Quantum phase transitions into Kondo states in bilayer graphene

We study a magnetic impurity intercalated in bilayer graphene. A representative configuration generates a hybridization function with strong dependence on the conduction-electron energy, including a full gap with one hard and one soft edge. Shifts of the chemical potential via gating or doping drive the system between non-Kondo (free-moment) and Kondo-screened phases, with strong variation of the Kondo scale. Quantum phase transitions near the soft edge are of Kosterlitz-Thouless type, while others are first order. Near the hard edge, a bound-state singlet appears inside the gap; although of single-particle character, its signatures in scanning tunneling spectroscopy are very similar to those arising from a many-body Kondo resonance.

Figure 1

(a) Lattice structure of Bernal-stacked BLG, showing impurity configuration considered here. (b) Dispersion $E_{\alpha {\alpha }^{\prime }}\left(q\right)$ of the four bands along a line through one of the Brillouin-zone corners $K_{\pm }$ (where $q=0$). (c) BLG density of states $\rho \left(E\right)$ (dashed line) and hybridization function $\Gamma \left(E\right)$ for the impurity configuration in (a) with $V_1=V_2$ (solid line). (d) Kondo temperature $T_K$ for $U=-2{\epsilon }_d=0.1$ and $V_1=V_2=0.21$ vs chemical potential $\mu$ outside (circles) and inside (squares) the hybridization gap. The (red) line shows the result of substituting $\Gamma \left(\mu \right)$ into Haldane's formula for $T_K$.

Figure 2

Ground-state impurity occupancy $\langle n_d\rangle$ on the $\mu$-${\epsilon }_d$ plane for $U=0.1$ and $V_1=V_2=0.21$. Thick black lines demarcate free-moment (FM) and strong-coupling (SC) phases. The occupancy varies continuously across a line of Kosterlitz-Thouless QPTs at $\mu =0$ (dashed). Elsewhere around the phase boundary, $\langle n_d\rangle$ jumps across a first-order line (solid). White lines at $\mu =0,-t_{\perp }$ delimit the hybridization gap; that at ${\epsilon }_d=0$ ($-U)$ shows where $\langle n_d\rangle$ would jump from 0 to 1 (1 to 2) for $V_1=V_2=0$. A through F schematically represent end points of paths discussed in the text.

Figure 3

(a)–(c) Impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ for ${\epsilon }_d=-U/2$ and different values of $\mu$ (increasing in the direction of the arrows) in the ranges (a) $0\le \mu \le 0.5$, (b) $\mu \le -t_{\perp }$, and (c) $-t_{\perp }\le \mu <-t_{\perp }+{10}^{-4}$. In (c), $\Delta \mu =\mu -{\mu }_c^{-}$ with ${\mu }_c^{-}\simeq -t_{\perp }+1.16\times {10}^{-6}$. (d) $T{\chi }_{\mathrm{imp}}\left(T\right)$ for $\mu =0$ and different values of ${\epsilon }_d$ between $-U/2$ and 0, with ${\epsilon }_{dc}^{+}\simeq -3.22\times {10}^{-4}$. Notice $t_{\perp }=0.12$.

Figure 4

Impurity spectral function $A_d(\omega ,T=0)$ for ${\epsilon }_d=-U/2$ and (a) four values of $\mu$ just below $-t_{\perp }$, and (b) two values of $\mu$ straddling ${\mu }_c^{-}$. Dashed lines denote poles inside the hybridization gap, corresponding to singlet bound states.