Magnetic field induced reentrance of superconductivity in the cage-type superconductor ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$

The morphological and/or chemical disorder is commonly known as detrimental to superconductivity and typically reduces the critical temperature $T_c$. We present research on the skutterudite-related ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ system, where local atomic disorder leads to novel disorder-enhanced superconductivity. Our present studies focus on the series of ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ compounds, where the metallic dopants $M=\mathrm{Ca}$, Ti, Sr, Zr, La, or Lu, when they are smaller than the atomic radius of Y (the case of Ti, Zr, Lu, or vacancies at the Y sites), generate the so-called peak effect at $T<T_c$ in the fields smaller than the critical field $H_{c2}$. The peak effect is well documented experimentally by measurements of the temperature variations in electric transport and ac magnetic susceptibility under external magnetic fields. In accordance with the commonly accepted explanation of the peak effect, we assume that the magnetic field induced reentrance of superconductivity in ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ results from a change in the structure of the vortex lattice close to $H_{c2}$. Using a simple theoretical model we argue that the effectiveness of this mechanism can depend on the size of the dopant. We also investigate the band structure properties by x-ray electron spectroscopy and ab initio calculations. It seems interesting that the density functional theory predicts a magnetic moment on the dopant Ti, which is a reason for the Kondo effect, confirmed experimentally.

Figure 1

Plot of Rietveld refinement for ${\mathrm{Y}}_{4.5}{\mathrm{Sr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. Black dots, observed pattern; red line, calculated; blue ticks, Bragg peak positions; magenta line, the difference between calculated and experimentally observed XRD patterns.

Figure 2

Cell volumes for the series of the ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ compounds vs atomic radius of metal $M$. The dashed blue line serves as a guide to the eye.

Figure 3

Variation in stoichiometry of ${\mathrm{Y}}_{4.1}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18.5}$ over the length of the sample. A very similar behavior was observed for all ${\mathrm{Y}}_{4.1}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18.5}$ samples investigated here.

Figure 4

Temperature dependencies of the real ${\chi }^{\prime }$ and imaginary ${\chi }^{\prime \prime }$ components of the ac mass magnetic susceptibility ${\chi }_{\mathrm{ac}}$ per 1 Oe for the series of ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ compounds at $\nu =100$ Hz. The used unit $4\pi \times {10}^{-3}{\mathrm{m}}^3$/kg (SI) is equivalent to 1 emu/g (cgs).

Figure 5

Temperature dependencies of the real ${\chi }^{\prime }$ and imaginary ${\chi }^{\prime \prime }$ components of the ac mass magnetic susceptibility ${\chi }_{\mathrm{ac}}$ per 1 Oe for ${\mathrm{Y}}_{4.5}{\mathrm{Sr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$, measured at different frequencies $\nu$, and derivative $d{\chi }^{\prime }/dT$ and $d{\chi }^{\prime \prime }/dT$. The main maximum in $d{\chi }^{\prime }/dT$ marked by an arrow is assigned to $T_c^{★}=3.63$ K of the inhomogeneous phase; the second weak maximum at $3.02$ K marked by an arrow is referred to the formation of the bulk $T_c$ phase. The inset shows a broad transition at 5.63 K to the state with separate superconducting regions which, however, do not form a continuous path across the system (the same ${\chi }_{\mathrm{ac}}$ units as in the main panel). Similar ${\chi }_{\mathrm{ac}}\left(T\right)$ behavior was observed for remaining compounds; cf. Table 2.

Figure 6

(a) Temperature dependence of specific heat, $C\left(T\right)/T$, for ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at various magnetic fields. The data are compared with the $C\left(T\right)/T$ measured for ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at the field of 2.4 T. The inset displays $\mathrm{\Delta }C\left(T\right)/T$ well approximated by function $f\left(\mathrm{\Delta }\right)$ with the following parameters: ${\mathrm{\Delta }}_0=3.04$ K and variance $D=0.5{\mathrm{K}}^2$, where $\mathrm{\Delta }C\left(T\right)=C_{\mathrm{Ti}}(T,B=2.4\mathrm{T})-C_{{\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}}(T,B=2.4\mathrm{T})$. (b) The $C$ isotherms as a function of magnetic field $B$ for ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. The inset shows the details in $C/T$ vs $B$ at $T_c$ in an extended scale; the units are the same.

