- Email: [email protected]
else431s of or dering in fcc alloys during codeposition
Surface Science 499 (2002) 174–182 www.elsevier.com/locate/susc
Kinetics of ordering in fcc alloys during codeposition Jun Ni *, Binglin Gu Department of Physics, Tsinghua University, Beijing 100084, China Received 3 August 2001; accepted for publication 24 October 2001
Abstract The kinetics of ordering in fcc alloys during codeposition is investigated by the kinetic mean field method in the pair approximation. The evolutions of the long-range order and short-range order parameters during codeposition are calculated. It is found that the kinetic path involves various stages of relaxation and there is transient ordered state during codeposition. The influence of the two characteristic times related to the adsorption and the atomic migration on the kinetics of the growth is analyzed. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Models of surface kinetics; Non-equilibrium thermodynamics and statistical mechanics; Growth; Surface thermodynamics (including phase transitions); Alloys
1. Introduction Epitaxial growth has been used with layer by layer (such as in Frank–van der Merwe growth mode) control over growth of materials. It is found that III–V semiconductor alloys exhibit long-range order (LRO) [1–5] when grown by epitaxial method despite the fact that the bulk phase diagram predicts phase separation at growth temperature due to the changes in the interatomic interactions on the surface and the substrate constraint. Epitaxial metal alloy shows various ordering behavior [6,7]. The amount of LRO and its presence depend sensitively on growth condition. Saito and M€ uler-Krumbhaar [8] used a kinetic mean field theory in pair approximation to discuss the crystal growth. Venkatasubramanian [9] de-
*
Corresponding author. Fax: +86-10-6278-1604. E-mail address: [email protected] (J. Ni).
veloped the kinetic mean field model to treat the growth of an alloy with short-range order (SRO). Smith and Zangwill [10] used the kinetic mean field to study the interplay between compositional ordering and surface roughening during the epitaxial growth of a binary alloy. It was shown that transient ordered states can be formed in various systems during the relaxation of a non-equilibrium state to the equilibrium state [11–14]. The occurrence of these transient ordered states is due to the multiple characteristic times during the relaxation process. When the evolution of the system is controlled by the multiple relaxation rates, the system can first reach a kind of transient state through the kinetics process with faster relaxation rate. This transient state is a relative stable state. As other relaxation processes become functional, the system gradually relaxes to the equilibrium state. In the kinetics of ordering in alloys, there are multiple characteristic times for the atomic migration that control the relaxation of
0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 8 1 7 - 9
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
175
different atomic species, which leads to transient states on its relaxation path to the final equilibrium phase [13,14]. For epitaxial growth, in addition to the diffusion process, there are other two relaxation processes, i.e., adsorption and evaporation process. This can result in transient states in the growth process due to multiple relaxation times. In this paper, we study the kinetics of ordering in alloys during codeposition. The evolutions of the LRO and SRO parameters during codeposition are calculated. We will show that there are various types of the kinetic path depending on the growth parameters. It is found that there is transient ordered state in the system during the codeposition. Several features in the kinetics of alloy growth are illustrated. In Section 2 we describe the methods. Section 3 presents the results of the kinetics of ordering during codeposition. In Section 4 we make the conclusions.
adsorption rate of atoms onto a vacant site is given by
2. Methods
We use the micro-master equation method [15,16] to describe the kinetics of ordering and disordering in the system. In the master equation, the configurations of the phases in an alloy are described by a set of cluster distribution functions. Pa1 ;...;an ðr1 ; . . . ; rn ; tÞ is defined as the probability that at a given time t1 the n lattice sites r1 ; . . . ; rn be simultaneously occupied by n atoms of type a1 . . . an . These cluster distribution functions should satisfy the normalization conditions. The cluster distribution functions will change with time in the evolution from the initial state to the final state as the three relaxation processes, surface migration, adsorption and evaporation take place on the lattice. In the pair approximation, the configurations of phases at a given temperature is described by point and pair distribution functions denoted as Pa ðrÞ and Pa1 a2 ðr1 ; r2 Þ, where a and r represent the type of species and lattice site, respectively. We assume that surface atomic migration occurs by surface diffusion whereby an A or B atom jumps to an empty nearest-neighbor adsorption site. We denote by Rai vfxg ðri ; rj ; fxgÞ the rate at which the ai atom at site ri jumps to a neighboring vacancy at site rj under the influence
We consider a (1 0 0) surface codeposition layer of an fcc alloy and the system consists of three components, two species (A and B) of atoms and vacancy (V) on the surface. The system is described by the stochastic lattice gas model with the Hamiltonian XX 0 H¼ Emm0 ðri rj Þcmi cmj : ð1Þ fijg
mm0
The variable cmi ¼ 1 if the site i is occupied by an mð¼ A; B; V Þ atom and is zero otherwise. The notation fijg means the summation over the pairs of neighboring lattice sites. Emm0 ðri rj Þ represents the interaction energy between an m atom at ri and an m0 atom at rj . We consider the growth of a monolayer of atoms on a given planar substrate. The surface kinetic processes comprise the relaxation processes such as the adsorption, the evaporation and the diffusion process. The adsorption and the evaporation are described by the single-site relaxation processes [8]. The adsorption is assumed to be determined by the chemical potential lai . The
waai ¼
1 La e i; sR
