This is the submitted version of the article: Aktor T., Garcia J.H., Roche S., Jauho A.-P., Power S.R.. Valley Hall effect and nonlocal resistance in locally gapped graphene. Physical Review B, (2021). 103. 115406: - . 10.1103/PhysRevB.103.115406. Available at: https://dx.doi.org/10.1103/PhysRevB.103.115406 APS/123-QED Valley Hall Effect and Non-Local Resistance in Locally Gapped Graphene Thomas Aktor,1 Jose H. Garcia,2 Stephan Roche,2, 3 Antti-Pekka Jauho,1 and Stephen R. Power4, ∗ 1 Center for Nanostructured Graphene (CNG), DTU Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona (Cerdanyola del Vallès), Spain 3 ICREA, Institució Catalana de Recerca i Estudis Avançats, 08070 Barcelona, Spain 4 School of Physics, Trinity College Dublin, Dublin 2, Ireland (Dated: October 1, 2020) We report on the emergence of bulk, valley-polarized currents in graphene-based devices, driven by spatially varying regions of broken sublattice symmetry, and revealed by non-local resistance (RNL ) fingerprints. By using a combination of quantum transport formalisms, giving access to bulk properties as well as multi-terminal device responses, the presence of a non-uniform local bandgap is shown to give rise to valley-dependent scattering and a finite Fermi surface contribution to the valley Hall conductivity, related to characteristics of RNL . These features are robust against disorder and provide an interpretation of controversial experiments based on Berry curvature arguments in graphene/hBN superlattices. Our findings suggest both an alternative mechanism for the generation of valley Hall effect in graphene, and a route towards valley-dependent electron optics, by materials and device engineering. Two-dimensional materials, and graphene in particu- currents are associated with Fermi sea contributions lar, are promising valleytronic candidates due to the K from Berry curvature hotspots below the Fermi energy, and K 0 valleys at the Dirac points, which can be ex- whereas within the relevant linear response regime, only ploited to encode, transport and process information [1]. Fermi surface contributions should play a role in device However, a key hurdle is the absence of external knobs, measurements [41, 42]. Experimental mapping of current analogous to magnetic fields and ferromagnetic contacts flow further suggested that edge currents may play a role in spintronics, to generate, manipulate and detect valley- [43], but recent theoretical studies cast doubt upon the polarized currents [2]. While optoelectronic access to the topological origin of such currents [42, 44, 45]. Finally, valley index is possible in certain materials [3–6], an all- the lattice mismatch between graphene and hBN, which electronic control would be highly desirable for device leads to a Moiré pattern for commensurate structures applications [7]. Certain defects or strain deformation [31, 46], also clearly indicates that electrons may not ex- fields have been theoretically proposed for achieving val- perience a uniform mass term [47], making the interpre- ley filtering, but the corresponding experimental imple- tation of experimental RNL signals in terms of VHE a mentation remains challenging [8–30]. Promising signa- true and unsolved conundrum. tures of valley phenomena have instead emerged from In this Letter, we demonstrate the emergence of valley- non-local resistace (RNL ) measurements in commensu- split bulk transport in the absence of a global band rately stacked graphene/hexagonal boron nitride (hBN) gap. This phenomenon requires instead the presence of systems [31–33]. Large RNL signals have been interpreted a spatially-varying mass term – a situation analogous to in terms of an intrinsic valley Hall effect (VHE), driven graphene-hBN heterostructures with commensurate layer by bulk Berry curvature [34–37], related to the mass term alignment. The valley-polarized current is a consequence induced by the interaction between graphene and hBN. of an extrinsic-like valley Hall effect that can be under- Under this mechanism, a quantized valley Hall conduc- stood by examining the scattering from a single, circular 2 v tivity, σxy = 2eh , within the band gap [2, 31, 36, 38], was mass dot. The exact solution to this case establishes a argued to generate a long-ranged valley current, enhanc- strong valley dependence in the scattered wavefunction at ing RNL beyond standard Ohmic contributions. Analo- low energies and a valley-splitting of incoming electronic gous behaviour has also been studied in bilayer graphene currents. Tight-binding simulations show that this effect [39, 40], where the VHE is tunable by an interlayer bias. is robust for various dot profiles and mass distributions. The valley Hall conductivity for a periodic array of dots However, the interpretation of experimental RNL in is then calculated using the Kubo-Bastin formalism, and terms of a bulk-driven VHE has been severely ques- confirms the formation of valley Hall signals and charge tioned by quantum transport simulations [2, 41, 42]. neutral currents. This system manifests a non-zero Fermi Bulk-driven RNL signals in gapped systems are found surface contribution to the valley Hall conductivity, con- to be strongly suppressed beyond evanescent contribu- sistent with the extrinsic contribution and in contrast tions, rendering them fully negligible at experimental to the homogeneous band gap case. Using Landauer- scales. One puzzling issue is that intrinsic valley Hall Buttiker simulations, we show that scattering-induced 2 a) b) c) for each mode. The Hamiltonian in Eq. (1) is invari- ant under VA ↔ VB , ψ1τ ↔ ψ2−τ , ψ2τ ↔ −ψ1−τ , so that the K valley result can be used to deduce the K 0 case. The local electron density n = ψ † ψ and particle current VA VB j = ψ † σψ are calculated for each valley separately, with K K′ the total electronic (valley) quantity given by the sum d) e) 90° (difference) of the two valleys. The local valley polar- 0.0 135° 45° isation at a point (r, φ) is the ratio of these quantities sc sc ξ(r, φ) ≡ jval (r, φ)/jtot (r, φ), and takes values −1, 0, 1 for avg 0 0.4 180° K 0° fully K polarized, valley neutral, and fully K polarized K' currents respectively. A figure-of-merit for the valley- 0.8 225° 315° splitting efficiency of a dot is 0.0 0.5 1.0 1.5 2.0 270° Z π E dφ sc ξavg = lim ξ (r, φ) , (2) r→∞ 0 π FIG. 1. (a) Incoming, scattered and transmitted waves for i.e the far-field limit of the scattered current polariza- scattering from a mass dot. (b) A and B sublattice sites in the dot have different onsite potentials. (c) K and K 0 tion, averaged over the upper-half plane. This quanti- valleys in the low-energy spectrum of graphene. (d) Valley fies the average valley polarisation of transverse currents, polarization of scattered currents: the peak indicates a strong with non-zero ξavg indicating VHE-type behaviour. We K 0 polarisation in the +y direction. (e) Total (bold) and consider a R = 4.5 dot with ∆ ˜ = 2 so that the band individual valley angular scattering profiles in the far-field edge inside the dot is at Ẽ = ±1, and focus here on limit at Ẽ = 0.12 (purple dot in (d)). valley-dependent effects – for general scattering proper- ties see [62]. In Fig 1(d), ξavg takes large negative val- valley-polarized currents significantly enhance RNL sig- ues as E increases from zero, before decaying to about nals, in contrast to the vanishing RNL predicted for uni- half its maximum magnitude, and then vanishing at en- form mass systems. Scattering-induced valley splitting, ergies above the band edge. This corresponds to K 0 elec- unlike Berry-phase induced deflection, is consistent with trons preferentially scattering in the +y-direction for all standard quantum transport methods, and provides an band gap energies, with the strongest effect near the gap alternative mechanism to interpret valleytronic phenom- centre. The angular dependence of scattering is deter- ena using RNL measurements. It also suggests a route mined from the radial component of jsc (r → ∞). Fig. towards valley engineering by using patterned hBN or 1(e) plots the total quantity (solid line) and its individ- other substrates [46–49], patterned gates [50], or dop- ual valley contributions (shaded and unshaded areas) at ing [51–58] to generate spatially-varying mass