- Email: [email protected]
Lattice distortion of an fcc crystal near a free surface
SURFACE
SCIENCE 21(1970) 109-122 0 North-Holland
LATTICE DISTORTION
Publishing Co.
OF AN fee CRYSTAL
NEAR
A FREE SURFACE A. CORCIOVEI,
M. CROITORU
and D. GRECU
Institute for Atomic Physics, Bucharest, Romania Received 17 September 1969 The problem of lattice distortion near a free surface is discussed for an fee crystal using the general invariance and symmetry properties of the crystal potential energy. The atomic force tensors were determined assuming a C4” symmetry in the surface region, when the atoms are in the ideal lattice configuration. All the anharmonic effects are neglected. From invariance requirements it results that the static displacements are directed along the normal to the surface. They can be calculated using the Green’s function method. Detailed calculations were done for a finite slab in the nearest neighbour approximation.
1. Introduction In a number of theoretical studies it was tried to calculate the lattice distortion near a free surface of a crystal. Theoretical calculations have been done for molecular and ionic crystals using explicit expressions for the two body interaction potential between the atoms1-3). Here the new equilibrium positions are determined from the minimum condition of the surface energy. Deformations normal to the crystal surface are found to be of the order of a few percent of the bulk lattice constant. Such calculations are not justified for metals or semiconductors where the contribution of the conduction electrons cannot be neglectedd). Especially in covalent crystals, i.e. Ge and Si, where the experimental LEED data have shown a strong rearrangement of the atoms in the surfaces), the electronic contribution could be very important 6). In the last years different attempts to solve this problem in a general form have been made7*s), starting from the general expression of the crystal energy. In the present paper similar ideas will be used to calculate the lattice distortion near a free surface in an fee crystal. Special attention is paid to the invariance and symmetry requirements of the total potential energy, from which the forces acting on the atoms in the perfect lattice equilibrium positions, and the form of the atomic force tensors are obtained. In the adopted model the deformation can be directed only along the normal to the free surface, but the arguments can be extended to include also the case of atomic 109
110
A. CORCIOVEI,
M. CROITORU
AND D. GRECU
rearrangements in the surface plane. In the harmonic approximation, the lattice distortion is given by the solution of an inhomogeneous linear system, which can be obtained by a Green’s function method as in the case of point defectss). In the present paper the nearest-neighbour approximation will be used for a slab with two free surfaces, and the obtained system will be solved taking into account the explicit form of a certain inverse matrixlo). Finally, different possible extensions will be discussed.
2. Invariance and symmetry conditions; model description Let us consider a semi-infinite In the harmonic approximation energy is given by
fee lattice bounded by a (001) free surface. the expression of the crystal potential
(1) Here u,(Z) represents the a-component of the displacement of the Ith atom from its equilibrium position, denoted by B(l); Y’e(.+. B(2)..*) is the static potential energy and Y,(l) and Yy,,(El’), defined by the relations
(Ia>
(lb) are the so-called atomic force constants or coupling parameters, c.p., of the first and second order. Obviously they are functions of all the equilibrium positions (B(Z)} and determine all the crystal properties. No assumptions about the explicit form of the potential energy will be made. Then the only information about the c.p.‘s can be obtained from invariance and symmetry requirements. From their definition
(lb) it results that Y,, (ff’) = Yp,
(f’f) .
(l’b)
Then, from the homogeneity and isotropy of the surrounding space it results that the crystal potential energy is invariant towards any rigid body translation and rotation of the crystal as a whole. These invariance requirements express the total momentum and total angular momentum conservation Iaws, and must be satisfied by any arbitrary set of mass points. They lead to
LA’lTKE ~~TOR~ON
OF AN FCC CRYSTAL
111
the following relations between the c.p.‘sll-14) 7 YY,(O=O,
(24
7 K,(~~‘) = 0,
i2b) W CW
These relations can be generalized to include the case when external forces are exerted on the crystal, provided the forces have zero resulting force and torque in the equilibrium stateis). If these forces are F(I), this means (9
F F,(l) = 0, 5] W)
B/?(O = T 44)
@*(I)*
Vb)
Then the equilibrium condition will be (5)
Y.%(l)= W).
As we shall not consider the crystal placed in an external field, the equilibrium condition (5) will become Y,(Z) = 0, (5’) and consequently, (3b) transforms into ; %,(El’) 9&‘)
= c Y&I’)
%(l’).
f3’b)
These relations are important in the transition from the microscopical to the macroscopical elastic theoryi4) and also in the problem of localized surface vibrational modesle). Here will be given another use of them, namely in the problem of lattice distortions near a free surface. In order to determine this deformation it is necessary to put into evidence the difference between the c.p.‘s of the semi-infinite crystal and those of an ideal one. For this, let us write W(Z) = R(Z) + n(z),
(6)
where R(I) represents the equilibrium position of the Ith atom in an ideal infinite crystal, and 1(Z) is its static displacement, due to the presence of the free surface. Expanding the equilibrium condition (5’) and retaining only the first two terms, we obtain @A(Z)+ 1 @&(ZE’)&(Z’) = 0. 1’sB
(7)
112
A. CORCIOVEI,
Here @L(Z)is the cr-component the non-equilibrium position
M. CROITORU
AND D. ORECU
of the force acting on the atom when it is in R(Z), and @&(ZZ’)are the second-order c.p.‘s
in this configuration, which can be considered as the starting point of the relaxation process towards the new equilibrium situation. Two facts contribute to the appearance of these forces namely: 1) the atoms of the surface region have a smaller number of neighbours, and 2) due to a possible redistribution of ionic and band electrons the potential between these atoms could be strongly changed. The previously mentioned invariance conditions have to be satisfied also in this nonequilibrium state and the following relations are obtained T @i(Z) = 0,
(ga)
These relations state that the net force and torque exerted on the semi-infinite crystal, when its atoms are in the ideal lattice configuration, is zero. They represent the initial conditions by means of which the system (7) has to be solved. Up to now, excepting the harmonic approximation no other hypothesis was used in deducing the above relations, thus they are quite general. In order to get a soluble model new simplifying assumptions are necessary. They are the following. a) Only nearest neighbour interactions are considered. That means that YJ’,,(ZZ’)and G&(K) are different from zero only if Z and I’ correspond to nearest neighbours. b) The presence of the free surface determines a changement of the interaction potential only between atoms of the first two layers. If we denote by Gola(ZZ’)the c.p.‘s of the ideal crystal then @&(ZZ’) = @@(II’) + d@,,(IZ’),
(10)
and A@,,(ZZ’) #O only for I, and Z; =0 and 1. Consequently the forces @L(Z) will be different from zero only in this region. c) All the anharmonic effects will be neglected, i.e. we assume that Y,,(ZZ’) g @&(ZZ’).
