Graphene Transistor Based on Tunable Dirac Fermion Optics

Graphene transistor based on tunable Dirac fermion optics

Ke Wanga,b, Mirza M. Elahic, Lei Wangd, K. M. Masum Habibc,1, Takashi Taniguchie, Kenji Watanabee, James Honed, Avik W. Ghoshc,f, Gil-Ho Leea,g,2, and Philip Kima,2

aDepartment of Physics, Harvard University, Cambridge, MA 02138; bSchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455; cDepartment of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904; dDepartment of Mechanical Engineering, Columbia University, New York City, NY 10027; eResearch Center for Functional Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan; fDepartment of Physics, University of Virginia, Charlottesville, VA 22904; and gDepartment of Physics, Pohang University of Science and Technology, Pohang 37673, South Korea

Edited by Tony F. Heinz, Stanford University, Stanford, CA, and accepted by Editorial Board Member Angel Rubio February 12, 2019 (received for review September 18, 2018) We present a quantum switch based on analogous Dirac fermion waveguiding (11–14), beam splitting (15), Veselago lensing (4), optics (DFO), in which the angle dependence of Klein tunneling is and negative refraction (5) in graphene. explicitly utilized to build tunable collimators and reflectors for the The strong angle dependence of Klein tunneling transmission quantum wave function of Dirac fermions. We employ a dual- T has been proposed to realize a type of switching device based source design with a single flat reflector, which minimizes diffu- on DF optics (DFO) (10, 14, 16–19). Fig. 1A shows a simple sive edge scattering and suppresses the background incoherent device scheme utilizing analogous electron optics. Here, a single- transmission. Our gate-tunable collimator–reflector device design layer graphene channel is controlled by several local gates with enables the quantitative measurement of the net DFO contribu- predetermined shapes, dividing up electron-doped (N) and hole- tion in the switching device operation. We obtain a full set of doped (P) regions in the channel. The electrons leaving the transmission coefficients between multiple leads of the device, source electrode pass through the first PN junction orthogonal to separating the classical contribution from the coherent transport the channel direction. This PN junction filters out electrons with contribution. The DFO behavior demonstrated in this work re- an oblique incident angle and collimates electron beams along SCIENCES

quires no explicit energy gap. We demonstrate its robustness the channel. The next PN junction, placed at an angle (∼45°), APPLIED PHYSICAL against thermal fluctuations up to 230 K and large bias current blocks the collimated electron beam due to the oblique incidence density up to 102 A/m, over a wide range of carrier densities. to the PN junction and reflects it along a path orthogonal to the The characterizable and tunable optical components (collimator– original. However, in this simplistic device design, the reflected reflector) coupled with the conjugated source electrodes devel- beam hitting the rough physical edge of the device would dif- oped in this work provide essential building blocks toward more fusively scatter (Fig. 1A), leading ultimately to a leakage cur- advanced DFO circuits such as quantum interferometers. The ca- rent into the drain electrode. On top of that, multiple bounces pability of building optical circuit analogies at a microscopic scale of electrons in between collimator and reflector junctions with highly tunable electron wavelength paves a path toward contribute to the leakage current. To circumvent these diffusive highly integrated and electrically tunable electron-optical components and circuits. Significance

