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Vacancies and small vacancy clusters in BCC transition metals : calculation of binding energy, atomic relaxation
and electronic and vibrational densities of states
K. Masuda
To cite this version:
K. Masuda. Vacancies and small vacancy clusters in BCC transition metals : calculation of binding energy, atomic relaxation and electronic and vibrational densities of states. Journal de Physique, 1982, 43 (6), pp.921-930. �10.1051/jphys:01982004306092100�. �jpa-00209470�
921
Vacancies and small vacancy clusters in BCC transition metals : calculation of binding energy, atomic relaxation and electronic and vibrational densities of states
K. Masuda
Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan
(Reçu le 30 décembre 1981, accepté le 16 fgvrier 1982)
Résumé. 2014 En utilisant la théorie des liaisons fortes, on calcule l’énergie de liaison Ebin, la relaxation des atomes et les densités d’états électroniques 03C1i(E) et de modes de vibration gi(03C9) pour des lacunes dans un métal de tran- sition cubique centré. Nous utilisons un potentiel Born-Mayer pour représenter la répulsion de coeur dur entre
voisins. Nous obtenons les énergies de liaison de bilacunes et de petits amas de lacunes (jusqu’a des tétralacunes)
par un procédé direct de minimisation de l’énergie et nous comparons ces résultats aux expériences existantes.
Avec la méthode de récursion de Haydock et al., nous calculons également les vibrations atomiques et la densité
d’état d près d’un défaut lacunaire. En général, la relaxation du réseau atomique autour du défaut joue un rôle
dominant dans la détermination des énergies de liaison et les modes de vibrations locaux. De plus, on montre
que Ebin et 03C1i(E) sont très sensibles à la structure du défaut (configuration des lacunes). Le pic principal de réso-
nance apparait au centre de la bande d de manière analogue à celui des états de surface observé sur W(001) ou 03B1-Fe(001 ) pour une tétralacune agrégée sur le plan (001).
Abstract 2014 A tight-binding type electronic theory is used to calculate the binding energy Ebin, atomic relaxation, and electronic and vibrational densities of states (DOS), 03C1i(E) and gi(03C9), for vacancy-type lattice defects in BCC transition metals : The short-range repulsive energies between neighbouring atomic sites are simulated by a Born-Mayer potential. Binding energies of di-vacancies and small vacancy clusters (up to tetra-vacancies) are
obtained using the direct energy minimization procedure, and compared with the available experimental results.
Atomic vibrations and d-electron DOS on atoms near the lattice defects are also investigated using the recursion method by Haydock et al. In general, atomic relaxation around the lattice defects plays a dominant role in deter-
mining the binding energies and vibrational properties (local vibrational DOS) of the defects. Furthermore, it
is shown that Ebin and electronic DOS 03C1i(E) are very sensitive to the structure of atomic defects (configuration of vacancies) : The prominent resonance peak appears near the centre of d-band, similar to the surface state peak
observed for W(001) or 03B1-Fe(001) surfaces, for the tetra-vacancy aggregated on the (001) plane.
J. Physique 43 (1982) 921-930 JUIN 1982,
Classification
Physics Abstracts
61. 70B
1. Introduction. - It is well known that vacancy- type lattice defects in metals play an important role
in the process of both self- and impurity-diffusions [1, 2]. In order to understand the diffusion and diffu- sion related phenomena from the theoretical point
of view, it is quite important to investigate the funda-
mental properties of vacancy-type lattice defects such
as vacancies, di-vacancies and small vacancy clusters.
We calculate the binding energy, atomic relaxation, and electronic and vibrational densities of states (DOS)
on the atoms near the vacancy-type lattice defects in BCC transition metals using a tight-binding (TB) type electronic theory : The atomic relaxation and binding energies are calculated using the simplified TB scheme (second moment approximation [3, 4]), while the
local vibrational and d-electron DOS are investi-
gated using the more rigorous approach (recursion
method by Haydock et al. [5]). To simulate the short- range repulsive energies, arising mainly from the
increase in sp kinetic energy upon compression, we
use a Born-Mayer potential [3]. This type of approach
has been developed by Friedel and his co-workers [1-9], and has been very successful for a wide variety
of lattice defect problems of transition metals [10, 11].
