- Email: [email protected]
Tunneling magnetoresistance of phase-separated manganites
Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
Tunneling magnetoresistance of phase-separated manganites A.O. Sboychakova,*, A.L. Rakhmanova, K.I. Kugela, M.Yu. Kaganb, I.V. Brodskyb a
Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya Str. 13/19, 125412 Moscow, Russia b Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygina Str. 2, 117334 Moscow, Russia
Abstract The mechanisms underlying the magnetoresistance of non-metallic phase-separated manganites are analyzed. The material is modeled by a system of small ferromagnetic metallic droplets (magnetic polarons or ferrons) in an insulating matrix. The concentration of metallic phase is assumed to be far from the percolation threshold. The electron tunneling between ferrons causes the charge transfer in such a system. The magnetoresistance is determined by the effect of the magnetic field H on the tunneling probability related to the changes both in the size of ferrons and in the mutual orientation of their magnetic moments. It is shown that the low-field magnetoresistance is proportional to H 2 and decreases with temperature as T n ; where n can vary from 2 up to 5 depending on the parameters of the system. In the strong-field limit, the tunneling magnetoresistance grows exponentially, but the crossover between these two regimes can correspond to a plateau. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Manganites; Phase separation; Magnetoresistance; Spin-dependent tunneling
1. Introduction Recent theoretical and experimental studies clearly demonstrated that the tendency toward phase separation is of fundamental importance for the physics manganites and seems to play a key role for the colossal magnetoresistance phenomenon [1–3]. The self-trapping of charge carriers is the most widely discussed type of phase separation first predicted in the seminal paper of Nagaev [4]. In such a phase-separated state, charge carriers become confined within small ferromagnetic metallic droplets (magnetic polarons or ferrons) located in an insulating antiferromagnetic matrix. In the limit of strong Coulomb interaction, each droplet contains one charge carrier in a potential well of ferromagnetically ordered local spins with a characteristic size of about several lattice constants. The radius of ferron can be determined by minimization of the energy EBtp2 d 2 =R2 þ 4pzJS 2 R3 =3d 3 ; where *Corresponding author. Tel.: +7-095-362-5147; fax: +7095-484-2633. E-mail address: [email protected] (A.O. Sboychakov).
the first term is the energy of an electron in a potential well of radius R and the second one corresponds to the energy of Heisenberg exchange between the localized spins S (see Refs. [5,6]). Minimization of the energy with respect to the 1=5 ferron radius gives the estimation: RBd t=zJS 2 ; where t is the hopping integral, J is the constant of the Heisenberg (antiferromagnetic) exchange, z is the number of nearest neighbors, and d is the lattice constant.
2. Formulation of the model Following Ref. [5], we can write the energy of ferron in magnetic field H in the form: p2 d 2 4p R3 þ zJS 2 3 2 3 R d % MðHcos y þ Ha cos2 cÞ sJ;
E1;s ¼ t
ð1Þ
where we take into account uniaxial magnetic anisotropy. Here, Ha is the anisotropy field, M ¼ 4pmB gSR3 3d 3 is the magnetic moment of a ferron, y
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 1 4 3 - 5
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
