ElSEVlER Physica B 193 (1994) 248-254

Electronic properties and Fermi surface of Ni,Pt

Arti Kashyap*, T. Nautiyal, S. Auluck

Department of Physics, University of Roorkec, Roorkee 237 667. India

Received 9 August 1993; revised 13 October lYY3

Abstract

We present calculations of the electronic properties of ordered Ni,Pt using the linear-muffin-tin-orbital method. Calculations are performed for the paramagnetic and ferromagnetic phases. Our total-energy calculations show that the ferromagnetic state has a lower energy. Hence the ferromagnetic phase is studied in detail. The calculations are done at a theoretical equilibrium lattice constant. The results for density of states, bulk modulus and magnetic moment at the equilibrium lattice constant are reported here. Also a brief discussion of the Fermi surface of ferromagnetic Ni,Pt is presented.

1. Introduction

In our previous work the transition-metal compounds N&Al [l], N&Fe [2] and Ni,Mn [3] have been studied and the results were found to be in good agreement with the available ex- perimental data. These compounds crystallize in the Cu,Au crystal structure. A member of this Ni,X series is the intermetallic compound Ni,Pt. In ordered form it crystallizes in a simple cubic Cu,Au structure with Ni atoms at the face centers and Pt atoms at the corners of the cube. We have studied the electronic structure and various ground-state properties of this compound using the linear-muffin-tin-orbital (LMTO) method of Andersen [4]. The plan of this paper is as follows. Section 2 contains a brief descrip- tion of the method of calculation. In Section 3 we present a brief discussion of the calculation of the paramagnetic state followed by the results

* Corresponding author.

for the ferromagnetic state in terms of density of states and cross-section of the Fermi surface along the high symmetry planes of the Brillouin zone. Finally, in Section 4, the main conclusions of this work are summed up.

2. Method of calculation

Calculations are performed using the LMTO method within the atomic sphere approximation (ASA) including the combined correction terms [4]. The Barth-Hedin exchange-correlation (X- C) potential [5] is used. The frozen atomic core charge densities are used throughout the calcula- tions. This frozen-core approximation is valid at ambient and low pressures where the core-states are highly localized. Since we are interested in ground state properties at ambient pressure, this approximation has been used in our work. With- in spheres around each atomic center the wave- function is expanded in muffin-tin-orbitals. The basis set includes s, p and d orbitals at each

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A. Kashyap et al. I Physica B 193 (1994) 248-254 249

atomic site. We have calculated the total energy as a function of lattice spacings in order to obtain the theoretical equilibrium lattice constant. The value of the equilibrium average Wigner-Seitz (WS) radius as determined from the minimum of the total energy is found to be 2.705 a.u. which corresponds to a lattice constant equal to 3.66 A. The spheres around the atom of the two con- stituent metals are taken to be of different radius using the method outlined by Andersen, Jepsen and Sob [6], which gives S,,/S,, = 0.898. The one-electron self-consistent potentials are ob- tained by performing iterations on a mesh of 84 k points in the irreducible one forty-eighth part of the Brillouin zone. We have used the tetrahedron method [7] to calculate the electronic density of states.

3. Results and discussion

Figure 1 displays the total-energy versus unit- cell-volume curve (solid curve for paramagnetic state and broken curve for ferromagnetic state). Minima occur at an equilibrium volume of 19.79 a.u. for both paramagnetic and ferromag- netic states. Corresponding to this volume the average WS equilibrium radius for the compound is 2.705 a.u. and the corresponding lattice con-

-294*66 #_I -294.66

- Pam. l .tu..tt Form.