The study of atomic vibration around the vacancy- type defects is also of great importance for the under-
standing of diffusion behaviour in metals. For iris- tance, in the case of point defects such as a vacancy
or an interstitial atom, resonance modes of phonon spectrum play an important role in the elementary
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004306092100
jumping process [12]. We investigate the atomic vibration around the vacancy-type lattice defects
using the recursion method within the harmonic
approximation : We take into account the nearest- neighbour and next-nearest-neighbour force cons-
tants, a and y, in the calculation of vibrational spec- trum, and neglect the angular dependent force
constants. This calculation allows us to roughly
estimate the vacancy migration temperature Tjjj (so-
called stage III temperature). To estimate Tjn we use
a simple manner as in references [13] and [14],
eD, wres and Wmax being the Debye temperature, the resonance-mode and maximum frequencies of the lattice, respectively.
The calculations of binding energies of di-vacancies and small vacancy clusters are of great significance
for the understanding of nucleation, growth or
removal processes of vacancy clusters [2, 15]. BCC
transition metals like a-Fe and Mo are known to show small void swelling [16]. Furthermore, these
calculations are of significance for the understanding
of vacancy conversion process into dislocation loops :
Jenkins et al. [17] observed both the (al2) I I I >
and the a ( 100 ) vacancy loops in BCC transition metals irradiated with heavy ions.
The present paper is organized as follows : In
section 2, we give the method of calculations for the atomic relaxation and binding energies of vacancy-
type lattice defects in BCC transition metals; TB type electronic theory and Born-Mayer repulsive potential
are used. Furthermore, we give the method of calcu- lations for the vibrational and electronic DOS on
atoms near the defects. In section 3, we present the results of numerical calculations and give the related discussions. The final section 4 is devoted to conclu- sions.
2. Method of calculations. - It is by now well
established that the major contribution to the cohe- sive energy of transition metals arises from the broa-
dening of the d-band in the solid. The transition metal d-band can be described in the Slater-Koster TB approximation neglecting s-electrons and s-d
mixing [6]. It is therefore of great interest to apply
this kind of TB approach to the calculation of elastic constants [3] and defect energies [4, 7, 11] of transi-
tion metals.
We use a TB approximation coupled to the moments
method [3, 4] to describe the transition metal d-band,
and estimate the change in the d-band energy due to the introduction of lattice defects. The local DOS
pi(E) of the d-electrons on atom i is approximated by
a Gaussian fitted to the first and second moments, J.lli and J.l2i [17]. We take the free atom level as the
origin of the energies and take into account the dia- gonal matrix element Vl = ,ui i) of the self-consistent
potential resulting from charge oscillations near the
lattice defects [18]. For the calculation of the second moments Jl2i, we assume that the resonance integral fly between atomic sites i and j varies with the inter- atomic distance Rij as
where the parameter q can be chosen so as to fit the elastic constants or phonon dispersion curve of the given transition metal [4].
One can now calculate the band structure energy
as a function of atomic displacements from the expres- sion
where /t2 denotes the second moment for the perfect
lattice and J12i = J12i - J1I i. In deriving equation (2b),
we have assumed that the self-consistent potential Vi completely screens the defect and each atom is neutral.
In this case, Vi can be given by [17]
To account for the repulsive core-core interaction
energies at small distances, arising mainly from the
increase in sp kinetic energy upon compression, we
take a Born-Mayer potential [3] :
This type of repulsive potential can stabilize the
crystal at short distances. The parameter p can be fitted to the bulk modulus (or phonon dispersion curve) with q, while Co is determined from the bulk
equilibrium condition [3, 4]. From equations (2) and (4), one can calculate the change in the total (electro-
nic and repulsive) energies due to introduction of lattice defects. In table I we present the parameter values qRo and pRo (Ro being the equilibrium inter-
atomic distance), obtained by using the above men-
tioned method for a-Fe, Mo and W, together with the
calculated vacancy formation energies Efv.
To obtain the local vibrational spectra gi(,co)
around vacancy-type lattice defects, we use the recur-
sion method originally introduced by Haydock et
al. [5]. Within the framework of the harmonic approxi- mation, local vibrational spectra gi(co) of the parti-
cular atom i can be expressed as
923
Table I. - Parameters qRo and pRo, and calculated vacancy formation energies Ef, (eV). Values in paren- theses are obtained for unrelaxed atomic configurations.