graphene | Dirac fermion | electron optics | quantum transport We report an electrically tunable graphene quantum switch based on Dirac fermion optics (DFO), with electrostatically defined – he linear energy momentum dispersion, coupled with pseu- analogies of mirror and collimators utilizing angle-dependent Tdospinors (1), makes graphene an ideal solid-state material Klein tunneling. The device design allows a previously unreported platform to realize an electronic device based on Dirac- quantitative characterization of the net DFO contribution and fermionic relativistic quantum mechanics. Employing local gate leads to improved device performance resilient to abrupt change control, several examples of electronic devices based on Dirac in temperature, bias, doping, and electrostatic environment. The fermion (DF) dynamics have been demonstrated, including electrically tunable collimator and reflector demonstrated in this Klein tunneling (2), negative refraction (3–5), and specular work, and the capability of accurate in situ characterization of Andreev reflection (6, 7). their performance, provide the building blocks toward more While the depletion region of conventional semiconducting complicated functional quantum device architecture such as PN junction blocks the electronic transport across the junction, highly integrated electron-optical circuits. the gapless band structure of the graphene facilitates electrically adjustable PN junctions and enables electronic optics. The Author contributions: K. Wang, L.W., J.H., A.W.G., G.-H.L., and P.K. designed research; transmission probability (T) across the PN junction is unity for K. Wang, L.W., and G.-H.L. performed research; L.W. contributed new reagents/analytic normal incident electrons due to the pseudospin conservation of tools; K. Wang, M.M.E., L.W., K.M.M.H., A.W.G., G.-H.L., and P.K. analyzed data; K. Wang, M.M.E., L.W., K.M.M.H., T.T., K. Watanabe, J.H., A.W.G., G.-H.L., and P.K. wrote the paper; DFs. This startling phenomenon known as Klein tunneling (8, 9) and T.T. and K. Watanabe provided hBN crystals. was first demonstrated in a graphene PNP junction (2). For the The authors declare no conflict of interest. DFs with an oblique incident angle (θ), a PN junction exhibits ’ This article is a PNAS Direct Submission. T.F.H. is a guest editor invited by the Snell s law like an electron beam path with a negative refraction Editorial Board. – medium (3 5) for incoming Dirac electron wavefunctions. Published under the PNAS license. However, T is exponentially suppressed with θ as T ∼ exp 1 2 Present address: Technology Computer Aided Design, Intel Corporation, Santa Clara, [−π(kF(d/2))sin θ] for the symmetric potential of P and N re- CA 95054. k d gions, where F is Fermi momentum and is characteristic 2To whom correspondence may be addressed. Email: [email protected] or pkim@ length scale of potential change across the junction (8, 9). A physics.harvard.edu. generalized equation for arbitrary junctions is in ref. 10. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. Depending on the value of kFd, the junction can be transparent 1073/pnas.1816119116/-/DCSupplemental. or reflective, a result that has been employed for electron

www.pnas.org/cgi/doi/10.1073/pnas.1816119116 PNAS Latest Articles | 1of5 Downloaded by guest on September 23, 2021 V A B gate region (controlled by gate 2) turns into the opposite carrier polarities of source and drain regions (controlled by gate V1), carriers injected from each source will either reflect back to the same source (oblique incident angle) or travel ballistically to the other source contact (perpendicular incident angle). This collimation–reflection results in suppressed conduction between the source and the drain, and the device is in “off” state. When V1 and V2 are at the same polarity, the carriers flow ballistically to the drain, and the device is in “on” state. This device operation scheme has an advantage compared with the aforementioned single-source collimator–reflector scheme (Fig. 1A) or a sawtooth-shaped gate structure (18, 20, 21), as there is no significant channel edge contribution and only one reflection can be used for the off operation. Even with a nonideal reflector, we thus expect considerably enhanced DFO of the switch. CDFig. 1B shows electron microscope image of the local gates used for the dual-source device before the integration of graphene channel with two-source and one-drain electrodes in place. Switching operation of our device can be demonstrated by measuring two terminal resistance RT between the drain electrode (1) and source electrodes (2 and 3). A common bias voltage VD is applied to the source electrodes while the drain electrode is grounded. Two gate regions, collimation gates and the central gate, are controlled by applied gate voltages V1 and V2, respectively. Fig. 1C shows the measured RT as a function of V1 and V2. The resistance map in (V1, V2) plane can be divided into four quadrants separated by the peak region of RT ∼ 8kΩ, corresponding to the charge neutral Dirac point, V1, V2 ∼ 0. These four distinctive quadrants represent the source collimation/central gate/drain collimation regions in the NNN, NPN, Fig. 1. Graphene quantum switch. (A) Schematics of the device in the off PPP, and PNP regimes, respectively. We note that the NNN V V mode. Central green area (gate voltage, 1) and the blue areas ( 2)are R ∼ Ω V ·V < regime has the lowest resistance T of 500 , while the PPP doped in different polarity ( 1 2 0). The collimated electron beams regime exhibits considerably larger resistance of ∼1.5 kΩ.Inan through vertical and horizontal junctions are reflected toward the device ideal device, we expect a P/N symmetry in the device gate op- edge in one-source geometry or back to the source in two-source geometry. – (B) Atomic force microscope image of bottom gates was taken before eration due to the particle hole symmetry in the graphene band transferring a stack of hBN/graphene/hBN. Overlaid broken lines guide the structure. However, the graphene channel can exhibit asymmetry

