Curie temperature of RE2Fe17 compounds — A Friedel model approach

Solid State Communications,

106, No. 6. pp. 379-383, 1998 0 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038- IO98/98 $ I9.00+.00

Pergamon

Vol.

PII: SOO38- 1098(98)00049-0

CURIE TEMPERATURE

OF RE2Fe l7 COMPOUNDS

- A FRIEDEL

MODEL APPROACH

K.G. Suresh, S.D. Mahanti? and K.V.S. Rama Rae” Magnetism

and Magnetic (Received

Materials Laboratory, Department of Physics, Indian Institute of Technology, Madras-600 036, India 28 November

1997; accepted

5 January

1998 by C.N.R. Rao)

The variations of Curie temperature (T,-) of RE2Fe17 compounds upon substitutional and interstitial modifications are qualitatively explained using a simple phenomenological equation Tc = C&ff(Md). Here peff is the effective magnetic moment/atom and f(Md) is a monotonically increasing function of (xl& where h is the Friedel length scale and d is the distance between nearest magnetic atoms. The Tc variation of substituted compounds can be understood by incorporating the effects of magnetic dilution and charge transfer whereas, in interstitially modified compounds the Tc variation can be explained primarily by incorporating the effects of band narrowing. Also, the effect of pressure on the Tc is understood by incorporating band broadening effects in the Friedel model. 0 1998 Elsevier Science Ltd. All rights reserved

RE2Fe17 systems form one of the major series of rare earth (RE)-transition metal (TM) intermetallic compounds, which find extensive applications in the field of permanent magnets [l]. These compounds exhibit anomalous behaviour in their Curie temperatures (Tc) under substitution, interstitial modification and application of pressure, compared with other RE-TM intermetallic compounds. The value of Tc of these compounds are generally low compared with other RE-TM intermetallic compounds and they are around room temperature [ 11. The main reason usually attributed to the low values of Tc of these iron-rich compounds is that some of the Fe-Fe distances are smaller than the critical distance 2.45 A needed for ferromagnetic exchange [2], leading to antiferromagnetic coupling between those Fe-Fe pairs. The resulting competition gives rise to low values of Tc. It has been recently reported [3-61 that there is a drastic increase in Tc upon interstitial modification using nitrogen (N), carbon (C) or hydrogen (H). On the substitution modified compounds, several reports are

* Author to whom all correspondence should be addressed. ’ On leave from the Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan48824, U.S.A. 379

available in literature which describe the effect of nonmagnetic elements like Al, Ga and Si as well as the magnetic elements like Co and Ni on the magnetic properties of these compounds [7-121. In all these substituted compounds, there is an initial increase in Tc with the concentration of substitutional elements followed by a decrease excepting in the Co-substituted compounds, where Tc increases monotonically for all Co concentrations. The maximum increase in Tc in the substituted compounds is around 200 K. This increase is puzzling considering the fact that the substituted atoms dilute the magnetism of these compounds. Furthermore, the increase in the density of sp-electrons (in the case of Al, Ga and Si) leads to an increase in the screening of the d-d exchange. Thus these two factors should bring about a reduction in Tc values with increase in the concentration of non-magnetic atoms, in contrast to what is observed. A first principle theoretical analysis of Tc in these rather complex systems is difficult at the present time. In this communication, we point out how the variation of Tc in different RE2Fe17 compounds can be understood qualitatively using a simple model proposed by Friedel et al. [13], which takes into account spatial fluctuations of the spin density. Although the magnetic properties of transition metals and their compounds at low temperatures (in particular the magnetic moment) are well described by spin-polarized band theory based on density functional formalism,

