PHYSICAL REVIEW B
VOLUME 58, NUMBER 22
1 DECEMBER 1998-II
Diffusion in generalized lattice-gas models
O. M. Braun
Institute of Physics, Ukrainian Academy of Sciences, UA-252022 Kiev, Ukraine
C. A. Sholl
Department of Physics and Electronics Engineering, University of New England, Armidale, NSW 2351, Australia
~Received 10 March 1998!
A technique has been developed to calculate the exact diffusion tensor for the lattice-gas model in the
low-concentration limit for any complex elementary cell. The analysis takes into account the detailed structure
of the elementary cell and the structure of the diffusion barrier. The technique is also applicable to nonMarkoff diffusion and, consequently, can include polaronic and inertia effects. The calculation reduces to
simple matrix-algebraic operations which may be carried out analytically in many cases. Applications to some
one- and two-dimensional models are described, including disordered systems and non-Markoff diffusion, in
structures with complex elementary cells. @S0163-1829~98!04346-X#
I. INTRODUCTION
Diffusion is a fundamental physical process which is of
significant practical importance in applications such as crystal growth, heterogeneous catalytic reactions, emission electronics, microelectronics, etc. ~see, for example, Refs. 1,2,
and references therein!. One widely used approach to diffusion is based on the lattice-gas model. This model assumes
that atoms occupy fixed lattice sites and can undergo jumps
to other vacant sites with some probability. When the system
consists of only a single atom which undergoes a random
walk on the lattice a key variable of the model is the conditional ~joint! probability g(l;t u l 0 ;t 0 ), which is the probability that an atom that was at the site l 0 at time t 0 , is at the
lattice site l at a later time t. The standard approach to the
diffusional problem assumes that the random walk of an
atom corresponds to a Poisson-Markoff process, so that the
distribution g(l;t u l 0 ;t 0 ) obeys the master equation ~e.g., see
Refs. 3,4!
]
g ~ l;t u l 0 ;t 0 ! 5
@ g l;l 8 g ~ l 8 ;t u l 0 ;t 0 ! 2 g l 8 ;l g ~ l;t u l 0 ;t 0 !#
]t
l8
(
52
(l D l;l 8g ~ l 8 ;t u l 0 ;t 0 ! ,
8
~1!
where g l;l 8 is the probability of the transition l 8 →l per unit
time, and D l;l 8 5 d ll 8 ( l 9 g l 9 ;l 8 2 g l;l 8 is the transfer matrix.
The matrix D l;l 8 should satisfy the constraint ( l D l;l 8 50,
which follows from the conservation of atoms. The jump
rates g l;l 8 are usually related to the corresponding energy
barriers through the Arrhenius law.
Because the dynamics of the lattice-gas model is purely
relaxational, it is useful to take the Laplace transform with
respect to time,
ḡ ~ z ! 5
E
`
0
dt e 2zt g ~ t ! ,
Re~ z ! .0,
~2!
so that Eq. ~1! reduces to
0163-1829/98/58~22!/14870~10!/$15.00
PRB 58
(l ~ z d ll 81D l;l 8 ! ḡ ~ l 8 ;l 0u z ! 5g ~ l;t 0u l 0 ;t 0 ! ,
8
~3!
where the right-hand side of Eq. ~3! incorporates the initial
condition.
For a periodic array of sites, it is convenient also to perform the spatial Fourier transform, g̃(k)5 ( lexp@ik(l
2l0 ) # g(l;l0 ), where the wave vector k is in the first Brillouin zone. Assuming the initial condition g(l;t 0 u l 0 ;t 0 )
5 d ll 0 , Eq. ~3! becomes
@ z1D̃ ~ k!# ˜ḡ ~ k;z ! 51.
~4!
Equation ~4! has a trivial solution which exhibits diffusional motion, i.e., in the limit t→` the mean-square atomic
displacement is proportional to time. In particular, in the
simplest model where the atom can jump only to the nearestneighboring sites with a rate g , and all jump directions are
equivalent ~i.e., the model is isotropic!, the result is ^ r 2 &
[ ^ x 2 1y 2 & .2 n Dt for t→`, where n is the dimensionality
of the system. The diffusion coefficient D in this case is
equal to D5qa 2 g , where q5 k /2n is the ‘‘geometrical’’ factor, and k is the coordination number ~the number of neighbor sites allowed for a jump!. Throughout this paper we will
only consider diffusion in one and two dimensions explicitly,
often with surface diffusion in mind, but the techniques discussed are equally applicable to three-dimensional systems.
The above approach has been studied in detail by Kutner
and Sosnowska5 and Kehr et al.;6 see also the review articles
of Dieterich et al.7 and Haus and Kehr.8 It may be extended
straightforwardly to describe anisotropic lattices and to include jumps to more distant neighbors. In addition the
method has been generalized to describe non-Markoff
processes,9,8 where the probability of a jump depends on the
history of previous jumps.
This theory has assumed that the elementary cell of the
lattice is primitive. Often, however, the lattice is characterized by a unit cell with several inequivalent sites. Although
there are some particular studies of models with complex
14 870
©1998 The American Physical Society
PRB 58
DIFFUSION IN GENERALIZED LATTICE-GAS MODELS
unit cells,5,6,8 there is no general solution to this problem.
