Coexistence of electron whispering-gallery modes and atomic collapse states in graphene/WSe2 heterostructure quantum dots

The relativistic massless charge carriers with a Fermi velocity of about c/300 in graphene enable us to realize two distinct types of resonances (here, c is the speed of light in vacuum). One is the electron whispering-gallery mode in graphene quantum dots arising from the Klein tunneling of the massless Dirac fermions. The other is the atomic collapse state, which has never been observed in experiment with real atoms due to the difficulty of producing heavy nuclei with charge Z > 170; however, they can be realized near a Coulomb impurity in graphene with a charge Z ≥ 1 because of the “small” velocity of the Dirac excitations. Here we demonstrate that both the electron whispering-gallery modes and atomic collapse states coexist in graphene/WSe2 heterostructure quantum dots due to the Coulomb-like potential near their edges. By applying a perpendicular magnetic field, we explore the evolution from the atomic collapse states to unusual Landau levels in the collapse regime.

The relativistic massless charge carriers with a Fermi velocity of about c/300 in graphene enable us to realize two distinct types of resonances (c, the speed of light in vacuum).One is electron whispering-gallery mode in graphene quantum dots arising from the Klein tunneling of the massless Dirac fermions.The other is atomic collapse state, which has never been observed in experiment with real atoms due to the difficulty of producing heavy nuclei with charge Z > 170, however, can be realized near a Coulomb impurity in graphene with a charge Z ≥ 1 because of the "small" velocity of the Dirac excitations.Here, unexpectedly, we demonstrate that both the electron whispering-gallery modes and atomic collapse states coexist in graphene/WSe2 heterostructure quantum dots due to the Coulomb-like potential near their edges.By applying a perpendicular magnetic field, evolution from the atomic collapse states to unusual Landau levels in the collapse regime are explored for the first time.
Here, we demonstrate the coexistence of the electron WGMs and ACSs in graphene/WSe2 heterostructure QDs.Because of the Coulomb-like potential near the edges of the QDs, we observe WGMs near the edge and ACSs in the center of the graphene/WSe2 heterostructure QDs.Moreover, the ACSs are further explored in the presence of high magnetic fields.The study about effect of magnetic fields on the ACSs has a long history (19).However, such a longstanding prediction remains highly controversial because that the theoretical results are contradictory (20)(21)(22)(23)(24)(25) and, more importantly, an experimental verification of this fundamental prediction is still lacking up to now (25,26).Our experiments demonstrate that the atomic collapse resonance effect still exist in magnetic fields and evolution from the ACSs to unusual Landau levels in the collapse regime are measured.
The graphene/WSe2 heterostructure was obtained by using a wet transfer fabrication of a monolayer graphene on mechanical-exfoliated thick WSe2 sheets (see methods for details of the sample preparation).In our experiment, nanoscale WSe2 QDs were surprisingly observed on surface of freshly mechanical exfoliated WSe2 sheets, as shown in atomic force microscopy (AFM) image of Fig. 1a.Usually, the thickness of the WSe2 QDs is the same as a WSe2 monolayer and the diameter is less than 10 nm.Such WSe2 QDs also can be observed when mechanical-exfoliated thick WSe2 is covered with graphene monolayer, i.e., in the graphene/WSe2 heterostructure, as shown in scanning tunneling microscope (STM) image of Fig. 1b (also see Fig. S1 for AFM images of the graphene/WSe2 heterostructure).At present, the exact origin for the emergence of the nanoscale WSe2 QDs is unclear.In our experiment, nanoscale monolayer-thick WSe2 anti-dots are usually observed around the WSe2 QDs (See Fig. S2), which suggests that the WSe2 QDs are generated from the anti-dots during the process of mechanical exfoliation.Figure 1c shows an atomic-resolved STM image around a graphene/WSe2 heterostructure QD.No atomic defect and strain structure (see Fig. S3 and the details of discussion) can be detected in graphene above the WSe2 QD.Fast Fourier transform (FFT) images inside and outside the QD are identical (see inset of Fig. 1c and fig.S2b) and the relative rotation angle between graphene and WSe2 is measured as about 20.4° (the rotation angle between the WSe2 QD and the WSe2 substrate is zero).A schematic side view of the graphene/WSe2 heterostructure is shown in Fig. 1d.
It is interesting to find that the WSe2 QD strongly modifies electronic properties of the graphene above it.Figure 1e shows two representative scanning tunneling spectroscopy (STS) spectra of the graphene on and off the WSe2 QD.The STS, i.e., dI/dV, measurement of the graphene off the WSe2 QD shows a typical V-shaped spectrum profile of graphene with the Dirac point at

