Molecular Collapse States in $\text{Graphene}/{\mathrm{WSe}}_2$ Heterostructure Quantum Dots

In relativistic physics, both atomic collapse in a heavy nucleus and Hawking radiation in a black hole are predicted to occur through the Klein tunneling process that couples particles and antiparticles. Recently, atomic collapse states (ACSs) were explicitly realized in graphene because of its relativistic Dirac excitation with a large “fine structure constant.” However, the essential role of the Klein tunneling in the ACSs remains elusive in experiment. Here we systematically study the quasibound states in elliptical graphene quantum dots (GQDs) and two coupled circular GQDs. Bonding and antibonding molecular collapse states formed by two coupled ACSs are observed in both systems. Our experiments supported by theoretical calculations indicate that the antibonding state of the ACSs will change into a Klein-tunneling-induced quasibound state revealing deep connection between the ACSs and the Klein tunneling.

Figure 1

(a) STM image of an elliptical GQD ($V=600\mathrm{mV}$, $I=100\mathrm{pA}$). (b) Top panel: schematic of the heterostructure. Bottom panel: enlargement of the black dashed square of panel (a). Inset of bottom panel: FFT of the heterostructure. The bright spots represent the reciprocal lattice of graphene (white dotted circles), ${\mathrm{WSe}}_2$ (green dotted circles), moiré structure of the heterostructure (red dotted circles). (c) Calculated charge density distributions for isolated circular and elliptical conductors. (d) Typical $dI/dV$ spectra taken on and off an elliptical GQD. Measured positions of the colored spectra are marked with the corresponding color in panel (a). (e) The radially $dI/dV$ spectroscopic maps along the $X$ axis and $Y$ axis of the elliptical GQD. (f) The calculated LDOS corresponds to (d). (g) The calculated space-energy maps of the LDOS along the $X$ axis and $Y$ axis of the elliptical GQD. The red dashed lines in (e) and (g) indicate Dirac point energy in theory calculation. The black solid (purple hollow) dots indicate the quasibound states via the WGMs (ACSs) confinement.

Figure 2

(a)–(d) Top panels: $dI/dV$ maps at $V_{\mathrm{bias}}=0.19\mathrm{V}$ ($N1$), 0.098 V ($N2$), 0.05 V ($N3$), and $-0.05\mathrm{V}$ ($N4$). N1–N4 are the quasibound states in the elliptical GQD. Bottom panels: calculated LDOS in the elliptical GQD at $E=0.19\mathrm{eV}$ ($N1$), 0.09 eV ($N2$), 0.04 eV ($N3$), and at $-0.059\mathrm{eV}$ ($N4$). The white dashed lines indicate the profile of the elliptical GQD. The green dotted lines indicate the cutoff area of the Coulomb-like electrostatic potential.

Figure 3

(a) STM image of two GQDs ($V=500\mathrm{mV}$, $I=100\mathrm{pA}$). (b) $dI/dV$ spectroscopic mapping along the black arrow in panel (a). (c) Energy-fixed $dI/dV$ mappings at $N1$ (bonding state) and $N2$ (antibonding state) of the molecular GQD in panel (a). (d) STM image of the two GQDs with a reduced separating distance $\sim 9\mathrm{nm}$ ($V=400\mathrm{mV}$,$I=80\mathrm{pA}$). (e) $dI/dV$ spectroscopic mapping along the black arrow in panel (d). (f) Energy-fixed $dI/dV$ mappings at $N1$ (bonding state) and $N2$ (antibonding state) of the molecular GQD in panel (d).

Figure 4

(a) Configurations of molecular Coulomb potential and elliptical Coulomb potential. (b) Potential profiles of the joint field $V(x,y)$ for different $\lambda$. (c)–(e) Top panels: the calculated LDOS space-energy maps along the $Y$ axis of the GQDs. Middle and bottom panels: the corresponding LDOS maps of the first two quasibound states $N1$ and $N2$.