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Coherent heterostructures with non-cubic components: strains and stresses for arbitrary interface orientations — theory and applications
surface science ELSEVIER
Surface Science 344 (1995) 276-282
Coherent heterostructures with non-cubic components: strains and stresses for arbitrary interface orientations theory and applications A.N. Efimov *, A.O. Lebedev loffe Institute, Polytechnitcheskaya Str. 26, St. Petersburg, 194021, Russian Federation Received 15 January 1995; accepted for publication 14 August 1995
Abstract
A method for calculating strain and stress tensors and the density of elastic energy in the framework of linear anisotropic elasticity theory has been developed. The method is valid for arbitrary symmetries of conjugated crystals and arbitrary interface orientations. Both a crystallographic tilt of an epitaxial layer with respect to the substrate and the phenomenon of layer triclinic distortion have been considered. The results are compared with experimental data. The method has been also used for an optimization of structural perfection of heterostructures with non-cubic components. Keywords: Epitaxy; Single crystal epitaxy; Surface energy; Surface stress
1. Introduction It is known that a pseudomorphically growing layer adapts itself to the substrate by small changes in the lattice parameters. This phenomenon has been widely studied for A3B 5 heterostructures. A complete description of this phenomenon is very important for a number of applications, for example, to interpret adequately X-ray diffraction data [1-3], to calculate the shift of phase equilibrium caused by an additional contribution of elastic energy to the chemical potential [4], and to determine the effect of anisotropic stress on important properties of heterostructures (e.g., on the band structure of semicon-
* Corresponding author. Fax: + 7 812 515 6747; E-mail: [email protected].
ductors or on the Tc of high-temperature superconductors). The problem has been partly solved in the framework of linear anisotropic elasticity theory by Hornstra and Bartels [5]. But these results are valid only if both components of a heteropair are cubic. However, the active materials used in modern functional electronics (opto-, acousto-, and, recently, high T~-superconducting electronics, etc.) are low-symmetry crystals in the majority of cases. Thus, it is necessary to consider this problem for arbitrary systems of conjugated crystals. The approach proposed in this paper for the first time allows us to calculate an equilibrium geometry (elastic strain in an epitaxial layer and misorientation of the layer lattice with respect to the substrate), stress tensor and the density of elastic energy for any symmetry of the crystals and an arbitrary orientation
0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 5 ) 0 0 8 4 3 - 8
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282 of the interfaces. Some applications for specific epitaxial systems are also presented.
2. Analysis The linear continuous anisotropic theory of elasticity is used here under the following assumptions: (1) The heterostructure is in a pseudomorphic state (without topological defects such as dislocations, broke bonds, etc., at the interface). (2) The mismatch in lattice parameters of the epitaxial layer and the substrate is small. (3) The bend of the structure is negligibly small. This is always true for small layer thicknesses. (4) The elastic properties and equilibrium lattice parameters do not change throughout the epitaxial layer. (5) The interface is a plane. Let the layer and substrate lattices be connected by a linear transformation M = {M/}. (The following conventional agreements are used here: top and bottom indexes are contravariant and covariant, respectively. Summing up for the same indexes at different levels is implied.) Since there are two crystallographic coordinate systems - the substrate system and the layer one - an arbitrary vector can be described by two sets of coordinates corresponding to these systems. The condition of a defect-free interface is equivalent to the condition of the coincidence of these coordinates for any vector r belonging to the interface [6]. Let h i be covariant components of the normal to the interface plane (i.e., the Miller indices of the interface plane). In this case the condition of pseudomorphism can be written as:
/ • r~
r,
hiri=O" ]
(1)
To describe fully a heterostructure (i.e. to calculate an equilibrium geometry, the stress tensor and the elastic energy) it is necessary to determine matrix M or, in other words, vector a j. Let F be the metric tensor of a non-strained layer crystal, S is the metric tensor of the substrate, /_ is the metric tensor of a strained layer. Assuming a small difference between F and S, it can be found for strain:
(2)
where 3 is the Kronecker delta ( 3 / = 1 if i = j, and 3 / = 0 if i ~ j), and a j is an arbitrary vector.
(3)
eij = ½ ( L i j - F i j ) .
Taking into account that:
L=Mt~.S-M,
(4)
where tr is the matrix transpose sign, and substituting Eq. (2) into Eq. (3), we can express the components of sij in terms of a: OOm.i = ½( Smj - Fmj + S m i a i h j + hmakSkd
(5)
+hmaiSikakhj).
For stress o-Pq, by changing the order of summation and neglecting the second-order quantities with respect to a i in Eq. (5) (assumption 2), one can write: o'Pq=½cPqmJ(Smj-Fmjq-2Smiaihj),
(6)
where C pqmy are the components of the elastic mouulus tensor. Using the condition of mechanical equilibrium for a strained body [7] in the case of a homogeneous deformation, we obtain: 0 "p
=
o'Pqhq :- O.
