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The role of phonons in the temperature dependence of magnetization in an itinerant electron ferromagnet
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 743-745 North-Holland
743
The role of phonons in the temperature dependence of magnetization in an itinerant electron ferromagnet Shinji Fukumoto, Shuji Ukon and D.l. Kim Department of Physics, Aoyama Gakuin Unicersity, Chitosedai, Setagaya-ku; Tokyo 157. Japan
From a numerical analysis on model itinerant electron ferromagnets, we show the importance of considering the effect of the electron-phonon interaction in accounting for the observed temperature dependence of magnetization of a ferromagnetic metal.
Quite contrary to the prevailing view [1], recently we pointed out the fundamental importance of taking into account the effect of the electron-phonon interaction (EPI) in understanding the magnetic properties of a metal [2]. We found, for instance, that the EPI can be responsible for the Curie-Weiss like temperature dependence of the magnetic susceptibility observed in itinerant electron ferromagnets. Also we already noted the possible importance of the role of phonons in determining the magnetization and its temperature dependence of a ferromagnetic metal. Our such earlier conclusion, however, was based on either a nonperturbational numerical analysis at zero temperature [3], or a perturbational discussion at finite temperatures [4]. In this paper we report our new result of a non perturbational numerical analysis for finite temperatures. We show how drastically the temperature dependence of magnetization can be modified by the effect of EPI in an itinerant electron ferromagnet. The equilibrium magnetization AI of a system at a given temperature is determined from the minimum with respect to AI of the free energy of the system. Note that the free energy F of a metal consists of the electron part Fel and the phonon part Fph' The foundation of our theory, then, is that while generally (Fph/ Fel) = (!J(lzwO/£F) = 10- 2 , £F and WD being the electron Fermi energy and the phonon Debye frequency, respectively, for their magnetization dependent parts, ~Fph and ~Fel' we have I ~Fph/~Fel 1 = (!J(I); see figs. 1 and 3 below. As for Fel here we take simply the mean field approximation, namely, the Stoner approximation. The AI dependence of Fph originates from that of the phonon frequency wq; wq depends on AI since the screening of the ion-ion interaction in a metal changes with the magnetization of the screening electrons. With the simple mean field approximation for a jellium-like model, in which we have only longitudinal acoustic phonons,
we obtain the following expression for such magnetization dependent phonon frequency [5]
(1) In the above [lq is the unscreened (bare) phonon frequency, g(q) is the electron-phonon interaction constant, v(q) = 4'lle 2/ q 2 is the Coulomb interaction, and in i ±(q) = F ±(q)/[l - V(q)F±(q)]. F ±(q) is the ordinary Lindhard response function of ± spin electrons and V(q) is the exchange interaction between electrons. In the second line of eq. (1) we noted 1g(q) 12/v(q) = [l~I' [lPI being the ionic plasma frequency. The effect of magnetization on the screening of phonon frequency comes in through the magnetization dependence of F±(q). Note that Iimq~oF±(q) = F±(O) "" N±(O), N ± (0) being the electronic density of states of ± spin electrons at the Fermi surface. For M = 0, we have N +(0) = N _(0) = N(O).
In numerically evaluating the phonon free energy of a metal with eq. (1), we use the Debye approximation. Then we have wq(AI) = s(AI)q, with the magnetization dependent sound velocity,
(
s(~f)
r =
~ + 2N(0) L(O)]
F+(O)
/ [ 1- V(O)F+(O)
+
1- V(O)F_(O) ,
(2) where
So
is the Bohm-Staver sound velocity and we put
[l~ - [l~1 = ~SJq2. The parameter ~ represents deviations
0304-8853/90/S03.50 II) 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
S. Fukum oto et al. / The role ofphonons in an itinerant f erromagnet
744
>' 0
'" §.
........
------ _--
,~
.. ..
U.
. . . . .-l~. ._ -, .
>,0 '" §.
u,
'
........
,, ,,
.
i
I
!
(b)
I
-20h-r~......._._-.,_J~~
0.4
0.6
0.0
0.2
M (pa/alom)
0 .04-~---.-_~,....-....,..J..~_-~_-'-I
o
06
0.4
M (p a/alom)
T=4oo K (b) .
0.6 ~----=~--------.----,
-----?~:_~:-
0.4
,,
E o
,,
,,
~
,,
:. ~0.2
,,
:2
,
.
