- Email: [email protected]
Formation energy of vacancies in transition metals
./. Phys.
Churn.
Solids
Pergamon
Press
FORMATION
I97 I. Vol.
ENERGY TRANSITION M.
Physique
32, pp. 637-652.
des Solidest-
(Received
in Great
Britain.
OF VACANCIES METALS:”
LANNOO
and
I.S.E.N.,
Printed
IN
G. ALLAN
3, rue FranGois
20 February
Bats-59
Lille,
France
1970)
Abstract-The
formation energies of vacancies are calculated in a simple tight-binding approximation. Friedel’s rule is introduced for self-consistency up to the nearest neighbours of the vacancy site. The first result is that self-consistency does not practically alter the energies. Comparison is made with the experimental results and leads to the following conclusions:-reasonable shape of the predicted curve -values higher than the experimental ones by an amount of I to 2 eV. This suggests that relaxation has to be important as is pointed out in some experimental works.
INTRODUCTION
IN THIS paper we would
like to estimate the order of magnitude of the formation energy for a vacancy in transition metals. To do this, it is necessary to choose a simple method for treating the d electrons in the perfect lattice. The easiest way is to use the tight-binding approximation which has proved to be of semi-quantitative value [ l] for the calculation of the cohesive energy. As the calculation of energies does not seem to depend very much on the details of the density of states curve n(E) and of the lattice structure, we shall simplify the problem by treating a simple cubic lattice, with five independent s orbitals per atom. We shall discuss this approximation at the end of the paper. We shall use here the Green’s ouerator method to relate the perturbed crystal to the unperturbed one. The perturbation will be assumed to extend up to the first neighbours of the vacancy site. In a first part we give a brief description of the band structure and of the choice of the parameters; then we detaiI how it is possible to describe the removal of an atom as we create a vacancy. *This work was done Harwell (England). tEquipe de Recherche
during
a stay
at the A.E.R.E.
of
In the second part we describe a non selfconsistent calculation of the vacancy introducing a variation of the Fermi level which is not physical for an isolated perturbation in a metal. In the third part we introduce the selfconsistency condition given by the Friedel Sum Rule[2]. From this condition we obtain the value of the perturbative potential on the first neighbours and we calculate the formation energy in the Hartree approximation. Finally we compare our results to the experimental values and discuss the approximations introduced. Let us however immediately point out that typical values of formation energies are about 2 to 4 eV for the middle of the transitional series and about 1.5 at the end. 1. BAND STRUCTURE AND PERTURBATIVE POTENTIAL SITE
DEFINITION ON THE
OF THE VACANCY
(a) Band structure parameters. We use the simplest tight-binding approximation, neglecting overlap integrals between the atomic wave functions, and taking into account interactions up to first neighbours only. If we call Ipj) the pseudo-orthonormal set of atomic functions centered on site Ri, the Bloch function can be written
du C.N.R.S.
kfk)
637
=
7
e ‘kIG
Iv,).
(1)
M.
638
The expression of the Hamiltonian perfect crystal is H=T+E
I
and G. ALLAN
LANNOO
v,
for the
(3)
A=-(cp,IVolvo). These lead to the band structure energy E, = EO- (Y- 2X(cos k,a + cos k,a + cos k,a) (4) where E0 is the atomic energy and a the lattice parameter. For our purposes it is convenient to change the origin of the energies and define E = E-
(E,,-CY).
EC =-;
x Ek+pE,, kocc
= 10J” Y(E-EO+cu)(E,,-E)dE
as a function of p/10, the filling up of the band. From the computed numerical values we have easily deduced the curve of Fig. I. The values obtained are in agreement with those of [3]. The results of Fig. 1 can be used to find the parameters which fit the experimental curves of the cohesive energy taken from [4]. The values of A and CYfor each transitional series are given in Table 1. They are not very different from those obtained in [I] assuming V(E) to be constant. Comparison between the theoretical and experimental curves is made in Fig. 2. We see that the overall shapes of the bands agree nicely except for the first series where there are complications corresponding to magnetic effects. Table I. Values of A and CY Is1series P series
(5)
The important quantity in our problem is the density of states per atom v(E) which is symmetric about E = 0. In terms of this we can write the cohesive energy per atom, if p is the number of electrons per atom
(6)
=- 10 j”’ v(e)edc+pa! EFo and epo defining the Fermi level of the perfect crystal, N being the number of atoms. Equations (6) are strictly valid only within a simple Hartree scheme. However, as pointed out in [l], they can be guessed to give a right order of magnitude of EC when one takes correlations into account in a rough way. As E is proportional to 2h, one can deter-
quantity
1 G” V(e)EdE 2h I
(2)
T kinetic energy operator, V1 lattice potential centered on site RI. With our approximations we have only to consider the two following parameters, R, and R0 being first neighbours a -6 = - (CpllVOld
mine the values of the dimensionless
lZh(eV) cr(eV)
5 0.25
6 0.30
jrd
series 7 0.35
(b) Perturbation on the vacancy site. In a first step we want to describe the perturbative potential due to the removal of the central atom. A first method for this is to proceed as in [5]. The perfect crystal Hamiltonian H can be represented in the atomic basis by a N X N matrix whose general element is (~i]H]~j). If H’ is the perturbed Hamiltonian, it can also be represented by a N X N matrix but with vanishing matrix elements on the first line and first column corresponding to ((oolH’ lpi). If we assume that the other matrix elements remain unchanged, this defines clearly the perturbation matrix. One can then solve the perturbed problem using the Green’s operator method. Here we want to introduce a completely
FORMATION
ENERGY
OF VACANCIES
639
P/IO
Fig. I. Curve of EJ20h (neglecting a) vs. p/IO.
b Ta
01
: : : : : :05 I.0 P/IO (a) (b) Fig. 2. Cohesive energy in the three transitional series. (a) fitted curves (A. (Y taken from Table I), (b) experimental curves.
equivalent method which has the advantage of giving immediately the results. Let us assume that we can describe the perturbation by one matrix element p localized on the vacancy site. The eigenvalues of the perturbed problem are given by
:
HOO+P--E HI0, j I
:
:
:
HOI-----H,,,,, = 0.
(7)
M.
640
If we develop this determinant the first line, we can write
LANNOO
with respect to
and
G.
ALLAN
(a) Potential localized at the origin. We first recall the definition of the Green’s operator for the perfect crystal
(Ho0 + p - E)A, + H,,A, . . . * . + H,,,,A, = 0. (8) Now we want to find the value of /3 for which A0 vanishes because this is the condition corresponding to the vacancy problem. We find H,,A, + H,,A, . . . . . + H,,,,A,,, A,, = (9) H,,+P-E .
G”(E)
= lim ’ q-*o+E-HHiq’
(10)
In terms of Go one can derive formally the wave function $(E) of the perturbed system $!J= p+GovlqJ
(11)
where cp is the solution of the unperturbed For finite values of E, the condition A,, = 0 is problem for the same energy E. always achieved for /3 * + w. We want first to examine the occurence of One can easily show that for this value of p bound states. Their energy is given by the the wave function of the perturbed problem relation has no contribution from cp,,.This is consistent det (I - G”V) = 0 with the fact that an infinite repulsive potential (12) on the central site will repel the atomic level which reduces in our simple case to * to plus infinity where it can no more interact with the system. The way we shall solve the problem will 1 -PC& = 0. (13) then be the following: we use a perturbative potential V,, = p localized on the vacancy site If p --, + 00 we find that there are bound states and use the Green’s operator method for only if Gg, vanishes outside the band. Comderiving the perturbed crystal quantities; to putations of G$, whose results are drawn in obtain the values for the vacancy problem we Fig. 3, show that this is not the case. finally let /3 tend to infinity. This we call step 1 We can now calculate %V(E) the variation corresponding to a perturbative potential VI of the total number of states of energy lower whose only matrix element is p. To treat the than E. In our simple model whole perturbative potential V, + V, where V, is the part extending on neighbouring atoms, Sh’l(~) =-AArg(l-PG$,) (14) we shall consider in a second step V, as the perturbative potential on the Hamiltonian H + VI whose solutions will be known after which is a standard result [6]. solving step 1. When p + + 03this is easily reduced to 2. NON
SELF-CONSISTENT
CALCULATION
We first consider here the simplest possible case where the perturbative potential k’, extends only on the vacancy site. We then take into account the fact that the removal of the central atom produces on each nearest neighbour (Fig. 4) of the vacancy site a perturbative potential whose matrix element is equal to (w/6, which corresponds to V,.
