Magnetovolume effect and finite-temperature theory of magnetism in transition metals and alloys

Physica B 161 (1989) 143-152 North-Holland, Amsterdam

MAGNETOVOLUME EFFECT AND FINITE-TEMPERATURE THEORY OF MAGNETISM IN TRANSITION METALS AND ALLOYS Y. KAKEHASHI Department of Physics, Hokkaido

Institute of Technology,

Teine-Maeda,

Nishi-ku, Sapporo 006, Japan

A review of recent developments in the theory of magnetovolume effects based on the Liberman-Pettifor virial theorem is presented. The genera1 expression of the electronic contribution to the thermal expansion is shown to cover a wide range of magnetovolume effects from the insulator to the weak ferromagnets. It consists of the positive term proportional to the specific heat and a term proportional to the temperature derivative of the amplitude of the local moment. By using the single-site spin fluctuation theory (SSF) the Fe-Ni as well as Fe,Pt invar alloys are shown to be understood from this viewpoint. The local environment effects and the electron correlations at finite temperatures improve the difficulties in the SSF.

1. Introduction Anomalous behavior of electrons associated with magnetism is considered to be responsible for the invar anomalies [l] in transition metals and alloys. In fact the Fe-Ni alloys show zero thermal expansion at room temperatures at c* = 65 at%Fe where the ferromagnetic instability takes place. If one subtracts the normal phonon contribution (a!,) from the observed thermal expansion coefficient (a) one obtains anomalous negative contribution which shows a minimum below T,, and gradually approaches zero above T,, with increasing temperature. The spontaneous volume magnetostriction (ws) obtained from this anomalous part is estimated to be about 2% [2]. This is roughly equal to the volume change of 1.6% due to the collapse of the ferromagnetism with increasing Fe concentration at low temperatures [3]. Since the concentration dependence of the volume at low temperatures should be explained by the electronic origin, this agreement in volume change implies that the invar anomaly is mainly due to the thermal excitations of electrons. The first attempt to understand the magnetovolume effect in metals and alloys on the basis of the itinerant-electron model was made by Wohlfarth and his coworkers [4]. They adop-

ted the Stoner model as an approach from the limit of very weak itinerant magnet. The model leads to w, proportional to M* (i.e., the square of the spontaneous magnetization). The result was however known to give too large w, as compared with the experimental data of typical transition metals [5]. Apart from the theoretical consideration Shiga [6] empirically found that the experimental data of the lattice parameters in many transition metal alloys are explained by assuming the volume change proportional to the magnitude of the local moment (LM) instead of the magnetization. Schlosser [7] modified the formula so that the volume change due to magnetism is proportional to an average of the squares of the local moments ((m’)). Later we derived the ShigaSchlosser formula at the ground state as the magnetic pressure in the weak interaction limit [8]. Moriya and Usami derived the same formula at finite temperatures by means of a phenomenological spin-fluctuation theory for weak ferromagnets [9]. On the other hand the electronic contribution to the thermal expansion coefficient ((Y,) in the insulators is known to be proportional to the specific heat, which is quite different from the case of the weak ferromagnets. Since the invar alloys are considered to be

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

between the strong and weak interaction limits the anomalous behaviors of magnetovolume effects have to be understood from a unified viewpoint. Among various theories of the magnetovolume effects [ 11, the virial theorem approach [l&13] which we proposed satisfies this requirement. In this presentation we review the recent development of our theory of magnetovolume effects [10-1.51 and finite-temperature magnetism in transition metals and alloys [ 16-201. We want to show a unified picture in magnetovolume effects to the best of our abilities. In the following section we briefly explain the virial theorem approach at finite temperatures. We derive from the general expression for cyc various formulas in the weak and strong interaction limits as well as the low density limit, and clarify the origin of the invar from the general viewpoint. Since the amplitude of LM is the most important quantity in the invar problem we will also argue this quantity from the theoretical viewpoint. In section 3 WC discuss the Fe-Ni and Fc,Pt invar alloys as a prototype in the magnetovolume effects by using the single-site theory of spin fluctuations (SSF) developed by Hubbard and Hasegawa 121, 22). We will show how the anomalous magnetovolume effects are realized with increasing Fe concentration with particular emphasis of the amplitude of LM and the specific heat. The phenomenological 2y model will also be discussed there. Section 4 is concerned with the improvements of the SSF. which are indispensable for the description of the magnetovolume effects in individual metals and alloys. We will discuss first the local environment effects (LEE) in Fe-Ni alloys, and next the local electron correlation effects on the thermal expansion in Fe and Ni. Finally we summarize our conclusion and discuss the condition of the invar in the last section.

