- Email: [email protected]
A new theory of the Invar behavior considering the role of phonon
Journal of Magnetism and Magnetic Materials MO-144 (199% 245-246
ELSEYIER
A new theoryof the Invar behaviorconsideringthe role of phonon T. Kizaki *, N. Miyai, N. Hino, H. Saita, N, Tateiwa, H, Ohta, H, Ito, J. Iwai, D.J. Kim Abstract By treating
the roles of electrons and phonons on the same footing, we numerically calculate thermal expansion coefficient of an itinerant electron ferromagnet for both T < T, and T > T,, with a simple model of the electronic density of states, and find thermal phonons to play a crucial role in the lnvar behavior.
In discussing the Invar behavior of an itinerant ferromagnet, it has been customary to assume that the phonon contribution to the thermal expansion, p, remains to be normal, independent of its magnetic property. Contrary to such prevailing view, we recently pointed out that the phonon Griineisen constant, yo, can become negative in the ferromagnetic state [l]. Further to explore such possible role of phonons, in this paper we make a first-principle mode1 calculation of the thermal expansion coefficient for an itinerant ferromagnet by treating electrons and phonons on the same footing. We thus find that thermal phonons indeed can play a major role in producing negative 4. Note that very recently negative ye’s were actually observed in Fe-Pt [2] and Fe-Ni [3] Invar systems, From the free energy F(T, V, M) of a system as the function of volume V, and magnetization M, at given temperature T, the equilibrium volume and magnetization are determined from aF/aV
I v=vq. M=M,, =Fv=O,
F,=O,
(1)
together with F, v > 0, F)MM > 0 and F, v FMM - F& > 0. Here we note that whereas the free energy of a metal consists of contributions from electrons and phonons, F = F,, I- Fp,,, I; h also depends on M in no less important way than F,, [l[ It is because the phonon frequency w&MM’) changes with M, since the screening of the ion-ion interaction changes with the spin splitting of the conduction electron energy bands. By the mean field approximation treatment of such magnetization dependent screening in a
jellium-like model, in which there is only a longitudinal acoustic phonon, we obtain the sound velocity, s = s(M)
asDl
where s,, = ( ~~V/(8ae2N(0)))‘/* is the Bohm-Staver sound velocity, with J2r,, the ionic plasma frequency, N(O) the electronic density of states per spin at the Fermi surface in the spin unsplit states, &to)
= F,(O)/(l
- liF,(O)),
with
f,(r) being the Fermi distribution of u spin electrons, v the exchange interaction between electrons, and we put 0; - 0s = lsiq2, with 0, the bare phonon frequency. 5 represents the effect of the direct interaction between ion cores, outside of the Coulomb interaction between point charge ions; t= 0 for the pure jellium model, and we anticipste O(e) = 1. In the Debye approximation, then, for the phonon free energy we obtain
3Nk,T
90
+9;
* Corresponding author. Fax: [email protected]. 0304~8853/95/$09.50 SSDI 0304.8853(94)005
-i-81-3-5384-6100;
(3)
/
dq
q2
tn(l
_
e-WW/W),
Q
cmail with N lhe total number of atoms, and q,, the Debye wave
Q 1995 Eisevier Science B.V. Al1 rights reserved 18-4
T. Kizaki et al. /Jourml
246
of Magnetism and Magnetic Materiuk 140-144 &W5) 245-246
number. For electrons, here we use the mean field approximation (Stoner) result, Fe,= -k,T~~deN(e)ln(l D
+e-8(r-fi~))
-$?rnz+
~gvn,,+~@‘M,s~, (r
Lp
(61
where p0 and no are the chemical potential and the total number of u spin electrons, and M, is the ionic mass; the last term represents the potential energy of the direct repulsion between ion cores. From the free energy, J3 in the ferromagnetic state can be calculated as
=&
+A*
(7)
where pr = -l-IF/N, and H,= -aF/aM, are respectively, the internal pressure and magnetic field; K is the compressibility, and K, is the magnetic expansivity (often called the forced magnetostriction) defined by the change in volume, AV, caused by an external magnetic field H as AV/V = K, H, which are given as [4] 1
K=-
FMM
V FVVFMM - Fiv’
w (9
While & represents the ordinary thermal expansion mechanism, pz represents the effect of the internal magnetic field on volume. If WC note in Eq. (7) that
