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In-plane misfit strains in heterostructures and superlattices: Arbitrary direction of growth
~
0038-1098/9153.00+.00 Pergamon Press plc
Solid State Communications, Vol. 78, No. 5, pp. 347-350, 1991. Printed in Great Britain.
IN-PLANE MISFIT STRAINS IN H E T E R O S T R U C T U R E S AND S U P E R L A T F I C E S : A R B I T R A R Y DIRECTION O F G R O W T H
E. Anastassakis
Department of Physics, National Technical University Zografou Campus, Athens 157 73, Greece (Received March 4 1991 by T. P. Martin) The in-plane isotropic strain due to lattice mismatch of artificial heterostructures and superlattices grown along an arbitrary direction, is derived by minimizing the elastic free energy. The results are expressed in terms of the lattice constants and elastic constants of the two material components. Specific applications are presented, involving standard and non-standard directions of growth in three of the most common superlattices.
is characterized
Cubic heterostructures and superlattices grown artificially along an arbitrary crystallographic direction are increasingly attracting interest, because of the extended range of physical and technical properties, imposed by the higher degree of anisotropy. The complete component array of the strains/stresses and the elastic energy density for each strained layer in such tailored structures were derived recently using• elasticity,1 theory, and were put in simple analyttcal forms. The strain (t;~) and stress (o~) tensor components were
by the bulk lattice constant a
O(t ~
ct relative to O, and a the elastic compliances Cil, thickness h a , ct=l,
2.
Unless necessary,
in
what
follows the material (layer) index a is dropped, from those parameters or equations that refer to each material individually in an obvious way; parameters carrying the index o, such as a o, G o, etc., refer to She syste~ O, otherwise they refer to the system O (e.g., a , As, G). It is also understood that because of the pseudomorphic growth assumed, the geometrical quantities N, l, m, n, are common to both materials, i.e., independent of a. At the heart of the present analysis is the tetragonal distortion as well as the elastic free energy density of each strained layer. These quantities have been derived for arbitrary N in Ref. 1, and are given by
t
expressed in the system of axes 0 : x~x~x~, where x~, x~ define the plane of the interface and x~ is normal to it. All strain/stress components w e ~ found tt{t depend linearly on the in-plane strain g , where ~ ~ ~:~- ~ . The elastic energy density was found to be vary a s (gll)2. The in-plane strain II, associated with the difference between the thermal expansion coefficients of the two materials, or their lattice mismatch, or both, was treated as a known parameter. It can be measured with X-rays or spectroscopic techniques, and has been calculated for the high symmetry directions [001], [111], [110] (Ref. 2). Otherwise, the necessary imput information for applying the results of Ref. 1 are the second-order elastic stiffnesses C..q of each participating material, expressed in the system O of the crystallographic axes [100], [010], [001], and some simple geometrical functions. For an arbitrary d~rection of growth, a theoretical estimate of s from elastic considerations is not trivial and, to our knowledge, this problem has not been dealt with so far. It is the p u r p o ~ of this work to obtain an analytical form for ~ m the case of an arbitrary direction of growth NIl[lmn]Ux~, where 1, m, n the direction co-
Ae
=
611-g-L
= - -3B~II -X---
[C2+CC,(I-T)+3C2(Imn) 2]
U(~") = ~ O (,)2,
(1)
(2)
where A = CIICL+
(CC,14/2)(C11 + C12)( 1-T)
+ C2(Cll + 2C 12+ C44)(lmn)2'
(3)
and ~± is the normal-to-the-plane strain component, B=(Ctt+2Ctz)/3 is the bulk modulus, C-=Ctt-C~z-2.C44 is the anisotropy factor, and T= 14+ m + n is a geometrical factor. G is a modulus given by G=3B(3-A~),
(4)
A c = At;/g II is the reduced tetragonal distortion, i.e., a dimensionless number which can
where
sines of N relative to O. For simplicity, the analysis is first carried out for a free-standing superlattice. The results are easily adapted to an epilayer grown pseudomorphically along N, on top of a substrate. Each layer of the free-standing superlattice
be computed directly from Eq. (1). For later use we rewrite Eq. (4) in a slightly different form, i.e., G = gO ° , 347
(5)
348
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
where g=
2(C11.C 1 2)
,
(6)
and Go=6B(1-C12/C 1 l). G
o
(7)
is the familiar from the literature "shear modu-