and c are angles between M and H and between M and easy axis, respectively, and s=2 is the projection of electron spin onto the direction of M; mB is the Bohr magneton, and g is the Lande factor. The last term in Eq. (1) is the energy of interaction between the confined % B generelectron and effective magnetic field Heff ¼ J=m ated by ferromagnetically ordered local spins, J% is the normalized exchange integral related to the double exchange mechanism. Strictly speaking, in Eq. (1) we 2 % should write mB ½B2 þ Heff þ 2ðB H eff Þ1=2 instead of J; where B is the magnetic induction inside the droplet with a due account of its demagnetization factor. We neglected, however induction B in comparison with the effective field, because Heff is if the order of 100 T [7]. We assume here that the direction of magnetic moments varies slowly enough and the usual thermodynamic approach is applicable. The ferron radius can be determined by minimization of energy (1), and we find in the linear approximation with respect to magnetic field b m g RðHÞDR0 1 þ H cos y þ Ha cos2 c ; b ¼ B : ð2Þ 5 zJS As a result of tunneling processes, ferrons with more than one electron are created and some ferrons become empty. The creation of a two-electron droplet is associated with an energy barrier of the order of the Coulomb repulsion energy of electrons in the droplet. Since the value of the ferron radius is of the order ( and dielectric constant eB10; we have of 10 A 2 V0 ¼ e =eR0 B0:1 eV. In the following, we assume the temperature is low enough in comparison to the Coulomb barrier and neglect the formation of ferrons with three or more electrons. The energy of an empty droplet is E0 ¼
4p R3 zJS 2 3 MðH cos y þ Ha cos2 cÞ: d 3
ð3Þ
Electrons in a two-electron droplet can form the states with total spin 0 or 1. In the latter case, however, the electrons occupy different energy levels in the potential well. It can be shown that the distance between these levels is about the Coulomb interaction energy (or exceeds it). Therefore, we consider only two-electron ferrons with antiparallel electron spins. The energy of such ferrons is E2 ¼ 2t
p2 d 2 4p R3 þ zJS 2 3 2 R d 3
MðH cos y þ Ha cos2 cÞ þ
e2 : eR
ð4Þ
The ferron radius in Eqs. (3) and (4) is given by expression (2). Thus, we have N1;s single-electron ferrons with spin projection s=2; N0 empty ferrons, and N2 ¼ N0 ferrons with two electrons. Below, we denote the states with
297
s ¼ þ1 and 1 as m and k, respectively. The thermal averages of numbers N0, N1,s and N2 can be found using the partition function Na X N N2N Y 1 Z Ea ðy:jÞ=kT 0 2 dOe Z¼ dN2 dN1;m þN1;k ; Na ! a fN g a
where subscript a with values 0, (1, m), (1, k), and 2 denotes different states of ferrons. The conventional averaging procedure leads to the following relationships expðV0 =2kTÞ N% 2 ¼ N% 0 ¼ N % 2 coshðJ=kTÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ð2=5; HÞF ð8=5 þ k; HÞ ; F 2 ð1; HÞ % expðsJ=kTÞ 3V0 R20 ; N% 1;s ¼ ðN 2N% 2 Þ ; k¼ % 10tp2 d 2 2 coshðJ=kTÞ where F ðx; HÞ ¼
Z
ð5Þ
M0 2 H cos y þ Ha cos cðy; jÞ dO exp x kT ð6Þ
and M0 is the magnetic moment of a ferron at zero field.
3. Magnetoresistance Within the framework of our model, the electron tunneling occurs via one of the four following tunneling processes: 1. a tunneling electron transforms two single-electron droplets to an empty droplet and a droplet with two electrons; for this process, we use notation (1s1 ; 1s2 )-(0, 2); 2. reverse process, i.e. (2, 0)-(1s1 ; 1s2 ); 3. an electron is being transferred from a single-electron droplet to an empty one: (1s1, 0)-(0, 1s2); 4. a two-electron droplet and a single-electron droplet exchange their positions by transferring an electron from one droplet to another: (2, 1s1 )-(1s2 ,2). The contribution to the current density j from each process can be written in the following form [8]: * + Nb Na X ri;j cos ai;j e X 0 0 ; ð7Þ jða ; b ; a; bÞ ¼ V i¼0 j¼0 ti;j ða0 ; b0 ; a; bÞ where ri;j is the distance between ferrons located at sites i and j; ai;j is the angle between radius vector ri;j and the direction of the electric field E; and h?i stands for statistical and space averaging. The summation in Eq. (7) is performed over all ferrons involved into the corresponding tunneling process. Parameter ti;j ðf ; iÞ is the characteristic time associated with process of tunneling from the initial state i ¼ ða; bÞ to final one
