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Variable range hopping conductivity in manganites
Solid State Communications 152 (2012) 1139–1141
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Variable range hopping conductivity in manganites Yu.Kh. Vekilov, Ya.M. Mukovskii n NITU ‘‘MISiS’’, Leninskii Prosp., 4, Moscow 119049, Russia
a r t i c l e i n f o
abstract
Article history: Received 10 March 2012 Accepted 2 April 2012 by Y.E. Lozovik Available online 12 April 2012
The nature of variable range hopping (VRH) conductivity which is observed in the insulating state of doped rare-earth manganites with perovskite structure is considered in the two component model of metallic-like droplets embedded in dielectric matrix. When the density of the metallic droplets is less than the percolation limit, the system falls into the insulating state with VRH conductivity defined by inter granular tunneling and electrostatic barriers. With temperature increasing the VRH regime is transforming into the hopping regime of small radius polarons. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Manganites B. Granularity D. Conductivity D. Tunneling
Rare-earth manganites RE1 xAxMnO3, where RE is rare-earth element, A is two valence element, A¼ Ca, Sr, Ba, Pb, are interesting materials for physics. They exhibit colossal magnetoresistance and are strong correlated systems, where lattice, charge and spin degrees of freedom are tightly connected one with other. They were studied intensively for a long period [1–14], and the corresponding field is very crowded up to now. But till now there are many problems in manganites have to be solved because in addition to the CMR effect itself other generic unusual features of the manganite compounds have been found in recent years. Among them is metal–insulator transition in doped manganites at temperature Tp, corresponding to the resistivity peak. The parent manganite compound LaMnO3 at ambient conditions is a paramagnetic insulator and is regarded as the archetypical cooperative Jahn–Teller and orbitally ordered system; it has the orthorhombic Pnma structure and transforms into the antiferromagnetic (AFM) insulator at TN 140 K [1–3]. Sufficient doping by two-valence ion makes it ferromagnetic (FM) and metallic in the FM state. The system show metallic-like behavior when reaching any critical x-value for doping (for example, x E0.16 for A ¼Sr), percolation threshold. So the percolating type of metal–insulator transition (MIT) takes place at some temperature Tp. The Tp corresponds to the maximum on the resistivity temperature dependence r(T) and is the transition temperature into a dielectric state at heating. This transition is nearby the Curie temperature TC, but can be as higher as lower. Below Tp at the metallic side of the transition, where dr/dT4 0, the power law of r(T) is observed [5,11,12]. On the insulating side of the MIT,
n
Corresponding author. E-mail address: [email protected] (Ya.M. Mukovskii).
0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.04.001
where dr/dT o0, in the temperature interval Tp oTo YD/2, YD is the Debye temperature, the activated by temperature VRH con1/4 ductivity of Mott type (s s0 exp( TM ) or Efros–Shklovsky 0 /T) 1/2 (ES) type (s s0 exp( TES /T) ) is observed [13,14]. At T4 YD/2 0 the activated conductivity of small radius polarons takes place [7,8], which is described by Arrhenius–like law (s s0T exp( U/ kBT)), where U is an activated energy, kB is the Boltzmann constant. The successive explanation of the VRH conductivity in manganites is absent. All authors [3,6,7,11–14] who have observed it tried to explain the VRH in analogy with doped semiconductors and Fermi glasses where at the finite density of states at the Fermi energy, N(EF)a0, the electronic states are localized due to disorder introduced by doping, and only hopping conductivity is possible. For states near the Fermi energy hops on large distances are more probable and this evolves tunneling and is reflected in the expression of the hopping rate w to a site at distance R, where the energy of the carrier is DE higher than at the origin, w exp( 2aR DE/kBT). For phonon assisted process the Mott law is obtained [15], and when electron–electron correlations are important (Coulomb gap) the Efros–Shklovsky law takes place [16]. But manganites are structurally ordered compounds, and the VRH regime in them is not connected with structural disorder appearing in the system when the RE atom is substituted by a two-valance one. Moreover, with increasing x the system becomes metallic-like. In contrast to semiconductors the VRH regime in manganites is observed at much higher temperatures and, that is especially essential, with finite residual conductivity, which indicates on the presence in insulating state of the second low resistive (metallic) shunting phase. Note also, that in contrast to semiconductors, in insulating state of manganites at first the ES
