Abstract
Electrostatic confinement of charge carriers in graphene is governed by Klein tunnelling, a relativistic quantum process in which particle–hole transmutation leads to unusual anisotropic transmission at p–n junction boundaries^{1,2,3,4,5}. Reflection and transmission at these boundaries affect the quantum interference of electronic waves, enabling the formation of novel quasibound states^{6,7,8,9,10,11,12}. Here we report the use of scanning tunnelling microscopy to map the electronic structure of Dirac fermions confined in quantum dots defined by circular graphene p–n junctions. The quantum dots were fabricated using a technique involving local manipulation of defect charge within the insulating substrate beneath a graphene monolayer^{13}. Inside such graphene quantum dots we observe resonances due to quasibound states and directly visualize the quantum interference patterns arising from these states. Outside the quantum dots Dirac fermions exhibit Friedel oscillationlike behaviour. Bolstered by a theoretical model describing relativistic particles in a harmonic oscillator potential, our findings yield insights into the spatial behaviour of electrostatically confined Dirac fermions.
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Main
Quantum confinement in graphene has previously been accomplished through lithographically patterned structures^{14,15,16,17}, graphene edges^{18}, and chemically synthesized graphene islands^{19,20,21,22}. These systems, however, are either too contaminated for direct wavefunction visualization or use metallic substrates that prevent electrostatic gating. Electron confinement in graphene has also been induced through high magnetic fields^{23} and supercritical impurities^{24}, but these methods are unwieldy for many technological applications. An alternative approach for confining electrons in graphene relies on using electrostatic potentials. However, this is notoriously difficult because Klein tunnelling renders electric potentials transparent to massless Dirac fermions at nonoblique incidence^{1,2,3,4,5}. Nevertheless, it has been theoretically predicted that a circular graphene p–n junction can localize Dirac electrons and form quasibound quantum dot states^{6,7,8,9,10,11}. A recent tunnelling spectroscopy experiment^{12} revealed signatures of electron confinement induced by the electrostatic potential created by a charged scanning tunnelling microscope (STM) tip. However, since the confining potential moves with the STM tip, this method allows neither spatial imaging of the resulting confined modes nor patterning control of the confinement potential.
Here we employ a new patterning technique that allows the creation of stationary circular p–n junctions in a graphene layer on top of hexagonal boron nitride (hBN). Figure 1a illustrates how stationary circular graphene p–n junctions are created. We start with a graphene/hBN heterostructure resting on a SiO_{2}/Si substrate. The doped Si substrate acts as a global backgate while the hBN layer provides a tunable local embedded gate after being treated by a voltage pulse from an STM tip^{13}. To create this embedded gate the STM tip is first retracted approximately 2 nm above the graphene surface and a voltage pulse of V_{s} = 5 V is then applied to the STM tip while simultaneously holding the backgate voltage to V_{g} = 40 V. The voltage pulse ionizes defects in the hBN region directly underneath the tip^{25} and the released charge migrates through the hBN to the graphene^{13}. This leads to a local spacecharge buildup in the hBN that effectively screens the backgate and functions as a negatively charged local embedded gate^{13} (using the opposite polarity gate voltage during this process leads to an opposite polarity space charge). Adjusting V_{g} afterwards allows us to tune the overall doping level so that the graphene is ndoped globally, but pdoped inside a circle centred below the location where the tip pulse occurred (it is also possible to control the charge carrier density profile as well as create opposite polarity p–n junctions by changing the V_{g} applied during the tip pulse). As shown schematically in Fig. 1b, the STM tip can then be moved to different locations to probe the electronic structure of the resulting stationary circular p–n junction.
To confirm that this procedure results in a circular p–n junction, we measured STM differential conductance (dI/dV_{s}) as a function of sample bias (V_{s}) on a grid of points covering the graphene area near a tip pulse. The Dirac point energy, E_{D}, was identified at every pixel, allowing us to map the charge carrier density, n, through the relation n(x, y) = −(sgn(E_{D})E_{D}^{2})/(π(ℏv_{F})^{2}), where v_{F} = 1.1 × 10^{6} m s^{−1} is the graphene Fermi velocity and ℏ is the reduced Planck constant. Figure 1c shows the resulting n(x, y) for a tip pulse centred in the top right corner (the carrier density n can be adjusted by changing V_{g}). The interior blue region exhibits positive charge density (ptype) whereas the red region outside has negative charge density (ntype).
