## Flicking the Berry phase switch

When an electron completes a cycle around the Dirac point (a particular location in graphene's electronic structure), the phase of its wave function changes by π. This so-called Berry phase is tricky to observe directly in solid-state measurements. Ghahari *et al.* built a graphene nanostructure consisting of a central region doped with positive carriers surrounded by a negatively doped background. Scanning tunneling spectroscopy revealed sudden jumps in conductivity as the external magnetic field was increased past a threshold value. The jumps occurred when electron orbits started encompassing the Dirac point, reflecting the switch of the Berry phase from zero to π. The tunability of conductivity by such minute changes in magnetic field is promising for future applications.

*Science*, this issue p. 845

## Abstract

The phase of a quantum state may not return to its original value after the system’s parameters cycle around a closed path; instead, the wave function may acquire a measurable phase difference called the Berry phase. Berry phases typically have been accessed through interference experiments. Here, we demonstrate an unusual Berry phase–induced spectroscopic feature: a sudden and large increase in the energy of angular-momentum states in circular graphene p-n junction resonators when a relatively small critical magnetic field is reached. This behavior results from turning on a π Berry phase associated with the topological properties of Dirac fermions in graphene. The Berry phase can be switched on and off with small magnetic field changes on the order of 10 millitesla, potentially enabling a variety of optoelectronic graphene device applications.

Geometric phases are a consequence of a phenomenon that can be described as “global change without local change.” A well-known classical example is the parallel transport of a vector around a path on a curved surface, which results in the vector pointing in a different direction after returning to its origin (Fig. 1A). Extending the classical phenomenon to the quantum realm by replacing the classical transport of a vector by the transport of a quantum state gave birth to the Berry phase, which is the geometric phase accumulated as a state evolves adiabatically around a cycle according to Schrödinger’s equation (*1*–*5*). Since its discovery (*1*), nontrivial Berry phases have been observed in many quantum systems with internal degrees of freedom, such as neutrons (*6*), nuclear spins (*7*), and photons (*8*, *9*), through a Berry phase–induced change in quantum interference patterns. Quantum systems in which a Berry phase alters the energy spectrum are comparatively rare; one example is graphene, where massless Dirac electrons carry a pseudospin ½, which is locked to the momentum . Because of the spin-momentum locking, the Berry phase associated with the state can take only two values, 0 or π, which gives rise to the unconventional “half-integer” Landau-level structure in the quantum Hall regime (*10*–*12*). Yet, in most cases studied to date, the Berry phase in graphene has played a “static” role because controlling the trajectories (and hence the phase of state ) of graphene electrons is experimentally challenging. Recently introduced graphene electron resonators (*13*–*15*) enable exquisite control of the electron orbits by means of local gate potentials and offer a unique opportunity to alter and directly measure the Berry phase of electron orbital states.

Here we report the control of the Berry phase of Dirac particles confined in a graphene electron resonator using a weak magnetic field, as recently suggested by theory (*16*). In this approach, a magnetic field enables fine control of the evolution of around the Dirac point for individual resonator states (Fig. 1B). A variation in the Berry phase, which is accumulated during the orbital motion of the confined states, can be detected from changes in the energy dispersion of electron resonances. For this reason, we use scanning tunneling spectroscopy (STS) to directly measure the resonator-confined electronic states, giving direct access to the shifts in the quantum phase of the electronic states.

Graphene resonators confine Dirac quasiparticles by Klein scattering from p-n junction boundaries (*13*–*25*). Circular graphene resonators composed of p-n junction rings host the whispering gallery modes analogous to those of acoustic and optical classical fields (*13*–*15*). These circular resonators can be produced by a scanning tunneling microscope (STM) probe in two different ways: (i) by using the electric field between the STM probe tip and graphene, with the tip acting as a movable top gate (*13*), or (ii) by creating a fixed charge distribution in the substrate, generated by strong electric field pulses applied between the tip and graphene/boron nitride heterostructure (*14*). Both methods are used in our study of Berry phase switching of resonator states.

Measurements were performed in a custom- built cryogenic STM system (*26*) on two fabricated graphene/hexagonal boron nitride (hBN)/SiO_{2} heterostructure devices, designed specifically for STM measurements (*27*). The field-induced switching of the graphene resonator states was initially observed in the movable tip–induced p-n junction resonators defined by the STM tip gating potential (*13*) [section II of (*27*)]. Subsequently, we created fixed p-n junction resonators, allowing us to study the spectroscopic properties and field dependence in a more controlled manner without a varying p-n junction potential; this report focuses mainly on the data obtained in this latter way. The fixed resonators display a pattern of Berry phase switching of the resonator modes that agrees with the one seen in the movable p-n junctions, demonstrating that different methods for creating the circular graphene resonators in separate devices lead to similar results.

