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Magnetoresistance due to domain wall bulging
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetoresistance due to domain wall bulging M.B. Fathi a, A. Phirouznia b, a b
Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran
a r t i c l e in fo
abstract
Article history: Received 6 July 2009 Received in revised form 2 November 2009 Available online 23 February 2010
The effect of bulging of domain wall (DW) on the magnetoresistance (MR) is investigated. With taking into account the auto-correlation between the points of the interface, one can formulate the mobility within the relaxation time approximation scheme. The results show that the bulging of DW, evaluated with the commonly accepted magnetic parameters for typical ferromagnetic materials of Co, Fe and Ni, has a countable role into the MR. & 2010 Elsevier B.V. All rights reserved.
Keywords: Magnetoresistance Domain wall Bulging Spintronics
1. Introduction Recent developments of nano-technology have enabled researchers to fabricate low dimensional devices. These developments have opened a new way for investigations directed on the nanoscale magnetoresistive devices. Magnetoresistance (MR) as a characteristic property of nanoscale devices, is of particular interest. Properties of ferromagnetic structures in nanometer scale have attracted much interest recently. Transport properties of these nano-structures are a powerful tool in the development of spintronics (spin electronics), the electronics in which the spin of electron plays the role of the active element for storage and transporting information. This has made an enormous interest for observing novel transport phenomena especially in the magnetic systems. The complexity of the local magnetic structures such as surfaces, interfaces and magnetic domain walls (DWs) offers a variety of novel features especially in the field of spin dependent transport phenomena. An outstanding problem in the field of spintronics is the effect of the magnetic structures such as magnetic DWs on the electronic transport. It was generally believed that DWs are sources of scattering, while experimental results suggest that, the DW magnetoresistance (MR) can be either positive or negative. Positive MR due to DW has been reported in striped domain structures [1] and the Ni wires [2]. In contrast, negative MR has been observed in some of experiments on very narrow wires and thin films [3–5]. Theoretically, the quantum decoherence caused
Corresponding author. Tel.: + 98 914 3221142.
E-mail address: [email protected] (A. Phirouznia). 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.02.045
by the DW has been mentioned as a source for reduction of the resistance and negative MR in the weakly localized regime (for example see [6]). On the other hand, spin-dependent impurity scattering which was proposed by Levy and Zhang was supposed to be responsible for mixing the spin channels and positive MR [7]. This paper is devoted to the calculation of the MR caused by DW bulging in a semiclassical approach based on the eigenvalue model.
2. Method Some effects such as pinning and current pressure cause the DW to change its shape leading to an effect known as DW bulging. 0 An electron with the wave vector k scatters to the state k due to the DW interface potential fluctuations. We consider the bulging of DW as a source of scattering similar to the roughness of interfaces. As the scattering is elastic, the spin dependent Þ could be evaluated within the relaxation relaxation times, tðEmðkÞ k time approximation. The relaxation time is given by [8,9] Z 1 1 2p X 0 0 0 d2 k jMðk;k Þj2 Sc ðqÞ ð1cosyÞdðEsk Esk0 Þ ð1Þ ¼ s 2 tðEk Þ 4p ‘ s0 0
where jMðk; k Þj is the matrix element for scattering from the k0 state to the k - state, Ek and Ek0 are the energies corresponding to 0 the wave vectors k and k , Sc(q) is the screening factor 0 corresponding to q ¼ k 7k and y represents the direction 0 between k and k .
ARTICLE IN PRESS 2402
M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
The electrons may be screened in 2D interface, and the screening factor, Sc(q), is simply chosen as [10] Sc ðqÞ ¼
q q þ qs
ð2Þ 0
where q is the norm of q ¼ k 7 k and qs ¼
m ðEsk Þe2 2pe‘
ð3Þ
2
The energy-dependent effective mass in the system is given by m*(Ek), e is the permittivity and e is the charge of an electron. The expression for the mobility of the electrons, m, reads [8] * + tðEmðkÞ Þ ð4Þ mmðkÞ ¼ jej kmðkÞ m ðEk Þ Since the electron gas is degenerate at low temperatures, the density of states (DOS) (which is responsible for charge transfer) is high around Fermi energy and therefore averaging over all energies, reduces to calculation of the relaxation time at Fermi energy. So,
mmðkÞ C jej
tðEmðkÞ Þ kf Þ m ðEmðkÞ k
Fig. 1. A schematic representation of bulging of the magnetic DW withits asperity parameter (FM: ferromagnet).
