Monovacancy Formation Energies in Cubic Crystals

U. KRAUSE et al. : Monovacancy Formation Energies in Cubic Crystals 479

phys. stat. sol. (b) 151, 479 (1989)

Subject classification: 61.70

Weierstrap- Institut f u r Mathematik der Akademie der Wissenschaften der DDR, Berlin1) ( a ) and Sektion Physik der Humboldt-Universitat zu Berlin2) (b)

Monovacancy Formation Energies in Cubic Crystals BY U. KRAUSE (a), J. P. KUSKA (b), and R. WEDELL (b)

Using the conception of an extended object (Umezawa et al.) based on the continuum theory of elasticity, monovacancy formation energies are calculated. Anisotropy is included for cubic symmetry by means of a first order approximation to the Green’s function tensor and some tensor algebraic calculations. The last considerations contribute to a better understanding of the meaning of “Voigt’s average isotropic elastic constants”. Further lattice structure effects are taken into account by a common parameter for every type of structure. A comparison with experimental data or results from other theories shows either satisfactory agreement or the inconsistency of the first order approximation to the anisotropy for some materials with common properties.

Unter Benutzung des Konzepts eines ausgedehnten Objekts (Umezawa u. a.), das auf der Konti- nuumstheorie der Elastizitlit beruht, werden Monovakanzformierungsenergien berechnet. Die An- isotropie wird fur kubische Symmetrie mit Hilfe einer Naherung erster Ordnung fur den Green- funktionstensor und einigen Tensoralgebrarechnungen berucksichtigt. Die letzteren Betrachtungen tragen zu einem besseren Versttindnis der Bedeutung der ,,gemittelten isotropen elastischen Kon- stanten nach Voigt“ bei. Weitere Gitterstruktureffekte werden durch einen gemeinsamen Para- meter fur jeden Strukturtyp berucksichtigt. Ein Vergleich mit experimentellen Daten oder Ergeb- nissen anderer Theorien zeigt entweder eine befriedigende Ubereinstimmung oder die Inkonsistenz der NLherung erster Ordnung fur die Anisotropie bei einigen Materialien mit gemeinsamen Eigen- schaften.

1. Introduction

Many important solid state properties are connected with imperfect structures of the crystal lattice [ 11. Whereas the conductivity of semiconductors is mainly influenced by point defects, mechanical properties such as the relation between elasticity and plasticity can be explained by the presence of dislocations. However, both kinds of defects interact with each other and their influence on solid state properties cannot be sharply separated. Especially during ion implantation different kinds of radiation damage are expected in dependence on ion energies, mass, dose, and flux [Z]. There- fore, experimental and theoretical studies of crystal defects are also of practical im- portance for semiconductor device fabrication. Such investigations were carried out already for years. According t o the diversity of materials (metals, covalent and ionic crystals, compounds) and effects quite different theories and methods have been developed in this field : Fully self-consistent quantum-mechanical calculations [3] yielding ground state properties of metals [4]; calculations [5 ] based on pseudo- potential theory [6] and used for defect calculations in alkali metals; theories starting from a dynamical theory of crystal lattices and constructing interatomic potentials [7] either by empirical parametrization or from more or less first principles and being especially successful for ionic and semi-ionic crystals ; theories which use interatomic

l) Mohrenstr. 39, DDR-1086 Berlin, GDR. 2, Invalidenstr. 110, DDR-1040 Berlin, GDR.

31 physira (b) 151/2

480 U. KRAUSE, J. P. KUSKA, and R. WEDELL

potentials together with density functional methods 181 and theories constructing interatomic potentials with regard to results of tight-binding calculations [9]. Green’s function techniques for point defects in semiconductors have also been developed [ 101.

Likely, the only common feature of all these concepts and theories is to use micro- scopic ideas and to avoid macroscopic continuum models. Nevertheless a comprehensive study of the condensed state of matter [ll] shows the usefulness of continuum theory of elasticity [ 121 for understanding defect properties of crystals.

In this paper we use the concephion of an extended object [12] to calculate monovacancy formation energies in cubic crystals. Extended object theory together with the boson transformation method and the spontaneous breakdown of symmetry are the main tools to handle with observable phenomena of solid state physics such as crystal defects, magnetic domains in a ferromagnet, or whirl lines in superconductors from the point of view of quantum field theory [ l l ] . In this paper we will not deal with quantum aspects of crystal defects, but we show in Section 2 that it is just the extended object theory [12] which allows the unification of the theory of crystal defects even on the level of a classical elastic continuum. Then the anisotropic cubic Green’s function tensor of the continuum is correctly developed and formulae for the displacement and energy due to a point defect are derived (Section 3). In Section 4 these formulae are suited for a practical use by a geometrical consideration. In Sec- tion 5 the obtained results are discussed in comparison with experimental data and results from other theories. Some concluding remarks are made in Section 6.