To spatially map the local electronic properties of such circular p–n junctions, we examined a rectangular sector near a p–n junction, as indicated in Fig. 2a. Figure 2b shows a topographic image of the clean graphene surface in this region. A 2.8 nm moiré pattern (corresponding to a 5° rotation angle between graphene and hBN) is visible^{26,27} and the region is seen to be free of adsorbates. A dI/dV_{s} map of the same region (Fig. 2c) reflects changes in the local density of states (LDOS) caused by the spatially varying charge density distribution. Since the p–n junction centre is stationary, we are able to move the STM tip to different locations inside and outside the p–n junction to spatially resolve the resulting electronic states. Figure 2d–g shows d^{2}I/dV_{s}^{2}(V_{g}, V_{s}) plots at four different locations, as denoted in Fig. 2c. We plot the derivative of dI/dV_{s} with respect to V_{s} to accentuate the most salient features, which are quasiperiodic resonances that disperse to lower energies with increasing V_{g} (see Supplementary Section 2 for dI/dV_{s} sweeps before differentiation). The energies of the observed resonances are seen to evolve as , where V_{CNP} is the local charge neutrality point, as expected for graphene’s relativistic band structure. We see that the energy spacing between observed resonances (Δɛ) decreases as we move away from the p–n junction centre until the resonances disappear outside. For example, Δɛ is 29 ± 2 mV at the centre, 16 ± 2 mV at 50 nm from the centre, and 13 ± 2 mV at 100 nm from the centre (for V_{g} = 32 V). A similar trend is also observed for p–n junctions that are ndoped in the centre and pdoped outside (Supplementary Section 3).
We have imaged these electronic states both inside and outside of circular p–n junctions. The dI/dV_{s} maps in Fig. 3a, b show eigenstate distributions mapped at two different energies within the same section of a circular p–n junction (similar to the boxed region of Fig. 2a, but with opposite heterojunction polarity). Circular quantum interference patterns resulting from confined Dirac fermions are clearly observed within the junction boundary, as well as scattering states exterior to the boundary. The junction boundary is demarcated by a dark band (low dI/dV_{s}) in the middle of each dI/dV_{s} map (and further marked by a dashed line). Comparing the overall spatial locations of the nodes and antinodes, the two eigenstate distributions in Fig. 3a, b are clearly different (for example, one has a node at the origin, whereas the other exhibits a central antinode). Figure 4a shows a more complete mapping of the energydependent eigenstates (within a p–n junction of the same polarity as Fig. 2a) along a line extending from the centre (left edge) to a point outside of the p–n junction (right edge) at a gate voltage of V_{g} = 32 V. The data are plotted as d^{2}I/dV_{s}^{2}(r, V_{s}) (where r is the radial distance from the centre) to accentuate the striking oscillatory features (see Supplementary Section 2 for dI/dV_{s}(r, V_{s}) before differentiation). The energy level structure and interior nodal patterns are clearly evident.
Our observations can be explained by considering the behaviour of massless Dirac fermions in response to a circular electrostatic potential. Due to Klein tunnelling, a graphene p–n junction perfectly transmits quasiparticles at normal incidence to the boundary, but reflects them at larger angles of incidence^{1,4,5}. In a potential well with circular symmetry, electrons with high angular momenta are obliquely incident on the barrier and are internally reflected, thus leading to particle confinement and the formation of quasibound quantum dot states^{7,8,9,10,11,12}. As angular momentum is increased, electrons are repelled from the centre of the potential by the centrifugal barrier, leading to an increase in the number of dI/dV_{s} resonances that should be observable in spectroscopy measured away from the centre^{28}. This is consistent with our observation that the apparent energy spacing between resonances (Δɛ) at the centre (Fig. 2d) is approximately double the apparent energy spacing at a point 100 nm away from the centre (Fig. 2f). Scattered quasiparticles (with nonzero angular momenta) external to the potential boundary contribute to Friedel oscillations that radiate outwards, as seen in Fig. 3. A circular graphene p–n junction with an ndoped interior thus acts as a quantum dot for electronlike carriers and a quantum antidot for holelike carriers (as in Fig. 3), whereas the reverse is true for p–n junctions of opposite polarity (as in Figs 2 and 4).
This qualitative picture can be confirmed by comparing the experimental results to a model based on the twodimensional massless Dirac Hamiltonian, , where U(r) is a scalar potential and σ = (σ_{x}, σ_{y}) are the pseudospin Pauli matrices. Since we are interested in the lowenergy eigenstates of the confinement potential, we use a parabolic model U(r) = −κr^{2} (that is, the lowest order approximation). The curvature of the potential, κ = 6 × 10^{−3} meV nm^{−2}, was extracted from measurements of the spatially dependent Dirac point energy (Supplementary Section 1). We solved the Dirac equation to obtain the eigenstates for Dirac fermions in this confinement potential (see Methods).