p, = - a~,,/av- aFp,/av =&I
W
+Pph,
and, similarity, as
that
H, = H,, f Hph,we can decompose fl
P = PC, + Pph:
(11)
Pph = KaPph/aT + K,aH,,/aT = PO1 =
&hl Pet,
+ +
&ah2 I%2
1 *
In carrying out numerical calcutation electronic density of states, N(e)=(l8N/W)[f-(e/W)*],
0
04
0.8
IO
TITC
0
04
n.n I 0 m-c
Fig. 1. The temperature dependence of p and its various components as defined in Eqs. (111, (12) etc. for the case of or / W, = 0.38 (the corresponding occupied region is shaded in the model density of states). dencies of relevant quantities, forms:
we assume the following
with positive parameters a, b and c; V, is some reference volume. Here, we set as @a, lo, a, b, c) = (1.02, 1.2,5/3, 1,4). v,, is the value of V= m(O) at the reference volume. Then, for different values of +/Wo., corresponding to different locations of the Fermi energy in the spin unsplit state, we calculate /3 and its various components. In Fig. l(a), for the case of c,JW,, = 0.38, certainly we observe a negative thermal expansion below T,. When Q/W’, is closer to 0.5, however, a negative thermal expansion is not obtained. The volume behavior sensitively depends on the value of cF/Wo. As can be seen in Fig. l(a), it is &,,, that is responsible for making p negative; more specifi&ly, it is &,,r& as shown in Fig. l(d). Fig. l(c) shows that although & < 0, as usually anticipated, it is compensated by the larger, positive &,t, and, then, j3,,, is much smaller than 1 fl,hl except near T = 0.
(12)
References
(13)
[tl D.J. Kim, Phys. Rept. 171 (1989) 129. [2] Lt. Maiiosa. G.A. Saunders, H. Rahdi, U. Kawald, J. Petzl aad H. Bach, J.Phys.: Condens Matter 3 (1991) 2273; G.A. Saunders, H.B. Seninin, H.A.A. Sedek and J. Pelzl, Phys. Rev. R 48 (1993) 15801. [3] V.A. Sidov and L.G. Khostantsev, J. Magn. Magn. Mater. 129 (1994) 356. [4] M. Shim&u, Rept. Progr. Phys. 44 (1981) 329; D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 239 (these Proceedings).
we use the model (14)
with the width W, which is illustrated in Fig. l(a) with the occupied region shaded. This band can accommodate up to 3 electrons per spin, per atom. As for the volume depen-
ELSEYIER
A new theoryof the Invar behaviorconsideringthe role of phonon T. Kizaki *, N. Miyai, N. Hino, H. Saita, N, Tateiwa, H, Ohta, H, Ito, J. Iwai, D.J. Kim Abstract By treating
the roles of electrons and phonons on the same footing, we numerically calculate thermal expansion coefficient of an itinerant electron ferromagnet for both T < T, and T > T,, with a simple model of the electronic density of states, and find thermal phonons to play a crucial role in the lnvar behavior.
In discussing the Invar behavior of an itinerant ferromagnet, it has been customary to assume that the phonon contribution to the thermal expansion, p, remains to be normal, independent of its magnetic property. Contrary to such prevailing view, we recently pointed out that the phonon Griineisen constant, yo, can become negative in the ferromagnetic state [l]. Further to explore such possible role of phonons, in this paper we make a first-principle mode1 calculation of the thermal expansion coefficient for an itinerant ferromagnet by treating electrons and phonons on the same footing. We thus find that thermal phonons indeed can play a major role in producing negative 4. Note that very recently negative ye’s were actually observed in Fe-Pt [2] and Fe-Ni [3] Invar systems, From the free energy F(T, V, M) of a system as the function of volume V, and magnetization M, at given temperature T, the equilibrium volume and magnetization are determined from aF/aV
I v=vq. M=M,, =Fv=O,
F,=O,
(1)
together with F, v > 0, F)MM > 0 and F, v FMM - F& > 0. Here we note that whereas the free energy of a metal consists of contributions from electrons and phonons, F = F,, I- Fp,,, I; h also depends on M in no less important way than F,, [l[ It is because the phonon frequency w&MM’) changes with M, since the screening of the ion-ion interaction changes with the spin splitting of the conduction electron energy bands. By the mean field approximation treatment of such magnetization dependent screening in a