2. Extended Objects in Crystals and Continuum Theory of Lattice Defects Elastic models of crystal defects are considered in the literature since 1907 (Volterra). Good reviews in this field are [13 to 191. The aim of the present section is to give the “extended object theory” [12] of Wadati, Matsumoto, and Umezawa a more specific crystallographic derivation. Notwithstanding its highly interesting quantum field theoretical foundation [ll, 12, 201 we can take up the extended object theory in its low momentum approximation as a straightforward and comprehensive picture of Somigliana’s conception on states of internal stress of a continuous homogeneous elastic body. Somigliana’s conception is very clear reported in [15], 0 4a: “To construct a Somigliana dislocation, mark out in the elastic body a surface S bounded by a curve C and make a cut coinciding with S. Give each pair of points adjacent to one another on opposite sides of the cut a relative displacement d, scraping away material where there would be interpenetration. Fill in the remaining gaps with additional material and cement together. This evidently leaves the material in a state of internal stress. . . . If d has a constant value, we have the usual dislocations of solid state theory, the dislocations of types 1, 2, 3, of Volterra. . . . To make a model of a point defect, we take for S a small sphere with a suitable distribution of d over the surface. . .”

If we take d constant in magnitude and directed radially, we get the familiar mis- fitting-sphere model for a point defect [15]. For a dislocation Somigliana’s internal surface S has a boundary. This boundary is the so-called dislocation line I, and Somi- gliana’s vector d is the constant Burgers vector. Using a formula we have

The essential point of (2.1) is that the integration path C goes through the internal surface S (at least one time):

u+(z) - u-(z) = d , x E s ) (2.2)

Monovacancy Formation Energies in Cubic Crystals 481

where u+(x) and u-(x) are the displacements on the outer and inner side of S, respectively. In Somigliana’s model of a point defect the internal surface S is a sphere, i.e. S is here a surface without boundary. We consider again the path integral $ aruf(x) dxr.

Because of the non-existence of a boundary of S, the integration path C must go a t least two times (or an even number of times) through S and we get if C crosses S two times

C

4 8ruf(x) dxr = u;(x,) - uT(~1) - (uj’(x2) - uF(~2)) , (2.3) C

= d&,) - d l ( X 2 ) >

= d?(q , x2); q, x2 E 8 .

If the vector d would be constant on the sphere S we have

d ( X J - d ( X J = 0

and the whole thing would be only a trivial translation. Therefore, in the case of a point defect, d must be a non-constant vector field on S. On the other hand, the difference between two vectors in (2.3) is again a vector, namely d * , and we see that there is no fundamental distinction between a dislocation and a point defect in Somigliana’s model. However, in the case of a dislocation the resulting state of internal stress is independent of the choice of S and depends only on the choice of the boundary L of S because the Burgers vector is constant on S. Therefore, a dislocation is an one- dimensional defect whereas a point defect is a two-dimensional defect in Somigliana’s model.

From (2.1) or from (2.3) we get using Stokes’s theorem

x E L x E S

for a dislocation , for a point defect . G ; ( ~ ) ( x ) := (aka, - Elat) u g ( ~ ) + o (2.4)

Equation (2.4) states that the displacement u(x) is multivalued or discontinuous in certain domains and in this way destroys the well-known elastic compatibility conditions, i.e.

EiktEjrnnarnak(anU1 + a~un) + 0 . (2.5) In (2.5) the summation convention is used, E~~~ is the total antisymmetric tensor of third degree.

We require that the destruction (2.5) of the compatibility conditions should be caused alone by the commutator (2.4). This requirement leads to the condition

(C?,a, - akarn) anui = 0;

because (2.5) transforms under (2.6) into

(EiklEjnzn. + E j k l E i r n n ) akanaanul =k 0 .

V x ; Vm, k , n, I (2.6)

(2.7) Thus the commutator (2.4) will be the only source term of internal stress and a well- defined mathematical problem is obtained. Furthermore, a state of internal stress has to fulfil the elastic equilibrium conditions

c<jkza,alu~ = 0 , (2.8) where Cijkz is the tensor of elastic constants. We consider a linear elastic infinite homogeneous medium and neglect outer body forces. Then we can use the elastic Green’s function tensor Gjk(x), defined by

482 U. KRAUSE, J. I?. KUSKA, and R. WEDELL

Acting with the operator Cijklai on both sides of (2.4) and using (2.6) and (2.9) the final result will be

a,,ul(z) = - J d3x' Gjk(X - d) C~k,&G$'"(d), (2.10)

all indices run from 1 to 3, 8; E a/axi. The foregoing discussion leads to the following definition of an extended object in crystals: An extended object in an infinite linear elastic homogeneous continuum is a state of internal stress oij, given by (2.10) by means of

Cfij = Cijkl 8jut 3

This state of internal stress is induced by the commutator (2.4) and fulfils the continuity conditions (2.6) and the elastic equilibrium condition (2.8). It should be mentioned that in the theory of continuous distributions of dislocations an expression similar to (2.10) was derived by Mura [17] (see [la], Q 18.3). A different theory of internal stress was given by Kroner [18] (see [16]) where the general incompatibility sij 9

