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Magnetic field effect on sound velocity below the curie temperature of an itinerant electron ferromagnet
Solid State Communications, Vol. 38, pp. 441-444. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038/1098/81/I 70441 --04 $02.00/0
MAGNETIC FIELD EFFECT ON SOUND VELOCITY BELOW THE CURIE TEMPERATURE OF AN ITINERANT ELECTRON FERROMAGNET D.J. Kim Department of Physics, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo 157, Japan (Received 4 November 1980 by W. Sasaki)
Starting from a general consideration on the exchange interaction effect on the screening of the ion-ion interaction we present a new theory concerning the magnetic field effect on the longitudinal acoustic sound velocity in the ferromagnetic state of a metal. IN A NUMBER of itinerant electron ferromagnets, most prominently in those which are called the Invars, the elastic properties of lattices appear to be very closely related to the magnetic properties of electrons [1]. At present, however, we are still far from a systematic and comprehensive understanding of those observations. One of such unexplained observations may be the anomalously large magnetic field dependence of the elastic constant in those systems; in FeNi alloys and FePt alloys [ 2 - 4 ] , for instance, a magnetic field of "~ 10 kG was found to change the elastic constant as measured through sound velocity as much as ~ 1%, both above and below the Curie temperature T c. Since the size of the relative change in the elastic constant due to a magnetic field H is expected to be O ( g B H / W ) 2 for T > T e and O ( # B H / W ) for T < T c, W being the width of the electron energy band which is generally of the order of a few eV, such observation is beyond our simple intuition; we need a quite large factor of 10410 6 for T > T c and that of 102-103 for T < T c to account for the observed result on FeNi and FePt alloys. Furthermore, we are required to explain the fact that the observed change in the elastic constant is negative for T > Te, and positive for T < T c. As for the problem for T > To, recently we presented a theory [5] based on our earlier formulation [6] of the exchange interaction effect on the screening of the ion-ion interaction in an itinerant electron ferromagnet; we pointed out that the magnetic field effect on the longitudinal acoustic sound velocity should be exchange enhanced by a factor proportional to the cubic power of the paramagnetic spin susceptibility of the metal which tends to diverge as T ~ T e. In this paper we extend the discussion to the temperature region below T c starting from the same basic consideration We obtain a result which seems to explain the observed magnetic field effect on the elastic constant of itinerant electron ferromagnets for T < T e both in sign and magnitude. From the nature of the present problem we are
required to treat the magnetic properties of electrons and the elastic properties of lattice exactly on the same footing. Under such requirement, in this paper we use essentially the jellium model, with some extension, and we consider only the longitudinal acoustic sound: We assume an effective single band model for the electrons of a metal and treat the effect of the exchange interaction between electrons by the mean field approximation (the Stoner model). Correspondingly, in handling the screening of the ion-ion interaction we include the effect of the exchange interaction between electrons by the same mean field approximation. In this way we discussed recently [7] how the velocity of the longitudinal acoustic sound would change with magnetization in the ferromagnetic state of a metal. It is quite straightforward to extend the discussion as to include the effectlof external magnetic field. Thus, for the sound velocity s(H) under a magnetic field H we obtain the following expression s(H)
1
= ~ + 2N(0)
\ So / q
1 -- VF+(O,H)
F_(O, H ) 1
-
[
]-1 "
(1)
In the above N(0) is the electronic density of states per spin at the Fermi surface in the paramagnetic state, F+_(0, H ) is the q -~ 0 limit of the static Lindhard function F±(q, H ) of + spin electrons under the magnetic field H, V is the effective exchange interaction between electrons, so = f2pl/[8rre2N(0)] 1/2 is the Bohm-Staver sound velocity, ~2pl being the ionic plasma frequency, is a parameter introduced by the relation ~2~-- I g(q)12/V(q) = ~s2oq2,
