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Phonon mechanism of the anomalous volume behavior of ferromagnetic metals
ELSEVIER
Physica B 219&220 (1996) 118 120
Phonon mechanism of the anomalous volume behavior of ferromagnetic metals N. Miyai, T. Kizaki, D.J. Kim* Physics Department, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo, 157, Japan
Abstract We explore the role of phonons in the volume behavior of an itinerant electron ferromagnet by taking into account the close relation between phonon and magnetic properties. From a model numerical calculation we find that phonon contribution to thermal expansion can be negative in the ferromagnetic state of a metal.
1. Introduction It is well known that in many metallic ferromagnets which are called Invars, such as F e - N i alloys, thermal expansion coeffÉcients become either very small or negative. As to the mechanism of such behaviors, however, although various mechanisms have been proposed, it is still very controversial [1]. The purpose of this paper is to show that a breakthrough might be obtained by considering the effects of the electron-phonon interaction. Concerning the role of phonons in the volume properties of an itinerant electron ferromagnet, it has been generally assumed that phonon contribution to thermal expansion is entirely independent of its magnetic properties and always positive. Contrary to such a prevailing view, from obtaining a negative phonon Grfineisen constant, ~'D,in a model calculation for a ferromagnetic metal at T = 0, we recently raised the possibility of phonon contribution to thermal expansion coefficient, fl, becoming negative [2]. The physical origin of such phonon behavior is that phonon frequency changes with magnetization since the screening of the ion-ion interaction changes with the spin splitting of the conduction electron
* Corresponding author.
energy bands [3]. Quite interestingly, very recently Mafiosa et al. found in Fe3-Pt alloys that in the temperature region where fl is negative, 70 also is negative [4]. In this paper we carry out a model calculation on the temperature dependence of fl for an itinerant electron ferromagnet by fully including such a hitherto neglected property of phonons. Thus, we reveal a new aspect of phonon effects on the volume properties of a ferromagnetic metal.
2. Thermal expansion coefficient of a ferromagnetic metal In order to calculate correctly fl of a ferromagnetic metal, particularly, that for below the Curie temperature, T o we need to use the correct thermodynamical expression for ft. From a Landau theoretic procedure it is obtained [5] as fl = Kapl/aT + K m a H x / a T
= fll -[- f12,
(1)
where rc is the compressibility, Xm is the magnetic expansivity, which is defined in terms of the change in volume caused by a magnetic field, H, as A V / V = gmH, and Pl = - Fv and Ht = - FM are, respectively, the internal pressure and magnetic field, where F is the free energy
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 6 6 9 - 9
N. Miyai et al. / Ph_ysicaB 219&220
of the system, and Fv and FM are, respectively, its derivatives with respect to volume and magnetization. Here, K and K, are given in terms of various differential coefficients of the free energy as F
1
J(-
V F,,r K,
=
---,
MM
FMM - F&s
Xhf FM, V
(3)
Fvv
with the high field magnetic %hf
=
(2)
’
F,,!I~,,
FM,
-
susceptibility,
Fiwl .
state, with 52,, the ionic plasma frequency, and we put Qz - sZ$ =
(4)
where pph = - aFph/aV and Hph = - a Fph/i3M are, respectively, the internal pressure and magnetic field due to phonons. BeI, Pell, and fleL2are similarly defined. F C1.Scan be calculated straightforwardly for a given form of the electronic density of states. Fph also can be calculated straightforwardly if the phonon frequency spectrum coph(q)is given. In this paper we use the jellium-like model in which we have only the longitudinal acoustic phonons. With the Debye approximation the frequency of such a phonon is given as wph (q) = sq, with the sound velocity s. If we treat the screening by the standard mean field approximation including the effects of the exchange interaction between electrons, P, and the spin splitting of the bands [3] we have =
< + 2N(O)/[F+(O)
+ F_(O)])
where e,(O) = F,,(O)/[l - PF,(O)], with F,,(O) = - Jda N(s) aji(a)/dc: z N,(O), wheref,(a) is the Fermi distribution of D spin electrons, N(E) is the electronic density of states per spin in the spin unsplit state, and N, (0) is that split states, at the Fermi surface in the spin s0 = 52,,,‘(8 rte’N(O)/V) li2 is the Bohm-Staver sound velocity of the pure jellium model in the paramagnetic
aw,
v IjD
In Eq. (1) fli represents the ordinary mechanism of thermal expansion produced by the temperature variation of the internal pressure. /I2 represents that caused by the temperature variation of the internal magnetic field. Note, however, that magnetic effect is present also in pi; both K and i?pi/aT sensitively reflect the magnetic property of a system. For T > Tc, B2 = 0, since F,, = 0. If the F of a system is given we can directly calculate from Eq. (1). Here we use p of the system where F(T, V, M) = F,Ls(T, V, M) + Fpt,(T, V, M), F,,,, is the mean field approximation (the Stoner model) contribution of electrons and Fph is that of phonons. /J then, may be divided into the contributions of the Stoner and phonon excitations: fl = bei + aph, where we abbreviated fie,,s by fi,,. From Eq. (1) we have
119
(1996) 118~120
=
-w,
ao,dM
T@+Gz
=701 +:ID2. [
1 (7)
Then if we note the relation K,/(dM/dV) find Eq. (5) to be rewritten as &h
with P(T/oD)
=
(NkBoDK/v)bDl
the
familiar =
+
function
(1 /Nw$h/awD
=
K
?DZiamoD)iac
in
the
[S], we
63)
Debye
model,
[31.
