Theory of Invar

Solid State Communications, Vol. 83, No. 9, pp. 739-743, 1992. Printed in Great Britain.

0038-1098/92 $5.00 + .00 Pergamon Press Ltd

THEORY OF INVAR V.L. Moruzzi IBM Research Division, Thomas J. Watson Research Center, P,O. Box 218, Yorktown Heights, NY 10598, USA

(Received 22 January 1992; in revisedform 17 March 1992 by R.H. Silsbee) A simple theory based on the high-spin, low-spin results of rigid-lattice band calculations for ordered FeaNi and the Debye-Griineisen approximation is presented. The theory qualitatively explains the anomalous temperature dependence of the thermal expansion, bulk modulus, magnetization, and high-field susceptibility, and the different pressure dependencies of the lattice constant at different temperatures of Invar.

INVAR, Fe65Ni35, which was discovered [1] about a hundred years ago, has a number of unusual properties including an anomalously low thermal expansion at room temperature. Although it was realized from the beginning that these properties were somehow related to magnetism, it was not until recently that a clear picture beganto emerge. Modern total-energy band calculations using a fixed-spinmoment (FSM) procedure have clearly demonstrated [2, 3] the existence of two types of solutions to the band equations. Low-spin (LS) solutions are found at low volumes, and high-spin (HS) solutions are found at high volumes. The zero-field results require the total-energy as a function of volume, E(I/), to be represented as two separate but crossing curves as shown in Fig. 1, where V = (47r/3)r ~vs and rws is the Wigner-Seitz radius. The transition from the LS to the HS solutions is first-order implying a magnetovolume instability [2]. The total-energy at the instability volume is only ,,~ I m Ry above the energy at the equilibrium or zero-pressure volume. The results near equilibrium shown in Fig. 1 clearly show that there are two solutions to the band equations, and that these two solutions correspond to two different magnetic states which have slightly different energy bands and densities-of-states. At the instability volume, t h e different HS and LS solutions are a direct consequence of an accidental degeneracy where the two states have different densities-of-states and different exchange splitting but identical total-energy. These multiple solutions are most easily found by using a constrained total moment technique such as that of the FSM procedure. Conventional spin-polarized calculations

which use a "floating moment" technique would not resolve the two solutions because, at the crossing, they have identical total-energies. A variational procedure would therefore be incapable of determining which of the two total moments is the stable solution because they are equally stable. We note that at very low volumes corresponding to rws < 2.48 a.u., the solutions to the band equations are nonmagnetic (NM). The present work does not attempt to find antiferromagnetic solutions (which requires consideration of a much larger magnetic unit cell). These band-theoretical results only apply to a rigid-lattice. At any finite temperature, lattice vibrations contribute to the free energy and lead to thermal expansion. For a "normal" metal, which we define as one which is not encumbered by the complexities implied by a magnetovolume instability, it is possible to account [4] for lattice vibrations and the resulting thermal expansion by coupling totalenergy band calculations with Debye-Griineisen theory. The derivatives of rigid-lattice total-energy binding curves at the zero-pressure volume yield a bulk modulus B (which implies a Debye temperature OD) and a Griineisen constant 7. These parameters are sufficient to define the free energy of the system as a function of volume for a given temperature. That is, the thermal evolution of the volume dependence of the free energy is completely defined. The Debye temperature, which is proportional to the square root of the bulk modulus, is a rigidlattice, zero-pressure quantity which we designate as O0. Within the approximations implied in the Debye-Griineisen theory, the volume dependence

739

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T H E O R Y OF INVAR

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Fig. 1. Zero-field rigid-lattice total energies and local magnetic moments for ordered Fe3Ni (from Ref. [3]) showing low-spin (LS) solutions at low volumes and high-spin (HS) solutions at large volumes. A critical pressure of 55 kbar is required to induce a transition from the HS to the LS branch as indicated by the common-tangent construction. of O is given by o = o0(v0/v)

where V0 is the rigid-lattice equilibrium volume. The free energy, F(T, V), is the sum of the rigid-lattice total-energy E(V), and the energy and entropy associated with the lattice vibrations. This is F(T, V) =E(V) - kBT{D(O/T ) - 3 In (1 - e - ° / r ) } + E0, where ke is the Boltzmann constant, D is the Debye function, and E0 accounts for zero-temperature lattice vibrations and is given by =

kBO.

