Magneto-volume effects in Fe-Ni invar: A first principles theory

Physica B 161 (1989) 153-156 North-Holland, Amsterdam

MAGNETO-VOLUME

EFFECTS

IN Fe-Ni

INVAR: A FIRST PRINCIPLES

THEORY

P. MOHN and K. SCHWARZ Technical University of Vienna, A-1060 Vienna, Austria

D. WAGNER Ruhr University Bochum,

D-4630 Bochum,

FRG

A complete theory of the magnetic properties, the magneto-volume effect, and the thermal properties of weak itinerant ferromagnets is proposed in terms of spin- and volume fluctuations. This theory combines the results of band structure calculations (only valid at OK) with fluctuation models including temperature effects. We use the results of previous spin-polarised band structure calculations performed with the fixed-spin-moment (FSM) method using the local spin density approximation for exchange and correlation, where the total energy is obtained as a function of volume and the magnetic moment. The fluctuations are treated classically within the frame of a Ginzburg-Landau theory. Numerical results are given for the Fe-Ni invar system.

1. Introduction

2. Theoretical

During the past few years a great deal of progress was made in the theory of spin-fluctuations in itinerant systems [l]. It has become clear that for finite temperature the Stoner model involving single particle excitations of the itinerant electrons is insufficient for most systems, but that collective excitations are inevitably present. At finite temperatures the early work of Murata and Doniach [2] had made it clear that in metals spin-fluctuation effects persist side by side with single particle excitations but that it is difficult to describe these theoretically for realistic models. Very successful attempts in this direction have been undertaken [3, 41 by applying a GinzburgLandau (G-L) theory of phase transitions to describe the Curie temperature of a wide variety of metals. Entel et al. [5] presented a generalisation of these ideas and they were the first to derive G-L parameters from band structure results. Band theory allows to obtain total energy surfaces in the M-V-plane calculated via the fixed-spinmoment (FSM) method [6, 71 employing the augmented spherical wave (ASW) formalism by Williams et al. [S]. For bee and fee Fe such FSM energy surfaces have been obtained by Moruzzi et al. [9].

The present investigation is based on a theory developed by Wagner [lo] who combined band structure results (FSM energy surfaces) and classical thermodynamics. He developed a general theory of the thermal and magneto-mechanical properties of weak itinerant ferromagnets in terms of spin- and volume (density) fluctuations. In contrast to ref. [5], where only longitudinal fluctuations of the magnetic moment are considered. Wagner included both longitudinal and transverse as well as volume fluctuations. The energy should be a functional of the noncentered random variables I and v(r) varying on a spatial scale larger than the range of the interactions. We apply this method and use a local approximation for the functional, while non-local effects are taken into account by the lowest order gradient terms. The Hamiltonian of the fluctuations is then given by

%‘=

& jd3r(

outline

E(M + m(r), V + u(r))

+ : z (vim,)’ + : i, j denote components

0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(oU(r))‘) . of the magnetic fluctua-

tion and the gradient. respectively. Using this Hamiltonian WC calculate the free cncrgy via a classical partition function integral over the phase space of the fluctuations. For details of the formalism WC refer to [Y1. The quantities calculated in this paper arc the two components of the fluctuating magnetisation (j?2f(r)) and the volume fluctuation ( LI’(1.)j:

where f’(,~) = I -- x ’ arctan( i = 1.2 for the transverse component and i _- 3 for the parallel component of the fluctuations. The energy function cp is defined by:

cp =

&

I

d’r( E(M + 111,v

+ U) - E(M.V)) (4)

The magnetic and mechanical equations of state are derived by requiring minimisation of E and

general

deviation

;I maximum

from “normal”

in the

lattice

bchaviour

constant

for the

lib Fe-Ni

an abrupt drop in the compressibility and a deviation of the average magnetic moment from the Slatcr-F’auling CLI~VC ['I. From all these experimental data it became clear that thcsc effects must be ruled by a delicate interplay between magnetic and mechanic properties. In this cast the usual thermally cxcited lattice-expansion must be compensated by a magneto-volume effect. Static models deal only with single particle (Stoner) excitations. They require vanishing local magnetic moments for a non-magnetic state. Consequently thev lcad to a Curie temperature which is several times the experimental value. Furthermore the equilibrium volume, V,. of the “non-magnetic” state (given by the saddle point at M = 0 in fig. 1) is too small with respect to experiment. This discrepancy leads to an unphysically large magneto-striction W, = AVIV (table I). Weiss (I?] proposed a two state model hetween a ferromagnetic ground and an antiferromagnetic excited state to explain the invar effect. Williams et al. [ 14) gave - on the basis of band structure calculations - an itinerant version xyslcm.