ð5Þ
f
Further development of the theory involves modeling of the scattering matrix: 0
Mðk;k Þ ¼ /Ck jDVjCk0 S
ð6Þ
which is the familiar expression for transition between the electron-states, Ck and Ck0 , with DV being the scattering potential.
3. Model
where L is the correlation length, and a is a parameter illustrating some estimate for roughness of f function, at deferent points. If we take f to be the roughness itself, then [8,13,14], /DrDr0 S ¼ D20 expððrr0 Þ2 =L2 Þ
ð10Þ
where the averaging is performed over the surface. In this expression D0 is called average asperity height, Fig. 1. Characteristically, one can suppose D0 and L to be some few angstroms in magnitude, for a typical DW surface. The spectral density for this distribution reads jf ðqÞj2 ¼ pD20 L2 expðq2 L2 =4Þ
ð11Þ
The Hamiltonian describing giant magnetoresistance (GMR) is described by [7] H0 ¼
‘ 2 r2 2m
^ þVðrÞ þ J r MðrÞ
ð7Þ
where J is the exchange splitting, V(r) is the nonmagnetic periodic ^ potential, and the unit vector MðrÞ points in the direction of local magnetization, M(r). Potential for scattering of electrons due to the magnetic impurities is given by X ½vþ jr MðrÞdðrri Þ ð8Þ Vscat ¼ i
4. Matrix element for the eigenvalue model In the eigenvalue model [14], the scattering potential is assumed to be caused by the change in energy eigenvalue due to the bulging, so the bulging potential is estimated by the deference between the energy eigenvalues,
DVðrÞ 7 ¼ jEðL 7 DrÞEðLÞj
ð12Þ
EðL 7 DrÞ and E(L) being the energy eigenvalues corresponding to the DW widths L 7 Dr and L, and 7 sign indicates increase or decrease of the width. The autocorrelation expression for this perturbing potential for a Gaussian distribution of bulging reads
where ri is the position of the impurities, v is the impurity potential and j represents the spin dependent interaction strength between the impurities and electrons. We choose the wavefunctions proposed by Levy and Zhang [7] for the DW, Cm ðk; rÞ and Ck ðk; rÞ, with the corresponding quantities. The important part of evaluating the effect of DW bulging is the formulating the bulging potential. Then we have to find the appropriate nonperturbed wave functions. First of all, we consider the bulging as the interface roughness, Fig. 1, and assume the correlation between the points r and r0 in 2D interface as being described by a function f ðx; yÞ ¼ f ðrÞ. This function may be supposed to describe the height of the bumps on the surface, i.e. the surface is at z ¼ f ðrÞ. Only the statistical properties of f is of importance. One employs, for example, a Gaussian distribution function or an exponential one. The correlation function for the Gaussian distribution is [11–13],
where the energy eigenvalues, EðL 7 D0 Þ and E(L), are the expectation values of the perturbation potential between the wavefunctions Cm ðk; rÞ and Ck ðk; rÞ. Using the perturbing potential, Eq. (12), and the corresponding spectral density, the square of the scattering matrix reads [14]
/f ðr0 þ rÞf ðr0 ÞS ¼ aexpðr2 =L2 Þ
jMðkk Þj2 ¼ A1 pðDVE0 Þ2 L2 expðq2 L2 =4Þ
ð9Þ
/DVðrÞDVðr0 ÞS ¼ ðDVE0 Þ2 expððrr0 Þ2 =L2 Þ
ð13Þ
The change in the energy eigenvalue for the average asperity height D0 , corresponding to L7 D0 , is chosen as the coefficient of the correlation function,
DVE0 ¼ jEðL 7 D0 ÞEðD0 Þj
0
ð14Þ
ð15Þ
ARTICLE IN PRESS M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
2403
where A is the surface of DW. At last one can obtain Z 1 1 2X 0 ¼ L d2 k ðDVE0 Þ2 expðq2 L2 =4Þ s tðEk Þ 2‘ s0 Sc ðqÞ ð1coshyÞdðEsk Esk0 Þ 0
ð16Þ
where the summation is over the two spin channels.
