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“Giant magnetoresistance” and the magnetic phase transition in TbNiSn
>-__ BB
a
ELSEWIER
13 January 1997 PHYSICS
LETTERS
A
Physics Letters A 224 (1997) 293-297
“Giant magnetoresistance” and the magnetic phase transition in TbNiSn Hidenobu
Hori”, Akio Oki”, Makio Kurisu”, Masahiro Furusawa”, Koichi Kindo ’
Yoshikazu Andohb,
a Sclmd of Materials Science, Japan Advanced Institute of Science and Technology (JAIST), Tatsunokuchi. Ishiwaka 923.12, Japan h Faculty of Education, Tottori University, Tottori 680. Japan c Research Center for Materials Science at Extreme Conditions. Osaka University, Toyonaka. Osaka 560, Japan
Received 2 September 1996; revised manuscript received 21 October 1996; accepted for publication 24 October 1996 Communicated by J. Flouquet
Abstract Extraordinary large changes are found in the magnetoresistance of TbNiSn single crystals at liquid helium temperatures. The large resistance changes correspond well to the magnetic phase transitions observed in multi-step magnetization processes. Such a magnetoresistance behavior of TbNiSn is explained well by the use of the semi-empirical equations based on the Boltzmann transport theory. The results of the analysis require that the magnetoresistance in the phase below 2.1 K is determined by the ground state electrons which form a “magnetic lattice”, while the other electrons in the excited state determine the phase above 2.1 K. PAC.? 72.15.Eb; 72.15.Qm; 73.50.Jt: 74.25.F~ Keywords: Magnetoresistance; Multi-step magnetization; Phase transition
1. Introduction
The rare-earth intermetallic equiatomic ternary compounds RNiSn crystals (R = rare-earth ion) show characteristic transport properties, which are discussed in connection with the recent topics, such as the highly correlated electron problem [ 11. As is seen in the discussion of the so-called magnetic superzone effect [ 2,3], the band structure near the Fermi level in the magnetically ordered rare-earth metal is modulated by the interaction between the Bloch electrons and the localized spins with the commensurate period [4]. The modulation affects the electric resistance through the variation of the effective mass and the density of carriers. Another important magnetic
origin of the resistance is the spin fluctuation effect, which causes the variation of the spin-dependent relaxation time through the s-f exchange interaction. Thus, the analysis of the magnetoresistance is considered to be one of the most important procedures to investigate highly correlated electron systems. But physically clear analysis of the magneto-transport problem has not appropriately been done for magnetic ordered crystals, so far. Some RNiSn-type crystals with large orbital moment have multi-step magnetization processes in magnetic fields. The variation of the spin fluctuation and magnetic superzone effects are expected in such materials and a systematic analysis can be made by using the variation of the phase transitions. As reported in
0375.9601/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00829-S
294
H. Hori et d/Physics
Fig. I. The temperature dependent phase transition in TbNiSn. (a) Temperature dependence of magnetic susceptibility along the three principal axes. (b) Temperature dependence of resistivity along the principal axes. The two insets in both data sets show the kinks in the susceptibility and resistivity curves near the phase transition at 2. I K, respectively. An applied field of 0.05 T in the susceptibility measurements causes a slight high temperature shift of the kink ISI.
the previous work [ 51, TbNiSn single crystal shows a distinctive multi-step magnetization process and a remarkably large giant magnetoresistance. The origin of the giant magnetoresistance, however, has not been explained so far, Against such a background, the experimental results and detailed analysis of the magnetoresistance are given in this work for TbNiSn at liquid helium temperatures. The first magnetic measurements of RNiSn, which has an orthorhombic TiNiSi-type crystal structure (Pnma), have been performed in the temperature range between 4.2 and 200 K by use of polycrystalline samples [ 61. Routsi et al. reported that the crystals of R = Tb and Dy are ordered antiferromagnetically below 10 K. No other phase transitions could be observed in the temperature range down to 4.2 K, although they suggested the existence of two successive phase transitions [ 61. As reported in our previous work, the temperature dependence of successive phase transitions has been observed for the single crystals of both TbNiSn and DyNiSn [ 53.