SW(E) =-+ArgGi,.
(15)
From this one can determine the formation energy of the vacancy. The method is analogous to that of [3]. We obtain the following result, which is proved in Appendix A. *We
wrik
GP,” for (q,IG’Jq,J.
FORMATION
ENERGY
OF
D i
JPCSVd32Na3-H
VACANCIES
641
642
M. LANNOO
and G. ALLAN
Fig. 4. Vacancy site and its first neighbours in the simple cubic lattice. P/IO
G” -=10
d
c$ 6N’ (E) de -
I
Fig. 5. I$/ZOA vs. p/IO (neglecting V&.
N(e) de (16) Letting /3 tend to infinity, we find
where N(E) is the total number of states of energy up to E. As before the quantity E&/20 A is dimensionless and in Fig. 5 we have plotted it vs. p/10, the filling of the band. Again we have a symmetric curve giving for instance for Mo (when 2h - 1 eV> a value of about 2.4eV which is of the order of the experimental value. In order to be able to solve the problem with V, we must know the new Green’s operator G* corresponding to H+ V,. It is related to Go through G’ = Go + G”I’lG’
Cl’, = Gym- GPoGkn G!o .
(20)
This allows us to determine in principle all the Green’s functions of H+ VI. In particular we find that all Cl,,, with I or m or both being zero identically vanish. This was intuitively evident. (b) Potential on the nearest neighbours. We have to treat now the V, part of the perturbative potential, which in this case gives a matrix element equal to a/6 for each of the six nearest neighbours (Fig. 4). The method we use is exactly the same as (17) for I’r. We can define 8N2(e) which is equal to
which gives for the matrix elements Gj, = G:,,, + G&PC:,,,. Solving first for Gj, we obtain
(18)
6N2(e) =-iArgdet(I-G’V,).
(21)
The difficulty is that we have to work out a 6 X 6 determinant. However at this stage one can simplify the problem by symmetry considerations.
FORMATION
In the cubic central atom functions on vacancy site representation reduced to
ENERGY
group 0,, the s function on the has A,, symmetry. The six s the first neighbours of the form a basis of a reducible r1 of Oh (8). This is easily
r1 =A,,+&+T,,,.
(22)
The basis functions for (22) are denoted ~~~~~~~~PO and cp,(&,), (ox, ‘pu, p,(T,,). They are derived in Appendix B. In this basis G’ is diagonal and has only three independent matrix elements. Gf,, G& = G:, called G&. G:, = G&,= G& called CL.
OF
VACANCIES
643
part 1.26. Then both the real and imaginary parts of 01/6 G’ are of maximum order 0.06. This allows us to write ~W(E)
= ;-fZm{G;s+2G&+3G&} = z ZmG:, (Appendix
B)
(26)
where G,, is an atomic Green’s function on the first neighbours. From what we have seen there can be no bound states. Then relation (16) still holds for the formation energy with [IN’
(23)
+ SN’(e)]
de
s#.o
The expression of the Green’s functions in terms of the atomic ones is also given in Appendix B. By symmetry, one immediately sees that only GiS differs from G,” (because P can only mix terms belonging to the same irreducible representation).’ From what we have seen before we obtain
In this basis, Vz is quite simple. It is a 6 x 6 unit matrix multiplied by a/6. From this one finds 8N2(e) =--b(Arg(l-fGjS)+2Arg(l--
-
N(E) de
E& is then increased with respect to E,L, by an amount
_ 10Y EPImG:, T
Arg(l--FGL)}.
(25)
From Table 1, we see that the OLwe have chosen are proportional to A with (Y= 0.6A. As we do the whole calculation in units of 2A, the term (u/6 has the value O-05. In units of (2A)-’ the maximum value for the real part of the G’ functions is 1.12 and for the imaginary
de.
W-9
To determine this quantity we can make use of (20). In three simple cases p = 0, p = 5, p = 10 we find that (28) reduces to aN (eFo). For p = 0 this vanishes. For p = 5 it gives 1.5 in units of 2A, i.e. I.5 eV for MO giving a total value of 3.9 eV for EFv. For p = 10 we obtain a value of order 3 eV which seems pretty high for the end of the series. In Fig. 6 we have plotted as usual the curve of the dimensionless quantity E&,/20h as a function of p/10. It has a shape quite similar to the cohesive energy curve of Fig. 2. 3. SELF-CONSISTENT
iGk,)f3
(27)
I
CALCULATION
We shall determine SN(+O) for the two cases of part 2 and comment the results. We shall then do a self-consistent calculation in two cases; (a) neglecting CY/~on the nearest neighbours, (b) taking it into account. (a) Non self-consistent 8N(q0). We know that in a metal the perturbative potential due to an isolated defect must be completely
644
M.
Y
0
:
LANNOO
:
and
:
;
G.
6.5
ALLAN
*
’
co
P/IO Fig. 6.
E320h
vs. p/IO
screened out over a few interatomic distances, especially for transition metals. The Fermi level is then the same as for the perfect crystal. A self-consistent calculation must satisfy Friedel’s rule, i.e. in the vacancy case 6N(+‘)
=-N(+‘).
(29)
An estimate of the failure to self-consistency for the preceeding calculations is then immediately given by the quantity {SN(eFO) +N(EF’)}. We have computed this term for the first case of part 2. We see from Fig. 7 that its maximum value is of order 0.1. Another interesting quantity to determine is the change in electronic density due to the creation of the vacancy. In the tight-binding approximation the main contribution comes from intraatomic terms, i.e. we can write, i being the electron index
I*(r) I2= 7 [ F M21qb(r)1’ + x alar’ cp[(r) cp; (r) 1.1’
I
and consider the first term as preponderant.
(30)
(including
V2J.
A measure of the charge variation Ch atom is then given by SN;=~~~lai(~)12-~IUiO(E)i’.
E
on the
(31)
Neglecting the l/N variation of l F one can easily show that this quantity is equal to &jI; a;
I
“‘{Zfl(E) -IB(e))dE
(32)
where I designs the imaginary part of G. Making use of (20) this reduces to SN: = + i Zm
fFoG&G,Oi Fde. I 00
(33)
We have plotted this quantity in Fig. 7 as a function of p/ 10 for the first and fourth nearest neighbours of the vacancy site. From all the results we see that the V, part of the perturbative potential is too repulsive when 0 < p < 5 and not repulsive enough for 5 < p < 10 which is the symmetric situation changing electrons to holes. If we add ~46 we have to evaluate (26). In Appendix B, we show that 8N2(e) is of the
FORMATION
ENERGY
645
OF VACANCIES
t
0.
l--
j--
0
P/IO
-c
Fig. 7. Curves representing as a function of p/10; (1) &V(Q) + N(Q), (2) (d/dE)GN; = - l/v(f!,,- I$) (first neighbours), (3) SN; (atom [lOO]), (4) SNV;(atom [200]).
order of -(Yv(E), i.e. at most equal to -0.2. It is always negative (repulsive potential). We then see the necessity of doing a self-
consistent calculation. This can alter the energies and change the repartition around the vacancy. For this we shall say that the
646
M.
LANNOO
and
V, part of the perturbative potential is the sum of a ‘bare’ part V,, (corresponding to a/6 on the first neighbours) and a ‘clothing part’ AV, self-consistently related to the charge density. We shall assume that AV, extends only on the nearest neighbours giving rise to one matrix element 6 on each of these. (b) Self-consistent calculation neglecting Vzb. We have only AVz to take into account. To satisfy Friedel’s rule one has to know ISN*(+~). As we have a 6 on each nearest neighbour, the problem is exactly the same as for the treatment of a/6 in part 2 except that we do not know if 6 is small. We then have simply to replace a/6 by 6 in (25). From Friedel’s rule
G.
