COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational
Materials Science 8 (1997) 79-86
Thermodynamics of binary alloy thin films A. Diaz-Ortiz
a,1,J.M. Sanchez a,*, J.L. Morti-L6pez
b
a Center jtir Materials Science and Engineering, The University of Texas at Austin, Austin, TX 78712, USA b Instituto de Fisk Universidud Autbnoma de San Luis Potosi, San Luis Potosi, S.L.P. 78000, Mexico
Abstract The thermodynamics of ordering binary alloy thin films with non-neutral or ‘polarizing’ boundary conditions is studied within the frame of an Ising-like Hamiltonian with nearest-neighbor interactions. The finite-temperature behavior is obtained by means of the pair approximation of the cluster variation method which is the lowest approximation that takes into account short-range correlations. The analysis focuses on the dependence upon film thickness of the phase diagrams and on the nature of the surface field (A) which acts close to the surface and leads to segregation of one of the components. The finite-size and surface effects along with the discreetness of the system entail novel features on the phase diagrams that are not present in continuous theories and/or in the usual mean-field theory (Bragg-Williams) that neglects short-range order
correlations. Temperature-chemical potential and temperature-concentration concentration and long-range order profiles.
1. Introduction Considerable effort has been devoted to the study of cooperative behavior at surfaces. The interest stems from their growing technological importance for various applications e.g., catalysis phenomena, well as for their importance in understanding phase transitions. For a recent review of phase transitions at surfaces see Ref. [l]. Several studies have shown that surfaces may significantly affect the thermodynamic properties of ordering alloys undergoing firstorder phase transitions. Lipowsky and co-workers [2-41 have carried out a detailed analysis of semi-infinite systems in the continuous limit of the Landau theory. They found that the surface may start to
* Corresponding author. E-mail:
[email protected]. ’ Permanent address: Instituto de Fisica, Universidad Autdnoma de San Luis Potosi, Alvaro Obregdn 64, San Luis Potosi, S.L.P., 78000, Mexico. 0927-0256/97/$17.00 Copyright PII SO927-0256(97)00019-O
0 1997 Published
phase diagrams are presented as well as
disorder (order) as the first-order transition is approached although the bulk is still in its ordered (disordered) state. Thus, a layer of the disordered (ordered) phase intervenes between the surface and the bulk, and the material undergoes the so-called surface-induced disorder @ID) (surface-induced order @IO)) transition *. The SID and SIO are good examples of the surface influence in the cooperative phenomena of semi-infinite systems. In the case of confined systems, such as fluids between walls or solid thin films (in which the confinement is between the substrate and the vacuum), the scenario is rather different because the interplay between the finite-size and surface effects leads to interesting and novel characteristics in the cooperative behavior of these systems.
* The surface-induced disorder and surface-induced order transitions have also been found in discrete Ising models using the cluster variation method. See for example [5].
by Elsevier Science B.V. All rights reserved.
80
A. DkOrrrz
YI al./ Con~puturiotzul Muteriuls Scierlce 8 (1997) 79-86
Generally, in the case of finite-size effects, a rounding and a shift of the critical point appear [6], while surface effects modify the spatial order in the system that leads to ‘surface critical exponents’ [7]. On physical grounds, it is natural to suppose that the walls, in the case of a fluid, or the substrate in the case of solid films, exert a ‘surface field’ over the system. However, this case has received less attention than the case in which the boundary conditions (the wall, the substrate, etc.) are ‘neutral’, that is, in which the surface field is neglected. In the case of magnetic films it is more plausible to consider films without surface (magnetic) fields. This case has been studied extensively (see, for example, Ref. [l] and references therein). However, if the Ising model is reinterpreted as a lattice-gas model for a fluid. the wall-particle interaction represents a surface magnetic field. Some investigations considering ‘nonneutral’ or ‘polarizing’ boundary conditions have been carried out for the case of a fluid confined between parallel plates [g-13]. Fisher and Nakanishi have addressed the problem of critical points in lattice-gas models and they derived scaling predictions for the shifts in temperature and chemical potential [8,9]. An example of a shift in the coexistence curve due to confinement is the case of capillary condensation, i.e. when the interaction between fluid atoms and the walls favors condensation at both walls at a lower pressure than the one needed to induce it in bulk [10,12]. All the cases mentioned above deal with systems that are phase-separating in the bulk and, to the best of the authors’ knowledge, the study of the effects of surface fields in confined systems with ordering interactions has not been treated previously. Here, we focus on phase transitions in binary thin films with nearest-neighbor ordering interactions, considering the effect of chemical surface fields. In the magnetic language we are dealing with antiferromagnetic films having magnetic fields applied on the surfaces. In the case of a binary alloy, even when such explicit surface fields are absent. the effect of disrupting translational symmetry contributes to a surface field, due to the reduction of the coordination number at the surfaces. Consequently, surface segregation will take place even in this case. Although the case of antisymmetric boundary conditions (i.e., the field in one surface is opposite to the field on the
other surface) is of great interest ‘, we will concentrates upon the symmetric case. The Hamiltonian for layered systems and the statistical mechanics approximations used to calculate the finite-temperature behavior are described in Section 2. The temperaturechemical potential and the temperature-composition phase diagrams as well as the concentration and order profiles are presented in Section 3. Finally, Section 4 is devoted to a summary of the main results and to an outlook for future work.