ð2Þ
where Lai ¼ lai =kB T , and sR is a characteristic time constant. This form of the adsorption rate represents approximately the growth rate for the case that chemical potential lai is very large. The evaporation rate depends on the local configuration. The evaporation rate then is defined as 1 DEai weai ¼ exp ; ð3Þ sR kB T where DEai ¼ Eai Ev . Eai is the energy before ai atom is evaporated and Ev is the energy after the evaporation. The transition probabilities are required to satisfy the detailed balanced condition between evaporation and adsorption weai lai Peq ð. . . ; v; . . .Þ DEai ¼ exp ¼ : ð4Þ waai Peq ð. . . ; ai ; . . .Þ kB T kB T
176
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
of the set of the neighboring atoms fxg at the corresponding sites fxg. The rate of change of the pair distribution function is given by d Pa aj ðri ; rj ; tÞ dt i X X Rvai fxg ðri ; rk ; fxgÞPvaj ai fxg ðri ; rj ; rk ; fxgÞ ¼ k
fxg
k6¼j
XX k
k6¼j
Rai vfxg ðri ; rk ; fxgÞPai aj vfxg ðri ; rj ; rk ; fxgÞ
fxg
XX Rvaj fxg ðrj ; rk ; fxgÞPai vaj fxg ðri ; rj ; rk ; fxgÞ k
k6¼j
fxg
XX Raj vfxg ðrj ; rk ; fxgÞPai aj vfxg ðri ; rj ; rk ; fxgÞ k
k6¼i
fxg
DEaj DEai 1 exp þ sR kB T kB T 1 Laj Pai aj ðri ; rj ; tÞ þ e Pai v ðri ; rj ; tÞ sR 1 La þ e i Pvaj ðri ; rj ; tÞ; sR
1 exp sR
ð5Þ
d Pa v ðri ; rj ; tÞ dt i X Rvai fxg ðri ; rj ; fxgÞPvai fxg ðri ; rj ; fxgÞ ¼ fxg
X
Rai vfxg ðri ; rj ; fxgÞPai vfxg ðri ; rj ; fxgÞ
fxg
XX Rvai fxg ðri ; rk ; fxgÞPvvai fxg ðri ; rj ; rk ; fxgÞ k
k6¼j
fxg
XX Rai vfxg ðri ; rk ; fxgÞPai vvfxg ðri ; rj ; rk ; fxgÞ k
k6¼j
k
X
k
fxg
X 1 DEak Pai ak ðri ; rj ; tÞ exp sR kB T ak 1 DEai Pai v ðri ; rj ; tÞ exp kB T sR X 1 1 eLak Pai v ðri ; rj ; tÞ eLai Pai v ðri ; rj ; tÞ; þ s s R R ak
ð7Þ
ð8Þ Pa1 a2 ðr1 ; r2 Þ ¼ Pa2 ðr2 Þ:
a1
fxg
XX Rvak fxg ðrj ; rk ; fxgÞPai vak fxg ðri ; rj ; rk ; fxgÞ k6¼i
Pai aj Pa a Pa7 ai Pa8 aj Pa14 aj ; Pa7i Pa7j 1 i
a2
fxg
XX Rak vfxg ðrj ; rk ; fxgÞPai ak vfxg ðri ; rj ; rk ; fxgÞ k6¼i
Pai aj a1 a14 ¼
where the sites ri and rj are occupied by interchanging atom or vacancy. A substitution of Eq. (7) into Eqs. (5) and (6) gives the closed kinetic equations in the pair approximation. It is easy to see the uncoupling of Eq. (7) leads to the fulfilment of the superposition equation obtained by the cluster variation method (CVM) for the equilibrium state. The point and pair distributions satisfy the following relations: X Pa1 a2 ðr1 ; r2 Þ ¼ Pa1 ðr1 Þ;
and
represents the summation of k over all the nearestneighbor sites of rj except ri . Formula relevant to Pvaj ðri ; rj ; tÞ can be written down similarly. In the approximation of first- and second-nearest-neighbor interactions, the function Rai vfxg ðr1 ; r2 ; fxgÞ is dependent on the configuration of species fxg located on the first- and second-coordination sphere of a chosen site pair. In this case, the kinetic equation for the pair distribution function given on the nearest-neighboring lattice sites, contains 16-particle distribution functions because there are four nearest neighbors and four second nearest neighbors for each lattice site. To obtain the closed kinetic equations in the pair approximation, the 16-particle distribution functions should be expressed in terms of the pair and point probability distribution functions.
þ
ð6Þ
P where k6¼j represents the summation of k over Pall the nearest-neighbor sites of ri except rj and k6¼i
They are statements of the conservation of probability. The pair distributions also satisfy P normalization condition P ðr1 ; r2 Þ ¼ 1. a a 1 2 a1 a2 Therefore, the point and pair distribution functions are not independent. Among them, we need choose an independent set. All other point and pair distribution functions can be obtained from this independent set according to the normalization condition. Following Vineyard [17], the jumping rate of constituent atoms to the neighboring vacancy sites are given by
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
Rai cfxg ðri ; rj ; fxgÞ ¼
1 exp sd
U DE kB T 2kB T
;
ð9Þ
where 1=sd is the vibration frequency associated with atom–vacancy interchange (assumed identical for A and B atoms). U is the average barrier height for the atomic interchange. DE is the energy difference for the atom–vacancy interchange. In order to describe the atomic configurations of ordered structures on the square lattice, we divide the square lattice into four square sublattices a, b, d, c. In the point approximation, the symmetry of the atomic configurations is determined by the occupation probability Pms of m species (m ¼ A; B; V ) in the sublattice s(¼ a, b, d, c). The atomic configurations of the phases are also generally described by the following independent order parameters: cm1 ¼ Pma þ Pmb Pmc Pmd ; cm2 ¼ Pma Pmb ; cm3
¼
Pmc
ð10Þ
Pmd ;
where m ¼ A and B. The order parameter cm1 indicates the atomic configuration between the nearest-neighbor sites of species m. Similarly, other order parameters cm2 and cm3 show the atomic configurations between the second nearest-neighbor sites. The values of these parameters define the following phases with different symmetry: (1) disordered phase: cmi ¼ 0; (2) O1 : cm1 ¼ 0, cm2 6¼ 0, cm3 6¼ 0; (3) O2 : cm1 6¼ 0, cm2 ¼ cm3 ¼ 0. In addition to the above order parameters, there are other order parameters related to the pair distribution functions. For the sublattices chosen to characterize the ordered phases in ternary system, there are 22 independent pair distribution functions which are characterized by 10 LRO parameters and 12 SRO parameters. In the point approximation, we have three independent probability distribution functions for binary system and six independent parameters for ternary system. There are no SRO parameters describing correlations and all the independent probability distribution functions correspond to LRO parameters. In the pair approximation, there are correlations. In addition to the LRO parameters
177
described by Eq. (10), there are other four LRO parameters represented by following equations: ac ac PCA ; cP1 ¼ PAC ad ad cP2 ¼ PAC PCA ;
ð11Þ
bc bc cP3 ¼ PAC PCA ; bd bd cP4 ¼ PAC PCA :
Twelve SRO parameters are defined by following equations: 0
ad ad gmm 2 ¼ Pmm0 þ Pm0 m ;
0
0
bd bd gmm 4 ¼ Pmm0 þ Pm0 m ;
ac ac gmm 1 ¼ Pmm0 þ Pm0 m ;
0
bc bc gmm 3 ¼ Pmm0 þ Pm0 m ;
ð12Þ