The volume dependence of the free energy is a consequence of the volume dependence of the rigidlattice total-energy, E(V), and the volume dependence of O. Since 7 is generally greater than unity, O decreases with increasing volume. Note also that the term which accounts for the lattice vibration and entropy increases (becomes more negative) with increasing volume, in part, because the Debye function increases with decreasing O (the entropy term involving the logarithm also increases with increasing volume). As a consequence of the above, the thermal evolution of the free energy for a normal metal takes the form shown [4] in Fig. 2 (we use potassium as an example because of its large thermal expansion). The zero-point lattice vibrations appropriate for T = 0

-Iq

I 4.8

I 5.0 rws (~..)

Fig. 2. Calculated free energy curves for b.c.c. potassium (from Ref. [4]) showing the progressive shift of the minimum to larger volume and the decrease in bulk modulus with increasing temperature. lead to a positive shift in the free energy and, because O decreases with volume, a shift in the zero-pressure volume to larger values. At finite temperatures, the lattice vibrations lead to a negative shift in the free energy, with the shift becoming more prominent for large volumes. As a direct consequence, the location of the minima in the free energy curves shifts progressively to larger volumes with increasing temperature. That is, the system expands. Another consequence of the theory is a gradual decrease in bulk modulus with increasing temperature, as evidenced by the decrease in curvature at the minima (this also leads to a decrease in O with increasing temperatures). This procedure, using only atomic numbers as inputs, yields temperature dependent coefficients of thermal expansions and bulk moduli for 14 nonmagnetic metals which are in good agreement with experiment. In Fig. 3 we show [5] a proposed thermal evolution of the free energy for ordered Fe3Ni, which we consider a model Invar system. The rigidlattice total-energy shown in Fig. 1 implies a different set of thermal parameters (V0, O0 and 7) for the two branches. For Fe3Ni we also expect significant contributions to the free energy from magnetic entropy. In particular, we expect a much larger contribution for the LS branch because of the possibility of longitudinal spin fluctuations towards the HS solutions. Thus the two sets of thermal parameters and the implied magnetic entropy define two free energy branches which must exhibit different

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THEORY OF INVAR

741

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VOLUME Fig. 3. Schematic representation of the thermal evolution of the two branches of the free energy for Invar. At low temperatures, the HS high volume branch determines the ground-state. At high temperature, the LS low volume branch determines the ground-state. The behaviour shifts from HS to LS at a cross-over temperature lying between 7"3 and T4.

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Fig. 4. Schematic representation of predicted Invar properties. The temperature dependencies of the volume (a), bulk modulus (b), magnetization (c), and high-field susceptibility (d) are all expected to be determined by HS solutions at low temperatures and LS solutions at high temperatures. At low temperatufts, the pressure dependence of the volume (e) is expected to be determined by HS bchaviour at low pressures and by LS behaviour at high pressures with a break defining the critical pressure needed to induce the HS to LS transition. The critical pressure is expected to decrease with increasing temperature (f) and go to zero at the cross-over temperature.