Figure 4b shows the results of our calculations in a plot of ∂LDOS/∂ɛ, the energy derivative of the LDOS, which corresponds to the experimental quantity d^{2}I/dV_{s}^{2}. The resulting eigenstate distribution (Fig. 4b) closely resembles the experimental eigenstate distribution (Fig. 4a). Both have a characteristic parabolic envelope due to the confinement potential, as well as a complex set of interior nodal patterns. The characteristic energy spacing seen experimentally is in good agreement with the characteristic energy scale ɛ^{∗} = (ℏ^{2}v_{F}^{2}κ)^{1/3} ≈ 15 meV that arises from the theoretical model.
Further insight into the nature of the observed resonances can be gained by directly comparing constantenergy experimental dI/dV_{s} linecuts (Fig. 4c) to the modulus square of the simulated quantum dot wavefunctions (Fig. 4d). Here it is useful to label the confined states by a radial quantum number n = 0,1,2, … and an azimuthal quantum number m = ±(1/2), ± (3/2), …, that is, HΨ_{n, m} = ɛ_{n, m}Ψ_{n, m}. To understand the experimentally observed behaviour, we note two important properties of the eigenstates Ψ_{n, m}. First, although each probability distribution Ψ_{n, m}^{2} features n + 1 maxima, most of the weight is concentrated in the first maximum. The position of this maximum is pushed further from the centre for larger values of m (Fig. 4d). Second, for massless Dirac fermions confined by a quadratic potential, we observe a nearperfect energy alignment of the states ɛ_{n, m}, ɛ_{n−1, m+2}, …at low quantum numbers, indicating an approximate degeneracy. This degeneracy explains why different resonances originating from different Ψ_{n, m} states form the horizontal rows seen in Fig. 4a, b (which are not perfectly horizontal because the degeneracy is not perfect). Combining these two observations, we are able to attribute each experimental dI/dV_{s} peak in Fig. 4c to a different Ψ_{n, m} state, wherein each eigenstate contributes most of its spectral weight to a single energy and radial position.
In addition to providing insight into the spatial and spectral distribution of the Ψ_{n, m} states, our simulations also explain other key aspects of the experimental data. In particular, the resonances in our simulation have finite widths, originating from Klein tunnelling of confined states into the Dirac continuum. The widths of these resonances lie within the range 4 meV to 10 meV for both the experimental data and the theoretical simulation (Supplementary Section 6). Furthermore, our simulation also explains the striking observation that the apparent energy spacing for the resonances close to the centre is nearly twice as large as the spacing away from the centre (see Fig. 2d). This occurs because only the lowest angular momentum states, m = ±1/2, have appreciable wavefunction density at the origin, whereas for all other m values the Ψ_{n, m} states contribute predominantly to offcentred measurements.
In conclusion, we have spatially mapped the electronic structure inside and outside of highly tunable quantum dots formed by circular graphene p–n junctions. In contrast to conventional semiconductor quantum dots, these new graphene quantum dots are fully exposed and directly accessible to realspace imaging tools. The techniques presented here might be extended to more complicated systems such as multiple quantum dots^{29,30} with variable coupling and arbitrary geometries.
Note added in proof: After acceptance of this paper, we became aware of a related manuscript (ref. 31) showing similar results to this work.
Methods
Sample fabrication.
We fabricated our samples using a transfer technique^{32} that uses 60–100nmthick hexagonal boron nitride (hBN) crystals (synthesized by Taniguchi and Watanabe^{33}) and 300nmthick SiO_{2} as the dielectric for electrostatic gating. Singlelayer graphene was mechanically exfoliated from graphite and deposited onto methyl methacrylate (MMA) before being transferred onto hBN previously exfoliated onto a heavily doped SiO_{2}/Si wafer. The graphene was electrically grounded through a Ti (10 nm)/Au (100 nm) electrode deposited via electron beam evaporation using a shadow mask. Devices were annealed in Ar/H_{2} gas at 350 °C before being transferred into our Omicron ultrahigh vacuum (UHV) lowtemperature scanning tunnelling microscope (STM). A second anneal was performed overnight at 250–400 °C and 10^{−11} torr.
STM and spectroscopy measurements.
STM measurements were performed at T = 4.8 K with a platinum iridium STM tip calibrated against the Au(111) Shockley surface state. STM topographic and dI/dV_{s} images were obtained at constant current with sample bias V_{s}, defined as the negative of the voltage applied to the STM tip with respect to the grounded graphene sample. A voltage V_{g} is applied to Si to electrostatically gate graphene. Scanning tunnelling spectroscopy (STS) measurements were performed by lockin detection of the a.c. tunnel current induced by a modulated voltage (1–6 mV at 613.7 Hz) added to V_{s}, while dI/dV_{s}(V_{g}, V_{s}) and dI/dV_{s}(r, V_{s}) measurements were acquired by sweeping V_{s} (starting from a fixed set of initial tunnelling parameters) and then incrementing V_{g} for dI/dV_{s}(V_{g}, V_{s}) and r for dI/dV_{s}(r, V_{s}). Measurements were restricted to −0.1 eV < V_{s} < 0.1 eV to avoid energy broadening induced by phononassisted inelastic tunnelling^{34}. All d^{2}I/dV_{s}^{2} figures are numerical derivatives of dI/dV_{s} with respect to V_{s}. These results were reproduced with numerous STM tips on more than 30 p–n junction structures.
