title: Bipolar electron waveguides in graphene
authors: R. R. Hartmann M. E. Portnoi
text: 
  
  
  
  
  
  
  
  richard.hartmann@dlsu.edu.ph
  
  Physics Department, De La Salle University,  2401 Taft Avenue, 0922 Manila, Philippines
  
  
  
  M.E.Portnoi@exeter.ac.uk
  Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
  ITMO University, St. Petersburg 197101, Russia
  
  
               
  
  
  
  We show analytically that the ability of Dirac materials to localize an electron in both a barrier and a well can be utilized to open a pseudo-gap in graphene's spectrum. By using narrow top-gates as guiding potentials, we demonstrate that graphene bipolar waveguides can create a non-monotonous one-dimensional dispersion along the electron waveguide, whose electrostatically controllable pseudo-band-gap is associated with strong terahertz transitions in a narrow frequency range.
  
  
  
  
  Bipolar electron waveguides in graphene
      M. E. Portnoi
      December 30, 2023
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  § INTRODUCTION
  
  Graphene's gapless, relativistic spectrum leads to many unusual transport properties <cit.>, such as Klein tunneling <cit.> and the suppression of back scattering <cit.>. Graphene also exhibits strong optical transitions, with a universal absorption of π e^2/ ħ c ≈ 2.3 % over a broad range of frequencies <cit.>. These defining features of graphene make electronic and optical control difficult to achieve. Indeed, for the realisation of optoelectronic devices which capitalize on the relative nature of graphene's dispersion, one must overcome certain undesirable relativistic features, without destroying all the attractive effects.
  
  
  For optoelectronic devices applications, it would be highly  desirable to modify graphene's spectrum such that it possesses valence-like and conduction-like bands, separated by an externally tunable pseudo-bandgap (a true gap would require a drastic modification of the material by either functionalization, cutting or rolling), as well as reduce the freedom of quasiparticle  motion from two-dimensions down to one. In essence, we wish to form an analog of a narrow-gap carbon nanotube spectrum, without physically deforming the sheet by cutting it to make a ribbon, or rolling it to form a nanotube. In this paper, we propose a setup based on a bipolar potential, which does not open a true bandgap. However, at certain parameters of the potential, the electron dispersion becomes non-monotonous, and has pronounced extrema associated with avoided crossings between the states confined within a barrier and a well. We call the energy separation at a local dispersion minimum a pseudo-gap. Such a pseudo-gap would result in the giant enhancement of the probability of optical transitions <cit.>. However, unlike a nanotube or ribbon, whose bandgap is predefined by geometry, we seek to create a pseudo-gap which is fully tunable, without the need for huge magnetic fields <cit.>.  
  
  
  
  Rather than achieving the quantization of momentum through geometry like a nanoribbon, or nanotube does, one may instead quantize momentum via the application a quasi-one-dimensional (1D) electrostatically defined potential, i.e., by using electron waveguides <cit.>. Unlike a physical tube whose radius cannot be changed, externally applied potentials can be easily varied. 
  
  
  
  There has been significant experimental progress since the pioneering work in the field of graphene electron waveguides <cit.> and the recent breakthrough of utilizing a nanotube as a top gates <cit.> enabled the detection of individual guided modes within a single waveguide. However, apart from graphene waveguides possessing a threshold in the characteristic potential strength required to observe a fully bound mode <cit.>, one could argue that a single graphene electron waveguides provides similar physics to that studied in quasi-1D channels within conventional semiconductor systems. Indeed, the absolute value of electron momentum along the direction of a waveguide formed by an attractive potential (quantum well) defines the electron's effective mass. Confined states of a deep well start with negative energy and for large values of momentum have a positive dispersion along the waveguide. Similarly, a potential barrier can also form a waveguide, where the confined electron states have negative dispersion, i.e., are hole-like. This raises the question what happens when branches of negative and positive dispersion meet? Answering this question is the focus of this paper. 
  In what follows we show that a bipolar electronic waveguide is a fully tunable quasi-1D system with a non-monotonous dispersion accompanied by pseudo-gaps, characterized by a giant enhancement of density of states and interband dipole transition probabilities in the energy range where graphene's own density of states is rather small. 
  In addition we present a general analytic formalism allowing us to find with a spectacular degree of accuracy the main features of a bipolar waveguide from the properties of a single quantum well.
  
