Simple approach to the band-structure problem

Solid State Communications,VoI. 13, PP. 133—136, 1973.

Pergamon Press.

Printed in Great Britain

SIMPLE APPROACH TO THE BAND-STRUCTURE PROBLEM O.K. Andersen Electrophysics Department, Technical University, Lyngby, Denmark (Received l6March 1973 by N.J. Meyer)

An approximate first-principles method is presented for calculating the band structure of solids, especially applicable to closely-packed structures. This scheme has the advantages of being physically transparent, and computionally very fast. In its simplest form it gives energies correct to within a few per cent of the band width, and the error may be substantially reduced without much increase in complexity or computing effort.

WITH the aim of calculating in a rapid and physically transparent way the electronic properties of solids, we have developed an approximate first-principles method which is especially applicable to closely-packed structures. In its simplest form, the scheme may be considered as an approximate form of the KKR method,1 in which the eigenvalues and eigenvectors are calculated from energy-independent structure constants SL’L (k) and the logarithimic derivatives, L 1(E) = pc/i/~of the l’th partial wave ~/,(E,r) at the atomic Wigner—Seitz sphere of radius p. The secular matrix of the corresponding 2method of linear combination (LCMTO) is linear in energy,of so muffin-tin orbitals that all eigenvalues and eigenvectors may be found simultaneously, and it is expressed entirely in terms of 5ilL (k) and the parameters of the function L 1(E). We utilize the method in the following to establish the 3 according validity ‘Wigner—Seitz rules’, in a closely to whichofa the bandempirical of predominantly 1-character packed solid extends between the energies where Ø(E, p) = 0 and Ø 1(E, p)in=Pd 0. and We compare also calculate gies at symmetry points themenerwith the results of earlier APW calculations. Our method is orders of magnitude faster computationally than the standard KKR and APW schemes, and the numerical results reported in this paper were carried out using only a desk calculator.

which in partial wave representation describe the scattering between muffin-tin (MT) spheres, is the dependence of the structure constants, BL’L (k, ic), on energy,E, through the relation K2 = E— VMTZ for the kinetic energy in the region of assumed free propagation between the MT spheres. Recent Fermi surface parametrizations4 have revealed that within dHvA-accuracy this s-dependence in the KKR equations is redundant and the choice of ~ at most influences the L-convergence. This is so for very much the same reason that the usual approximation of a constant MT floor, works, namely 27r/K that, isinseveral the interstitial region,VMTZ, a typical wavelength times the distance between the inscribed and circumscribed spheres of the atomic polyhedron. Before considering the MT model we therefore consider the ‘atomic sphere model (ASM) in which thesame atomic polyhedra are potenreplaced by spheres of the volume, and the tial is spherically symmetric within each sphere. This model has no interstitial region, so the only condition 2 is that it be close to the kinetic energy in the on K regions of the atomic spheres, in order to ensure outer rapid convergence of the energies. We shall choose K equal to zero, whereby the KKR equations, after a renormalization, become particularly simple. ~ +S ‘k~’A 0 L i P,~ I L L L L’~ I] L The potential function is ‘

— —

The major complication of the KKR equations p,(E)

=

.

p 1{Lj(E)}

~LEKcotTh{L1(E),K}~L’L +BL’L(k,K)IAL

=

0 =

133

(41+ 2)[L1(E)+l+

1] [L,(E)—l]~,

(2)

.134

SIMPLE APPROACH TO THE BAND-STRUCTURE PROBLEM

and the energy independent structure constants are SL’L(k) =

—2(21”—’) ~‘ ~‘(L’,L) (2!’— 1)!! (21—1)!! c

e~~(p/R)”~’[4ir(2l”+ l)]~YL’,(R),

~

R*O

(3)

Vol. 13, No. 2

Table 1. F.C.C. structure constants and energies I

k

S(k)

