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Vacancies in transition metals
I. Phys. Chem. Solids, 1976, Vol. 31, pp. 699-709.
Pergamon Press.
Printed in Great Britain
VACANCIES IN TRANSITION METALS FORMATION ENERGY AND FORMATION VOLUME G. ALLAN
and M.
LANNOO
Physique des Solides,? I.S.E.N. 3, rue F. Bats,
59646Lille Cedex, France
(Received 20 February 1973;accepted in revisedform
12December 1975)
Abstract-This
work describes a calculation of the formation energy and volume for a vacancy in transition metals. One uses a tight-binding scheme for the d band and a Born-Mayer type potential to account for the repulsive part of the energy at small distances. The results show that the relaxation energy is small in all cases, less than 0.1 eV. This seems to be coherent with the good agreement obtained for the theoretical and experimental values of the formation energy E,” of the vacancy, without including relaxation. The center of the transitions series is found to give a contraction (formation volume of order - 0.4 at. vol.) whereas the edges are found to produce dilatations.
INTRODUCTION This
work is devoted to a calculation of the formation energy and formation volume for a vacancy in transition metals. It follows a previous calculation [ 1,2] which was concerned only with the purely electronic part of the formation energy of the vacancy. Since that time a model has been developed for this class of solids[3] which by addition of a Born-Mayer type repulsive potential to the electronic attractive energy resulted in a good overall agreement between the theoretical and experimental values of the elastic constants. Our aim is then to include such terms for the vacancy and determine the relaxation around the defect. We have chosen the vacancy case not only for its experimental interest but also because it is the simplest problem where we shall be able to test the validity of some approximations which are necessary for more complicated defects of the same nature. For instance the same type of calculations can be used for vacancy complexes, migration energies of vacancies and also for studies of surface relaxation and reconstruction. Such theoretical studies have already been attempted. We shall not review the work on continuum elasticity theory. This is done for instance in[4,5]. More realistic atomic models have been used especially for the noble metals[6-121 and also for Pb[13], Ni, Fe[l4] and Fecu[I& 161. All these models suffer from the fact that they do not take an explicit account of the redistribution of valence electrons due to vacancy formation, when this redistribution seems to be a leading effect in the case of surface relaxation. Here we shall analyze this change in electronic properties within the framework of a tight-binding approximation, which works fairly well for transition metals, especially for the prediction of average properties along tr~sitional series [ 17,2]. We shall also use a method of moments[l8] which allows a fairly simple evaluation of the total energy of the system with a quite good approximation. We shall expand the energy of the perturbed system to second order in the atomic displace*Research ~uipment supplied by
CNRS.
ments and finally use the method of lattice statics [ 19-211 to evaluate the relaxed configuration. Section 1 gives a brief description of the tight-binding parameters, the Born-Mayer repulsive terms, the use of a method of moments and their application to the perfect crystal. The model is then extended in a non selfconsistent approximation to any defect problem and a general expansion of the energy in terms of the atomic displacements is derived. In Section 2 the validity of the two main approximations of the preceeding model is discussed. An exact expansion of the energy is derived within a Green’s function treatment. It will be compared to the approximate results in the simple cubic system. The neglect of selfconsistency effects is then partly justified by a detailed analysis showing that these can only Iead to second order corrections. The application to the vacancy is finally done in Section 3. A first part briefly recalls the principles of the methods of lattice statics and shows that one can obtain equivalent results by a direct minimization procedure. The numerical values of the formation energy, the atomic displacements and the formation volume are given versus the number of electrons in the d band. They are compared with available experimental observations and other theoretical results and finally the numerical errors introduced by our approximations are estimated. 1. APPROXIMA~ T~AT~NT In this section we shall describe the approximate treatment to be used in the following. It will first be applied to the perfect crystal and then to the perturbed system for which the energy will be expanded to second order in the atomic displacements. 1.1 ~esc~~?ion
of the model
; application
to
the perfect
crystal
It has been shown since a long time[ 171 that a very simple tight-binding scheme gives at least semiquantitative results for the cohesive energy and the surface tension[2] in transition metals. This follows from the fact that the d band is narrow (above S-8eV wide) 699
700
G. ALLAN and M. LANNOO
and that the cohesion is mainly due to d electrons. The most simple theoretical shape obtained for the cohesive energy along transitional series is a parabola which fits very well to the experimental curve[l, 2,171. Ducastelle[3] has extended this simple model to account for the elastic constants of the transition metals. He calculates the electronic part of the energy in a tight-binding approximation which includes resonance integrals between nearest neighbours d orbitals. This can most easily be done using a method of moments which we briefly recall. One can define a partial density of states ni(e) on atom i as: IZ,(E)= Tr8(e -H)
V(27W)
Co e-Y The repulsive energy per atom z is then given by N E~=-C~~~~~. 2
-
In this model one can then write the following set of simple relations
2
1
(7)
one is able to calculate from (6) and (7) the properties of the electronic part of the energy as a function of the atomic configuration, which provides the main contribution to the total energy at large distances. However at distances smaller than the equilibrium one there must exist a repulsive part which is mainly due to the increase in kinetic energy corresponding to the compression of the wave functions. Such an effect has been discussed in great detail by Slater[23] for simple molecules. To account for this term one can take as Ducastelle a Born-Mayer repulsive term:
(1)
where H is the Hamiltonian, the trace being taken on the atomic orbitals centered on site i. It has been shown [ 181 that this can roughly be approximated by a Gaussian curve fitted to the second moment pzl of the exact function. One then obtains
ni(e)=p
P(R)=p,exp-qR
exp-&
-q;-pR=o taking the free atom energy level as the origin of the energies. Now pL2!can be expressed as[18]
P2’ = g
P:,
E,=-,-; E,=-P
(3)
To that degree of approximation the problem is completely equivalent to that of a fivefold degenerate s band in the same structure with an effective resonance integral P. For the perfect crystal, with a given interatomic distance R, all p:j have a common value P*(R) so that one obtains /.~;i= N/3*(R)
(5)
where N is the coordination number (number of nearest neighbours). It is then a simple matter to evaluate the attractive energy per atom &
E‘ l-;
-
where the summation in (3) extends over all the N nearest neighbours and pij is an effective resonance integral which is simply related to the usual two-center integrals dda, ddrr, dd6[22,23] by (4)
(W
qE,
The first one is the equilibrium condition; the second one gives the cohesive energy per atom I?,; the final two express z and G in terms of E,. The cohesive energy per atom can be written EC = +10 = Ecu exp-m.
(CFT
(8b)
If one makes the assumption that /l and q/p are constant along a given transitional series, (8b) gives a good fit to the cohesive energy versus the number of electrons Nd in the d band which fixes E: through
The important parameters p and 4 should in fact vary along the transitional series. However it is possible to show that if one takes where eFo is the perfect crystal Fermi level, no(e) the unperturbed density of states. If now one makes the reasonable assumption [3] that p varies exponentially with the interatomic distance R:
qR,,-3 pRo=9 where R. is the equilibrium interatomic
(9) distance at the
middle of the series for instance, one obtains good agreement for the elastic constants[3] especially in the case of face-centered cubic systems. The choice for q comes from theoretical calc~ations [24]and for p by fitting to experimental data for the compressibility. We shall choose the average values given by (9) t~oughout this work. From these considerations one then obtains a very simple model which gives a correct account of the trends of the elastic constants at least for the f.c.c crystals. It is this model we intend to use in the following to estimate the relaxation energy for the vacancy. 1.2 The perfurbed system The perturbation due to any type of defect at site 0 changes all the partial densities of states n!(e). In a non selfconsistent calculation, this introduces an unphysic~ shift of the Fermi level [ 1,2,18] (6: --, EF)in order to keep the total number of particles constant, i.e.
Using the fact that ds;el=_ dp2i one directly obtains from (16) and (6%) -
N\-N-
&,i = + --@-
ee.
(17)
In the vacancy case this is only non zero for the nearest nei~bours where (17) gives -(112Nf&5]. We shall use this result later to determine the formation energy of the unrelaxed vacancy. 1.2.2 Relaxed system. In a relaxation process the atomic displacements will modify all the resonance integrals. One can write to second order pii = j%l-+ c-hi)
f@
8, is a quantity which contains the displacements and can be written where n’(r) and n(e) are now the total densities of states. As fF - lFDis of order l/&f for a point defect, where &f is the total number of atoms (10) can be expanded to first order Q I
[n(~f-n~(~)~d~+(et;-~~)n(t.~~)=O.
(11)
The total attractive energy is itself changed by an amount SE equal to ‘F 6E, = 10
J
n(e)6 da - 10
!
