- Email: [email protected]
Negative high field magnetoresistance in 3d ferromagnets
Physica B 294}295 (2001) 102}106
Negative high "eld magnetoresistance in 3d ferromagnets B. Raquet *, M. Viret, P. Warin, E. Sondergard, R. Mamy Laboratoire de Physique de la Matie% re Condense& e de Toulouse, LPMC-INSA, av. de Rangueil 31077 Toulouse, France CEA Saclay, Service de l+e& tat condense& , F-91191, Gif sur Yvette, France Laboratoire National des Champs Magne& tiques Pulse& s, LNCMP-INSA, av. de Rangueil, 31077 Toulouse, France
Abstract We report on the purely magnetic contribution to the electronic transport in 3d ferromagnets. Negative and linear high magnetic "eld magnetoresistance, well above the technical magnetization saturation, probes the magnetic disorder and is modeled in term of spin scattering due to spin-#ip s}d inter-band electronic transitions. 2001 Elsevier Science B.V. All rights reserved. Keywords: Band ferromagnetism; Magnetoresistance; Magnon scattering
1. Introduction 3d transition metals Fe, Co, Ni are well-known model systems in which much work has been devoted to the basic understanding of itinerant ferromagnetism. Current research is widely focused on thin "lm structures of Fe, Co and Ni in which low-"eld spin-dependent scattering on nanoscale magnetic inhomogeneities is intensively studied [1]. High-"eld e!ects on the resistivity were studied in the 1970s on high-quality whiskers. At low temperatures, Shubnikov}de Haas oscillations in the quantum limit have provided direct information on the Fermi surface [2]. However, little has been reported at higher temperature on the magnetic
* Correspondence address: Laboratoire de Physique de la Matie`re CondenseH e de Toulouse, LPMC-INSA, av. de Rangueil 31077 Toulouse, France. Tel.: #33-61-55-99-67; fax: #33-6155-99-50. E-mail address: [email protected] (B. Raquet).
contribution to the electronic transport in high magnetic "eld, well above the technical magnetization saturation. In the high-temperature regime, but still well below ¹ , the high "eld magnetoresis tance (MR) on Fe, Co, Ni exhibits a remarkable linear negative signal with no departure toward saturation up to 40 T. The MR slopes are in the range of 10\ cm T\ at 300 K and strongly increase with temperature. Our systematic MR study of Fe, Co and Ni thin "lms provides new data from which the spin-dependent scattering processes can be re-analyzed. The high-"eld negative MR is related to the magnetic contribution to the resistivity, (¹). However, its origin, magnitude and
temperature dependence are still unresolved in the high-temperature regime [3]. We propose an extension of the two current conduction model in which the negative and linear MR originates from the magnetic "eld e!ect on the spin mixing resistivity. Our simulations support the idea that the main contribution is from spin-#ip s}d inter-band scattering. In particular, we put forward
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 6 1 8 - 9
B. Raquet et al. / Physica B 294}295 (2001) 102}106
103
an estimate of the spin-dependent e\}magnon scattering and its dumping under high "eld. Comparison with experiment is discussed in term of conduction band splitting due to ferromagnetic exchange and intrinsic magnon properties in Fe, Co, Ni.
2. Samples and experiments Magnetoresistance has been measured on Fe, Co and Ni thin "lms deposited on MgO and Al O substrates by molecular bean epitaxy (for Fe and Ni) and by thermal evaporation in ultra-high vacuum deposition chamber (for Co). The "lm thicknesses range from 7 to 80 nm. A residual resistance ratio above 20 for the thicker "lms attests their high structural quality. We present a systematic study of the magnetoresistance measured on patterned "lms, in the longitudinal con"guration (iH and in plane) between 4 and 500 K, up to a 40 T pulsed "eld. We focus on the high-"eld regime where the intrinsic anisotropic MR and any potential giant MR due to grain boundaries have no more e!ects on the magnetoresistive signal.
Fig. 1. High-"eld magnetoresistance, in the longitudinal con"guration and at various temperatures measured on Fe (a), Co (b) and Ni (c) patterned thin "lms. The low-"eld resistivity drop in Co is an extrinsic MR e!ect due to grain boundaries.