(a) Kimura, H., Progress in the Study of Point Defects,
Edited by Doayama, M. and Yoshida, S., University of Tokyo Press (1977), p. 119.
(’) Kornblit, L., Physica 106B (1981) 351-355.
with
where a denotes the component a (= x, y and z) of
the vibrational state of the atom i. The recursion coefficients an and bn can be obtained using the
recursion method [5] :
where D = k- 112 ø£1-1/2. M and 4i are the mass tensor and force constant matrix, respectively. In
the following, gi(co) is always normalized as
Using the gi(o)) function, we also calculate the
Debye parameters from the formula [19]
where
From the Debye parameter calculations it may be
possible to discuss the softening (or hardening) of
atomic vibrations near the lattice defects.
The local d-electron DOS on atoms near the vacancy-type lattice defects can be calculated using
a similar manner (but with replacement of W2 by
electron energy E). This calculation allows us to
investigate the change in the fine structure of the
local d-electron DOS due to introduction of the lattice defect. We determine the two centre integrals dd 0", ddir and ddb between the neighbouring d-
orbitals in this paper to fit the previous d-band
calculations of transition metals [20, 21]. The hopping integrals up to the second nearest-neighbours are
calculated using the Slater-Koster table [22].
3. Numerical results and discussions. - The pre- sent numerical calculations are performed using parameter values presented in table I. To calculate
the atomic configurations around the vacancy-type lattice defects, we have relaxed the atoms in a spheri-
cal region with radius
4 - 2 Ro,
73 using a directenergy minimization procedure [23]. The choice of the relaxation region is made by considering the
size dependence of the defect formation energies.
Even for the larger crystallite with radius 6. fi Ro,
/
we have obtained almost the same defect formation
energies; the difference between the results is found to be less than - 2 %.
3.1 ATOMIC RELAXATIONS AND BINDING ENERGIES. -
Using the above mentioned method (direct energy minimization procedure and Gaussian DOS), we
have calculated the atomic relaxations and binding energies of di-, tri-, and tetra-vacancies in a-Fe, Mo and W, and presented the results in tables II-VI : The atomic relaxations around a single- and di-
vacancies are given in tables II and III, respectively.
Table II. - Lattice relaxation around a vacancy.
The direction and length of the arrow are schematic.
Positive (negative) bi value indicates inward (outward)
relaxation.
Table III. - Lattice relaxation around di-vacancy.
The direction and length of the arrow are schematic.
6i (i = A, B, C, ---) represents only the magnitude of the
atomic relaxation , while 6? (a = x, y and z) denotes the a-component of the 6i and positive (negative) ði indicates
the inward (outward) component with respect to the centre of di-vacancy.
Table IV. - Atomic relaxation around tri-vacancies.
Notations 6; and ðf are similar to those used in table III.
Table V. - Lattice relaxation around tetra-vacancy.
Notations bi and 6? are similar to those used in table III.
The corresponding results for tri- and tetra-vacancies
[(110) planer type] are presented in tables IV and V, respectively. The magnitudes of atomic relaxations for other type tri- or tetra-vacancies are comparable
to those presented in tables IV and V. The calculated
binding energies are summarized in table VI : Energies
are given in units of eV. In general, the magnitude of
lattice relaxation is small and always less than 5 % (in units of Ro). This is consistent with the previous
theoretical calculations of lattice relaxations around
a single vacancy [4, 24] or of the relaxations near the surface [7, 25] of d-band metals.
However, the effects of atomic relaxation on the calculated binding energies Ebin are important : one
notices in table VI that Ebin for the relaxed atomic
configurations are significantly different from those of the unrelaxed atomic configurations (values in parentheses). In particular, one notices the following
marked features in the Ebin calculations (table VI).
For di-vacancies in BCC transition metals, unrelaxed binding energies Ebin for the first-nearest-neighbour pairs are greater than those for the next-nearest-neigh-
bour pairs, while the reversed results are obtained for the relaxed di-vacancy configurations. This indicates that the atomic relaxations around the vacancy-type lattice defects play a dominant role in determining
the binding energies. In this respect, we point out
that satisfactory results could not be expected from
the binding energy calculations of the unrelaxed
925
Table VI. - Calculated binding energies Ebin f or small
vacancy clusters. Energies are given in units of eV.
