THE PSEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE AND TRANSITION METALS

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THE PSEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE AND TRANSITION

METALS

C. Demangeat, F. Gautier, R. Riedinger

To cite this version:

C. Demangeat, F. Gautier, R. Riedinger. THE PSEUDOPOTENTIAL THEORY AND THE IMPU-

RITY STATES IN NOBLE AND TRANSITION METALS. Journal de Physique Colloques, 1972, 33

(C3), pp.C3-251-C3-258. �10.1051/jphyscol:1972339�. �jpa-00215073�

JOURNAL DE PHYSIQUE Colloque C3, suppldment au no 5-6, Tome 33, Mai-Juin 1972, page C3-251

THE P SEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE

AND TRANSITION METALS

C . DEMANGEAT, F. GAUTIER and R. RIEDINGER Laboratoire de Structure Electronique des Solides (E. R. A. NO 100) UniversitC Louis-Pasteur, 4, rue Blaise-Pascal, 67-Strasbourg, France

RBsumB.

-

Nous montrons que la theorie du pseudo-potentiel permet de justifier et de calcu- ler a partir des premiers principes l'utilisation de bandes c< d ⁾⁾ traites en liaisons fortes et hybridees avec des bandes s-p traitees en ondes planes orthogonalis6es. Nous appliquons ce mod6le au calcul des Btats d'impuretB dans les mBtaux nobles et de transition (variation de densit6 d'etats, aniso- tropie des temps de relaxation...). Nous discutons brikvement les rksultats obtenus pour les alliages CuNi et NiCr ; nous montrons en particulier que la largeur des Btats lies virtuels provient princi-

-

pa~ementde l'hybridization s-d de la matrice.

Abstract. - We show that the pseudopotential theory applied to transition metals allows the justification and the calculation from the first principles of tight binding << d ⁾⁾ bands hybridized with s-p 0. P. W. We apply this model to the computation of the impurity states in noble and transition metals (density of states, anisotropy of relaxation time...). We discuss briefly the results obtained for NiCr and CuNi alloys ; we show that the virtual bound state width arises mainly from the hosts-d hybridization.

I. Introduction.

-

In this paper, we show that the electronic structure of the transition metals and of the dilute alloys with transition elements can be obtain- ed from the scattering theory expressed in the tight binding (T. B.) nearly free electron (NFE) represen- tation. We use and extend the pseudopotential theory for transition metals introduced by Harrison [I] and Kanamori et al. 121. We apply this theory for the compu- tation of the electronic properties of the virtual bound states in copper and in nickel (density of states relaxa- tion times...).

Various semi empirical pseudopotential schemes [3], [4] for pure metals were developped recently ; they assume that the s-d bands of the transition metals can be described as a set of T. B. ⁽⁽ d >> and N. F. E.

<( sp ⁾⁾ bands. This model was first justified from a

modification of the K. K. R. equations taking into account the resonant part of the << d ^)> phase shift [5], [6], [7], [8]. Here, we use another approach [1] : the simplest description of the wave functions

I

Y

>

is obtained by taking them as linear combinations of localized << d ⁾⁾ atomic like orbitals (labelled

I

di

>)

centered on the transition nuclei and << sp n like func- tions

I

^Yo

>

similar to the functions of normal metals :

1 Y > = I F , > + C X d i I d i > ; (1.1)

i

IYo >

is a linear combination of the pseudofunction

l(g

>

and of the core orbitals

1

ci

>

:

In the present case, we have to use, for the description of

I

Y

>,

an overcomplete set of functions

I

a

>

made o f : (1) plane waves

I

k

>

(for the expansion of

I

^(g

>),

(2) core orbitals

I

ci

>,

^{(3) (( d >> states}

^I

^di

>

whose energies fall into the continuum of the conduc- tion states. However, the ⁽⁽ d ^>) states are not eigen- functions of the pure crystal so that we have to extend the classical pseudopotential theory to this case.

