Transport, magnetic and superconducting properties of RuSrCuO ( Eu, Gd) doped with Sn
RuSnSrEuCuO and RuSnSrGdCuO have been comprehensively studied by microwave and dc resistivity and magnetoresistivity and by the dc Hall measurements. The magnetic ordering temperature is considerably reduced with increasing Sn content. However, doping with Sn leads to only slight reduction of the superconducting critical temperature accompanied with the increase of the upper critical field , indicating an increased disorder in the system and a reduced scattering length of the conducting holes in CuO layers. In spite of the increased scattering rate, the normal state resistivity and the Hall resistivity are reduced with respect to the pure compound, due to the increased number of itinerant holes in CuO layers, which represent the main conductivity channel. Most of the electrons in RuO layers are presumably localized, but the observed negative magnetoresistance and the extraordinary Hall effect lead to the conclusion that there exists a small number of itinerant electrons in RuO layers that exhibit colossal magnetoresistance.
pacs:74.70.Pq 74.25.Fy 74.25.Nf 74.25.Ha 75.47.Gk
Despite the intensive investigation of ruthenate-cuprates in the last ten years, a lot of questions about the nature of magnetic order and superconductivity, in particular their coexistence, are still unanswered Nachtrab:06 (). It is well established that in RuSrCuO (Ru1212, Eu, Gd, Y) compounds the magnetic ordering of ruthenium sublattice occurs at about 130 K. The ordering is predominantly antiferromagnetic (AFM) with an easy axis perpendicular to the layers and with a weak ferromagnetic (FM) component parallel to the layers.
While both, RuO and CuO layers may participate in the normal-state conductivity, the superconducting (SC) properties of Ru1212 compounds are associated only with the charge carriers in the CuO planes, and are, thus, strongly dependent on the concentration of these carriers. The related superconducting critical temperature ranges between 15 and 50 K, depending on the sample composition and/or sample preparation conditions. According to the thermopower and Hall coefficient measurements, pure Ru1212 samples are expected to be intrinsically underdoped, with the effective hole concentration in the CuO planes nearly equal to McCrone:03 ().
In high- superconductors, SC properties are strongly dependent on the effective hole concentration in the CuO planes. In the optimally doped systems, the effective hole concentration is estimated to be holes/Cu. In this respect, in recent years there has been a lot of experimental effort to adjust the number of charge carriers in the CuO planes in Ru1212 samples using different substitutional impurities. The largest changes in are found when Ru is partially replaced by Cu ( K for 40% of Ru replaced by Cu Klamut:01 ()). In several studies ruthenium was replaced by Nb,V Sn, Ti and Rh McCrone:03 (); McLaughlin:01 (); Williams:03 (); Hassen:06 (); McLaughlin:99 (); Malo:00 (); Hassen:03 (); Yamada:04 (); Steiger:07 (). Substitutions have also been made for other ions. For example, trivalent gadolinium was replaced by Ca and Ce Klamut:01a () or by isovalent Y and Dy McCrone:03 (). Finally, divalent strontium was replaced by La and Na Williams:03 (); Hassen:06 (). If the substitutional impurity lowers the total number of holes (La for Sr, Ce for Gd, Nb for Ru), is strongly depressed, indicating that is reduced well below . However, in the cases when valence counting of impurities would suggest a strong increase in (Na for Sr, Sn for Ru), the increase in was not experimentally observed, contrary to the observation in common high- systems with substitutional impurities. This is rather similar to the observation in the underdoped YBaCuO compounds where is nearly constant in a wide range of . The increase of was only observed in Ca substitution for Gd Klamut:01a (), but it can be attributed to the formation of RuSrGdCuO phase. The magnetic ordering is, however, strongly influenced by the substitutions in all of the mentioned doping studies.