ALLAN
6/2h as a function of p/10. We find that no bound states can occur. The curve of 6/2h is given in Fig. 8. The values are antisymmetric with respect to p/l0 = 0.5. To derive the formation energy we shall use the self-consistent Hartree approximation for both the perturbed and unperturbed crystal. For this let us define the following quantities for the perfect crystal: po(rj) the electronic density of theJth electron and V,(r,) the potential acting on this electron and due to all others. In the perturbed crystal these quantities are changed into po(rj) + Ap (rj) and V&j)+AV2(rj). If we now calculate the formation energy, we find: Ew=C
6N(e,O) = 6N’(ej?0) +SN*(ejP) =-iv(egy we can numerically
compute
Ej-iz
I [po(rj)+Ap(r~)l
: [Vo(rj)‘+AV2(r,)l
drj
(34)
the values of
Fig. 8.6/2h
-E
vs. p/ IO.
j
EjO+iE
j
J [po(rj)Vo(rj)Id~,+& (35)
FORMATION
ENERGY
where C Ej and z Ej” are the sums of oneelectron) energies: E, is the energy due to the atom put on the surface and the other terms come from the fact that, when summing the one-electron energies, we count twice the Coulomb interactions. Now it is possible to show (Appendix C) that: 2 4-Z j
I
OF
VACANCIES
647
fact that ]a,o]2 is equal to l/N and from the expression of A~u,~~ (Appendix C) we can compute the formation energy from the formula: E FI’10 --
I +’ [N(E) +SN(e)]
de-6N(e,O)6
Ej”+Es =-
lOlEd [N(E) +SN(E)]
de. (36)
The two other terms combine to give i
+b(rhA~z/,(l;)ld7j.
(37)
At this stage we must point out that AI’, and V,(r,) are in principle different for each electron j. However this difference being of order l/N can be neglected. From this it is easy to show that the two first terms in (37) give an equal contribution. (37) can be rewritten
/4(q)AV,(rj) drj. (38) I Here we must notice that in p. and V, we do not consider the contribution coming from the central atom, because it is expected to cancel with the similar contribution due to the same atom when it is put on the surface. From this and using the fact that the wave function is a combination of atomic orbitals, we can rewrite (3 8) under the form:
The numerical results show practically no difference with the non self-consistent case within 1 per cent. Again EFv tends to zero for p = 0 and p = 10. We have complete symmetry for electrons and holes. We want now to take into account the fact that EFv tends towards an appreciable value for p nearly equal to ten. This is probably due to the V,, part of the perturbative potential. (c) Self-consistent calculation including VZb. The only change is that we have now a/6 +6 on each first neighbour. It is this quantity which we call 6’ which has now to displace the right number of states at the Fermi level. S’ has then to take exactly the same values as 6 before (Fig. 8). For a given Fermi level the first term in (40) remains unchanged. The change occurs in the last two terms where 6 is now different, i.e.
-42
where la,o]2 is the perfect crystal contribution to ‘pl and A]u,]~ is the change of this quantity in the perturbed crystal (the factor 6 occurs because we have an equal contribution from the 6 nearest neighbours). From the
S=S’-;
(41)
6’ giving the same value as before, we can say that the formation energy is simply given by the value obtained when neglecting V,, to which we must add the following contribution N(+O)a+;
C A~u~~~. C-X+.0
(42)
Again we find no difference between the numerical values obtained in this way and the results obtained in the non se&consistent case taking V, into account. In order to know the charge variation on the first neighbours we give in Fig. 9 the curve of
648
M. LANNOO
and G. ALLAN
0.2
0.1 % sa L
c
t+
-0.1
-0.2
,-.
Fig. 9. ,c”, Ala, Iz vs. p/l0 (1) non self-consistent, (2) self-consistent. F
the term C A]u,]~, which is analogous with SiV; in (3 2j<” It can be written in a similar form -- 1 “’ {I;, -Z;,}de. 7r J In Fig. 9, we have plotted this quantity as well as &Vi obtained in the non self-consistent case. Comparison shows that self-consistency has the effect of reducing the change in charge density on the first neighbours. 4. DISCUSSION
If we neglect the term Vz6, we get values of the same order of magnitude as other authors [ 131 but it is difficult to compare with the experiment. There are very few experimental results on the formation energy of vacancies in transition metals and some of them are
still controverted. However we compare those which are available at the moment with our predicted values in Table 2. We can notice that the calculations neglecting the Vz6 term give a good agreement at the middle of the series (MO, W) but not at all at the end (Ni, Pt). One could then think that the formation energy could be completely due to the s band or to s-d mixing but this is hard to believe in view of the rather high formation energies (of the order of 1.5 eV). On the other hand calculations including Vz6 lead to quite large values. There are different possibilities to be considered. The crudeness of the approximations: tight-binding method, Hartree approximation, 5 s bands in a simple cubic lattice. We do not believe this is the main reason because the same considerations applied to the calculation of the surface tension of Pt, Pd, Ni would
FORMATION
ENERGY
lead to a value of about O-5 eV the experimental value being about O-4 eV. The occurence of an important relaxation: this is probably the case as shown by the comparison with the surface tension problem.
OF VACANCIES
649
argument is true for (Ywhich is an average of the three independant ai occuring in the exact d band. (c) Hartree approximation. This is perhaps the most serious approximation here. What
Table 2. Comparison between the experimental EFv in eV and the predicted one (a) Vz,, = 0, (b) V,, # 0. We also indicate the number of d electrons :p
EXP
Fe 1*5[7]
Ni = 1.4[7,81=
3%.8*91
(a) N.S.C. (b)
1.5 3.4
0.7 3.1 9.4
2.8 4.5 5
P
7.8
=4’.5,,1 0.7 4.1 9.6
MO -2t[7,8,9] 2.4 3.9 5
Nb - 1~8[10,11] 2.3 3.4 4
*One value of 4.4 eV has been reported [Ref. [ 1211. tThe values in that case go from 1.5 to 2.9 eV. The last value appears to be more plausible when comparing to W and Pt and Ni.
The evaluation of the energy gained when putting the vacancy atom on the surface: it is perhaps slightly different of the cohesive energy. It could also be interesting to evaluate the energy when the atom is put in an interstitial site. Let us now examine some of the approximations which have been used: (a) Influence of lattice. Let us compare our case with five s bands in a f.c.c. structure. For the V, part the result would be formally the same the differences coming from minor changes in the shape of Y(E). We then expect for Ei, values quite similar to ours. Now the term LYwill be spread over 12 nearest neighbotus (al 12 on each of them) adding also a total amount of OLto E;,, and giving the same value at the end of the series. (b) Influence of degeneracy. If we now take five d orbitals instead of five independent s ones, they are now coupled together in the band structure. However for the VI part of the perturbative potential, one can by symmetry split the problem in five independent parts. We expect that our simplified problem gives an average of the total contribution of these five parts, as for the cohesive energy. The same
we can hope is that the effect of correlations cancel when we substract the energies of the perturbed and unperturbed crystal. Let us now return to the comparison between the predicted and experimental results. We came to the conclusion that the relaxation energy was probably large. This seems also to be the experimental conclusion [Hoch Ref. [7], p. 51 from the large values taken by the entropy of formation of the vacancy. From our values the relaxation energy would be of the order of 0.9 to 1.5 eV. We can also notice that the shape of the predicted EFv curve as a function of the filling of the band seems to follow the experimental values satisfactorily. It tends to zero for zero filling as seems to be shown by the weak values for K and Na which are respectively O-4 and 0.6eV. It gives a maximum value near the middle of the series (highest experimental values available) and ends up to a large value for Pt and Ni. We must still mention that there is uncertainty in the experimental results and that, for instance for MO and W measured values respectively as high as 2.9 and 4.4 eV have been reported.