Such an approach was first used by Anderson and MacMillan [9] for the study of transition impurities in normal metals. Later on, using wave functions (1.1) (1.2), Harrison [I] extended the Phillips Kleinmann theory [lo] for the study of the pseudopotentials in transition metals. Independently, Kanamori et al. [2]

generalized the work of Anderson and MacMillan [9]

for the calculation of the density of states of pure and impure metals. They started from the same overcom- plete set of functions

I

a

>

and introduced a pseudo- greenian GK (defined in this overcomplete space X) whose trace over this set is equal to the trace of the true greenian (E

-

^H')-I over a complete set of the original Hilbert space Je, : it's then possible to compute the density of the states from the pseudogreenian GK as if the basic set

I

a

>

was complete ; the authors deduced a general pseudohamiltonian and showed that the Hubbard's schemes are only special cases of this general pseudohamiltonian. They also developed an approximate calculation of the density of states for alloys and applied this theory to the study of the screening of NiAl alloys [I 11.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972339

C3-252 C. DEMANGEAT, F. GAUTIER AKD R. RIEDINGER In the paper, we summarize the results we

obtained for the electronic structure of metals and dilute alloys with transition elements. We genera- lize 1131 the Austin Heine Sham theorem 1121 when the wave function is given by (1.1). We recover the pseudohamiltonian introduced by Kanamori [2] but, we relate directly the pseudofunctions ( cp

>

to the eigenfunctions

I

Y

>

(11). Then, we can generalize the scattering theory in the overcomplete space and show that the perturbed wave function can be comput- ed as if the set

I

^a

>

was complete. The perturbed wave functions, the t matrix, the relaxation times, the electronic density and the density of states can then be computed in a general way. Finally, we summarize two studies of transition impurities in copper [14]

and in nickel 1151. The impurity potentials are deter- mined self-consistently in such a way that the Friedel sum rule is satisfied. The present computations allow the calculation of the virtual bound state shapes and show that the width of the resonance arises mainly from the host hybridization ; the term initially intro- duced by Anderson (Anderson's model [16]), and which comes from the impurity s-d hybridization is then negligible. The extension of such a work to impu- rity clusters [I81 and to the study of magnetic proper- ties is straightforward.

In the range of concentrated alloys, the C. P. A.

computations have to be used [17] : this interpolation formula allows the determination of the electronic structure of alloys in an intermediate range of concen- trations ; however, the s-d hybridization can be taken only in an approximate way [17], [18] and the present computation has to be used for the study of highly diluted alloys (isolated impurities, pairs of impuri- ties...).

Finally, let us note that the extension of the K. K. R.

equations to a metal with an impurity atom allows the computation of some properties of the impurity state (resistivity [21], energies of the bound states, density of states in the impurity muffin tin sphere, soft X ray absorption or emission) when the impurity potential is restricted in the impurity muffin tin sphere [19], [20].

11. The pseudopotential theory for metals and alloys with transition elements. - 11.1 INTRODUC-

TION. - In the present section, we are seeking a rapidly convergent expansion for the eigenfunc- tions

I

Y

>

^{(HI Y}

>

^{= E}

I

Y

>)

of a crystal with transition atoms. This will be easily achieved by an extension of the classical pseudopotential theory we briefly recall in the following.

In such a case,

I

Y

>

is chosen as a linear combina- tion of core functions

I

ci

>

and of a pseudofunc- tion

I

cp

>

:

From their high localization around each nucleus, the

core functions are assumed to be eigenfunctions of the crystal :

H is the crystal hamiltonian.

I

ci

>

and

I

Y

>

are then orthogonal and the coefficients

Xci

are given by :

The pseudofunction

I

^cp

>

is then shown to be (Austin Heine Sham theorem [12]) the eigenfunction of a pseudohamiltonian HA with the same eigenvalue ^E as the corresponding eigenfunction [ Y

>

:

The functions

I

^4PCi

>

are arbitrary : this arbitrariness can be used to determine a pseudofunction as smooth as possible. The plane wave expansion of

1

cp

>,

is then rapidly convergent and, for pure metals, the coefficients

X,,

(kt # k) of a Bloch function

I

^cpk

>

can be determined by perturbation theory as shown by numerous calculations [22].

We want to extend such an expansion to metals with transition atoms (transition impurities in normal metals, pure and impure transition metals) [13]. In such a case, the convergence of the expansion (2.5) is not good from the high d >) character of

I

Y

>

^near

the nuclei of transition atoms. To recover a good convergence, we have to include ^<( d >> orbitals in the subset of auxiliary functions

1

i

>

which defines

I

Y

>.