2. Experimental The magnetization and magnetoresistance measurements in this work were performed in high magnetic
Letters A 224 (1997) 293-297
Fig. 2. Magnetization K, and its anisotropy
process along the h-axis at 1.4 K and 4.2 at I .4 K.
produced by the pulsed magnet in The Center for Materials Science at Extreme Conditions, Osaka University, and in the steady high magnetic field produced by a superconducting magnet in JAIST. Fig. la shows the data of the temperature dependence of the magnetic susceptibility. The temperature dependence of the electric resistance is also given in Fig. lb. Three successive phase transitions with increasing temperature are seen in these data. Recently, another new phase transition at 2.1 K with a small peak in the specific heat was observed [ 71. The preliminary neutron experiments show that the phase transition does not change the spin structure in zero field [ 8 1. As shown in the inset to Figs. la and b, the very small kinks around 2 K both on the resistance and the susceptibility curves are also confirmed by our group. The changes in these quantities are, however, very small at the transition temperature, while the change in magnetoresistance is quite large, as shown in Fig. 3. This result means that the phase transition arises from the difference in the process of electron scattering. To investigate the phase transition, our attention is focused on the data with the magnetic field ,LLOHor Bc parallel to the b-axis. Fig. 2 shows the anisotropic magnetization process in the magnetic field Bc along the b-axis at 1.4 and 4.2 K. A good correspondence between the abrupt changes in magnetoresistance and magnetization are found, as shown in Fig. 3. This shows that the change of the magnetoresistance comes from the spin structure. The transition fields B,I , Bc2, fields
H. Hori et d/Physics
Letters A 224 (1997) 293-297
shows a positive magnetoresistance nature while the data at 4.2 K indicates a negative result means that the change in the basic phenomena is related to the phase transition
295
at 1.4 K, one. This physical at 2.1 K.
3. Discussions
Fig. 3. Magnetoresistance at 1.4 and 4.2 K. The solid lines show the theoretical curves. Bc3 and B,j at 1.4 K are given by 0.6, l&4.3 and 5.3 T, respectively. The magnetization at each step in 1.4 K are estimated to be 0, &MS, iM,, $Ms and M, for the phases I: (O-&I), II: (B,l-Bc2), III: (Bc2-Bc3), IV: ( Bc3-Bc4) and V: ( Bc4 <), respectively. The M, value of 8.1,~~ at 1.4 K is approximately consistent with the theoretically expected saturation gJZ value of 9.0,~ at 0 K. The slight difference of the saturation moment can be explained by conduction electron spin polarization, which antiferromagneticaliy shields the 4f-localized spin and possibly crystal field effects. Taking account of these effects, the observed magnetization M per Tb’+ is given by the localized 4f-moment MT and the parameter A, which represents the mean spin polarization per one Tb3+ spin of the conduction electron. Thus M can be written as M =
AM,.
(1)
The value of A is 0.9. So a large reduction of M is not expected in the small crystalline field with an overall splitting value of 124 K [5]. The spin-structure dependence of the parameter A can be considered, because the spin fluctuation or the stability of the spin structure for each phase is different. As shown in Fig. 3, an abrupt increase of the magnetoresistance is noticeable in phases II and IV. In particular in phase II, a remarkable difference is found between the data of both temperatures, although the magnetization curves at both temperatures are similar. Moreover, it should be noticed that the resistance does not always increase with increasing magnetization and it is also remarkable that phase II
The problems in the experimental results are summarized as follows: (I) Why does the localized moment at thedeep 4f-level sensitively affect the conduction electron, although the 4f-level is separated from the Fermi level &F by about one eV? (2) How can we explain the relation between the magnetoresistance and magnetization data? (3) What kind of phase transition is produced at 2.1 K? Since a theoretically rigorous or physically clear formula has not been given for the problems so far, the following magnetic model is based on a semiempirical formula. The spin on each Tb3+ induces a spin polarization on the conduction electrons near the Fermi level &Fby s-f exchange interaction. Moreover. the induced spin moments around the localized spins interfere with each other and form a kind of lattice of spin density at T = 0 K. The structure of such a “spin density lattice” follows the localized 4f-spin structure of Tb3+. Then, the spatial period of the field induced phase is not always consistent with the spatial period of the crystallographic lattice. Then. the coupling between the phonons and the spin system is varied in each magnetic phase, because phonons reflect the real lattice structure. The spin fluctuation of the conduction electrons affects the resistance through the spin dependent relaxation time. When the “spin density lattice” has a period commensurate with the real lattice, some deformation of the conduction band near EF is expected. The deformation effect changes the effective mass m’ and it affects the resistivity. The discussion of the “magnetic superzone effect” by Miwa [ 41 strongly supports this model. In order to use the one-particle theory [ 9 1, the following model is assumed: the conduction electrons are separated into ground state electrons and the other excited state ones. The electrons which form the “spin density lattice” are in the ground state and have the periodicity of the localized spin structure at 0 K, while the other electrons are produced by thermal excitation
296
H. Hori et al./ Physics Letters A 224 (1997) 293-297
from the “magnetic lattice state” and the periodicity is independent from the magnetic structure. Because the ground state electron has the periodicity of the magnetic lattice, they do not make a large contribution to the resistance. The excited electrons, however, make a large contribution for the resistance through the magnetic scattering, because they do not have the periodicity of the magnetic lattice. Each excited electron sees the 4f-spin with the spin polarization as the isolated spin scatterers with the moment M. Then, the formula in Ref. [ 91 is applicable for the excited electrons. The total resistance R in every phase is given by the summation of the magnetic part R( T, Ho) and another part Ro(T) which originates from the usual non-magnetic scattering: R = Ro(T)
+ R(T, Ho) .