M. LANNOO
650 5. CONCLUSION
and G. ALLAN REFERENCES
In this paper we have determined the FRIEDEL J. in “The Physics ofMeto/s” (Edited by Ziman). p. 340. Cambridge University Press (1969). formation energy of a vacancy in transition FRIEDFL J.. Nuouo Cik. Suppl.. 7.187 (1958). metals using a tight-binding method and CYROT-LACKMANN F.. J. Phvs. C/rem. So/ids approximating the true d band to five indepen29.123S(l968). GSCHNEIDER K. A.. Solid State Phys. 16. 275 dent s bands in a simple cubic lattice. This (1965). allowed us to do a self-consistent calculation LANNOO M. and LENGLART P. (to be pubup to the first neighbours in the Hartree lished in J. Phvs. Chem.Solids, LANNOO M., These, Ann. Phys. 3, no 5 ( 1968). approximation. 6. GAUTIER F. and LENGLART P.. Phys. Rev. 139 Basically the results we find are larger than 3A,70S (1965). the few available experimental values. This 7. International Conference on vacancies and interstitials in metals (preprints) Jiilich (I 968). we think is probably due to a large amount of 8. CO-II-ERILL R. M. J.. DOYAMA M.. JACKSON relaxation in the neighbourhood of the J. J.. JACKSON J. J.. and MESH11 M.. “Laffice vacancy site. This conclusion seems to be in defects in quenched metals”. Academic Press. New York(l965). agreement with experimental observations. 9. THOMSON M. W., “Defecrsandliadiarion Damage The curves we predict for the vacancy formain Mefals”. Cambridge University Press ( 1969). tion energy as a function of the filling up of 10. CHEKHOVSKOY V. YA.. ZHUKOVA I. A.. Fiz. rverd. Tela. 8.9 ( 1966). the band seem to show the right experiI I. KRAFTMAKHER YA. A.. Fiz. tverd. Tela.~.~. 5.950~~ mental behaviour: They start at small values (1963). 12. FRIEDEL J.. in “The Interaction for small filling, having a maximum value of radiation with solids”. Proceedings of the Mol Summer School. about the middle of the series and ending North Holland Amsterdam, p. I35 ( 1964). at a finite large value (typically 3 eV for the 13. GERL M..J. Plrvs. Chem. Solids (to be published). predicted values, 1.5 eV for the experimental ones). The next step in such calculations would be APPENDIXA to use the true d band in tight-binding. HowThe formation energy 4,. can be formally written ever it will probably not affect too much the values obtained here and we are not sure it E;,,,= 10~E’{N.u(E-EO+a)+Gvl(E-E~+a)}EdE can be worth doing such an improvement - 10IEI”N. v(E-E,,+a)EdE+Es (A.11 owing to the complications which are then introduced. Certainly at the moment the 6~’ being the variation of the density of states, i.e. the more desirable calculation would be an been put on the surface, and EF is the new Fermi level defined by estimate of the relaxation energy which could give useful points of comparison. (N- I) $= IE’{N. v(E-E,+(Y) Let us finally notice that our simplified +6v’(E-&+a)}dE (A.21 model could in a straightforward way be used to derive energies of more complicated defects, 8~’ being the variation of the density of states, i.e. the of 6Nr. binding energy of vacancy complexes for derivative Now one can simplify (A. 1) and (A.?) neglecting terms instance. of order l/N. This gives, as EF- Eye is a l/N quantity It should be pointed out that though selfE;,r= ION .~(E~O)(EF-~~O)(E~~+E”.-~) consistency seems to affect very little the energies for vacancies, it must give rise to an +IO~~“SY~(E)((+E~-~)+E~ (A.31 essential change for substitutional impurities. -p = ION . v(eFo) (eF-eFo) + 10 I”’ W(E) de. Along the lines given here, one could easily (A.41 derive for instance the heat of solution of To obtain A.3 and A.4. we have used relation (5) which transition impurities in transition metals in defines E. Putting (A.4) into (A.3) we obtain: the limit of zero dilution.
FORMATION
ENERGY
From
Ej.,. = -p(epO+EEU-a) + 1o~+“W(e)(~-e~~~) Integrating obtain:
by parts.
OF
de+&.
and as there
VACANCIES this
we can derive
states.
we
which
is used
order
(B.3)
Gs+2G,,+3G,
ualue
ofSN2(eF0)
6N*(e)
of the From
relation
in (26).
(B.3) Appraximare We start from
NOW. one can expect pE, - Es to be of the cohesive energy E,. From (6) we then obtain
the following
6G,.,=
(A.5)
are no bound
651
ia (26)
= ;I;,.
(20) we can write (B.4)
If we now
integrate
the 2” term
by parts
we find: From
8N’(e)
I
N(e)
de-
de.
of the perfect
(A.81
APPENDIX
B
.
BN*(e)
We start from the atomic orbitals vpi(i = k 1, -C 2, * 3 in the notations of Fig. 5). Standard symmetry considerations easily lead to
i= -av(eF~).
91 APPENDIX
E,,: cp, =
2VX
a,=-&
C
(C.1) Sum of the one-electron energies in the se& consistenl case As in Appendix A, we want to determine the sum of the one-electron energies plus the surface term, but now in the self-consistent case i.e. without variation of the Fermi level.
91+9-1-9?-9--2
T,,,: 9, = -!v5
2 (91-9-r)
C E,+E,= J
(9p?-9-z)
10jEro{N
.v(E-E,+(y)
+&@--E,,+a))EdE+E,. 9* = Jx4
(9s - 9-3).
(B.1)
Now, crystal z
(8.2) Symmetric Green’s Jimctions The relations we shall derive are valid for every Green’s operator we consider (GO. Cr. C*). We shall then omit the index. If we call GU the matrix elements of G between atomic orbitals centered on the nearest neighbours, there are obviously three independent Gu which are for instance G,,,. G,.-, and Cl.,. In terms of these, and using (B. I) we can write
clearly energy.
the first Then
E,+E,-+
E,O=
term
(B.2)
integral
is the
perfect
lO~%v(E-E,,+a)EdE+E, =
101E”6v(E)(e+E,--)de+E,
=
lOj?h+)edr + 10(EO--~)SN(rFo)
From
Friedel’s
We then
+4c,.*
in the
(Cl)
J
+ E,.
cc.21
rule N(Q)
GM = G,,, +GL-, GEE=GII+GI.-l-2G1.* Grr= G,,-G,.-,.
(B.6)
In Fig. 7 we have plotted the second term (B.6) which is at most equal to O+t. It is always negligible compared to Y(E) which is given in Fig. 3. We have then to a good approximation SN*(e$‘)
3($9,-t+,)--
(B.5)
we obtain
@-I-,
9r =
crystal
/p, = 1;” =-TV(E). Then
(B. I ) Busisfunrrions
the periodicity
cp
6”
E&=IO
= --p/10.
(C.3)
obtain: lOl~~‘Sv(r)tda+E,--p(E,--a).
(C.4)
652
M.
We now form
follow
Appendix
10 I”’ Integrating
A and can write
[&(E)
this by parts
+Y(e)]
easily
lO[N(+‘)
this under
ede.
and the
Now given
+6N(e)]dc. first
term
vanishes
and
out to be the same
to determine~~~~~A~a,~*.
6G:,
the Each
turns
let us try by:
This
quantity
is
where I:, is the imaginary part of the matrix element of the Green’s operator of the perturbed problem. We now want to show how to compute the first term from the functions we already know. We first have (Appendix B)
(C.6)
-lOl”“[N(~)+bN(r)]de which
ALLAN
(C.5)
-t6N(+“)]Q
the
G.
gives
-lO~“‘[N(e) From self-consistency final expression is
LANNOO
of these
three
= G:, + 2G& Green’s
+ 3G&.
function
(Cl
is simply
given
I)
by
(C.7) as in Appendix
(C.2) Determination of]a,,]* andA]a,]* From the fact that the perfect crystal of Bloch type we have
A. I$,, can then be written
wave
function
is
Gm = CC.81
(C.13)
We can then write
1:s GE G-r q., = , ~1--SC:,~‘+211-6G1,1~+31*--SG~I’~
giving (C.9)
Gnl (1--8G$,p
As term
we know
the
Gr quantities
we
can
compute
the I:,
Churn.
