Orbital hybridization in graphene-based artificial atoms

a and b. The distribution of states (0, 1) and (1, 1), respectively. Left panel: The LDOS maps from the analytically solved wavefunctions. Middle panel: The numerically calculated LDOS maps for a circular QD. Right panel: The numerically calculated LDOS maps for an elliptical QD.

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a. Left panel: A STM image (V_b = 600 mV, I = 130 pA) of the almost circular QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.03\) embedded in the graphene/WSe₂ heterostructure. The radius \({r}_{0}\approx 9\,{\rm{nm}}\). The yellow dashed line shows outline of the circular QD. The height profile along the red dashed line is shown with solid red line. Top right panels: Atomic-resolved STM image on (1 T’ phase) and off (2H phase) the QD, respectively. Bottom right panels: The FFT image obtained from the STM image on and off the QD, respectively. The white and green circles show reciprocal lattices of graphene and WSe₂, respectively. The unlabeled bright spots correspond to the reciprocal moiré superlattices and higher-order scattering. b. The dI/dV spectroscopic map versus the spatial position along axis of the almost circular QD. Orbital states can be clearly observed in the QD. The first four states are labelled by wavefunctions and black dashed lines. The two yellow dashed lines mark the size of the QD.

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a. A STM image (V_b = 600 mV, I = 100 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.07\) embedded in the graphene/WSe₂ heterostructure. The minor radius r₁ is approximately 12.5 nm, and the major radius r₂ is approximately 14.5 nm. The height profile along the red dashed line is shown with solid red line. b. The dI/dV spectroscopic maps versus the spatial position along the major axis (left panel) and minor axis (right panel) of the elliptical QD, respectively. Orbital states can be clearly observed in the QD. The first four states are labelled by wavefunctions and black dashed lines in the right panel. The yellow dashed lines mark the size of the QD. c. dI/dV maps of different orbital states. For an elliptical confinement, the anisotropy of confining potential results in orbital hybridization between the s-orbital and d-orbital states, giving rise to new states sd+ \({\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and sd- \(-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\), which exhibit θ-shaped and rotated θ-shaped features marked by the green dotted lines, respectively. The yellow dashed lines show outlines of the elliptical QD.

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a. A STM image (V_b = 450 mV, I = 300 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.08\) embedded in the graphene/WSe₂ heterostructure. The minor radius r₁ is approximately 6 nm, and the major radius r₂ is approximately 7 nm. The height profile along the red dashed line is shown with solid red line. b. The \(-{d}^{3}I/{{dV}}^{3}\) spectroscopic maps versus the spatial position along the major axis (left panel) and minor axis (right panel) of the elliptical QD, respectively. Orbital states can be clearly observed in the QD. The first four states are labelled by wavefunctions and black dashed lines in the right panel. The yellow dashed lines mark the size of the QD.

Source Data

a. The dI/dV spectroscopic map versus the spatial position along the major axis of the elliptical QD in Fig. 2c. Orbital states can be clearly observed in the QD. The two yellow dashed lines mark the size of the QD.

Source Data

a. Left panel: A STM image (V_b = 500 mV, I = 100 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.17\) embedded in the graphene/WSe₂ heterostructure. The minor radius r₁ is approximately 7.6 nm, and the major radius r₂ is approximately 10.7 nm. The height profile along the red dashed line is shown with solid red line. Top right panels: Atomic-resolved STM image on (1 T’ phase) and off (2H phase) the QD, respectively. Bottom right panels: The FFT image obtained from the STM image on and off the QD, respectively. The white and green circles show reciprocal lattices of graphene and WSe₂, respectively. The unlabeled bright spots correspond to the reciprocal moiré superlattices and higher-order scattering. b. The dI/dV spectroscopic maps versus the spatial position along the major axis (left panel) and minor axis (right panel) of the elliptical QD, respectively. Orbital states can be clearly observed in the QD. The first four states are labelled by wavefunctions and black dashed lines in the right panel. The yellow dashed lines mark the size of the QD. c. dI/dV maps of different orbital states. For an elliptical confinement, the anisotropy of confining potential results in orbital hybridization between the s-orbital and d-orbital states, giving rise to new states sd+ \({\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and sd- \(-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\), which exhibit θ-shaped and rotated θ-shaped features marked by the green dotted lines, respectively. The yellow dashed lines show outlines of the elliptical QD.