boundaries of graphene. (C) Color-coded total resistance (RT) as a function of in contact resistance due to the metal-induced contact doping V V D 1 and 2.( ) Slide cut of the resistance shows the on/off ratio of 6 at fixed (22), which prefers N channel to have lower contact resistance in V = 2 5 V. Semiclassical ray tracing simulation matches experimental data our devices. The best device performance, therefore, is shown jV j especially for higher 1 (on or off state). To fit the off state (P/N/P), we use along the PNP–NNN regime, because there arise additional σ = edge roughness parameter e 15° (SD of Gaussian distribution of added angled PN junctions between contacts and graphene in PNP (off) random angles to specular edge reflections). regime. Fig. 1D shows a slice cut of RT along V1 at a fixed V2 = 5 V, crossing the PNP (off) to NNN (on) regimes. We choose this edge scattering and multiple bouncing events, one may design particular gate operation scheme for a pragmatic demonstration the collimator–reflector to minimize the channel edge scattering. of a large on/off ratio achieved in our device, although the on/off For example, a sawtooth-shaped top gate, which can create ratio defined in this way contains not only the DFO contribution double reflections sending the incoming DF beam back to source but also the contact and P/N junction resistance as we will below. To benchmark our experimental data, we perform semiclassical electrode, has been theoretically conceived based on DFO (18, ray tracing simulation (5) utilizing a billiard model (23–25) 20). The previous experimental study on such device architecture + coupled with analytical Klein tunnelling equations at junctions exhibited a signature of DFO behavior with the NP N/NPN on/ (simulation details in Methods). For Fig. 1D, channel resistance off ratio of 1.3 (21). However, the definition and value of the on/ (R ) is calculated from simulation and R (contact resistance) is off ratio can largely depend on the device operation scheme and ch C calculated from Fig. 1C diagonal elements (V1 = V2) (as for every other device specifics. Therefore, it can be challenging to use the V contact resistance is changing). Then total resistance, R = on/off ratio as a universal and accurate metrics for quantifying 1 T 2RC + Rch. To fit the off state (P/N/P), we include a random the DFO contribution. The necessity for establishing device- scattering angle around a specular trajectory (following a independent methodology to measure the DFO contribution Gaussian distribution with SD σe = 15°) at the edges. Our motivates us to develop a series of experimental designs that analysis shows that on/off ratio degrades with increasing σe as it allow two independent methods of accurately characterizing creates more and more states inside the transport gap. We em- DFO contribution. phasize that the switching operation based on our DFO does not In this work, we isolate the net DFO contribution from rela- require a “bandgap” in the channel material, since the device tivistic carriers and measure their full transmission coefficients in operation relies on Klein tunneling of DFs, which in turn keeps a dual-source design with a single flat reflector. The same device the high mobility of graphene intact in the on state and uses a design also reduces diffusive scattering at edges and multiple gate-tunable transport gap for off state. bounces that are otherwise responsible for high off-state current To realize complete collimation-filter DFO switch, alignment leakage. Fig. 1B shows a schematic diagram of the proposed between the collimated beam and the reflected beam is neces- device and the overall operational procedure. When the central sary. Random scatterers in the channel can alter the propagation

