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Extraordinary hall effect (EHE) in amorphous transition-metal ferromagnets
Journal
Extraordinary ferromagnets
of Magnetism
and Magnetic
Hall Effect (EHE) in amorphous
Materials
99 (1991) 190-192 North-Holland
transition-metal
A.V. Vedyayev, A.B. Granovsky and A.D. Arsenieva Moscow State University, Physics Department, Received
The effect imply bands.
21 March
119899 Moscow,
USSR
1991
expression for the EHE coefficient R, is derived for a transition-metal originates from the skew-scattering of d-type electrons by the spin-orbit different relaxation times of elastic scattering and densities of states at Linear terms in the R, temperature dependence are shown to arise from
1. Introduction The Hall voltage of ferromagnetic alloys can be written in the form
metals and
Ey = ROB, j, + Rs4TMz j,,
(1)
where B, - magnetic induction, M, - magnetization, and the EHE coefficient R, is at least one order of magnitude larger than that of the ordinary Hall effect R,. R, is proportional to the non-diagonal part of the conductivity tensor dXY :
ferromagnet under the assumption that the interaction. The spin-splitted d-type carriers the Fermi level for spin-up and spin-down these properties.
where /.3is a coefficient dependent on the alloy composition but not on the temperature. The analysis of EHE in the framework of the coherent potential approximation theory [l] brought to the understanding that the skew scattering of spin-splitted d-type carriers by the spin-orbit interaction plays a dominant role for EHE in transition metal alloys. Some details of the R, temperature dependence at high temperatures in amorphous ferromagnets are studied in ref. [2], where the elastic scattering of the d-type states on a structure disorder is considered basing on the Faber-Ziman model and with account for the skew-scattering mechanism.
(2)
where p is the full resistivity. The main cause of EHE is the spin-orbit interaction. Two mechanisms are usually considered, namely, the “skewscattering” and the “side-jump”. The last leads to the following (universal for all temperatures and scattering mechanisms) temperature dependence of R,:
(3)
R, = ,+l’t 030~8853/91/$03.50
0 1991 - Elsevier Science Publishers
2. Formulation
and results
In this work we derive the expression for R, in the low temperature region (T CK T,, T, is the Curie temperature) treating the interaction of carriers with the magnetic subsystem of the alloy in the spin-wave approximation. As in ref. [2], we assume that the effect originates from the skew-
B.V. All rights reserved
A. V. Vehayev
191
et al. / EHE in amorphous TM ferromagnets
scattering of d-type electrons by the spin-orbit interaction. The Hamiltonian of the system is
R:’ = Apo( J/V)‘(
- 1)2( y(‘) - 1)
b,( T/T,)@
+ b, ( T/T,)2( y(i) - 1 - 0.25y’2’)) H=~E
+a
k.oak,o
k,o
+ Ri”( J/V)2b,(
k,o
+N-’
C
(L,(k,
R2)=R~o’(J,V)2~(1
k’)aLt.akt,
k,k’ n,a +L,+(k,
+ N-3/2
+
exp[i(k
+ c2( T/Q2
k’)allaktl)
(6)
+)(c,(T/r,) - 1)))
+ c3( T/Tc)3’2(o.5y(1)
exp[i(k-k’)Rz]
(7) where Rio’ is the residual coefficient of EHE, (Y= p,‘/pk , p. is the residual part of the resistivity, /I = N;/NG , R(,O) ari ses from scattering on the structure disorder and contains the third-order correlation function
C exp[i(k-k’-q)Rz](2S)“2 k,k’,q n,a
- k’ + q)RE](2S)“2
F$i(k,
k’, k”) =Np2
(4) where u = t or J, Rz is the radius vector of an atom of the sort (Y, L: = V, T J,S, + Hg; F$(k, k’), J,(k, k’), H,*(k, k’) are the matrix elements of Coulomb, exchange and spin-orbit interactions. H,“)(k, k’) = iX”,“(k, k’)[k, k’],M,, where AT is the spin-orbit interaction parameter. The spin-splitted carriers form two subbands with different densities of states at the Fermi level Nr! # Nk and different relaxation times of the elastic scattering 7 T # TV. (The last are well defined in amorphous systems because the inelastic processes of electron-magnon scattering leading to spin transfer are much weaker than the elastic ones.) The third-order terms of the scattering potential are relevant for the linear in the spinorbit interaction part of a,,. In this approximation, there are no non-diagonal terms of the statistical matrix in quantum kinetic equations [3], and they have the form of Boltzrnann equations. We take into account in the collision terms both the elastic scattering of electrons on the structure disorder and the electron-magnon scattering. The equations are solved by expansion in the ratio of the elastic and inelastic relaxation times, this ratio being small for amorphous materials. Our result can be written in the form R = R(o) + R(l) + R(2) s 9 S S S
T/T,)3’2,
(5)
c exp{i[(k-k’)Rt n,m,m’
+ (k’ - k”)Rf, +(k”-k)RY,t]}. This term and its possible temperature dependence at T > TD (TD is the Debye temperature) is discussed in ref. [2]. Parameters y(“) are defined as the ratios y(“) = M’“‘(2k,)/M’3’(2k,), where M’“‘(2k,) M’“‘(2k,)
=
(8)
are the moments dq
1 +S,8(2k,q)q” jc 0 %B (J)
(9)
of the structure factors Saa of the alloy SaB(k) = N-’
c exp[ik(RE n,m
- R!)]
