Relationship between critical current and flux-flow resistivity in the mixed state of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$

TL;DR

This study investigates the relationship between critical current and flux-flow resistivity in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ superconductors, revealing a universal relation and abnormal flux-flow behavior across different doping levels.

Contribution

It introduces a simple relationship between critical current density and flux-flow resistivity in unconventional superconductors, highlighting doping-dependent flux-flow behavior.

Findings

All E vs j curves merge at a single point ($j_0,E_0$).

Flux-flow resistivity $ ho_{ff}$ increases with decreasing magnetic field in these materials.

Parameter $E_0$ varies with doping and temperature, indicating coexistence of phases.

Abstract

We studied the temperature and magnetic field dependence of vortex dissipation and critical current in the mixed-state of unconventional superconducting alloys Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ ($0.044 \leq x \leq 0.100$) through current-voltage measurements. Our results reveal that all the electric field $E$ vs current density $j$ curves in the Ohmic regime merge to one point ($j_0,E_0$) and that there is a simple relationship between the critical current density $j_c$ and flux-flow resistivity $\rho_{\rm ff}$: $\rho_{\rm ff}/\rho_{\rm n} = (1- j_{c}/j_{0})^{-1}$, where $\rho_{\rm n}=E_0/j_0$ is the normal-state resistivity just above the superconducting transition. In addition, $E_0$ is positive for all five dopings, reflecting the abnormal behavior of the flux-flow resistivity $\rho_{\rm ff}$: it increases with decreasing magnetic field. In contrast, $E_0$ is negative for the conventional superconductor Nb since, as expected, $\rho_{\rm ff}$ decreases with decreasing magnetic field. Furthermore, in the under-doped and over-doped single crystals of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, the parameter $E_0$ remains temperature independent, while it decreases with increasing temperature for the single crystals around optimal doping ($ 0.060\leq x\leq 0.072 $). This result points to the co-existence of superconductivity with some other phase around optimal doping.

Equations

$${\rho }_{\mathrm{ff}}^{-1}=\frac{j_0}{E_0}\cdot \left(1-\frac{j_c}{j_0}\right),$$

$$\frac{{\rho }_{\mathrm{ff}}}{{\rho }_{\mathrm{n}}}\propto \frac{H}{H_{c2}},$$

$${\rho }_{\mathrm{ff}}=\frac{E_0}{j_0-j_c},$$

Full text

I Introduction

The cobalt-doped superconducting iron-arsenide material Ba(Fe1-xCox)2As2  has been widely studied in part due to its rich phase diagram. Of special interest is the region of the temperature-doping ($T$-$x$) phase diagram in which the spin-density-wave (SDW) phase Fernandes et al. (2010); Rotundu and Birgeneau (2011); Chu et al. (2010, 2012); Fernandes et al. (2010) is in macroscopic coexistence with the superconducting (SC) phase Dioguardi et al. (2013). With an increase in cobalt concentration, the SDW order is suppressed providing the stage for the possible existence of a quantum phase transition under the superconducting dome Ning et al. (2009); Fernandes et al. (2013); Arsenijević et al. (2013). In addition, neutron scattering and nuclear magnetic resonance studies of BaFe2(As1-xPx)2 as a function of isovalent phosphorous doping have shown that the second-order SDW phase transition present at low phosphorous doping changes into a weakly first order transition in the vicinity of the optimally-doped sample Hu et al. (2015). These experimental facts have been used to propose a scenario in which quenched order gives rise to a spatially inhomogeneous emulsion with puddles of SC and SDW phases Chowdhury et al. (2015).

Critical current density ($j_{c}$) and flux-flow resistivity ($\rho_{\rm ff}$) have each been widely studied both in conventional type-II superconductors Dew-Hughes (1974); Strnad et al. (1964); Kim et al. (1965) and in unconventional superconductors such as iron-pnictide systems Prozorov et al. (2008); Tanatar et al. (2010); Hanisch et al. (2011); Okada et al. (2012, 2013, 2014, 2015). Nevertheless, to our knowledge there has not been any study revealing and discussing the presence of a relationship between these physical quantities. To address this issue, we carried out current-voltage ($I$-$V$) measurements on five superconducting Ba(Fe1-xCox)2As2 single crystals with Co doping $x$ within the range $0.044\leq x\leq 0.100$. We have discovered the following linear relationship between $j_{c}$ and inverse of flux-flow resistivity $\rho_{\rm ff}$-1:

[TABLE]

where $E_{0}>0$ for all five dopings. Moreover, the analysis of our data shows that $E_{0}/j_{0}$ is the normal-state resistivity. We further show that such a relationship also exists in conventional type-II superconductors such as niobium, in which $E_{0}$, in contrast with Ba(Fe1-xCox)2As2, has a negative value. In addition, the value of $E_{0}$ is temperature independent in under-doped and over-doped single crystals of Ba(Fe1-xCox)2As2, whereas it decreases with increasing temperature for samples around the optimal doping ($0.060\leq x\leq 0.072$). This latter experimental fact is consistent with the existence of a secondary phase such as SDW, glass phase, or recently proposed micro-emulsion phase present in this region of the phase diagram.

II Experimental details

Single crystals of Ba(Fe1-xCox)2As2 were grown using the FeAs self-flux method Chu et al. (2009); Ni et al. (2008). Their actual Co-doping value $x$ was determined by comparing the values of their superconducting critical temperature $T_{c0}$ at zero magnetic field with values from well-established $T_{c0}$-$x$ phase diagrams Ni et al. (2008); Chu et al. (2009); Reid et al. (2010). The Co-doping values of the single crystals discussed in this paper are $x=$0.044, 0.056, 0.060, 0.072, and 0.100, which cover the under-doped, optimally-doped, and over-doped regions. The temperature ($T$) and magnetic field ($H$) dependences of $\rho_{\rm ff}$ were obtained from $I$-$V$ measurements carried out on thin samples using the standard four-probe method with current flowing in the $ab$-plane and magnetic fields applied along the crystallographic $c$-axis. Due to the high current required to depin the strongly pinned flux vortices in the mixed state of these superconductors, a combination of Linear Research, Inc. LR700 resistance bridge with extended current limit and Physical Property Measuring System (PPMS) were used to carry out the $I$-$V$ measurements.

We reduced Joule heating of the single crystals due to the high current used in the $I$-$V$ measurements as follows: (1) the cross-section area of the single crystals was reduced down to $0.17\times 0.04$ mm2 in order to increase the actual current density. The reason is that, for a given heating power per unit length $p\equiv(I^{2}R)/L=(j^{2}A^{2})\cdot(\rho L/A)/L=j^{2}\rho A$, a maximum current density $j$ is accomplished for an achievable minimum cross-section area $A$; (2) multiple short thick gold current leads were used for the two current terminals to minimize the Joule heating in the gold wires and also to increase the heat transport from the single crystal to the thermal bath, since, for a given applied current, the dissipated power $P=I^{2}R=I^{2}\rho L/A$; (3) we used Sn, instead of silver paste, in order to decrease the contact resistance between the single crystal and the current leads down to less than 10 $\mu\Omega$ Tanatar et al. (2010); (4) an additional temperature sensor mounted on the top of the sample using $N$-type grease was used to control and measure the temperature of the sample and long folded manganin wires were used as the terminal leads of the thermometer to decrease the heat transport between the thermometer and puck since manganin has poor thermal conduction; (5) after the temperature was deemed stable, a 60 sec wait time (with the persistent current flowing through the single crystal) was included into the measurement sequence and only then the $I$-$V$ data were collected.

The upper limit of the current used in our measurements was sample dependent. We determined this value experimentally by monitoring the temperature of the single crystal with the temperature sensor mounted on the crysta,l as discussed above, and by stopping the current sweep when this temperature increased by more than 0.1 K from the desired temperature, an increase due to Joule heating. Hence, with all the improvements discussed above, we were able to get reliable $I$-$V$ data for these Ba(Fe1-xCox)2As2 single crystals by reducing the temperature instability due to Joule heating in the mixed state to less than 0.1 K. In addition, due to the high current limit imposed by our experimental condition, all the $I$-$V$ measurements were done at temperatures 0.85 $T_{c0}\leq T\leq$ 0.98 $T_{c0}$, i.e. about 2 K below the $H$-$T$ phase boundary.

III Results and discussion

III.1 Electric field - current density curves below $T_{c}$

We show in inset to Fig. 1 schematically a type-II superconductor in an external magnetic field: when the magnetic field is applied along the $c$ crystallographic direction (smallest dimension in the figure) and the applied current $I$ is along the $a$ crystallographic axis, the flux vortices are driven in the $b$ crystallographic direction resulting in a longitudinal dissipative voltage along the current direction. An $I$-$V$ curve is generated by increasing the applied current and measuring the longitudinal voltage.

