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Spin dynamics in Heusler alloys
Journal of Magnetism and Magnetic Materials 31-34 (1983) 47-48
47
SPIN DYNAMICS IN HEUSLER ALLOYS Y. K U B O * A N D S. I S H I D A
Department of Physics. Faculty of Science, Kagoshima University. Kagoshima, Japan The transverse dynamical spin susceptibility of ferromagnetic Heusler alloys X2MnY(X = Ni, Cu, Pd, Y= AI, Sn) is evaluated within RPA with the use of the energy values and the wave functions determined by the T.B.-OPW method. The magnetic interactions in these alloys are investigated through the results and compared with experimental results.
In the ferromagnetic Heusler alloys X2MnY(X = Ni, Cu, Pd, Y = AI, Sn), Ishikawa et al. [1] have studied the magnetic interactions in the alloys by the neutron spin wave scattering and obtained the following results. (a) The nearest neighbor interaction depends strongly on X-atoms and the interaction is largest for Cu-Heusler and it decreases in the sequence of C u - N i - P d . (b) The exchange integrals at large distances have an oscillatory character which can be interpreted by the s - d type interaction based on the nearly free electron model. In this situation, we have evaluated the spin wave spectra in the three alloys (Cu2MnA1, Ni2MnSn and Pd2MnSn) by using RPA theory with the use of the energy values and the wave functions determined by the T.B.-OPW method [2]. The spin wave spectrum Im X- +(q, go~)can be deduced from the pole of X- +(q, go). In the case of XzMnY, the X-+(q, go) is given as an approximate expression [3] by
X-+(q, go)=lF(q)12~'~X~(q, go) (/~ = 1-30),
(/L, ~ = 1-15, p = 1-30),
(3)
e(q, ~o) = I + A(q, go),
(4)
Ape(q , go) = U~,~(q)x°e(q, go) (/x, ~ = 1-15),
(5)
where 1 is the unit matrix. Thus, the evaluation of the pole of X- +(q, go) is reduced to the determination of the pole of e-l(q, go). The Ueff.~ d-d (q) have been given so as to satisfy the condition det(c(q, 0~)) = 0. In the procedure, as approximations, U~n.Mn( q o - d ) = U~ef.ed-a(q)for ~ = 1-5 and U~n.x(q d-n ) = U~f(q) for ~ = 6-15 are used. Now, the value of U~ff, d-d M,(0) corresponds to the exchange-splitting value AE between majority-spin and minority-spin
A
~'~(~,,.,)
(1)
X~,(q, ~o) = E X ° , ( q , 60) - EX°e(q, gO) U~ff.d-d~ (q) ×Xe(q, go)
(/~, ~' = 1-30, ~ = 1-15),
(2)
where F(q) is a form factor, #, ~, and ~ the symmetry orbital indices [2] (d-orbitals of Mn-atom, those of X-atoms and conduction states (OPW's) correspond to the indices (1-5), (6-15) and (16-30), respectively) and X°,(q, go) is an unenhanced magnetic susceptibility. The values U~f(q) of the effective exchange interaction are taken as adjustable parameters. The main source of spin wave excitations is due to the transitions between the d-orbital states represented by the T.B-OPW method, and the pole of X- + (q, go) is mainly determined by the d-orbital parts in (2) (these parts correspond to the cases # = 1-15 in (2) and are denoted by X~(q, go)). The function X~(q, o~1 is written in terms of the dielectric function c(q, go) as
,,,) = E
1.0
2D
Irn~L°n(qow)
3.0fiw(eV)/d) 1:~12Mn7
5,0
60
'/I)
8.0
/~
,\
go)xL(q, go)
* Present address: University of Library and Information Science, Yatabe-macbi, Tsukuba-gun, Ibaraki 305, Japan.
Fig. 1. Imaginary parts of the unenhanced dynamical susceptibility only due to the diagonal terms of the d-electronic spin-flip excitations of Mn.