where m 6¼ m0 . The coverage during the codeposition is defined as h ¼ CA þ CB , where CA and CB are the concentrations of A and B atoms respectively. The energy of the system are expressed in terms of the pairwise interactions. We consider the firstðkÞ and second-nearest-neighbor interactions Eij (k ¼ 1; 2 for the first- and second-nearest-neighbors respectively). It is easy to show that the interaction ðkÞ energy Eij affects the kinetics of the system only through first- and second-neighbor interchange energies ðkÞ
ðkÞ
ðkÞ
ðkÞ
wij ¼ 2Eij þ Eii þ Ejj :
ð13Þ
The differential equations describing the kinetics of the system were integrated numerically using the Runge-Kutta method. 3. Results We choose the interaction energies of the system ðkÞ as wmm0 ¼ 0:8. The system with such energy parameters is expected to exhibit ordering of the ordered phase O1 and no phase separation. Fig. 1 shows the variation of the coverage, the concentrations, the order parameters and the site probability Pms as a function of growth time at the temperature kB T ¼ 0:32. Fig. 1(a) displays that the coverage and the concentrations of each atomic species increase exponentially with the growth time in the initial stage of growth. As shown in Fig. 1(b), the ordering of A and B atoms onto different sublattices leads to opposite signs for the order parameters cA2 and cA3 . The initial stage of ordering involves very fast increase of the LRO parameters. When the layer is
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
Fig. 1. Concentrations and order parameters as a function of growth time at temperature kB T ¼ 0:32, sR =sd ¼ 103 : (a) curves for s coverage h and concentration xi ; (b) curves for cA1 ,cA2 and cA3 ; (c) curves for cP1 ,cP2 , cP3 and cP4 ; (d) curves for cAB i ; (e) curves for PA ; and (f) curves for PBs .
almost filled, the absolute values of the LRO parameters cA2 and cA3 are approximately equal to 1, which corresponds to an almost fully ordered phase. We have investigated the evolution of LRO and SRO with the variation of temperature. In the high temperature, the phase formed during depo-
sition become disordered due to the entropy effect. For layer growth scenario, there is kinetic freezing due to the low atomic jumping rates in the low temperature. Some partial ordered states and metastable disordered phase can be formed during a growth process. The occurrence of the metastable
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
179
Fig. 2. Concentrations and order parameters as a function of growth time at temperature kB T ¼ 0:27, sR =sd ¼ 103 : (a) curves for s coverage h and concentration xi ; (b) curves for cA1 , cA2 and cA3 ; (c) curves for cP1 , cP2 , cP3 and cP4 ; (d) curves for cAB i ; (e) curves for PA ; and (f) curves for PBs .
state depends on the difference of the two characteristic times. One characteristic time in the growth process is the adsorption rate and the other is the atomic jumping rate. Fig. 2 shows the variation of the coverage, the concentrations, the order parameters and the site probability Pms as a function of the growth time at the temperature kB T ¼ 0:27. From Fig. 2(a), we can see that the change rate of the concentrations becomes faster when tempera-
ture is decreased. Fig. 2(b) shows that there are two stages of evolution in the LRO during the growth process. In the first stage of the evolution, the absolute values of the LRO parameters cA2 and cA2 increase as the growth proceeds. However the absolute values of the order parameters cA2 and cA3 are not equal, which is different with the case of ordered phase O1 and leads to a transient ordered state. In the second stage, since the concentration
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
of vacancy decreases exponentially as growth proceeds, the diffusion rates for the atomic species A and B become very small due to low vacancy concentration in the late stage of the growth and the atomic configuration become frozen up. The order parameters reach stable values and the system relaxes to a metastable state. In the growth process, there are also two stages of the evolution of the LRO related to the correlation as shown in Fig. 2(c). Figs. 1(d) and 2(d) show the evolution of the SRO parameters at the temperature kB T ¼ 0:32 and kB T ¼ 0:27 respectively. The four SRO parameters cAB i should be fourfold degenerate for the disordered state in high temperature and nondegenerate for the ordered state. The evolution of SRO displays different features at the two temperatures. It can be seen that the SRO parameters cAB i are non-degenerate for the temperature kB T ¼ 0:32 while cAB is almost equal for the temperature i kB T ¼ 0:27. This feature of the SRO at the temperature kB T ¼ 0:27 is similar to that of disordered phase in the high temperature, which means that the freezing due to low diffusion rate made the metastable state come close to the disordered state. Smith and Zangwill [18] studied the ordering after the codeposition with different coverage for a square lattice on surface. They found that the SRO changes very rapidly to a quasi-equilibrium values followed by a stationary stage in which the value of the LRO parameters is still zero and the SRO is essentially a constant as a function of the time. The LRO starts to grow exponentially at a much later time. In our calculations, we consider the ordering during the codeposition. It can be seen that the LRO and SRO changes at almost same time during codeposition. Both the LRO and SRO involve a very fast increase because the ordering occurs only at certain coverage and deposition process speed up the initial ordering. We have investigated the effect of the adsorption rate and the atomic jumping rate on the kinetics of the system. Fig. 3 shows the variation of the LRO parameters of the system as a function of the ratio sR =sd between the atomic jumping rate and the adsorption rate. It can be seen that the adsorption rate has a significant effect on the phase configuration in a growth process. For a large adsorption rate, the LRO parameters of the phase
Fig. 3. LRO parameters as a function of ratio sR =sd between the atomic jumping rate and the adsorption rate, kB T ¼ 0:32: (a) curves for cA1 , cA2 and cA3 , and (b) curves for cP1 , cP2 , cP3 and cP4 .