The aim of the present work is to develop a systematic and
practical method for evaluating the diffusion tensor for the
low-concentration limit of the lattice gas model for any complex unit cell. The approach of Bookout and Parris,11 which
uses the mapping of a random-walk problem onto the
imaginary-time Schrödinger equation, is a possible approach
but it seems to be more complicated than the method described below, and it does not allow a direct generalization to
incorporate non-Markoff effects.
A quite direct method of obtaining the diffusion tensor for
Bravais lattices is to take the continuum limit of Eq. ~1! ~see,
for example, Ref. 10!. This leads to the diffusion equation
with the diffusion tensor given by
1
Di j5
2
~5!
where the summation is over the k allowed jumps and l a ,i is
the i Cartesian component of the jump of length l a which
occurs with frequency g a . However, for complex unit cells
the continuum limit of Eq. ~1! gives a set of coupled partial
differential equations, rather than a single diffusion equation,
because of the inequivalence of sites in a cell. It is not therefore possible simply to deduce the diffusion tensor in the
same way as for the Bravais lattice case.
Another useful approach to diffusion is to rewrite the
long-time equation ^ x 2 & 52Dt in the form
2D5
^ x 2&
t
5
II. TECHNIQUE
A. Non-Bravais lattice
The sites in a periodic structure with s sites per unit cell
may be labeled by two indices l and a , where l labels the
elementary cell and a labels the different sites within the
cell, a 51, . . . ,s. In this case Eq. ~1! will have the index l
replaced by two indices l and a , so that the master equation
for the joint probability takes the form
]
g ~ l, a ;t u l0 , a 0 ;t 0 !
]t
5
k
( l a ,i l a , j g a ,
a 51
^ x 2& N
N
t
14 871
( @ g l, a ;l8, a 8 g ~ l8 , a 8 ;t u l0 , a 0 ;t 0 !
l8 , a
2 g l8 , a 8 ;l, a g ~ l, a ;t u l0 , a 0 ;t 0 !# .
~7!
The limit t→` of the solution of Eq. ~7! must be the
equilibrium distribution,
lim g ~ l, a ;t u ••• ! 5N 21
lat r a ,
~8!
t→`
where r a is the average thermal population of the site a ,
and the factor N 21
lat (N lat is the total number of unit cells in
the system! is introduced for convenience.
In order to define the parameters g l, a ;l8 , a 8 in Eq. ~7!, we
associate with each site within the unit cell an energy « a .
The equilibrium occupation numbers should satisfy the Boltzmann distribution
s
5a 20 f corr^ g & ,
~6!
where a 0 is the mean distance between the sites, N is the
total number of jumps of the atom in the time t, ^ g & is the
average jump rate, and f corr is the correlation factor defined
as f corr5 ^ x 2 & /(Na 20 ). For a system with a complex unit cell
f corr<1, and the equality f corr51 only holds for a Bravais
lattice. The average jump rate ^ g & is straightforward to calculate for a complex structure, but the correlation factor f corr
is not. A calculation of the diffusion tensor, as described
below, combined with a calculation of the average jump rate
provides a means of obtaining the correlation factor through
Eq. ~6!.
The method developed here to find the diffusional tensor
for a complex unit cell is based on the properties of the
solution of an equation of the form of Eq. ~4! in the small k
limit. The problem is reduced to a set of algebraic equations,
which in many cases can be solved analytically, and in any
case may simply be solved with a computer. In addition, the
approach may easily be extended to study non-Markoff processes as well as Langmuir-type lattice-gas models.
The paper is organized as follows. The technique is described in Sec. II. Then in Sec. III it is applied to the onedimensional lattice, where we obtain the exact result for any
lattice structure, including disorder. In Sec. IV we consider
some applications to non-Markov processes, including the
triangular lattice and the ‘‘split’’ ~110! surface of the body
centered cubic crystal. Finally, Sec. V concludes the paper
with a short discussion of the results.
r a 5Z
21
exp~ 2 b « a ! ,
Z5
(
a 51
exp~ 2 b « a ! ,
~9!
where b 51/k B T, T is the temperature, k B is Boltzmann’s
constant, and the normalization factor Z corresponds to one
particle per elementary cell
s
(
a 51
r a 51.
~10!
The simplest ~but not unique! way to satisfy the condition
~8! is to explore the detailed balance condition ~e.g., see Ref.
12! g l, a ;l8 , a 8 r a 8 5 g l8 , a 8 ;l, a r a , which leads to the following
constraint for possible values of the rate parameters:
g l, a ;l8 , a 8 5 g l8 , a 8 ;l, a ~ r a / r a 8 !
5 g l8 , a 8 ;l, a exp@ b ~ « a 8 2« a !# .
~11!
Otherwise the parameters g l, a ;l8 , a 8 are arbitrary; their values
are determined by the activation energies of the transitions.
The transfer matrix becomes
D l, a ;l8 , a 8 5 d ll8 d aa 8
(
l9 , a 9
g l9 , a 9 ;l8 , a 8 2 g l, a ;l8 , a 8 .
~12!
From the periodicity of the lattice it follows that D l, a ;l8 , a 8
5D l2l8 , a ;0,a 8 , and we may perform the Fourier transform
with respect to the cell index to obtain
D̃ a ; a 0 ~ k! 5
(l exp@ ik~ l2l0 !# D l, a ;l , a
0
0
,
~13!
14 872
O. M. BRAUN AND C. A. SHOLL
g̃ a ; a 0 ~ k,t ! 5
(l exp@ ik~ l2l0 !# g ~ l, a ;t u l0 , a 0 ;0 ! .
~14!