(10’)
These assumptions, although restrictive, permit to cover a large enough variety of lattice distortion in the surface vicinity. The next hypothesis
LATIKE
DISTORTION
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OF AN FCC CRYSTAL
referring to the symmetry properties of the c.p.‘s reduces considerably the field of application of the model. But any changement made in it can lead to new models and physical situations. d) We assume that the potential energy of the crystal with a free surface and in its ideal lattice configuration still remains invariant towards any symmetry operation (translation, rotation or reflection) which takes the surface plane (and any plane parallel to it) into itself. Symmetry operations which take one crystallografic plane parallel to the surface into another are excluded. It is a strong restriction and will lead, as it will be shown below, to a lattice distortion only along the normal to the free surface. From the invariance of the potential energy with respect to any lattice vector translation in the surface plane one obtains
In accordance with assumption (d) any inversion made in the surface plane will leave invariant the crystal potential energy. The consequences of this fact can be obtained immediately. Let us denote by S a symmetry operation which takes the ideal lattice configuration into itself. It may be composed of a rotation or reflection Q and a translation t S=JZ+t. (12) If we discard the translations t, in the surface region the rotations (and reflections) {a} represent a subgroup of the point group of the infinite ideal lattice; in our case it will be the subgroup C,, of the full group 0,. The transformation rules of the c.p.‘slsy l7) result from the invariance of the crystal potential energy towards such a symmetry operation
@is (1Z’)= c s,, s,,, G&y (IL’)
)
(13b)
a’, 8’
where R(L) = S-R(Z), If now S represents K(11,
an inversion
R(L’) = S-R(V).
(13’)
Z in the surface plane, using (13a) we obtain
12, &) = - @A(- 11, - 12, Z3) = + @i(-
z,, - Z,, Z3)
a=xory a=z.
(14)
114
A. CORCIOVEI,
M. CROITORU
AND
D. GRECU
Combining (14) with (1 la) we find Q;(t) = @h(E)= 0,
(14’)
i.e. the forces acting on the atoms in the ideal lattice configuration are directed along the normal to the free surface. This is a direct consequence of hypothesis (d) and in order to obtain other situations it is necessary to modify it. If S is a symmetry operation which does not change the position of the atoms R(I) and R(Z), the relation (13b) becomes @&(1z) = c S,, S,,
(13’b)
and in this form it can be used to get the force tensors ~~~~(~~)~of the surface region. For instance if E=(OOO)and I’ =(l 10) the only symmetry operation is x-+y and one obtains: cP!_=@;,,=a,,
(li:,=
b,,
~8:~= @;* = d, ,
cP&=@~~=c~,
@;, = @ip:,= d; .
Then by making an inversion /'-(iTO) and by means of (13b) we find al
c1
(@LB (0,iio)j = ( - ;f
_ ;I
But tli:, (0,~ lo) = cp;lcl (110,o) =
- d, -d, . b, >
aj;,(o,iio),
so that finally d; = -dl and
W,@WO)~
= (_
5;
_ 91
2;)
In the same way we can determine all the other force tensors and the results are given in table 1, where also the force tensors of the ideal crystal are presented. From table 1, we see that there is a number of 21 unknown c.p.‘s of second order. Their number can be reduced by using @b), which for different Eand a becomes c(o +8/?+4a=O, a,, + 4a, + 2a, + 2bz = 0, b,+4b,
+4c,=O,
a~+4a3+2a,f2b,-t2a+2/?=0, b;+4b3+4c,+4j3=0.
(15)
LATTICE
DISTORTION
OF AN FCC CRYSTAL
i i E
> o”o 800
I ,I
115
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A. CORCIOVEI,
M. CROITORU
AND D.GRECU
Finally, the forces acting on the atoms of the first two layers can be determined from (9b). One gets Q:(O) = ~(a, + b, - 2d, - dJ, @p:(l) = - ~(a, + bz - CI- P + 2d3 - e2 + y).
(16)
In order to satisfy @a) it is necessary to have cr + p - y = 2d, + d2 + 2d, - e2.
U5’)
Thus the number of unknown c.p.‘s is reduced to 15. In accordance with (14’) the forces are directed along the z-axis and the relation (9a) is automatically satisfied. Starting from the system (7) and using (1 la, b) it is easy to show that &(!I, L &) = &(l, + h,, I, + h,, Q, (17a) whatever the lattice vector R(h,, hz, 0) in the surface plane would be. This relation tells us that all the atoms of the same layer have the same displacement. By multiplying (7) in a suitable manner, we can put it into the form c s,,. @,k(I) + c a’
I’. B a’.8’
s,, s,,
@& (It’) c s,, L, (I’) = 0, Y
which, by using (13a) and (13b) becomes @i(L) + c @&L’)C s&(a) = 0. (18) L’,B Y Here L and L’ are related to I and 1’ by (13’). From (18) we obtain the transformation rule of the displacements under a symmetry operation (17b) W) = c S,,W)Y the same as for the force components. Considering an inversion in the surface plane, from (17a) and (17b) we obtain J,(l) = n,(l) = 0,
&(i,, L &I = &(l,).
