Lattice dynamics in ferromagnetic invar alloys

Journal of Magnetism and Magnetic Materials 10 (1979) 177-182 0 North-Holland Publishing Company

LATTICE DYNAMICS IN FERROMAGNETIC INVAR ALLOYS Y. ENDOH Department

of Physics, Tohoku University, Sendai, 980, Japan

The lattice dynamical properties of the Invar alloys Fe6sNiJs and Fe,zPtza are discussed. The experiments on the lattice vibration by inelastic neutron scattering have shown an apparent correlation of phonon anomalies with the ferromagnetic long range order. The elastic softening below the Curie temperature is attributed to the dynamical aspects, at least for the [SSO]TA1 mode. The lattice dynamical features are derived from the electronic structure of the Invar alloys. They are consistent with the theoretical model that Invar characters are closely related to the detailed band structure of the d electrons near the Fermi level. The present studies indicate significant contributions of phonon anomalies to the Invar problem.

1. Introduction martensitic transformation in these crystals [ 111. The elastic softening is recognized to be closely related to the Invar characteristics since the elastic anomaly is associated with the ferromagnetic order; moreover, it is largest in the Fe6sNi3s alloy, where the Invar character is strongest among FeNi alloys. Even though it is an acceptable microscopic mechanism of lattice softening, complete understanding is beyond our reach. Therefore lattice dynamical information bearing on the Invar properties are useful for a better understanding. There are, however, few experimental results on the lattice dynamics of these Invar alloys. The only reliable result is the INS experiment by

The modem concepts of the Invar characteristics are so widely scattered that the definition of Invar properties will be a big subject of debate in this symposium. Our opinion is that the temperature dependence of the thermal expansion, i.e., the anomalously small thermal expansion coefficient in the wide range pf the temperature, which is shown in fig. 1, is a specific Invar property. This Invar characteristic is closely related to the magnetic peculiarities, such as the large volume magnetostriction, the large pressure dependence of the Curie temperature, the enhancement of the high field susceptibility, and so forth. We recognize that the Invar problem cannot be settled unless these magnetic problems are understood completely, But it seems to us to be more significant to comprehend the unusual thermal behaviour of the lattice in Invar alloys as shown in fig. 1. We have made investigations of the dynamical features of ferromagnetic transition metals by using inelastic neutron scattering (INS) for several years [l-7]. This paper is focussed on the elastic properties of the classical Invar alloys, namely Fee5Nis5 and Fe4’t2sP which may have a close connection with the thermal behaviour of Invar. The elastic softening in these classical Invars was originally observed by ultrasonic experiments [8-lo]. This important result, however, has been interpreted in various ways; for instance, the elastic softening has been considered to be driven by the structural phase transition of the

2015-

dw

-T

-5 1 Thermal

Expansion

Kb-’

(KI

Coefficient

Fig. 1. Thermal expansion coefficient of typical Invar alloy. Fe6 s Nig 5, together with those of other 3d transition metals. Anomalous features occur in Invar; FeMn is between the two cases of Invar and Ni. 177

178

Y. Endoh

/ Lattice dynamics

Hallman and Brockhouse [12], as far as we know. They found no apparent elastic anomalies in a FeToNi3o alloy. However, the elastic constants clearly decrease below the ferromagnetic Curie temperature Tc, measured by the ultrasonic method. Their argument to reconcile the discrepancy between the two different results is the following. For the neutron scattering, thermal neutrons have much larger frequencies of above 10 ~2 Hz than the ultrasonic frequencies which are around 10 7 Hz. Furthermore, if the frequency of the phonon mode, co, is faster than the collision frequency, v, which is in turn the inverse collision time, r -~ , the wave propagation of such phonons is the coUisionless regime of 'zero sound'. On the other hand one observes sound propagation in thermal equilibrium under the condition ~or < < 1. Although the significance of thermodynamics was pointed out for the Invar problem, the introduction of two physically different regimes in this problem seems to be difficult to understand, because the idea is generally accepted in the anharmonic lattice, in which frequent collisions between quasi-particles cause noticeable differences. The characteristic Invar property is the unusually small thermal expansion below Tc and the thermal expansion anomaly becomes more pronounced at very low temperatures. Therefore we have started careful INS experiments of such alloys in order to determine the zero sound elastic constants at several temperatures above and below T c. In this paper the experimental details are not documented since the results were reported previously [ 4 - 6 ] . The remainder of this paper is devoted to discussions about the zero sound anomaly in FeNi and FePt Invars, the lattice dynamical features developed by introducing an electron-phonon coupling [13], and finally the specific relation with the Invar problem.