Solids
Pergamon
Press
FORMATION
I97 I. Vol.
ENERGY TRANSITION M.
Physique
32, pp. 637-652.
des Solidest-
(Received
in Great
Britain.
OF VACANCIES METALS:”
LANNOO
and
I.S.E.N.,
Printed
IN
G. ALLAN
3, rue FranGois
20 February
Bats-59
Lille,
France
1970)
Abstract-The
formation energies of vacancies are calculated in a simple tight-binding approximation. Friedel’s rule is introduced for self-consistency up to the nearest neighbours of the vacancy site. The first result is that self-consistency does not practically alter the energies. Comparison is made with the experimental results and leads to the following conclusions:-reasonable shape of the predicted curve -values higher than the experimental ones by an amount of I to 2 eV. This suggests that relaxation has to be important as is pointed out in some experimental works.
INTRODUCTION
IN THIS paper we would
like to estimate the order of magnitude of the formation energy for a vacancy in transition metals. To do this, it is necessary to choose a simple method for treating the d electrons in the perfect lattice. The easiest way is to use the tight-binding approximation which has proved to be of semi-quantitative value [ l] for the calculation of the cohesive energy. As the calculation of energies does not seem to depend very much on the details of the density of states curve n(E) and of the lattice structure, we shall simplify the problem by treating a simple cubic lattice, with five independent s orbitals per atom. We shall discuss this approximation at the end of the paper. We shall use here the Green’s ouerator method to relate the perturbed crystal to the unperturbed one. The perturbation will be assumed to extend up to the first neighbours of the vacancy site. In a first part we give a brief description of the band structure and of the choice of the parameters; then we detaiI how it is possible to describe the removal of an atom as we create a vacancy. *This work was done Harwell (England). tEquipe de Recherche
during
a stay
at the A.E.R.E.
of
In the second part we describe a non selfconsistent calculation of the vacancy introducing a variation of the Fermi level which is not physical for an isolated perturbation in a metal. In the third part we introduce the selfconsistency condition given by the Friedel Sum Rule[2]. From this condition we obtain the value of the perturbative potential on the first neighbours and we calculate the formation energy in the Hartree approximation. Finally we compare our results to the experimental values and discuss the approximations introduced. Let us however immediately point out that typical values of formation energies are about 2 to 4 eV for the middle of the transitional series and about 1.5 at the end. 1. BAND STRUCTURE AND PERTURBATIVE POTENTIAL SITE
DEFINITION ON THE
OF THE VACANCY
(a) Band structure parameters. We use the simplest tight-binding approximation, neglecting overlap integrals between the atomic wave functions, and taking into account interactions up to first neighbours only. If we call Ipj) the pseudo-orthonormal set of atomic functions centered on site Ri, the Bloch function can be written
du C.N.R.S.
kfk)
637
=
7
e ‘kIG
Iv,).
(1)
M.
638
The expression of the Hamiltonian perfect crystal is H=T+E
I
and G. ALLAN
LANNOO
v,
for the
(3)
A=-(cp,IVolvo). These lead to the band structure energy E, = EO- (Y- 2X(cos k,a + cos k,a + cos k,a) (4) where E0 is the atomic energy and a the lattice parameter. For our purposes it is convenient to change the origin of the energies and define E = E-
(E,,-CY).
EC =-;
x Ek+pE,, kocc
= 10J” Y(E-EO+cu)(E,,-E)dE
as a function of p/10, the filling up of the band. From the computed numerical values we have easily deduced the curve of Fig. I. The values obtained are in agreement with those of [3]. The results of Fig. 1 can be used to find the parameters which fit the experimental curves of the cohesive energy taken from [4]. The values of A and CYfor each transitional series are given in Table 1. They are not very different from those obtained in [I] assuming V(E) to be constant. Comparison between the theoretical and experimental curves is made in Fig. 2. We see that the overall shapes of the bands agree nicely except for the first series where there are complications corresponding to magnetic effects. Table I. Values of A and CY Is1series P series
(5)
The important quantity in our problem is the density of states per atom v(E) which is symmetric about E = 0. In terms of this we can write the cohesive energy per atom, if p is the number of electrons per atom
(6)
=- 10 j”’ v(e)edc+pa! EFo and epo defining the Fermi level of the perfect crystal, N being the number of atoms. Equations (6) are strictly valid only within a simple Hartree scheme. However, as pointed out in [l], they can be guessed to give a right order of magnitude of EC when one takes correlations into account in a rough way. As E is proportional to 2h, one can deter-
quantity
1 G” V(e)EdE 2h I
(2)
T kinetic energy operator, V1 lattice potential centered on site RI. With our approximations we have only to consider the two following parameters, R, and R0 being first neighbours a -6 = - (CpllVOld
mine the values of the dimensionless
lZh(eV) cr(eV)
5 0.25
6 0.30
jrd
series 7 0.35
(b) Perturbation on the vacancy site. In a first step we want to describe the perturbative potential due to the removal of the central atom. A first method for this is to proceed as in [5]. The perfect crystal Hamiltonian H can be represented in the atomic basis by a N X N matrix whose general element is (~i]H]~j). If H’ is the perturbed Hamiltonian, it can also be represented by a N X N matrix but with vanishing matrix elements on the first line and first column corresponding to ((oolH’ lpi). If we assume that the other matrix elements remain unchanged, this defines clearly the perturbation matrix. One can then solve the perturbed problem using the Green’s operator method. Here we want to introduce a completely
FORMATION
ENERGY
OF VACANCIES
639
P/IO
Fig. I. Curve of EJ20h (neglecting a) vs. p/IO.
b Ta
01
: : : : : :05 I.0 P/IO (a) (b) Fig. 2. Cohesive energy in the three transitional series. (a) fitted curves (A. (Y taken from Table I), (b) experimental curves.
equivalent method which has the advantage of giving immediately the results. Let us assume that we can describe the perturbation by one matrix element p localized on the vacancy site. The eigenvalues of the perturbed problem are given by
:
HOO+P--E HI0, j I
:
:
:
HOI-----H,,,,, = 0.
(7)
M.
640
If we develop this determinant the first line, we can write
LANNOO
with respect to
and
G.
ALLAN
(a) Potential localized at the origin. We first recall the definition of the Green’s operator for the perfect crystal
(Ho0 + p - E)A, + H,,A, . . . * . + H,,,,A, = 0. (8) Now we want to find the value of /3 for which A0 vanishes because this is the condition corresponding to the vacancy problem. We find H,,A, + H,,A, . . . . . + H,,,,A,,, A,, = (9) H,,+P-E .
G”(E)
= lim ’ q-*o+E-HHiq’
(10)
In terms of Go one can derive formally the wave function $(E) of the perturbed system $!J= p+GovlqJ
(11)
where cp is the solution of the unperturbed For finite values of E, the condition A,, = 0 is problem for the same energy E. always achieved for /3 * + w. We want first to examine the occurence of One can easily show that for this value of p bound states. Their energy is given by the the wave function of the perturbed problem relation has no contribution from cp,,.This is consistent det (I - G”V) = 0 with the fact that an infinite repulsive potential (12) on the central site will repel the atomic level which reduces in our simple case to * to plus infinity where it can no more interact with the system. The way we shall solve the problem will 1 -PC& = 0. (13) then be the following: we use a perturbative potential V,, = p localized on the vacancy site If p --, + 00 we find that there are bound states and use the Green’s operator method for only if Gg, vanishes outside the band. Comderiving the perturbed crystal quantities; to putations of G$, whose results are drawn in obtain the values for the vacancy problem we Fig. 3, show that this is not the case. finally let /3 tend to infinity. This we call step 1 We can now calculate %V(E) the variation corresponding to a perturbative potential VI of the total number of states of energy lower whose only matrix element is p. To treat the than E. In our simple model whole perturbative potential V, + V, where V, is the part extending on neighbouring atoms, Sh’l(~) =-AArg(l-PG$,) (14) we shall consider in a second step V, as the perturbative potential on the Hamiltonian H + VI whose solutions will be known after which is a standard result [6]. solving step 1. When p + + 03this is easily reduced to 2. NON
SELF-CONSISTENT
CALCULATION
We first consider here the simplest possible case where the perturbative potential k’, extends only on the vacancy site. We then take into account the fact that the removal of the central atom produces on each nearest neighbour (Fig. 4) of the vacancy site a perturbative potential whose matrix element is equal to (w/6, which corresponds to V,.
