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Phase diagrams of FCC alloy films
ARTICLE IN PRESS
Physica B 373 (2006) 198–205 www.elsevier.com/locate/physb
Phase diagrams of FCC alloy films Jiangling Pan, Yanlin Xu, Jun Ni Department of Physics and Key Laboratory of Atomic and Molecular Nanoscience, Tsinghua University, Beijing 100084, PR China Received 11 July 2005; received in revised form 18 October 2005; accepted 15 November 2005
Abstract We have studied the order–disorder phase transitions of FCC alloy thin films in the (0 0 1) direction using the mean-field method. Surface field is used to describe the surface effects such as surface segregation. All the different types of the phase diagrams of alloy thin films under the surface field are calculated. It is shown that the order–disorder phase transitions of alloy films are much more complex than those of bulk due to the combined effects of surface field and finite size confinement. The main features of phase diagrams are as follows: (1) for the thin films with even number layers, there are some middle-temperature phases due to the anti-phase boundary phenomena induced by the surface field, which leads to more complex phase behavior than those of odd number layer films. (2) For the quasi-two-dimensional thin films with odd number layers, there is an ABðABÞn A phase which does not show order–disorder phase transition with the increase of temperature due to the finite size of film. (3) For the thin films with both even and odd number layers, there are some critical points in the phase diagrams. r 2005 Elsevier B.V. All rights reserved. PACS: 64.60.Cn; 05.50.þq; 68.65.þg; 81.30.Bx Keywords: Phase diagrams; Thin films; Low-dimensional structures; Order–disorder transition
1. Introduction Experimental and theoretical studies of surface phenomena have shown that the surface behavior is significantly different from that of bulk systems [1–10]. Due to the influence of surface field, the surface shows many different phenomena from the bulk, such as correlation of ordering and surface segregation [1–4], surface-induced ordering and surface-induced disordering phenomena [5–7]. The development of the molecular-beam-epitaxy (MBE) has permitted the growth of thin film with monoatomic-layer accuracy. These thin films have two surfaces. The two surfaces, which are contacted with substrate and vacuum, respectively, can be considered as under two different surface fields. The thin films can be also considered as a quasi-two-dimensional structures. Due to the combined effects of surface field and finite size confinement, the thin films have different structures and phase transition Corresponding author. Tel.: +8610 62772781; fax: +8610 62781604.
E-mail address: [email protected] (J. Ni). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.146
behavior from the bulk. The phase transition behavior and the surface effects of binary alloy thin films have attracted much interest [11–16]. Studies on the phase transitions of BCC (1 1 0) Ax B1x alloys show that both surface field and film thickness have strong effect on the transition temperature [11]. In the thin films of BCC binary alloys with ordering interaction, the large confining field describing segregation can result in an unusual phase diagram with a disordered phase lying between the ordered phases down to zero temperature [12–14]. For the FCC thin film alloys, there have been some researches on the ground states of multilayer thin films [14,17]. Ni and Liu have studied the order–disorder phase transition of FCC (0 0 1) A3 B alloy thin films. They found that the order–disorder phase transitions between the even number layer films and the odd number layer films are significantly different. There is an oscillatory phase behavior as a function of film thickness due to the combined effects of surface field and finite size confinement [16]. There are many phase diagram calculations for the FCC bulk systems [18–22] and the surface systems [8–10], Mohri
ARTICLE IN PRESS J. Pan et al. / Physica B 373 (2006) 198–205
and Sanchez have calculated the phase diagrams of FCC lattice with nearest- and next nearest-neighbor pair interactions based on the Ising model [18]. The study on the (0 0 1) surface of binary FCC A3 B alloys shows that the order–disorder phase transitions of the surfaces coupled weakly to the bulk could be of second order even when the bulk has first order transition [8]. Although the FCC alloy film systems show a rich variety of phase phenomena, little is known on the phase diagrams of FCC thin films. In this paper, we present a study of the globe phase diagrams of FCC (0 0 1) alloy thin films under surface confinement. We have calculated the various types of phase diagrams classified according to the ground state sequences. We show that there is a rich variety of phase behavior in the alloy films due to the combined effects of surface field and finite size confinement. The outline of this paper is as follows: Section 2 describes the methods we used. Section 3 presents the results on the phase transitions and the phase diagrams of the thin films. Section 4 is the conclusion.
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alloys. We have also chosen other different values of a, such as 0.1 and 0.3 (a is also considered to be small), which gives the similar phase diagrams as a ¼ 0:2 due to the dominant effect of the nearest-neighbor interactions. For the bulk with the ordered L12 structure, four interpenetrating simple-cubic sublattices are used to describe the ordered phases. If we consider the planes in the (0 0 1) direction, each plane is composed of two sublattices. Therefore, for a film, we can divide each film layer into two equivalent sublattices a and b. In the mean-field approximation, the configurations of the system are described by the site probabilities of the A species denoted by Pn;i , where n holds for the sublattice ða; bÞ, and i ¼ 1; . . . ; N represents the plane number. Due to the probability normalization condition, the site probabilities of the B species are then 1 Pn;i . The energy of the thin film described by the first two terms in Eq. (1) can be written as a sum of intra-layer and inter-layer interaction energies X U¼ ð12 U i2;i þ 12 U i1;i þ U i;i þ 12 U iþ1;i þ 12 U iþ2;i Þ, ð2Þ i
2. Methods where We consider the FCC (0 0 1) alloy films and use an Isinglike Hamiltonian to describe the binary Ax B1x alloy on a lattice consisting of N two-dimensional parallel layers. The nearest-neighbor and next nearest-neighbor interactions are included. Generally, the nearest-neighbor pair interaction energy for intra-layer atomic pairs is different with that for inter-layer atomic pairs for films. In our calculations, we do not consider their difference and the nearestneighbor pair interaction energy is defined equally for intra-layer and inter-layer atomic pairs as an approximation. The surface effects, such as surface segregation, are described by surface fields [17,23]. The Hamiltonian is given as X X X H¼J si sj aJ si sj m si NN
h
NNN
X i2f1g
si þ
X
!
si ,
all
ð1Þ
i2fNg
where J is the nearest-neighbor interaction, a is the ratio between the nearest-neighbor and next nearest-neighbor interactions. The first sum runs over all the nearestneighbor sites and the second sum runs over all the next nearest-neighbor sites. The energy parameter m is the chemical potential. h represents the surface field, which describes the effects of confinement on the surface atoms. We consider the surface field only acting on the two outmost layers. si ¼ 1 corresponds to A atoms and si ¼ 1 corresponds to B atoms. We only consider the antiferromagnetic nearest-neighbor and ferromagnetic next nearest-neighbor interactions, i.e. J40 and a40. In our calculations, we choose the interaction energy a ¼ 0:2 as in Refs. [18,23–26], which corresponds to an alloy system with L10 and L12 bulk structures such as in CuAu or Co–Pt
X
U i;i ¼ 2N JZ ab ii Pai Pbi X
U i;iþ1 ¼
N aJZ nn ii Pni Pni ,
n¼a;b
2N
1 n2 JZniiþ1 Pn1 i Pn2 iþ1 ,
n1 ¼a;b n2 ¼a;b
X
U i;iþ2 ¼
2N aJZnn iiþ2 Pni Pniþ2 ,
ð3Þ
n¼a;b
where N is the number of lattice sites in each plane. Z ab ij is the number of neighbors between the sites of a sublattice in ith plane and the sites of b sublattice in jth plane. In the ab particular case that we considered here, Z aa ii ¼ Z ii ¼ 4, ab aa Zaa iiþ1 ¼ Z iiþ1 ¼ 2, and Z iiþ2 ¼ 1. The energy can be written as ( N X X N aJ Znn U¼ iiþ2 Pni2 Pni n
i¼1
þJ
X
1 n2 Z niiþ1 Pn1 i1 Pn2 i
þ 2 JZab ii Pai Pbi
n1 ;n2
þ aJ
X n
þaJ
X
Z nn ii Pni Pni þ J )
X
1 n2 Z niiþ1 Pn1 i Pn2 iþ1
n1 ;n2
Z nn iiþ2 Pni Pniþ2 .