formed by the growth process is small and the phase become disordered. When the adsorption rate decreases, the LRO increases and the phase become more ordered. This shows that decrease of adsorption rate has the similar effect on the LRO of the phase formed during the growth process as increase of temperature. In the case of higher adsorption rate, the deposition process becomes faster. The atoms have less time to adjust their configuration and the phase become more disordered. In case of higher atomic jumping rate, the atoms move faster and the system becomes more ordered due to faster relaxation. Therefore,
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
increase of the atomic jumping rate has the similar effect on the LRO of the phase formed during the growth process as decrease of the adsorption rate. Fig. 4 shows the variation of the LRO parameters as a function of temperature. From the figure, we can see that the LRO parameters approach to zero in low temperature, which means that the phase formed by the growth process becomes a frozen metastable disordered state due to the low atomic jumping rate in low temperature. When the temperature increases, the LRO parameters increase and approach steady values. In the transition region from the disordered state to the ordered state.
181
There is an intermediate phase in which the absolute values of the order parameters cA2 and cA3 are different and order parameters cP2 and cP3 related to the correlation functions become non-zero. This feature is different with that of order parameters of the ordered phase O1 for which the order parameters cP2 and cP3 are zero. In the high temperature, the absolute values of the atomic jumping rate is large enough for the system to become equilibrium. Then the order parameters cA2 and cA3 become equal while the order parameters cp2 and cp3 approaches zero. At the temperature kB T ¼ 1:12, there is an order–disorder phase transition. This transition is related to the surface order–disorder phase transition due to entropy induced disordering and does not depend strongly on the growth parameters such as the adsorption rate and the atomic jumping rate.
4. Summary
Fig. 4. LRO parameters as a function of temperature, sR =sd ¼ 103 : (a) curves for cA1 , cA2 and cA3 , and (b) curves for cP1 , cP2 , cP3 and cP4 .
We have studied the kinetics of ordering in alloys during codeposition. We consider monolayer codeposition of two atomic species onto an fcc (0 0 1) crystal surface. The system is described by the stochastic lattice gas model with nearest neighbor and next nearest-neighbor interactions. In epitaxial growth, the surface kinetic processes comprises the relaxation processes such as adsorption and evaporation, and surface diffusion processes. The adsorption and the evaporation are described by the single-site relaxation processes. The kinetics of ordering is described by the micro-master equation method in the pair approximation. The evolutions of the LRO and SRO parameters during codeposition are calculated. There are different types of kinetic path depending on the interaction and relaxation parameters. The kinetics of the growth is controlled by two kinds of characteristic times, the adsorption rate and the atomic jumping rates. It is found that there is transient ordered state during the codeposition. The occurrence of the transient state depends on the growth parameters. The kinetic path of the growth involves various stages of relaxation. We have investigated the effect of the adsorption rate and the atomic jumping rate on the kinetics of the
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
growth. For a low adsorption rate, the phase formed during the growth process is an ordered phase. With the increase of the adsorption rate, the LRO parameters become smaller and the phase become disordered. Increase of the atomic jumping rate has the similar effect on the LRO of the phase formed during the growth process as decrease of the adsorption rate. The evolution of the LRO and SRO parameters displays various features with the variation of temperature. The LRO parameters approach to zero in low temperature and the phase become a frozen metastable disordered state due to the low atomic jumping rate in low temperature. There is an intermediate phase in the transition region from the disordered state to the ordered state as temperature is increased. In high temperature, the atomic jumping rate is large enough for the system to become equilibrium and there is an order–disorder phase transition. This transition is related to the entropy induced disordering and does not depend on the growth parameters. Acknowledgements This research was supported by National Key Program of Basic Research development of China under Grant no. G2000067107) and the National Natural Science Foundation of China under Grant no. 19804007.
