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>
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LE JOURNAL DE
PHYSIQUE
ET LE
RADIUM
TOME
23,
OCTOBRE
1962,
616.
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.
present paper treats point defects, in particular vacancies in monovalent
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.
Abstract.
metals, by
2014
The
means
In a generalized sense, a metal containing pointdefects (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 problem to the theory. On the contrary, a vacant site
is a stronger perturbation of the electronic structure 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 copper [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
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 presented 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 ") potential due to the defect. The second stage is to
solve the Schrôdinger equation with the additional
potential. Finally, as a third stage, we must compute 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
impurities [7], [8], [9].
The early work on the electronic structure of screened by rearrangements of the electron distriimpurities and point-defects in metals was based bution, the problem of finding the perturbing
on the model of spherical energy surfaces and of
potential is one of self-consistency. Already Hunfree or quasi-free electrons. The tendency of the tington [4] attempted a self-consistent treatment
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 wavetunctions.
In the main, the present paper is a short sum-
vacancy in a free electron model. From the
view-point of doing actual computations, the selfconsistency problem was greatly simplified by the
introduction of Friedel’s charge condition [7], [8],
[9] into the vacancy problem [11], [12], [5], [6].
of
a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010061600
617
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 characterized 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
Eq. (3) connects a very large number of unknowns !7(ruz) with each other and is therefore difficult to solve. We shall now discuss two general
methods that have been devised to handle eq. (3).
We mula) The Green’s function method.
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 (*)
-
-
appro-
ximately equal to the Thomas-Fermi screening
length), whereas for others (e.g. for the reliable calculation of the binding energy of a divacancy, see
sect. 2b) a more detailed treatment of the selfconsistency 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 functions 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
In order to simplify
over the energy bands n.
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 equation
The function
is the Green’s function of the unperturbed
and satisfies the difference equation
problem
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 unknowns and can be handled by standard computational methods. This approach is particularly
suit able for carryingthrough a self-,consistency procedure, since the perturbing potential is characterized by a finite number of matrix éléments
rather than by the continuous function V(r).
If V(r) is a
b) The Wannier-Slater method.
slowly varying function (not an operator) over the
region of extension of a(r), eq. (5) may be approximated by
-
We may then replace the différence equation (3)
by the differential equation
where
Here É(R) are the Fourier coefficients of the energies E(k) of the Bloch-waves, defined by
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 :
or
Vij are the matrix elements characterizing the perturbation
according to
Hi(r) may
tial V(r).
be
an
operator
or an
ordinary poten-
The
c) Comparison of the two-methods.
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.
618
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 exponentially with increasing distance from their
centre, contrary to an earlier statement in the literature [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 allowed 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 wavefunction. 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 reference may be made to a paper by Friedel [16].
2.
Application
of the
quasi-free électron picture.
showed that this quantity, taken
k,,, is equal to the number Z_ of electrons attracted by the potential V(r) :
Friedel
[7], [9]
at the Fermi surface k
z
The increase of the sum of the one-electron energies 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 potential of a C;u+-ion [19] plus a correction for the
screening action of the conduction electrons. Jongenburger [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
Wigner-Seitz sphere of copper is
radius of the
2.675 au.
r,,
=
,
Jongenburger’s potential (16) does not satisfy
Friedel’s
condition (13), i.e. Z_ is not equal to -1.
vacaneies.
We shall base our disa) Single
cussion on eq. (11), i.e. spherical energy surfaces, Jongenburger [11] states that for his potential
and shall confine ourselves to potentials V(r) with Z(kp) = - 0.95. This is not correct, however,
the Born approximation was used to calculate
spherical symmetry. We may then separate since
shifts 1J¡ for 1 &#x3E; 1. If the exact phase
all
in
phase
and
coordinates
obtain
spherical
phaseeq. (11)
shifts ~l(k) for the partial waves of U(r). k is the shifts are used for both 1Jo and 1Jl, Z(k,) = - 0 . 755
wave-number of the solution obtaining in the region is found [10]. This means that the screening assu0. It is convenient to introduce the med by Jongenburger is too strong, and that in
where V(r)
reality the electron originally located in the vacant
quantity
cell is not completely expelled from it.