  
  
  
  
  
  
  
  
  Transmission through single and multiple barriers in graphene has been a subject of extensive research <cit.> including periodic potentials <cit.> and sinusoidal multiple-quantum-well systems <cit.>. 
  
  Despite this significant body of research, the phenomenon of pseudo-gap formation in bipolar waveguides has been hitherto overlooked. 
  
  It should be emphasized that the idea of using bipolar waveguides stems directly from the essential feature of graphene, as a gapless material, that a potential barrier can contain guided electron modes, effectively acting as a potential well. For non-relativistic particles, the probability of tunneling between two wells results in the splitting of energy levels, to form a doublet state. In contrast, the probability of tunneling between a well and a barrier in graphene results in the bands forming an avoided crossing at finite k_y, where ħ k_y, is the momentum along the waveguide. As we demonstrate below, by modulating the applied voltage, bipolar waveguides have fully controllable pseudo-gaps, which exhibit extremely strong optical transitions. Not only this, but these transitions occur in the highly elusive and desirable THz frequency range. This part of the electromagnetic spectrum is notoriously difficult to generate and manipulate <cit.>. Therefore, by using a suitably chosen combination of guiding potentials, one can transform a graphene sheet into a narrow-gap nanotube without rolling, where the effective nanotube radius is controlled by the strength of the applied potential.
  
  
  Inspired by the most advanced experimentally attainable waveguides <cit.>, our proposed bipolar waveguide is defined by two nanotubes (top gates), of radius r_0, separated by a distance D, as shown in Fig. <ref> (a). Both nanotubes are placed at a height, h, above the metallic substrate. The potential profile in the graphene plane separated from the same substrate by another distance t is given by the expression:
  
      U(x)=eϕ_1/2ln[(x+D/2)^2+(h-t)^2/(x+D/2)^2+(h+t)^2]+eϕ_2/2ln[(x-D/2)^2+(h-t)^2/(x-D/2)^2+(h+t)^2],
  
  where ϕ_1,2=ϕ_1,2/ln(2h-r_0/r_0), and ϕ_1 (ϕ_2) is the applied voltage between the left (right) nanotube and the back gate. The expression above can be easily modified when the top gates are fully embedded in a dielectric material. In the above-mentioned recent work <cit.>, a smooth electron waveguide was fabricated using a carbon nanotube as a top gate and the graphene sheet was sandwiched in between two layers of hexagonal boron nitride (h-BN). The top h-BN layer had a thickness, h, of between 4 and 100 nm, and the bottom layer had a thickness, t of around 20 nm. It can be seen from Fig. <ref> (b) that the potential in the plane of the graphene sheet, derived using image charges in the substrate, can be very well approximated by a linear combination of two shifted hyperbolic secant functions. It is convenient to use this particular approximation because for a single well, the hyperbolic secant potential possesses quasi-exact solutions to the Dirac equation <cit.>. This will enable us to treat both the size of the pseudo-gap and the optical transitions across it analytically.
  
  
  
  
  § NEGATIVE DISPERSION AND PSEUDO-GAPS
  
  
  
  The low-energy quasiparticle behaviour in graphene is known to be described with spectacular accuracy by the 2D Dirac equation for massless fermions <cit.>. In the presence of a confining electrostatic potential, U(x), the effective 1D matrix Hamiltonian for confined modes in a graphene waveguide can be written in the standard basis of graphene's two sub-lattices as
  
      Ĥ=ħ v_F(
      k̂_xσ_x+s_Kk_yσ_y) +IU(x),
  
  where k̂_x=-i∂/∂ x, k_y is a wavenumber corresponding to the motion along the waveguide, σ_x, y, z are the Pauli spin matrices, I is the 2 by 2 unit matrix, v_F is the Fermi velocity, which is approximately ≈10^6 m/s, and s_K is the valley quantum number, which has the value of +1 and -1 for the K and K' valley respectively. 
  
  We are interested in the situation when U(x) is a combination of two fast decaying potentials separated by a distance d. For a better understanding of the underlying physics it is instructive to look both at the same and different sign constituent potentials. This allows a comparison with the familiar non-relativistic results for the double quantum well. The results are especially transparent for a quasi-one dimensional potential formed by a combination of either; a well and a barrier of the same strength, or, two equal wells:
  
      U(x)=
      u(x+d/2) ±  
      u(x-d/2),
  
  where, u(x) is an individual symmetric potential well, for which the Hamiltonian, Eq. (<ref>), admits exact zero-energy eigenfunctions, which we denote as Ψ_0(x). The function Ψ_0(x) is normalized and assumed to rapidly decay outside of the well. The tunneling between wells, or between a well and a barrier, results in the energy level, E=0, splitting into two levels, E_1 and E_2. At E=0, the barrier wave function, denoted Ψ_-, is the complex conjugate of the well wave function, Ψ_-=Ψ_0^⋆(x-d/2), reflecting the fact that the barrier for electrons is a well for holes; whereas, for the two-well case Ψ_+=Ψ_0(x-d/2).
  