Fixed circular-shaped p-n junctions were created by applying an electric field pulse between the STM probe and the graphene device to ionize impurities in the hBN insulator, as described in (*28*). This method creates a stationary screening charge distribution in the hBN insulating underlayer, resulting in a fixed circular doping profile in the graphene sheet (Fig. 2, A and B) (see fig. S10 for schematic of the method). We probed the quantum states in the graphene resonator by measuring the tunneling differential conductance, as a function of tunneling bias,, back gate potential, , spatial position, , and magnetic field, . The quasibound resonances, originating from Klein scattering at the p-n ring, are seen in measurements made at as a function of radial distance (Fig. 2D). The eigenstates in Fig. 2D form a series of resonant levels vertically distributed in energy and are seen to exist within an envelope region of the confining p-n junction potential outlined by a high-intensity state following the p-n junction profile. Because of rotational symmetry, the corresponding quantum states are described by radial quantum numbers, *n*, and angular momentum quantum numbers, *m* [section I of (*27*)]. Only degenerate states with have nonzero wave function amplitude in the center of the resonator, as shown by the calculated wave functions in Fig. 2C, and dominate the measurements at in Fig. 2D (*13*–*15*). Higher–angular momentum states are observed off center in the spatial distribution of the resonator eigenstates in Fig. 2D but can be difficult to distinguish because of overlapping degenerate levels (*14*, *15*). Several calculated wave functions and corresponding eigenstates are indicated by color-coded circles in Fig. 2, C and D, respectively, to illustrate some patterns of the states in the spectroscopic map. States with a specific radial quantum number *n* follow arcs trailing the parabolic outline of the confining potential in Fig. 2D [see fig. S1 for the enumeration of the various (*n*,* m*) quantum states].

We probe the magnetic field dependence of the resonator states (Fig. 3A) for the states *n* = 1 to *n* = 5 corresponding to the energy range indicated by the yellow dashed line in Fig. 2D. We see the degenerate states in the center of the map at ** B** = 0, corresponding to the states seen in the center of Fig. 2D at . However, as the magnetic field is increased, new resonances suddenly appear between the

*n*th quantum levels, where none previously existed. The spacing between the new magnetic field–induced states, δε

*, is about one-half the spacing, Δε, at . A magnified view of the map around the*

_{m}*n*= 4 level (Fig. 3C) shows the appearance of new resonances switching on at a critical magnetic field, T. As explained below, the switching transition corresponds to the sudden separation of the sublevels, which is very sharp (see Fig. 3D, and see fig. S11 for more detail). This energy separation, on the order of 10 meV at , is much larger than other magnetic field splittings, i.e., Zeeman splitting (, with the Bohr magneton) or orbital effects [; see (

*16*)]. As we discuss below, these results correspond to the energy shift of particular quantum states, which suddenly occurs owing to the switching on of a π Berry phase when a weak critical magnetic field is reached.

The origin of the sudden jump in energy observed in the experiment (Fig. 3A) can be seen from the Bohr-Sommerfeld quantization rule, which determines the energy levels :(1)where the contour *C _{R}* is defined in Fig. 1. Here and are the canonical coordinates and momenta, respectively; is Planck’s constant divided by 2π; is the Berry phase accumulated in each orbital cycle; and γ is a constant, the so-called Maslov index (

*5*). The orbits are obtained from the semiclassical Hamiltonian , with the confining potential and the kinetic momentum (

*e*is the electron charge and the vector potential). At zero magnetic field, states with opposite angular momenta are degenerate, and their orbits are time-reversed images of one another (Fig. 1C, left). When a small magnetic field is turned on, the Lorentz force bends the paths of the and charge carriers in opposite directions, breaking the time-reversal symmetry and slightly lifting the orbital degeneracy. Crucially, at a critical magnetic field

*B*

_{c}, the orbit with angular momentum antiparallel to the field is twisted by the Lorentz force into a “skipping” orbit with loops (Fig. 1C, right). At this transition, the momentum-space trajectory, defined below, encloses the Dirac point. At this instant, the Berry phase discontinuously jumps from to π. As a result, states with , which are degenerate at , are abruptly pulled apart by half a period. The sudden change in Berry phase therefore gives rise to a (giant) energy shift equal to half the Bohr-Sommerfeld level spacing.

The intuitive picture described above can be made rigorous by using the Einstein-Brillouin-Keller (EBK) quantization (*29*, *30*). Besides being more rigorous, EBK quantization facilitates visualization of the trajectory of along a semiclassical orbit, particularly because the orbits are quasiperiodic. In central force motion, the particle’s orbit takes place in an annulus between the two classical turning points, and because the momentum in this annulus is two-valued, one can define a torus on which the momentum is uniquely determined [section I of (*27*)]. The EBK quantization rules are formed by evaluating along the two topologically distinct loops on this torus, and (Fig. 1D). The EBK rule along gives the half-integer quantization of angular momentum; that along (Eq. 1) determines the energy levels for a given angular momentum. The Berry phase term in Eq. 1 is determined by the winding number of about the origin, evaluated on Below , the azimuthal component of has the same sign along (Fig. 1E, left): The corresponding **-**space loop (Fig. 1B, blue curve) does not enclose the origin and the Berry phase is zero. Above , however, has a sign change along (Fig. 1E, right): The loop encircles the Dirac point and the Berry phase is π (Fig. 1B, red curve). Figure 1B provides an intuitive visualization of the switching mechanism: The changing magnetic field shifts the -space contour, and at it slips it over the Dirac cone apex, instantly changing the right side of Eq. 1 by π and shifting the energy levels accordingly (see movie S1).