Variations in the screening charge will affect
V(r) only in its outer region, say for r &#x3E; a$.
-
=
619
Stehle [10] uses therefore the following expression
for the potential of a vacancy in copper :
change
the final results for
Llp1J
and Eei
hardly
at all.
where
and
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 -&#x3E; 0.
Since the conduction electrons are unable
to penetrate into the core of the repulsive potential anyway, the detailed shape of the potential in
this region is irrelevant for the present problem,
6owewer the following results [10] are obtained
m.
for the free electron case in*
Replacing
1.45 nl (applicable to copper) would
this by m*
03BC* is
=
=
0°
-
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
=
determined from the Born approximation (which
applicable,y since the phase-shifts for 1 &#x3E; 2 are
small, comp. table 1), and finally the quantity
as
is
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 addition to the quantity (d 1)1/dk)k=-O, which is different
0. Eq. (15) gives
from zero only for 1
=
The maximum value which eq. (15) can yield for
Z(kF) = :1: 1 is 0394pmax 3.81lL ohm cm/% vacan0 for
cies, corresponding to 1)0 = :t 7c/2, 1)1
=
=
(*) It is interesting to compare y* with the ThomasFermi sereeningparamater03BCTh.jr = (4kp J aH)U2 = O.95/aH.
Stehle’s potential is twice as concentrated as the ThomasFermi potential.
"
Friedel’s sum and phase shifts for Stehle’s potenFIG. 1.
tial eq. (17) as function of the screening parameter [1.*.
1 &#x3E; 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 classical scattering calculation leads to the
concentration of vacant sites)
(c
expression
=
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
=
’
620
For n
1, we find indeed a value close to Apmax.
=
it may be mentioned that for n &#x3E; 30
the
eq. (19) agrees better than within 15
which
a
of
Dexter
uses
shift
calculation
[20],
phase
repulsive square well potential, the height of
11.4 eV) is determined as the sum of
which (Vo
the free electron Fermi energy and the workfunction 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
In
passing,
% with
As we would expect, eq. (18a) agrees well with
Stehle’s value (*). A good theoretical value for
the vacancy resistivityPin copper within the framework of the quasi-free electron approximation
appears to be
=
FIG. 2. - Effective nuclear charges
to IV explained in the text.
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
p(r) for the potentials 1
Electronic energy Eej and electrical resistivity
vacancy in the quasi-free electron model for a
monovalent f. c. c. metal. (Repulsive square well potential 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.
Fic. 3.
Apv for
-
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
Figure
a
*
such
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 elementary charge uniformly distributed in a sphere
3.26 aa. The latter potenof radius rq) with rq
tial agrees quite closely with curve III (*). Such
Jongenburger type potentials, with rq as a parameter adjustable to the charge condition, were considered 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
Curve I
=
=
1.8] that oiie posi(*) This result indicates [(rq/rs)3
tive charge uniformly distributed over a volume approximately twice that of the Wigner-Seitz cell is able to give
qu,alitatively the correct screening for a vacancy.
potentials as a function of ro kp, together with
VuJl (kolkF)2 where
=
the height of the potential. It is seen
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 resistivity than concentrated ones. (The opposite stameasures
that the
tement is true for the calculation of
repulsive potentials, see below.)
Stehle’s potential with fil * au
the electronic energy eq. (14)
=
.Eel due
1.88
gives
to
for
=
(*) Uriginally, Blatt [22] gave
however the erratum [23].
a
smaller value.