3. Results and analysis For Fe, Co and Ni, the high-"eld MR changes from positive to negative when the electronic mean free path is reduced by increasing the temperature (Fig. 1). In the high-temperature regime, but still well below ¹ , a surprisingly high linear negative MR is observed with no departure toward saturation. A signi"cant enhancement of this negative slope is observed with increasing temperature. In the metallic regime, the magnetic "eld dependence of the resistivity can be empirically expressed as a function of the relaxation time : ()" C (1/)#C (w )\. The "rst term originates from the electronic scattering processes and varies inversely with the mean free path. The second term is the well-known normal MR due to the Lorentz force which, in a classical schematic view, curves the electron trajectories. The induced increase of resistance is very sensitive to the sample purity and dominates the MR at low temperature. For high
quality "lms, the purely magnetic MR is only observable at rather high temperature and preferentially in the longitudinal con"guration where the Lorentz force is negligible. Following the Matthiessen's rule, we de"ne the resistivity by: (¹, H)" # (¹)# (¹, H), .
where is the residual resistivity, (¹) the . phonon scattering and (¹, H), the spin-depen dent scattering due magnetic disorder. The high"eld MR probes the variations of (H)"
(H)! (0)where the absolute (¹, H)
value remains unclear. Fig. 1 shows our high-"eld MR measurements where the negative and linear MR slope appears at higher temperature. The extracted slopes R (H, ¹)/RH for Fe, Co Ni at
several temperatures are presented in Fig. 2. If we normalize the temperature by the respective Curie temperatures, we notice that the high-"eld scattering for the three ferromagnets roughly scales on a same curve (inset of Fig. 2). This approach
104
B. Raquet et al. / Physica B 294}295 (2001) 102}106
Fig. 2. Temperature dependence of the high-"eld MR slope (H, ¹)/H for Fe, Co and Ni thin "lms. In the inset, the
temperature is normalized by ¹ .
con"rms that the resistivity variation originates in spin disorder. We point out that these data were taken on several Fe, Co and Ni thin "lms with various thichnesses. The high magnetic "eld spindependent scattering processes are, therefore, weakly dependent on surface and structural e!ects. Generally speaking, spin disorder resistivity (¹) originates from spin-#ip processes both
intra and inter band [s!}s8], [d!}d8] and [s!}d8]. These electronic transitions are due to e\}e\ Coulomb interactions and e\-spin wave scattering. Considering a simpli"ed band structure for strong itinerant ferromagnets, previous calculations have shown the major resistive contribution of inelastic [s!}d8] transitions assisted by annihilation or creation of magnon [3}6], above some temperature of the order of 20 K. Unfortunately, a theoretical estimate of (¹) is only tractable
under drastic assumptions on the band structure. Its temperature dependence in the high temperature regime remains unclear and strong e\}phonon scattering prevents any accurate experimental determination of (¹). In that re spect, we argue that the high-"eld dependence of the spin disorder resistivity provides a new interesting insight on magnetic disorder acting as spindependent scattering centers. In order to understand the physical origin of (¹, H), we propose an extension of the two
current conduction model where the high magnetic "eld is explicitly introduced. In ferromagnetic 3d
metals, the current is essentially carried by the s electrons which are scattered into spin polarized d bands split by the exchange interaction [7]. The s band can then be decomposed in two spin subbands with distinct relaxation times > and \. Spin mixing processes are described by an extra relaxation time ! which tends to equalize the two band currents. We consider here that the spin mixing resistivity (H) mainly results from spin-#ip ! s}d transitions due to e\}magnon scattering which a!ects ! much more than > and \ [3,7]. Theoretical estimates of the spin mixing resistivity (H) ! have to be compared to the spin disorder term (¹) related to our measured MR curves. We
introduce the e!ect of a magnetic "eld on the inelastic electronic transition rate with annihilation of a spin wave q, considering a "eld-induced gap in the magnon dispersion relation: (q)"Dq#H, where is the magnon susceptibility equal to "2g . In the spirit of previous calculations [7], we obtain the expression of the spin mixing resistivity including explicitly temperature and "eld dependences (detailed calculations will be published elsewhere): m NJS (¹, H)" k ¹ ! ne k $
;
z exp z !Ln(exp z!1) exp z!1
(1)
with
z"
(m NJS) m 2 H ; # . (k ) k ¹ $
(2)
Here, NJS is the band conduction splitting resulting from the ferromagnetic exchange coupling. Calculations of (¹, H) performed for Fe, Co ! and Ni with the values in Table 1 for the physical parameters are displayed in Figs. 3a}c. The temperature dependence of (¹, H"0) (Fig. 3a) and its ! magnitude for the 3d elements are consistent with previous calculations [4], in the range of few cm at RT. The strength of the spin disorder resistivity and the di!erence between $C (¹), !M (¹) and
B. Raquet et al. / Physica B 294}295 (2001) 102}106 Table 1 Physical parameters commonly used for Fe, Co and Ni [5]
Fe Co Ni
(10\ kg)
k (As \) $
S
N (10 m\)
12.4 8.46 8.23
1.35 1.38 1.39
1.06 0.77 0.27
8.47 9.09 9.17
Fig. 3. (a) Theoretical temperature dependence of (H) based ! on the e\}magnon scattering model for Fe, Co and Ni, deduced from Eq. (1) with the physical parameters of Table 1. (b) Simulations of the high-"eld magnetic "eld dependence of (H) for Fe, ! Co and Ni. (c) Theoretical estimate of the high-"eld sensitivity (H, ¹)/H. The inset is the same data plotted with the ! ¹ normalization. The dashed line is related to the account of the temperature dependence of the magnon mass in Ni.
