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The electronic structure of point defects in metals
A. Seeger
To cite this version:
A. Seeger. The electronic structure of point defects in metals. J. Phys. Radium, 1962, 23 (10),
pp.616-626. �10.1051/jphysrad:019620023010061600�. �jpa-00236649�
THE ELECTRONIC STRUCTURE OF POINT DEFECTS IN METALS
By A. SEEGER,
Max-Planck-Institut für Metallforschung, Stuttgart,
and Institut für theoretische und angewandte Physik der Technischen Hochschule Stuttgart, Stuttgart, Germany.
Résumé. 2014 On calcule l’effet de défauts ponctuels, en particulier des lacunes, sur une matrice monovalente, par deux méthodes : 1) une méthode utilisant l’approximation des électrons libres ; 2) une méthode de fonctions de Green reposant sur un développement en fonctions de Wannier.
On discute les questions de self-consistence, le domaine d’application des deux méthodes et le choix du potentiel perturbateur. Pour divers potentiels on calcule en détail les valeurs numériques
des énergies de formation, de liaison, et la résistivité électrique. Les calculs ont été faits par H. Stehle, E. Mann, H. Bross, et l’auteur.
Abstract.
2014The present paper treats point defects, in particular vacancies in monovalent metals, by means of the electron theory of metals. Two theoretical methods are discussed : The free or quasi-free electron model, and the Green’s function method, based on the use of Wannier
functions. Problems of self-consistency, range of applicability of the models, and choice of the
perturbing potential are considered. For a number of potentials detailed numerical results for formation and binding energies and for the electrical resistivity are given. The work reported
here is mainly due to H. Stehle, E. Mann, H. Bross, and the author.
LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, OCTOBRE 1962,
In a generalized sense, a metal containing point-
defects (vacancies, divacancies, interstitials) may be considered as a dilute alloy. The fundamental
problems in determining the electronic structure of
an impurity and of a vacancy in, say, copper are very similar. We may therefore hope to learn something useful for the study of dilute alloys from
the investigations of point-defects. This is certainly
not because the point defects present a simpler pro- blem to the theory. On the contrary, a vacant site
is a stronger perturbation of the electronic struc- ture of copper than a substitutional Zn or Au atom.
An interstitial Cu atom in a Cu crystal distorts the
lattice much more than a typical interstitial impu- rity, e.g. a hydrogen or an oxygen atom. However,
as a result of the recent interest in point defects, so
much more experimental and theoretical work has been done on vacancies in simple metals than on
ary particular impurity that bur knowledge of
vacancies is considerably more detailed. This was
not always so. The early electron theory work on alloys [1], [2] preceded that on vacancies in cop-
per [3], [4]. Fumi’s approach [5], [6] to the calcu-
lation of the energy of formation of vacancies in monovalent metals is based on Friedel’s work on
impurities [7], [8], [9].
The early work on the electronic structure of
impurities and point-defects in metals was based
on the model of spherical energy surfaces and of free or quasi-free electrons. The tendency of the
last few years has been to use more realistic energy surfaces (e.g. multiply connected Fermi surfaces in
the noble metals) and to employ better wave-
tunctions.
In the main, the present paper is a short sum-
mary of work done by a group consisting of
H. Stehle, E. Mann, H. Bross, and the author.
The work on non-spherical energy surfaces and the
use of Wannier-functions (section 3) is unpublished.
A considérable amount of the earlier work on the
quasi-free electron picture (section 2) has hitherto
been available only as a thesis [10]. Although
some of our results are general, much of the more
detailed discussion refers to monovalent f. c. c.
metals, unless stated otherwise.
1. General discussion.
In view of the additional level of difficulty pre- sented by the defect problem, it is customary to
consider the corresponding problem for the perfect
metal as solved, at least in principle. The first stage in the èlectron theory treatment of a defect
is then to find the additional (" perturbing ") po- tential due to the defect. The second stage is to
solve the Schrôdinger equation with the additional potential. Finally, as a third stage, we must com- pute the interesting physical quantities, such as energies or resistivities, from the solutions of the Schrôdinger equation.