References [1] T.S. Kuan, T.F. Kuech, W.I. Wang, E.L. Wilkie, Phys. Rev. Lett. 54 (1985) 201. [2] G.B. Stringfellow, G.S. Chen, J. Vac. Sci. Technol. B 9 (1991) 2182. [3] J.E. Bernard, S. Froyen, A. Zunger, Phys. Rev. B 44 (1991) 11178. [4] A. Zunger, S. Mahajan, in: T.S. Moss, S. Mahajan (Eds.), Handbook on Semiconductors, vol. 3, Elsevier, Amsterdan, 1994, p. 1399. [5] B.L. Gu, Z.F. Huang, J. Ni, J.Z. Yu, K. Ohno, Y. Kawazoe, Phys. Rev. B 51 (1995) 7104. [6] C. Ern, W. Donner, H. Dosch, B. Adams, D. Nowikow, Phys. Rev. Lett. 85 (2000) 1926. [7] J.A. Pitney, I.K. Robinson, J.A. Vartaniants, R. Appleton, C.P. Flynn, Phys. Rev. B 62 (2000) 13084. [8] Y. Saito, H. M€ uler-Krumbhaar, J. Chem. Phys. 70 (1979) 1078. [9] R. Venkatasubramanian, J. Mater. Res. 7 (1992) 1235. [10] J.R. Smith, A. Zangwill, Phys. Rev. Lett. 76 (1997) 2097. [11] L.Q. Chen, A.G. Khachaturyan, Phys. Rev. B 44 (1991) 4681. [12] L. Reinhard, P.E.A. Turchi, Phys. Rev. Lett. 72 (1994) 120. [13] J. Ni, B.L. Gu, T. Ashino, S. Iwata, Phys. Rev. Lett. 79 (1997) 3922. [14] J. Ni, B.L. Gu, J. Chem. Phys. 113 (2000) 10272. [15] L.Q. Chen, J.A. Simmons, Acta Metall. Mater. 42 (1994) 2943. [16] A.S. Bakai, M.P. Fateev, Phys. Status Solidi B 158 (1990) 81. [17] G.H. Vineyard, Phys. Rev. 102 (1956) 981. [18] J.R. Smith, A. Zangwill, Surf. Sci. 316 (1994) 359.
Kinetics of ordering in fcc alloys during codeposition Jun Ni *, Binglin Gu Department of Physics, Tsinghua University, Beijing 100084, China Received 3 August 2001; accepted for publication 24 October 2001
Abstract The kinetics of ordering in fcc alloys during codeposition is investigated by the kinetic mean field method in the pair approximation. The evolutions of the long-range order and short-range order parameters during codeposition are calculated. It is found that the kinetic path involves various stages of relaxation and there is transient ordered state during codeposition. The influence of the two characteristic times related to the adsorption and the atomic migration on the kinetics of the growth is analyzed. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Models of surface kinetics; Non-equilibrium thermodynamics and statistical mechanics; Growth; Surface thermodynamics (including phase transitions); Alloys
1. Introduction Epitaxial growth has been used with layer by layer (such as in Frank–van der Merwe growth mode) control over growth of materials. It is found that III–V semiconductor alloys exhibit long-range order (LRO) [1–5] when grown by epitaxial method despite the fact that the bulk phase diagram predicts phase separation at growth temperature due to the changes in the interatomic interactions on the surface and the substrate constraint. Epitaxial metal alloy shows various ordering behavior [6,7]. The amount of LRO and its presence depend sensitively on growth condition. Saito and M€ uler-Krumbhaar [8] used a kinetic mean field theory in pair approximation to discuss the crystal growth. Venkatasubramanian [9] de-
*
Corresponding author. Fax: +86-10-6278-1604. E-mail address: [email protected] (J. Ni).
veloped the kinetic mean field model to treat the growth of an alloy with short-range order (SRO). Smith and Zangwill [10] used the kinetic mean field to study the interplay between compositional ordering and surface roughening during the epitaxial growth of a binary alloy. It was shown that transient ordered states can be formed in various systems during the relaxation of a non-equilibrium state to the equilibrium state [11–14]. The occurrence of these transient ordered states is due to the multiple characteristic times during the relaxation process. When the evolution of the system is controlled by the multiple relaxation rates, the system can first reach a kind of transient state through the kinetics process with faster relaxation rate. This transient state is a relative stable state. As other relaxation processes become functional, the system gradually relaxes to the equilibrium state. In the kinetics of ordering in alloys, there are multiple characteristic times for the atomic migration that control the relaxation of
0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 8 1 7 - 9
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
175
different atomic species, which leads to transient states on its relaxation path to the final equilibrium phase [13,14]. For epitaxial growth, in addition to the diffusion process, there are other two relaxation processes, i.e., adsorption and evaporation process. This can result in transient states in the growth process due to multiple relaxation times. In this paper, we study the kinetics of ordering in alloys during codeposition. The evolutions of the LRO and SRO parameters during codeposition are calculated. We will show that there are various types of the kinetic path depending on the growth parameters. It is found that there is transient ordered state in the system during the codeposition. Several features in the kinetics of alloy growth are illustrated. In Section 2 we describe the methods. Section 3 presents the results of the kinetics of ordering during codeposition. In Section 4 we make the conclusions.