jellium-like model, in which there is only a longitudinal acoustic phonon, we obtain the sound velocity, s = s(M)
asDl
where s,, = ( ~~V/(8ae2N(0)))‘/* is the Bohm-Staver sound velocity, with J2r,, the ionic plasma frequency, N(O) the electronic density of states per spin at the Fermi surface in the spin unsplit states, &to)
= F,(O)/(l
- liF,(O)),
with
f,(r) being the Fermi distribution of u spin electrons, v the exchange interaction between electrons, and we put 0; - 0s = lsiq2, with 0, the bare phonon frequency. 5 represents the effect of the direct interaction between ion cores, outside of the Coulomb interaction between point charge ions; t= 0 for the pure jellium model, and we anticipste O(e) = 1. In the Debye approximation, then, for the phonon free energy we obtain
3Nk,T
90
+9;
* Corresponding author. Fax: [email protected]. 0304~8853/95/$09.50 SSDI 0304.8853(94)005
-i-81-3-5384-6100;
(3)
/
dq
q2
tn(l
_
e-WW/W),
Q
cmail with N lhe total number of atoms, and q,, the Debye wave
Q 1995 Eisevier Science B.V. Al1 rights reserved 18-4
T. Kizaki et al. /Jourml
246
of Magnetism and Magnetic Materiuk 140-144 &W5) 245-246
number. For electrons, here we use the mean field approximation (Stoner) result, Fe,= -k,T~~deN(e)ln(l D
+e-8(r-fi~))
-$?rnz+
~gvn,,+~@‘M,s~, (r
Lp
(61
where p0 and no are the chemical potential and the total number of u spin electrons, and M, is the ionic mass; the last term represents the potential energy of the direct repulsion between ion cores. From the free energy, J3 in the ferromagnetic state can be calculated as
=&
+A*
(7)
where pr = -l-IF/N, and H,= -aF/aM, are respectively, the internal pressure and magnetic field; K is the compressibility, and K, is the magnetic expansivity (often called the forced magnetostriction) defined by the change in volume, AV, caused by an external magnetic field H as AV/V = K, H, which are given as [4] 1
K=-
FMM
V FVVFMM - Fiv’
w (9
While & represents the ordinary thermal expansion mechanism, pz represents the effect of the internal magnetic field on volume. If WC note in Eq. (7) that
p, = - a~,,/av- aFp,/av =&I
W
+Pph,
and, similarity, as
that
H, = H,, f Hph,we can decompose fl
P = PC, + Pph:
(11)
Pph = KaPph/aT + K,aH,,/aT = PO1 =
&hl Pet,
+ +
&ah2 I%2
1 *
In carrying out numerical calcutation electronic density of states, N(e)=(l8N/W)[f-(e/W)*],
0
04
0.8
IO
TITC
0
04
n.n I 0 m-c
Fig. 1. The temperature dependence of p and its various components as defined in Eqs. (111, (12) etc. for the case of or / W, = 0.38 (the corresponding occupied region is shaded in the model density of states). dencies of relevant quantities, forms:
we assume the following
with positive parameters a, b and c; V, is some reference volume. Here, we set as @a, lo, a, b, c) = (1.02, 1.2,5/3, 1,4). v,, is the value of V= m(O) at the reference volume. Then, for different values of +/Wo., corresponding to different locations of the Fermi energy in the spin unsplit state, we calculate /3 and its various components. In Fig. l(a), for the case of c,JW,, = 0.38, certainly we observe a negative thermal expansion below T,. When Q/W’, is closer to 0.5, however, a negative thermal expansion is not obtained. The volume behavior sensitively depends on the value of cF/Wo. As can be seen in Fig. l(a), it is &,,, that is responsible for making p negative; more specifi&ly, it is &,,r& as shown in Fig. l(d). Fig. l(c) shows that although & < 0, as usually anticipated, it is compensated by the larger, positive &,t, and, then, j3,,, is much smaller than 1 fl,hl except near T = 0.
(12)
References
(13)
[tl D.J. Kim, Phys. Rept. 171 (1989) 129. [2] Lt. Maiiosa. G.A. Saunders, H. Rahdi, U. Kawald, J. Petzl aad H. Bach, J.Phys.: Condens Matter 3 (1991) 2273; G.A. Saunders, H.B. Seninin, H.A.A. Sedek and J. Pelzl, Phys. Rev. R 48 (1993) 15801. [3] V.A. Sidov and L.G. Khostantsev, J. Magn. Magn. Mater. 129 (1994) 356. [4] M. Shim&u, Rept. Progr. Phys. 44 (1981) 329; D.J. Kim, J. Magn. Magn. Mater. 140-144 (1995) 239 (these Proceedings).
we use the model (14)
with the width W, which is illustrated in Fig. l(a) with the occupied region shaded. This band can accommodate up to 3 electrons per spin, per atom. As for the volume depen-