SW(E) =-+ArgGi,.
(15)
From this one can determine the formation energy of the vacancy. The method is analogous to that of [3]. We obtain the following result, which is proved in Appendix A. *We
wrik
GP,” for (q,IG’Jq,J.
FORMATION
ENERGY
OF
D i
JPCSVd32Na3-H
VACANCIES
641
642
M. LANNOO
and G. ALLAN
Fig. 4. Vacancy site and its first neighbours in the simple cubic lattice. P/IO
G” -=10
d
c$ 6N’ (E) de -
I
Fig. 5. I$/ZOA vs. p/IO (neglecting V&.
N(e) de (16) Letting /3 tend to infinity, we find
where N(E) is the total number of states of energy up to E. As before the quantity E&/20 A is dimensionless and in Fig. 5 we have plotted it vs. p/10, the filling of the band. Again we have a symmetric curve giving for instance for Mo (when 2h - 1 eV> a value of about 2.4eV which is of the order of the experimental value. In order to be able to solve the problem with V, we must know the new Green’s operator G* corresponding to H+ V,. It is related to Go through G’ = Go + G”I’lG’
Cl’, = Gym- GPoGkn G!o .
(20)
This allows us to determine in principle all the Green’s functions of H+ VI. In particular we find that all Cl,,, with I or m or both being zero identically vanish. This was intuitively evident. (b) Potential on the nearest neighbours. We have to treat now the V, part of the perturbative potential, which in this case gives a matrix element equal to a/6 for each of the six nearest neighbours (Fig. 4). The method we use is exactly the same as (17) for I’r. We can define 8N2(e) which is equal to
which gives for the matrix elements Gj, = G:,,, + G&PC:,,,. Solving first for Gj, we obtain
(18)
6N2(e) =-iArgdet(I-G’V,).
(21)
The difficulty is that we have to work out a 6 X 6 determinant. However at this stage one can simplify the problem by symmetry considerations.
FORMATION
In the cubic central atom functions on vacancy site representation reduced to
ENERGY
group 0,, the s function on the has A,, symmetry. The six s the first neighbours of the form a basis of a reducible r1 of Oh (8). This is easily
r1 =A,,+&+T,,,.
(22)
The basis functions for (22) are denoted ~~~~~~~~PO and cp,(&,), (ox, ‘pu, p,(T,,). They are derived in Appendix B. In this basis G’ is diagonal and has only three independent matrix elements. Gf,, G& = G:, called G&. G:, = G&,= G& called CL.
OF
VACANCIES
643
part 1.26. Then both the real and imaginary parts of 01/6 G’ are of maximum order 0.06. This allows us to write ~W(E)
= ;-fZm{G;s+2G&+3G&} = z ZmG:, (Appendix
B)
(26)
where G,, is an atomic Green’s function on the first neighbours. From what we have seen there can be no bound states. Then relation (16) still holds for the formation energy with [IN’
(23)
+ SN’(e)]
de
s#.o
The expression of the Green’s functions in terms of the atomic ones is also given in Appendix B. By symmetry, one immediately sees that only GiS differs from G,” (because P can only mix terms belonging to the same irreducible representation).’ From what we have seen before we obtain
In this basis, Vz is quite simple. It is a 6 x 6 unit matrix multiplied by a/6. From this one finds 8N2(e) =--b(Arg(l-fGjS)+2Arg(l--
-
N(E) de
E& is then increased with respect to E,L, by an amount
_ 10Y EPImG:, T
Arg(l--FGL)}.
(25)
From Table 1, we see that the OLwe have chosen are proportional to A with (Y= 0.6A. As we do the whole calculation in units of 2A, the term (u/6 has the value O-05. In units of (2A)-’ the maximum value for the real part of the G’ functions is 1.12 and for the imaginary
de.
W-9
To determine this quantity we can make use of (20). In three simple cases p = 0, p = 5, p = 10 we find that (28) reduces to aN (eFo). For p = 0 this vanishes. For p = 5 it gives 1.5 in units of 2A, i.e. I.5 eV for MO giving a total value of 3.9 eV for EFv. For p = 10 we obtain a value of order 3 eV which seems pretty high for the end of the series. In Fig. 6 we have plotted as usual the curve of the dimensionless quantity E&,/20h as a function of p/10. It has a shape quite similar to the cohesive energy curve of Fig. 2. 3. SELF-CONSISTENT
iGk,)f3
(27)
I
CALCULATION
We shall determine SN(+O) for the two cases of part 2 and comment the results. We shall then do a self-consistent calculation in two cases; (a) neglecting CY/~on the nearest neighbours, (b) taking it into account. (a) Non self-consistent 8N(q0). We know that in a metal the perturbative potential due to an isolated defect must be completely
644
M.
Y
0
:
LANNOO
:
and
:
;
G.
6.5
ALLAN
*
’
co
P/IO Fig. 6.
E320h
vs. p/IO
screened out over a few interatomic distances, especially for transition metals. The Fermi level is then the same as for the perfect crystal. A self-consistent calculation must satisfy Friedel’s rule, i.e. in the vacancy case 6N(+‘)
=-N(+‘).
(29)
An estimate of the failure to self-consistency for the preceeding calculations is then immediately given by the quantity {SN(eFO) +N(EF’)}. We have computed this term for the first case of part 2. We see from Fig. 7 that its maximum value is of order 0.1. Another interesting quantity to determine is the change in electronic density due to the creation of the vacancy. In the tight-binding approximation the main contribution comes from intraatomic terms, i.e. we can write, i being the electron index
I*(r) I2= 7 [ F M21qb(r)1’ + x alar’ cp[(r) cp; (r) 1.1’
I
and consider the first term as preponderant.
(30)
(including
V2J.
A measure of the charge variation Ch atom is then given by SN;=~~~lai(~)12-~IUiO(E)i’.
E
on the
(31)
Neglecting the l/N variation of l F one can easily show that this quantity is equal to &jI; a;
I
“‘{Zfl(E) -IB(e))dE
(32)
where I designs the imaginary part of G. Making use of (20) this reduces to SN: = + i Zm
fFoG&G,Oi Fde. I 00
(33)
We have plotted this quantity in Fig. 7 as a function of p/ 10 for the first and fourth nearest neighbours of the vacancy site. From all the results we see that the V, part of the perturbative potential is too repulsive when 0 < p < 5 and not repulsive enough for 5 < p < 10 which is the symmetric situation changing electrons to holes. If we add ~46 we have to evaluate (26). In Appendix B, we show that 8N2(e) is of the
FORMATION
ENERGY
645
OF VACANCIES
t
0.
l--
j--
0
P/IO
-c
Fig. 7. Curves representing as a function of p/10; (1) &V(Q) + N(Q), (2) (d/dE)GN; = - l/v(f!,,- I$) (first neighbours), (3) SN; (atom [lOO]), (4) SNV;(atom [200]).
order of -(Yv(E), i.e. at most equal to -0.2. It is always negative (repulsive potential). We then see the necessity of doing a self-
consistent calculation. This can alter the energies and change the repartition around the vacancy. For this we shall say that the
646
M.
LANNOO
and
V, part of the perturbative potential is the sum of a ‘bare’ part V,, (corresponding to a/6 on the first neighbours) and a ‘clothing part’ AV, self-consistently related to the charge density. We shall assume that AV, extends only on the nearest neighbours giving rise to one matrix element 6 on each of these. (b) Self-consistent calculation neglecting Vzb. We have only AVz to take into account. To satisfy Friedel’s rule one has to know ISN*(+~). As we have a 6 on each nearest neighbour, the problem is exactly the same as for the treatment of a/6 in part 2 except that we do not know if 6 is small. We then have simply to replace a/6 by 6 in (25). From Friedel’s rule
G.