(7)
The non-homogeneous system of linear equations (7) allows us to obtain the unique solution for a i and to determine the matrix of the linear transformation M. Finally, the system (7) can be given as: A ~ a " = B p,
where
A p = cPqmJhqhjSmn,
n p = 1cPqmJhq(Fmj - am j ) ,
Taking into account (1), the tensor M/ can be given by: M / = 3 j + aJhi,
277
(8) and for a: a n = Dpn p ,
where D is the inverse matrix of A.
(9)
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
278
Then, from Eq. (4) tensor /_ can be written as: tpq = Spq "~- hpamSmq + SpjaJhq.
(10)
Now, 8mj can be determined from Eq. (3), orpq can be found from Eq. (6), and the density of elastic energy of the strained layer is given by:
E = orPq•pq//2.
(11)
Misorientation of an epitaxial layer with respect to the substrate (the angle between the (hlh2h3)-
plane of the layer and the same plane of the substrate) can be expressed as:
hiLiJMfhk COS q 0 =
v/(hmLmnhn)(hpMPLqrMfhs)
.
Thus, Eq. (3) allows us to calculate the elastic deformation of the epitaxial layer, which should be triclinic even for cubic crystals in the general case of an arbitrary interface orientation. Misorientation can
Table 1 Mirorientation for some A3B 5 compounds; calculated and experimental data Subs~a~
GaAs
Layer and its thickness h (/xm)
(001), projection of normal on (001) oriented at 65 ° counter clockwise
A1 xGal _ x As ( x = 0.986) h = 0.5
26.2
24.9
[11]
A1 xGal - x As ( x = 0.724) h = 0.6 A1 xGa i - x As ( x = 0.651) h = 0.76 AlAs h = 1.0
19.1
17.4
[11]
17.1
16.0
[11]
2.6
3.5
[31
AlAs h = 0.98
21.1
22.7
[31
AlAs h = 0.5
32.3
32.0
[3]
GaAs
GaAs
as above
GaAs
(001), angular deviation 0.25 ° towards [1101 (001), angular deviation 1.98 ° towards [1101 (001), angular deviation 3.03 ° towards [1101 (001), 4.5 ° along ( 1101 (001), 8.8 ° along (110> (111), 3.0 ° along <211> (111), 5.0 ° along
G a x l n I _xAS ( x = 0.88) h = 7.0 GaxIn I_xAs ( x = 0.88) h =7.0 Gaxlnl_xAS ( x = 0.9) h = 8.0 Gafln~ _ ~ A s ( x = 0.9)
<211>
h =
GaAs
GaAs
GaAs
GaAs
GaAs
GaAs
8.0
Misorientation of planes
(HKL) in arc
Its orientation (HKL) and deviation
from (0111, deviation angle 2.5 ° as above
(12)
Calc.
Exp.
sec Ref.
265
540
[12]
516
1100
[12]
113
187
[12]
187
238
[121
A.N. Efimov, A.O. Lebedev/ Surface Science 344 (1995)276-282 be given by (12), stress and density of elastic energy are described by (6) and (11), respectively. As a result, the problem has been solved completely. For faster computing o - P q , ~mj and C pqmj should be represented as co- and contravariant components of vectors a n d second order tensor, respectively, in 6-dimensional space [8]. In conclusion, it is interesting to note that the elastic energy can be equal to 0, if the following conditions are satisfied:
( F i / - Sij) 2 ~ ( F i i - Sii)( F j j - Sjj).
(13)
40 I
3. Applications 3.1. Misorientation phenomenon and triclinic deformation There are no reliable experimental data on misorientation between crystallographic planes of the epitaxial layer and the substrate and on triclinic deformation of the layer for pseudomorphic non-cubic heterostructures. So we are forced to use experimental data for cubic heterostructures. For these epitaxial systems, the misorientation phenomenon is of great practical interest and has been discussed, for example, in Refs. [3,5,9-15]. A comparison of the calculated and experimental data for some A3B 5 compounds is given in Table 1 and is plotted in Fig. 1. For solid solutions, the calculations have been made under the assumption that the elastic moduli and lattice parameters are linear functions of composition. One can see that there is a good agreement between the calculated and experimental data for A1As/GaAs and A l x G a ] _ x A s / G a A s . For G a x l n l _ ~ A s / G a A s heterosmactures, the agreement is worse due to breaks of pseudomorphism at the great thickness of the epitaxial layer and substantial misfit in the system. As regards the G a P / S i system, we have obtained lower values of misorientations for substrate deviations from 0 ° to 10 °. Most likely, the break of pseudomorphism in this case results from
i
i
I [
i
i
o
by Igarashi
--
o~lr
l'e~Inlt8
50 °n
~"
o
20 i O
o
I -5
I -5
These conditions are similar to the conditions of the defect-free conjugation of crystals described in Ref. [6].