profiles in the energy shown by the dot in Fig. 1(d)). K and K 0 graphene to direct the flow of valley currents. have equal contributions at φ = 0, π, corresponding to Scattering from a circular mass dot is considered using a valley-neutral situation for exact forward- and back- a Dirac Hamiltonian near the K and K 0 (τ = ±1) points scattering from the dot. At other angles the two valleys scatter anti-symmetrically with respect to the x-axis. At   ṼA (r) −iτ ∂x − ∂y the energy shown, scattering is largest in the transverse Hτ (r) = ~vF , (1) directions so that electrons from different valleys scatter −iτ ∂x + ∂y ṼB (r) almost entirely to opposite sides of the dot. with scaled variables X̃ ≡ ~vXF . Here VA (r), VB (r) are Fig. 2(a)-(c) shows the current flow near the dot at the A and B sublattice potentials. Within a dot of radius this energy, with arrows showing the current magnitude R we set VA/B (r < R) = ± ∆ 2 , where the sign depends and direction in each case. Fig. 2(a) shows the total on the sublattice. The mass term ∆ leads to band-gap charge flow, with the color scale showing valley polari- opening in the range − ∆ ∆ 2 < E < 2 in the region where sation, whereas Fig. 2(b),(c) show the current flow and VA/B 6= 0. Following Refs. [59–61], we switch to po- electron density for each valley separately. K and K 0 lar coordinates (r, φ) and consider an incoming (ψinc ), electrons flow in opposite directions around the dot, lead- scattered (ψsc ) and transmitted (ψtr ) waves, as shown in ing to the same antisymmetry around the x-axis that Fig. 1(a). ψinc and ψsc are expanded in terms of an- was noted for angular scattering. Valley-polarized cur- gular momentum basis states m using Bessel and Han- rents are observed in the immediate vicinity of the dot kel functions respectively, as in the potential dot case in Fig. 2(a), but quickly recombine to give a largely [60, 61]. Inside the dot, ψtr is expanded in terms of Bessel valley-neutral forward current. This is also evident from functions with sublattice-dependent coefficients [62]. A Fig. 2(b),(c), where the large K (K 0 ) current near the closed form expression for the wavefunction is found by top (bottom) of the dot diffuses quickly behind it. The enforcing continuity at the dot boundary and solving for far-field behaviour emerges from currents further away, the scattering and transmission coefficients, csm and ctm , namely the K (K 0 ) current flowing in the bottom (top) 3 uniform WC = 5R WC = 10R WC = 20R Valley Polarization K Electron density K' a) b) 1.0 c) 2 WC = 5R 2 K' K 0 high xy v (e 2/h) v, 0 a) b) c) xy 0 1 0.5 d) / uniform xy xy 2 v v xy v 0 0.0 0 1 0 1 1 0 1 1 0 1 E (eV) E (eV) E (eV) d) e) v v FIG. 3. (a) Valley Hall conductivity σxy , and (b) σxy normalized by its peak value, for three different supercell v sizes. (c),(d) σxy for WC = 5R and uniform mass distribu- tions, together with their Fermi surface contribution (shaded). Dashed vertical lines show the band edges in mass regions. disordered mass G(E), and lead broadening term, Γ, give the spectral FIG. 2. (a) Total and (b), (c) individual valley current flows density of injected states A(E) = GΓG+ and local cur- near a mass dot at low energy. Electrons from different valleys rent flow in real-space. Projecting A onto the graphene flow in opposite directions around the dot. (d) Similar valley- splitting behaviour for a disordered mass region. (e) Phase basis |ψk i measures the local distribution of scattered space map for the average valley polarization ξavg for different states in k-space, ρ(k) = hψk | A |ψk i, from which Ẽ and ∆˜ values, with the dotted line denoting the band edge. the real-space valley polarisation is calculated ξT B = P P .P k∈K ρ(k) − k∈K 0 ρ(k) k∈K,K 0 ρ(k). The ana- lytic polarisation from Fig. 2(a) is reproduced numer- left of these panels. ically [62] not only for a perfect dot, but also for non- Counterpropagating flows of K and K 0 currents emerge uniform mass distributions. Fig. 2(d) shows a disordered from the scattering coefficients csm̄ (K) of the individual case, where VA/B 6= 0 only on 1% of sites within the dot, modes, whose analytic form is