0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
48
Y. Kubo, S. Ishida / Spin dynamics in Heusler alloys
d-bands of Mn, then we can take AE as the criterion for the evaluation of U~ff.Mn(q). d-d Thus, the value U~ff.Mn(q d- d ) is estimated from the condition U~f~(q)Re X°~(q, ~)--- 1 (# = ~ = 1-5) in (5). However, we have no such a d- d(q) criterion for the value U~ff, d-ax (q), then, values for Ueff,X have been taken rather tentatively [4-6]. Because the values of Re X°~(q, o~) for X-atom (/x = ~ = 6-15) are about 1 / 2 0 - 1 / 1 0 0 of those for Mn, as long as reasonable values are taken for U~dx(q), there is no noticeable difference among the obtained results. The spin wave energies ktos thus o b t ~ n e d [4,5] are in a fairly good correspondence with the experimental results [ 1]. The contributions to the spin wave spectrum Im X - + ( q , ~s) from partial spin correlation functions Im X~,(q, O~s)for lg = 1-30 in (2) have been evaluated. It was found that the cross terms between the d-electrons of M n and conduction electrons contribute to Im X - +(q, ~0s) for small q(~O,O,1/4)2zt/a) as well as those between the d-electrons of Mn[4-6]. This is consistent with the experimental result (b). Furthermore, the obtained values ued~dn(q) have a larger q-dependence for X = Cu. The spectra Im X°n(q, t 0 ) = 5-'.~_l l m X°~,(q, o~) (/~ = 1-5) for Cu2MnA1 and Pd2MnSn are shown in
fig. 1. In the case of Cu2MnAI, it should be noted that the vertical scale in q = (O,O,l/4)2~r/a is two times larger than that in the others. As seen in fig. l, Im X°n(q, o~) in X = Cu significantly decreases with increasing value of k/I in contrast with the case of X = P d . In the case of X = Ni, the situation is nearly intermediate. This is the reason why the q-dependence of the UeJ~dn(q) for X = Cu is particularly larger than those of the other alloys. This indicates that an inter-atomic effective exchange interaction in the T.B.-part of the Hamiltonian [6] is important. This situation is consistent with the experimental result (a).
References [1] Y. Ishikawa and Y. Noda, AlP Conf. Proc. 24 (1975) 145. [2] S. Ishida, H. Asato, E. Iwashima, Y. Kubo and J. Ishida, J. Phys. FI 1 (1981) 1035. [3] J.F. Cooke, Phys. Rev. B7 (1973) 1108. [4] Y. Kubo, S. Ishida, J. Ishida and S. Asano, J. Phys. Soc. Japan 48 (1980) 407. [5] Y. Kubo, S. Ishida and J. Ishida, J. Phys. Soc. Japan 50 (1980 47. [6] Y. Kubo and S. Ishida, J. Phys. Soc. Japan to be published.
47
SPIN DYNAMICS IN HEUSLER ALLOYS Y. K U B O * A N D S. I S H I D A
Department of Physics. Faculty of Science, Kagoshima University. Kagoshima, Japan The transverse dynamical spin susceptibility of ferromagnetic Heusler alloys X2MnY(X = Ni, Cu, Pd, Y= AI, Sn) is evaluated within RPA with the use of the energy values and the wave functions determined by the T.B.-OPW method. The magnetic interactions in these alloys are investigated through the results and compared with experimental results.
In the ferromagnetic Heusler alloys X2MnY(X = Ni, Cu, Pd, Y = AI, Sn), Ishikawa et al. [1] have studied the magnetic interactions in the alloys by the neutron spin wave scattering and obtained the following results. (a) The nearest neighbor interaction depends strongly on X-atoms and the interaction is largest for Cu-Heusler and it decreases in the sequence of C u - N i - P d . (b) The exchange integrals at large distances have an oscillatory character which can be interpreted by the s - d type interaction based on the nearly free electron model. In this situation, we have evaluated the spin wave spectra in the three alloys (Cu2MnA1, Ni2MnSn and Pd2MnSn) by using RPA theory with the use of the energy values and the wave functions determined by the T.B.-OPW method [2]. The spin wave spectrum Im X- +(q, go~)can be deduced from the pole of X- +(q, go). In the case of XzMnY, the X-+(q, go) is given as an approximate expression [3] by
X-+(q, go)=lF(q)12~'~X~(q, go) (/~ = 1-30),
(/L, ~ = 1-15, p = 1-30),
(3)
e(q, ~o) = I + A(q, go),
(4)
Ape(q , go) = U~,~(q)x°e(q, go) (/x, ~ = 1-15),
(5)