,G (¹) are mainly governed by the e!ective mag non mass, i.e. the spin wave sti!ness constant and the exchange splitting in the conduction band. Heavy magnons and large band splitting enhance spin disorder scattering and consistently explain $C (¹)'!M (¹)',G (¹). More relevant is the
105
"eld dependence of this spin disorder resistivity: our model predicts a linear high "eld decrease of (H) (Fig. 3b), in qualitative agreement with our
experimental observations. Simulations also reveal a higher negative MR slope for Ni (Fig. 3c), in accordance with our data (Fig. 2). The magnetic "eld sensitivity is mostly determined by the e!ective magnon mass and the minimum accessible spinwave q vector related to the minimum distance between the Fermi surfaces of the two sub-bands: the spin disorder resistivity is more e!ectively reduced by an applied "eld if the energy gap in the magnon spectrum is large compared to the minimum magnon energy Dq involved in the scatter ing process and the magnon mass is heavy. This argument fully explains the higher "eld dependence we measure for ,G (H). Moreover, the reasonable
scaling we obtain (insets in Figs. 2 and 3c) by normalizing the temperature by ¹ (which roughly scales the energy band splitting to a unique value) is an indication of the validity of the relevant physical parameters. Finally, we emphasize that, despite the qualitative agreement, the theoretical negative MR slopes deduced from the model are smaller than the experimental ones for the three elements: Fe, Co and Ni. The experimental temperature dependence of the high "eld R (H, ¹)/RH values is also
higher than that predicted by the model which hints that e\}magnon is perhaps not the only relevant spin-dependent scattering process. For ¹/¹ (0.6, an interesting improvement of the model comes from the increase of the e!ective magnon mass with temperature. Considering the standard polynomial expression for the magnon mass as a function of T de"ned by " /[1! / ¹# / ¹ ], where / and / are constants of the order of 10\ and 10\, respectively [8], leads to a signi"cant increase of (¹, H), R (H, ¹)/RH and
its temperature dependence, in better agreement with the measurements (dashed lines in Fig. 3c). We conclude that probing the high-"eld magnetic contribution to the resistivity in the high-temperature regime provides an interesting insight into some fundamental aspects of spin di!usion scattering in band ferromagnetism.
106
B. Raquet et al. / Physica B 294}295 (2001) 102}106
Experimental work is in progress to study the high"eld MR in weak ferromagnets for which the magnetic susceptibility at high "eld is large. More work is also needed to improve the high-"eld e\}magnon scattering model to explicitly include the s and d conduction electrons. Calculations are in progress to study in which respect, s}d electronic transition with spin-#ip due to incoherent magnetic disorder may a!ect the spin disorder resistivity and its magnetic "eld dependence.
References [1] M. Viret et al., Phys. Rev. B 53 (1996) 8464. [2] R.V. Coleman, R.C. Morris, D.J. Sellmeyer, Phys. Rev. B 8 (1973) 317. [3] B. Loegel, F. Gautier, J. Phys. Chem Solids 32 (1971) 2723. [4] D.A. Goodings, Phys. Rev. 132 (1963) 542. [5] L. Colquitt, Phys. Rev. 139 (1965) A1857. [6] G.R. Taylor, A. Isin, V. Coleman, Phys. Rev. 165 (1968) 621. [7] A. Fert, J. Phys. C 2 (1969) 1784. [8] M.W. Stringfellow, J. Phys. C 1 (1968) 950.