lus" for growth along [001] and can be easily derived from Eqs. (1), (3), (4) simply by setting l--m--0, n = l . Thus, Eq. (6) provides the necessary factor for converting G O to G. Before proceeding to the main objective of this analysis, ~.e., the determination of the inplane strain ca' it is necessary first to comment on the definition of the lattice constant which is appropriate for an arbitrarily oriented crystallographic plane, i.e., one that is normal to N. The lattice constant a introduced earlier refers to a oct
unit cell oriented according to the system O of the unstrained cubic lattice. The lattice constant a
Ix
which describes an appropriate unit cell oriented according to the system O before the layer is strained, is scaled relative to a through a geooct
metrical factor k which depends only on N, i.e., act.= haoct" (8) Explicit knowledge of I~ is not really necessary for obtaining the strain e , as will be shown, but it is appropriate to introduce it in the analysis in order to make the latter more concise. [See also a relevant remark in Ref. 2, following Eq. (2)]. It . Suppose now the (unknown) strain e is prect
sent, that is, the two types of layers have been interfaced as a free-standing dislocation-free superlattice. The lattice constant a now becomes anisotropic. In the plane normal to N, it exhibits a H
common value for both layers, i.e., a II ~ a~ = a~,
Vol. 78, No. 5
heffG + h2ff 1 o 1 ao I Go2ao2 = ell h e f fG + h 2 k.i t
ol
(12)
02
with heft , (13) a = gixna II Thus aeff can be calculated directly from Eqs. (11) or
(12). The latter equation has an interesting II with the inphysical meaning: it identifies aeff plane lattice constant of a free-standing superlattice, growneff along ~01], with effective layer thtcknesses h 1 and h 2 . We call this the equi-
valent [001] superlattice. In view of Eqs. (10), (8), the in-plane strain of Eq. (9) now becomes eix = (a ff-aoa)/aoc t,
(14)
and can be calculated explicitly. Equation (14) is the key result of the present work. All other quantities such as, tetragonal distortion, strain and stress components, free energy density, can now be determined, and the only input information necessary consists of N, caj, and aoa. The normal-to-the-plane lattice constant is, by definition, aX a =aa(l+e&a )=a a [l+e~ (1-A~ix)]
(15a)
= ~.[a~ff( l-~ ~'ct)+ aoct A ~"a l
(15b)
_=ha-L
(15c)
eff,a
where
'
a -L
eff,ct
is the
normal-to-the-plane lattice con-
stant of the equivalent [001] superlattice. Thus, in the presence of misfit strains, Eq. (8) is replaced by the pair of Eqs. (10) and (15c). Since the entire computation is based on A~', we give below A e and the main results for some standard and non-standard directions N:
whereas parallel to N each layer exhibits its own value, i.e., aJi1 ~ a-~ ~* a II. Only the in-plane value a
II II
.
ts necessary for determining six, .
ect = [a
II
.,
-aix)/aix.
(9)
In order to determine a II it is necessary to minimize the elastic free energy. For this purpose we consider a rectangular parallelepiped of volume V = V l + v 2 , where vct=A(all)2hct and A is the number of periods of. the superlattice. The elastic ener.gy stored in v is U=UI+U2,where u (It =v IXU IX and U ~s given by Eq. (2). The requirement that (du/da I) =0 combined with Eqs. (2), (8) and (9), leads to the following equation, which characterizes the system as a whole, and not lust the individual layers, "
"
a =~. aeff ,
(i) For N=[001] we have, from Ref. (1), A ~ = 3B/Cll and from Eqs. (8), (6), and (5), k---1, g = l and G-=G°,
II
0o)
where
respectively. These equations hold for each material independently. Then, a
II
=
II
aeff
---
hlG la o l+h2G2ao2
(16a)
h i G 1 +h2G 2
which is ide~i~al to the usual expression found the l i t e r a t u r e . The explicit form of G Ix Eq. (7), also agrees with that in Refs. 2-4 but basically different from that used in Refs. 5, The in-plane strain for layer 1 is II neff
II
E1
" not
h2G2Aa/a o 1
(16b) aol hiG1+ h2G 2 where Aa=ao2-aotiS the lattice misfit, and likewise for layer 2. =
,
(ii) For N=[I10]/v~ we have I, biG1 a ol +h202ao2 aeff= hlG1 +h2G 2 II
(11)
in i.e. is 6.
A e -- 6B/(Clt+C~2+2C44 ) and, from Eq. (4),
Vol. 78, No. 5
349
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
O = 3B(C11-C12+6C44)/(C11+C12+2C44). It and e ct II are obtained from equations analogous aeff to Eqs. (16a) and (16b) respectively.
Table I. Numerical values for the various quantities described in the text for superlattice Si/Ge, i.e., ct=l(2) corresponds to Si(Ge). The lattice constants aol(o 2)= 5.4310 (5.6579) in ,g,, and the elastic constants CIt =
(iii) For N= [l l l]/C'J we have 1, A ~" = 9B/(Ct! + 2C12 + 4C44)
(17a)
G = 36BC44/(CtI+2C12+4C44 ).