Pd E/mRy

Free electons p2E

ASM*

APWt ASM*

11.1 9.0 4.1 —2.7 x3 —14.6 x1 —16.6 X~ —6.17

539 503 422 316 153 1271: 788

537 504 427 315 158 118 782

(x1

1.767

X1

—2.30

15201: 1290 0 0 l325~ 1290 129111 1290 1l7~ 118

—

Exact

X where (—1)!! 1, l”~l’+ 1, and m”=m’ —m. The Gaunt coefficients, c, are tabulated in reference 5, the spherical harmonics, Y, have the phase of reference 6, and Rare the atomic positions. The lattice summation may be performed by the Ewald technique or, for d—dinteractions, even by direct summation in real space. The structure constants (3) do not depend on the lattice constant, and the potential function,p,(E), only appears in the diagonal of (l)and is independent of rn. For a given structure, S~’, (k) can therefore be diagonalized once and for all yielding ‘canonical’ s-, p- and d-bands, S,~(k),interms of which the Ufl~ hybridized energy band structure for a given potential follows from the implicit equation p,(E) = S1~(k), where p,(E) are increasing functions of E. The f.c.c. Sd-bands are shown in Fig. 1. It is convenient for instance in self-consistent calculations that if s—p—d hybridization is neglected ‘canonical’ eigenvectors and partial state densities may also be obtained. The s--p—d hybridization can only be included if the widths and relative positions of the s-, p-, and d-bands, as given by p,(E), are taken into account by requiring that the determinant of the entire matrix (1) be zero, Numerical values of some f.c.c. structure constants are shown in Table 1 together with a comparison of eigenvalues obtained by those the APW method, using a MT 7 with obtained by the present potential for Pd, method, using the same potential for r ~ PMT and the constant VMTZ for ~MT
—

results are shown, and it is seen that the discrepancies are at most a few per cent of the respective bandwidths, which are about 0.4 and 1.5 Ry for the d- and ~for 2andPd 22.0 ~ bands, and about 9.9 ~ 15.9 ~o~ the free electron s-, p-, and d-bands respectively. Having established the accuracy of(l) we see from the structure constants in Table 1 that the Wigner—Seitz rules hold quite well, since, if these were correct, we should according to (2) find S~(F 1) —°°,Sd(XlorX3)= —15 and Sd(Xs) = 10. Further, for the bottom of the f.c.C. p-band S~(L2’)= —10.39 from (3), while the Wigner—Seitz rule yields the value —12. For the dband extrema in the b.c.c. structure we find from (3) 5d (H Sd (H12) = —16.2, and 25’) = 108. In the h.cp.

5 X2 ~12 F25’

d

~

~ sd

*

25.1 17.8

24.2 18.1

6.00 6.1011 7.921: 0 6.47~ 6.1311

6.03 6.03 6.03 0 6.03 6.03

From equations (1) or (5), and (4) with (l’ 1” 1) ~ 2.

t 1: §

Reference 7. sd-hybridization neglected. sd-hybridization included. II LCMTO including the combined correction term. 5d

structure

(M

5d

1+) = —15.1 and

(M 4+)

=

10.4.

A parametrization of p1(E) orL1(E) may be obtamed by performing a generalized expansion of ~ r) in the set ~~~(r) Ø,(E~,r) which is defined through the boundary condition pçb’~(p)/Ø~j(p) = Lu,, whereenergy, L~1is an or E~1is an arbitrary andarbitrary where ~number, is normalized to unity in the atomic sphere. The result is 2(p) [E~

[L 1(E)

L~11’ =

~n

P4~ni

1—E]~

which in the range of the vl.band, and to an accuracy of orderbyI 0~times the band width, may beinapproximated the truncated Laurent expansion e (E E~,)p2,i.e. —

[L,(E) —L~,]~

—

[m~,e]’’ + a~,+ b~,e. (4)

The essential band parameters, E~orL~jand m~1= ~ ~ (p), may be estimated quite reliably from atomic eigenvalues and wavefunctions. The contributions a~1and b~1from the excited states cannot be neglected but their variation among similar metals is negligible. In (5) below we shall make use of the fact

Vol. 13, No. 2

SIMPLE APPROACH TO THE BAND-STRUCTURE PROBLEM

135

2

is~

~

~

i

__

i

FIG. 1. ‘Canonical’ f.c.c.d-bands. that ~ and b~,also may be expressed in terms of the function a~(E,r)/aElE~l.In connection with (2) the choice L~,= —(I + 1) is convenient and we name the corresponding energy, E~1,the position of the band, C~,and the corresponding value of 2m1,~the intrinsic, relative band mass, .~, since for free electrons this parameter is always unity. theFor Pd partial potential used 1d =For 7.89. waves above p~ = 0.759 and / contributing only through weak hybridization, the choice L 1,, = I is convenient and the corresponding energies and intrinsic, relative masses are respectively V~iand(2l+3)m~:~r~,.

to the probability density in the outer region of the polyhedron and thereby to the band width. This correction is conveniently combined with a correction for the neglect of the partial waves with 1> 2. An even greater advantage of the LCMTO method over (I) is that its secular matrix is linear in energy to almostbethe accuracy byFor which logarithmic derivatives can fitted by (4). ic = the 0 we obtain explicitly8 H —EQ — ~ E ~ LL