1140
Yu.Kh. Vekilov, Ya.M. Mukovskii / Solid State Communications 152 (2012) 1139–1141
law is observed that with temperature decreasing is followed by the Mott law [13,14]. Following to these facts we give another explanation of the VRH conductivity in manganites. The appearance of the VRH regime in manganites is connected with two phase state of the system (metallic and dielectric) [9,10]. Depending on density of metallic droplets, the metallic or insulating state is realized, and in the insulating matrix the presence the metallic droplets can lead to the VRH conductivity. The picture is similar to the appearance of VRH regime in granular metals [17] and polygrain quasicrystals [18], and we can use the granular electronic system model, which was successfully used for these objects. The model presents the system of metallic granules with size from several to hundred nm embedded in a dielectric matrix. In such systems size quantization of electronic states due to electron confinement in the granule is important. The mean spacing d between electronic levels in the granule is inversely proportional to the granule volume V and the density of electronic states at the Fermi energy EF d ¼(N(EF)V) 1. The granularity (i.e. the size quantization) effects are not important when T4 d/kB. Two features define electronic transport in such system: the first one is the inter granular tunneling conductance g, corresponding to one spin component and measured in units of quantum conductance e2/g, g ¼G/(2e2/_). The samples with g41 reveal the metallic properties, and samples with go1 are insulating ones. The conception of the ‘‘granular system’’ means g 5g0, where g0 is dimensionless inter granular conductance. The case g g0 suits to uniformly disordered system. The second important feature is the electrostatic (Coulomb) energy Ec(Ec 4 d), which is needed to remove or replace one electron from or on the granule. This energy is important in the insulating state when electrons are localized on the granule. The physics of the insulating state is closely related to the well known Coulomb blockade phenomenon of a single grain connected to a metallic reservoir [19]. In the ‘‘metallic’’ regime, gb1, the electrons freely move through the sample, and Coulomb interaction is screened. In the opposite limit, g51, an electron has to overcome electrostatic barrier Ec in order to be placed on the granule, and this mediates the electronic transport at ToEc/kB. It is possible by changing the size and the shape of the granules (in dependence of inter granular tunneling conductance) to get a good metal or insulator. In the insulating state tunneling and activated hopping will lead to VRH conductivity. In quasiclassical approximation the probability pffiffiffiffiffiffiffiffiffiffiffi of tunneling is proportional to exp( ba), where b ¼ 2 2mU =_, a and U is the width and the height of the barrier correspondingly. Each charged granule creates an electric field in the inter granular space and acts as a plate of micro capacitor with capacitance C eb, where e is the dielectric constant and b is the width of the boundary. The energy of the charged granule is equal e2/2C e2/2eb. Correspondingly, the conductivity s is equal to s exp ðe2 =erTÞbr . Finding the extremum we get pffiffiffiffiffiffiffiffi 2 r min ¼ e = beT and finally we get ES law s expððT 0 =TÞÞ1=2 with T0 ¼ 2eO(be). The result means that the charge of the granule acts as a point one. This is true at large distances from the granule, but at small distances we should take into account the multipole effects. In the dipole approximation we get for the conductivity the Mott-like law " # T 1=4 s ¼ s0 exp 0 T Both laws can be obtained from the expression for the hopping probability w ( 2rij/x–Eij/kBT), where x is the localization length for the electron hopping to distance rij between two localized