2 -294.87

B s = -294.69

I

-294.70

Volume (au.) Fig. 1. Total-energy versus volume curve for paramagnetic

and ferromagnetic Ni,Pt.

stant is 3.66 A, which is very close to the ex- perimental value of 3.646 A [8]. We performed all calculations at this theoretical equilibrium lattice constant. Earlier leading theoretical work on this compound was done by Pisanty et al. [9] using the LMTO method. Their method of calculation, exchange-correlation potential, ratio of WS radii for the two atomic species are the same as used in the present work. However, they have obtained the equilibrium lattice constant by interpolation to p(V) = 0 and got a value of 3.597 A. This value is quite different from ours (3.66 A). As electronic properties are very sensi- tive to the lattice constant used, we expect this difference of lattice constants to be the main reason for differences between our results and those from Ref. [9]. Besides this, a small differ- ence may also be registered due to the difference of convergence criterion used in the two works. This work presents a more detailed study on Ni,Pt. Results for the paramagnetic state fol- lowed by those for the ferromagnetic state are discussed and a comparison is made with Pisan- ty’s calculations and experimental data wherever available.

3.1. Paramagnetic state

The total density of states (DOS) for the compound is shown in Fig. 2. Also the l-project- ed DOS are displayed in Table 1. From Fig. 2 and Table 1 it is obvious that at and near the Fermi energy E, most of the contribution comes

Energy (Ry) Fig. 2. Total density of states in paramagnetic Ni,Pt. The

Fermi energy is marked (E,).

250

Table 1

A. Kashyap et al. I Physica B 193 (1994) 248-254

l-projected density of states at the Fermi energy, in states per Ry unit cell, the coefficient of electronic specific heat y. in

mJ/mol K’, charge occupancies, in electrons, the magnetic moment, k/atom, the total energy, in mRy, and the bulk modulus.

in MBar, at equilibrium WS volume for Ni,Pt

Ni-s Ni-p Ni-d Pt-s Pt-p Pt-d Total Y

l-projected DOS PM

FMJ

t Theory”

Expt !’

0.44 0.71 47.66 0.50 1.21

0.10 0.23 23.95 0.10 0.39

0.13 0.27 6.65 0.14 0.45

29.72 177.86 30.81

9.36 110.15 19.08

5.72 20.00

54.80

Charge occupancies PM

FM1

T

Magnetic moment FM

Theory”

Expt’

0.62 0.74 8.63 0.87 0.88 8.27 39.99

0.31 0.37 4.09 0.43 0.44 3.98 19.16

0.31 0.37 4.54 0.44 0.44 4.30 20.84

0.443’ 0.324d 1.653

0.700’ 0.300” 2.400

0.490’ 0.250* 1.720

Total energy PM -294.701

FM -294.707

Bulk modulus PM 2.19

FM 2.17

“Ref. [9]; ‘Ref. [21]; ‘Ni-site; dPt-site

from Ni d-states. However, at lower energies, Pt d-states are dominant. The total DOS, N(E,), at E, is 177.86 states per Ry unit cell. Using the expression y = (1/3)1r’k~N(E,), we obtain the coefficient of electronic specific heat y, equal to 30.81 mJ/K2 mol. The experimental value of y for magnetic Ni,Pt [lo] is 54.8 mJ/K2 mol which is very large compared to our value and that from Pisanty et al. 191. The large value of N(E,) shows that this state is unstable for the formation of a ferromagnetic state, since the Stoner criter- ion for magnetism IN(&) > 1 is satisfied. Here I is the Stoner parameter which may be written as

I= c mxE,)~~(EF)12 , where Z, is the Stoner parameter for d orbitals at site t and N,(E,) is the d-DOS at site t. For Z, we have used the values from the work of Janak

[Ill. The important band parameters for paramag-

netic Ni,Pt are given in Table 2. We have tabulated the self-consistent band width A-B (where A and B are the top and bottom of the band) as well as the estimate W of the band width based on the canonical band theory. A comparison of these two indicates the influence

Table 2

/.L is the effective d-band mass at site t (in units of free-electron

mass), A-B is the self-consistently calculated d-bandwidth at site t (in mRy), W the d-bandwidth estimated on the basis of canonical band theory (in mRy), n(= (A-B)IW) and ?j

