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The strain tensor in a general pseudomorphic epitaxial film
Pergamon
0022-3697(94)00160-X
THE STRAIN TENSOR IN A GENERAL EPITAXIAL FILM P. M. MARCUS?
J. Phys. Chem. Solids Vol. 55. No. 12, pp. 1513-1519, 1994 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/94 $7.00 + 0.00
PSEUDOMORPHIC
and F. JONA
Department of Materials Science and Engineering, State University of New York, Stony Brook, New York 11794-2275, U.S.A. Abstract-The use of strain analysis, in particular, the measurement and calculation of strain ratios in the study of ultrathin pseudomorphic epitaxial films is discussed. The general solution is given of the problem in crystal-elasticity theory of finding the three unknown out-of-plane strain components from the three known in-plane strain components of a pseudomorphic ultrathin film growing on a substrate with a unit mesh different in both size and shape from the film’s unit mesh. It is shown that the experimental determination, e.g. by quantitative analysis of low-energy electron diffraction intensities, of the strains in the epitaxial ultrathin film of a cubic material grown on a general surface yields values of the ratios of two shear moduli to the bulk modulus of the film material. Keywords: A. thin films, B. epitaxial growth, D. elastic properties.
I. INTRODUCTION Epitaxial
growth
is the process
grows on top of another
by which
a crystal
crystal in such a way that the
orientations of the two crystals bear a well-defined relationship with one another. The two crystals may have similar or different atomic structures. In most experimental studies and in many technical applications the growing crystal is kept thin, and is appropriately referred to as an epitaxial film. When the thickness involves only a small number of atomic layers (say, 10 or 20) the film is called an ultrathin film. The crystal on which the film is growing is called the substrate. The most interesting case of epitaxial growth is the one in which the growing film assumes the unit mesh of the substrate in all planes parallel to the interface. This case is referred to as pseudomorphism, and the film is then said to be pseudomorphic with the substrate. Since in general the unit mesh of the growing crystal is different from that of the substrate, the film is strained by being forced to assume the unit mesh of the substrate at the interface. We will refer to the structure of the growing crystal in its unstrained state as the equilibrium phase. The strain in the plane of the film is called the epitaxial strain and is determined by the mismatch (or misfit) between the unit mesh of the equilibrium phase of the film and the unit mesh of the substrate. In the simple case when both the equilibrium phase and the substrate are
tIBM Research Emeritus.
cubic, calling a_ and a the lattice constants of the film’s equilibrium phase and the substrate, respectively, the misfit is defined as (a - ~_,)/a_. In general, if the misfit is small the film may grow pseudomorphically to be very thick. If the misfit is large the film may grow pseudomorphically for only a few layers and then develop misfit dislocations to relieve the strains, or it may not grow pseudomorphitally at all. Just how thick a pseudomorphic film can grow dislocation-free for a given misfit depends on a number of material parameters and growth conditions, and cannot be predicted with certainty. But what can be predicted is the strain which the film will have when the film is pseudomorphic. In a thin pseudomorphic film the strain is uniform throughout the film except for relaxation effects near the substrate-film interface and near the film-vacuum interface. Of the six components tik of the strain tensor t’, three are known at the outset from the misfit tg the substrate, namely, the compressive (or tensile) strains along two axes in the plane of the interface (c ;, and t&), and the shear strain in the plane (c;~). The three remaining strain components, the one along the axis perpendicular to the film (E;~) and the two associated shear strains (6 i3 and cij) can then be calculated, within the limits of linear elasticity theory, as we will show below. These same strain components can also be measured by a quantitative low-energy electron diffraction (QLEED) analysis: since ultrathin films thicker than 6-8 atomic layers include layers with bulk properties, QLEED can determine the interlayer spacings and the registries of 1513
1514
P. M. MARCUS and F. JONA
both bulk and surface regions, and hence, if the equilibrium phase is known, the strains [l]. In this paper we give the general solution of the problem in crystal-elasticity theory of finding the three unknown strain components t;,, E;, and t$ from the three known in-plane components of strain. We also show that, for cubic materials, a QLEED measurement of the strains in a single general epitaxial film allows the determination of ratios of elastic moduli of the film. In a recent review on the application of QLEED analysis to the study of ultrathin films [l] we give two ways of finding the relations between strains (more precisely, strain ratios) and elastic constant ratios. One way uses the strain energy, which is a function of the elastic constants and the strain components. Minimization of the strain energy with respect to ti3/~;,, t;r/~;, and t;,/c;, leads to relations between these strain ratios and ratios of elastic constants of the equilibrium phase. The second way is essentially the procedure given here in Section IV, but generalized here to unit meshes of films and substrate that are not similar. Another useful application of the strain energy, also demonstrated in the review [l], is in finding the most favorable orientation of an epitaxial film on the substrate in the case of so-called row-matching epitaxy-a case, to be defined in the next section, in which particular atomic rows of the film and the substrate are made parallel to each other and the inter-row spacing in the film is strained to match the inter-row spacing on the substrate surface. In a number of studies of ultrathin epitaxial films we have demonstrated (see references in [l]) that the measurement of the perpendicular strain, and the associated strain analysis are useful for: (1) identification of the equilibrium phase of the epitaxial film, e.g. whether it is bee or fee; (2) estimate of elasticconstant ratios of the film; (3) estimate of the equilibrium lattice constants of metastable phases which may be grown epitaxially on suitable substrates, but are not available as macroscopic crystals. The present paper is an extension of the material discussed in our review [l]. It reproduces in succinct form some of the results derived in that review (to which the reader is referred for more extended discussion and detailed applications), but the main goal is to discuss the general problem of finding the whole strain tensor in an epitaxial film growing on a substrate with a unit mesh different in both size and shape-in particular, this paper sketches the general solution for the out-of-plane components of the strain tensor in the film (this general problem is not solved in the review). In Section II we recall the distinctions among three different cases of pseudomorphic epitaxy; in Section
III we summarize the formulae useful for the strain analysis of an epitaxial film in the simplest most common case encountered in experimental studies. In Sections IV and V we introduce and solve the problem of finding the complete strain tensor in the most general case of pseudomorphic epitaxy. Application of the solution is made to an fcc(210) film in Case 1 epitaxial strain (see Section II) to show how c& and 6; vary with the elastic constants of the film. Application to Cu{210} shows how measured strains determine the ratios of two shear moduli to the bulk modulus of Cu.