states i and j with energy difference Eij. When Eij [N(EF) r3ij] 1we get the Mott law, when Eij e2/erij—the ES law. Because the energies are additive ones it is possible to join these contributions Eij ¼a1[N(EF) r3ij] 1 þa2e2/erij, where a1 and a2 are of order of unit. For large rij the second contribution will dominate, and for small distances the first one will. When the finite sizes of granules are important we get the Mott law. In general case it is possible to observe s ¼ s0 expððT 0 =TÞÞ1=p , with p laying in the interval from 1/4 to 1/2. From the above consideration it follows also that the ES law should be observed at higher temperatures (where the concentration of metallic droplets is less) than the Mott law. The hopping length in the VRH mechanism should exceed the granule size and it decreases with temperature increasing. When at some temperature it becomes equal to the granule size, the VRH regime ceases working, and the system falls into another activated regime. (for example, in granular conductors, in regime with the Arrhenius law, in doped manganites, hopping conductivity of small radius polarons). The hops to large distances in a system where each granule is separated from neighboring one by tunnel junction involve a lot of grains. And such cooperative tunneling (cotunneling) can be elastic or inelastic [17]. At elastic cotunneling the same electron is always stays on the same energetic level. The inelastic cotunneling is accompanied by electron-hole pair excitation. The elastic cotunneling is effective at temperatures T oTn O(dEc), and for common metals these temperatures are very low (order of several Kelvin). In manganites Tn is as minimum an order higher. When inter granular interaction is strong and barriers are not important, manganites turn into metallic-like state with power law temperature dependence of conductivity. As experiments show [11,12] the resistivity r can be represented as r ¼ r0 þ r2T2 þ r4.5T4.5 (r0 is the residual resistivity due to defects, the second contribution is due to electron-electron scattering, and the third one is connected with electron-magnon scattering). The granularity in metallic system restricts screening and as a result in a definite temperature interval the logarithmic temperature dependence of conductivity can appear s lnT [17]. Note that power law temperature dependent conductivity s Ta can be closed to the logarithmic one, when alnTo1 because Ta exp(alnT). The experimental information about logarithmic temperature dependence conductivity in manganites is absent. In conclusion, we have shown that the appearance of VRH conductivity in manganites can be explained as a consequence of the two phase state (metallic and dielectric) of the system. In turn, the experimental manifestation of VRH conductivity proofs the presence of metallic droplets in insulating state of manganites.
Acknowledgments This work was supported by the Joint RFBR-DST grant 11-0292706-Ind_a and the RFBR grant 12-02-00717_a. We thank Prof. M.A. Chernikov for his attention to the present work. References [1] E.O. Wollan, W.C. Koehler, Phys. Rev. 100 (1955) 545–563. [2] J.B. Goodenough, M. Longo, in: V.H. Hellenwege, A.M. Hellenwege (Eds.), Magnetic Properties Magnetic and Other Properties of Oxides and Related Compounds. Part A, Springer-Verlag, Berlin, 1970, pp. 315–407. [3] M. Viret, L. Runno, J.M.D. Coey, Phys. Rev. B 55 (1997) 8067–8070. [4] F. Moussa, M. Hennion, J. Rodriguez-Carvajal, et al., Phys. Rev. B 54 (1996) 15149–15155. [5] V.E. Arkhipov, V.P. Dyakina, V.V. Marchenkov, Ya.M. Mukovskii, et al., Phys. Met. Met. (Rus.) 12 (1997) 638–644. [6] S.K. Hassanian, M. Nadeem, W.H. Shah, et al., J. Phys.: Cond. Mat. 12 (2000) 9007–9018.
Yu.Kh. Vekilov, Ya.M. Mukovskii / Solid State Communications 152 (2012) 1139–1141
[7] Y. Sun, X. Xu, Y. Zhang, J. Phys.: Cond. Mat. 12 (2000) 10475–10480. [8] Himanshu Jain, A.K. Raychaudhuri, Ya.M. Mukovskii, D. Shulyatev, Solid State Commun. 138 (2006) 318–323. [9] L.P. Gor’kov, Physics-Uspekhi (Advances in Physical Sciences) 41 (1998) 589–595. [10] M.O. Drero, L.P. Gorkov, V.Z. Kresin, Eur. Phys. B14 (2000) 459–471. [11] M.Y. Baazaoui, S. Zemni, M. Bondard, et al., Physica B 405 (2010) 1470–1476. [12] A.G. Mostafa, E.K. Abdel-Khalek, W.M. Davush, S.F. Mostafa, J. Magn. Magn. Mater. 320 (2008) 3356–3360. [13] R. Laiho, K.G. Lisunov, E. Lahderanta, et al., J. Phys.: Cond. Mater. 17 (2005) 3429–3444.