2. Virial theorem approach volume effects One of the magnetovolume

Pettifor virial theorem with the surface integral form [ IO-12.23-251. In this approach WC divide a solid into any type of cells. which arc taken usually so that each nucleus is at the center of a cell. The electronic pressure p, is then exactly expressed by three terms. the pressure given by the wave functions at the cell boundary, the pressure due to the Madelung force between cells, and the pressure due to the charge fluctuations between cells (see cq. (2.6) in ref. [ 1 I]). The last two terms can be neglected in transition metals and alloys because of the charge neutrality and strong suppression of the charge fluctuations in each cell. By making use of the atomic sphere approximation and the linear method to the radial wave function at the Wigner-Seitz sphere (see Appendix in ref. [ 111). we obtain the electronic contribution to the thermal expansion coefficient:

(1) Here the s-d charge transfer with increasing temperature has been neglected. B and V denote the bulk modulus and volume respectively, fi (Xl) is the proportionality constant given by the radial wave function at the Wigner-Seitz sphere. E,,(T) is the bonding energy defined by

(annihilation) Here c;,,,, (c,,,,,) is the creation operator for an electron with the orbital v and spin o-, l,L,,,,. is the transfer integral between (site i, orbital V) and (site j. orbital v’). The brackets () denote the thermal average. In the following we adopt the degenerate-band Hubbard model:

to the magneto-

most useful approaches to the effects is to use the Liberman-

Here U,(/,) denotes the Coulomb (exchange) energy parameter. n,(S,) is the total charge (spin) density operator on site i. Equation (1) leads to well-known formulas in

Y. Kakehashi

I Magnetism in transition metals and alloys

14.5

I

various limits because of its generality. In the insulator limit we obtain a relation E, = 2((H) - E(atoms)) by using the second order perturbation theory. Here E(atoms) denotes the energy for isolated atoms whose temperature dependence can be neglected far below the Hund’s rule coupling energy. Then eq. (1) reduces to

I

20 ,..’

...”

,..’

,....$ /..“‘&’ ......

/....’ ,._..” ,..’

,..’ . ..’

,:’ ,:’

2d ff”=FV

C V’

12 ;;

where C, is the specific heat for electrons. The “electronic” Griineisen parameter 2fi reduces to 2(21+ 1) (10 for d, and 14 for f electrons). Since the Anderson super exchange integral 9 is proportional to the square of the transfer integrals whose dependence is given by volume eq. (4) is nothing else but the wellV- 2(21+1)/3 known formula for an insulator [26]:

50

E : >4 ” ;--: Oo

cl

-0

00

, Ii

400

,120(

1200

800

T(K) Fig. 1. Thermal expansion coefficient (Y[28] and specific heat C, [29] in Ni. Dotted curve shows a - a,, (Y, being electronic contribution obtained from eq. (6) with use of the theoretical value d = 3.98 [8], experimental C, [29] and bulk modulus

The relation (5) is experimentally well-known as the 10/3 law for the linear thermal expansion in transition metal oxides [27]. There is another limit in which (Y, is proportional to the specific heat C,. It is the low density limit, in which the electron (or hole) number on a site is limited to zero or one. Then it is shown that the thermal average of the interaction term (the last term in eq. (3)) is temperature independent. The resulting thermal expansion is given by

d a, = TV

cv

(6)

Note that the proportionality constant is different from the case of the insulator by a factor of two. Figure 1 shows the experimental data of (Yand C, in Ni. If we subtract the theoretical value (6) from (Ywe obtain a smooth curve which hardly shows the anomaly around T,. The result verifies the relation (6) and no anomaly in cyr(= CY-(Y,) around T, in Ni. The forced volume magnetostriction also reduces to a simple formula in this limit (see eq.