(2)
where ~ a is the bare phonon frequency, g(q) is the electron-phonon interaction constant, and V(q) is the Coulomb repulsion between electrons. Note that the parameter ~ conveniently describes deviations from the
441
442
MAGNETIC FIELD EFFECT ON SOUND VELOCITY
jellium model: ~ = 0 for the jellium model since there ~22q= ~,1 = I g(q)12/V(q); if ~ ;> 0 ( < 0), the lattice is harder (softer) than in the jellium model. If we simply put H = 0 in equation (1) it reduces to our earlier result
Y(eF, M) = -- ½N(O) W [
-
VN+(0)i
(3)
Y+(0, H ) = k
where f(e) is tile Fermi distribution function, and ek+ = ek T 11kM +_H
(4)
is the one particle energy of spin electrons under the magnetic field, M = n+ -- n being the total magnetization of the system where n+ are the total numbers of ± spin electrons (we put/l B = 1). Equations (1)--(4) comprise the basis of our discussion. From them we can calculate the magnetic field dependence of sound velocity for any temperatures if the electronicdensity of states N(e) together with the values of %,, V, and ~ are given; the calculation of the temperature dependence of magnetization is included in the above procedure. Since, however, the primary objective of this paper is only to show the plausibility of our consideration, in our actual numerical calculation on equation (1), we confine ourselves to the simple case of tile low temperature limit. In such low temperature limit, by expanding in H and retaining the lowest order terms we can reduce equation (3) to F+(0, H) m N±(0) ± N.',(0)[1 +
~V×hrlH
- ' ,v . . .tu. l,v+T0)+ N_(0)/[i +
H,
(5)
where N~ (0) are the densities of states of +- spin electrons at the Fermi surface in the ferromagnetic state, N~ (0) being their derivatives with respect to energy, and Xhf is what is called the high field susceptibility, 4N+(0)N_(0) ×hr(M) = N+(0) + N_(0) -- 2VN+(0)N_(0)"
(6)
With the simplification of equation (5) it is straightforward to obtain the following result for As(H) = s ( H ) - s(O) from equation (1),
As(H)/s(O) ~=A(eF, ~, M )(H/W ),
(7)
with
A = (So/S(O)) 2 y ( e y , M ) x h f ( M ) ,
(8)
~
1[ N+(o)
liTVN_m)I'III-:VN+(O) +
In further carrying out our discussion on equation (1) note that
where
y'( lJV+(O)
Ll]
[s(O)ls,,] :.
for
Vol. 38, No. 5
1-2 I
1 -
VN_(O)]
(9)
If we neglect the exchange enhancement effect by putting 1/[1 -- I~N+(0)] ~ 1 in estimating the magnitude of A, we obtain I A[--~ 1. As remarked already, in order to account for the observed results on FeNi and FePt alloys, A is required to be positive and of the order of 102 or larger. In the following we carry out a numerical calculation of A by assuming a sinrple form of electronic density of states and show that actually A is positive and can be exchange enhanced to that extent. In our numerical calculation on equations (6)- (91, we use tile R)llowing form of model electronic density of states, ,N(e) = (6NIW3)e(W-- e),
(10)
which is illustrated by the bottonr figure of Fig. l, where W is the width of the band and N is the total nunlber of atoms in the system. With the above density of states and given total number of electrons n+ + n_, or, eF, we realize different magnetization M by changing the value
of/;. In Fig. 1, by the real lines we present tire restilt of our numerical calculation on A(eu, ~, M) for different occupations, eF/W, of the band as the function of the parameter ~ and the magnetization M (normalized by the maximum possible magnetization Mo). For convenience, the sound velocity [s(O)/so] without tile external magnetic field is also shown by dotted lines. Note that with the density of states of equation (10), symmetric with respect to the center of the band, both A(ev, ~, M ) and s(O)lso are the same for ev/W = x and 1 x. The result of Fig. 1 shows that the magnitude of A can be indeed as large as 10 a - 10 s, with positive sign, just as required to account for the experimental observations on FeNi and FePt alloys [2 4]. Depending upon the values of ev/W and ~, however, the detailed behavior of A can be quite different; if the Fermi energy in the paranlagnetic state is at or near the nlaxinmm of the density of states (eF/W ~ 0.5) the niagnitude of A is small; for the same value of er,/W, the magnitude of A is smaller for a larger value of ~. Quite differing from the case of FeNi and FePt alloys, in pure Ni the observed magnetic field effect on the sound velocity is very small [2]. Within our present model calculation, such behavior of Ni may be associated with the case (a) with ~ = 2 or even larger in Fig. 1. In this respect, it is interesting to note that in Ni the sound velocity s(0) without external magnetic field was observed to increase with increasing magnetization 12]
