Phys. Rev. B 21 (1980) 2482. I61 R. Fuchs and K.L. Khewer, Phys. Rev. 140 (1965) 140; W.E. Jones and R. Fuchs, Phys. Rev. B 4 (1971) 3481. 171 G. KanneIIis, J.F. Morhange and M. Balkan&i, Phys. Rev. B 28 (1983) 3390; 3398, 3406. PI W. Weber, Phys. Rev. B 15 (1977) 4789; K.C. Rustagi and W. Weber, Solid State Commun. 18 (1976) 673. E.W. Montroll, G.H. Weiss and I.P. Ipatova, Theory of Lattice Dynamics 191 A.A. Maradudm, in the Harmonic Approximation, Solid State Physics, Suppl. 3 (Academic Press, New York, 1971). WI P. Bruesh, Phonons: Theory and Experiments, Vol. I (Springer, Heidelberg, 1982). 1111 M. Born and K. Huang, DynamicaI Theory of Crystal Lattices (Oxford Univ. Press, Oxford, 1954). WI K. Kunc, R.M. Martin, Phys. Rev. B 24 (1981) 2311. u31 A. Ontan and R.J. Chicotka, Phys. Rev. B 10 (1974) 591. 1141 S. Yip and Y.C. Chang, Phys. Rev. B 30 (1984) 7037. P51 L. Migho, P. Ruggerone, G. Benedek and L. Colombo, Phys. Scripta 37 (1988) 768. WI L. Colombo and L. Migho, Vuoto 28 (1988) 37. P71 E. Molinari, A. FasoIino and K. Kunc, Superlattices Microstruct. 2 (1986) 397. WI F.W. de Wette and G.P. Alldredge, in: Vibrational Properties of Solids, Ed. G. Gilat, Vol. 15 of Methods in Computational Physics (Academic Press, New York, 1976) p. 163. 1191 A. Dereux, J.P. Vigneron, P. Lambin and A.A. Lucas, Phys. Scripta 38 (1988) 462. PO1 P. HaIevi, Opt. Commun. 20 (1977) 167.