2of5 | www.pnas.org/cgi/doi/10.1073/pnas.1816119116 Wang et al. Downloaded by guest on September 23, 2021 that is, RJ,1(V1,V1) = 0. As a result, RJ,1 can be expressed in terms of RT: RJ,1(V1,V2) = [RT(V1,V2) + RT(V2,V1) − RT(V1,V1) − RT(V2,V2)]/2. Note that, in this expression, all of the terms of RC values and RG values are canceled out and RJ,1 can be obtained from the measured RT map. Similarly, RJ,2 can be extracted from configuration B, where the beam goes through only the reflector junction in front of the drain electrode. Now when the beam from the source goes through both collimator and reflector junctions, we need to introduce an effective resistance RSJ(V1, V2), which describes the effect of the collimation-filtering. RT in this situation (configuration C) can be written as RT(V1,V2) = RC,1(V1) + RG,1(V1) + RSJ(V1,V2) + RG,2(V2) + RC,2(V1); here, considering the electron–hole symmetry of the graphene channel, R2(V2) ∼ R2(−V2). We also assume a negligible series junction resistance in the unipolar regime, that is, RSJ(V1,V2) ∼ 0 for V1·V2 > 0. Note that RSJ(V1,V2) becomes precisely zero for V1 = V2. Summing up, we can rewrite RSJ in terms of RT: RSJ(V1, V2) = RT(V1,V2) − RT(V1,−V2) for V1·V2 < 0. Fig. 2 shows RJ,1, RJ,2 and RSJ as a function of V1 with a fixed voltage at V2 = −5 V. The finite RJ,1 and RJ,2 in the bipolar regime (V1 > 0) is a consequence of the reflected electrons at the Fig. 2. Extractions of PN junction resistances. Resistance contributions from PN junctions, whereas small value of RJ for V1 < 0 indicates that the collimation junctions (RJ,1), reflection junction (RJ,2), and series of both junctions (R ) as a function of V at V = −5 V. The resistance contributions the junctions are transparent in the unipolar regime. We also SJ 2 1 R + R R are extracted by symmetrizing total resistance to eliminate the contribution plot J,1 J,2 to compare with SJ. As we discussed above, if V ·V < R R + R from contact and bulk resistances. When PN junction are established ( 1 2 the DFO contribution exists, SJ would be larger than J,1 J,2.