In metals, where extra or missing charges will be
screened by rearrangements of the electron distri- bution, the problem of finding the perturbing potential is one of self-consistency. Already Hun- tington [4] attempted a self-consistent treatment of a vacancy in a free electron model. From the
view-point of doing actual computations, the self- consistency problem was greatly simplified by the
introduction of Friedel’s charge condition [7], [8], [9] into the vacancy problem [11], [12], [5], [6].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010061600
This condition states that in metals the defect plus
the electron screen must be electrically neutral.
If the perturbing potential of a vacancy is assumed to be spherically symmetrical and to be charac-
terized by two parameters only (height and width),
the Friedel condition gives a relation between
height and width that must necessarily be fulfilled.
For some purposes it may suffice to fix the remai-
ning adjustable parameter by inspection (e.g. to
choose the width of the potential of a vacancy of the order of one or two atomic diameters or appro-
ximately equal to the Thomas-Fermi screening length), whereas for others (e.g. for the reliable cal- culation of the binding energy of a divacancy, see
sect. 2b) a more detailed treatment of the self-
consistency problem is required. This means that
the first and the second stage of our approach have
to be treated simultaneously. We shall corne back
to these questions later.
As emphasized by Slater [13], a good starting point to handle the Schrôdinger equation for the
one-ele,ctron wave function T(r) of the perturbed crystal
(where Ho is the Hamiltonian of the perfect crystal [periodic potential] and Hl its perturbation by a point defect) is the introduction of Wannier func- tions an(r
-Ri). The wave-function is written as
where the first summation extends over all lattice
points Ri of the perfect lattice and the second one over the energy bands n. In order to simplify
matters, in eq. (2) we have confined ourselves to Bravais lattices. Henceforth, we shall drop the
band index n and consider one band of conduction electrons only.
The coefficients U (Ri) satisfy the difference equa- tion
Here É(R) are the Fourier coefficients of the ener-
gies E(k) of the Bloch-waves, defined by
or
Vij are the matrix elements characterizing the per- turbation according to
Hi(r) may be an operator or an ordinary poten- tial V(r).
Eq. (3) connects a very large number of un-
knowns !7(ruz) with each other and is therefore diffi- cult to solve. We shall now discuss two general
methods that have been devised to handle eq. (3).
a) The Green’s function method.
-We mul-
tiply eq. (3) by exp (- ik(Ri
-RI), sum over Ri,
use eq. (4), divide by [ E( k) - E], and sum over
all k-vectors of the Brillouin zone. The result is (*)
The function
is the Green’s function of the unperturbed problem
and satisfies the difference equation
For localized potentials, only few of the matrix elements Yij are different from zero. In such a
case, eq. (6) contains only a small number of un-
knowns and can be handled by standard compu- tational methods. This approach is particularly
suit able for carryingthrough a self-,consistency pro-
cedure, since the perturbing potential is charac-
terized by a finite number of matrix éléments rather than by the continuous function V(r).
b) The Wannier-Slater method.
-If V(r) is a slowly varying function (not an operator) over the region of extension of a(r), eq. (5) may be approxi-
mated by
We may then replace the différence equation (3) by the differential equation
where e( k) stands now for the operator e(2013 ip).
Eq. (10) is particularly useful if e(k) is a qua- dratic function of k. If furthermore the effective
mass tensor m* is isotropic, eq. (10) takes the
form of a Schrôdinger equation with m* replacing
the electron mass m :
c) Comparison of the two-methods.
-The
Green’s function method, which is exact, is the
easier to apply the smaller the number of the
important matrix elements V;j. This number is
(*) Eq. (6) was first derived by Koster and Slater [14],
although in a less direct way. The present derivation is
due to Mann-and Bross.
small, if both the perturbing potential V(r) and
the Wannier functions are strongly localized.
Kohn [15] has shown that, save for thé limiting
case of free electrons, the Wannier functions can
be chosen in such a way that they fall off expo-
nentially with increasing distance from their centre, contrary to an earlier statement in the lite- rature [13]. A rough idea of the rapidity of this
fall-off can be obtained from the variation of the Fourier coefficients e( Ri) with increasing Ri. Since
very good representations of the Fermi surfaces of the noble metals can be obtained by including in
eq. (4) terms only up to next-nearest neighbours of
the origin, the Wannier functions of the conduction electrons in these metals must be fairly localized.