SF0 ~“(E)Ede
Here p is the value at the normal internuclear equilibrium distance Rn and dii, uij are functions of the displacements UCof atom i and u, of atom j given by Z&j=
jUi-IQ/
(12)
where the factor IQ takes into account the degeneracy of the d band and the spin degeneracy. expanding (12) to first order in (EF- 6:) and using (11) one obtains:
where Rii is the vector joining atom j to atom i in the perfect crystat, To expand 8; to second order in the atomic displacements, one must first expand it to second order in the change of second moment &zi using (16) and its derivative, then write Spzi in terms of f?, from (18). This leads to
One can write this as a sum of atomic contributions given by CP s;= 10 [n;(e) - $(E)](E - E:) de. (14) To evaluate (14) we shal1 approximate ni(e) by (2) where ptt; will be the second moment of the atom in the perturbed system. 1.2.1 Unrelaxed system. If the atoms are kept at their perfect crystal positions it is clear that
where N: is the new coordination number ofatom i. As this one is equal to N or N - 1 one can with a good approximation expand each SE to first order in pzi - p;i,
To S& must be added a repulsive part Sf, which is half the sum of the terms which connect it to its nearest neighbours. It can be expande< in terms of the dij, u+ After that one expresses l/S}and the repulsive coefficient c in terms of the cohesive energy per atom EC using (6) and (8). After some simple m~~pulations, one then finds the total energy change 8; for atom i, to second order in the displacements
702
G. ALLAN and l&. LANNOO
perturbation due to the atomic displacements. One can write G =g+gVG
(22)
(26) where j and j‘ only involve nearest neighbours of atom i. Here we have from (8)
if this expansion is convergent. The change in density of states &i(e) can be written h(s)=-~fTr(G--g)
(27)
and
[!$Z(k-i)].
Ft =exp-
From (22) one can outline the following properties: There are linear terms only for atoms which have N! # N or their nearest neighbours. There are quadratic coupling terms only for pairs of atoms which are nearest neighbours or have one common nearest neighbour. 2. VALIDITY OF THE A~ROX~ATR TREATMENT
The treatment we have described in Section 1 suffers from two important approximations. The first one is due to the use of a method of moments restricted to the second one. We shall analyze its validity by comparison to the results of an exact Green’s function treatment which will be applied to the simple cubic case. The second one comes from the neglect of selfconsistency effects which through charge redistribution can affect the energy of the system. 2.1 Exact Green’s firnction treatment We shall develop here a general treatment of the atomic displacements. This is done by the Green’s function technique and is valid for any structure. However we shall apply it only to the unphysical case of the simple cubic lattice. This is done to test the approximate treatments of Section 1. The difficulty here comes from the eval~tion of the change in the total attractive energy 6E. This can be written, using (13) after integrating by parts
where @J(E) is the change in the total number of states of energy less than E, due to the relaxation process. To determine this let us define g and G the resolvents of the unrelaxed system and the relaxed one respectively: 1 g=-
E-h
where Tr for the trace of the matrix. With (26) one can write:
Using the cyclic properties of the trace and the fact that
one obtains:
which for the change in the total number of states #J(E) gives
(28b) Using (24) and (28) one can write the total change in attractive SFe due to atomic displacements
X
I',,VM(gthgki
t
gik&
de-.
(29)
In (29) we have considered terms up to second order in V only because we want linear and quadratic terms in the displacements. Use has also been made of symmetry properties and fi’ means a sum over pairs of orbitals i,j C’ ,: over pairs ij and hk. The factor 4 arises from spin and also from the fact that pairs occur twice in the general expansion. Now, to simplify the procedure we shall use an effective resonance integral p to represent the whole set of interactions between d orbitals as in (4) thereby reducing the problem to that of a fivefold degenerate s band. With this approximation and using the exponential dependence of fi defined in (7) one can expand V,, to second order in the displacements as
G--L-
E-h-V
h being the unrelaxed problem hamiltonian, while V is the
)
the IA,,and dii being defined in (20).
703
Vacanciesin tr~s~ti~n metals From this one obtains displacements
to
second order in the
due to 6p0+ Sp.It acts to maintain the Fermi energy at its perfect crystal vaiue e:, and must satisfy the Friedel sum rule [2&J. The selfconsistent change in energy ESCis given by % =
I’:‘[svn(r)tfiv(r)]r dr -;j (pot spot 8~) x(V,+ V)dT +$
To this must be added the repulsive part which gives to second order, fir a given pair ij
c being the value at the normal internuclear distance. It is important to notice that in (31) the summation extends over pairs of atoms and that in (29) it was over pairs of d orbitals. This has introduced an extra factor of 5 in (31). One can then combine (3 I) and (32) to obtain the totat change in energy SE due to dis~Iacem~nts. When doing this one can make use of the perfect crystal equilibrium condition which relates c and jp / and express the result in terms of the maximum cohesive energy E
f
poVod7.
(34)
The first term is due to the sum of one efectron energies while the two last terms account for the electron interactions which were counted twice[l, 21. One can rewrite (34) by integrating by parts. using the relation
J
@PO + sp)Vod7 =
J
poV d7
the fact that the nonselfconsistent energy (24) as
eNSC
is given by
and the cunservat~on of the total number of states at the Fermi level E: (Friedet sum rule)[26].
This allows to obtain
-j Here J,, f;, and J$ are ~mensionless integrals defined in the Appendix. They depend on the Fermi level position and involve summations over Green’s functions. Equation (33) is quite simiiar to B &i obtained from (22). In section 3 we shall compare ~or~es~ond~ngcoeficients of the two expansions for the simple cubic lattice and show that (22) is a fair approximation to the exact result. This can already be checked on Figs. 6 and 7.
Here we shaff analyze formally the digerence in energy between a non selfconsistent and a selfconsistent tr~tment of the same pert~bed system. For this we shall first carefully define the notations. We shaIl take the perfect crystal with charge density pa(r) and one electron electrostatic potentiaf VO(P)as reference. The charge density will be modi~ed by an nmonnt Sp&) in the non seffconsistent perturbed problem, and S&r) + Sp(r) in the selfconsistent one. We use the same notations for the density of states vO,8~0,SVO t Su respectively, and for the total number of states NO, NO, Si%+ Sh? Finally we call V(r) the change in potential between the perfect crystal and the selfconsistent perturbed system. In the Hartree approximation V(r) is purely electrostatic and
i f
(Spot Sp)V d7.
(35)
To evaluate the di~erent terms of f35) we shall use the Green’s function technique. We shall now call g the resolvent of the non self~onsistent problem, G of the selfconsistent one. With this notation N(r) is given by (28b), alt~ongh the meaning of the quantities is di~erent. Now standard manipulations allow us to write:
J (p,r,tspo)v
J
6pVdT=-+
d7 = -;
I
J’”rrwgck
‘?FTrV(G-g)dc
In this way (35) reduces to lEsc -
ENSC = ;
J
&?,v dr
where Y means the imaginary part of the quantity which
704
C. ALUN and M. LANNW
follows. In a ~rturbat~on expansion the Leading term is $_fSpoV d7. Now Sp, is of first order in the perturbation created by the defect, as well as V so that the whole term is only of second order. The correction (36) then turns out to be small in many cases. This is the case for the surface tension as well as the vacancy where it turns out to be smaller than 0.01 eV [ 1,2]. Furthermore such a correction will not affect the linear terms in the expansion of the energy versus the atomic displacements. As these are essential in the relaxation problem one can conclude that a non selfconsistent calculation will provide a quite accurate treatment for that problem. 3. ~~~RR~~.~L
simple to make a basis change which d~ago~a~izesDo. This corresponds to
where Iqo) is a Rtoch sum [I; q) being the Ith eigenvector of DO for wavevector q, with eigenvalue D*(I; q). One immed~ateIy obtains
RESULTS AND DlSCWX?N
As we have now obtained the expansion of the energy to second order in the atomic displacements, the remaining problem consists in a minimization procedure which will yield the equilibrium atomic positions and the relaxation energy. In a first part we shall recall the minimization procedure and the way the formation volume can be obtained. The second part will be devoted to a presentation and a discussion of the numericaf results. 3.1. ~~nj~~~~~~o~~~o~e~~r~ and ~5r~ff~~5~ ~~~~~~ One cm minimize the energy with respect to the atomic dispIacements by different techniques: the method of lattice statics[ 19-211which in principle gives exact results or either directly within a given sphere around the defect using suitable boundary conditions. We have used both of them and will give a brief summary of the corresponding techniques. Let us first recall the principle of the method of lattice statics[l9-211. If one minimizes the second order expansion of the energy with respect to one atomic coordinate kla (the Ruth component of the ith atom displacement) one obtains the fotmal equations:
O=-(iap)+(iaplu).