Source Data

a. Left panel: A STM image (V_b = 400 mV, I = 120 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.18\) embedded in the graphene/MoSe₂ heterostructure. The minor radius r₁ is approximately 8.4 nm, and the major radius r₂ is approximately 12.1 nm. The height profile along the red dashed line is shown with solid red line. Top right panels: Atomic-resolved STM image on (1 T’ phase) and off (2H phase) the QD, respectively. Bottom right panels: The FFT image obtained from the STM image on and off the QD, respectively. The white and green circles show reciprocal lattices of graphene and MoSe₂, respectively. The unlabeled bright spots correspond to the reciprocal moiré superlattices and higher-order scattering. b. The dI/dV spectroscopic map versus the spatial position along the major axis of the elliptical QD. Orbital states can be clearly observed in the QD. The two yellow dashed lines mark the size of the QD. c. dI/dV maps of different orbital states. For an elliptical confinement, the anisotropy of confining potential results in orbital hybridization between the s-orbital and d-orbital states, giving rise to new states sd+ \({\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and sd- \(-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\), which exhibit θ-shaped and rotated θ-shaped features marked by the green dotted lines, respectively. The yellow dashed lines show outlines of the elliptical QD.

Source Data

a. Left panel: A STM image (V_b = 500 mV, I = 100 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.19\) embedded in the graphene/WSe₂ heterostructure. The minor radius r₁ is approximately 7.6 nm, and the major radius r₂ is approximately 11.2 nm. The height profile along the red dashed line is shown with solid red line. Top right panels: Atomic-resolved STM image on (1 T’ phase) and off (2H phase) the QD, respectively. Bottom right panels: The FFT image obtained from the STM image on and off the QD, respectively. The white and green circles show reciprocal lattices of graphene and WSe₂, respectively. The unlabeled bright spots correspond to the reciprocal moiré superlattices and higher-order scattering. b. The dI/dV spectroscopic map versus the spatial position along the major axis of the elliptical QD. Orbital states can be clearly observed in the QD. The two yellow dashed lines mark the size of the QD. c. dI/dV maps of different orbital states. For an elliptical confinement, the anisotropy of confining potential results in orbital hybridization between the s-orbital and d-orbital states, giving rise to new states sd+ \({\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and sd- \(-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\), which exhibit θ-shaped and rotated θ-shaped features marked by the green dotted lines, respectively. The yellow dashed lines show outlines of the elliptical QD.

Source Data

a. A STM image (V_b = −140 mV, I = 150 pA) of the elliptical QD with anisotropy degree \({{dr}}^{{\prime} }=\,0.21\) embedded in the graphene/WSe₂ heterostructure. The minor radius r₁ is approximately 7.8 nm, and the major radius r₂ is approximately 12 nm. b. The dI/dV spectroscopic map versus the spatial position along the minor axis of the elliptical QD. Orbital states can be clearly observed in the QD. The first four states are labelled by wavefunctions and black dashed lines. The two yellow dashed lines mark the size of the QD. c. dI/dV maps of different orbital states. For an elliptical confinement, the anisotropy of confining potential results in orbital hybridization between the s-orbital and d-orbital states, giving rise to new states sd+ \({\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and sd- \(-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\), which exhibit θ-shaped and rotated θ-shaped features marked by the green dotted lines, respectively. The yellow dashed lines show outlines of the elliptical QD.

Source Data

From the leftmost panel to the rightmost panel: \({{dr}}^{{\prime} }=0,0.05,0.10,0.15,0.20\), the other parameters are the same and shown in Supplementary information. a1-a5. The space-energy LDOS for the five potentials, along the minor axis of ellipse. The red, yellow, green, and blue color bars respectively indicate the energy of quasibound states s1, p1, sd+ , sd-. b1-b5, c1-c5. The numerically calculated spatial distributions of hybridized states sd+ , sd- evolved as the QD is deformed from circle to ellipse. d1-d5, e1-e5. The spatial distributions of sd+ and sd- obtained from the combination of analytical wavefunctions \({\psi }_{{sd}+}={\psi }_{(\mathrm{0,2})}+\alpha {\psi }_{(\mathrm{2,1})}\) and \({\psi }_{{sd}-}=-\alpha {\psi }_{(\mathrm{0,2})}+{\psi }_{(\mathrm{2,1})}\). From left to right: α = 0.0, 0.3, 0.5, 0.7, 0.9.

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