ð4Þ
n
We have made the definition that the probabilities with io1 and i4N are zero. The surface term with surface field describing the segregation is U s ¼ NhðPa1 þ Pb1 þ PaN þ PbN Þ:
ð5Þ
The total energy of the system is U t ¼ U þ U s . In the site approximation, the entropy in the layered structure is
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given by S¼
N X X N kB ðPni ln Pni þ ð1 Pni Þ lnð1 Pni ÞÞ 2 n i¼1
ð6Þ
Table 1 Structures of the possible ground states for the six and seven layer alloy films Phase no.
with n ¼ a and b. The concentration of A in the system is given by N 1 X x¼ ðPai þ Pbi Þ=2. N i¼1
(7)
The concentration for each layer is given by xi ¼ ðPai þ Pbi Þ=2. The equilibrium state of the system is calculated from the minimization of the free energy F ¼ U t TS mNNðxA xB Þ for given values of the temperature T and the chemical potential m. The conjugate gradient method is used for the minimization of the free energy and the penalty Lagrangean multiplier method is used to fulfill the concentration constraint. In order to describe the order–disorder phase transitions, we define order parameters in all planes of the film. The single-layer order parameter of ith plane is Zi ¼ Pai Pbi .
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12
Structure N¼6
N¼7
AAAAAA ACACAA ACACCA ACCCCA ABCCBA ABCBBA ABBBBA CBBBBC CBCBBC CBCBCC CBCBCB CCCCCC
AAAAAAA ACACACA ABABABA ABCBCBA ABBBBBA CBBBBBC CBCBCBC CCCCCCC — — — —
(8)
3. Results 3.1. Ground states Before the calculations of phase diagrams, we have determined the ground states of FCC alloy films in the (0 0 1) direction using the Monte Carlo annealing method. In the calculations, we allow the system to anneal down from the state at high temperature to the ordered state at low temperature. During the process of annealing, spin flipping mechanism is used to get the configuration with minimal free energy. When temperature approaches zero, the system will be annealed into the ground state. In this way, we get all the possible ground states under different energy parameters. The supercell of 24 (in the (1 0 0) direction)24 (in the (0 1 0) direction)N (in the (0 0 1) direction) is taken in our calculations. (0 0 1) direction is set to be the stack direction of the atomic layer of the alloy films. We use the periodic boundary conditions in the (1 0 0) and (0 1 0) directions. The calculations on other larger geometries such as 30 30 N and 70 70 N are also performed and the same results are obtained. In our calculations, we consider the annealing process of alloy films from high temperature ðkB T=J ¼ 2:0Þ to low temperature ðkB T=J ¼ 106 Þ to get the ground states. We have determined all the possible ground states for the films with layer number N ¼ 6 and 7. The structures of the ground states for the six layer and seven layer films are described in Table 1. From our calculations, it is concluded that there are three basic layer structures for all ordered phases. These three basic layer structures are as following: (a) all sites in the layer are occupied by A atoms (or mostly by A atoms if temperature is not zero), which is denoted as A;
(a)
(b) Fig. 1. Phase diagrams of ground states for FCC alloy films in the (0 0 1) direction. (a) N ¼ 6, (b) N ¼ 7.
(b) all sites in the layer are occupied by B atoms (or mostly by B atoms if temperature is not zero), which is denoted as B; (c) the occupation on two sublattices is different, which is denoted as C. We use Oi ði ¼ 1; 2; . . .Þ to denote different phase structures. we only consider the case for A atom segregation ðh40Þ since the case for B atom segregation ðho0Þ is similar. If we exchange all A atoms with B atoms in the film, we can get the other ground states. The corresponding ground states by exchanging all A atoms with B atoms in Oi is denoted by O0i . The ground state diagrams of chemical potential vs surface field are shown in Fig. 1.
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3.2. Phase diagrams of thin alloy films From the phase diagrams of ground states, we can classify the phase diagrams of chemical potential vs temperature as follows: in the calculations of the phase diagrams, we change the chemical potential m from 1 to þ1, which means that the alloys Ax B1x will change from pure B to pure A. The sequence of ground states along the line parallel to the m axis corresponds to those of low temperature phases in the T m phase diagram when the chemical potential m is varied from 1 to þ1. Thus one may analyze the types of the phase diagrams according to the sequence of ground states along a line parallel to the m axis. We have calculated all the types of phase diagrams for six and seven layer alloy films. To obtain the phase diagrams, we minimize the free energy for different m=J and kB T=J. We change the chemical potential m=J from 1 to þ1 and kB T=J from 0.01 to 6.00. We have calculated the phase diagrams for the six layer thin films with the surface fields equal to 17:0; 10:0; 6:0; 4:0; 2:0, and 0.0 and the seven layer thin films with the surface fields equal to 17:0; 10:0; 4:2, and 2.0, respectively. The reason why we choose these different surface field values is that these values give all the different types of phase diagrams under surface confinement. Moreover, we can compare the even number layer films with the odd number layer films almost at the same condition. The phase structures of the six and seven layer films are listed in Tables 1 and 2. The phase structures listed in Table 1 correspond to the ground states. The phase structures listed in Table 2 are middletemperature phases, which are different from the ground state structures and only emerge in the region of kB T=J40. We use C 0 and C 00 to denote the layer structures which have different values of order parameters with C. Oim and O0im ði ¼ 1; 2; . . .Þ are used to denote the middletemperature phase structures with same symmetry but different values of order parameters. Since O0im has the same phase symmetry with Oim , we only list the Oim phase structures in Table 2. From our calculations, there are some middle-temperature phases in the even number layer films due to the effect of anti-phase boundary [27,28] occurring in the even number layer films [17]. We have
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denoted the middle-temperature phases in the phase diagrams of Figs. 2 and 4(a). Figs. 2 and 3 show the different types of phase diagrams under different surface fields for the six layer and seven layer films, respectively. For N ¼ 6, there are six types of phase diagrams, in which the phase sequences in low temperature are O01 -O8 -O7 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O8 O9 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O8 -O9 -O10 -O5 -O4 -O3 -O2 -O1 , O01 -O11 -O10 -O12 -O4 -O3 -O2 -O1 , O01 -O02 -O11 -O10 -O12 -O4 -O3 O2 -O1 , and O01 -O02 -O11 -O12 -O011 -O2 -O1 as chemical potential m=J changes from 1 to 1. For N ¼ 7, there are four types of phase diagrams, in which the phase sequences in low temperature are O01 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O6 -O7 O4 -O3 -O2 -O1 , O01 -O7 -O3 -O2 -O1 , and O01 -O02 -O7 -O8 -O3 -O2 O1 as chemical potential m=J changes from 1 to 1. It can be seen that there is a significant difference between even number layer films and odd number layer films. The phase diagrams and phase sequences for the even number layer films are more complex than those for the odd number layer films due to the effect of the anti-phase boundary occurring in even number layer thin films. Although different surface fields will result in different ground states and different types of phase diagrams, all the phase diagrams for the even number layer alloy films or the
(a)
(b)
(c)
(d)
(e)
(f)
Table 2 Structures of the possible middle-temperature phases for six layer alloy films Phase no.
Structure N¼6
O1m O2m O3m O4m O5m O6m O7m
ACAC0 AA ACC0 C0 CA ABCC0 BA CBCC0 C0 C CBC0 C0 BC CCCCC0 C00 C0 CC0 CC0 C
µ
µ
Fig. 2. Chemical potential–temperature phase diagrams for six layer FCC alloy films in the (0 0 1) direction under different surface fields. The values of the surface field are: (a) h=J ¼ 17:0; (b) h=J ¼ 10:0; (c) h=J ¼ 6:0; (d) h=J ¼ 4:0; (e) h=J ¼ 2:0; (f) h=J ¼ 0:0.