II. THE THREE CASES OF PSEUDOMORPHISM When pseudomorphism occurs, i.e. when an epitaxial film assumes various lattice constants of the substrate in the plane of the surface, we distinguish three possible cases of pseudomorphism. Case 1 is characterized by the fact that the unit mesh of the substrate and the unit mesh of the film’s equilibrium phase are similar, e.g. both square, i.e. the unit meshes differ only in scale. The distances between atomic rows must be matched in both crystallographic directions in the surface plane-the two sides of the film’s unit mesh are resealed by the same factor, but the angle between the sides is unchanged. The planar epitaxial strain is therefore isotropic, and the distances between atomic layers parallel to the film plane are altered by the Poisson ratio. We will be interested in the ratio between the out-of-plane and the in-plane components of strain. Case 2 also involves the matching of two sets of rows at the film-substrate interface, but the unit mesh of the substrate and the unit mesh of the film’s equilibrium phase are not similar, they may differ in both shape and size. The sides of the unit mesh of the equilibrium phase may differ from their counterparts on the substrate by different scales, and the angles between the sides may be different in the two unit meshes. Pseudomorphism can still occur if the differences in sides and angles between the sides amount to only a few percent. In this case the epitaxial strain is anisotropic, and the calculation of the whole strain tensor is more complicated than in case 1. Case 3 occurs when the inter-row distances of only one set of atomic rows are closely matched between equilibrium phase and substrate-the inter-row distances in the other crystallographic direction on the substrate surface can be quite different in the two nets. In this case pseudomorphism is only partial, in that row spacings are made equal in one direction, but not in the other. There is a known strain in the direction perpendicular to the matched rows, and only zero stress in the direction along these rows.
The strain tensor in a general pseudomorphic
Case 3 has been observed experimentally, e.g. in the case of growth of body-centered-cubic {1lo} films on face-centered-cubic { 111) or hexagonal-close-packed (0001) substrates (see, e.g. Ref. [l]). Among the orientations that can be encountered in case 3 are those known under the names of Kurdjumov-Sachs (abbreviated KS) and Nishiyama-Wasserman (abbreviated NW). We do not consider case 3 in this paper, but we refer to our review for a quantitative discussion.
epitaxial film
1515
where y is a function of elastic constants of the film’s equilibrium phase which depends on the crystallographic orientation of the film plane (see, e.g. Ref. [l]). For cubic {OOl}, y = 2c,,/c,, = 2v/(l - v). where v is the Poisson ratio for (001) planes. Hence, in the case of small strains, 63 _
(c/c,,)
-
1
1= -I’?
<-(a/a,,)-
(4)
which we rewrite as III. THE STRAIN RATIO The quantities of interest, when growing an epitaxial pseudomorphic film, are the ratios of the out-ofplane strain components to the strain (or strains) in the plane of the film. The reason that these strain ratios are interesting is that, on the one hand, they can be measured with an experiment, e.g. QLEED, and on the other hand they can be calculated from the elastic properties of the equilibrium phase, assuming, of course. that linear elasticity theory applies and that both the equilibrium phase and its elastic constants are known. Consider for example the simple case of a cubic equilibrium phase (lattice constant ae4) growing onto the {OOI ) surface of a cubic substrate (lattice constant u # +). If the film grows pseudomorphically (case I). the macroscopic strain in the plane of the film is isotropic and is equal to the misfit (a - (‘eqVu,, = (a/a,, ) - 1. The elastic response of the film will cause a change of the periodicity in the perpendicular direction-the structure of the film will no longer be cubic, but rather tetragonal, more precisely, body-centered tetragonal. Accordingly, we call the film’s lattice constant in the perpendicular direction c, and the macroscopic strain in the perpendicular direction will be (c - c,~)/c,~ = (c/c,,) - 1. If these macroscopic strains are both small we can equate them to the infinitesimal strain components cr in the plane of the film and L, perpendicular to the film, respectively. i.e. we put “%
1 =t,,
(1)
and cC,q
1 =tj.
(2)
Now the strain ratio c,/e, can be written, assuming linear elasticity theory. as tj Cl
C=l_, %
(3)
a-1.
Under the conditions stated, namely, when we know the equilibrium phase of the film and its elastic constants, and when the macroscopic strains are small, eqn (5) allows us to calculate the perpendicular periodicity c for pseudomorphic growth of the film on a substrate with cubic lattice constant a. The periodicity c can also be determined experimentally by techniques such as QLEED (see Ref. 111). If the macroscopic strains are large, however, eqns (1) and (2) are no longer valid. The macroscopic strain ratio must be calculated from the microscopic strain ratio (3) by integration. The infinitesimal strains are c, = da/a and t3 = de/c and so t3 de/c ,=-d&=
--Y.
which, when integrated between the limits ccs to c and that y is independent of the magnitude of the strain), yields aeq to a (under the assumption
a -
C
-=
Qq
-7
0aeq
.
Hence, the ratio rs of macroscopic strains is. rather than eqn (4)
(c/c,,>- 1 (a/a,,)P - 1
” = (a/a,)
- 1=
(a/a,,) - 1
(8)
Equations (7) and (8) are therefore the correct equations to use for the calculation of c, whereas eqns (4) and (5) are approximates. In fact, if we expand the right-hand side of eqn (7) using the binomial theorem (1 + u)” = 1 + nu + [n(n - l)u’]/2! + , with u = (a/a,,) - 1 and n = -_‘r’,we obtain C
i*
(aeq >
__=l_y
%
(a,, > “-1
+...,
1516
P. M. MARCUS and F. JONA
which for small strains becomes eqn (5). In practice, the error encountered in using the approximation (5) rather then the correct eqn (7) is not very large: for example, for a film of Cu{OOl} (y = 1.44) grown on a substrate with 15% misfit (a/a, = 1.15) the difference in the values of c calculated with eqns (5) or (7) is only 4.3%. A number of applications of eqns (4) and (8) can be found in our review article [l].