-NC?,,,. (10)
The terms linear and quadratic in temperature arise from the inelastic coherent (momentum-conserving) scattering, while the term T312 comes from the incoherent inelastic processes.
3. Discussion The new and most important result is the appearance of the linear terms (when (Y# 1 in R’,”
192
A. V. Vedyayev et al. / EHE in amorphow
and (Y# 1, fi # 1 in Ri2)), which can dominate in the ferromagnetic region. When (Y= 1 and j3 = 1, (5)-(8) turn into the expression R, = R~“‘+A,po(T/Tc)2
+A2R$o)(T/c)3’2
(11)
similar to that derived for the crystalline alloys by the variational method without accounting for the difference in the “elastic” relaxation times [4]. This difference, characterised in our case by the parameter (Y= p:/pk 0: T~/T ?, seems to be quite natural for the d-type carriers (for s-type electrons it can arise, for example, from the standard Mott’s mechanism which leads to the well-known twocurrent model). While the linear term in R’,” is “ruled” by the elastic relaxation times ratio a and, generally, appears both in the cases of s- and d-type carriers, there are two necessary conditions for the non-zero Ri2’: a # 1 and p = Ni/NG # 1, the last being held for the d-type carriers only. The linear term in R(,‘) and Rg’ are roughly proportional to the ratio (r t - r 1 )/t, where t is the effective relaxation time for the inelastic scattering processes. The same ratio is considered [5] to be the main reason of extremum in the temperature dependence of thermoelectric power (TEP) of amorphous and disordered crystalline ferromagnets in the middle temperature range 0 < T-c T,. That is why one can expect the occurrence of correlating extrema in temperature dependences of TEP and the EHE coefficient measured on the same alloy sample. The contributions to R, are rather sensitive, besides the above discussed parameters (r and p, to the details of the ionic structure of an alloy. That is, the parameters y(“) dependent on the
TM ferromagnets
alloy composition can strongly influence the values and signs of relevant terms. It is necessary to mention that R’,O’is proportional to the third-order correlation function (which can be roughly approximated by the structure factors product) and hence at low temperatures it depends as (1 + p(T/TD)2) on T. For this reason the term quadratic in the temperature of R, arises both from the scattering on phonons and on magnons. An experimental identification of different contributions to various terms of R, is possible due to the difference in their concentration dependence. The precised measurements of the temperature dependence of R, at low (T-SK T,) temperatures have not yet been fulfilled. The ratio of the temperature dependent to the temperature independent terms in R, is a small value of the order (J/ V)2. But the experimental identification of the temperature dependent terms, especially those linear in temperature, could bring to understanding whether the skew scattering dominates in EHE and enlighten the type of carriers in the amorphous ferromagnetic alloys.
References [l] Ye.1. Kondorsky, A.V. Vedyayev and A.B. Granovsky, Fiz. Met. Metallov. 40 (1975) 455, 903, 688. A.V. Vedyayev, A.N. Voloshinsky, A.B. Granovsky and N.V. Ryzhanova, Izv. VU2 ser. Fizika (1987) 66. [2] A.V. Vedyayev and A.B. Granovsky, Fiz. Met. Metallov. 58 (1984) 1084. [3] J.M. Luttinger, Phys. Rev. 112 (1958) 739. [4] A.N. Voloshinsky and N.V. Ryzhanova, Fii. Met. Metal. 35 (1973) 269. [5] LYa. Korenblit, J. Phys. F 12 (1982) 1259.