If jph dominates over Be,, we would have p < 0 for yD < 0. The result of Manosa et al. can be understood in this way.
3. A model numerical calculation and conclusion In our numerical calculation on F,,,, and Fphr and then M(T), and p, etc. we use the model electronic density of states, N(E) = 30(N/W)[ l/4 - (E/W)‘] with the bandwidth W. This band can accommodate up to 5 electrons per spin per atom. We assume the following volume dependencies for narrow range of V around some reference volume V,: V(T),
w = w,(V/v”)mo, < = So(V/VJC,
B = B,(v/Vo)yb,
(9)
with positive constants, a, h, and c. In the following numerical example, we fix the values of various parameters as follows: W. = 6 eV, M, = IO5 m,,, MI and m, being, respectively, the masses of an ion and an electron, to = 1.2, u = 1, h = l/2, and c = 3. We then look into how the results would depend on the values of or/ W,, + being the Fermi energy in the spin unsplit state. In Fig. 1 we show the result for the case of apI W. = 0.64 (with NIV,, = 4.60057 x 102’/cm3 and PN(0)lv=v, = 1.07052). The model electronic density of states, with the occupied region shaded, is shown in panel (a) of the figure. We note in Fig. l(a) that whereas flph dominates over bei at not too low temperatures, the behavior of aph for T < T, is not normal at all; /I becomes negative by the effect of phonons. The behavior
120
N. Miyai et aL /Physica B 119&220 (1996) 118 120
21(a)~'''
~
1
........
4
(if ....... r~., ..........
I'" '
. . . .
/---'"1
i
J
l /
i
i
0 6 [-[(c) .
i
i
400 .
.
800 .
I/ T(K)
.
0
i
400
2 I-l(d)' . . . . . '
i
i/
800 '
'
T(K) '
'
................1 -2 0
400
800
i
-2 ~
"~'i......... i'""
0
400
I
I 800
I T(K)
Fig. 1. The temperature dependences of fl and its various components as defined in the text and ?'D defined in Eq. (7) for the case of eV/W0 = 0.64 in the model electronic density of states shown in (a), with the occupied region shaded.
of flph is analyzed in panels (b) and (d). Certainly, it is ~'D2 that is responsible for making 70 and, then, flph and fl negative. Our result can explain the recent concurrent observation of negative thermal expansion and negative phonon Griineisen constant [4]. In panel (c) we present the behavior of fla. Although /Jel2 certainly becomes negative as usually anticipated, it is overcompensated by the positive flea1. Note also that the size of flo~is generally smaller than that of fiph by one order of magnitude except near T = 0.
The behavior of fl and its various components depend very sensitively, upon the location of Sv in a given N(e) [6]. In the present model, for e,v/W = 0.71, slightly different from the above case of Fig. 1, we have flph > 0 and fl > 0 [7]. These results in this paper strongly suggest the necessity of fundamentally reconsidering the role of phonons in the volume behavior of a metallic ferromagnet. This work is partially supported by the Research Institute of Aoyama G a k u i n University.
References [1] For a recent review and references, see E.F. Wassermann, in: Ferromagnetic Materials, Vol. 5, eds. E.P. Wohlfarth and K.H.J. Buschow (North-Holland, Amsterdam, 1990) p. 237. [2] D.J. Kim, Phys. Rev. B 39 (1989) 6844. [3] D.J. Kim, Phys. Rev. B 25 (1982) 6919; Phys. Rep. 171 (1988) 129. [4] LI. Mafiosa, G.A. Saunders, H. Radhi, U. Kawald, J. Pelzl and H. Bach, J. Phys.: Condens. Matter 3 (1991) 2273; G.A. Saunders, H.B. Senin, H.A.A. Sedek and J. Pelzl, Phys. Rev. B 48 (1993) 15801. [5] D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 239. [6] T. Kizaki, N. Miyai, N. Hino, H. Saita, N. Tateiwa, H. Ohta, H. Ito, J. Iwai and D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 245. [7] N. Miyai and D.J. Kim (unpublished).