(19)
This relation states that the static displacements n(l) are directed along the normal to the free surface only, and are the same for all atoms of the same layer. As it was mentioned above this is a direct consequence of our last hypothesis. At the same conclusion we could arrive by observing that due to hypothesis (c)the deformed crystal must have the same symmetry properties as the ideal lattice configuration with free surface. Then, from the invariance of the deformation energy with respect to all symmetry operations of the subgroup C4,_ the relations (19) will result immediately. Taking into account all the above results the complexity of the system (7) is considerable reduced.
LATTICE
DISTORTION
OF AN FCC CRYSTAL
117
It is suitable to introduce the notation (20) where the summation is made over the nearest neighbours of the origin. Now the translational invariance condition (8b) transforms into 5 @%(I,, IL) = 0.
(21)
These quantities can be interpreted as a type of c.p.‘s between adjacent planes. With this notation the system (7) will become (22) + t: @L(L &)3,(G) = 0, 1’3 which now corresponds to a one-dimensional problem (a linear chain with a free end). It is much more convenient to put (22) into the form KG)
; @p(V)At, = - @p;< - c d@(ll’) A,, ,
(22’)
where for the sake of simplicity, we have denoted I, and 1; by I and I’, respectively and the indices z have been omitted. The summation is extended over three values of I’, namely : 2’ = I- I, I, I-t I, and for the surface atoms over two values, I’= 1 and 2 (we slightly change the notations and label the surface plane with I= 1). By @(li’) we understand the c.p.‘s of the infinite ideal lattice and d@pIII’)are the deviations due to the presence of the free surface. From the explicit expressions of the force tensors one obtains @(I, Z)= - sp,
@(I, 11: 1) = 48,
(23a)
where the second relation results from (21) applied to an ideal lattice. If we denote d@(l,2) = LlQi(2,l) = 4&, (23b) in order to have satisfied (21) we must have d@(2,2) = - 4p1,
A@(l, 1) = 4P(l - ?>,
(23~)
and ? = (G/a) - 1. Then the system (22’) written in a matricial form will be -2
0 ........ 1 -2 1 0 *.... 0 1 -2 1 o.... . . . . . . . . . ..*............
. . .* _. .
1
I,‘ I, A,
II~ :
-.
=
CW
118
A.CORClOVEI,
M. CROITORU
AND
D. GRECU
-l-;_~~~~..~::~~~~~~~
(24)
where f = - @p; /4p = CP;/4j3. Before we start to solve this equation, fulfilment of the invariance conditions displacements given by the solutions really satisfied. Indeed, using (9b) we
We)
let us discuss one point related to the in the final deformed state. With the of (24) it is not obvious that (3’b) is can write (3’b) in the form
c Q@‘,(U) A,(Y) - ~~~(~~‘~;1,(1’)> d 1’ = - T {Q&,(11’)R&‘) - zP&(ll’) R,(Z’)}
= - @l;(l) S,, + Q;(Z) 6,,.
Taking into account (14’) and (19) one obtains a new system, namely T Q:,(P)
&(I’) = - @l(r),
t2-57
from which one can determine also the displacements A(/). Finally, we get an equation similar to (24), except that now
j-=_-!L
a, + b2
-__-1,
2(” t-P>”
?==cc+p
(26)
To have only one solution for both equations it is necessary that cI=p,
cz==f(a,+b,).
(27)
3. Lattice distortion: the finite slab The matrix equation (24) can be written in a more condensed form, namely @*A=$-d@nL,
(28)
where @ and A@ are infinite square matrices and Aandfare column matrices having as elements the displacements of the different layers and respectively the forces acting on them. If we denote by G the inverse matrix of @ we have A=G.f
[email protected].
(28’)
Once G is known the displacements can be determined following the partitioned technique extensively used in the problems of localized vibration in
LATTICE
DlSTORTlON
119
OF AN FCC CRYSTAL
solidsr4J*-so). This method requires the knowledge of the static Green’s function. It was used by Flinn and Maraduding) in the similar problem of lattice distortion near a point defect. In this paper another method will be developed. For this purpose, instead of a semi-infinite crystal, we shall consider a partial finite crystal with n layers bounded by two parallel free surfaces. Then the matrices appearing in (28’) become finite n-dimensional matrices. In our case the inverse matrix G was determined many years ago by Rutherfordlo). It is a symmetrical matrix, and its elements are given by G,, = -
r(n + 1 - s)
r
n+l (29)
- s(n + 1 - r)
=
r>s.
n+l Now writing
(28’) in the partitioned
-($
$Zb
Z.>
form we have
c;
i
jj
(i>-
(30)
Here by a and a’ we have labelled the boundary regions where the forces and elements of A@ are different from zero, and by b the remaining space. In the present case the boundary regions are limited to the first two layers near the free surface,
so that
Equation (30) is equivalent rank, namely A, = G,, - f,
with a system of three matrix equations
+ G,,, - f,,, - (G,, -A@,, - 1, + G,,. 9A@,ro, -Jo.) ,
+ . f, +
of smaller
(324
Iz, = G,, . f,
G,,,, . f,,
- (Gbo - AQa, +.I, + Gbo, A@,,, -&) ,
Wb)
Iz,, = G,.,
G,.,, . far
- (G,, . A@,, . La + G,,,,, -A@,,‘,, -La.).