279
r
i
q
~
i
0
5
I0
15
20
25
Substrate orientation, angle from [111] toward [tl0], degrees Fig. 1. Dependence of angular misalignment between the [111] axes on subslrate orientation (GaP on Si).
the difference in the types of chemical bonds of the conjugated crystals.
3.2. Scalar cost function for design of defect-free heterostructures with non-cubic components The problem of how to optimize both composition and crystallographic orientation of the substrate for the case of non-cubic heterostructures has been discussed in detail in our recent paper [16]. Here we consider this problem briefly to demonstrate an application of the above approach for optimization of YBa2Cu207/PrGaO3 heterostructure. The crystal structure and dielectric parameters of PrGaO 3 as a promising material for the substrate have been reported in Ref. [17]. YBa2Cu307 and PrGaO 3 crystal structures are based on reduced pseudocubic perovskite-type cells with nearly equal cell parameters. But since the substrate and film lattices are distorted and ordered in different ways, the nonprimitive unit cells of these phases in Bravais settings have greatly differing lattice parameters [18]. Therefore for all further calculations we have recalculated unit cell lattice parameters for the settings based on reduced pseudocubic perovskite cells. Lattice parameters corresponding to both conventional and our settings are shown in Table 2. For optimization of the heterostructure a wellgrounded scalar cost function should be defined. It is
A.N. Efirnov, A.O. Lebedev / Surface Science 344 (1995) 276-282
280
Table 2 Lattice parameters of YBaCuO and PIGaO 3 Phase
YBa2Cu30 7
PrGaO 3
Table 3 Different types of orientation relationships layer-substrate
Reduced pseudocubic perovskite cell
Bravais cell
a = 3.82 b = 3.87 c = 3.90
a = 3.82 b = 3.87 c = 3.90
a = 3.873 b = 3.870 c = 3.873 13 = 90.32 °
a = 5.462 b = 5.493 c = 7.740
known that when the mismatch between the layer and the substrate lattices is small during the initial stage of growth (when the layer is still thin) the layer accommodates to the substrate by elastic strain. Structural defects (misfit dislocations) in this case are absent. The elastic energy increases with the layer thickness up to a critical value. Then the stressed structure relaxes and, as a result, structural defects arise. Since the elastic energy is the basic cause of the initiation of defects, it is evident that the energy, or an arbitrary increasing function of this energy, can be taken as a cost function. We define
f= ( E1/Eo) °'5,
Type of relationship
Substrate directions
Parallel layer directions
1
[100], [010], [001]
[1001, [010], [0011
2
[100], [010], [0011
[1001, [001], [010]
3
[100], [010], [001]
[001], [010], [100]
4 5 6
[1001, [010], [001] [100], [010], [001] [100], [010], [001]
[0101, [100], [0011 [001], [100], [010] [010], [001], [100]
and equal to 24 for a triclinic system for given enantiomorphous form and is minimal and equal to unity for a cubic system. These settings correspond to various types of "layer-substrate" orientation relationships. It is obvious that the misfit (14) is different for various relationship types. Therefore the misfit should be calculated for all possible orientation relationships. It can be assumed that, in reality, only epitaxial relationships with minimum elastic energy (minimum misfit) can be observed. In our case since the layer cell is rhombic there are six different types of orientation relationships, as specified in Table 3. The " m i s f i t " as a function of substrate misorientation from (001) to (100) for these six orientation
,2 i
(14)
I
where f is the cost function, E 1 is the elastic energy of the epitaxial layer, calculated from Eq. (11), E o is a coefficient equal to the energy of the model "cubic-on-cubic" epitaxial pair if lattice mismatch is equal to 1% and the thicknesses and elastic modules of the model and real layers are equal to each other. For cubic heterostructures, such a cost function corresponds to the traditional definition of lattice mismatch. Below, the function defined by Eq. (14) is called the "misfit". There are many nonequivalent ways of selecting coordinate systems based on the translation vectors. But only one of them corresponds to the commonly accepted setting of the crystallographic axes. Its choice is in a certain sense a matter of convention. The number of the settings is determined for a given lattice by its symmetry. For example, the number of various settings of crystallographic axes is maximal
1.o
-
I
-]
.,°.o-°.o
e~.l
v Rel.:~
s--.0
8
711
~
•
"
F.eL~
" .....
o.o-00
~'
~
\v/
-60 -30 O 30 60 SUBSTPATEORIENTATION,/d, IGLEFROM(COl) TOWAI~DS(IDO).DEG~ES
I /
~0
Fig. 2. Misfit as a function of substrate misorientation from (001) to (100) for layer Y B a z C u 3 0 x and PrGaO 3 substrate (6 orientation relationshipsare presented).