Physica B 219&220 (1996) 118 120
Phonon mechanism of the anomalous volume behavior of ferromagnetic metals N. Miyai, T. Kizaki, D.J. Kim* Physics Department, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo, 157, Japan
Abstract We explore the role of phonons in the volume behavior of an itinerant electron ferromagnet by taking into account the close relation between phonon and magnetic properties. From a model numerical calculation we find that phonon contribution to thermal expansion can be negative in the ferromagnetic state of a metal.
1. Introduction It is well known that in many metallic ferromagnets which are called Invars, such as F e - N i alloys, thermal expansion coeffÉcients become either very small or negative. As to the mechanism of such behaviors, however, although various mechanisms have been proposed, it is still very controversial [1]. The purpose of this paper is to show that a breakthrough might be obtained by considering the effects of the electron-phonon interaction. Concerning the role of phonons in the volume properties of an itinerant electron ferromagnet, it has been generally assumed that phonon contribution to thermal expansion is entirely independent of its magnetic properties and always positive. Contrary to such a prevailing view, from obtaining a negative phonon Grfineisen constant, ~'D,in a model calculation for a ferromagnetic metal at T = 0, we recently raised the possibility of phonon contribution to thermal expansion coefficient, fl, becoming negative [2]. The physical origin of such phonon behavior is that phonon frequency changes with magnetization since the screening of the ion-ion interaction changes with the spin splitting of the conduction electron
* Corresponding author.
energy bands [3]. Quite interestingly, very recently Mafiosa et al. found in Fe3-Pt alloys that in the temperature region where fl is negative, 70 also is negative [4]. In this paper we carry out a model calculation on the temperature dependence of fl for an itinerant electron ferromagnet by fully including such a hitherto neglected property of phonons. Thus, we reveal a new aspect of phonon effects on the volume properties of a ferromagnetic metal.
2. Thermal expansion coefficient of a ferromagnetic metal In order to calculate correctly fl of a ferromagnetic metal, particularly, that for below the Curie temperature, T o we need to use the correct thermodynamical expression for ft. From a Landau theoretic procedure it is obtained [5] as fl = Kapl/aT + K m a H x / a T
= fll -[- f12,
(1)
where rc is the compressibility, Xm is the magnetic expansivity, which is defined in terms of the change in volume caused by a magnetic field, H, as A V / V = gmH, and Pl = - Fv and Ht = - FM are, respectively, the internal pressure and magnetic field, where F is the free energy
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 6 6 9 - 9
N. Miyai et al. / Ph_ysicaB 219&220
of the system, and Fv and FM are, respectively, its derivatives with respect to volume and magnetization. Here, K and K, are given in terms of various differential coefficients of the free energy as F
1
J(-
V F,,r K,
=
---,
MM
FMM - F&s
Xhf FM, V
(3)
Fvv
with the high field magnetic %hf
=
(2)
’
F,,!I~,,
FM,
-
susceptibility,
Fiwl .
state, with 52,, the ionic plasma frequency, and we put Qz - sZ$ =
(4)
where pph = - aFph/aV and Hph = - a Fph/i3M are, respectively, the internal pressure and magnetic field due to phonons. BeI, Pell, and fleL2are similarly defined. F C1.Scan be calculated straightforwardly for a given form of the electronic density of states. Fph also can be calculated straightforwardly if the phonon frequency spectrum coph(q)is given. In this paper we use the jellium-like model in which we have only the longitudinal acoustic phonons. With the Debye approximation the frequency of such a phonon is given as wph (q) = sq, with the sound velocity s. If we treat the screening by the standard mean field approximation including the effects of the exchange interaction between electrons, P, and the spin splitting of the bands [3] we have =
< + 2N(O)/[F+(O)
+ F_(O)])
where e,(O) = F,,(O)/[l - PF,(O)], with F,,(O) = - Jda N(s) aji(a)/dc: z N,(O), wheref,(a) is the Fermi distribution of D spin electrons, N(E) is the electronic density of states per spin in the spin unsplit state, and N, (0) is that split states, at the Fermi surface in the spin s0 = 52,,,‘(8 rte’N(O)/V) li2 is the Bohm-Staver sound velocity of the pure jellium model in the paramagnetic
aw,
v IjD
In Eq. (1) fli represents the ordinary mechanism of thermal expansion produced by the temperature variation of the internal pressure. /I2 represents that caused by the temperature variation of the internal magnetic field. Note, however, that magnetic effect is present also in pi; both K and i?pi/aT sensitively reflect the magnetic property of a system. For T > Tc, B2 = 0, since F,, = 0. If the F of a system is given we can directly calculate from Eq. (1). Here we use p of the system where F(T, V, M) = F,Ls(T, V, M) + Fpt,(T, V, M), F,,,, is the mean field approximation (the Stoner model) contribution of electrons and Fph is that of phonons. /J then, may be divided into the contributions of the Stoner and phonon excitations: fl = bei + aph, where we abbreviated fie,,s by fi,,. From Eq. (1) we have
119
(1996) 118~120
=
-w,
ao,dM
T@+Gz
=701 +:ID2. [
1 (7)
Then if we note the relation K,/(dM/dV) find Eq. (5) to be rewritten as &h
with P(T/oD)
=
(NkBoDK/v)bDl
the
familiar =
+
function
(1 /Nw$h/awD
=
K
?DZiamoD)iac
in
the
[S], we
63)
Debye
model,
[31.