Taking the Laplace transform, with the initial condition that
the atom was at the site a 0 within the cell l0 at t50
g ~ l, a ;0 u l0 , a 0 ;0 ! 5 d ll0 d aa 0 ,
@ z d aa 8 1D̃ a ; a 8 ~ k!#˜ḡ a 8 ; a ~ k,z ! 5 d aa .
(
a
0
8
0
~16!
B. Eigenvalue problem
To solve Eq. ~16! it is convenient to introduce a new
matrix L(k) with elements
L a ; a 8 ~ k! 5 r a21/2D̃ a ; a 8 ~ k! r a 8 ,
1/2
~17!
so that Eq. ~16! takes the form
21/2
@ z d aa 8 1L a ; a 8 ~ k!# @ r a 8 ˜ḡ a 8 ; a ~ k,z ! r a1/2# 5 d aa .
(
a
0
8
transfer matrix cannot be transformed to the Jordan form!.
The lowest eigenvalue will be denoted l 1 (k).
~c! The eigenvector u(1) (k) is a continuous function of k
around the point k50, and for small values of k the eigenvalue l 1 (k) has the expansion
~15!
we obtain the transformed rate equation
0
l 1 ~ k! .D xx k 2x 12D xy k x k y 1D y y k 2y ,
The matrix L(k) for each k is the Hermitian square s3s
matrix, L a ; a 8 (k)5L a*8 ; a (k).
The eigenvalues and eigenvectors of L(k) are the solutions of
C. General solution
To solve Eq. ~19! for the lowest eigenvalue, let us rewrite
it in the form
L~ k! v~ k! 5l 1 ~ k! v~ k! ,
L a ; a 8 ~ k! u a 8 ~ k! 5l s ~ k! u a
(
a
8
~ k! ,
L~ k! .L0 1L1 k x 1 21 L2 k 2x ,
v~ k! .v0 1v1 k x 1 21 v2 k 2x ,
L0 5 lim L~ k! ,
~21!
(s @ u ~as !~ k!# * u ~as8!~ k! 5 d aa 8 .
~22!
(s
u ~as ! ~ k!@ u ~as ! ~ k!# *
0
z1l s ~ k!
r a21/2
,
0
~23!
which may be verified by direct substitution of Eq. ~23! into
Eq. ~18!.
It has been proved5,12 that the eigenvalues l s (k) satisfy
the following conditions.
~a! All l s (k)>0, and l s (k) is an even function of k.
~b! One, and only one, eigenvalue l s (k) is equal to zero
when the vector k is equal to zero ~assuming that the lattice
cannot be separated into independent sublattices, so that the
~26!
~27!
L1 5 lim
k→0
]2
] k 2x
]
L~ k! ,
]kx
~28!
L~ k! .
Substitution of Eqs. ~26!, ~27!, and ~24! into Eq. ~25! yields
~ L0 1L1 k x 1 21 L2 k 2x !~ v0 1v1 k x 1 21 v2 k 2x !
5D xx k 2x ~ v0 1v1 k x 1 21 v2 k 2x !
Using the eigenvectors us (k) and the eigenvalues l s (k), the
solution of Eq. ~18! can be expressed as
r a1/2
k x →0, k y 50,
k→0
L2 5 lim
~20!
(a @ u ~as !~ k!# * u ~as 8 !~ k! 5 d ss 8 ,
k x →0, k y 50,
where
s 51, . . . ,s.
Because the matrix L is Hermitian, the eigenvectors u( s )
form a complete orthonormal basis
˜ḡ
a ; a 0 ~ k,z ! 5
~25!
where v is an unnormalized eigenvector. Consider diffusion
along the x axis, i.e. put k y 50, and evaluate Eq. ~25! in the
limit k x →0. The Taylor expansions of L(k) and v(k) are
k→0
~s!
~24!
As shown below, the coefficients D••• in Eq. ~24! are the
corresponding diffusion coefficients. Thus, the calculation of
diffusion coefficients reduces to evaluating the k→0 expansion of the lowest eigenvalue l 1 (k).
The above theory has followed the work of Kutner and
Sosnowska5 and Kehr et al.6 The following analysis describes a method which allows the coefficients D••• to be
found in general.
~19!
where u(k)[ $ u a (k) % is a vector with s components. Labeling the s eigenvalues by the index s , Eq. ~19! may be rewritten as
~s!
k→0.
0
~18!
L~ k! u~ k! 5l ~ k! u~ k! ,
PRB 58
~29!
which is satisfied if
L0 v0 50,
~30!
L0 v1 1L1 v0 50,
~31!
L0 v2 12L1 v1 1L2 v0 52D xx v0 .
~32!
Multiplying both sides of Eq. ~32! on the left by
and
taking into account that v†0 L0 50 according to Eq. ~30! and
that the matrix L0 is Hermitian, we get
v†0 ,
1
D xx 5v*
0 ~ L1 v1 1 2 L2 v0 ! / ~ v*
0 v0 ! .
~33!
The normalized vector v0 is
~ v0 ! a 5 r a1/2 ,
~34!
PRB 58
DIFFUSION IN GENERALIZED LATTICE-GAS MODELS
since it satisfies the normalization condition and Eq. ~30!:
L0 v0 5
8
(
(l
a
9
2 r a21/2
9
2D
g l9 2l8 , a 9 ;0,a r a
(
(l
a
z2
L0 v1 52L1 v0 ,
k→0
52i
~35!