The-derivation of eq. (10) is only valid, if V(r)
varies slowly over the extension of the Wannier functions. Since the modulus of the Wannier
functions of free electrons falls off as the inverse first power of the distance from the centres of the
functions, eq. (10) seems to be not very powerful
for nearly free electrons. On the other hand, we
can solve eq. (10) most easily in the quasi-free
electron case in which the energy surfaces can be
approximated by spheres. It is therefore not a
priori clear whether eq. (11) can be applied to any consistent model of a metal with a half-filled band of conduction electrons, in which we are not allo-
wed to break off the Taylor expansion of z(k) after
thé quadratic terms in k. However, if we use the
free electron model to describe such a metal, we are
also lead to eq. (10), with U(r) now being the wave-
function. This suggests that the Wannier-Slater method has a wider applicability to metals than
its standard derivation would suggest, at least if m*
differs not too much from m. It would certainly be interesting to check this conclusion by comparing
with each other the solutions of eq. (6) and eq. (11)
for the same problem. This has not yet been done.
In addition to the two methods discussed here,
further approaches to the approximate solution of
the Schrôdinger equation in a perturbed crystal
have been proposed. For a critical review refe-
rence may be made to a paper by Friedel [16].
2. Application of the quasi-free électron picture.
a) Single vacaneies.
-We shall base our dis- cussion on eq. (11), i.e. spherical energy surfaces,
and shall confine ourselves to potentials V(r) with spherical symmetry. We may then separate
eq. (11) in spherical coordinates and obtain phase-
shifts ~l(k) for the partial waves of U(r). k is the
wave-number of the solution obtaining in the region
where V(r) = 0. It is convenient to introduce the
quantity
Friedel [7], [9] showed that this quantity, taken
at the Fermi surface k z k,,, is equal to the num-
ber Z_ of electrons attracted by the potential V(r) :
The increase of the sum of the one-electron ener-
gies due to the introduction of V(r) into the crystal
is given by
For later use, we also give the expression for the
extra electrical resistivity Ap due to the scattering
of the quasi-free electrons from a concentration c of
randomly distributed scattering centres with the ’ potential V(r) [17], [11], [18] :
Let us now consider the specific case of a vacancy in copper (1 conduction electron per atom) and let
us assume that we may neglect the displacements
of the neighbouring ions. A good approximation
to V(r) should be the negative Ilartree-Fock poten-
tial of a C;u+-ion [19] plus a correction for the
screening action of the conduction electrons. Jon-
genburger [11] assumed that the screening can be
allowed for by distributing one positive elementary charge uniformly inside the Wigner-Seitz sphere of
the vacant site. He showed that the resulting potential can be represented by
where e is the electronic charge,
is Bohr’s hydrogen radius, and
denotes the Rydberg unit. For comparison, the
radius of the Wigner-Seitz sphere of copper is
r,, = 2.675 au.
,Jongenburger’s potential (16) does not satisfy
Friedel’s condition (13), i.e. Z_ is not equal to -1.
Jongenburger [11] states that for his potential Z(kp) = - 0.95. This is not correct, however,
since the Born approximation was used to calculate
all phase shifts 1J¡ for 1 > 1. If the exact phase
shifts are used for both 1Jo and 1Jl, Z(k,) = - 0 . 755
is found [10]. This means that the screening assu-
med by Jongenburger is too strong, and that in
reality the electron originally located in the vacant
cell is not completely expelled from it.
Variations in the screening charge will affect
V(r) only in its outer region, say for r > a$.
Stehle [10] uses therefore the following expression
for the potential of a vacancy in copper :
where
and
03BC* is determined from the condition Z(kp) = -1.
The potential eq. (17) varies so rapidly over the
interatomic distance that in this particular case the
Wannier-Slater method for the derivation of eq. (10) is invalid. It may be better to consider
the potential eq. (17) within the framework of the .free electron model for the conduction electrons.
It is true that then the potential V(r) should
not tend to infinite values, as eq. (17) does for
r -> 0. Since the conduction electrons are unable
to penetrate into the core of the repulsive poten-
tial anyway, the detailed shape of the potential in
this region is irrelevant for the present problem,
6owewer the following results [10] are obtained
for the free electron case in* = m. Replacing
this by m* = 1.45 nl (applicable to copper) would
change the final results for Llp1J and Eei hardly
at all.