(17b)
II a n d aeff
165.8 (128.5), Ct2=
63.9 (48.3), C44= 79.6 (66.8) in GPa are taken from
t~ct II are obtained from equations analogous
to Eqs. (16a) and (16b), respectively. The above expressions for Gltl01, GIlnl coincide with those stated in Ref. 2.
Ref. 8 (300K). The first, second, and third row of results in the three right-most columns correspond to h 1
1-A ~'. a
Nti
A~
G
II
a eff
e
II
a -L e ff,ot
GPa (iv) For NIi[11~]/¢'6 we have 1, A = [3(Cu-C12 +C44)(C11 +C12+2C44)-C 2 ]
Si
1.7708
Ge
1.7517
Si
1.5098
Ge
1.4503
Si
1.4392
Ge
1.3717
Si
1.5336
Ge
1.4812
Si
1.6216
Ge
1.5820
t001]
×(C11"C12+ 4C44 ) / 1 0 8 C 2
A~" = (B/6C2A)(Cll-C12+C44)(Cll-C12+4C44). G = 3B(3-A~ ). II
and ~
aeff
are obtained from equations analogous
[ 1101
to Eqs. (16a) and (16b), respectively. (v) For N=[120]/V'5 we have, 3 B ( 8C1 1 " 8 C 1 2 + 9 C 4 4 )
A~ =
[111l
4(C211- C2I 2)+C4 4 (1 7C l 1-8C12 ) G = 3B(3-A~). II a n d aeff
t~ctII are obtained from equations analogous
to Eqs. (16a) and (16b) respectively. The present analysis is readily applicable to strained epilayers (ht) grown on top of substrates (h2), along N. If h2~h I then II aeff= a ea-ff,2=ao2
[ l 1~1
[ 1 20]
and, from Eq. (15b, c), a-Left,1 = ao2"AaA 81
1.829 5. 6579 0 5. 5303 5. 4310 0 -2. 255 281.0 -4.010 4. 178 1.853 437.5 5. 6579 0 5.5316 5. 4310 0 -2.231 348.8 -4.010 4. 178 1.857 458.2 5. 6579 0 5.5318 5. 4310 0 -2. 228 366.5 -4.010 4. 178 430.5 1.849 5. 6579 0 5.5314 5.4310 0 341.9 -2.235 -4.010 4. 178 404.7 1.842 5. 6579 0 5.5310 5. 4310 0 3 19.2 -2.242 -4.010
360.9
5.3544 5. 4310 5.6579 5.7538 5.8284 5.3153 5.3796 5.4310 5.6579 5.7147 5.7600 5.3313 5.3867 5. 4310 5.6579 5.7047 5.7422 5.3099 5.3774 5. 4310 5.6579 5.7187 5. 7670 5.2899 5.3688 5 . 4310 5.6579 5.7317 5.7899
Accordingly, ~2II = ~-~ = 0 and for
~1II =
Table I coincide with the results tabulated in Ref. 2. The calculated values for the lattice constants in Ref. (2) appear to correspond to the present lattice constants all and a -l-. Actually they should it
Aa/aol
all directions N, whereas =
be
•
All these results have been applied to three standard types of superlattices, i.e., Si/Ge, GaSb/AISb and ZnTe/ZnSe. We consider three combinations of h t, h2: (i) ht~h 2, i.e., Si epilayer on top of a much thicker Ge substrate, (ii) h t= h 2, i.e., the free-standing symmetric superlattice, and
(iii) hl)h2,
i.e.,
Ge
epilayer on
top
of a much thicker Si substrate, and likewise for the other two superlattices. The results of these calculations are compiled in Tables 1-1II. Some of, the entries in the first three directions N of
interpreted as coinciding with our effective lattice constants a eUf f and a -L eft,c? respectively. It is interesting to notice in Tables I-III that, for all directions N the calculated parameters exhibit critical values (maxima or minima) for N along either [001] or [111], all other directions of N yielding values in between. In other words, Eqs. (16) and (17) each lead to the upper or lower bound of a II and e II Specits'ing the eff
ct"
type of critical point in each case depends on the relative sign and magnitude of C-=C11-C12-2C44 for rt=l, 2. Such behavior is dictated by the critical
350
Vol. 78, No. 5
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
Table II. Same as Table I for GaSb/AlSb. Here, ao1(o2)= 6.0959 (6•1355) in )~, Cll = 88.34 (87.7),
Table III. Same as Table I for ZnTe/ZnSe. Here, ao1(o2)= 6.1037 (5.6676) in .~, C l l = 71.3 (90.0),
C12 = 40.23 (43.40), C44= 43.22 (40.80) in GPa.
