Introduction

It can be shown using supersymmetric methods that whenever the Schrödinger equation can be solved exactly for a one-dimensional (1D) potential, there exists a corresponding potential for which the two-dimensional (2D) Dirac equation admits exact eigenvalues and eigenfunctions1. A broad class of quasi-1D potentials can also be solved quasi-exactly by transforming the 2D Dirac equation into the Heun equation2 or one of its confluent forms3,4,5, or via the application of Darboux transformations6,7,8,9,10,11. These exact and quasi-exact bound-state solutions have direct applications to electronic waveguides in 2D Dirac materials2,3,4,5,12,13, such as graphene, where the low-energy spectrum of the charge carriers can be described by a Dirac Hamiltonian14, and the guiding potential can be generated via a top gate15,16,17,18,19,20. Recent advances in device fabrication, utilizing carbon nanotubes as top gates, has enabled the detection of individual guided modes21, opening the door to several new classes of devices such as THz emitters5,13, transistors12, and ultrafast electronic switching devices22. These advances in electron waveguide fabrication technology make the need for analytic solutions all the more important, since they are highly useful in: determining device geometry, finding the threshold voltage required to observe a zero-energy mode12, calculating the size of the THz pseudogap in bipolar waveguides13, as well as ascertaining the optical selection rules5,13 in graphene heterostructures.

In extension to the well-known case of graphene, Dirac cones can in general possess valley-dependent tilt23. There are only a handful of 2D electronic systems that have been predicted to host these tilted cones24,25,26,27,28,29,30,31,32,33, one of which is 8-Pmmn borophene23,34, which has attracted considerable attention. In general, boron-based nanomaterials are a growing field of interest35,36,37; indeed, exploring the tilt of 2D Dirac cones in the context of 8-Pmmn borophene has recently led to a plethora of theoretical works spanning many fields of research38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84 from optics to transport, and many more; for a comprehensive review, see Ref.85. The spectacular rise of borophene, and the growing interest in tilted Dirac materials, has led to the revisiting of several well-known problems in graphene, e.g., Klein tunneling47 and transport across quasi-1D heterostructures43,52,73,86, within the context of tilted Dirac materials. As mentioned previously, methods such as supersymmetry and reducing the Dirac equation to the Heun equation, utilize solutions of known problems, to generate solutions to new ones. This begs the question: does a simple mapping exist which would allow us to harness the large body of exact and quasi-exact solutions for 1D waveguides in graphene and then apply them to materials with tilted Dirac cones?

In what follows, we show that the differential equations governing guided modes in an anisotropic tilted Dirac waveguide (orientated at an arbitrary angle) can, via a simple transformation, be mapped onto the graphene problem, i.e., transformed into the massless 2D Dirac equation14, for the same potential, but of modified strength, effective momentum, and modified energy scale. After outlining the transformation, we study the particular case of the hyperbolic secant potential, which in graphene is known to admit quasi-exact solutions to the eigenvalue problem; but nevertheless, the whole spectrum can be obtained via a semi-analytic approach2,12. We use this known graphene waveguide spectrum to generate the corresponding tilted waveguide spectrum, which is verified using a transfer matrix method. Finally, we discuss valleytronic applications.

Transformation

The Hamiltonian describing the guided modes contained within a smooth electron waveguide in a tilted Dirac material can be written as

$$\begin{aligned} {\hat{H}}=\hbar \left( v_{x}\sigma _{x}{\hat{k}}_{x}+sv_{y}\sigma _{y}{\hat{k}}_{y}+sv_{t}\sigma _{0}{\hat{k}}_{y}\right) +\sigma _{0}U\left( x,y\right) , \end{aligned}$$
(1)

where \({\hat{k}}_x=-i \partial _x\), \({\hat{k}}_y=-i \partial _y\), \(\sigma _{x,y}\) are the Pauli spin matrices, \(\sigma _{0}\) is the identity matrix, \(v_{x}\) and \(v_{y}\) are the anisotropic velocities, \(v_{t}\) is the tilt velocity and \(s=\pm 1\) is the valley index number; here \(s=1\) and \(s=-1\) are analogous to the K and \(K'\) valley, respectively. This Hamiltonian is of the same form as the low-energy two-band effective Hamiltonian used to describe 8-Pmmn borophene23, 2B:Pmmn borophane87, and \(\alpha \text {-(BEDT-TTF)}_2\text {I}_3\)24,25. In what follows, we set \(s=1\), but it should be noted that the other valley’s eigenvalues can be obtained by replacing \(v_y\) and \(v_t\) with \(-v_y\) and \(-v_t\). In general, the crystallographic orientation is not known, nor is it currently possible to deposit the top gate at a selected angle relative to the crystallographic axis. Therefore, we shall solve for the case of a waveguide at an arbitrary angle relative to the crystallographic axis (\(x-y\)). The electrostatic potential, \(U\left( x,y\right)\), is 1D, directed along the \(y'\)-axis, and varies along the \(x'-\)axis (see Fig. 1 for geometry), i.e., \(U=U\left( x'\right)\). We rotate the \(x-y\) axes counterclockwise through an angle \(\theta\). The new axes \(x'-y'\) are defined by the original coordinates via the transformation:

$$\begin{aligned} x'&=x\cos \theta +y\sin \theta , \nonumber \\ y'&=- x\sin \theta +y\cos \theta . \end{aligned}$$
(2)