PM

FM

Site CL A-B W 77 -rl

Ni 12.49 284 190 1.49 1.21 Pt 5.44 493 87 5.66 7.92

Nit 12.59 282 190 1.48 - Ni, 12.44 286 190 1.50 - Pt’ 5.45 492 87 5.65 - Pt, 5.40 490 87 5.63 -

A. Kashyap et al. I Physica B 193 (1994) 248-254 251

of hybridization on the band width. The expres- sion for the band width of the unhybridized band of atom t and orbital 1 may be found in Ref. [12] and details of such calculations are given by Podgorny [13]. As observed for N&Fe and Ni,Mn, here also we find that Ni d-bands broaden in the presence of a Pt atom (nNi > jjNi and npt < +&. However, broadening of Ni bands due to the presence of Pt atoms is found to be larger in Ni,Pt as compared to N&Fe and Ni,Mn, suggesting a comparatively higher de- gree of hybridization in Ni,Pt.

The variation of pressure with volume is also studied (Fig. 3(a) for the paramagnetic state). From the pressure-versus-volume curve the bulk modulus is calculated and found to be 2.19 Mbar for paramagnetic Ni,Pt. As no experimental data is available we compare our results with N&Al for which the theoretical bulk modulus [14] was calculated (using the LMTO method) to be 2.12 Mbar, which was in good agreement with the experimental value [15] of 2.4Mbar.

3.2. Ferromagnetic state

The total energy E of Ni,Pt for paramagnetic (P) and ferromagnetic (FM) phases at an equilibrium WS radius of 2.705 a.u. are E(P) = -294.701 Ry and E(FM) = -294.707 Ry giving E(FM) < E(P). We have checked the conver- gence up to the third decimal. Thus our calcula- tions predict that Ni,Pt is ferromagnetic, which is confirmed by the magnetic measurements of Kuentzler [ 161. Our total-energy calculations

-0.30 15.00 18.00 21.00

Volume (au.)

(b)

“; IS.00 18.00 21.00

Volume (au.)

Fig. 3. Variation of pressure with volume for (a) paramagnet- ic and (b) ferromagnetic Ni,Pt.

indicate that at volumes lower than 17.9 a.u. it is the paramagnetic phase which is more stable. The Stoner parameter Z is also found to be consistently less than one in this volume range. At equilibrium volume, the difference AE = E(P) - E(FM) is 6 mRy, which is small in com- parison with N&Fe (AE = 15 mRy) and Ni,Mn (AE = 70 mRy), which explains why the two curves cross each other at a certain point. Fur- ther, the pressure-versus-volume variation is also studied for the FM state (shown in Fig. 3(b)). From this curve, the bulk modulus was found to be 2.17 Mbar which is only slightly different from the value for the paramagnetic phase.

One can write the Stoner criterion [17] as AU = ml, for both the atomic species, where AU is the split between the center of the up and down d-band for an atomic site, and Z is the Stoner parameter for the same site. Using ZNi = O.O69Ry, Z,, = 0.043 Ry (Ref. [ll]) and our mNi = 0.45b/atom, mpt = 0.31*latom for re- spective d-bands, we obtain AUNi = 0.031 Ry and AU,,, = 0.013 Ry. Our self-consistent band struc- ture calculations give [18] AU = C’ - C, (where C is the center of the respective d-band) equal to 0.026 Ry and 0.037 Ry for Ni-d and Pt-d bands. These values are quite close to AU from the Stoner criterion. This reflects the consistency of our band structure calculations.