R R + R SCIENCES 0), we find that SJ is always larger than J,1 J,2, by an amount that cor- Indeed, as shown in Fig. 2, RSJ is larger than RJ,1 + RJ,2 for V1 > responds to the contribution from optical behavior of charge carriers (col- 0.7 V, where the two PN junctions are well developed. RSJ is APPLIED PHYSICAL limation + reflection). (Inset) RSJ/(RJ,1 + RJ,2) plotted as a function of V1 and larger than RJ,1 + RJ,2 for PNP regime as well (Fig. 2, Inset), V2. As PN junction height becomes higher, the contribution from collimation and reflections increases up to ∼30%. directly confirming the DFO switching occurs at both polarities. In contrast, a control device of one-source geometry in Fig. 1A showed no difference between RSJ and RJ,1 + RJ,2 for PNP re- direction of the beam after collimation, directing beams with gime, implying no DFO contribution (SI Appendix, Fig. S4). wrong incident angles to the reflector. Disorder will thus reduce the filtering efficiency of the collimator–reflector pair. We follow analysis similar to ref. 26 to probe the disorder-induced degradation of collimation-filter DFO switch. We first assign the re- AB sistance of single PN junction RJ in the diffusive transport limit, by summing over all incident angles to the junction (8). For completely diffusive transport, we can write the total resistance of the device as a sum of serially connected local resistances, including the contributions from the junction, contact, and graphene channels. In a ballistic graphene channel where the DFO collimation-filter switching is effective, we then expect the measured RT to be substantially larger than the sum of all of the local resistance contributions (26). We emphasize that the trivial PN junction resistances them- selves contribute to the on/off as well; therefore, it is important to isolate the DFO contribution from RT. We have implemented several crucial device designs that enabled us to isolate the DFO contribution from trivial background junction resistances. Spe- cifically, independent control of carrier density in each gate region in our device design allows us to measure and eliminate the trivial resistance contributions from the collimator junctions (RJ,1) and the reflector junction (RJ,2) by using different gating schemes (SI Appendix, Fig. S1). As shown in Fig. 2, the collimation junction governs RT in the gate configuration A, while the reflector junction governs RT in the configuration B. Thus, Fig. 3. Extractions of transmission coefficients. (A) Extracted transmission R R T V V B J,1 and J,2 can be probed independently. In configuration A, coefficients ( ij) as a function of 1 and 2.( ) Relative transmission coeffi- T = T T + T V V V = the beam from the source crosses only the collimator junction cients, R 2 23/( 12 13) as a function of 1 and 2, and a 1D cut at 2 5 V. In the absence of PN junctions (NNN regime), T is very close to 1, and before reaching the drain electrode. In this configuration, RT is R currents injected from any contacts are equally split toward the other two expressed as RT(V1,V2) = RC,1(V1) + RG,1(V1) + RJ,1(V1,V2) + R V + R V R R reservoirs. When PN junctions are established, the optical behavior of DFs G,2( 2) C,2( 2), where C,1 and C,2 represent the contact T R leads to an enhancement of R value that is significantly higher than 1 resistance of source and drain electrodes, respectively, and G,1 ∼ R ( 1.4). Near zero gate voltage, carrier paths are no longer ballistic due to and G,2 do the graphene channel resistance of blue and green electron–hole puddles at charge neutrality point. DFO breaks down and R V regions, respectively. Here, J,1 is symmetric with exchanging 1 renders our method of extracting TR (SI) inaccurate. This leads to emergence V R V V = R V V V = V V = and 2, J,1( 1, 2) J,1( 2, 1), and vanishes when 1 2, of artifact seen at 1 0V.