(
a,a
8
x
k→0
S
8
(
a,a8
˜ḡ
a ; a 8 ~ k;z ! r a 8 ,
~40!
Substituting the expression ~23! for ˜ḡ into the right-hand side
of Eq. ~40! and taking into account Eq. ~24!, the normalization ~10!, and also the equation
lim u~ 1 ! ~ k! 5v0 5 $ r a1/2% ,
~41!
k→0
we finally obtain D5D xx .
To find the diffusion coefficient in an arbitrary direction,
we need to use the complete expansions instead of the expansions ~26! and ~27!:
~42!
1 yy 2
2
xy
v~ k! .v0 1vx1 k x 1v1y k y 1 21 vxx
2 k x 1v2 k x k y 1 2 v2 k y ,
D
k→0,
g 0,a 8 ;2l x ,l y , a r a 50,
where we have also used Eqs. ~17!, ~13!, ~12!, and ~11!.
Thus, the problem reduces to the calculation of the matrices L0 , L1 , and L2 defined by Eqs. ~28!, the solution of the
system of (s21) linearly independent equations ~35!, and
finally the calculation of D xx as
D xx 5v*
0 L1 v1 1 v*
0 L2 v0 .
] k 2x
k→0,
g l x ,l y , a ;0,a 8 r a 8
1
2
]2
1
2
xy
yy 2
L~ k! .L0 1Lx1 k x 1L1y k y 1 21 Lxx
2 k x 1L2 k x k y 1 2 L2 k y ,
1/2
r a1/2 L a ; a 8 ~ k ! r a 8
(l l(.0 l x a(, a
y
2
(
a,a8
~39!
z→0.
which is a set of s linear equations for the components of v1 .
If v1 is a solution of Eq. ~35! then v1 1cv0 , where c is an
arbitrary constant, will also be a solution due to Eq. ~30!.
Consequently, one component of the vector v1 , for example
(v1 ) 1 , may be chosen arbitrarily. This choice does not effect
D xx because in Eq. ~33!
]
]kx
52 lim
g l8 2l, a 8 ;0,a r a 50,
8
where we have used Eqs. ~17!, ~13!, ~11!, and ~12!.
The vector v1 can be found by solving Eq. ~31!
v†0 L1 v0 5 lim
^ x 2 ~ t ! & .2Dt as t→`.
Equating the right-hand sides of Eqs. ~38! and ~39! and taking the Laplace transforms, we get
1/2
L a ; a 8~ 0 ! r a 8
(
a
5 r a21/2
14 873
~43!
2
where Lxy
2 5limk→0 ] L(k)/ ] k x ] k y , etc.
Instead of Eqs. ~30! to ~32! we now get the equations
L0 v0 50,
~44!
L0 vx1 52Lx1 v0 ,
L0 v1y 52L1y v0 ,
~36!
~45!
and
D. Diffusion coefficient
It remains to prove that D xx in Eq. ~36! is indeed the
diffusion coefficient along the x axis. To prove this, recall
that according to the initial condition ~15!, the expression
( a , a 0 g(l, a ;t u l0 , a 0 ;0) r a 0 is equal to the probability that
the atom at the origin in the cell l0 at t50, where it occupies
the sites a 51, . . . ,s with probabilities r a so that
( a , a 0 g(l, a ;0 u l0 , a 0 ;0)5 d ll0 , will be found in any of the
sites in the cell l at a later time t. Consequently, the meansquare displacement along the x axis is equal to
^ x 2~ t ! & 5 (
l
(
a,a
8
~ l2l0 ! 2x g ~ l, a ;t u l0 , a 8 ;0 ! r a 8 . ~37!
Using Eq. ~14!, this may be rewritten as
^ x ~ t ! & 52 lim
2
]2
(
2
k→0 ] k x a , a 8
g̃ a ; a 8 ~ k;t ! r a 8 .
~38!
On the other hand, according to the definition of the diffusion coefficient,
1 xx
1
L2 v0 1Lx1 vx1 1 L0 vxx
2 5D xx v0 ,
2
2
y x
x y
xy
Lxy
2 v0 1L1 v1 1L1 v1 1L0 v2 52D xy v0 ,
~46!
1 yy
1
L v 1L1y v1y 1 L0 v2y y 5D y y v0 .
2 2 0
2
Multiplying Eqs. ~46! by v†0 from the left, using the fact that
L0 is Hermitian so that v†0 L0 50, and taking into account
the normalization, we obtain
1
D xx 5 v*
Lxx v 1v* Lx vx ,
2 0 2 0 0 1 1
1
Ly y v 1v* Ly vy ,
D y y 5 v*
2 0 2 0 0 1 1
1
1
D xy 5 v*
Lxy v 1 ~ v* Ly vx 1v* Lx vy ! .
2 0 2 0 2 0 1 1 0 1 1
~47!
14 874
PRB 58
O. M. BRAUN AND C. A. SHOLL
FIG. 1. Calculation of the anisotropic diffusion coefficient.
Consider the atomic displacement along the x 8 direction,
which is at an angle u to the x axis. From Fig. 1 it follows
that x 8 5r cos(a2u)5x cosu1y sinu, so that the mean-square
displacement along the x 8 direction is equal to
^ r 2u & 5 ^ ~ x 8 ! 2 &
k→0
HF
cos2 u
]2
]2
]kx
] k 2y
12 sin u cos u
1sin2 u
2
] ]
]kx ]ky
G(
a,a8
g̃ a , a 8 ~ k;t ! r a 8
J
~49!