thermal evolutions. The large longitudinal spin fluctuations associated with the LS branch are expected to lead to a crossing of the two branches. That is, the energy difference between the minima associated with the two branches must be temperature dependent and, depending upon the detailed to have two different types of bchaviour, a HS values of the thermal parameters and the magnetic behaviour at low temperatures and a LS behaviour at entropy, this energy difference can change sign. For high temperatures. Figure 4(a) is a schematic Invar, this sign change must occur near room representation of the expected bchaviour. The temperature. Thus the H S solutions define the straight line extrapolations to zero-temperature are ground-state at low temperatures and the L S solutions indicative of the rigid-lattice volumes associated with define the ground-state at high temperatures. Note that the minima of the HS and LS branches which, as the volume associated with the instability shifts shown in Fig. 1, differ by about 0.05a.u. The progressively to larger values with increasing tem- coefficient of thermal expansion, which is a derivaperature. tive of Fig. 4(a), must decrease and tend towards zero The hypothetical thermal evolution shown in Fig. at or near our cross-over temperature. As another consequence of the theory, the low3 implies interesting thermal properties which are summarized in Fig. 4. As a direct consequence of the temperature bulk modulus is determined by the HS free energy evolution, Invar should expand like a branch and is expected to exhibit normal metallic normal metal at low temperatures with the expansion behaviour with a gradual decrease with increasing completely determined by the HS solutions. At some temperature. Above the cross-over temperature, we finite temperature determined by the details of the would expect a shift to a higher bulk modulus thermal parameters, the LS solutions correspond to determined by the LS branch. At still higher the ground-state and the thermal expansion should be temperatures, we again expect normal metallic determined by the LS solutions. At the "cross-over" behaviour and a gradualdecrease in the bulk temperature there must be a pause in the thermal modulus. That is, w e would expect the temperature expansion (there may even be a contraction). Thus variation of the bulk modulus to have two different we expect the temperature dependence of the volume types of bchaviour, a HS behaviour at low tempera-

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T H E O R Y OF INVAR

tures and a LS behaviour at high temperatures as shown in Fig. 4(b). Figure 1 clearly shows [2] that the bulk modulus associated with our LS solutions is larger than that associated with our HS solutions. The straight line extrapolations to zero temperature are consistent with these rigid-lattice results. The magnetic properties are complicated by temperature dependent (transverse) spin fluctuations. However, our theory also predicts that the magnetization must be determined by HS behaviour at low temperatures and by LS behaviour at high temperatures. The theory again predicts the temperature dependence of the magnetic properties to have HS or LS behaviour depending upon whether the temperature is below or above the cross-over temperatures. This temperature dependence is shown in Fig. 4(c). The magnetization is expected to be characterized by large apparent deviations from normal Brillouin behaviour. Our theory predicts that the low-temperature magnetization should obey a HS Brillouin behaviour while the high-temperature magnetization should obey a LS Brillouin behaviour as indicated in the figure. That is, in addition to the usual transverse spin fluctuations, we expect longitudinal (HS to LS) changes in the spin. As shown in Fig. 1, the magnetic moment at the minimum of the LS branch is derived only from iron, and has an average value of ,,~ 0.4#B per atom (averaged over the four atoms in the unit cell). The value corresponding to the minimum of t h e HS branch is ,,~ 1.6#a per atom. Apparent deviations from normal Brillouin behaviour as a direct consequence of the transition from HS to LS behaviour are expected. Note that although we require t h e behaviour to be determined by our HS and LS solutions at low and high temperatures, there is only one Curie temperature for the system. The high-field susceptibility also has different characteristics in different temperature ranges. We again expect the low temperature behaviour to be determined by the HS branch and the hightemperature behaviour to be determined by the LS branch. At low temperatures the high-field susceptibility should show a gradual increase with increasing temperatures as expected for a normal magnetic system. We expect a shift to a different behaviour at our cross-over temperature as indicated in Fig. 4(d). The FSM procedure used in our band calculations yields the total energy vs. magnetic moment, E(M), at given volumes. Stable solutions correspond to local minima in the E(M) curves. At the instability volume, two local minima are observed. High-field susceptibilities are determined from the curvature at the minima. The rigid-lattice results show that the zero-