[301.

(2.2) in ref. [13]).

aw -= ah

[ 1 b -am ah

T=O

+

aM

3BV T 3

’

(7)

where [aolah],=, denotes the value at T =O. Figure 2 again shows that the magnetovolume effects in Ni are explained as the low density limit. It is known that Sc,In [32] shows negative w, in contradiction to the prediction of the phenomenological theory for weak ferromagnets. This problem might be solved from the viewpoint of the low density limit. Equation (1) also leads to the Shiga-Schlosser formula in the weak-interaction and classical limit (see eq. (3.10) in ref. [ll]).

(8) Here mi = 2S,, and the Fermi distribution function was replaced by a step function in the derivation. If one further neglects the spin fluc-

rt

---

Y t

Fig. 2. Experimental curve.

cq. (6))

(open circles (311) and theoretical

forced volume

(solid

magnetostriction.

tuations the result reduces to that of the Stoner model. Since most of the transition metals and alloys arc in the intermediate regime between the strong and weak interaction limits. neither cq. (4) nor (8) holds true in general. We then substitute the bonding part in eq. (3) into eq. (1).

Here we neglected the temperature change of the on-site charge fluctuations because of the strong suppression of the charge fluctuations due to the Coulomb repulsion. Equation (9) shows that the cy, in the intermediate regime is determined by a competition between the specific heat term (the first term) and the local moment term (the second term). In particular the large negative CY,(the invar effect) is due to the decrease of the amplitude of LM with increasing temperature because the C,. term is always positive. Other anomalies of the invar in dwldh, B and dT,.ldP have also been shown to be caused by a large change of the amplitude of LM with the magnetic field and the volume change [12,13.33].

Although the specific heat is well-known. at least experimentally, the temperature variation of the amplitude of LM has not been well undcrstood yet in the intermediate regime. We therefore briefly discuss this problem in the following. The amplitude of a LM at the ground state is determined by the competition between the Hund’s rule coupling which tends to parallel the spins on a site and the electron hopping which destroys such an order on a site. When the former is much stronger than the latter the maximum amplitude is realized. while the minimum amplitude occurs in the opposite case. Thus the amplitude at the ground state in the intermediate regime is expected to be between the two. When the temperature is elevated the thermal excitations generally destroy the Hund’s rule coupling as well as the correlated motion of electrons. Finally, the independent motion of the electrons should be recovered at high temperatures. WC therefore expect that the lower bound of the amplitude of LM at finite temperatures is just the same as the amplitude in non-interacting system. Figure 3 shows the upper and lowe bounds of the amplitude of LM at finite tempcraturcs. Theoretical results [ 19,201 of calculations at T, with a quantum effect on the amplitude of LM are certainly between the two. Experimental

I

Fig. 3. Amplitude peratures curves

of local moments

as a function

show the upper

of

and lower

values at 7‘, are shown by 11 neutron

scatterings

[X5], Ni (36))

(V (m’))

d-electron

hounds.

119.Xl].

at finite tem-

number The

(n).

Solid

theoretical

while the data from the

are shown hv open circles (Cr [%I]. WI-C

and open triangle

(y-Fe

[3S]),

I Magnetism in transition metals and alloys

Y. Kakehashi

data [34-361 obtained by neutron scattering are far below the lower bound because of too small a range of integration with respect to the energy [37]. Recently, we have shown that the second moment of the inner core 3s spectra has a term proportional to (m*) [38]. The experimental data [39] below and above T, in cu-Fe hardly show the temperature dependence, suggesting the temperature independent amplitude of LM.