2. Model We consider an Ising-like model describing a binary alloy on a three dimensional lattice consisting of N parallel lattice layers (see Fig. 1). Each layer has ~5” lattice points, labeled by i, j, . . . , and to each lattice site we assign an occupation operator a, that takes the values 1 and - 1 for A and B atoms, respectively. Vij is the effective pair interaction between nearest-neighbor lattice sites i and j. The Hamiltonian is then 2Y=;~V,,aiaj+p~o;+A, i.j
&T,+A,~ i
iE
I
c
CT,.
itN (1)
where p is the effective chemical potential and A, (k = 1, N) are the chemical surface fields acting on the surfaces. Although from the asymmetric case (A, # A,> arise interesting theoretical questions [ 141, for simplicity we restrict our investigation to the case in which A, = A, = A. Thus, A represents the effective surface segregation energy (in the case of a fluid confined between two parallel plates, A corresponds to the binding energy between the molecules and the walls). In principle, owing to elastic effects as well as differences in the local environment, it is generally expected that the atomic interactions in the surface region will differ from those in the bulk. Although the model contemplates this possibility, we assume that the pair interactions at the surface are the same as those in the bulk.
3 The case of antisymmetric Refs. [9, I I, 141 (and references arating confined systems.
surface fields has been treated in therein), for the case of phase-sep-
A. Diaz-Ortiz et al./ Compututiond
Fig. 1. Schematic representation of two bee unit cells displaying two adjacent (110) planes. Solid and open circles represent the two interpenetrating simple-cubic sublattices a and p.
The free energy of the system was determined using the cluster variation method within the pair approximation [15], which is the lowest approximation that takes into account short-range order correlations. The pair approximation is chosen for the sake of simplicity since the treatment allows analytical results for selected properties, as for example, the percolation threshold in the limit of large surface fields. We note that preliminary calculations in higher approximations, e.g., the tetrahedron approximation, display the same qualitative behavior and are discussed elsewhere [ 161. The calculations, presented and discussed in the next section, were performed for bee films with surfaces oriented in the (110) direction. The orderdisorder transitions are described in the usual manner by subdividing the bee lattice into two interpenetrating simple-cubic sublattices (a and p>. For the particular case of the (1101 direction in the bee lattice, each parallel layer contains the same number of lattice sites belonging to each sublattice (see Fig. 1). The phase transition is then described by the intralayer long-range order (LRO) parameters defined by q(k)
= P:< A) - P;( A),
(2)
Materials
Science 8 (1997) 79-86
81
fixed nominal composition x (hereafter x denotes the total concentration of A atoms), temperature (T), and thickness (Nl is determined by the value of A which, in turn, determines the component that segregates to the surface. Indeed, the relevant parameter controlling the thermodynamics is the ratio between the surface field and the nearest-neighbor pair interaction (V= b
0 relating to the A species hold for the B component when the surface chemical field has the value of -A). This preferential enhancement of one of the components on the surface drives an asymmetry in the temperaturechemical potential phase diagrams leading to an enhancement (or depression) of the transition temperatures with respect to the corresponding bulk value 4. If the surface segregation is so strong (large A) as to saturate the surface layer with A atoms, a structural separation between the disordered surface layer (pure A) and an ordered bulk (at T = 0) may take place, reducing the effective thickness from N to N - 2. It is worth emphasizing that the surface fields compete with the ordering interactions in the alloy. In this sense one can think of the surface field as a clustering field acting on the surfaces. In the following we analyze first the case of weak surface field (small A), followed by the case of intermediate values of A, and finally some comments are made concerning the case of strong surface field (large A). 3.1. Weak sur$ace field
3. Results and discussion
The effects of small A on the thermodynamics of bcc(l10) alloy films are summarized in the temperature-chemical potential (Fig. 2(a)) and temperaturecomposition (Fig. Z(b)) phase diagrams for a value of A equal to 1.5 and several thicknesses of the film up to five hundred layers thick. The first feature
As the surface field acts only on the surfaces of the film, the equilibrium state of the film for a given value of the chemical potential or, equivalently, for a
4 Higher transition temperatures than the corresponding temperatures in the bulk are also present in the case of A = 0 (Ref. [161).
