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Exact results for imperfect crystals obtained by a newmatrix method

P. Audit

To cite this version:P. Audit. Exact results for imperfect crystals obtained by a new matrix method. Journal de Physique,1986, 47 (3), pp.473-481. �10.1051/jphys:01986004703047300�. �jpa-00210227�

473

Exact results for imperfect crystals obtained by a new matrix method

P. Audit

Laboratoire PMTM, Université Paris-Nord, 93430 Villetaneuse, France

(Reçu le 12 août 1985, accepté le 24 octobre 1985)

Résumé. 2014 A partir d’une représentation dans l’espace réel, sous forme de matrices infinies, de l’hamiltonien« liaisons fortes » décrivant un cristal perturbé par la présence d’une lacune ou d’une surface, sont obtenues lesexpressions analytiques exactes : des éléments diagonaux et non diagonaux de la fonction de Green, de la densitéd’états totale et locale, de l’énergie libre. Sur ces exemples apparait clairement la possibilité offerte par la méthodede traiter d’une façon simple et rigoureuse les défauts simples dans les cristaux.

Abstract. 2014 The exact analytical expressions for the diagonal and off-diagonal elements of the Green function,the local and total density of states, and the free-energy are derived from the real space representation, in the formof infinite matrices, of the tight-binding Hamiltonian describing a crystal perturbed by a vacancy or a surface. Thefeasibility of this matrix method, which offers a most simple and rigourous treatment of simple defect models incrystals, is thus established.

J. Physique 47 (1986) 473-481 MARS 1986,

Classification

Physics Abstracts71.55 - 61.70B - 73.20C

1. Introduction.

Formulating the physical properties of a crystalperturbed by the presence of a defect is a challenge,regarding the necessity to meet two conflicting requi-rements : to fully exploit the translational symmetryof the host crystal and to describe properly the defectitself. In this respect, many methods which have beenintroduced and used are not fully satisfactory, forexample : the cluster and defect-molecule calculationsfail to give band-edges relative to which the bond-state energy (if any) can be referred [1]; supercellcalculations introduce some spurious effects [2];numerical calculations from the method of moments[3] appear to be somewhat unstable; ...The Green’sfunction formulation suffers from none of thesedrawbacks. This function can be formally obtainedvia a perturbation expansion of the Dyson equationwhose complexity increases with the size of the per-turbed space and the dimensionality of the crystal,so that heretofore closed analytical expressions forthe Green function matrix elements have been obtain-ed in the cases : where the perturbation is diagonalonly, i.e. vibrations of a linear chain containing a singlesubstitutional isotopic impurity [4] or the related

problem of a diagonal perturbation in a one-dimen-sional TBM (tight-binding model) [5]; or when theperturbation is off diagonal only, i.e. an ideal surfacein a (simple cubic) SC crystal [6].

Therefore the direct and natural description of theimperfect crystal in real space which is offered by aninfinite matrix representation of the Hamiltonianseems attractive; especially if it can directly yieldanalytical closed expressions for the crystal properties,not limited to the Green function, without appealingto Bloch’s theorem, Elucidation of FIGCM (functionsof infinite generalized cyclic matrix) properties [7, 8]allowed recently to achieve this goal, in the case ofperfect crystals; generalization of the method to

imperfect crystals is initiated in the present paper, byconsidering two simple models, namely the nearest-neighbour’s band TBM of a linear chain with a

vacancy and the SC lattice with a surface, in order toprove its feasibility. In both considered cases, anyfunction of the Hamiltonian could be obtained ininfinite matrix form, from which were directly derivedexact analytical expressions; so hereafter are givensuch expressions for the diagonal and off-diagonalelements of the Green function, for the local andtotal density of states, for the free-energy.Those results are illuminating with regard to the

basic properties of the crystal vacancy and surface.Moreover, the method appears to make simple modelsof imperfect crystals so easily accessible to exact

analysis, that it can be reasonably expected to bepowerful in dealing with more realistic defect pro-blems, such as multiband, self-consistent... calcula-tions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703047300

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2. Some useful matrix properties.

The one-particle Hamiltonian of a perfect cubic crystalcan be represented in terms of cyclic matrices mp,By introducing some sparse diagonal matrices Dpto account for the presence of the defects, the repre-sentation of imperfect crystals is readily obtained.Therefore, we first discuss some useful properties ofclasses of matrices mp and Dp, and their commutationrelations too.