5. Results We choose commonly accepted values of kf =1 A˚ 1 and J= 0.5 eV for typical ferromagnetic materials of Co, Fe and Ni [7]. The two spin channels can be considered as two parallel current channels so that the inverse of mobility can be understood as a summation of inverse of two spin-dependent mobilities [8], 1
m
¼
1
mm
þ
1
mk
ð17Þ Fig. 4. Partial resistivity, as a function of asperity height, D0 .
Therefore resistivity, r, can be considered to be summation of two resistivities, r ¼ rm þ rk . In the present calculations, we have found small difference in quantitative behavior of these two spin channels. We define partial resistivity due to bulging as follows:
dr
r0
Fig. 2. The resistivity, r, due to bulging of DW, as a function of k/kf.
100%;
ð18Þ
where dr ¼ rr0 , r0 is the DW resistivity, and r is the resistivity due to the DW bulging. Fig. 2 shows the partial resistivity as a function of wave vector, k. As the figure shows, the maximum partial resistivity as a function of k occurs around the Fermi wave vector, kf. The partial resistivity as a function of correlation length, L, is shown in Fig. 3. As shown in this figure, with the increasing of the correlation length and average asperity height the contribution of the DW bulging into the partial resistivity is suppressed and for a typical range of correlation length [15] the maximum of the partial resistivity due to the DW bulging is about 2–3%. For higher correlation lengths which can be considered up to several 100 nm the partial resistivity due to the DW bulging is negligible. Fig. 4 shows the partial resistivity versus the average asperity height, D0 . As is obvious from the figure, for acceptable values of asperity height for a microscopic bulging which is assumed to be higher than 1 A˚ the maximum value of the partial resistivity due to roughness of the DW is about 2%. However, it should be noted that the partial resistivity is not a monotonic function of asperity height and at low average asperities the partial resistivity increases by increasing the asperity height and then decreases smoothly after a maximum is passed. For the present case only a realistic range has been chosen for microscopic asperity height.
6. Conclusion
Fig. 3. The partial resistivity, due to bulging of DW, as a function of the correlation length, L.
A semiclassical approach has been employed to take into account the effect of bulging of magnetic DW into the resistivity. The effect of bulging can be considered as the roughness of the heterojunctions by introducing an effective interaction for scattering potential. Calculations of the partial resistivity for different types of bulgings show that the partial resistivity of the sample decreases with increasing of the correlation length and asperity height. The results show that the bulging of the DW has a countable contribution into MR which is about 1–3% of DW MR, therefore providing more smooth and rigid DWs has a countable effect on transport of electrons.
ARTICLE IN PRESS 2404
M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
References [1] J.F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S.M. Thompson, J.M.D. Coey, Phys. Rev. Lett. 77 (1996) 1580. [2] Y. Shimazu, K. Sakai, T. Noda, I. Yamamoto, M. Yamaguchi, Physica B 284–288 (2000) 1239. [3] K. Hong, N. Giordano, J. Phys. Condens. Matter 8 (1996) L301. [4] U. Ruediger, J. Yu, S. Zhang, A.D. Kent, S.S.P. Parkin, Phys. Rev. Lett. 80 (1998) 5639. ¨ [5] A.D. Kent, J. Yu, U. Rudiger, S.S.P. Parkin, J. Phys. Condens. Matter. 13 (2001) R461. [6] G. Tatara, H. Fukuyama, Phys. Rev. Lett. 73 (1997) 3773.
[7] P.M. Levy, S. Zhang, Phys. Rev. Lett. 79 (1997) 5110. [8] B.R. Nag, Electron Transport in Compound Semiconductors, Springer-Verlag, New York, 1980. [9] G.D. Mahan, Mnay-Particle Physics, third ed. Kulwer Academic/Plenum Publisher, 2000. [10] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. [11] R.E. Prange, T.-W. Nee, Phys. Rev. 168 (1968) 779. [12] B.R. Nag, S. Mukhopadhyay, M. Das, J. Appl. Phys. 86 (1999) 459. [13] P. Roblin, W. Liao, Phys. Rev. B 46 (1993) 2416. [14] B.R. Nag, Semicond. Sci. Technol. 19 (2004) 162–166. [15] R. Skomski, J. Phys. Condens. Matter 15 (2003) R841–R896.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetoresistance due to domain wall bulging M.B. Fathi a, A. Phirouznia b, a b
Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran
a r t i c l e in fo
abstract
Article history: Received 6 July 2009 Received in revised form 2 November 2009 Available online 23 February 2010
The effect of bulging of domain wall (DW) on the magnetoresistance (MR) is investigated. With taking into account the auto-correlation between the points of the interface, one can formulate the mobility within the relaxation time approximation scheme. The results show that the bulging of DW, evaluated with the commonly accepted magnetic parameters for typical ferromagnetic materials of Co, Fe and Ni, has a countable role into the MR. & 2010 Elsevier B.V. All rights reserved.