Figure 7

(a) Temperature dependence of specific heat, $C\left(T\right)/T$, for ${\mathrm{Y}}_{4.5}{\mathrm{Sr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at various magnetic fields. The inset shows the details in $C/T$ vs $T$ near $T_c^{★}$ in an extended scale; the units are the same as in panel (a). Panel (b) exhibits temperature dependencies of the upper critical field for the bulk ($T_c$) and disorder ($T_c^{★}$) phases. The green points in the H-T plot indicate formation of disconnected superconducting regions, which give the superconducting transition a percolation path upon forming. The H-T data are approximated by the Ginzburg-Landau (GL) model: $H_{c2}\left(T\right)=H_{c2}\left(0\right)\frac{1-t^2}{1+t^2}$, and $t=T/T_c$ (lines). Panel (c) shows heat capacity $C$ isotherms as a function of magnetic field. The arrows indicate the respective critical fields for the phases described in panel (b).

Figure 8

For the purpose of comparison, the ac susceptibility, electrical resistivity, and specific heat are displayed for ${\mathrm{Y}}_{4.5}{\mathrm{Ca}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. The data from various experiments clearly show the superconducting bulk phase below $T_c$ and the inhomogeneous $T_c^{★}$ phase. Similar results were obtained for other dopants $M$.

Figure 9

Electrical resistivity $\rho$ as a function of temperature at $B=0$ for the series of ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ superconductors, where $M=\mathrm{Y}$, Ca, Ti, Sr, Zr, La, and Lu. The inset displays the resistivity data in coordinates $ln\rho =f\left(T^{-1/4}\right)$. The lines approximate the linear $ln\rho$ vs $T^{-1/4}$ behavior in the temperature range between $\sim 80$ K and $\sim 300–350$ K, depending on the sample.

Figure 10

Low-temperature electrical resistivity, $\rho$, for ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ for a measuring current of 10 mA. Panel (a) shows ${\rho }_B\left(T\right)$; panel (b) exhibits isotherms of $\rho$ as a function of the magnetic field. Analogous behaviors in ${\rho }_B\left(T\right)$ and ${\rho }_T\left(B\right)$ have been demonstrated in the temperature region $T<T_c^{★}$ for ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ with $M=\mathrm{Ca}$ and Lu. Here, $T_c^{★}$ is defined as the temperature corresponding to midpoint of the resistive transition at zero magnetic field.

Figure 11

(a) Electrical resistivity for ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ for a measuring current of 10 mA. Panel (a) shows the low-temperature dependence of $\rho$ when the magnetic field is in the range of $0\le B\le 2.4$ T. The resistivity drop at $B=0$ indicates the superconducting transition at $T_c^{★}$, while the weak change of $\rho$ at $\sim 2.8$ K is related to the bulk transition at $T_c$. For the fields $B>0.4$ T the resistivity shows an anomalous rise in the SC state. (b) Magnetoresistance isotherms of ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ as a function of applied magnetic field for the temperature changing from 1.8 to 4 K with the appearance of the anomalous peak effect in $\rho$. At $T=1.8$ K $\rho$ first increases with rise of the magnetic field, then a reentrance of superconductivity happens at $B\approx 2.2$ T. As the temperature increases, the “reentrance” behavior becomes less and less obvious and finally disappears at $T>2.6$ K. At the lowest temperatures the effect is expected to be observed for the magnetic field $B<2.4$ T. The dashed line shows the estimated critical field, below which the peak effect coexists with the SC state.

Figure 12

${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$; electrical resistivity ${\rho }_0$ approximated from the normal state near $T_c^{★}$ to $T=0$ vs magnetic field. The field of 2.4 T separates different slopes in ${\rho }_0\left(B\right)$ behavior; cf. Fig. 11.

Figure 13

Electrical resistivity $\rho$ for ${\mathrm{Y}}_{4.5}{\mathrm{Zr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ below $T_c^{★}$ for a measuring current of 10 mA. Panel (a) shows ${\rho }_B\left(T\right)$; panel (b) displays isotherms of $\rho$ as a function of the magnetic field.