IV. THE GENERAL PROBLEM OF PSEUDOMORPHIC EPITAXY We now consider the general problem of determining the strain tensor in a pseudomorphic film grown under the conditions of case 2 epitaxy as defined in Section II. The conditions are that the film’s equilibrium phase and the substrate have different unit meshes-different in both size and shape. This difference fixes the macroscopic strains in the plane of the film: recall that in case 2 the epitaxial strain is anisotropic, i.e. in the plane of the film there are two dzjkent principal strains in the two crystallographic directions and one shear strain. We will assume that these macroscopic strains are small, so that we can set them equal to the corresponding components of the strain tensor. Thus, the difference between the substrate unit mesh and the unit mesh of the equilibrium phase fixes three components of the strain tensor, namely, taking a carthesian coordinate system with 2; and jz; in the plane and 2; perpendicular to it, the components E;, , t iz and t;r. The problem is to find the other three components: tix, t& and c;, . In order to solve the problem we need, first, to express the in-plane strain components in terms of the unit-mesh parameters of substrate and film equilibrium phase, and, second, to recall the formulae for the transformation of tensor components from one coordinate system to another, the so-called tensor transformation law. The need for this law in the present case arises from the fact that we must be able to describe the strain tensor in two coordinate systems: the system defined by the crystal axes and the system defined by the surface axes. The atom displacements that give rise to the strains in the film are most conveniently described in the surface system, but the relations between stresses and strains involving the elastic constants are given in all treatments of crystal elasticity in the crystal system. We discuss the in-plane strains in Section 1V.A and the tensor transformation law in Section 1V.B. IV.A. Derivation of the in-plane strain components We recall first the general definition of a twodimensional strain (see, e.g. Ref. [2], p. 94). Given a
plane delined by two perpendicular axes 1 and 2, when a vector with components x, , x2 is deformed to have components xi + ul, x2 + u, (u,, u2 are the components of the displacement vector u) the strain components in the plane are defined as
au, &=-, a4 ax, ax,
E;,=--,
E;z=-
i au, -+ax,
2(
au, . (10) ax, >
For our purposes it is convenient to express the strain components in terms of the linear transformation matrix which transforms the vector (x,, x2) into the vector (x, + u, , x2 + IL*). This linear transformation matrix T is found from the relation: =
(11) whence ul= (T,, - 1)x, + T,zxz a, = T2,x, + (T,, - 1)x,.
(12)
We see therefore that
au, T,, -
E;, =-= ax,
1,
(13)
&=-=
au, Tz2-
ax,
I.
Now the problem is to find T for the general case 2, namely, the transformation=matrix that deforms the unit mesh of the film’s equilibrium phase into the unit mesh of the substrate. Using the general symbols a, and a2 and 0 for the basic vectors of the unit meshes and the angles between them, respectively, we will add the superscript eq for the in-plane unit mesh of the equilibrium phase of the film and the superscript s for the in-plane unit mesh of the substrate. Figure 1 shows the relation between the two, where we have chosen a? collinear with a;, and the surface coordinate system n;, 2; with 2; parallel to these a, vectors. The x, y components of the a,, a2 vectors in the xi - xi system will be designated a,, , al2 and a*, , az2, respectively. Now the components of the T matrix can be found = from the equation
The strain tensor in a general pseudomorphic epitaxial film
ks’
a
l-
21’ /
ap, I
c
/ .- ‘@ tPs .’ ’ b-
4
a”1
Fig. 1. Relation between substrate unit mesh (basis vectors a;, a;, angle 0%)and equilibrium-phase unit mesh (basis vectors a?, a?, angle P) for case-2 pseudomorphism.
1517
with 9; and ft; in the plane of the film and P; perpendicular to it. In order to express the components of a tensor in the surface axes in terms of the components in the crystal axes, or vice versa, we need the angular relations between these axes, which are given by the direction cosines, i.e., the cosines of the angles between a,! and ii (i = 1 to 3). The cosine of the angle between Si and 2, is given by the magnitude of the component xj, (i,j = 1 to 3) of the 2; vector in the 2j system. The transformation rule for a second-rank tensor is then (see, e.g. Ref. [2], p. 11). 3
which upon consideration
of Fig. 1 can he rewritten
Sb = 1
i,j = 1-3,
x~,x~,S,,,
(19)
k.l= I
as where Sk, is a component
(15) Carrying
out the matrix multiplication
T,,a;q=a”
1,
x
of a second-rank tensor in the “old” system (say, the crystal system $) and S,> is a component of the same tensor in the “new” system (say, the surface system 2;). The inverse transformation in terms of the components x, is
we find: 3 S,=
3
C x~kxj~~~~= k,,= I
1 xi,~~S;,, k,l= I
(20)
T,,a~cosP+T,,a~sinP=a~costl’, which uses the relation
T,, a;q = 0, x T2, ap cos 0
+ T,,a;q sin P
= ai sin OS,
(16)
T,z=
I
ai cos 19’ a? sin
e- -
ai cos P af4 sin
I
xki
1
(21)
both quantities being the cosine of the angle between x, and xi (which is not the same as the angle between xk and xl).
whence
T,, =$,
xik =
eq 1V.C. Calculation of strain ratios for case 2
T,, = 0,
a3 sin 0”
T,, =
(17)
a? sin 6q .
Using eqn (13) we finally obtain
c;,=--4
1,
a? 1
L;2 = -
a3cosP
a; cos Oq
2 ( a~sinOq-a~sinP a$ sin 8”
622
=
a? sin
eq
-
>’
1.