[ (M,)2 + F (T)(SM:) @Md2
-
1 + FtWuBffo/bT)
(M;) = M2 2i l”?,
(AM,)2
=
(aM:)/(M;)
; M2),
N (M2 - (cY(AM~)>~).
(3)
(5)
’
(6M:) = (g,urd2(W+ 1) -
1’
(4)
CM+- M-)2 2
Phase
p (Y Rn
1.4 K 4.2 K 1.4 K 4.2 K 1.4K 4.2 K
I
11
111
IV
V
0.080 0.11 1.1” 1.1” 0 0.74
0.052 0.12 1.1’ 3.09 I.17 1.31
0.035 0.094 1.1” 1.1” 0.86 0.74
0.073 0.11 1.1” 1.1” 0 0.74
0.086 0.15 I.15 1.15 0 0.74
a The value is the theoretical one for the isotropic case. because the fit is very insensitive to the [Y parameter in this phase.
(2)
The resistivity Ro(T) depends on the ratio between the ground and excited electron numbers, because both kinds of electrons have different scattering manners, respectively. Then, Ro is considered to be a measure for the number of excited electrons. The resistance R(T, Ho) is produced by excited electrons that are scattered by the magnetically shielded net moment M. The resistance formula is given by M, of the localized moment, the up-spin staggered magnetic moment M+ and the down-spin moment M_ and the ratio between Zeeman and thermal energies of gpnHn/kgT. Considering the difference in the excited electron scattering is not necessary because of the low energy excitation at low temperature, the formula [ 91 based on the Boltzmann transport equation is given by
R(T,Ho) N p(m*)
Table I Fitting parameters in magnetoresistance. R,I in phase V is given by zero, subtracting the residual resistivity of 0.13 x IOeh Rem determined from the data in phase V at 1.4 K. The error of the fitting parameters are estimated to be &3% except for the value with a footnote. P’/~ in lo-? 12’/” cm’/*, R,j in IOeh 0 cm
s:, (6)
F(x) = p(m*)
&,
=
f-@(G)‘.
(8)
where r, J, N and S are the numerical constant, exchange parameter, number of electrons and magnitude of the electron spin, respectively. The new additional fitting parameter LYin Eq. (6) is introduced to express the anisotropy effect of the Tb’+ spin. The anisotropy effect of the band structure is ignored, because the present case shows no clear anisotropic effects on the resistivity. The first term in parentheses in Eq. (3) originates from the spin-conservation scattering and the second term arises from the spin-reversed scattering process. Although the spin structure of this crystal has not been reported so far, the resistance for all phases is characterized by these four parameters. The field dependence in the resistance at 1.4 and 4.2 K are calculated by the use of the magnetization data shown in Fig. 2. The fitting parameter p(m*) is proportional to (wz* J/N) 2 which includes the information of the conduction band near &F. The value of p’/2, which is directly proportional to rn*J/N, is shown in Table 1. As is seen in Fig. 3, the theoretical solid curves at 1.4 and 4.2 K are in good agreement with the experimental data. The values of the fitting parameters used in the calculation are listed in Table 1. The parameter LYdetermined as 1.15 at the saturation field is consistent with the value of A in Eq. ( 1). The large value of cyin phase II at 4.2 K is remarkable.