I
I
I
500
1000
1500
T (K)
Fig. 2. The temperature dependence of magnet ization as determ ined from the minima of free energies as in fig. 1. The Stoner behavior of magnetization given by the broken line is mod ified by the phonon effect into that of the solid line.
s-
_._._
.s 0.0 Ol
,. ..,
.., .,
-2.0 (a)
-4.0 0.0
0.2
., .,
·· ·•· ,· ··· .
.,-
.. .
400
600
800
1000
Fig. 4. The temperature dependence of magnetization corresponding to fig. 3, The broken line is for the magnetization behavior determined from Fe1 alone, and the solid line is for that determined from Fel + Fph '
from the pure jellium model; ~ = 0 for the pure jellium model. Figs. 1 and 2 show such numerical result for the case of the parabolic electron energy dispersion. We chose k F = 1.17 X lOs/cm. corresponding to II/N = 0.6, II and N being, respect ively, the total number of electrons and atoms in the system, 111*/111 = 16, III and 111* being, respectively, the free and the effective electron masses, approximately simulating the case of 3d holes in Ni. We put V = V(O)N(O) ~ 1.2 and ~ = 2.8. As shown in fig. 1, the position of the minimum of free energy shifts appreciably by the effect of EPI. Fph shows discontinuous behavior at a larger M; Fph , and , therefore, Fel + Fph beyond such an M may be outside of the validity of the present approach. Such calculations for different temperatures are summarized in fig. 2. Without the phonon effect, the Curie temperature would be as high as == 10 3 K. Such a too high Curie temperature in the Stoner theory is signifi-
·
,,
··· , ··· ...· .
0.4.,---------r--r--___,_-1
4.0,------=--------.--,r-" fflW =0.5, V=1.1. ; =1 .., ,/ T=200(K) • 2.0
.'.'
200
T (K)
Fig. 1. The magnetization dependent parts of Fel (broken line). Fp h (dot-dashed line) and Fel + Fph (solid line) for the p~a bolic free electrons with k F = 1.17 X 108/cm, m·/m = 16, V = i~(O)N(O) = 1.2 and ~ = 2.8. respectively. at T = 200 K (a). and
U. -
,, ,,
0.2
i I
i i
0.2
,,
04
a
; ;
~=2.8!I
V. l .2. (a)
. .
~
"
" il~ .. ;
-10
i
K, .1.17xl0·(lIcm).
......"",
E o
.. .......
<1
· 10
0. 8 . , . - - - - - - - - - - - - - . f.IW=0 .5,V=1.1. ;=1 ... ... " . 0.6
10r------n----,
10r------..."...---, T=200 (K)
,
0.2
U.
I
-2.0
, I
I I
.
0.4 0.6 M (liB latom)
I
-4.0
_-~'
0.8
1.0
0.0
(b)
0.2
0.4 0.6 0.8 M (~IB latom)
1.0
Fig. 3. The magnetization dependent parts of Fel (broken line), Fp h (dot-dashed line) and Fel + Fp h (solid line) for n/ N = z = 1, or, I F/1I' = 1/2, in the electronic density of star s of eq. (3), for T= 200 K (a). a nd 400 K (b). We put 11'= leV, V = 1.1 and ~ = 1.
s.
Fukumoto et al. / The role ofphonons in an itinerant ferromagnet
cantly reduced to a value close to the observed Curie temperature of Ni by the phonon effect. Figs. 3 and 4 show another numerical example for the model electronic density of states,
745
various itinerant electron ferromagnets. Thus, some additional excitations have been looked for as the origin of decreasing magnetization. Our present study strongly suggests phonon can be such an excitation.
(3) where JV is the electron energy band width. We took JV= 1 eV, V = 1.1 and ~ = 1. The general behaviors in figs. 3 and 4 are very similar to that of the figs. 1 and 2. The decrease with temperature of magnetization is much enhanced by the phonon effect. Note that such an effect has long been anticipated from experiments [6]. It was noted that the spin wave excitations alone are not sufficient to account for the observed decrease with temperature of magnetization in
References [1) C. Herring, in: Magnetism, vol, 4, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1966) p. 290. (2) D.l. Kim, Phys. Rep. 171 (1988) 129. (3) D.l. Kim, Phys. Rev. Lett. 47 (1981) 1213. (4) D.l. Kim, Phys. Rev. B 25 (1982) 6919. (5) D.l. Kim, r, Phys. Soc. Jpn, 40 (1976) 1250. (6) Y. Ishikawa, S. Onodera and K. Tajima, J. Magn, Magn. Mat. 10 (1979) 183.