in ferromagnetic Invar alloys {( (0]

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Fe72 P128

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Fe65 Ni35

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.15 .20 0 .05 PNONON WAVE VECTOR (~)

10

Fig. 2. Phonon dispersion curves of [~'~0]TA1 mode in both FeTzPt28 and Fe6sNi3s in the small momentum region at several temperatures.

elastic constants were determined from the slope of the dispersion curves and direct comparisons with the first sound were made. The temperature dependence of two typical modes in Fe6sNi3s is shown in figs. 3 and 4. It is remarkable that the temperature dependence is quite different in these two modes, since .&5 .AO

r

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Tc

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/

~ .30

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~c, .25 I.Z

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~ .lO 2. Zero sound anomaly and bulk properties .05

The frequencies of the transverse acoustic phonon modes are depressed below Tc in two Invar alloys [4,5], which is consistent with the result of the elastic softening detected by the ultrasonic measurements [9,10]. The shift is most pronounced for [110]TA~ mode, as is illustrated in fig. 2. Zero-sound

,15

t

t

200

t

/.00

--

-

f

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600

T (K)

Fig. 3. Temperature dependence of the elastic constant (C 11 -C 12) in Fe 6 s Nia s. The solid curve is the ultrasonic result taken from Hausch and Warlimont [9].

Y. Endoh / Lattice dynamics in ferromagnetic lnvar alloys I

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1

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exist noticeable time effects on the spin waves at low temperatures. The same discrepancy between dynamic spin wave velocity and static velocity has been found in Invar alloys [7]. Since the increase of the negative pressure is an important subject in the Invar problem, the frequency dependent characters might also be important. The anisotropy of the crystal forces and the deviations from the Cauchy relation are enhanced at very low temperatures [ 13]. Since the electronic contribution to the lattice forces is dominant at these temperature regions in general, it is reasonable to suppose that the Invar properties are electronically driven. The elastic softening below Tc is also to be attributed to dynamical effects and this will be discussed later, following the description of lattice vibration effects in Invar.

Fe6sNi3s

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2,6

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J

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O0

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I

200

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J

300

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400

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179

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500

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600

~T(K)

Fig. 4. Temperature dependence of ~(C11 + C12 + 2C44) in Fe 6 s Ni35. The solid and broken curves are the ultrasonic results for Fe 6 s Ni3 s and Fe 49Nis 1, respectively [9 ].

the transverse modes only show elastic softening in the zero sound region. It is also noteworthy that the zero sound values for [ 110] L mode give an excellent fit to the first sound curve in Fe49Nisl [9], which has no Invar character. This fact immediately provides the important result that the elastic softening at least for the transverse mode below Tc is essential for the Invar effect. Nevertheless, as already discussed, it was not observed in the previous INS studies on FeToNi3o [12]. As for the longitudinal modes, thermodynamical considerations are necessary to reconcile the discrepant values between the two physically different sound regions. The zero sound anomaly as well as the discrepant results between zero and first sound measurements for the longitudinal modes cause significant differences for both dynamic and static bulk properties. For instance, the zero sound bulk modulus in FeNi Invar increases below Tc, contrary to the unusual decrease of the first sound bulk modulus below Tc. This new feature observed by the INS experiments may lead to future progress in this problem. In this respect, the experimental investigations on the spin dynamics from the same crystals revealed that there

3. Lattice dynamics of FeNi Invar

3.1. Phonon dispersion relations [6, 7] The phonon dispersion curves of Fe6sNi3s at room temperature (293 K) are shown in fig. 5. These curves are similar to those of FeToNi3o, except for the [ 110]TAI mode. The unusual dispersion of this mode in Fe6sNi3s with a positive curvature (d2~/dq 2 > 0) is due to the fact that the frequency shift is extended widely up to the middle of the Brillouin zone. As is seen in fig. 6, 4 neighbour force constants (12 parameters) are necessary in order to fit this anomalous dispersion to the Born-von K~rm~n model. Further investigations at several temperatures across Tc revealed that the frequency shift depends on the temperature. This temperature dependent perturbation has a close connection to the ferromagnetic long range order. We emphasize therefore that the elastic softening is attributed to the dynamical aspects.

3.2. Electron-phonon interaction in FeNi Invar The temperature dependent dispersion relation of [ 110] TA~ mode and the related elastic softening are discussed in terms of the phenomenological electronphonon interaction [ 14],

Hep = ~ tz

Igq~l QquPk ,

(1)

Y. Endoh / Lattice dynamics in ferromagnetic Invar alloys

180 ' ~ (mev)

Fe~s Ni.3s

293 K

/.,o

.