LL

—

2

P~( — ) L’L V1 —E I”,’ —E

~°

+ ~.j

From the eigenvectors,A~,the wavefunction can be expressed either as a slowly convergent one-center expansion in partial waves, or as a rapidly convergent multi-center expansion in MT orbitals. The KKR equations express the2 If, condition thesethetwo expaninstead,that we use variational sions are equivalent. principle for the Hamiltonian in the basis of Bloch sums of MT orbitals, we arrive at the LCMTO secular matrix, which, for a MT potential, may written as (K). 2 +beVMTZ —E),A H(E, K) —EO(E, K) + (K Here H and 0 are respectively the Hamiltonian and overlap matrices calculated as the sum of integrals over all atomic spheres, and ~ is the difference between the atomic polyhedron- and atomic sphere-overlap matrices. For any value of 2ir/~>>(p ~PM,I,), the KKR secular matrix is a factor of H-EO, which therefore also yields the KKR energies, E~Thus the atomic sphere model may be corrected by the s-term, which is proportional

rj

~

r

~

v1’

S~(k)

r1”(V1” E) 2(21”+ l)2(21”+ 3) —

+

EL” SL’L”(k)

which exllibits the one-, two-, and three-center terms. We have suppressed the indexfrom v, and parameters may be obtained L the potential 1(E) via p Cr

= = — —

=

2 {m [1 + a(L~+ 1 + 1)12 + b(L~+ / + 1)2 (2/ —+ 3){m[ 1~ ++ a(L~ F -2 h~ 1)] 2 + b(L~—1)2 } ~ p (m,~ b ) F ~r2(mh + bg~)‘ —‘

—

p

V

=

E~— p2 (mg + bg’

—

V g h

-

-

=

Er—p 2(mg + bh —(L — l)1 a

=

~

=

-

1)1,

—

+ I + 11’

)‘,

a.

},

136

SIMPLE APPROACH TO THE BAND-STRUCTURE PROBLEM

Vol. 13, No. 2

This form has been obtained by augmenting tails of the MT orbitals centered at R0 inside all spheres cen-

correction term are shown in Table 1.

tered at R ~r R0 continuously and differentiably by the functions aØ(E, Ir — Rl)/aEJE , which are orthogonal to the core states. The MT orbitais are now

Acknowledgements This work was partially performed at the Research Establishment Risö and useful discussions with Professor A.R. Mackintosh are gratefully acknowledged. Dr. R. Nevald kindly performed the computations illustrated in Fig. 1.

independent of energy to first order in (E E~1). Results obtained from (5) and including the combined —

—

REFERENCES 1.

SEGALL B. and HAM F.S.,Methods in Computational Physics, Vol. 8, Ch. 7, Academic Press, New York(l968).

2.

ANDERSEN O.K. and KASOWSKI R.V., Phys. Rev. B4, 1063 (1971), and KASOWSKI R.V. and ANDERSEN O.K., Solid State Commun. 11, 799 (1972).

3.

HODGES L.,WATSON R.E. and EHRENREICH H.,Phys. Rev. B5, 3953 (1972).

4.

SHAW J.C., KETTERSON J.B. and WINDMILLER L.R., Phys. Rev. B5, 3894 (1972); DEVILLERS MAC., Solid State Commun. 11, 395 (1972); ANDERSON O.K., Phys. Rev. Lett. 27, 1211(1971), and references contained therein.

5.

CONDON E.V. and SHORTLEY G.H., The Theory ofAtomic Spectra, p. 175, Cambridge University Press (1951).

6.

LANDAU L.D. and LIFSHITZ E.M., Quantum Mechanics, Pergamon Press, Oxford (1965).

7.

ANDERSON O.K,Phys. Rev. B2, 883 (1970).

8.

ANDERSON O.K., Winter College on Electrons in Crystalline Solids 1972, IAEA, Vienna (Unpublished).

Eine auf ersten Prinzipien basierte Naherungsmethode fur die Berechnung von Bandstrukturen in Festkorpern, für dicht-packte Strukturen besonders geeignet, wird vorgelegt. Das Verfahren hat den Vorteil physikalischer Durchsichtigkeit und bietet eine sehr schnell durchführbare Rechenmetode. In einfachster Form gibt es Energiewerten korrekt bis aufeinige Prozent der Bandweiten, und der Fehier kann ohne grosse Schwierigkeiten und mit geringer Vergrosserung der Rechenzeit erheblich reduziert werden.