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
298
f ¼ ða0 ; b0 Þ: It can be written in the following form [8]: % coshðJcosn 1 i;j =kTÞ ¼ o0 % ti;j ða0 ; b0 ; a; bÞ coshðJ=kTÞ ri;j eðE ri;j Þ exp þ l kT Ea0 þ Eb0 Ea Eb ; 2kT
MRðHÞ ¼
ð8Þ
where l and o0 are the tunneling length and the characteristic frequency for the electron motion in the potential well, and ni;j is the angle between magnetic moments of the droplets. The pre-exponential factor in Eq. (8) is related to the spin-dependent tunneling. Substituting Eq. (8) into Eq. (7) we perform the space averaging assuming a random distribution of ferrons. Combining then the current densities corresponding to all tunneling processes, we obtain the following expression for conductivity s ¼ j=E sðHÞ ¼
% 32pe2 l 5 N 2 cosh2 ðJ=2kTÞ 3 2 % V kT cosh ðJ=kTÞ Z Z dO1 dO2 exp V0 =2kT % coshðJcosn 12 =kTÞ F 2 ð1; HÞ 7M0 H cos y1 þ Ha cos2 c1 exp 10kT 7M0 Hcosy2 þ Ha cos2 c2 exp 10kT sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 F ð5 þ k; HÞ ð13 þ 5kÞM0 þ 2exp 10kT F ð25; HÞ 2 ½Hcosy1 þHa cos c1 7M0 Hcosy2 þ Ha cos2 c2 exp 10kT sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ð25; HÞ ð13 þ 5kÞM0 exp þ 10kT F ð85 þ k; HÞ ½H cos y1 þHa cos2 c1 ð13 þ 5kÞM0 exp 10kT ½H cos y2 þHa cos2 c2 ;
obtain
ð9Þ
where k and F ðx; HÞ are given by Eqs. (5) and (6). Based on formula (9), we can analyze the behavior of magnetoresistance MRðHÞ ¼ sðHÞ=sð0Þ 1: In general, the integration in Eq. (9) can be performed only numerically. However, in some limiting cases, it is possible to find explicit relationships for the magnetoresistance, in particular, when the temperature is not too low and the parameters involved in Eq. (9) obey the % M0 H; M0 Ha okT: inequality J; First, consider the low-field limit. Representing % and Ha ; we Eq. (9) as a series expansion in H; J;
3 M02 H 2 2 M03 J%2 Ha H 2 þ ðcos2 b 13Þ; ð10Þ 2 2 100 k T 225 k5 T 5
where b is the angle between the easy axis and magnetic field. The first term is related to the magnetic field induced change in the droplet size and the second one stems from the spin-dependent tunneling. The relative contributions of each mechanism depend on specific properties of the material. When the first term is dominant, the temperature dependence of the magnetoresistance behaves as 1=T 2 : In the opposite case, the magnetoresistance decreases with temperature as 1=T 5 : The magnetoresistance in Eq. (10) depends on the angle between the easy axis and the magnetic field. This means that expression (10) is valid only for single crystals. We assumed above that ferrons are spherical and field Ha is determined only by the crystallographic anisotropy. It can be shown, however, that effect of the ferron shape becomes significant even at small deviations from sphericity. For elliptical droplets, the same formula (10) remains valid, but b denotes the angle between the longer axis of the ellipsoid and the magnetic field. It is natural to suppose that the longer axes of the droplets are mainly oriented along the applied magnetic field. If the measuring current is also parallel to magnetic field, the tunneling in this direction becomes more favorable enhancing the effect of orientation on the magnetoresistance. Thus, under usual experimental conditions we have cos bE1 and rather high effective values of Ha : This can lead to the substantial contribution of the second term in Eq. (10) to the total magnetoresistance. Note, for example that MRpH 2 =T 5 was observed in a rather wide range of fields and temperatures for the (La1x Prx)0.7Ca0.3 MnO3 system [9]. At high fields (10–20 T), the magnetoresistance grows exponentially [10] bV0 H J% % MRðHÞ ¼ 1 ð11Þ cothðJ=kTÞexp 10kT kT and its value can be as high as several hundred percents even far from the percolation threshold. The numerical analysis of the intermediate field range demonstrates that the crossover between the low- and high-field regimes can correspond to a plateau in the MRðHÞ curve.
