MAGNETOVOLUME EFFECT IN FERROMAGNETIC METALS : NUMERICAL RESULTS FOR BCC Fe AND FCC Ni

The magnetovolume effect in ferromagnetic bcc Fe and fcc Ni is estimated numerically in the fixed moment method by using the LMTO method and the local spin density approximation. The obtained results are compared with the experimental results. The magnetovolume effect is discussed by using the local spin density approximation [I-31 or by using the Stoner model where the volume dependence of the molecular field coefficient is taken into account [4]. The magnetovolume effects in the ferromagnetic metals can be estimated by using two function; b(M, V) = BE(M, V)/aM and p(M, V) = -BE (M, V) /aV, where E (M, V) is the energy of the system with the magnetic moment M and the volume V. It is not necessary to obtain the total energy itself as the functions of M and V. The accurate estimation of the total energy may be cumbersome as there is a cancellation of large terms with opposite signs. A method to calculate the magnetovolume effect in ferromagnetic metals is given, which is within the framework of the standard band calculation. The method is equivalent to the fixed moment method given by Moruzzi et al. [5] but they did not discuss the magnetovolume effect. Two functions b (M, V) and p (M, V) are obtained numerically by using the linear muffin-tin orbital method [6, 7] and the local spin density approximation and the m&netovolume effect in the ferromagnetic bcc Fe and fcc Ni are estimated numerically. For an electron system having a given total number of electrons N, a given total spin magnetic moment M and a given volume V, the generalized chemical potentials /I, are determined in the local spin density approximation by [8]. d E Du (E) = ( N + ~ M / P B ) 12, (1) where D, (E) is the density of states for a-spin electron under the constrains that the values of N, M and V are fixed beforehand. It should be noted that the D, (E) is implicitly a function of M and V. One can show that the function b (M, V) is given by [91 The electronic contribution pel to p (M, V) is given in the atomic sphere approximation by [ll] u where NA is the total number of atoms. And Dz (E) , nz (r) , nu (r) , v, (r) and vxcu (r) are the density of states of valence electrons, the number density of valence electrons, the total density of electrons, the effective potential and the exchange correlation potential for an electron in the a-spin state, respectively. In equation (3) s is the radius of the atomic sphere and Exc (n+, n-) is the exchange correlation energy density in the local spin density approximation. The contribution from the zero point lattice vibration plat to p (M, V) is estimated by using the Debye model [3]. The dependence of plat on the magnetic moment is neglected for simplicity. The sum of pel and plat gives P (M, V). When an external magnetic field Hand an external pressure P are applied to the system, the equilibrium spin magnetic moment and the equilibrium volume in the external magnetic field and the pressure are determined by the solutions of the equations b (M, V) = H and p (M, V) = P. The magnetovolume effect in the ferromagnetic metals is estimated by these relations. The magnetic spin susceptibility at constant V and the compressibility at constant M can be written as Xv= l / b ~ and KM = -1/Vpv, respectively, where b~ = ab (M, V) /aM and pv = ap (M, V) /aV. The magnetic spin susceptibility at constant pressure and the compressibility at constant magnetic field can be written as X p = vXy and KH = ~ K M I respectively, where 7 is the magnetovolume enhancement factor introduced by Shimizu [4] and is given by This formalism is equivalent to that discussed by Shimizu [lo] in the Stoner model. v = (1 + d / b ~ p v ) ' = (1 v&J?x,.M)-'. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888130 C8 292 JOURNAL DE PHYSIQUE Here gl = bb (M, V) /bV and it is the so-called the magnetovolume coupling [4]. The forced volume magnetostriction h is given as h = ~ ~ I X V I E M . The magnetovolume effect in the ferromagnetic bcc Fe and fcc Ni at zero temperature is calculated numerically. The self-consistent band structure is calculated for given values of N, M and V by the linear muffin-tin orbital method [6, 71, where the von Barth and Hedin [12] exchange correlation potential is used. The practical method of the calculation of the band structure is the same as that used before [9]. The values for the Debye temperature and the Griineisen constant, which are necessary in the calculation of plat, are taken as 467 K and 1.66 for Fe and 450 K and 1.88 for Ni, respectively [3]. The obtained equilibrium values of M and the lattice constant a are 2.22 p~ /a tom and 5.345 au for Fe and 0.56 p ~ / a t o m and 6.593 au for Ni, respectively. Their observed values are 2.12 p~ /a tom and 5.4057 au for Fe and 0.56 pB/atom and 6.6590 au for Ni, respectively. The corresponding values obtained by Andersen et al. [2] are 2.17 p~ /a tom and 5.280 au for Fe and 0.68 pB/atom and 6.704 au for Ni, respectively. The values obtained by Moruzzi et al. [5] are 2.15 p ~ l a t o m and 5.341 au for Fe and 0.60 p~ /a tom and 6.602 au for Ni, respectively. The difference between the theoretical values of M may cqme from the different method of the calculation and the difference between the values of a. The estimated values for the Xv are 3.79 x emu/mole for Fe and 2 . 6 7 ~ lo-' emu/mole for Ni. The experimental values are 2.66 x emu/mole for Fe and 1.10 x emu/mole for Ni [4]. Even if one takes into account the contribution from the orbital moment the calculated value is to small as pointed out before [9]. The bulk modulus BM = 62' is estimated as 2.59 Mbar for Fe and 2.26 Mbar for Ni. The experimental values are 1.73 Mbar for Fe and 1.87 Mbar for Ni. The values obtained by Andersen et al. [2] are 2.60 Mbar for Fe and 2.09 Mbar for Ni. The values obtained by Moruzzi et al. 131 are 2.17 Mbar for Fe and 2.27 Mbar for Ni. The estimated values for gl and 7 are -3.338 x lo7 mole 0e/cm3 and 1.12 for Fe and -0.784 x lo7 mole 0e/cm3 and 1.01 for Ni, respectively. The values for h are estimated as 5.49 x lo-'' 0e-' for Fe and 0.93 x lo-'' 0e-I for Ni. The experimental values of h at 0 K are 4.5 x lo-'' 0e-' for Fe and 1.2 x lo-'' 0e-' for Ni [4]. The values obtained by Andersen et al. [2] are 4.4 x lo-'' oe-' for Fe and 0.96 x lo-" 0e-' for Ni. The values obtained by Moruzzi et al. [3] are 9.0 x lo-'' 0e-' for Fe and 1.1 x lo-'' 0e-I for Ni. The obtained results for the lattice constant and the magnetic moment are in agrement with the experimental results. The obtained value of a is a little smaller than the experimental value and it seems to be a general trend obtained by the local spin density functional formalism. The calculated results of the magnetovolume effect in the ferromagnetic state do not give satisfactory agreement with the experimental results. The reason of the discrepancy between the calculated values and the experimental values in the ferromagnetic metals may come from the defect of the local spin density approximation. The magnetovolume effect is e s sentially the second order derivative of the total energy. The details of the exchange correlation potential may have a large effect on the estimated values of the magnetovolume effect. It seems necessary to improve the expression of the local spin density approximation. The contribution from the orbital motion of itinerant electrons to the magnetovolume effect is not taken into account in the present numercial calculation [4]. The contribution from the orbital motion of electrons to the magnetovolume effect in ferromagnetic metals should be considered and it is a future problem. [l] Janak, J. F. and Williams, A. R., Phys. Rev. B 14 (1976) 4199. [2] Andersen, 0. K., Madsen, J., Poulsen, U. K., Jepsen, 0. and Kollard, J., Physica 8688B (1977) 249. [3] Moruzzi, V. L., Janak, J. F. and Williams, A. R., Calculated Electronic Properties of Metals (Pergamon Press, New York) 1978; Phys. Rev. B 12 (1974) 1257. [4] Shimizu, M., Rep. Prog. Phys. 44 (1981) 329. [5] Moruzzi, V. L., Marcus, P. M., Schwarz, K. and Mohn, P., Phys. Rev. B 34 (1986) 1784. [6] Andersen, 0. K., Phys. Rev. B 12 (1975) 3060. [I Skriver, H. L., The LMTO Method (SpringerVerlag, New York) 1984. [8] Lundqvist, S. and March, N. H., Theory of the Inhomogeneous Electron Gas (Plenum Press, New York, London) 1983. 191 Takahashi, I. and Shimizu, M., J. Phys. Soc. Jpn 56 (1987) 4540; J. Phys. F (1988) in press. [lo] Shimizu, M., Proc. Phys. Soc. 84 (1964) 397. [ll] Janak, J. F., Phys. Rev. B 9 (1974) 3985. [12] von Barth, U. and Hedin, L., J. Phys. C 5 (1972) 1629.