3. Magnetovolume

effect in Fe-Ni

and

Fe,Pt alloys

We discuss in this section the Fe-Ni and Fe,Pt invars as a typical example showing anomalously large magnetovolume effects, and explain the mechanism along the picture presented in the previous section. Since these alloys belong to the class with the electron-electron interactions comparable to the bandwidth, we have to rely on the interpolation theories of magnetism [40]. The obtained results are then regarded as a test of the finite-temperature theory of magnetism at the same time. Here we adopted the SSF [22]. The theory interpolates in the simplest way between the weak and strong interaction limits by using the alloy analogy and the static approximation to the functional integral method. We first show in fig. 4 the calculated C, and (m*) in various concentrations in Fe-Ni alloys. --

1

‘I

The peak of the specific heat becomes larger with increasing Fe concentration because the increasing amplitude of LM in the strong ferromagnet enhances the magnetic entropy, but gradually decreases at more than 40 at% Fe since the collapse of the strong ferromagnetism with increasing temperature decreases the magnetic entropy. This behavior is consistent with the experimental data [41,42]. A small and broad peak around T, near c* is explained as the suppression of increasing magnetic entropy with temperature. This is caused by a considerable reduction of the amplitude of Fe LM due to thermal excitations and the appearance of a feed-back effect that the surrounding effective medium acts to reduce the thermal reversal of Fe LM when the magnetization decreases with increasing temperature [ 121. On the other hand the amplitude of LM simply increases, and the reduction of the amplitude due to thermal excitations becomes stronger as the Fe concentration approaches c*. Resulting (Y, curves therefore show anomalous negative thermal expansion near c* (see fig. 5). In fig. 6 we show various contributions to the thermal expansion near c*. The result confirms the origin of the invar. A serious difficulty in the present calculations is that the theory does not describe a positive peak of ay, in Ni. We will come back to this problem in the next section. Other characteristics of the invar are qualitatively well explained by the SSF.

10

6

S

147

^m t

IE

10 4

0

1000

500

2 500

T (K)

Fig. 4. Calculated specific heats (solid curves) and tudes of local moments (dotted curves) in Fe,Ni,_,

1121.

1000 T

amplialloys

(K)

Fig. 5. Calculated electronic contribution to the linear thermal expansion coefficients (q/3) in various concentrations [lo, 121.

IO

-

7

0

Mugnetism

1

v6 _._.

._,_._

__-_-

x

‘p 0 -

1

Y. Kukehushi

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-cIc.

_________________---------

>

\

-10

” \

a” -20

-30

c

‘3 5 :

’ ,’ *’ ,,:

Lb*

10 20 0 -

40~

,’

aI

50

-7--

CL--

,000 I

2

a-a,

Fig. 6. Various contributions to q at SOat%Fe in Fe-Ni alloys 1121. -.-: the C,. term (the first term in eq. (9)); ----: the d(m’)/aT term (the second term in eq, (Y)); total a,i3. The -. .-: the s-d charge transfer term; -: inset shows the experimental data for Fe,, dNi,, h allovs [2].

In recent investigations the phenomenological 2y-model [43] which assumes the high-spin state with large volume, and the low-spin state with small volume has revived since the first-principle band calculations have shown the existence of multiple solutions for the ground state magnetization in y-Fe [44]. In what follows we show that our theory does not cause any difficulty in explaining the experimental and theoretical facts associated with the 2y-model. We first point out that the theory reproduces all the characteristics which are regarded as the evidence of the phenomenological 2y-model: (i) the existence of the ferromagnetic instability [ 121 which was interpreted as a crossing of the highspin and low-spin states in energy with increasing Fe concentration; (ii) the large difference in (Y between a-Fe and y-Fe [lo] which was regarded as an excitation from the low-spin to high-spin state in y-Fe, and (iii) persistence of the anomaly in (Y(T) above T,. (121 which was not explained by the Landau-Belov type mean-field theory [4]. Secondly, our theory reduces to the Stoner theory at the ground state in pure metals. Thus the theory can reproduce the multi-solutions in y-Fe [45] wh‘ICh was regarded as theoretical evidence for the 2y-model. We also examined the