  
  In the weak wavefunction overlap approximation, resembling the tight-binding (Hückel molecular orbital) methods widely used in solid-state and molecular physics, we may write the wave functions corresponding to eigenvalues E_1 and E_2 as:
  
      Ψ_1=1/√(2)[Ψ_±(x-d/2)+Ψ_0(x+d/2)],
  
  
      Ψ_2=1/√(2)[Ψ_±(x-d/2)-Ψ_0(x+d/2)].
  
  Following an approach similar to the non-relativistic case <cit.>, but for a matrix Hamiltonian, the energy level splitting, E_g=|E_2-E_1|, can be shown to be
  
  
      E_g=2ħ v_F|Ψ_±^†(-d/2)σ_xΨ_0(d/2)|.
  
  It should be noted that there is a striking difference between the relativistic and non-relativistic case. For the non-relativistic case the splitting is proportional to the product of the single well function and its derivative <cit.>; whereas, in the relativistic case, it depends only on the individual well and barrier functions (or the two shifted well functions for the double well). For simplicity, we considered above two potentials of equal strengths, and estimated the splitting of the E=0 state. These results can be easily generalized for non-equal potentials and non-zero values of energies as long as the energy levels in the individual potentials coincide (see Appendix). This theorem demonstrates the utility of quasi-exact solutions to the Dirac equation <cit.> for bipolar waveguides. Indeed, the exact solutions often correspond to the case where there is symmetry between the positive and negative energy solutions, allowing all pseudo-gaps to be treated within this formalism. Furthermore, knowledge of the exact wave-functions allows the matrix element of optical transitions across the pseudo-gaps to be calculated analytically. 
  
  
  
  
  
  
  
  In what follows we shall model a bipolar waveguide as
  
  
      U(x)=-u_1/cosh[(x+d/2)/L]+u_2/cosh[(x-d/2)/L],
  
  where u_1 and u_2 are the depth of the well and the height of the barrier, respectively, and L is the effective width of the potential, which for now we assume to be the same for the well and the barrier. For u_1=u_2=u_0>0, this smooth potential indeed provides an excellent fit to the realistic potential generated by two oppositely charged nanotubes above the surface of graphene, see Fig. <ref> (b). Also each of the individual secant potentials supports exact analytic solutions at E=0 <cit.>, allowing the comparison of the numerical results with the approximate formula, Eq. (<ref>). It is convenient to introduce similar dimensionless parameters as Ref. <cit.>, namely ω=u_0 L / ( ħ v_F) and Δ=|k_y| L. It should be noted that the number of bound states contained within a realistic confining potential is also defined by the product of the characteristic potential depth and its width <cit.> rather than its exact form. Effects such as non-linear screening and the renormalization of the Fermi velocity <cit.>, can be accommodated within the same dimensionless parameter, u_0 L/(ħ v_F). The particular choice of the potential we use does not influence the physical picture. The use of this dimensionless parameter also allows the application of our results to other 2D systems with linear dispersion beyond graphene, e.g., surface states of topological insulators.
  
  In the absence of inter-potential tunneling, the dispersion lines of the well and barrier (indicated in Fig. <ref> by blue and red lines respectively) cross when Δ_n=ω - n -1/2, where n is a positive integer <cit.>. The corresponding exact wavefunctions when substituted into Eq. (<ref>) yield the following approximate expression for the n=0 pseudo-gap (see Appendix)
  
  
  
  
      E_g/(ħ v_F/L)≈2exp(-Δ_0d/L)/B(1+Δ_0,Δ_0),
  
  where B(m,n) is the Beta function. This formula gives an extremely good approximation to the numerical solution in the limit when the ratio between the wire separation and the effective width of the potential is large, i.e., d/L ≫ 1. 
  