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
relationships, is shown in Fig. 2. For the first minimum, at - 4 5 °, "misfits" (i.e. energies) for two different orientations of epitaxial layer are almost equal. Growth of an epitaxial layer on the substrate with such orientation must lead to polysynthetic twinning of the epitaxial layer. As for other minima, at 27 ° and 61 °, the misfits of the dominant orientation relationships are less than 0.1%, meanwhile the energies of competitive orientations are larger. This can give rise to growth of high-perfect, twin-free epitaxial layers for such substrate orientations.
4. Discussion (1) We have considered compensation of a small mismatch between parameters of unit cells of an epitaxial layer and substrate. When the mismatch is large, the optimal orientation of the substrate is typically determined using Coincidence Site Lattice concept [20,21]. Our method of calculating of the mismatch can be extended easily to this concept if unit cell parameters of the layer and the substrate are replaced by the parameters of reduced supercells (i.e., the parameters of multiple cells forming the lattice of near-CSLs). It should be noted that existence of near-CSL as a mathematical structure does not mean that this CSL makes physical sense. In our opinion (see the recent paper of the authors [22]), in many papers where the CSL-concept is used to predict an orientation relationship or to interpret experimentally obtained orientation relationships for systems with a large difference in the lattice parameters, there is no proof of its physical meaning. Just a verification of Eqs. (3)-(1 I) or, at least, a some evidence for this would justify the application of the CSL-concept. But unfortunately we do not know of such experimental data yet. (2) The geometrical aspects of selection of the compositions and orientations of the substrates discussed by us are necessary but not sufficient for obtaining perfect heterocompositions. Aspects, such as the physical properties of the substrate, the chemical interaction between the layer and substrate, the possibility of accidental doping of the layer with elements contained in the substrate, the probability of polycentric refaceting of the substrate surface in
281
the case of an unfavorable orientation of the interface, require a separate study. However, it should be pointed out that if the optimal orientation corresponds to a deep local minimum of the elastic energy (which is true in the examples given above with a small mismatch), after the deposition of the epitaxial layer refaceting of the interface will be energetically disadvantageous. (3) The approach discussed in this paper can be completely applied to multilayer heterocompositions and superlattices. One can show that for any intermediate layer of pseudomorphic multilayer structure the stress and slxain tensors are equal to those for a single layer lying directly on the substrate. Therefore the approach presented in Section 2 can be applied to each intermediate layer without taking into account any sub- or overlayer. Moreover, the approach is valid also for layers with continuously varying composition in the direction normal to the interface. In this case the layer may be regarded as consisting of many thin sublayers, each of them having practically constant composition. (4) Specific requirements on the crystallographic orientations of both the epitaxial layer and the substrate are imposed sometimes in functional electronic devices. For example, the squares of the electromechanical coupling coefficients for aluminum nitride films vary from 0.05% to 0.9% as a function of layer orientation [23], velocity and modes of surface acoustic wave (SAW) also significantly depend on the layer and substrate orientations. Optimal orientation in these cases may not correspond to the minimum of the misfit. However, the approach proposed above is useful for finding a compromise when many phases are considered as candidates for the substrate.
Acknowledgements The authors would like to thank Dr. M. Krasin'kova for helpful discussions. This work is supported, in part, by the US Defense Department and by the International Science Foundation (Grant N NUH000). The program for computer-aided Solving of the problem is written in PASCAL and is realized on an IBM-compatible computer. The program is available from the authors.
282
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
References [1] V.L. Indenbom and V.M. Kaganer, Phys. Status Solidi (a) 118 (1990) 71. [2] Yu.P. Hapachev and F.N. Chuhovskij, Soy. Crystallogr. 734 (1989) 776. [3] P. Auvray, M. Baudet and A. Regreny, J. Cryst. Growth 95 (1989) 288. [4] V.V. Kuznetsov, W. Sadowski and V.S. Sorokin, Cryst. Res. Technol. 20 (1985) 1373. [5] J. Hornstra and W.J. Bartels, J. Cryst. Growth 44 (1978) 513. [6] A.N. Efimov and A.O. Lebedev, Sov. Supercon.: Physics, Chemistry, Technol. (SPCT) 6 (1993) 203. [7] L.D. Landau and M. Liftshits, Theory of Elasticity (Nauka, Moscow, 1987) p. 61. [8] Yu.l. Sirotin and M.P. Shaskolskaya, Principles of Crystal Physics (Nauka, Moscow, 1975) p. 666. [9] A. Ohki, N. Shibata and S. Zembutsu, J. Appl. Phys. 64 (1988) 694. [10] J.E. Ayers, S.K. Ghandhi and L.J. Schowalter, J. Cryst. Growth 113 (1991) 430. [11] A. Leiberich and J. Levkoff, J. Cryst. Growth 100 (1990) 330.