Acknowledgements This work was supported by RFBR (grants 00-1596570, 00-02-16255, and 02-02-16708), CRDF (grant RP2-2355-MO-02), INTAS (grant 01-2008), NWO (grant 047-008-017), and by Grant of President of Russia (00-15-9654).
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
References [1] E. Dagotto, T. Hotta, A. Moreo, Phys. Rep. 344 (2001) 1. [2] M.Yu. Kagan, K.I. Kugel, Usp. Fiz. Nauk 171 (2001) 577; [Phys. Usp. 44 (2001) 553]. [3] E.L. Nagaev, Phys. Rep. 346 (2001) 387. [4] E.L. Nagaev, Pis’ma v ZhETF 6 (1967) 484; [Sov. Phys. JETP Lett. 6 (1967) 18]. [5] M.Yu. Kagan, D.I. Khomskii, M.V. Mostovoy, Eur. Phys. J. B 12 (1999) 217. [6] M.Yu. Kagan, K.I. Kugel, D.I. Khomskii, Zh. Eksp. Teor. Fiz. 120 (2001) 470; [JETP 93 (2001) 415].
299
[7] Yu.A. Izyumov, Yu.N. Skryabin, Usp. Fiz. Nauk. 171 (2001) 121; [Phys. Usp. 44 (2001) 109]. [8] N.F. Mott, E.A. Davis, Electronic Processes in NonCrystalline Materials, Clarendon Press, Oxford, 1979. [9] N.A. Babushkina, et al., J. Phys.: Condens. Matter, submitted. [10] A.L. Rakhmanov, K.I. Kugel, Ya.M. Blanter, M.Yu. Kagan, Phys. Rev. B 63 (2001) 174424.
Tunneling magnetoresistance of phase-separated manganites A.O. Sboychakova,*, A.L. Rakhmanova, K.I. Kugela, M.Yu. Kaganb, I.V. Brodskyb a
Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya Str. 13/19, 125412 Moscow, Russia b Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygina Str. 2, 117334 Moscow, Russia
Abstract The mechanisms underlying the magnetoresistance of non-metallic phase-separated manganites are analyzed. The material is modeled by a system of small ferromagnetic metallic droplets (magnetic polarons or ferrons) in an insulating matrix. The concentration of metallic phase is assumed to be far from the percolation threshold. The electron tunneling between ferrons causes the charge transfer in such a system. The magnetoresistance is determined by the effect of the magnetic field H on the tunneling probability related to the changes both in the size of ferrons and in the mutual orientation of their magnetic moments. It is shown that the low-field magnetoresistance is proportional to H 2 and decreases with temperature as T n ; where n can vary from 2 up to 5 depending on the parameters of the system. In the strong-field limit, the tunneling magnetoresistance grows exponentially, but the crossover between these two regimes can correspond to a plateau. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Manganites; Phase separation; Magnetoresistance; Spin-dependent tunneling
1. Introduction Recent theoretical and experimental studies clearly demonstrated that the tendency toward phase separation is of fundamental importance for the physics manganites and seems to play a key role for the colossal magnetoresistance phenomenon [1–3]. The self-trapping of charge carriers is the most widely discussed type of phase separation first predicted in the seminal paper of Nagaev [4]. In such a phase-separated state, charge carriers become confined within small ferromagnetic metallic droplets (magnetic polarons or ferrons) located in an insulating antiferromagnetic matrix. In the limit of strong Coulomb interaction, each droplet contains one charge carrier in a potential well of ferromagnetically ordered local spins with a characteristic size of about several lattice constants. The radius of ferron can be determined by minimization of the energy EBtp2 d 2 =R2 þ 4pzJS 2 R3 =3d 3 ; where *Corresponding author. Tel.: +7-095-362-5147; fax: +7095-484-2633. E-mail address: [email protected] (A.O. Sboychakov).
the first term is the energy of an electron in a potential well of radius R and the second one corresponds to the energy of Heisenberg exchange between the localized spins S (see Refs. [5,6]). Minimization of the energy with respect to the 1=5 ferron radius gives the estimation: RBd t=zJS 2 ; where t is the hopping integral, J is the constant of the Heisenberg (antiferromagnetic) exchange, z is the number of nearest neighbors, and d is the lattice constant.