The important band parameters are given in Table 2. For Ni-d states we find that the spin-up bands are wider, compared with spin-down bands. The DOS for spin-up and spin-down electrons at Ni sites and Pt sites are shown in Fig. 4(a) (total DOS) and Fig. 4(b) (d-DOS). The partial and total DOS are listed in Table 1. The exchange splitting of spin-up and spin-down bands has resulted in a drop of total DOS at the Fermi energy, N(E,), to 110.15 states per Ry unit cell. Our N(E,) corresponds to a y coeffi- cient of 19.08 mJ/K2 mol which is close to that from Pisanty et al. [9] and much less than the experimental value [lo] of 54.8 mJ/K2 mol. The partial charge occupancies are given in Table 1. A total charge of 0.049e is transferred from three Ni atoms to the Pt atom which is in accordance with the electronegativity difference [20]. We obtain the magnetic moment at Ni and Pt sites to be equal to 0.443blatom and 0.324hlatom,

252 A. Kashyap et 111. I Physica B 193 (1994) 248-254

Fig. 4. (a) d-density of states and (b) total density of states

for Ni and Pt atoms in ferromagnetic Ni,Pt.

respectively, amounting to a total magnetic mo- ment of 1.653~~ per unit cell. These values are in better agreement with experiment 1211 as compared to those reported in Ref. [9] as shown in Table 1.

Figures 5 and 6 display the Fermi surface (FS) cross-sections for majority and minority carriers,

respectively. For majority carriers six (19-24)

Fig. 5. Majority-carrier Fermi surface for ferromagnetic

Ni,Pt.

Fig. 6. Minority-carrier Fermi surface for ferromagnetic

Ni,Pt

bands intersect the Fermi level and for minority carriers eight (14-21) bands do so. We have calculated areas and masses of numerous Fermi surface orbits as done previously for N&Fe [2] and Ni,Mn [3]. As there is no experimental data

to compare with, we have given the areas and masses of the orbits in the (0 0 1) plane (Table 3) only.

4. Conclusions

Our total energy calculations show the stability of the formation of the ferromagnetic state. The theoretical value of the lattice constant comes out to be 3.66 A in good agreement with the

experimental value of 3.646 A. Pisanty et al. [9] get a value of the equilibrium lattice constant 3.597 A by interpolation to p(V) = 0. Compari- son of the calculated electronic specific-heat coefficient from this work and Ref. [9] with experiment gives a large value of the enhance- ment factor. This hints at Ni,Pt being a highly normalized system. We have obtained a magnetic moment of 1.653& per cell for this compound which is very close to the experimental value of 1.72/_+, . The bulk modulus was found to be B(P) = 2. 19 Mbar and @FM) = 2.17 Mbar. We have studied the Fermi surface of ferromagnetic Ni,Pt and calculated the areas and masses for many FS orbits. However, nonavailability of experimental as well as theoretical results for FS studies prohibits any comparison. To sum up we

A. Kashyap et al. I Physica B 193 (1994) 248-254 253

Table 3 Fermi surface orbit areas, in MG, and effective band masses, in free-electron units, for majority and minority carriers in Ni,Pt in the (0 0 1) plane

Majority carriers

Orbit Area

F(l9) 87.90 I(20) 13.24 r(21) 77.12 X (22) 122.20 X (22) 25.01 XM (22) 6.12 M (21) 26.35 R (22) 4.24 R (23) 1.87 R (24) 0.92

a In the FXM plane.

Mass

-0.47 -0.99 -2.47

2.58 -1.29

0.55 -0.76

0.53 0.17 0.10

Minority carriers

Orbit Area Mass

I- (14) 5.53 -0.54 I(l5) 5.56 -0.52 I(l6) 20.46 -0.89 r(l7) 32.12 -1.91 I(l8) 46.12 -2.52 I(l9) 52.03 -3.47 X (20) 59.81 2.22 x (20)” 60.39 2.23 X (21) 55.20 2.83 X (21) 27.53 -3.44 XM (21) 0.72 0.37 TM (21) 1.98 0.44 M (20)” 68.76 2.30 M (20) 60.82 2.02 R (18) 0.47 0.16 R (18) 33.28 -4.88 R (19) 0.41 0.12 R(19) 56.28 -3.67

have presented a detailed study of the electronic structure of Ni,Pt. Our results are in better agreement with experiments compared to those of Pisanty et al. [9]. The reason is that the value of the equilibrium lattice constant as obtained by us, by a total-energy criterion, is very close to the experimental value compared to that by Pisanty et al. [9], using interpolation to p(V) = 0. This suggests that the total-energy approach is a better way of estimating the equilibrium lattice constant.