8.. - y - &ikl&j?nnamakUnl + 0 i

with the strain unl serves as a source term for the internal stress. As mentioned above, Wadati et al. [12] built up the extended objects in four-dimensional space-time under very general propositions, using quantum field theoretical methods. In this way they gave a generalization and unification of the theory of elastic models for crystal defects. It is shown in [ 121 that the description of a dislocation, a grain boundary, a free surface or of a point defect is only a matter of making the right ansatz for G;("(z) whereas the four-dimensional analogues of (2.4), (2.6), (2.9), and (2.10) work without any modification. An important feature of this extended object theory is the fact that the GL'"(s) can be constructed syst,ematically for the different types of defects. Using the boson transformation method Wadati et al. (12],[11] came also to the commutator

Q,v +(j) (z) = (apav - ad,) %(X) (2.4 a)

and to the continuity condition

@,a, - a v a p ) a&= 0 ("x; Ypu, v, @ > i ) (2.6a)

in order to determine the multivalued displacement u of an extended object, where latin indices run from 1 to 3 and greek indices run from 0 to 3 with 8, as time derivation a /a t = a/t?x,, z = (q,, 2). The domains in which some components of G~" ' (X) do not vanish are the domains of discontinuity or multivalueness of u. The possibilities of discontinuities in time include for example instantaneous jumps of defects. Further- more, in general the multivalued displacement has to satisfy the wave equation (elast,ic equilibrium condition) of an infinite homogeneous linear elastic continuum

(2.8a)

: = Ciljm,

(-Pi& + cizjmazam) U d X ) = 0 ,

where ~ i j is called the effective density 1121. Setting C:! : = eij, Ciy =

Ci: = Cg := 0, (2.8a) reads as

c:;"a,a,u,(x) = : A ~ ~ ~ , ( ~ ) = o . Let be Gjk t,he Green's function tensor of (2.8a),

&jGj,(~) = -8i,$4)(~) .


Acting with C;:aA on both sides of (2.4a) and using (2.6a) and the Green’s function tensor of (2.8a) one gets finally

~ , U , ( X ) = - d42’ G;.~(x - 5’) C$a;G:‘”(x’) . (2.10a) In (2.10) the greek indices run only from 1 to 3, but in (2.10a) from 0 to 3, so that the difference between (2.10a) and (2.10) is only formal. For the dual value of G&(j) we have by definition

(j) /1y +(j) @A, (4 = f EIeGflv (2) ,

cfi”,” is the total antisymmetric tensor of fourth degree and from (2.4a) and (2.6a) the divergenceless condition

I (j) (2.11) a GIe ( x ) = 0

follows. Equation (2.10a) is the basic equation of the extended object theory. It describes the multivalued displacement caused by an extended object which is represented by G:(”(z) in four-dimensional space-time. The above given definition of an extended object can easily carried over to the general time-dependent case. Practi- cally the representations G$‘) of extended objects are given by

(2.12)

for a dislocation. Ma is really the Burgers vector [ E l , T a time parameter, and a the arc length of the dislocation line I,;

(2.13)

for a model of a plane surface or a grain boundary [ 121, al, a2 parametrize this surface in space, z in time, and

for a model with a general surface. e is a weight function, T as before a time parameter, 4, 0, are parameters on the surface, and q is a parameter of the surface in space. In

is the Jacoby determinant, for discussion see [ l l , 121. a(Yfl, yv, ... ) 3 ( t , o, ...)

these formulae

The model of a vacancy is a static sphere of radius R on which the displacement is multivalued, i.e. 11 = dy; + yi + y$, p(7) := d(7 - R), yo = t and al, vz are the angles in spherical co-ordinates. For simplification we set as in [12] HeS := M&@. Inserting this spherical parametrization into (2.14) and using GL(i) = - I E ~ , , G A ~ I e (j)

one has

~,t(’) = x(dl,a: - sl,a;) a(r’) (2.15)

with 8; = a/az;, r’ = I/xT + xiz + xf and a(?) defined by - o(r) = e ( r ) , i.e. a(r) = a ar

- - -e(R - r ) [12]. The tensor of elastic constants C$jk, in a cubic material is given by

(2.16)


In the static case the Green’s function tensor is defined by

C+j&aZGka(~) = - d ~ i n d ( ~ ) ( ~ ) . (2.17)

With (2.15), (2.16), and (2.17) the basic formula (2.10) reads as

C&U,(X) = M(3a + 2b + ,u) 8” J d3z’(a(r’) 8 ,Gj l (~ - z’)) - Mdj,,~(r) . (2.18)

Equation (2.18) gives the displacement caused by a spherical extended object (vacancy) within a cubic infinite homogeneous linear elastic medium.

