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Bulk and surface ordering in cubic crystals
Surface Science 110 (1981) 523-532 North-Holland Publishing Company
523
BULK AND SURFACE ORDERING IN CUBIC CRYSTALS
3. VAN KRIEKEN and J.P. VAN DER EERDEN RIM Laboratory Netherlands
of Solid State Physics, Catholic University, Toernooiveld,
Received 2 March 1981; accepted for publi~tion
Nijmegen,
The
9 June 1981
The order-disorder transition of face and body centred cubic crystals is studied by a Monte Carlo simulation of a two-component lattice gas with nearest neighbour interactions, and of different compositions. It is shown that the degree of surface order may both be higher and lower than that of the bulk. General guide lines are drawn from the results to predict such behaviour in experimental situations.
1. Introduction
The order-disorder behaviour of the surface of multi component crystals is the object of both experimental and theoretical studies. It has been realized for a long time that in this respect the surfaces will behave differently from the bulk in general, which is of primary importance for, e.g., alloys which are to be used as catalysts. Until now most studies of surface effects have been devoted to surface enrichment, which may be caused either by surface energy [l-6], by relaxation of surface strain [7] or by combined effects [8-l 11. In this paper, however, we intend to present some results on what we call surface ordering. It will be shown that surfaces may be either more or less ordered than the bulk of the crystals [12] depending on the crystal structure and the particular crystal face. The related subject of surface miscibility is discussed in refs. {6,f 31. Monte Carlo calculations of face centered (fee) and body centered cubic (bee) crystals and their (001) and (011) faces have been carried out. Somewhat larger systems were chosen than in the earlier simulations of ordering fee [14] and bee [ 151 bulk crystals. The equivalent bee antiferromagnet in a field has been simulated in detail by Landau [ 161. These simulations show that the type of order-disorder phase transition is different for fee and bee crystals and depends for the latter moreover on the composition. When the fee (001) and the bee (01 I) surfaces are approximated by one-layer models they both are equivalent to square Ising systems. Their degree of ordering on top of a crystal, however, is largely determined by that crystal, except close to the bulk transition temperature where the surface order (in the fee case) or disorder (for bee) may penetrate into the crystal. 0039-6028/8I/OOOO-0000~$02.50
0 1981 North-Ho~and
524
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
In order to study surface effects we chose the thin film representation, given by Sundaram and Wynblatt [ 171 who investigated surface segregation. The alternative description with a “semi-infinite” crystal is not suitable when phase transitions are of interest. A Monte Carlo study of the surface of an ordering alloy has been carried out by Donelly and King [18] with the object to obtain quantitative agreement with Auger data on Cu-Ni alloys. it is the aim of this paper to study the relation between bulk order-disorder phase transitions and the degree of order of the surface. This will lead to general qualitative guidelines to estimate these effects in a real physical system.
2. Monte Carlo procedure The Monte Carlo computations are carried out on 4000 atomic positions. Each of the positions is either fluid, (F), or of solid type A, or of solid type B and interacts with its 12 (fee) or 8 (bee) nearest neighbours. In each Monte Carlo step the exchange of an A and B atom is attempted. They are exchanged when the system decreases its total energy, and moreover with a probability exp(-m/H) when a total energy increase of Af? would be the result. Thus the initially smooth interfaces remain smooth during the whole simulations, and the average composition of the crystal is constant. Data points are obtained from at least 6000 exchanges per site. In principle 6 different interaction energies Gii can be defined [19] between the three types of particles (i, j is A, B or F). For the present investigation, however, the solid-fluid interactions can be neglected (4 AF = @nF = 0). Since for the present purpose, surface enrichment is not of interest we take moreover $JAA= @un. Chosing, finally, GFF = 0 only one energy parameter, @= ~AB--(~AA+~~B)P
y
(1)
is sufficient to determine the transition probabilities [ 191. Periodic boundary conditions are chosen in all directions. An interface has been generated by adding a few completely fluid atomic layers, in such a way that a crystal slab of 18 atomic layers was represented. We concentrate on four different bulk systems which we think are representative for different classes of more realistic models. Indeed we investigated fee and bee lattices with an equal number of A and B atoms (fee l/l and bee l/l), and with three times as much A atoms as B atoms (fee 3/l and bee 3/l). The (001) surface on fee and the (011) surface on bee are simulated in detail in order to understand surface ordering. 3. Definition
of order parameters
Since we mean with ordered crystal in the present context a crystal in which each atom is surrounded by a large number of different atoms it is natural to use
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