(37)
Here /u) is the displacement vector whose components along an atomic basis I&X)are the ui,, jF) a force whose components are the linear terms of SE and D an operator whose matrix elements (ioliD/jp) are those of the dynamical matrix of the perturbed system. Let us first write D=fiO+A Da corresponding to the perfect crystal expansion. If one defines IF’) = IF) f Ala)
(384
(i@‘) = (icYID”/rr).
(38b)
where SF is the relaxation energy. If A was equal to zero, eqns (39) directly give the solution of the system, by a typical Green’s function technique. The problem becomes more complicated for nonvanishing A. Aowever as A is of local character Alec) only concerns the immediate neighbourh~d of the point defect and /F’f has still only a restricted number of components. Then in (39b) one has to consider two classes of components: those corresponding to /iof for which (h]F’) is non zero and which are in the vicinity of the defect and those for which the (iculF’) vanish. The (iolu) of the first class whose number is m can be determined by solving m linear equations with m unknowns~21]” From these all other (ifulu) can be determined by repeated apphcation of (39b) and Gnafly SEF determined from (39a). The procedure which has just been described can be greatly simplified by symmetry considerations which we shah not describe because they are quite well known for cubic systems. These help in reducing the number of independent components of !F’f and ju)[Zlf. We have also preferred to use an iteration procedure for (39b) instead of solving the m equations with m unknowns. Let us mention that we have also used a direct minimization procedure for atoms inside a given sphere. We have taken two types of boundary conditions: one with all displacements vanishing outside the sphere, the other taking instead the dij defined in (20) to be zero outside the same sphere. In both cases we have obtained essentially the same results as with the method of lattice statics for the displacements near the defect as well as the energy. The best convergence is obtained in the second case (Q = Of and the cahculation is much simpler than with the method of lattice statics. This is interesting for catculations of more complex defect systems. The calculation of the formation volume VF can now be done using the well-known expression[271 which gives for the change in volume AV around the vacancy
(37) gives:
At this stage it is necessary to invert (3Xb).For this it is
AV=&xRiFi
(40)
where K is the compressibility, Ri the vector joining the
705
Vacancies in transition metals
origin to atom i and Fi the force on atom i. This one is directly obtained from (38a) once the displacements are known. The total formation volume is obtained by adding the missing atom on the surface which increases the volume by one atomic volume VO.This gives VP = Vo+AV.
(411
3.2 Results and discussion Let us first begin with the formation energy of the unrelaxed system Ef which one can directly obtain from Section 1 with a good approximation. When removing the atom at the central site one necessary loses its average energy which one regains when putting it on the surface. The change in attractive energy comes from (17) and only concerns the nearest neighbours. Their total contribution is simply -f ee. One must add to this the suppression of half the repulsive terms which connected these atoms to the central site, i.e. a total mount of -& The formation energy in this simple model turns out to be
Fig. 1. Totalgain in energySE,/EcM vs IV,,.-.-.-.;
simple cubic, body centered cubic, ------; face centered cubic.
-;
due to relaxation in units of Ecu is plotted on Fig. 1 vs Nd for the three cubic systems. Some particular values are quoted in Table 1. It decreases in the sequence simple cubic, body centered cubic, face-centered cubic. This is roughly related to the fact that in (22), for Nd = 5, i.e. EQ= 0, the linear term is roughly proportional to l/N, so lthat the gain in energy approximatively varies as l/N*. Ep=-~-CR One can also notice that the relaxation energy vanishes near the beginning and the end of the series where the E1-? c linear term in (22) is zero. =--. In Table 1 we give the results for solids where there are 2 ,-; experimental data. We give EcM and Nd which allow the calculation of S& for a given structure. Then we combine To evaluate EF we shall use this relation, taking for EC the SE, with EF for the two values of q/p and give the final experiemental value. We shall also choose for q/p two result E, in these two cases. It can be seen that the agreement with experiment is very good. extreme values l/3 and l/5 as was done by Ducastelle [3]. The main conclusion to draw from these results is that However, it must be noted that in our theoretical model leading to (42) EC and then EF would tend to zero at the the relaxation energy turns out to be small in all cases, end of the series. The case of filled bands is complicated being always smaller than 0.1 eV. We have also plotted the displacements of atoms in the by s-d interaction which is probably at the origin of the cohesion of noble metals[17]. If one then uses (42) immediate vicinity of the vacancy site. We have taken the semi~mpirically as in Table 1 taking the experiments EC, radial components A, chosen to be positive for an inwards one can see that the agreement remains surprisingly good displacement. The detailed labelling is given in Table 2. for EF in the noble metals. This suggests that the effect of One can see in Figs 2-4 that these are weak, mostly s-d interaction could lead to a theoretical formulation inwards for Nd = 5. They all change their sign where the similar to the one used in this work. This would certainly linear terms vanish. It must be noticed that the order of magnitude of the be worth studying in more details. displacements is always small, i.e. of maximum order One can now analyze the influence of the relaxation process on the energy of the system. The gain in energy 0.01 Ro except for an almost completely empty or filled Table 1. Comparison of the theoretical and experimental values of EFV Fe
Ni
EcMeV
N‘i
SEp”eV
E, eV
q-1 ___EFV P
5 1.5 1281
12&q
W
Pt
MO
Nb
8.2 5 C.C. -0.041 8.2
8.2 9.6 f.c.c.
7.2 5
7.2 4
6
0.035 7
3
2.1
2.6
2
1.4
3.15? f28-30,341
fone value of 4.4 eV has been measured[331. $-IIIthis case the values range from 1.5 to 2.9 eV.
[2iY5
Ag
Au
CC. -
--0.1
Cu
-%28 7.2
3.8
3.2
2.7
2.7
1.42
1.2
1.01
1.7
1.8
0.85
0.8
0.7
f28zOl
1.8 [31,321
1.15 BOI
1.1 [301
0.99 [301
G. ALLAWand M, LANNOO
Table 2. Definition of the inwards displacements A,. (r, m, n) define the corresponding atom
hi A2 A3 A4
simple cubic
f.c.c.
b.c.c.
WQ) (2W (110)
(tfo) (200) (21t)
(tll) (2W (220) (222)
(220)
5/R,
10
’ Nd
0.c
Fig. 4. Values of the displacement A/R, vs N,. Curve i corres~nds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic. 0.c
L 10
> Nd
-0.0
Fig. 2. Values of the displacement A/R0 vs iVd. Curve i corresponds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic.
Fig. 3. Values of the displacement A/& vs Nd. Curve i corresponds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic.
band (Figs. 2-4). This seems to be coherent with an experimental evidence that the change in parameter at tungsten [l lo] and Ni [MO] surfaces[35] is extremely small, as would also be predicted by our modei.[41]. The order of magnitude of our displacements is found to be similar to those obtained by Tewordt [IO] for instance. He finds that for Cu the nearest neighbours relax inwards by an amount of 0.016 Ro. Although the comparison is not meaningful we obtain similar values for half-filled bands. However in the case of Ni; Johnson~l4] finds + 0.021 RO inwards which is completely in disagreement with our value which turns out to be outwards because we are in a case where Nd is equal to 9.4 and one has a change in signfor all the displacements with respect to the half-filled band. This considerable discrepancy certainly comes from the fact that we have taken into account the effect of the d electrons. It could perhaps be partly removed if one incorporate s-d mixing. Finally we have found experimental values of volume changes due to vacancy formation only for Au[36-381 where AV/ V,, is of order - 0.54. The maximum negative value we find for AVI Va is of order - 0.4 for a half-filled band as shown on Fig. 5. We also find a change in sign for almost filled or empty bands. It should be interesting to have experimental confirmation of this. On the other hand in such limits a study of the influence of the s electrons and of s-d interaction would be worth doing, to see if they could alter the results obtained for the d electrons. The validity of the second moment approximation has already been analyzed for unrelaxed systems. The results obtained for the unrelaxed formation energy EF can then be considered as correct. To test the second order expansion obtained in Section 1 we have used the exact treatment of 2.1 applied to the simple cubic lattice. We give a comparison in two typical cases, one linear term in Fig. 6, and one quadratic coefficient in Fig. 7. In each case we have plotted the exact result, an approximation to the second moment as in Section 1, and another approximation up to the fourth moment using an Edgeworth’s series[l8]. One can see that the second moment
Vacancies
in transitionmetals
707
affect the formation energy EF of the unrelaxed problem[l, 21.The only modification will then concern the second order terms. One can be more precise about this by looking at the situation for Nd = 5, which is already self-consistent. For this EQ= 0 and then one can conclude that the whole effect of selfconsistency will concern the term in (EQ)‘/N:~’ which in fact seems to lead to spurious effects for almost filled or empty bands (for instance some quadratic terms can become negative). CONCLUSION
In this work we have calculated the formation energy and formation volume for a vacancy in transition metals. We have used a simple tight-binding model for the attractive part of the energy and a Born-Mayer type potential for the repulsive energy at short distances. To estimate the first part we have used a very simple non selfconsistent treatment where the density of states is
Fig. 5. Formation volume A V/ V, vs Nd, the number of d electrons. The wings of the curves are not given, because they do not correspond to a minimum of the energy.