Hence, the wave vector operators in the non-rotated coordinate system \({\hat{k}}=\left( {\hat{k}}_{x},{\hat{k}}_{y}\right)\) are expressed in the rotated coordinate frame, \({\hat{k}}'=\left( {\hat{k}}_{x'},{\hat{k}}_{y'}\right)\), via the relations:

$$\begin{aligned} {\hat{k}}_{x}&=\cos \theta {\hat{k}}_{x'}-\sin \theta {\hat{k}}_{y'}, \nonumber \\ {\hat{k}}_{y}&=\sin \theta {\hat{k}}_{x'}+\cos \theta {\hat{k}}_{y'}. \end{aligned}$$
(3)

The Hamiltonian, Eq. (1), acts on the two-component Dirac wavefunction \(\Psi =\left( \psi _{A}\left( x'\right) ,\,\psi _{B}\left( x'\right) \right) ^{\intercal } e^{ik_{y'} y'}\) to yield the coupled first-order differential equations \({\hat{H}} \Psi = \varepsilon \Psi\), where \(\psi _{A}\) and \(\psi _{B}\) are the wavefunctions associated with the A and B sublattices of the tilted Dirac material. These coupled first-order differential equations can be recast into the same equations used to describe guided modes propagating along a smooth electron waveguide in graphene:

$$\begin{aligned} \left( \sigma _{x}{\hat{k}}_{x'}+\sigma _{y}{\widetilde{\Delta }}+\sigma _{0}{\widetilde{V}}\left( x'\right) \right) \Phi \left( x'\right) ={\widetilde{E}}\Phi \left( x'\right) , \end{aligned}$$
(4)

where the effective potential \({\widetilde{V}}\), energy \({\widetilde{E}}\), and momentum \({\widetilde{\Delta }}\) are obtained from the original tilted case via the relations:

$$\begin{aligned} {\widetilde{V}}\left( x'\right)&=\frac{lV\left( x'\right) }{l^{2}-t^{2}\sin ^{2}\theta }, \nonumber \\ {\widetilde{E}}&=\frac{l}{l^{2}-t^{2}\sin ^{2}\theta }\left( E-\frac{t\cos \theta }{l^{2}}\Delta \right) , \nonumber \\ {\widetilde{\Delta }}&=\frac{T\Delta }{l\sqrt{l^{2}-t^{2}\sin ^{2}\theta }}, \end{aligned}$$
(5)

where \(V\left( x'\right) = U\left( x'\right) L/\hbar v_x\), \(E=\varepsilon L/\hbar v_x\) and \(\Delta =k_y'L\), and L is a constant, associated with the effective width of the potential. We define the tilt and anisotropy parameters as \(t=v_{t}/v_{x}\) and \(T=v_{y}/v_{x}\), respectively, and \(l=\sqrt{1-\left( 1-T^{2}\right) \sin ^{2}\theta }\). The eigenfunctions of the guided modes in the effective graphene sheet, \(\Phi\), can be mapped onto the tilted Dirac spinor components, \(\psi _{A}\) and \(\psi _{B}\), via the expression:

$$\begin{aligned} \Phi \left( x'\right) =\left( \begin{array}{c} \left( 1+\mu \right) \psi _{A}+\left( 1-\mu \right) \psi _{B}e^{-i\varphi }\\ \left( 1-\mu \right) \psi _{A}+\left( 1+\mu \right) \psi _{B}e^{-i\varphi } \end{array}\right) e^{-i\sin \theta {\displaystyle \int } \frac{t \left( V-E\right) +k_{y'} \left( 1+t^{2}-T^{2}\right) \cos \theta }{l^{2}-t^{2}\sin ^{2}\theta }dx'}, \end{aligned}$$
(6)

where \(\varphi =\arctan \left( T\tan \theta \right)\) and \(\mu =\left( l-t\sin \theta \right) ^{\frac{1}{2}}\left( l+t\sin \theta \right) ^{-\frac{1}{2}}\). It then follows that if the eigenfunctions and eigenvalues are known for the potential \({\widetilde{V}}\) in graphene, one can immediately write down the eigenfunctions and eigenvalues of a 1D confining potential of the same form, orientated at an arbitrary angle, in a tilted Dirac material. Conversely, if a quasi-1D potential readily admits exact or quasi-exact solutions for the tilted case, and no solutions are known for the graphene problem, then our mapping method can be used to obtain the eigenfunctions and eigenvalues for the case of graphene. This mapping also reveals the angular dependence of the number of guided modes contained within the waveguide. Namely, it can be seen from Eq. (5) that rotating the orientation of the waveguide is equivalent to varying the effective depth of the potential (see Fig. 2b). Indeed, the effective potential’s depth, \({\widetilde{V}}_{0}\), is equal to the actual potential’s depth, \(V_0\) at \(\theta =0\) and rises to a maximum value of \({\widetilde{V}}_0 / V_0= T/(T^{2}-t^{2})\) at \(\theta =\pi /2\).