Table 1 Short range (T-
order
parameters
for different
bulk
crystals
in the completely
disordered
52.5
phase
-)
PAB PC;‘)
bee l/l
bee 3/l
fee l/l
fee 3/l
0.5 0.5
0.375 0.375
0.5 0.5
0.375 0.375
the probability PAB that a certain nearest neighbour pair of atoms consists of two different ones as a “short range order parameter”, although it is not an order parameter in the sense of the Landau theory of phase transitions [20]. In the disordered state the value ofPAn is 0.5 for l/l and 0.375 for 3/l systems as shown in table 1. Perfect order by definition corresponds to PAB = 1.Ofor l/l and PAB = 0.5for 3/l systems. This perfect order can, however, not always be realized due to the lattice geometry. Indeed fee l/l is an example of such a “frustrated” model [21] where the maximal value of PAn is only 8/12, i.e. only eight out of twelve nearest neighbours of an A atom is a B atom. Analogous to this bulk order parameter one may define a “layer short range order parameter” @i’)(n) which is the probability that a pair of atoms in the nth layer from a surface with orientation (h/cl) consists of two different atoms. In table 2 the values of some of these order parameters are given for the low temperature phase, i.e. the phase with maximal, but not always perfect, order. The low temperature phase of the bee systems is most easily understood if one describes the bee lattice as two interwoven simple cubic sublattices. In bee l/l one of these is
Table 2 Bulk (PAB) and layer (P@)(m)) short range order parameters for different bulk crystals in a phase of maximal order (T-+ 0). For the fee systems the phase with ordered (001) layers is chosen. When not all layers have the same structure, we arbitrarily chose the most ordered ones to be at an odd (m = 2n + 1) level bee l/l
bee 3/l
fee l/l
fee 3/l
1.0
0.5
0.667
0.5
+ 1)
0.0
O.O/l.O
0.5
0.5
O) (2n + 1)
0.5
0.5
1.0
O.O/l.O
PAB P$$0)(2n)/P~~0)(2n PfA”)(2n)/P$$
0.0
O.O/l.O
Pa0,0’)(2n)/Pa0,0’)(2n
+ 1)
0.0
O.O/l.O
PFi
+ 1)
1.0
0.5
0.5
0.5
P$$1)(2n)/Pgi1)(2n
+ 1)
1.0
0.5
0.5
0.5
P$$‘)(2n)/P!$‘)(2n
+ 1)
1.0
0.5
1.0
O.O/l.O
‘)(2n)/PFi
‘)(2n
.
526
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
filled with A atoms, the other with B atoms. In bee 3/l one is filled with A atoms, the other randomly with 50% A atoms and 50% B atoms. The fact that the second sublattice is filled at random implies a finite entropy S = ik In 2 per atom in the ground state, which is important to understand the phase transition behaviour. Of course this ground state degeneracy will be removed in a physical situation by next nearest neighbour interactions. The fee lattice can be described as a pile of square lattices perpendicular to the [OOl] direction in such a way that one can be shifted to the next by a transition over [S, S, j/2]. In one of the ordered phases of fee l/l all of these (001) layers are perfectly ordered and the ground state entropy is k In 2 per layer, the (010) and (100) layers, however, are disordered. In the two other ordered phases the (010) or (100) layers are ordered. In one of the ordered phases of fee 3/l half of the (001) layers is filled with A atoms and the others are ordered, leading to an entropy of k In 2 per two layers. These brief descriptions should suffice to understand table 2.
4. Phase transitions In this section we briefly go through the phase transition behaviour and show that the Monte Carlo results are in line with results from analytical and renormalization theories. The bee l/l system has zero entropy in its ground state, as mentioned in section 3, and its ordering behaviour can be described in a satisfactory way with cluster variation methods [22] which predict a second-order phase transition [23]. The value of the critical temperature T, is in reasonable agreement with the values @/kT, = 0.3 14 from high temperature series expansions [24], @/kT, = 0.305 from renormalization [24] and 0.33 from Monte Carlo studies [ 15,161. In accordance with these results we find in fig. 1 that the slope of the P&T) curve, which is proportional to the specific heat, has its maximum at ck/kT, = 0.33. The situation with the bee 3/l system is less clear. Cluster variation methods with next nearest neighbour interactions [23] predict a second-order transition at a lower temperature than the bee l/l system. Guttman [ 151 interpreted his Monte Carlo data as exhibiting such a transition at Q/kT, = 0.57. However, as shown in fig. 1 it is questionable whether a maximum slope in the PAN versus T curve is indicating an actual transition. Hence we suggest that at this deviation from stoichiometry no transition exists any more, or, alternatively that T, = 0 for CA/C, = 3/l, due to the large degree of disorder already at T = 0. This interpretation follows the results of renormalization [25], high temperature series expansion [26], and of Monte Carlo calculations on the antiferromagnetic bee Ising model [16]. Indeed there would be a maximal value for the CdCu ratio for which T, > 0. From ref. [24] we estimate that this value should be somewhat larger than 2.8, but ref. [16] gives a critical value of 4.4 for this ratio, and a critical temperature @/kT, = 0.47 for the present 3/l ratio.
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
521
Fig. 1. Bulk short range order parameters PAB for bee and fee systems with compositions CA/ CB = 3/l and l/l as a function of inverse temperature.