Vol. 38, No. 5 i
MAGNETIC FIELD EFFECT ON SOUND VELOCITY
EF:O5W
200
~:c,/
I00 _ ~ _ i i i i
05///
exchange enhancement factors. As for the sign of A. however, there seems to be no obvious reason why it should be positive as we found for our model density of 2O states of equation (10);A may become negative for s(M) some other form of density of states. This point should So be studied further. Finally, remember that the result of Fig. 1 is for the zero temperature limit; with a proper finite temperature treatment the divergence of A for M ~ 0 is expected to be rounded out. IO It is rather surprising that our simple consideration can explain the essential aspects of the experimental observations as shown in the above. This fact seems to indicate the fundamental adequacy of the present consideration on the interaction between electrons and 122M~o) lattice in an itinerant electron ferromagnet. Note that the present result is closely related to the problem of the possible effect of the electron-phonon interaction on the magnetization of an itinerant electron ferromagnet. As will be discussed in detail separately [8], if we put the contribution of the electron-phonon interaction to magnetization, Mp, as I0 Mp/Nla n = B ( h c o o / W ), (11)
EF=040rO6W (b)
(O)
A /{:o~
20 200
°/
I0 I00
I0
-0-5-0
, , , ,~'" 0
0
05
20() I
~/[. )
.
.
.
I
00
0
05
2© 200, (4000)
05 M/Mo
I.O
0
IO
~1,5
.
0
I0
I
I
0
I
05 M/M o
[
I
I
I
( c (b)) (9)~(b) )
-
I
I
i
443
I
J 0.5
i
B is found to be related to A of equation (7) in a simple way,
I
N+(0)N (0) B = -- ]VN(O) N(0)[N+(0) + N_(0)] A.
-
J
~
I.O
W Fig. I. The magnetization (34) dependence of A defined by equation (8) for different occupations er/Wof the band given by equation (10) and for different values of is shown by the real lines. Note that in (d) a different scale shown with parentheses is used for the ordinate for the case of ~ = 1. The dotted lines are for the magnetization dependence of the sound velocity without the external magnetic field. In the bottom figure of the density of states the different locations of Fermi energy at the paramagnetic state are marked correspondingly to the different cases in the top figure. in accordance with such behavior of A in Fig. 1 ; in FeNi and FePt alloys s(0) decreases with increasing magnetization [ 2 - 4 ] . Our result that the magnitude of A can be as large as 102-103 may be understandable if we note that in the expression for A there appear the square powers of the
(12)
Thus, except the case of full magnetization, M/Mo ~-- l, where N+(0) or N_(0) goes to zero, we find B ~--- - A . Our result that in certain situations A is positive and as large as ~ 10 2 implies that the electron-phonon interaction contribution to the magnetization is negative (destructive) and can be as large I BI "~ 10 2, or -- 1 # B per atom. The above result on B seems to have a rather drastic impact on our understanding of itinerant electron ferromagnetism. Here note, however, that the important quantity B can not be measured directly; only through the relation of equation (12) we can infer on the sign and magnitude of B. In this respect, experiments on the magnetic field effect on sound velocity are very important. REFERENCES 1.
2. 3. 4.
For recent reviews and references, see E.P. Wohlfarth, IEEE Trans. NAG-11,1638 (1975); Y. Nakamura, IEEE Trans. MAG-12,278.(1976); M. Shimizu, J. Magn. Magn. Mater. 20, 47 (1980). G.A. Alers, J.R. Neighbours & H. Sato, J. Phys. Chem. Solids 13, 40 (1960). G. Hausch and H. Warlimont, Phy~ Lett. 41A, 437 (1972). G. Hausch, J. Phys. Soc. Japan 37, 819 (1974).
444 5. 6.
MAGNETIC FIELD EFFECT ON SOUND VELOCITY D.J. Kim, Solid State Commun. 36, 3113 (1980). D.J. Kim, J. Phy~ Soc. Japan 40, 1244, 1250 (1976);Physica 91B, 281 (1977);Phys. Rev. Lett.
7. 8.
Vol. 38, No. 5
39, 98, 511(E) (1977). D.J. Kim, Solid State Commun. 30, 249 (1979). D.J. Kim, Solid State Commun. 38,441 (1981).