However, from their extension, these orbitals (atomic orbitals, for example) are not eigenfunctions of the hamiltonian H and are not orthogonal. Thus, we have to generalize the pseudopotential theorem (2.4) when the functions i

>

which appear in the expansion o f I Y > :

are no more (necessarily) eigenstates of H. It's easy to prove that a pseudofunction ] cp

>

corresponding to the eigenfunction

I

Y

>

by (2.6) is an eigenfunction of the pseudohamiltonian Hw with the same eigen- value E

where Hw is defined by :

I

^Qi

>

is an arbitrary vector associated with the i-th orbital.

The proofs of (2.4) [12] and (2.8) [I 31 are obtained in the same way by an expansion of

I

cp

>

^following

THE PSEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE C3-253

the eigenfunctions

I

^Yn

>

^{of H ;} Xi is then determined from the Schrodinger equations for

I

^Y

>,

from (2.6) and (2.8), as :

(2.6), (2.7), (2.8), (2.9) are the basic equations for a generalized theory of the pseudopotential ; the auxi- liary functions

1

i

>

are chosen by convenience in order to obtain the best convergence for the expan- sion (2.5) of the pseudofunction ( q

>.

11.2 THE PSEUDOPOTENTIAL THEOREM AND THE S-d

HYBRIDIZATION MODELS.

-

11.2.1 The expressions of the pseudopotential theorem in the overcomplete space X.

-

We can try to solve (2.7) by the expan- sion (2.5) of the pseudofunction following a complete orthonormal set of functions

1

k

>

for the Hilbert space XI (spanned by the eigenfunctions of H ) :

However, the generalization of the perturbation theory used for normal metals and alloys is quite difficult for transition metals [I]. It is more convenient to intro- duce a new (and equivalent) system of linear equations in the overcomplete space X = X1 @ X, defined as the direct sum of the initial Hilbert space X1 and of the subspace X, spanned by the orbitals

I

i

>

: such a system will describe the s-d mixing in the same way as introduced by the interpolation schemes. Then, to each vector

I

Y) of X whose components in the representation { k, i ) are (X,, Xi) corresponds one vector (and only one) of XI :

From the expression (2.9) of Xi, (2.10) is equivalent to the system :

The definition (2.12b) of X j was written in a more symmetric way multiplying (2.9) by the matrix ele- ments A, of an arbitrary matrix A of X, and summing over all j.

<

Fi

I

is then defined from

<

Qi

I

^{by :}

The energy spectrum is determined by det = 0 where

li?

is defined in X by :

This result was obtained first by Kanamori et al. [2]

from their study of the pseudogreenian.

11.2.2 The hybridized tight binding nearly free electron models.

-

In this section, we discuss briefly some models which can be used for the study of pure metals and alloys. The system (2.12) allows the theo- retical justification of the interpolation schemes using hybridized tight binding free electron models, if the matrix A i j is of the tight binding form :

A i j = < i l ~ - H l j > . (2.15) Then, following [I], we can eliminate the core states

I

ci

>

which are assumed to be eigenfunctions of the crystal : multiplying (2.12b) for the core state

I

ci

>

by

<

k

I

ci

>,

summing over all the core states and substracting from (2.12a), we obtain (2.16b) and the generalized Hodges Ehrenreich Lang model [4] :

HA is the Austin Heine Sham pseudopotential (2.4).

The s-s block is not really a nearly free electron hamil- tonian but the functions

<

^Fdi

^I

have to be chosen (as in the usual pseudopotential theory) so that the cc d ⁾⁾ states are repelled far from the energy range we consider.

The s-d block is energy dependent from the non orthogonality of the plane waves

I

k

>

and of the

I

di

>

states. For practical purposes, it seems simpler to orthogonalize the plane waves to the d states. The basis states of this representation a are the

1

di

>

^func-

tions and the 0. P. W.

I

Qi

>

defined as :

The wave function ( Y

>

can then be written as :

where X: is deduced from X, by (see (2.6), (2.17)) :

x % = x d i +

< i I c p > . (2.19) The Mueller's model [3] can then be obtained as follows :

(1) Multiplying (2.16b) by

<

^k

1

di

>,

summing over all ⁽⁽ d n states and substracting from (2.16a), we obtain (2.20~) and the system :

C3-254 C. DEMANGEAT, F. GAUTIER AND R. RIEDINGER (2) We choose the states

(<

Fdi

I

⁼

<

^di

I

a

-

H)

so that the pseudohamiltonian (2.20) is hermitic when this condition is satisfied for the ⁽⁽ s-s D block.