where p:(A) and $(A> are the concentrations of A atoms on the sublattices LYand p in layer k, respectively.
A. Dia,--0rri: et al./ Con~putatronul Mutrria1.s Science 8 (1997) 79-86 BCCI I IO). DELTA=
-10
-8
-6
-1
-2
0
CHEMICAL BCC(I
I .5
2
4
6
8
07
08
09
IO
POTENTIAL
IO). DELTA=I.S
6 t
0’ 0
; 01
_.
02
03 ATO!vlIC
04
05
Oh
CONCENTRATION
Fio0’ 2. (a) Temperature-chemical potential, and (h) temperatureconcentration phase diagrams, for bee (I IO) films for a value of the chemical surface field, A = IS, and thickness from three to five hundred layers.
noted from these figures is an asymmetry of the phase diagrams, which vanishes as the number of layers increases. For example, for a fixed thickness, the transition temperatures are lower for x < 0.5 than for x > 0.5. This asymmetry in the phase diagrams can be explained in following way: at low temperatures, for nominal concentrations of A atoms less than 0.5 and A > 0, but small, the ordering interactions will rule the equilibrium state of the film favoring the A-B bonds. Note that this does not necessarily imply that the system develops long-range order. The long-range order will not be established until the nominal concentration of A atoms reaches the percolation threshold (x,). The maximum number of A-B bonds is attained when the total concentration of A atoms is 0.5. In the case of xP
A-B bonds (the long-range order is developed by the inner layers), and the remaining B atoms segregate to the surface. When the temperature increases the A atoms migrate from the inner layers to the surface, due to the presence of the non-vanishing surface field, destroying the long-range order in the film. In the case of nominal concentrations of A atoms higher than 0.5 but lower than the percolation concentration and in the low temperature regime, the surface layers will be populated chiefly by A atoms. The effect of the surface field is to retain the A atoms on the surfaces thus enhancing the order in the inner layers. Consequently, for a given thickness and small positive values of A, it is easier to disorder a film with nominal concentration of A atoms of 0.3 than a film with nominal concentration of A atoms equal to 0.7 (See Fig. 2(b)). As pointed out above, for small values of 4. the long-range order is developed by the inner layers of the film; thus for very thick films the asymmetry in the phase diagrams will disappear, since the corresponding bulk phase diagram is symmetric. Another feature that can be observed from Fig. 2(a), (b) is that the percolation concentration is not a uniform function of the film thickness and, consequently, there are three regions on which the critical temperature of the film is lower, equal. or greater that the corresponding temperature in the bulk. The explanation for this resides in the following arguments: when the total atomic concentration of A atoms in the film is greater than 0.5 and the surface layers are saturated with A atoms, the inner layers behave as a film of effective thickness N - 2 with effective nominal concentration of A atoms ( x,r,). Nx - 2
x.--= N-2’ cl,
(3)
and subject to a vanishing surface field. This can be observed in Fig. 3(b) for the percolation limit of a ten-layer film with A = I .5 (the percolation limit for an eight-layer film and A = 0 is 0.8978 [17]). The effect of the surface field is to retain A atoms at the surfaces and, consequently, increase long-range order in the inner layers when x > 0.5. Small values of A correspond to a weak surface segregation. Then the qualitative behavior of the system can be explained in terms of the case of A = 0. The percolation concentrations for a three-
A. Diaz-Ortiz
et al./ Computational
BCC(I IO), DELTA=IS 1.8
0.6 0.4 0.2 0.0
3
0 08
4 0.09
0.1 ATOMIC
0.11
0.12
013
0.14
CONCENTRATION
BCC( I IO). DELTA=
1.5
1
Materiuls
Sciencr 8 (1997) 79-86
83