'\ /‘ '-.
'\._./'
Fig. 6. Coefficients a,, (Y,,, vs the number of electrons Nd in the d band for the simple cubic lattice.-; treatment, ------; approximate treatment to p2, - .- .-. ; approximate treatment to p4.
0
5
10 Nd
Fig. 7. Coefficients a,, (Y,,, vs the number of electrons Nd in the d band for the simple cubic lattice. -; treatment, ------; approximate treatment to ~2, - .- .-. ; approximate treatment to F.+
is in fact quite good, at least to obtain an order of magnitude of relaxation effects. Let us finally consider the effects of selfconsistency. It has already been shown elsewhere that this does not approximation
JPCSVOL.37NO.7-E
exact
exact
approximated by a Guassian curve having the same second moment as the true curve. We have then justified the use of this simplified model by showing that it reproduces nicely, in the simple cubic case, the results
G. ALLAN and M. LANNOO
708
obtained from an exact Green’s function treatment. This fact has allowed us to apply it to the b.c.c. and f.c.c. lattices where the results are quite similar. We have finally shown that selfconsistency will not modify the energy to first order which justifies its neglect except near the beginning and the end of the transitional series. The main conclusion obtained here is that the relaxation energy is always small, less than 0.1 eV for all cases of interest. This is coherent with the results of Table 2 where it is shown that the unrelaxed formation energy is of the same order of magnitude than the experimental one. The formation volume is of order - 0.3 to - 0.4 VOfor a half-filled band. It continuously increases when one departs from this case, vanishing for about 8.5 and 1.5 electrons in the d band. It then changes it sign and one obtains a dilatation for nearly filled or empty bands. This effect is found to be in contradiction with other evaluations, for Ni for instance, where the effect of d electrons was not directly incoporated. We hope to be able in the future to study the influence of s electrons and s-d mixing in those extreme cases. We have already extended such a study to the problems of relaxation near transition metal surfaces [39]. We shall also consider the problem of reconstruction, where there is some experimental evidence for f.c.c. systems 1401. REFERENCES 1. Lannoo M. and Allan G., J. Phys. Chem. Sol. 32(3), 637 1971). 2. Allan G., Thesis, Orsay (1969), Ann. Phys. 5, 169 (16 0). 3. Ducastelle F., J. de Phys. 31, 1055 (1970). 4. Thomson M. W., Dejects and Radiation Damage in Metals. Academic Press, New York (1969). 5. Quere Y., Dtijauts ponctuels dans les mbtaux. Masson, Paris (1967). 6. Huntington H. B. and Seitz F., Phys. Rev. 61, 315 (1942). 7. Huttington H. B. and Seitz F., Phys. Reu. 61, 324 (1942). 8. Huttington H. B., Phys. Rev. 91, 1092 (1953). 9. Eshelby J. D., J. Appl. Phys. 25, 255 (1954). 10. Tewordt L., Phys. Rev. 109, 61 (1958). Il. Seeger A. and Mann E., J. Phys. Chem. Solids 12,326 (1960). 12. Johnson R. A. and Brown E., Phys. Rev. 127, 446 (1962). 13. Girifalco L. A. and Weizer V. G., J. Phys. Chem. Solids 12, 260 (1960). 14. Johnson R. A. Phys. Rev. 145, 423 (1966). 1s. Erginsoy C., Vineyard G. H. and Englert A., Phys. Rev. 133, 595 (1964). 16. Johnson R. A., Phys. Rev. 134, 1329 (1964). 17. Friedel J., The Physics of Metals (Edited by Ziman) .pp. 340-408. Cambridge University Press (1969). 18. Cvrot-Lackmann F.. J. Phvs. Chem. Solids 29, 1235 (1968). 19. K&zaki H., J. Phys. Chek Solids 2, 24 (1957). 20. Hardy J. R., J. Phys. Chem. Solids 15, 39 (1968). 21. Stoneham A. M. and Bartram R. H., Phys. Rev. B 2, 3403 (1970). 22. Ducastelle F. and Cvrot-Lackmann F.. J. Phvs. Chem. Sol. 31, 6 (1970). . 23. Slater J. C., Quantum Theory ofMolecules and Solids Vol. 1 McGraw Hill, New York (1963). 24. Ducastelle F. and Cyrot-Lackmann F., J. Phys. Chem. Sol. 32, 1, 285 (1971). 25. Gerl M., J. Phys. Chem. Sol. 31, 315 (1970). 26. Friedel J., N&o Cimento, Suppl. 7, 287 (1958). 27. Temkin D. E., Sou. Phvs. Sol. State 11, 1614 (1970); Deplante J. L., Thesis, Orsay (1469); Hardy .I. R. J. Phys. Chem. Solids 15, 39 (1960). 28. International Conference on Vacancies and interstitials in Metals, Jiilich (1968). 29. Cotterill R. M. J., Doyama M., Jackson .I. J. and Meshii M.,
30. 31. 32. 33.
34. 35. 36. 37. 38. 39. 40.
Lattice Defectsin Quenched Metals. CambridgeUniversity Press, New York (1569). Thomson M. W., Dejects and Radiation Damage in Metals. Academic Press, New York (1969). Chekhovskoy V. Ya. and ZhukovaI. A., Fiz. Tuerdogo Tela 8, 9 (1966). Kraftmakher Ya. A., Fiz. ‘fuerdogo Tela 5, 950 (1963). Friedel .I., The Interaction of Radiation in Solids, p. 135. Proceedings of the Mol. Summer School, North Holland, Amsterdam (1964). Kraftmakher Ya. A., Sou. Phys. Sol. State 14,2,325 (1972). Stern R. M. and Sinharoy S., Surf. Sci. 33, 131 (1972). Bauerle J. E. and Koehler J. S., Phys. Rec. 107,1493 (1957). Takamura .I., Acta Met. 9, 547 (1961). Simmons R. 0. and Balluffi R. W., Phys. Rev. 125,862 (1962). Allan G. and Lannoo M., Surf. Sci. 40, 375 (1973). Somorjai G. A., Principles of Surface Chemistry. Prentice Hall, Englewood Cliffs, New Jersey (1972). APPENDIX
We want to start from (31) and (32) and obtain the final result (33). We define go the resolvent of the perfect crystal. It can be shown without trouble that for an s band in a system with cubic symmetry the following relationship holds
where 0 and 1 are two nearest neighbours and N is the coordination number of the system, n(e) the density of states for the s band. It is now easy to show that the attractive parl of the average energy per atom in the perfect crystal is
The average repulsive energy per atom is N ER= - c. 2 From the exponential dependence of IpI and c the equilibrium condition introduces the relationship -qc,-pen=0 which combined with the definition of the cohesive energy
_ -
E, =--t,-in
(AS)
gives le=--
E<
1-4 P
,=+_-_q JZ PI-9
L
(-46)
P
Then (A6) with (A2) and (A3) relates IpI and c to the cohesive energy per atom E,. After that to transform (31) we express the Green’s functions g,, in terms of gip of the perfect crystal. In the vacancy case we have shown previously [ 1,2] that one has
(A7) if the vacancy
is at site 0.
Vacancies in transition metals From this one can then transform the first term of (31) because there gz = g& between nearest neighbours. A number of terms cancel with the repulsive part because of the equilibrium condition (A4) which related 161,c and J’j Tgi, da If one does all the simplifications one obtains:
709
where E%, is the Fermi energy giving I& which is zero for symmetric bands, one can then derive the final eqn (33) with the following definitions:
Expressing now IpI and c in terms of E, and using the fact that J$=
Q Ec = Ec, x
I
Tg:, dc (A9) %g:,
fc
I
‘OFM5g:,de
which are dimensionless quantities independent of p.