Wang et al. PNAS Latest Articles | 3of5 Downloaded by guest on September 23, 2021 We deliberately divided the source into two terminals and A separated the injection and reflection paths to systematically characterize the DFO contribution by measuring the corresponding transmission coefficients. This measurement provides another independent metric for characterizing DFO contribution, consistent with our previous finding in Fig. 2. We analyze a full set of transmission coefficients Tij between the ith and jth terminal in our device as a function of two gate voltages (V1,V2). Note that the i and j indices can represent all three electrodes including two source and one drain electrodes. We employ a scattering matrix model in conjunction with the Landauer–Buttiker formalism to compute currents in all possible source–drain and gate configurations to determine Tij (SI Appendix,Figs.S2andS3). Fig. 3A shows Tij as a function of V1 and V2. The diagonal matrix elements (i = j) represent the fraction of carriers reflected back to the same B electrode from which they were injected. In the absence of PN junctions (along with the diagonal line for V1 = V2), the main contribution to the diagonal element Tii represents the probability of carriers being reflected right back at the contact interface in their unsuccessful attempts of getting through. Therefore, 1 – Tii is the contact transparency for the ith contact. We find each Tii approaches 0.6 in the NNN regime, consistent with the contact transparency of ∼0.4 estimated in the two-terminal resistance (SI Appendix,Fig.S5). The off-diagonal matrix elements of Tij contain the quality of DFO switching. In particular, in the presence of PN junctions, we expect the T23 = T32 (source-to-source reflection) is maximized, and T12 and T13 (source-to-drain transmission) are minimized. To quantify the quality of the DFO switching, we define the relative transmission coefficients, TR = 2T23/(T12 + T13). Fig. 3B Fig. 4. Temperature and bias current dependence. (A) On/off behavior of shows TR as a function of V1 and V2. TR is expected to be larger total resistance (RT) with a central gate (V2) at a fixed collimation gates (V1) in the off regime, while it becomes smaller in the on regime. A I Inset I T V V V = at 6 V with various bias current ( bias). ( ) On/off ratio as a function of bias horizontal line cut of R map in ( 1, 2) plane at 2 5 V shows B R V V = T shows no appreciable degradation. ( ) T as a function of 2 at 1 6 V also the evolution of R from the NPN regime to the NNN regime. shows a robust behavior against to the temperature (T) variation. (Inset) On/ The contact transparency along this line is kept high (>0.4) to off ratio stays the same up to T = 230 K. minimize its influence on TR. In the absence of PN junctions (NNN regime, V1 > 0), TR is close to 1, and the injected currents from one of the source contacts are equally split toward the other wide range of bias currents (up to 150 mA) and temperatures two electrodes. However, when the PN junctions are established, (1.8–230 K). We indeed confirm the stability of the device per- Klein tunneling across the junction establishes a collimation– formance over the entire measured range. The small change in T B reflection effect, increasing R above the unity. Fig. 3 shows RT near the peak around V2 ∼−1 V is due to the thermally that TR in the fully developed PNP regime can reach up to 1.4, excited electrons and holes across the Dirac point. However, the indicating that DFO switching is effective. Near zero gate voltage device characteristics for high values of jV2j are not affected by (charge neutrality point), carrier motions becomes nonballistic operating temperatures up to 230 K and channel current density due to the enhanced effect from disordered electron–hole pud- up to 102 A/m, suggesting the robustness of DFO process in our dles. In this regime, DFO picture breaks down and our method device. Further quantitative experimental confirmation of the T of extracting R becomes inaccurate. This leads to the strongly DFO contribution requires temperature-dependent evaluation fluctuating values of TR near V1 = 0V. of RSJ/(RJ,1 + RJ,2) and TR in the future studies. We also dem- Viewed as a transistor, our DFO switching device exhibits onstrate that the on/off device performance can be further im- modestly low on/off ratio due to the absence of any energy gaps, proved by engineering the geometric shape of gate electrodes and therefore due to the lack of carrier depletion and device and optimizing DFO (SI Appendix, Fig. S6). insulation. Instead of bandgap, we have introduced a transport In conclusion, a quantum switch based on DFO has been in- gap utilizing angle-dependent filtering by the collimator–reflector vestigated, utilizing angle dependence of Klein tunneling to re- pair. This transport gap is robust against temperature variations alize optical analogies of the tunable collimator and reflector. or bias voltages with ideal edges for Klein tunneling. Even in the presence of diffusive edge scattering, it turns into a pseudogap Experimental evidence of DFO characteristics has been dem- with a nonzero floor. Thus, it provides stability of the device onstrated and quantitatively characterized with two independent against temperature and bias voltages up to pseudogap range, metrics by isolating the net DFO contribution to device re- which depends on gate voltages (16). The critical device pa- sistance and by measuring a full set of transmission coefficients. rameters that govern the charge transport characteristics, in- Our analysis establishes multiple methodologies for the quan- cluding contact transparencies, Klein-tunneling probability, titative analysis of the DFO effect, leading to further device carrier densities, and quantum conductance for the channels are designing optimization (29, 30). The fully characterizable and all insensitive to the temperature and applied bias voltage below tunable DFO collimator and reflector provide the foundations critical values, presumably set by inelastic scattering processes. In for future microscopic-scale electron-optical components, to- the graphene channel with hBN, we expect such critical energy ward highly integrated and electrically controllable optical scale to be ∼100 meV, due to optical or substrate-induced circuits with variable wavelength and functionality for device phonons (27, 28). Fig. 4 shows the device characteristic with a operation.