There is still an open debate whether RuO layers are conducting or not. The magnetization Papageorgiou:07 () and NQR Tokunaga:01 (); Kumagai:01 () results suggest that most of the electrons are localized resulting in Ru ions while magnetoresistivity and Hall measurements McCrone:03 (); Pozek:02 () suggest the existence of conductivity in magnetically ordered RuO layers. We shall deal with this question in our doping study, where the number of charge carriers is modified by the replacement of Ru ions with Sn. In this paper a comprehensive microwave study on several Ru1212Eu and Ru1212Gd samples doped with Sn is reported. One of the samples (RuSnSrGdCuO) is also characterized by transport, magnetoresistance and Hall measurements, and the results are compared with our previous measurements Pozek:02 (); Pozek:07 ().
In this context, it is important to point out the controversy related to the role of the Sn substitution on Ru sites in Ru1212Gd samples. McLaughlin et al. McLaughlin:01 (); McLaughlin:99 () and Hassen and Mandal Hassen:06 () reported, respectively, the increase and decrease of with doping. It is possible that this controversy reflects different ways to experimentally determine , or might be related to different methods of sample preparation. Moreover, we shall revise the observation in the Hall study (Fig. 2 in Ref. McCrone:03 ()) in which the increase of the Hall coefficient with increased Sn content was reported.
Ii Experimental details
RuSnSrCuO polycrystalline samples, where Eu or Gd, were prepared by solid state synthesis, as described elsewhere Williams:03 ().
An elliptical TE copper cavity operating at 9.3 GHz was used for the microwave measurements. The sample was placed in the center of the cavity on a sapphire holder. At this position microwave electric field has its maximum. External dc magnetic field perpendicular to the microwave electric field was varied from zero up to 8 T. The temperature of the sample could be varied from liquid helium to room temperature. The measured quantity was , the total losses of the cavity loaded by the sample. It is simply related to the surface resistance of the material which comprises both, nonresonant resistance and resonant spin contributions. The details of the detection scheme are given elsewhere Nebendahl:01 ().
Resistivity, magnetoresistance and Hall effect measurements were carried out in the standard six-contact configuration using the rotational sample holder and the conventional ac technique (22 Hz, 1 mA), in magnetic fields up to 8 T. Temperature sweeps for the resistivity measurements were performed with carbon-glass and platinum thermometers, while magnetic field dependent sweeps were done at constant temperatures which was controlled with a capacitance thermometer.
Iii Results and analyses
The temperature dependence of the microwave surface resistance of various Ru1212Eu (Sn doping , 0.1, 0.2, 0.3 and 0.4) and Ru1212Gd (, 0.2) compounds is shown in Figs. 1(a) and 1(b), respectively. The data are normalized at K, for comparison. We found that, for the samples of similar geometry, the absolute level of the normal-state absorption in the Eu-based samples was systematically higher than the absorption in the Gd-based samples. Moreover, the surface resistance of the Eu-based samples shows more pronounced rise as the temperature decreases towards (, hereafter) than it is the case in Gd-based samples. The related crossover to the SC state occurs at lower temperatures. The SC crossover temperature region is rather broad in these data, resulting in a rather uncertain estimation of . This problem can be easily solved by taking into account that the superconductivity in the vicinity of is strongly dependent on magnetic fields. For this purpose, the difference between the microwave absorption in zero field and in T is plotted in Fig. 2. [A considerably smaller signal of Sn-doped sample in Figs. 2(b) and 4(d), with respect to the pure Gd-based sample, is only due to the smaller geometry of the sample.] Both the magnetic and SC transitions can be detected here with a much better resolution than in Fig. 1.
The magnetic ordering temperature of the Ru lattice corresponds with small peaks clearly seen in Fig. 2. The dependence of on the Sn content is shown in Figs. 3(a) and 3(b). is strongly suppressed with increasing in both the Gd- and Eu-based samples (the magnetic critical temperature is reduced nearly by a factor of 2 for ). This suppression seems to be a consequence of the reduced content of the Ru magnetic ions (dilution of the magnetic lattice) and the increased disorder in the Ru lattice. Similar decrease in is found in the samples in which Ru is replaced by nonmagnetic Nb ions McCrone:03 () ( K for McLaughlin:01 ()).