adsorption rate of atoms onto a vacant site is given by
2. Methods
We use the micro-master equation method [15,16] to describe the kinetics of ordering and disordering in the system. In the master equation, the configurations of the phases in an alloy are described by a set of cluster distribution functions. Pa1 ;...;an ðr1 ; . . . ; rn ; tÞ is defined as the probability that at a given time t1 the n lattice sites r1 ; . . . ; rn be simultaneously occupied by n atoms of type a1 . . . an . These cluster distribution functions should satisfy the normalization conditions. The cluster distribution functions will change with time in the evolution from the initial state to the final state as the three relaxation processes, surface migration, adsorption and evaporation take place on the lattice. In the pair approximation, the configurations of phases at a given temperature is described by point and pair distribution functions denoted as Pa ðrÞ and Pa1 a2 ðr1 ; r2 Þ, where a and r represent the type of species and lattice site, respectively. We assume that surface atomic migration occurs by surface diffusion whereby an A or B atom jumps to an empty nearest-neighbor adsorption site. We denote by Rai vfxg ðri ; rj ; fxgÞ the rate at which the ai atom at site ri jumps to a neighboring vacancy at site rj under the influence
We consider a (1 0 0) surface codeposition layer of an fcc alloy and the system consists of three components, two species (A and B) of atoms and vacancy (V) on the surface. The system is described by the stochastic lattice gas model with the Hamiltonian XX 0 H¼ Emm0 ðri rj Þcmi cmj : ð1Þ fijg
mm0
The variable cmi ¼ 1 if the site i is occupied by an mð¼ A; B; V Þ atom and is zero otherwise. The notation fijg means the summation over the pairs of neighboring lattice sites. Emm0 ðri rj Þ represents the interaction energy between an m atom at ri and an m0 atom at rj . We consider the growth of a monolayer of atoms on a given planar substrate. The surface kinetic processes comprise the relaxation processes such as the adsorption, the evaporation and the diffusion process. The adsorption and the evaporation are described by the single-site relaxation processes [8]. The adsorption is assumed to be determined by the chemical potential lai . The
waai ¼
1 La e i; sR
ð2Þ
where Lai ¼ lai =kB T , and sR is a characteristic time constant. This form of the adsorption rate represents approximately the growth rate for the case that chemical potential lai is very large. The evaporation rate depends on the local configuration. The evaporation rate then is defined as 1 DEai weai ¼ exp ; ð3Þ sR kB T where DEai ¼ Eai Ev . Eai is the energy before ai atom is evaporated and Ev is the energy after the evaporation. The transition probabilities are required to satisfy the detailed balanced condition between evaporation and adsorption weai lai Peq ð. . . ; v; . . .Þ DEai ¼ exp ¼ : ð4Þ waai Peq ð. . . ; ai ; . . .Þ kB T kB T
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
of the set of the neighboring atoms fxg at the corresponding sites fxg. The rate of change of the pair distribution function is given by d Pa aj ðri ; rj ; tÞ dt i X X Rvai fxg ðri ; rk ; fxgÞPvaj ai fxg ðri ; rj ; rk ; fxgÞ ¼ k
fxg
k6¼j
XX k
k6¼j
Rai vfxg ðri ; rk ; fxgÞPai aj vfxg ðri ; rj ; rk ; fxgÞ
fxg
XX Rvaj fxg ðrj ; rk ; fxgÞPai vaj fxg ðri ; rj ; rk ; fxgÞ k
k6¼j
fxg
XX Raj vfxg ðrj ; rk ; fxgÞPai aj vfxg ðri ; rj ; rk ; fxgÞ k
k6¼i
fxg
DEaj DEai 1 exp þ sR kB T kB T 1 Laj Pai aj ðri ; rj ; tÞ þ e Pai v ðri ; rj ; tÞ sR 1 La þ e i Pvaj ðri ; rj ; tÞ; sR
1 exp sR
ð5Þ
d Pa v ðri ; rj ; tÞ dt i X Rvai fxg ðri ; rj ; fxgÞPvai fxg ðri ; rj ; fxgÞ ¼ fxg
X
Rai vfxg ðri ; rj ; fxgÞPai vfxg ðri ; rj ; fxgÞ
fxg
XX Rvai fxg ðri ; rk ; fxgÞPvvai fxg ðri ; rj ; rk ; fxgÞ k
k6¼j
fxg
XX Rai vfxg ðri ; rk ; fxgÞPai vvfxg ðri ; rj ; rk ; fxgÞ k
k6¼j
k
X
k
fxg
X 1 DEak Pai ak ðri ; rj ; tÞ exp sR kB T ak 1 DEai Pai v ðri ; rj ; tÞ exp kB T sR X 1 1 eLak Pai v ðri ; rj ; tÞ eLai Pai v ðri ; rj ; tÞ; þ s s R R ak
ð7Þ
ð8Þ Pa1 a2 ðr1 ; r2 Þ ¼ Pa2 ðr2 Þ:
a1
fxg
XX Rvak fxg ðrj ; rk ; fxgÞPai vak fxg ðri ; rj ; rk ; fxgÞ k6¼i
Pai aj Pa a Pa7 ai Pa8 aj Pa14 aj ; Pa7i Pa7j 1 i
a2
fxg
XX Rak vfxg ðrj ; rk ; fxgÞPai ak vfxg ðri ; rj ; rk ; fxgÞ k6¼i
Pai aj a1 a14 ¼
where the sites ri and rj are occupied by interchanging atom or vacancy. A substitution of Eq. (7) into Eqs. (5) and (6) gives the closed kinetic equations in the pair approximation. It is easy to see the uncoupling of Eq. (7) leads to the fulfilment of the superposition equation obtained by the cluster variation method (CVM) for the equilibrium state. The point and pair distributions satisfy the following relations: X Pa1 a2 ðr1 ; r2 Þ ¼ Pa1 ðr1 Þ;
and
represents the summation of k over all the nearestneighbor sites of rj except ri . Formula relevant to Pvaj ðri ; rj ; tÞ can be written down similarly. In the approximation of first- and second-nearest-neighbor interactions, the function Rai vfxg ðr1 ; r2 ; fxgÞ is dependent on the configuration of species fxg located on the first- and second-coordination sphere of a chosen site pair. In this case, the kinetic equation for the pair distribution function given on the nearest-neighboring lattice sites, contains 16-particle distribution functions because there are four nearest neighbors and four second nearest neighbors for each lattice site. To obtain the closed kinetic equations in the pair approximation, the 16-particle distribution functions should be expressed in terms of the pair and point probability distribution functions.
