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Resistivity in two-dimensional doped antiferromagnet
Physics Letters A 323 (2004) 138–147 www.elsevier.com/locate/pla
Resistivity in two-dimensional doped antiferromagnet A.M. Belemuk, A.F. Barabanov ∗ Institute for High Pressure Physics, 142190 Troitsk, Moscow region, Russia Received 1 December 2003; accepted 28 December 2003 Communicated by V.M. Agranovich
Abstract The temperature dependence of the resistivity ρ(T ) in two-dimensional doped antiferromagnet is investigated within the single band Kondo lattice model. To take into account the strong temperature dependent scattering anisotropy the kinetic equation for the nonequilibrium carrier distribution function is derived through the density matrix formalism in two-moment approach. Obtained ρ(T ) dependence demonstrates qualitative agreement with experimental data for optimally doped high temperature superconductors in a broad temperature range. It is found that a gap in the spin excitation spectrum is of crucial importance for the behavior of ρ(T ). 2004 Published by Elsevier B.V. PACS: 72.10.Di; 74.25.Fy; 71.10.Ay
1. Introduction Since the discovery of high-Tc superconductors the anomalous properties of their normal state have been the subject of intensive theoretical study. Although a broad range of models and theories have been proposed, none has successfully explained all available experimental data. One of the most unusual properties of high-Tc cuprates is the temperature dependence of the resistivity [1–5]. The optimal doped samples are characterized by in-plane resistivity ρ(T ) that grows nearly linearly with temperature from just above the superconducting transition temperature Tc up to some hundreds degrees Kelvin. For example, in the optimum composition range near x ∼ 0.15 for La1−x Srx CuO4 the T -linear resistivity has been seen from just above Tc = 35 K to near 1000 K [4]. The value of Tc is well below the Debye temperature. At so low temperature standard Fermi-liquid theory predicts ρ(T ) ∼ T 2 for electron–electron scattering and ρ ∼ T 5 for three-dimensional electron–phonon scattering. In the overdoped regime, the exponent γ of ρ ∝ T γ changes from γ = 1 to 2. In La1−x Srx CuO4 above x ∼ 0.2 a power-law dependence with γ ∼ 1.5 was found [4]. The underdoped samples exhibit deviations from the linear behavior at low temperatures. For example, the resistivity in YBa2 Cu3 O6+x displays strong curvature below TK ∼ 320 K [1] and follows a power-law above TK (with γ ∼ 1.2). With increasing hole doping, the curvature * Corresponding author.
E-mail address: [email protected] (A.F. Barabanov). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.12.064
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below TK evolves into the linear-T behavior. The reason for this behavior is not clear. Possible explanation that it reflects a subtle change in the structure of the hole spectrum. The influence of the pseudogap opening on the normal state properties is still under discussion [2,5]. Theoretical models relevant to the understanding of the pseudogap state in high-temperature superconductors (HTSC) are discussed in [6]. There are two broad classes of theoretical approaches for explaining the anomalous of cuprates. The first one postulates the presence of two independent relaxation time-scales for the Hall and electric current [7,8]. The second one deals with the Fermi liquid concept involving an anomalous momentum dependent scattering time [9–13]. Fermi-liquid approaches are based on distinct temperature dependence of quasiparticle scattering time into different regions of the Fermi surface (FS). “Hot” regions (near X points (±π, 0) and (0, ±π) points) and “cold” regions (near the Brillouin zone (BZ) diagonal) correspond respectively strong and weak quasiparticle scattering. They differ by values of the Fermi velocity and temperature dependences of the scattering time. “Hot/cold” region models [10–12] have not led to general agreement on what regions dominate the transport. Multi-patch model [13] seems to be the best systematic analysis of normal state transport properties compatible with the experiments. The scattering operator of the linearized Boltzmann equation is projected on the patches and temperature dependences of its coefficients are assigned. In this model the linear temperature dependence of the resistivity is associated to the inter-patch scattering of carriers. From the models covered it is apparent that severe speculations with many phenomenological parameters are introduced at analysis of transport properties. In this Letter we present a study of the resistivity of the twodimensional doped antiferromagnet for the optimally doped case sticking to the models of the second type— nearly antiferromagnetic (AFM) two-dimensional Fermi liquid. Our analysis is less phemenological relative to close approaches. This mainly concerns the spin-polaron carrier spectrum and the spin excitation propagator which were calculated previously. The unconventional normal state transport properties are usually viewed as a manifestation of the strongly correlated nature of the charge dynamics in CuO2 of HTSC. Relevant physics of high-Tc compounds is fairly well described by the three-band Emery model [14–16]. This model was previously studied on the basis of the spin polaron concept in a broad doping interval [17]. The concept gives rise to an elementary excitation spectrum appropriate to angular resolved photoemission spectroscopy (ARPES) experiments. In particular, it captures such subtle features as the minimum of the spectrum near momentum (±π/2, ±π/2) and the remnant FS for undoped compounds, the large FS and the extended saddle point close to the FS for the optimally doped cuprates and the presence of a pseudogap for intermediate doping [21–23]. Due to excessive complication of the three-band model we consider more simple one, the single-band Kondo lattice model on a square lattice with intra-site coupling of the carriers and the spin subsystem. Below for the carrier spectrum we adopt the spectrum of the lowest spin-polaron band [17] which reflects the main features of ARPES. As to the spin excitation propagator, we treat it in the framework of the spherically symmetric self-consistent approach [18] for 2D S = 1/2 frustrated AFM Heisenberg model. In particular, it means that the average one-site spin values are zero SRα = 0 and the AFM spin–spin correlation functions S0α SRα at fixed α do not depend on the Cartesian index α. It is important that such a treatment gives a temperature dependent gap ∆ at AFM momentum Q = (π, π) in the spin-excitation spectrum. In Section 2 we briefly introduce a model for the carriers scattered by the spin subsystems. In Section 3 we consider a multi-moment method [19] for the kinetic equation solving based on a nonequilibrium stastistical operator. The method allows to take into account a strongly anisotropic temperature dependent scattering. In Section 4 we present our results for the dependence ρ(T ) given by the two-moment realization of a multi-moment method and discuss the distinctions of our treatment from close approaches.