There is some controversy in the literature about the nature of the transition of the fee systems. Cluster variation techniques [27,28] predict a first-order transition at @/kT, = 1.05 and 1.03 for the fee l/l and fee 3/l system. The system size of Fosdicks fee 3/l Monte Carlo system was probably too small to find out which type of transition takes place at @/kT, = 1.02. And renormalization results, finally, do not reveal a transition in the fee l/l system and a second-order transition at @/kT, = 1.23 for fee 3/l. Our results, displayed in fig. 1 clearly support the existence of a first-order transition for both fee systems, taking place at @/kT, = 1.12 and 1.09 for the l/l and the 3/l system.
5. Surface ordering
In order to understand surface ordering effects it is useful to consider the onelayer model as a first, crude, approximation. This oneqayer model will have its own critical behaviour and an order-disorder phase transition fichkl). The first question which arises is whether this critical behavour would be visible still for a layer on top of a bulk crystal in terms of, e.g., specific heat anomalies. In accordance with expectations, our Monte Carlo calculations never indicated such anomalies and
528
J. van Krieken, J.F. van der Eerden /Bulk and surface ordering irz cubic crystals
hence we conclude that the bulk is dominant over the surface as far as transitions are concerned. The second question is whether the suppressed phase transition in the monolayer would still announce itself by less pronounced anomalies. Here the answer is affirmative. Indeed we find that if l$ hkz) < T, (7$(chk') > jr,.), then the surface is less (more) ordered than the bulk. At first sight one might expect that 7(,hk’) is always less then the bulk T,,since the bulk coordination number is higher, and generally the critical temperature is an increasing function of the number of interacting neighbours. However, this is only true in case of the perfectly ordering bee l/l and bee 3/l systems. In the maximal ordered state of the fee systems there are 12 interacting neighbours in the crystal and less on all surfaces, but in the bulk at least four are repulsive. in conclusion we expect that surfaces on a perfectly ordering crystal will be less ordered than the bulk, and that some “non-frustrated” surfaces on “frustrated” crystals may be more ordered. In order to clarify this we investigated the (110) surface of bee l/l and (001) of fee l/l as characteristic representatives of both classes. We used the following method. First we determined with Monte Carlo simulations the layer order parameter pahI3k’)(n)for each layer with (h/cl) orientation in our thin crystal. Next we fit these numbers numeric~y with a function which describes a symmetric ordering prophile
(2) We give no theoretical justi~cation of this functional dependence, but it was found that it fitted the measured P$$“(n) values within their accuracy. The two terms in square brackets arise from the fact that a symmetric profile between the lowest layer at Y2= PZminand the highest layer at n = IZ,, is caused by the thin crystal slab geometry of the simulated system. From the numbers A, B and d we derive physical quantities. The surface order parameter @f, = pahBk”(nmirJ and the bulk layer order parameter pk”B = ~~‘(~(~~i~ f n,,)) are obtained from the fitted parameters by P(s\)n =A+B,
l’$iA =A
+ 2B
(3) exp
, -nmin2inmilx
(4)
The number d measures how many layers deviate significantly from bulk layers, and we hence refer to d as the penetration depth of the surface disorder (bee) or surface order (fee). Clearly simulation results should be rejected if the thickness nmax nmin of the slab is larger than about 2d since then both surfaces interact and the middle layers can not be considered as bulk layers any more. In fig. 2 we compare bulk and surface order. It is seen that for both systems the
J. van Krieken, J.P. van der Eerden / Bulk and surface ordering in cubic crystals
529
p(hkl) AB
/o
p’ I
/
P b
P I
‘bee
,
!,
, (011), I0 bulk/
p ,‘surface
Q/k1
d
Oi2
Oi4
0,6
Oi8
li0
1,2
Fig. 2. Layer short range order parameters for a bee l/l crystal slab with two (011) surfaces and an fee l/l slab with two (001) surfaces. The value for the surface layers deviates from that for bulk layers in the direction of the value for the exactly solved square Ising model.
surface order deviates from the bulk order in the direction of the order in the twodimensional square Ising model, as expected. The maximal deviation is a factor two for the bee and a factor three for the fee system, and is found close to the bulk transition temperature. The position of this maximal deviation is caused by the fact that any system is most sensitive to perturbations close to its transition temperature. In the present case the surface acts as an external perturbation. The range over which a perturbation is felt (the correlation length) is here the penetration depth d. Following the definitions of Pfeuty and Toulouse [29] d diverges at T, if the transition is second order and increases but remains finite if the transition is first order. This behaviour has been found indeed and is represented in fig. 3. For the bee systems the peak is
530
I.0
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals I
2.0
I
I
015
0.7
0.4
(PlkT
d
C
I
b
I
I
; \o
/ bcclll (OllJ,'
] o-f#cc l/l(OOl) \o \
.l.O I
I 20
I I
/
\
I \
I
0
P
TL
,O/
.d
/O’
’
\
T: O,
0’
&-‘o’ 0
1.0
_-o2.0
Fig. 3. Dependence of the penetration depth of surface ture. First-order behaviour for fee and second-order data.