Let mp be the N x N topological cyclic matrix oforder p, whose elements are given by

for p >, 0, other rows being obtained by cyclic per-mutation ; and m-P is the transpose and inverse of mp.They satisfy the following identities

Another identity which probed to be very useful indealing with perfect periodic crystals [8] is

where

is the symmetric topological cyclic matrix of order pand Cp(x) is the pth order Chebysheff polynomialof the second kind [9], obeying the recurrence for-mulae

Note that the relation (5) is valid for p N only,because from equations (2) and (4), we have

When a N x N matrix A is expected to retainsomething of the property expressed by (5); it isconvenient to consider an expansion in the set of

orthogonal Chebysheff polynomials in the form

where the inner product is

making use of the relation Cq(2 cos 0) = 2 cos qO; the terms q > p in equation (10) cancel out, just as all termswith q and p having different parities. Now considering a function of A expressible as an infinite power seriesexpansion, we have from (10)

Generalization to three dimensions is readily obtained by means of the following relations

with the inner product

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Other useful identities are derived from

Cq(2 cos 0) = 2 cos q8 as

and

Turning now to the diagonal N x N matricesdefined by

they clearly satisfy the following identities

Finally, the matrices mp and Dp do not commute,but rather obey the rule

3. Matrix solution for a tight-binding model of aninfinite-chain with a vacancy.

Let us consider a one-dimensional chain of N sites,

with periodic boundary conditions and nearest neigh-bours interactions only. A vacancy is located at theorigin, so that the Hamiltonian of the system is

where e is the atomic energy at the occupied sitesand P the transfer energy. Changing the zero andscale of energy, it is convenient to use the reducedHamiltonian

where a = - 8/fl.We now proceed to derive an expression of an

arbitrary matrix function f(H) in the form of equa-tion (12). The CP(H) matrices exhibit different formsaccording to the parity of p. Clearly we have first

and making use of equations (2-4; 7, 8; 18-22), wefind the recurrence relations

whose first terms are proved to be correct in appendix A. Now substitution of equations (25, 26, 27) into equa-tion (12) yields, for N -+ oo as understood subsequently,

We note that the first term in equation (28) is just the contribution of the chain without defect, that should beobtained upon setting Do = D -1 = 0 in equation (24b); whereas the following terms represent the contribu-tion of the vacancy to f(H).

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The matrix Green’s function G(z) = (zI - H)-1 is readily obtained from equation (28) in the form

yielding the following expressions for the diagonal elements

and for the first off-diagonal elements

The local density of states is defined by

taking into account the relation

we have from equations (30) and (31)

The total density of states is obtained by summing over I expressions (35) and (36) as

Making use of equation (17), simplification occurs in the form

Analysing this explicit expression of the spectrum, we note that the density of states of the perfect chain is recover-ed in the first term, whereas the second term represents a localized bound state at zero energy (rememberingthe change of variables in equation (24b)), which lies outside the band, for any realistic assumption concerningthe 8 and fl parameters; the two negative 6 functions in the third term are found to be exactly situated at theband edges, so that they tend to smooth out these Van Hove singularities. Finally, when integrated, equation (40)contains N electrons, as required.

Many thermodynamic functions may be written as the trace of some matrix function, which from equa-tion (28), has the form

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and taking into account equation (17)

Bypassing expression of the spectrum, the latter equation yields directly, for example, the free energy with respectto the chemical potential 14 which may be written from the spectral theorem as

we have from (42), for the variation in the free energy due to the vacancy

or turning back to the inertial energy units of equation (23)

it should be straightforward to obtain other thermodynamic functions in the same way.

4. Matrix solution for a tight-binding model of asurface in a SC lattice

Now, as a second example, we consider a one-band,nearest-neighbour, tight-binding model of a simplecubic semi-infinite crystal, which is obtained fromthe infinite periodic crystal by breaking the bondsbetween the atoms of two adjacent planes. The cal-culation is non self-consistent, but very simple analy-tical expressions are derived for the main physicalproperties of the crystal, which illuminate the funda-mental properties of an ideal surface.

Shifting the origin of the energy to eliminate theconstant diagonal matrix elements and changingthe scale, the Hamiltonian of the one-dimensionallattice with one broken bond between sites 0 and 1can be represented by the N x N matrix

The SC crystal with broken bonds between twoneighbouring planes (100) can be described by theN 3 x N 3 matrix

where M1 has been defined in (6). As the matrix termsin the sum (47) commute, we have

So that taking into account equations (13), (14) and(5) is obtained the matrix function expression (wherethe limit N -+ oo is implied)

which can be transformed by substitution of explicit expressions for ChebysheTs polynomials of the matrix.Obviously, we have

then by using the rules (2, 3, 4 ; 8 ; 18-22) of section 1, are easily found after some algebra the recurrence relationssatisfied by the Chebysheff polynomials of even and odd order respectively

