Slater-Koster parametrization of the band structure of TiNi

J. Phys. Chem. Sdi& Vol. 45, No. 4. pp. 439-445. Printed in Great Britain.

ow-3697/84

1984

SLATER-KOSTER

53.00 + .@I

Pergamon

PARAMETRIZATION STRUCTURE OF TiNi

Press Ltd.

OF THE BAND

J. D. SHOREand D. A. PAPACONSTANTOPOULOS Naval Research Laboratory, Washington, DC 20375, U.S.A. (Received 8 August 1983; accepted 15 September 1983) Abstract-A very accurate Slater-Koster par~et~~tion of the band structure of the intermetallic compound TiNi is presented. Various technical aspects of optimizing the Slater-Koster interpolation method for the CsCl structure are discussed.

I. INTRODUCTION The results of first principles band structure calculations using augmented plane wave (APW), Green’s function (KKR), linear combinations of Gaussian orbitals, or pseudopotential approaches can be recast in a linear combination of atomic orbitals (LCAO) basis using the Slater-Koster (SK) method as an interpolation scheme[l]. There are at least three important virtues in presenting the band structure of solids in this SKLCAO form. First, using the SK parameters, one can easily diagonalize a small size matrix (9 x 9 for a typical transition metal or 18 x 18 for a compound) to obtain the energy bands and densities of states for a given material. This avoids the complexity of the APW or KKR techniques which only the experts can use reliably and efficiently. Hence, the SK method is readily usable for the analysis of experiments by the experimentalists themselves. Second, a great variety of theories utilize the SK parameters to construct tightbinding (TB) Hamiltonians for the study of phonon spectra, defects, disordered materials, surfaces, interfaces, etc. The availability of such parameters is particularly useful to scientists working in these areas. At present, people who study such phenomena spend considerable time creating these TB Hamiltonians. Often in determining the SK parameters they invoke approximations that limit the accuracy of their results. Third, there is a clear educational value in this approach; it enables students of solid state physics courses to easily generate the band structure of a material and gain a better understanding of the concepts involved. As a demonstration of this technique for compounds, we present here an SK parametrization of the APW calculation of Papaconstantopoulos et aI.[2] for TiNi. Previously Connolly[3] stressed the accuracy of this approach in antiferromagnetic Cr. In a recent paper Bruinsma[4] calculated the contribution of the electron-phonon interaction to the phonon dispersion curves of TiNi. He applied the method proposed by Varma and Weber [5] with the modification of using an orthogonalized TB Hamiltonian based on an SK fit to the APW calculation of Papaconstantopoulos et aZ.[2]. KS

Vol. 45. No. 4-F

2. SLATER-KOSTERINTERPOLATION We have applied the SK interpolation method to fit the APW energy bands of Papaconstantopoulos et a[.[21 for TiNi. The resulting TB Hamiltonian has the form of an 18 x IS matrix which includes interactions between orthogonal s, p and d-orbitals of both atoms. The matrix elements contain interactions with first and second neighbors that require the determination of 48 three-center-integral parameters to obtain a fit to the APW results. A close examination of the SK matrix [ I] reveals that certain off-diagonal matrix elements involvingp-orbital interactions with s and dare imaginary quantities so, at first sight, it appears that one has to deal with a complex matrix. However, by changing the sign of the following SK parameters (see Table 1): E,,(200) for both Ti-Ti and Fe-Fe interactions; E;,,Y(ll1) for the Ti-Fe interactions; and E,.,( 11l), E,,J 11 l), and Ex.d,(111)for the Fe-Ti interactions, we obtain an equivalent real matrix which is more convenient for the computations. In order to avoid incorrect assignments of states, it is essential to use group theory[6] to reduce the 18 x 18 matrix to smaller matrices at high symmetry points or lines in the irreducible Brillouin zone. This block-diagonalization of the 18 x 18 matrix can be accomplished by using the symmetrization shown in Table 2. Using these reduced size matrices we have performed a least-squares fit to the APW results at the high symmetry points F, X, M, and R and also at three points along each of the symmetry lines A, Ic, and Z. Therefore, we have fitted to the APW results at a total of 16 k-points. At each k-point we have fitted 10 bands with the exception of F and R where we have also fitted the higher levels, Fis, RF, and &. As we will point out in the next section the inclusion of at least a few levels from higher than 10th bands is essential for determining accurately the p-like component of the density of states (DOS). 3. DISCUSSION Our 48 SK parameters are listed in Table 1 following the usual notation of the original SK paper[ 11.In Table 3 we list therms fitting errors per band together

440

J. D. SHORE and D. A. PAPACONSTANTOPOULOS Table 1. Slater-Koster parameters for TiNi expressed in Ry. The standard notation of Ref. 1 is used with abbreviations d, = x2 - y2 and d2 = 32* - r’. Note that the Ti atom is at the origin and that the imaginary i has been omitted from the matrix (see text) Ti-Ti

Ni-Ni

E ,,,(OOO)

1.2273

E ,,,(OOO)

1.3198

1.8540

E

0.7343

0.5751

0.7398

0.5305

,y,xy(OOO)