3.0
kT/Q
1
order (fee) or disorder (bee) on temperabehaviour for bee is suggested by these
rather broad, and appreciable penetration of the surface disorder (d > 1) is found over a temperature interval AT which is about 20% of the absolute temperature T,. For the fee system the peak is much narrower, being less than 3% of T, at the high temperature side and less than 1% at the low temperature side. Within this temperature interval large penetration depths are found, probably due to finite size effects.
6. Conclusion We have done Monte Carlo simulations on various three-dimensional cubic lattice gas models, with the object to understand the relations between bulk and surface
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
531
order. Although the systems under consideration were too much simplified to enable direct comparison with experiments we have drawn some general conclusions. We have shown that for lattices which allow perfect ordering a second-order phase transition from the ordered to the disordered state takes place. Surfaces of such crystals will have a lower degree of order since their atoms have a smaller coordination number. This surface disorder penetrates into the crystal over a fairly wide range of temperatures around the transition temperature. For the frustrated systems, on the other hand, where the lattice structure is incompatible with perfect ordering, the order-disorder transition is found to be a first-order one. Now there may exist surfaces which, in a one layer approximation, are not frustrated and have a larger transition temperature than the bulk. Such surfaces will have a higher degree of order than the bulk, but, in contrast to the nonfrustrated bulk crystals, this order will, even close to T,, not penetrate appreciably in the bulk. In experimental situations many of the presuppositions of this paper are not precisely satisfied. The effect of next and further neighbour attractive interactions will be to shift and rescale the effective value of @/kT and to reduce the ground state degeneracy of bee 3/l. Different surface and bulk pair interactions [18] will similarly shift the effective @/kT value and also change the difference between bulk and surface order. Surface segregation, finally, changes the surface layer composition. It will, therefore, tend to relax the frustrated surfaces, and make the penetration depth curve (fig. 3) less different from the curve for non-frustrated surfaces. In conclusion, if, as will be often the case, these effects can be considered as second order effects, the surface ordering in experimental situations can probably be understood along the lines developed in this paper.
Acknowledgement We express our gratitude to the department of Computer Graphics whose computer facilities we could freely use. We thank Professor Bennema for stimulating the investigations, Dr. H. Knops for clarifying discussion about the phase transitions in ths model and Dr. R. van Santen for providing the initial ideas of this paper.
Reierences [l] [2] [3] [4]
R.A. van Santen and W.M.H. Sachtler, J. Catalysis 33 (1974) 202. R.A. van Santen and M.A.M. Boersma, J. Catalysis 34 (1974) 13. J.J. Burton, E. Hyman and D.G. Fedak, J. Catalysis 37 (1975) 106. R. Bouwman, L.H. Toneman, M.A.M. Boersma and R.A. van Santen, (1976) 72.
Surface
Sci. 59
532
[S] [6] [7] [ 81 [9] [lo] [11] [ 121 [13] [14] [lS] [16] [17] [ 181 [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
F.L. Williams and D. Nason, Surface Sci. 45 (1974) 377. J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) 2549. D. McLean, Grain Boundaries in Metals (Oxford Univ. Press, London, 1957). P. Wynblatt and R.C. Ku, Surface Sci. 65 (1977) 5 11. V. Kumar, Surface Sci. 84 (1979) L231. C. Molinari, J.C. Joud and P. Desre, Surface Sci. 84 (1979) 141. Y.W. Lee and H.I. Aaronson, Surface Sci. 95 (1980) 227. J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) 2542. C.R. Helms, Surface Sci. 69 (1977) 689. L.D. Fosdick, Phys. Rev. 116 (1959) 565. L. Guttman, J. Chem. Phys. 34 (1961) 1024. D.P. Landau, Phys. Rev. B16 (1977) 4164. V.S. Sundaram and P. Wynblatt, Surface Sci. 52 (1975) 569. R.G. Donelly and T.S. King, Surface Sci. 74 (1978) 89. J.P. van der Eerden, P. Bennema and T.A. Cherepanova, Progr. Crystal Growth Characterization l(l978) 219. H.E. Stanley, Introduction to Phase Transitions and to Critical Phenomena (Clarendon, Oxford, 1978). See, e.g., Proc. 14th Intern. Conf. on Thermodynamics and Statistical Mechanics, Physica 106A (1981). R. Kikuchi, J. Chem. Phys. 60 (1974) 1071. R. Kikuchi and C.M. van Baal, Scripta Met. 8 (1974) 425. M.F. Sykes, D.S. Gaunt, P.D. Robertsand J.A. Wyles, J. Phys. A5 (1972) 640. G.D. Mahan and F.H. Claro, J. Phys. Cl0 (1977) L73; Phys. Rev. B16 (1977) 1168. A. Bienenstock and J. Lewis, Phys. Rev. 160 (1967) 393. N.S. Golosow, L.E. Popov and L.Ya Pudan, J. Phys. Chem. Solids 34 (1973) 1149; 1157. CM. van Baal, Physica 64 (1973) 571. P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena (Wiley, London, 1977) pp. 4-6.