H. Hori et al./ Physics Letters A 224 (1997) 293-297
The following reason can be considered: phase II with the magnetization of iMS shows the large magnetic cell, and many Brillouin zone boundaries intersect the Fermi surface. Then the intersection reduces the number of “effective” carriers. The reduction is anisotropic in phase II, because of the intersection between the band and large magnetic cell. Thus the anisotropic effect can reflect on the value of CY. As the conclusion of this work, the difference in Ro between phase I and II means that the number of ground state electrons at 4.2 K is remarkably larger than that at 1.4 K and the difference in p suggests a temperature variation of the conduction band near &F, which is related to the superzone effect. Taking account of the fact that there is no substantial change in the spin structure between 1.4 and 4.2 K [ 81, the magnetoresistance data lead to the model that the phase below 2.1 K is ruled by “magnetic lattice” electrons. All estimated values are physically appropriate and such a consistent result strongly supports the model and analysis in this work. This experiment and the analysis method are expected to lead to a new analysis method for the resistance in the highly correlated spin system related with the spin fluctuation.
297
Acknowledgement This work was performed through the financial support by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of the Japanese Government and is also partly supported by the Special Coordination Funds of the Science and Technology Agency of the Japanese Government.
References T. Takabatake,Y. Nakazawa and M. Ishikawa, Jpn. J. Appl. Phys. 26 Suppl. 3 (1987) 547. R.J. Elliot and EA. Wedgwood, Proc. R. Sot. 81 (1963) 846. A.E. Dwight. J. Less-Common Met. 93 ( 1983) 41 I. H. Miwa. Prog. Theor. Phys. 29 (1963) 477. M. Kurisu, H. Hori, M. Furusawa. M. Miyake, Y. Andoh, 1. Oguro, K. Kindo, T. Takeuchi and A. Yamagishi. Physica B 201 (1994) 107. J. Magn. [61 C. Routsi, J.K. Yakinthos and H. Gamari-Seale, Magn. Mater. 117 ( 1992) 79. [71 T. Suzuki, M. Kitamura, T. Fujita, Y. Andoh and M. Kurisu. in: Abstr. Meeting Phys. Sot. Japan, 50th Annual Meeting, Part 3 (1995) 170 [in Japanese]. US1 Y. Andoh and M. Kurisu et al., in preparation. 191 K. Yosida, Phys. Rev. 107 (1957) 396.
a
ELSEWIER
13 January 1997 PHYSICS
LETTERS
A
Physics Letters A 224 (1997) 293-297
“Giant magnetoresistance” and the magnetic phase transition in TbNiSn Hidenobu
Hori”, Akio Oki”, Makio Kurisu”, Masahiro Furusawa”, Koichi Kindo ’
Yoshikazu Andohb,
a Sclmd of Materials Science, Japan Advanced Institute of Science and Technology (JAIST), Tatsunokuchi. Ishiwaka 923.12, Japan h Faculty of Education, Tottori University, Tottori 680. Japan c Research Center for Materials Science at Extreme Conditions. Osaka University, Toyonaka. Osaka 560, Japan
Received 2 September 1996; revised manuscript received 21 October 1996; accepted for publication 24 October 1996 Communicated by J. Flouquet
Abstract Extraordinary large changes are found in the magnetoresistance of TbNiSn single crystals at liquid helium temperatures. The large resistance changes correspond well to the magnetic phase transitions observed in multi-step magnetization processes. Such a magnetoresistance behavior of TbNiSn is explained well by the use of the semi-empirical equations based on the Boltzmann transport theory. The results of the analysis require that the magnetoresistance in the phase below 2.1 K is determined by the ground state electrons which form a “magnetic lattice”, while the other electrons in the excited state determine the phase above 2.1 K. PAC.? 72.15.Eb; 72.15.Qm; 73.50.Jt: 74.25.F~ Keywords: Magnetoresistance; Multi-step magnetization; Phase transition
1. Introduction
The rare-earth intermetallic equiatomic ternary compounds RNiSn crystals (R = rare-earth ion) show characteristic transport properties, which are discussed in connection with the recent topics, such as the highly correlated electron problem [ 11. As is seen in the discussion of the so-called magnetic superzone effect [ 2,3], the band structure near the Fermi level in the magnetically ordered rare-earth metal is modulated by the interaction between the Bloch electrons and the localized spins with the commensurate period [4]. The modulation affects the electric resistance through the variation of the effective mass and the density of carriers. Another important magnetic
origin of the resistance is the spin fluctuation effect, which causes the variation of the spin-dependent relaxation time through the s-f exchange interaction. Thus, the analysis of the magnetoresistance is considered to be one of the most important procedures to investigate highly correlated electron systems. But physically clear analysis of the magneto-transport problem has not appropriately been done for magnetic ordered crystals, so far. Some RNiSn-type crystals with large orbital moment have multi-step magnetization processes in magnetic fields. The variation of the spin fluctuation and magnetic superzone effects are expected in such materials and a systematic analysis can be made by using the variation of the phase transitions. As reported in
0375.9601/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00829-S
294
H. Hori et d/Physics
Fig. I. The temperature dependent phase transition in TbNiSn. (a) Temperature dependence of magnetic susceptibility along the three principal axes. (b) Temperature dependence of resistivity along the principal axes. The two insets in both data sets show the kinks in the susceptibility and resistivity curves near the phase transition at 2. I K, respectively. An applied field of 0.05 T in the susceptibility measurements causes a slight high temperature shift of the kink ISI.
