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Charge order and resistivity transition of Zn-doped cuprate superconductors
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 095702 (4pp)
doi:10.1088/0953-8984/27/9/095702
Charge order and resistivity transition of Zn-doped cuprate superconductors ¨ E V L de Mello and David Mockli Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi, RJ 24210-340, Brazil E-mail: [emailprotected] Received 27 November 2014 Accepted for publication 19 January 2015 Published 17 February 2015 Abstract
Impurity doping using Zn atoms was largely studied in cuprates because this process substantially reduces the superconducting critical temperature Tc without any effect on the pseudogap onset T ∗ . Earlier theories missed the recently established ubiquitous presence of incommensurate charge modulations in these materials. The charge order is a consequence of a phase separation transition which we describe by a continuity equation of the local free energy density. The Zn atoms generate a local magnetic moment, freezing their neighbors’ spins, slowing down the electronic segregation process. Then the Zn-doped properties are that of a granular superconductor whose size of the charge order modulations are dictated by the degree of phase separation. Keywords: inhom*ogeneous superconductors, theories and models of superconducting state, phase separation and segregation in nanoscale systems (Some figures may appear in colour only in the online journal)
La2 CuO4+y [20]. The latter established a direct relation between the interstitial oxygen degree of phase separation controlled by the time of x-ray illumination t on a compound and its superconducting transition temperature Tc (t) [20]. These observations suggest that the origin of the incommensurate charge order is an electronic phase separation in the CuO planes. Such a transition can be simulated by a continuity equation of the local free energy density, or Cahn– Hilliard (CH) differential equation [21]. Solving numerically the CH equation it is possible to follow the time evolution of a conserved order parameter that yields the local charge density distribution. The calculations show that the local free energy follows the charges and also has incommensurate modulations. The average modulation height was used to scale the two-body pair formation and precisely reproduced the values of Tc (t) measured by Poccia et al [20, 22]. Recently, this approach was also used to derive the doping dependence of the field distribution above Tc measured by muon spin relaxation (µ-SR) in Bi2212 compounds [23]. The induced magnetic moment around the Zn dopant atoms creates small regions where the antiferromagnetic order is suppressed avoiding local hole segregation [1]. In this microscopic scenario, the Zn doping diminishes the phase separation process and its concentration is related with the
1. Introduction
The effect of non-magnetic Zn impurity doping substituting planar Cu was studied in several cuprate superconductors for more than twenty years. It was established that each Zn atom induces a local magnetic moment in the four neighboring Cu sites by suppressing local antiferromagnetic correlation [1]. These local moments cause pair breaking that substantially reduces Tc [2] and the onset of the Nernst signal Ton , [3] but does not significantly modify the pseudogap temperature T ∗ [3–6]. On the other hand, the size of the local and average superconducting gap was significantly suppressed in Bi2 Sr2 CaCu2 O8+δ (Bi2212) [7, 8]. All these properties are still waiting for a generally accepted theory. An important piece of information, missing in the past, was the planar incommensurate charge order measured in different materials by many recent experiments [9–16]. These results confirmed earlier indications of a microscopic granular structure in the CuO planes by scanning tunneling microscopy (STM) [17, 18]. Altogether, these experiments suggest an intrinsic charge inhom*ogeneity but there are also extrinsic cases, where the in-plane electronic distribution follows the out of plane oxygen dopant, as observed directly in YBa2 Cu3 O7−δ (YBCO) compounds [11, 19] and indirectly on optimum 0953-8984/15/095702+04$33.00
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¨ E V L de Mello and D Mockli
J. Phys.: Condens. Matter 27 (2015) 095702
and u(ri , t → ∞) = ±1 corresponding to the extreme case of complete phase separation. In the simulations shown here we used A = B = 1 and c = 0.008. To deal with the Zn-doped system we use the same approach used to interpret the measurements of the time of phase separation enhanced by x-ray irradiation and the respective variations of Tc [20, 22]. As mentioned, the presence of Zn concentration x in a given sample controls the degree of phase separation. In this way we relate x with the time evolution t of the CH simulations. The free energy results for various time of phase separations is shown in figure 1. After t = 6000 the pattern does not change appreciably, only the domain walls tend to increase. In fact, since each Zn atom affects five Cu atoms, the one it substitutes and the four neighbors, we relate the percentage x of Zn with a decrease of 5 × x% of the GL free-energy modulations height. We argued recently [23] that the local GL free energy modulations shown in the inset of figure 1 causes the charges to lose part of their kinetic energy, which favors the formation of local Cooper pairs. This is the mechanism by which cuprates form electronic granular superconductors, with local pair amplitudes above Tc without long range order [22, 23, 27– 29]. We use these procedures in the next sections.