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temperature high-field susceptibility associated with the HS solutions is greater than that associated with the LS solutions is indicated by the straight line extrapolation to zero-temperature. There are a large number of direct experimental observations of the behaviour predicted in Fig. 4, The thermal expansion shown in Fig. 4(a) bears a striking resemblance to the published experimental thermal expansion data of Fe-Ni alloys of Hayase, Shiga and Nakamura [6]. Alloys near the Invar composition clearly exhibit the predicted thermal expansion. Recent determinations of the temperature dependence of the bulk modulus by Shiga et al. [7] and by Renaud and Steinemann [8] take the form shown in Fig. 4(b) except at very low temperatures where we believe antiferromagnetic solutions may occur [2]. Crangle and Hallam [9] have shown that the temperature dependence of the magnetization of Invar exhibits large deviations from Brillouin behaviour. Their experimental results can be explained on the basis of a change in the amplitude of the local moments as a function of temperature and clearly resemble Fig. 4(c). The high-field susceptibility measured by Yamada et al. [10] is reproduced in Ref. [7]. The data points can be represented by two connecting straight lines as depicted in Fig. 4(d). The thermal evolution of the two free energy branches shown in Fig. 3 implies an interesting pressure dependence of the volume (or lattice constant) for different temperatures. For low temperatures, where the ground-state is given by our HS branch, the lattice constant would decrease with increasing pressure. At some critical pressure determined by the indicated common-tangent construction, the system must undergo a transition from HS to LS behaviour. As shown in Fig. 1, our total-energy FSM electronic structure calculations yield a critical pressure of 55 kbar for ordered Fe3Ni. At higher pressures, the decrease in lattice constant is determined by the LS branch (which has a higher bulk modulus and is therefore more resistant to volume changes). The expected behaviour is shown in Fig. 4(e). With increasing temperature, the energy difference between the LS and HS minimum decreases along with the slope of the commontangent so that the pressure required to drive the system from HS to LS behaviour must also decrease. That is, with increasing temperature, the critical pressure must decrease as shown in Fig. 4(f). At and above our cross-over temperature where the groundstate is given by our LS branch, the system is already in a LS state so that the critical pressure is zero. Although the agreement between our theoretical predictions and experiment for the temperature

743 T H E O R Y OF INVAR dependence of the thermal expansion, bulk modulus, Invar data are not anomalous, and that it can be magnetization and high-field susceptibility is impress- explained on the basis of band calculations and ive, the pressure dependence of the volume (or lattice theories which account for lattice vibrations and constant) at different temperatures is even more magnetic entropy. convincing and should leave no doubt that our theory captures the essentials of Invar behaviour. Oomi and Acknowledgements - Helpful discussions with M. Acet, P. Entel, P.M, Marcus and T. Penney are Mori [11] have measured the lattice constants of gratefully acknowledged. various Fe-Ni and Fe-Pt Invar alloys as a function of pressure. They find the same general behaviour REFERENCES shown in Fig. 4(e). At low temperatures, they find a 1. C.E. Guillaume, Compt. Rend. Acad. Sci. 125, critical pressure at which the behaviour changes. 235 (1897). Although they interpret the critical pressure as a 2. V.L. Moruzzi, Physica B161, 99 (1989). pressure needed to induce a transition from ferro3. V.L. Moruzzi, Phys. Rev. 1141, 6939 (1990). magnetic to paramagnetic behaviour, our theory 4. V.L. Moruzzi, J.F. Janak & K. Schwarz, Phys. Rev. B37, 790 (1988). dearly identifies this as the pressure needed to 5. First-principles calculations of the free energy induce the HS to LS transition. In agreement with for Invar are currently intractable. our theory, the critical pressure goes to zero near 6. M. Hayase, M. Shiga & Y. Nakamura, J. Phys. room temperature or at our estimated cross-over Soc. Jpn 34, 925 (1973). temperature. 7. M. Shiga, K. Makita, K. Uematsu, Y. Muraoka In summary, we have proposed that the multiple & Y. Nakamura, J. Phys. Condens. Matter 2, magnetic states defined by different but near-lying 1239 (1990). 8. Ph. Renaud & S.G. Steinemann, Physica B161, E(V) curves, and a thermally excited transition from 75 (1989). the lower to the upper state, yield a qualitative 9. J. Crangle & G.C. Hallam, Proc. Phys, Soc. understanding of Invar behaviour. We suggest a (London) A272, 119 (1963). plausible thermal evolution of the free energy based 10. O. Yamada, F. Ono & I. Nakai, Physica 86on first-principle electronic structure calculations, a 88B, 311 (1977); O. Yamada, F. Ono, I. Nakai, Debye-Griineisen treatment, and magnetic entropy H. Maruyama, K. Ohta & M. Suzuki, J. Magn. changes which qualitatively explains a large body of Magn. Mater. 31-34, 105 (1983). experimental Invar data which has previously been 11. G. Oomi & N. Mori, J. Phys. Soc. Jpn 50, 2917 (1981~. difficult to understand. Our theorv shows that these Vol. 83, No. 9

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