Ed2,d2(000) E ,,,(200) E ,,,(200)

-0.0706 0.0741

l.lb58

0.0138 -0.0977

E*s,d2'(oo2)

0.0246

E ,,,(200)

0.1127

0.3285

E y,y(200)

-0.0112

0.0033

E E E E

x,xy(O2O) z,d2(oo2) xy,xy(~OO) xy,xy(oO*)

Ed2,d2(002) Edl,d1(002)

Es

,(lll)

E ;,w

0.0341

0.0123

0.0032

-0.0679

0.0525

0.0207 0.0001 -0.0463 0.0015

-0.0021 0.0002 -0.0197 O.OOOj

Ti-Ni

Ni-Ti

0.1144

0.1144

0.0827

-0.0868

-0.0457

-0.0460

E ,~,Wl)

0.0639

0.0639

Ex

0.0571

0.0571

0.0259

-0.0304

E ,',yull)

y(lll)

E ,~,yw) E E E E E

x,yz(lll) x,dl("') xy,xy(lll' xy,xz(lll) xv.dZclll)

0.0331

-0.04uo

-0.0061

0.0214

0.0114

0.0114

0.0196

0.0196

0.0109

-0.0182

with the rms errors of Bruinsma[4] as we calculated them from the information given in his paper. Bruinsma made the two-center approximation and the assumption that (spa) = ((ssa) x @PG))“~ which decreased the total number of his SK parameters to 26. At this point we wish to point out a typographical error in Bruinsma’s paper for the ppn parameters for Ti-Ti and Ni-Ti. The correct values are 0.2041 Ry and -0.0338 Ry, respectively. In Table 3 we compare the rms errors taken on a regular mesh of 35 k-points (while the fitting was done at 16 k-points in our calculation and at 7 k-points in Bruinsma’s). Our calculation gives a better fit to the APW results than that of Bruinsma for the following three reasons: (a) We have not made the two-center approximation nor used any simplification for determining the (spa) parameters. (b) We have fitted to the APW results at 16 k-points instead of 7. (c) We have included in the fit higher bands representative of the p-orbitals. The main deficiency of Bruinsma’s calculation is the fact that he has not fit the p-levels correctly. For example, he obtains the T,s at 0.888 Ry while the APW value is 1.288 Ry and our SK value is 1.289 Ry. His calculation is also inaccurate for the high s-levels (for example, his Ry is at 1.024 Ry while both the

0.0107

-0.OlbZ

APW and our SK values are at 1.103 Ry). On the other hand, the d-levels are described well by his calculation and are in good agreement with ours. This leads to the realization that since at the Fermi level the states are predominantly d-like (as shown in Fig. 1) his calculation of the phonon dispersion curves and the association of the anomalies with Fermi surface areas are probably reliable. The question of accurately fitting high levels such as T,s and R,. is an important one for the following reason. Although the lower bands fit very well if the high bands are ignored, the eigenvectors of the TB Hamiltonian are in error. This inaccuracy shows up when one calculates the angular momentumcomponents of the DOS. The purpose of this paper is to show how the SK interpolation scheme is applied to the CsCl structure and demonstrate its accuracy, rather than to use the results to interpret experiments. For completeness, however, we present in Figs. l-3 our calculated DOS decomposed per atom and angular momentum. These DOS were computed by the following procedure. After we determined the SK parameters, we diagonalized our 18 x 18 Hamiltonian at a grid of 969 k-points in the irreducible Brillouin zone. The

Slater-Koster

parametrization

Table 2. Symmetrized combinations

441

of the band structure of TiNi

of atomic orbitals for the various symmetry points and directions in the cubic Brillouin zone

-~ Synnnetrization

Representation s2 h2-Y2)2 (XY)p

x2

A1(OOz)

5

A2fOOd

b2-Y2)2

A28(002)

(XY),

A5(OOd

h),

x1mw

(3z2-r2)1

(3z2-r2) 1

52

x2

(2x)2

22

(3z2-r2)2

22

X*(002) X3(O@J Xs(OOZ)

x2

x2, (002) x3, (002)

X4~(O02) X5’ (002)

(3Ar2) 2

s2 (zx)2 (xl+Y1)/&

(3Ar2) 1

(XY)l

52

(x2+y&/(ir

(XY)*

(322J) 2

(xy-zx)*/fl (xy+zx)ll~~ (x2-Y2)1

(xY+zx)2/e

X2

(x2-Y2)

(x,-Y,)/E

(3z2-r2) 1

(XYI2

52

(3z2-r2) 2

2

(xy-2x)2/E (y2-22)/F fxl+Y1+Llfl6 (Yt-zx)ll@

fxY+Yz+zxl~/

@-

(x2+Y2+2*)/6 (x:fY,u @

(x2-Y2) 1

(YZ-ZX)2/6

(xYtyz+zx)2j~ (x2-Y2) 2

(Yd, x2 *1

% (xY-zx)l/e

5 X2 (YZ)l (ZX)l

(3zL2) 1

bYI

(3~2~r2) (Y&

G

(x2-Y2) 1 w2

bY-zx12: iE (3z2-r2J1

z2 (x:f,

YZ (x2-Y?2

(YZI2 (3zW)

2

442

J. D.

SHORE

and D. A.