(18)
1V.B. The tensor transformation law The crystal axes are denoted by the unit vectors RI, S,, 9,; the surface axes by the unit vectors n;, 2;, 2;,
The problem, as mentioned above, is the following: given the in-plane strain components E;, , ei2 and .z;r produced by the matching of dissimilar unit meshes as discussed in Section IV.A, calculate the remaining three components ~;r, E& and 6G3 of the strain tensor. We outline the calculation as follows. We first apply three unit stresses (which we will call canonical stresses) in the plane of the film, namely, two with only the normal components a;, and a& along 2; and n$, respectively, and one with only the shear component oi2 = a;, . We then proceed to find the strain tensors (which we call the canonical strains) produced by these canonical stresses (index h). For this purpose we transform the canonical stresses to the crystal axes S, , ?Z2,S3 using the tensor transformation formula (19) obtaining the components
P. M. MARCUS and F. JONA
1518
(unprimed because referred to crystal axes) in both single- and double-index notation: $’
= q
=
x;ix;j,
h = 1,2;
oc3) P = a$) = xiixij + xijxij,
i,j = 1 to 3; p = l-6; i, j = 1 to 3; p = 1 to 6. (22)
Then we solve the stress-strain equations
h = 1 to 3, p = 1 to 6
(23)
for tch) the strains produced by the three canonical unit it;esses in the crystal-axes system (the cp4are the elastic constants of the film’s equilibrium phase). We thus have found the three canonical strain tensors in the crystal axes. After conversion of the strain components from the one-index form E:) to the two-index form c$) = t$‘) (recall that c,, = 2t,, , c5= 2~,, and Lo= 2~,,), we transform the strain components back to the surface axes, again by using the tensor transformation law (primed quantities refer to surface axes): 3 d$”
=
c
x;~x@),
h, k, 1 = 1 to 3.
(24)
t,j= 1
Now the three strain tensors in the surface axes are superimposed with three different weight factors fi, fi, & to yield the total strain tensor in the film:
(@), = i J&y, h=,
h, k, 1 = 1 to 3.
(25)
The appropriate weight factors are found in terms of the unit-mesh parameters by setting the two longitudinal strain components (E?)’ and (&m>’ and the shear component (c$“‘)’ equal to the corresponding quantities determined by the mismatch and derived in Section 1V.A. The total strain tensor (25) has the correct in-plane components and contains the threedesired out-of-plane components. Our review article [1] contains a FORTRAN program for calculation of the strain ratios for given elastic constants and mismatch 6p in case 1 for an arbitrary surface. In case 1, .siI = ci2 = &p and L12=&conditions which can be derived from the general formula (18) by noting that for similar unit meshes as required by case 1, 8” = Be9 and a; /a;” = as/Q. A small modification of that program involving eqns (18) for the in-plane strain components will solve case 2 numerically. Reference [1] also has tables of the xys for the 20 most closely packed surfaces of the bee and fee lattices.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.0 G/9 Fig. 2. Contours of constant strain ratio C&/E;, from - 1.6 to -0.4 for a cubic (210) film on the plane of elastic-moduli ratios, axes Q/B and G/B. The circle marks the position of fee cu.
V. APPLICATIONS OF STRAIN ANALYSIS OF ULTRATHIN FILMS We show that the determination of the “bulk” structure of an epitaxial ultrathin film, which allows the determination of strain ratios, gives quantitative information about the elastic properties of the film. We limit our discussion to cubic crystals, which have 3 independent elastic constants (c,, , cIz and Q) and two so-called elastic-moduli ratios, namely, the ratio G/B of the shear modulus G = (c,, - c,,)/2 to the bulk modulus B = (q, + 2c,,)/3, and the
0.7 0.6 e4Je 0.5 0.4 0.3 0.2
0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 G/9 Fig. 3. Contours of constant strain ratio e&/c;, from -0.8 to 0.6 for a cubic (210) film on the plane of elastic-moduli ratios. The strain ratio vanishes for isotropic materials with _. G=c,. The circle marks the position of fee Cu.
The strain tensor in a general pseudomorphic epitaxial film fore find contour
1.519
lines of constant
plane whose axes are elastic-moduli of such constant
0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 G/B Fig. 4. The two lines are contours of constant strain ratios c&/r;, = -1.171 and tGj/t;, = -0.638 on the elastic-moduli ratio plane for a film of fee Cu(210). The strain ratios correspond to Cu elastic constants c,, = 1.7620Mbar, c,? = 1.2494 Mbar, c, = 0.8177 Mbar). The two contours cross at the Cu point (G/B = 0.1805, cM/B = 0.576) thereby showing that a structure determination of an fcc(210) film which determined both tij/t ;, and t;& ;, would determine the elastic-moduli ratios G/B and Q/B for the film material.
ratio
of the C~ elastic constant
Q/B
to the bulk
modulus B. Given a specific surface of a cubic crystal (which is identified
by the plane of the film) we can calculate,
from the elastic constants
f I and t,3/tl, (or t&/t;,, elastic-moduli
ratios
cik, the strain ratios eij/c ;,
see Ref. [l]), as well as the two We can there-
c,,,,/B and G/B.
strain-ratio
strain-ratio
on a
ratios. Examples
contours
for a cubic
1210) surface [the surface axes for cubic {210}, indexed as the (420) surface in our review [l], are, in terms of the cubic lattice constant: xi = (0,O. l), xi = (l/,,/?, - 2/$, 0) and x; = (2/,/?, 1/fi, 0) (see Ref. [l])] are given in Fig. 2 for tiz/e;,, and in Fig. 3 for L&/L;, . In both figures the circles mark the position for Cu (for which G/B = 0.1805, c,/B = 0.576, and c&/t;, = -1.1707, c&/e;, = -0.6380). In general, when film and substrate have similar primitive unit meshes, a crystalline film with a particular surface will exhibit up to 3 strain ratios, i.e. ratios of the out-of-plane strains to the in-plane (isotropic) strain. Measurement of the bulk structure of the film (e.g. by QLEED) will yield these strain ratios. If the equilibrium phase is cubic, any two strain ratios will determine the two elastic-moduli ratios. Such a determination is illustrated in Fig. 4 for face-centered-cubic Cu with the strain ratios for the (210) surface. Each of the two strain ratios for Cu(210) fixes a line on the elastic-moduli-ratio plane. and the intersection of the two lines determines the elastic-moduli ratios for Cu. work was sponsored in part by the National Science Foundation with Grant DMR9404421.
Acknowledgement-This
REFERENCES 1. Jona F. and Marcus P. M., Crirical Reviews on Surface Chemistry, in press (1994). 2. Nye J. F., Physical Properties of Crystals. Clarendon Press, Oxford (1957).