2. Model The effective Hamiltonian of the regular Kondo model type on a square lattice is
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Fig. 1. The carrier spectrum εk (in eV) is presented by equal-energy lines εk = const, the thick solid line εk = 0.08 corresponds to the Fermi surface for the optimally doped case. The arrow indicates the hot-point scattering by the AFM vector Q = (π, π ) from a state k1 below the FS into a state k2 above the FS.
Hˆ tot = Hˆ 0 + Hˆ 1 ,
Hˆ 0 = Hˆ h + Iˆ,
Hˆ 1 = Hˆ f + Jˆ.
The term Hˆ 0 describes the motion of the fermionic carriers (holes) † εk akσ akσ Hˆ h = k,σ
and the system of localized spins 1 α 1 α Iˆ = I1 SR+g SRα + I2 SR+d SRα . 2 2 R,g
R,d
Here g and d are the vectors of the nearest and second nearest neighbors. The exchange Hamiltonian Iˆ corresponds to AFM frustrated interaction between spins, p (0 p 1)—a frustration parameter, I1 = (1 − p)I and I2 = pI are exchange constants for the first and the second nearest neighbors. We use the following form for the hole spectrum (see Fig. 1) εk = τ a1 γg (k) + a2 γg2 (k) + a3 γd (k) + a4 γd2 (k) + a5 γg (k)γd (k) (1) with the values of the coefficients ai and the quadratic harmonics γi (k) given by a1 = 1.5, a2 = 1.8, a3 = −0.5, a4 = 0.7, a5 = 0.5, γg (k) = (cos(kx ) + cos(ky ))/2, γd (k) = cos(kx ) cos(ky ).
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The term Jˆ is the intrasite exchange Hamiltonian
1 Jˆ = J √ N
† ak+q,γ S α σˆ α akγ2 , 1 q γ1 γ2
k,q,γ1 ,γ2
1 −iqR Sq = √ e SR , N q
where σˆ α are the Pauli matrices (summation is assumed over repeated Cartesian indices α). Coupling with an external homogeneous electric field E is given by ˆ Pˆ = e k1 |ˆr|k2 ak†1 σ ak2 σ Hˆ f = −PE, k1 ,k2 ,σ
where P is the carrier polarization operator. The spin subsystem is treated in the spherically symmetric way (in spin space) [18]. In this approach the spin excitation spectrum consists of three degenerate branches described by the retarded Green’s function
α Sqα S−q ω
−i = h¯
∞ 0
Aq 1 α = 2 2 eiωt Sqα (t), S−q . ω − ωq2 h¯
(2)
The numerator Aq and the spin wave excitation spectrum ωq were calculated selfconsistently in terms of frustration parameter p and a finite number of spin–spin correlation functions Cr = SR SR+r and have the form Aq = −8 I1 1 − γg (q) Cg + I2 1 − γd (q) Cd ,
8 (1 − γg ) B1 + (1 + γg )B2 + (1 − γd ) B3 + (1 + γd )B4 + γg (1 − γd )B5 , ωq = I 3
(3)
parameters Bi are expressed in terms of Cg , C2g , Cd , Cg+d , C2d . Functions Aq and ωq vanish as q → 0. In the limit q → Q the numerator Aq tends to a positive constant AQ , a gap ∆ ≡ ωQ appears in the spin excitation spectrum α takes the form ωq2 ≈ ∆2 + c2 (q − Q)2 and the spin susceptibility χ(q, ω) = −Sqα |S−q ω χ(q, ω) ≈
χQ 2 1 + ξ (q − Q)2
− ω2 ξ 2 /c2
,
χQ =
AQ 2 ξ , c2
where we introduced the correlation length ξ = c/∆. A similar form of phenomenological spin susceptibility with additional term in the denominator −iω/ωsf (ωsf is a typical energy scale for the antiferromagnetic magnons describing the spin dynamics) is used in the nearly antiferromagnetic Fermi-liquid theory [9]. α yields the correct value of The principal distinction of our approach lies in the fact that our form of Sqα |S−q ω 2 α α SR = 3/4 at arbitrary temperature and frustration. It means that Sq |S−q ω must obey the sum rule +∞ α 1 −h¯ 3 nB (h¯ ω) Im Sqα S−q dω = . ω N q π 4 −∞
α S α respecting to (2) has the form The spin–spin correlation function Cq = S−q q
Cq =
Aq 1 + 2nB (h¯ ωq ) , 2h¯ ωq
−1 nB (h¯ ω) = eh¯ ω/T − 1 .
It is sharply peaked at q = Q. It means that the conducting carriers are strongly coupled to a resonance structure in the spectrum of spin fluctuations.
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3. Kinetic equation To take into account the scattering anisotropy we use a multi-moment approach developed for an appropriate description of low temperature dependence of resistivity and Hall coefficient in polyvalent metals [19,20]. In this case a strong electron–phonon scattering anisotropy becomes temperature dependent when the umklapp processes freeze out rapidly. The deviation from equilibrium in the steady-state case can be represented in terms of the density matrix. General form of such a matrix is ρˆ 0 = ρˆ 00 (1 + Fˆ ),
ρˆ 00 =
1 −Hˆ 0 /T e , Z
Z = Sp{Hˆ 0 },
(4)
where the operator Fˆ satisfies the conditions [Fˆ , Hˆ 0 ] = 0,
Fˆ = Fˆ † ,
Fˆ ≡ Sp ρˆ 00 Fˆ = 0.