3. Approximation of the Elastic Cubic Green’s Function Tensor and Vacancy Displacement

The problem to solve (2.17) numerically or analytically is discussed in [13], [21], and [22]. An analytical derivation of the cubic Green’s function tensor is restricted due to difficulties to find the roots of sextic equations [23]. A nlrmerical treatment [21] leads to a huge amount of data which are hardly to handle in further calculations. Therefore, we follow the treatment of [22] where the problem

CiklmakamUl = -fit@) ( 3 4

for an infinite linear elastic homogeneous medium is considered. In (3.1) u means the displacement caused by a point force f acting on the co-ordinate origin. The bomdary condition is

lim u(z) = 0 I*l-+c=

and the transition between (3.1) and (2.17) is given by replacing f,, by ddn and u, by Gin. The solution of (3.1) is generally [22] given by

2n

where e is a unit vector and ye is an angle, both in Fourier space, r = Iri. A power series development of wt(e) up to first order in p/b gives

(3.3)

with (ef) = elfi + ezf2 + e3f3; a, b, p are the numbers from (2.16). It should be mentioned that the term


is missing in [22] . This term leads in the following to cumbersome expressions. Carrying out the integration (3.2) we have the Green's function tensor in the same approximation as (here no summation over equal indices)

(3.4) In (3.4) the notation ni = xt l r is used and A,, denotes the following matrix:

0

-(?a," - nf) n$ A - (ng - ng) n$

- Bn,

ngn;) ,

Azm =

with A : = n,n2(n2,ng - f n:n$ -

B := (n: - nz) n,n,n, . Furthermore the divergency of (3.4) will be

(3.5)

a + b 3 p 1 a - 3 b 5 26 -- (n: + nd +nil nz + - n;] - +8nb(a + 2 b ) % 7 [ ( 2 ( a + b ) 2 ) a + b

R, :=a (ng + ng) n , ,

R, := (n: + ng) n, , (3.7)

R, is approximated by

R, = f ($ + ng) n, + Bn?ngn,.


From (2.18) together with (3.6) and (3.8) one obtains the displacement

a,,u,(z) = -Mdjya(r) - J Z ~ , . ~ , I ( Z ) +

(3.9)

with (a + c, := a f b c, :=- a + 2b b ’ b(a + 26) ’

In (3.9) I , J , and K represent the following integrals:

d32’ a(r’) I(z) := __ s 4n IZ - Z” ’

over the sphere volume (r’ 5 R). Let A denote the Laplacian a = 8: + 8; + a:, then the following identity holds:

1; r < R , { 0; r > R . &AAaK(x ) = + LAJ(x) = ~ I ( z ) = -G(v) =

In order to obtain vacancy formation energies one has to integrate the energy density w = f Cijkl a,u, &u, and according to (3.9) the result will be

E = E< + E’ = J d32 W ( X ) + J d32 w ( x ) . r < R r > R

Finally, one gets

(3.10)

The field (3.9) is illustrated on Fig. 1 to 3 for five spherical, extended objects (vacancies) arranged in the (loo)-, (110)-, and (111)-planes of a diamond lattice, respectively. These spheres are furnished with meridians and parallels of latitude. Then we


Fig. 1. The displacement field for five vacancies in Xi arranged in the (100) plane (further explanation in the text)

calculated the displacement field lim u(x) on all spheres of each figure by super- Ixl--LR+O

position. The obtained displacement field vectors were added to the meridians and parallels of latitude gettingin this way new meridians and degrees of longitude which are shown on the figures. Therefore, the distortion of the initial spheres by the interacting displacement fields leads t o such new closed surfaces. We used silicon elastic constants, distances of atoms and silicon atom radius, but the factor M = 312 for reproducing a remarkable effect. Central perspective is used. I n Fig. 1 and 2 the

1

Fig. 2. The same as on Fig. 1 but for the (110) plane (further explanation in the text)


Fig. 3. The same as Fig. 1 but for the (111) plane (further explanation in the text)

reader looks perpendicular to the z-x plane, whereas Fig. 3 includes a rotation of 60” around the z-axis. The pictures are enlarged as far as possible. Consequently, the sizes of the objects in the figures are not comparable. As seen from the figures the interaction is the weakest for the (100)-plane. This may be explained by the relatively large distances between the interacting objects. In both other cases, next neighbours are included into the interaction due to the configuration of the (110)- and (111)-plane; respectively.

4. Geometry of Tensor Space

Formulae (3.10) t o (3.12) make sense only in the case if ,u/b is really a small quantity. From experimental data it is well known that this condition is not fulfilled in most cases. But a careful consideration of the geometry of tensor spaces will give us an idea how to proceed.