C12= 40.7 (53.4), C44= 31.2 (39.6) in GPa.
Ntl
ct
A~
G
a
II
~
II
e ff
a-t. e f f ,~x
a
Ntl
A~"
GPa 'GaSb
1.9 107
1 83.9
[OOl] AISb
1.9897
176.3
IGaSb
1.5701
241.4
[1 10] 'AISb
1.6408
237.2
IGaSb
1.4820
256.2
[111] AISb
1.5501
253.0
GaSb
1.6 103
234.6
[1121 AISb
1.6839
229.7
,GaSb
1.7247
2 15.3
AISb
1.8015
209.1
[ 12o1
GPa
5 0.318 0
6.1355 6.1152 6. 0959 0 -0•329 -0.646 0.650 6.1355 0,322 6.1155 0 6. 0959 0 -0.326 -0.646 0.650 6 . 1 3 5 5 0.323 0 6 . 1 155 6. 0959 0 -0•325 -0.646 0.650 6 . 1 3 5 5 0.321 0 6. 1154 6. 0959 0 -0.326 -0.646 0.650 0.320 6.1355 0 6. 1154 6. 0959 0 -0.327 -0.646
6. 0598 6.0782 6. 0959 6.1355 6. 1555 6.1746 6.0733 6. 0847 6.0959 6.1355 6. 1482 6. 1608 6.0768 6.0864 6. 0959 6.1355 6. 1464 6.1572 6.0717 6.0839 6. 0959 6.1355 6. 1491 6.1625 6.0672 6.0817 6. 0959 6.1355 6. 1516 6. 1672
points that the frse energy density exhibits along these two directions'. In conclusion, we have reached explicit forms for the in-plane isotropic strain in superlattices and heterostructures grown pseudomorphically along an arbitrary crystallographic direction. Essential to this calculation is the reduced tetragonal distortion A~ for which an analytical expression is already available. Knowing AT allows the calculation of the corresponding "shear modulus" G, from which the in-plane and normal-to-the-
1. 2. 3. 4. 5.
G
II aef f X
It E; aif f .1102) ,ct • 145
'ZnTe
2.1416
1 31.1
Iznse
2.1866
160.1
[ZnTe
1.7511
1 90.7
ZnSe
1.7681
242.4
ZnTe
1.6508
206.0
ZnSe
1.6 621
263.3
ZnTe
1.8076
1 82.1
ZnSe
1.8335
229.6
ZnTe
1.9401
1 61.8
ZnSe
1.9757
201.6
[OOll
[1 lO1
[1111
[ 1 12]
5. 6676 -3.928 0 5. 8639 6.1037 0 3. 464 7. 695 -7. 145 -3. 999 5. 6676 0 5. 8596 6.1037 0 3. 388 7. 695 -7.145 -4. 008 5. 6676 0 5. 8590 6. 1037 0 3. 378 7. 695 -7. 145 5. 6676 -3.985 5. 8604 • 0 6.1037
[ 1 2o1
0 3.403 7. 695 -7. 145 -3. 963 5 . 6 676 0 5.8618 6.1037 0 3. 427 7. 695
6.6015
6.3774 6. 1037 5.6676 5.4346 5.1500 6.4312 6.2870 6. 1037 5.6676 5.5202 5.3325 6.3875 6.2629 6. 1037 5.6676 5.5408 5.3788 6.4559 6.3001 6. 1037 5.6676 5.5068 5.3040 6.5136 6.3311 6. 1037 5.6676 5.4780 5.2}120
phme lattice constants are determined within a geometrical factor. It would be interesting to compare the results of such calculationsh with the corresponding experimental values of ~ obtained from high resolution X-ray diffraction data. Supported in part by the General Secretariat for Research and Technology, Greece. Useful discussions are acknowledged with 1. P. Ipatova, A.Yu. Maslov, and V. Shchukin.
REFERENCES E. Anastassakis, J. Appl. Phys. 68, 4561 6. (1990). 7. Chris G. Van de Walle and Richard M. Martin, Phys. Rev. B34, 5621 (1986). 8. G . C . Osbourn, J, Vac. Sei. Technol. B1, 379 (1983). Takashi Nakayama, J. Phys. Soc. Jpn. 59, 1029 (1990). Le Hong Shon, Koichi Inoue, and Kazuo Murase, Solid State Commun. 62, 621 (1987)•
A. Yu. Maslov, private communications. R. M. Abdelouhab, R. Braunstein, M. A. Rao, and H. Kroemer, Phys. Rev. 39, 5857 (1989). Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technolog)', Vol. 22a, edited by O. Madelung and M. Schuiz (Springer, New York, 1987).