Shown in parentheses are the values obtained for unre-
laxed atomic configurations. Symbols D represent vacancies.
vacancy-impurity [26], impurity-impurity [27] or
vacancy-vacancy [28] pairs.
The above mentioned tendency (importance of
lattice relaxation) becomes much more pronounced
for the tri- and tetra-vacancies. As shown in table VI, Ebin for tri- and tetra-vacancies aggregated on the (111) plane are negative (strongly repulsive inter- actions) for the relaxed atomic configurations, while the ) I Ebin 1 valuers are sufficiently small for the unrelaxed defect configurations. In view of this we can state that the gain in binding energy per additional vacancy is not a simple and monotonical function of the cluster size as one may expect intuitively. Indeed, the calculated Ebin is very sensitive to the type of the lattice defects, i.e., configuration of vacancies, and quite different binding energies are obtained for tri- and tetra-vacancies depending on their configuration :
For tri-vacancies, the most stable configuration is
the clusters aggregated on the (110) plane, while it
is the non-planer clusters (tetrahedron) for the tetra-
vacancies.
Recent perturbed angular correlation (PAC) tech-
nique, though impurities are associated, has enabled
us to get informations on the configuration of vacancy- type lattice defects in BCC transition metals [29].
Mossbauer spectroscopy experiments can also give
the similar informations [30]. These experiments
showed that second-nearest-neighbour di-vacancies
are more stable than the first-neighbour di-vacancies in BCC transition metals. Furthermore, tetragonal configuration of four vacancies was found to be stable in BCC transition metals [29, 30]. These experi-
mental results are consistent with the present Ebin calculations in table VI.
Another important consequence drawn from the
Ebin calculations are the followings : It is worth pointing out that for tri- and tetra-vacancies Ebin depends sensitively on the vacancy aggregation mor- phology, i.e., on the vacancy aggregation plane.
Ebin increases as going (I 11) --+ (100) -+ (110) planes.
This is common to all BCC transition metals consi- dered here and indicates that vacancies tend to
aggregate on the (110) planes rather than (100) or (111) planes. This conclusion is in agreement with the recent experimental results [31, 32] on the damage
structure of BCC transition metals (a-Fe and Mo)
irradiated with heavy ions. Jenkins et al. [32], using
a transmission electron microscope, observed small vacancy loops in a-iron irradiated with Fe+, Kr+, Ge +, Xe + and W + ions; they demonstrated that both the (al2) 111 ) and a ( 100 ) loops are pro- duced by the Eyre-Bullough mechanism [33], with
initial vacancy aggregation on a { 110 } plane.
Finally, we compare the present Ebin results with other theoretical calculations. Using the pair poten- tial method, Johnson and Wilson [34] calculated the
di-vacancy binding energies in BCC transition metals
(a-Fe, W, Mo, Ta and V). Their results are partly
consistent with the present calculations : They
obtained stronger binding energies for the second-
nearest-neighbour pairs than those for the first-
nearest-neighbour pairs, and negative (repulsive) binding energies for the third-nearest-neighbour pairs.
This is consistent with the present calculations.
However, their values of di-vacancy binding energies
are systematically larger (factor 4 - 5) than those of the present calculations. On the other hand, Beeler
and Johnson [35] calculated the binding energies of
small vacancy clusters in a-Fe and concluded that the binding energy of any vacancy configuration can
be closely approximated (within 10 %) by the sum-
mation of all the vacancy-pair interaction energies
in the configuration. This criterion can hardly be applied to the present Ebin results, since Eb;n values
in table VI depend sensitively on the configuration
of vacancies.