Pergamon Press.
Printed in Great Britain
VACANCIES IN TRANSITION METALS FORMATION ENERGY AND FORMATION VOLUME G. ALLAN
and M.
LANNOO
Physique des Solides,? I.S.E.N. 3, rue F. Bats,
59646Lille Cedex, France
(Received 20 February 1973;accepted in revisedform
12December 1975)
Abstract-This
work describes a calculation of the formation energy and volume for a vacancy in transition metals. One uses a tight-binding scheme for the d band and a Born-Mayer type potential to account for the repulsive part of the energy at small distances. The results show that the relaxation energy is small in all cases, less than 0.1 eV. This seems to be coherent with the good agreement obtained for the theoretical and experimental values of the formation energy E,” of the vacancy, without including relaxation. The center of the transitions series is found to give a contraction (formation volume of order - 0.4 at. vol.) whereas the edges are found to produce dilatations.
INTRODUCTION This
work is devoted to a calculation of the formation energy and formation volume for a vacancy in transition metals. It follows a previous calculation [ 1,2] which was concerned only with the purely electronic part of the formation energy of the vacancy. Since that time a model has been developed for this class of solids[3] which by addition of a Born-Mayer type repulsive potential to the electronic attractive energy resulted in a good overall agreement between the theoretical and experimental values of the elastic constants. Our aim is then to include such terms for the vacancy and determine the relaxation around the defect. We have chosen the vacancy case not only for its experimental interest but also because it is the simplest problem where we shall be able to test the validity of some approximations which are necessary for more complicated defects of the same nature. For instance the same type of calculations can be used for vacancy complexes, migration energies of vacancies and also for studies of surface relaxation and reconstruction. Such theoretical studies have already been attempted. We shall not review the work on continuum elasticity theory. This is done for instance in[4,5]. More realistic atomic models have been used especially for the noble metals[6-121 and also for Pb[13], Ni, Fe[l4] and Fecu[I& 161. All these models suffer from the fact that they do not take an explicit account of the redistribution of valence electrons due to vacancy formation, when this redistribution seems to be a leading effect in the case of surface relaxation. Here we shall analyze this change in electronic properties within the framework of a tight-binding approximation, which works fairly well for transition metals, especially for the prediction of average properties along tr~sitional series [ 17,2]. We shall also use a method of moments[l8] which allows a fairly simple evaluation of the total energy of the system with a quite good approximation. We shall expand the energy of the perturbed system to second order in the atomic displace*Research ~uipment supplied by
CNRS.
ments and finally use the method of lattice statics [ 19-211 to evaluate the relaxed configuration. Section 1 gives a brief description of the tight-binding parameters, the Born-Mayer repulsive terms, the use of a method of moments and their application to the perfect crystal. The model is then extended in a non selfconsistent approximation to any defect problem and a general expansion of the energy in terms of the atomic displacements is derived. In Section 2 the validity of the two main approximations of the preceeding model is discussed. An exact expansion of the energy is derived within a Green’s function treatment. It will be compared to the approximate results in the simple cubic system. The neglect of selfconsistency effects is then partly justified by a detailed analysis showing that these can only Iead to second order corrections. The application to the vacancy is finally done in Section 3. A first part briefly recalls the principles of the methods of lattice statics and shows that one can obtain equivalent results by a direct minimization procedure. The numerical values of the formation energy, the atomic displacements and the formation volume are given versus the number of electrons in the d band. They are compared with available experimental observations and other theoretical results and finally the numerical errors introduced by our approximations are estimated. 1. APPROXIMA~ T~AT~NT In this section we shall describe the approximate treatment to be used in the following. It will first be applied to the perfect crystal and then to the perturbed system for which the energy will be expanded to second order in the atomic displacements. 1.1 ~esc~~?ion
of the model
; application
to
the perfect
crystal
It has been shown since a long time[ 171 that a very simple tight-binding scheme gives at least semiquantitative results for the cohesive energy and the surface tension[2] in transition metals. This follows from the fact that the d band is narrow (above S-8eV wide) 699
700
G. ALLAN and M. LANNOO
and that the cohesion is mainly due to d electrons. The most simple theoretical shape obtained for the cohesive energy along transitional series is a parabola which fits very well to the experimental curve[l, 2,171. Ducastelle[3] has extended this simple model to account for the elastic constants of the transition metals. He calculates the electronic part of the energy in a tight-binding approximation which includes resonance integrals between nearest neighbours d orbitals. This can most easily be done using a method of moments which we briefly recall. One can define a partial density of states ni(e) on atom i as: IZ,(E)= Tr8(e -H)
V(27W)
Co e-Y The repulsive energy per atom z is then given by N E~=-C~~~~~. 2
-
In this model one can then write the following set of simple relations
2
1
(7)
one is able to calculate from (6) and (7) the properties of the electronic part of the energy as a function of the atomic configuration, which provides the main contribution to the total energy at large distances. However at distances smaller than the equilibrium one there must exist a repulsive part which is mainly due to the increase in kinetic energy corresponding to the compression of the wave functions. Such an effect has been discussed in great detail by Slater[23] for simple molecules. To account for this term one can take as Ducastelle a Born-Mayer repulsive term:
(1)
where H is the Hamiltonian, the trace being taken on the atomic orbitals centered on site i. It has been shown [ 181 that this can roughly be approximated by a Gaussian curve fitted to the second moment pzl of the exact function. One then obtains
ni(e)=p
P(R)=p,exp-qR
exp-&
-q;-pR=o taking the free atom energy level as the origin of the energies. Now pL2!can be expressed as[18]
P2’ = g
P:,
E,=-,-; E,=-P
(3)
To that degree of approximation the problem is completely equivalent to that of a fivefold degenerate s band in the same structure with an effective resonance integral P. For the perfect crystal, with a given interatomic distance R, all p:j have a common value P*(R) so that one obtains /.~;i= N/3*(R)
(5)
where N is the coordination number (number of nearest neighbours). It is then a simple matter to evaluate the attractive energy per atom &
E‘ l-;
-
where the summation in (3) extends over all the N nearest neighbours and pij is an effective resonance integral which is simply related to the usual two-center integrals dda, ddrr, dd6[22,23] by (4)
(W
qE,
The first one is the equilibrium condition; the second one gives the cohesive energy per atom I?,; the final two express z and G in terms of E,. The cohesive energy per atom can be written EC = +10 = Ecu exp-m.
(CFT
(8b)
If one makes the assumption that /l and q/p are constant along a given transitional series, (8b) gives a good fit to the cohesive energy versus the number of electrons Nd in the d band which fixes E: through
The important parameters p and 4 should in fact vary along the transitional series. However it is possible to show that if one takes where eFo is the perfect crystal Fermi level, no(e) the unperturbed density of states. If now one makes the reasonable assumption [3] that p varies exponentially with the interatomic distance R:
qR,,-3 pRo=9 where R. is the equilibrium interatomic
(9) distance at the
middle of the series for instance, one obtains good agreement for the elastic constants[3] especially in the case of face-centered cubic systems. The choice for q comes from theoretical calc~ations [24]and for p by fitting to experimental data for the compressibility. We shall choose the average values given by (9) t~oughout this work. From these considerations one then obtains a very simple model which gives a correct account of the trends of the elastic constants at least for the f.c.c crystals. It is this model we intend to use in the following to estimate the relaxation energy for the vacancy. 1.2 The perfurbed system The perturbation due to any type of defect at site 0 changes all the partial densities of states n!(e). In a non selfconsistent calculation, this introduces an unphysic~ shift of the Fermi level [ 1,2,18] (6: --, EF)in order to keep the total number of particles constant, i.e.
Using the fact that ds;el=_ dp2i one directly obtains from (16) and (6%) -
N\-N-
&,i = + --@-
ee.
(17)
In the vacancy case this is only non zero for the nearest nei~bours where (17) gives -(112Nf&5]. We shall use this result later to determine the formation energy of the unrelaxed vacancy. 1.2.2 Relaxed system. In a relaxation process the atomic displacements will modify all the resonance integrals. One can write to second order pii = j%l-+ c-hi)
f@
8, is a quantity which contains the displacements and can be written where n’(r) and n(e) are now the total densities of states. As fF - lFDis of order l/&f for a point defect, where &f is the total number of atoms (10) can be expanded to first order Q I
[n(~f-n~(~)~d~+(et;-~~)n(t.~~)=O.
(11)
The total attractive energy is itself changed by an amount SE equal to ‘F 6E, = 10
J
n(e)6 da - 10
!