Tc OF REzFei7 COMPOUNDS

380

- A FRIEDEL

theories for Tc are not so well developed, pa~icularly for the alloy systems. The earliest theory for Tc was given by Stoner [I41 and Wohlfarth [l5]. The fundamental difficulty with the Stoner-Wohlfarth (SW) theory [14, 151, which is a straightforward extension of the band theory to finite temperatures is that it fails to take into account the correct low energy excitations properly. The limitation of the SW theory was realized in the seventies in many pioneering papers by Moriya and Takahashi [ 16], Hubbard [ 171, Hasegawa [ 181, Korenman et al. [19]. In recent years, several groups have taken proper account of these low energy excitations using mode coupling theories, where parameters of the Hamiltonian are obtained from first principle density functional calculations [20]. However, these methods have been applied only to elemental metals such as Fe, Co and Ni and cannot be easily extended to ternary and quarternary compounds. A simpler theory within the purview of the band theory, but which takes into account the effects of spin fluctuations on Tc was given by Mohn and Wohlfarth (MW) [21]. Recently, Jaswal et al. [22] have used this theory to explain the observed changes in the Tc in RE2Fe17N, compounds. According to this theory, the Tc is given by

M2

Tc a -

x0

where M is the magnetization x0 is given by

(1) at 0 K and the susceptibility,

In equation (2) ~~(~~) [o = l,l] are the densities of states at the Fermi level with two spin orientations and I is the Stoner exchange parameter. According to the calculations of Jaswal er al. [22], the unit cell expansion after nitrogenation causes a reduction in the value of x0 and combined with an increase in M leads to an increase in Tr. If one attempts to extend the MW spin fluctuation theory to the case of substituted compounds in order to explain the observed initial increase in T, one has to assume a drastic reduction in the densities of states at the Fermi level, both for Iarge and small atom substitutions. This is because of the reduction in M under substitution and therefore, according to equation (l), a large reduction in ~0 is needed to give the observed increase in Tc. Calculation of M and ~0 for substitutional compounds, which are needed in MW theory to obtain the Tc is difficult. Therefore, in order to understand the Tc variation in these systems, we suggest a simple unified phenomenological approach based on the model

MODEL APPROACH

Vol. 106, No. 6

proposed by Friedel er al. [13]. In contrast to the SW model, the Friedel model incorporates the effect of spatial fluctuations of magnetic moments on Tc. The basic ideas underlying both these models are the same, i.e. the main ingredient is the finite energy difference (AE) between the paramagnetic and the ordered ferromagnetic state. If it is energetically favorable, the system goes to the ordered state at T = 0 K. The transition temperature in SW model is proportional to AE and the ordered moment vanishes uniformly above Tc caused by spin-flip excitations to the Stoner continuum. The deficiency of the Stoner model (it gives extremely high values for T,) is that it is concerned only with the energy fluctuations of the magnetic electrons and ignores the spatial fluctuations of the spin density, both in magnitude and direction. Friedel model which is discussed next, removes this shortcoming. Friedel model, introduces both an energy scale (same as the Stoner parameter y defined below) and a length scale, h. According to this model, the itinerant 3d electrons are initially non-magnetic and one introduces local spin fluctuations (LSFs) associated with these electrons, each LSF extending over a finite region of space. Because of the LSF, a spin imbalance is created say, near site A (Fig. I), which in turn gives rise to an exchange potential acting on other electrons in the neighbouring region. The condition for this spontaneous polarization is identical to the Stoner criterion viz. y = IN(E,G) 7 1, [I is the Stoner parameter and N(Er) is the total density of states at the Fermi level]. Thus, y > 1 is the criterion for the formation of a LSF or local moment in the Friedel model. This is also the criterion for the stability of the ferromagnetic state in the SW model. This LSF lives long enough, polarizing neigh~uring region and the process spreads through the entire lattice, leading to a self-consistent polarization at every

Fig. 1. Schematic representation of the Friedel oscillations of the spin density in transition metals.