Repeating the procedure described above, we obtain the anisotropic diffusion coefficient in the form
D ~ u ! 5D xx cos2 u 1D y y sin2 u 1D xy sin 2 u
xx
yy
xy
2
2
5 12 v*
0 @ L2 cos u 1L2 sin u 1L2 sin 2 u # v0
x x
y y
2
2
1v*
0 @ L1 v1 cos u 1L1 v1 sin u
1 ~ L1y vx1 1Lx1 v1y ! sin u cos u # .
g l, a ,m;l8 , a 8 ,m 8 g eq ~ a 8 ,m 8 !
5g eq ~ a ,m !
(
l8 , a 8 ,m 8
g l8 , a 8 ,m 8 ;l, a ,m ,
(m g eq~ a ,m ! 5N 21
lat r a .
~48!
.2D ~ u ! t as t→`.
(
l8 , a 8 ,m 8
~51!
and to test whether this solution satisfies the Boltzmann distribution
5 ^ x 2 & cos2 u 1 ^ y 2 & sin2 u 12 ^ xy & sin u cos u
52 lim
memory index could take two values m j 561, where the
value m j 521 corresponds to the atom arriving at site j from
the left-hand side, and m j 511 corresponds to arrival from
the right-hand side. Thus, in the non-Markoff case the transition rates depend on three indices l, a and m. Contrary to
the Markoff case, a proper choice of the values g l8 , a 8 ,m 8 ;l, a ,m
cannot now be guaranteed by the detailed balance condition
because this condition is not necessarily satisfied for a nonMarkoff process. In the non-Markoff case it is necessary to
find the stationary solution g eq ( a ,m) of the master equation
~50!
E. Jumps with a memory
We now consider the generalization of the technique to
non-Markoff processes ~e.g., see Ref. 9 and the survey of
Haus and Kehr,8 and references therein!; namely, to random
walks for which the memory is not lost after each step, but
only after a finite number of steps.
There are at least two reasons for such a generalization.
~a! Inertia effect. After a jump, the inertia of an atom
results in persistent motion in the same direction. Therefore,
the probability of a jump in the same direction, g f , may
exceed the probabilities of backward and sideways jumps.
~b! Polaronic effect. Just after an atomic jump, the surrounding substrate atoms do not immediately adjust to the
new position of the atom. As a result, the energy of the atom
in the new site will exceed the energy which it had in the old
site before the jump. This may result in the probability of the
backward jump, g b , exceeding the probabilities of the forward and sideways jumps.
To generalize the above technique to include memory effects, we introduce a ‘‘memory’’ index m j for each site j.
For example, in the case of the one-dimensional lattice, the
~52!
Another problem in the non-Markoff case is that the matrix L(k) is not Hermitian. This situation is analogous to that
for the Fokker-Planck equation, where the Smoluchowsky
equation corresponds to the Markoff case, while a more general Fokker-Planck-Kramers equation, which takes into account inertia effects, corresponds to the non-Markoff case.3,4
However, the technique described above was based on two
requirements: first, on the completeness of the base of eigenvectors of the matrix L0 which is satisfied provided L0 is
Hermitian at least in the limit k→0, and second, on the
second order expansion ~24! which follows from the law of
conservation of atoms. If both these factors are satisfied for a
non-Markoff process, the technique can still be applied to
find the diffusion tensor.
III. ONE-DIMENSIONAL LATTICE
As examples of applications, we begin with a onedimensional Markoff model with nearest neighbor jumps.
The sites may be enumerated in increasing order by an index
i, and the atom can jump to the left-hand and right-hand sites
with the rates g i71;i . The elementary cell of the lattice has
lattice constant a and consists of s sites enumerated by the
index a ( a 51, . . . ,s) with site energies « a . The site index
i may be written as i5ls1 a , where l enumerates the elementary cells. It is convenient to define the rates g̃ a , a 8 as
g̃ a 61; a 5 g l, a 61;l, a
if a 52,3, . . . ,s21,
g̃ 1;s 5 g l11,1;l,s ,
~53!
g̃ s;1 5 g l21,s;l,1
and it is assumed that they satisfy the detailed balance condition ~11!. It is also convenient to assume that the index a is
cyclic, so that the value a 5s11 is equivalent to a 51, and
a 50 to a 5s.
Applying the theory of Sec. II gives
PRB 58
DIFFUSION IN GENERALIZED LATTICE-GAS MODELS
1
v* L v 5a 2 g̃ 1;s r s ,
2 0 2 0
~54!
~ L1 v1 ! 1 5 ~ L1 ! 1;s w s 52a 2 g̃ 1;s r s AZ exp~ b « 1 /2! ,
~ L1 v1 ! a 50,
a 52, . . . ,s21,
~55!
~56!
where 1/G is the residence time averaged over all sites
s
1
1
5
.
G a 51 r a g̃ a 11; a
(
~57!
The diffusion coefficient ~36! is therefore
D5a G.
2
~58!
s
a 51
~59!
it follows that
S(
s
f 21
corr5
1
1
s a 51 r a g a 11; a
DF
s
G
1
r ~g
1 g a 21; a ! ,
2s a 51 a a 11; a
~60!
(
taking into account that a 0 5a/s.
