Engineering the Kondo state in two-dimensional semiconducting phosphorene

Correlated interaction between dilute localized impurity electrons and the itinerant host conduction electrons in metals gives rise to the conventional many-body Kondo effect below sufficiently low temperature. In sharp contrast to these conventional Kondo systems, we report an intrinsic, robust, and high-temperature Kondo state in two-dimensional semiconducting phosphorene. While absorbed at a thermodynamically stable lattice defect, Cr impurity triggers an electronic phase transition in phosphorene to provide conduction electrons, which strongly interact with the localized moment generated at the Cr site. These manifest into the intrinsic Kondo state, where the impurity moment is quenched in multiple stages and at temperatures in the 40–200 K range. Further, along with a much smaller extension of the Kondo cloud, the predicted Kondo state is shown to be robust under uniaxial strain and layer thickness, which greatly simplifies its future experimental realization. We predict the present study will open up new avenues in Kondo physics and trigger further theoretical and experimental studies.

Figure 1

(a) Top and side views of Cr absorption at DV($5\left|8\right|5$) on SLP. The Cr impurity (red ball) is absorbed in between the two half layers indicated with black and white balls, at the valley site (see Appendix pp1). (b) The Cr impurity is found to have competing spin-3/2 and spin-2 states. While the conventional PBE exchange correlation predicts a spin-3/2 ground state, inclusion of on-site Coulomb interaction and the hybrid exchange correlation anticipate the spin-2 state as the ground state.

Figure 2

(a) Unfolded band structure and the corresponding density of states (DOS) of Cr@DV-SLP with a very small ($\sim 0.7\%$) Cr impurity concentration, calculated with the $\mathrm{DFT}+U_{\mathrm{Cr}}$ formalism with $U_{\mathrm{Cr}}=4$ eV. Red (blue) represents the majority (minority) channel. While the anisotropic dispersion along the armchair and zigzag directions is retained, the defected Cr@DV-SLP becomes half metallic. Further, the conduction electrons are of P $p$ character. (b) The magnetization density indicates that the Cr magnetic moment is mainly localized at the impurity site. (c) The corresponding charge density indicates that the conduction electrons are delocalized along the armchair direction and over the entire supercell ($\sim 23\AA$).

Figure 3

Schematic band structure of Cr@DV-SLP under moderate uniaxial tensile and compressive strains along armchair (${\varepsilon }_a=\pm 4\%$) and zigzag (${\varepsilon }_z=\pm 4\%$) directions. Similar to that of the unstrained Cr@DV-SLP, the electronic structure remains half metallic under uniaxial strains, and the conduction electrons are delocalized along the armchair directions. Further, the conventional PBE functional and supplemented on-site Coulomb interaction of $U_{\mathrm{Cr}}=2$ and 4 eV indicate a qualitatively similar trend.

Figure 4

(a) While the defected DV-BLP is semiconducting, the Cr impurity absorption at DV transforms the electronic structure to metallic. (b) The P $p$ conduction electrons are delocalized along the armchair direction in the defected layer and over the entire supercell considered. Although the results are shown for $U_{\mathrm{Cr}}=4$ eV, PBE and $U_{\mathrm{Cr}}=2$ eV provide qualitatively similar results.

Figure 5

Unfolded band structure and the corresponding density of states of Cr@DV-SLP calculated with (a) the PBE exchange-correlation functional, (b) the on-site Coulomb interaction $U_{\mathrm{Cr}}=2$ eV, and (c) the density of states of Cr@DV-SLP calculated with the HSE06 hybrid exchange-correlation functional. All calculations are performed for an $S=2$ spin ground state. The half-metallic character is consistent within the entire considered theoretical hierarchy.

Figure 6

Unfolded band structure and the corresponding density of states of Cr@DV-SLP for the low-lying $S=3/2$ state, calculated for an on-site Coulomb interaction $U_{\mathrm{Cr}}=4$ eV. The Cr@DV-SLP remains half metallic regardless of the spin state.

Figure 7

(a) Projected crystal orbital Hamilton population (pCOHP) between Cr $d$ and P $p$ electrons evaluated for $U_{\mathrm{Cr}}=4$ eV. The negative (positive) pCOHP values refer to the bonding (antibonding) nature of the interaction between Cr $d$ and P $p$ electrons. The presence of the bonding interaction for all magnetic $d$ orbitals with the P $p$ electrons favors screening of all moments during the Kondo state formation. (b) Hybridization ${\mathrm{\Delta }}_{\nu }$ evaluated using the Kramers-Kronig relation for Cr $d$ orbitals in the case of $U_{\mathrm{Cr}}=4$ eV.

Figure 8

Strain energy of pristine SLP under uniaxial strain applied along the armchair and zigzag directions. Straining the phosphorene lattice along the armchair direction is easier than straining along the zigzag direction.

Figure 9

Calculated energy gaps for pristine SLP (left) and DV-SLP (right) under uniaxial strain applied along the armchair and zigzag directions using the conventional PBE functional.

Figure 10

Binding energy for Cr@DV-SLP. The applied strain does not affect the binding energy significantly, which remains in the 3.80–3.50 eV range.

Figure 11

(a) Unfolded band structure and the corresponding density of states of Cr@DV-BLP calculated with the PBE exchange-correlation functional. Unlike the half-metallic nature of Cr@DV-SLP, the Cr impurity absorption in DV-BLP leads to a metallic solution.