† akσ } the operator Fˆ can be taken as an oneIn order to find one particle distribution function fk = Sp{ρˆ 0 akσ particle one, i.e. † F (k)akσ akσ . Fˆ = k,σ
Then fk takes the form fk = fk0 + gk ,
fk0 = 1/ e(εk −µ)/T + 1 ,
gk =
−∂f 0 Φk , ∂εk
Φk = T F (k),
µ is the chemical potential. ˆ The main problem in dealing with density matrix (4) is to construct the relevant operator F . We can represent operator Fˆ as a linear superposition of a full set of operators Fˆl : Fˆ = l ηl Fˆl . Moments Fˆl are assumed to be one-particle operators commuting with the Hamiltonian Hˆ 0 † Fˆl = Fl (k)akσ akσ , [Fˆl , Hˆ 0 ] = 0, Fˆl = 0. k,σ
There are two small parameters in the problem. They are c—the deviation of the system from the equilibrium characterized by coefficients ηl and operators Fˆl , and λ—the hole–spin interaction characterized by the term Jˆ. We build perturbation theory in parameter cλ2 1. Solving the evolution equation for the density matrix to first order in cλ2 one gives rise to the system of equations Xl = Pˆll ηl , where Xl =
Fl (k)eEvk
k
Pll =
(5) −∂f 0 , ∂εk
vk = h¯ −1 ∇k εk ,
(6)
π 2 Aq 0 J Fl (k) − Fl (k + q) Fl (k) − Fl (k + q) fk0 1 − fk+q nB (h¯ ωq )δ(εk+q − εk − h¯ ωq ). h¯ h¯ ωq k,q
System (5) is equivalent to the linearized Boltzmann equation if {Fˆl } is a full set of operators commuting with Hˆ 0 . If we restrict ourself to a finite number N of the operators Fˆl then N is dictated by the effect of the anisotropy on carrier scattering and its interplay with the shape of the FS.
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Common practice in solving the Boltzmann equation is to introduce relaxation time approximation Xk =
gk . τ
It is one-moment approximation (OMA) with Φk = τ eEvk ,
Fˆ = η1 Fˆ1 ,
η1 = τ eE/T ,
F1 (k) = nvk ,
where n is a unit vector in the direction of the applied electric field. In the following we consider a minimal realization of this multi-moment method limiting our analysis to N = 2 moments. The two-moment approximation (TMA) permits to distinguish between hot and cold regions of the BZ. Besides it reflects properties of the carrier scattering by a spin mode peaked at AFM vector Q. On the other hand, in order to include the forward scattering (scattering with small momenta transferred), which is important for the transport in the cold region, a larger number of moments is required. Apparently, the same takes place if we consider the Hall effect, then the kinetic equation contains derivatives of gk with respect to kx and ky and we need a detail description of gk . We deserve the case N > 2 for further investigation. In two-moment approximation Φk = T η1 F1 (k) + η2 F2 (k) , F1 (k) = nvk , F2 (k) = (nvk )3 . A comparison between representation of ρ(T ) in terms of OMA and TMA provides us with an understanding of anomalous temperature dependence of resistivity in high-Tc superconductors. Current density j α = σ αβ E β
1 α −∂f 0 α = evk Φk V ∂εk k,σ
−1 defines the conductivity tensor σ αβ and the resistivity tensor ραβ = σαβ . At quadratic symmetry we have ρ = ρxx = ρyy .
4. Results and discussion We present the results of our calculations for the spectrum parameter τ = 0.15 eV. Fig. 1 shows the spectrum εk (1) based on a spin polaron concept [17], which we relate to the optimally doped compounds. As can be seen the value of bandwidth W ∼ = 0.66 eV is typical of the lowest quasiparticle band in HTSC [21–23]. The spectrum demonstrates a saddle point (SP) which is a characteristic feature of the optimally doped HTSC. The SP is located close to the Fermi line (FL) with an energy difference εSP − εF ≈ 0.042 eV ≈ 460 K. This is a scale for an essential change in the spectrum topology and carrier velocities. This value is close to the experimental one, for example, in Bi2212 at optimal doping εSP − εF ∼ = 0.035 eV [24]. For the spin excitation spectrum we use the AFM exchange integral I = 0.7τ ≈ 1200 K and the frustration parameter p = 0.08. The adopted value of I is typical of HTSC. The frustration parameter p governs the temperature dependence of the spin gap ∆ = ωQ mainly at low T , specifically ∆(T = 100 K) = 72 K at p = 0.08 and I = 1200 K. In [10] and [9] ∆(T = 100 K) equals to 165 K and 120 K respectively. At high temperatures T > 200 K and p 0.1 the gap increases approximately linear in T . The selfconsistent approach for the spin subsystem is equivalent to some variant of a mean-field approach and in contrast to the phenomenological overdoped treatment of the susceptibility [9] gives a zero damping of the spin excitations. We show that a slight modification of the self-consistent spin spectrum can lead to a good description of the ρ(T )behavior. Nearby 100 K the optimally doped compounds undergo the transition to the superconducting state. We represent our results for normal state resistivity for temperatures above 50 K.
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Fig. 2. Resistivity ρ(T ) and ρ ∗ (T ) (in arbitrary units) calculated for the spin spectrum with ωq and ωq∗ , respectively (see Fig. 3). The dotted line is for the one-moment approximation (ρ1 ) and the solid line is for the two-moment approximation (ρ2 ).
Fig. 3. Left panel: the spin spectrum ωq and ωq∗ at T = 250 K, p = 0.08, I = 1200 K for Γ → M direction. Right panel: the spin gap temperature dependence ∆(T ) and ∆∗ (T ).
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Fig. 4. One-moment approximation: contours of constant value for deviation g1 (k) (in arbitrary units) of the hole distribution function f (k) from the equilibrium Fermi function f0 (k) at T = 100 K (the electric field directed along the BZ-diagonal Γ → M).