[12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23]
H. Nagai, J. Appl. Phys. 45 (1974) 3789. O. Igarashi, Jpn. J. Appl. Phys. 15 (1976) 1435. Yu.N. Drozdov, Sov. Crystallogr. 38 (1993) 189. T. Fukui and H. Saito, J. Vac. Sci. Technol. B 6 (1988) 1373. A.N. Efimov and A.O. Lebedev, Sov. Solid State Phys. 36 (1994) 328. M. Sasaura, M. Mukaida and Sh. Miyazawa, Appl. Phys. Lett. 57 (1990) 2728. E.G. Fesenko, The Perovskite Family and Ferroelectricity (Atomizdat, Moscow, 1972) p. 17. W. Bollmann, Crystal Defects and Crystalline Interfaces (Springer, Berlin, 1970) p. 254. V.M. Kosevitch, V.M. Ievlev, L.S. Palatnik and A.I. Fedorenko, Structure of Intergrain and Interphase Boundaries (Metallurgiya, Moscow, 1980) p. 256. A. Zur and T.C. McGill, J. Appl. Phys. 55 (1984) 378. A.N. Efimov and A.O. Lebedev, Thin Solid Films 260 (1995) 111. A.O. Lebedev, Proc. Leningrad Electrical Engineering Institute 380 (1987) 74.
Surface Science 344 (1995) 276-282
Coherent heterostructures with non-cubic components: strains and stresses for arbitrary interface orientations theory and applications A.N. Efimov *, A.O. Lebedev loffe Institute, Polytechnitcheskaya Str. 26, St. Petersburg, 194021, Russian Federation Received 15 January 1995; accepted for publication 14 August 1995
Abstract
A method for calculating strain and stress tensors and the density of elastic energy in the framework of linear anisotropic elasticity theory has been developed. The method is valid for arbitrary symmetries of conjugated crystals and arbitrary interface orientations. Both a crystallographic tilt of an epitaxial layer with respect to the substrate and the phenomenon of layer triclinic distortion have been considered. The results are compared with experimental data. The method has been also used for an optimization of structural perfection of heterostructures with non-cubic components. Keywords: Epitaxy; Single crystal epitaxy; Surface energy; Surface stress
1. Introduction It is known that a pseudomorphically growing layer adapts itself to the substrate by small changes in the lattice parameters. This phenomenon has been widely studied for A3B 5 heterostructures. A complete description of this phenomenon is very important for a number of applications, for example, to interpret adequately X-ray diffraction data [1-3], to calculate the shift of phase equilibrium caused by an additional contribution of elastic energy to the chemical potential [4], and to determine the effect of anisotropic stress on important properties of heterostructures (e.g., on the band structure of semicon-
* Corresponding author. Fax: + 7 812 515 6747; E-mail: [email protected].
ductors or on the Tc of high-temperature superconductors). The problem has been partly solved in the framework of linear anisotropic elasticity theory by Hornstra and Bartels [5]. But these results are valid only if both components of a heteropair are cubic. However, the active materials used in modern functional electronics (opto-, acousto-, and, recently, high T~-superconducting electronics, etc.) are low-symmetry crystals in the majority of cases. Thus, it is necessary to consider this problem for arbitrary systems of conjugated crystals. The approach proposed in this paper for the first time allows us to calculate an equilibrium geometry (elastic strain in an epitaxial layer and misorientation of the layer lattice with respect to the substrate), stress tensor and the density of elastic energy for any symmetry of the crystals and an arbitrary orientation
0039-6028/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 3 9 - 6 0 2 8 ( 9 5 ) 0 0 8 4 3 - 8
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282 of the interfaces. Some applications for specific epitaxial systems are also presented.
2. Analysis The linear continuous anisotropic theory of elasticity is used here under the following assumptions: (1) The heterostructure is in a pseudomorphic state (without topological defects such as dislocations, broke bonds, etc., at the interface). (2) The mismatch in lattice parameters of the epitaxial layer and the substrate is small. (3) The bend of the structure is negligibly small. This is always true for small layer thicknesses. (4) The elastic properties and equilibrium lattice parameters do not change throughout the epitaxial layer. (5) The interface is a plane. Let the layer and substrate lattices be connected by a linear transformation M = {M/}. (The following conventional agreements are used here: top and bottom indexes are contravariant and covariant, respectively. Summing up for the same indexes at different levels is implied.) Since there are two crystallographic coordinate systems - the substrate system and the layer one - an arbitrary vector can be described by two sets of coordinates corresponding to these systems. The condition of a defect-free interface is equivalent to the condition of the coincidence of these coordinates for any vector r belonging to the interface [6]. Let h i be covariant components of the normal to the interface plane (i.e., the Miller indices of the interface plane). In this case the condition of pseudomorphism can be written as:
/ • r~
r,
hiri=O" ]
(1)
To describe fully a heterostructure (i.e. to calculate an equilibrium geometry, the stress tensor and the elastic energy) it is necessary to determine matrix M or, in other words, vector a j. Let F be the metric tensor of a non-strained layer crystal, S is the metric tensor of the substrate, /_ is the metric tensor of a strained layer. Assuming a small difference between F and S, it can be found for strain:
(2)
where 3 is the Kronecker delta ( 3 / = 1 if i = j, and 3 / = 0 if i ~ j), and a j is an arbitrary vector.