Let 3 be the group of rotations in the three-dimensional real euclidean space R3. 91 is the totality of all rotation matrices R = (Rcj). T p denotes the space of tensors of p-th degree over R3. Any rotation of 6 E T, will be denoted by k ++ C E Tp that means in the case that c^ E T4,

h

R * ̂ c = Rp&jRrkRslCijk~ . (4.1)

A tensor 5 E T, is called an isotropic tensor, if and only if fi * 6 = 6 for all R E 3, otherwise the tensor is called anisotropic. Let us define with I241 a scalar product “0” for all C, D E T,,

h 6 0 D : = cij ... IDij _.. 1 7 (4.2)


that means summation over all indices; This _scalar pro_duct generates the norm 161 = (60 @I2 and the distance g(C, D) := IC - 61, C, 6 E T,. Scalar product, norm, and distance are invariant under rotations. The tensor space T, is a direct sum of the linear subspace J, of isotropic tensors and the linear subspace A, of anisotropic tensors [24],

J, and A, are invariant subspaces of T, under rotations and A, is the orthogonal complement of J, in T,. Therefore, we have a unique decomposition of C E T, into the isotropic part CJ E Jp and the anisotropic part

with ~

It holds

h

T p = Jp + A , , (4.3)

A

A

E A,,

(j = ijJ + (4.4)

e(c, CJ) = min (6,fi) CAo CJ = 0 and = 1 6 J l 2 + 1 6 A l Z .

A *

(4.5) 6SJ,

h ,. and the physical meaning of (4.5) is: CJ is the isotropic approximation of C [24].

For the tensor of elastic constants Cijkl (2.16) we have the norm-square,

1Q2 = 3c?iii + 12G212 + 6G122 = 3(3a2 + 8b2 + 4 d + 2ap + 4bp + p2) .

Following [24] we get the isotropic part CJ as

and its norm-square as IeJl2 = 3(3a2 + 8b2 + 4ab + 2ap + 4bp + +p2) . (4.8)

Then the anisotropic part is given by

and A

The result (4.7) is well-known [25] (see also [26]) but somewhat ambiguously presented: eJ has the extremal property (4.5) and therefore the term averaging (in German: “Mittelung”) used in 1251 is misleading. It is crucial t o mention that the full isotropic part (4.7) of (2.16) is not obtained by setting p equal to zero.

A c i j k l = C i j k l - CCkl

ICAI2 = f p2 .

h

Let us construct an auxilary tensor C* of elasticity

c& = a&j& f bo(6idjl + M j k ) + &I E dpidpjdpkdpl

lim c&$ = C&l (4.10)

(4.9) having the properties

and pe4O

It?*I2 = Icy. (4.11) We have clearly a, = a + p15 and b, = b + p15, while p, is obtained from the quadratic equation (4.11) as

(4.12) po= - @ + 2 b +$/4 +((a + 2 b + $ P I 2 ++/.A po > 0 for physical reasons.

490 U. HRAUSE, J. P. KUSKA, and R. WEDELL

The basic idea of the ansatz (4.10) is straightforward to see from (3.4). The isotropic Green’s function tensor is given by setting p / b = 0 in (3.4), i.e. p = 0. But p = 0 does not supply the whole isotropic part of Cijkl, as seen above. The replacement of a, b, p by a,, b,, pa will cancel this discrepancy.

However, the conservation of the norm does not automatically preserve the values of the elastic coefficients. But the above-presented procedure is in conformity with a first-order approximation for the anisotropy of the Green’s function tensor given in Section 3 (cf. (3.3)). Throughout the following section we perform all calculations ((3.11) and (3.12)) with constants a,, b,, p,. That means we use (4.9) instead of (2.16). The constants a,, b,, pa are calculated in the described manner from the original physical constants of elasticity given in [39], [ la].

5. Results and Discussion

Tables 1 and 2 contain the results of vacancy formation energy calculations using (3.10) to (3.12). The elastic constants are mainly taken from [14], the lattice constants and the radii of atoms in crystals from [39]. We used covalent, ionic, and metallic radii, respectively. Calculations were performed for a larger number of materials than presented in the tables. However, for Cu, Ag, Au (group I-B), Pb, Th, Pd, for the alkali metals (Li, Na, K, Rb, Cs), and for PbTe, KBr, KCI, KI, RbBr, BbC1, RbF, RbI the energy E’ from (3.11) was negative. Thus in the field of cubic monatomic crystals the failure of the presented method is for some reasons not accidental, but reflects common properties of the group I elements of the periodic table except Pb, Th, Pd. In the case of cubic diatomic crystals we have only, but not necessary, negative energies E’ if one of these elements forming a monatomic crystal shows also negative energies.

Let us first consider Table 1. The value 1/M = 3.591 is fitted to the monovacancy formation energy of A1 (0.66 eV [29]), then 2.932 = c/(2/3) . 3.591 is the l/M-value for b.c.c. lattices and 1.7955 = 112. 3.591 will be the 1/M-value for the diamond lattice. Thus, each structural type needs its own $!-value. The agreement between our and experimental values is quite good, the largest relative errors appear for Nb and V, roughly equal to 30%. In [40] using apparantly isotropic theory of elasticity and treating b.c.c. metals good values are reported for the single vacancy formation energies of the alkali metals, but sharp overestimations, especially for Cr, V, and Mo, are observed. In [32] the largest relative error (15%) appears for Mo, For the diamond structure the reported data are wide-spread. If the higher values should have more evidence as i t seems to be (cf. [all, p. 220 of the Russian edition), then we have to change the l/M-fit, setting for example 1/34 = 1.635.