Fig. 2 shows the temperature dependence of ρ1,2 (T ) corresponding to OMA and TMA calculations in framework of self-consistent spin spectrum ωq . In Fig. 3 ωq is given in Γ (0, 0) → M(π, π) direction at T = 250 K on the left panel and its spin gap temperature dependence ∆(T ) = ωQ is shown on the right panel by solid curves. Both curves ρ1,2 (T ) are close to the liner temperature dependence above T ≈ 100 K with a slope ρ2 (400 K)/ρ2 (100 K) ≈ 9. The liner law ρ(T ) is in agreement with the experiment but the value of the slope is greater then the experimental one ρ(400 K)/ρ(100 K) ≈ 5 [1,3,4]. ∗ (T ) calculated with spin spectrum ω∗ slightly different In Fig. 2 we also depict the temperature dependence ρ1,2 q ∗ from ωq . The spectrum ωq is given in Fig. 3 by dashed lines. A linear law for ρ2∗ (T ) starts from T ≈ 60 K and has a slope ρ2 (400 K)/ρ2 (100 K) ≈ 5 in accordance with the experiment. We see a strong sensitivity of ρ(T ) with respect to the slight changes in the spin-fluctuation gap parameter as it was pointed out in [9]. In both cases at T ≈ 100 K the TMA gives ρ2 (T ) approximately 1.5 times smaller than ρ1 (T ). To clarify this point, in Figs. 4 and 5 we give functions g1 (k) in OMA and g2 (k) in TMA at T = 100 K (the electric field is directed along the BZ-diagonal Γ → M). We see that there is a strong suppression of g(k) in k-regions near the FS which are close to X–M BZ boundary in the first and third quandrants. As may be seen from Fig. 1 the carriers from theses regions are strongly scattered by spin excitations with momenta q ≈ Q and these regions correspond to the “hot points” regions. The strong scattering takes place due to the following. Firstly, the spin gap value at q = Q is equal to ∆ ≈ 70 K at T ∼ 100 K. As a result the magnons with q ≈ Q are strongly excited and give a large factor Aq ωq−1 nB (h¯ ωq ) in scattering integrals Pll . Secondly, the topology of εk close to the “hot” regions allows to fulfill the conservation laws for momentum and energy for the scattering
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Fig. 5. Two-moment approximation: contours of constant value for deviation g2 (k) (in arbitrary units) of the hole distribution function f (k) from the equilibrium Fermi function f0 (k) at T = 100 K (the electric field directed along the BZ-diagonal Γ → M).
process. This can be seen qualitatively for the scattering process from k1 to k2 (see Fig. 1). This scattering leads to great difference |vk1 − vk2 |. As a result these scattering process gives a main contribution to the scattering integrals. It is natural that the conductivity will be dominated by the part of the Fermi line where the scattering is weakest. As it is clear from the variational principal [25] g2 (k) allows to describe such a g1 (k) function rearrangement which decreases g(k) in “hot” regions and increases g(k) in the regions with weaker scattering—“cold” regions. A comparison of Figs. 4 and 3 demonstrates such a nonequilibrium carrier transfer where the “cold” regions correspond to the patches close to the BZ diagonal Γ → M in the first and third BZ quadrants. At the same time there is an essential transfer from the “hot” regions to the regions close to X–M boundary in the second and third BZ quandrants. Such a transfer increases the derivatives of g(k) in k and it must be important in treating the Hall effect. In summary, we demonstrated that it is possible to describe HTSC ρ(T ) dependence without reference to phenomenological approaches involving overdamped magnons (see Refs. [9,10]). We have got a correct ρ(T )dependence with the carrier spectrum which is based on the lowest spin-polaron band and which differs essentially from the spectra adopted in close treatments of the problem. The main distinctions are related to the energy difference between the Fermi energy εF and the top and bottom of the band energies εΓ and εbottom. Our spectrum has the value of εΓ − εF ∼ = 0.5 eV which is more than two times greater than spectra used from local density approximation [9,10]. We also presented a multi-moment method for solving of the kinetic equation which is alternative to the multipatch approach [13]. For a finite number of patches the multi-patch method leads to a discontinuous changes of
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g(k) between regions and needs an artificial special procedure to smooth these changes. Our method avoids this problem with the discontinuous g(k). Of course, both methods must coincide in the limit N → ∞.
Acknowledgements This work was supported by Russian Fund of Fundamental Investigations.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
J.M. Harris, Y.F. Yan, N.P. Ong, Phys. Rev. B 46 (1992) 14293. B. Bucher, P. Steiner, J. Karpinski, E. Kaldis, P. Wachter, Phys. Rev. Lett. 70 (1993) 2012. A. Carrington, D.J.C. Walker, A.P. Mackenzie, J.R. Cooper, Phys. Rev. B 48 (1993) 13051. H. Takagi, B. Batlogg, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski, W.F. Peck, Phys. Rev. Lett. 69 (1992) 2975. A. Malinowski, M.Z. Cieplak, S. Guha, Q. Wu, B. Kim, A. Krickser, A. Perali, K. Karpinska, M. Berkowski, C.H. Shang, P. Lindenfeld, Phys. Rev. B 66 (2002) 104512. M.V. Sadovskii, Phys. Usp. 44 (5) (2001). P.W. Anderson, Phys. Rev. Lett. 67 (1991) 2092. P. Coleman, A.J. Schofield, A.M. Tsvelik, J. Phys.: Condens. Matter 8 (1996) 9985. B.P. Stojkovic, D. Pines, Phys. Rev. B 55 (1997) 8576. R. Hlubina, T.M. Rice, Phys. Rev. B 51 (1995) 9253. L.B. Ioffe, A.J. Millis, Phys. Rev. B 58 (1998) 11631. A.T. Zheleznyak, V.M. Yakovenko, H.D. Drew, Phys. Rev. B 59 (1999) 207. A. Perali, M. Sindel, G. Kotliar, Eur. Phys. J. B 24 (2001) 487. V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. V.J. Emery, G. Reiter, Phys. Rev. B 38 (1988) 4547. F.C. Zhang, T.M. Rice, Phys. Rev. B 37 (1988) 3759. A.F. Barabanov, R. Hayn, A.A. Kovalev, O.V. Urazaev, A.M. Belemouk, JETP 92 (2001) 677. A.F. Barabanov, V.M. Berezovsky, Phys. Lett. A 186 (1994) 175; A.F. Barabanov, V.M. Berezovsky, JETP 79 (1994) 627. A.F. Barabanov, L.A. Maksimov, Fiz. Met. Metalloved. 29 (1970) 471, Phys. Met. Metalogr. 29 (1970) 471 (in English). J. Black, D.L. Mills, Phys. Rev. B 9 (1974) 1458. B.O. Wells, Z.X. Shen, A. Matsuura, et al., Phys. Rev. Lett. 74 (1995) 964. D.S. Marshall, D.S. Dessau, A.G. Loeser, et al., Phys. Rev. Lett. 76 (1996) 4841. F. Ronning, C. Kim, D.L. Feng, et al., Science 282 (1998) 2067. M.R. Norman, M. Randeria, H. Ding, J.C. Campuzano, Phys. Rev. B 52 (1995) 615. J.M. Ziman, Electrons and Phonons, The Theory of Transport Phenomena in Solids, Oxford Univ. Press, Oxford, 1960.