the temperature-chemical potential and temperaturecomposition phase diagrams for a three-layer film with A = 7.15. This simple case illustrates well the novel characteristics displayed by the system when the competition between finite-size and surface effects is taken into account. In these plots the continuous lines denote second-order phase transitions while the dashed lines stand for first-order phase transitions. In the present case, the asymmetry in the phase diagrams and the differences from the bulk phase diagram are striking. First, we note that the phase diagram has two humps of different heights. In the first hump the transitions are determined exclusively by the surface, while for the second hump the orderdisorder transitions are driven by the inner layers. When the concentration in A atoms of the films is
N=3, DELTA=7.15,
7
0.0 0 86
0.87
0.88 ATOMIC
0.89
0.9
091
092
BCC(I 10)
0 93
CONCENTRATION
Fig. 3. Low-temperature detail of Fig. 2(b) around percolation thresholds showing the asymmetry in the phase diagrams when A # 0. The arrows denote the location of the percolation concentrations in the case of A = 0 and films with three and infinite layers: (a) at x = l/12 and I /8, and (b) at 7/8 and I 1,’ 12, respectively.
, -10
-13
0
-5 CHEMICAL
IO
5
POTENTIAL
N=3, DELTA=7.15,
BCC( I IO)
4.5,
layer film in the case of zero surface field are: k and E, while for the film with N + m the concentrations are a and a (Ref. [17]). This is seen in Fig. 3(a), (b) showing the low-temperature region of the phase diagram of Fig. 2(b) in a concentration range around the percolation thresholds. The arrows denote the location of the percolation concentration when A = 0 for films of three and an infinite number of layers
t171. 3.2. Intermediate
3.5 3.0 2.5 ‘s
2.0
0.0
surface fields
More interesting is the case of intermediate chemical surface fields, in which the interdependence between the finite-size and surface effects can be appreciated more clearly. In Fig. 4(a), (b) we show
,
4.0
‘0.1
0.2
03
0.4 ATOMIC
0.5
-0.6
0.7
0.8
0.9
J I
CONCENTRATION
Fig. 4. (a) Phase diagram in the chemical potential-temperature plane for a three-layer film and intermediate surface fields (A = 7.15). (b) Phase diagram in the concentration temperature. The solid lines denotes second-order phase transitions while dashed lines stand for first-order phase transitions. The ‘hole’ at low temperature is a disordered phase. See the text.
A. Diuz-Orb
84
et al./ Compututional
small, almost all the A atoms are localized at the surfaces and the system does not display long-range order until the concentration in the surface layers is equal to + (in the pair approximation, a square lattice percolates at concentrations 4 and :), which corresponds to a total concentration of i for the film. The second hump is explained using analogous considerations: when the film is rich in A species, both surfaces are completely filled with A atoms, and spontaneous order appears only when the composition in the central layer is : or, equivalently, when the overall concentration is g. Quite unexpected features are the low-temperature disordered phase and the two-phase region (denoted by the dashed lines in Fig. 4(a), (b)). Both phenomena have the same source: at low temperatures and intermediate surface fields, the A component is localized at the surfaces of the film and, in the low composition regime, the spatial order is established when the surface concentration is +. If the overall concentration of A atoms continues to increase, the order must cease when we have the central layer full of B atoms and coated by two layers with concentrations of $, that is, when the overall concentration of A atoms i. In order to re-establish the long-range order it is necessary to increase the composition of the central layer to at least f , or equivalently, to increase the overall concentration of A atoms up to $. At this stage the film has the following structure: a central layer with composition of a coated by two layers of pure A. When the temperature is increased,
N=lO. DELTA=l0.43 7 6 5 4 ‘s 3 2 1
J
0 0
01
0.2
03
ATOMIC
0.4
0.5
0.6
0.7
0.8
0.9
CONCENTRATION
Fig. 5. Phase diagram in the x-T plane for a ten-layer binary alloy thin film for intermediate surface field (A = 10.43). See the text.