0038/1098/81/I 70441 --04 $02.00/0
MAGNETIC FIELD EFFECT ON SOUND VELOCITY BELOW THE CURIE TEMPERATURE OF AN ITINERANT ELECTRON FERROMAGNET D.J. Kim Department of Physics, Aoyama Gakuin University, Chitosedai, Setagaya-ku, Tokyo 157, Japan (Received 4 November 1980 by W. Sasaki)
Starting from a general consideration on the exchange interaction effect on the screening of the ion-ion interaction we present a new theory concerning the magnetic field effect on the longitudinal acoustic sound velocity in the ferromagnetic state of a metal. IN A NUMBER of itinerant electron ferromagnets, most prominently in those which are called the Invars, the elastic properties of lattices appear to be very closely related to the magnetic properties of electrons [1]. At present, however, we are still far from a systematic and comprehensive understanding of those observations. One of such unexplained observations may be the anomalously large magnetic field dependence of the elastic constant in those systems; in FeNi alloys and FePt alloys [ 2 - 4 ] , for instance, a magnetic field of "~ 10 kG was found to change the elastic constant as measured through sound velocity as much as ~ 1%, both above and below the Curie temperature T c. Since the size of the relative change in the elastic constant due to a magnetic field H is expected to be O ( g B H / W ) 2 for T > T e and O ( # B H / W ) for T < T c, W being the width of the electron energy band which is generally of the order of a few eV, such observation is beyond our simple intuition; we need a quite large factor of 10410 6 for T > T c and that of 102-103 for T < T c to account for the observed result on FeNi and FePt alloys. Furthermore, we are required to explain the fact that the observed change in the elastic constant is negative for T > Te, and positive for T < T c. As for the problem for T > To, recently we presented a theory [5] based on our earlier formulation [6] of the exchange interaction effect on the screening of the ion-ion interaction in an itinerant electron ferromagnet; we pointed out that the magnetic field effect on the longitudinal acoustic sound velocity should be exchange enhanced by a factor proportional to the cubic power of the paramagnetic spin susceptibility of the metal which tends to diverge as T ~ T e. In this paper we extend the discussion to the temperature region below T c starting from the same basic consideration We obtain a result which seems to explain the observed magnetic field effect on the elastic constant of itinerant electron ferromagnets for T < T e both in sign and magnitude. From the nature of the present problem we are
required to treat the magnetic properties of electrons and the elastic properties of lattice exactly on the same footing. Under such requirement, in this paper we use essentially the jellium model, with some extension, and we consider only the longitudinal acoustic sound: We assume an effective single band model for the electrons of a metal and treat the effect of the exchange interaction between electrons by the mean field approximation (the Stoner model). Correspondingly, in handling the screening of the ion-ion interaction we include the effect of the exchange interaction between electrons by the same mean field approximation. In this way we discussed recently [7] how the velocity of the longitudinal acoustic sound would change with magnetization in the ferromagnetic state of a metal. It is quite straightforward to extend the discussion as to include the effectlof external magnetic field. Thus, for the sound velocity s(H) under a magnetic field H we obtain the following expression s(H)
1
= ~ + 2N(0)
\ So / q
1 -- VF+(O,H)
F_(O, H ) 1
-
[
]-1 "
(1)
In the above N(0) is the electronic density of states per spin at the Fermi surface in the paramagnetic state, F+_(0, H ) is the q -~ 0 limit of the static Lindhard function F±(q, H ) of + spin electrons under the magnetic field H, V is the effective exchange interaction between electrons, so = f2pl/[8rre2N(0)] 1/2 is the Bohm-Staver sound velocity, ~2pl being the ionic plasma frequency, is a parameter introduced by the relation ~2~-- I g(q)12/V(q) = ~s2oq2,
(2)