(32a’)
l
We see that Iz, does not enter in any right hand side of these equations and is completely determined once 1, and 12,sare known. As the two free surfaces are equivalent an inversion operation with respect to the central plane of the slab will transform the slab into itself. Let us denote by T this inversion operation. It can be written in the partitioned
A. CORCIOVEI,
120
M. CROITORU
AND D. GRECU
form (33)
where T,,, and Tap, are 2 x 2 matrices
(-; The invariance requirement different elements of G
(33’)
-3.
leads us to the following
G,, = T,,. . G,,. . T,., , G,, = Tao,. Go,, TM > G,,, = T,,. . G,, . ToaT, l
which obviously
of the form
are automatically f,, = T,,
between
G,, = TM . GM . Tk 3 G,, = Tb,,. G,,! . To,, >
satisfied,
- fa 7
relations
(34)
and to
A,. = T,., L, , l
A@,,,,, = T,., . A@,, . T,,! .
(34’)
Then it is easily seen that (32a’) is a direct consequence left with the equations
of (32a) and we are
A, = (G,, + Go,, . %,I. (f, - A@,, . Aa>2 2, = (Ga + %a, . Lx). Taking
into account
the
(fa - A@,, . Aa) .
(29) we obtain
n+l
n+lI
2
1 -.2---n+l
n+l
’ I r
G,,, =
Solving (35a) with respect to 1, one obtains
n+
LATIKE
and introducing
DISTORTION
121
OF AN FCC CRYSTAL
it into (35b) we find
r=3,4
,..., (n-2).
W’b)
4. Discussion In this paper we have tried to calculate the static deformation near a free surface in an fee crystal, starting from the expression of the potential energy and taking into account the general symmetry and invariance requirements. The method is easily extended to other cubic symmetries and even to other lattices. Some main conclusions can be drawn. The strong connection between the lattice distortion and the hypothesis about the symmetry properties in the ideal configuration was put forward. Assuming the CbV symmetry, as was done in this paper, the deformations can be only along the normal direction. In order to obtain a rearrangement of the atoms in the surface plane a lower symmetry, as for instance CzV, must be considered. The problem can be solved in the harmonic approximation, although this involves an extra relation [see (27)] between the c.p.‘s of the perfect lattice. This difficulty can be overpassed taking into account the anharmonic effects, but this would considerably complicate the solution of the problem. We have restricted ourselves to the nearest neighbour approximation, because in this case the problem of a slab with two parallel free surfaces is easily solved using the known expression for the inverse matrix G. It would be possible to consider also the next nearest neighbourslo), but the formulas will become too intricate. To solve the problem of a semi-infinite crystal it is necessary to calculate the static Green’s functions. Assuming the C,, symmetry, as all the atoms in a given plane have the same deformation along z-axis, we are led to a one-dimensional problem. In order to obtain the solution of the nonhomogeneous system (28’) it is suitable to write it in the partitioned form (30). In this paper the boundary region was limited to the first two layers, but this restriction can be relaxed to a larger space. For instance, assuming this region to be set up from the first three layers, it is only necessary to replace (31) by
122
A. CORCIOVEI,
M. CROITORU
AND D. GRECU
with f1 +f2
+f3
=o,
(38’)
and the solution will proceed in the same way. Perhaps some of the unexpected results quoted above will be improved.
Acknowledgements The authors would like to thank attention on Rutherford’s results.
Mr. G. Costache
for drawing
their
References 1) R. Suttleworth, Proc. Phys. Sot. (London) 62 (1949) 167. 2) B. J. Alder, J. R. Vaisnys and G. Jura, J. Phys. Chem. Solids 11 (1959) 182. 3) J. Vail, Can. J. Phys. 45 (1967) 2661. 4) L. I. Ahmad, Surface Sci. 12 (1968) 437; K. Huang and G. Wyllie, Proc. Phys. Sot. (London) 62 (1949) 180. 5) R. E. Schlier and H. E. Farnsworth, J. Chem. Phys. 30 (1959) 917; J. J. Lander and J. Morrison, J. Chem. Phys. 37 (1962) 729; J. Appl. Phys. 34 (1963) 2298 ; J. J. Lander, G. W. Gobeli and J. Morrison, J. Appl. Phys. 34 (1963) 1403; E. A. Wood. J. Appl. Phys. 35 (1964) 1306. 6) P. Ducros, Surface Sci. 10 (1968) 295. 7) I. Babuska, E. Vitasek and K. Kroupa, Czech. J. Phys. 10 (1960) 419, 488. 8) T. E. Feuchtwang, Phys. Rev. 155 (1967) 715. 9) P. A. Flinn and A. A. Maradudin, Ann. Phys. (N.Y.) 18 (1962) 81. 10) D. E. Rutherford, Proc. Roy. Sot. Edinburgh A 63 (1951) 232. 11) M. Born, K. Huang, Dynamic Theory of Crystal Lattices (Oxford Univ. Press, 1954). 12) G. Leibfried, in: Handbuch derPhysik, Band 7/I (Springer-Verlag, Berlin, 1955) p.104. 13) A. A. Maradudin, E. W. Montroll and G. H. Weiss, Solid State Phys. Suppl. (1963). 14) W. Ludwig, Ergeb. Exakt. Naturwiss. 43 (1967) I. 15) D. C. Wallace, Rev. Mod. Phys. 37 (1965) 57. 16) W. Ludwig and B. Lengeler, Solid State Commun. 2 (1964) 83. 17) G. Leibfried and W. Ludwig, Solid State Phys. 12 (1961) 275. 18) G. Lehmann and R. E. De Wames, Phys. Rev. 131 (1963) 1008. 19) A. A. Maradudin, Rept. Progr. Phys. 28 (1965) 331; Solid State Phys. 18 (1966) 273; 19 (1967) 1. 20) Yu. A. Izyumov, Advan. Phys. 14 (1965) 569.