  
  
  
  In Fig. <ref> we plot the numerically obtained energy dependence on Δ, i.e. the momentum along the barrier in dimensionless units, for the cases of ω=0.75 (panel (a)) and ω=1.75 (panel (b)). In both cases the two oppositely charged nanotubes are separated by a distance d/L=8, which corresponds to approximately 175 nm for the case of h=40 nm and t=20 nm. 
  At this distance the energy-level splitting formula, Eq. (<ref>), accurately predicts the value of the n=0 pseudo-gap within a few per cent error. This error becomes one order of magnitude smaller when d/L=12. It is instructive to compare these electrostatically induced pseudo-gaps with curvature-induced gaps in carbon nanotubes. For a narrow-gap carbon nanotube, it is well established that the larger is its radius, the smaller is the curvature-induced gap <cit.>. Therefore, the strength of the applied voltage, for a particular guided mode, can be mapped to the radius of the nanotube. Increasing the voltage results in more tightly confined guided modes, characterised by a higher value of Δ_0 entering Eq. (<ref>), therefore we arrive to a smaller value of the pseudo-gap, which moves to the right in Fig. <ref> (b).
  
  
  It can also be seen from panel (b) of this figure that the deeper the well and higher the barrier, the more states are contained within each channel, increasing correspondingly the number of avoided crossings appearing in the dispersion. It should also be noted that although increasing the voltage leads to stronger confinement for lower order modes, it also results in the appearance of higher order modes which are more spread out, leading to additional wider pseudo-gaps. In Fig. <ref> (b), which corresponds to the case of a deeper well and higher barrier, we can see two pseudo-gaps at E=0 as well as additional pseudo-gaps at non-zero energy since there are more guided modes in the low-energy part of the spectrum.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  It can be seen by comparing Fig. <ref> (c) to (a) that increasing the distance between the nanotubes, decreases the size of the gap. This is a result of the decrease in the overlap between the well and barrier functions. Technologically it is quite difficult to have exact control over the precise tube separation. However, this is not so important since it is possible to control the value of the pseudo-gap by the applied voltage. Furthermore, it can be see from Fig. <ref> (d) that the effect of pseudo-gap opening is robust against asymmetry in the system. In Fig. <ref> (d), the dispersion is recalculated for two tubes separated at the same distance as in Fig. <ref> (a), but with the depth of the well increased to ω=1.9, while the size of the barrier decreased to ω=0.6. This demonstrates that top gates of mismatched radius, or dissimilar magnitudes of applied voltage will just shift the value in energy in which the avoided crossing occurs. Although Eqs. (<ref>,<ref>) give for the double well and bipolar waveguide the same value of energy-level splitting at the value of k_y corresponding to E=0 in a single well, for the double well there is no pseudo-gap and the dispersion remains monotonous, this case is considered in depth elsewhere <cit.>.
  
  
  
  § INTERBAND TRANSITIONS
  
  In what follows we shall demonstrate that much like in narrow-gap nanotubes <cit.>, the 
  wavefunction intermixing leads to strongly allowed optical transitions across the pseudo-gaps. 
  
  The probability of optical transitions is proportional to the squared modulus of the matrix element of velocity operator between the relevant states. The velocity operator written in the same basis as the Hamiltonian given in Eq. (<ref>) is <cit.> 
  
  
      v̂=v_F(σ_xx̂+s_Kσ_yŷ).
  
  The probability of a dipole transition is proportional to |⟨Ψ_f|v̂·e|Ψ_i⟩|^2,
  where Ψ_i and Ψ_f are the initial and final states, respectively, and e=(e_x,e_y) is the light polarization vector. For linearly polarized light, e=(cos(φ_0),sin(φ_0)), while for right- and left-handed polarized light e=(1,-i)/√(2) and e=(1,i)/√(2). Within the small wavefunction overlap approximation, for a bipolar waveguide defined by the potential given by Eq. (<ref>) with u_1=u_2=u_0>0, the matrix element of velocity of the n=0 mode at Δ=Δ_0 is (see Appendix)
  
      |⟨Ψ_2|v̂|Ψ_1⟩|/v_F≈|e_xB(1-e^-d/L;1/2+Δ_0, 0)e^-Δ_0d/L/B(1+Δ_0, Δ_0)-ik_y/|k_y|e_yB(1/2+Δ_0, 1/2+Δ_0)/B(1+Δ_0, Δ_0)|,
  
  where B(x;m,n) is the incomplete Beta function. For the case of the double well, at the same value of momentum, the matrix element of velocity of the n=0 mode is the first term of Eq. (<ref>), whereas v_y=0. It reflects the fact that transitions in a double well are only caused by light polarized along the x-direction (normal to the waveguide), similar to the non-relativistic case and the 1D square-well potential in graphene <cit.>.
  