The SC ordering temperature is clearly seen in Fig. 2 for all the samples, and its dependence on is further shown in Fig. 3. The sudden drop in the difference of the microwave absorption in zero field and in T indicates the onset of the SC state in the grains. This effect can be associated with the intrinsic value of . On the other hand, the whole sample is expected to become superconducting at much lower temperatures where the 3D network of intergranular Josephson junctions is established. It is clear that the influence of Sn doping on is not nearly as dramatic as the influence on . In the Gd-based samples, for example, the replacement of 20% of Ru by Sn does not appreciably change . In the Eu-based samples, on the other hand, a slight reduction of with increasing is observed, but it is negligible in comparison with the related change in .
The SC state is further analyzed by measuring the magnetic field dependence of the microwave absorption. The results taken at different temperatures are shown in Fig. 4 for two undoped samples and two Sn-doped samples. At lower fields (below 0.5 T), the microwave absorption is strongly field-dependent due to the intergranular superconductivity dominated by the 3D Josephson network. At higher fields, all the Josephson junctions are driven to the normal state, and superconductivity is localized only in the grains. The microwave absorption grows slowly in this field range as the number of vortices increases, and it is expected to reach the normal-state absorption at the upper critical field .
The microwave magnetoresistance is therefore a useful tool for the estimation of upper critical fields. If one assumes that the depinning frequency is lower than the driving microwave frequency, it suffices to extrapolate the microwave absorption curves to the normal-state values and determine . However, this simple determination of is not possible in cases where the depinning frequency is comparable to the driving frequency. Then, one has to take into account also the frequency shift. The details of the complete procedure are explaned in Ref. Janjusevic:06 (). The shape of the low-temperature microwave magnetoresistance curves of the Gd-based samples [ K curves in Figs. 4(c) and 4(d)] points at the regime where the two frequencies are comparable to each other. At higher temperatures, on the other hand, the driving microwave frequency seems to be larger than the depinning frequency allowing the direct estimation of the upper critical fields. In the Eu-based samples, the depinning frequency seems to be low enough even at low temperatures, so that the upper critical fields can be simply determined in the complete temperature region of Figs. 4(a) and 4(b).
The upper critical fields estimated from the curves in Fig. 4 (and similar curves for other samples not shown here) are plotted in Fig. 5. The comparison of estimated in two Eu-based samples, and 0.3, is shown in Fig. 5(a). The analogous comparison of in the Gd-based samples with and 0.2 is given in Fig. 5(b). From these figures one can see that the upper critical fields in the Gd-based samples are lower than in the Eu-based samples, and, in both cases, the upper critical fields are higher in the doped samples than in the undoped samples. Having in mind that higher means lower Ginzburg-Landau coherence length, which, in turn, depends on the electron mean free path, one concludes that the disorder in the undoped Eu-based samples is higher than in the undoped Gd-based samples, and that the Sn doping introduces an additional disorder in both systems. The low-temperature ( and 10 K) dependence of the upper critical field on the Sn concentration in the Eu-based samples is also shown [Fig. 5(c)]. From Fig. 5(c), we can see that the upper critical field increases with for low to moderate concentrations, while for the highest concentration () is suppressed. The drop of in the sample is related to the drop of in Fig. 3(a).
In order to clarify the role of the disorder introduced by Sn doping, we extend the experimental study and analysis to transport coefficients. We have chosen the doped sample RuSnSrGdCuO which exhibits a sharp SC onset in Fig. 2(b), and measured its resistivity, magnetoresistivity and Hall resistivity.