SCIENCE 21(1970) 109-122 0 North-Holland
LATTICE DISTORTION
Publishing Co.
OF AN fee CRYSTAL
NEAR
A FREE SURFACE A. CORCIOVEI,
M. CROITORU
and D. GRECU
Institute for Atomic Physics, Bucharest, Romania Received 17 September 1969 The problem of lattice distortion near a free surface is discussed for an fee crystal using the general invariance and symmetry properties of the crystal potential energy. The atomic force tensors were determined assuming a C4” symmetry in the surface region, when the atoms are in the ideal lattice configuration. All the anharmonic effects are neglected. From invariance requirements it results that the static displacements are directed along the normal to the surface. They can be calculated using the Green’s function method. Detailed calculations were done for a finite slab in the nearest neighbour approximation.
1. Introduction In a number of theoretical studies it was tried to calculate the lattice distortion near a free surface of a crystal. Theoretical calculations have been done for molecular and ionic crystals using explicit expressions for the two body interaction potential between the atoms1-3). Here the new equilibrium positions are determined from the minimum condition of the surface energy. Deformations normal to the crystal surface are found to be of the order of a few percent of the bulk lattice constant. Such calculations are not justified for metals or semiconductors where the contribution of the conduction electrons cannot be neglectedd). Especially in covalent crystals, i.e. Ge and Si, where the experimental LEED data have shown a strong rearrangement of the atoms in the surfaces), the electronic contribution could be very important 6). In the last years different attempts to solve this problem in a general form have been made7*s), starting from the general expression of the crystal energy. In the present paper similar ideas will be used to calculate the lattice distortion near a free surface in an fee crystal. Special attention is paid to the invariance and symmetry requirements of the total potential energy, from which the forces acting on the atoms in the perfect lattice equilibrium positions, and the form of the atomic force tensors are obtained. In the adopted model the deformation can be directed only along the normal to the free surface, but the arguments can be extended to include also the case of atomic 109
110
A. CORCIOVEI,
M. CROITORU
AND D. GRECU
rearrangements in the surface plane. In the harmonic approximation, the lattice distortion is given by the solution of an inhomogeneous linear system, which can be obtained by a Green’s function method as in the case of point defectss). In the present paper the nearest-neighbour approximation will be used for a slab with two free surfaces, and the obtained system will be solved taking into account the explicit form of a certain inverse matrixlo). Finally, different possible extensions will be discussed.
2. Invariance and symmetry conditions; model description Let us consider a semi-infinite In the harmonic approximation energy is given by
fee lattice bounded by a (001) free surface. the expression of the crystal potential
(1) Here u,(Z) represents the a-component of the displacement of the Ith atom from its equilibrium position, denoted by B(l); Y’e(.+. B(2)..*) is the static potential energy and Y,(l) and Yy,,(El’), defined by the relations
(Ia>
(lb) are the so-called atomic force constants or coupling parameters, c.p., of the first and second order. Obviously they are functions of all the equilibrium positions (B(Z)} and determine all the crystal properties. No assumptions about the explicit form of the potential energy will be made. Then the only information about the c.p.‘s can be obtained from invariance and symmetry requirements. From their definition
(lb) it results that Y,, (ff’) = Yp,
(f’f) .
(l’b)
Then, from the homogeneity and isotropy of the surrounding space it results that the crystal potential energy is invariant towards any rigid body translation and rotation of the crystal as a whole. These invariance requirements express the total momentum and total angular momentum conservation Iaws, and must be satisfied by any arbitrary set of mass points. They lead to
LA’lTKE ~~TOR~ON
OF AN FCC CRYSTAL
111
the following relations between the c.p.‘sll-14) 7 YY,(O=O,
(24
7 K,(~~‘) = 0,
i2b) W CW
These relations can be generalized to include the case when external forces are exerted on the crystal, provided the forces have zero resulting force and torque in the equilibrium stateis). If these forces are F(I), this means (9
F F,(l) = 0, 5] W)
B/?(O = T 44)
@*(I)*
Vb)
Then the equilibrium condition will be (5)
Y.%(l)= W).
As we shall not consider the crystal placed in an external field, the equilibrium condition (5) will become Y,(Z) = 0, (5’) and consequently, (3b) transforms into ; %,(El’) 9&‘)
= c Y&I’)
%(l’).
f3’b)
These relations are important in the transition from the microscopical to the macroscopical elastic theoryi4) and also in the problem of localized surface vibrational modesle). Here will be given another use of them, namely in the problem of lattice distortions near a free surface. In order to determine this deformation it is necessary to put into evidence the difference between the c.p.‘s of the semi-infinite crystal and those of an ideal one. For this, let us write W(Z) = R(Z) + n(z),
(6)
where R(I) represents the equilibrium position of the Ith atom in an ideal infinite crystal, and 1(Z) is its static displacement, due to the presence of the free surface. Expanding the equilibrium condition (5’) and retaining only the first two terms, we obtain @A(Z)+ 1 @&(ZE’)&(Z’) = 0. 1’sB
(7)
112
A. CORCIOVEI,
Here @L(Z)is the cr-component the non-equilibrium position
M. CROITORU
AND D. ORECU
of the force acting on the atom when it is in R(Z), and @&(ZZ’)are the second-order c.p.‘s
in this configuration, which can be considered as the starting point of the relaxation process towards the new equilibrium situation. Two facts contribute to the appearance of these forces namely: 1) the atoms of the surface region have a smaller number of neighbours, and 2) due to a possible redistribution of ionic and band electrons the potential between these atoms could be strongly changed. The previously mentioned invariance conditions have to be satisfied also in this nonequilibrium state and the following relations are obtained T @i(Z) = 0,
(ga)
These relations state that the net force and torque exerted on the semi-infinite crystal, when its atoms are in the ideal lattice configuration, is zero. They represent the initial conditions by means of which the system (7) has to be solved. Up to now, excepting the harmonic approximation no other hypothesis was used in deducing the above relations, thus they are quite general. In order to get a soluble model new simplifying assumptions are necessary. They are the following. a) Only nearest neighbour interactions are considered. That means that YJ’,,(ZZ’)and G&(K) are different from zero only if Z and I’ correspond to nearest neighbours. b) The presence of the free surface determines a changement of the interaction potential only between atoms of the first two layers. If we denote by Gola(ZZ’)the c.p.‘s of the ideal crystal then @&(ZZ’) = @@(II’) + d@,,(IZ’),
(10)
and A@,,(ZZ’) #O only for I, and Z; =0 and 1. Consequently the forces @L(Z) will be different from zero only in this region. c) All the anharmonic effects will be neglected, i.e. we assume that Y,,(ZZ’) g @&(ZZ’).