SF0 ~“(E)Ede
Here p is the value at the normal internuclear equilibrium distance Rn and dii, uij are functions of the displacements UCof atom i and u, of atom j given by Z&j=
jUi-IQ/
(12)
where the factor IQ takes into account the degeneracy of the d band and the spin degeneracy. expanding (12) to first order in (EF- 6:) and using (11) one obtains:
where Rii is the vector joining atom j to atom i in the perfect crystat, To expand 8; to second order in the atomic displacements, one must first expand it to second order in the change of second moment &zi using (16) and its derivative, then write Spzi in terms of f?, from (18). This leads to
One can write this as a sum of atomic contributions given by CP s;= 10 [n;(e) - $(E)](E - E:) de. (14) To evaluate (14) we shal1 approximate ni(e) by (2) where ptt; will be the second moment of the atom in the perturbed system. 1.2.1 Unrelaxed system. If the atoms are kept at their perfect crystal positions it is clear that
where N: is the new coordination number ofatom i. As this one is equal to N or N - 1 one can with a good approximation expand each SE to first order in pzi - p;i,
To S& must be added a repulsive part Sf, which is half the sum of the terms which connect it to its nearest neighbours. It can be expande< in terms of the dij, u+ After that one expresses l/S}and the repulsive coefficient c in terms of the cohesive energy per atom EC using (6) and (8). After some simple m~~pulations, one then finds the total energy change 8; for atom i, to second order in the displacements
702
G. ALLAN and l&. LANNOO
perturbation due to the atomic displacements. One can write G =g+gVG
(22)
(26) where j and j‘ only involve nearest neighbours of atom i. Here we have from (8)
if this expansion is convergent. The change in density of states &i(e) can be written h(s)=-~fTr(G--g)
(27)
and
[!$Z(k-i)].
Ft =exp-
From (22) one can outline the following properties: There are linear terms only for atoms which have N! # N or their nearest neighbours. There are quadratic coupling terms only for pairs of atoms which are nearest neighbours or have one common nearest neighbour. 2. VALIDITY OF THE A~ROX~ATR TREATMENT
The treatment we have described in Section 1 suffers from two important approximations. The first one is due to the use of a method of moments restricted to the second one. We shall analyze its validity by comparison to the results of an exact Green’s function treatment which will be applied to the simple cubic case. The second one comes from the neglect of selfconsistency effects which through charge redistribution can affect the energy of the system. 2.1 Exact Green’s firnction treatment We shall develop here a general treatment of the atomic displacements. This is done by the Green’s function technique and is valid for any structure. However we shall apply it only to the unphysical case of the simple cubic lattice. This is done to test the approximate treatments of Section 1. The difficulty here comes from the eval~tion of the change in the total attractive energy 6E. This can be written, using (13) after integrating by parts
where @J(E) is the change in the total number of states of energy less than E, due to the relaxation process. To determine this let us define g and G the resolvents of the unrelaxed system and the relaxed one respectively: 1 g=-
E-h
where Tr for the trace of the matrix. With (26) one can write:
Using the cyclic properties of the trace and the fact that
one obtains:
which for the change in the total number of states #J(E) gives
(28b) Using (24) and (28) one can write the total change in attractive SFe due to atomic displacements
X
I',,VM(gthgki
t
gik&
de-.
(29)
In (29) we have considered terms up to second order in V only because we want linear and quadratic terms in the displacements. Use has also been made of symmetry properties and fi’ means a sum over pairs of orbitals i,j C’ ,: over pairs ij and hk. The factor 4 arises from spin and also from the fact that pairs occur twice in the general expansion. Now, to simplify the procedure we shall use an effective resonance integral p to represent the whole set of interactions between d orbitals as in (4) thereby reducing the problem to that of a fivefold degenerate s band. With this approximation and using the exponential dependence of fi defined in (7) one can expand V,, to second order in the displacements as
G--L-
E-h-V
h being the unrelaxed problem hamiltonian, while V is the
)
the IA,,and dii being defined in (20).
703
Vacanciesin tr~s~ti~n metals From this one obtains displacements
to
second order in the
due to 6p0+ Sp.It acts to maintain the Fermi energy at its perfect crystal vaiue e:, and must satisfy the Friedel sum rule [2&J. The selfconsistent change in energy ESCis given by % =
I’:‘[svn(r)tfiv(r)]r dr -;j (pot spot 8~) x(V,+ V)dT +$
To this must be added the repulsive part which gives to second order, fir a given pair ij
c being the value at the normal internuclear distance. It is important to notice that in (31) the summation extends over pairs of atoms and that in (29) it was over pairs of d orbitals. This has introduced an extra factor of 5 in (31). One can then combine (3 I) and (32) to obtain the totat change in energy SE due to dis~Iacem~nts. When doing this one can make use of the perfect crystal equilibrium condition which relates c and jp / and express the result in terms of the maximum cohesive energy E
f
poVod7.
(34)
The first term is due to the sum of one efectron energies while the two last terms account for the electron interactions which were counted twice[l, 21. One can rewrite (34) by integrating by parts. using the relation
J
@PO + sp)Vod7 =
J
poV d7
the fact that the nonselfconsistent energy (24) as
eNSC
is given by
and the cunservat~on of the total number of states at the Fermi level E: (Friedet sum rule)[26].
This allows to obtain
-j Here J,, f;, and J$ are ~mensionless integrals defined in the Appendix. They depend on the Fermi level position and involve summations over Green’s functions. Equation (33) is quite simiiar to B &i obtained from (22). In section 3 we shall compare ~or~es~ond~ngcoeficients of the two expansions for the simple cubic lattice and show that (22) is a fair approximation to the exact result. This can already be checked on Figs. 6 and 7.
Here we shaff analyze formally the digerence in energy between a non selfconsistent and a selfconsistent tr~tment of the same pert~bed system. For this we shall first carefully define the notations. We shaIl take the perfect crystal with charge density pa(r) and one electron electrostatic potentiaf VO(P)as reference. The charge density will be modi~ed by an nmonnt Sp&) in the non seffconsistent perturbed problem, and S&r) + Sp(r) in the selfconsistent one. We use the same notations for the density of states vO,8~0,SVO t Su respectively, and for the total number of states NO, NO, Si%+ Sh? Finally we call V(r) the change in potential between the perfect crystal and the selfconsistent perturbed system. In the Hartree approximation V(r) is purely electrostatic and
i f
(Spot Sp)V d7.
(35)
To evaluate the di~erent terms of f35) we shall use the Green’s function technique. We shall now call g the resolvent of the non self~onsistent problem, G of the selfconsistent one. With this notation N(r) is given by (28b), alt~ongh the meaning of the quantities is di~erent. Now standard manipulations allow us to write:
J (p,r,tspo)v
J
6pVdT=-+
d7 = -;
I
J’”rrwgck
‘?FTrV(G-g)dc
In this way (35) reduces to lEsc -
ENSC = ;
J
&?,v dr
where Y means the imaginary part of the quantity which
704
C. ALUN and M. LANNW
follows. In a ~rturbat~on expansion the Leading term is $_fSpoV d7. Now Sp, is of first order in the perturbation created by the defect, as well as V so that the whole term is only of second order. The correction (36) then turns out to be small in many cases. This is the case for the surface tension as well as the vacancy where it turns out to be smaller than 0.01 eV [ 1,2]. Furthermore such a correction will not affect the linear terms in the expansion of the energy versus the atomic displacements. As these are essential in the relaxation problem one can conclude that a non selfconsistent calculation will provide a quite accurate treatment for that problem. 3. ~~~RR~~.~L
simple to make a basis change which d~ago~a~izesDo. This corresponds to
where Iqo) is a Rtoch sum [I; q) being the Ith eigenvector of DO for wavevector q, with eigenvalue D*(I; q). One immed~ateIy obtains
RESULTS AND DlSCWX?N
As we have now obtained the expansion of the energy to second order in the atomic displacements, the remaining problem consists in a minimization procedure which will yield the equilibrium atomic positions and the relaxation energy. In a first part we shall recall the minimization procedure and the way the formation volume can be obtained. The second part will be devoted to a presentation and a discussion of the numericaf results. 3.1. ~~nj~~~~~~o~~~o~e~~r~ and ~5r~ff~~5~ ~~~~~~ One cm minimize the energy with respect to the atomic dispIacements by different techniques: the method of lattice statics[ 19-211which in principle gives exact results or either directly within a given sphere around the defect using suitable boundary conditions. We have used both of them and will give a brief summary of the corresponding techniques. Let us first recall the principle of the method of lattice statics[l9-211. If one minimizes the second order expansion of the energy with respect to one atomic coordinate kla (the Ruth component of the ith atom displacement) one obtains the fotmal equations:
O=-(iap)+(iaplu).