given in [62]. The m and but with increased magnitudes to preserve the average m̄ ≡ −m − 1 modes are closely connected, and we find mass. The result, for this and other mass distributions, csm̄ (K) = csm (K 0 ), so that every K valley mode has a cor- is almost identical to the analytic, perfect dot predic- responding K 0 mode which contributes with equal mag- tion. Insensitivity to the strength, size or composition of nitude but opposite angular momentum. At low ener- mass regions suggests that realisation of valleytronic de- gies, only m, m̄ = 0 modes contribute, and the clearest vices may be easier to achieve using mass dot engineering valley-splitting is observed. As E increases, there is a than strain-based proposals. gradual onset of contributions from m, m̄ > 0 modes, Mass dots induce equal and opposite deflections of and valley-splitting effects from different modes partially K and K 0 electrons in the y direction, giving rise to cancel, leading to a decrease in the magnitude of ξavg a pure valley current in the transverse direction, which (Fig. 1(d)). However, the presence of higher-order modes is reminiscent of Berry curvature deflections in globally maintains the sign of ξavg within the gap, so that the de- gapped systems. [2, 31, 36, 38]. To determine possi- flection direction is consistent. This is clear from Fig. ble experimental signatures of this phenomenon, we ana- 2e), where we show ξavg across a range of mass strengths v lyze the valley Hall conductivity σxy and non-local resis- and electron energies. The dotted line denotes the band tance RNL using both the Kubo and Landauer-Buttiker edge, and a uniform preference for enhanced K 0 scatter- formalisms. We employ the single-particle Kubo-Bastin ing is seen inside the gap, with a weak oscillatory pat- formula [23, 65–72] to calculate both the complete and tern outside due to resonances with bound states. The Fermi surface contributions to σxy v [62] for a R = 10 strongest valley polarization (dark red region) is at low nm mass dot (VA/B = ±0.1|t|) placed into one of energies for all sizes and strengths. For further discussion three different square supercells with side lengths WC = of the energy dependence, see [62]. 5R, 10R, 20R. This allows us to examine the competition To test the robustness of the valley-splitting predicted between scattering and Berry curvature effects, which above, we perform tight-binding simulations using the can arise in periodic systems due to a finite concentration patched Green’s function approach [21, 63]. A finite c of sites with mass ∆, giving rise to an effective global v dot with R = 10 nm and VA/B (r < R) = ±0.1|t| is mass ≈ c∆. For a global mass, the height of the σxy embedded in an infinite graphene sheet, and we per- peak should remain constant, with wider peaks expected form transport calculations using a point probe 250 nm for larger ∆eff (smaller WC ). However, Fig. 3(a) shows away so that incoming electrons resemble a plane wave, that instead the height decreases with dot density, while as in Fig. 1(a). The system Green’s functions [64], the width is independent of WC , so that the three curves 4 a) b) No dots c) finite device size, and emerge from configuration-specific 1 3 0.2 0.2 TAL v RNL (h/2e 2) Mass dots TDI resonances in the pairwise transmissions Tpq . Unlike σxy , the RNL signal is not mediated solely by Hall effects, but T 0.1 0.0 5 6 is sensitive to the device size, edge types, probe place- 2 4 0.0 ments, and the distances between scatterers and edges. 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 E(|t|) E(|t|) Such effects are difficult to remove from the simulations, 75% K' 75% K d) but average out in experimental-scale systems to give a vanishing contribution to the RNL signal. This repre- 60 sents the current taking a direct path between source 50 and drain, and not probing the non-local region of the 40 device. With mass dots present (solid blue curve), RNL still displays rapid oscillations due to finite size effects, nm 30 but a wide positive peak feature now emerges at low ener- 20 gies. Similar behaviour is seen for different dot sizes and 10 separations, mass strengths, and disorder types, but does 0 not occur in pristine systems or for dots with VA = VB . 