the previous work [ 51, TbNiSn single crystal shows a distinctive multi-step magnetization process and a remarkably large giant magnetoresistance. The origin of the giant magnetoresistance, however, has not been explained so far, Against such a background, the experimental results and detailed analysis of the magnetoresistance are given in this work for TbNiSn at liquid helium temperatures. The first magnetic measurements of RNiSn, which has an orthorhombic TiNiSi-type crystal structure (Pnma), have been performed in the temperature range between 4.2 and 200 K by use of polycrystalline samples [ 61. Routsi et al. reported that the crystals of R = Tb and Dy are ordered antiferromagnetically below 10 K. No other phase transitions could be observed in the temperature range down to 4.2 K, although they suggested the existence of two successive phase transitions [ 61. As reported in our previous work, the temperature dependence of successive phase transitions has been observed for the single crystals of both TbNiSn and DyNiSn [ 53.
2. Experimental The magnetization and magnetoresistance measurements in this work were performed in high magnetic
Letters A 224 (1997) 293-297
Fig. 2. Magnetization K, and its anisotropy
process along the h-axis at 1.4 K and 4.2 at I .4 K.
produced by the pulsed magnet in The Center for Materials Science at Extreme Conditions, Osaka University, and in the steady high magnetic field produced by a superconducting magnet in JAIST. Fig. la shows the data of the temperature dependence of the magnetic susceptibility. The temperature dependence of the electric resistance is also given in Fig. lb. Three successive phase transitions with increasing temperature are seen in these data. Recently, another new phase transition at 2.1 K with a small peak in the specific heat was observed [ 71. The preliminary neutron experiments show that the phase transition does not change the spin structure in zero field [ 8 1. As shown in the inset to Figs. la and b, the very small kinks around 2 K both on the resistance and the susceptibility curves are also confirmed by our group. The changes in these quantities are, however, very small at the transition temperature, while the change in magnetoresistance is quite large, as shown in Fig. 3. This result means that the phase transition arises from the difference in the process of electron scattering. To investigate the phase transition, our attention is focused on the data with the magnetic field ,LLOHor Bc parallel to the b-axis. Fig. 2 shows the anisotropic magnetization process in the magnetic field Bc along the b-axis at 1.4 and 4.2 K. A good correspondence between the abrupt changes in magnetoresistance and magnetization are found, as shown in Fig. 3. This shows that the change of the magnetoresistance comes from the spin structure. The transition fields B,I , Bc2, fields
H. Hori et d/Physics
Letters A 224 (1997) 293-297
shows a positive magnetoresistance nature while the data at 4.2 K indicates a negative result means that the change in the basic phenomena is related to the phase transition
295
at 1.4 K, one. This physical at 2.1 K.
3. Discussions
Fig. 3. Magnetoresistance at 1.4 and 4.2 K. The solid lines show the theoretical curves. Bc3 and B,j at 1.4 K are given by 0.6, l&4.3 and 5.3 T, respectively. The magnetization at each step in 1.4 K are estimated to be 0, &MS, iM,, $Ms and M, for the phases I: (O-&I), II: (B,l-Bc2), III: (Bc2-Bc3), IV: ( Bc3-Bc4) and V: ( Bc4 <), respectively. The M, value of 8.1,~~ at 1.4 K is approximately consistent with the theoretically expected saturation gJZ value of 9.0,~ at 0 K. The slight difference of the saturation moment can be explained by conduction electron spin polarization, which antiferromagneticaliy shields the 4f-localized spin and possibly crystal field effects. Taking account of these effects, the observed magnetization M per Tb’+ is given by the localized 4f-moment MT and the parameter A, which represents the mean spin polarization per one Tb3+ spin of the conduction electron. Thus M can be written as M =
AM,.