  In stark contrast, the transition across the pseudo-gap of a bipolar waveguide is strongly polarized along the y-axis (waveguide direction). The situation changes away from the pseudo-gap. For small values of |k_y|, the transitions are polarized normally to the y-direction, as expected from the momentum alignment phenomenon in graphene <cit.>. For large values of |k_y|, the overlap between the well and barrier wavefunctions becomes very small leading to vanishing transition probabilities for both polarizations. This effect very much resembles the situation in a narrow-gap carbon nanotube, where optical transitions polarized along the nanotube axis are allowed in the narrow energy interval around the curvature-induced gap <cit.>. The main difference from a nanotube can be clearly seen from Fig. <ref>, namely both transitions polarizations along and normal to the waveguide are allowed across the pseudo-gap. The contribution of the x-component exponentially decreases with an increase in top gate potential or the separation between the gates, as can be seen from Fig. <ref>. 
  
  
  
  The presence of both polarizations for pseudo-gap transitions leads to an effect absent in both graphene and non-chiral nanotubes. Namely, right- and left-handed polarized light produces different populations of pseudo-valleys with opposite signs of k_y. This can be clearly seen from Eq. (<ref>).  This feature does not depend on the model describing the potential (see Appendix).
  
  The discussed transitions across the pseudo-gap, fully controlled by the top gate voltages, can be easily brought into the highly desirable THz frequency range, which is usually extremely difficult to control. The presence of the van Hove singularity at the pseudo-gap edge enhances the strength of these transitions. These effects opens the avenue for novel gate-controlled polarized sensitive THz detectors based on bipolar waveguides in graphene.
  
  
  
  
  
  
  § CONCLUSIONS
  
  To conclude, we have shown that bipolar waveguides allows for the creation of a non-monotonous 1D dispersion along the electron waveguide. The repulsion of well and barrier states results in the appearance of pseudo-gaps in the spectrum, whose size and symmetry can be fully controlled by the top gate voltages. These gaps can be estimated analytically for exactly-solvable potentials. The opening of these pseudo-gaps results in strongly allowed THz transitions with non-trivial optical selection rules. 
  The predicted negative dispersion of the guided modes may lead to various other physical effects ranging from solitary waves <cit.> to Gunn-diode type current oscillations <cit.>.
  
  
  
  
  § ACKNOWLEDGEMENTS
  
  We thank C.A. Downing for his critical reading of the manuscript. This work was supported by the EU H2020 RISE project TERASSE (H2020-823878). RRH acknowledges financial support from URCO (71 F U 3TAY18-3TAY19). The work of MEP was supported by the Ministry of Science and Higher Education of Russian Federation, Goszadanie no. 2019-1246. 
  
  
  
  
  
  § DERIVATION OF THE APPROXIMATE EXPRESSION FOR THE PSEUDO-GAP
  
  In what follows, we show how the bound state eigenfunctions of an individual quasi-1D quantum well in graphene can be used to obtain the approximate size of the pseudo-gaps present in a graphene bipolar waveguide's energy spectrum. Let us consider a bipolar waveguide created by the 1D guiding potential,
  
      U(x)=u_I(x+d/2)+u_II(x-d/2),
  
  built from two separate guiding channels u_I and u_II, separated a distance d apart. The single well and barrier 2D Dirac equations are
  
      [Ĥ_0+Iu_I(x+d/2)]Ψ_d/2=EΨ_d/2,
  
  and
  
      [Ĥ_0+Iu_II(x-d/2)]Ψ_-d/2=EΨ_-d/2,
  
  respectively, and Ĥ_0=ħ v_F(-iσ_x∂/∂ x+σ_yk_y), where σ_x,y are the Pauli spin matrices, I is the 2 by 2 unit matrix, and k_y is the electron wavenumber along the direction of the waveguide, and Ψ_±d/2 is a two component wavefunction of the form
  
      Ψ_±d/2 
      =1/N_±d/2([ ψ_A(x±d/2); ψ_B(x±d/2) ]),
  
  where the subscripts d/2 and -d/2
  correspond to the left and right guiding potential, respectively. Finally, the spinor components ψ_A and ψ_B are associated with graphene's two sub-lattices and N^2_±d/2=∫_-∞^∞(|ψ_A(x±d/2)|^2+|ψ_B(x±d/2)|^2)dx. 
  