eV
Off D E  (n-doping).Whereas, spectrum of graphene on the WSe2 QD displays a series of resonance peaks with the Dirac point estimated as 0.3 eV WSe2 QD generates a circular p-n junction, i.e., a GQD, on graphene.The almost equally spaced peaks in the spectrum are the quasibound states confined in the GQD via the WGMs (9)(10)(11)(12)(13)(14)(15).Such a result is further confirmed by carrying out STS mapping at different resonance energies (Fig. 1f and Fig. S4).For the WGMs, the quasibound states, except the lowest one, display shell structures and are progressively closer to the GQD edge with increasing the energy (Fig. S7), as observed in our experiment.In high magnetic fields, these quasibound states are condensed into Landau levels of massless Dirac fermions in graphene monolayer.These experimental results indicate that the observed peaks in the GQD are due to the confinement of the massless Dirac fermions of graphene, rather than the electronic states hosted in edges of WSe2 (27).According to our experiment, the potential difference (U) on and off the GQDs shows positive correlation to the ratio () of the number of edge atoms to the number of inner atoms in the WSe2 QD (see Fig. S5).Therefore, the large potential difference on and off the GQD may arise from the edge of the WSe2 QD.The dangling bonds at the edge can significantly change the electronic property and work function of the WSe2 QD (28)(29)(30), generating the large circular electrostatic potential on graphene covering it.
To further explore electronic properties of the GQDs, we performed the radially dI/dV spectroscopic maps of several GQDs with different sizes and potentials, as shown in Fig. 2 (see insets of Fig. S6 for the STM images of the GQDs).By following the spatial dependence of global local density of states (LDOS) in the maps, it is interesting to note that the WSe2 QDs generate a Coulomb-like electrostatic potential: , where ħ is the reduced Planck constant, v F is the Fermi velocity, r is the distance from the center of GQD, Z   with α ~ 2.5 the fine structure constant of graphene ( 8), and r0 is the cutoff radius of Coulomb potential.We obtained different values of  and r0 for different GQDs.As shown in Fig. 2, the GQDs with different  exhibit quite different features of the quasibound states.For the  = 2 GQD, there is only one resonance peak at the center of the GQD (Fig. 2a, Top).Whereas, for the  = 4.3 GQD, besides several quasibound states confined via the WGMs at the edge of the GQD, there are three unequally spaced resonance peaks located at the center of the GQD (Fig. 2c, Top panels).To fully understand these unusual quasibound states, we numerically solved the problem for a Coulomb-like electrostatic potential with different values of  and r0 in the graphene monolayer (the values of  and r0 are extracted from our experimental results) (see supplementary information for the details).Bottom panels of Fig. 2 show the theoretical space-energy maps of the LDOS of the GQDs, which are in good agreement with the experimental results (see Fig. S6 for dI/dV spectra and corresponding simulated LDOS at different positions of the GQDs, see Fig. S8 for the spatial distribution of the quasibound states).According to our analysis, the resonance peaks located at the edge of GQDs arise from the quasibound states via the WGMs confinement.The energy levels of these quasibound states at the edge can be estimated as / F eff vR (here eff R is the effective radius of the GQD), as observed in our experiment (see Fig. S5d) and reported in previous studies (10,12,13).Whereas the energy levels of the quasibound states at the center of the GQD follow an exponential function 0 , where . This is a characteristic feature of the ACSs in the supercritical regime due to the Coulomb-like electrostatic potential (see Fig. S9 and the details of discuss.The lowest quasibound state at the center of the GQD is not arising from the WGM confinement, it is the ACS, see section 10 of supplementary information for discussion) (6)(7)(8).
Therefore, our experimental results, supported by our theoretical calculation, strongly indicate the coexistence of the WGMs and ACSs in the graphene/WSe2 heterostructure GQDs.In previous studies of the Coulomb impurity in graphene with a supercritical charge, only the ACSs are observed because of the small r0 ~ 0.5 nm (16,17).In this work, the Coulomb-like potential near the edges of the GQDs and the increase of about one order of magnitude of the r0 allow us to observe both the WGMs and the ACSs.
The Coulomb-like potential also strongly affects the electronic properties of the GQDs in the presence of magnetic fields.By applying a perpendicular magnetic field, we can observe well defined Landau levels (LLs) of massless Dirac fermions at positions away from the GQD (see Fig. S11a).When approaching the GQD, the Coulomb-like potential generates pronounced bending of the LLs (see Fig. S11a for the experimental result and theoretical simulation).Figure 3 shows radially spectroscopic maps around the  = 2.4 GQD in three different magnetic fields.Near the edge of the GQD, the bending of the LLs follows the Coulomb-like electrostatic potential.Inside the GQD, complex evolution of LDOS due to the transition from the confinement of the electrostatic potential to confinement of magnetic field is observed with increasing the magnetic field (see Fig. S12 for more experimental data).At B = 10 T, we can observe LLs inside the GQD.
However, the N = -1 LL is split into three peaks: two of them with higher energies are localized in the center of the GQD and the third one is mainly located at the edge of the GQD.The splitting does not occur in pairs and the energy spacing of the splitting is as large as ~40 meV (Fig. S13), which removes valley and spin splitting as the origin of the observed phenomenon.The splitting LLs should be attributed to lifting the orbital degeneracy of LLs, which can be understood by considering the quantum-mechanical electron motion in the presence of a magnetic field and a Coulomb-like electrostatic potential.Considering the effect of the magnetic field and the electrostatic potential, the equation thus reads:


where mN  ] independent of m because of translational invariance.The GQD lifts this orbital degeneracy m and the LLs are split into a series of sublevels, which exhibit similar behavior as that observed around charged impurities (17,26), due to the Coulomb-like electrostatic potential.However, previous experiments (17,26) in the presence of a magnetic field were limited to a charge impurity in the subcritical regime.Further, the small cutoff radius of a charge impurity prohibits to explore the evolution from the ACSs to the LLs in experiment.Such difficulties can be naturally overcome in the studied GQDs.
The detailed comparison between experiment and theory can be made by numerically solving the problem for two dimensional massless Dirac fermion of graphene monolayer in the presence of Coulomb-like electrostatic potential Vr  ( ) and a magnetic field B (see supplementary information for the details).The calculated radially LDOS maps in the different magnetic fields display that the orbital degeneracy is lifted, which is well consistent with our experimental results (Fig. 3).Based on the calculated results, we can identify the orbital states of the split -1 LL (Marked in Fig. 3).Thanks to the high-quality LLs in the GQD, m = -1 orbital state of -1 LL can be clearly identified and exhibits some characteristics distinguished from that observed in the subcritical regime (17,26).The most important feature is that the m = -1 orbital state can be viewed as the evolution of the ACS with increasing magnetic field.At zero magnetic field, the broad ACS is located at the center of the GQD and, interestingly, the narrower m = -1 orbital state appears in the same energy region in the presence of high magnetic field.Such a result indicates directly connection of the ACS and the lowest orbital state (m = -1) of the -1 LL.
To better explore the evolution of the ACS in the presence of magnetic fields, we summarize the measured LLs at the center of the  = 2.4 GQD as a function of the square root of the magnetic field B (red dots in Fig. 4, see Fig. S14a for the corresponding STS spectra).The evolution of LLs displayed a nonlinear dependence on the square root of the magnetic field, which is quite different from the feature of pristine graphene monolayer under magnetic field.The theoretical map of LDOS at the center of the  = 2.4 GQD is also plotted as a function of B (see supplementary information for the details, more calculated maps for different energies and magnetic fields are shown in Fig. S14), as shown in Fig. 4. With increasing magnetic field, the perturbed LLs (N = 0, N = -1, N = -2) display nonlinear dependence on B .At a higher B , the -1 LL and -2 LL are well distinctive, which split into low-energy orbital states (m = -1, m = 0).However, we did not observe the splitting of the -2 LL in the experiment, which is probably due to the large full width at half maximum (FWHM) of the LL peaks, prohibiting the observation of the splitting in the experiment.Furthermore, the ACS-R1 resonance is obvious at lower B , and is well connected to the m = -1 orbital state of -1 LL (see section 14 for the detailed discussion).Similarly, ACS-R2 resonance has the similar characteristic, connected to the m = -1 orbital state of -2 LL.However, such a feature is harder to be recognized in the experiment due to the broadening peak of the -2 LL.
Our experiments, complemented by theoretical calculations, explicitly demonstrated the existence of atomic collapse resonance effect in the presence of high magnetic fields and revealed the close connection between the ACS and the lowest orbital state (m = -1) of the LLs.
then the underlying Cu foil was etched away.The graphene/PMMA film was cleaned in deionized water for hours.
The WSe2 crystal was separated into thick-layer WSe2 sheets by traditional mechanical exfoliation technology and then transferred to is the vector potential(21,25,26), e is the electron charge.Due to the axial symmetry of the electrostatic potential in the GQD, we can describe the eigenstates by the orbital quantum number m (here, we neglect spin).In the absence of the GQD, the eigen-energies Nm E