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(a)
(b)
(c)
µ
(d)
µ
Fig. 3. Chemical potential–temperature phase diagrams for seven layer FCC alloy films in the (0 0 1) direction under different surface fields. The values of the surface field are: (a) h=J ¼ 17:0; (b) h=J ¼ 10:0; (c) h=J ¼ 4:2; (d) h=J ¼ 2:0.
odd number layer ones have the same phase sequences and almost same shape in low temperatures when chemical potential m=J40. But the phase diagrams and phase sequences of the alloy films are drastically different for different surface fields when chemical potential m=Jo0. We can get the similar results if we draw the phase diagrams with surface field h=Jo0. For the real systems, such as a thin film growing on a substrate, the substrate and the vacuum contacted with the surfaces of the thin films act as two surface fields to the thin films. Surface fields are considered as interactions in the surface planes to offset the reduction of surface stresses or missing neighbors. Schweika have found that the surface field for the (0 0 1) surfaces of Cu3 Au as well as for CuAu is about 100–200 meV which are considerably larger than the bulk ordering energy [29]. The reduced value of the surface field for the Cu3 Au (0 0 1) surface is h=J ¼ 4:0 approximately [30]. Since the NiPt alloy has similar phase diagrams to the CuAu alloy, it is evaluated that the surface field for Ni3 Ptð0 0 1Þ alloys is approximately, h=J ¼ 4:0 [31,32]. We consider that Cu3 Au; CuAu; Ni3 Pt, and NiPt even number layer and odd number layer thin films could be described by the phase diagrams of Fig. 2(d) and 3(c), respectively. The even number layer alloy thin films of CuAu and NiPt would show many different stable structures as compared to the bulk and odd number layer ones. It is easy to simulate the thin films under different surface fields. However, it is rather difficult to determine the surface field h experimentally. It has been achieved for only a few systems, e.g., epitaxial FeCo(0 0 1) films grown on MgO(0 0 1) substrates display interface-induced ordering [33]. The pinning of a Co-rich first layer at the interface is mediated by an interface field h ¼ 9 meV. Compared to
the J ð10:75 meVÞ of the FeCo alloys, the value of the surface field for FeCo is approximate h=J ¼ 0:84. We have also calculated the concentration vs temperature phase diagrams for the six and seven layer alloy films when the surface field is h=J ¼ 0:0. As shown in Fig. 4, the phase diagram of the seven layer thin film is quite similar to the FCC binary alloy bulk phase diagram [18]. The phase structures and phase sequences are the same as the bulk. While the phase diagram for the six layer film is more complex than those of the bulk and seven layer films. Except for those phase structures corresponding to the bulk ones, the six layer films also have O2 ðACACAAÞ, O02 ðBCBCBBÞ and other middle-temperature phases. The differences in the phase behavior between the films with even number layers and the films with odd number layers are due to the anti-phase boundary effects. The formation of the anti-phase boundary in the FCC alloy films with even number layers is related to the effect of the surface segregation. We take an A3 B alloy with L12 ordered bulk as an example. The ordered L12 structure is a stack of two kinds of atomic layers along a cubic axis, one is the layer containing only A atoms (layer A), the other is the layer that contains equal A and B atoms (layer C). In the following, we regard A atoms as segregation atoms. When the surface field is weak, the ground states of the films have the truncated bulk structures with an alternative stack of A and C layers and have no surface segregation. For the film with even-number layers, one surface is A and the other is C. For the films with oddnumber layers, both surfaces are the same kind of layers.
(a)
(b)
Fig. 4. Concentration–temperature phase diagrams for six and seven layer FCC (0 0 1) alloy films. The value of surface field is h=J ¼ 0:0. (a) N ¼ 6, (b) N ¼ 7.
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When the surface field is strong, the ground states have the surface segregation of A atoms. The two surfaces of A3 B films should be A and thus the layer stack of the film is ACA . . . ACA. If the layer number of the film is even, there should be AA or CC anti-phase boundary structures in the ground states. The ground states of the odd layer films are still the truncated bulk structures with an alternative stack of A and C layers. This leads to the difference in the phase behavior between the film with even number layers and the film with odd number layers. Some middle-temperature phases are also related to the antiphase boundary effects. With the increase of temperature, both the tendency of ordering and surface segregation will change. Since the formation of the anti-phase boundary is correlated with the intensity of the surface segregation, the middle-temperature phases will occur near the order–disorder transition boundary where the ordering is weakened and surface segregation is dominated. In the order–disorder phase transitions of the alloy films, there are two types of phase transitions, the first order and the second order phase transitions. In the phase diagrams of Figs. 2–4, the heavy solid boundaries are used to denote the first order phase transitions. The thin solid boundaries are used to denote the second order phase transitions. We have determined the types of phase transitions through the changes of order parameters and specific heats as functions of temperature. For example, we have analyzed the phase transitions from O3 to O2m and O2m to the disordered phase in Fig. 2(b). Fig. 5 shows the variation of order parameter and specific heat as functions of temperature. It can be seen that the system experiences two kinds of phase
Fig. 5. Temperature dependence of the order parameter and specific heat for six layer alloy films when h=J ¼ 10:0 and m=J ¼ 6. There are two kinds of phase transitions, the first order phase transition occurs at temperature kB T=J ¼ 3:51 and the second order phase transition occurs at temperature kB T=J ¼ 4:86.
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transitions with the increase of temperature. There is a phase transition from O3 to O2m at temperature kB T=J ¼ 3:51 and the type of the transition is first order because both the order parameter and specific heat change discontinuously. At kB T=J ¼ 4:86, there is a phase transition from O2m to disordered phase with the transition being second order because the order parameters change continuously to zero and the specific heat changes discontinuously. 3.3. Quasi-thermodynamic phase transitions in alloy films For the odd number layer films, there is a phase structure ABðABÞn A ðn ¼ 1; 2; . . .Þ in a certain chemical potential region when the surface field h=J40 or BAðBAÞn B when h=Jo0. The monolayer order parameters of ABðABÞn A phase are zero, Thus the thin films which have ABðABÞn A phase structure do not undergo order–disorder phase transition with the increase of temperature due to the finite size in the (0 0 1) direction of films. But the alloy films will gradually approach to the bulk with the increase of layer number of the films and the corresponding bulk phase structure has order–disorder phase transition as the temperature increases. We have calculated the thermodynamic parameters such as the specific heat C v to analyze the change of the ABðABÞn A phase structure. We have calculated the variation of the specific heat as a function of temperature for the alloy films with different layer numbers to show the phase behavior of the ABðABÞn A phase structure. Fig. 6 shows the specific heat as a function of temperature for the films with different layer numbers and also for the bulk. It is shown that the specific heat curve of the bulk is discontinuous near temperature kB T=J ¼ 5:21, so that there is a phase transition from the ordered L10 structure to the disordered phase. Circles, triangles and inverse triangles correspond to the specific heat for the 21; 11 and 7 layers films, respectively. It is shown that the specific heat curves for the films with different layer numbers are continuous as functions of temperature, but each of the specific heat curves for the different layer films has a temperature peak when the temperature is near the
Fig. 6. The variation of the specific heats as functions of temperature. m=J ¼ 0:0. Squares correspond to bulk. Circles, triangles and inverse triangles correspond to 21, 11 and 7 layer alloy films, respectively, with surface field h=J ¼ 6:0.