3.2 LOCAL d-ELECTRON DOS. - We have also calculated the local d-electron DOS pf(E), Â = xy,
yz, zx, X2 _y2 and 3 z2 - r2 on atom i near the
vacancy-type lattice defects, using the recursion method by Haydock et al. [5]. In figure 1 (left side)
Fig. 1. - Local d-electron density of states on atoms (o sites) nearest to single and di-vacancies (next-nearest-neighbour pair) in a-Fe. Energies are given in units of eV.
we show the local DOS pf{E) on the first-nearest-
neighbour atom of a single vacancy in a-Fe. Dashed
curves represent the DOS for perfect lattice : The
two-centre integrals dda, dd7c and ddb are chosen to fit the Pettifor’s values [20], and atomic relaxations around vacancy (obtained in section 3.1) are taken
into account. The corresponding results for the di- vacancy (next-nearest-neighbour pair) in a-Fe are
shown in figure 1 (right-hand). One notices that the local DOS for h = xy, yz and zx orbitals are more
influenced by a vacancy or by a di-vacancy than those
for A = x2 - y2 and 3 z2 - r2 orbitals. This is related to the fact that change in the second moment
flJ.l2i for the dg orbital is greater than that for the dy
orbital. In general, the change in the local electronic DOS is not so large for these defects.
In contrast, however, the change in the electronic DOS is much more pronounced for the atoms nearest to tri- and tetra-vacancies as shown in figure 2. The
most stricking feature in figure 2 is that the resonance
peaks appear near the centre of the d-band for tri- vacancies (x2 - y2 orbital) and tetra-vacancies (yz,
zx, x2 - y2 and 3 z2 - r2 orbitals). These peaks are
similar in nature to the surface state peaks observed
for instance for W(001) and a-Fe(001) surfaces [21, 35].
It is remarkable that surface-state like peak appears in the local electronic DOS on the atoms around the small vacancy clusters like tri- or tetra-vacancies.
For other type tri- and tetra-vacancies, the surface- state like peak does not appear so clearly. This indi-
cates that local d-electron DOS is quite sensitive to
the type of vacancy clusters, i.e., to the configuration
of vacancies. Furthermore, in view of the fact that surface-state peak plays an important role in deter-
mining the reactivity (strength of chemisorption bond)
of metal surfaces, vacancy clusters are expected to
have different interaction (binding) energies with impurity atoms, depending on the vacancy configu-
rations. In real crystals, this is the important problem,
and similar discussions have already been given in
reference [36], dealing with the electronic structure of
grain boundaries [36].
3. 3 VIBRATIONAL SPECTRA. - As mentioned
before, the atomic vibrations around vacancy-type lattice defects are of significance for understanding
of diffusion behaviour. Firstly we investigate the
local vibrational spectra go(w) for the unrelaxed
927
Fig. 2. - Local d-electron density of states on atoms (o sites) nearest to tri- and tetra-vacancies in a-Fe. Energies are given
in units of eV.
defect configuration assuming the same force constants
as in the perfect lattice. Secondly, we calculate go(m) taking into account the full atomic relaxation (elec-
tronic effects) around the defect, and investigate the
effects of atomic relaxation and electronic rearrange- ment on the go(co) behaviour. The parameters used for the calculations are presented in table VII, together
with the Debye parameters calculated from equa- tion (9) and bulk phonon spectrum.
In figure 3b, we show the local vibrational spectra go(w) at the nearest-neighbour site of the unre-
laxed vacancy » in W ; the bulk phonon spectrum of W is shown in figure 3c. go(m) for the «relaxed vacancy » is shown in figure 3a. Comparing the go(co)
curves in figure 3b and figure 3a, we come to the conclusion that the effects of atomic relaxation and electronic rearrangement (narrowing of d-band) on
the go(w) are large : When the changes in force
constants resulting from atomic relaxation and elec- tronic rearrangement are taken into account, one observes a drastic softening in the go(m) curves. A prominent resonance peak appears near the frequency
Table VII. - Calculated Debye parameters for the nearest-neighbour atoms of vacancies and di-vacancies
(next-nearest-neighbour pair).
OA, represents the average Debye parameter calculated from equation (9) [average value over n = - 2, - 1, 1
and 2]. ø A v values in parentheses are obtained for the
unrelaxed defect configurations.
0.6.(Omax. The corresponding results of go(co) for Mo
and a-Fe are qualitatively the same as that for W.
According to Refs. [12] and [13] the lower resonance peak in go(ro) is responsible for the migration of a single vacancy, and its position relative to ( Omax