We plot in Fig. 1 the electric field - current density ($E$-$j$) curves obtained, by taking into account the sample geometry, from the $I$-$V$ characteristics at $T=22.2$ K under fields of $1.6\leq H\leq 3.0$ T for the Ba(Fe1-xCox)2As2 single crystal with $x=0.060$. These data are typical for all the single crystals with different Co doping that we have measured. A voltage is first detected when vortices start to creep. By increasing the applied current, the flux vortices are driven harder and harder. A linear (Ohmic) regime in the $E$-$j$ curve appears when the vortices are fully de-pinned, which corresponds to the flux-flow regime. The slope of the linear regime in the $E$-$j$ curve represents the flux-flow resistivity, i.e., $\rho_{\rm ff}$ $=dE/dj$. This physical quantity is dominated by the scattering of the quasiparticles in/around the vortex cores. The intercept of this linear regime with the $j$ axis gives the critical-current density $j_{c}$. With increasing $H$, the Ohmic regime increases and it covers the whole current range for $H\geq H_{c2}$ at the measured temperature.

III.2 Relationship between $j_{c}$ and $\rho_{\rm ff}$

It is well known that the flux-flow resistivity in the mixed state of conventional type-II superconductors is proportional with the magnetic field in the low-field and low-temperature regimes, i.e., there is the following empirical relationship Kim et al. (1965):

[TABLE]

where $\rho_{n}\equiv\rho_{\rm ff}(H_{c2})$ Bardeen and Stephen (1965), and it saturates near $H_{c2}$ Kopnin and Lopatin (1995); Kopnin and Volovik (1997); Kambe et al. (1999). Indeed, the flux-flow resistivity of, for example, niobium increases with increasing $H$ Huebener et al. (1970); Peroz and Villard (2005): Fig. 2(a) shows the increase of flux-flow resistivity with increasing $H$ up to 126 mT ($h=H/H_{c2}=0.3$, $H_{c2}=420~{}m$T), measured at a high reduced temperature ($t=T/T_{c0}\sim 0.9$) Peroz and Villard (2005).

Now we turn our attention to the experimental results obtained on the Ba(Fe1-xCox)2As2 single crystals. We show $\rho_{\rm ff}$($H$) measured on Ba(Fe1-xCox)2As2 with $x=0.056$ at $T=21.5$ K and $0.2$ T $\leq$ $H\leq 1.2$ T in Fig. 2(b). These measurements are done at a reduced temperature $t\approx 0.9$ and reduced field $0.05\leq h\leq 0.29$ since $T_{c0}=23.2$ K and $H_{c2}=4.2$ T at this temperature. Therefore, the data shown for Nb in Fig. 2(a) and for Ba(Fe1-xCox)2As2 with $x=0.056$ in Fig. 2(b) were collected at the same reduced magnetic field and temperature. Nevertheless, notice that, in contrast with Nb, the flux-flow resistivity of Ba(Fe1-xCox)2As2 decreases with increasing magnetic field. This behavior is typical for all the samples we have measured. Hence, these two systems show opposite field dependences of flux-flow resistivity.

We note that a similar abnormal field dependence of flux-flow resistivity was observed in unconventional heavy-fermion superconductor CeCoIn5 and it was shown to be the result of critical spin fluctuations Hu et al. (2012). We have discussed the origin of the abnormal \rho_{\rm ff}$$(H) and its doping dependence in the Ba(Fe1-xCox)2As2 system elsewhere Huang et al. (2017).

In contrast to different $\rho_{\rm ff}$($H$) measured in Nb on one hand and Ba(Fe1-xCox)2As2  and CeCoIn5 on the other hand, the insets to Figs. 2 show similar magnetic field dependences of the critical current densities of Nb and Ba(Fe1-xCox)2As2: $j_{c}$ decreases monotonically with increasing $H$. In fact, such $j_{c}(H)$ behavior is typical of type-II superconductors Summers et al. (1991); Attanasio et al. (1995); Traito et al. (2006); Hänisch et al. (2015): the critical current density required to depin the flux vortices decreases with increasing $H$.