ALLAN
6/2h as a function of p/10. We find that no bound states can occur. The curve of 6/2h is given in Fig. 8. The values are antisymmetric with respect to p/l0 = 0.5. To derive the formation energy we shall use the self-consistent Hartree approximation for both the perturbed and unperturbed crystal. For this let us define the following quantities for the perfect crystal: po(rj) the electronic density of theJth electron and V,(r,) the potential acting on this electron and due to all others. In the perturbed crystal these quantities are changed into po(rj) + Ap (rj) and V&j)+AV2(rj). If we now calculate the formation energy, we find: Ew=C
6N(e,O) = 6N’(ej?0) +SN*(ejP) =-iv(egy we can numerically
compute
Ej-iz
I [po(rj)+Ap(r~)l
: [Vo(rj)‘+AV2(r,)l
drj
(34)
the values of
Fig. 8.6/2h
-E
vs. p/ IO.
j
EjO+iE
j
J [po(rj)Vo(rj)Id~,+& (35)
FORMATION
ENERGY
where C Ej and z Ej” are the sums of oneelectron) energies: E, is the energy due to the atom put on the surface and the other terms come from the fact that, when summing the one-electron energies, we count twice the Coulomb interactions. Now it is possible to show (Appendix C) that: 2 4-Z j
I
OF
VACANCIES
647
fact that ]a,o]2 is equal to l/N and from the expression of A~u,~~ (Appendix C) we can compute the formation energy from the formula: E FI’10 --
I +’ [N(E) +SN(e)]
de-6N(e,O)6
Ej”+Es =-
lOlEd [N(E) +SN(E)]
de. (36)
The two other terms combine to give i
+b(rhA~z/,(l;)ld7j.
(37)
At this stage we must point out that AI’, and V,(r,) are in principle different for each electron j. However this difference being of order l/N can be neglected. From this it is easy to show that the two first terms in (37) give an equal contribution. (37) can be rewritten
/4(q)AV,(rj) drj. (38) I Here we must notice that in p. and V, we do not consider the contribution coming from the central atom, because it is expected to cancel with the similar contribution due to the same atom when it is put on the surface. From this and using the fact that the wave function is a combination of atomic orbitals, we can rewrite (3 8) under the form:
The numerical results show practically no difference with the non self-consistent case within 1 per cent. Again EFv tends to zero for p = 0 and p = 10. We have complete symmetry for electrons and holes. We want now to take into account the fact that EFv tends towards an appreciable value for p nearly equal to ten. This is probably due to the V,, part of the perturbative potential. (c) Self-consistent calculation including VZb. The only change is that we have now a/6 +6 on each first neighbour. It is this quantity which we call 6’ which has now to displace the right number of states at the Fermi level. S’ has then to take exactly the same values as 6 before (Fig. 8). For a given Fermi level the first term in (40) remains unchanged. The change occurs in the last two terms where 6 is now different, i.e.
-42
where la,o]2 is the perfect crystal contribution to ‘pl and A]u,]~ is the change of this quantity in the perturbed crystal (the factor 6 occurs because we have an equal contribution from the 6 nearest neighbours). From the
S=S’-;
(41)
6’ giving the same value as before, we can say that the formation energy is simply given by the value obtained when neglecting V,, to which we must add the following contribution N(+O)a+;
C A~u~~~. C-X+.0
(42)
Again we find no difference between the numerical values obtained in this way and the results obtained in the non se&consistent case taking V, into account. In order to know the charge variation on the first neighbours we give in Fig. 9 the curve of
648
M. LANNOO
and G. ALLAN
0.2
0.1 % sa L
c
t+
-0.1
-0.2
,-.
Fig. 9. ,c”, Ala, Iz vs. p/l0 (1) non self-consistent, (2) self-consistent. F
the term C A]u,]~, which is analogous with SiV; in (3 2j<” It can be written in a similar form -- 1 “’ {I;, -Z;,}de. 7r J In Fig. 9, we have plotted this quantity as well as &Vi obtained in the non self-consistent case. Comparison shows that self-consistency has the effect of reducing the change in charge density on the first neighbours. 4. DISCUSSION
If we neglect the term Vz6, we get values of the same order of magnitude as other authors [ 131 but it is difficult to compare with the experiment. There are very few experimental results on the formation energy of vacancies in transition metals and some of them are
still controverted. However we compare those which are available at the moment with our predicted values in Table 2. We can notice that the calculations neglecting the Vz6 term give a good agreement at the middle of the series (MO, W) but not at all at the end (Ni, Pt). One could then think that the formation energy could be completely due to the s band or to s-d mixing but this is hard to believe in view of the rather high formation energies (of the order of 1.5 eV). On the other hand calculations including Vz6 lead to quite large values. There are different possibilities to be considered. The crudeness of the approximations: tight-binding method, Hartree approximation, 5 s bands in a simple cubic lattice. We do not believe this is the main reason because the same considerations applied to the calculation of the surface tension of Pt, Pd, Ni would
FORMATION
ENERGY
lead to a value of about O-5 eV the experimental value being about O-4 eV. The occurence of an important relaxation: this is probably the case as shown by the comparison with the surface tension problem.
OF VACANCIES
649
argument is true for (Ywhich is an average of the three independant ai occuring in the exact d band. (c) Hartree approximation. This is perhaps the most serious approximation here. What
Table 2. Comparison between the experimental EFv in eV and the predicted one (a) Vz,, = 0, (b) V,, # 0. We also indicate the number of d electrons :p
EXP
Fe 1*5[7]
Ni = 1.4[7,81=
3%.8*91
(a) N.S.C. (b)
1.5 3.4
0.7 3.1 9.4
2.8 4.5 5
P
7.8
=4’.5,,1 0.7 4.1 9.6
MO -2t[7,8,9] 2.4 3.9 5
Nb - 1~8[10,11] 2.3 3.4 4
*One value of 4.4 eV has been reported [Ref. [ 1211. tThe values in that case go from 1.5 to 2.9 eV. The last value appears to be more plausible when comparing to W and Pt and Ni.
The evaluation of the energy gained when putting the vacancy atom on the surface: it is perhaps slightly different of the cohesive energy. It could also be interesting to evaluate the energy when the atom is put in an interstitial site. Let us now examine some of the approximations which have been used: (a) Influence of lattice. Let us compare our case with five s bands in a f.c.c. structure. For the V, part the result would be formally the same the differences coming from minor changes in the shape of Y(E). We then expect for Ei, values quite similar to ours. Now the term LYwill be spread over 12 nearest neighbotus (al 12 on each of them) adding also a total amount of OLto E;,, and giving the same value at the end of the series. (b) Influence of degeneracy. If we now take five d orbitals instead of five independent s ones, they are now coupled together in the band structure. However for the VI part of the perturbative potential, one can by symmetry split the problem in five independent parts. We expect that our simplified problem gives an average of the total contribution of these five parts, as for the cohesive energy. The same
we can hope is that the effect of correlations cancel when we substract the energies of the perturbed and unperturbed crystal. Let us now return to the comparison between the predicted and experimental results. We came to the conclusion that the relaxation energy was probably large. This seems also to be the experimental conclusion [Hoch Ref. [7], p. 51 from the large values taken by the entropy of formation of the vacancy. From our values the relaxation energy would be of the order of 0.9 to 1.5 eV. We can also notice that the shape of the predicted EFv curve as a function of the filling of the band seems to follow the experimental values satisfactorily. It tends to zero for zero filling as seems to be shown by the weak values for K and Na which are respectively O-4 and 0.6eV. It gives a maximum value near the middle of the series (highest experimental values available) and ends up to a large value for Pt and Ni. We must still mention that there is uncertainty in the experimental results and that, for instance for MO and W measured values respectively as high as 2.9 and 4.4 eV have been reported.