Figure 6(a) shows the resistivity measured in zero magnetic field and in . In both cases, the resistivity at 200 K is 9.5 mcm, which is much lower than 15.5 mcm of the undoped sample Pozek:02 (). In zero field, one observes a relatively broad SC transition characterized by the change in the magnitude from 90% to 10% within the temperature interval K. Such temperature dependence of the resisitivity in the vicinity of is usually attributed to the spontaneous vortex phase. The temperature interval characterizing the SC transition in the 8 T resistivity data is much broader than that of the zero-field data, and the zero resistance occurs only at 10 K, indicating that the pinning above 10 K is not strong enough to suppress the vortex motion, even for dc driving currents. The absence of strong pinning forces above 10 K justifies the estimation of from the magnetic field dependence of the microwave absorption given above [Fig. 4(d)].
The onset of superconductivity in the grains could be seen better if the resistivity in is subtracted from the zero-field resistivity, as shown in Fig. 6(b). The subtraction also reveals the existence of a broad maximum at the magnetic ordering temperature K. This peak can be further analyzed using magnetic field dependent resistivity measurements.
The relative transversal magnetoresistivity is shown in Fig. 7 in a large temperature range from 140 K down to the superconducting transition. The curves are grouped in five subsets according to temperature intervals in Fig. 6(b) labeled by (a), (b), (c), (d), and (e), exhibiting several qualitatively different physical situations.
At temperatures well above [Fig. 7(a)], the magnetoresistivity is characterized by the negative quadratic-in-field behaviour similar to the magnetic-field dependence observed in dilute alloys containing uncorrelated magnetic impurities Yosida:57 (). With decresing temperature the correlations between magnetic Ru ions start to play important role and the quadratic-in-field magnetoresistance transform into the linear-in-field behaviour. The same trend can also be seen for temperatures below [Fig. 7(b)], but not too close to . Not surprisingly, the same linear-in-field behaviour was already observed in undoped and in La-doped samples in the vicinity of . Just below , superposed to the negative linear magnetoresistivity, a small positive component develops at low fields, which is related to the AFM ordering of the Ru lattice. The same positive contribution is also observed in the longitudinal configuration (i.e. ). Finally, it should be noticed that the negative magnetoresistivity of Figs. 7(a) and 7(b) does not show any sign of saturation up to .
Figures 7(c) and 7(d) reveal a relatively complicated behaviour of the magnetoresistivity at temperatures between 57 and 47 K. The positive AFM contribution decreases rapidly and vanishes below 53 K. Below 49 K one observes positive contribution, presumably related to the SC fluctuations. The resulting magnetoresistivity, which is the sum of the positive SC and negative FM contribution, yields a local minimum. Finally, below K, the positive contribution to the magnetoresistivity related to the SC ordering of the conducting layers starts to dominate. The sharp increase of the magnitude of the relative magnetoresistivity in this temperature region, shown in Fig. 7(e), reflects the sharp decrease of in Fig. 6(a).
So far we have established two seemingly opposed observations: the resisitivity of the Sn-doped sample is lowered with respect to the pure compound, while the electron relaxation rates (proportional to the resisitivity) increases with the Sn concentration (as estimated from the dependence of ). In order to resolve the problem, we have measured the Hall resistance in which the relaxation rates cancell out in the first approximation, allowing in this way the estimation of the dependence of the effective number of charge carriers.
The Hall resistivity of RuSnSrGdCuO as a function of magnetic field is shown in Fig. 8(a), for two temperatures below , and compared to the results of the undoped sample Pozek:02 (). A nonlinear increase of the Hall resistivity, which is the combination of the ordinary and extraordinary Hall contributions, and is a characteristic of magnetic metals, is observed at low fields. At high applied fields, a linear increase dominates, representing the ordinary Hall contribution only. The magnitude of the extraordinary contribution is smaller than that in the undoped sample [dotted line in Fig. 8(a)].