Keywords: Magnetoresistance Domain wall Bulging Spintronics
1. Introduction Recent developments of nano-technology have enabled researchers to fabricate low dimensional devices. These developments have opened a new way for investigations directed on the nanoscale magnetoresistive devices. Magnetoresistance (MR) as a characteristic property of nanoscale devices, is of particular interest. Properties of ferromagnetic structures in nanometer scale have attracted much interest recently. Transport properties of these nano-structures are a powerful tool in the development of spintronics (spin electronics), the electronics in which the spin of electron plays the role of the active element for storage and transporting information. This has made an enormous interest for observing novel transport phenomena especially in the magnetic systems. The complexity of the local magnetic structures such as surfaces, interfaces and magnetic domain walls (DWs) offers a variety of novel features especially in the field of spin dependent transport phenomena. An outstanding problem in the field of spintronics is the effect of the magnetic structures such as magnetic DWs on the electronic transport. It was generally believed that DWs are sources of scattering, while experimental results suggest that, the DW magnetoresistance (MR) can be either positive or negative. Positive MR due to DW has been reported in striped domain structures [1] and the Ni wires [2]. In contrast, negative MR has been observed in some of experiments on very narrow wires and thin films [3–5]. Theoretically, the quantum decoherence caused
Corresponding author. Tel.: + 98 914 3221142.
E-mail address: [email protected] (A. Phirouznia). 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.02.045
by the DW has been mentioned as a source for reduction of the resistance and negative MR in the weakly localized regime (for example see [6]). On the other hand, spin-dependent impurity scattering which was proposed by Levy and Zhang was supposed to be responsible for mixing the spin channels and positive MR [7]. This paper is devoted to the calculation of the MR caused by DW bulging in a semiclassical approach based on the eigenvalue model.
2. Method Some effects such as pinning and current pressure cause the DW to change its shape leading to an effect known as DW bulging. 0 An electron with the wave vector k scatters to the state k due to the DW interface potential fluctuations. We consider the bulging of DW as a source of scattering similar to the roughness of interfaces. As the scattering is elastic, the spin dependent Þ could be evaluated within the relaxation relaxation times, tðEmðkÞ k time approximation. The relaxation time is given by [8,9] Z 1 1 2p X 0 0 0 d2 k jMðk;k Þj2 Sc ðqÞ ð1cosyÞdðEsk Esk0 Þ ð1Þ ¼ s 2 tðEk Þ 4p ‘ s0 0
where jMðk; k Þj is the matrix element for scattering from the k0 state to the k - state, Ek and Ek0 are the energies corresponding to 0 the wave vectors k and k , Sc(q) is the screening factor 0 corresponding to q ¼ k 7k and y represents the direction 0 between k and k .
ARTICLE IN PRESS 2402
M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
The electrons may be screened in 2D interface, and the screening factor, Sc(q), is simply chosen as [10] Sc ðqÞ ¼
q q þ qs
ð2Þ 0
where q is the norm of q ¼ k 7 k and qs ¼
m ðEsk Þe2 2pe‘
ð3Þ
2
The energy-dependent effective mass in the system is given by m*(Ek), e is the permittivity and e is the charge of an electron. The expression for the mobility of the electrons, m, reads [8] * + tðEmðkÞ Þ ð4Þ mmðkÞ ¼ jej kmðkÞ m ðEk Þ Since the electron gas is degenerate at low temperatures, the density of states (DOS) (which is responsible for charge transfer) is high around Fermi energy and therefore averaging over all energies, reduces to calculation of the relaxation time at Fermi energy. So,
mmðkÞ C jej
tðEmðkÞ Þ kf Þ m ðEmðkÞ k
Fig. 1. A schematic representation of bulging of the magnetic DW withits asperity parameter (FM: ferromagnet).