FIG. 1.
-Friedel’s sum and phase shifts for Stehle’s poten-
tial eq. (17) as function of the screening parameter [1.*.
Figure 1 shows (for k = kp) as a function of 03BC*/aH, the phase-shifts 1)0 and ~1, both computed by numerical integration, the expression
as determined from the Born approximation (which
is applicable, y since the phase-shifts for 1 > 2 are small, comp. table 1), and finally the quantity
0° TABLE 1
-Z(kp). [1.* aH = 1.88 satisfies the condition Z- == - 1 with sufficient accuracy, yielding Z(kF) = - 0.997 (*). For this value of z
table 1 shows for two différent values of k a number of numerical values for the phase shifts 1)1, in addi- tion to the quantity (d 1)1/dk)k=-O, which is different from zero only for 1 = 0. Eq. (15) gives
The maximum value which eq. (15) can yield for Z(kF) = :1: 1 is 0394pmax = 3.81lL ohm cm/% vacan- cies, corresponding to 1)0 = :t 7c/2, 1)1 = 0 for (*) It is interesting to compare y* with the Thomas-
Fermi sereeningparamater03BCTh.jr = (4kp J aH)U2 = O.95/aH.
Stehle’s potential is twice as concentrated as the Thomas- Fermi potential.
"
1 > 1. The corresponding potential must be
rather high and concentrated, so that a classical consideration should be possible. If a cluster of n vacancies is treated as an impenetrable sphere
with a volume equal to n atomic volumes, a clas-
sical scattering calculation leads to the expression
(c = concentration of vacant sites)
’
where a is the length of the edge of the elementary
cube of an f. c. c. metal. Inserting the numerical - value for copper (haie2 = 1.651 X 10-16 el.st.
units = 1 .486 X 10 2 y ocm) gives
For n = 1, we find indeed a value close to Apmax.
In passing, it may be mentioned that for n > 30
eq. (19) agrees better than within 15 % with the
phase shift calculation of Dexter [20], which uses a repulsive square well potential, the height of
which (Vo = 11.4 eV) is determined as the sum of
the free electron Fermi energy and the work- function of copper. The agreement is not so good
with the result of Asdente and Friedel [21].
These authors ùsed a somewhat lower square well
potential satisfying eq. (13) for Z- === -ne
FIG. 2. - Effective nuclear charges p(r) for the potentials 1
to IV explained in the text.
Figure 2 gives some of the potentials that le
liave discussed. It was found convenient to plot
on a logarithmic scale twice the effective charge Q(r), which is defined by
Curve I gives Hartree’s [19] self-consistent
Hartree-Fock-potential of the Cu+-ion. Curve II
shows Jongenburger [11] potential eq. (16),
curve III shows Stehle’s potential [17] with Il* aH = 1.88, and curve IV shows a Jongenburger type potential (Cu + -potential plus one positive ele- mentary charge uniformly distributed in a sphere
of radius rq) with rq = 3.26 aa. The latter poten-
tial agrees quite closely with curve III (*). Such Jongenburger type potentials, with rq as a para- meter adjustable to the charge condition, were con-
sidered by Blatt [22]. For the choice of rq which
satisfies Friedel’s conditions (the value of r, is not stated) Blatt finds by numerical integration
(*) This result indicates [(rq/rs)3 = 1.8] that oiie posi-
tive charge uniformly distributed over a volume approxi- mately twice that of the Wigner-Seitz cell is able to give qu,alitatively the correct screening for a vacancy.
As we would expect, eq. (18a) agrees well with Stehle’s value (*). A good theoretical value for the vacancy resistivityPin copper within the frame- work of the quasi-free electron approximation
appears to be
An experimental value which could be compared directly with eq. (21) is not available at present.
Jongenburger [11], Abelès [12] and Bross [24]
have considered repulsive square well potentials of
radius ro and height Vo, satisfying the condition Z(kp) = - 1. Figure 3 gives the resistivity for
Fic. 3.