where 1 is the unit matrix. Thus, the evaluation of the pole of X- +(q, go) is reduced to the determination of the pole of e-l(q, go). The Ueff.~ d-d (q) have been given so as to satisfy the condition det(c(q, 0~)) = 0. In the procedure, as approximations, U~n.Mn( q o - d ) = U~ef.ed-a(q)for ~ = 1-5 and U~n.x(q d-n ) = U~f(q) for ~ = 6-15 are used. Now, the value of U~ff, d-d M,(0) corresponds to the exchange-splitting value AE between majority-spin and minority-spin
A
~'~(~,,.,)
(1)
X~,(q, ~o) = E X ° , ( q , 60) - EX°e(q, gO) U~ff.d-d~ (q) ×Xe(q, go)
(/~, ~' = 1-30, ~ = 1-15),
(2)
where F(q) is a form factor, #, ~, and ~ the symmetry orbital indices [2] (d-orbitals of Mn-atom, those of X-atoms and conduction states (OPW's) correspond to the indices (1-5), (6-15) and (16-30), respectively) and X°,(q, go) is an unenhanced magnetic susceptibility. The values U~f(q) of the effective exchange interaction are taken as adjustable parameters. The main source of spin wave excitations is due to the transitions between the d-orbital states represented by the T.B-OPW method, and the pole of X- + (q, go) is mainly determined by the d-orbital parts in (2) (these parts correspond to the cases # = 1-15 in (2) and are denoted by X~(q, go)). The function X~(q, o~1 is written in terms of the dielectric function c(q, go) as
,,,) = E
1.0
2D
Irn~L°n(qow)
3.0fiw(eV)/d) 1:~12Mn7
5,0
60
'/I)
8.0
/~
,\
go)xL(q, go)
* Present address: University of Library and Information Science, Yatabe-macbi, Tsukuba-gun, Ibaraki 305, Japan.
Fig. 1. Imaginary parts of the unenhanced dynamical susceptibility only due to the diagonal terms of the d-electronic spin-flip excitations of Mn.
0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
48
Y. Kubo, S. Ishida / Spin dynamics in Heusler alloys
d-bands of Mn, then we can take AE as the criterion for the evaluation of U~ff.Mn(q). d-d Thus, the value U~ff.Mn(q d- d ) is estimated from the condition U~f~(q)Re X°~(q, ~)--- 1 (# = ~ = 1-5) in (5). However, we have no such a d- d(q) criterion for the value U~ff, d-ax (q), then, values for Ueff,X have been taken rather tentatively [4-6]. Because the values of Re X°~(q, o~) for X-atom (/x = ~ = 6-15) are about 1 / 2 0 - 1 / 1 0 0 of those for Mn, as long as reasonable values are taken for U~dx(q), there is no noticeable difference among the obtained results. The spin wave energies ktos thus o b t ~ n e d [4,5] are in a fairly good correspondence with the experimental results [ 1]. The contributions to the spin wave spectrum Im X - + ( q , ~s) from partial spin correlation functions Im X~,(q, O~s)for lg = 1-30 in (2) have been evaluated. It was found that the cross terms between the d-electrons of M n and conduction electrons contribute to Im X - +(q, ~0s) for small q(~O,O,1/4)2zt/a) as well as those between the d-electrons of Mn[4-6]. This is consistent with the experimental result (b). Furthermore, the obtained values ued~dn(q) have a larger q-dependence for X = Cu. The spectra Im X°n(q, t 0 ) = 5-'.~_l l m X°~,(q, o~) (/~ = 1-5) for Cu2MnA1 and Pd2MnSn are shown in
fig. 1. In the case of Cu2MnAI, it should be noted that the vertical scale in q = (O,O,l/4)2~r/a is two times larger than that in the others. As seen in fig. l, Im X°n(q, o~) in X = Cu significantly decreases with increasing value of k/I in contrast with the case of X = P d . In the case of X = Ni, the situation is nearly intermediate. This is the reason why the q-dependence of the UeJ~dn(q) for X = Cu is particularly larger than those of the other alloys. This indicates that an inter-atomic effective exchange interaction in the T.B.-part of the Hamiltonian [6] is important. This situation is consistent with the experimental result (a).
References [1] Y. Ishikawa and Y. Noda, AlP Conf. Proc. 24 (1975) 145. [2] S. Ishida, H. Asato, E. Iwashima, Y. Kubo and J. Ishida, J. Phys. FI 1 (1981) 1035. [3] J.F. Cooke, Phys. Rev. B7 (1973) 1108. [4] Y. Kubo, S. Ishida, J. Ishida and S. Asano, J. Phys. Soc. Japan 48 (1980) 407. [5] Y. Kubo, S. Ishida and J. Ishida, J. Phys. Soc. Japan 50 (1980 47. [6] Y. Kubo and S. Ishida, J. Phys. Soc. Japan to be published.