In general, one would not expect a relationship between $\rho_{\rm ff}$ and $j_{c}$ to exist since the former represents the dissipation in the free-flux-flow regime due to the quasiparticles present in/around the vortex cores and is independent of the pinning strength, while the latter reveals the strength of the pinning centers. However, plots of $\rho_{\rm ff}$-1 vs $j_{c}$ shown in the insets to Figs. 3 reveal the presence of a linear relationship between these two quantities in Ba(Fe1-xCox)2As2, Nb, and CeCoIn5 systems, with a positive slope for the Nb sample and negative slopes for the Ba(Fe1-xCox)2As2  and CeCoIn5 samples.

We show in the main panels of Figs. 3 the $E$ vs $j$ data for these three systems. For the Ba(Fe1-xCox)2As2  system we show the data for the $x=0.056$ single crystal as an example (the other Co-doped samples studied display a similar behavior). The linear flux-flow regime is given by $E=\rho_{\rm ff}\cdot(j-j_{c})$. In addition, notice that all the fitting lines in the flux-flow regime merge at the same point ($j_{0},E_{0}$) for all three systems. [The data do not extend all the way to this merging point ($j_{0},E_{0}$) since we performed all the $E$-$j$ measurements at $H<H_{c2}$.] This implies that there is the following relationship between $\rho_{\rm ff}$ and $j_{c}$ :

[TABLE]

from which Eq. (1) directly follows. Therefore, the linear relationship between $\rho_{\rm ff}$-1 and $j_{c}$, shown by the insets of Figs. 3, is a result of the fact that the Ohmic regimes measured at different $H$ values merge in one point, denoted here ($j_{0},E_{0}$). The merging point has a positive coordinate ($j_{0},E_{0}$) for the Ba(Fe1-xCox)2As2  and CeCoIn5 samples, and a negative coordinate for the Nb sample. This result is a consequence of the fact that \rho_{\rm ff}$$(H) decreases with increasing $H$ in Ba(Fe1-xCox)2As2  and CeCoIn5, while it increases with increasing $H$ in Nb.

The merging point ($j_{0},E_{0}$) gives the resistivity in the normal state, i.e., $E_{0}/j_{0}\approx\rho_{n}$. Indeed, Eq. (1) gives $\rho_{\rm ff}=E_{0}/j_{0}\approx\rho_{n}$ at $H=H_{c2}$ of a certain temperature since here $j_{c}=0$ and $\rho_{\rm ff}\approx\rho_{n}$ [see Eq. (2)]. We also plot $E_{0}/j_{0}$ vs $x$ along with the experimentally-obtained $\rho_{n}$ vs $x$ in Fig. 4. Notice the excellent agreement between these quantities at the same value of $x$. Also notice the change in the value of $\rho_{n}$ around optimal doping. With regards to Nb, the value of $E_{0}/j_{0}=2.9\times 10^{-8}$ $\Omega\textrm{cm}$, obtained from Fig. 3(b), is also in excellent agreement with the experimentally obtained $\rho_{n}=2.66\times 10^{-8}$ $\Omega\cdot\textrm{cm}$ Huebener et al. (1970).

Based on the experimental fact that $E_{0}/j_{0}\approx\rho_{n}$, we conclude that the Ohmic $E$-$j$ regimes measured at different $H$ values merge at the same point ($j_{0},E_{0}$) in systems with very small normal-state magnetoresistivity. Therefore, the presence of the linear relationship between $\rho_{\rm ff}$-1 and $j_{c}$, given by Eq. (1) and insets to Figs. 3, is the consequence of small magnetoresistance near $T_{c}$ in the systems discussed here.