(37)
Here /u) is the displacement vector whose components along an atomic basis I&X)are the ui,, jF) a force whose components are the linear terms of SE and D an operator whose matrix elements (ioliD/jp) are those of the dynamical matrix of the perturbed system. Let us first write D=fiO+A Da corresponding to the perfect crystal expansion. If one defines IF’) = IF) f Ala)
(384
(i@‘) = (icYID”/rr).
(38b)
where SF is the relaxation energy. If A was equal to zero, eqns (39) directly give the solution of the system, by a typical Green’s function technique. The problem becomes more complicated for nonvanishing A. Aowever as A is of local character Alec) only concerns the immediate neighbourh~d of the point defect and /F’f has still only a restricted number of components. Then in (39b) one has to consider two classes of components: those corresponding to /iof for which (h]F’) is non zero and which are in the vicinity of the defect and those for which the (iculF’) vanish. The (iolu) of the first class whose number is m can be determined by solving m linear equations with m unknowns~21]” From these all other (ifulu) can be determined by repeated apphcation of (39b) and Gnafly SEF determined from (39a). The procedure which has just been described can be greatly simplified by symmetry considerations which we shah not describe because they are quite well known for cubic systems. These help in reducing the number of independent components of !F’f and ju)[Zlf. We have also preferred to use an iteration procedure for (39b) instead of solving the m equations with m unknowns. Let us mention that we have also used a direct minimization procedure for atoms inside a given sphere. We have taken two types of boundary conditions: one with all displacements vanishing outside the sphere, the other taking instead the dij defined in (20) to be zero outside the same sphere. In both cases we have obtained essentially the same results as with the method of lattice statics for the displacements near the defect as well as the energy. The best convergence is obtained in the second case (Q = Of and the cahculation is much simpler than with the method of lattice statics. This is interesting for catculations of more complex defect systems. The calculation of the formation volume VF can now be done using the well-known expression[271 which gives for the change in volume AV around the vacancy
(37) gives:
At this stage it is necessary to invert (3Xb).For this it is
AV=&xRiFi
(40)
where K is the compressibility, Ri the vector joining the
705
Vacancies in transition metals
origin to atom i and Fi the force on atom i. This one is directly obtained from (38a) once the displacements are known. The total formation volume is obtained by adding the missing atom on the surface which increases the volume by one atomic volume VO.This gives VP = Vo+AV.
(411
3.2 Results and discussion Let us first begin with the formation energy of the unrelaxed system Ef which one can directly obtain from Section 1 with a good approximation. When removing the atom at the central site one necessary loses its average energy which one regains when putting it on the surface. The change in attractive energy comes from (17) and only concerns the nearest neighbours. Their total contribution is simply -f ee. One must add to this the suppression of half the repulsive terms which connected these atoms to the central site, i.e. a total mount of -& The formation energy in this simple model turns out to be
Fig. 1. Totalgain in energySE,/EcM vs IV,,.-.-.-.;
simple cubic, body centered cubic, ------; face centered cubic.
-;
due to relaxation in units of Ecu is plotted on Fig. 1 vs Nd for the three cubic systems. Some particular values are quoted in Table 1. It decreases in the sequence simple cubic, body centered cubic, face-centered cubic. This is roughly related to the fact that in (22), for Nd = 5, i.e. EQ= 0, the linear term is roughly proportional to l/N, so lthat the gain in energy approximatively varies as l/N*. Ep=-~-CR One can also notice that the relaxation energy vanishes near the beginning and the end of the series where the E1-? c linear term in (22) is zero. =--. In Table 1 we give the results for solids where there are 2 ,-; experimental data. We give EcM and Nd which allow the calculation of S& for a given structure. Then we combine To evaluate EF we shall use this relation, taking for EC the SE, with EF for the two values of q/p and give the final experiemental value. We shall also choose for q/p two result E, in these two cases. It can be seen that the agreement with experiment is very good. extreme values l/3 and l/5 as was done by Ducastelle [3]. The main conclusion to draw from these results is that However, it must be noted that in our theoretical model leading to (42) EC and then EF would tend to zero at the the relaxation energy turns out to be small in all cases, end of the series. The case of filled bands is complicated being always smaller than 0.1 eV. We have also plotted the displacements of atoms in the by s-d interaction which is probably at the origin of the cohesion of noble metals[17]. If one then uses (42) immediate vicinity of the vacancy site. We have taken the semi~mpirically as in Table 1 taking the experiments EC, radial components A, chosen to be positive for an inwards one can see that the agreement remains surprisingly good displacement. The detailed labelling is given in Table 2. for EF in the noble metals. This suggests that the effect of One can see in Figs 2-4 that these are weak, mostly s-d interaction could lead to a theoretical formulation inwards for Nd = 5. They all change their sign where the similar to the one used in this work. This would certainly linear terms vanish. It must be noticed that the order of magnitude of the be worth studying in more details. displacements is always small, i.e. of maximum order One can now analyze the influence of the relaxation process on the energy of the system. The gain in energy 0.01 Ro except for an almost completely empty or filled Table 1. Comparison of the theoretical and experimental values of EFV Fe
Ni
EcMeV
N‘i
SEp”eV
E, eV
q-1 ___EFV P
5 1.5 1281
12&q
W
Pt
MO
Nb
8.2 5 C.C. -0.041 8.2
8.2 9.6 f.c.c.
7.2 5
7.2 4
6
0.035 7
3
2.1
2.6
2
1.4
3.15? f28-30,341
fone value of 4.4 eV has been measured[331. $-IIIthis case the values range from 1.5 to 2.9 eV.
[2iY5
Ag
Au
CC. -
--0.1
Cu
-%28 7.2
3.8
3.2
2.7
2.7
1.42
1.2
1.01
1.7
1.8
0.85
0.8
0.7
f28zOl
1.8 [31,321
1.15 BOI
1.1 [301
0.99 [301
G. ALLAWand M, LANNOO
Table 2. Definition of the inwards displacements A,. (r, m, n) define the corresponding atom
hi A2 A3 A4
simple cubic
f.c.c.
b.c.c.
WQ) (2W (110)
(tfo) (200) (21t)
(tll) (2W (220) (222)
(220)
5/R,
10
’ Nd
0.c
Fig. 4. Values of the displacement A/R, vs N,. Curve i corres~nds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic. 0.c
L 10
> Nd
-0.0
Fig. 2. Values of the displacement A/R0 vs iVd. Curve i corresponds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic.
Fig. 3. Values of the displacement A/& vs Nd. Curve i corresponds to Ai. 2; simple cubic, 3; cubic centered, 4; face centered cubic.
band (Figs. 2-4). This seems to be coherent with an experimental evidence that the change in parameter at tungsten [l lo] and Ni [MO] surfaces[35] is extremely small, as would also be predicted by our modei.[41]. The order of magnitude of our displacements is found to be similar to those obtained by Tewordt [IO] for instance. He finds that for Cu the nearest neighbours relax inwards by an amount of 0.016 Ro. Although the comparison is not meaningful we obtain similar values for half-filled bands. However in the case of Ni; Johnson~l4] finds + 0.021 RO inwards which is completely in disagreement with our value which turns out to be outwards because we are in a case where Nd is equal to 9.4 and one has a change in signfor all the displacements with respect to the half-filled band. This considerable discrepancy certainly comes from the fact that we have taken into account the effect of the d electrons. It could perhaps be partly removed if one incorporate s-d mixing. Finally we have found experimental values of volume changes due to vacancy formation only for Au[36-381 where AV/ V,, is of order - 0.54. The maximum negative value we find for AVI Va is of order - 0.4 for a half-filled band as shown on Fig. 5. We also find a change in sign for almost filled or empty bands. It should be interesting to have experimental confirmation of this. On the other hand in such limits a study of the influence of the s electrons and of s-d interaction would be worth doing, to see if they could alter the results obtained for the d electrons. The validity of the second moment approximation has already been analyzed for unrelaxed systems. The results obtained for the unrelaxed formation energy EF can then be considered as correct. To test the second order expansion obtained in Section 1 we have used the exact treatment of 2.1 applied to the simple cubic lattice. We give a comparison in two typical cases, one linear term in Fig. 6, and one quadratic coefficient in Fig. 7. In each case we have plotted the exact result, an approximation to the second moment as in Section 1, and another approximation up to the fourth moment using an Edgeworth’s series[l8]. One can see that the second moment
Vacancies
in transitionmetals
707
affect the formation energy EF of the unrelaxed problem[l, 21.The only modification will then concern the second order terms. One can be more precise about this by looking at the situation for Nd = 5, which is already self-consistent. For this EQ= 0 and then one can conclude that the whole effect of selfconsistency will concern the term in (EQ)‘/N:~’ which in fact seems to lead to spurious effects for almost filled or empty bands (for instance some quadratic terms can become negative). CONCLUSION
In this work we have calculated the formation energy and formation volume for a vacancy in transition metals. We have used a simple tight-binding model for the attractive part of the energy and a Born-Mayer type potential for the repulsive energy at short distances. To estimate the first part we have used a very simple non selfconsistent treatment where the density of states is
Fig. 5. Formation volume A V/ V, vs Nd, the number of d electrons. The wings of the curves are not given, because they do not correspond to a minimum of the energy.