0022-3697(94)00160-X
THE STRAIN TENSOR IN A GENERAL EPITAXIAL FILM P. M. MARCUS?
J. Phys. Chem. Solids Vol. 55. No. 12, pp. 1513-1519, 1994 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/94 $7.00 + 0.00
PSEUDOMORPHIC
and F. JONA
Department of Materials Science and Engineering, State University of New York, Stony Brook, New York 11794-2275, U.S.A. Abstract-The use of strain analysis, in particular, the measurement and calculation of strain ratios in the study of ultrathin pseudomorphic epitaxial films is discussed. The general solution is given of the problem in crystal-elasticity theory of finding the three unknown out-of-plane strain components from the three known in-plane strain components of a pseudomorphic ultrathin film growing on a substrate with a unit mesh different in both size and shape from the film’s unit mesh. It is shown that the experimental determination, e.g. by quantitative analysis of low-energy electron diffraction intensities, of the strains in the epitaxial ultrathin film of a cubic material grown on a general surface yields values of the ratios of two shear moduli to the bulk modulus of the film material. Keywords: A. thin films, B. epitaxial growth, D. elastic properties.
I. INTRODUCTION Epitaxial
growth
is the process
grows on top of another
by which
a crystal
crystal in such a way that the
orientations of the two crystals bear a well-defined relationship with one another. The two crystals may have similar or different atomic structures. In most experimental studies and in many technical applications the growing crystal is kept thin, and is appropriately referred to as an epitaxial film. When the thickness involves only a small number of atomic layers (say, 10 or 20) the film is called an ultrathin film. The crystal on which the film is growing is called the substrate. The most interesting case of epitaxial growth is the one in which the growing film assumes the unit mesh of the substrate in all planes parallel to the interface. This case is referred to as pseudomorphism, and the film is then said to be pseudomorphic with the substrate. Since in general the unit mesh of the growing crystal is different from that of the substrate, the film is strained by being forced to assume the unit mesh of the substrate at the interface. We will refer to the structure of the growing crystal in its unstrained state as the equilibrium phase. The strain in the plane of the film is called the epitaxial strain and is determined by the mismatch (or misfit) between the unit mesh of the equilibrium phase of the film and the unit mesh of the substrate. In the simple case when both the equilibrium phase and the substrate are
tIBM Research Emeritus.
cubic, calling a_ and a the lattice constants of the film’s equilibrium phase and the substrate, respectively, the misfit is defined as (a - ~_,)/a_. In general, if the misfit is small the film may grow pseudomorphically to be very thick. If the misfit is large the film may grow pseudomorphically for only a few layers and then develop misfit dislocations to relieve the strains, or it may not grow pseudomorphitally at all. Just how thick a pseudomorphic film can grow dislocation-free for a given misfit depends on a number of material parameters and growth conditions, and cannot be predicted with certainty. But what can be predicted is the strain which the film will have when the film is pseudomorphic. In a thin pseudomorphic film the strain is uniform throughout the film except for relaxation effects near the substrate-film interface and near the film-vacuum interface. Of the six components tik of the strain tensor t’, three are known at the outset from the misfit tg the substrate, namely, the compressive (or tensile) strains along two axes in the plane of the interface (c ;, and t&), and the shear strain in the plane (c;~). The three remaining strain components, the one along the axis perpendicular to the film (E;~) and the two associated shear strains (6 i3 and cij) can then be calculated, within the limits of linear elasticity theory, as we will show below. These same strain components can also be measured by a quantitative low-energy electron diffraction (QLEED) analysis: since ultrathin films thicker than 6-8 atomic layers include layers with bulk properties, QLEED can determine the interlayer spacings and the registries of 1513
1514
P. M. MARCUS and F. JONA
both bulk and surface regions, and hence, if the equilibrium phase is known, the strains [l]. In this paper we give the general solution of the problem in crystal-elasticity theory of finding the three unknown strain components t;,, E;, and t$ from the three known in-plane components of strain. We also show that, for cubic materials, a QLEED measurement of the strains in a single general epitaxial film allows the determination of ratios of elastic moduli of the film. In a recent review on the application of QLEED analysis to the study of ultrathin films [l] we give two ways of finding the relations between strains (more precisely, strain ratios) and elastic constant ratios. One way uses the strain energy, which is a function of the elastic constants and the strain components. Minimization of the strain energy with respect to ti3/~;,, t;r/~;, and t;,/c;, leads to relations between these strain ratios and ratios of elastic constants of the equilibrium phase. The second way is essentially the procedure given here in Section IV, but generalized here to unit meshes of films and substrate that are not similar. Another useful application of the strain energy, also demonstrated in the review [l], is in finding the most favorable orientation of an epitaxial film on the substrate in the case of so-called row-matching epitaxy-a case, to be defined in the next section, in which particular atomic rows of the film and the substrate are made parallel to each other and the inter-row spacing in the film is strained to match the inter-row spacing on the substrate surface. In a number of studies of ultrathin epitaxial films we have demonstrated (see references in [l]) that the measurement of the perpendicular strain, and the associated strain analysis are useful for: (1) identification of the equilibrium phase of the epitaxial film, e.g. whether it is bee or fee; (2) estimate of elasticconstant ratios of the film; (3) estimate of the equilibrium lattice constants of metastable phases which may be grown epitaxially on suitable substrates, but are not available as macroscopic crystals. The present paper is an extension of the material discussed in our review [l]. It reproduces in succinct form some of the results derived in that review (to which the reader is referred for more extended discussion and detailed applications), but the main goal is to discuss the general problem of finding the whole strain tensor in an epitaxial film growing on a substrate with a unit mesh different in both size and shape-in particular, this paper sketches the general solution for the out-of-plane components of the strain tensor in the film (this general problem is not solved in the review). In Section II we recall the distinctions among three different cases of pseudomorphic epitaxy; in Section
III we summarize the formulae useful for the strain analysis of an epitaxial film in the simplest most common case encountered in experimental studies. In Sections IV and V we introduce and solve the problem of finding the complete strain tensor in the most general case of pseudomorphic epitaxy. Application of the solution is made to an fcc(210) film in Case 1 epitaxial strain (see Section II) to show how c& and 6; vary with the elastic constants of the film. Application to Cu{210} shows how measured strains determine the ratios of two shear moduli to the bulk modulus of Cu.