Figure 1. Detail of the time evolution of the GL free-energy domain wall between two patches along a line in the x-direction. t = 40 000 represents the ‘infinity’ time phase separation or the system without Zn. The Zn doped system is simulated by lowering the potential barrier. The inset shows the free energy modulations along a line of 42 sites in the x-direction.
relative decrease of the average free energy modulations, shown in the inset of figure 1. In this way we can estimate the average effect of Zn doping on the local d-wave superconducting gap [7, 8]. Then the Zn-doped properties are that of a granular superconductor whose size of the charge order modulations are dictated by the degree of phase separation. In this paper we show that the Zn-doped data, even in different systems like YBCO and La2−x Srx CuO4 (LASCO), are consistent with this scenario.
3. Calculations of local pairing
We use the size of the free energy modulations described above (figure 1) and called Vgb as a scale to the two-body average attraction in the superconducting self-consistent Bogoliubov– deGennes method [22, 23, 27–29]. The CH calculations furnish an inhom*ogeneous charge density that is kept fixed during the self-consistent calculations, yielding local dependent chemical potential and d-wave pair potential d (r). The details and the choice of kinetic parameters of the Hubbard Hamiltonian, derived from angleresolved photoemission spectroscopy (ARPES) experiments, were discussed in previous publications [22, 27]. The Coulomb repulsion U does not influence the values of d . The values of Vgb are chosen to yield results compatible with the local energy gap measured by STM [17, 18]. Such calculations were done on a 28 × 28 square lattice with a nearest neighbor hopping integral t = 0.15 eV. The YBCO pure system (x = 0 Zn doping) is known to have an average d (x = 0, T ≈ 0) ∼ 41 meV from many experiments [30] and this is simulated by an attractive pair potential at T = 0 of Vgb = 2.30 t. In the simulations, the phase separation is essentially completed with 40 000 time steps of simulations and this is taken as equivalent to x = 0. As the Zn concentration x increases, the time t decreases and the the free energy modulations become shorter as shown in figure 1. Decreasing VGL the corresponding value of the d (x, T ) also decreases, as listed in table 1. Before finishing this section, we should comment on the ARPES measurements on Bi2212 films [31] since they did not see any appreciable change in d (x, T ) with x in opposition to the STM [7] and ARPES [8] results. Their ∂T c(x)/∂x is unusual small, less than half of the YBCO and LASCO
2. Charge order modulations
The incommensurate charge order modulations occur through a diffusive transport or phase separation that is described by a conserved order parameter since the total charge is constant. Such an order parameter is the normalized difference between the local (p(ri , t)) and the average charge density p, that is, u(ri , t) = [p(ri , t) − p]/p. Then the Ginzburg–Landau (GL) free energy density is given by 1 f (u) = ε 2 |∇u|2 + VGL (u, T ), (1) 2 where the potential is defined as VGL (u, T ) = −A2 (T )u2 /2 + B 2 u4 /4 + ..., A2 (T ) = α[TPS (p) − T ], α and B are constants, and ε controls the spatial separation of the charge-segregated modulations [24, 25]. The diffusive transport of the order parameter leads to a continuity equation for the local free energy current density J = M∇(δf/δu), [21, 26] ∂u = − ∇.J ∂t = − M∇ 2 [ε 2 ∇ 2 u − A2 (T )u + B 2 u3 ]
(2)
where M is the mobility or the charge transport coefficient that sets the phase separation time scale. The order parameter varies between u(ri , t) ≈ 0 corresponding to the hom*ogeneous system above the phase separation critical temperature TPS , 2
¨ E V L de Mello and D Mockli
J. Phys.: Condens. Matter 27 (2015) 095702
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Table 1. The first three columns are experimental results of optimum YBCO Tc from two experiments [3, 5].
0 0.5 1.0 2.0 3.0 7.0
91 84 79 67 56 25
(K)
ρab (µm)
t Steps
Av d (0) (meV)
7.0 7.7 8.0 9.4 11.0 19.0
40 000 26 000 25 730 20 000 9500 6000
42.8 41.5 39.4 36.4 33.2 22.0
140
Tctheo (K)
120
91.0 84.0 79.0 67.2 55.0 24.0
100
EJ /kB (K)
Zn(%)
exp Tc
EJ(0.0%) EJ(0.5%) EJ(1.0%) EJ(0.2%) EJ(0.3%) EJ(0.7%) T(K)
80 60 40 20
Note: The last column has our calculations of Tc using the results of Av d (T ) in equation (3) and TPS ∼ 300 K. ‘Steps’ refers to time steps of the CH simulations as also shown in figure 1.
results [3–5]. It appears that the Zn doping on the Bi2212 film yields smaller effect than in the bulk.
50
100 150 T (K)
200
Figure 2. The estimation of Tc (x) from equation (3) with the calculated values of Av d (T ) and the the values of Rn taken from the experimental ρab (p, T Tc ). The relation between the time step and the Zn percentage x is listed in table 1. The arrow indicates T = Tc for x = 0, the undoped system.