Resistivity in two-dimensional doped antiferromagnet A.M. Belemuk, A.F. Barabanov ∗ Institute for High Pressure Physics, 142190 Troitsk, Moscow region, Russia Received 1 December 2003; accepted 28 December 2003 Communicated by V.M. Agranovich
Abstract The temperature dependence of the resistivity ρ(T ) in two-dimensional doped antiferromagnet is investigated within the single band Kondo lattice model. To take into account the strong temperature dependent scattering anisotropy the kinetic equation for the nonequilibrium carrier distribution function is derived through the density matrix formalism in two-moment approach. Obtained ρ(T ) dependence demonstrates qualitative agreement with experimental data for optimally doped high temperature superconductors in a broad temperature range. It is found that a gap in the spin excitation spectrum is of crucial importance for the behavior of ρ(T ). 2004 Published by Elsevier B.V. PACS: 72.10.Di; 74.25.Fy; 71.10.Ay
1. Introduction Since the discovery of high-Tc superconductors the anomalous properties of their normal state have been the subject of intensive theoretical study. Although a broad range of models and theories have been proposed, none has successfully explained all available experimental data. One of the most unusual properties of high-Tc cuprates is the temperature dependence of the resistivity [1–5]. The optimal doped samples are characterized by in-plane resistivity ρ(T ) that grows nearly linearly with temperature from just above the superconducting transition temperature Tc up to some hundreds degrees Kelvin. For example, in the optimum composition range near x ∼ 0.15 for La1−x Srx CuO4 the T -linear resistivity has been seen from just above Tc = 35 K to near 1000 K [4]. The value of Tc is well below the Debye temperature. At so low temperature standard Fermi-liquid theory predicts ρ(T ) ∼ T 2 for electron–electron scattering and ρ ∼ T 5 for three-dimensional electron–phonon scattering. In the overdoped regime, the exponent γ of ρ ∝ T γ changes from γ = 1 to 2. In La1−x Srx CuO4 above x ∼ 0.2 a power-law dependence with γ ∼ 1.5 was found [4]. The underdoped samples exhibit deviations from the linear behavior at low temperatures. For example, the resistivity in YBa2 Cu3 O6+x displays strong curvature below TK ∼ 320 K [1] and follows a power-law above TK (with γ ∼ 1.2). With increasing hole doping, the curvature * Corresponding author.
E-mail address: [email protected] (A.F. Barabanov). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.12.064
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below TK evolves into the linear-T behavior. The reason for this behavior is not clear. Possible explanation that it reflects a subtle change in the structure of the hole spectrum. The influence of the pseudogap opening on the normal state properties is still under discussion [2,5]. Theoretical models relevant to the understanding of the pseudogap state in high-temperature superconductors (HTSC) are discussed in [6]. There are two broad classes of theoretical approaches for explaining the anomalous of cuprates. The first one postulates the presence of two independent relaxation time-scales for the Hall and electric current [7,8]. The second one deals with the Fermi liquid concept involving an anomalous momentum dependent scattering time [9–13]. Fermi-liquid approaches are based on distinct temperature dependence of quasiparticle scattering time into different regions of the Fermi surface (FS). “Hot” regions (near X points (±π, 0) and (0, ±π) points) and “cold” regions (near the Brillouin zone (BZ) diagonal) correspond respectively strong and weak quasiparticle scattering. They differ by values of the Fermi velocity and temperature dependences of the scattering time. “Hot/cold” region models [10–12] have not led to general agreement on what regions dominate the transport. Multi-patch model [13] seems to be the best systematic analysis of normal state transport properties compatible with the experiments. The scattering operator of the linearized Boltzmann equation is projected on the patches and temperature dependences of its coefficients are assigned. In this model the linear temperature dependence of the resistivity is associated to the inter-patch scattering of carriers. From the models covered it is apparent that severe speculations with many phenomenological parameters are introduced at analysis of transport properties. In this Letter we present a study of the resistivity of the twodimensional doped antiferromagnet for the optimally doped case sticking to the models of the second type— nearly antiferromagnetic (AFM) two-dimensional Fermi liquid. Our analysis is less phemenological relative to close approaches. This mainly concerns the spin-polaron carrier spectrum and the spin excitation propagator which were calculated previously. The unconventional normal state transport properties are usually viewed as a manifestation of the strongly correlated nature of the charge dynamics in CuO2 of HTSC. Relevant physics of high-Tc compounds is fairly well described by the three-band Emery model [14–16]. This model was previously studied on the basis of the spin polaron concept in a broad doping interval [17]. The concept gives rise to an elementary excitation spectrum appropriate to angular resolved photoemission spectroscopy (ARPES) experiments. In particular, it captures such subtle features as the minimum of the spectrum near momentum (±π/2, ±π/2) and the remnant FS for undoped compounds, the large FS and the extended saddle point close to the FS for the optimally doped cuprates and the presence of a pseudogap for intermediate doping [21–23]. Due to excessive complication of the three-band model we consider more simple one, the single-band Kondo lattice model on a square lattice with intra-site coupling of the carriers and the spin subsystem. Below for the carrier spectrum we adopt the spectrum of the lowest spin-polaron band [17] which reflects the main features of ARPES. As to the spin excitation propagator, we treat it in the framework of the spherically symmetric self-consistent approach [18] for 2D S = 1/2 frustrated AFM Heisenberg model. In particular, it means that the average one-site spin values are zero SRα = 0 and the AFM spin–spin correlation functions S0α SRα at fixed α do not depend on the Cartesian index α. It is important that such a treatment gives a temperature dependent gap ∆ at AFM momentum Q = (π, π) in the spin-excitation spectrum. In Section 2 we briefly introduce a model for the carriers scattered by the spin subsystems. In Section 3 we consider a multi-moment method [19] for the kinetic equation solving based on a nonequilibrium stastistical operator. The method allows to take into account a strongly anisotropic temperature dependent scattering. In Section 4 we present our results for the dependence ρ(T ) given by the two-moment realization of a multi-moment method and discuss the distinctions of our treatment from close approaches.
