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Spatial and Magnetic Confinement of Massless Dirac Fermions

The massless Dirac fermions and the ease to introduce spatial and magnetic confinement in graphene provide us unprecedented opportunity to explore confined relativistic matter in this condensed-matter system. Here we report the interplay between the confinement induced by external electric fields and magnetic fields of the massless Dirac fermions in graphene. When the magnetic length lB is larger than the characteristic length of the confined electric potential lV, the spatial confinement dominates and a relatively small critical magnetic field splits the spatial-confinement-induced atomic-like shell states by switching on a pi Berry phase of the quasiparticles. When the lB becomes smaller than the lV, the transition from spatial confinement to magnetic confinement occurs and the atomic-like shell states condense into Landau levels (LLs) of the Fock-Darwin states in graphene. Our experiment demonstrates that the spatial confinement dramatically changes the energy spacing between the LLs and generates large electron-hole asymmetry of the energy spacing between the LLs. These results shed light on puzzling observations in previous experiments, which hitherto remained unaddressed.

Graphene, with massless Dirac fermions as the low-energy quasiparticles, provides a unique solid-state system to explore many oddball effects of quantum-relativistic matter [1][2][3][4][5]. More importantly, the one-atom-thick structure of graphene is uniquely amenable to introduce electric and magnetic fields to tune the behaviors of the relativistic fermions. For example, recent experiments demonstrated the confinement of the massless Dirac fermions in graphene into quasibound states via whispering-gallery modes by using circular electronic junctions (graphene quantum dot) [6][7][8][9][10][11], which enable the demonstration of the so-called Klein tunneling [12] at the atomic scale. By further introducing the magnetic fields, it is interesting to find that a small critical magnetic field could lift the angular momentum quantum numbers of the quasibound states by switching on a  Berry phase [13] and a large magnetic field can reshape the electric potentials into wedding cake-like structures through electron-electron interactions [14]. These results indicate that the magnetic fields can strongly modify the effects of the spatial confinements and realize more exotic electronic properties in graphene.
In this Letter, we use scanning tunneling microscopy (STM) to directly probe the interplay between the spatial and magnetic confinements of massless Dirac fermions in graphene. In our experiment, a circular electronic junction confirms the massless Dirac fermions into atomic-like shell states and a relatively small critical magnetic field splits the quasibound states with different angular momentum quantum numbers by switching on a  Berry phase. Not only the spatial confinement is modified by the magnetic field, our result indicates that the spatial confinement also can strongly affect the effects of the magnetic confinement. When the magnetic length lB changes from lB > lV to lB < lV (lV, the characteristic length of the confined electric potential), the transition from spatial confinement to magnetic confinement occurs and the quasibound states gradually condense into Landau levels (LLs) of the Fock-Darwin states in graphene [15]. Our experiments demonstrate that the spatial confinement can change the energy spacing between the LLs and introduce large electron-hole asymmetry between the energy spacing of the LLs.
Our experiments were carried out on monolayer graphene on a 0.7% Nb-doped SrTiO3(001) substrate by using a high-magnetic-field STM at T = 4.2 K (see Supplemental Material for details [16]). The work function difference between the metal STM tip and graphene leads to an electric field acting on graphene and results in the confining potential on the hole, i.e., a circular electronic p-n junction, in graphene below the tip [6,13,14,[17][18][19][20][21][22], as schematically shown in Fig. 1(a). During the STM measurements, the confining potential also depends on the tunneling bias between graphene and the probe tip Vb, as demonstrated explicitly in Ref. [6]. This allows us to tune the spatial confining potential by slightly varying the doping of the graphene monolayer (see Supplemental Material Fig. S2 for details [16]). Figure 1 shows two representative scanning tunneling spectroscopy (STS) spectra recorded in two different regions of the monolayer graphene (labelled as D1 and D2 respectively).