1141
[14] V.S. Zakhvalinskii, R. Laiho, A.V. Lashkul, et al., J. Phys.: Cond. Mater. 23 (2011) 015802. [15] N.F. Mott, E.A. Davis, Electronic Properties in Non-Crystalline Materials, Clarendon Press, Oxford, 1979. [16] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, Heidelberg, 1984. [17] I.S. Beloborodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469–518. [18] Yu.Kh. Vekilov, M.A. Chernikov, Europhys. Lett. 87 (2009) 17010. [19] D.A. Averin, Y.V. Nazarov, Phys. Rev. Lett. 65 (1990) 2446–2449.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Variable range hopping conductivity in manganites Yu.Kh. Vekilov, Ya.M. Mukovskii n NITU ‘‘MISiS’’, Leninskii Prosp., 4, Moscow 119049, Russia
a r t i c l e i n f o
abstract
Article history: Received 10 March 2012 Accepted 2 April 2012 by Y.E. Lozovik Available online 12 April 2012
The nature of variable range hopping (VRH) conductivity which is observed in the insulating state of doped rare-earth manganites with perovskite structure is considered in the two component model of metallic-like droplets embedded in dielectric matrix. When the density of the metallic droplets is less than the percolation limit, the system falls into the insulating state with VRH conductivity defined by inter granular tunneling and electrostatic barriers. With temperature increasing the VRH regime is transforming into the hopping regime of small radius polarons. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Manganites B. Granularity D. Conductivity D. Tunneling
Rare-earth manganites RE1 xAxMnO3, where RE is rare-earth element, A is two valence element, A¼ Ca, Sr, Ba, Pb, are interesting materials for physics. They exhibit colossal magnetoresistance and are strong correlated systems, where lattice, charge and spin degrees of freedom are tightly connected one with other. They were studied intensively for a long period [1–14], and the corresponding field is very crowded up to now. But till now there are many problems in manganites have to be solved because in addition to the CMR effect itself other generic unusual features of the manganite compounds have been found in recent years. Among them is metal–insulator transition in doped manganites at temperature Tp, corresponding to the resistivity peak. The parent manganite compound LaMnO3 at ambient conditions is a paramagnetic insulator and is regarded as the archetypical cooperative Jahn–Teller and orbitally ordered system; it has the orthorhombic Pnma structure and transforms into the antiferromagnetic (AFM) insulator at TN 140 K [1–3]. Sufficient doping by two-valence ion makes it ferromagnetic (FM) and metallic in the FM state. The system show metallic-like behavior when reaching any critical x-value for doping (for example, x E0.16 for A ¼Sr), percolation threshold. So the percolating type of metal–insulator transition (MIT) takes place at some temperature Tp. The Tp corresponds to the maximum on the resistivity temperature dependence r(T) and is the transition temperature into a dielectric state at heating. This transition is nearby the Curie temperature TC, but can be as higher as lower. Below Tp at the metallic side of the transition, where dr/dT4 0, the power law of r(T) is observed [5,11,12]. On the insulating side of the MIT,
n
Corresponding author. E-mail address: [email protected] (Ya.M. Mukovskii).
0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.04.001
where dr/dT o0, in the temperature interval Tp oTo YD/2, YD is the Debye temperature, the activated by temperature VRH con1/4 ductivity of Mott type (s s0 exp( TM ) or Efros–Shklovsky 0 /T) 1/2 (ES) type (s s0 exp( TES /T) ) is observed [13,14]. At T4 YD/2 0 the activated conductivity of small radius polarons takes place [7,8], which is described by Arrhenius–like law (s s0T exp( U/ kBT)), where U is an activated energy, kB is the Boltzmann constant. The successive explanation of the VRH conductivity in manganites is absent. All authors [3,6,7,11–14] who have observed it tried to explain the VRH in analogy with doped semiconductors and Fermi glasses where at the finite density of states at the Fermi energy, N(EF)a0, the electronic states are localized due to disorder introduced by doping, and only hopping conductivity is possible. For states near the Fermi energy hops on large distances are more probable and this evolves tunneling and is reflected in the expression of the hopping rate w to a site at distance R, where the energy of the carrier is DE higher than at the origin, w exp( 2aR DE/kBT). For phonon assisted process the Mott law is obtained [15], and when electron–electron correlations are important (Coulomb gap) the Efros–Shklovsky law takes place [16]. But manganites are structurally ordered compounds, and the VRH regime in them is not connected with structural disorder appearing in the system when the RE atom is substituted by a two-valance one. Moreover, with increasing x the system becomes metallic-like. In contrast to semiconductors the VRH regime in manganites is observed at much higher temperatures and, that is especially essential, with finite residual conductivity, which indicates on the presence in insulating state of the second low resistive (metallic) shunting phase. Note also, that in contrast to semiconductors, in insulating state of manganites at first the ES