4of5 | www.pnas.org/cgi/doi/10.1073/pnas.1816119116 Wang et al. Downloaded by guest on September 23, 2021 Methods of incident electrons that are transmitted and 1 – T reflected back. To cal- T E Sample Fabrication. The local bottom gates were fabricated by electron beam culate transmission ( ), we consider a generalized expression considering pseudospin conservation for angle dependent transmission across asym- lithography on SiO2 substrate with palladium–gold metallic alloy. Vacuum annealing of the metallic gates produces a surface roughness of ∼0.37 nm, metric PN junctions (10, 17). In this calculation, we use split distance between gates d = 60 nm, which is consistent with experimental data (∼50–80 nm which was limited by SiO2 substrate roughness. After fabrication of the local gates, a stack of hBN/graphene/hBN van der Waals heterostructure prepared from SEM and atomic force microscopy images). The contact i-to-contact j T =N N by dry transfer technique (31) was transferred onto the local gates. The flat transmission ij( j/ Total) is obtained by counting the number of electrons N j surface of the local gate ensures spatially uniform electrostatic gating, hence ( j) reaching the contact divided by total number of injected electrons well-defined straight PN junctions. High-contact transparency of electrodes (NTotal) from contact i. Then terminal current I is calculated from Landauer– to the graphene is critical for our experiments as opaque contacts with low Buttiker formula by summing up the terminal transmissions. transparency would hinder electrons to enter or exit electrodes and lower the visibility of the electronic optical phenomena happening in the gra- ACKNOWLEDGMENTS. The experimental work and theoretical analysis were phene channel. Here, we adopted an in situ etching technique (4, 32, 33) to partly supported by INDEX, a funded center of Nanoelectronics Research achieve highly transparent contacts. Initiative, a Semiconductor Research Corporation program sponsored by Nanoelectronics Research Corporation and National Institute of Standards and Simulation Method. Semiclassical ray tracing simulation considers electrons as Technology. P.K. acknowledges support from Office of Naval Research (ONR) Award N00014-16-1-2921 and the Lloyd Foundation. K. Wang acknowledges noninteracting point particles with speed v and mass m = (E − qV)/v 2 F F F partial support from ONR Award N00014-15-1-2761. G.-H.L. acknowledges following classical trajectories (billiard model) (23–25). Here, v is the Fermi F partial support from the National Research Foundation of Korea Grant funded E q velocity, F is the Fermi energy, and is the electrical charge. This has been by the Korean Government (Grant 2016R1A5A1008184). K. Watanabe and benchmarked against experiments on graphene PN junctions (5). Electrons T.T. acknowledge support from the Elemental Strategy Initiative conducted are injected from the source at random angles, weighted by a cosine dis- by the Ministry of Education, Culture, Sports, Science and Technology, Japan, tribution (34). Away from PN junctions, the electron trajectories are calcu- and Creating the Seeds for New Technology (Award JPMJCR15F3), Japan lated using classical laws of motion. At the junction, we estimate a fraction T Science and Technology Agency.

1. Castro Neto AH, et al. (2009) The electronic properties of graphene. Rev Mod Phys 81: 18. Quentin W, et al. (2014) A Klein-tunneling transistor with ballistic graphene. 2D 109–162. Materials 1:011006. 2. Young AF, Kim P (2009) Quantum interference and Klein tunnelling in graphene 19. Tan Y, et al. (2017) Graphene Klein tunnel transistors for high speed analog RF ap- heterojunctions. Nat Phys 5:222–226. plications. Sci Rep 7:9714. 3. Cheianov VV, Fal’ko V, Altshuler BL (2007) The focusing of electron flow and a Ve- 20. Jang MS, Kim H, Son YW, Atwater HA, Goddard WA, 3rd (2013) Graphene field effect SCIENCES selago lens in graphene p-n junctions. Science 315:1252–1255. transistor without an energy gap. Proc Natl Acad Sci USA 110:8786–8789.