(1)
The value of A is 0.9. So a large reduction of M is not expected in the small crystalline field with an overall splitting value of 124 K [5]. The spin-structure dependence of the parameter A can be considered, because the spin fluctuation or the stability of the spin structure for each phase is different. As shown in Fig. 3, an abrupt increase of the magnetoresistance is noticeable in phases II and IV. In particular in phase II, a remarkable difference is found between the data of both temperatures, although the magnetization curves at both temperatures are similar. Moreover, it should be noticed that the resistance does not always increase with increasing magnetization and it is also remarkable that phase II
The problems in the experimental results are summarized as follows: (I) Why does the localized moment at thedeep 4f-level sensitively affect the conduction electron, although the 4f-level is separated from the Fermi level &F by about one eV? (2) How can we explain the relation between the magnetoresistance and magnetization data? (3) What kind of phase transition is produced at 2.1 K? Since a theoretically rigorous or physically clear formula has not been given for the problems so far, the following magnetic model is based on a semiempirical formula. The spin on each Tb3+ induces a spin polarization on the conduction electrons near the Fermi level &Fby s-f exchange interaction. Moreover. the induced spin moments around the localized spins interfere with each other and form a kind of lattice of spin density at T = 0 K. The structure of such a “spin density lattice” follows the localized 4f-spin structure of Tb3+. Then, the spatial period of the field induced phase is not always consistent with the spatial period of the crystallographic lattice. Then. the coupling between the phonons and the spin system is varied in each magnetic phase, because phonons reflect the real lattice structure. The spin fluctuation of the conduction electrons affects the resistance through the spin dependent relaxation time. When the “spin density lattice” has a period commensurate with the real lattice, some deformation of the conduction band near EF is expected. The deformation effect changes the effective mass m’ and it affects the resistivity. The discussion of the “magnetic superzone effect” by Miwa [ 41 strongly supports this model. In order to use the one-particle theory [ 9 1, the following model is assumed: the conduction electrons are separated into ground state electrons and the other excited state ones. The electrons which form the “spin density lattice” are in the ground state and have the periodicity of the localized spin structure at 0 K, while the other electrons are produced by thermal excitation
296
H. Hori et al./ Physics Letters A 224 (1997) 293-297
from the “magnetic lattice state” and the periodicity is independent from the magnetic structure. Because the ground state electron has the periodicity of the magnetic lattice, they do not make a large contribution to the resistance. The excited electrons, however, make a large contribution for the resistance through the magnetic scattering, because they do not have the periodicity of the magnetic lattice. Each excited electron sees the 4f-spin with the spin polarization as the isolated spin scatterers with the moment M. Then, the formula in Ref. [ 91 is applicable for the excited electrons. The total resistance R in every phase is given by the summation of the magnetic part R( T, Ho) and another part Ro(T) which originates from the usual non-magnetic scattering: R = Ro(T)
+ R(T, Ho) .
[ (M,)2 + F (T)(SM:) @Md2
-
1 + FtWuBffo/bT)
(M;) = M2 2i l”?,
(AM,)2
=
(aM:)/(M;)
; M2),
N (M2 - (cY(AM~)>~).
(3)
(5)
’
(6M:) = (g,urd2(W+ 1) -
1’
(4)
CM+- M-)2 2
Phase
p (Y Rn
1.4 K 4.2 K 1.4 K 4.2 K 1.4K 4.2 K
I
11
111
IV
V
0.080 0.11 1.1” 1.1” 0 0.74
0.052 0.12 1.1’ 3.09 I.17 1.31
0.035 0.094 1.1” 1.1” 0.86 0.74
0.073 0.11 1.1” 1.1” 0 0.74
0.086 0.15 I.15 1.15 0 0.74
a The value is the theoretical one for the isotropic case. because the fit is very insensitive to the [Y parameter in this phase.