1140
Yu.Kh. Vekilov, Ya.M. Mukovskii / Solid State Communications 152 (2012) 1139–1141
law is observed that with temperature decreasing is followed by the Mott law [13,14]. Following to these facts we give another explanation of the VRH conductivity in manganites. The appearance of the VRH regime in manganites is connected with two phase state of the system (metallic and dielectric) [9,10]. Depending on density of metallic droplets, the metallic or insulating state is realized, and in the insulating matrix the presence the metallic droplets can lead to the VRH conductivity. The picture is similar to the appearance of VRH regime in granular metals [17] and polygrain quasicrystals [18], and we can use the granular electronic system model, which was successfully used for these objects. The model presents the system of metallic granules with size from several to hundred nm embedded in a dielectric matrix. In such systems size quantization of electronic states due to electron confinement in the granule is important. The mean spacing d between electronic levels in the granule is inversely proportional to the granule volume V and the density of electronic states at the Fermi energy EF d ¼(N(EF)V) 1. The granularity (i.e. the size quantization) effects are not important when T4 d/kB. Two features define electronic transport in such system: the first one is the inter granular tunneling conductance g, corresponding to one spin component and measured in units of quantum conductance e2/g, g ¼G/(2e2/_). The samples with g41 reveal the metallic properties, and samples with go1 are insulating ones. The conception of the ‘‘granular system’’ means g 5g0, where g0 is dimensionless inter granular conductance. The case g g0 suits to uniformly disordered system. The second important feature is the electrostatic (Coulomb) energy Ec(Ec 4 d), which is needed to remove or replace one electron from or on the granule. This energy is important in the insulating state when electrons are localized on the granule. The physics of the insulating state is closely related to the well known Coulomb blockade phenomenon of a single grain connected to a metallic reservoir [19]. In the ‘‘metallic’’ regime, gb1, the electrons freely move through the sample, and Coulomb interaction is screened. In the opposite limit, g51, an electron has to overcome electrostatic barrier Ec in order to be placed on the granule, and this mediates the electronic transport at ToEc/kB. It is possible by changing the size and the shape of the granules (in dependence of inter granular tunneling conductance) to get a good metal or insulator. In the insulating state tunneling and activated hopping will lead to VRH conductivity. In quasiclassical approximation the probability pffiffiffiffiffiffiffiffiffiffiffi of tunneling is proportional to exp( ba), where b ¼ 2 2mU =_, a and U is the width and the height of the barrier correspondingly. Each charged granule creates an electric field in the inter granular space and acts as a plate of micro capacitor with capacitance C eb, where e is the dielectric constant and b is the width of the boundary. The energy of the charged granule is equal e2/2C e2/2eb. Correspondingly, the conductivity s is equal to s exp ðe2 =erTÞbr . Finding the extremum we get pffiffiffiffiffiffiffiffi 2 r min ¼ e = beT and finally we get ES law s expððT 0 =TÞÞ1=2 with T0 ¼ 2eO(be). The result means that the charge of the granule acts as a point one. This is true at large distances from the granule, but at small distances we should take into account the multipole effects. In the dipole approximation we get for the conductivity the Mott-like law " # T 1=4 s ¼ s0 exp 0 T Both laws can be obtained from the expression for the hopping probability w ( 2rij/x–Eij/kBT), where x is the localization length for the electron hopping to distance rij between two localized