2. Formulation of the model Following Ref. [5], we can write the energy of ferron in magnetic field H in the form: p2 d 2 4p R3 þ zJS 2 3 2 3 R d % MðHcos y þ Ha cos2 cÞ sJ;
E1;s ¼ t
ð1Þ
where we take into account uniaxial magnetic anisotropy. Here, Ha is the anisotropy field, M ¼ 4pmB gSR3 3d 3 is the magnetic moment of a ferron, y
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 1 4 3 - 5
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
and c are angles between M and H and between M and easy axis, respectively, and s=2 is the projection of electron spin onto the direction of M; mB is the Bohr magneton, and g is the Lande factor. The last term in Eq. (1) is the energy of interaction between the confined % B generelectron and effective magnetic field Heff ¼ J=m ated by ferromagnetically ordered local spins, J% is the normalized exchange integral related to the double exchange mechanism. Strictly speaking, in Eq. (1) we 2 % should write mB ½B2 þ Heff þ 2ðB H eff Þ1=2 instead of J; where B is the magnetic induction inside the droplet with a due account of its demagnetization factor. We neglected, however induction B in comparison with the effective field, because Heff is if the order of 100 T [7]. We assume here that the direction of magnetic moments varies slowly enough and the usual thermodynamic approach is applicable. The ferron radius can be determined by minimization of energy (1), and we find in the linear approximation with respect to magnetic field b m g RðHÞDR0 1 þ H cos y þ Ha cos2 c ; b ¼ B : ð2Þ 5 zJS As a result of tunneling processes, ferrons with more than one electron are created and some ferrons become empty. The creation of a two-electron droplet is associated with an energy barrier of the order of the Coulomb repulsion energy of electrons in the droplet. Since the value of the ferron radius is of the order ( and dielectric constant eB10; we have of 10 A 2 V0 ¼ e =eR0 B0:1 eV. In the following, we assume the temperature is low enough in comparison to the Coulomb barrier and neglect the formation of ferrons with three or more electrons. The energy of an empty droplet is E0 ¼
4p R3 zJS 2 3 MðH cos y þ Ha cos2 cÞ: d 3
ð3Þ
Electrons in a two-electron droplet can form the states with total spin 0 or 1. In the latter case, however, the electrons occupy different energy levels in the potential well. It can be shown that the distance between these levels is about the Coulomb interaction energy (or exceeds it). Therefore, we consider only two-electron ferrons with antiparallel electron spins. The energy of such ferrons is E2 ¼ 2t
p2 d 2 4p R3 þ zJS 2 3 2 R d 3
MðH cos y þ Ha cos2 cÞ þ
e2 : eR
ð4Þ
The ferron radius in Eqs. (3) and (4) is given by expression (2). Thus, we have N1;s single-electron ferrons with spin projection s=2; N0 empty ferrons, and N2 ¼ N0 ferrons with two electrons. Below, we denote the states with
297
s ¼ þ1 and 1 as m and k, respectively. The thermal averages of numbers N0, N1,s and N2 can be found using the partition function Na X N N2N Y 1 Z Ea ðy:jÞ=kT 0 2 dOe Z¼ dN2 dN1;m þN1;k ; Na ! a fN g a
where subscript a with values 0, (1, m), (1, k), and 2 denotes different states of ferrons. The conventional averaging procedure leads to the following relationships expðV0 =2kTÞ N% 2 ¼ N% 0 ¼ N % 2 coshðJ=kTÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ð2=5; HÞF ð8=5 þ k; HÞ ; F 2 ð1; HÞ % expðsJ=kTÞ 3V0 R20 ; N% 1;s ¼ ðN 2N% 2 Þ ; k¼ % 10tp2 d 2 2 coshðJ=kTÞ where F ðx; HÞ ¼
Z
ð5Þ
M0 2 H cos y þ Ha cos cðy; jÞ dO exp x kT ð6Þ
and M0 is the magnetic moment of a ferron at zero field.