The average values of the Hall coefficient are further plotted in Fig. 8(b) for low and high fields. Both, the low- and high-field slopes are smaller than for the pure compound. In particular, the absolute value of the ordinary is roughly 30% smaller than in the pure compound, indicating a substantial increase of the effective number of charge carriers with Sn doping. Before discussing these data in more detail (Sec. IV B), it is useful to recall the prediction for of the single-component free-hole model with the carrier concentration , which is found to explain well the low-temperature Hall measurements in LaSrCuO in the wide doping range Ando:04 (). With Å being the primitive cell volume of Ru1212Gd, the result is mC. The measured high-field Hall coefficients indicated in Fig. 8(b) lead to , and for the pure, Sn-doped and La-doped samples, respectively.
In the following, we shall first show that the main changes in the zero-field resistivity, and in the (high-field) ordinary Hall coefficent with Sn doping (which amounts to several tens of percent according to the inset of Fig. 6(a) and Fig. 8), can be explained by using a single-component model for the conductivity tensor (subsection IV.1). Within this model, only CuO layers contribute to the observed conductivity, while the direct contributions of the RuO layers to the total conductivity are small and can be safely neglected. Then, we shall argue, in subsection IV.2, that this single-component model fails to account for the observed contribution to the magnetoresistivity (changes of the order of a few percent in Fig. 6), and the (low-field) Hall coefficient. In order to successfully explain these experimental features, one has to go beyond the single-component model and to take into account a presumably small number of itinerant electrons in the RuO layers.
The motivation for such two-step analysis of our data comes from the results of the zero-field NMR (NQR) experiments in the pure Ru1212Gd Tokunaga:01 (); Kumagai:01 () and the related band structure calculations Nakamura:01 (). The number of electrons in RuO layers is equal to the number of holes doped into the CuO plane. Naively, Ru ion can be visualized as a composition of Ru ion and one extra electron. In this picture, the basic question is as to which extent this extra electron is localized to the parent Ru ion. In Ru-NQR experiments, two well-distinguished signals are found, corresponding to Ru with the spin state and Ru with the spin state . These results suggest that most of the electrons are localized (leading to Ru with the spin state ), which means that the number of itinerant (delocalized) electrons in the RuO layers is small. In NQR, it is also found that the magnetic order in the RuO layers coexists with the SC order in the CuO planes down to K. Finally, the NQR frequency measured on Cu nuclei Kumagai:01 (); Kramer:02 (), which is close to measured in the underdoped YBaCuO compounds Yasuoka:89 (), puts Ru1212 systems into the high- superconductors with an intermediate copper-oxygen hybridization and, consequently, an intermediate value of KupcicEFG (). The band structure calculations Nakamura:01 (), on the other hand, suggest the itinerant character of electrons in the RuO layers, rather than the localized character, but all other conclusions of Ref. Nakamura:01 () are consistent with the NQR observations. Namely, there are three bands crossing , the first two are related to two CuO layers, while the third one is related to the RuO layer. The magnetic ordering and the calculated magnitude of the magnetic moments of Ru ions are also consistent with the local magnetic fields measured by NQR Tokunaga:01 (); Kumagai:01 ().
In the present analysis, the splitting between two hole-like CuO bands is neglected and the AFM fluctuations in the CuO planes are taken into account KupcicRaman (). This gives the effective concentration of charge carriers (holes) in the CuO layers to be proportional to () rather than to , and with the band mass close to the free electron mass. Thus, nearly measures the hole concentration in the CuO bands with respect to the half-filling. The effective concentration of itinerant electrons in the RuO layers is assumed to be small in comparison with the value characterizing the completely delocalized case.
iv.1 Resistivity and ordinary Hall coefficient
The first term in the off-diagonal component comes from the Lorentz force and is proportional to ( is the cyclotron frequency). The second term, , includes all extraordinary contributions and is proportional to the magnetization of the Ru lattice, which means that the dependence on is given in terms of the Brillouin function . According to Ref. Nakamura:01 (), the contributions of two bands in Eqs. (1) can be regarded as decoupled (i.e. the hybridization between the CuO and RuO bands is negligible). The structure of the effective numbers , Cu, Ru and , for this case is given in the Appendix. In the absence of magnetic fields, for example, we obtain and , with and defined above. and are, respectively, the relaxation times of holes and electrons averaged over two spin projections (see Appendix). In the absence of magnetic fields, it is reasonable to assume that not only is small, when compared to , but also the relaxation rate in the RuO layers is much larger than , due to the presence of magnetic ions and the substitutional impurities in the RuO layers. Thus, for . But, as mentioned above, the main effects in the magnetoresistivity are expected to come from the field dependence of [labeled hereafter by ].