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bulk transition temperature kB T=J ¼ 5:21. The thicker the thin films, the sharper the peak. We can conclude that the continuous C v peak of the films will transform to a discontinuous curve and the phase behavior for the alloy films will approach to the phase transition for the bulk when the alloy films approach from quasi two-dimensional thin films to three-dimensional bulk with the increase of layer numbers. We use the temperature peak value of the specific heat curve to describe the transition temperature for such quasi-thermodynamic phase transitions in finite size thin films and use the dash line to denote it in Fig. 3. For the CuAu(0 0 1) alloys, the surface energy difference drive Au to the surface layer, then the tendency to favor bonds between unlike atoms in the system drives Cu to the second layer which lead to ð. . . AuCuAuCu . . .Þ structure [34]. We consider that the quasi two-dimensional thin films of CuAu(0 0 1) structure will show quasi-thermodynamic phase transitions. 3.4. Critical points There are some critical points in the phase diagrams of the thin films. The critical points are determined by singularity in the specific heat curve. For example, for the critical point A in Fig. 2(b), we denote the chemical potential and temperature of the A point by mA and T A , respectively. The specific heat as a function of temperature is discontinuous at A point. Now we analyze the variation of the order parameter as a function of temperature around the critical point A. We take two chemical potential points: m ¼ 0:0 and m ¼ 0:3. Fig. 7(a) shows that the equilibrium state is O4 ðACCCCAÞ at low temperature. As the temperature increases, the order parameters of the two outmost layers are always zero and the order parameters of
(a)
(b)
Fig. 7. Temperature dependence of the order parameters around the critical point. The value of surface field is h=J ¼ 10:0. (a) m=J ¼ 0:0, (b) m=J ¼ 0:3.
the inner layers decrease continuously. It is noted that the order parameters of the second and fifth layers decrease faster than those of the third and fourth layers with the increase of temperature. Fig. 7(b) shows that the equilibrium state is O5 ðABCCBAÞ at low temperature. The changes of the order parameters of the two outmost layers and the third and fourth layers for the O5 phase are similar to those of the corresponding ones for the O4 phase with the increase of temperature. The order parameters of the second and fifth layers for the O5 phase have a peak and gradually change to zero with the increase of temperature. We have also calculated the specific heat as a function of temperature for m ¼ 0:0 and m ¼ 0:3, the results show that the specific heats are continuous near temperature T A and discontinuous at T A . In fact, it can be seen more clearly from the concentration vs temperature phase diagram. There is a phase separation region between O011 phase and O2 phase below the critical temperature in the X T phase diagram. It can be seen that the O011 and O2 phases are only distinguishable below the critical temperature and the same is for the O4 and O5 phases. B point in Fig. 2(b) is also determined by specific heat. We denote the chemical potential and temperature of B point by mB and T B , respectively. When mXmB , the specific heat as a function of temperature is discontinuous near T B and there is phase transition from O3 to O2m ; when momB , the specific heat C v is continuous near T B and there is no phase transition between O3 and O2m , thus B is a critical point. 3.5. Strong surface field effects in FCC alloy films For a very large surface field h, an N layer film undergoes a separation between its surfaces and the ðN 2Þ inner layers in the BCC (1 1 0) alloy films [12,13]. In our calculations of the FCC (0 0 1) alloy films, we have observed the same phenomena. From the phase diagrams of Figs. 2(a) and 3(a), it can be seen that there is a disordered phase lying between the ordered phases down to zero temperature. In case of N ¼ 6, there is a surfaceordered phase O8 for a large surface field, which has the structure that the two outmost layers are ordered and the inner four layers are disordered. When surface field h=J412:25, there is a disordered phase O7 , which has the structure that the two outmost layers are mostly A atoms and the inner four layers are mostly B atoms. On the right side of O7 in Fig. 2(a), there is another ordered phase O6 . The disordered O7 phase separates the ordered phases O8 and O6 down to zero temperature in the phase diagram. For N ¼ 7, when the surface field h=J414:40, the disordered phase O5 separates the ordered phases O6 and O4 down to zero temperature in the phase diagram. Generally, for the FCC alloy films, when the surface field is greater than a certain critical surface field, an N layer system undergoes a phase separation between its surface and the inner ðN 2Þ layers. In the phase diagrams, both the ordered phase and the disordered phase can exist down to the temperature zero.
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4. Conclusion
References
In summary, we have studied the order–disorder phase transitions of the FCC alloy thin films in the (0 0 1) direction. In the model we used, we consider nearest- and next nearest-neighbor interactions. We have analyzed the phase behavior and calculated the phase diagrams of the alloy films classified according to the ground state sequences. Due to the anti-phase boundary effects caused by surface segregation for the even number layer films, the basic phase structures for even number layer films and odd number layer films are different under the effects of the surface field and there are some middle-temperature phases emerged in even number layer films when the temperature is nonzero. Thus the types of phase diagrams and phase behaviors for even number layer films and the odd number layer films are significantly different. We consider that the even number layer thin films for CuAu and NiPt alloys will show many different stable structures as compared to the bulk and odd number layer ones if we control the appropriate concentrations and temperatures. For the odd number layer thin films, There is an ABðABÞn A phase which does not show phase transition with the increase of temperature due to the finite size of the films, such as the (0 0 1) thin films of CuAu and NiPt alloys with odd number layer structures. We have discussed the phase behavior of the ABðABÞn A phase with different layer numbers to show how the change of the ABðABÞn A phase crosses over to the bulk phase FTRANSITION as the quasi-twodimensional thin films approach to the three-dimensional bulk structure with the increase of layer number of the thin films. In the phase diagrams of the alloy films, we show there are some critical points. For a very large surface field, an N layer system undergoes a separation between its surface and the inner ðN 2Þ layers, there is a disordered phase lying between the ordered phases down to zero temperature.
[1] M. Polak, L. Rubinovich, Surf. Sci. Rep. 38 (2000) 127. [2] H. Reichert, P.J. Eng, H. Dosch, I.K. Robinson, Phys. Rev. Lett. 74 (1995) 2006. [3] H. Reichert, H. Dosch, Surf. Sci. 345 (1996) 27. [4] For an experimental review on surface critical phenomena, see H. Dosch, Critical phenomena at surfaces and interfaces, Springer Tracts in Modern Physics, vol. 126, Springer, Berlin, 1992. [5] R. Lipowsky, Phys. Rev. Lett. 49 (1982) 1575. [6] R. Lipowsky, J. Appl. Phys. 55 (1984) 2485. [7] F.F. Haas, F. Schmid, K. Binder, Phys. Rev. B 61 (2000) 15077. [8] J.M. Sanchez, J.L. Mora´n-Lo´pez, Phys. Rev. B 32 (1985) 3534. [9] J.L. Mora´n-Lo´pez, L.M. Falicov, Phys. Rev. B 18 (1978) 2542; J.L. Mora´n-Lo´pez, L.M. Falicov, Phys. Rev. B 18 (1978) 2549. [10] J.L. Mora´n-Lo´pez, K.H. Bennemann, Phys. Rev. B 15 (1977) 4769. [11] E.M. Sosa-Hernandez, F. Aguilera-Granja, J.L. Mora´n-Lo´pez, Phys. Rev. B 52 (1995) 5392. [12] A. Dı´ az-Ortiz, J.M. Sanchez, J.L. Mora´n-Lo´pez, Phys. Rev. Lett. 81 (1998) 1146. [13] A. Dı´ az-Ortiz, J.M. Sanchez, J.L. Mora´n-Lo´pez, Phys. Stat. Sol. (b) 220 (2000) 389. [14] A. Dı´ az-Ortiz, J.M. Sanchez, Phys. Rev. B 62 (2000) 1148. [15] F.L. Castillo Alvarado, A. Sukiennicki, L. Wojtczak, I. Zasada, Physica B 344 (2004) 477. [16] J. Ni, H. Liu, B.L. Gu, Chin. Phys. Lett. 18 (2001) 1504. [17] H. Liu, J. Ni, B.L. Gu, Eur. Phys. J. B 26 (2002) 261. [18] T. Mohri, J.M. Sanchez, D. de Fontaine, Acta Metall. 33 (1985) 1171. [19] J.M. Sanchez, D. de Fontaine, Phys. Rev. B 25 (1982) 1759. [20] K. Binder, Phys. Rev. Lett. 45 (1980) 811. [21] J.M. Sanchez, D. de Fontaine, W. Teitler, Phys. Rev. B 26 (1982) 1465. [22] K. Binder, W. Kinzel, W. Selke, J. Magn. Magn. Mater. 31–34 (1983) 1445. [23] D.M. Kroll, G. Gompper, Phys. Rev. B 36 (1987) 7078. [24] G. Gompper, D.M. Kroll, Phys. Rev. B 38 (1988) 459. [25] V.S. Sundaram, B. Farrell, R.S. Alben, W.D. Robertson, Phys. Rev. Lett. 31 (1973) 1136. [26] Y. Le Bouar, A. Loiseau, A. Finel, Phys. Rev. B 68 (2003) 224203. [27] H.M. Polatoglou, Mater. Sci. Eng. B 37 (1996) 177. [28] A. Finel, V. Mazauric, F. Ducastelle, Phys. Rev. Lett. 65 (1990) 1016. [29] W. Schweika, H. Reichert, W. Babik, O. Klein, S. Engemann, Phys. Rev. B 70 (2004) 41401. [30] M. Kessler, W. Dieterich, A. Majhofer, Phys. Rev. B 64 (2001) 125412. [31] M. Sanati, L.G. Wang, A. Zunger, Phys. Rev. Lett. 90 (2003) 45502. [32] J. Tersoff, Phys. Rev. B 42 (1990) 10965. [33] B. Nikel, F. Schlesener, W. Donner, H. Dosch, C. Detlefs, J. Chem. Phys. 117 (2002) 902. [34] W. Schweika, K. Binder, D.P. Landau, Phys. Rev. Lett. 65 (1990) 3321.