2 8 8
mm  highly N-doped Si wafer [(100) oriented, 500 µm thick].We placed graphene/PMMA onto Si wafer which has been transferred with WSe2 sheets in advance.Finally, the PMMA was removed by acetone and then annealed in low pressure with Ar flow of 50 sccm and H2 flow of 50 sccm at ~300 °C for 1 hours.AFM, STM and STS Measurements.The topographical images are measured by atomic force microscope (AFM, Bruker Multimode 8) with a tapping mode.We employed the n-doped Si tip coated with Platinum-Iridium (Bruker, SCM-PIT-V2, frequency 50-100KHz, spring constant 1.5-6 N/m) to characterize WSe2 and graphene/WSe2 heterostructure samples.STM/STS measurements were performed in low-temperature (77 K for Fig.S5a and c, 4.2 K for Fig.S5b) and ultrahigh-vacuum (~10 -10 Torr) scanning probe microscopes [USM-1400 (77 K) and USM-1300 (4.2 K)] from UNISOKU.The tips were obtained by chemical etching from a Pt/Ir (80:20%) alloy wire.The differential conductance (dI/dV) measurements were taken by a standard lock-in technique with an ac bias modulation of 5 mV and 793 Hz signal added to the tunneling bias.

Fig. 1│The structures
Fig. 1│The structures and general dI/dV features of Graphene/WSe2 heterostructure.a, A

Fig. 2│Coexistence of
Fig. 2│Coexistence of WGMs confinement and ACSs in the GQDs.Top of a to c, The radially dI/dV

Fig. 3│Lifting the
Fig. 3│Lifting the orbital degeneracy of LLs in the GQD.Top panels of a to c, the radially dI/dV

Fig. 4│Evolution from
Fig. 4│Evolution from the ACSs to unusual LLs in the GQD.The measured LLs at the center of