523
BULK AND SURFACE ORDERING IN CUBIC CRYSTALS
3. VAN KRIEKEN and J.P. VAN DER EERDEN RIM Laboratory Netherlands
of Solid State Physics, Catholic University, Toernooiveld,
Received 2 March 1981; accepted for publi~tion
Nijmegen,
The
9 June 1981
The order-disorder transition of face and body centred cubic crystals is studied by a Monte Carlo simulation of a two-component lattice gas with nearest neighbour interactions, and of different compositions. It is shown that the degree of surface order may both be higher and lower than that of the bulk. General guide lines are drawn from the results to predict such behaviour in experimental situations.
1. Introduction
The order-disorder behaviour of the surface of multi component crystals is the object of both experimental and theoretical studies. It has been realized for a long time that in this respect the surfaces will behave differently from the bulk in general, which is of primary importance for, e.g., alloys which are to be used as catalysts. Until now most studies of surface effects have been devoted to surface enrichment, which may be caused either by surface energy [l-6], by relaxation of surface strain [7] or by combined effects [8-l 11. In this paper, however, we intend to present some results on what we call surface ordering. It will be shown that surfaces may be either more or less ordered than the bulk of the crystals [12] depending on the crystal structure and the particular crystal face. The related subject of surface miscibility is discussed in refs. {6,f 31. Monte Carlo calculations of face centered (fee) and body centered cubic (bee) crystals and their (001) and (011) faces have been carried out. Somewhat larger systems were chosen than in the earlier simulations of ordering fee [14] and bee [ 151 bulk crystals. The equivalent bee antiferromagnet in a field has been simulated in detail by Landau [ 161. These simulations show that the type of order-disorder phase transition is different for fee and bee crystals and depends for the latter moreover on the composition. When the fee (001) and the bee (01 I) surfaces are approximated by one-layer models they both are equivalent to square Ising systems. Their degree of ordering on top of a crystal, however, is largely determined by that crystal, except close to the bulk transition temperature where the surface order (in the fee case) or disorder (for bee) may penetrate into the crystal. 0039-6028/8I/OOOO-0000~$02.50
0 1981 North-Ho~and
524
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
In order to study surface effects we chose the thin film representation, given by Sundaram and Wynblatt [ 171 who investigated surface segregation. The alternative description with a “semi-infinite” crystal is not suitable when phase transitions are of interest. A Monte Carlo study of the surface of an ordering alloy has been carried out by Donelly and King [18] with the object to obtain quantitative agreement with Auger data on Cu-Ni alloys. it is the aim of this paper to study the relation between bulk order-disorder phase transitions and the degree of order of the surface. This will lead to general qualitative guidelines to estimate these effects in a real physical system.
2. Monte Carlo procedure The Monte Carlo computations are carried out on 4000 atomic positions. Each of the positions is either fluid, (F), or of solid type A, or of solid type B and interacts with its 12 (fee) or 8 (bee) nearest neighbours. In each Monte Carlo step the exchange of an A and B atom is attempted. They are exchanged when the system decreases its total energy, and moreover with a probability exp(-m/H) when a total energy increase of Af? would be the result. Thus the initially smooth interfaces remain smooth during the whole simulations, and the average composition of the crystal is constant. Data points are obtained from at least 6000 exchanges per site. In principle 6 different interaction energies Gii can be defined [19] between the three types of particles (i, j is A, B or F). For the present investigation, however, the solid-fluid interactions can be neglected (4 AF = @nF = 0). Since for the present purpose, surface enrichment is not of interest we take moreover $JAA= @un. Chosing, finally, GFF = 0 only one energy parameter, @= ~AB--(~AA+~~B)P
y
(1)
is sufficient to determine the transition probabilities [ 191. Periodic boundary conditions are chosen in all directions. An interface has been generated by adding a few completely fluid atomic layers, in such a way that a crystal slab of 18 atomic layers was represented. We concentrate on four different bulk systems which we think are representative for different classes of more realistic models. Indeed we investigated fee and bee lattices with an equal number of A and B atoms (fee l/l and bee l/l), and with three times as much A atoms as B atoms (fee 3/l and bee 3/l). The (001) surface on fee and the (011) surface on bee are simulated in detail in order to understand surface ordering. 3. Definition
of order parameters
Since we mean with ordered crystal in the present context a crystal in which each atom is surrounded by a large number of different atoms it is natural to use
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
Table 1 Short range (T-
order
parameters
for different
bulk
crystals
in the completely
disordered
52.5
phase
-)
PAB PC;‘)
bee l/l
bee 3/l
fee l/l
fee 3/l
0.5 0.5
0.375 0.375
0.5 0.5
0.375 0.375