The composition of the vacancy formation energy as a sum of positive structural formation energy and negative relaxation energy [5] does not appear in our method. Although we have also two energies, E’ and E< (see (3.10)), both come from the quadratic energy density w = Cijk, aguz alu, and therefore none of them can be negative, i.e. there is no correspondence.

In Table 2 vacancy formation energies for diatomic crystals are collected. Each lattice structure type (Bl, B,, B3) has again its own M-value obtained in a averaging procedure for the data given in the last column. Generally, for the structure types B, and B, only Schottky defect data were available. In these cases we fitted the term

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8 16

.11

8.24

9 2.

73

2.8 f 0.

69)

p1.

V

2.62

66

1.33

27

23.0

98

12.0

17

4.37

6 1.

50

2.1 f 0.

29)

Cr

A29

2.49

24

1.22

49

35.0

6.

78

10.0

8 2.

932

2.03

1.

91*)

a-

Fe

b.c.

c.

2.47

12

1.25

44

23.0

1 13

.46

11.6

6 1.

98

1.6 f 0.

Z9)

; 1.6

& 0

.l6)

Nb

2.85

27

1.44

05

24.6

5 13

.45

2.87

3 1.

78

2.6

+ 0.

39)

w %

w 2.

7349

1.

3818

52

.327

20

.453

16

.072

4.

57

3.65

); ca

. 3.

6*);

4.0

0.39

)

C A4

7 1.

5415

0.

7583

10

7.9

12.4

51

.8

1.79

55

4.35

4.

16, 3

.681

°); ~

5.2

5~

~)

Ge

diam

ond

2.44

22

1.22

57

12.8

53

4.82

6 6.

68

2.52

2.

07,

l.91

lo);

2.45

11);

2.3 f 0.

412)

Si

2.

3469

1.

1816

16

.577

6.

392

7.96

2 2.

84

2.13

, 2.3

51°)

; 2.7

511)

; 2.5

, 4.5

'2)

3,

Exp

erim

enta

l va

canc

y fo

rmat

ion

ener

gies

, ci

ted

afte

r [2

7].

4,

Exp

erim

enta

l m

onov

acan

cy fo

rmat

ion

ener

gy f

rom

[28]

. 5,

E

xper

imen

tal

vaca

ncy

form

atio

n en

thal

py f

rom

[29]

. 6,

E

xper

imen

tal

mon

ovac

ancy

form

atio

n en

ergi

es f

rom

[30]

. ')

Exp

erim

enta

l m

onov

acan

cy fo

rmat

ion

ener

gy f

rom

[31

]. T

heor

etic

al m

onov

acan

cy fo

rmat

ion

ener

gy f

rom

[32

]. g

, E

xper

imen

tal

mon

ovac

ancy

form

atio

n en

ergi

es,

cite

d af

ter

[32]

. lo

) A

ctiv

atio

n en

ergi

es,

cite

d af

ter

[33]

. 11)

Sing

le v

acan

cy f

orm

atio

n en

ergi

es f

rom

[34

]. 12)

Exp

erim

enta

l va

canc

y fo

rmat

ion

enth

alpi

es, c

ited

aft

er [

35].

Tab

le 2

M

onov

acan

cy f

orm

atio

n en

ergi

es E

in

diat

omic

cub

ic c

ryst

als

publ

ishc

d da

ta (

eV)

1/M

E

m

ater

ial

latt

ice

dist

ance

ra

dii

(lo-

' nm

) C

,,,,

CllZ

Z ~

12

12

AB

of

nex

t R~

RB

(1

0'0 N

/m2)

(l

olo N

/m2)

(l

olo N

/m2)

(e

V)

neig

hbou

rs

( 10-

1 nm

)

AgB

r €3

, 2.

885

1.26

1.

95

5.61

3.

27

0.72

4 1.

6328

2.

01

1.13

13)

AgC

l 2.

77

1.26

1.

81

5.98

5 3.

611

0.62

4 1.

70

1.45

13)

LiB

r 2.

745

0.68

1.

95

3.92

1.

89

1.88

5 2.

02

1.80

13);

1.80

14); 1

.531

5)

LiF

2.

01

0.68

1.

36

11.3

97

4.76

7 6.

364

2.25

2.

6814

); 2.

51 +

0.171

6)

Li J

3.00

0.

68

2.16

2.

907

1.42

1 1.

407

2.02

1.

3414

); 1.

0613

); 1.2

915)

N

aBr

2.98

0.

97

1.95

4.

037

1.01

3 1.

015

1.69

1.

66

0.02

14);

1.72

13);

NaC

l 2.

81

0.97

1.

81

4.93

6 1.

288

1.27

8 1.

71

2.07

+0.0

514)

; 2.4

7 +0

.271

6)

NaF

2.

31

0.97

1.

36

9.7

2.38

2.

822

1.72

2.

4213

) ; 3.

2415

) N

aJ

3.23

0.