Negative high "eld magnetoresistance in 3d ferromagnets B. Raquet *, M. Viret, P. Warin, E. Sondergard, R. Mamy Laboratoire de Physique de la Matie% re Condense& e de Toulouse, LPMC-INSA, av. de Rangueil 31077 Toulouse, France CEA Saclay, Service de l+e& tat condense& , F-91191, Gif sur Yvette, France Laboratoire National des Champs Magne& tiques Pulse& s, LNCMP-INSA, av. de Rangueil, 31077 Toulouse, France
Abstract We report on the purely magnetic contribution to the electronic transport in 3d ferromagnets. Negative and linear high magnetic "eld magnetoresistance, well above the technical magnetization saturation, probes the magnetic disorder and is modeled in term of spin scattering due to spin-#ip s}d inter-band electronic transitions. 2001 Elsevier Science B.V. All rights reserved. Keywords: Band ferromagnetism; Magnetoresistance; Magnon scattering
1. Introduction 3d transition metals Fe, Co, Ni are well-known model systems in which much work has been devoted to the basic understanding of itinerant ferromagnetism. Current research is widely focused on thin "lm structures of Fe, Co and Ni in which low-"eld spin-dependent scattering on nanoscale magnetic inhomogeneities is intensively studied [1]. High-"eld e!ects on the resistivity were studied in the 1970s on high-quality whiskers. At low temperatures, Shubnikov}de Haas oscillations in the quantum limit have provided direct information on the Fermi surface [2]. However, little has been reported at higher temperature on the magnetic
* Correspondence address: Laboratoire de Physique de la Matie`re CondenseH e de Toulouse, LPMC-INSA, av. de Rangueil 31077 Toulouse, France. Tel.: #33-61-55-99-67; fax: #33-6155-99-50. E-mail address: [email protected] (B. Raquet).
contribution to the electronic transport in high magnetic "eld, well above the technical magnetization saturation. In the high-temperature regime, but still well below ¹ , the high "eld magnetoresis tance (MR) on Fe, Co, Ni exhibits a remarkable linear negative signal with no departure toward saturation up to 40 T. The MR slopes are in the range of 10\ cm T\ at 300 K and strongly increase with temperature. Our systematic MR study of Fe, Co and Ni thin "lms provides new data from which the spin-dependent scattering processes can be re-analyzed. The high-"eld negative MR is related to the magnetic contribution to the resistivity, (¹). However, its origin, magnitude and
temperature dependence are still unresolved in the high-temperature regime [3]. We propose an extension of the two current conduction model in which the negative and linear MR originates from the magnetic "eld e!ect on the spin mixing resistivity. Our simulations support the idea that the main contribution is from spin-#ip s}d inter-band scattering. In particular, we put forward
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 6 1 8 - 9
B. Raquet et al. / Physica B 294}295 (2001) 102}106
103
an estimate of the spin-dependent e\}magnon scattering and its dumping under high "eld. Comparison with experiment is discussed in term of conduction band splitting due to ferromagnetic exchange and intrinsic magnon properties in Fe, Co, Ni.
2. Samples and experiments Magnetoresistance has been measured on Fe, Co and Ni thin "lms deposited on MgO and Al O substrates by molecular bean epitaxy (for Fe and Ni) and by thermal evaporation in ultra-high vacuum deposition chamber (for Co). The "lm thicknesses range from 7 to 80 nm. A residual resistance ratio above 20 for the thicker "lms attests their high structural quality. We present a systematic study of the magnetoresistance measured on patterned "lms, in the longitudinal con"guration (iH and in plane) between 4 and 500 K, up to a 40 T pulsed "eld. We focus on the high-"eld regime where the intrinsic anisotropic MR and any potential giant MR due to grain boundaries have no more e!ects on the magnetoresistive signal.
Fig. 1. High-"eld magnetoresistance, in the longitudinal con"guration and at various temperatures measured on Fe (a), Co (b) and Ni (c) patterned thin "lms. The low-"eld resistivity drop in Co is an extrinsic MR e!ect due to grain boundaries.