  
  
  Let us consider the case when at energy, E, the bound state energy level contained within each guiding potential coincides, and splits into the levels E_1 and E_2. In this instance their corresponding wavefunctions are
  
      Ψ_1=1/√(2)(Ψ_-d/2+Ψ_d/2),
  
  
      Ψ_2=1/√(2)(Ψ_-d/2-Ψ_d/2),
  
  
  which obey the differential equations
  
      [Ĥ_0+IU(x)]Ψ_1=E_1Ψ_1,
  
  
      [Ĥ_0+IU(x)]Ψ_2=E_2Ψ_2.
  
  In the small wavefunction overlap approximation, for the rightmost waveguide, the function Ψ_d/2 is assumed to be vanishingly small in comparison to Ψ_-d/2 and vice versa, i.e., _0^∞Ψ_-d/2^†Ψ_d/2dx≈0. Multiplying the above expressions by Ψ_-d/2^†, subtracting Eq. (<ref>) from Eq. (<ref>), and integrating from x=0 to ∞ yields 
  
      E_g=2|_0^∞[Ψ_d/2^†Iu_II(x-d/2)Ψ_-d/2]^⋆dx|,
  
  where E_g=|E_2-E_1|. Using Eq. (<ref>) and Eq. (<ref>) allows one to write
  
      E_g=2|_0^∞(Ψ_d/2^†Ĥ_0Ψ_-d/2)^⋆dx|.
  
  Substituting the wavefunctions, Eq. (<ref>), into Eq. (<ref>), followed by integration by parts yields
  
  
      E_g=2|ħ v_FΨ_-d/2^†(0)σ_xΨ_d/2(0)+i_0^∞Ψ_-d/2^†Iu_I(x+d/2)Ψ_d/2dx|.
  
  From x=0 to ∞, the integral appearing in the above expression is negligibly small compared to the lead term. 
  Therefore, the size of the pseudo-gap, E_g, can be expressed as
  
      E_g=2ħ v_F|Ψ_-d/2^†(0)σ_xΨ_d/2(0)|.
  
  
  We shall now show that for the case of a bipolar waveguide composed of a barrier potential, u_II(x), which is equal and opposite in sign to the well potential, u_I(x), i.e., u_II(x)=-u_I(x-d), that it is sufficient to use the bound state solutions of the well alone to calculate the size of the pseudo-gaps. Let us consider the case of a single potential well, defined by the electrostatic potential u(x), whose exact bound state solutions for graphene are known. The massless 2D Dirac equation of a barrier defined by the quasi-1D potential -u(x) can be written as
  
      [-iσ_x∂/∂ x+k_yσ_y-Iu(x)]Ψ=EΨ,
  
  which is formally equivalent to 
  
      [-iσ_x∂/∂ x+k_yσ_y+Iu(x)]Ψ^⋆=-EΨ^⋆.
  
  Therefore, at E=0 the wavefunctions of the barrier are simply the complex conjugate of the identical well, and at non-zero energy, the barrier wavefunctions can be obtained from the well wavefunctions by taking its complex conjugate accompanied by an exchange of the sign of E. Let us now consider the particular case of a bipolar waveguide described by the potential given in Eq. (<ref>), where
  
      u_I=-u_0/cosh[(x+d/2)/L],     u_II=u_0/cosh[(x-d/2)/L],
  
  and u_0>0. For the hyperbolic secant potential the bound state zero-energy solutions to Eq. (<ref>) (obtained in Ref. <cit.>) for k_y>0 and s_K=1 are
  
      Ψ_A=i(-1)^n+1 _2F_1(-n+2ω, -n; 1-n+ω; t)t^-n/2+ω/2+1/4(1-t)^-n/2+ω/2-1/4
  
  and
  
      Ψ_B= _2F_1(-n+2ω, -n; 1-n+ω; 1-t)t^-n/2+ω/2-1/4(1-t)^-n/2+ω/2+1/4
  
  where _2F_1(a,b;c;x) is the hypergeometric function, 2t= 1-tanh[(x+d/2)/L], ω=u_0/(ħ v_F/L) and n is a positive integer. It should be noted that for the same valley, the negative k_y solutions can be obtained by simply exchanging Ψ_A for Ψ_B and vice-versa. Similarly for positive k_y, the opposite valley sub-lattice functions can be obtained from Eqs. (<ref>-<ref>) via the interchange of Ψ_A and Ψ_B. Substituting the exact solution for the n=0 mode into Eq. (<ref>) yields 
  