(10’)
These assumptions, although restrictive, permit to cover a large enough variety of lattice distortion in the surface vicinity. The next hypothesis
LATIKE
DISTORTION
113
OF AN FCC CRYSTAL
referring to the symmetry properties of the c.p.‘s reduces considerably the field of application of the model. But any changement made in it can lead to new models and physical situations. d) We assume that the potential energy of the crystal with a free surface and in its ideal lattice configuration still remains invariant towards any symmetry operation (translation, rotation or reflection) which takes the surface plane (and any plane parallel to it) into itself. Symmetry operations which take one crystallografic plane parallel to the surface into another are excluded. It is a strong restriction and will lead, as it will be shown below, to a lattice distortion only along the normal to the free surface. From the invariance of the potential energy with respect to any lattice vector translation in the surface plane one obtains
In accordance with assumption (d) any inversion made in the surface plane will leave invariant the crystal potential energy. The consequences of this fact can be obtained immediately. Let us denote by S a symmetry operation which takes the ideal lattice configuration into itself. It may be composed of a rotation or reflection Q and a translation t S=JZ+t. (12) If we discard the translations t, in the surface region the rotations (and reflections) {a} represent a subgroup of the point group of the infinite ideal lattice; in our case it will be the subgroup C,, of the full group 0,. The transformation rules of the c.p.‘slsy l7) result from the invariance of the crystal potential energy towards such a symmetry operation
@is (1Z’)= c s,, s,,, G&y (IL’)
)
(13b)
a’, 8’
where R(L) = S-R(Z), If now S represents K(11,
an inversion
R(L’) = S-R(V).
(13’)
Z in the surface plane, using (13a) we obtain
12, &) = - @A(- 11, - 12, Z3) = + @i(-
z,, - Z,, Z3)
a=xory a=z.
(14)
114
A. CORCIOVEI,
M. CROITORU
AND
D. GRECU
Combining (14) with (1 la) we find Q;(t) = @h(E)= 0,
(14’)
i.e. the forces acting on the atoms in the ideal lattice configuration are directed along the normal to the free surface. This is a direct consequence of hypothesis (d) and in order to obtain other situations it is necessary to modify it. If S is a symmetry operation which does not change the position of the atoms R(I) and R(Z), the relation (13b) becomes @&(1z) = c S,, S,,
(13’b)
and in this form it can be used to get the force tensors ~~~~(~~)~of the surface region. For instance if E=(OOO)and I’ =(l 10) the only symmetry operation is x-+y and one obtains: cP!_=@;,,=a,,
(li:,=
b,,
~8:~= @;* = d, ,
cP&=@~~=c~,
@;, = @ip:,= d; .
Then by making an inversion /'-(iTO) and by means of (13b) we find al
c1
(@LB (0,iio)j = ( - ;f
_ ;I
But tli:, (0,~ lo) = cp;lcl (110,o) =
- d, -d, . b, >
aj;,(o,iio),
so that finally d; = -dl and
W,@WO)~
= (_
5;
_ 91
2;)
In the same way we can determine all the other force tensors and the results are given in table 1, where also the force tensors of the ideal crystal are presented. From table 1, we see that there is a number of 21 unknown c.p.‘s of second order. Their number can be reduced by using @b), which for different Eand a becomes c(o +8/?+4a=O, a,, + 4a, + 2a, + 2bz = 0, b,+4b,
+4c,=O,
a~+4a3+2a,f2b,-t2a+2/?=0, b;+4b3+4c,+4j3=0.
(15)
LATTICE
DISTORTION
OF AN FCC CRYSTAL
i i E
> o”o 800
I ,I
115
116
A. CORCIOVEI,
M. CROITORU
AND D.GRECU
Finally, the forces acting on the atoms of the first two layers can be determined from (9b). One gets Q:(O) = ~(a, + b, - 2d, - dJ, @p:(l) = - ~(a, + bz - CI- P + 2d3 - e2 + y).
(16)
In order to satisfy @a) it is necessary to have cr + p - y = 2d, + d2 + 2d, - e2.
U5’)
Thus the number of unknown c.p.‘s is reduced to 15. In accordance with (14’) the forces are directed along the z-axis and the relation (9a) is automatically satisfied. Starting from the system (7) and using (1 la, b) it is easy to show that &(!I, L &) = &(l, + h,, I, + h,, Q, (17a) whatever the lattice vector R(h,, hz, 0) in the surface plane would be. This relation tells us that all the atoms of the same layer have the same displacement. By multiplying (7) in a suitable manner, we can put it into the form c s,,. @,k(I) + c a’
I’. B a’.8’
s,, s,,
@& (It’) c s,, L, (I’) = 0, Y
which, by using (13a) and (13b) becomes @i(L) + c @&L’)C s&(a) = 0. (18) L’,B Y Here L and L’ are related to I and 1’ by (13’). From (18) we obtain the transformation rule of the displacements under a symmetry operation (17b) W) = c S,,W)Y the same as for the force components. Considering an inversion in the surface plane, from (17a) and (17b) we obtain J,(l) = n,(l) = 0,
&(i,, L &I = &(l,).