3. Results and analysis For Fe, Co and Ni, the high-"eld MR changes from positive to negative when the electronic mean free path is reduced by increasing the temperature (Fig. 1). In the high-temperature regime, but still well below ¹ , a surprisingly high linear negative MR is observed with no departure toward saturation. A signi"cant enhancement of this negative slope is observed with increasing temperature. In the metallic regime, the magnetic "eld dependence of the resistivity can be empirically expressed as a function of the relaxation time : ()" C (1/)#C (w )\. The "rst term originates from the electronic scattering processes and varies inversely with the mean free path. The second term is the well-known normal MR due to the Lorentz force which, in a classical schematic view, curves the electron trajectories. The induced increase of resistance is very sensitive to the sample purity and dominates the MR at low temperature. For high
quality "lms, the purely magnetic MR is only observable at rather high temperature and preferentially in the longitudinal con"guration where the Lorentz force is negligible. Following the Matthiessen's rule, we de"ne the resistivity by: (¹, H)" # (¹)# (¹, H), .
where is the residual resistivity, (¹) the . phonon scattering and (¹, H), the spin-depen dent scattering due magnetic disorder. The high"eld MR probes the variations of (H)"
(H)! (0)where the absolute (¹, H)
value remains unclear. Fig. 1 shows our high-"eld MR measurements where the negative and linear MR slope appears at higher temperature. The extracted slopes R (H, ¹)/RH for Fe, Co Ni at
several temperatures are presented in Fig. 2. If we normalize the temperature by the respective Curie temperatures, we notice that the high-"eld scattering for the three ferromagnets roughly scales on a same curve (inset of Fig. 2). This approach
104
B. Raquet et al. / Physica B 294}295 (2001) 102}106
Fig. 2. Temperature dependence of the high-"eld MR slope (H, ¹)/H for Fe, Co and Ni thin "lms. In the inset, the
temperature is normalized by ¹ .
con"rms that the resistivity variation originates in spin disorder. We point out that these data were taken on several Fe, Co and Ni thin "lms with various thichnesses. The high magnetic "eld spindependent scattering processes are, therefore, weakly dependent on surface and structural e!ects. Generally speaking, spin disorder resistivity (¹) originates from spin-#ip processes both
intra and inter band [s!}s8], [d!}d8] and [s!}d8]. These electronic transitions are due to e\}e\ Coulomb interactions and e\-spin wave scattering. Considering a simpli"ed band structure for strong itinerant ferromagnets, previous calculations have shown the major resistive contribution of inelastic [s!}d8] transitions assisted by annihilation or creation of magnon [3}6], above some temperature of the order of 20 K. Unfortunately, a theoretical estimate of (¹) is only tractable
under drastic assumptions on the band structure. Its temperature dependence in the high temperature regime remains unclear and strong e\}phonon scattering prevents any accurate experimental determination of (¹). In that re spect, we argue that the high-"eld dependence of the spin disorder resistivity provides a new interesting insight on magnetic disorder acting as spindependent scattering centers. In order to understand the physical origin of (¹, H), we propose an extension of the two
current conduction model where the high magnetic "eld is explicitly introduced. In ferromagnetic 3d
metals, the current is essentially carried by the s electrons which are scattered into spin polarized d bands split by the exchange interaction [7]. The s band can then be decomposed in two spin subbands with distinct relaxation times > and \. Spin mixing processes are described by an extra relaxation time ! which tends to equalize the two band currents. We consider here that the spin mixing resistivity (H) mainly results from spin-#ip ! s}d transitions due to e\}magnon scattering which a!ects ! much more than > and \ [3,7]. Theoretical estimates of the spin mixing resistivity (H) ! have to be compared to the spin disorder term (¹) related to our measured MR curves. We
introduce the e!ect of a magnetic "eld on the inelastic electronic transition rate with annihilation of a spin wave q, considering a "eld-induced gap in the magnon dispersion relation: (q)"Dq#H, where is the magnon susceptibility equal to "2g . In the spirit of previous calculations [7], we obtain the expression of the spin mixing resistivity including explicitly temperature and "eld dependences (detailed calculations will be published elsewhere): m NJS (¹, H)" k ¹ ! ne k $
;
z exp z !Ln(exp z!1) exp z!1
(1)
with
z"
(m NJS) m 2 H ; # . (k ) k ¹ $
(2)
Here, NJS is the band conduction splitting resulting from the ferromagnetic exchange coupling. Calculations of (¹, H) performed for Fe, Co ! and Ni with the values in Table 1 for the physical parameters are displayed in Figs. 3a}c. The temperature dependence of (¹, H"0) (Fig. 3a) and its ! magnitude for the 3d elements are consistent with previous calculations [4], in the range of few cm at RT. The strength of the spin disorder resistivity and the di!erence between $C (¹), !M (¹) and
B. Raquet et al. / Physica B 294}295 (2001) 102}106 Table 1 Physical parameters commonly used for Fe, Co and Ni [5]
Fe Co Ni
(10\ kg)
k (As \) $
S
N (10 m\)
12.4 8.46 8.23
1.35 1.38 1.39
1.06 0.77 0.27
8.47 9.09 9.17
Fig. 3. (a) Theoretical temperature dependence of (H) based ! on the e\}magnon scattering model for Fe, Co and Ni, deduced from Eq. (1) with the physical parameters of Table 1. (b) Simulations of the high-"eld magnetic "eld dependence of (H) for Fe, ! Co and Ni. (c) Theoretical estimate of the high-"eld sensitivity (H, ¹)/H. The inset is the same data plotted with the ! ¹ normalization. The dashed line is related to the account of the temperature dependence of the magnon mass in Ni.