0038-1098/9153.00+.00 Pergamon Press plc
Solid State Communications, Vol. 78, No. 5, pp. 347-350, 1991. Printed in Great Britain.
IN-PLANE MISFIT STRAINS IN H E T E R O S T R U C T U R E S AND S U P E R L A T F I C E S : A R B I T R A R Y DIRECTION O F G R O W T H
E. Anastassakis
Department of Physics, National Technical University Zografou Campus, Athens 157 73, Greece (Received March 4 1991 by T. P. Martin) The in-plane isotropic strain due to lattice mismatch of artificial heterostructures and superlattices grown along an arbitrary direction, is derived by minimizing the elastic free energy. The results are expressed in terms of the lattice constants and elastic constants of the two material components. Specific applications are presented, involving standard and non-standard directions of growth in three of the most common superlattices.
is characterized
Cubic heterostructures and superlattices grown artificially along an arbitrary crystallographic direction are increasingly attracting interest, because of the extended range of physical and technical properties, imposed by the higher degree of anisotropy. The complete component array of the strains/stresses and the elastic energy density for each strained layer in such tailored structures were derived recently using• elasticity,1 theory, and were put in simple analyttcal forms. The strain (t;~) and stress (o~) tensor components were
by the bulk lattice constant a
O(t ~
ct relative to O, and a the elastic compliances Cil, thickness h a , ct=l,
2.
Unless necessary,
in
what
follows the material (layer) index a is dropped, from those parameters or equations that refer to each material individually in an obvious way; parameters carrying the index o, such as a o, G o, etc., refer to She syste~ O, otherwise they refer to the system O (e.g., a , As, G). It is also understood that because of the pseudomorphic growth assumed, the geometrical quantities N, l, m, n, are common to both materials, i.e., independent of a. At the heart of the present analysis is the tetragonal distortion as well as the elastic free energy density of each strained layer. These quantities have been derived for arbitrary N in Ref. 1, and are given by
t
expressed in the system of axes 0 : x~x~x~, where x~, x~ define the plane of the interface and x~ is normal to it. All strain/stress components w e ~ found tt{t depend linearly on the in-plane strain g , where ~ ~ ~:~- ~ . The elastic energy density was found to be vary a s (gll)2. The in-plane strain II, associated with the difference between the thermal expansion coefficients of the two materials, or their lattice mismatch, or both, was treated as a known parameter. It can be measured with X-rays or spectroscopic techniques, and has been calculated for the high symmetry directions [001], [111], [110] (Ref. 2). Otherwise, the necessary imput information for applying the results of Ref. 1 are the second-order elastic stiffnesses C..q of each participating material, expressed in the system O of the crystallographic axes [100], [010], [001], and some simple geometrical functions. For an arbitrary d~rection of growth, a theoretical estimate of s from elastic considerations is not trivial and, to our knowledge, this problem has not been dealt with so far. It is the p u r p o ~ of this work to obtain an analytical form for ~ m the case of an arbitrary direction of growth NIl[lmn]Ux~, where 1, m, n the direction co-
Ae
=
611-g-L
= - -3B~II -X---
[C2+CC,(I-T)+3C2(Imn) 2]
U(~") = ~ O (,)2,
(1)
(2)
where A = CIICL+
(CC,14/2)(C11 + C12)( 1-T)
+ C2(Cll + 2C 12+ C44)(lmn)2'
(3)
and ~± is the normal-to-the-plane strain component, B=(Ctt+2Ctz)/3 is the bulk modulus, C-=Ctt-C~z-2.C44 is the anisotropy factor, and T= 14+ m + n is a geometrical factor. G is a modulus given by G=3B(3-A~),
(4)
A c = At;/g II is the reduced tetragonal distortion, i.e., a dimensionless number which can
where
sines of N relative to O. For simplicity, the analysis is first carried out for a free-standing superlattice. The results are easily adapted to an epilayer grown pseudomorphically along N, on top of a substrate. Each layer of the free-standing superlattice
be computed directly from Eq. (1). For later use we rewrite Eq. (4) in a slightly different form, i.e., G = gO ° , 347
(5)
348
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
where g=
2(C11.C 1 2)
,
(6)
and Go=6B(1-C12/C 1 l). G
o
(7)
is the familiar from the literature "shear modu-