      E_g=2^1-2ωħ v_F/N_d/2N_-d/2|(1-ρ)^ω+1/2(1+ρ)^ω-1/2+(1-ρ)^ω-1/2(1+ρ)^ω+1/2|,
  
  where ρ=tanh(d/2L), and since d≫ L Eq. (<ref>) becomes
  
      E_g≈2ħ v_Fe^-(ω-1/2)d/L/LB(ω+1/2,ω-1/2),
  
  where B(m,n) is the Beta function.
  
  
  
  
  § APPROXIMATE EXPRESSION FOR THE TRANSITION MATRIX ELEMENT
  
  In the presence of an electromagnetic field, the particle momentum operator, p̂, is modified such that p̂→p̂+eA/c, where e is the elementary charge, and A is the magnetic vector potential, which is related to e=(e_x,e_y), the unit vector describing the polarization of the electromagnetic wave, via the relation A=Ae. For linearly polarized light, the polarization vector is expressed as (cos(φ_0),sin(φ_0)), while for right- and left-handed polarized light it is
  (1,-i)/√(2) and (1,i)/√(2), respectively. The general form of the perturbation due to an electromagnetic wave impinging normally to a Dirac material is
  
      δ H=
      eA v_F/c(σ_xe_x+s_Kσ_ye_y),
  
  which is related to the velocity operator, v̂, given by Eq. (<ref>) in the main text, by the simple relation δ H=(eA/c)v̂·e <cit.>. The transition matrix element is proportional to |⟨Ψ_f|v̂·e|Ψ_i⟩|^2, where Ψ_i and Ψ_f are the initial and final states, respectively. Using the functions given in Eq. (4) and Eq. (5) of the main text, the transition matrix element at the pseudo-gap edge (or at the same value of momentum between the doublet states for two wells) in the small wavefunction overlap approximation is given by:
  
      |⟨Ψ_2|v̂·e|Ψ_1⟩|/v_F=δ_k_y,i,k_y,f|_-∞^∞(Φ_0+Φ_±)dx|,
  
  where the subscripts + and - correspond to the case of the double well and bipolar waveguide, respectively, k_y,i and k_y,f are the initial and final wavenumbers along the direction of the waveguide, and the terms Φ_0 and Φ_± are defined as:
  
      Φ_0   =i Im[Ψ_±^†(x-d/2)(σ_xe_x+s_Kσ_ye_y)Ψ_0(x+d/2)],
      Φ_-   =Ψ_-^†(x-d/2)s_Kσ_ye_yΨ_-(x-d/2),
      Φ_+   =0.
  
  Since the potentials u_I and u_II are symmetric,  the solutions of ψ_A(x±d/2) and ψ_B(x±d/2) can be constructed to be purely imaginary and real respectively. Therefore, Φ_0,  which appears in the transition matrix element, Eq. (<ref>), reduces to 
      Φ_0=i Im[Ψ_±^†(x-d/2)σ_xe_xΨ_0(x+d/2)].
  
  It should be noted that for pseudo-gaps arising from the repulsion of well and barrier zero-energy guided modes, Ψ_-(x-d/2)=Ψ_0^⋆(x-d/2), this coupled with the fact that ψ_A is purely imaginary, and ψ_B, real, means that Φ_0 appearing on the left hand side of Eq. (<ref>) neither depends on the sign of k_y nor on s_K; whereas Φ_- is a function of k_y, i.e., Φ_-=Φ_-(k_y), which obeys the relation Φ_-(-k_y)=-Φ_-(k_y), and is independent of s_K.
  
  We will now determine the dipole matrix element for transitions across the n=0 pseudo-gap edge of a bipolar graphene waveguide, composed of a well and barrier of equal strength, described by the hyperbolic secant functions given in Eq. (<ref>). In the limit that d/L ≫ 1, Ψ_A(x-d/2)Ψ_B(x+d/2) 
  ≫Ψ_A(x+d/2)Ψ_B(x-d/2). Upon changing to the variable t̃={ 1-tanh[(x-d/2)/L]} /2, integrating Φ_0 across the domain of x, and retaining only the Ψ_A(x-d/2)Ψ_B(x+d/2) terms, one obtains
  
      _-∞^∞Φ_0 dx≈-ie_xe^-Δ_0d/L/B(1+Δ_0, Δ_0)_0^1-e^-d/Lt̃^-1/2+Δ_0(1-t̃)^-1dt̃.
  
