Superconducting ground state of quasi-one-dimensional ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ investigated using $\mu \text{SR}$ measurements

The superconducting state of the newly discovered superconductor ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$, with a quasi-one-dimensional crystal structure $(T_{\mathbf{c}}\sim 6\text{K})$, is investigated using magnetization and muon-spin relaxation or rotation $\left(\mu \mathrm{SR}\right)$ measurements. Our analysis shows that the temperature dependence of the superfluid density obtained from transverse-field $\mu \mathrm{SR}$ measurements fits either to an isotropic $s$-wave character for the superconducting gap or to a $d$-wave model with line nodes. Furthermore, the goodness-of-fit $\left({\chi }^2\right)$ values indicate that our data fit better to the $d$-wave model $\left({\chi }^2\sim 1\right)$ than the $s$-wave model $\left({\chi }^2\sim 1.38\right)$. Therefore our $\mu \mathrm{SR}$ analysis is more consistent with having line nodes than being fully gapped, which is in agreement with the results of the penetration depth measured using a tunnel diode oscillator technique. Our zero-field $\mu \mathrm{SR}$ measurements do reveal very weak evidence of the spontaneous appearance of an internal magnetic field below the transition temperature, which might indicate that the superconducting state is not conventional. This observation suggests that the electrons are paired via unconventional channels such as spin fluctuations, as proposed on the basis of theoretical models of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$. Furthermore, from our transverse-field $\mu \mathrm{SR}$ study the magnetic penetration depth ${\lambda }_L$, superconducting carrier density $n_s$, and effective-mass enhancement $m^{*}$ have been estimated to be ${\lambda }_L\left(0\right)=432\left(4\right)$ nm, $n_s=2.7\times {10}^{27}\text{carriers}/{\mathrm{m}}^3$, and $m^{*}=1.75m_e$, respectively.

Figure 1

(a) Quasi-1D crystal structure of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ and (b) temperature dependence of the dc magnetic susceptibility measured in the zero-field–cooled (ZFC) and field-cooled (FC) states of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ in the presence of an applied magnetic field of 10 G. Inset: Magnetization vs field at 1.8 K.

Figure 2

Transverse-field $\mu \mathrm{SR}$ time spectra (one component) for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ collected (a) at $T=7.5$ K and (b) at $T=2$ K in an applied magnetic field $H=400$ G for the zero-field-cooled (ZFC) state.

Figure 3

(a) Temperature dependence of the muon depolarization rate ${\sigma }_{\mathrm{sc}}\left(T\right)$ of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ collected in an applied magnetic field of 400 G in the zero-field-cooled (ZFC) and field-cooled (FC) modes. (b) ${\sigma }_{\mathrm{sc}}\left(T\right)$ of the FC mode (symbols); lines are fits to the data using Eq. (2). The short-dashed (blue) line shows the fit using an isotropic single-gap $s$-wave model with ${\mathrm{\Delta }}_0/k_B{T}_{\mathbf{c}}=1.6\left(1\right)$ and the solid (red) line and dotted (purple) lines show fits to the $d$-wave model with ${\mathrm{\Delta }}_0/k_B{T}_{\mathbf{c}}=3.2\left(2\right)$ and ${\mathrm{\Delta }}_0/k_B{T}_{\mathbf{c}}=2.40$, respectively. Inset: Plot of quality of fit ${\chi }^2$ vs ${\mathrm{\Delta }}_0/k_B{T}_{\mathbf{c}}$.

Figure 4

(a) Zero-field $\mu \mathrm{SR}$ time spectra for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ collected at 0.6 K [(red) circles] and 8.0 K [(blue) squares] are shown together with lines that are least squares fits to the data using Eq. (3). These spectra, collected below and above $T_{\mathbf{c}}$, are representative of the data collected over a range of $T$. (b) Temperature dependence of the electronic relaxation rate of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ measured in zero magnetic field, where $T_{\mathbf{c}}=5.8$ K is shown by the dotted vertical line.