Materials Science 8 (1997) 79-86 N=IO. DELTA=l0.43.
kT=l
I
3
* ,:
k=l k=2
\,
,’
‘\
: ,:
“\ “,,
-
k=5
‘..,
j,,:
“‘,;:,,,, I (a)
\f_ ____... -Y,j ,__.-._ 0.0
0.1
02
__.. . ..-_
0.3
04
ATOMIC
:
:
0.5
0.6
0.7
08
09
CONCENTRATION
N=lO. DELTA=l0.43.
kTzl.3
1.0
j:
i/:/m 0.4
k=$. _/
0.3 02
-
0.1 o.
7 0.0
0 1
_,/
_ ,’ ,_I’ ,,,’ ___
0.2
‘.
/ i
,,.’
0.3
ATOMIC
k=2 -
@I
J,, 0.4
I 0.5
0.6
07
0.8
0.9
CONCENTRATION
Fig. 6. (a) Isothermal long-range order and (b) concentration profiles for the case of intermediate chemical surface fields (A = 10.43) corresponding to the phase diagram of Fig. 5. The surface, subsurface, and the center layer profiles are denoted by solid, dashed, and dotted lines, respectively. The calculation was carried out at kT = 1.3.
this sandwich structure became unstable and an ordered phase is developed, mainly because the B atoms of the central layer begin to occupy sites in the surface layers. A region of two phases will emerge when 4 < x < f in which the disordered and the ordered phases coexist. The segment of secondorder phase transitions that intrudes into this twophase region corresponds to the points at which the determinant of the Hessian matrix of the free energy vanishes. With the exception of the presence of a two-phase region at low temperatures, the general features displayed by the three-layer films are present in thicker films. In Fig. 5 we show the phase diagram in the x-T plane for a ten-layer film with a chemical surface field A = 10.43. The ordered phase appears
A. Diaz-Ortiz
et ai./Computurional
when x = l/(2 N) and a low-temperature disordered phase extends between x = 3/(2N) and the corresponding effective percolation concentration of an eight-layer film (Eq. (3)) with A = 0, that is, x = 0.28176. Note that the appearance of the disordered phase at low temperatures is dictated by the values of this percolation limit. Typical long-range order and concentration profiles are displayed in Fig. 6(a), (b) and support the above arguments. It is worth emphasizing that in the regime of small compositions the transitions are driven by the surfaces while for large concentrations the transitions are ruled by the inner layers, as seen in the long-range order profile of Fig. 6(a). 3.3. Strong surface fields The case of strong surface fields is described by the limit A -+ ~0. In such a case the surface separates from the bulk and the phase diagram splits into two unconnected ordered regions. For one of these regions, the boundaries at T = 0 are located at x = l/(2 N) and x = 3/(2N). The highest T, occurs at x = l/N, corresponding to the maximum temperature in the order-disorder phase diagram of a square lattice binary alloy. The other ordered region will be bounded by the effective percolation concentrations corresponding to a film of thickness N - 2 and with A = 0, and with a maximum determined by the height of the phase diagram at equiatomic composition. In other words, the maximum will be at the overall concentration of A atoms x,,, = (N + 2)/ (2N) (See Eq. (3)).