where ~ a is the bare phonon frequency, g(q) is the electron-phonon interaction constant, and V(q) is the Coulomb repulsion between electrons. Note that the parameter ~ conveniently describes deviations from the
441
442
MAGNETIC FIELD EFFECT ON SOUND VELOCITY
jellium model: ~ = 0 for the jellium model since there ~22q= ~,1 = I g(q)12/V(q); if ~ ;> 0 ( < 0), the lattice is harder (softer) than in the jellium model. If we simply put H = 0 in equation (1) it reduces to our earlier result
Y(eF, M) = -- ½N(O) W [
-
VN+(0)i
(3)
Y+(0, H ) = k
where f(e) is tile Fermi distribution function, and ek+ = ek T 11kM +_H
(4)
is the one particle energy of spin electrons under the magnetic field, M = n+ -- n being the total magnetization of the system where n+ are the total numbers of ± spin electrons (we put/l B = 1). Equations (1)--(4) comprise the basis of our discussion. From them we can calculate the magnetic field dependence of sound velocity for any temperatures if the electronicdensity of states N(e) together with the values of %,, V, and ~ are given; the calculation of the temperature dependence of magnetization is included in the above procedure. Since, however, the primary objective of this paper is only to show the plausibility of our consideration, in our actual numerical calculation on equation (1), we confine ourselves to the simple case of tile low temperature limit. In such low temperature limit, by expanding in H and retaining the lowest order terms we can reduce equation (3) to F+(0, H) m N±(0) ± N.',(0)[1 +
~V×hrlH
- ' ,v . . .tu. l,v+T0)+ N_(0)/[i +
H,
(5)
where N~ (0) are the densities of states of +- spin electrons at the Fermi surface in the ferromagnetic state, N~ (0) being their derivatives with respect to energy, and Xhf is what is called the high field susceptibility, 4N+(0)N_(0) ×hr(M) = N+(0) + N_(0) -- 2VN+(0)N_(0)"
(6)
With the simplification of equation (5) it is straightforward to obtain the following result for As(H) = s ( H ) - s(O) from equation (1),
As(H)/s(O) ~=A(eF, ~, M )(H/W ),
(7)
with
A = (So/S(O)) 2 y ( e y , M ) x h f ( M ) ,
(8)
~
1[ N+(o)
liTVN_m)I'III-:VN+(O) +
In further carrying out our discussion on equation (1) note that
where
y'( lJV+(O)
Ll]
[s(O)ls,,] :.
for
Vol. 38, No. 5
1-2 I
1 -
VN_(O)]
(9)
If we neglect the exchange enhancement effect by putting 1/[1 -- I~N+(0)] ~ 1 in estimating the magnitude of A, we obtain I A[--~ 1. As remarked already, in order to account for the observed results on FeNi and FePt alloys, A is required to be positive and of the order of 102 or larger. In the following we carry out a numerical calculation of A by assuming a sinrple form of electronic density of states and show that actually A is positive and can be exchange enhanced to that extent. In our numerical calculation on equations (6)- (91, we use tile R)llowing form of model electronic density of states, ,N(e) = (6NIW3)e(W-- e),
(10)
which is illustrated by the bottonr figure of Fig. l, where W is the width of the band and N is the total nunlber of atoms in the system. With the above density of states and given total number of electrons n+ + n_, or, eF, we realize different magnetization M by changing the value
of/;. In Fig. 1, by the real lines we present tire restilt of our numerical calculation on A(eu, ~, M) for different occupations, eF/W, of the band as the function of the parameter ~ and the magnetization M (normalized by the maximum possible magnetization Mo). For convenience, the sound velocity [s(O)/so] without tile external magnetic field is also shown by dotted lines. Note that with the density of states of equation (10), symmetric with respect to the center of the band, both A(ev, ~, M ) and s(O)lso are the same for ev/W = x and 1 x. The result of Fig. 1 shows that the magnitude of A can be indeed as large as 10 a - 10 s, with positive sign, just as required to account for the experimental observations on FeNi and FePt alloys [2 4]. Depending upon the values of ev/W and ~, however, the detailed behavior of A can be quite different; if the Fermi energy in the paranlagnetic state is at or near the nlaxinmm of the density of states (eF/W ~ 0.5) the niagnitude of A is small; for the same value of er,/W, the magnitude of A is smaller for a larger value of ~. Quite differing from the case of FeNi and FePt alloys, in pure Ni the observed magnetic field effect on the sound velocity is very small [2]. Within our present model calculation, such behavior of Ni may be associated with the case (a) with ~ = 2 or even larger in Fig. 1. In this respect, it is interesting to note that in Ni the sound velocity s(0) without external magnetic field was observed to increase with increasing magnetization 12]