III.3 Relationship between $\rho_{\rm ff}$ and $j_{c}$ at different temperatures

To study the effect of temperature on the relationship between $\rho_{\rm ff}$ and $j_{c}$  given by Eq. (1), we performed $E$-$j$ measurements on all five samples of Ba(Fe1-xCox)2As2 studied here over the temperature range $0.85~{}T_{c0}\leq T\leq 0.98~{}T_{c0}$ and extracted \rho_{\rm ff}$$(T) and $j_{c}(T)$. We show plots of $\rho_{\rm ff}^{-1}$ vs $j_{c}$ data measured at different $T$ and $x$ values in Figs. 5. The linear relationship between $\rho_{\rm ff}^{-1}$ and $j_{c}$ holds for all the dopings [with a positive slope $1/E_{0}$, see Eq. (1)], implying that all these samples have negligible magnetoresistivity close to $T_{c}$. Moreover, the slopes $1/E_{0}$ of these plots are independent of temperature for the under-doped ($x=0.044$ and 0.056) and over-doped ($x=0.100$) samples since all the $\rho_{\rm ff}^{-1}$ vs $j_{c}$ data for different temperatures overlap. However, the slope $1/E_{0}$ increases slightly (strongly) with increasing temperature for the $x=0.060$ ($x=0.072$) single crystals. Notice that these two Co-doped samples are in the optimum-doping region of the phase diagram (see Fig. 6). It is reasonable to conclude that a phase or phase boundary that is sensitive to changes in temperature exists around the optimal doping and it affects the flux-flow behavior. Unfortunately, our data do not allow us to identify the precise nature of this additional phase. However, recent theoretical and experimental studies have shown that around optimal Co concentration there is a crossover between the SDW and SC coexisting phases, and the pure SC phase Fernandes et al. (2010); Huang et al. (2017). Hence, the additional phase with pronounced temperature dependence revealed by our data may well be the SDW phase, although either spatially inhomogeneous microemulsion Chowdhury et al. (2015) or spin-glass Dioguardi et al. (2013) phases cannot be ruled out.

IV Conclusion

We carried out current-voltage measurements as a function of temperature and applied magnetic field on Ba(Fe1-xCox)2As2 single crystals with five different Co concentrations in the range $0.044\leq x\leq 0.100$. We compare and contrast these results with results obtained by plotting published data on the canonical single band conventional superconductor Nb and multiband unconventional superconductor CeCoIn5.

We find that $j_{c}$ decreases with increasing $H$ in all these three systems, indicating that the de-pinning mechanism is the same in these conventional and unconventional superconductors. On the other hand, $\rho_{\rm ff}$ shows opposite field dependences for the conventional Nb and unconventional Ba(Fe1-xCox)2As2  and CeCoIn5: $\rho_{\rm ff}$ increases with increasing $H$ in Nb, while it decreases sharply with increasing $H$ in Ba(Fe1-xCox)2As2  and CeCoIn5. The former result is typical of conventional type-II superconductors. Since in the flux-flow regime the dominant contribution to $\rho_{\rm ff}$  comes from the quasiparticle scattering in/around the vortex cores, the sharp increase in $\rho_{\rm ff}$  at low fields indicates the presence of a strong scattering mechanism in the unconventional superconductors Ba(Fe1-xCox)2As2  and CeCoIn5, as discussed previously Hu et al. (2012); Huang et al. (2017).

We also revealed a relationship between the critical current density and flux-flow resistivity in all three systems studied: $\rho_{\rm ff}^{-1}=(j_{0}/E_{0})\cdot(1-j_{c}/j_{0})$, where $E_{0}/j_{0}$ is the normal-state resistivity just above $T_{c}$ and $1/E_{0}$ is the slope of $\rho_{\rm ff}$-1 vs $j_{c}$. The above relationship is a consequence of the negligible magnetoresistance just above $T_{c}$ in all these three systems. The parameter $E_{0}$ is positive for all five Co concentrations studied and for CeCoIn5, reflecting the abnormal increases of $\rho_{\rm ff}$  with decreasing $H$ and, as expected, it is negative for Nb. In addition, $E_{0}$ is temperature independent in under-doped and over-doped single crystals of Ba(Fe1-xCox)2As2, while $E_{0}$ decreases with increasing temperature for the single crystals around the optimal Co concentration ($0.060\leq x\leq 0.072$). This latter result reflects the presence of a temperature-dependent secondary, to superconductivity, phase, most likely the spin density wave phase, although either spatially inhomogeneous micro-emulsion Chowdhury et al. (2015) or spin-glass Dioguardi et al. (2013) phases cannot be ruled out. This work has also shown that $I$-$V$ is a powerful transport technique that can probe the presence of different phases in the superconducting state.

Acknowledgments We gratefully acknowledge Alan Baldwin’s assistance with electronics and software development. This work was supported at KSU by the National Science Foundation grant Nos. DMR-1505826 and DMR-1506547. T. H. acknowledges the support of NSFC, grant No. 11574338, and XDB04040300. H. X. acknowledges the support of NSFC, grant No. U1530402.