The effect of impurity and the suppression of superconductivity in Na(Fe_0.97−xCo_0.03T_x)As (T = Cu, Mn)

In figure 3, we present the temperature dependence of the in-plane resistivity from 2 K to 300 K for Na(Fe$_{0.97-x}$Co$_{0.03}$Cu$_{x}$)As single crystals. Obviously, ${{T}_{c}}$ drops down and the residual resistivity goes up with increase of the Cu concentration. In the low doping region, the resistivity decreases upon cooling, and this is followed by a superconducting transition. For the highly doped sample, an upturn is observed at low temperatures. It is interesting to note that, for the sample x = 0.03, the transition seems to be broad; an initial drop of resistivity starts at about 16 K, which is followed by a sharper drop at about 8–9 K. This may be induced by a chemical segregation. But the strange point is that this broadened transition occurs in most of the measured curves for this doping level, even from the samples of different batches. We notice that actually the normal state starts to show a low T upturn at this doping point. The low T upturn gets stronger with increase of the Cu content, which may be induced by a stronger localization effect. A strong semiconducting-like temperature dependence of the resistivity is observed for the sample with x = 0.05–0.06. Interestingly, it is found that ${{T}_{c}}$ is suppressed to zero at the threshold of strong semiconducting behavior. For the sample with x = 0.05, a downward trend of resistivity is observed at about 2.5 K. So the doping level x = 0.05 is regarded as the critical doping concentration which suppresses ${{T}_{c}}$ to zero. We define the residual resistivity ${{\rho }_{0}}$ by extrapolating normal state data in a linear way in the low temperature region to zero temperature, as shown in figure 4(a). One can see that ${{T}_{c}}$ decreases monotonically with increase of ${{\rho }_{0}}$, and it is suppressed to zero at a residual resistivity of 0.87 mΩ cm. Figure 4(b) shows the temperature dependence of the DC magnetization taken at 20 Oe after the zero-field-cooling (ZFC) and field-cooling (FC) procedures for the superconducting Na(Fe$_{0.97-x}$Co$_{0.03}$Cu$_{x}$)As single crystals. In this study, ${{T}_{c}}$ is defined on the magnetization curve using the crossing point of the normal state background line and the extrapolated linear line of the steep transition, which is nearly consistent with the point where the resistivity reaches zero. For the pristine sample, ${{T}_{c}}$ reaches about 20.5 K.

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The temperature dependence of the in-plane resistivity and the DC magnetization of Na(Fe$_{0.97-x}$Co$_{0.03}$Mn$_{x}$)As are shown in figure 5 and figure 6(b), respectively. As for the Cu-doped samples, the residual resistivity is determined by a linear extrapolation of normal state data to zero temperature, as shown in figure 6(a). It is clear that the residual resistivity increases with increasing doping level, and consequently ${{T}_{c}}$ is also suppressed monotonically. Interestingly, a similar broadening effect of the resistive transition occurs at about x = 0.035; this is similar to the case for doping with Cu, and seems to be some kind of intrinsic feature. Compared with Cu-doped samples, Mn-doped ones show much weaker localization even to a very high doping. The enhancements of the residual resistivity versus the doping are similar in the two systems below about x = 0.03, but it gets much faster in the Mn-doped system beyond this doping. Interestingly, superconductivity still exists in the sample with a residual resistivity, even up 2.86 m$\Omega$ cm. One may argue that the Mn dopants may not be successfully doped into the system when the concentration is higher than a certain value. However, this argument cannot be supported by the doping dependence of the c-axis lattice constant, as shown in figure 2(c). Moreover, the residual resistivity of the Mn-doped samples also increases continuously, which again indicates a successful doping of Mn into the material. Clearly, the Mn impurity shows a weaker ${{T}_{c}}$ suppression effect than Cu in the high doping regime. This remains an interesting and unresolved observation.

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In order to study the impurity scattering mechanism, it is crucial to determine whether the impurity is magnetic or nonmagnetic. To evaluate the magnetic moments induced by Cu and Mn dopants, we have done the magnetization measurements under high magnetic fields. The raw data from the magnetization measurements at 1 T are shown in figures 7(a) and (b). The clear divergence of the magnetic susceptibility at low temperatures can be understood as indicating the existence of some local magnetic moments. One can see that the low temperature upturn gets enhanced clearly by the Mn doping, but not enhanced by the Cu doping, indicating that Mn is a magnetic impurity, while Cu behaves as a nonmagnetic or very weak magnetic impurity. To further investigate the magnetic moments induced by Cu and Mn dopants, we assume that the low temperature magnetization can be written as the Curie–Weiss law

$$\begin{eqnarray}&&\chi =M/H={{\chi }_{0}}+{{C}_{0}}/\left( T+{{T}_{\theta }} \right),\end{eqnarray} \tag{ 1 }$$