the probability PAB that a certain nearest neighbour pair of atoms consists of two different ones as a “short range order parameter”, although it is not an order parameter in the sense of the Landau theory of phase transitions [20]. In the disordered state the value ofPAn is 0.5 for l/l and 0.375 for 3/l systems as shown in table 1. Perfect order by definition corresponds to PAB = 1.Ofor l/l and PAB = 0.5for 3/l systems. This perfect order can, however, not always be realized due to the lattice geometry. Indeed fee l/l is an example of such a “frustrated” model [21] where the maximal value of PAn is only 8/12, i.e. only eight out of twelve nearest neighbours of an A atom is a B atom. Analogous to this bulk order parameter one may define a “layer short range order parameter” @i’)(n) which is the probability that a pair of atoms in the nth layer from a surface with orientation (h/cl) consists of two different atoms. In table 2 the values of some of these order parameters are given for the low temperature phase, i.e. the phase with maximal, but not always perfect, order. The low temperature phase of the bee systems is most easily understood if one describes the bee lattice as two interwoven simple cubic sublattices. In bee l/l one of these is
Table 2 Bulk (PAB) and layer (P@)(m)) short range order parameters for different bulk crystals in a phase of maximal order (T-+ 0). For the fee systems the phase with ordered (001) layers is chosen. When not all layers have the same structure, we arbitrarily chose the most ordered ones to be at an odd (m = 2n + 1) level bee l/l
bee 3/l
fee l/l
fee 3/l
1.0
0.5
0.667
0.5
+ 1)
0.0
O.O/l.O
0.5
0.5
O) (2n + 1)
0.5
0.5
1.0
O.O/l.O
PAB P$$0)(2n)/P~~0)(2n PfA”)(2n)/P$$
0.0
O.O/l.O
Pa0,0’)(2n)/Pa0,0’)(2n
+ 1)
0.0
O.O/l.O
PFi
+ 1)
1.0
0.5
0.5
0.5
P$$1)(2n)/Pgi1)(2n
+ 1)
1.0
0.5
0.5
0.5
P$$‘)(2n)/P!$‘)(2n
+ 1)
1.0
0.5
1.0
O.O/l.O
‘)(2n)/PFi
‘)(2n
.
526
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
filled with A atoms, the other with B atoms. In bee 3/l one is filled with A atoms, the other randomly with 50% A atoms and 50% B atoms. The fact that the second sublattice is filled at random implies a finite entropy S = ik In 2 per atom in the ground state, which is important to understand the phase transition behaviour. Of course this ground state degeneracy will be removed in a physical situation by next nearest neighbour interactions. The fee lattice can be described as a pile of square lattices perpendicular to the [OOl] direction in such a way that one can be shifted to the next by a transition over [S, S, j/2]. In one of the ordered phases of fee l/l all of these (001) layers are perfectly ordered and the ground state entropy is k In 2 per layer, the (010) and (100) layers, however, are disordered. In the two other ordered phases the (010) or (100) layers are ordered. In one of the ordered phases of fee 3/l half of the (001) layers is filled with A atoms and the others are ordered, leading to an entropy of k In 2 per two layers. These brief descriptions should suffice to understand table 2.
4. Phase transitions In this section we briefly go through the phase transition behaviour and show that the Monte Carlo results are in line with results from analytical and renormalization theories. The bee l/l system has zero entropy in its ground state, as mentioned in section 3, and its ordering behaviour can be described in a satisfactory way with cluster variation methods [22] which predict a second-order phase transition [23]. The value of the critical temperature T, is in reasonable agreement with the values @/kT, = 0.3 14 from high temperature series expansions [24], @/kT, = 0.305 from renormalization [24] and 0.33 from Monte Carlo studies [ 15,161. In accordance with these results we find in fig. 1 that the slope of the P&T) curve, which is proportional to the specific heat, has its maximum at ck/kT, = 0.33. The situation with the bee 3/l system is less clear. Cluster variation methods with next nearest neighbour interactions [23] predict a second-order transition at a lower temperature than the bee l/l system. Guttman [ 151 interpreted his Monte Carlo data as exhibiting such a transition at Q/kT, = 0.57. However, as shown in fig. 1 it is questionable whether a maximum slope in the PAN versus T curve is indicating an actual transition. Hence we suggest that at this deviation from stoichiometry no transition exists any more, or, alternatively that T, = 0 for CA/C, = 3/l, due to the large degree of disorder already at T = 0. This interpretation follows the results of renormalization [25], high temperature series expansion [26], and of Monte Carlo calculations on the antiferromagnetic bee Ising model [16]. Indeed there would be a maximal value for the CdCu ratio for which T, > 0. From ref. [24] we estimate that this value should be somewhat larger than 2.8, but ref. [16] gives a critical value of 4.4 for this ratio, and a critical temperature @/kT, = 0.47 for the present 3/l ratio.
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
521
Fig. 1. Bulk short range order parameters PAB for bee and fee systems with compositions CA/ CB = 3/l and l/l as a function of inverse temperature.
There is some controversy in the literature about the nature of the transition of the fee systems. Cluster variation techniques [27,28] predict a first-order transition at @/kT, = 1.05 and 1.03 for the fee l/l and fee 3/l system. The system size of Fosdicks fee 3/l Monte Carlo system was probably too small to find out which type of transition takes place at @/kT, = 1.02. And renormalization results, finally, do not reveal a transition in the fee l/l system and a second-order transition at @/kT, = 1.23 for fee 3/l. Our results, displayed in fig. 1 clearly support the existence of a first-order transition for both fee systems, taking place at @/kT, = 1.12 and 1.09 for the l/l and the 3/l system.