irz trtmsirmn

mcra1.s crntl crllov.~

possibility of 2-y solutions above 7‘,. within the single-site theory assuming various densities of states, but we did not tind such solutions because the directional disorder of LM’s smooth the density of states above 7;. [II]. In the recent photoemission experiments Kisker. Wassermann and Carbone [46] measured the difference in energy distribution curves in ordered Fe,Pt at 0.6 T(. and 1.22 T<.. respectively. They found that the difference curve is cxplained considerably well by the curve obtained from the difference of the density of states between the high-spin and low-spin solutions at T =0 in y-Fe. Thus they claimed that the 2ystates are realized in the ordered Fe,Pt. However such a difference curve is again explained better by the SSF (15,471 as shown in fig. 7. Overestimates in the minima and maxima of the peaks on our calculations should be improved by taking into account the magnetic short range order above T,. which is completely neglected in the single-site theory. More detailed calculations [48] have recently shown that the magnetic short range order certainly reduces the magnitude of the difference curves by about 30%. In conclusion Fe,Pt is explained by the simple transition from the strong ferromagnetism to the paramagnetism with considerable reduction of the amplitude of LM. Recent experimental d&a

-10

-20

I t

1 6 Energy

below

EF

(eV)

Fig. 7. Difference-energy distribution curves In t;e,Pt. The experimental data (0) are taken at 500 and 270 K [46]. ----: 2y-model [46]: --: the single-site theory [ 151 with use of the density of states taken from [47].

Y. Kakehashi

I Magnetism in transition metals and alloys

of the specific heat in Fe,Pt show a small peak at T, [l]. Therefore the Fe,Pt invar should also be explained by the same mechanism as in Fe-Ni alloys: a large reduction of the amplitude of Fe LM and a small contribution of the specific heat term.

4. LEE and electron correlations

at finite

temperatures

Although the SSF explains many aspects of the finite-temperature magnetism and the magnetovolume effects, there are some difficulties inherent in the approximations. First the theory gives the first order transition at c* in Fe-Ni in contradiction to the experimental fact, thus, it does not describe various anomalies in the vicinity of c*. Second, the magnetovolume effects in the typical transition metals (Fe and Ni) are not explained by the theory based on the static approximation to the functional integral method. In the following we briefly discuss the local environment effects and electron correlations at finite temperatures which improve these difficulties. The single-site theory self-consistently treats the thermal spin fluctuations of an impurity LM in an effective medium by making use of the alloy analogy approximation (i.e., the coherent potential approximation). No fluctuation of LM’s with respect to the surrounding atomic configurations (((m) - (m))‘) is taken into account (here the upper bar denotes the configurational average). If we extend an impurity to a cluster with neighboring atoms, the central LM is determined by the atomic and magnetic configurations on the surrounding atoms. Then the fluctuations due to the local environments are described. By using the method of a distribution function we obtain the self-consistent equations for (m>, (m>2, and an effective medium. This is called the finite-temperature theory of the LEE [49], and is in particular important for the magnetic alloys near the ferromagnetic instability where the energy difference between up and down LM’s is very small.

149

When the theory is applied to Fe-Ni alloys the following points are improved [16,17], (i) rapid but continuous decrease of the magnetization near c* at low temperatures; (ii) downward deviation of the magnetization from the Brillouin curve due to the random atomic configuration, and (iii) broad internal-field distribution consistent with the Mossbauer experiments. An important concept for understanding the anomalies in close packed Fe alloys is the nonlinear magnetic couplings between Fe LM’s: Fe LM’s with large amplitude favor the ferromagnetic coupling, while Fe LM’s with small amplitude favor the anti-ferromagnetic coupling. Because of this nature Fe-Ni alloys near c* cause strong disorder effects such as the broad internal-field distribution and the spin glass, once the amplitudes of Fe LM are disturbed by the local environments, thermal excitations and pressure. Another effect missing in the SSF are the electron-correlation effects at finite temperatures. The theory uses the static approximation. This approximation reduces to the Hartree-Fock one at T = 0 K. Thus the electron correlations are not taken into account. This holds true even at finite temperatures because the energy associated with the correlated motion of electrons is much higher than the Curie and Niel temperatures in transition metals and alloys. A conventional scheme to avoid the difficulty is to introduce the reduced Coulomb and exchange energy parameters so as to reproduce the ground-state magnetization. Such a phenomenological procedure, however, generally enhances the itinerant character. In particular the electron hopping, therefore, the bonding energy E, is overestimated. Since E, is directly connected with the thermal expansion via eq. (1) the electron correlations are important for the magnetovolume effects. Figures 8 and 9 show the numerical results for cr-Fe and Ni with and without electron correlations at finite temperatures. The variational approach (VA) [18] which goes beyond the static approximation is adopted in the correlation calculations, and the single-band Hubbard model is used for brevity. A positive peak around T, in (Y,