Tc OF REzFe i7 COMPOUNDS

Vol. 106, No. 6

- A FRIEDEL

site. But the interesting point is that when the exchange potential polarizes the neighbouring region, it attracts only those spins, which are in the same direction as the LSF and repels the spins having the opposite direction. Thus there will be a spatially non-uniform spin density distribution, as shown in Fig. 1. This implies that, not only there is a redistribution of spins with respect to energy, but also with respect to space. These spatial changes in the spin density as a characteristic features of the Friedel model and are characterized by a length scale X as shown in Fig. 1. The polarization of the neighbourhood critically depends on the ratio (A/d), where d is the distance between the neighbouring magnetic atoms. When A/d 2 1, there is strong ferromagnetic coupling between these two LSFs and it has been shown by Friedel et al. [ 131 that X is inversely proportional to the d-band Fermi wave vector (Ak) of electrons or holes, according to whether 3d band is less than or more than half-filled. Based on the above picture, it is clear that the Tc critically depends on the values of y and Nd, the former determining the strength of LSF and the ordered moment. In order to isolate the effects of magnetic moment and the exchange coupling (through xld) on the variation in T,-, we have plotted a graph between Tc/p& and (xld) for Fe, Co and Ni, as shown in Fig. 2. Here pcff is the effective magnetic moment per transition metal atom, measured from the paramagnetic susceptibility. The values of ()\ld) have been obtained from [ 131. From the linear nature of the graph, we propose an empirical relation between Tc, peff and (xld) as Tc = C/.&&X, d),

(3)

where, C is the constant of proportionality having the units K/pi and _/Q/d) is an increasing linear function of (xl& From the equation (3) we see that a small variation in peff can largely influence the Curie temperature because of the quadratic dependence.

MODEL APPROACH

381

In the following, we use equation (3) to explain the behavior of Tc in RE2Fe I, compounds upon substitutions and interstitial modifications. In addition, we also explain the reduction in Tc with hydrostatic pressure. 1. EFFECT OF SUBSTITUTIONS TEMPERATURE

ON THE CURIE

Non-magnetic elements like Al, Ga and Si can substitute for Fe in RE*Fe 17compounds up to a maximum concentration of 10 per formula unit [7-lo]. In all these cases, there is an initial increase in Tc of about 200 K and a monotonic decrease at higher concentrations. However, this increase is accompanied by a reduction in the Fe magnetic moment [23]. RE*Fe i7 compounds are classified as weak ferromagnets because of the fact that both the 3d sub-bands are incompletely filled. Also, since the band is more than half-filled, it has a hole character. The reduction in the Fe magnetic moment is because of the transfer of valence electrons of the substituted atom to the 3d band of Fe [23, 241. Consequently, the 3d band gets progressively filled up and hence the Fermi level goes up. Using a simple one dimensional E vs k diagram, it can be shown that the 3d band Fermi wave vector, Ak (ksz - kF) decreases as the concentration of the substituted atom increases, leading to an increase in h. In Fig. 3, we give the variation of the lattice parameters a and c in (Er,5Pr0.5)zFeI,-xAlx. It can be seen that the values of a and c almost remain a constant upto a value of x = 2 and increases afterwards. Since d is proportional to a and c, the ratio (x/d) initially increases and according to equation (3), the Tc increases. However, there is a reduction in the Fe magnetic moment with increase of Al concentration [23]. This along with the

250

200

$1

-2

150

IO0

-

Fe ,I

8.50 1.0

1.5

2.0

2.5

3.0

0

/ z 2

4

6

8

x

h/d

Fig. 2. Plot of TC+zff vs (u/6) for Fe, Co and Ni.

Fig. 3. Lattice parameter variation in (Ero.sPro.s)IFe I,-xAl.t compounds as a function of Al concentration.