-Electronic energy Eej and electrical resistivity Apv for a vacancy in the quasi-free electron model for a
monovalent f. c. c. metal. (Repulsive square well poten-
tial of height Vo and radius ro. The dotted line gives
the asymptote for Vo --7 oo.) The numerical values for the resistivity refer to copper.
such potentials as a function of ro kp, together with VuJl = (kolkF)2 where
*
measures the height of the potential. It is seen
that the resistivity decreases with increasing ro and
decreasing Vo. Qualitatively it may be said that the electrical resistivity responds to the variation
of the potential Y( r), and not its absolute magnitude.
Therefore spread-out potentials give a lower resis- tivity than concentrated ones. (The opposite sta-
tement is true for the calculation of .Eel due to repulsive potentials, see below.)
Stehle’s potential with fil * au = 1.88 gives for
the electronic energy eq. (14)
(*) Uriginally, Blatt [22] gave a smaller value. See
however the erratum [23].
where
is the Fermi energy of the conduction electrons.
This agrees closely with the value obtained earlier
by Fumi for the square-well potential ro = rs. For the model used in figure 3, Eei has been given as a
fùnction of kF ro by Seeger and Bross [25]. It was
found that Eel grows gradually from the minimum value 0 . 5715 03B6 for,o ko = 00 to a maximum value
203B6/3 in the range ot validity of the Born approxi- mation, i.e. for spread-out potentials. It is seen
that Eel depends much less on the details of the
potential than Ap. This is because the main con-
tribution to the integral in eq. (14) comes from its
upper limit, and this is fixed by the charge condi-
tion. It should be remarked, however, tbat the
total electronic contribution to the energy of a vacancy is given [6] by
Unfortunately, Etot is therefore considerably
more sensitive to the details of the potential
than Eel. In order to obtain reliable theoretical values for the energy of formation of vacancies
(and other point defects), it is therefore essential
to have a good self-consistent potential V(r).
Stehle [10] has checked the self-consistency of
his potential in the following way. Let us suppose
that the potential eq. (17) is cut off at an exterior
FIG; 4.
-Phase shifts and Friedel’s sum at the Fermi surface as a function of the radius r for Stehle’s potential
with y* off = 1.88.
-Z- = Z_(kF ; r) is the effec- tive charge of the screening as calculated from the diffe-
rence of the Cu-ion potential and the screened potential
eq. (17).
radius r, and let us consider the phase-shifts for
k = k, as a function of r. Figure 4 shows -10(k.r ; r), ~1(kF ; r), and an estimate of
.- Z-(r) is calculated from eq. (12)..
is calculated from the curvés III and I of figure 2
and gives the screening charge of Stehle’s potential
as a function of the radius. It is seen that the condition eq. (13) is satisfied with an accuracy of at least 10 %. In order to correct f or the devia-
tion, V(r) would have to be increased near r = 2.5 aH and decreased correspondingly for lar-
ger values of r. It should be remarked that the
self-consistency considered here is more restricted than that of the Hartree-Fock scheme, since we
are using the same " ordinary " potential V(r) for
all the wave-functions.
b) Multiple vacaneies. --’Whereas the discussion’
°of single vacancies was detailed, we shall be rather brief in our discussion of multiple vacancies and
mainly give the results only. The reason for
this is that the theoretical method of the preceding
section can only partially be carried over to the
more complicated geometries of multiple vacancies,
and that it gets rather involved. For the treat- ment of divacancies and trivacancies spheroidal
coordinates seem to be appropriate. Useful solu- tions in terms of spheroidal wave-functions can
only be obtained for a spheroidal region of an infi- nitely high répulsive potential, with V(r) = 0 eve- rywhere else. This corresponds to an extremely
concentrated potential. Fortunately, the other extreme, namely that of a low, widely spread-out potential, can be handled by Born’s approximation.
Treating the concentrated potential for a diva-
cancy and comparing it with the corresponding potential for single vacancies, Seeger and Bross [25]
showed that the electronic contribution to the bin-
ding energy of a divacancy is
,On the other hand, Seeger and Bross [26], Friedel,
and Blandin found that within the range of the validity of Born’s approximation b.Eel is zero.
The reason for this is that independent of the shape
of Y(r), Born’s approximation together with the charge condition gives a unique relation between Lei
and Z_ (*). Z_ is not changed when vacancies
cluster to multiple vacancies.