'\ /‘ '-.
'\._./'
Fig. 6. Coefficients a,, (Y,,, vs the number of electrons Nd in the d band for the simple cubic lattice.-; treatment, ------; approximate treatment to p2, - .- .-. ; approximate treatment to p4.
0
5
10 Nd
Fig. 7. Coefficients a,, (Y,,, vs the number of electrons Nd in the d band for the simple cubic lattice. -; treatment, ------; approximate treatment to ~2, - .- .-. ; approximate treatment to F.+
is in fact quite good, at least to obtain an order of magnitude of relaxation effects. Let us finally consider the effects of selfconsistency. It has already been shown elsewhere that this does not approximation
JPCSVOL.37NO.7-E
exact
exact
approximated by a Guassian curve having the same second moment as the true curve. We have then justified the use of this simplified model by showing that it reproduces nicely, in the simple cubic case, the results
G. ALLAN and M. LANNOO
708
obtained from an exact Green’s function treatment. This fact has allowed us to apply it to the b.c.c. and f.c.c. lattices where the results are quite similar. We have finally shown that selfconsistency will not modify the energy to first order which justifies its neglect except near the beginning and the end of the transitional series. The main conclusion obtained here is that the relaxation energy is always small, less than 0.1 eV for all cases of interest. This is coherent with the results of Table 2 where it is shown that the unrelaxed formation energy is of the same order of magnitude than the experimental one. The formation volume is of order - 0.3 to - 0.4 VOfor a half-filled band. It continuously increases when one departs from this case, vanishing for about 8.5 and 1.5 electrons in the d band. It then changes it sign and one obtains a dilatation for nearly filled or empty bands. This effect is found to be in contradiction with other evaluations, for Ni for instance, where the effect of d electrons was not directly incoporated. We hope to be able in the future to study the influence of s electrons and s-d mixing in those extreme cases. We have already extended such a study to the problems of relaxation near transition metal surfaces [39]. We shall also consider the problem of reconstruction, where there is some experimental evidence for f.c.c. systems 1401. REFERENCES 1. Lannoo M. and Allan G., J. Phys. Chem. Sol. 32(3), 637 1971). 2. Allan G., Thesis, Orsay (1969), Ann. Phys. 5, 169 (16 0). 3. Ducastelle F., J. de Phys. 31, 1055 (1970). 4. Thomson M. W., Dejects and Radiation Damage in Metals. Academic Press, New York (1969). 5. Quere Y., Dtijauts ponctuels dans les mbtaux. Masson, Paris (1967). 6. Huntington H. B. and Seitz F., Phys. Rev. 61, 315 (1942). 7. Huttington H. B. and Seitz F., Phys. Reu. 61, 324 (1942). 8. Huttington H. B., Phys. Rev. 91, 1092 (1953). 9. Eshelby J. D., J. Appl. Phys. 25, 255 (1954). 10. Tewordt L., Phys. Rev. 109, 61 (1958). Il. Seeger A. and Mann E., J. Phys. Chem. Solids 12,326 (1960). 12. Johnson R. A. and Brown E., Phys. Rev. 127, 446 (1962). 13. Girifalco L. A. and Weizer V. G., J. Phys. Chem. Solids 12, 260 (1960). 14. Johnson R. A. Phys. Rev. 145, 423 (1966). 1s. Erginsoy C., Vineyard G. H. and Englert A., Phys. Rev. 133, 595 (1964). 16. Johnson R. A., Phys. Rev. 134, 1329 (1964). 17. Friedel J., The Physics of Metals (Edited by Ziman) .pp. 340-408. Cambridge University Press (1969). 18. Cvrot-Lackmann F.. J. Phvs. Chem. Solids 29, 1235 (1968). 19. K&zaki H., J. Phys. Chek Solids 2, 24 (1957). 20. Hardy J. R., J. Phys. Chem. Solids 15, 39 (1968). 21. Stoneham A. M. and Bartram R. H., Phys. Rev. B 2, 3403 (1970). 22. Ducastelle F. and Cvrot-Lackmann F.. J. Phvs. Chem. Sol. 31, 6 (1970). . 23. Slater J. C., Quantum Theory ofMolecules and Solids Vol. 1 McGraw Hill, New York (1963). 24. Ducastelle F. and Cyrot-Lackmann F., J. Phys. Chem. Sol. 32, 1, 285 (1971). 25. Gerl M., J. Phys. Chem. Sol. 31, 315 (1970). 26. Friedel J., N&o Cimento, Suppl. 7, 287 (1958). 27. Temkin D. E., Sou. Phvs. Sol. State 11, 1614 (1970); Deplante J. L., Thesis, Orsay (1469); Hardy .I. R. J. Phys. Chem. Solids 15, 39 (1960). 28. International Conference on Vacancies and interstitials in Metals, Jiilich (1968). 29. Cotterill R. M. J., Doyama M., Jackson .I. J. and Meshii M.,
30. 31. 32. 33.
34. 35. 36. 37. 38. 39. 40.
Lattice Defectsin Quenched Metals. CambridgeUniversity Press, New York (1569). Thomson M. W., Dejects and Radiation Damage in Metals. Academic Press, New York (1969). Chekhovskoy V. Ya. and ZhukovaI. A., Fiz. Tuerdogo Tela 8, 9 (1966). Kraftmakher Ya. A., Fiz. ‘fuerdogo Tela 5, 950 (1963). Friedel .I., The Interaction of Radiation in Solids, p. 135. Proceedings of the Mol. Summer School, North Holland, Amsterdam (1964). Kraftmakher Ya. A., Sou. Phys. Sol. State 14,2,325 (1972). Stern R. M. and Sinharoy S., Surf. Sci. 33, 131 (1972). Bauerle J. E. and Koehler J. S., Phys. Rec. 107,1493 (1957). Takamura .I., Acta Met. 9, 547 (1961). Simmons R. 0. and Balluffi R. W., Phys. Rev. 125,862 (1962). Allan G. and Lannoo M., Surf. Sci. 40, 375 (1973). Somorjai G. A., Principles of Surface Chemistry. Prentice Hall, Englewood Cliffs, New Jersey (1972). APPENDIX
We want to start from (31) and (32) and obtain the final result (33). We define go the resolvent of the perfect crystal. It can be shown without trouble that for an s band in a system with cubic symmetry the following relationship holds
where 0 and 1 are two nearest neighbours and N is the coordination number of the system, n(e) the density of states for the s band. It is now easy to show that the attractive parl of the average energy per atom in the perfect crystal is
The average repulsive energy per atom is N ER= - c. 2 From the exponential dependence of IpI and c the equilibrium condition introduces the relationship -qc,-pen=0 which combined with the definition of the cohesive energy
_ -
E, =--t,-in
(AS)
gives le=--
E<
1-4 P
,=+_-_q JZ PI-9
L
(-46)
P
Then (A6) with (A2) and (A3) relates IpI and c to the cohesive energy per atom E,. After that to transform (31) we express the Green’s functions g,, in terms of gip of the perfect crystal. In the vacancy case we have shown previously [ 1,2] that one has
(A7) if the vacancy
is at site 0.
Vacancies in transition metals From this one can then transform the first term of (31) because there gz = g& between nearest neighbours. A number of terms cancel with the repulsive part because of the equilibrium condition (A4) which related 161,c and J’j Tgi, da If one does all the simplifications one obtains:
709
where E%, is the Fermi energy giving I& which is zero for symmetric bands, one can then derive the final eqn (33) with the following definitions:
Expressing now IpI and c in terms of E, and using the fact that J$=
Q Ec = Ec, x
I
Tg:, dc (A9) %g:,
fc
I
‘OFM5g:,de
which are dimensionless quantities independent of p.