M. LANNOO
650 5. CONCLUSION
and G. ALLAN REFERENCES
In this paper we have determined the FRIEDEL J. in “The Physics ofMeto/s” (Edited by Ziman). p. 340. Cambridge University Press (1969). formation energy of a vacancy in transition FRIEDFL J.. Nuouo Cik. Suppl.. 7.187 (1958). metals using a tight-binding method and CYROT-LACKMANN F.. J. Phvs. C/rem. So/ids approximating the true d band to five indepen29.123S(l968). GSCHNEIDER K. A.. Solid State Phys. 16. 275 dent s bands in a simple cubic lattice. This (1965). allowed us to do a self-consistent calculation LANNOO M. and LENGLART P. (to be pubup to the first neighbours in the Hartree lished in J. Phvs. Chem.Solids, LANNOO M., These, Ann. Phys. 3, no 5 ( 1968). approximation. 6. GAUTIER F. and LENGLART P.. Phys. Rev. 139 Basically the results we find are larger than 3A,70S (1965). the few available experimental values. This 7. International Conference on vacancies and interstitials in metals (preprints) Jiilich (I 968). we think is probably due to a large amount of 8. CO-II-ERILL R. M. J.. DOYAMA M.. JACKSON relaxation in the neighbourhood of the J. J.. JACKSON J. J.. and MESH11 M.. “Laffice vacancy site. This conclusion seems to be in defects in quenched metals”. Academic Press. New York(l965). agreement with experimental observations. 9. THOMSON M. W., “Defecrsandliadiarion Damage The curves we predict for the vacancy formain Mefals”. Cambridge University Press ( 1969). tion energy as a function of the filling up of 10. CHEKHOVSKOY V. YA.. ZHUKOVA I. A.. Fiz. rverd. Tela. 8.9 ( 1966). the band seem to show the right experiI I. KRAFTMAKHER YA. A.. Fiz. tverd. Tela.~.~. 5.950~~ mental behaviour: They start at small values (1963). 12. FRIEDEL J.. in “The Interaction for small filling, having a maximum value of radiation with solids”. Proceedings of the Mol Summer School. about the middle of the series and ending North Holland Amsterdam, p. I35 ( 1964). at a finite large value (typically 3 eV for the 13. GERL M..J. Plrvs. Chem. Solids (to be published). predicted values, 1.5 eV for the experimental ones). The next step in such calculations would be APPENDIXA to use the true d band in tight-binding. HowThe formation energy 4,. can be formally written ever it will probably not affect too much the values obtained here and we are not sure it E;,,,= 10~E’{N.u(E-EO+a)+Gvl(E-E~+a)}EdE can be worth doing such an improvement - 10IEI”N. v(E-E,,+a)EdE+Es (A.11 owing to the complications which are then introduced. Certainly at the moment the 6~’ being the variation of the density of states, i.e. the more desirable calculation would be an been put on the surface, and EF is the new Fermi level defined by estimate of the relaxation energy which could give useful points of comparison. (N- I) $= IE’{N. v(E-E,+(Y) Let us finally notice that our simplified +6v’(E-&+a)}dE (A.21 model could in a straightforward way be used to derive energies of more complicated defects, 8~’ being the variation of the density of states, i.e. the of 6Nr. binding energy of vacancy complexes for derivative Now one can simplify (A. 1) and (A.?) neglecting terms instance. of order l/N. This gives, as EF- Eye is a l/N quantity It should be pointed out that though selfE;,r= ION .~(E~O)(EF-~~O)(E~~+E”.-~) consistency seems to affect very little the energies for vacancies, it must give rise to an +IO~~“SY~(E)((+E~-~)+E~ (A.31 essential change for substitutional impurities. -p = ION . v(eFo) (eF-eFo) + 10 I”’ W(E) de. Along the lines given here, one could easily (A.41 derive for instance the heat of solution of To obtain A.3 and A.4. we have used relation (5) which transition impurities in transition metals in defines E. Putting (A.4) into (A.3) we obtain: the limit of zero dilution.
FORMATION
ENERGY
From
Ej.,. = -p(epO+EEU-a) + 1o~+“W(e)(~-e~~~) Integrating obtain:
by parts.
OF
de+&.
and as there
VACANCIES this
we can derive
states.
we
which
is used
order
(B.3)
Gs+2G,,+3G,
ualue
ofSN2(eF0)
6N*(e)
of the From
relation
in (26).
(B.3) Appraximare We start from
NOW. one can expect pE, - Es to be of the cohesive energy E,. From (6) we then obtain
the following
6G,.,=
(A.5)
are no bound
651
ia (26)
= ;I;,.
(20) we can write (B.4)
If we now
integrate
the 2” term
by parts
we find: From
8N’(e)
I
N(e)
de-
de.
of the perfect
(A.81
APPENDIX
B
.
BN*(e)
We start from the atomic orbitals vpi(i = k 1, -C 2, * 3 in the notations of Fig. 5). Standard symmetry considerations easily lead to
i= -av(eF~).
91 APPENDIX
E,,: cp, =
2VX
a,=-&
C
(C.1) Sum of the one-electron energies in the se& consistenl case As in Appendix A, we want to determine the sum of the one-electron energies plus the surface term, but now in the self-consistent case i.e. without variation of the Fermi level.
91+9-1-9?-9--2
T,,,: 9, = -!v5
2 (91-9-r)
C E,+E,= J
(9p?-9-z)
10jEro{N
.v(E-E,+(y)
+&@--E,,+a))EdE+E,. 9* = Jx4
(9s - 9-3).
(B.1)
Now, crystal z
(8.2) Symmetric Green’s Jimctions The relations we shall derive are valid for every Green’s operator we consider (GO. Cr. C*). We shall then omit the index. If we call GU the matrix elements of G between atomic orbitals centered on the nearest neighbours, there are obviously three independent Gu which are for instance G,,,. G,.-, and Cl.,. In terms of these, and using (B. I) we can write
clearly energy.
the first Then
E,+E,-+
E,O=
term
(B.2)
integral
is the
perfect
lO~%v(E-E,,+a)EdE+E, =
101E”6v(E)(e+E,--)de+E,
=
lOj?h+)edr + 10(EO--~)SN(rFo)
From
Friedel’s
We then
+4c,.*
in the
(Cl)
J
+ E,.
cc.21
rule N(Q)
GM = G,,, +GL-, GEE=GII+GI.-l-2G1.* Grr= G,,-G,.-,.
(B.6)
In Fig. 7 we have plotted the second term (B.6) which is at most equal to O+t. It is always negligible compared to Y(E) which is given in Fig. 3. We have then to a good approximation SN*(e$‘)
3($9,-t+,)--
(B.5)
we obtain
@-I-,
9r =
crystal
/p, = 1;” =-TV(E). Then
(B. I ) Busisfunrrions
the periodicity
cp
6”
E&=IO
= --p/10.
(C.3)
obtain: lOl~~‘Sv(r)tda+E,--p(E,--a).
(C.4)
652
M.
We now form
follow
Appendix
10 I”’ Integrating
A and can write
[&(E)
this by parts
+Y(e)]
easily
lO[N(+‘)
this under
ede.
and the
Now given
+6N(e)]dc. first
term
vanishes
and
out to be the same
to determine~~~~~A~a,~*.
6G:,
the Each
turns
let us try by:
This
quantity
is
where I:, is the imaginary part of the matrix element of the Green’s operator of the perturbed problem. We now want to show how to compute the first term from the functions we already know. We first have (Appendix B)
(C.6)
-lOl”“[N(~)+bN(r)]de which
ALLAN
(C.5)
-t6N(+“)]Q
the
G.
gives
-lO~“‘[N(e) From self-consistency final expression is
LANNOO
of these
three
= G:, + 2G& Green’s
+ 3G&.
function
(Cl
is simply
given
I)
by
(C.7) as in Appendix
(C.2) Determination of]a,,]* andA]a,]* From the fact that the perfect crystal of Bloch type we have
A. I$,, can then be written
wave
function
is
Gm = CC.81
(C.13)
We can then write
1:s GE G-r q., = , ~1--SC:,~‘+211-6G1,1~+31*--SG~I’~
giving (C.9)
Gnl (1--8G$,p
As term
we know
the
Gr quantities
we
can
compute
the I:,