4. Summary
and outlook
The phase transitions occurring in binary alloy thin films with polarizing boundary conditions were studied. It is found that the interdependence between the surface and finite-size effects entails novel features in the phase equilibria of these systems. In particular, the phase diagrams display characteristics not observed when the chemical surface fields are neglected: a low-temperature disordered phase, reentrance of the ordered phase, the shifting in the critical temperature, etc. A vital characteristic in obtaining these phenomena is the discreetness of the
Materids
Science 8 (1997) 79-86
85
system. A continuous theory will be incapable of displaying for example, the low-temperature disordered phase. The same is true for the case of meanfield theory (Bragg-Williams), since the Bragg-Williams approximation does not give infinite percolation limits and most of the present result depend on the existence of finite percolation thresholds. Although the results presented here were obtained within the pair approximation, we believe that the main features found for phase equilibria in ordering binary alloy films will remain true in higher approximations and exact calculations, e.g. Monte Carlo simulations (insofar as the Hamiltonian contains only nearest-neighbor interactions), because they are dictated by the nature of the low-energy states. When long-range and/or many-body interactions are included most of the salient characteristics will be lost. An Ising model represents a good approximation for binary alloys, and semi-empirical methods can be used along with the cluster expansion [ l&19] in order to compute the effective cluster interactions for the case of thin films of real binary alloys. On the other hand, an experimental realization of the chemical surface fields may be obtained by choosing a binary compound that orders in the bulk and has large a surface segregation energy. Control of the surface field may be achieved by absorbing, for example, hydrogen. Experiments in this kind of system would be highly desirable to check the predictions of the model. Future work includes: (i> The study of the asymmetric boundary conditions, when A, # A, in the case when the Hamiltonian is short-ranged; (ii) the more general problem of phase equilibria in which next-nearest-neighbor interactions are included in the Hamiltonian (prototype phase diagrams); (iii) calculations for surfaces with high Miller indices (stepped surfaces); (iv> the inclusion of magnetic degrees of freedom in the Hamiltonian, that are necessary to describe the surface effects of magnetic binary alloys; and (v) the study of complex systems, such as surfaces of oxides.
Acknowledgements This work was partially supported in Mexico by Consejo National de Ciencia y Tecnologia through
A. Diaz-Ortiz
86
C, al./ Conymtutiond
grants No. 485100-5-3883E and No. 4920-E9406. One of us (A.D.-0.) acknowledges the partial support of the U.S.A.-Mexico Foundation for Science.
References [II K. Binder, in: D.G. Pettifor (Ed.), Cohesion and Structure of Surfaces,
Elsevier, New York, 1995, ch. 3.
M R. Lipowsky, Phys. Rev. Lett. 49 (1982) 1575. [31 R. Lipowsky, W. Speth, Phys. Rev. B 28 (1983) 3983. [41 R. Lipowsky, J. AppI. Phys. 55 (1984) 2485. 151J.M. Sanchez, J.L. Moran-Lopez, Phys. Rev. B 32 (1985) 3534.
[61 M.E. Fisher, in: M.S. Green (Ed.), Proceedings Enrico Fermi School of Physics, Academic, New York, 1971.
of the 1970 course LI, Varenna, Italy,
Mutrrials
Science 8 (19971 79-86
[7] M.E. Fisher, J. Vat. Sci. Technol.
10 (1973) 665.
[81 M.E. Fisher, H. Nakanishi, J. Chem. Phys. 75 (1981) 5857. [9] H. Nakanishi, M.E. Fisher, J. Chem. Phys. 78 (1983) 3279. [lo] R. Evans, U.M.B. Marconi, P. Tarazona. J. Chem. Phys. 84 (1986) 2376. [I 11 A.O. Parry, R. Evans, Phys. Rev. Lett. 64 (1990) 439. [121 K. Binder, D.P. Landau, J. Chem. Phys. 96 (1992) 1444. [I31 K. Binder, D.P. Landau, A.M. Ferrenberg, Phys. Rev. Len. 74 (1995) 298. [14] M.E. Fisher, P.G. de Gennes, Compt. Rend. Acad. Sci. (Paris) B 287 (1978) 207. [I51 R. Kikuchi, Phys. Rev. 81 (1951) 998. (161 A. Diaz-Ortiz, J.M. Sanchez, J.L. MoranLopez, to be published. [17] A. Diaz-Ortiz, J.M. Sanchez, F. Aguilera-Granja, J.L. Moran-Lopez, unpublished. [I81 J.M. Sanchez, F. Ducastelle, D. Gratias, Physica 128A (I 984) 334. [I91 J.M. Sanchez, Phys. Rev. B 48 (1993) 14013.