(3)
eij = ½ ( L i j - F i j ) .
Taking into account that:
L=Mt~.S-M,
(4)
where tr is the matrix transpose sign, and substituting Eq. (2) into Eq. (3), we can express the components of sij in terms of a: OOm.i = ½( Smj - Fmj + S m i a i h j + hmakSkd
(5)
+hmaiSikakhj).
For stress o-Pq, by changing the order of summation and neglecting the second-order quantities with respect to a i in Eq. (5) (assumption 2), one can write: o'Pq=½cPqmJ(Smj-Fmjq-2Smiaihj),
(6)
where C pqmy are the components of the elastic mouulus tensor. Using the condition of mechanical equilibrium for a strained body [7] in the case of a homogeneous deformation, we obtain: 0 "p
=
o'Pqhq :- O.
(7)
The non-homogeneous system of linear equations (7) allows us to obtain the unique solution for a i and to determine the matrix of the linear transformation M. Finally, the system (7) can be given as: A ~ a " = B p,
where
A p = cPqmJhqhjSmn,
n p = 1cPqmJhq(Fmj - am j ) ,
Taking into account (1), the tensor M/ can be given by: M / = 3 j + aJhi,
277
(8) and for a: a n = Dpn p ,
where D is the inverse matrix of A.
(9)
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
278
Then, from Eq. (4) tensor /_ can be written as: tpq = Spq "~- hpamSmq + SpjaJhq.
(10)
Now, 8mj can be determined from Eq. (3), orpq can be found from Eq. (6), and the density of elastic energy of the strained layer is given by:
E = orPq•pq//2.
(11)
Misorientation of an epitaxial layer with respect to the substrate (the angle between the (hlh2h3)-
plane of the layer and the same plane of the substrate) can be expressed as:
hiLiJMfhk COS q 0 =
v/(hmLmnhn)(hpMPLqrMfhs)
.
Thus, Eq. (3) allows us to calculate the elastic deformation of the epitaxial layer, which should be triclinic even for cubic crystals in the general case of an arbitrary interface orientation. Misorientation can
Table 1 Mirorientation for some A3B 5 compounds; calculated and experimental data Subs~a~
GaAs
Layer and its thickness h (/xm)
(001), projection of normal on (001) oriented at 65 ° counter clockwise
A1 xGal _ x As ( x = 0.986) h = 0.5
26.2
24.9
[11]
A1 xGal - x As ( x = 0.724) h = 0.6 A1 xGa i - x As ( x = 0.651) h = 0.76 AlAs h = 1.0
19.1
17.4
[11]
17.1
16.0
[11]
2.6
3.5
[31
AlAs h = 0.98
21.1
22.7
[31
AlAs h = 0.5
32.3
32.0
[3]
GaAs
GaAs
as above
GaAs
(001), angular deviation 0.25 ° towards [1101 (001), angular deviation 1.98 ° towards [1101 (001), angular deviation 3.03 ° towards [1101 (001), 4.5 ° along ( 1101 (001), 8.8 ° along (110> (111), 3.0 ° along <211> (111), 5.0 ° along
G a x l n I _xAS ( x = 0.88) h = 7.0 GaxIn I_xAs ( x = 0.88) h =7.0 Gaxlnl_xAS ( x = 0.9) h = 8.0 Gafln~ _ ~ A s ( x = 0.9)
<211>
h =
GaAs
GaAs
GaAs
GaAs
GaAs
GaAs
8.0
Misorientation of planes
(HKL) in arc
Its orientation (HKL) and deviation
from (0111, deviation angle 2.5 ° as above
(12)
Calc.
Exp.
sec Ref.
265
540
[12]
516
1100
[12]
113
187
[12]
187
238
[121
A.N. Efimov, A.O. Lebedev/ Surface Science 344 (1995)276-282 be given by (12), stress and density of elastic energy are described by (6) and (11), respectively. As a result, the problem has been solved completely. For faster computing o - P q , ~mj and C pqmj should be represented as co- and contravariant components of vectors a n d second order tensor, respectively, in 6-dimensional space [8]. In conclusion, it is interesting to note that the elastic energy can be equal to 0, if the following conditions are satisfied:
( F i / - Sij) 2 ~ ( F i i - Sii)( F j j - Sjj).