Due to the different charge transfer between the substrate and the graphene, the Dirac points ED in the D1 and D2 regions are at about (147  5) meV and (74  5) meV, respectively. Although the work function difference between the metal STM tip and graphene can generate band bending of graphene beneath the tip in both the regions, the dependence of the band bending on the Vb makes us almost unable to detect the effect of the confining potential in the zero-magnetic-field STS measurements in the D2 region (see Supplemental Material for details [16]). Therefore, no resonance can be measured in tunneling spectrum of the D2 region [bottom panel of Fig. 1 Whereas, clear whispering-gallery resonances, corresponding to tunneling into the circular p-n junction modes at energy  = μ0 +eVb (μ0 is the local Fermi level) [6], are observed in the tunneling spectrum of the D1 region [top panel of Fig. 1(b)]. The energy spacing between the quasibound states is consistent well with that observed previously in the tip-induced graphene resonators [6]. The ability to tune the spatial confining potential allows us to systematically explore the interplay between the spatial and magnetic confinements of the massless Dirac fermions in monolayer graphene, as illustrated below.
A prominent effect of the magnetic fields on the quasibound states of the massless Dirac fermions is the  shift of the Berry phase of the confined quasiparticles [13,23]. To further understand our experimental result, we theoretically calculated local density of states (LDOS) in the center of the graphene resonator as a function of magnetic fields and energy. In our calculations, the following Hamiltonian is adopted Here half-integer number. The LDOS of the graphene resonator can be expressed as the sum of -state contributions [6,23]. The shift of Berry phase and the sudden decrease of energy, as discussed above, can be explained using the Einstein-Brillouin-Keller rule [24][25][26] starting from the Hamiltonian here (see Supplemental Material for details [16]). Therefore, for the case that the lV is smaller than the lB, the spatial confinement dominates and we can observe the quasibound states, as shown in Fig. 1(e). When the lB becomes smaller than the lV, a transition from spatial confinement to magnetic confinement occurs and the LLs of massless Dirac fermions begin to be observed. In the D1 region, the characteristic Landau quantization of monolayer graphene is observed for B > 1.4 T, which also agrees well with the estimated lV above. In the D2 region, the effect of the tip-induced confining potential in the zero-field measurement is almost negligible. Therefore, we observe the characteristic LLs of monolayer graphene for the magnetic field on the order of 0.1 T [ Fig. 1(f), left]. Such a feature is also reproduced quite well in our calculation of the Landau quantization in monolayer graphene when we consider a very weak confining potential with  = 9 eV/μm 2 and ED = 74 meV, as well as vF = 1.1×10 6 m/s. [ Fig. 1(f), right, see Supplemental Material [16] for details]. the LLs for holes is systematically larger than that for electrons, i.e., the deduced "Fermi velocity" for holes is larger than that for electrons, because that the tip-induced potential is a confining potential for the hole but it is a repulsive potential for the electron. In the D2 region, the electron-hole asymmetry, as defined by / he FF  , could be as large as 110%. In the D1 region, the electron-hole asymmetry increases to about 130% due to the relatively stronger tip-induced potential. Generally, the electron-hole asymmetry is much more pronounced in small magnetic fields. Such a result is quite reasonable because that the spatial confinement is expected to change the spacing of the LLs, especially for the case that the lV is comparable to the lB. In previous STM studies of the Landau quantization in monolayer graphene, similar large electron-hole asymmetry has been frequently observed and is mainly attributed to the enhanced next-nearest-neighbor hopping [19,20,[24][25][26][27]. To account for the electron-hole asymmetry observed in the experiment, an unreasonable large next-nearest-neighbor hopping ~ 1 eV is needed to be used as a fitting parameter in the tight-binding model. 8 By taking into account the effects of the tip-induced potential, our calculation can reproduce quite well the above experimental phenomena (Figs. 2 and 3), indicating that the large electron-hole asymmetry observed in STM studies is mainly arising from the tip-induced potential. In our experiment, the tip-induced potential promotes and resists the formation of the Landau quantization of the hole and electron respectively, resulting in the giant electron-hole asymmetry.
To further explore the effects of spatial confinement on the magnetic confinement of the massless Dirac fermions, we calculated the Landau quantization of monolayer graphene with different strengths of the confining potential (see Supplemental Material [16] for details). In summary, the interplay between the spatial and magnetic confinements of massless Dirac fermions in graphene is systematically explored in this work. Our result demonstrates that a relatively small critical magnetic field lifts the angular momentum degeneracy of the spatial-confined quasibound states by switching on a  Berry phase. In turn, the spatial potential also strongly modifies the magnetic-field-induced Landau quantization and generates giant electron-hole asymmetry in the energy spacing between the LLs.