(2)
The resistivity Ro(T) depends on the ratio between the ground and excited electron numbers, because both kinds of electrons have different scattering manners, respectively. Then, Ro is considered to be a measure for the number of excited electrons. The resistance R(T, Ho) is produced by excited electrons that are scattered by the magnetically shielded net moment M. The resistance formula is given by M, of the localized moment, the up-spin staggered magnetic moment M+ and the down-spin moment M_ and the ratio between Zeeman and thermal energies of gpnHn/kgT. Considering the difference in the excited electron scattering is not necessary because of the low energy excitation at low temperature, the formula [ 91 based on the Boltzmann transport equation is given by
R(T,Ho) N p(m*)
Table I Fitting parameters in magnetoresistance. R,I in phase V is given by zero, subtracting the residual resistivity of 0.13 x IOeh Rem determined from the data in phase V at 1.4 K. The error of the fitting parameters are estimated to be &3% except for the value with a footnote. P’/~ in lo-? 12’/” cm’/*, R,j in IOeh 0 cm
s:, (6)
F(x) = p(m*)
&,
=
f-@(G)‘.
(8)
where r, J, N and S are the numerical constant, exchange parameter, number of electrons and magnitude of the electron spin, respectively. The new additional fitting parameter LYin Eq. (6) is introduced to express the anisotropy effect of the Tb’+ spin. The anisotropy effect of the band structure is ignored, because the present case shows no clear anisotropic effects on the resistivity. The first term in parentheses in Eq. (3) originates from the spin-conservation scattering and the second term arises from the spin-reversed scattering process. Although the spin structure of this crystal has not been reported so far, the resistance for all phases is characterized by these four parameters. The field dependence in the resistance at 1.4 and 4.2 K are calculated by the use of the magnetization data shown in Fig. 2. The fitting parameter p(m*) is proportional to (wz* J/N) 2 which includes the information of the conduction band near &F. The value of p’/2, which is directly proportional to rn*J/N, is shown in Table 1. As is seen in Fig. 3, the theoretical solid curves at 1.4 and 4.2 K are in good agreement with the experimental data. The values of the fitting parameters used in the calculation are listed in Table 1. The parameter LYdetermined as 1.15 at the saturation field is consistent with the value of A in Eq. ( 1). The large value of cyin phase II at 4.2 K is remarkable.
H. Hori et al./ Physics Letters A 224 (1997) 293-297
The following reason can be considered: phase II with the magnetization of iMS shows the large magnetic cell, and many Brillouin zone boundaries intersect the Fermi surface. Then the intersection reduces the number of “effective” carriers. The reduction is anisotropic in phase II, because of the intersection between the band and large magnetic cell. Thus the anisotropic effect can reflect on the value of CY. As the conclusion of this work, the difference in Ro between phase I and II means that the number of ground state electrons at 4.2 K is remarkably larger than that at 1.4 K and the difference in p suggests a temperature variation of the conduction band near &F, which is related to the superzone effect. Taking account of the fact that there is no substantial change in the spin structure between 1.4 and 4.2 K [ 81, the magnetoresistance data lead to the model that the phase below 2.1 K is ruled by “magnetic lattice” electrons. All estimated values are physically appropriate and such a consistent result strongly supports the model and analysis in this work. This experiment and the analysis method are expected to lead to a new analysis method for the resistance in the highly correlated spin system related with the spin fluctuation.
297
Acknowledgement This work was performed through the financial support by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of the Japanese Government and is also partly supported by the Special Coordination Funds of the Science and Technology Agency of the Japanese Government.
References T. Takabatake,Y. Nakazawa and M. Ishikawa, Jpn. J. Appl. Phys. 26 Suppl. 3 (1987) 547. R.J. Elliot and EA. Wedgwood, Proc. R. Sot. 81 (1963) 846. A.E. Dwight. J. Less-Common Met. 93 ( 1983) 41 I. H. Miwa. Prog. Theor. Phys. 29 (1963) 477. M. Kurisu, H. Hori, M. Furusawa. M. Miyake, Y. Andoh, 1. Oguro, K. Kindo, T. Takeuchi and A. Yamagishi. Physica B 201 (1994) 107. J. Magn. [61 C. Routsi, J.K. Yakinthos and H. Gamari-Seale, Magn. Mater. 117 ( 1992) 79. [71 T. Suzuki, M. Kitamura, T. Fujita, Y. Andoh and M. Kurisu. in: Abstr. Meeting Phys. Sot. Japan, 50th Annual Meeting, Part 3 (1995) 170 [in Japanese]. US1 Y. Andoh and M. Kurisu et al., in preparation. 191 K. Yosida, Phys. Rev. 107 (1957) 396.