3. Magnetoresistance Within the framework of our model, the electron tunneling occurs via one of the four following tunneling processes: 1. a tunneling electron transforms two single-electron droplets to an empty droplet and a droplet with two electrons; for this process, we use notation (1s1 ; 1s2 )-(0, 2); 2. reverse process, i.e. (2, 0)-(1s1 ; 1s2 ); 3. an electron is being transferred from a single-electron droplet to an empty one: (1s1, 0)-(0, 1s2); 4. a two-electron droplet and a single-electron droplet exchange their positions by transferring an electron from one droplet to another: (2, 1s1 )-(1s2 ,2). The contribution to the current density j from each process can be written in the following form [8]: * + Nb Na X ri;j cos ai;j e X 0 0 ; ð7Þ jða ; b ; a; bÞ ¼ V i¼0 j¼0 ti;j ða0 ; b0 ; a; bÞ where ri;j is the distance between ferrons located at sites i and j; ai;j is the angle between radius vector ri;j and the direction of the electric field E; and h?i stands for statistical and space averaging. The summation in Eq. (7) is performed over all ferrons involved into the corresponding tunneling process. Parameter ti;j ðf ; iÞ is the characteristic time associated with process of tunneling from the initial state i ¼ ða; bÞ to final one
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
298
f ¼ ða0 ; b0 Þ: It can be written in the following form [8]: % coshðJcosn 1 i;j =kTÞ ¼ o0 % ti;j ða0 ; b0 ; a; bÞ coshðJ=kTÞ ri;j eðE ri;j Þ exp þ l kT Ea0 þ Eb0 Ea Eb ; 2kT
MRðHÞ ¼
ð8Þ
where l and o0 are the tunneling length and the characteristic frequency for the electron motion in the potential well, and ni;j is the angle between magnetic moments of the droplets. The pre-exponential factor in Eq. (8) is related to the spin-dependent tunneling. Substituting Eq. (8) into Eq. (7) we perform the space averaging assuming a random distribution of ferrons. Combining then the current densities corresponding to all tunneling processes, we obtain the following expression for conductivity s ¼ j=E sðHÞ ¼
% 32pe2 l 5 N 2 cosh2 ðJ=2kTÞ 3 2 % V kT cosh ðJ=kTÞ Z Z dO1 dO2 exp V0 =2kT % coshðJcosn 12 =kTÞ F 2 ð1; HÞ 7M0 H cos y1 þ Ha cos2 c1 exp 10kT 7M0 Hcosy2 þ Ha cos2 c2 exp 10kT sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 F ð5 þ k; HÞ ð13 þ 5kÞM0 þ 2exp 10kT F ð25; HÞ 2 ½Hcosy1 þHa cos c1 7M0 Hcosy2 þ Ha cos2 c2 exp 10kT sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ð25; HÞ ð13 þ 5kÞM0 exp þ 10kT F ð85 þ k; HÞ ½H cos y1 þHa cos2 c1 ð13 þ 5kÞM0 exp 10kT ½H cos y2 þHa cos2 c2 ;
obtain
ð9Þ
where k and F ðx; HÞ are given by Eqs. (5) and (6). Based on formula (9), we can analyze the behavior of magnetoresistance MRðHÞ ¼ sðHÞ=sð0Þ 1: In general, the integration in Eq. (9) can be performed only numerically. However, in some limiting cases, it is possible to find explicit relationships for the magnetoresistance, in particular, when the temperature is not too low and the parameters involved in Eq. (9) obey the % M0 H; M0 Ha okT: inequality J; First, consider the low-field limit. Representing % and Ha ; we Eq. (9) as a series expansion in H; J;
3 M02 H 2 2 M03 J%2 Ha H 2 þ ðcos2 b 13Þ; ð10Þ 2 2 100 k T 225 k5 T 5