The expressions ~58! and ~57! for the one-dimensional
model may be rewritten in the form
D5D f F,
~61!
where D f 5a 20 g 0 , a 0 is the mean distance between sites, g 0
is some average transition rate, and the dimensionless correlation factor F is defined by the expression
F 21 5 ^ e 2 b « a & ^ e b « a g 0 / g a 11; a & ,
~62!
where the operation ^ ••• & is defined as s 21 ( as 51 •••. Taking
the limit s→`, we can apply the expressions ~61!, ~62! to a
disordered lattice. The result ~61!, ~62! has been obtained
previously by Lyo and Richards15 using another technique.
According to Kramers’ theory,16 the transition rate g a 11,a
is mainly determined by the activation energy « a 11,a which
is to be overcome by the atom, and depends on the latter
exponentially, g a 11,a 5G 0 exp(2b«a11,a ) where the prefactor G 0 will be assumed independent of a . The activation
~63!
where d « a*11,a is the saddle energy relative to its mean and
d « a is the well energy relative to its mean, the correlation
factor ~62! may be rewritten as
F 21 5 ^ exp~ 2 b d « a ! & ^ exp~ b d « a*11; a ! & .
~64!
In this formulation the frequency g 0 in D f is
G 0 exp(2b^«a11,a & ), which is the jump rate for the mean
activation energy barrier.
If the fluctuations of the well energies are Gaussian so
that
Prob ~ d « a ! 5
The result ~58!, ~57! has been obtained previously by
Kehr et al.6 ~see also Refs. 13 and 8! for a simpler valley
model for which all barriers have the same height. The same
result has also been obtained for the general case recently
using a different technique ~see Ref. 18, and references
therein! but the present method is simpler and more general.
The result ~57! enables the calculation of the correlation
factor f corr for this one-dimensional model. Using the definition of f corr , the general expression for the diffusion coefficient given above, and calculating the average jump rate as
^ g & 5 ( r a ~ g a 11; a 1 g a 21; a ! ,
barrier is equal to the difference between the saddle energy
« a*11,a and the well energy « a . Writing the activation energy as
« a 11; a 5 ^ « a 11; a & 1 d « a*11; a 2 d « a ,
~ L1 v1 ! s 5 ~ L1 ! s;1 w 1 51a 2 G AZ exp~ b « s /2! ,
v0* L1 v1 5a 2 @ 2 g̃ 1;s r s 1G # .
14 875
1
d « A2 p
F
exp 2
~ d«a!2
2~ d« !2
G
,
~65!
where d « is the dispersion, and the saddle energies fluctuate
in the same manner but with the dispersion d « * , then
^ e 2b d«a& 5
Prob ~ d « a ! e
E
2bd«a
`
2`
d~ d«a!,
F
G
1
5exp ~ b d « ! 2 ,
2
~66!
and, consequently,
H
J
1
F5exp 2 b 2 @~ d « ! 2 1 ~ d « * ! 2 # ,1.
2
~67!
In particular, if we put d «' d « * 'k B T, we obtain F'e 21 ,
so that thermal fluctuations in the shape of the potential
lower the diffusion coefficient by a factor of e in the case of
the one-dimensional model. Mak et al.17 have shown that
Eqs. ~61! to ~62! are also true for a two-dimensional valley
model for which all the barriers have the same energy but the
well energies are random.
IV. NON-MARKOFF JUMPS
A. Bravais one-dimensional lattice
In order to include a memory of one previous step into the
theory, we associate with each site an index m561 such
that m521 for an atom arriving at a site from the left-hand
neighboring site, and m51 for arrival from the right-hand
site. Denoting the transition rate of an atomic jump in the
same direction as the previous jump by g f , and the rate of
the backward jump by g b , the transition rates take the following form:
g l,21;l21,21 5 g f ,
g l,21;l21,11 5 g b ,
g l,11;l11,11 5 g f ,
g l,11;l11,21 5 g b .
The Fourier transform of the transfer matrix is
~68!
O. M. BRAUN AND C. A. SHOLL
14 876
L~ k ! 5
5
S
S
L 22
L 21
L 12
L 11
PRB 58
D
~ g f 1 g b ! 2 g f e 2ika
2 g b e 2ika
2 g b e 1ika
~ g f 1 g b ! 2 g f e 1ika
D
,
~69!
and the matrices L0 , L1 and L2 are equal to
L0 5
S
L1 5ia
gb
2gb
2gb
gb
S
D
gf
gb
2gb
2g f
L2 5a 2
S
gf
gb
gb
gf
D
FIG. 2. Non-Markoff diffusion on the triangular lattice.
,
D
~70!
,
.
B. Triangular lattice
It can be seen that while the matrix L(k) is non-Hermitian,
the matrices L0 and L2 are Hermitian, and the technique of
Sec. II may be applied. It is easy to show that
v0 5
v1 5
A S D
1
1
,
2 1
~71!
SD
ia g f 1 g b 0
,
A2 g b 1
and the diffusion coefficient is now equal to9
1 ~ g f 1gb!g f
D5 a 2
.
2
gb
~72!
If we introduce a parameter h using the relations g f
5 g exp h and g b 5 g exp(2h), so that h determines the
L~ k! 5
1
strength of the memory of the previous step, then D may be
represented as the product of D f 5a 2 g ~the diffusion coefficient for uncorrelated random walks! times the correlation
factor F5cosh(h)exp(2h).
A second example of non-Markoff diffusion is the triangular lattice, for which the memory index m can take six
different values, m51, . . . ,6, corresponding to jumps from
the six nearest-neighboring sites. Taking into account the
symmetry of the lattice, we introduce four transition rates
g b , g b8 , g 8f and g f as shown in Fig. 2. Depending on the
model under investigation, the following variants may occur.