The energy dependence of the ds and sd block disappear when the

I

di

>

states are orthogonal

(<

di

I

d j

>

⁼ dij) as assumed later in the study of alloys. In the following, we will use this pseudo- hamiltonian for the study of dilute alloys and the exponent M in X? will be omitted.

11.2.3 The choice of the basis functions. Pure metals.

-

For pure metals, we can simplify (2.20) taking into account the translational invariance. For a given k we choose as basis functions of the represen- tation (

a k )

: (1) the 0. P. W.

I a:+, >

^{where K is}

a vector of the reciprocal lattice, (2) the Bloch sums Idmk

>

^:

I

dmk

>

^{= ---} 1 eik" m 1 >

JN

a

built from the orbitals

<

r

1

ma

>

⁼

<

^{r - 1}

1

m

>

centered on the crystalline sides 1. The functions

I

m

>

are chosen for simplicity as the basis functions of the irreducible representations of the point group (&(m = 1, 2, 3),

TI,

(m = 4,5) for the cubic group).

It's natbral to choose the functions ] m

>

^{as atomic}

grbitals ; in that case, the basis functions overlap and the ⁽⁽ dd ⁾⁾ block itself describes ⁽⁽ d ⁾⁾ bands, whose width is of the good order of magnitude. However, this choice is not unique and, for example, it was shown that if

I

m

>

is a solution of the Schrodinger equation in the corresponding muffin tin sphere (r

c

r,) and zero, otherwise the Hubbard's interpo- lation scheme [6] can be deduced from (2.16) [ll].

This choice represents as a starting point ⁽⁽ d D bands with a zero width : all the width of the ⁽⁽ d >> bands arises from the mixing with the plane waves of large k vectors as described by Heine [5] and the conver- gence of the development (2.5) is slow.

111. The impurity problem in the pseudopotential theory. Application to copper and nickel alloys.

-

In this section, we discuss briefly the impurity problem as described in the over complete space X by the pseudohamiltonian. First, we point out that the pro- perties of the alloy can be computed as if the basis states of the representation {ak} would define a complete set of orthonormal functions. Then, we sum- inarize briefly some results which are obtained, in this representation, for the electronic density, the residual resistivity and the density of states (111.2) [14]. Finally, we discuss the results which are deduced from the study of C U N ~ and NiCr - alloys.

111. THE IMPURITY PROBLEM IN THE OVER COMPLETE SPACE [13].

-

The Bloch states

I

Yzk) and the eigen- Values azk for the pure metals are obtained from the equation (2.20), which can be written formally as M 0 X(k) = 0. However, let us note that the eigen-

states of the pseudohamiltonian

I

Y:k) are : (1) the states

I

Y:k) which determine the eigenstates of H , (see (2. ll)), (2) the spurious ^D states

I

^Y&) which arise from the fact that the dimension of the represen- tative space is increased by the number of the

I

di

>

functions. It's easily seen from (2.12) that the corres- ponding energies an. and components Xn, are given by :

These states will be shown to be unimportant.

TO each Bloch state of the pure metal ( !Pik) is corresponding a perturbed state

I

Ynk) :

Vp is the perturbing potential and Go is the Green operator defined by Go M0 = I where I is the unit in X and M O is the matrix which appears in (2.20) for pure metal. If, for simplicity, we assume that M O is hermitic (the generalization to non hermitic opera- tions is straightforward) then, the Green operator is represented in { ak } by :

The perturbed state can then be determined in the usual way as :

and, from the corresponding equation written in X, for

I

Ynk

>,

the scattering amplitude is given by :

The contribution of the spurious states is shown to be zero using (3.1) and the scattering amplitude is equal to the corresponding quantity defined in the overcom- plete space. The density of states per unit energy n ( ~ ) is also easily expressed in terms of the pseudogree- nian G (GM = 1) [13] :

The first term is the pseudopotential density of states calculated from the pseudohamiltonian as if the basis set

I

^a

>

was orthonormal and complete Tr B=

[

^m Baa

) .

no,,,(&) is the orthogonality term arising from the fact that the set

I

a

>

is over complete in XI ; it is corres- ponding to the density of states of the <( spurious ^)>

states and can be neglected when F is chosen so that these states do not appear in the energy range of interest. The density of states can then be computed as if the set

I

a

>

was complete and orthonormal.