where ${{C}_{0}}={{\mu }_{0}}\mu _{eff}^{2}/3{{k}_{B}}$, ${{\chi }_{0}}$ and ${{T}_{\theta }}$ are the fitting parameters, and ${{\mu }_{eff}}$ is the local magnetic moment per Fe site. The first term, ${{\chi }_{0}}$, comes from the Pauli paramagnetism of the conduction electrons, which is related to the density of states (DOS) at the Fermi energy. The second term, ${{C}_{0}}/\left( T+{{T}_{\theta }} \right)$, is given by the local magnetic moments of the ions at the Fe sites (including dopants like Co, Cu, and Mn). The fitting process is not straightforward; for a precise evaluation of the local magnetic moments, we adjust the ${{\chi }_{0}}$ value to get a linear function of $1/\left( \chi -{{\chi }_{0}} \right)$ versus T in the low temperature limit. Then we fit the data with a linear function, as shown in figure 8. The slope of the linear line gives $1/{{C}_{0}}$, and the intercept delivers the value of ${{T}_{\theta }}/{{C}_{0}}$. Once ${{C}_{0}}$ is obtained, we can get the average magnetic moment of a single Fe site (including the contribution of Fe and the dopants). The results are shown in figure 7(c). Clearly, doping with Mn induces strong local magnetic moments, especially when x exceeds 0.35, while doping with Cu seems to even weaken the average local moments. These facts suggest that Mn ions here play the role of magnetic impurities, while Cu dopants act as nonmagnetic or weak magnetic impurities. One possible picture in which to interpret this is that the Cu dopant may have a full shell of the d$^{10}$ configuration with the ionic state of Cu$^{1+}$ as predicted by the theoretical calculations [40, 41]. Similar results are observed for Cu-doped Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ [31].

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The residual resistivity of ${\mbox{Na}}({\mbox{F}}{{{\mbox{e}}}_{0.97-x}}{\mbox{C}}{{{\mbox{o}}}_{0.03}}{{{\mbox{T}}}_{x}}){\mbox{As}}({\mbox{T}}={\mbox{Cu}},{\mbox{Mn}})$ plotted as a function of doping concentration x is shown in figure 9. One can see that both doping with Cu and doping with Mn can enhance the residual resistivity. In the low doping region (x = 1%, 2%, 3%), the residual resistivity increases with the same ratio, 0.18 m$\Omega$ cm/%, for the Cu-doped and the Mn-doped samples. However, the residual resistivity increases rapidly for highly Mn-doped samples. Surprisingly, superconductivity still exists even up to a residual resistivity of 2.86 m$\Omega$ cm for the case of Mn doping. A similar phenomenon is observed for Co-doped Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ superconductor [31], where superconductivity is maintained even up to a residual resistivity of 6 m$\Omega$ cm. We must emphasize that the Mn elements have been doped into the Fe sites without doubt, because the lattice constant changes monotonically, and the high residual resistivity increases monotonically with the doping level of Mn up to 4%. For a simple S$^{\pm }$ pairing model, this is very difficult to understand. A further in-depth understanding is highly desired.

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It seems that both Cu and Mn act as charge donors. In this case, one may argue that the suppression of ${{T}_{c}}$ may result from the change of the charge carrier density. However, we have reason to believe that the ${{T}_{c}}$ suppression of the ${\mbox{Na}}({\mbox{F}}{{{\mbox{e}}}_{0.97-x}}{\mbox{C}}{{{\mbox{o}}}_{0.03}}{{{\mbox{T}}}_{x}}){\mbox{As}}({\mbox{T}}={\mbox{Cu}},{\mbox{Mn}})$ system is due to impurity scattering. If Cu behaves as an electron dopant, the carrier density will be increased, and result in a decrease of the residual resistivity. Our results showed that the residual resistivity increases rapidly with increase of the doping content for both Cu- and Mn-doped samples. So it is reasonable to say that Cu behaves more like random scatterers than an electron dopant. And we believe that such a rapid increase of the residual resistivity in Mn-doped samples is mainly attributable to impurity scattering. Plenty of studies on some other iron-based superconductors have revealed that Cu 3d states do not provide additional free carriers to the Fermi level [31, 41, 42]. Similarly, it is proved that in Mn-doped BaFe$_{2}$As$_{2}$, Mn dopants act as magnetic impurities rather than hole donors [43].