Extraordinary ferromagnets
of Magnetism
and Magnetic
Hall Effect (EHE) in amorphous
Materials
99 (1991) 190-192 North-Holland
transition-metal
A.V. Vedyayev, A.B. Granovsky and A.D. Arsenieva Moscow State University, Physics Department, Received
The effect imply bands.
21 March
119899 Moscow,
USSR
1991
expression for the EHE coefficient R, is derived for a transition-metal originates from the skew-scattering of d-type electrons by the spin-orbit different relaxation times of elastic scattering and densities of states at Linear terms in the R, temperature dependence are shown to arise from
1. Introduction The Hall voltage of ferromagnetic alloys can be written in the form
metals and
Ey = ROB, j, + Rs4TMz j,,
(1)
where B, - magnetic induction, M, - magnetization, and the EHE coefficient R, is at least one order of magnitude larger than that of the ordinary Hall effect R,. R, is proportional to the non-diagonal part of the conductivity tensor dXY :
ferromagnet under the assumption that the interaction. The spin-splitted d-type carriers the Fermi level for spin-up and spin-down these properties.
where /.3is a coefficient dependent on the alloy composition but not on the temperature. The analysis of EHE in the framework of the coherent potential approximation theory [l] brought to the understanding that the skew scattering of spin-splitted d-type carriers by the spin-orbit interaction plays a dominant role for EHE in transition metal alloys. Some details of the R, temperature dependence at high temperatures in amorphous ferromagnets are studied in ref. [2], where the elastic scattering of the d-type states on a structure disorder is considered basing on the Faber-Ziman model and with account for the skew-scattering mechanism.
(2)
where p is the full resistivity. The main cause of EHE is the spin-orbit interaction. Two mechanisms are usually considered, namely, the “skewscattering” and the “side-jump”. The last leads to the following (universal for all temperatures and scattering mechanisms) temperature dependence of R,:
(3)
R, = ,+
0 1991 - Elsevier Science Publishers
2. Formulation
and results
In this work we derive the expression for R, in the low temperature region (T CK T,, T, is the Curie temperature) treating the interaction of carriers with the magnetic subsystem of the alloy in the spin-wave approximation. As in ref. [2], we assume that the effect originates from the skew-
B.V. All rights reserved
A. V. Vehayev
191
et al. / EHE in amorphous TM ferromagnets
scattering of d-type electrons by the spin-orbit interaction. The Hamiltonian of the system is
R:’ = Apo( J/V)‘(
- 1)2( y(‘) - 1)
b,( T/T,)@
+ b, ( T/T,)2( y(i) - 1 - 0.25y’2’)) H=~E
+a
k.oak,o
k,o
+ Ri”( J/V)2b,(
k,o
+N-’
C
(L,(k,
R2)=R~o’(J,V)2~(1
k’)aLt.akt,
k,k’ n,a +L,+(k,
+ N-3/2
+
exp[i(k
+ c2( T/Q2
k’)allaktl)
(6)
+)(c,(T/r,) - 1)))
+ c3( T/Tc)3’2(o.5y(1)
exp[i(k-k’)Rz]
(7) where Rio’ is the residual coefficient of EHE, (Y= p,‘/pk , p. is the residual part of the resistivity, /I = N;/NG , R(,O) ari ses from scattering on the structure disorder and contains the third-order correlation function
C exp[i(k-k’-q)Rz](2S)“2 k,k’,q n,a
- k’ + q)RE](2S)“2
F$i(k,
k’, k”) =Np2
(4) where u = t or J, Rz is the radius vector of an atom of the sort (Y, L: = V, T J,S, + Hg; F$(k, k’), J,(k, k’), H,*(k, k’) are the matrix elements of Coulomb, exchange and spin-orbit interactions. H,“)(k, k’) = iX”,“(k, k’)[k, k’],M,, where AT is the spin-orbit interaction parameter. The spin-splitted carriers form two subbands with different densities of states at the Fermi level Nr! # Nk and different relaxation times of the elastic scattering 7 T # TV. (The last are well defined in amorphous systems because the inelastic processes of electron-magnon scattering leading to spin transfer are much weaker than the elastic ones.) The third-order terms of the scattering potential are relevant for the linear in the spinorbit interaction part of a,,. In this approximation, there are no non-diagonal terms of the statistical matrix in quantum kinetic equations [3], and they have the form of Boltzrnann equations. We take into account in the collision terms both the elastic scattering of electrons on the structure disorder and the electron-magnon scattering. The equations are solved by expansion in the ratio of the elastic and inelastic relaxation times, this ratio being small for amorphous materials. Our result can be written in the form R = R(o) + R(l) + R(2) s 9 S S S
T/T,)3’2,
(5)
c exp{i[(k-k’)Rt n,m,m’
+ (k’ - k”)Rf, +(k”-k)RY,t]}. This term and its possible temperature dependence at T > TD (TD is the Debye temperature) is discussed in ref. [2]. Parameters y(“) are defined as the ratios y(“) = M’“‘(2k,)/M’3’(2k,), where M’“‘(2k,) M’“‘(2k,)
=
(8)
are the moments dq
1 +S,8(2k,q)q” jc 0 %B (J)
(9)
of the structure factors Saa of the alloy SaB(k) = N-’
c exp[ik(RE n,m
- R!)]