The semiclassical picture additionally allows us to estimate the strength of the critical field necessary to switch the Berry phase (*16*). The value of , which is sensitive to the confining potential profile, is obtained by finding the field strength necessary to bend the electron orbit into a skipping orbit at the outer return point , thus resulting in zero azimuthal momentum: . For a parabolic potential , which accurately describes the low-energy resonances (*16*), one obtains a simple expression:

(2)where ε is the energy of the orbit. Using obtained from a parabolic fitting of the potential profile (see Fig. 2D) and (relative to the position of the Dirac point), which corresponds to the state in Fig. 3A, we find , in excellent agreement with the magnetic field values measured in the experiment.

A detailed comparison between experiment and theory can be made by solving the two-dimensional (2D) Dirac equation describing graphene electrons in the presence of a confining potential and a magnetic field , . Following the semiclassical discussion above, we use a simple parabolic potential model , with , to fit the experimental data. The calculated local density of states (LDOS) at the center of the resonator [see section I of (*27*) for details] is in good agreement with the experiment (Fig. 3, A and B) and exhibits the half-period jumps between time-reversed states at similar magnetic field values.

Equation 2 also predicts a higher critical field for larger *m* states, as a larger Lorentz force is needed to induce skipping orbits. These larger angular momentum states have zero wave function weight at (see Fig. 2C) but can be probed at positions away from the center of the resonator (*14*, *15*). Figure 4A shows the spectral conductance map measured at a position 70 nm away from the center of the p-n junction resonator (see fig. S12 for additional measurements off center), and Fig. 4B shows the calculated LDOS at the same position. Much more complex spectral features are seen in comparison with the on-center measurement (Fig. 3A), which is sensitive only to the states. First, at , twice as many resonances are observed within the same energy range when compared to Fig. 3A. This confirms that additional states are contributing to the STM maps. Second, both the theory and experimental maps exhibit a “staircase” pattern. Such an arrangement results from an overlap between the field-jumped *m* state and the next higher non-jumped *m* state, which are nearly degenerate in energy. For example, at positive fields, the field-jumped state overlaps in energy with the non-jumped state, giving rise to a strong intensity at this energy. Figure 4, C and D, illustrates this effect by showing separately the and contributions to the total density of states, respectively; is indicated with a dashed line. In a similar fashion, this behavior continues for all adjacent *m* levels, resulting in the series of staircase steps in the measured and calculated conductance maps in Fig. 4, A and B. Because the states jump by half of the energy spacing, the staircase patterns can visually form upward- and downward-looking lines, depending on subtle intensity variations at the transitions. For instance, whereas Fig. 4A shows mostly downward staircase patterns, other measurements (fig. S12) show both down and upward connections at the transitions; the parabolic potential model, Fig. 4B, shows an upward staircase. These behaviors depend on subtle effects, such as the potential shape in the off-center measurement, which are not relevant for our main discussion. More importantly we plot, as a guide to the eye, in Eq. 2 (dashed lines) for different values of , showing excellent agreement with the semiclassical estimation above (upon summation in Fig. 4B, the positions of the steps, i.e., the white fringes, seem shifted to higher values; this is expected and analogous to the peak shift observed when summing two Lorentzian functions). Overall, our one-parameter Dirac equation gives a very good description of the resonance dispersion, the critical field , and its dependence on energy and momentum.

Implications of these results include possible use in helicity-sensitive electro-optical measurements at terahertz frequencies with the ability to switch circular-polarized optical signals with small modulations of magnetic fields on the order of 10 mT [section III.C of (*27*)]. These applications, in conjunction with the fidelity of fabricating a variety of p-n junction and other electrostatic boundaries with impurity doping of boron nitride in graphene heterostructures, will expand the quantum tool box of graphene-based electron optics for future studies and applications.

## SUPPLEMENTARY MATERIALS

## References and Notes

**Acknowledgments:**F.G., D.W., C.G., and Y.Z. acknowledge support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, grant no. 70NANB10H193, through the University of Maryland. J.W. acknowledges support from the National Research Council Fellowship. F.D.N. greatly appreciates support from the Swiss National Science Foundation under project number PZ00P2_167965. Y.Z. acknowledges support by the National Science Foundation of China under project number 11674150 and the National 1000 Young Talents Program. J.F.R.-N. acknowledges support from the NSF grant DMR-1507806. K.W. and T.T. acknowledge support from JSPS KAKENHI grant no. JP15K21722. L.S.L. acknowledges support by the Center for Integrated Quantum Materials (CIQM) under NSF award 1231319 and by the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under award no. DESC0001088. We thank S. Blankenship and A. Band for their contributions to this project and M. Zwolak and M. Stiles for valuable discussions.