743
The role of phonons in the temperature dependence of magnetization in an itinerant electron ferromagnet Shinji Fukumoto, Shuji Ukon and D.l. Kim Department of Physics, Aoyama Gakuin Unicersity, Chitosedai, Setagaya-ku; Tokyo 157. Japan
From a numerical analysis on model itinerant electron ferromagnets, we show the importance of considering the effect of the electron-phonon interaction in accounting for the observed temperature dependence of magnetization of a ferromagnetic metal.
Quite contrary to the prevailing view [1], recently we pointed out the fundamental importance of taking into account the effect of the electron-phonon interaction (EPI) in understanding the magnetic properties of a metal [2]. We found, for instance, that the EPI can be responsible for the Curie-Weiss like temperature dependence of the magnetic susceptibility observed in itinerant electron ferromagnets. Also we already noted the possible importance of the role of phonons in determining the magnetization and its temperature dependence of a ferromagnetic metal. Our such earlier conclusion, however, was based on either a nonperturbational numerical analysis at zero temperature [3], or a perturbational discussion at finite temperatures [4]. In this paper we report our new result of a non perturbational numerical analysis for finite temperatures. We show how drastically the temperature dependence of magnetization can be modified by the effect of EPI in an itinerant electron ferromagnet. The equilibrium magnetization AI of a system at a given temperature is determined from the minimum with respect to AI of the free energy of the system. Note that the free energy F of a metal consists of the electron part Fel and the phonon part Fph' The foundation of our theory, then, is that while generally (Fph/ Fel) = (!J(lzwO/£F) = 10- 2 , £F and WD being the electron Fermi energy and the phonon Debye frequency, respectively, for their magnetization dependent parts, ~Fph and ~Fel' we have I ~Fph/~Fel 1 = (!J(I); see figs. 1 and 3 below. As for Fel here we take simply the mean field approximation, namely, the Stoner approximation. The AI dependence of Fph originates from that of the phonon frequency wq; wq depends on AI since the screening of the ion-ion interaction in a metal changes with the magnetization of the screening electrons. With the simple mean field approximation for a jellium-like model, in which we have only longitudinal acoustic phonons,
we obtain the following expression for such magnetization dependent phonon frequency [5]
(1) In the above [lq is the unscreened (bare) phonon frequency, g(q) is the electron-phonon interaction constant, v(q) = 4'lle 2/ q 2 is the Coulomb interaction, and in i ±(q) = F ±(q)/[l - V(q)F±(q)]. F ±(q) is the ordinary Lindhard response function of ± spin electrons and V(q) is the exchange interaction between electrons. In the second line of eq. (1) we noted 1g(q) 12/v(q) = [l~I' [lPI being the ionic plasma frequency. The effect of magnetization on the screening of phonon frequency comes in through the magnetization dependence of F±(q). Note that Iimq~oF±(q) = F±(O) "" N±(O), N ± (0) being the electronic density of states of ± spin electrons at the Fermi surface. For M = 0, we have N +(0) = N _(0) = N(O).
In numerically evaluating the phonon free energy of a metal with eq. (1), we use the Debye approximation. Then we have wq(AI) = s(AI)q, with the magnetization dependent sound velocity,
(
s(~f)
r =
~ + 2N(0) L(O)]
F+(O)
/ [ 1- V(O)F+(O)
+
1- V(O)F_(O) ,
(2) where
So
is the Bohm-Staver sound velocity and we put
[l~ - [l~1 = ~SJq2. The parameter ~ represents deviations
0304-8853/90/S03.50 II) 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
S. Fukum oto et al. / The role ofphonons in an itinerant f erromagnet
744
>' 0
'" §.
........
------ _--
,~
.. ..
U.
. . . . .-l~. ._ -, .
>,0 '" §.
u,
'
........
,, ,,
.
i
I
!
(b)
I
-20h-r~......._._-.,_J~~
0.4
0.6
0.0
0.2
M (pa/alom)
0 .04-~---.-_~,....-....,..J..~_-~_-'-I
o
06
0.4
M (p a/alom)
T=4oo K (b) .
0.6 ~----=~--------.----,
-----?~:_~:-
0.4
,,
E o
,,
,,
~
,,
:. ~0.2
,,
:2
,
.