ð5Þ
f
Further development of the theory involves modeling of the scattering matrix: 0
Mðk;k Þ ¼ /Ck jDVjCk0 S
ð6Þ
which is the familiar expression for transition between the electron-states, Ck and Ck0 , with DV being the scattering potential.
3. Model
where L is the correlation length, and a is a parameter illustrating some estimate for roughness of f function, at deferent points. If we take f to be the roughness itself, then [8,13,14], /DrDr0 S ¼ D20 expððrr0 Þ2 =L2 Þ
ð10Þ
where the averaging is performed over the surface. In this expression D0 is called average asperity height, Fig. 1. Characteristically, one can suppose D0 and L to be some few angstroms in magnitude, for a typical DW surface. The spectral density for this distribution reads jf ðqÞj2 ¼ pD20 L2 expðq2 L2 =4Þ
ð11Þ
The Hamiltonian describing giant magnetoresistance (GMR) is described by [7] H0 ¼
‘ 2 r2 2m
^ þVðrÞ þ J r MðrÞ
ð7Þ
where J is the exchange splitting, V(r) is the nonmagnetic periodic ^ potential, and the unit vector MðrÞ points in the direction of local magnetization, M(r). Potential for scattering of electrons due to the magnetic impurities is given by X ½vþ jr MðrÞdðrri Þ ð8Þ Vscat ¼ i
4. Matrix element for the eigenvalue model In the eigenvalue model [14], the scattering potential is assumed to be caused by the change in energy eigenvalue due to the bulging, so the bulging potential is estimated by the deference between the energy eigenvalues,
DVðrÞ 7 ¼ jEðL 7 DrÞEðLÞj
ð12Þ
EðL 7 DrÞ and E(L) being the energy eigenvalues corresponding to the DW widths L 7 Dr and L, and 7 sign indicates increase or decrease of the width. The autocorrelation expression for this perturbing potential for a Gaussian distribution of bulging reads
where ri is the position of the impurities, v is the impurity potential and j represents the spin dependent interaction strength between the impurities and electrons. We choose the wavefunctions proposed by Levy and Zhang [7] for the DW, Cm ðk; rÞ and Ck ðk; rÞ, with the corresponding quantities. The important part of evaluating the effect of DW bulging is the formulating the bulging potential. Then we have to find the appropriate nonperturbed wave functions. First of all, we consider the bulging as the interface roughness, Fig. 1, and assume the correlation between the points r and r0 in 2D interface as being described by a function f ðx; yÞ ¼ f ðrÞ. This function may be supposed to describe the height of the bumps on the surface, i.e. the surface is at z ¼ f ðrÞ. Only the statistical properties of f is of importance. One employs, for example, a Gaussian distribution function or an exponential one. The correlation function for the Gaussian distribution is [11–13],
where the energy eigenvalues, EðL 7 D0 Þ and E(L), are the expectation values of the perturbation potential between the wavefunctions Cm ðk; rÞ and Ck ðk; rÞ. Using the perturbing potential, Eq. (12), and the corresponding spectral density, the square of the scattering matrix reads [14]
/f ðr0 þ rÞf ðr0 ÞS ¼ aexpðr2 =L2 Þ
jMðkk Þj2 ¼ A1 pðDVE0 Þ2 L2 expðq2 L2 =4Þ
ð9Þ
/DVðrÞDVðr0 ÞS ¼ ðDVE0 Þ2 expððrr0 Þ2 =L2 Þ
ð13Þ
The change in the energy eigenvalue for the average asperity height D0 , corresponding to L7 D0 , is chosen as the coefficient of the correlation function,
DVE0 ¼ jEðL 7 D0 ÞEðD0 Þj
0
ð14Þ
ð15Þ
ARTICLE IN PRESS M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
2403
where A is the surface of DW. At last one can obtain Z 1 1 2X 0 ¼ L d2 k ðDVE0 Þ2 expðq2 L2 =4Þ s tðEk Þ 2‘ s0 Sc ðqÞ ð1coshyÞdðEsk Esk0 Þ 0
ð16Þ
where the summation is over the two spin channels.