II. THE THREE CASES OF PSEUDOMORPHISM When pseudomorphism occurs, i.e. when an epitaxial film assumes various lattice constants of the substrate in the plane of the surface, we distinguish three possible cases of pseudomorphism. Case 1 is characterized by the fact that the unit mesh of the substrate and the unit mesh of the film’s equilibrium phase are similar, e.g. both square, i.e. the unit meshes differ only in scale. The distances between atomic rows must be matched in both crystallographic directions in the surface plane-the two sides of the film’s unit mesh are resealed by the same factor, but the angle between the sides is unchanged. The planar epitaxial strain is therefore isotropic, and the distances between atomic layers parallel to the film plane are altered by the Poisson ratio. We will be interested in the ratio between the out-of-plane and the in-plane components of strain. Case 2 also involves the matching of two sets of rows at the film-substrate interface, but the unit mesh of the substrate and the unit mesh of the film’s equilibrium phase are not similar, they may differ in both shape and size. The sides of the unit mesh of the equilibrium phase may differ from their counterparts on the substrate by different scales, and the angles between the sides may be different in the two unit meshes. Pseudomorphism can still occur if the differences in sides and angles between the sides amount to only a few percent. In this case the epitaxial strain is anisotropic, and the calculation of the whole strain tensor is more complicated than in case 1. Case 3 occurs when the inter-row distances of only one set of atomic rows are closely matched between equilibrium phase and substrate-the inter-row distances in the other crystallographic direction on the substrate surface can be quite different in the two nets. In this case pseudomorphism is only partial, in that row spacings are made equal in one direction, but not in the other. There is a known strain in the direction perpendicular to the matched rows, and only zero stress in the direction along these rows.
The strain tensor in a general pseudomorphic
Case 3 has been observed experimentally, e.g. in the case of growth of body-centered-cubic {1lo} films on face-centered-cubic { 111) or hexagonal-close-packed (0001) substrates (see, e.g. Ref. [l]). Among the orientations that can be encountered in case 3 are those known under the names of Kurdjumov-Sachs (abbreviated KS) and Nishiyama-Wasserman (abbreviated NW). We do not consider case 3 in this paper, but we refer to our review for a quantitative discussion.
epitaxial film
1515
where y is a function of elastic constants of the film’s equilibrium phase which depends on the crystallographic orientation of the film plane (see, e.g. Ref. [l]). For cubic {OOl}, y = 2c,,/c,, = 2v/(l - v). where v is the Poisson ratio for (001) planes. Hence, in the case of small strains, 63 _
(c/c,,)
-
1
1= -I’?
<-(a/a,,)-
(4)
which we rewrite as III. THE STRAIN RATIO The quantities of interest, when growing an epitaxial pseudomorphic film, are the ratios of the out-ofplane strain components to the strain (or strains) in the plane of the film. The reason that these strain ratios are interesting is that, on the one hand, they can be measured with an experiment, e.g. QLEED, and on the other hand they can be calculated from the elastic properties of the equilibrium phase, assuming, of course. that linear elasticity theory applies and that both the equilibrium phase and its elastic constants are known. Consider for example the simple case of a cubic equilibrium phase (lattice constant ae4) growing onto the {OOI ) surface of a cubic substrate (lattice constant u # +). If the film grows pseudomorphically (case I). the macroscopic strain in the plane of the film is isotropic and is equal to the misfit (a - (‘eqVu,, = (a/a,, ) - 1. The elastic response of the film will cause a change of the periodicity in the perpendicular direction-the structure of the film will no longer be cubic, but rather tetragonal, more precisely, body-centered tetragonal. Accordingly, we call the film’s lattice constant in the perpendicular direction c, and the macroscopic strain in the perpendicular direction will be (c - c,~)/c,~ = (c/c,,) - 1. If these macroscopic strains are both small we can equate them to the infinitesimal strain components cr in the plane of the film and L, perpendicular to the film, respectively. i.e. we put “%
1 =t,,
(1)
and cC,q
1 =tj.
(2)
Now the strain ratio c,/e, can be written, assuming linear elasticity theory. as tj Cl
C=l_, %
(3)
a-1.
Under the conditions stated, namely, when we know the equilibrium phase of the film and its elastic constants, and when the macroscopic strains are small, eqn (5) allows us to calculate the perpendicular periodicity c for pseudomorphic growth of the film on a substrate with cubic lattice constant a. The periodicity c can also be determined experimentally by techniques such as QLEED (see Ref. 111). If the macroscopic strains are large, however, eqns (1) and (2) are no longer valid. The macroscopic strain ratio must be calculated from the microscopic strain ratio (3) by integration. The infinitesimal strains are c, = da/a and t3 = de/c and so t3 de/c ,=-d&=
--Y.
which, when integrated between the limits ccs to c and that y is independent of the magnitude of the strain), yields aeq to a (under the assumption
a -
C
-=
-7
0aeq
.
Hence, the ratio rs of macroscopic strains is. rather than eqn (4)
(c/c,,>- 1 (a/a,,)P - 1
” = (a/a,)
- 1=
(a/a,,) - 1
(8)
Equations (7) and (8) are therefore the correct equations to use for the calculation of c, whereas eqns (4) and (5) are approximates. In fact, if we expand the right-hand side of eqn (7) using the binomial theorem (1 + u)” = 1 + nu + [n(n - l)u’]/2! + , with u = (a/a,,) - 1 and n = -_‘r’,we obtain C
i*
(aeq >
__=l_y
%
(a,, > “-1
+...,
1516
P. M. MARCUS and F. JONA
which for small strains becomes eqn (5). In practice, the error encountered in using the approximation (5) rather then the correct eqn (7) is not very large: for example, for a film of Cu{OOl} (y = 1.44) grown on a substrate with 15% misfit (a/a, = 1.15) the difference in the values of c calculated with eqns (5) or (7) is only 4.3%. A number of applications of eqns (4) and (8) can be found in our review article [l].