In the above, we have discussed the influence of doping with Cu and with Mn on the lattice parameter, magnetization and resistivity. We have also discovered that Mn doping gives rise to a continuous increase of the local magnetic moments, and that the higher dopings with Mn can especially enhance the local magnetic moments greatly, whereas Cu behaves as a nonmagnetic or very weakly magnetic impurity. On the basis of these results, in the following, we focus on the discussion of the pair breaking mechanism for ${\mbox{Na}}({\mbox{F}}{{{\mbox{e}}}_{0.97-x}}{\mbox{C}}{{{\mbox{o}}}_{0.03}}{{{\mbox{T}}}_{x}}){\mbox{As}}({\mbox{T}}={\mbox{Cu}},{\mbox{Mn}})$ superconductors. According to the S$^{\pm }$ scenario with equal gaps of opposite signs on different Fermi surfaces, ${{T}_{c}}$ is expected to be markedly suppressed due to the potential scattering by substituted nonmagnetic impurities, and it obeys a universal Abrikosov–Gorʼkov formula [5], $-{\rm ln} t=\psi \left( 1/2+\alpha /2t \right)-\psi \left( 1/2 \right)$, where $t={{T}_{c}}/{{T}_{c0}}$ with ${{T}_{c0}}$ and ${{T}_{c}}$ the transition temperatures of the pristine and the doped samples respectively, $\psi \left( x \right)$ the digamma function, and α the pair breaking parameter [44]. The Abrikosov–Gorʼkov formula would lead to ${{T}_{c}}$ vanishing at $\alpha _{c}^{theory}=0.28$. However, plenty of previous studies of the impurity effect on the iron-based superconductor have revealed that the critical value of α is much larger than 0.28, which means that the rate of ${{T}_{c}}$ suppression found previously is too small to be explained by the S$^{\pm }$ scenario [22–25, 27, 33]. According to the theoretical calculations based on the five-orbital model [11], $\alpha =z\hbar \Gamma /2\pi {{k}_{B}}{{T}_{c0}}$, where $z=m/{{m}^{*}}$ is the renormalization factor and $\Gamma$ is the scattering rate. We use the relation $\Gamma =n{{e}^{2}}\;\Delta {{\rho }_{0}}/{{m}^{*}}=e\;\Delta {{\rho }_{0}}/{{m}^{*}}{{R}_{H}}$, where n is the carrier density, ${{R}_{H}}$ is the Hall coefficient, and ${{m}^{*}}$ is the effective mass. In this study, we use $z=1/4.2$ obtained from optical spectroscopy experiments for Na-111 superconductor [45, 46], and ${{R}_{H}}=-7\times {{10}^{-9}}$ m$^{3}$/C is obtained from the transport measurements by extrapolating ${{R}_{H}}$ data to zero temperature, which is consistent with the result reported before [39]. The critical value of α obtained, for ${{T}_{c}}$ to vanish, based on our experimental data for Cu-doped samples, is about 0.64, which is still much larger than the theoretically expected value ${{\alpha }_{c}}=0.28$, but the gap between the experimental and the theoretical values becomes much smaller compared with the case for previous studies on the Ba-122 system [23–25].

To further study the pair breaking mechanism in the ${\mbox{Na}}({\mbox{F}}{{{\mbox{e}}}_{0.97-x}}{\mbox{C}}{{{\mbox{o}}}_{0.03}}{{{\mbox{T}}}_{x}}){\mbox{As}}({\mbox{T}}={\mbox{Cu}},{\mbox{Mn}})$ system, we calculated the critical residual resistivity for ${{T}_{c}}$ to vanish on the basis of the S$^{\pm }$ scenario; it is proportional to the pair breaking parameter α. The relationship between ${{T}_{c}}$ and α can be transformed into a relationship between ${{T}_{c}}$ and the residual resistivity by using the relation ${{\rho }_{0}}=2\pi \alpha {{k}_{B}}{{m}^{*}}{{R}_{H}}{{T}_{c0}}/z\hbar e+\rho _{0}^{pri}$, where $\rho _{0}^{pri}$ represents the residual resistivity of the pristine sample. As shown in figure 10, ${{T}_{c}}$ was plotted as a function of ${{\rho }_{0}}$. The dashed line represents the relationship between ${{T}_{c}}$ and the residual resistivity based on the S$^{\pm }$ scenario, which follows a universal Abrikosov–Gorʼkov formula. One can see that the critical residual resistivity for ${{T}_{c}}$ to vanish in the S$^{\pm }$ model is 0.44 m$\Omega$ cm. As we can see, in the low doping region, our data can be roughly fitted to this model, which means that the impurity effect in Na(Fe$_{0.97}$Co$_{0.03}$)As induced either by Mn or by Cu can be explained by the S$^{\pm }$ scenario in the low doping regime. In the case of Mn doping, the suppression of ${{T}_{c}}$ gets much weaker beyond x = 0.03 and the superconductivity is retained even up to a residual resistivity of 2.86 m$\Omega$ cm, indicating that the magnetic Mn impurities are even not as detrimental as the nonmagnetic Cu impurities to the superconductivity in the high doping region. This is an interesting observation; further theoretical and experimental efforts are expected to examine why the superconductivity can be so robust under Mn doping.

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