See
621
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 approximation, 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 contribution to the integral in eq. (14) comes from its
upper limit, and this is fixed by the charge condition. It should be remarked, however, tbat the
total electronic contribution to the energy of a
vacancy is given [6] by
=
is therefore considerably
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
Unfortunately, Etot
more
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 deviation, V(r) would have to be increased near
2.5 aH and decreased correspondingly for larr
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
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 treatment of divacancies and trivacancies spheroidal
coordinates seem to be appropriate. Useful solutions in terms of spheroidal wave-functions can
only be obtained for a spheroidal region of an infinitely high répulsive potential, with V(r) = 0 everywhere 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 divacancy and comparing it with the corresponding
potential for single vacancies, Seeger and Bross [25]
showed that the electronic contribution to the binding energy of a divacancy is
brief in
our
,
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.
Phase shifts and Friedel’s sum at the Fermi
surface as a function of the radius r for Stehle’s potential
Zwith y* off
1.88.
Z_(kF ; r) is the effective charge of the screening as calculated from the difference of the Cu-ion potential and the screened potential
eq. (17).
FIG; 4.
-
=
-
=
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)..
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
3
= - j 03BEZ-.
Bloch-electrons 2/3
For quasi-free electrons this relation is Eel
Blandin [27] has shown that for
be replaced by an other numerical factor.
has to
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 formation 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 calculating the energy of formation of a single vacancy.
of the charge density.
A number of authors have discussed the existence of long-range radial oscillations of the charge
density of the conduction electrons around impurities 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 calculation of Eei, since the magnitude of these oscillations is small compared with the main contribution to the charge density. The electrical resistivity may be affected more strongly, since it responds 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 obtaining the phase-shifts of table 1 (Hartree approximation 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 oscillations :
c) Long-range oscillations
-
(*) 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 formation of two single vacancies and of a divacancy just
cancel.
where
Inserting the numerical data of table 1 for ~l(kf)
correcting for the higher phases gives
and
these results to the interaction of
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--3variation 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
Let
us
apply
two vacancies.
-
integer.
With the numerical results of eq. (28) we find
that the interaction between two vacancies is repulsive 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
6.94 and 2kp ao = 9.82.
ao, we find 2kF b
This means that the long-range oscillations increase
the energy of two neighbouring vacancies. The
corresponding eff ect on the binding energy is presumably small and within the uncertainty dis=
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 addition to the normal energy of migration) for the formation 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 divacancies from single vacancies in annealing experiments. Indications for the existence of this
energy barrir have been found in the analysis of
quenching experiments on silver [32]. A similar
cussed in sect. b.
623
barrier of 0.24 eV height proposed earlier for
gold [33] appears to be too large to be compatible
with the present theory.
stitials
namely
0.9
uut smaller tan the value of eq.
(21),
,ohm cm/% interstitials.
Application
3.
d) Miscellaneous topics. In the preceding discussion we had in mind mainly metals with one
conduction electron per atom. The numerical factors have been given for an f. c. c. crystal and should
therefore be applicable to the noble metals, in particular copper, to wbich some of the detailed calculations pertain directly. Due to the special electronic structure of some ferromagnetic metals and
alloys, in particular nickel and cobalt, the techniques 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
cornes
of the Green’s function method,
-
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 calculation 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 therefore 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-interstitial 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 redistribution 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-
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 neutrality 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 wouldlike, however, to call attention to the pioneering work ouf
I. M. Lifshitz and his collaborators on the application of the Green’s function method to defect problems 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.
fine
-
We de-
strongly localized perturbations those which
give non-vanishing matrix elements Vii =1= 0 only
if Ri
0. For simple lattices they have
Ri
been considered in some detail by Koster and
Slater [38]. We shall denote the only non-vanishing matrix element of the perturbing potentiel
by Voo. The secular determinant eq. (29) takes
then the simple form
as
=
=
If we are interested in localized states, for
which E falls outsde the energy band e(k), we may
624
Friedel’s condition may be written in the followreplace the summation in eq. (7) by an integration
the energies of the band. We obtain
ing way :
.