IV. THE GENERAL PROBLEM OF PSEUDOMORPHIC EPITAXY We now consider the general problem of determining the strain tensor in a pseudomorphic film grown under the conditions of case 2 epitaxy as defined in Section II. The conditions are that the film’s equilibrium phase and the substrate have different unit meshes-different in both size and shape. This difference fixes the macroscopic strains in the plane of the film: recall that in case 2 the epitaxial strain is anisotropic, i.e. in the plane of the film there are two dzjkent principal strains in the two crystallographic directions and one shear strain. We will assume that these macroscopic strains are small, so that we can set them equal to the corresponding components of the strain tensor. Thus, the difference between the substrate unit mesh and the unit mesh of the equilibrium phase fixes three components of the strain tensor, namely, taking a carthesian coordinate system with 2; and jz; in the plane and 2; perpendicular to it, the components E;, , t iz and t;r. The problem is to find the other three components: tix, t& and c;, . In order to solve the problem we need, first, to express the in-plane strain components in terms of the unit-mesh parameters of substrate and film equilibrium phase, and, second, to recall the formulae for the transformation of tensor components from one coordinate system to another, the so-called tensor transformation law. The need for this law in the present case arises from the fact that we must be able to describe the strain tensor in two coordinate systems: the system defined by the crystal axes and the system defined by the surface axes. The atom displacements that give rise to the strains in the film are most conveniently described in the surface system, but the relations between stresses and strains involving the elastic constants are given in all treatments of crystal elasticity in the crystal system. We discuss the in-plane strains in Section 1V.A and the tensor transformation law in Section 1V.B. IV.A. Derivation of the in-plane strain components We recall first the general definition of a twodimensional strain (see, e.g. Ref. [2], p. 94). Given a
plane delined by two perpendicular axes 1 and 2, when a vector with components x, , x2 is deformed to have components xi + ul, x2 + u, (u,, u2 are the components of the displacement vector u) the strain components in the plane are defined as
au, &=-, a4 ax, ax,
E;,=--,
E;z=-
i au, -+ax,
2(
au, . (10) ax, >
For our purposes it is convenient to express the strain components in terms of the linear transformation matrix which transforms the vector (x,, x2) into the vector (x, + u, , x2 + IL*). This linear transformation matrix T is found from the relation: =
(11) whence ul= (T,, - 1)x, + T,zxz a, = T2,x, + (T,, - 1)x,.
(12)
We see therefore that
au, T,, -
E;, =-= ax,
1,
(13)
&=-=
au, Tz2-
ax,
I.
Now the problem is to find T for the general case 2, namely, the transformation=matrix that deforms the unit mesh of the film’s equilibrium phase into the unit mesh of the substrate. Using the general symbols a, and a2 and 0 for the basic vectors of the unit meshes and the angles between them, respectively, we will add the superscript eq for the in-plane unit mesh of the equilibrium phase of the film and the superscript s for the in-plane unit mesh of the substrate. Figure 1 shows the relation between the two, where we have chosen a? collinear with a;, and the surface coordinate system n;, 2; with 2; parallel to these a, vectors. The x, y components of the a,, a2 vectors in the xi - xi system will be designated a,, , al2 and a*, , az2, respectively. Now the components of the T matrix can be found = from the equation
The strain tensor in a general pseudomorphic epitaxial film
ks’
a
l-
21’ /
ap, I
c
/ .- ‘@ tPs .’ ’ b-
4
a”1
Fig. 1. Relation between substrate unit mesh (basis vectors a;, a;, angle 0%)and equilibrium-phase unit mesh (basis vectors a?, a?, angle P) for case-2 pseudomorphism.
1517
with 9; and ft; in the plane of the film and P; perpendicular to it. In order to express the components of a tensor in the surface axes in terms of the components in the crystal axes, or vice versa, we need the angular relations between these axes, which are given by the direction cosines, i.e., the cosines of the angles between a,! and ii (i = 1 to 3). The cosine of the angle between Si and 2, is given by the magnitude of the component xj, (i,j = 1 to 3) of the 2; vector in the 2j system. The transformation rule for a second-rank tensor is then (see, e.g. Ref. [2], p. 11). 3
which upon consideration
of Fig. 1 can he rewritten
Sb = 1
i,j = 1-3,
x~,x~,S,,,
(19)
k.l= I
as where Sk, is a component
(15) Carrying
out the matrix multiplication
T,,a;q=a”
1,
x
of a second-rank tensor in the “old” system (say, the crystal system $) and S,> is a component of the same tensor in the “new” system (say, the surface system 2;). The inverse transformation in terms of the components x, is
we find: 3 S,=
3
C x~kxj~~~~= k,,= I
1 xi,~~S;,, k,l= I
(20)
T,,a~cosP+T,,a~sinP=a~costl’, which uses the relation
T,, a;q = 0, x T2, ap cos 0
+ T,,a;q sin P
= ai sin OS,
(16)
T,z=
I
ai cos 19’ a? sin
e- -
ai cos P af4 sin
I
xki
1
(21)
both quantities being the cosine of the angle between x, and xi (which is not the same as the angle between xk and xl).
whence
T,, =$,
xik =
eq 1V.C. Calculation of strain ratios for case 2
T,, = 0,
a3 sin 0”
T,, =
(17)
a? sin 6q .
Using eqn (13) we finally obtain
c;,=--4
1,
a? 1
L;2 = -
a3cosP
a; cos Oq
2 ( a~sinOq-a~sinP a$ sin 8”
622
=
a? sin
eq
-
>’
1.