-NC?,,,. (10)
The terms linear and quadratic in temperature arise from the inelastic coherent (momentum-conserving) scattering, while the term T312 comes from the incoherent inelastic processes.
3. Discussion The new and most important result is the appearance of the linear terms (when (Y# 1 in R’,”
192
A. V. Vedyayev et al. / EHE in amorphow
and (Y# 1, fi # 1 in Ri2)), which can dominate in the ferromagnetic region. When (Y= 1 and j3 = 1, (5)-(8) turn into the expression R, = R~“‘+A,po(T/Tc)2
+A2R$o)(T/c)3’2
(11)
similar to that derived for the crystalline alloys by the variational method without accounting for the difference in the “elastic” relaxation times [4]. This difference, characterised in our case by the parameter (Y= p:/pk 0: T~/T ?, seems to be quite natural for the d-type carriers (for s-type electrons it can arise, for example, from the standard Mott’s mechanism which leads to the well-known twocurrent model). While the linear term in R’,” is “ruled” by the elastic relaxation times ratio a and, generally, appears both in the cases of s- and d-type carriers, there are two necessary conditions for the non-zero Ri2’: a # 1 and p = Ni/NG # 1, the last being held for the d-type carriers only. The linear term in R(,‘) and Rg’ are roughly proportional to the ratio (r t - r 1 )/t, where t is the effective relaxation time for the inelastic scattering processes. The same ratio is considered [5] to be the main reason of extremum in the temperature dependence of thermoelectric power (TEP) of amorphous and disordered crystalline ferromagnets in the middle temperature range 0 < T-c T,. That is why one can expect the occurrence of correlating extrema in temperature dependences of TEP and the EHE coefficient measured on the same alloy sample. The contributions to R, are rather sensitive, besides the above discussed parameters (r and p, to the details of the ionic structure of an alloy. That is, the parameters y(“) dependent on the
TM ferromagnets
alloy composition can strongly influence the values and signs of relevant terms. It is necessary to mention that R’,O’is proportional to the third-order correlation function (which can be roughly approximated by the structure factors product) and hence at low temperatures it depends as (1 + p(T/TD)2) on T. For this reason the term quadratic in the temperature of R, arises both from the scattering on phonons and on magnons. An experimental identification of different contributions to various terms of R, is possible due to the difference in their concentration dependence. The precised measurements of the temperature dependence of R, at low (T-SK T,) temperatures have not yet been fulfilled. The ratio of the temperature dependent to the temperature independent terms in R, is a small value of the order (J/ V)2. But the experimental identification of the temperature dependent terms, especially those linear in temperature, could bring to understanding whether the skew scattering dominates in EHE and enlighten the type of carriers in the amorphous ferromagnetic alloys.
References [l] Ye.1. Kondorsky, A.V. Vedyayev and A.B. Granovsky, Fiz. Met. Metallov. 40 (1975) 455, 903, 688. A.V. Vedyayev, A.N. Voloshinsky, A.B. Granovsky and N.V. Ryzhanova, Izv. VU2 ser. Fizika (1987) 66. [2] A.V. Vedyayev and A.B. Granovsky, Fiz. Met. Metallov. 58 (1984) 1084. [3] J.M. Luttinger, Phys. Rev. 112 (1958) 739. [4] A.N. Voloshinsky and N.V. Ryzhanova, Fii. Met. Metal. 35 (1973) 269. [5] LYa. Korenblit, J. Phys. F 12 (1982) 1259.