97

2.16

3.

035

0.91

5 0.

742

1.67

1.

8413

); 1.

8715

) C

aO

2.40

0.

99

1.40

22

.3

5.9

8.1

4.55

-

MgO

2.

10

0.65

1.

40

29.6

64

9.50

8 15

.581

5.

79

7.6,

8.818)

SrO

2.

57

1.13

1.

40

17.3

4.

5 5.

6 3.

85

-

LiC

l 2.

565

0.68

1.

81

4.89

2.

22

2.49

2.

07

2.12

13);

2.12

14);

1.72

15)

2.17

, 2.

7215

); 2.3

4l3)

2.23

15)

2.44

13) ; 2

.481

5); 2

.221

7)

CsB

r B

, 3.

715

1.80

4 2.

106

3.06

3 0.

807

0.75

1.

671

2.23

1.

8013

) CS

Cl

3.56

1.

804

1.95

5 3.

664

0.88

2 0.

804

2.29

1.

7713

) C

sJ

3.95

1.

804

2.33

3 2.

446

0.66

1 0.

629

1.38

1.

8313

) 2.

0266

1.

99

2.31

, 1.

612)

for

Ga-

atom

s 2.

30

2.31

, 2.0

12) f

or A

s-at

oms

lnS

b 2.

797

1.45

5 1.

419

6.7

3.64

9 3.

019

1.76

2.

05,

1.21

2) fo

r In

-ato

ms

1.63

2.

05,

1.41

2) fo

r Sb

-ato

ms

GaS

b 2.

637

1.24

3 1.

419

8.97

4.

12

4.48

1.

50

1.85

, 1.3

12) f

or G

a-at

oms

2.24

2.

33,

1.71

2) fo

r Sb

-ato

ms

GaA

s B,

2.

438

1.24

3 1.

305

11.8

77

5.37

2 5.

944

0.24

O

.02l

9) fo

r bo

th

12)

See

Tab

le 1

. la

) E

xper

imen

tal

Scho

ttky

def

ect

ener

gies

, ci

ted

afte

r

14)

Exp

erim

enta

l va

canc

y pa

ir f

orm

atio

n en

ergi

es,

cite

d af

ter

[19]

. 17)

Exp

erim

enta

l Sch

ottk

y de

fect

ene

rgy

from

[37

]. 15) C

alcu

late

d va

lues

of f

orm

atio

n en

ergi

es o

f Sch

ottk

y de

fect

s, c

ited

afte

r [36

] (p.

236

). la) T

heor

etic

al S

chot

tky

defe

ct e

nerg

y fr

om [

7] (p

. 150

).

19)

Exp

erim

enta

l vac

ancy

for

mat

ion

ener

gy fr

om [

38].

13

) E

xper

imen

tal

form

atio

n en

thal

pies

of

Scho

ttky

def

ects

, cite

d af

ter

[36]

. [7

1 (P

. 177

).

d e 'd w 3 "F 8 s M


higher (Na+: 5.12 eV, C1-: 5.01 eV) than the corresponding experimental Schottky energy (2.2 to 2.5 eV) because the lattice cohesive energy had to be subtracted (see [37]). For the structure type B,, on the other hand, mainly elemental vacancy energies are available. Measurements [38] of monovacancy formation energies in InSb (without distinction between the components) provide a considerably lower value (cf. Table 2). The difference may be probably also connected with the above-mentioned lathice cohesive energy.

6. Concluding Remarks

In the method of an extended object the crystal is represented by its elastic constants and treated as a linear elastic continuum. I n this theory, the simplest model for a point defect is a sphere with a given radius on which the displacement is multivalued. Such a model corresponds to the muffin-tin approximation to the spatial form of the electron density used successfully in modern calculations of solid state properties (see e.g. [3]). Therefore, we can evaluate also interstitials, provided their radii are known.

Using the superposition principle different point defect configurations can be calculated. Already the figures in [lo] show for Si that the idea of a ‘‘radius” of a vacancy is a rough model having the great advantage to allow analytical calculations (see Section 3). It seems physically more adequate to determine the boundary for the vacancy through a closed surface formed e.g. by local minima of the electron density in each radial direction in the defect-free crystal.

Another important problem is, however, the appearance of negative energies for some metals in the presented formalism. The single way to remove this obstacle is a more sophisticated approximation for the cubic Green’s function tensor. An instruc- tive test of the accuracy of extended object theory will be the evaluation of vacancy energies in hexagonal materials for which the exact Green’s function tensor is known t221, [421.

References

[l] CH. KITTEL, Introduction to Solid State Physics, 4th Ed., Wiley, 1974. [2] K. HOLLDACK and H. KERKOW, phys. stat. sol. (a) 98, 527 (1986).