(19)
This relation states that the static displacements n(l) are directed along the normal to the free surface only, and are the same for all atoms of the same layer. As it was mentioned above this is a direct consequence of our last hypothesis. At the same conclusion we could arrive by observing that due to hypothesis (c)the deformed crystal must have the same symmetry properties as the ideal lattice configuration with free surface. Then, from the invariance of the deformation energy with respect to all symmetry operations of the subgroup C4,_ the relations (19) will result immediately. Taking into account all the above results the complexity of the system (7) is considerable reduced.
LATTICE
DISTORTION
OF AN FCC CRYSTAL
117
It is suitable to introduce the notation (20) where the summation is made over the nearest neighbours of the origin. Now the translational invariance condition (8b) transforms into 5 @%(I,, IL) = 0.
(21)
These quantities can be interpreted as a type of c.p.‘s between adjacent planes. With this notation the system (7) will become (22) + t: @L(L &)3,(G) = 0, 1’3 which now corresponds to a one-dimensional problem (a linear chain with a free end). It is much more convenient to put (22) into the form KG)
; @p(V)At, = - @p;< - c d@(ll’) A,, ,
(22’)
where for the sake of simplicity, we have denoted I, and 1; by I and I’, respectively and the indices z have been omitted. The summation is extended over three values of I’, namely : 2’ = I- I, I, I-t I, and for the surface atoms over two values, I’= 1 and 2 (we slightly change the notations and label the surface plane with I= 1). By @(li’) we understand the c.p.‘s of the infinite ideal lattice and d@pIII’)are the deviations due to the presence of the free surface. From the explicit expressions of the force tensors one obtains @(I, Z)= - sp,
@(I, 11: 1) = 48,
(23a)
where the second relation results from (21) applied to an ideal lattice. If we denote d@(l,2) = LlQi(2,l) = 4&, (23b) in order to have satisfied (21) we must have d@(2,2) = - 4p1,
A@(l, 1) = 4P(l - ?>,
(23~)
and ? = (G/a) - 1. Then the system (22’) written in a matricial form will be -2
0 ........ 1 -2 1 0 *.... 0 1 -2 1 o.... . . . . . . . . . ..*............
. . .* _. .
1
I,‘ I, A,
II~ :
-.
=
CW
118
A.CORClOVEI,
M. CROITORU
AND
D. GRECU
-l-;_~~~~..~::~~~~~~~
(24)
where f = - @p; /4p = CP;/4j3. Before we start to solve this equation, fulfilment of the invariance conditions displacements given by the solutions really satisfied. Indeed, using (9b) we
We)
let us discuss one point related to the in the final deformed state. With the of (24) it is not obvious that (3’b) is can write (3’b) in the form
c Q@‘,(U) A,(Y) - ~~~(~~‘~;1,(1’)> d 1’ = - T {Q&,(11’)R&‘) - zP&(ll’) R,(Z’)}
= - @l;(l) S,, + Q;(Z) 6,,.
Taking into account (14’) and (19) one obtains a new system, namely T Q:,(P)
&(I’) = - @l(r),
t2-57
from which one can determine also the displacements A(/). Finally, we get an equation similar to (24), except that now
j-=_-!L
a, + b2
-__-1,
2(” t-P>”
?==cc+p
(26)
To have only one solution for both equations it is necessary that cI=p,
cz==f(a,+b,).
(27)
3. Lattice distortion: the finite slab The matrix equation (24) can be written in a more condensed form, namely @*A=$-d@nL,
(28)
where @ and A@ are infinite square matrices and Aandfare column matrices having as elements the displacements of the different layers and respectively the forces acting on them. If we denote by G the inverse matrix of @ we have A=G.f
[email protected].
(28’)
Once G is known the displacements can be determined following the partitioned technique extensively used in the problems of localized vibration in
LATTICE
DlSTORTlON
119
OF AN FCC CRYSTAL
solidsr4J*-so). This method requires the knowledge of the static Green’s function. It was used by Flinn and Maraduding) in the similar problem of lattice distortion near a point defect. In this paper another method will be developed. For this purpose, instead of a semi-infinite crystal, we shall consider a partial finite crystal with n layers bounded by two parallel free surfaces. Then the matrices appearing in (28’) become finite n-dimensional matrices. In our case the inverse matrix G was determined many years ago by Rutherfordlo). It is a symmetrical matrix, and its elements are given by G,, = -
r(n + 1 - s)
r
n+l (29)
- s(n + 1 - r)
=
r>s.
n+l Now writing
(28’) in the partitioned
-($
$Zb
Z.>
form we have
c;
i
jj
(i>-
(30)
Here by a and a’ we have labelled the boundary regions where the forces and elements of A@ are different from zero, and by b the remaining space. In the present case the boundary regions are limited to the first two layers near the free surface,
so that
Equation (30) is equivalent rank, namely A, = G,, - f,
with a system of three matrix equations
+ G,,, - f,,, - (G,, -A@,, - 1, + G,,. 9A@,ro, -Jo.) ,
+ . f, +
of smaller
(324
Iz, = G,, . f,
G,,,, . f,,
- (Gbo - AQa, +.I, + Gbo, A@,,, -&) ,
Wb)
Iz,, = G,.,
G,.,, . far
- (G,, . A@,, . La + G,,,,, -A@,,‘,, -La.).