4. Lee G-H, Park G-H, Lee H-J (2015) Observation of negative refraction of Dirac fermions 21. Sei M, et al. (2017) Dirac fermion reflector by ballistic graphene sawtooth-shaped npn APPLIED PHYSICAL Nat Phys – in graphene. 11:925 929. junctions. Semicond Sci Technol 32:045010. Science 5. Chen S, et al. (2016) Electron optics with p-n junctions in ballistic graphene. 22. Giovannetti G, et al. (2008) Doping graphene with metal contacts. Phys Rev Lett 101: – 353:1522 1525. 026803. 6. Efetov DK, et al. (2015) Specular interband Andreev reflections at van der Waals in- 23. Beenakker CWJ, van Houten H (1989) Billiard model of a ballistic multiprobe con- Nat Phys – terfaces between graphene and NbSe2. 12:328 332. ductor. Phys Rev Lett 63:1857–1860. 7. Beenakker CWJ (2008) Colloquium: Andreev reflection and Klein tunneling in gra- 24. Milovanovic SP, Ramezani Masir M, Peeters FM (2013) Spectroscopy of snake states phene. Rev Mod Phys 80:1337–1354. using a graphene Hall bar. Appl Phys Lett 103:233502. 8. Cheianov VV, Fal’ko VI (2006) Selective transmission of Dirac electrons and ballistic 25. Milovanovic SP, Ramezani Masir M, Peeters FM (2014) Magnetic electron focusing and magnetoresistance of n-p junctions in graphene. Phys Rev B Condens Matter Mater tuning of the electron current with a pn-junction. J Appl Phys 115:043719. Phys 74:041403. 26. Stander N, Huard B, Goldhaber-Gordon D (2009) Evidence for Klein tunneling in 9. Katsnelson MI, Novoselov KS, Geim AK (2006) Chiral tunnelling and the Klein paradox graphene p-n junctions. Phys Rev Lett 102:026807. in graphene. Nat Phys 2:620–625. 27. Hwang EH, Das Sarma S (2008) Acoustic phonon scattering limited carrier mobility in 10. Sajjad RN, Ghosh AW (2013) Manipulating chiral transmission by gate geometry: two-dimensional extrinsic graphene. Phys Rev B 77:115449. Switching in graphene with transmission gaps. ACS Nano 7:9808–9813. 28. Chen J-H, Jang C, Xiao S, Ishigami M, Fuhrer MS (2008) Intrinsic and extrinsic per- 11. Williams JR, Low T, Lundstrom MS, Marcus CM (2011) Gate-controlled guiding of Nat Nanotechnol – electrons in graphene. Nat Nanotechnol 6:222–225. formance limits of graphene devices on SiO2. 3:206 209. 12. Rickhaus P, et al. (2015) Guiding of electrons in a few-mode ballistic graphene 29. Paris MGA (1999) Entanglement and visibility at the output of a Mach-Zehnder in- Phys Rev A – channel. Nano Lett 15:5819–5825. terferometer. 59:1615 1621. Nat Commun 13. Kim M, et al. (2016) Valley-symmetry-preserved transport in ballistic graphene with 30. Bøggild P, et al. (2017) A two-dimensional Dirac fermion microscope. 8: gate-defined carrier guiding. Nat Phys 12:1022–1026. 15783. 14. Liu M-H, Gorini C, Richter K (2017) Creating and steering highly directional electron 31. Wang L, et al. (2013) One-dimensional electrical contact to a two-dimensional ma- beams in graphene. Phys Rev Lett 118:066801. terial. Science 342:614–617. 15. Rickhaus P, et al. (2015) Gate tuneable beamsplitter in ballistic graphene. Appl Phys 32. Calado VE, et al. (2015) Ballistic Josephson junctions in edge-contacted graphene. Nat Lett 107:251901. Nanotechnol 10:761–764. 16. Sajjad RN, Ghosh AW (2011) High efficiency switching using graphene based electron 33. Ben Shalom M, et al. (2015) Quantum oscillations of the critical current and high-field “optics.” Appl Phys Lett 99:123101. superconducting proximity in ballistic graphene. Nat Phys 12:318–322. 17. Sajjad RN, et al. (2012) Manifestation of chiral tunneling at a tilted graphene p-n 34. Molenkamp LW, et al. (1990) Electron-beam collimation with a quantum point conjunction. Phys Rev B Condens Matter Mater Phys 86:155412. tact. Phys Rev B Condens Matter 41:1274–1277.

Wang et al. PNAS Latest Articles | 5of5 Downloaded by guest on September 23, 2021