Voltage-driven quantum oscillations in graphene

We predict unusual (for non-relativistic quantum mechanics) electron states in graphene, which are localized within a finite-width potential barrier. The density of localized states in the sufficiently high and/or wide graphene barrier exhibits a number of singularities at certain values of the energy. Such singularities provide quantum oscillations of both the transport (e.g., conductivity) and thermodynamic properties of graphene - when increasing the barrier height and/or width, similarly to the well-known Shubnikov-de-Haas (SdH) oscillations of conductivity in pure metals. However, here the SdH-like oscillations are driven by an electric field instead of the usual magnetically-driven SdH-oscillations.

The Shubnikov-de-Haas effect, i.e., the oscillations of the magneto-resistance of metals when increasing an external magnetic field, was one of the first macroscopic manifestations of the quantum-mechanical nature of matter. The key to understanding this remarkable phenomenon was pointed out by Landau and Onsager and it is described in many textbooks on solid state physics (see, e.g., Ref. 1). Namely, electrons in the conduction band of a metal in a strong magnetic field behave like simple harmonic oscillators. The resulting energy spectrum is made up of equidistant Landau levels separated by the cyclotron energy. The density of electron states has singularities at the Landau levels. When the magnetic field is changed, the positions of the Landau levels move and pass periodically through the Fermi energy. As a result, the population of electrons at the Fermi surface also oscillates and, in turn, leads to quantum oscillations of the transport and thermodynamic properties of a metal. The quantum oscillations also manifest themselves in the thermoconductivity, magnetization, sound attenuation, magnetostriction, and other quantities.
These quantum oscillations are pronounced in conductors with a long mean free path of charge carriers. This can occur in pure metals, semimetals, and narrow band-gap semiconductors at low temperatures, as well as in graphene, a one-atom-thick sheet of carbon. The Shubnikov-de-Haas oscillations of the magneto-resistivity were observed in graphene [2,3] soon after its discovery [4]. Due to the monolayer honeycomb-lattice struc-ture of graphene, its electrons obey a massless Dirac-like equation (see, e.g., Refs. 5,6). This is responsible for the unusual properties of graphene. In particular, the Landau levels in graphene are not equidistant and these influence the period of the Shubnikov-de-Haas oscillations [7,8]. Graphene has another striking property: it has unusual relativistic effects which are counterintuitive for electrons with speeds much slower than the speed of light [9]. For example, it has been recently shown [5] that graphene could be used for experimentally testing the so-called Klein paradox [10]. This quantum-mechanical effect of relativistic particles penetrating through high and wide potential barriers can be illustrated with massless Dirac fermions in graphene with a potential barrier controlled by an applied voltage. A high potential energy barrier in graphene, as was shown in Ref. 11, can also act as an unusual electron lens, due to the negative refraction of electron waves at the edge of the barrier, in analogy to the negative refraction of 3D [12] and 2D [13,14] electromagnetic waves.
Our goal here is to show that, due to the Dirac-like Hamiltonian of graphene with a potential energy barrier, quantum oscillations similar to the Shubnikov-de-Haas effect can be observed without an applied magnetic field. Below we prove that the density of electron states in a graphene sheet with a potential barrier should display quantum oscillations if the strength of the barrier (i.e., the product V 0 D of the barrier's height V 0 and width D) exceeds some threshold value. In these oscillations, the barrier strength V 0 D, that can be controlled by a gate voltage, plays the same role as the external magnetic field in the Shubnikov-de-Haas effect.
The quantum oscillations predicted here originate from a new type of electron states in graphene. Contrary to non-relativistic quantum mechanics, where localized states can only exist inside quantum wells, we find that the electron states in graphene can be localized within the barrier. We show that the energy E(q y ) of the localized states (versus the wave vector component q y along the barrier) becomes non-monotonic if V 0 D > π v F (v F is the Fermi velocity). This produces singularities of the density of electron states for energies where dE/dq y = 0. When the magnitude and/or width of the barrier changes, the locations of the singularities move and periodically cross the Fermi level, generating quantum oscillations of both thermodynamic and transport properties, e.g., of the conductance in the y-direction (along the barrier).
Electron states localized in a barrier.-The tunnelling of relativistic particles through a finite-width potential barrier has recently been studied in Refs. 5, 15, 16. Here we consider another type of electron waves that propagate strictly along the barrier and damp away from it. Our analysis shows that a step-like barrier (i.e., a single edge of an infinitely-wide barrier) does not support such electron waves, which would be an analog to surface electromagnetic waves (plasmon-polaritons) at the interface between two different media. Therefore, even though the potential barrier in graphene could act as an electron lens [11], it cannot provide the perfect lensing (i.e., subwavelength image reconstruction) that is possible for Veselago's lens in optics [17]. However, as we show in this section, the electron waves in graphene can be localized inside a finite-width potential barrier.