states i and j with energy difference Eij. When Eij [N(EF) r3ij] 1we get the Mott law, when Eij e2/erij—the ES law. Because the energies are additive ones it is possible to join these contributions Eij ¼a1[N(EF) r3ij] 1 þa2e2/erij, where a1 and a2 are of order of unit. For large rij the second contribution will dominate, and for small distances the first one will. When the finite sizes of granules are important we get the Mott law. In general case it is possible to observe s ¼ s0 expððT 0 =TÞÞ1=p , with p laying in the interval from 1/4 to 1/2. From the above consideration it follows also that the ES law should be observed at higher temperatures (where the concentration of metallic droplets is less) than the Mott law. The hopping length in the VRH mechanism should exceed the granule size and it decreases with temperature increasing. When at some temperature it becomes equal to the granule size, the VRH regime ceases working, and the system falls into another activated regime. (for example, in granular conductors, in regime with the Arrhenius law, in doped manganites, hopping conductivity of small radius polarons). The hops to large distances in a system where each granule is separated from neighboring one by tunnel junction involve a lot of grains. And such cooperative tunneling (cotunneling) can be elastic or inelastic [17]. At elastic cotunneling the same electron is always stays on the same energetic level. The inelastic cotunneling is accompanied by electron-hole pair excitation. The elastic cotunneling is effective at temperatures T oTn O(dEc), and for common metals these temperatures are very low (order of several Kelvin). In manganites Tn is as minimum an order higher. When inter granular interaction is strong and barriers are not important, manganites turn into metallic-like state with power law temperature dependence of conductivity. As experiments show [11,12] the resistivity r can be represented as r ¼ r0 þ r2T2 þ r4.5T4.5 (r0 is the residual resistivity due to defects, the second contribution is due to electron-electron scattering, and the third one is connected with electron-magnon scattering). The granularity in metallic system restricts screening and as a result in a definite temperature interval the logarithmic temperature dependence of conductivity can appear s lnT [17]. Note that power law temperature dependent conductivity s Ta can be closed to the logarithmic one, when alnTo1 because Ta exp(alnT). The experimental information about logarithmic temperature dependence conductivity in manganites is absent. In conclusion, we have shown that the appearance of VRH conductivity in manganites can be explained as a consequence of the two phase state (metallic and dielectric) of the system. In turn, the experimental manifestation of VRH conductivity proofs the presence of metallic droplets in insulating state of manganites.
Acknowledgments This work was supported by the Joint RFBR-DST grant 11-0292706-Ind_a and the RFBR grant 12-02-00717_a. We thank Prof. M.A. Chernikov for his attention to the present work. References [1] E.O. Wollan, W.C. Koehler, Phys. Rev. 100 (1955) 545–563. [2] J.B. Goodenough, M. Longo, in: V.H. Hellenwege, A.M. Hellenwege (Eds.), Magnetic Properties Magnetic and Other Properties of Oxides and Related Compounds. Part A, Springer-Verlag, Berlin, 1970, pp. 315–407. [3] M. Viret, L. Runno, J.M.D. Coey, Phys. Rev. B 55 (1997) 8067–8070. [4] F. Moussa, M. Hennion, J. Rodriguez-Carvajal, et al., Phys. Rev. B 54 (1996) 15149–15155. [5] V.E. Arkhipov, V.P. Dyakina, V.V. Marchenkov, Ya.M. Mukovskii, et al., Phys. Met. Met. (Rus.) 12 (1997) 638–644. [6] S.K. Hassanian, M. Nadeem, W.H. Shah, et al., J. Phys.: Cond. Mat. 12 (2000) 9007–9018.
Yu.Kh. Vekilov, Ya.M. Mukovskii / Solid State Communications 152 (2012) 1139–1141
[7] Y. Sun, X. Xu, Y. Zhang, J. Phys.: Cond. Mat. 12 (2000) 10475–10480. [8] Himanshu Jain, A.K. Raychaudhuri, Ya.M. Mukovskii, D. Shulyatev, Solid State Commun. 138 (2006) 318–323. [9] L.P. Gor’kov, Physics-Uspekhi (Advances in Physical Sciences) 41 (1998) 589–595. [10] M.O. Drero, L.P. Gorkov, V.Z. Kresin, Eur. Phys. B14 (2000) 459–471. [11] M.Y. Baazaoui, S. Zemni, M. Bondard, et al., Physica B 405 (2010) 1470–1476. [12] A.G. Mostafa, E.K. Abdel-Khalek, W.M. Davush, S.F. Mostafa, J. Magn. Magn. Mater. 320 (2008) 3356–3360. [13] R. Laiho, K.G. Lisunov, E. Lahderanta, et al., J. Phys.: Cond. Mater. 17 (2005) 3429–3444.
1141
[14] V.S. Zakhvalinskii, R. Laiho, A.V. Lashkul, et al., J. Phys.: Cond. Mater. 23 (2011) 015802. [15] N.F. Mott, E.A. Davis, Electronic Properties in Non-Crystalline Materials, Clarendon Press, Oxford, 1979. [16] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, Heidelberg, 1984. [17] I.S. Beloborodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469–518. [18] Yu.Kh. Vekilov, M.A. Chernikov, Europhys. Lett. 87 (2009) 17010. [19] D.A. Averin, Y.V. Nazarov, Phys. Rev. Lett. 65 (1990) 2446–2449.