.

]

i

30

v/~

,

.

30

20

20

10

0

r'

x

i

n

. . . .

i

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Fig. 5. Dispersion relation curves along high symmetry directions in Fe6sNias at room temperature, 293 K. The curves are obtained by fitting the data to the B-vK model including 4 neighbour force constants. where gqv and Qqv represent, respectively, the coupling coefficient and the normal coordinates of a phonon with wave number q and polarization #; pg

F-e65Ni3~ 25

is the electron density. The bare phonon frequency £2qu is given by the Hamiltonian without Hep and the renormalized frequency of the Hamiltonian with Hep is defined as Wq~. The symbol/2 is dropped in the further discussions, since the mode dependence is not considered. Then COqis given as 2 _- ~ q2 - Igql2X~/, (.Oq

[~;~0] T1

where Xt~ denotes the charge susceptibility at the Fermi energy. We further postulate that the frequencies at T > Tc are taken as ~2q, although the phonons in the paramagnetic states are dressed. In order to extract the electron-phonon interaction effects, we derive the quantity Aco~, which is obtained by rewriting eq. (2),

- - 20

>-

~15 LIJ Z O Z "T' D..

(2)

o 513 K

lC

•

78 K

- -

1St ~ 3 r d n. { lst.~L.th, n.

---

(;:L~ ~:

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~o~(r) = ~ ( r ) ~ a~ ~ IgqIs x~(r). ~2q

(3)

We f'md in fig. 7 that the shift, Acorn, approaches a linear function of the order parameter of the magnetization as q + 0, which eventually gives the elastic softening below Tc mentioned in the previous section. Our analysis gave the approximate relation

x~(T) o: ~(T) =M(T)/M(O) . It is remarkable that X~ -~ 0 is proportional to mag-

(4)

Y. Endoh / Lattice dynamics in ferromagnetic lnvar alloys Temperature 100 ,__..,~ .....

-,1 -,2

-.3 I

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/./

.................

~ "~

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"

-,6

(K)

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~

[110]

a/T < 0 [ 18]. This negative thermal expansion is

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o

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the result of the large contribution of the electronic energy to the total energy, which is also primarily related to the large n(EF). We think that both anomalous phonon dispersion and negative thermal expansion are derived from the same physical origin in both metals, namely the enhanced electron-phonon interaction due to the large density of states, which contributes both to the Invar properties in Fe6sNi3s as well as to the high T c superconductivity in Nb (NbZr).

q//q Z.B

Fig. 7. Temperature shift of the phonon frequencies of the [g'I'0]TAI mode in Fe6sNias. The amount of shift is represented by (t~/s2~ - 1). The solid curve is proportional to the temperature dependence of the magnetization in the same crystal.

netization. The results show that the renormalization of phonon frequencies is spin dependent, which is quite consistent to the theory by Kim [15] : The charge susceptibility in a ferromagnetic metal can be transformed to the spin dependent charge density. We next consider the enhancement of the elect r o n - p h o n o n coupling constant, gqu- In order to discuss this we point out the significant similarity between the phonon dispersion curves from FeNi Invar and those from the high Tc superconducting metal Nb [16]. Apart from the large dips in the dispersion curves for the longitudinal modes the unusual dispersion curves having positive curvature may be noted. The effect is more pronounced in comparison with the observations on Mo [17], which is a weak superconductor located next to Nb in the periodic table. The positive curvature is considered to be due to the enhanced electron-phonon interaction as has already been discussed. The enhancement is thus general because gqu in eq. (1) depends on the density of states at the Fermi energy, n(EF). gkU o: n ( E F X I 2 ) ,

where (/2) is the matrix element averaged over the Fermi surface, and the d electron density of states is large in both metals. We also notice the fact that the thermal expansion coefficient at the lowest temperature in Nb rich NbZr alloys is negative,

(5)