For weak magnetic fields, the general expressions for the Hall resistivity and magnetoresistivity
The Hall coefficient is given by . The anomalous contributions (the negative magnetoresistivity and the extraordinary Hall coefficent) are hidden here in and , and we put in the rest of this subsection.
In the Sn doped samples, the replacement of any Ru atom by Sn introduces one extra hole in the system, which is redistributed between RuO and CuO layers. The part of the extra charge which remains in the RuO layers decreases the effective number of itinerant electrons (), thus increasing both the resistivity and Hall resistivity. One may say that it gives rise to the conversion of Ru into Ru. The part of the extra charge transferred into the CuO layers increases the effective number (. The experimentally determined decrease of the normal-state resistivity and the (high-field) Hall coefficent with the Sn doping (Figs. 6 and 8) shows that a substantial portion of the doped holes are indeed transferred to the CuO layers and that the small changes in already small can be neglected. This scenario is expected to hold for not too large number of Sn impurities (). Our results should be contrasted to the Sn doping study of McCrone et. al (Fig. 2 in Ref. McCrone:03 ()), where the increase of resistivity and the Hall coefficient was observed for .
As mentioned above, the correlation between and the superconducting critical temperature is expected to be similar to the correlations found in the isostructural YBaCuO compounds. The main difference is that a very complicated role of the CuO chains in supporting superconductivity in the CuO layers is played here by the magnetically ordered RuO layers. Indeed, the dependence of on the Sn doping (shown in Fig. 3) is similar to the observation in the underdoped YBaCuO compounds ( is nearly constant in a wide range of , K in YBaCuO).
Similar scenario holds also for other Ru1212 systems. For example, the replacement of Sr by La increases the total number of conduction electrons in the system. A substantial part of the doped electrons are transferred to the CuO layers, reducing the effective number below the critical value required for superconductivity, in agreement with experiments. In RuSrLaGdCuO, the high temperature resistivity and the high-field Hall resistivity are found to be increased by roughly 60-70% with respect to the pure compound, and the superconductivity is completely suppressed Pozek:07 (). The same effect of reducing the effective number is obtained by the partial replacement of Ru ions by Nb McCrone:03 (); McLaughlin:01 ().
iv.2 Magnetic properties
The magnetoresistivity data of Fig. 7 give us the indirect way to study the magnetic properties of the Ru lattice. In the model described above, we can write
with and being, respectively, the zero-field total conductivity and the zero-field conductivity of the RuO subsystem. The first factor is nearly proportional to , and represents an enhancement factor dependent on Sn doping. It grows with decreasing , as can be easily seen from our data taken in three compounds: RuSnSrGdCuO, RuSrGdCuO, and RuLaSrGdCuO. The second factor in Eq. (5), , is the magnetoresistivity of the isolated RuO subsystem, and it is expected to be similar to that of magnetic metals Yosida:57 (); Fert:77 (); Kataoka:01 (). Precisely, the negative quadratic-in-field magnetoresistivity observed at temperatures well above has to be contrasted to the magnetoresistivity of common high- superconductors which is very small and positive Harris:95 (). This high temperature behaviour can be understood as a clear evidence of the exchange interaction between itinerant electrons in the RuO layers and the localized Ru spins Yosida:57 (). The AFM interactions between localized spins become important at low enough temperatures leading to the linear-in-field behaviour with maximal slope at [Figs. 7(a)-(c)]. We recall that exactly the same temperature evolution of the magnetoresistivity was found in Ref. Kataoka:01 () for small concentration of conduction electrons and low stability of the FM state of the localized spins. Hence, we believe that expression (5) includes all relevant processes in the magnetoresistivity of ruthenate cupartes in the nonsuperconducting temperature range.