Acknowledgements

Financial support from the Council of Sci- entific and Industrial Research (CSIR), India, Scheme No. 3(698)/91-EMR-II, is gratefully acknowledged.

References

[l] T. Nautiyal and S. Amuck, Phys. Rev. B 45 (1992) 13930.

PI

131 (41 151 bl

[71

RI

[91

WI

1111

WI

[I31 1141

T. Nautiyal and S. Amuck, Phys. Rev. B 47 (1992) 1726. T. Nautiyal and S. Amuck, Phys. Rev. B, to appear. O.K. Andersen, Phys. Rev. B 12 (1975) 3060. U. Von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. O.K. Andersen, 0. Jepsen and M. Sob, in: Electronic Band Structure and its Applications, ed. M. Yussouff, Springer Lecture Notes in Physics Vol. 283 (Springer- Verlag, Berlin, 1987). 0. Jepsen and O.K. Andersen, Solid State Commun. 9 (1971) 1763; G. Lehmann and M. Taut, Phys. Stat. Sol. B 54 (1972) 469. C.E. Dahmani, Thesis, Universiti Louis Pasteur, Stras- bourg (1985). A. Pisanty, C. Amador, Y. Ruiz and Martha de la Vaga, Z Phys. B 80 (1990) 237. R. Kuentzler and R.W. Waterstrat, in: LT-17 Proc., eds. U. Eckem, A. Schmid, W. Weber and H. Wuhl (Elsevier, Amsterdam, 1984). J.F. Janak, Phys. Rev. B 16 (1977) 255; O.K. An- dersen, 0. Jepsen and D. Glotzel, in: Highlights of Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North-Holland, Amsterdam, 1985). O.K. Andersen, H.L. Skriver, H. Nohl and B. Johan- son, Pure Appl. Chem. 52 (1979) 93; O.K. Andersen, W. Klose and H. Nohl, Phys. Rev. B 17 (1978) 1209. M. Podgorny, Acta Phys. Polon. A 78 (1990) 941. B.I. Min, A.J. Freeman and H.J.F. Jansen, Phys. Rev. B 37 (1988) 6757.

254 A. Kashyap et al. I Physica B 193 (1994) 248-254

[15] J.J.M. Frame, M. Rosen and E.P. Wohlfarth, Physica

B + C 86-88 (1977) 317.

[16] R. Kuentzler, Inst. Phys. Conf. Ser. 55 (1981) 397.

[17] U.K. Poulsen, J. Kollar and O.K. Andersen, J. Phys. F

6 (1976) L241.

[18] H.L. Skriver, The LMTO Method: Muffin-Tin-Orbital

and Electronic Structure (Springer-Verlag, New York,

1984).

[19] It appears that Pisanty et al. have calculated the y

values (expressed in mJ/mol K*) from their density of

states/atom, as can be verified from their Fig. 1 for

Ni,Pt, which gives a total DOS of about 120s/(Ryfor-

mula unit) and should, therefore, give y = 20 mJ/mol K’

using y = n*kiN(E,)/3. However, they report (Fig. 4) a

value close to 5 which shows that the hJ(E,) value used

by Pisanty et al. in the calculation of y is the one

calculated in states/(Ry atom). Hence we have put a

factor of 4 in their values reported in our Table 1.

[20] L. Pauling, The Nature of the Chemical Bond and the

Structure of Molecules and Crystals (Oxford and IBH,

New Delhi, 1967; 3rd ed.).

121) G. Fischer and M.J. Besnus. Solid State Commun. 7

(1969) 1527; M.J. Besnus and A. Herr, Phys. Lett. 39A

(1972) 83.