where b is the angle between the easy axis and magnetic field. The first term is related to the magnetic field induced change in the droplet size and the second one stems from the spin-dependent tunneling. The relative contributions of each mechanism depend on specific properties of the material. When the first term is dominant, the temperature dependence of the magnetoresistance behaves as 1=T 2 : In the opposite case, the magnetoresistance decreases with temperature as 1=T 5 : The magnetoresistance in Eq. (10) depends on the angle between the easy axis and the magnetic field. This means that expression (10) is valid only for single crystals. We assumed above that ferrons are spherical and field Ha is determined only by the crystallographic anisotropy. It can be shown, however, that effect of the ferron shape becomes significant even at small deviations from sphericity. For elliptical droplets, the same formula (10) remains valid, but b denotes the angle between the longer axis of the ellipsoid and the magnetic field. It is natural to suppose that the longer axes of the droplets are mainly oriented along the applied magnetic field. If the measuring current is also parallel to magnetic field, the tunneling in this direction becomes more favorable enhancing the effect of orientation on the magnetoresistance. Thus, under usual experimental conditions we have cos bE1 and rather high effective values of Ha : This can lead to the substantial contribution of the second term in Eq. (10) to the total magnetoresistance. Note, for example that MRpH 2 =T 5 was observed in a rather wide range of fields and temperatures for the (La1x Prx)0.7Ca0.3 MnO3 system [9]. At high fields (10–20 T), the magnetoresistance grows exponentially [10] bV0 H J% % MRðHÞ ¼ 1 ð11Þ cothðJ=kTÞexp 10kT kT and its value can be as high as several hundred percents even far from the percolation threshold. The numerical analysis of the intermediate field range demonstrates that the crossover between the low- and high-field regimes can correspond to a plateau in the MRðHÞ curve.
Acknowledgements This work was supported by RFBR (grants 00-1596570, 00-02-16255, and 02-02-16708), CRDF (grant RP2-2355-MO-02), INTAS (grant 01-2008), NWO (grant 047-008-017), and by Grant of President of Russia (00-15-9654).
A.O. Sboychakov et al. / Journal of Magnetism and Magnetic Materials 258–259 (2003) 296–299
References [1] E. Dagotto, T. Hotta, A. Moreo, Phys. Rep. 344 (2001) 1. [2] M.Yu. Kagan, K.I. Kugel, Usp. Fiz. Nauk 171 (2001) 577; [Phys. Usp. 44 (2001) 553]. [3] E.L. Nagaev, Phys. Rep. 346 (2001) 387. [4] E.L. Nagaev, Pis’ma v ZhETF 6 (1967) 484; [Sov. Phys. JETP Lett. 6 (1967) 18]. [5] M.Yu. Kagan, D.I. Khomskii, M.V. Mostovoy, Eur. Phys. J. B 12 (1999) 217. [6] M.Yu. Kagan, K.I. Kugel, D.I. Khomskii, Zh. Eksp. Teor. Fiz. 120 (2001) 470; [JETP 93 (2001) 415].
299
[7] Yu.A. Izyumov, Yu.N. Skryabin, Usp. Fiz. Nauk. 171 (2001) 121; [Phys. Usp. 44 (2001) 109]. [8] N.F. Mott, E.A. Davis, Electronic Processes in NonCrystalline Materials, Clarendon Press, Oxford, 1979. [9] N.A. Babushkina, et al., J. Phys.: Condens. Matter, submitted. [10] A.L. Rakhmanov, K.I. Kugel, Ya.M. Blanter, M.Yu. Kagan, Phys. Rev. B 63 (2001) 174424.