lus" for growth along [001] and can be easily derived from Eqs. (1), (3), (4) simply by setting l--m--0, n = l . Thus, Eq. (6) provides the necessary factor for converting G O to G. Before proceeding to the main objective of this analysis, ~.e., the determination of the inplane strain ca' it is necessary first to comment on the definition of the lattice constant which is appropriate for an arbitrarily oriented crystallographic plane, i.e., one that is normal to N. The lattice constant a introduced earlier refers to a oct
unit cell oriented according to the system O of the unstrained cubic lattice. The lattice constant a
Ix
which describes an appropriate unit cell oriented according to the system O before the layer is strained, is scaled relative to a through a geooct
metrical factor k which depends only on N, i.e., act.= haoct" (8) Explicit knowledge of I~ is not really necessary for obtaining the strain e , as will be shown, but it is appropriate to introduce it in the analysis in order to make the latter more concise. [See also a relevant remark in Ref. 2, following Eq. (2)]. It . Suppose now the (unknown) strain e is prect
sent, that is, the two types of layers have been interfaced as a free-standing dislocation-free superlattice. The lattice constant a now becomes anisotropic. In the plane normal to N, it exhibits a H
common value for both layers, i.e., a II ~ a~ = a~,
Vol. 78, No. 5
heffG + h2ff 1 o 1 ao I Go2ao2 = ell h e f fG + h 2 k.i t
ol
(12)
02
with heft , (13) a = gixna II Thus aeff can be calculated directly from Eqs. (11) or
(12). The latter equation has an interesting II with the inphysical meaning: it identifies aeff plane lattice constant of a free-standing superlattice, growneff along ~01], with effective layer thtcknesses h 1 and h 2 . We call this the equi-
valent [001] superlattice. In view of Eqs. (10), (8), the in-plane strain of Eq. (9) now becomes eix = (a ff-aoa)/aoc t,
(14)
and can be calculated explicitly. Equation (14) is the key result of the present work. All other quantities such as, tetragonal distortion, strain and stress components, free energy density, can now be determined, and the only input information necessary consists of N, caj, and aoa. The normal-to-the-plane lattice constant is, by definition, aX a =aa(l+e&a )=a a [l+e~ (1-A~ix)]
(15a)
= ~.[a~ff( l-~ ~'ct)+ aoct A ~"a l
(15b)
_=ha-L
(15c)
eff,a
where
'
a -L
eff,ct
is the
normal-to-the-plane lattice con-
stant of the equivalent [001] superlattice. Thus, in the presence of misfit strains, Eq. (8) is replaced by the pair of Eqs. (10) and (15c). Since the entire computation is based on A~', we give below A e and the main results for some standard and non-standard directions N:
whereas parallel to N each layer exhibits its own value, i.e., aJi1 ~ a-~ ~* a II. Only the in-plane value a
II II
.
ts necessary for determining six, .
ect = [a
II
.,
-aix)/aix.
(9)
In order to determine a II it is necessary to minimize the elastic free energy. For this purpose we consider a rectangular parallelepiped of volume V = V l + v 2 , where vct=A(all)2hct and A is the number of periods of. the superlattice. The elastic ener.gy stored in v is U=UI+U2,where u (It =v IXU IX and U ~s given by Eq. (2). The requirement that (du/da I) =0 combined with Eqs. (2), (8) and (9), leads to the following equation, which characterizes the system as a whole, and not lust the individual layers, "
"
a =~. aeff ,
(i) For N=[001] we have, from Ref. (1), A ~ = 3B/Cll and from Eqs. (8), (6), and (5), k---1, g = l and G-=G°,
II
0o)
where
respectively. These equations hold for each material independently. Then, a
II
=
II
aeff
---
hlG la o l+h2G2ao2
(16a)
h i G 1 +h2G 2
which is ide~i~al to the usual expression found the l i t e r a t u r e . The explicit form of G Ix Eq. (7), also agrees with that in Refs. 2-4 but basically different from that used in Refs. 5, The in-plane strain for layer 1 is II neff
II
E1
" not
h2G2Aa/a o 1
(16b) aol hiG1+ h2G 2 where Aa=ao2-aotiS the lattice misfit, and likewise for layer 2. =
,
(ii) For N=[I10]/v~ we have I, biG1 a ol +h202ao2 aeff= hlG1 +h2G 2 II
(11)
in i.e. is 6.
A e -- 6B/(Clt+C~2+2C44 ) and, from Eq. (4),
Vol. 78, No. 5
349
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
O = 3B(C11-C12+6C44)/(C11+C12+2C44). It and e ct II are obtained from equations analogous aeff to Eqs. (16a) and (16b) respectively.
Table I. Numerical values for the various quantities described in the text for superlattice Si/Ge, i.e., ct=l(2) corresponds to Si(Ge). The lattice constants aol(o 2)= 5.4310 (5.6579) in ,g,, and the elastic constants CIt =
(iii) For N= [l l l]/C'J we have 1, A ~" = 9B/(Ct! + 2C12 + 4C44)
(17a)
G = 36BC44/(CtI+2C12+4C44 ).