Acknowledgements This research was supported by the National Natural Science Foundation of China under Grant no. 10274036 and National Key Program of Basic Research Development of China.
Physica B 373 (2006) 198–205 www.elsevier.com/locate/physb
Phase diagrams of FCC alloy films Jiangling Pan, Yanlin Xu, Jun Ni Department of Physics and Key Laboratory of Atomic and Molecular Nanoscience, Tsinghua University, Beijing 100084, PR China Received 11 July 2005; received in revised form 18 October 2005; accepted 15 November 2005
Abstract We have studied the order–disorder phase transitions of FCC alloy thin films in the (0 0 1) direction using the mean-field method. Surface field is used to describe the surface effects such as surface segregation. All the different types of the phase diagrams of alloy thin films under the surface field are calculated. It is shown that the order–disorder phase transitions of alloy films are much more complex than those of bulk due to the combined effects of surface field and finite size confinement. The main features of phase diagrams are as follows: (1) for the thin films with even number layers, there are some middle-temperature phases due to the anti-phase boundary phenomena induced by the surface field, which leads to more complex phase behavior than those of odd number layer films. (2) For the quasi-two-dimensional thin films with odd number layers, there is an ABðABÞn A phase which does not show order–disorder phase transition with the increase of temperature due to the finite size of film. (3) For the thin films with both even and odd number layers, there are some critical points in the phase diagrams. r 2005 Elsevier B.V. All rights reserved. PACS: 64.60.Cn; 05.50.þq; 68.65.þg; 81.30.Bx Keywords: Phase diagrams; Thin films; Low-dimensional structures; Order–disorder transition
1. Introduction Experimental and theoretical studies of surface phenomena have shown that the surface behavior is significantly different from that of bulk systems [1–10]. Due to the influence of surface field, the surface shows many different phenomena from the bulk, such as correlation of ordering and surface segregation [1–4], surface-induced ordering and surface-induced disordering phenomena [5–7]. The development of the molecular-beam-epitaxy (MBE) has permitted the growth of thin film with monoatomic-layer accuracy. These thin films have two surfaces. The two surfaces, which are contacted with substrate and vacuum, respectively, can be considered as under two different surface fields. The thin films can be also considered as a quasi-two-dimensional structures. Due to the combined effects of surface field and finite size confinement, the thin films have different structures and phase transition Corresponding author. Tel.: +8610 62772781; fax: +8610 62781604.
E-mail address: [email protected] (J. Ni). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.146
behavior from the bulk. The phase transition behavior and the surface effects of binary alloy thin films have attracted much interest [11–16]. Studies on the phase transitions of BCC (1 1 0) Ax B1x alloys show that both surface field and film thickness have strong effect on the transition temperature [11]. In the thin films of BCC binary alloys with ordering interaction, the large confining field describing segregation can result in an unusual phase diagram with a disordered phase lying between the ordered phases down to zero temperature [12–14]. For the FCC thin film alloys, there have been some researches on the ground states of multilayer thin films [14,17]. Ni and Liu have studied the order–disorder phase transition of FCC (0 0 1) A3 B alloy thin films. They found that the order–disorder phase transitions between the even number layer films and the odd number layer films are significantly different. There is an oscillatory phase behavior as a function of film thickness due to the combined effects of surface field and finite size confinement [16]. There are many phase diagram calculations for the FCC bulk systems [18–22] and the surface systems [8–10], Mohri
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and Sanchez have calculated the phase diagrams of FCC lattice with nearest- and next nearest-neighbor pair interactions based on the Ising model [18]. The study on the (0 0 1) surface of binary FCC A3 B alloys shows that the order–disorder phase transitions of the surfaces coupled weakly to the bulk could be of second order even when the bulk has first order transition [8]. Although the FCC alloy film systems show a rich variety of phase phenomena, little is known on the phase diagrams of FCC thin films. In this paper, we present a study of the globe phase diagrams of FCC (0 0 1) alloy thin films under surface confinement. We have calculated the various types of phase diagrams classified according to the ground state sequences. We show that there is a rich variety of phase behavior in the alloy films due to the combined effects of surface field and finite size confinement. The outline of this paper is as follows: Section 2 describes the methods we used. Section 3 presents the results on the phase transitions and the phase diagrams of the thin films. Section 4 is the conclusion.
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alloys. We have also chosen other different values of a, such as 0.1 and 0.3 (a is also considered to be small), which gives the similar phase diagrams as a ¼ 0:2 due to the dominant effect of the nearest-neighbor interactions. For the bulk with the ordered L12 structure, four interpenetrating simple-cubic sublattices are used to describe the ordered phases. If we consider the planes in the (0 0 1) direction, each plane is composed of two sublattices. Therefore, for a film, we can divide each film layer into two equivalent sublattices a and b. In the mean-field approximation, the configurations of the system are described by the site probabilities of the A species denoted by Pn;i , where n holds for the sublattice ða; bÞ, and i ¼ 1; . . . ; N represents the plane number. Due to the probability normalization condition, the site probabilities of the B species are then 1 Pn;i . The energy of the thin film described by the first two terms in Eq. (1) can be written as a sum of intra-layer and inter-layer interaction energies X U¼ ð12 U i2;i þ 12 U i1;i þ U i;i þ 12 U iþ1;i þ 12 U iþ2;i Þ, ð2Þ i
2. Methods where We consider the FCC (0 0 1) alloy films and use an Isinglike Hamiltonian to describe the binary Ax B1x alloy on a lattice consisting of N two-dimensional parallel layers. The nearest-neighbor and next nearest-neighbor interactions are included. Generally, the nearest-neighbor pair interaction energy for intra-layer atomic pairs is different with that for inter-layer atomic pairs for films. In our calculations, we do not consider their difference and the nearestneighbor pair interaction energy is defined equally for intra-layer and inter-layer atomic pairs as an approximation. The surface effects, such as surface segregation, are described by surface fields [17,23]. The Hamiltonian is given as X X X H¼J si sj aJ si sj m si NN
h
NNN
X i2f1g
si þ
X
!