,Since neither one of the two limiting cases is very realistic, the best result for AEei appears to lie in between. It is believed that a reasonable estimate is
For the Fermi energies of the noble metals, this
leads to divacancy binding energies from 0.1 eV
to 0.2 eV. This seems to be the right magnitude
2
(*) For quasi-free electrons this relation is Eel = - j 3 03BEZ-.
Blandin [27] has shown that for Bloch-electrons 2/3 has to
be replaced by an other numerical factor.
622
to account f or the available experimental data (*) (see however sect. c).
For scattering potentials of spheroidal shape it is
no longer possible to give a closed formula for the electrical rêsistivity such as eq. (15). Rather, the
variational principle has to be used [28]. In this
way, Bross and Seeger [29] showed that the for- mation of a divacancy from two isolated single
vacancies is expected to lead to a reduction of the
residual resistance by about 10 %. The effects of
the anisotropy, in particular positive deviations
from Matthiessen’s rule and magnetoresistivity,
are expected to be small.
.
The electronic contribution to the binding energy of larger multiple vacancies has been discussed along similar lines. Here we have the additiQnal
difficulty that the geometrical configuration of
these clusters is not well known. We refer the reader to the original paper [30].
As the main result of the present section we may state that the requirements in the knowledge of the potential are much higher for calculations of the
binding energy of vacancy complexes than for cal-
culating the energy of formation of a single vacancy.
c) Long-range oscillations of the charge density.
-
A number of authors have discussed the exis- tence of long-range radial oscillations of the charge density of the conduction electrons around impu-
rities and their importance in physical and chemical problems, in particular the spin coupling between impurities the Knight-shift, and the interaction between impurities (see other contributions to this
symposium). Since the potentials V(r) we have
considered sofar do not show such oscillations, it is
clear that they cannot be self-consistent in regions
where these oscillations dominate. This deficiency
is expected to have very little effect on the calcu- lation of Eei, since the magnitude of these oscil- lations is small compared with the main contri- bution to the charge density. The electrical resis-
tivity may be affected more strongly, since it res- ponds more to the variation in thé potential than
to its absolute magnitude. Numerical results do not yet seem to be available in the literature.
In the approximation which we used for obtai-
ning the phase-shifts of table 1 (Hartree approxi-
mation for free electrons), Blandin [31] gives the following expression for the change dptl(r) in the density of the electrons due to the long-range oscil-
lations :
(*) Seeger and Bross [25] have shown that the correction for the relaxation of the neighbouring ions is unimportant
for copper, since the contributions to the energies of for-
mation of two single vacancies and of a divacancy just
cancel.
where
Inserting the numerical data of table 1 for ~l(kf)
and correcting for the higher phases gives
Let us apply these results to the interaction of two vacancies. The potential around a vacancy
repels electrons. We obtain therefore a repulsive
interaction between two vacancies separated by a
distance R, if the oscillations around one vacancy result on the average in an increase of the electron
density at the location of the other vacancy. If we consider the potential around a vacancy as highly
localized (in order to be able to neglect the r--3-
variation of the charge density across the vacancy),
the condition for repulsive interaction is that the
quantity [2k,, R + cpF -(2m + 1) 7t] lies between
-
rJ2 and + n/2. Here m is a suitably chosen integer.
With the numerical results of eq. (28) we find
that the interaction between two vacancies is re-
pulsive between 2kF .R = 6.08 and 2k,, R = 9 .23
with a maximum of Appel at 2kp Rm = 7.24. The
density of the extra electrons at the maximum is
0394pel(Rm) = 0.015 electrons/atom. Since
figure 2 shows that in this region the charge density corresponding to the potential eq. (17) is negligible compared with Apei(Rm). Denoting the distance
between neighbouring vacancies in an f. c. c. metal
by b and that between next-nearest neighbours by
ao, we find 2kF b = 6.94 and 2kp ao = 9.82.
This means that the long-range oscillations increase the energy of two neighbouring vacancies. The
corresponding eff ect on the binding energy is pre-
sumably small and within the uncertainty dis-
cussed in sect. b. The energy of the next-nearest
neighbour configuration is slightly decreased.