2. Model The effective Hamiltonian of the regular Kondo model type on a square lattice is
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Fig. 1. The carrier spectrum εk (in eV) is presented by equal-energy lines εk = const, the thick solid line εk = 0.08 corresponds to the Fermi surface for the optimally doped case. The arrow indicates the hot-point scattering by the AFM vector Q = (π, π ) from a state k1 below the FS into a state k2 above the FS.
Hˆ tot = Hˆ 0 + Hˆ 1 ,
Hˆ 0 = Hˆ h + Iˆ,
Hˆ 1 = Hˆ f + Jˆ.
The term Hˆ 0 describes the motion of the fermionic carriers (holes) † εk akσ akσ Hˆ h = k,σ
and the system of localized spins 1 α 1 α Iˆ = I1 SR+g SRα + I2 SR+d SRα . 2 2 R,g
R,d
Here g and d are the vectors of the nearest and second nearest neighbors. The exchange Hamiltonian Iˆ corresponds to AFM frustrated interaction between spins, p (0 p 1)—a frustration parameter, I1 = (1 − p)I and I2 = pI are exchange constants for the first and the second nearest neighbors. We use the following form for the hole spectrum (see Fig. 1) εk = τ a1 γg (k) + a2 γg2 (k) + a3 γd (k) + a4 γd2 (k) + a5 γg (k)γd (k) (1) with the values of the coefficients ai and the quadratic harmonics γi (k) given by a1 = 1.5, a2 = 1.8, a3 = −0.5, a4 = 0.7, a5 = 0.5, γg (k) = (cos(kx ) + cos(ky ))/2, γd (k) = cos(kx ) cos(ky ).
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The term Jˆ is the intrasite exchange Hamiltonian
1 Jˆ = J √ N
† ak+q,γ S α σˆ α akγ2 , 1 q γ1 γ2
k,q,γ1 ,γ2
1 −iqR Sq = √ e SR , N q
where σˆ α are the Pauli matrices (summation is assumed over repeated Cartesian indices α). Coupling with an external homogeneous electric field E is given by ˆ Pˆ = e k1 |ˆr|k2 ak†1 σ ak2 σ Hˆ f = −PE, k1 ,k2 ,σ
where P is the carrier polarization operator. The spin subsystem is treated in the spherically symmetric way (in spin space) [18]. In this approach the spin excitation spectrum consists of three degenerate branches described by the retarded Green’s function
α Sqα S−q ω
−i = h¯
∞ 0
Aq 1 α = 2 2 eiωt Sqα (t), S−q . ω − ωq2 h¯
(2)
The numerator Aq and the spin wave excitation spectrum ωq were calculated selfconsistently in terms of frustration parameter p and a finite number of spin–spin correlation functions Cr = SR SR+r and have the form Aq = −8 I1 1 − γg (q) Cg + I2 1 − γd (q) Cd ,
8 (1 − γg ) B1 + (1 + γg )B2 + (1 − γd ) B3 + (1 + γd )B4 + γg (1 − γd )B5 , ωq = I 3
(3)
parameters Bi are expressed in terms of Cg , C2g , Cd , Cg+d , C2d . Functions Aq and ωq vanish as q → 0. In the limit q → Q the numerator Aq tends to a positive constant AQ , a gap ∆ ≡ ωQ appears in the spin excitation spectrum α takes the form ωq2 ≈ ∆2 + c2 (q − Q)2 and the spin susceptibility χ(q, ω) = −Sqα |S−q ω χ(q, ω) ≈
χQ 2 1 + ξ (q − Q)2
− ω2 ξ 2 /c2
,
χQ =
AQ 2 ξ , c2
where we introduced the correlation length ξ = c/∆. A similar form of phenomenological spin susceptibility with additional term in the denominator −iω/ωsf (ωsf is a typical energy scale for the antiferromagnetic magnons describing the spin dynamics) is used in the nearly antiferromagnetic Fermi-liquid theory [9]. α yields the correct value of The principal distinction of our approach lies in the fact that our form of Sqα |S−q ω 2 α α SR = 3/4 at arbitrary temperature and frustration. It means that Sq |S−q ω must obey the sum rule +∞ α 1 −h¯ 3 nB (h¯ ω) Im Sqα S−q dω = . ω N q π 4 −∞
α S α respecting to (2) has the form The spin–spin correlation function Cq = S−q q
Cq =
Aq 1 + 2nB (h¯ ωq ) , 2h¯ ωq
−1 nB (h¯ ω) = eh¯ ω/T − 1 .
It is sharply peaked at q = Q. It means that the conducting carriers are strongly coupled to a resonance structure in the spectrum of spin fluctuations.
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3. Kinetic equation To take into account the scattering anisotropy we use a multi-moment approach developed for an appropriate description of low temperature dependence of resistivity and Hall coefficient in polyvalent metals [19,20]. In this case a strong electron–phonon scattering anisotropy becomes temperature dependent when the umklapp processes freeze out rapidly. The deviation from equilibrium in the steady-state case can be represented in terms of the density matrix. General form of such a matrix is ρˆ 0 = ρˆ 00 (1 + Fˆ ),
ρˆ 00 =
1 −Hˆ 0 /T e , Z
Z = Sp{Hˆ 0 },
(4)
where the operator Fˆ satisfies the conditions [Fˆ , Hˆ 0 ] = 0,
Fˆ = Fˆ † ,
Fˆ ≡ Sp ρˆ 00 Fˆ = 0.