Vol. 38, No. 5 i
MAGNETIC FIELD EFFECT ON SOUND VELOCITY
EF:O5W
200
~:c,/
I00 _ ~ _ i i i i
05///
exchange enhancement factors. As for the sign of A. however, there seems to be no obvious reason why it should be positive as we found for our model density of 2O states of equation (10);A may become negative for s(M) some other form of density of states. This point should So be studied further. Finally, remember that the result of Fig. 1 is for the zero temperature limit; with a proper finite temperature treatment the divergence of A for M ~ 0 is expected to be rounded out. IO It is rather surprising that our simple consideration can explain the essential aspects of the experimental observations as shown in the above. This fact seems to indicate the fundamental adequacy of the present consideration on the interaction between electrons and 122M~o) lattice in an itinerant electron ferromagnet. Note that the present result is closely related to the problem of the possible effect of the electron-phonon interaction on the magnetization of an itinerant electron ferromagnet. As will be discussed in detail separately [8], if we put the contribution of the electron-phonon interaction to magnetization, Mp, as I0 Mp/Nla n = B ( h c o o / W ), (11)
EF=040rO6W (b)
(O)
A /{:o~
20 200
°/
I0 I00
I0
-0-5-0
, , , ,~'" 0
0
05
20() I
~/[. )
.
.
.
I
00
0
05
2© 200, (4000)
05 M/Mo
I.O
0
IO
~1,5
.
0
I0
I
I
0
I
05 M/M o
[
I
I
I
( c (b)) (9)~(b) )
-
I
I
i
443
I
J 0.5
i
B is found to be related to A of equation (7) in a simple way,
I
N+(0)N (0) B = -- ]VN(O) N(0)[N+(0) + N_(0)] A.
-
J
~
I.O
W Fig. I. The magnetization (34) dependence of A defined by equation (8) for different occupations er/Wof the band given by equation (10) and for different values of is shown by the real lines. Note that in (d) a different scale shown with parentheses is used for the ordinate for the case of ~ = 1. The dotted lines are for the magnetization dependence of the sound velocity without the external magnetic field. In the bottom figure of the density of states the different locations of Fermi energy at the paramagnetic state are marked correspondingly to the different cases in the top figure. in accordance with such behavior of A in Fig. 1 ; in FeNi and FePt alloys s(0) decreases with increasing magnetization [ 2 - 4 ] . Our result that the magnitude of A can be as large as 102-103 may be understandable if we note that in the expression for A there appear the square powers of the
(12)
Thus, except the case of full magnetization, M/Mo ~-- l, where N+(0) or N_(0) goes to zero, we find B ~--- - A . Our result that in certain situations A is positive and as large as ~ 10 2 implies that the electron-phonon interaction contribution to the magnetization is negative (destructive) and can be as large I BI "~ 10 2, or -- 1 # B per atom. The above result on B seems to have a rather drastic impact on our understanding of itinerant electron ferromagnetism. Here note, however, that the important quantity B can not be measured directly; only through the relation of equation (12) we can infer on the sign and magnitude of B. In this respect, experiments on the magnetic field effect on sound velocity are very important. REFERENCES 1.
2. 3. 4.
For recent reviews and references, see E.P. Wohlfarth, IEEE Trans. NAG-11,1638 (1975); Y. Nakamura, IEEE Trans. MAG-12,278.(1976); M. Shimizu, J. Magn. Magn. Mater. 20, 47 (1980). G.A. Alers, J.R. Neighbours & H. Sato, J. Phys. Chem. Solids 13, 40 (1960). G. Hausch and H. Warlimont, Phy~ Lett. 41A, 437 (1972). G. Hausch, J. Phys. Soc. Japan 37, 819 (1974).
444 5. 6.
MAGNETIC FIELD EFFECT ON SOUND VELOCITY D.J. Kim, Solid State Commun. 36, 3113 (1980). D.J. Kim, J. Phy~ Soc. Japan 40, 1244, 1250 (1976);Physica 91B, 281 (1977);Phys. Rev. Lett.
7. 8.
Vol. 38, No. 5
39, 98, 511(E) (1977). D.J. Kim, Solid State Commun. 30, 249 (1979). D.J. Kim, Solid State Commun. 38,441 (1981).