Between these two configurations, the long-range
oscillations give rise to an energy barrier (in addi-
tion to the normal energy of migration) for the for- mation of a divacancy, the height of which can be
estimated to be of the order 1/20 eV. Suc a
barrier will eff ect the rate of formation of diva- cancies from single vacancies in annealing expe- riments. Indications for the existence of this
energy barrir have been found in the analysis of
quenching experiments on silver [32]. A similar
623 barrier of 0.24 eV height proposed earlier for
gold [33] appears to be too large to be compatible
with the present theory.
d) Miscellaneous topics.
-In the preceding dis-
cussion we had in mind mainly metals with one
conduction electron per atom. The numerical fac- tors have been given for an f. c. c. crystal and should
therefore be applicable to the noble metals, in par- ticular copper, to wbich some of the detailed calcu- lations pertain directly. Due to the special elec-
tronic structure of some ferromagnetic metals and alloys, in particular nickel and cobalt, the tech- niques of the resistivity calculations can also be
applied to them. For details see ref. [34]. We
should like to mention that the application of the quasi-free electron model to nickel and cobalt is
more justified than that to the noble metals, since
the number of electrons per atom in the conduction band is smaller and therefore the Fermi-surfaces
are more nearly spherical.
We have neglected sofar the relaxation of the atoms surrounding vacancies or other point defects.
A unified simultaneous treatment of this relaxation and the electronic effects has not yet been given
and remains an important task for the future. At
present, we have to be satisfied with calculations in which classical models .for the ion-ion interaction
are used. A model which supplements the approach to the electron redistribution effects
given here has been developed by Seeger and
Mann [35], and has been applied to the calculation of vacancy formation and interstitial formation and migration energies in copper [35, 36].
If the relaxation ôf the neighbouring ions is
taken into account, difficulties arise in the calcu- lation of the electrical resistivity. The strains surrounding the defect also scatter the conduction
electrons, and the interference with the scattering
from the centre of the defect has to be taken into account. On the other hand, the charge of the relaxing ions will in general help to screen the
extra charge at the centre of the defect and there- fore reduce the scattering. For vacancies in the
noble metals, both effects are small and of opposite sign. It appears therefore justified to heglect
them and to use the value given in eq. (21).
For interstitial atoms in copper, however, the
situation is rather différent [36]. The outward
relaxation of the atoms surrounding a Cu-inter-
stitial has such a magnitude that in spite of the
introduction of an extra atom the average density
of the positive charge remains the same as in the ideal metal. This means that much less redis- tribution of the conduction electrons is required
for self-consistency than in the vacancy case. The
scattering of the electrons is mainly due to the
strain field. It is therefore not surprising that experimentally the electrical resistivity of inter-
stitials cornes uut smaller tan the value of eq. (21), namely 0.9 ,ohm cm/% interstitials.
3. Application of the Green’s function method, The Green’s function method seems to be the
appropriate starting point to remove the principal
deficiencies of the techniques described in section 2.
It is possible to treat arbitrary energy surfaces, including open Fermi surfaces. The method can
be made self-consistent, an aspect being studied at present by E. Mann. The perturbïng potentials
due to electron redistribution and those due to the
displacement of ions may be treated by the same formalism, since both enter through the matrix elements Vij. The expression for the charge neu- trality in terms of the Vz?’s is therefore generally valid, and an analogous statement holds for the electrical resistivity. While there is every reason to hope that the method will eventually lead to a satisfactory general solution for the electronic structure of point defects in metals, the difficulties that have to be overcome are formidable. -The work of our group is still preliminary in a number
of respects. We shall only give a few selected results that are of particular interest in connection with the discussions of sect. 2, without detailed
références to the literature. We would like, how-
ever, to call attention to the pioneering work ouf
I. M. Lifshitz and his collaborators on the appli-
cation of the Green’s function method to defect pro- blems in crystal lattice dynamics. An éasily
accessible account of this work may be found in
ref. [37].
The energy eigenvalues E for the perturbed crystal (the perturbation being characterized by
the matrix elements Vij) are obtained from the secular determinant of eq. (6)
Eq. (29) is an implicit equation for E, since the
Green’s function Gg(Ri - Ri) depends on E. We
shall discuss the solution of eq. (29) in a number of special cases.
a) Strongly localized perturbations.