(17b)
II a n d aeff
165.8 (128.5), Ct2=
63.9 (48.3), C44= 79.6 (66.8) in GPa are taken from
t~ct II are obtained from equations analogous
to Eqs. (16a) and (16b), respectively. The above expressions for Gltl01, GIlnl coincide with those stated in Ref. 2.
Ref. 8 (300K). The first, second, and third row of results in the three right-most columns correspond to h 1
1-A ~'. a
Nti
A~
G
II
a eff
e
II
a -L e ff,ot
GPa (iv) For NIi[11~]/¢'6 we have 1, A = [3(Cu-C12 +C44)(C11 +C12+2C44)-C 2 ]
Si
1.7708
Ge
1.7517
Si
1.5098
Ge
1.4503
Si
1.4392
Ge
1.3717
Si
1.5336
Ge
1.4812
Si
1.6216
Ge
1.5820
t001]
×(C11"C12+ 4C44 ) / 1 0 8 C 2
A~" = (B/6C2A)(Cll-C12+C44)(Cll-C12+4C44). G = 3B(3-A~ ). II
and ~
aeff
are obtained from equations analogous
[ 1101
to Eqs. (16a) and (16b), respectively. (v) For N=[120]/V'5 we have, 3 B ( 8C1 1 " 8 C 1 2 + 9 C 4 4 )
A~ =
[111l
4(C211- C2I 2)+C4 4 (1 7C l 1-8C12 ) G = 3B(3-A~). II a n d aeff
t~ctII are obtained from equations analogous
to Eqs. (16a) and (16b) respectively. The present analysis is readily applicable to strained epilayers (ht) grown on top of substrates (h2), along N. If h2~h I then II aeff= a ea-ff,2=ao2
[ l 1~1
[ 1 20]
and, from Eq. (15b, c), a-Left,1 = ao2"AaA 81
1.829 5. 6579 0 5. 5303 5. 4310 0 -2. 255 281.0 -4.010 4. 178 1.853 437.5 5. 6579 0 5.5316 5. 4310 0 -2.231 348.8 -4.010 4. 178 1.857 458.2 5. 6579 0 5.5318 5. 4310 0 -2. 228 366.5 -4.010 4. 178 430.5 1.849 5. 6579 0 5.5314 5.4310 0 341.9 -2.235 -4.010 4. 178 404.7 1.842 5. 6579 0 5.5310 5. 4310 0 3 19.2 -2.242 -4.010
360.9
5.3544 5. 4310 5.6579 5.7538 5.8284 5.3153 5.3796 5.4310 5.6579 5.7147 5.7600 5.3313 5.3867 5. 4310 5.6579 5.7047 5.7422 5.3099 5.3774 5. 4310 5.6579 5.7187 5. 7670 5.2899 5.3688 5 . 4310 5.6579 5.7317 5.7899
Accordingly, ~2II = ~-~ = 0 and for
~1II =
Table I coincide with the results tabulated in Ref. 2. The calculated values for the lattice constants in Ref. (2) appear to correspond to the present lattice constants all and a -l-. Actually they should it
Aa/aol
all directions N, whereas =
be
•
All these results have been applied to three standard types of superlattices, i.e., Si/Ge, GaSb/AISb and ZnTe/ZnSe. We consider three combinations of h t, h2: (i) ht~h 2, i.e., Si epilayer on top of a much thicker Ge substrate, (ii) h t= h 2, i.e., the free-standing symmetric superlattice, and
(iii) hl)h2,
i.e.,
Ge
epilayer on
top
of a much thicker Si substrate, and likewise for the other two superlattices. The results of these calculations are compiled in Tables 1-1II. Some of, the entries in the first three directions N of
interpreted as coinciding with our effective lattice constants a eUf f and a -L eft,c? respectively. It is interesting to notice in Tables I-III that, for all directions N the calculated parameters exhibit critical values (maxima or minima) for N along either [001] or [111], all other directions of N yielding values in between. In other words, Eqs. (16) and (17) each lead to the upper or lower bound of a II and e II Specits'ing the eff
ct"
type of critical point in each case depends on the relative sign and magnitude of C-=C11-C12-2C44 for rt=l, 2. Such behavior is dictated by the critical
350
Vol. 78, No. 5
IN-PLANE MISFIT STRAINS IN HETEROSTRUCTURES AND SUPERLATTICES
Table II. Same as Table I for GaSb/AlSb. Here, ao1(o2)= 6.0959 (6•1355) in )~, Cll = 88.34 (87.7),
Table III. Same as Table I for ZnTe/ZnSe. Here, ao1(o2)= 6.1037 (5.6676) in .~, C l l = 71.3 (90.0),
C12 = 40.23 (43.40), C44= 43.22 (40.80) in GPa.