382

Tc OF RE2Fe {, COMPOUNDS - A FRIEDEL

magnetic dilution effect reduces peff. This is the reason for the relatively smaller increase (about 150 K) in these cases, compared with the larger increase of about 400 K seen in the interstitially modified compounds [3-61. Identical situation exists in the case of Ga-substituted compounds [9]. Shen et al. [lo] have studied the effect of Si substitution in RE2Fe1, compounds and found that there is a reduction in the Fe magnetic moment, which was attributed to the charge transfer from the valence band of Si to the 3d band of Fe. The lattice parameters are found to decrease with increase in Si concentration. Hence the increase in the Tc for small Si concentration is again a consequence of the increase in X and a decrease in d values. Similar to the case of Al or Ga, there is a reduction in ,uerf with Si, which contribute to the reduction in Tc at higher Si concentrations, according to the equation (3). Substitutions of Fe by Co and Ni [ 11, 121 in RE?Fe ,, compounds have also been found to increase Tc. In the case of Co substituted compounds, there is a monotonic increase in Tc upto about 1100 K over the full concentration range. There is however a reduction in the 3d magnetic moment with increase of Co concentration [ 11J. Co substitution also brings about a reduction in the lattice parameters like Si and hence there will be an increase in (xl&. But the difference between Co and Si substituted compounds is that in the former case, the reduction in the magnetic moment is much less compared to the latter because of the magnetic moment associated with Co. This coupled with the quadratic dependence of Tc on perr can dramatically increase Tc in Co systems. Substitution with Ni brings about only a small initial increase in Tc values [12], exactly similar to the nonmagnetic elements. Since the 3d band in Ni compounds are more nearly filled compared to Fe and Co, a small amount of charge transfer wit1 result in the complete filling of the Ni 3d band. In this respect, Ni is similar to the non-magnetic elements.

MODEL APPROACH

Vol. 106, No. 6

can be altered by two effects. The first is the effect of 3d-band narrowing caused by the lattice expansion; there is an increase in the Fe magnetic moment 123, 241. The second effect is the charge transfer from the interstitial atom to the Fe d-orbitals, which tend to reduce the Fe moment. Our Mossbauer studies show an increase in the hyperfine field at the Fe site, indicating that the band narrowing effect is dominant, thereby increasing the Fe effective moment [23]. Band structure calculations carried out by Uebele et al. 1251 and Beuerie and Fahnle 1261 have also confirmed that the 3d band narrowing effect is dominant over the charge transfer effect. To see the effect of interstitial modification on the ratio (x/d), we have used a simple tight-binding model to find out how this ratio changes with lattice expansion. As long as the number of electron in the d-band remains constant, this ratio does not change. However, there will be an increase in o\/d) due to the small charge transfer from the interstitial atoms to the Fe 3d band. This increase in (xl&) in conjunction with an increase in ~~rr can explain the observed large increase in Tc following equation (3). 3. EFFECT OF PRESSURE ON THE CURIE TEMPERATURE Studies on the effect of hydrostatic pressure on the jr, of RE?Fe 17compounds have revealed [27] that there is a monotonic reduction in T, with pressure. as shown in Fig. 4. Application of pressure alters the band structure of these compounds, as a consequence of the unit cell contraction. However, the effect of pressure is exactly opposite to that of interstitial occupancy of N, C and H. Because of the contraction of the unit cell, width of the

2. EFFECT OF INTERSTITIAL ATOMS ON THE CURIE TEMPERATURE As has been discussed at the beginning of this note, RE2Fe17 compounds absorb elements like N, C and H, which occupy the interstitial sites. There is an increase in both the lattice parameters and Tc, with increasing concentration of these atoms [4]. It should be pointed out that in these systems the increase in Tc is quite high, reaching a maximum of about 425 K for RE2Fei7N2,,. The maximum increase in Tc in the Fe-substituted compounds is about 200 K. To understand the large increase in Tc in the interstitially modified compounds we look at p,rr and (xld) separately. in these systems, Feff

Pressure (k bar)

Fig. 4. Variation of Tc as a function pressure in RE2Fe r7 compounds.

of hydrostatic

Vol. 106, No. 6

Tc OF RE*Fe I, COMPOUNDS

- A FRIEDEL

3d band increases (band broadening) and the 3d magnetic moment decreases [28]. As a result of band broadening, the density of states near the Fermi surface decreases and this will lead to transfer of charges from d-band to the sp-band. As a result, the ratio (x/d) may decrease. This reduction in (hid), accompanied by the reduction in the ordered magnetic moment explains the decrease in Tc under application of pressure, according to the equation (3). In summary, we have shown that it is possible to understand the behaviour of Curie temperature in RE*Fe ,, systems with substitutional and interstitial modifications, using a simple empirical relation involving perr and (xld), which characterize the Friedel model. It is also possible to understand the pressure dependence of Tc using the same relation. Ac~~~w~edge~e~f~-One of the authors (K.G.S.) is grateful to the Indian Institute of Technology, Madras for financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

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