4. Calculations of Tc
At low temperatures the charge modulations form independent SC small grains with local pair amplitudes d (ri , T ) interacting with one another via Josephson coupling, [23, 27] exactly as a granular superconductor [32]. Detailed studies of the d-wave weak links between superconductors have shown that the tunneling current depends mainly on the maximum lobe amplitude and qualitatively resembles that of a s-wave superconductor [33, 34]. Under this approximation we assume the average Josephson coupling energy EJAv between the puddles to be the simple analytical expression derived for coupling between two similar s-wave superconductors [35] Av d (x, T ) πhAv d (x, T ) EJAv (x, T ) = tanh , (3) 2e2 Rn (x) 2kB T N where Av d (x, T ) = i d (ri , x, T )/N is the average of the local d-wave gaps, and Rn (x) is the average normal resistance between neighboring patches at a temperature just above the phase coherence temperature Tc (x). Rn (x) is proportional to the normal state in-plane resistivity ρab (x, T Tc ) just above Tc (x) measured in the experiments [3–5]. When the temperature is lowered, thermal fluctuations diminish, and long-range phase coherence is achieved when kB T ≈ EJAv (T ) at Tc . The Josephson coupling of equation (3) and kB Tc are plotted in figure 2. Taking the measured values of the normal resistivity ρab (x, T Tc ) [3–5] to scale the normal resistance Rn (x) in equation (3), we derive the theoretical values of Tc (x). We list measured and calculated results to optimum YBCO in table 1. In figure 3 we plot our main results; the theoretical values of Tc (x) extracted from equation (3) and figure 2 with the experimental data listed in table 1. We also plot a ‘pairbreaking’ estimation based on the resistivity ratio just above Tc , i.e. Tc (x) = Tc (0)×ρab (0)/ρab (x). This linear approximation works well only for x < 3%. At the bottom of figure 3 we show results for a film of LASCO with Tc (x = 0) = 34 K with experimental data from [4] and similar calculations but with TPS ∼ 200 K.
Figure 3. The experimental critical temperature [3, 5] (red), the
calculations assuming only the ‘pair breaking’ ratio of the normal resistivity ρab (p, T Tc ) (blue) and the theoretical Tc using equation (3) (green) and the parameters listed in table 1 for optimum YBCO. At bottom we plot the same for LASCO for x = 0.02, 0.025 and 0.055.
In our approach, the measured onset of a Nernst signal [3] Tco (x) occurs due to the vortex motion between the patches with higher electronic density that have the larger local d-wave amplitude. The Zn doping effect on Tco (x) is the same as that on Tc because both depend on the superconducting properties. Concerning the behavior of T ∗ (x), we do not have a way to calculate this temperature, but if its origin is related to the phase separation, recalling that we use only one single value of TPS ∼ 300 K for Zn doped optimum YBCO, it is in our approach independent of x, as it is measured [1, 3]. On the other hand, if it depends on the superconducting properties it should also decrease, like Tc (x) and the onset of the Nernst temperature Tco (x) [3, 5, 6]. A more accurate way than resistivity measurements to resolve between these two possibilities is to perform TF-µSR, similar to what was done on cuprate, [16] to determine if the onset of diamagnetic response depends on the Zn concentration. 3
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J. Phys.: Condens. Matter 27 (2015) 095702
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5. Conclusion
We argued that there is enough evidence that cuprates undergo a phase separation transition that leads to the formation of incommensurate charge order, breaking spatial invariance. Contrary to some current ideas on competing orders, [36] the charge segregation transition favors the formation of local Cooper pairs through the free energy modulations and it is taken as the origin of the superconducting state. The incommensurate charge order modulations also play the role of grains in a granular superconductor in which Tc is reached by Josephson coupling [22, 27]. This scenario provided a complete picture of cuprates and a consistent interpretation of the Zn-doped experiments Acknowledgments
We acknowledge partial financial aid from the Brazilian agencies FAPERJ and CNPq and CAPES. References [1] Alloul H, Bobroff J, Gabay M and Hirschfeld P J 2009 Rev. Mod. Phys. 81 45 [2] Bernhard C, Tallon J L, Bucci C, DeRenzi R, Guidi G, Williams G V M and Niedermayer Ch 1996 Phys. Rev. Lett. 77 2304 [3] Xu Z A, Shen J Q, Zhao S R, Zhang Y J and Ong C K 2005 Phys. Rev. B 72 144527 [4] Cieplak M Z et al 1998 Appl. Phys. Lett. 73 2823 [5] Walker D J C, Mackenzie A P and Cooper J R 1995 Phys. Rev. B 51 15653 [6] Abe Y, Segawa K and Ando Y 1999 Phys. Rev. B 60 R15055 [7] Pan S H, Hudson W W, Lang K M, Eisaki H, Uchida S and Davis J C 2002 Nature 413 746 [8] White P J et al 2000 arXiv:cond-mat/9901349 [9] Wise W D et al 2008 Nat. Phys. 4 696
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