Broken time-reversal symmetry in superconducting ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$

The superconducting state of the filled skutterudite alloy series ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$ has been systematically studied by specific heat and zero-field muon spin relaxation ($\mu \mathrm{SR}$), while the critical fields have been determined by magnetization measurements. An additional inhomogeneous local magnetic field, indicative of broken time-reversal symmetry, is observed in the superconducting states of the alloys. For $x\lesssim 0.5$, the broken-TRS phase sets in below a temperature $T_m$, distinctly lower than the superconducting transition temperature $T_c$. For $x\gtrsim 0.5,T_m\approx T_c$. The local field strength decreases as $x\to 1$, where ${\mathrm{LaPt}}_4{\mathrm{Ge}}_{12}$ is characterized by conventional pairing. The lower critical field $H_{c1}\left(T\right)$ of ${\mathrm{PrPt}}_4{\mathrm{Ge}}_{12}$ shows the onset of a second quadratic temperature region below $T_q\sim T_m$. Upper critical field $H_{c2}\left(T\right)$ measurements suggest multiband superconductivity, and point gap nodes are consistent with the specific heat data. In ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$, only a single specific-heat discontinuity is observed at $T_c$, in contrast to the second jump seen in ${\mathrm{PrOs}}_4{\mathrm{Sb}}_{12}$ below $T_c$. These results suggest that superconductivity in ${\mathrm{PrPt}}_4{\mathrm{Ge}}_{12}$ is characterized by a complex order parameter.

Figure 1

Powder x-ray diffraction results for ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$ ($x=0$, 0.1, 0.3, 0.5, 0.7, 0.8, and 0.9). (a) X-ray diffraction pattern for the [h k l] = [2 2 0] peak. A gradual shift of $2\mathrm{\Theta }$ indicates a continuous change in the lattice parameter of ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$. (b) Lattice constant of ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$ calculated from XRD patterns, Vegard's law is obeyed. Dashed lines are guide to the eye.

Figure 2

Representative $M-{\mu }_{0}H$ at several temperatures. $H_{c1}$ is typically defined as the field value where the $M-{\mu }_{0}H$ deviates from the linear initial shielding curve.

Figure 3

(a) Temperature dependence of the electronic specific heat $C_e$ plotted as $ln(C_e/{\gamma }_e{T}_c)$ vs $T_c/T$, with offsets for clarity. Curves represent power law fits to $C_e$ vs $T$. (b) La doping dependence of $n$, where $n$ is the power of the $C_e\propto T^n$ fit below $T_c$. (c) La doping dependence of $\mathrm{\Delta }{C}_p/{\gamma }_e{T}_c$. For all $x$, the values of $\mathrm{\Delta }{C}_p/{\gamma }_e{T}_c$ are slightly larger than the BCS value of 1.43.

Figure 4

(a) and (b) Temperature dependence of $C_e/T$ in zero applied field for $x=0.3$ and 0.7, respectively. Insets: Fits at low $T$. Curves are fits to Eq. (4): (i) Low-$T$ weighted sum of asymptotic BCS term (see text) and $T^3$. (ii) Low-$T$ single asymptotic BCS term [36]. (iii) Low-$T$ weighted sum of two power-law terms $T^2$ and $T^3$. Fits up to $T_c$: (iv) Weighted sum of power-law term and BCS empirical term. (v) Single BCS empirical term. (vi) Weighted sum of two BCS empirical terms.

Figure 5

Critical-field study of ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$ ($x=1,0.9,0.7$, and 0.3). Arrows: Values of $T_m$. (a) Lower critical field $H_{c1}\left(T\right)$ of $x=0$. Two quadratic temperature regions are observed with lower $T^2$ onset temperature $T_q=6\mathrm{K}$. (b)–(e) Upper critical field $H_{c2}\left(T\right)$ of $x=0,0.1,0.3$ and 0.7, respectively. Dashed curves are fit to Eq. (6). Solid curves are a linear $T/T_c$ fit to $H_{c2}\left(T\right)$, assuming ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$ is a two-band superconductor in the dirty limit.

Figure 6

Representative muon asymmetry spectra $A_{\mu }\left(t\right)$ for ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}, x=0.3$ and 0.7, measured above and below $T_c$ (squares and circles, respectively). A background signal has been subtracted from the data. Solid curves: Fits using the exponentially damped Kubo-Toyabe function [Eqs. (1) and (2)].

Figure 7

(a)–(f): Temperature dependence of the ZF KT static Gaussian relaxation rate $\mathrm{\Delta }$ in ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}, x=0$, 0.3, 0.5, 0.7, 0.8, and 0.9 samples, respectively. Curves: Fits of Eqs. (7) and (8) assuming a BCS-like temperature dependence. Arrows: $T_c$ from specific heat, $T_m$ from onset of broken TRS.

Figure 8

(a) Temperature-La concentration $x$ phase diagram for ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}$. Squares: $T_c$ from specific heat data. Circles: Onset temperature $T_m$ of broken TRS. (b) Circles: $x$ dependence of local spontaneous magnetic field distribution width ${\mathrm{\Delta }}_e\left(0\right)/{\gamma }_{\mu }$. Triangles: Pointlike node gap fraction $1-f$ from specific-heat results [Eq. (4)]. Curves are guides to the eye.

Figure 9

(a)–(f): Temperature dependence of the zero-field exponential damping rate $\lambda$ in ${\mathrm{Pr}}_{1-x}{\mathrm{La}}_x{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}, x=0$, 0.3, 0.5, 0.7, 0.8, and 0.9.

Figure 10

Longitudinal field $H_{\mathrm{L}}$ dependence of ${\mu }^{+}$ polarization time spectra from ${\mathrm{Pr}}_{0.3}{\mathrm{La}}_{0.7}{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}, T=\text{10}\text{K}>T_c$. Curves: Fits of Eqs. (1) and (3) to the spectra.

Figure 11

Field dependence of the relaxation rate ${\lambda }_{\mathrm{LF}}$ in ${\mathrm{Pr}}_{0.3}{\mathrm{La}}_{0.7}{\mathrm{Pt}}_4{\mathrm{Ge}}_{12}, T=10\mathrm{K}$. Curve: Fit of Eq. (9) to the data for ${\mu }_0{H}_L⩾0.5\mathrm{mT}$.