† akσ } the operator Fˆ can be taken as an oneIn order to find one particle distribution function fk = Sp{ρˆ 0 akσ particle one, i.e. † F (k)akσ akσ . Fˆ = k,σ
Then fk takes the form fk = fk0 + gk ,
fk0 = 1/ e(εk −µ)/T + 1 ,
gk =
−∂f 0 Φk , ∂εk
Φk = T F (k),
µ is the chemical potential. ˆ The main problem in dealing with density matrix (4) is to construct the relevant operator F . We can represent operator Fˆ as a linear superposition of a full set of operators Fˆl : Fˆ = l ηl Fˆl . Moments Fˆl are assumed to be one-particle operators commuting with the Hamiltonian Hˆ 0 † Fˆl = Fl (k)akσ akσ , [Fˆl , Hˆ 0 ] = 0, Fˆl = 0. k,σ
There are two small parameters in the problem. They are c—the deviation of the system from the equilibrium characterized by coefficients ηl and operators Fˆl , and λ—the hole–spin interaction characterized by the term Jˆ. We build perturbation theory in parameter cλ2 1. Solving the evolution equation for the density matrix to first order in cλ2 one gives rise to the system of equations Xl = Pˆll ηl , where Xl =
Fl (k)eEvk
k
Pll =
(5) −∂f 0 , ∂εk
vk = h¯ −1 ∇k εk ,
(6)
π 2 Aq 0 J Fl (k) − Fl (k + q) Fl (k) − Fl (k + q) fk0 1 − fk+q nB (h¯ ωq )δ(εk+q − εk − h¯ ωq ). h¯ h¯ ωq k,q
System (5) is equivalent to the linearized Boltzmann equation if {Fˆl } is a full set of operators commuting with Hˆ 0 . If we restrict ourself to a finite number N of the operators Fˆl then N is dictated by the effect of the anisotropy on carrier scattering and its interplay with the shape of the FS.
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Common practice in solving the Boltzmann equation is to introduce relaxation time approximation Xk =
gk . τ
It is one-moment approximation (OMA) with Φk = τ eEvk ,
Fˆ = η1 Fˆ1 ,
η1 = τ eE/T ,
F1 (k) = nvk ,
where n is a unit vector in the direction of the applied electric field. In the following we consider a minimal realization of this multi-moment method limiting our analysis to N = 2 moments. The two-moment approximation (TMA) permits to distinguish between hot and cold regions of the BZ. Besides it reflects properties of the carrier scattering by a spin mode peaked at AFM vector Q. On the other hand, in order to include the forward scattering (scattering with small momenta transferred), which is important for the transport in the cold region, a larger number of moments is required. Apparently, the same takes place if we consider the Hall effect, then the kinetic equation contains derivatives of gk with respect to kx and ky and we need a detail description of gk . We deserve the case N > 2 for further investigation. In two-moment approximation Φk = T η1 F1 (k) + η2 F2 (k) , F1 (k) = nvk , F2 (k) = (nvk )3 . A comparison between representation of ρ(T ) in terms of OMA and TMA provides us with an understanding of anomalous temperature dependence of resistivity in high-Tc superconductors. Current density j α = σ αβ E β
1 α −∂f 0 α = evk Φk V ∂εk k,σ
−1 defines the conductivity tensor σ αβ and the resistivity tensor ραβ = σαβ . At quadratic symmetry we have ρ = ρxx = ρyy .
4. Results and discussion We present the results of our calculations for the spectrum parameter τ = 0.15 eV. Fig. 1 shows the spectrum εk (1) based on a spin polaron concept [17], which we relate to the optimally doped compounds. As can be seen the value of bandwidth W ∼ = 0.66 eV is typical of the lowest quasiparticle band in HTSC [21–23]. The spectrum demonstrates a saddle point (SP) which is a characteristic feature of the optimally doped HTSC. The SP is located close to the Fermi line (FL) with an energy difference εSP − εF ≈ 0.042 eV ≈ 460 K. This is a scale for an essential change in the spectrum topology and carrier velocities. This value is close to the experimental one, for example, in Bi2212 at optimal doping εSP − εF ∼ = 0.035 eV [24]. For the spin excitation spectrum we use the AFM exchange integral I = 0.7τ ≈ 1200 K and the frustration parameter p = 0.08. The adopted value of I is typical of HTSC. The frustration parameter p governs the temperature dependence of the spin gap ∆ = ωQ mainly at low T , specifically ∆(T = 100 K) = 72 K at p = 0.08 and I = 1200 K. In [10] and [9] ∆(T = 100 K) equals to 165 K and 120 K respectively. At high temperatures T > 200 K and p 0.1 the gap increases approximately linear in T . The selfconsistent approach for the spin subsystem is equivalent to some variant of a mean-field approach and in contrast to the phenomenological overdoped treatment of the susceptibility [9] gives a zero damping of the spin excitations. We show that a slight modification of the self-consistent spin spectrum can lead to a good description of the ρ(T )behavior. Nearby 100 K the optimally doped compounds undergo the transition to the superconducting state. We represent our results for normal state resistivity for temperatures above 50 K.
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Fig. 2. Resistivity ρ(T ) and ρ ∗ (T ) (in arbitrary units) calculated for the spin spectrum with ωq and ωq∗ , respectively (see Fig. 3). The dotted line is for the one-moment approximation (ρ1 ) and the solid line is for the two-moment approximation (ρ2 ).
Fig. 3. Left panel: the spin spectrum ωq and ωq∗ at T = 250 K, p = 0.08, I = 1200 K for Γ → M direction. Right panel: the spin gap temperature dependence ∆(T ) and ∆∗ (T ).
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Fig. 4. One-moment approximation: contours of constant value for deviation g1 (k) (in arbitrary units) of the hole distribution function f (k) from the equilibrium Fermi function f0 (k) at T = 100 K (the electric field directed along the BZ-diagonal Γ → M).