Y. Kukehushi

-0.01

I Magnetwn

Ni

’

1

T/Tc

Fig. 8. Electronic contribution to the thermal expansion coefficients in Fe and Ni with (solid curves) and without (dotted curves) local electron-correlations [ 141.

r--

--7

itt tramitiwt

nwtu1.s uttd alloys

and negative divergence of dwidh at T(. arc obtained in Ni by taking into account the local electron correlations [14]. This is because the electron correlations suppress the two-hole states so that the eys. (6) and (7) are realized. In CY-Fe negative aC and positive dwiah are both suppressed because of the enhancement of the atomic character. The w, in the VA arc 0.015 for cu-Fe and -0.007 for Ni. while the static approximation gives 0.035 and -0.007. respectively [ 141. The reduction by a factor of two in a-Fe is explained as follows. The electron correlation effect on the volume in U-Fe results from the suppression of clcctron hopping with energy loss due to the formation of the doubly occupied state on an orbital (see eq. (I)). In the ferromagnetic spin arrangement at the ground state, the electron hopping to the neighboring site is suppressed by the Pauli principle, thus the correlation effect is less important. When the temperature is clevated and the random spin configuration takes place above T,., the number of holes with the same spin increases at the surrounding sites. Then more electron hopping and therefore more suppression of the electron hopping rate due to correlations are expected. This unbalance of correlation effects below and above T, causes the reduction of w, in a-Fe. Note that the unbalance of electron correlation effects due to the spin arrangement is not expected in Ni because of the low density of holes. Although the theoretical values of w, are comparable to the experimental data (roughly 0.005 for a-Fe and -0.003 for Ni (141). the theory seems to still overestimates w,. It might be partly attributed to too simplified a model in the theoretical calculations.

Fe

5. Summary

0

0.5

1.0

T/Tc

Fig. 9. Forced volume local electron-correlations.

magnetostriction

with

and

without

We presented our theory of magnetovolume effects on the basis of the Liberman-Pettifor virial theorem, and discussed the present status of the theoretical description based on the finitetemperature theory of magnetism in transition metals and alloys.

Y. Kakehashi

I Magnetism in transition metals and alloys

We have shown that the general expression for (Y, reproduces the well-known formulae m various limits. In the insulator and low-density limits, (Y,is proportional to the specific heat C,, while (Y, is proportional to the temperature change of the amplitude a( m2) /dT in the weak and classical limit. In the intermediate regime the CX,is understood as the competition between the C, term and a(m*) laT term. The invar anomaly occurs when a large contraction of the amplitude of LM’s with increasing temperature dominates (Y,. Experimentally this situation should be expected if the following conditions are satisfied: (i) a large ground-state magnetization which manifests a potential for large contraction of (m*), and (ii) a small peak of specific heat at T, as compared with that expected from the LM model. This implies a gradual transition from a strong to a weak magnetism due to thermal excitations. From the theoretical viewpoint the fee DOS with the Fermi level just below the sharp peak at the top of the d-band is favorable for the occurrence of the invar anomaly. In fact a sharp peak at the top of the d-band makes it easy to produce the strong ferromagnetism with a large magnetization at T = 0 K even if the exchange energy parameter J is not so large. However such a strong ferromagnetism easily collapses with the thermal excitations, pressure and magnetic field once holes are created at the top of the up spin band. This satisfies the above mentioned conditions .

The SSF qualitatively describes the anomalies of the invar. The theory is consistent with the band theory at T = 0 K, and also explains all the phenomena regarded as the evidence of the 2ystates. The finite-temperature theory of LEE describes the strong disorder effects near c* which often appear in the invar alloys. Electron correlations at finite temperatures remove the difficulty in explaining the magnetovolume effects in o-Fe and Ni. Further development of the finite-temperature theory of magnetism should be desired particularly in the low-temperature regime for the purpose of more quantitative understanding of the magnetovolume effects.

151

References [II For a review on recent developments

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Magnetism

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in transition

metals

und allob’\

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