si ,
all
ð1Þ
i2fNg
where J is the nearest-neighbor interaction, a is the ratio between the nearest-neighbor and next nearest-neighbor interactions. The first sum runs over all the nearestneighbor sites and the second sum runs over all the next nearest-neighbor sites. The energy parameter m is the chemical potential. h represents the surface field, which describes the effects of confinement on the surface atoms. We consider the surface field only acting on the two outmost layers. si ¼ 1 corresponds to A atoms and si ¼ 1 corresponds to B atoms. We only consider the antiferromagnetic nearest-neighbor and ferromagnetic next nearest-neighbor interactions, i.e. J40 and a40. In our calculations, we choose the interaction energy a ¼ 0:2 as in Refs. [18,23–26], which corresponds to an alloy system with L10 and L12 bulk structures such as in CuAu or Co–Pt
X
U i;i ¼ 2N JZ ab ii Pai Pbi X
U i;iþ1 ¼
N aJZ nn ii Pni Pni ,
n¼a;b
2N
1 n2 JZniiþ1 Pn1 i Pn2 iþ1 ,
n1 ¼a;b n2 ¼a;b
X
U i;iþ2 ¼
2N aJZnn iiþ2 Pni Pniþ2 ,
ð3Þ
n¼a;b
where N is the number of lattice sites in each plane. Z ab ij is the number of neighbors between the sites of a sublattice in ith plane and the sites of b sublattice in jth plane. In the ab particular case that we considered here, Z aa ii ¼ Z ii ¼ 4, ab aa Zaa iiþ1 ¼ Z iiþ1 ¼ 2, and Z iiþ2 ¼ 1. The energy can be written as ( N X X N aJ Znn U¼ iiþ2 Pni2 Pni n
i¼1
þJ
X
1 n2 Z niiþ1 Pn1 i1 Pn2 i
þ 2 JZab ii Pai Pbi
n1 ;n2
þ aJ
X n
þaJ
X
Z nn ii Pni Pni þ J )
X
1 n2 Z niiþ1 Pn1 i Pn2 iþ1
n1 ;n2
Z nn iiþ2 Pni Pniþ2 .
ð4Þ
n
We have made the definition that the probabilities with io1 and i4N are zero. The surface term with surface field describing the segregation is U s ¼ NhðPa1 þ Pb1 þ PaN þ PbN Þ:
ð5Þ
The total energy of the system is U t ¼ U þ U s . In the site approximation, the entropy in the layered structure is
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given by S¼
N X X N kB ðPni ln Pni þ ð1 Pni Þ lnð1 Pni ÞÞ 2 n i¼1
ð6Þ
Table 1 Structures of the possible ground states for the six and seven layer alloy films Phase no.
with n ¼ a and b. The concentration of A in the system is given by N 1 X x¼ ðPai þ Pbi Þ=2. N i¼1
(7)
The concentration for each layer is given by xi ¼ ðPai þ Pbi Þ=2. The equilibrium state of the system is calculated from the minimization of the free energy F ¼ U t TS mNNðxA xB Þ for given values of the temperature T and the chemical potential m. The conjugate gradient method is used for the minimization of the free energy and the penalty Lagrangean multiplier method is used to fulfill the concentration constraint. In order to describe the order–disorder phase transitions, we define order parameters in all planes of the film. The single-layer order parameter of ith plane is Zi ¼ Pai Pbi .
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12
Structure N¼6
N¼7
AAAAAA ACACAA ACACCA ACCCCA ABCCBA ABCBBA ABBBBA CBBBBC CBCBBC CBCBCC CBCBCB CCCCCC
AAAAAAA ACACACA ABABABA ABCBCBA ABBBBBA CBBBBBC CBCBCBC CCCCCCC — — — —
(8)
3. Results 3.1. Ground states Before the calculations of phase diagrams, we have determined the ground states of FCC alloy films in the (0 0 1) direction using the Monte Carlo annealing method. In the calculations, we allow the system to anneal down from the state at high temperature to the ordered state at low temperature. During the process of annealing, spin flipping mechanism is used to get the configuration with minimal free energy. When temperature approaches zero, the system will be annealed into the ground state. In this way, we get all the possible ground states under different energy parameters. The supercell of 24 (in the (1 0 0) direction)24 (in the (0 1 0) direction)N (in the (0 0 1) direction) is taken in our calculations. (0 0 1) direction is set to be the stack direction of the atomic layer of the alloy films. We use the periodic boundary conditions in the (1 0 0) and (0 1 0) directions. The calculations on other larger geometries such as 30 30 N and 70 70 N are also performed and the same results are obtained. In our calculations, we consider the annealing process of alloy films from high temperature ðkB T=J ¼ 2:0Þ to low temperature ðkB T=J ¼ 106 Þ to get the ground states. We have determined all the possible ground states for the films with layer number N ¼ 6 and 7. The structures of the ground states for the six layer and seven layer films are described in Table 1. From our calculations, it is concluded that there are three basic layer structures for all ordered phases. These three basic layer structures are as following: (a) all sites in the layer are occupied by A atoms (or mostly by A atoms if temperature is not zero), which is denoted as A;
(a)
(b) Fig. 1. Phase diagrams of ground states for FCC alloy films in the (0 0 1) direction. (a) N ¼ 6, (b) N ¼ 7.
(b) all sites in the layer are occupied by B atoms (or mostly by B atoms if temperature is not zero), which is denoted as B; (c) the occupation on two sublattices is different, which is denoted as C. We use Oi ði ¼ 1; 2; . . .Þ to denote different phase structures. we only consider the case for A atom segregation ðh40Þ since the case for B atom segregation ðho0Þ is similar. If we exchange all A atoms with B atoms in the film, we can get the other ground states. The corresponding ground states by exchanging all A atoms with B atoms in Oi is denoted by O0i . The ground state diagrams of chemical potential vs surface field are shown in Fig. 1.
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3.2. Phase diagrams of thin alloy films From the phase diagrams of ground states, we can classify the phase diagrams of chemical potential vs temperature as follows: in the calculations of the phase diagrams, we change the chemical potential m from 1 to þ1, which means that the alloys Ax B1x will change from pure B to pure A. The sequence of ground states along the line parallel to the m axis corresponds to those of low temperature phases in the T m phase diagram when the chemical potential m is varied from 1 to þ1. Thus one may analyze the types of the phase diagrams according to the sequence of ground states along a line parallel to the m axis. We have calculated all the types of phase diagrams for six and seven layer alloy films. To obtain the phase diagrams, we minimize the free energy for different m=J and kB T=J. We change the chemical potential m=J from 1 to þ1 and kB T=J from 0.01 to 6.00. We have calculated the phase diagrams for the six layer thin films with the surface fields equal to 17:0; 10:0; 6:0; 4:0; 2:0, and 0.0 and the seven layer thin films with the surface fields equal to 17:0; 10:0; 4:2, and 2.0, respectively. The reason why we choose these different surface field values is that these values give all the different types of phase diagrams under surface confinement. Moreover, we can compare the even number layer films with the odd number layer films almost at the same condition. The phase structures of the six and seven layer films are listed in Tables 1 and 2. The phase structures listed in Table 1 correspond to the ground states. The phase structures listed in Table 2 are middletemperature phases, which are different from the ground state structures and only emerge in the region of kB T=J40. We use C 0 and C 00 to denote the layer structures which have different values of order parameters with C. Oim and O0im ði ¼ 1; 2; . . .Þ are used to denote the middletemperature phase structures with same symmetry but different values of order parameters. Since O0im has the same phase symmetry with Oim , we only list the Oim phase structures in Table 2. From our calculations, there are some middle-temperature phases in the even number layer films due to the effect of anti-phase boundary [27,28] occurring in the even number layer films [17]. We have
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denoted the middle-temperature phases in the phase diagrams of Figs. 2 and 4(a). Figs. 2 and 3 show the different types of phase diagrams under different surface fields for the six layer and seven layer films, respectively. For N ¼ 6, there are six types of phase diagrams, in which the phase sequences in low temperature are O01 -O8 -O7 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O8 O9 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O8 -O9 -O10 -O5 -O4 -O3 -O2 -O1 , O01 -O11 -O10 -O12 -O4 -O3 -O2 -O1 , O01 -O02 -O11 -O10 -O12 -O4 -O3 O2 -O1 , and O01 -O02 -O11 -O12 -O011 -O2 -O1 as chemical potential m=J changes from 1 to 1. For N ¼ 7, there are four types of phase diagrams, in which the phase sequences in low temperature are O01 -O6 -O5 -O4 -O3 -O2 -O1 , O01 -O6 -O7 O4 -O3 -O2 -O1 , O01 -O7 -O3 -O2 -O1 , and O01 -O02 -O7 -O8 -O3 -O2 O1 as chemical potential m=J changes from 1 to 1. It can be seen that there is a significant difference between even number layer films and odd number layer films. The phase diagrams and phase sequences for the even number layer films are more complex than those for the odd number layer films due to the effect of the anti-phase boundary occurring in even number layer thin films. Although different surface fields will result in different ground states and different types of phase diagrams, all the phase diagrams for the even number layer alloy films or the
(a)
(b)
(c)
(d)
(e)
(f)
Table 2 Structures of the possible middle-temperature phases for six layer alloy films Phase no.