-We de-
fine as strongly localized perturbations those which
give non-vanishing matrix elements Vii =1= 0 only
if Ri = Ri = 0. For simple lattices they have
been considered in some detail by Koster and Slater [38]. We shall denote the only non-vani- shing matrix element of the perturbing potentiel by Voo. The secular determinant eq. (29) takes
then the simple form
If we are interested in localized states, for
which E falls outsde the energy band e(k), we may
replace the summation in eq. (7) by an integration
over the energies of the band. We obtain
where
is the density of states per atomic volume Q (not allowing for spin). In eq. (32) the integration is
to be extended over the energy surface E( k) = s in k-space. Eqs. (30) and (31) are very convenient for a discussion of the energies of bound states. It
can be seen that with increasing Voo the energy E
moves away from the band edge. The position of
an eigenvalue E close to a band edge (taken as
E = 0) is determined mainly by the density of
states near to that band edge, whereas the position
of an eigenvalue far away from it depends on the density of states over the whole band. It can fur- thermore be seen that in three dimensions, where
the density of states near a band-edge varies as e.l/2,
a finite minimum value of 1 V 001 is necessary to obtain a solution to eq. (30), i.e. a localized state.
Let us now consider the effect of Voo on an
energy value lying in the unperturbed crystal at an energy E within the band. It is no longer permis-
sible to replace the summation by an integration throughout. For energy values close to E the summation in eq. (7) has to be carried out expli- citly. E. Mann (unpublished) has shown that for arbitrary energy surfaces E( k) and with full allow-
ance for a possible degeneracy of the eigenvalues,
the following result is obtained :
Here Gs(0) is the principal value of eq. (31), and
In eq. (34) AE denotes the average increase of the energy eigenvalues on the energy surface
e;(k) = E. Solving eqs. (30) and (33) for AE, we get
’
The total increases of the energy coming from the eigenvalues within the band is given by
In addition to eq. (36), we may have contri- butions from bound states. Eq. (36) may also be derived in an entirely différent way by employing
the methods of Wentzel’s meson pair theory [39].
.
Friedel’s condition may be written in the follow-
ing way :
It expresses the fact that the number of electrons
repelled by the perturbation must be equal to the
number of eigenvalues pushed through the Fermi
surface (allowing for spin), since the introduction of a sufficiently low concentration of imperfections
leaves the Fermi energy 03B6 of the metal unchanged.
Using eq. (34), we can give the following simple
form to the charge condition :
For Z- = :!: 1 (interstitial or vacancy in a
monovalent metal), eq. (37a) is equivalent to
For Do(E)-curves actually occuring for conduc- tion electrons, Gs(0) will be zero or negative at
the Fermi level of a monovalent metal. This
means that eq. (39) cannot have a solution for a
positive Voo, i.e. for a perturbation such as a
vacancy that repels the electrons. This is ana-
logous to the finding in sect. 3 (fig. 3), where it
was shown that for a potential that is too loca-
lized not even an infinitely high repulsive potential
is able to repel enough electrons to satisfy Friedel’s
condition in a monovalent metal. In contrast to
this, an attractive potential, such as that of an
interstitial atom, will for sufficiently large 1 V oof always satisfy Friedel’s condition. However, in
this case 1 V 001 will have to be so large that a bound
state is formed. To give an example, for a band of quasi-free electrons [D,(F,) N e:lj2], eq. (39) is satis-
fied for Voo = - 3.8 (, whereas a bound state
occurs for Voo = 20130.53 (.
The coefficient U(R¡), which determine the wave- function, are given by
In eq. (40) the eigenvalues z(k) within the band
enter explicitly. In contrast .to the preceding dis- cussion, the knowledge of the density of states does
no longer suffice.
Clogston [40] and E. Mann (unpublished) have shown, how for spherical energy surfaces eq. (40) is
related to the phase shift analysis of the type employed in sect. 3. In the present case there is only one partial wave, having s-symmetry. The only non-vanishing phase shift is therefore "1)0.
Mann has furthermore shown that at large dis-
tances from the perturbation the charge density
625
calculated from eq. (40) follows eq. (27), where ocp
and Cf>p are functions of Voo, Do(() and Gç(0).
Furthermore, for a general energy function e(k), Mann was able to show explicitly that the charge density integrated over the .fondamental block satisfies the charge condition exactly. Finally, it
may be mentioned that for large Ri an asymptotic expression for GE(R,,) and thereby an explicit form
for U(R.) has been derived [37], [41].
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