  The integral ∫_0^1-e^-d/Lt̃^-1/2+Δ_0(1-t̃)^-1dt̃ is of the form of the incomplete Beta function, which we shall denote as B(1-e^-d/L;1/2+Δ_0,0). 
  
  The integration of Φ_- from x=-∞ to ∞ yields
  
      _-∞^∞Φ_- dx=-k_y/|k_y|e_yB(1/2+Δ_0,1/2+Δ_0)/B(1+Δ_0, Δ_0).
  
  Therefore, for the bipolar waveguide, the transition matrix element at the n=0 pseudo-gap edge is
  
      |⟨Ψ_2|v̂·e|Ψ_1⟩|/v_F≈|e_xB(1-e^-d/L;1/2+Δ_0, 0)e^-Δ_0d/L/B(1+Δ_0, Δ_0)-ik_y/|k_y|e_yB(1/2+Δ_0, 1/2+Δ_0)/B(1+Δ_0, Δ_0)|,
  
  while for the case of the double well, at the same value of momentum, the transition matrix element is 
  
      |⟨Ψ_2|v̂·e|Ψ_1⟩|/v_F≈|e_xB(1-e^-d/L;1/2+Δ_0, 0)e^-Δ_0d/L/B(1+Δ_0, Δ_0)|.
  
  The striking feature of Eq. (<ref>), which also appears in the main text as Eq. 10, is the apparent dependence of the pseudo-valley population on the handedness of the excitation. This is indeed allowed by symmetry, since the inversion symmetry is lifted in a bipolar waveguide by opposite top-gate potentials. This feature does not depend on the particular model chosen to describe the bipolar waveguide, as we show below. 
  
  In the most general form the transition matrix element can be written as
  
      |⟨ f|
      v̂·e|i⟩|/v_F=|_-∞^∞[Ψ_B,f^⋆(e_x+ie_ys_K)Ψ_A,i+Ψ_A,f^⋆(e_x-ie_ys_K)Ψ_B,i]dx|,
  
  where the subscripts i and f correspond to the initial and final states. The spinor components Ψ_A and Ψ_B depend on both the valley index number, s_K, and the sign of k_y, i.e., Ψ_B,f=Ψ_B,f(s_K,s), where s=k_y/|k_y|. From Eq. (2) of the main text it is clear that the change of the sign of k_y, as well as s_K, lead to the swapping of the A and B indices in Eq. (<ref>). For s=1 and s_K=1, Eq. (<ref>) yields
  
      |⟨ f|v̂·e|i⟩|/v_F
      =|_-∞^∞[Ψ_B,f^⋆(e_x+ie_y)Ψ_A,i+Ψ_A,f^⋆(e_x-ie_y)Ψ_B,i]dx|.
  
  For s=1 and s_K=-1,
  
      |⟨ f|v̂·e|i⟩|/v_F
      =|_-∞^∞[Ψ_B,f^⋆(e_x+ie_y)Ψ_A,i+Ψ_A,f^⋆(e_x-ie_y)Ψ_B,i]dx|.
  
  For s=-1 and s_K=1, 
  
      |⟨ f|v̂·e|i⟩|/v_F=|_-∞^∞[Ψ_B,f^⋆(e_x-ie_y)Ψ_A,i+Ψ_A,f^⋆(e_x+ie_y)Ψ_B,i]dx|.
  
  For s=-1 and s_K=-1,
  
      |⟨ f|v̂·e|i⟩|/v_F=|_-∞^∞[Ψ_B,f^⋆(e_x-ie_y)Ψ_A,i+Ψ_A,f^⋆(e_x+ie_y)Ψ_B,i]dx|.
  
  combining Eqs (<ref>-<ref>) together results in an expression which does not depend on the sign of s_K:
  
      |⟨ f|v̂·e|i⟩ /v_F|=|_-∞^∞[Ψ_B,f^⋆(e_x+ise_y)Ψ_A,i+Ψ_A,f^⋆(e_x-ise_y)Ψ_B,i]dx|,
  
  where the functions Ψ_A and Ψ_B entering Eqs (<ref>-<ref>) are evaluated for s_K=s=1. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  