PAPACONSTANTOPOULOS

resulting eigenvalues and the corresponding eigenvectors were used to calculate the DOS by the tetrahedron method[7]. In Fig. 1, we show the total DOS together with the Ti and Ni d-components. The bonding states are separated from the antibonding states by a minimum, to the right of which lies the Fermi level. It should be noted that the bonding levels consist predominantly of Ni d-states while the antibonding levels consist mainly of Ti d-states. In Fig. 2, we show a further decomposition of the d-DOS into its t,,(d,) and e,(dz)

za

components, where we note that for Ti the two components have comparable contributions while for the Ni site, the tt symmetry dominates. In Fig. 3, we show the s- and p-DOS for the two sites. As we mentioned before, the p-DOS are sensitive to fitting high bands above the Fermi level EF. An interesting check of our calculation can be provided by measurements of the K X-ray spectra. In Table 4, we list the Fermi level values, N(E,), of the DOS as well as the integrated quantities up to EF. We note that since EF lies to the right of the minimum

100.0

;;; 80.0 ! ;r 60.0 ” L 40.0 lh

20.0

cl 0.0 0.0

-I

.2

.4

.6

ENERGY

EF

.6

1.0

1.2

.6

1.0

1.2

.6

1.0

1.2

(Ry)

80.0

Y y

60.0

E 5 L

40.0

D

I ;;’

20.0

:: 0 0.0 0.0

.2

.4

.6

ENERGY

^z c

EF (Fly)

60.

E : :,

40.

0. 0.0

.2

.4 ENERGY

.6

EF (Ry)

Fig. 1. Total DOS of TiNi (top panel). Ti d-like DOS (middle panel). Ni d-like DOS (bottom panel).

Slater-Koster

parametrization

443

of the band structure of TiNi

(AH/SLllVlS)

ZP-TN-soa

N -A

’

I

’

’

’

’

’

0 r(

t

-7

al

\

.

c

w”

t

m

I-

i-’

P

N

t

tc I

0

.

-d-

0

.

0

.

1

0

I

0 .

.

I

0

I

0

.

0

0 .

t 0

I

0

I

0

(~tuS3lVlSl

t

0

I

0

I

0

F P-TN-soa

I

0

J. D. SHORE and D. A. PAPACONSTANTOPOULOS

I

I

I

.

0

0

.

0

I\

(D

In

(ntvs3~vs)

0

v

.

I

1

0

I

.

.

0

0

03

N

-3

0

s--FL--soa

90 0

Slater-Koster

parametrization

Table 3. Comparison of rms fitting errors expressed in Ry Band 1

Present Calculation .0024

Ref. 4

of the band structure of TiNi

Table 4. Values of the densities of states at the Fermi level together with the corresponding integrated DOS up to EF Character

.0207

2

.0032

.0153

3

.a017

.0115

4

.0024

.0093

5

.0020

.0086

6

.0037

.0131

7

.0040

.0078

8

.0032

.0076

9

.0039

.0125

10

.0029

.0099

11

.0059

.0539

12

.0359

.I599

3d*4s2 respectively, it is interesting to compare them with the electron configuration of the compound TiNi as shown in Table 4. The number of Ti delectrons has increased approximately by 1.0, half of which comes from Ti s-electrons and the other half from Ni s-electrons. Our conclusion from this work is that the SK parametrization of the energy bands for CsCl structure materials can give a very accurate band structure and a good starting point for studying disorder effects, impurities, and phonon spectra.

00s statesfRylcel1

Integrated DOS

47.14

14.00

Ti-s

0.32

0.52

Ti-p

2.47

0.87

Ti-dl

14.13

2.04

Ti-d2

17.50

1.05

Ti

34.42

4.48

Ni-s

0.09

0.62

Ni-p

1.98

0.61

Ni-dl

2.15

4.94

Total

Ni-d2 Ni

in the DOS and in the antibonding states, Ti-d states are the larger contributors to Iv’(&). Conside~ng that the atomic configuration of Ti and Ni are 3d24s2 and

445

8.50

3.35

32.72

9.52

Acknowledgements-The authors wish to thank Dr. A. C. Switendick for many helpful discussions, and Mrs. L. N. Blohm for technical assistance.

REFERENCES 1. Slater J. C. and Koster G. F., P&r. Rev. 94, 1498 (1954).

2. Papaconstantopoulos D. A., Kamm G. N. and Poulopoulos P. N., Solid St. Common. 41, 93 (1982). 3. Connolly J. W. D., Electronic density of states, p. 27. NBS Spec. Publ. No. 323. US GPO Washington, D.C. (1971). 4. Bruinsma R., P&r. ROD.BU, 2951 (1982). 5. Vanna C. M. and Weber W., Phys. Rev. B19,6142 (1979). 6. Bell D. G., Rev. Mod. Phys. 26, 311 (I 954). 7. Lehmann G. and Taut M., Phys. Status Solidi (b) 54,469 ( 1972).