5. Surface ordering
In order to understand surface ordering effects it is useful to consider the onelayer model as a first, crude, approximation. This oneqayer model will have its own critical behaviour and an order-disorder phase transition fichkl). The first question which arises is whether this critical behavour would be visible still for a layer on top of a bulk crystal in terms of, e.g., specific heat anomalies. In accordance with expectations, our Monte Carlo calculations never indicated such anomalies and
528
J. van Krieken, J.F. van der Eerden /Bulk and surface ordering irz cubic crystals
hence we conclude that the bulk is dominant over the surface as far as transitions are concerned. The second question is whether the suppressed phase transition in the monolayer would still announce itself by less pronounced anomalies. Here the answer is affirmative. Indeed we find that if l$ hkz) < T, (7$(chk') > jr,.), then the surface is less (more) ordered than the bulk. At first sight one might expect that 7(,hk’) is always less then the bulk T,,since the bulk coordination number is higher, and generally the critical temperature is an increasing function of the number of interacting neighbours. However, this is only true in case of the perfectly ordering bee l/l and bee 3/l systems. In the maximal ordered state of the fee systems there are 12 interacting neighbours in the crystal and less on all surfaces, but in the bulk at least four are repulsive. in conclusion we expect that surfaces on a perfectly ordering crystal will be less ordered than the bulk, and that some “non-frustrated” surfaces on “frustrated” crystals may be more ordered. In order to clarify this we investigated the (110) surface of bee l/l and (001) of fee l/l as characteristic representatives of both classes. We used the following method. First we determined with Monte Carlo simulations the layer order parameter pahI3k’)(n)for each layer with (h/cl) orientation in our thin crystal. Next we fit these numbers numeric~y with a function which describes a symmetric ordering prophile
(2) We give no theoretical justi~cation of this functional dependence, but it was found that it fitted the measured P$$“(n) values within their accuracy. The two terms in square brackets arise from the fact that a symmetric profile between the lowest layer at Y2= PZminand the highest layer at n = IZ,, is caused by the thin crystal slab geometry of the simulated system. From the numbers A, B and d we derive physical quantities. The surface order parameter @f, = pahBk”(nmirJ and the bulk layer order parameter pk”B = ~~‘(~(~~i~ f n,,)) are obtained from the fitted parameters by P(s\)n =A+B,
l’$iA =A
+ 2B
(3) exp
, -nmin2inmilx
(4)
The number d measures how many layers deviate significantly from bulk layers, and we hence refer to d as the penetration depth of the surface disorder (bee) or surface order (fee). Clearly simulation results should be rejected if the thickness nmax nmin of the slab is larger than about 2d since then both surfaces interact and the middle layers can not be considered as bulk layers any more. In fig. 2 we compare bulk and surface order. It is seen that for both systems the
J. van Krieken, J.P. van der Eerden / Bulk and surface ordering in cubic crystals
529
p(hkl) AB
/o
p’ I
/
P b
P I
‘bee
,
!,
, (011), I0 bulk/
p ,‘surface
Q/k1
d
Oi2
Oi4
0,6
Oi8
li0
1,2
Fig. 2. Layer short range order parameters for a bee l/l crystal slab with two (011) surfaces and an fee l/l slab with two (001) surfaces. The value for the surface layers deviates from that for bulk layers in the direction of the value for the exactly solved square Ising model.
surface order deviates from the bulk order in the direction of the order in the twodimensional square Ising model, as expected. The maximal deviation is a factor two for the bee and a factor three for the fee system, and is found close to the bulk transition temperature. The position of this maximal deviation is caused by the fact that any system is most sensitive to perturbations close to its transition temperature. In the present case the surface acts as an external perturbation. The range over which a perturbation is felt (the correlation length) is here the penetration depth d. Following the definitions of Pfeuty and Toulouse [29] d diverges at T, if the transition is second order and increases but remains finite if the transition is first order. This behaviour has been found indeed and is represented in fig. 3. For the bee systems the peak is
530
I.0
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals I
2.0
I
I
015
0.7
0.4
(PlkT
d
C
I
b
I
I
; \o
/ bcclll (OllJ,'
] o-f#cc l/l(OOl) \o \
.l.O I
I 20
I I
/
\
I \
I
0
P
TL
,O/
.d
/O’
’
\
T: O,
0’
&-‘o’ 0
1.0
_-o2.0
Fig. 3. Dependence of the penetration depth of surface ture. First-order behaviour for fee and second-order data.
3.0
kT/Q
1
order (fee) or disorder (bee) on temperabehaviour for bee is suggested by these
rather broad, and appreciable penetration of the surface disorder (d > 1) is found over a temperature interval AT which is about 20% of the absolute temperature T,. For the fee system the peak is much narrower, being less than 3% of T, at the high temperature side and less than 1% at the low temperature side. Within this temperature interval large penetration depths are found, probably due to finite size effects.