(32a’)
l
We see that Iz, does not enter in any right hand side of these equations and is completely determined once 1, and 12,sare known. As the two free surfaces are equivalent an inversion operation with respect to the central plane of the slab will transform the slab into itself. Let us denote by T this inversion operation. It can be written in the partitioned
A. CORCIOVEI,
120
M. CROITORU
AND D. GRECU
form (33)
where T,,, and Tap, are 2 x 2 matrices
(-; The invariance requirement different elements of G
(33’)
-3.
leads us to the following
G,, = T,,. . G,,. . T,., , G,, = Tao,. Go,, TM > G,,, = T,,. . G,, . ToaT, l
which obviously
of the form
are automatically f,, = T,,
between
G,, = TM . GM . Tk 3 G,, = Tb,,. G,,! . To,, >
satisfied,
- fa 7
relations
(34)
and to
A,. = T,., L, , l
A@,,,,, = T,., . A@,, . T,,! .
(34’)
Then it is easily seen that (32a’) is a direct consequence left with the equations
of (32a) and we are
A, = (G,, + Go,, . %,I. (f, - A@,, . Aa>2 2, = (Ga + %a, . Lx). Taking
into account
the
(fa - A@,, . Aa) .
(29) we obtain
n+l
n+lI
2
1 -.2---n+l
n+l
’ I r
G,,, =
Solving (35a) with respect to 1, one obtains
n+
LATIKE
and introducing
DISTORTION
121
OF AN FCC CRYSTAL
it into (35b) we find
r=3,4
,..., (n-2).
W’b)
4. Discussion In this paper we have tried to calculate the static deformation near a free surface in an fee crystal, starting from the expression of the potential energy and taking into account the general symmetry and invariance requirements. The method is easily extended to other cubic symmetries and even to other lattices. Some main conclusions can be drawn. The strong connection between the lattice distortion and the hypothesis about the symmetry properties in the ideal configuration was put forward. Assuming the CbV symmetry, as was done in this paper, the deformations can be only along the normal direction. In order to obtain a rearrangement of the atoms in the surface plane a lower symmetry, as for instance CzV, must be considered. The problem can be solved in the harmonic approximation, although this involves an extra relation [see (27)] between the c.p.‘s of the perfect lattice. This difficulty can be overpassed taking into account the anharmonic effects, but this would considerably complicate the solution of the problem. We have restricted ourselves to the nearest neighbour approximation, because in this case the problem of a slab with two parallel free surfaces is easily solved using the known expression for the inverse matrix G. It would be possible to consider also the next nearest neighbourslo), but the formulas will become too intricate. To solve the problem of a semi-infinite crystal it is necessary to calculate the static Green’s functions. Assuming the C,, symmetry, as all the atoms in a given plane have the same deformation along z-axis, we are led to a one-dimensional problem. In order to obtain the solution of the nonhomogeneous system (28’) it is suitable to write it in the partitioned form (30). In this paper the boundary region was limited to the first two layers, but this restriction can be relaxed to a larger space. For instance, assuming this region to be set up from the first three layers, it is only necessary to replace (31) by
122
A. CORCIOVEI,
M. CROITORU
AND D. GRECU
with f1 +f2
+f3
=o,
(38’)
and the solution will proceed in the same way. Perhaps some of the unexpected results quoted above will be improved.
Acknowledgements The authors would like to thank attention on Rutherford’s results.
Mr. G. Costache
for drawing
their
References 1) R. Suttleworth, Proc. Phys. Sot. (London) 62 (1949) 167. 2) B. J. Alder, J. R. Vaisnys and G. Jura, J. Phys. Chem. Solids 11 (1959) 182. 3) J. Vail, Can. J. Phys. 45 (1967) 2661. 4) L. I. Ahmad, Surface Sci. 12 (1968) 437; K. Huang and G. Wyllie, Proc. Phys. Sot. (London) 62 (1949) 180. 5) R. E. Schlier and H. E. Farnsworth, J. Chem. Phys. 30 (1959) 917; J. J. Lander and J. Morrison, J. Chem. Phys. 37 (1962) 729; J. Appl. Phys. 34 (1963) 2298 ; J. J. Lander, G. W. Gobeli and J. Morrison, J. Appl. Phys. 34 (1963) 1403; E. A. Wood. J. Appl. Phys. 35 (1964) 1306. 6) P. Ducros, Surface Sci. 10 (1968) 295. 7) I. Babuska, E. Vitasek and K. Kroupa, Czech. J. Phys. 10 (1960) 419, 488. 8) T. E. Feuchtwang, Phys. Rev. 155 (1967) 715. 9) P. A. Flinn and A. A. Maradudin, Ann. Phys. (N.Y.) 18 (1962) 81. 10) D. E. Rutherford, Proc. Roy. Sot. Edinburgh A 63 (1951) 232. 11) M. Born, K. Huang, Dynamic Theory of Crystal Lattices (Oxford Univ. Press, 1954). 12) G. Leibfried, in: Handbuch derPhysik, Band 7/I (Springer-Verlag, Berlin, 1955) p.104. 13) A. A. Maradudin, E. W. Montroll and G. H. Weiss, Solid State Phys. Suppl. (1963). 14) W. Ludwig, Ergeb. Exakt. Naturwiss. 43 (1967) I. 15) D. C. Wallace, Rev. Mod. Phys. 37 (1965) 57. 16) W. Ludwig and B. Lengeler, Solid State Commun. 2 (1964) 83. 17) G. Leibfried and W. Ludwig, Solid State Phys. 12 (1961) 275. 18) G. Lehmann and R. E. De Wames, Phys. Rev. 131 (1963) 1008. 19) A. A. Maradudin, Rept. Progr. Phys. 28 (1965) 331; Solid State Phys. 18 (1966) 273; 19 (1967) 1. 20) Yu. A. Izyumov, Advan. Phys. 14 (1965) 569.