I
I
I
500
1000
1500
T (K)
Fig. 2. The temperature dependence of magnet ization as determ ined from the minima of free energies as in fig. 1. The Stoner behavior of magnetization given by the broken line is mod ified by the phonon effect into that of the solid line.
s-
_._._
.s 0.0 Ol
,. ..,
.., .,
-2.0 (a)
-4.0 0.0
0.2
., .,
·· ·•· ,· ··· .
.,-
.. .
400
600
800
1000
Fig. 4. The temperature dependence of magnetization corresponding to fig. 3, The broken line is for the magnetization behavior determined from Fe1 alone, and the solid line is for that determined from Fel + Fph '
from the pure jellium model; ~ = 0 for the pure jellium model. Figs. 1 and 2 show such numerical result for the case of the parabolic electron energy dispersion. We chose k F = 1.17 X lOs/cm. corresponding to II/N = 0.6, II and N being, respect ively, the total number of electrons and atoms in the system, 111*/111 = 16, III and 111* being, respectively, the free and the effective electron masses, approximately simulating the case of 3d holes in Ni. We put V = V(O)N(O) ~ 1.2 and ~ = 2.8. As shown in fig. 1, the position of the minimum of free energy shifts appreciably by the effect of EPI. Fph shows discontinuous behavior at a larger M; Fph , and , therefore, Fel + Fph beyond such an M may be outside of the validity of the present approach. Such calculations for different temperatures are summarized in fig. 2. Without the phonon effect, the Curie temperature would be as high as == 10 3 K. Such a too high Curie temperature in the Stoner theory is signifi-
·
,,
··· , ··· ...· .
0.4.,---------r--r--___,_-1
4.0,------=--------.--,r-" fflW =0.5, V=1.1. ; =1 .., ,/ T=200(K) • 2.0
.'.'
200
T (K)
Fig. 1. The magnetization dependent parts of Fel (broken line). Fp h (dot-dashed line) and Fel + Fph (solid line) for the p~a bolic free electrons with k F = 1.17 X 108/cm, m·/m = 16, V = i~(O)N(O) = 1.2 and ~ = 2.8. respectively. at T = 200 K (a). and
U. -
,, ,,
0.2
i I
i i
0.2
,,
04
a
; ;
~=2.8!I
V. l .2. (a)
. .
~
"
" il~ .. ;
-10
i
K, .1.17xl0·(lIcm).
......"",
E o
.. .......
<1
· 10
0. 8 . , . - - - - - - - - - - - - - . f.IW=0 .5,V=1.1. ;=1 ... ... " . 0.6
10r------n----,
10r------..."...---, T=200 (K)
,
0.2
U.
I
-2.0
, I
I I
.
0.4 0.6 M (liB latom)
I
-4.0
_-~'
0.8
1.0
0.0
(b)
0.2
0.4 0.6 0.8 M (~IB latom)
1.0
Fig. 3. The magnetization dependent parts of Fel (broken line), Fp h (dot-dashed line) and Fel + Fp h (solid line) for n/ N = z = 1, or, I F/1I' = 1/2, in the electronic density of star s of eq. (3), for T= 200 K (a). a nd 400 K (b). We put 11'= leV, V = 1.1 and ~ = 1.
s.
Fukumoto et al. / The role ofphonons in an itinerant ferromagnet
cantly reduced to a value close to the observed Curie temperature of Ni by the phonon effect. Figs. 3 and 4 show another numerical example for the model electronic density of states,
745
various itinerant electron ferromagnets. Thus, some additional excitations have been looked for as the origin of decreasing magnetization. Our present study strongly suggests phonon can be such an excitation.
(3) where JV is the electron energy band width. We took JV= 1 eV, V = 1.1 and ~ = 1. The general behaviors in figs. 3 and 4 are very similar to that of the figs. 1 and 2. The decrease with temperature of magnetization is much enhanced by the phonon effect. Note that such an effect has long been anticipated from experiments [6]. It was noted that the spin wave excitations alone are not sufficient to account for the observed decrease with temperature of magnetization in
References [1) C. Herring, in: Magnetism, vol, 4, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1966) p. 290. (2) D.l. Kim, Phys. Rep. 171 (1988) 129. (3) D.l. Kim, Phys. Rev. Lett. 47 (1981) 1213. (4) D.l. Kim, Phys. Rev. B 25 (1982) 6919. (5) D.l. Kim, r, Phys. Soc. Jpn, 40 (1976) 1250. (6) Y. Ishikawa, S. Onodera and K. Tajima, J. Magn, Magn. Mat. 10 (1979) 183.