þ
ð6Þ
P where k6¼j represents the summation of k over Pall the nearest-neighbor sites of ri except rj and k6¼i
They are statements of the conservation of probability. The pair distributions also satisfy P normalization condition P ðr1 ; r2 Þ ¼ 1. a a 1 2 a1 a2 Therefore, the point and pair distribution functions are not independent. Among them, we need choose an independent set. All other point and pair distribution functions can be obtained from this independent set according to the normalization condition. Following Vineyard [17], the jumping rate of constituent atoms to the neighboring vacancy sites are given by
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
Rai cfxg ðri ; rj ; fxgÞ ¼
1 exp sd
U DE kB T 2kB T
;
ð9Þ
where 1=sd is the vibration frequency associated with atom–vacancy interchange (assumed identical for A and B atoms). U is the average barrier height for the atomic interchange. DE is the energy difference for the atom–vacancy interchange. In order to describe the atomic configurations of ordered structures on the square lattice, we divide the square lattice into four square sublattices a, b, d, c. In the point approximation, the symmetry of the atomic configurations is determined by the occupation probability Pms of m species (m ¼ A; B; V ) in the sublattice s(¼ a, b, d, c). The atomic configurations of the phases are also generally described by the following independent order parameters: cm1 ¼ Pma þ Pmb Pmc Pmd ; cm2 ¼ Pma Pmb ; cm3
¼
Pmc
ð10Þ
Pmd ;
where m ¼ A and B. The order parameter cm1 indicates the atomic configuration between the nearest-neighbor sites of species m. Similarly, other order parameters cm2 and cm3 show the atomic configurations between the second nearest-neighbor sites. The values of these parameters define the following phases with different symmetry: (1) disordered phase: cmi ¼ 0; (2) O1 : cm1 ¼ 0, cm2 6¼ 0, cm3 6¼ 0; (3) O2 : cm1 6¼ 0, cm2 ¼ cm3 ¼ 0. In addition to the above order parameters, there are other order parameters related to the pair distribution functions. For the sublattices chosen to characterize the ordered phases in ternary system, there are 22 independent pair distribution functions which are characterized by 10 LRO parameters and 12 SRO parameters. In the point approximation, we have three independent probability distribution functions for binary system and six independent parameters for ternary system. There are no SRO parameters describing correlations and all the independent probability distribution functions correspond to LRO parameters. In the pair approximation, there are correlations. In addition to the LRO parameters
177
described by Eq. (10), there are other four LRO parameters represented by following equations: ac ac PCA ; cP1 ¼ PAC ad ad cP2 ¼ PAC PCA ;
ð11Þ
bc bc cP3 ¼ PAC PCA ; bd bd cP4 ¼ PAC PCA :
Twelve SRO parameters are defined by following equations: 0
ad ad gmm 2 ¼ Pmm0 þ Pm0 m ;
0
0
bd bd gmm 4 ¼ Pmm0 þ Pm0 m ;
ac ac gmm 1 ¼ Pmm0 þ Pm0 m ;
0
bc bc gmm 3 ¼ Pmm0 þ Pm0 m ;
ð12Þ
where m 6¼ m0 . The coverage during the codeposition is defined as h ¼ CA þ CB , where CA and CB are the concentrations of A and B atoms respectively. The energy of the system are expressed in terms of the pairwise interactions. We consider the firstðkÞ and second-nearest-neighbor interactions Eij (k ¼ 1; 2 for the first- and second-nearest-neighbors respectively). It is easy to show that the interaction ðkÞ energy Eij affects the kinetics of the system only through first- and second-neighbor interchange energies ðkÞ
ðkÞ
ðkÞ
ðkÞ
wij ¼ 2Eij þ Eii þ Ejj :
ð13Þ
The differential equations describing the kinetics of the system were integrated numerically using the Runge-Kutta method. 3. Results We choose the interaction energies of the system ðkÞ as wmm0 ¼ 0:8. The system with such energy parameters is expected to exhibit ordering of the ordered phase O1 and no phase separation. Fig. 1 shows the variation of the coverage, the concentrations, the order parameters and the site probability Pms as a function of growth time at the temperature kB T ¼ 0:32. Fig. 1(a) displays that the coverage and the concentrations of each atomic species increase exponentially with the growth time in the initial stage of growth. As shown in Fig. 1(b), the ordering of A and B atoms onto different sublattices leads to opposite signs for the order parameters cA2 and cA3 . The initial stage of ordering involves very fast increase of the LRO parameters. When the layer is
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
Fig. 1. Concentrations and order parameters as a function of growth time at temperature kB T ¼ 0:32, sR =sd ¼ 103 : (a) curves for s coverage h and concentration xi ; (b) curves for cA1 ,cA2 and cA3 ; (c) curves for cP1 ,cP2 , cP3 and cP4 ; (d) curves for cAB i ; (e) curves for PA ; and (f) curves for PBs .
almost filled, the absolute values of the LRO parameters cA2 and cA3 are approximately equal to 1, which corresponds to an almost fully ordered phase. We have investigated the evolution of LRO and SRO with the variation of temperature. In the high temperature, the phase formed during depo-
sition become disordered due to the entropy effect. For layer growth scenario, there is kinetic freezing due to the low atomic jumping rates in the low temperature. Some partial ordered states and metastable disordered phase can be formed during a growth process. The occurrence of the metastable
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
179
Fig. 2. Concentrations and order parameters as a function of growth time at temperature kB T ¼ 0:27, sR =sd ¼ 103 : (a) curves for s coverage h and concentration xi ; (b) curves for cA1 , cA2 and cA3 ; (c) curves for cP1 , cP2 , cP3 and cP4 ; (d) curves for cAB i ; (e) curves for PA ; and (f) curves for PBs .
