Kondo effect in monolayer transition metal dichalcogenide Ising superconductors

Many recent studies show that the hole-doped monolayer transition metal dichalcogenides (TMDs) exhibit Ising superconductivity. The Ising-type spin-orbit coupling (SOC) results in spin-momentum locking perpendicular to the two-dimensional crystal near the $K$ and $-K$ valleys, strongly protecting the superconducting state against in-plane magnetic fields. In this work, we theoretically study the Kondo effect of a magnetic impurity doped in monolayer TMDs with the Ising SOC protected superconducting state. Based on behaviors of spin-induced Yu-Shiba-Rusinov bound states and localized magnetic susceptibility, we show that the Ising SOC plays an important role in enhancing or suppressing the Kondo screening of the magnetic impurity, depending on the relative position of the Fermi level with respect to the Ising splitting. More importantly, the Ising SOC exhibits opposite influences on the quantum phase transition between magnetic doublet and Kondo singlet ground states in electron- and hole-doped cases. These features in Kondo screening can be tuned by ionic liquid-gating in experiments.

Figure 1

(a) Schematic diagram of a magnetic impurity doped in monolayer TMD Ising superconductors. The monolayer TMDs consist of one layer of TM atoms sandwiched between two layers of $S$ atoms, forming the trigonal prismatic structure. (b) The Ising-SOC-induced spin-splitting and spin-momentum locking near the $K(-K)$ valleys at the Fermi surface with carrier doping. (c) and (d) illustrate the Ising split band structure of the electron- and hole-doped TMDs coupled with a magnetic impurity, respectively. The red dashed lines indicate the bands around the valleys without Ising SOC. The Fermi level is fixed at ${\varepsilon }_{k\sigma }=0$, and the chemical potential $\mu$ can be tuned by the carrier doping through ionic liquid-gating in experiment.

Figure 2

(a) The local density of states (LDOS) of a magnetic impurity coupled with electron-doped conduction bands [see the red-dashed line in Fig. 1] in the absence of Ising SOC and $\mathrm{\Delta }=0.0$. The Kondo resonance is enhanced by increasing the chemical potential $\mu$, with the coupling strength ${\mathrm{\Gamma }}_0=0.02$. (b) The YSR bound states in the LDOS are tuned crossing the Fermi level by increasing $\mu$, with the energy gap $\mathrm{\Delta }=0.01$, and the coupling strength ${\mathrm{\Gamma }}_0=0.04$. Other parameters are the impurity level ${\varepsilon }_{d\sigma }=-0.05$, the Coulomb interaction $U=2$, the Ising SOC $I_{so}=0$, and the temperature $T=0$.

Figure 3

(a) The Kondo resonance in the LDOS of a magnetic impurity in electron-doped monolayer TMDs in a normal state. (b) The Ising-dependent YSR bound states induced by the magnetic impurity in electron-doped Ising superconductor. Correspondingly, the susceptibility ${\chi }_d\left(T\right)$ of the magnetic impurity is shown in (c). (d) The ground-state phase diagram dominated by the competition of Kondo effect and Ising superconductivity. The ratio of the energy scales $\left(T_K^0/\mathrm{\Delta }\right)$, underlying the phase transition between the magnetic doublet and the Kondo singlet ground states, shows depending on the Ising SOC $I_{so}$. Other parameters are the impurity level ${\varepsilon }_{d\sigma }=-0.05$, the Coulomb interaction $U=2$, and the chemical potential $\mu =0.2$.

Figure 4

(a) The variation of Kondo resonance with $I_{so}$ in hole-doped monolayer TMDs in a normal state. (b) The YSR bound states induced by the magnetic impurity in Ising superconductor. (c) The Ising-dependent behaviors of magnetic susceptibility ${\chi }_d\left(T\right)$. (d) The ratio of the energy scales $\left(T_K^0/\mathrm{\Delta }\right)$ underlying the level-crossing quantum phase transition varies with Ising SOC $I_{so}$. The chemical potential is evaluated with $\mu =0.2$, and other parameters used are the same as that in Fig. 3.