Structure N¼6
O1m O2m O3m O4m O5m O6m O7m
ACAC0 AA ACC0 C0 CA ABCC0 BA CBCC0 C0 C CBC0 C0 BC CCCCC0 C00 C0 CC0 CC0 C
µ
µ
Fig. 2. Chemical potential–temperature phase diagrams for six layer FCC alloy films in the (0 0 1) direction under different surface fields. The values of the surface field are: (a) h=J ¼ 17:0; (b) h=J ¼ 10:0; (c) h=J ¼ 6:0; (d) h=J ¼ 4:0; (e) h=J ¼ 2:0; (f) h=J ¼ 0:0.
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(a)
(b)
(c)
µ
(d)
µ
Fig. 3. Chemical potential–temperature phase diagrams for seven layer FCC alloy films in the (0 0 1) direction under different surface fields. The values of the surface field are: (a) h=J ¼ 17:0; (b) h=J ¼ 10:0; (c) h=J ¼ 4:2; (d) h=J ¼ 2:0.
odd number layer ones have the same phase sequences and almost same shape in low temperatures when chemical potential m=J40. But the phase diagrams and phase sequences of the alloy films are drastically different for different surface fields when chemical potential m=Jo0. We can get the similar results if we draw the phase diagrams with surface field h=Jo0. For the real systems, such as a thin film growing on a substrate, the substrate and the vacuum contacted with the surfaces of the thin films act as two surface fields to the thin films. Surface fields are considered as interactions in the surface planes to offset the reduction of surface stresses or missing neighbors. Schweika have found that the surface field for the (0 0 1) surfaces of Cu3 Au as well as for CuAu is about 100–200 meV which are considerably larger than the bulk ordering energy [29]. The reduced value of the surface field for the Cu3 Au (0 0 1) surface is h=J ¼ 4:0 approximately [30]. Since the NiPt alloy has similar phase diagrams to the CuAu alloy, it is evaluated that the surface field for Ni3 Ptð0 0 1Þ alloys is approximately, h=J ¼ 4:0 [31,32]. We consider that Cu3 Au; CuAu; Ni3 Pt, and NiPt even number layer and odd number layer thin films could be described by the phase diagrams of Fig. 2(d) and 3(c), respectively. The even number layer alloy thin films of CuAu and NiPt would show many different stable structures as compared to the bulk and odd number layer ones. It is easy to simulate the thin films under different surface fields. However, it is rather difficult to determine the surface field h experimentally. It has been achieved for only a few systems, e.g., epitaxial FeCo(0 0 1) films grown on MgO(0 0 1) substrates display interface-induced ordering [33]. The pinning of a Co-rich first layer at the interface is mediated by an interface field h ¼ 9 meV. Compared to
the J ð10:75 meVÞ of the FeCo alloys, the value of the surface field for FeCo is approximate h=J ¼ 0:84. We have also calculated the concentration vs temperature phase diagrams for the six and seven layer alloy films when the surface field is h=J ¼ 0:0. As shown in Fig. 4, the phase diagram of the seven layer thin film is quite similar to the FCC binary alloy bulk phase diagram [18]. The phase structures and phase sequences are the same as the bulk. While the phase diagram for the six layer film is more complex than those of the bulk and seven layer films. Except for those phase structures corresponding to the bulk ones, the six layer films also have O2 ðACACAAÞ, O02 ðBCBCBBÞ and other middle-temperature phases. The differences in the phase behavior between the films with even number layers and the films with odd number layers are due to the anti-phase boundary effects. The formation of the anti-phase boundary in the FCC alloy films with even number layers is related to the effect of the surface segregation. We take an A3 B alloy with L12 ordered bulk as an example. The ordered L12 structure is a stack of two kinds of atomic layers along a cubic axis, one is the layer containing only A atoms (layer A), the other is the layer that contains equal A and B atoms (layer C). In the following, we regard A atoms as segregation atoms. When the surface field is weak, the ground states of the films have the truncated bulk structures with an alternative stack of A and C layers and have no surface segregation. For the film with even-number layers, one surface is A and the other is C. For the films with oddnumber layers, both surfaces are the same kind of layers.
(a)
(b)
Fig. 4. Concentration–temperature phase diagrams for six and seven layer FCC (0 0 1) alloy films. The value of surface field is h=J ¼ 0:0. (a) N ¼ 6, (b) N ¼ 7.
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When the surface field is strong, the ground states have the surface segregation of A atoms. The two surfaces of A3 B films should be A and thus the layer stack of the film is ACA . . . ACA. If the layer number of the film is even, there should be AA or CC anti-phase boundary structures in the ground states. The ground states of the odd layer films are still the truncated bulk structures with an alternative stack of A and C layers. This leads to the difference in the phase behavior between the film with even number layers and the film with odd number layers. Some middle-temperature phases are also related to the antiphase boundary effects. With the increase of temperature, both the tendency of ordering and surface segregation will change. Since the formation of the anti-phase boundary is correlated with the intensity of the surface segregation, the middle-temperature phases will occur near the order–disorder transition boundary where the ordering is weakened and surface segregation is dominated. In the order–disorder phase transitions of the alloy films, there are two types of phase transitions, the first order and the second order phase transitions. In the phase diagrams of Figs. 2–4, the heavy solid boundaries are used to denote the first order phase transitions. The thin solid boundaries are used to denote the second order phase transitions. We have determined the types of phase transitions through the changes of order parameters and specific heats as functions of temperature. For example, we have analyzed the phase transitions from O3 to O2m and O2m to the disordered phase in Fig. 2(b). Fig. 5 shows the variation of order parameter and specific heat as functions of temperature. It can be seen that the system experiences two kinds of phase
Fig. 5. Temperature dependence of the order parameter and specific heat for six layer alloy films when h=J ¼ 10:0 and m=J ¼ 6. There are two kinds of phase transitions, the first order phase transition occurs at temperature kB T=J ¼ 3:51 and the second order phase transition occurs at temperature kB T=J ¼ 4:86.