THE PSEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE C3-255 111.2 SOLUTION OF THE SCATTERING EQUATIONS.

-

The impurity problem can be solved, as a first step, neglecting the modification of the s-s part of the pseudohamiltonian by the impurity potential. The inclusion of such terms can be done using the Born's approximation but, it will be of small importance on the structure of the resonant states that we want to compute [23]. The impurity potential is then deter- mined by its matrix elements in the ( ak } representa- tion :

<

dmkl Vp/,I dm' kt

>

⁼

v;?'

⁼

The Dyson equation (3.2) and the corresponding equation fox the perturbed Green function G :

can be solved [14]. Let us summarize below some of these results.

1. Variation of the density of states introduced by the impurity atom.

-

The variation of the density of states per unit energy and per spin, An(&), is given by :

I d

An(&) = -

-

Arg D(E) .n d&

D(E) = det [(I

-

L') (1 - L) -

Godd(l - L)-

'

^(vdd

+

^{M) (1}

^-

L)] (3.9b) where the corresponding matrices L, L+, God* and M are defined by :

c ; ~ ~ m ~ m -

-

- 1

1

Goddm'm(q) eiq(a

-

^A')

.

^(3.10d)

N ,

The matrix elements of Vdd are defined by ( 3 . 7 ~ ) . The total displaced charge Z(E) for energies ^8'

<

e is obtained by integration of (3.9) and the Friedel sum rule can now be written as :

Z = Z(+) = - 1 Arg Dm(+) (3.11)

n o

where ^o is the spin index and Z is the impurity excess charge.

2. t matrix, relaxation times and resistivity.

-

The solution of (3.2) allows the determination of the t matrix in the { ak } representation :

The physical interpretation of these expressions is quite simple : some transitions dk' + Qk due to the d-s part of the impurity potential are ineffective from the transitions Qk ⁴ dk introduced by the hybri- dization in the pure metal.

The impurity relaxation time can be easily computed if we note that tkn!'; is an even function of k and

k'

: by the application of the optical theorem, the relaxa- tion time can be computed in each point k of the Fermi surface as :

and the resistivity can be obtained by integration on the Fermi surface.

3. Variation of the electronic density around the impurity.

-

The electronic density in the alloy can be computed directly from

<

^r

¹

^G

I

r

>

^; it's expressed in terms of the matrixes (3.10) [24] and can be comput- ed exactly near the impurity without use of some asymptotic form [18].

111.3 APPLICATION TO THE COMPUTATION OF VIRTUAL BOUND STATES IN COPPER AND NICKEL.

-

Such a for- malism was recently used for the study of impurities in copper and nickel ; however, up till now, only interpolation schemes as introduced by Mueller and Zornberg [3], [25] were used for the determination of the energies c;(k) and of the coefficients Xa(k).

1. CuNi alloys : the virtual bound state width and the anisotropy of the relaxation times.

-

For the impurity potential, we can choose a difference of configurations between nickel and copper atoms.

However, in the present case, it's more convenient to choose a Yukawa's potential V(r) = Z(exp

-

yr)[r where y is adjusted so that the Friedel sum rule (3.11) is verified. Such a (( self consistent ^>) calculation gives y = 2.98 a. u. when the wave functions for Ni3d 94s are used. It's assumed that the same truncation of the infinite determinant (2.20) can be used for the pure metal and for the alloy : we keep only the four vectors

I

^k

+

K

>

which are the nearest from the considered 1/48th of the Brillouin zone. Such a truncation elimi- nates the spurious solutions

I

^Y,

>

which are built from plane waves of high k vectors.

The results we discuss in the following are quite intensitive to the choice of the wave functions. Let us now summarize briefly the main conclusions (Fig. 1) :

a) Two virtual bound states are obtained for fhe symmetries T',, and

r;5

; they are quite different in shape and position. This difference arises mainly

C3-256 C. DEMANGEAT, F. GAUTIER AND R. RIEDINGER from the difference between the partial density of

states a(&, i) ⁼

<

i

I

8(&

-

Ho) li

>.

FIG. la. - Density of states for pure copper.

b) The virtual bound state are not lorentzian as predicted by an Anderson model [I61 : the proximity of the d band structure requires an asymmetry.

FIG. lb.