9: SEiSM Equations tities.

+ i%p/SM = 0, SEISV + &+3/sv= 0. (2-5)

3. Application

determine

to Fe,Ni

all

relevant

(5) quan-

invar

The Fe-Ni invar system is a random substitutional alloy of an fee structure. Although the ideal invar composition is Fe,,Ni,, , we approximate it by Fe,Ni assuming the ordered Cu,Au lattice. This composition is close to the invar case and order/disorder should not dominate the physics involved. The most striking feature of this system is the extremely small thermal expansion coefficient [ 111. This anomaly is not isolated but is rather one particular aspect of a more

Fig.

1. Contour

70

75

volume/atom

(bohr3)

plot

of

the

fitted

E(M,V)

based on the results of Moruzzi

between

two contour

show the equilibrium temperature

volume

positions

for different

tions only; ----

q< = 5.0.

lines is 0.5 mRy.

FSM

The additional

I/, and M,

models:

(-.-

energy

surface

1151. The difference curves

as a function

.-.-)

Q, = 0.0 spin fluctuations

Stoner

only: ~-~-

q1 = 7.5. -x--x--x+~ qc = 10.0. (spin

fluctuations).

of

cxcita-

-~

and

P. Mohn and K. Schwarz Table

I Magneto-volume

I

4c

T,.

V,

0,

(m2)rc

(v2)r,

ckK

Stoner 0.0 5.0 7.5 10.0

? 501 561 609 677

69.70 71.17 72.35 73.27 74.57

0.065 0.043 0.026 0.013 -0.004

0 0.299 0.340 0.375 0.423

0 0.0 30.65 52.57 80.90

? -27.3 -15.1 -7.4 + 0.8

Calculated magnetic and mechanic parameters for different values of qc (in bohr -‘) (C = 0.55, D = 0.002, k, = 14.5). T, is the Curie temperature (K), V, is the volume at T,. o, = (V, ~ V,.)/V, is the spontaneous volume magnetostriction. (m’) and ( vZ) are the averaged square of the fluctuating magnetisation and volume in & and bohrh respectively. a300K = V’(SV/ST),i3 is the coefficient of the thermal expansion at room temperature (300 K) in units of 10-O Km’. V, = 74.25 is the equilibrium volume at T = 0.

of this idea. They found for the proper composition a ferromagnetically ordered ground state (at a larger volume) and a non-magnetic state (at a smaller volume) which have about the same energy. If the energy difference is such that the non-magnetic (or antiferromagnetic) state is accessible by thermal excitation then this mechanism can explain the small thermal expansion. The unsolved problem of the too large magnetostriction remains, since fluctuations are completely neglected. Instead of using just two isolated equilibrium states as above, the FSM method [6, 71 allows one to compute an energy surface as a function of magnetic moment and volume E(M, V). Fe,Ni is the first system for which such results have been obtained [6]. A detailed investigation of the energy surface has been undertaken by Moruzzi [15] and is the basis of this paper. This ground state function is fitted with bivariate Chebyshev polynomials up to third power in the volume V and up to sixth power in the magnetic moment M. The inherent symmetry, that the free energy has to be a polynomial in even powers of M only, is obeyed (fig. 1). With this additional information on the system it is possible to go beyond the quantummechanical results at T = 0 K and to introduce fluctuations. There are 4 parameters in eqs. (2) and (3)) namely the cutoff-wave vectors kc and qc and the real space parameters of the fluctuations for

effects in Fe-Ni

invar

155

magnetisation C and volume D (C is related to spin wave stiffness). Both C and D depend on cp via the relations: C = 2&p/6( m;) t2, D = 2&p/8( u’) 5’ .