~a! The forward jump model, where g b 5 g 8b 5 g 8f , g f , so
that the atom has a larger-than-average probability of making
a transition in the same direction as the previous transition,
while the probability for a transition in any other direction is
reduced compared to the average value.
~b! The reduced reversal model, where g b , g 8b 5 g 8f
5 g f , so that the atom has a less-than-average probability of
returning to the site visited at the previous step.
~c! The backward jump model , where g b . g 8b 5 g 8f
5 g f , so that the atom has a larger-than-average probability
of returning to the previous site.
Using the notation shown in Fig. 2, after a long but
straightforward calculation we get for the transfer matrix
L(k) the expression
6g2g fe1
1
2 g 8f e 1
1
2 g b8 e 1
1
2 g be 1
1
2 g b8 e 1
1
2 g 8f e 1
1
2 g 8f e 12
2
6 g 2 g f e 12
2
2 g 8f e 12
2
2 g 8b e 12
2
2 g b e 12
2
2 g 8b e 12
2
2 g b8 e 22
2
2 g 8f e 22
2
6 g 2 g f e 22
2
2 g 8f e 22
2
2 g b8 e 22
2
2 g b e 22
2
2 g be 2
1
2 g 8b e 2
1
2 g 8f e 2
1
6g2g fe2
1
2 g 8f e 2
1
2 g 8b e 2
1
2 g b8 e 21
2
2 g b e 21
2
2 g b8 e 21
2
2 g 8f e 21
2
6 g 2 g f e 21
2
2 g 8f e 21
2
2 g 8f e 11
2
2 g 8b e 11
2
2 g b e 11
2
2 g 8b e 11
2
2 g 8f e 11
2
6 g 2 g f e 11
2
2
,
~73!
DIFFUSION IN GENERALIZED LATTICE-GAS MODELS
PRB 58
14 877
66
where e 6
1 5exp(6ikxa), e 2 5exp(6ikxa/26ik y a y ) and g
1
5 6 ( g f 12 g 8f 12 g 8b 1 g b ).
The vectors v0 and v1 are now equal to
SD
SD
1
1
v0 5
1
A6
1
1
,
1
1
~74!
1
FIG. 3. Non-Markoff model for the ‘‘split’’ ~110! bcc surface.
1/2
v1 5
ia g A6
21/2
3 g 8b 12 g b 1 g 8f
21
model, since polaronic and inertia effects could play a significant role, especially for jumps between the adsites within
a cell.
To include the memory of a previous jump, we associate
with each adsite a memory index m in addition to the index
a 51,2 enumerating the sites within the cell. Let m51 for an
atom arriving at a given site from an upper site, m521 for
arrival from a lower site, and m50 for arrival from the leftor right-hand sites. It is now necessary to introduce five transition rates g f , g 8f , g 9f , g b , and g 8b as shown in Fig. 3.
It is useful to introduce a single index j 5 a 2 1m11,
which varies from 1 to 6, instead of the two indices a and m.
Using the symmetry of the model, the stationary solution
r j [g eq ( a ,m) of the master equation ~51! is r 1 5 r 3 5 r 4
5 r 6 5 g 9f /2( g 8f 12 g 9f ) and r 2 5 r 5 5 g 8f /2( g 8f 12 g 9f ), so that
the vector v0 is equal to
,
21/2
1/2
and the isotropic diffusion coefficient is given by the expression
~75!
D5D f ~ 11F ! ,
where D f 5 a g is the diffusion coefficient for the Markoff
model, and F describes the non-Markoff correlation
3
2
2
F5
2 ~ g f 2 g b 1 g 8f 2 g 8b !
3 g 8b 12 g b 1 g 8f
~76!
.
C. The ‘‘split’’ „110… bcc surface
As a final example, we consider the ~110! face of the bcc
structure which gives a rhomboidal lattice for an adatom, and
suppose that owing to local substrate distortion around the
adatom, the original single adsorption site is split into two
symmetric sites separated by a small energy barrier. Such a
distorted honeycomb lattice was introduced and studied by
Ala-Nissila et al.14 to described the H-W(110) adsystem in
the framework of the Markoff random process. However, it
seems reasonable to include also the memory effects into this
L~ k! 5
1
v0 5
1
A2 ~ g 8f 12 g 9f !
SD
Ag 9f
Ag 8f
Ag 9f
Ag 9f
Ag 8f
Ag 9f
.
~77!
Omitting a lengthy calculation, the matrix L(k) is
3g
0
0
2 g f e 11
2 g̃ f e 11
2 g b e 11
0
3g8
0
2 g̃ f
2 g 8b
2 g̃ f
0
0
3g
2 g b e 12
2 g̃ f e 12
2 g f e 12
2 g f e 21
2 g̃ f e 21
2 g b e 21
3g
0
0
2 g̃ f
2 g 8b
2 g̃ f
0
3g8
0
2 g b e 22
2 g̃ f e 22
2 g f e 22
0
0
3g
2
,
~78!
O. M. BRAUN AND C. A. SHOLL
14 878
where g 5( g f 1 g 8f 1 g b )/3, g 8 5(2 g 9f 1 g 8b )/3, g̃ f 5 Ag 8f g 9f
and e 66 5exp(6ikxax6ikyay).