If jph dominates over Be,, we would have p < 0 for yD < 0. The result of Manosa et al. can be understood in this way.
3. A model numerical calculation and conclusion In our numerical calculation on F,,,, and Fphr and then M(T), and p, etc. we use the model electronic density of states, N(E) = 30(N/W)[ l/4 - (E/W)‘] with the bandwidth W. This band can accommodate up to 5 electrons per spin per atom. We assume the following volume dependencies for narrow range of V around some reference volume V,: V(T),
w = w,(V/v”)mo, < = So(V/VJC,
B = B,(v/Vo)yb,
(9)
with positive constants, a, h, and c. In the following numerical example, we fix the values of various parameters as follows: W. = 6 eV, M, = IO5 m,,, MI and m, being, respectively, the masses of an ion and an electron, to = 1.2, u = 1, h = l/2, and c = 3. We then look into how the results would depend on the values of or/ W,, + being the Fermi energy in the spin unsplit state. In Fig. 1 we show the result for the case of apI W. = 0.64 (with NIV,, = 4.60057 x 102’/cm3 and PN(0)lv=v, = 1.07052). The model electronic density of states, with the occupied region shaded, is shown in panel (a) of the figure. We note in Fig. l(a) that whereas flph dominates over bei at not too low temperatures, the behavior of aph for T < T, is not normal at all; /I becomes negative by the effect of phonons. The behavior
120
N. Miyai et aL /Physica B 119&220 (1996) 118 120
21(a)~'''
~
1
........
4
(if ....... r~., ..........
I'" '
. . . .
/---'"1
i
J
l /
i
i
0 6 [-[(c) .
i
i
400 .
.
800 .
I/ T(K)
.
0
i
400
2 I-l(d)' . . . . . '
i
i/
800 '
'
T(K) '
'
................1 -2 0
400
800
i
-2 ~
"~'i......... i'""
0
400
I
I 800
I T(K)
Fig. 1. The temperature dependences of fl and its various components as defined in the text and ?'D defined in Eq. (7) for the case of eV/W0 = 0.64 in the model electronic density of states shown in (a), with the occupied region shaded.
of flph is analyzed in panels (b) and (d). Certainly, it is ~'D2 that is responsible for making 70 and, then, flph and fl negative. Our result can explain the recent concurrent observation of negative thermal expansion and negative phonon Griineisen constant [4]. In panel (c) we present the behavior of fla. Although /Jel2 certainly becomes negative as usually anticipated, it is overcompensated by the positive flea1. Note also that the size of flo~is generally smaller than that of fiph by one order of magnitude except near T = 0.
The behavior of fl and its various components depend very sensitively, upon the location of Sv in a given N(e) [6]. In the present model, for e,v/W = 0.71, slightly different from the above case of Fig. 1, we have flph > 0 and fl > 0 [7]. These results in this paper strongly suggest the necessity of fundamentally reconsidering the role of phonons in the volume behavior of a metallic ferromagnet. This work is partially supported by the Research Institute of Aoyama G a k u i n University.
References [1] For a recent review and references, see E.F. Wassermann, in: Ferromagnetic Materials, Vol. 5, eds. E.P. Wohlfarth and K.H.J. Buschow (North-Holland, Amsterdam, 1990) p. 237. [2] D.J. Kim, Phys. Rev. B 39 (1989) 6844. [3] D.J. Kim, Phys. Rev. B 25 (1982) 6919; Phys. Rep. 171 (1988) 129. [4] LI. Mafiosa, G.A. Saunders, H. Radhi, U. Kawald, J. Pelzl and H. Bach, J. Phys.: Condens. Matter 3 (1991) 2273; G.A. Saunders, H.B. Senin, H.A.A. Sedek and J. Pelzl, Phys. Rev. B 48 (1993) 15801. [5] D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 239. [6] T. Kizaki, N. Miyai, N. Hino, H. Saita, N. Tateiwa, H. Ohta, H. Ito, J. Iwai and D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 245. [7] N. Miyai and D.J. Kim (unpublished).