(6)

The derivatives are taken at T = 0 K, and (for all calculations) 5, the correlation length, is assumed to be 1.5 A independent of temperature; k, can be determined from the Curie temperature and takes the value of about 14.5 bohr~‘. The location of the energy minimum of the system as a function of temperature can be displayed in the M-V-plane starting at M,, V,, at T = 0 K. These functions M(T), V(T) are added to the contour plot E(M,V) in fig. 1. Since qc is not known explicitly we present calculations for 4 different values of qc = 0.0, 5.0, 7.5, 10.0 bohr-‘, respectively. For qc = 0.0 (i.e., magnetic fluctuations only) the main effect is already a dramatic increase of the volume V, at T,. with respect to the Stoner volume Vs. The amplitude of volume fluctuations is controlled by qc. Enhancing qc increases the average volume V, accordingly. As a consequence the magnetostriction o,~ (table 1) drops and reaches the experimental value of 0, = 0.019 for qc z 6.0 bohrr’. It is found that volume fluctuations elevate the Curie temperature moderately, since enhancing (u’) increases the average volume. Generally magnetic systems tend to be more stable at larger volumes, so that a higher temperature is needed to destroy long range order. The value of the fluctuating magnetic moment deviates somewhat from the “classical” Moriya formula ( rn* ) r, = Mt/5 (M, = 1.6~~). This formula holds only for expansions linear in (rn*). When higher powers are included one can approximate this relation by (m’) Tc z M$7. The thermal expansion coefficient at room temperature is denoted as (car,,,k. Wassermann has estimated the magnetic part of (Y~(,,, k to be - 15 x 10-hK-l [9]. For qc = 6.0 bohr-’ about our theory yields a value of - 11 X 10-6K-‘. This value consists of a negative magnetic and a positive volume contribution, where the latter

accounts for the skewing of the energy surface towards higher volumes. Correction of the volume part would even improve the agreement.

Acknowledgement The authors wish to thank Dr. V.L.. Moruzzi for providing us with the accurate energy surface of Fe,Ni.

4. Conclusion References Our theoretical approach has combined the quantummechanical results by band theory with the treatment of fluctuations by statistical mechanics. This allows us to derive results at finite temperature so that we find a semiquantitative description of the invar behaviour. It is essential to include both spin and volume fluctuations. The prerequisite for the occurrence of an invar behaviour is a strong magneto-volume coupling which causes a specially curved energy surface. In order to find the invar effect at moderate temperatures, the energy surface must be shallow as in Fe,Ni (see fig. 1). This present theory provides a new interpretation in the form of a continuous variation of the mean value of the total energy as given by eq. (1). This contrasts the earlier two state models which assume a transition between a high-spin-high-volume and a low-spin-low-volume state. In our picture invar is not an isolated phenomenon but is just one specific manifestation of the usual interplay between magnetism. volume and temperature.

r. Moriya. Spin Fluctuations in Itinerant Electron Magnetism (Springer. lY85). K.K. Murata and S. Don&h. Phys. Rev. Lett. ?Y (1972) 285. G. Lonzarich and L. Taillefcr, J. Phys. c‘ IX (IYXS) 433. P. Mohn and E.P. Wohlfarth. J. Phys. F 17 (IYN7) 2321. I’. Entel. M. Schriiter. in: Proc. Int. Conf. on Magnctism, D Givord, cd. (Le\ iditionq de Physique. Pari\. IYXY) p. 33. A.R. Williams. V.L.. Moruzzi. J. Ktibler and K. Schwarz. Bull. Am. Phys. Sot. 24, (1984) 27X. K. Schwarz and P. Mohn, J. Phvs. F II (10X4) LIZ)). A.R. Williams, J. Ktihler and d.D. Gelntt Jr., Phys. Rev. B I’) (lY7Y) 6094. V.L. Moruzzi. P.M. Marcus. K. Schwarz and P. Mohn, Phys. Rev. B 33 (I’)%) 17X-l. D. Wagner, J. Phys. Cond. Matter. in press. E.F. Wassermann. in: “Festkbrperprohleme” 27 (Vicwcg. Braunschweig. 1987). J. Kaspar and D.R. S&hub. Phys. Rev. Lctt 47 (1481)

51. R.J. Weiss. Proc. Phys. Sot. X2 (I%?) 281. A.R. Williams, V.L. Moruzzi. C.D. Gelatt Jr., J. Kilhler and K. Schwarz, J. Appl. Phys. 53 (19X2) 7010. V.L. Moruzzi. private communication.