The solution of the key equation ~35! leads to the following expression for the vector v1 :
vx1 5
where
SD SD
w x1
w 1y
w x2
0
w x1
2w x1
,
v1y 5
2w 1y
w 1y
2w x2
0
2w x1
2w 1y
,
3 ~ g 9f 1 g 8b ! 2 g 8f g 9f # ,
~80!
w x2 5a x A g̃ f / @~ 2 g f 1 g 8f 12 g b !~ g 9f 1 g 8b ! 2 g 8f g 9f # , ~81!
w 1y 52a y A/ ~ g 8f 12 g b ! ,
~82!
and
A52i
~ g f 1 g 8f 1 g b ! Ag 9f
A2 ~ g 8f 12 g 9f !
~83!
.
As a result, we obtain the diffusion coefficients
D xx 5a 2x
g 8f g 9f ~ g f 1 g 8f 1 g b !~ 2 g 9f 1 g 8b !
,
~ g 8f 12 g 9f !@~ 2 g f 1 g 8f 12 g b !~ g 9f 1 g b8 ! 2 g 8f g 9f #
~84!
D y y 5a 2y
g 9f ~ g f 1 g 8f 1 g b ! 2
~ g 8f 12 g 9f !~ g 8f 12 g b !
,
~85!
and their ratio is
d5
S D
ax
D xx
5
Dyy
ay
3
d5
D xx
2g8
5
.
D y y g 8 12 g
~87!
V. CONCLUSION
A technique has been developed to calculate the diffusion
tensor for the lattice-gas model at low particle concentration
for any complex elementary cell and which can also include
memory effects. The calculation reduces to simple matrixalgebraic operations, with the only step that could hinder a
complete analytic solution being the solution of the matrix
equation ~35!. For many cases analytic expressions for the
diffusion tensor may be obtained. Some one- and twodimensional models were analyzed as examples, some of
which have been studied previously by other techniques,
while others such as the one-dimensional model with an arbitrary elementary cell and the two-dimensional nonMarkoff models, have not been treated previously. Although
we have concentrated on one- and two-dimensional examples, having in mind applications to surface diffusion, the
technique described is equally applicable three-dimensional
models, so that it will also be useful in applications to bulk
diffusion problems.5–9
The technique developed in the present work may be programed for a computer calculation in cases where an analytic
solution is not possible, so that the exact diffusion tensor for
walks of a single atom may be found in practice for a wide
class of lattice-gas models. Moreover, the technique allows
the study of diffusion of a pair of interacting atoms, or a trio
of atoms, etc., so that it may be used for the calculation of
the exact diffusion tensor for stochastic motion of adsorbed
islands.
2
ACKNOWLEDGMENTS
g 8f ~ g 8f 12 g b !~ 2 g 9f 1 g 8b !
~ g f 1 g 8f 1 g b !@~ 2 g f 1 g 8f 12 g b !~ g 9f 1 g 8b ! 2 g 8f g 9f #
.
~86!
1
For Markoff jumps this ratio reduces to
~79!
w x1 52a x A ~ g 9f 1 g b8 ! / @~ 2 g f 1 g 8f 12 g b !
PRB 58
A. G. Naumovets and Yu.S. Vedula, Surf. Sci. Rep. 4, 365
~1985!.
2
R. Gomer, Rep. Prog. Phys. 53, 917 ~1990!.
3
C. W. Gardiner, Handbook of Stochastic Methods ~SpringerVerlag, Berlin, 1983!.
4
H. Risken, The Fokker-Planck Equation ~Springer, Berlin, 1984!.
5
R. Kutner and I. Sosnowska, Acta Phys. Pol. A 47, 475 ~1975!; J.
Phys. Chem. Solids 38, 741 ~1977!.
6
K. W. Kehr, D. Richter, and R. H. Swendsen, J. Phys. F 8, 433
~1978!.
O. Braun was partially supported by a grant from the
Australian Government Department of Industry, Science and
Technology.
7
W. Dieterich, P. Fulde, and I. Peschel, Adv. Phys. 29, 527
~1980!.
8
J. W. Haus and K. W. Kehr, Phys. Rep. 150, 263 ~1987!.
9
J. W. Haus and K. W. Kehr, J. Phys. Chem. Solids 40, 1019
~1979!.
10
A.R. Allnatt and A.B. Lidiard, Atomic Transport in Solids ~Cambridge University Press, Cambridge, 1993!.
11
B. D. Bookout and P. E. Parris, Phys. Rev. Lett. 71, 16 ~1993!.
12
N.G. van Kampen, in Stochastic Processes in Chemistry and
Physics ~North-Holland, Amsterdam, 1981!.
PRB 58
13
DIFFUSION IN GENERALIZED LATTICE-GAS MODELS
J. W. Haus, K. W. Kehr, and J. W. Lyklema, Phys. Rev. B 25,
2905 ~1982!.
14
T. Ala-Nissila, J. Kjoll, S. C. Ying, and R. A. Tahir-Kheli, Phys.
Rev. B 44, 2122 ~1991!.
15
S. K. Lyo and P. M. Richards, Phys. Rev. B 32, 301 ~1985!.
16
H. A. Kramers, Physica ~Amsterdam! 7, 284 ~1940!; P. Hänggi,
14 879
P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 ~1990!.
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4052 ~1988!.
18
K. W. Kehr and T. Wichmann, in Disordered Materials, edited
by D. K. Chaturvedi and G. E. Murch ~Transtec. Switzerland,
1996!, p. 151.
17