-

Variation of the density of states per impurity of nickel in copper.

c) The widths are equal to about 0.3 eV.

d ) The contributions of the virtual bound states arises from three terms :

(i)

vdd

G~~~ introduces a broadening from the non zero cc d ⁾⁾ density of states (Im GOdd(&,) # 0) at the virtual bound states energy s, : a d-d resonance arises between these extended states and the localized one repelled from the d band. Such a contribution was introduced by Kanamori [26] and Sokoloff [27].

(ii) The contribution to the width introduced by the L terms arises both from the host and impurity s-d hybridization terms (cf. (3.10)) ; it can be as large as the previous one.

(iii) The last term, G~~~ M is the only one which appeared in the Anderson's model : it arises from the impurity hybridization term alone and it's only a

THE PSEUDOPOTENTIAL THEORY AND THE IMPURITY STATES IN NOBLE C3-257

tenth of the other terms in the present calculations (for copper and nickel alloys).

The main origin of the width of the virtual bound states is the host s-d hybridization term.

e) The computation shows the existence of a large anisotropy of the relaxation times [24]. The relaxation times are two or three times larger on the neck than on the belly orbits and they are highly anisotropic even on the neck itself. Detailed computations are in progress in order to allow the detailed comparison with De Hass Van Alphen experiments.

2. Extension of the calculation to nickel alloys [IS].

- The extension to magnetic alloys is straightforward.

This was achieved for the calculation of dilute alloys of nickel with virtual bound states introduced in the majority spin band. The following assumptions on the impurity potential matrix elements are done :

(i) The matrix elements between << d ⁾⁾ orbitals and 0. P. W.,

I

^@:

>

are spin independent.

(ii) The impurity potential is built from a configu- ration mixing determined by the Friedel sum rule :

V,

and V, are respectively the potentials related to the configurations 3 d9+' 4 s and 3 d9+'-I 4 s ; 0

< 1 <

1 is determined by (3.11) and by the exchange condi- tion :

6md is the variation of magnetization introduced by the impurity and J is the exchange parameter used for the pure metal. The main conclusions of the study for NiCr alloys can be summarized as follows (Fig. 2) :

-

FIG. 2. -Variation of the density of states Ant(&, Tis) for the majority spins in EiCr alloys (et is the energy of the top of the

(( d ⁾⁾ band).

a) For the majority spin bands, the largest part of the virtual bound state is above the Fermi level and the screening charge Zt(+) is about equal to Z ; for the minority spin bands, the self consistent solution is thus almost zero.

b) The main origin for the width of the virtual bound state arises from the host hybridization terms and the corresponding discussion we presented for CuNi alloys is valid. _-

c) The proximity of the c< d ^>) bands is introducing a strong asymmetry in the virtual bound state width : this state is defined only for 8

>

8, where 8, is the top of the ^<( d ^>) bands. FO; E

<

8 , the strongly negative values for An(&) define the ⁽⁽ antiresonant state >> from which the virtual bound state was built.

d) The calculated value for the variation of the magnetic moment per impurity

is in qualitative agreement with experiment

e) However, the variation of the specific heat intro- duced by the impurity (67

--

15 mJ/mole) is smaller than the experimental one (6y

--

45 mJ/mole) so that a more detailed computation has to be done for a quantitative comparison with the experiment.

IV. Conclusion.

-

In this paper, we presented a general method for the computation of the electronic properties of alloys with transition elements. The s-d method we used was founded on a hybridization generalization of the pseudopotential theorem for cc d ⁾⁾ orbitals (11). It can be applied for the exact computation of the properties of dilute alloys (111) ; variation of the density of states, anisotropy of the relaxation time, variation of the electronic density around each impurity, electronic structure of pairs [18]

<< s )) and << d ^>) hyperfine fields in dilute alloys [29].

The present model allows a computation of the impurity states taking into account a realistic band structure. The results obtained for the density of states by this pseudopotential approach are very similar to the results of a multiple scattering theory [20].

However, the pseudopotential method allows in a much easier way, the computation of the physical properties ; it's not limited to localized (muffin tin) potentials and is self consistent as far as the screening charge is concerned. However, up to now, the nume- rical calculations we presented (111.3) are not fully self consistent ; they can be improved using a first principle pseudopotential scheme (2.20) with atomic basis orbitals, Moreover the good convergence (and the validity) of the truncation have to be verified. These points are now under study [28].

C3-258 C. DEMANGEAT, F. GAUTIER AND R. RIEDINGER

References

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