Fig. 2 shows the temperature dependence of ρ1,2 (T ) corresponding to OMA and TMA calculations in framework of self-consistent spin spectrum ωq . In Fig. 3 ωq is given in Γ (0, 0) → M(π, π) direction at T = 250 K on the left panel and its spin gap temperature dependence ∆(T ) = ωQ is shown on the right panel by solid curves. Both curves ρ1,2 (T ) are close to the liner temperature dependence above T ≈ 100 K with a slope ρ2 (400 K)/ρ2 (100 K) ≈ 9. The liner law ρ(T ) is in agreement with the experiment but the value of the slope is greater then the experimental one ρ(400 K)/ρ(100 K) ≈ 5 [1,3,4]. ∗ (T ) calculated with spin spectrum ω∗ slightly different In Fig. 2 we also depict the temperature dependence ρ1,2 q ∗ from ωq . The spectrum ωq is given in Fig. 3 by dashed lines. A linear law for ρ2∗ (T ) starts from T ≈ 60 K and has a slope ρ2 (400 K)/ρ2 (100 K) ≈ 5 in accordance with the experiment. We see a strong sensitivity of ρ(T ) with respect to the slight changes in the spin-fluctuation gap parameter as it was pointed out in [9]. In both cases at T ≈ 100 K the TMA gives ρ2 (T ) approximately 1.5 times smaller than ρ1 (T ). To clarify this point, in Figs. 4 and 5 we give functions g1 (k) in OMA and g2 (k) in TMA at T = 100 K (the electric field is directed along the BZ-diagonal Γ → M). We see that there is a strong suppression of g(k) in k-regions near the FS which are close to X–M BZ boundary in the first and third quandrants. As may be seen from Fig. 1 the carriers from theses regions are strongly scattered by spin excitations with momenta q ≈ Q and these regions correspond to the “hot points” regions. The strong scattering takes place due to the following. Firstly, the spin gap value at q = Q is equal to ∆ ≈ 70 K at T ∼ 100 K. As a result the magnons with q ≈ Q are strongly excited and give a large factor Aq ωq−1 nB (h¯ ωq ) in scattering integrals Pll . Secondly, the topology of εk close to the “hot” regions allows to fulfill the conservation laws for momentum and energy for the scattering
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Fig. 5. Two-moment approximation: contours of constant value for deviation g2 (k) (in arbitrary units) of the hole distribution function f (k) from the equilibrium Fermi function f0 (k) at T = 100 K (the electric field directed along the BZ-diagonal Γ → M).
process. This can be seen qualitatively for the scattering process from k1 to k2 (see Fig. 1). This scattering leads to great difference |vk1 − vk2 |. As a result these scattering process gives a main contribution to the scattering integrals. It is natural that the conductivity will be dominated by the part of the Fermi line where the scattering is weakest. As it is clear from the variational principal [25] g2 (k) allows to describe such a g1 (k) function rearrangement which decreases g(k) in “hot” regions and increases g(k) in the regions with weaker scattering—“cold” regions. A comparison of Figs. 4 and 3 demonstrates such a nonequilibrium carrier transfer where the “cold” regions correspond to the patches close to the BZ diagonal Γ → M in the first and third BZ quadrants. At the same time there is an essential transfer from the “hot” regions to the regions close to X–M boundary in the second and third BZ quandrants. Such a transfer increases the derivatives of g(k) in k and it must be important in treating the Hall effect. In summary, we demonstrated that it is possible to describe HTSC ρ(T ) dependence without reference to phenomenological approaches involving overdamped magnons (see Refs. [9,10]). We have got a correct ρ(T )dependence with the carrier spectrum which is based on the lowest spin-polaron band and which differs essentially from the spectra adopted in close treatments of the problem. The main distinctions are related to the energy difference between the Fermi energy εF and the top and bottom of the band energies εΓ and εbottom. Our spectrum has the value of εΓ − εF ∼ = 0.5 eV which is more than two times greater than spectra used from local density approximation [9,10]. We also presented a multi-moment method for solving of the kinetic equation which is alternative to the multipatch approach [13]. For a finite number of patches the multi-patch method leads to a discontinuous changes of
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g(k) between regions and needs an artificial special procedure to smooth these changes. Our method avoids this problem with the discontinuous g(k). Of course, both methods must coincide in the limit N → ∞.
Acknowledgements This work was supported by Russian Fund of Fundamental Investigations.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
J.M. Harris, Y.F. Yan, N.P. Ong, Phys. Rev. B 46 (1992) 14293. B. Bucher, P. Steiner, J. Karpinski, E. Kaldis, P. Wachter, Phys. Rev. Lett. 70 (1993) 2012. A. Carrington, D.J.C. Walker, A.P. Mackenzie, J.R. Cooper, Phys. Rev. B 48 (1993) 13051. H. Takagi, B. Batlogg, H.L. Kao, J. Kwo, R.J. Cava, J.J. Krajewski, W.F. Peck, Phys. Rev. Lett. 69 (1992) 2975. A. Malinowski, M.Z. Cieplak, S. Guha, Q. Wu, B. Kim, A. Krickser, A. Perali, K. Karpinska, M. Berkowski, C.H. Shang, P. Lindenfeld, Phys. Rev. B 66 (2002) 104512. M.V. Sadovskii, Phys. Usp. 44 (5) (2001). P.W. Anderson, Phys. Rev. Lett. 67 (1991) 2092. P. Coleman, A.J. Schofield, A.M. Tsvelik, J. Phys.: Condens. Matter 8 (1996) 9985. B.P. Stojkovic, D. Pines, Phys. Rev. B 55 (1997) 8576. R. Hlubina, T.M. Rice, Phys. Rev. B 51 (1995) 9253. L.B. Ioffe, A.J. Millis, Phys. Rev. B 58 (1998) 11631. A.T. Zheleznyak, V.M. Yakovenko, H.D. Drew, Phys. Rev. B 59 (1999) 207. A. Perali, M. Sindel, G. Kotliar, Eur. Phys. J. B 24 (2001) 487. V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. V.J. Emery, G. Reiter, Phys. Rev. B 38 (1988) 4547. F.C. Zhang, T.M. Rice, Phys. Rev. B 37 (1988) 3759. A.F. Barabanov, R. Hayn, A.A. Kovalev, O.V. Urazaev, A.M. Belemouk, JETP 92 (2001) 677. A.F. Barabanov, V.M. Berezovsky, Phys. Lett. A 186 (1994) 175; A.F. Barabanov, V.M. Berezovsky, JETP 79 (1994) 627. A.F. Barabanov, L.A. Maksimov, Fiz. Met. Metalloved. 29 (1970) 471, Phys. Met. Metalogr. 29 (1970) 471 (in English). J. Black, D.L. Mills, Phys. Rev. B 9 (1974) 1458. B.O. Wells, Z.X. Shen, A. Matsuura, et al., Phys. Rev. Lett. 74 (1995) 964. D.S. Marshall, D.S. Dessau, A.G. Loeser, et al., Phys. Rev. Lett. 76 (1996) 4841. F. Ronning, C. Kim, D.L. Feng, et al., Science 282 (1998) 2067. M.R. Norman, M. Randeria, H. Ding, J.C. Campuzano, Phys. Rev. B 52 (1995) 615. J.M. Ziman, Electrons and Phonons, The Theory of Transport Phenomena in Solids, Oxford Univ. Press, Oxford, 1960.