Taking into account that , one can rewrite Eq. (5) in the form:
where . Figure 9 shows the field dependence of the quantity for the three aforementioned samples at their respective magnetic ordering temperatures. If one assumes, based on the previous subsection, that and , the magnetoconductivity of ruthenium layers is found to exceed 18% at . The magnitude of this effect is comparable to the colossal magnetoresistivity of the manganite perovskites.
In conclusion, a microwave study of Sn doping in Ru1212 samples has shown that magnetic ordering is strongly influenced by introduction of tin into RuO layers, while the superconducting critical temperature remains practically unchanged.
The magnetic field dependence of the microwave absorption has shown that the upper critical field in doped samples is increased, probably due to the increased disorder. However, in spite of increased disorder, the dc resistivity measurement in RuSnSrGdCuO is lower than in the respective pure compound. We conclude that it is due to the increased number of holes in the CuO layers. This conclusion is also confirmed by Hall resistivity measurements presented in this paper.
In RuSnSrGdCuO magnetoresistivity develops from quadratic-in-field behavior (characteristic for uncorrelated magnetic impurities) at high temperatures to linear-in-field behavior close to , indicating significant correlations between magnetic Ru ions. Below , a small positive component of magnetoresistivity, related to the AFM ordering of the RuO lattice, is superposed to the negative linear magnetoresistivity. As the temperature is lowered below , the field-induced ferromagnetic order seems to prevail. At the onset of superconductivity () both, positive superconducting and negative ferromagnetic contributions to the same magnetoresistivity curve occur.
We have shown that the single component model for the conductivity tensor suffices to explain the variation of the normal-state resistivity and the high-field Hall coefficient with doping. The main conductivity channel is through CuO layers and the various dopings simply change the number of conducting holes in these layers. However, the low-field Hall resistivity and the magnetoresistivity provide evidence that there is a small conduction in magnetically ordered RuO layers. Quantitative analysis of the observed negative magnetoresistance, leads to the conclusion that small number of delocalized electrons display a colossal magnetoresistance which exceeds 18%. This is a remarkable result.
We acknowledge funding support from the Croatian Ministry of Science, Education and Sports (projects no. 119-1191458-1022 “Microwave Investigations of New Materials”, no. 119-1191458-0512 “Low-dimensional strongly correlated conducting systems” and no. 119-1191458-1023 “Systems with spatial and dimensional constraints: correlations and spin effects”), the New Zealand Marsden Fund, the New Zealand Foundation for Research Science and Technology, and the Alexander von Humboldt Foundation.
Appendix A Conductivity tensor
In the multiband models with the interband hybridization negligible (Ru1212 systems are the example), the elements of the conductivity tensor are given by the sum of the intraband terms ( Cu, Ru in the present case). The presence of magnetic background and external magnetic fields makes, in the first place, that the relaxation rates of conduction electrons depend on spin, and also introduces the extraordinary contributions to the off-diagonal elements of the conductivity tensor , as discussed in the main text. In the first approximation (the case of noninteracting magnetic impurities Yosida:57 ()) we can write , where is a constant and is the total magnetization of the RuO layer). Thus, in the Ru1212 systems, at temperatures well above , we have and , with and given by Ziman (); KupcicRaman ()
is the electron group velocity, , and are the elements of the inverse effective-mass tensor .
If magnetic fields are weak and the exchange interaction between the conduction holes and localized Ru spins is negligible, the field dependence in the denominators in (7) can be neglected, and the sum over spin projections gives for the averaged relaxation times of itinerant electrons the expression , resulting in Yosida:57 (). According to Eq. (5), this also leads to the total magnetoresistivity which is negative and quadratic-in-field. The effects of interactions between localized spins on the conductivity tensor at different temperatures are expected to be similar to that found in Ref. Kataoka:01 ().
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