(13)
40 I
3. Applications 3.1. Misorientation phenomenon and triclinic deformation There are no reliable experimental data on misorientation between crystallographic planes of the epitaxial layer and the substrate and on triclinic deformation of the layer for pseudomorphic non-cubic heterostructures. So we are forced to use experimental data for cubic heterostructures. For these epitaxial systems, the misorientation phenomenon is of great practical interest and has been discussed, for example, in Refs. [3,5,9-15]. A comparison of the calculated and experimental data for some A3B 5 compounds is given in Table 1 and is plotted in Fig. 1. For solid solutions, the calculations have been made under the assumption that the elastic moduli and lattice parameters are linear functions of composition. One can see that there is a good agreement between the calculated and experimental data for A1As/GaAs and A l x G a ] _ x A s / G a A s . For G a x l n l _ ~ A s / G a A s heterosmactures, the agreement is worse due to breaks of pseudomorphism at the great thickness of the epitaxial layer and substantial misfit in the system. As regards the G a P / S i system, we have obtained lower values of misorientations for substrate deviations from 0 ° to 10 °. Most likely, the break of pseudomorphism in this case results from
i
i
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i
i
o
by Igarashi
--
o~lr
l'e~Inlt8
50 °n
~"
o
20 i O
o
I -5
I -5
These conditions are similar to the conditions of the defect-free conjugation of crystals described in Ref. [6].
279
r
i
q
~
i
0
5
I0
15
20
25
Substrate orientation, angle from [111] toward [tl0], degrees Fig. 1. Dependence of angular misalignment between the [111] axes on subslrate orientation (GaP on Si).
the difference in the types of chemical bonds of the conjugated crystals.
3.2. Scalar cost function for design of defect-free heterostructures with non-cubic components The problem of how to optimize both composition and crystallographic orientation of the substrate for the case of non-cubic heterostructures has been discussed in detail in our recent paper [16]. Here we consider this problem briefly to demonstrate an application of the above approach for optimization of YBa2Cu207/PrGaO3 heterostructure. The crystal structure and dielectric parameters of PrGaO 3 as a promising material for the substrate have been reported in Ref. [17]. YBa2Cu307 and PrGaO 3 crystal structures are based on reduced pseudocubic perovskite-type cells with nearly equal cell parameters. But since the substrate and film lattices are distorted and ordered in different ways, the nonprimitive unit cells of these phases in Bravais settings have greatly differing lattice parameters [18]. Therefore for all further calculations we have recalculated unit cell lattice parameters for the settings based on reduced pseudocubic perovskite cells. Lattice parameters corresponding to both conventional and our settings are shown in Table 2. For optimization of the heterostructure a wellgrounded scalar cost function should be defined. It is
A.N. Efirnov, A.O. Lebedev / Surface Science 344 (1995) 276-282
280
Table 2 Lattice parameters of YBaCuO and PIGaO 3 Phase
YBa2Cu30 7
PrGaO 3
Table 3 Different types of orientation relationships layer-substrate
Reduced pseudocubic perovskite cell
Bravais cell
a = 3.82 b = 3.87 c = 3.90
a = 3.82 b = 3.87 c = 3.90
a = 3.873 b = 3.870 c = 3.873 13 = 90.32 °
a = 5.462 b = 5.493 c = 7.740
known that when the mismatch between the layer and the substrate lattices is small during the initial stage of growth (when the layer is still thin) the layer accommodates to the substrate by elastic strain. Structural defects (misfit dislocations) in this case are absent. The elastic energy increases with the layer thickness up to a critical value. Then the stressed structure relaxes and, as a result, structural defects arise. Since the elastic energy is the basic cause of the initiation of defects, it is evident that the energy, or an arbitrary increasing function of this energy, can be taken as a cost function. We define
f= ( E1/Eo) °'5,
Type of relationship
Substrate directions
Parallel layer directions
1
[100], [010], [001]
[1001, [010], [0011
2
[100], [010], [0011
[1001, [001], [010]
3
[100], [010], [001]
[001], [010], [100]
4 5 6
[1001, [010], [001] [100], [010], [001] [100], [010], [001]
[0101, [100], [0011 [001], [100], [010] [010], [001], [100]
and equal to 24 for a triclinic system for given enantiomorphous form and is minimal and equal to unity for a cubic system. These settings correspond to various types of "layer-substrate" orientation relationships. It is obvious that the misfit (14) is different for various relationship types. Therefore the misfit should be calculated for all possible orientation relationships. It can be assumed that, in reality, only epitaxial relationships with minimum elastic energy (minimum misfit) can be observed. In our case since the layer cell is rhombic there are six different types of orientation relationships, as specified in Table 3. The " m i s f i t " as a function of substrate misorientation from (001) to (100) for these six orientation
,2 i
(14)
I
where f is the cost function, E 1 is the elastic energy of the epitaxial layer, calculated from Eq. (11), E o is a coefficient equal to the energy of the model "cubic-on-cubic" epitaxial pair if lattice mismatch is equal to 1% and the thicknesses and elastic modules of the model and real layers are equal to each other. For cubic heterostructures, such a cost function corresponds to the traditional definition of lattice mismatch. Below, the function defined by Eq. (14) is called the "misfit". There are many nonequivalent ways of selecting coordinate systems based on the translation vectors. But only one of them corresponds to the commonly accepted setting of the crystallographic axes. Its choice is in a certain sense a matter of convention. The number of the settings is determined for a given lattice by its symmetry. For example, the number of various settings of crystallographic axes is maximal
1.o
-
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-]
.,°.o-°.o
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8
711
~
•
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" .....