C12= 40.7 (53.4), C44= 31.2 (39.6) in GPa.
Ntl
ct
A~
G
a
II
~
II
e ff
a-t. e f f ,~x
a
Ntl
A~"
GPa 'GaSb
1.9 107
1 83.9
[OOl] AISb
1.9897
176.3
IGaSb
1.5701
241.4
[1 10] 'AISb
1.6408
237.2
IGaSb
1.4820
256.2
[111] AISb
1.5501
253.0
GaSb
1.6 103
234.6
[1121 AISb
1.6839
229.7
,GaSb
1.7247
2 15.3
AISb
1.8015
209.1
[ 12o1
GPa
5 0.318 0
6.1355 6.1152 6. 0959 0 -0•329 -0.646 0.650 6.1355 0,322 6.1155 0 6. 0959 0 -0.326 -0.646 0.650 6 . 1 3 5 5 0.323 0 6 . 1 155 6. 0959 0 -0•325 -0.646 0.650 6 . 1 3 5 5 0.321 0 6. 1154 6. 0959 0 -0.326 -0.646 0.650 0.320 6.1355 0 6. 1154 6. 0959 0 -0.327 -0.646
6. 0598 6.0782 6. 0959 6.1355 6. 1555 6.1746 6.0733 6. 0847 6.0959 6.1355 6. 1482 6. 1608 6.0768 6.0864 6. 0959 6.1355 6. 1464 6.1572 6.0717 6.0839 6. 0959 6.1355 6. 1491 6.1625 6.0672 6.0817 6. 0959 6.1355 6. 1516 6. 1672
points that the frse energy density exhibits along these two directions'. In conclusion, we have reached explicit forms for the in-plane isotropic strain in superlattices and heterostructures grown pseudomorphically along an arbitrary crystallographic direction. Essential to this calculation is the reduced tetragonal distortion A~ for which an analytical expression is already available. Knowing AT allows the calculation of the corresponding "shear modulus" G, from which the in-plane and normal-to-the-
1. 2. 3. 4. 5.
G
II aef f X
It E; aif f .1102) ,ct • 145
'ZnTe
2.1416
1 31.1
Iznse
2.1866
160.1
[ZnTe
1.7511
1 90.7
ZnSe
1.7681
242.4
ZnTe
1.6508
206.0
ZnSe
1.6 621
263.3
ZnTe
1.8076
1 82.1
ZnSe
1.8335
229.6
ZnTe
1.9401
1 61.8
ZnSe
1.9757
201.6
[OOll
[1 lO1
[1111
[ 1 12]
5. 6676 -3.928 0 5. 8639 6.1037 0 3. 464 7. 695 -7. 145 -3. 999 5. 6676 0 5. 8596 6.1037 0 3. 388 7. 695 -7.145 -4. 008 5. 6676 0 5. 8590 6. 1037 0 3. 378 7. 695 -7. 145 5. 6676 -3.985 5. 8604 • 0 6.1037
[ 1 2o1
0 3.403 7. 695 -7. 145 -3. 963 5 . 6 676 0 5.8618 6.1037 0 3. 427 7. 695
6.6015
6.3774 6. 1037 5.6676 5.4346 5.1500 6.4312 6.2870 6. 1037 5.6676 5.5202 5.3325 6.3875 6.2629 6. 1037 5.6676 5.5408 5.3788 6.4559 6.3001 6. 1037 5.6676 5.5068 5.3040 6.5136 6.3311 6. 1037 5.6676 5.4780 5.2}120
phme lattice constants are determined within a geometrical factor. It would be interesting to compare the results of such calculationsh with the corresponding experimental values of ~ obtained from high resolution X-ray diffraction data. Supported in part by the General Secretariat for Research and Technology, Greece. Useful discussions are acknowledged with 1. P. Ipatova, A.Yu. Maslov, and V. Shchukin.
REFERENCES E. Anastassakis, J. Appl. Phys. 68, 4561 6. (1990). 7. Chris G. Van de Walle and Richard M. Martin, Phys. Rev. B34, 5621 (1986). 8. G . C . Osbourn, J, Vac. Sei. Technol. B1, 379 (1983). Takashi Nakayama, J. Phys. Soc. Jpn. 59, 1029 (1990). Le Hong Shon, Koichi Inoue, and Kazuo Murase, Solid State Commun. 62, 621 (1987)•
A. Yu. Maslov, private communications. R. M. Abdelouhab, R. Braunstein, M. A. Rao, and H. Kroemer, Phys. Rev. 39, 5857 (1989). Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technolog)', Vol. 22a, edited by O. Madelung and M. Schuiz (Springer, New York, 1987).