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transitions with the increase of temperature. There is a phase transition from O3 to O2m at temperature kB T=J ¼ 3:51 and the type of the transition is first order because both the order parameter and specific heat change discontinuously. At kB T=J ¼ 4:86, there is a phase transition from O2m to disordered phase with the transition being second order because the order parameters change continuously to zero and the specific heat changes discontinuously. 3.3. Quasi-thermodynamic phase transitions in alloy films For the odd number layer films, there is a phase structure ABðABÞn A ðn ¼ 1; 2; . . .Þ in a certain chemical potential region when the surface field h=J40 or BAðBAÞn B when h=Jo0. The monolayer order parameters of ABðABÞn A phase are zero, Thus the thin films which have ABðABÞn A phase structure do not undergo order–disorder phase transition with the increase of temperature due to the finite size in the (0 0 1) direction of films. But the alloy films will gradually approach to the bulk with the increase of layer number of the films and the corresponding bulk phase structure has order–disorder phase transition as the temperature increases. We have calculated the thermodynamic parameters such as the specific heat C v to analyze the change of the ABðABÞn A phase structure. We have calculated the variation of the specific heat as a function of temperature for the alloy films with different layer numbers to show the phase behavior of the ABðABÞn A phase structure. Fig. 6 shows the specific heat as a function of temperature for the films with different layer numbers and also for the bulk. It is shown that the specific heat curve of the bulk is discontinuous near temperature kB T=J ¼ 5:21, so that there is a phase transition from the ordered L10 structure to the disordered phase. Circles, triangles and inverse triangles correspond to the specific heat for the 21; 11 and 7 layers films, respectively. It is shown that the specific heat curves for the films with different layer numbers are continuous as functions of temperature, but each of the specific heat curves for the different layer films has a temperature peak when the temperature is near the
Fig. 6. The variation of the specific heats as functions of temperature. m=J ¼ 0:0. Squares correspond to bulk. Circles, triangles and inverse triangles correspond to 21, 11 and 7 layer alloy films, respectively, with surface field h=J ¼ 6:0.
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bulk transition temperature kB T=J ¼ 5:21. The thicker the thin films, the sharper the peak. We can conclude that the continuous C v peak of the films will transform to a discontinuous curve and the phase behavior for the alloy films will approach to the phase transition for the bulk when the alloy films approach from quasi two-dimensional thin films to three-dimensional bulk with the increase of layer numbers. We use the temperature peak value of the specific heat curve to describe the transition temperature for such quasi-thermodynamic phase transitions in finite size thin films and use the dash line to denote it in Fig. 3. For the CuAu(0 0 1) alloys, the surface energy difference drive Au to the surface layer, then the tendency to favor bonds between unlike atoms in the system drives Cu to the second layer which lead to ð. . . AuCuAuCu . . .Þ structure [34]. We consider that the quasi two-dimensional thin films of CuAu(0 0 1) structure will show quasi-thermodynamic phase transitions. 3.4. Critical points There are some critical points in the phase diagrams of the thin films. The critical points are determined by singularity in the specific heat curve. For example, for the critical point A in Fig. 2(b), we denote the chemical potential and temperature of the A point by mA and T A , respectively. The specific heat as a function of temperature is discontinuous at A point. Now we analyze the variation of the order parameter as a function of temperature around the critical point A. We take two chemical potential points: m ¼ 0:0 and m ¼ 0:3. Fig. 7(a) shows that the equilibrium state is O4 ðACCCCAÞ at low temperature. As the temperature increases, the order parameters of the two outmost layers are always zero and the order parameters of
(a)
(b)
Fig. 7. Temperature dependence of the order parameters around the critical point. The value of surface field is h=J ¼ 10:0. (a) m=J ¼ 0:0, (b) m=J ¼ 0:3.
the inner layers decrease continuously. It is noted that the order parameters of the second and fifth layers decrease faster than those of the third and fourth layers with the increase of temperature. Fig. 7(b) shows that the equilibrium state is O5 ðABCCBAÞ at low temperature. The changes of the order parameters of the two outmost layers and the third and fourth layers for the O5 phase are similar to those of the corresponding ones for the O4 phase with the increase of temperature. The order parameters of the second and fifth layers for the O5 phase have a peak and gradually change to zero with the increase of temperature. We have also calculated the specific heat as a function of temperature for m ¼ 0:0 and m ¼ 0:3, the results show that the specific heats are continuous near temperature T A and discontinuous at T A . In fact, it can be seen more clearly from the concentration vs temperature phase diagram. There is a phase separation region between O011 phase and O2 phase below the critical temperature in the X T phase diagram. It can be seen that the O011 and O2 phases are only distinguishable below the critical temperature and the same is for the O4 and O5 phases. B point in Fig. 2(b) is also determined by specific heat. We denote the chemical potential and temperature of B point by mB and T B , respectively. When mXmB , the specific heat as a function of temperature is discontinuous near T B and there is phase transition from O3 to O2m ; when momB , the specific heat C v is continuous near T B and there is no phase transition between O3 and O2m , thus B is a critical point. 3.5. Strong surface field effects in FCC alloy films For a very large surface field h, an N layer film undergoes a separation between its surfaces and the ðN 2Þ inner layers in the BCC (1 1 0) alloy films [12,13]. In our calculations of the FCC (0 0 1) alloy films, we have observed the same phenomena. From the phase diagrams of Figs. 2(a) and 3(a), it can be seen that there is a disordered phase lying between the ordered phases down to zero temperature. In case of N ¼ 6, there is a surfaceordered phase O8 for a large surface field, which has the structure that the two outmost layers are ordered and the inner four layers are disordered. When surface field h=J412:25, there is a disordered phase O7 , which has the structure that the two outmost layers are mostly A atoms and the inner four layers are mostly B atoms. On the right side of O7 in Fig. 2(a), there is another ordered phase O6 . The disordered O7 phase separates the ordered phases O8 and O6 down to zero temperature in the phase diagram. For N ¼ 7, when the surface field h=J414:40, the disordered phase O5 separates the ordered phases O6 and O4 down to zero temperature in the phase diagram. Generally, for the FCC alloy films, when the surface field is greater than a certain critical surface field, an N layer system undergoes a phase separation between its surface and the inner ðN 2Þ layers. In the phase diagrams, both the ordered phase and the disordered phase can exist down to the temperature zero.
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4. Conclusion
References
In summary, we have studied the order–disorder phase transitions of the FCC alloy thin films in the (0 0 1) direction. In the model we used, we consider nearest- and next nearest-neighbor interactions. We have analyzed the phase behavior and calculated the phase diagrams of the alloy films classified according to the ground state sequences. Due to the anti-phase boundary effects caused by surface segregation for the even number layer films, the basic phase structures for even number layer films and odd number layer films are different under the effects of the surface field and there are some middle-temperature phases emerged in even number layer films when the temperature is nonzero. Thus the types of phase diagrams and phase behaviors for even number layer films and the odd number layer films are significantly different. We consider that the even number layer thin films for CuAu and NiPt alloys will show many different stable structures as compared to the bulk and odd number layer ones if we control the appropriate concentrations and temperatures. For the odd number layer thin films, There is an ABðABÞn A phase which does not show phase transition with the increase of temperature due to the finite size of the films, such as the (0 0 1) thin films of CuAu and NiPt alloys with odd number layer structures. We have discussed the phase behavior of the ABðABÞn A phase with different layer numbers to show how the change of the ABðABÞn A phase crosses over to the bulk phase FTRANSITION as the quasi-twodimensional thin films approach to the three-dimensional bulk structure with the increase of layer number of the thin films. In the phase diagrams of the alloy films, we show there are some critical points. For a very large surface field, an N layer system undergoes a separation between its surface and the inner ðN 2Þ layers, there is a disordered phase lying between the ordered phases down to zero temperature.
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Acknowledgements This research was supported by the National Natural Science Foundation of China under Grant no. 10274036 and National Key Program of Basic Research Development of China.