Creation of graphene p–n junction.
The STM tunnelling bias and current are set to V_{s} = −0.5 V and I = 0.5 nA, respectively. To create a circular graphene p–n junction that is pdoped (ndoped) at the centre, we set V_{g} = 40 V (V_{g} = −40 V). The STM feedback loop is opened, and the STM tip is withdrawn by Δz ∼ 1.5–2 nm. The sample bias is increased to V_{s} = 5 V for 1 min. The sample bias is then decreased to V_{s} = −0.5 V.
Theoretical modelling.
The eigenstates of the Dirac equation are obtained by solving , where U(r) is the electrostatic potential and p = −iℏ∇_{r}. Because U(r) is radially symmetric, we use the polar decomposition ansatz
where m is a halfinteger. By inserting the ansatz into the eigenvalue equation we obtain
To make direct connection with the STS measurements we calculate the local density of states LDOS(ɛ) as a function of r. The LDOS can be written as the sum of mstate contributions, LDOS(ɛ) = ∑_{m}LDOS_{m}(ɛ), with
where ν labels the radial eigenstates for fixed m and 〈u_{v}(r)^{2}〉_{λ} = ∫_{0}^{∞}dr′u_{ν}(r′)^{2}{\text{e}}^{\u2013{(r\u2013{r}^{\prime})}^{2}/2\lambda} represents a spatial average of the wavefunction centred at r with a Gaussian weight λ/r^{∗} = 0.01. To solve the radial Dirac equation, we use the finite difference method discretized in 1,200 lattice sites in the interval 0 < r < L, with a large repulsive potential at r = L = 12r^{∗}. Spurious states arising from the finite potential jumps at the boundaries, localized within a few lattice sites of r = 0 and r = L, are excluded. We sum over eigenstates with azimuthal quantum numbers −401/2 ≤ m ≤ 401/2, which is sufficient to accurately represent the states in the energy range of interest. The delta function is approximated as a Lorentzian with width 0.3ɛ^{∗}, which is sufficiently small that the intrinsic linewidths of the quantum dot eigenstates are preserved.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.
Change history
30 September 2016
In the version of this Letter originally published, in the Acknowledgements section, the descriptions of the funding sources for STM measurement and instrumentation and graphene characterization were incorrect and should have read: 'This research was supported by the sp2 program (KC2207) (STM measurement and instrumentation) funded by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the US Department of Energy under Contract No. DEAC0205CH11231. For the graphene characterization we used the Molecular Foundry at LBNL, which is funded by the Director, Office of Science, Office of Basic Energy Sciences, Scientific User Facilities Division, of the US Department of Energy under Contract No. DEAC0205CH11231.' This has now been corrected in all versions of the Letter.
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Acknowledgements
The authors thank A. N. Pasupathy, J. A. Stroscio, N. B. Zhitenev and J.Wyrick for stimulating discussions. This research was supported by the sp^{2} program (KC2207) (STM measurement and instrumentation) funded by the Director, Office of Science, Office of Basic Energy Sciences Materials Sciences and Engineering Division, of the US Department of Energy under Contract No. DEAC0205CH11231. For the graphene characterization we used the Molecular Foundry at LBNL, which is funded by the Director, Office of Science, Office of Basic Energy Sciences, Scientific User Facilities Division, of the US Department of Energy under Contract No. DEAC0205CH11231. Support was also provided by National Science Foundation award DMR1206512 (device fabrication, image analysis). L.S.L. was supported, in part, by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR1231319 (theoretical modelling). D.W. was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program, 32 CFR 168a.
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J.V.Jr, D.W. and J.L. conceived the work and designed the research strategy. D.W., J.L., J.F.R.N. and J.V.Jr performed data analysis. S.K., J.V.Jr and A.Z. facilitated sample fabrication. J.L., D.W. and J.V.Jr carried out STM/STS measurements. K.W. and T.T. synthesized the hBN samples. J.F.R.N. and L.S.L. performed theoretical calculations. M.F.C. supervised the STM/STS experiments. J.L., D.W., J.V.Jr, J.F.R.N., L.S.L. and M.F.C. cowrote the manuscript. D.W. and M.F.C. coordinated the collaboration. All authors discussed the results and commented on the paper.
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Lee, J., Wong, D., Velasco Jr, J. et al. Imaging electrostatically confined Dirac fermions in graphene quantum dots. Nature Phys 12, 1032–1036 (2016). https://doi.org/10.1038/nphys3805
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DOI: https://doi.org/10.1038/nphys3805
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