40 60 80 100 120 140 160 To understand the origin of the positive RNL signal, nm we analyse the pairwise transmission profiles between probes. The sign of RNL [62] is a competition between FIG. 4. (a) Six-terminal device for RNL simulations. (b) transmissions along an edge of the device (TAL ) and diag- RNL with and without mass dots, showing a robust positive trend in the presence of dots. (c) Transmission along one edge onally across the device (TDI ). Fig 4(c) confirms the pos- (TAL ) and diagonally across the device (TDI ). (d) Map of itive RNL feature corresponds to an enhancement of TAL local current flow (arrows) and valley polarisation (color) for relative to TDI for our system. To better interpret the the energy shown in orange in (b) and (c). RNL is mediated dominant RNL signal, we map the local current through- by valley-polarized currents in the device. out the device. In multi-terminal devices, the current flow is not simply an injection at the source and absorp- tion at the drain. It results from a superposition of cur- coincide when normalized (Fig. 3(b)). This suggests a rents injected and absorbed by each probe [77–81]. The valley-splitting mechanism analogous to an extrinsic spin arrows in Fig. 4(d) show the current in our system for the Hall effect induced by skew-scattering [73, 74]. In this energy marked by orange line in Fig. 4(b),(c). Due to fi- v case, the magnitude of σxy should vary with the dot den- nite size effects, the flow pattern is complex and sensitive sity, but with an energy dependence following the scat- to the chosen energy. Streams and vortices appear, dis- tering profile of a single dot, as observed. To further sub- appear and relocate as the energy is varied. However, at stantiate this hypothesis, we examine the Fermi surface low energies, strong transverse scattering from the dots v (FS) contributions [68, 75] to σxy , which remain finite leads to indirect current paths which probe large portions throughout the band gap for the mass dot system in Fig. of the device. This is in stark contrast to pristine sys- v,FS v 3(c), with σxy ≈ 0.4 σxy at E = 0. For a uniform mass, tems, where more direct paths between source and drain this contribution vanishes (aside from broadening effects are observed [62]. The valley polarization of the current, [62]) in the gap, as shown in Fig. 3(d). Similar behaviour calculated in the multi-terminal setup, is shown here by is found for each WC , demonstrating that a robust Fermi color shading. We note that strong valley splitting at surface contribution emerges in the presence of spatially dots near the source and drain contacts, reinforced by distributed mass dots. subsequent scattering from other dots, leads to valley- The absence of a global gap enables straightforward polarized currents throughout the entire device. RNL simulations using Landauer-Buttiker methods. We Electrons from the two valleys tend to flow in opposite consider the six-terminal setup in Fig. 4(a), where R = 5 directions near the edge, for example, near the bottom nm dots (VA/B = ±0.4|t|) with periodicity WC ∼ 6R edge, K (K 0 ) currents flow primarily to the right (left). are embedded in a 64 nm wide zigzag ribbon. The cur- This may appear analogous to the quantum spin Hall ef- rents Ip Pand potentials Vp in each lead p are related by fect, where electrons of different spin counterpropagate Ip = 2eh (Tqp Vp − Tpq Vq ) , where transmissions Tpq are along an edge, there are fundamental differences. First calculated using recursive Green’s function techniques the studied system does not have an insulating bulk, but [63, 76, 77]. The potential difference between leads 1 rather strong transverse scattering sources which deflect and 2 is fixed, and the net current in the remaining leads valley-polarized current towards the device edges. An im- set to zero. Solving for the current, I1 = −I2 , and the portant aspect here is that significant current still flows in potentials V3−6 yields RNL = V3 −V I 4 . 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