6. Conclusion We have done Monte Carlo simulations on various three-dimensional cubic lattice gas models, with the object to understand the relations between bulk and surface
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
531
order. Although the systems under consideration were too much simplified to enable direct comparison with experiments we have drawn some general conclusions. We have shown that for lattices which allow perfect ordering a second-order phase transition from the ordered to the disordered state takes place. Surfaces of such crystals will have a lower degree of order since their atoms have a smaller coordination number. This surface disorder penetrates into the crystal over a fairly wide range of temperatures around the transition temperature. For the frustrated systems, on the other hand, where the lattice structure is incompatible with perfect ordering, the order-disorder transition is found to be a first-order one. Now there may exist surfaces which, in a one layer approximation, are not frustrated and have a larger transition temperature than the bulk. Such surfaces will have a higher degree of order than the bulk, but, in contrast to the nonfrustrated bulk crystals, this order will, even close to T,, not penetrate appreciably in the bulk. In experimental situations many of the presuppositions of this paper are not precisely satisfied. The effect of next and further neighbour attractive interactions will be to shift and rescale the effective value of @/kT and to reduce the ground state degeneracy of bee 3/l. Different surface and bulk pair interactions [18] will similarly shift the effective @/kT value and also change the difference between bulk and surface order. Surface segregation, finally, changes the surface layer composition. It will, therefore, tend to relax the frustrated surfaces, and make the penetration depth curve (fig. 3) less different from the curve for non-frustrated surfaces. In conclusion, if, as will be often the case, these effects can be considered as second order effects, the surface ordering in experimental situations can probably be understood along the lines developed in this paper.
Acknowledgement We express our gratitude to the department of Computer Graphics whose computer facilities we could freely use. We thank Professor Bennema for stimulating the investigations, Dr. H. Knops for clarifying discussion about the phase transitions in ths model and Dr. R. van Santen for providing the initial ideas of this paper.
Reierences [l] [2] [3] [4]
R.A. van Santen and W.M.H. Sachtler, J. Catalysis 33 (1974) 202. R.A. van Santen and M.A.M. Boersma, J. Catalysis 34 (1974) 13. J.J. Burton, E. Hyman and D.G. Fedak, J. Catalysis 37 (1975) 106. R. Bouwman, L.H. Toneman, M.A.M. Boersma and R.A. van Santen, (1976) 72.
Surface
Sci. 59
532
[S] [6] [7] [ 81 [9] [lo] [11] [ 121 [13] [14] [lS] [16] [17] [ 181 [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
J. van Krieken, J.P. van der Eerden /Bulk and surface ordering in cubic crystals
F.L. Williams and D. Nason, Surface Sci. 45 (1974) 377. J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) 2549. D. McLean, Grain Boundaries in Metals (Oxford Univ. Press, London, 1957). P. Wynblatt and R.C. Ku, Surface Sci. 65 (1977) 5 11. V. Kumar, Surface Sci. 84 (1979) L231. C. Molinari, J.C. Joud and P. Desre, Surface Sci. 84 (1979) 141. Y.W. Lee and H.I. Aaronson, Surface Sci. 95 (1980) 227. J.L. Moran-Lopez and L.M. Falicov, Phys. Rev. B18 (1978) 2542. C.R. Helms, Surface Sci. 69 (1977) 689. L.D. Fosdick, Phys. Rev. 116 (1959) 565. L. Guttman, J. Chem. Phys. 34 (1961) 1024. D.P. Landau, Phys. Rev. B16 (1977) 4164. V.S. Sundaram and P. Wynblatt, Surface Sci. 52 (1975) 569. R.G. Donelly and T.S. King, Surface Sci. 74 (1978) 89. J.P. van der Eerden, P. Bennema and T.A. Cherepanova, Progr. Crystal Growth Characterization l(l978) 219. H.E. Stanley, Introduction to Phase Transitions and to Critical Phenomena (Clarendon, Oxford, 1978). See, e.g., Proc. 14th Intern. Conf. on Thermodynamics and Statistical Mechanics, Physica 106A (1981). R. Kikuchi, J. Chem. Phys. 60 (1974) 1071. R. Kikuchi and C.M. van Baal, Scripta Met. 8 (1974) 425. M.F. Sykes, D.S. Gaunt, P.D. Robertsand J.A. Wyles, J. Phys. A5 (1972) 640. G.D. Mahan and F.H. Claro, J. Phys. Cl0 (1977) L73; Phys. Rev. B16 (1977) 1168. A. Bienenstock and J. Lewis, Phys. Rev. 160 (1967) 393. N.S. Golosow, L.E. Popov and L.Ya Pudan, J. Phys. Chem. Solids 34 (1973) 1149; 1157. CM. van Baal, Physica 64 (1973) 571. P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena (Wiley, London, 1977) pp. 4-6.