131 V. L. MORUZZI, J. F. JANAK, and A. R. WILLIAMS, Calculated Electronic Properties of Metals,

[4] V. L. MORUZZI, A. R. WILLIAMS, and J. F. JANAK, Phys. Rev. B 15, 2854 (1977). [5] A. SUGIYANIA, J. Phys. SOC. Japan 55, 4272 (1986). [S] W. A. HARRISON, Pseudopotentials in the Theorie of Metals, W. A. Benjamin, Inc., New

[7] Computer Simulation of Solids, Ed. C. R. A. CATLOW and W. C. MACKRODT, Lecture Notes

[8] M. S. DAW and M. I. BASKES, Phys. Rev. B 29, 6443 (1984). [9] M. W. FINNIS and 5. E. SINCLAIR, Phil. Mag. A 50, 45 (1984).

K. HOLLDACE, Thesis, Berlin 1987.

Pergamon Press, 1978.

York 1966.

in Physics, Vol. 166, Springer-Verlag, 1982.

[lo] J. BERNHOLC, N. 0. LIPAFU, and S. T. PANTELIDES, Phys. Rev. B 21, 3545 (1980). [ll] H. UMEZAWA, H. MATSUMOTO, and M. TACHIKI, Thermo Field Dynamics and Condensed

[12] M. WADATI, H. MATSUMOTO, and H. UMEZAWA, Phys. Rev. B 18, 4077 (1978). [13] D. 5. BACON, D. M. BARNETT, and R. 0. ScATTERaooD, Anisotropic Continuum Theorie of

[la] C. TEODOSIU, Elastic Models of Crystal Defects, Springer-Verlag, 1982. [15] J. D. ESHELBY, Solid State Physics 3, 79 (1956).

States, North-Holland Publ. Co., 1982.

Lattice Defects, Prog. Mater. Sci. 23, 51 (1979).

494

1161 R. DE WIT, Solid State Physics 10, 249 (1960). [17] T. MURA, Phil. Mag. 8, 843 (1963). [18] E. KRONER, Z. Phys. 139, 175 (1954). [I91 A. SEEQER, Theorie der Gitterfehlstellen, Handbuch Physik, Vol. VII/l, Ed. S. FLUQGE,

1201 H. MATSUMOTO, N. J. PAPASTAMATIOU, and H. UMEZAWA, Nuclear Phys. B 97, 90 (1975). [21] D. M. BARNETT, phys. stat. sol. (b) 49, 741 (1972). [22] I. M. LIFSHITS and L. N. ROSENZWEIG, Zh. eksper. teor. Fiz. 1 7 , 783 (1947). [23] R. BIRKELAND, Math. 2. 26, 566 (1927). [24] J. RYCHLEWSKI, ZAMM 66, 256 (1985). [25] G. LEIBITRIED, Z. Phys. 135,23 (1953). [26] J. D. ESHELRY, Acta metall. 3, 487 (1955). [27] A. A. SMIRNOV, Teorija splavov vnedrenih, Izd. Nauka, 1979 (p. 112). [28] R. W. BALLUFFI, J. nuclear Mater. 69 & 70, 240 (1978). [29] R. W. SIEQEL, J. nuclear Mater. 69 & 70, 117 (1978). [30] H. MATTER, J. WINTER, and W. TRIFTSHAUSER, Appl. Phys. 20, 135 (1979). [31] R. M. EMRICK, J. Phys. F 12, 1327 (1982). 1321 J. M. HARDER and D. J. BACON, Phil. Mag. A 64, 651 (1986). [33] V. S. VAVILOV, A. E. KIT, and 0. R. NIJASOVA, Mekhanismy obrasovanija i migrazii defektov

v poluprovodnikakh, Izd. Nauka, 1981 (p. 48). [34] R. HASIQUTI (Ed.), Lattice defects in Semiconductors, University of Tokyo Press, Tokyo 1968

(p. 135). [35] M. G. MILVIDSKII and W. B. OSVENSKII, Strukturnye defekty v monokristallakh polupro-

vodnikov, Izd. Nauka, Moscow 1984. [36] J. CORISH, P. W. M. JACOBS, and S. RADHAKRISHNA, Point Defects in Ionic Crystals, p. 224,

in Surface and Defect properties of Solids, Vol. 6, Ed. M. W. ROBERTS and J. M. THOMAS, Pergamon Press, 1977.

U. KRAUSE et al.: Monovacancy Formation Energies in Cubic Crystals

Springer-Verlag 1955.

[37] W. C. MACKRODT and R. F. STEWART, J. Phys. C 12,431 (1979). [38] A. S. GUPTA, S. V. NAIDU, R. ROY, and P. SEN, Solid State Commun. 58, 219 (1986). [39] LANDOLT-BORNSTEIN, Zahlenwerte und Funktionen, Ed. K. H. HELLWEQE, Springer-Verlag,

[40] L. KORNBLIT, Physica B + C 106, 351 (1981). [41] M. LANNOO and J. BOURQEOIN, Point Defects in Semiconductors I, Springer-Verlag, 1981. [42] E. KRONER, Z. Phys. 136,402 (1953).

1955.

(Received February 13, 1987; in revised forw December 12, 1988)

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Monovacancy Formation Energies in Cubic Crystals