We consider electron states in graphene with a potential barrier located in a single-layer graphene occupying the xy-plane (see Fig. 1). For simplicity, we assume that the barrier V (x) has sharp edges, Electrons in monolayer graphene obey the Dirac-like equation, where v F is the Fermi velocity and σ = (σ x , σ y ) are Pauli matrices. We seek stationary spinor solutions of the form, with energy E and momentum q along the barrier. We focus on the states with In this case, the electron waves satisfying Eq. (2) damp away from the barrier, and the components ψ 1 and ψ 2 of the Dirac spinor can be written in the form, with real k x = (q 2 − ǫ 2 ) 1/2 and q x = [(ǫ − V/D) 2 − q 2 ] 1/2 . Here V = V 0 D/ v F is the effective barrier strength and tan θ = q/q x .
Matching the functions ψ 1 (x) and ψ 2 (x) at the points x = ±D/2, we obtain a set of four homogeneous algebraic equations for the constants a, b, c, and d. Equating the determinant of this set to zero, we obtain a dispersion relation for the localizes electron states, Figure 1 illustrates the behavior of the probability distribution W (x) for a localized state. Note that W (x) is an even function with continuous derivative W ′ (x), in spite of the fact that each one of the functions |ψ 1 | 2 and |ψ 1 | 2 are not even (the chirality of the Dirac spinors) and both have discontinuous derivatives at the points x = ±D/2. Note that similar localized states can also be observed in a 2D electron gas when a voltage is applied to produce a potential well. For electrons with a quadratic dispersion law, this spectrum is E n = y 2 n + 4Q 2 − V, where E = mD 2 E/2 2 , V = mD 2 V 0 /2 2 , Q = qD, m is the electron mass. Here y n is the nth root of the equation y tan(y) = (V − y 2 ) 1/2 for even states, and y cot(y) = −(V − y 2 ) 1/2 for odd states.
The spectrum Eq. (7) of localized states in graphene is shown by the solid black curves in Fig. 2, for dimensionless variables Q = qD and E = ǫD. This spectrum consists of an infinite number of branches E n (Q). Each of these branches starts from the lines E = ±|Q| (red solid straight lines in Fig. 1b) at E = V/2 − π 2 n 2 /2V and tends asymptotically to the lines E = V ± Q (dashed red lines). Moreover, a particular branch of the spectrum starts at the point (Q = 0, E = 0) and also tends to the line E = V − Q, when increasing Q. The behavior of different branches of the spectrum depends on the barrier strength V. If V < π/2, all branches satisfy E < 0 (see inset in Fig. 2). Localized states with positive energies appear only for V > π/2. When V increases, new branches in the spectrum with positive energies appear. When V is within the interval (n + 1/2)π < V < (n + 3/2)π, the number of branches with E > 0 is (n + 1), n = 1, 2, 3, . . . . We emphasize that each of the branches with positive energy has a maximum E max n at a certain wave number Q = Q max n . Near these points, the group velocity of localized electron waves tends to zero. This effect is similar to the stop-light phenomenon [18] found in various media, including superconductors [19].
Note that defect-induced localized electron states in graphene and the enhancement of conductivity due to an increase of the electron density of states localized near the graphene edges were recently reported [20,21,22]. Contrary to these examples, the electron states studied here are localized within the barrier and also these are tunable, i.e., the energy levels can be shifted by changing the barrier strength (e.g., via tuning a gate voltage).
Density of localized electron states.-To calculate the density N (E) of electron states, we use the general formula N (E) = α δ(E − E α ), where α labels the quantum state and δ(x) is Dirac's delta-function. Using where L x and L y are the lengths of the graphene sheet in the x and y directions, respectively. For localized states, we obtain Here n runs over the number of positive roots of the equation E(Q) = E.
The dimensionless density of states N (E)/N 0 is shown in Fig. 3. The localized electron states exhibit two types of peculiarities. First, increasing E, the jumps or steps (each one of magnitude 2D/L x ) in N (E)/N 0 occur at the points E = V/2 − π 2 n 2 /2V, where new branches of the spectrum arise or disappear. More importantly, singularities are observed when E = E max n , where |dE n /dQ| −1 in Eq. (9) diverges. The locations of the singularities shift when changing the barrier strength V. Therefore, they periodically cross the Fermi level E F . This produces quantum oscillations of the density of states at the Fermi energy. They are seen in Fig. 4, showing N (E F )/N 0 versus the effective barrier strength V. The periodic change in the number of electron states near the Fermi level, increasing the barrier strength (e.g., by varying a gate voltage), necessarily results in quantum oscillations of the transport and thermodynamic properties of graphene. For example, the conductance of graphene along the barrier qualitatively mimics the quantum oscillations of the density of electron states. Figure 4 shows the quantum oscillations of the density of electron states in e-type graphene (with a positive Fermi energy). For p-type graphene, with E F < 0, quantum oscillations of the density of states at the Fermi level can also be observed, if the (now opposite-bias) applied voltage forms a potential well instead of a barrier. Indeed, the Dirac equation (2) is invariant with respect to the transformation: E → −E, V → −V, x → −x, y → −y.
In conclusion, we predict an unusual type of electron states in graphene localized within a potential barrier. For barriers with sufficiently high magnitude and width, the density of localized states has singularities. This feature of localized states can result in quantum oscillations of the thermodynamic and transport properties (e.g., the conductance along the barrier) of graphene when changing the barrier strength (e.g., by varying a gate voltage).
These electric-field driven quantum oscillations are similar to the Shubnikov-de-Haas oscillations of conductivity, which are produced in standard metals when changing the external magnetic field.