Kondo Effect in High-Tc Cuprates

We study the Kondo effect due to the nonmagnetic impurity, e.g., Zn, in high-Tc cuprates based on the spin-change separated state. In the optimal or overdoped case with the Kondo screening, the residual resistivity is dominated by the spinons while the T-dependent part determined by the holons. This gives ρ(T ) = 4h̄ e2 nimp. 1−x + αT x (x: hole concentration,nimp.: impurity concentration, α: constant ), which is in agreement with experiments. In the underdoped region with the pseudo spin gap, an SU(2) formulation predicts that the holon phase shift is related to the formation of the local spin moment, and hence the residual resistivity is given by ρres. = 4h̄ e2 nimp. x , which is also consistent with the experiments. The magnetic impurity case, e.g., Ni, is also discussed. 74.25.Fy, 74.25.Ha, 74.72.-h, 75.20.Hr Typeset using REVTEX 1 Kondo effect is a phenomenon shown by a magnetic impurity put into a nonmagnetic metal [1]. As the temperature is decreased, the magnetic moment is screened by the conduction electrons and finally the Kondo singlet, i.e., the singlet of the localized spin and the conduction electrons, is formed. In the usual Kondo effect, the conduction electrons are assumed to be non-interacting or to form a Fermi liquid, which is magnetically inert due to Fermi degeneracy. When the conduction electrons are strongly correlated and magnetically active, it is expected that Kondo effect is also modified. High-Tc cuprates offer a unique opportunity to study such an effect. In the undoped high-Tc cuprates, the valency of Cu atom is Cu (d) and the system is a Mott insulator. By the hole doping, the system becomes metallic and shows superconductivity with high Tc. We believe that the Kondo effect in this system is actually observed for the nonmagnetic impurity, e.g., Zn, replacing Cu atom in the conducting plane. The valency of Zn is Zn (d) and compared with the Cu case one electron is trapped by one additional positive charge of the neucleus, which forms a singlet on the Zn site. In the underdoped cuprates with spin gap, it is found experimentally that a local moment of S = 1/2 appears on neighboring Cu sites [2–11]. We believe this localized spin is not screened by the conduction electron spins because of the reduced density of states for spins at the Fermi energy EF in the presence of the spin gap [12,13]. Once the spin gap collapse with the increased hole concentration, the density of states for spins at EF recovers and also the Kondo screening, i.e., the singlet formation between localized spin and conduction spins, occurs. Associated with the formation of local moments, it is found that the residual resistivity ρres. is very large in high-Tc cuprates. For example ρres. at 1% Zn doping in La2−xSrxCuO4 (x = 0.15) amounts to ∼ 100μΩcm/%. This value should be compared with ρres. = 0.32μΩcm/% for Zn doping in the Cu metal [14]. The latter is understood in terms of the Born approximation using the screened Coulomb potential, and hence ρres. is proportional to Z (Z: the difference of the valence between the host and impurity atoms) as observed for Zn, Ga, Ge, As in Cu metal, and this explains the small ρres. for Zn with Z = 1 [14]. Because the d-orbitals of the impurity atoms are completely occupied in these cases, the resonant scattering is absent and the phase shifts are distributed to various partial 2 wave components l. On the other hand, d-obitals of Cr, Mn, Fe, Co, Ni atoms in Cu metal are partially filled, and cause the resonant scattering. Then ρres. is dominated by the d-wave component l = 2, and the residual resistivity can be analysed in terms of the Friedel sum rule [15] and the Kondo effect. Friedel sum rule is the expression of the charge neutrality, and is given explicitly by Z = 1 π ∑ l,σ (2l+ 1)δl,σ (1) where δl,σ is the phase shift for the partial wave component l with spin σ. Let S be the spin of the impurity, and the phase shifts are given by Z = 2l+ 1 π (δ↑ + δ↓) 2S = 2l+ 1 π (δ↑ − δ↓) (2) with l = 2. The spin S becomes zero below the Kondo temperature TK due to Kondo screening, and these two equations determine the phase shifts and hence the residual resistivity below and above TK [16]. This prediction is consistent with the experiments. For examples Fe in Cu ( Z = 3 ) shows ρres. = 18.5μΩcm/% below TK corresponding to S = 0 and Z = 3.3, whcih is more than 4 times larger than the Si case (ρres. = 3.95μΩcm/% with Z = 3). Then it is generally true that the larger residual resistivity is expected when only one l component contributes. Now let us consider the case of high-Tc cuprates. It has been convincingly discussed that the degeneracy of the d-orbitals is lifted by the crystal field and only dx2−y2 orbital is relevant to the conduction. Furthermore, the single band t-J model is the low energy effective model [17]. Then it is expected that the s-wave (l = 0) scattering dominates the residual resistivity. Assuming s-wave scattering, the residual resistivity in the limit of small nimp. is given in two dimensions as ρres. = 2h̄ e2 nimp. n (sin δ0↑ + sin 2 δ0↓) (3) where nimp. is the impurity concentration, n the carrier concentration, and δ0σ is the phase shift [9,14]. Theoretically it is not a trivial problem whether the carriers are the electrons 3 with the concentration n = 1−x or the doped holes with n = x. The experimentally observed values of ρres. in the underdoped region are quantitatively fitted by eq.(3) by putting the carrier concentration n = x, and the phase shift δ0σ = π/2 ( unitarity limit) [9–11]. Note that this is the largest value theoretically expected form eq.(3). As the doping proceeds to the optimal and overdoped regions, the residul resistivity decreases and fits the formula eq.(3) with n being increasing to 1 − x with δ0σ = π/2 unchanged [11]. This crossover seems to correspond to the disappearance of the local moment. It is noted that ρres. in the optimal and overdoped regions is consistent with the Fermi liquid picture, where S = 0 and Z = 1 in eqs. (2) and (3). However considering the fact that the conductivity without the impurities is proportional to x and hence is dominated by the hole carriers in the optimal doping region, the residual resistivity coresponding to n = 1 − x is a mystery. Even more unconventional is the underdoped case, where the phase shift is not for the electrons. Can one consider the phase shift for the holes ? Then what determines that phase shift ? These are the questions to which we give solutions below. The transport properties in high Tc cuprates have been analyzed in terms of the gauge model based on the mean field theory of RVB states [18–20]. Let us first consider the U(1) theory which is applicable to optimally doped and overdoped regions. In this formalism the electron (C iσ) is described as the composite particle of spinon (f † iσ) and holon (bi), i.e., C iσ = f † iσbi (4)