Figure 14

Field-dependent normalized resistance $R/R_n$ of ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ for different measuring currents at 1.9 K.

Figure 15

${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$; real (${\chi }^{\prime }$) and imaginary (${\chi }^{\prime \prime }$) parts of the ac susceptibility as a function of applied dc magnetic field. The curves are reversible for increasing as well as decreasing fields.

Figure 16

Electrical resistivity $\rho$ for ${\mathrm{Y}}_{4.5}{\mathrm{Sr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ for a measuring current of 10 mA. Panel (a) shows ${\rho }_B\left(T\right)$; panel (b) displays isotherms of $\rho$ as a function of the magnetic field. The change of $\rho$ between $T_c^{★}=3.63$ and $T=5.63$ K ($B=0$) indicates formation of disconnected superconducting regions. Similar ${\rho }_B\left(T\right)$ characteristics are observed for ${\mathrm{Y}}_{4.5}{\mathrm{La}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$.

Figure 17

Magnetoresistance isotherms for a measuring current of 10 mA as a function of applied magnetic field at different temperatures for ${\mathrm{Lu}}_{5-\delta }{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ with small Lu deficiency $\delta$ (a), and for off-stoichiometry ${\mathrm{Lu}}_{4.6}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (b). The peak in ${\rho }_T\left(B\right)$ in the superconducting state of ${\mathrm{Lu}}_{4.6}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ occurs in the magnetic fields $B>1$ T, while for the ${\mathrm{Lu}}_{5-\delta }{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ sample with the composition similar to $5:6:18$ this behavior is not clearly observed.

Figure 18

The band structure calculated along high-symmetry $k$ lines in the Brillouin zone for stoichiometric ${\mathrm{Lu}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (a) and ${\mathrm{Lu}}_{4.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (b). The bands are calculated for $U_{4f}=6.8$ eV and $U_d=3$ eV and spin-orbit (SO) coupling.

Figure 19

The comparison of TDOS for ${\mathrm{Lu}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ and ${\mathrm{Lu}}_{4.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. The inset shows details near the Fermi level. The DOSs were calculated for SO coupling and for $U_{4f}=6.8$ eV and $U_d=3$ eV giving the best comparison to the XPS valence band spectra (see Ref. [42]).

Figure 20

Valence band XPS spectra for ${\mathrm{Y}}_{4.5}{M}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ are compared with the calculated total DOS for pristine ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (TDOS for ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ is taken from Ref. [41]). The intensities of all measured XPS bands are normalized to the background for ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at $E>{\epsilon }_F$. Solid brown line represents the Lu $4f$ VB XPS spectra obtained for ${\mathrm{Lu}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (see Ref. [42]), and normalized to VB XPS spectra of ${\mathrm{Y}}_{4.5}{\mathrm{Lu}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at $E=-12$ eV.

Figure 21

(a) The total DOS per formula unit within GGA approximation (SO; $U_d=3$ eV) for the both spin orientations (spin-up and spin-down) for ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ and ${\mathrm{Y}}_{4.5}{\mathrm{Sr}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. Panel (b) shows the spin-polarized TDOS for Ti per one Ti atom. The inset displays resistivity $\rho$ as a function of $lnT$ at various magnetic fields near $T_c$ for ${\mathrm{Y}}_{4.5}{\mathrm{Ti}}_{0.5}{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$.

Figure 22

Illustration of the vortex configurations used to calculate the pinning energy according to Eq. (10). Panel (a) shows the situation when the vortex is pinned at the impurity (represented by a small red sphere; $r=0$), whereas in panel (b) the vortex is located away from the impurity ($r=\infty$).

Figure 23

Schematic illustration of local atom displacements around an impurity smaller (a) and larger (b) than Y atoms (not to scale). The dashed circles represent a perfect square lattice without the impurity-induced distortion. The light gray shaded area shows the range of the influence of the impurity.

Figure 24

The pinning energy [Eq. (10)] as a function of parameter $a$. $a<0$ ($a>0$) corresponds to impurities smaller (larger) than Y atoms. The dashed line shows $a=0$, i.e., a situation when there is no impurity-induced deformation.