(18)
1V.B. The tensor transformation law The crystal axes are denoted by the unit vectors RI, S,, 9,; the surface axes by the unit vectors n;, 2;, 2;,
The problem, as mentioned above, is the following: given the in-plane strain components E;, , ei2 and .z;r produced by the matching of dissimilar unit meshes as discussed in Section IV.A, calculate the remaining three components ~;r, E& and 6G3 of the strain tensor. We outline the calculation as follows. We first apply three unit stresses (which we will call canonical stresses) in the plane of the film, namely, two with only the normal components a;, and a& along 2; and n$, respectively, and one with only the shear component oi2 = a;, . We then proceed to find the strain tensors (which we call the canonical strains) produced by these canonical stresses (index h). For this purpose we transform the canonical stresses to the crystal axes S, , ?Z2,S3 using the tensor transformation formula (19) obtaining the components
P. M. MARCUS and F. JONA
1518
(unprimed because referred to crystal axes) in both single- and double-index notation: $’
= q
=
x;ix;j,
h = 1,2;
oc3) P = a$) = xiixij + xijxij,
i,j = 1 to 3; p = l-6; i, j = 1 to 3; p = 1 to 6. (22)
Then we solve the stress-strain equations
h = 1 to 3, p = 1 to 6
(23)
for tch) the strains produced by the three canonical unit it;esses in the crystal-axes system (the cp4are the elastic constants of the film’s equilibrium phase). We thus have found the three canonical strain tensors in the crystal axes. After conversion of the strain components from the one-index form E:) to the two-index form c$) = t$‘) (recall that c,, = 2t,, , c5= 2~,, and Lo= 2~,,), we transform the strain components back to the surface axes, again by using the tensor transformation law (primed quantities refer to surface axes): 3 d$”
=
c
x;~x@),
h, k, 1 = 1 to 3.
(24)
t,j= 1
Now the three strain tensors in the surface axes are superimposed with three different weight factors fi, fi, & to yield the total strain tensor in the film:
(@), = i J&y, h=,
h, k, 1 = 1 to 3.
(25)
The appropriate weight factors are found in terms of the unit-mesh parameters by setting the two longitudinal strain components (E?)’ and (&m>’ and the shear component (c$“‘)’ equal to the corresponding quantities determined by the mismatch and derived in Section 1V.A. The total strain tensor (25) has the correct in-plane components and contains the threedesired out-of-plane components. Our review article [1] contains a FORTRAN program for calculation of the strain ratios for given elastic constants and mismatch 6p in case 1 for an arbitrary surface. In case 1, .siI = ci2 = &p and L12=&conditions which can be derived from the general formula (18) by noting that for similar unit meshes as required by case 1, 8” = Be9 and a; /a;” = as/Q. A small modification of that program involving eqns (18) for the in-plane strain components will solve case 2 numerically. Reference [1] also has tables of the xys for the 20 most closely packed surfaces of the bee and fee lattices.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.0 G/9 Fig. 2. Contours of constant strain ratio C&/E;, from - 1.6 to -0.4 for a cubic (210) film on the plane of elastic-moduli ratios, axes Q/B and G/B. The circle marks the position of fee cu.
V. APPLICATIONS OF STRAIN ANALYSIS OF ULTRATHIN FILMS We show that the determination of the “bulk” structure of an epitaxial ultrathin film, which allows the determination of strain ratios, gives quantitative information about the elastic properties of the film. We limit our discussion to cubic crystals, which have 3 independent elastic constants (c,, , cIz and Q) and two so-called elastic-moduli ratios, namely, the ratio G/B of the shear modulus G = (c,, - c,,)/2 to the bulk modulus B = (q, + 2c,,)/3, and the
0.7 0.6 e4Je 0.5 0.4 0.3 0.2
0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 G/9 Fig. 3. Contours of constant strain ratio e&/c;, from -0.8 to 0.6 for a cubic (210) film on the plane of elastic-moduli ratios. The strain ratio vanishes for isotropic materials with _. G=c,. The circle marks the position of fee Cu.
The strain tensor in a general pseudomorphic epitaxial film fore find contour
1.519
lines of constant
plane whose axes are elastic-moduli of such constant
0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 G/B Fig. 4. The two lines are contours of constant strain ratios c&/r;, = -1.171 and tGj/t;, = -0.638 on the elastic-moduli ratio plane for a film of fee Cu(210). The strain ratios correspond to Cu elastic constants c,, = 1.7620Mbar, c,? = 1.2494 Mbar, c, = 0.8177 Mbar). The two contours cross at the Cu point (G/B = 0.1805, cM/B = 0.576) thereby showing that a structure determination of an fcc(210) film which determined both tij/t ;, and t;& ;, would determine the elastic-moduli ratios G/B and Q/B for the film material.
ratio
of the C~ elastic constant
Q/B
to the bulk
modulus B. Given a specific surface of a cubic crystal (which is identified
by the plane of the film) we can calculate,
from the elastic constants
f I and t,3/tl, (or t&/t;,, elastic-moduli
ratios
cik, the strain ratios eij/c ;,
see Ref. [l]), as well as the two We can there-
c,,,,/B and G/B.
strain-ratio
strain-ratio
on a
ratios. Examples
contours
for a cubic
1210) surface [the surface axes for cubic {210}, indexed as the (420) surface in our review [l], are, in terms of the cubic lattice constant: xi = (0,O. l), xi = (l/,,/?, - 2/$, 0) and x; = (2/,/?, 1/fi, 0) (see Ref. [l])] are given in Fig. 2 for tiz/e;,, and in Fig. 3 for L&/L;, . In both figures the circles mark the position for Cu (for which G/B = 0.1805, c,/B = 0.576, and c&/t;, = -1.1707, c&/e;, = -0.6380). In general, when film and substrate have similar primitive unit meshes, a crystalline film with a particular surface will exhibit up to 3 strain ratios, i.e. ratios of the out-of-plane strains to the in-plane (isotropic) strain. Measurement of the bulk structure of the film (e.g. by QLEED) will yield these strain ratios. If the equilibrium phase is cubic, any two strain ratios will determine the two elastic-moduli ratios. Such a determination is illustrated in Fig. 4 for face-centered-cubic Cu with the strain ratios for the (210) surface. Each of the two strain ratios for Cu(210) fixes a line on the elastic-moduli-ratio plane. and the intersection of the two lines determines the elastic-moduli ratios for Cu. work was sponsored in part by the National Science Foundation with Grant DMR9404421.
Acknowledgement-This
REFERENCES 1. Jona F. and Marcus P. M., Crirical Reviews on Surface Chemistry, in press (1994). 2. Nye J. F., Physical Properties of Crystals. Clarendon Press, Oxford (1957).