5. Results We choose commonly accepted values of kf =1 A˚ 1 and J= 0.5 eV for typical ferromagnetic materials of Co, Fe and Ni [7]. The two spin channels can be considered as two parallel current channels so that the inverse of mobility can be understood as a summation of inverse of two spin-dependent mobilities [8], 1
m
¼
1
mm
þ
1
mk
ð17Þ Fig. 4. Partial resistivity, as a function of asperity height, D0 .
Therefore resistivity, r, can be considered to be summation of two resistivities, r ¼ rm þ rk . In the present calculations, we have found small difference in quantitative behavior of these two spin channels. We define partial resistivity due to bulging as follows:
dr
r0
Fig. 2. The resistivity, r, due to bulging of DW, as a function of k/kf.
100%;
ð18Þ
where dr ¼ rr0 , r0 is the DW resistivity, and r is the resistivity due to the DW bulging. Fig. 2 shows the partial resistivity as a function of wave vector, k. As the figure shows, the maximum partial resistivity as a function of k occurs around the Fermi wave vector, kf. The partial resistivity as a function of correlation length, L, is shown in Fig. 3. As shown in this figure, with the increasing of the correlation length and average asperity height the contribution of the DW bulging into the partial resistivity is suppressed and for a typical range of correlation length [15] the maximum of the partial resistivity due to the DW bulging is about 2–3%. For higher correlation lengths which can be considered up to several 100 nm the partial resistivity due to the DW bulging is negligible. Fig. 4 shows the partial resistivity versus the average asperity height, D0 . As is obvious from the figure, for acceptable values of asperity height for a microscopic bulging which is assumed to be higher than 1 A˚ the maximum value of the partial resistivity due to roughness of the DW is about 2%. However, it should be noted that the partial resistivity is not a monotonic function of asperity height and at low average asperities the partial resistivity increases by increasing the asperity height and then decreases smoothly after a maximum is passed. For the present case only a realistic range has been chosen for microscopic asperity height.
6. Conclusion
Fig. 3. The partial resistivity, due to bulging of DW, as a function of the correlation length, L.
A semiclassical approach has been employed to take into account the effect of bulging of magnetic DW into the resistivity. The effect of bulging can be considered as the roughness of the heterojunctions by introducing an effective interaction for scattering potential. Calculations of the partial resistivity for different types of bulgings show that the partial resistivity of the sample decreases with increasing of the correlation length and asperity height. The results show that the bulging of the DW has a countable contribution into MR which is about 1–3% of DW MR, therefore providing more smooth and rigid DWs has a countable effect on transport of electrons.
ARTICLE IN PRESS 2404
M.B. Fathi, A. Phirouznia / Journal of Magnetism and Magnetic Materials 322 (2010) 2401–2404
References [1] J.F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S.M. Thompson, J.M.D. Coey, Phys. Rev. Lett. 77 (1996) 1580. [2] Y. Shimazu, K. Sakai, T. Noda, I. Yamamoto, M. Yamaguchi, Physica B 284–288 (2000) 1239. [3] K. Hong, N. Giordano, J. Phys. Condens. Matter 8 (1996) L301. [4] U. Ruediger, J. Yu, S. Zhang, A.D. Kent, S.S.P. Parkin, Phys. Rev. Lett. 80 (1998) 5639. ¨ [5] A.D. Kent, J. Yu, U. Rudiger, S.S.P. Parkin, J. Phys. Condens. Matter. 13 (2001) R461. [6] G. Tatara, H. Fukuyama, Phys. Rev. Lett. 73 (1997) 3773.
[7] P.M. Levy, S. Zhang, Phys. Rev. Lett. 79 (1997) 5110. [8] B.R. Nag, Electron Transport in Compound Semiconductors, Springer-Verlag, New York, 1980. [9] G.D. Mahan, Mnay-Particle Physics, third ed. Kulwer Academic/Plenum Publisher, 2000. [10] T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437. [11] R.E. Prange, T.-W. Nee, Phys. Rev. 168 (1968) 779. [12] B.R. Nag, S. Mukhopadhyay, M. Das, J. Appl. Phys. 86 (1999) 459. [13] P. Roblin, W. Liao, Phys. Rev. B 46 (1993) 2416. [14] B.R. Nag, Semicond. Sci. Technol. 19 (2004) 162–166. [15] R. Skomski, J. Phys. Condens. Matter 15 (2003) R841–R896.