over
where
is the
density of states
allowing for spin). In
per atomic volume Q (not
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 furthermore be seen that in three dimensions, where
the density of states near a band-edge varies as e.l/2,
a finite minimum value of1 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 permissible to replace the summation by an integration
throughout. For energy values close to E the
summation in eq. (7) has to be carried out explicitly. E. Mann (unpublished) has shown that for
arbitrary energy surfaces E( k) and with full allowance for a possible degeneracy of the eigenvalues,
the following result is obtained :
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
monovalent metal), eq. (37a) is equivalent to
a
=
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
For Do(E)-curves actually occuring for conduction 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 analogous to the finding in sect. 3 (fig. 3), where it
was shown that for a potential that is too localized 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 large1 V oof
always satisfy Friedel’s condition. However, in
this case1 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,) e:lj2], eq. (39) is satisfied for Voo = - 3.8 (, whereas a bound state
occurs for Voo = 20130.53 (.
The coefficient U(R¡), which determine the waveN
function,
are
given by
=
’
get
(40) the eigenvalues z(k) within the band
explicitly. In contrast .to the preceding discussion, 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 distances from the perturbation the charge density
In eq.
enter
The total increases of the energy coming from the
eigenvalues within the band is given by
In addition to eq. (36), we may have contributions 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].
625
calculated from eq. (40) follows eq. (27), where ocp
and Cf&#x3E;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].
-
b) Moderately localized perturbations.
By a
moderately localized perturbation we mean one in
which in addition to Voo a limited number of
matrix elements Vij are different from zero.
--
quantities not mentioned
following meaning
The
before have the
E. Mann has investigated in some detail the case in
which two more matrix elements are non-vanishing,
namely Ylo (Wannier functions centred at the origin
and of its nearest neighbours) and Y11 (both Wannier
functions centred at one of the nearest neighbours
to the atom at the origin). In addition to solutions
with s-symmetry, we now find also solutions with
p- and d-symmetry. For the sake of simplicity, we
take into account only the s-functions, which are
the only ones that do not have a node at the centre
of the perturbation. The results are general, however, with respect to the shape of the energy surfaces z(k).
Eq. (36) is to be replaced by
1
Within the p
Vi
V oo
présent approximation
pP
= -F Voo.
functions
with
(neglecting
p- and d-symmetry)
Friedel’s condition is satisfied for a vacancy if
Voo 5 03B6, and for an interstitial if Voo = -1.1 03B6.
For an attractive potential we can have up to two
bound states, the first one occuring at
=
and the second one at Voo = - 2 . 603B6. For a
finds Eel
0.6 03B6, which is in good
agreement with the values obtained from the
theory of sect. 3. For an interstitial, he fmds
(including the energy of the bound electron)
Eel = -1. 6 03B6. Heie a considerable différence
exists to the case of a square well potential of
radius r. in a quasi-free electron gas, which does
not give a bound state and which leads to a higher
energy, namely Eel =- 0. 8 03B6 [35].
vacancy, Mann
D(k) is the local density of states in k-space, i.e.
In the
eq.
tight-binding approximation
(37b) simplifies to
=
The author would like to
Acknowledgment.
thank his collaborators, in particular Dozent Dr.
H. Bross, Dr. H. Stehle, and Dipl. Phys. E. Mann
not only for making available to him their unpublished results, but also for many interesting and
stimulating discussions. The support of the
Deutsche Forschungsgemeinschaft for the work
reported here is gratefully acknowledged. Last but
not least, my thanks are due to Prof. Jacques
Friedel, with whom I have had stimulating discussions related to the present topic for more than
a decade, and who through his scientific work and.
his encouragement has indirectly also influenced
the present contribution.
--
For vacancies and interstitials in monovalent
metals the charge condition is given by the vanishing of the denominator of the inverse tangent
in eq. (37), taken at the Fermi surface. The condition for the occurence of a bound state is the vanishing of the same quantity, taken at the band
edge. The perturbation is now sufficiently extended to enable us to satisfy the charge condition for
As an example,
a vaeancy in a monovalent metal.
let us for quasi-free electrons consider Ylo
0,
=
626
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