4. lnvar property We conclude with a brief discussion of the specific relationship of lattice dynamics to the Invar property. The unusual elastic softening as well as the lattice dynamical anomaly particularly for the [~'~'0]TA1 mode in the classical Invar alloys are reasonably understood by the electron-phonon interaction. This interaction is enhanced by the detailed structure of the d-band in Invar alloys. The same argument that the d band has a very important role in explaining Invar property was originally made by Kanamori et al. [19], who developed a theory to relate the electron structure with the crystal volume. We have found a correlation of the negative thermal expansion at the lowest temperatures, a[T < 0,with the positive curvature of the phonon dispersion from the experiments on both Nb and FeNi. We think that the two phenomena are due to the enhanced electronphonon interaction. The Invar property may therefore appear as being due to the effect of a rather narrow range of energies near the Fermi level rather than that of the change of the total energy width of the 3d band. It is not so easy to understand the frequency dependent features of the longitudinal modes. This result also implies that the negative pressure in the ferromagnetic state includes time effects. It may be possible to assess the effects of spin fluctuations on the longitudinal modes in Invar, since the dynamical response of the magnetic excitations at low temperatures indicates the necessity of such a term. This will be discussed further by Ishikawa in this symposium [7]. We need to examine the mode dependence as well as the frequency dependence

182

Y. Endoh / Lattice dynamics in ferromagnetic Invar alloys

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metals by Ishikawa's group performed at Tohoku University, the Tokai Research Laboratory of the Japan Atomic Energy Research Institute and the Institute Laue Langevin. The author is indebted to Y. Ishikawa, M. Kohgi, Y. Noda, M. Onodera, S. Onodera, W.G. Stifling and K. Tajima for their numerous discussions and collaborations. He would particularly like to thank Y. Ishikawa for his encouragement to undertake this project. The author is grateful to J. Kanamori, and E.P. Wohlfarth for a critical reading of the manuscript. The work was partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education.

References

0

,5

1

1.5

2

2.5

~' (THz) Fig. 8. Phonon density of states in the low energy region. The solid line represents the paramagnetic density of states, while the dotted curve the ferromagnetic one at room temperature.

of the e l e c t r o n - p h o n o n interaction before reaching definite conclusions. We may also remark that the decrease of the Debye temperature can be derived from the phonon density of states. In particular, the anomalous decrease at low temperature is due to the energy shift of the transverse modes with decrease of temperature. The density of states in the low frequency region which contributes predominantly to the lattice specific heat changes with temperatures as is shown typically in fig. 8. Before closing this paper, we emphasize the significance of the contribution of phonon excitations to the lattice expansion. The dispersion of transverse modes is temperature dependent at temperatures below Tc, where the Invar properties occur.

Acknowledgements The present paper is based on the experimental investigations on the dynamics in the transition

[1] Y. Ishikawa, Physica 91B (1977) 130. [2] Y. Ishikawa, J. Appl. Phys. 49 (1978) 2130. [3] M. Kohgi, Y. Ishikawa and N. Wakabayashi, Solid State Commun. 18 (1976) 509. [4] K. Tajima, Y. Endoh, Y. Ishikawa and W.G. Stirling, Phys. Rev. Lett. 37 (1976) 519. [5] Y. Endoh, Y. Noda and Y. Idhikawa, Solid State Commun. 23 (1977) 951. [6] Y. Endoh, Y. Noda and Y. Ishikawa, in: Transition Metals 1977, eds. M.J.G. Lee, J.M. Perz and E. Fawcett (The Institute of Physics, Bristol, 1978) p. 164. [7] Y. Ishikawa, S. Onodera and K. Tajima, J. Magn. Magn. Mat. 10 (1979) 183. [8] G.A. Alers, J.R. Neighbours and H. Sato. J. Phys. Chem. Solids 13 (1960) 40. [9] G. Hausch and H. Warlimont, Acta Met. 21 (1973) 401. [10] G. Hausch, J. Phys. Soc. Japan. 37 (1974) 819. [ 11 ] M. Shimotomai, R.R. Hashiguchi and K. Karasawa, in: Proc. 1st JIM Int. Symp. New Aspects of Martensitic Transformation (Kobe, 1976) p. 129. [12] E. Hallman and B. Brockhouse, Can. J. Phys. 47 (1969) 1117. [13] Y. Endoh and Y. Noda, submitted to J. Phys. Soc. Japan. [14] D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964). [15] D.J. Kim. J. Phys. Soc. Japan 40 (1976) 1244. [16] Y. Nakagawa and A.D.B. Woods, Lattice dynamics, ed. E.F. Wallis (Pergamon Press, Oxford, 1965) p. 39. [17] A.D.B. Woods and S.H. Chen, Solid State Commun. 2 (1962) 233. [18] T.F. Smith and T.R. Finlayson, J. Phys. F6 (1976) 709. [ 19] J. Kanamori, Y. Teraoka and T. Jo, Proc. AlP Conf. 24 (1975) 16.