state depends on the difference of the two characteristic times. One characteristic time in the growth process is the adsorption rate and the other is the atomic jumping rate. Fig. 2 shows the variation of the coverage, the concentrations, the order parameters and the site probability Pms as a function of the growth time at the temperature kB T ¼ 0:27. From Fig. 2(a), we can see that the change rate of the concentrations becomes faster when tempera-
ture is decreased. Fig. 2(b) shows that there are two stages of evolution in the LRO during the growth process. In the first stage of the evolution, the absolute values of the LRO parameters cA2 and cA2 increase as the growth proceeds. However the absolute values of the order parameters cA2 and cA3 are not equal, which is different with the case of ordered phase O1 and leads to a transient ordered state. In the second stage, since the concentration
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
of vacancy decreases exponentially as growth proceeds, the diffusion rates for the atomic species A and B become very small due to low vacancy concentration in the late stage of the growth and the atomic configuration become frozen up. The order parameters reach stable values and the system relaxes to a metastable state. In the growth process, there are also two stages of the evolution of the LRO related to the correlation as shown in Fig. 2(c). Figs. 1(d) and 2(d) show the evolution of the SRO parameters at the temperature kB T ¼ 0:32 and kB T ¼ 0:27 respectively. The four SRO parameters cAB i should be fourfold degenerate for the disordered state in high temperature and nondegenerate for the ordered state. The evolution of SRO displays different features at the two temperatures. It can be seen that the SRO parameters cAB i are non-degenerate for the temperature kB T ¼ 0:32 while cAB is almost equal for the temperature i kB T ¼ 0:27. This feature of the SRO at the temperature kB T ¼ 0:27 is similar to that of disordered phase in the high temperature, which means that the freezing due to low diffusion rate made the metastable state come close to the disordered state. Smith and Zangwill [18] studied the ordering after the codeposition with different coverage for a square lattice on surface. They found that the SRO changes very rapidly to a quasi-equilibrium values followed by a stationary stage in which the value of the LRO parameters is still zero and the SRO is essentially a constant as a function of the time. The LRO starts to grow exponentially at a much later time. In our calculations, we consider the ordering during the codeposition. It can be seen that the LRO and SRO changes at almost same time during codeposition. Both the LRO and SRO involve a very fast increase because the ordering occurs only at certain coverage and deposition process speed up the initial ordering. We have investigated the effect of the adsorption rate and the atomic jumping rate on the kinetics of the system. Fig. 3 shows the variation of the LRO parameters of the system as a function of the ratio sR =sd between the atomic jumping rate and the adsorption rate. It can be seen that the adsorption rate has a significant effect on the phase configuration in a growth process. For a large adsorption rate, the LRO parameters of the phase
Fig. 3. LRO parameters as a function of ratio sR =sd between the atomic jumping rate and the adsorption rate, kB T ¼ 0:32: (a) curves for cA1 , cA2 and cA3 , and (b) curves for cP1 , cP2 , cP3 and cP4 .
formed by the growth process is small and the phase become disordered. When the adsorption rate decreases, the LRO increases and the phase become more ordered. This shows that decrease of adsorption rate has the similar effect on the LRO of the phase formed during the growth process as increase of temperature. In the case of higher adsorption rate, the deposition process becomes faster. The atoms have less time to adjust their configuration and the phase become more disordered. In case of higher atomic jumping rate, the atoms move faster and the system becomes more ordered due to faster relaxation. Therefore,
J. Ni, B. Gu / Surface Science 499 (2002) 174–182
increase of the atomic jumping rate has the similar effect on the LRO of the phase formed during the growth process as decrease of the adsorption rate. Fig. 4 shows the variation of the LRO parameters as a function of temperature. From the figure, we can see that the LRO parameters approach to zero in low temperature, which means that the phase formed by the growth process becomes a frozen metastable disordered state due to the low atomic jumping rate in low temperature. When the temperature increases, the LRO parameters increase and approach steady values. In the transition region from the disordered state to the ordered state.
181
There is an intermediate phase in which the absolute values of the order parameters cA2 and cA3 are different and order parameters cP2 and cP3 related to the correlation functions become non-zero. This feature is different with that of order parameters of the ordered phase O1 for which the order parameters cP2 and cP3 are zero. In the high temperature, the absolute values of the atomic jumping rate is large enough for the system to become equilibrium. Then the order parameters cA2 and cA3 become equal while the order parameters cp2 and cp3 approaches zero. At the temperature kB T ¼ 1:12, there is an order–disorder phase transition. This transition is related to the surface order–disorder phase transition due to entropy induced disordering and does not depend strongly on the growth parameters such as the adsorption rate and the atomic jumping rate.
4. Summary
Fig. 4. LRO parameters as a function of temperature, sR =sd ¼ 103 : (a) curves for cA1 , cA2 and cA3 , and (b) curves for cP1 , cP2 , cP3 and cP4 .
We have studied the kinetics of ordering in alloys during codeposition. We consider monolayer codeposition of two atomic species onto an fcc (0 0 1) crystal surface. The system is described by the stochastic lattice gas model with nearest neighbor and next nearest-neighbor interactions. In epitaxial growth, the surface kinetic processes comprises the relaxation processes such as adsorption and evaporation, and surface diffusion processes. The adsorption and the evaporation are described by the single-site relaxation processes. The kinetics of ordering is described by the micro-master equation method in the pair approximation. The evolutions of the LRO and SRO parameters during codeposition are calculated. There are different types of kinetic path depending on the interaction and relaxation parameters. The kinetics of the growth is controlled by two kinds of characteristic times, the adsorption rate and the atomic jumping rates. It is found that there is transient ordered state during the codeposition. The occurrence of the transient state depends on the growth parameters. The kinetic path of the growth involves various stages of relaxation. We have investigated the effect of the adsorption rate and the atomic jumping rate on the kinetics of the
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J. Ni, B. Gu / Surface Science 499 (2002) 174–182
growth. For a low adsorption rate, the phase formed during the growth process is an ordered phase. With the increase of the adsorption rate, the LRO parameters become smaller and the phase become disordered. Increase of the atomic jumping rate has the similar effect on the LRO of the phase formed during the growth process as decrease of the adsorption rate. The evolution of the LRO and SRO parameters displays various features with the variation of temperature. The LRO parameters approach to zero in low temperature and the phase become a frozen metastable disordered state due to the low atomic jumping rate in low temperature. There is an intermediate phase in the transition region from the disordered state to the ordered state as temperature is increased. In high temperature, the atomic jumping rate is large enough for the system to become equilibrium and there is an order–disorder phase transition. This transition is related to the entropy induced disordering and does not depend on the growth parameters. Acknowledgements This research was supported by National Key Program of Basic Research development of China under Grant no. G2000067107) and the National Natural Science Foundation of China under Grant no. 19804007.
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