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The first order terms in those expressions, which are subsequently used, are checked by recurrence in appendix B.Now substitution of equations (50, 51, 52) into equation (49) yields the following general expression

where, for simplicity, we have just written the first off-diagonal elements of the non-cyclic blocks. From thisexpression follow readily the elements of the matrix Green’s function

in the form

for the diagonal elements.Moreover, any interesting off-diagonal element of the matrix Green’s function can also be easily obtained

from (53), as for example

Denote the local density of states per site for the two and three dimensional perfect crystal respectively asV3(E) and v2(E); then, in the present case, from equations (55) and (56) and definition (33), the local densityof states is

Whereas the total density of states is obtained by summing the above expressions in the form

which, by virtue of equation (17) can be rewritten as

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Considering this exact expression of the density of states, the influence of the surface shows up obviously. Thetotal number of states is easily checked to be equal to N 3, and no new state appears outside the three-dimensionalband In fact the distribution of the new states is just the sum of a positive 3d-band and two negative translated2d-bands. The Van Hove singularities being more pronounced in the 2d than in the 3d spectrum, and due to thecoincidence of the singularities in V3(E) and v2(E ± 2), it follows that the presence of the surface tends to erasethe characteristic singularities of the 3d-spectrum at E = + 2 and E = ± 6 (band edges).

Following the same line of argument as in section 3, the variation in the thermodynamic functions can beeasily obtained Considering, from equation (53),

we have, by taking equation (17), into account

from which follows, for example, the variation of the free energy AF due to the presence of the surface, with res-pect to the periodic crystal’s free energy F as

with p expressed in the same units as Hl in equation (46).

5. Conclusion.

The main objective of the present paper was to demonstrate the ability of the FIGCM method to deal withcrystals, exhibiting limited lack of periodicity, in the most natural and rigorous way; consequently the obtainedresults have only been briefly discussed here. But the characteristic of analytic formulae is to be self-explanatory ;the displacements of the spectral weight due to the defect has been clearly stated, whereas the answers to otherimportant questions concerning, for example, the localization and the charge transfers are given by the expres-sions of the Green’s function elements, and will be analysed elsewhere. Moreover, it is clear from the treatedexample of the free-energy calculation that the determination of the density of states is not a necessary prerequi-site to obtain the global physical properties of interest in closed form, by this method

From these results, cyclic matrices appear to show more flexibility than Bloch’s waves to be perturbed bydefects. This property is worth being fully exploited in future by considering, for example, self-consistent treat-ments, high defect concentrations, ...

Appendix A.

Here, our purpose is to prove the validity of the lower order terms of the recurrence relations (26) and (27),which are used in the main text

First making use of equations (2), (3), (4); (20), (21), (22) we find

Dropping irrelevant terms of order p > 2, we have successively

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and from equation (26) we have

substituting equations (A. 2, A. 3) in equation (8), we finally find

as required.Considering now the Chebysheff polynomials of odd order, we have to check the following formula

by recurrence, so we first calculate

and from equation (27) we have

Substituting equations (A. 7, A. 8) in equation (8), we finally obtain

as required

Appendix B.

To prove that the diagonal elements of the matrix (51) are correct, we first calculate

then evaluating the diagonal elements of the matrix

and getting

from equation (51 ), we have by substituting equations (B. 2) and (B. 3) in equation (8)

as required.

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Let us now check the first off-diagonal elements of the matrix (52), by considering first

and then

obtained from equation (52), making use of equation (8), we find

which completes the proof by recurrence.

References

[1] CASTLING, B., J. Physique Colloq. 36 (1975) C8-3183.[2] LOUIE, S. G., SCHLÜTER, M., CHELIKOWSKY, J. R. and

COHEN, M. L., Phys. Rev. B 13 (1976) 1654.[3] CYROT-LACKMANN, F., Adv. Phys. 16 (1967) 393.[4] MARADUDIN, A. A., MONTROLL, E. W., WEISS, G. H.,

IPATOVA, I. P., Theory of lattice dynamics in theharmonic approximation (Acad. Press., N.Y.) 1971.

[5] ECONOMOU, E. N., Green’s functions in Quantum Physics(Springer-Verlag, Berlin) 1983.

[6] ALLAN, G., Ann. Phys. 5 (1970) 169.

[7] AUDIT, Ph., J. Physique 42 (1981) 903.[8] AUDIT, Ph., J. Math. Phys. 26 (1985) 361.[9] ABRAMOWITZ, M. and STEGUN, J., Handbook of Mathe-

matical functions (Dover) 1970.