o.o-00
~'
~
\v/
-60 -30 O 30 60 SUBSTPATEORIENTATION,/d, IGLEFROM(COl) TOWAI~DS(IDO).DEG~ES
I /
~0
Fig. 2. Misfit as a function of substrate misorientation from (001) to (100) for layer Y B a z C u 3 0 x and PrGaO 3 substrate (6 orientation relationshipsare presented).
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
relationships, is shown in Fig. 2. For the first minimum, at - 4 5 °, "misfits" (i.e. energies) for two different orientations of epitaxial layer are almost equal. Growth of an epitaxial layer on the substrate with such orientation must lead to polysynthetic twinning of the epitaxial layer. As for other minima, at 27 ° and 61 °, the misfits of the dominant orientation relationships are less than 0.1%, meanwhile the energies of competitive orientations are larger. This can give rise to growth of high-perfect, twin-free epitaxial layers for such substrate orientations.
4. Discussion (1) We have considered compensation of a small mismatch between parameters of unit cells of an epitaxial layer and substrate. When the mismatch is large, the optimal orientation of the substrate is typically determined using Coincidence Site Lattice concept [20,21]. Our method of calculating of the mismatch can be extended easily to this concept if unit cell parameters of the layer and the substrate are replaced by the parameters of reduced supercells (i.e., the parameters of multiple cells forming the lattice of near-CSLs). It should be noted that existence of near-CSL as a mathematical structure does not mean that this CSL makes physical sense. In our opinion (see the recent paper of the authors [22]), in many papers where the CSL-concept is used to predict an orientation relationship or to interpret experimentally obtained orientation relationships for systems with a large difference in the lattice parameters, there is no proof of its physical meaning. Just a verification of Eqs. (3)-(1 I) or, at least, a some evidence for this would justify the application of the CSL-concept. But unfortunately we do not know of such experimental data yet. (2) The geometrical aspects of selection of the compositions and orientations of the substrates discussed by us are necessary but not sufficient for obtaining perfect heterocompositions. Aspects, such as the physical properties of the substrate, the chemical interaction between the layer and substrate, the possibility of accidental doping of the layer with elements contained in the substrate, the probability of polycentric refaceting of the substrate surface in
281
the case of an unfavorable orientation of the interface, require a separate study. However, it should be pointed out that if the optimal orientation corresponds to a deep local minimum of the elastic energy (which is true in the examples given above with a small mismatch), after the deposition of the epitaxial layer refaceting of the interface will be energetically disadvantageous. (3) The approach discussed in this paper can be completely applied to multilayer heterocompositions and superlattices. One can show that for any intermediate layer of pseudomorphic multilayer structure the stress and slxain tensors are equal to those for a single layer lying directly on the substrate. Therefore the approach presented in Section 2 can be applied to each intermediate layer without taking into account any sub- or overlayer. Moreover, the approach is valid also for layers with continuously varying composition in the direction normal to the interface. In this case the layer may be regarded as consisting of many thin sublayers, each of them having practically constant composition. (4) Specific requirements on the crystallographic orientations of both the epitaxial layer and the substrate are imposed sometimes in functional electronic devices. For example, the squares of the electromechanical coupling coefficients for aluminum nitride films vary from 0.05% to 0.9% as a function of layer orientation [23], velocity and modes of surface acoustic wave (SAW) also significantly depend on the layer and substrate orientations. Optimal orientation in these cases may not correspond to the minimum of the misfit. However, the approach proposed above is useful for finding a compromise when many phases are considered as candidates for the substrate.
Acknowledgements The authors would like to thank Dr. M. Krasin'kova for helpful discussions. This work is supported, in part, by the US Defense Department and by the International Science Foundation (Grant N NUH000). The program for computer-aided Solving of the problem is written in PASCAL and is realized on an IBM-compatible computer. The program is available from the authors.
282
A.N. Efimov, A.O. Lebedev / Surface Science 344 (1995) 276-282
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