,G (¹) are mainly governed by the e!ective mag non mass, i.e. the spin wave sti!ness constant and the exchange splitting in the conduction band. Heavy magnons and large band splitting enhance spin disorder scattering and consistently explain $C (¹)'!M (¹)',G (¹). More relevant is the
105
"eld dependence of this spin disorder resistivity: our model predicts a linear high "eld decrease of (H) (Fig. 3b), in qualitative agreement with our
experimental observations. Simulations also reveal a higher negative MR slope for Ni (Fig. 3c), in accordance with our data (Fig. 2). The magnetic "eld sensitivity is mostly determined by the e!ective magnon mass and the minimum accessible spinwave q vector related to the minimum distance between the Fermi surfaces of the two sub-bands: the spin disorder resistivity is more e!ectively reduced by an applied "eld if the energy gap in the magnon spectrum is large compared to the minimum magnon energy Dq involved in the scatter ing process and the magnon mass is heavy. This argument fully explains the higher "eld dependence we measure for ,G (H). Moreover, the reasonable
scaling we obtain (insets in Figs. 2 and 3c) by normalizing the temperature by ¹ (which roughly scales the energy band splitting to a unique value) is an indication of the validity of the relevant physical parameters. Finally, we emphasize that, despite the qualitative agreement, the theoretical negative MR slopes deduced from the model are smaller than the experimental ones for the three elements: Fe, Co and Ni. The experimental temperature dependence of the high "eld R (H, ¹)/RH values is also
higher than that predicted by the model which hints that e\}magnon is perhaps not the only relevant spin-dependent scattering process. For ¹/¹ (0.6, an interesting improvement of the model comes from the increase of the e!ective magnon mass with temperature. Considering the standard polynomial expression for the magnon mass as a function of T de"ned by " /[1! / ¹# / ¹ ], where / and / are constants of the order of 10\ and 10\, respectively [8], leads to a signi"cant increase of (¹, H), R (H, ¹)/RH and
its temperature dependence, in better agreement with the measurements (dashed lines in Fig. 3c). We conclude that probing the high-"eld magnetic contribution to the resistivity in the high-temperature regime provides an interesting insight into some fundamental aspects of spin di!usion scattering in band ferromagnetism.
106
B. Raquet et al. / Physica B 294}295 (2001) 102}106
Experimental work is in progress to study the high"eld MR in weak ferromagnets for which the magnetic susceptibility at high "eld is large. More work is also needed to improve the high-"eld e\}magnon scattering model to explicitly include the s and d conduction electrons. Calculations are in progress to study in which respect, s}d electronic transition with spin-#ip due to incoherent magnetic disorder may a!ect the spin disorder resistivity and its magnetic "eld dependence.
References [1] M. Viret et al., Phys. Rev. B 53 (1996) 8464. [2] R.V. Coleman, R.C. Morris, D.J. Sellmeyer, Phys. Rev. B 8 (1973) 317. [3] B. Loegel, F. Gautier, J. Phys. Chem Solids 32 (1971) 2723. [4] D.A. Goodings, Phys. Rev. 132 (1963) 542. [5] L. Colquitt, Phys. Rev. 139 (1965) A1857. [6] G.R. Taylor, A. Isin, V. Coleman, Phys. Rev. 165 (1968) 621. [7] A. Fert, J. Phys. C 2 (1969) 1784. [8] M.W. Stringfellow, J. Phys. C 1 (1968) 950.