Figure 1
figure 1

A schematic diagram of an electrostatic potential, \(U(x')\), created by an applied top-gate voltage in a tilted 2D Dirac material. The waveguide is orientated at an angle of \(\theta\), relative to the x-axis of the crystal. The potential is invariant along the \(y'\)-axis, and varies in strength along the \(x'\)-axis. The \(x'-y'\) axes are denoted by the solid black arrows, whereas the crystallographic axes \(x-y\) are shown by the light gray arrows.

Full size image
Figure 2
figure 2

(a) The black curve shows the hyperbolic secant potential \(V(x')=-V_0/\cosh (x'/L)\), for the case of \(V_0=3.2\). The dashed and solid horizontal lines are the bound-state energy levels for the \(s=1\) and \(s=-1\) chirality, respectively (which corresponds to the K and \(K'\) valley in the effective graphene sheet), for the case of \(\Delta =k_{y'} L=1\), when the waveguide is orientated at angle \(\theta =0\). The tilted Dirac material is defined by parameters \(v_x=0.86\,v_{\mathrm {F}}\), \(v_y=0.69\,v_{\mathrm {F}}\) and \(v_t=0.32\,v_{\mathrm {F}}\). (b) The relative strength of the effective potential’s depth, \({\widetilde{V}}_{0}\), compared to the actual potential depth \(V_0\).

Full size image

It should be noted that to perform the same transformations for the other chirality (i.e., the \(s=-1\), or graphene \(K'\) valley analog), one must exchange t with \(-t\), and T with \(-T\). Therefore, the eigenvalue spectrum of one valley can be obtained by a reflection of the other valley’s eigenvalue spectrum about the \(k_{y'}\)-axis. It can be seen from Eq. (5) that in the absence of a tilt term, i.e., \(t=0\), the eigenvalue spectrum of a given valley is symmetric with respect to \(k_{y'}\). Thus, both chiralities have the same band structure. Similarly for \(t \ne 0\), if the waveguide is orientated such that \(\cos {\theta }=0\), the eigenvalue spectrum is chirality-independent. In all other cases, the energy spectrum for a given valley lacks \(E(k_{y'})=E(-k_{y'})\) symmetry. This gives rise to the possibility of utilizing smooth electron waveguides in tilted Dirac materials as the basis of valleytronic devices. This will be discussed in the penultimate section.

Quasi-exact solution to the tilted Dirac equation for the hyperbolic secant potential

In this section, we shall apply our simple transformations, given in Eq. (5), to generate the energy spectrum of a smooth electron waveguide in a tilted Dirac material for a potential which has been studied in depth in graphene2,12:

$$\begin{aligned} V\left( x'\right) =-\frac{V_0}{\cosh \left( x'/L\right) }. \end{aligned}$$
(7)

This potential, shown in Fig. 2a, belongs to the class of quantum models which are quasi-exactly solvable2,88,89,90,91,92,93,94, where only some of the eigenfunctions and eigenvalues are found explicitly. The depth of the well is given by \(V_0\), and the potential width is characterized by the parameter L. Here \(V_0\) and L are taken to be positive parameters. For realistic top-gated structures, the width of the potential is defined by the geometry of the top gate structure, and the strength of the potential is defined by the voltage applied to the top gate. In graphene, the wavefunctions can be solved in terms of Heun polynomials, which reduce to hypergeometric functions for the case of zero energy2,12. For zero-energy modes (\({\widetilde{E}}=0\)), the permissible values of \({\widetilde{\Delta }}\) are given by the simple relation \({\widetilde{\Delta }}={\widetilde{V}}_0-n-\frac{1}{2}\), where \({\widetilde{V}}_0\) is the depth of the effective potential and n is a non-negative integer. For non-zero energies, exact energy eigenvalues can be obtained when the Heun polynomials are terminated2. To illustrate the power of our mapping method, we apply the transformations given in Eq. (5) to the well-known zero-energy solutions to the 2D Dirac equation for the 1D hyperbolic secant potential. The corresponding tilted Dirac equation solutions become:

$$\begin{aligned} E=\frac{t\cos \theta }{T\sqrt{l^{2}-t^{2}\sin ^{2}\theta }}\left[ V_{0}-\left( n+\frac{1}{2}\right) \frac{l^{2}-t^{2}\sin ^{2}\theta }{l}\right] , \end{aligned}$$
(8)

and their corresponding \(n=0\) eigenfunctions for two different waveguide orientations are shown in Fig. 3, for the case of 8-Pmmn borophene, which is described by the parameters \(v_x=0.86\,v_{\mathrm {F}}\), \(v_y=0.69\,v_{\mathrm {F}}\) and \(v_t=0.32\,v_{\mathrm {F}}\)23.

In Fig. 4 we plot the 8-Pmmn borophene eigenvalue spectrum for the potential defined by \(V_0=3.2\) for two orientations, \(\theta =0\) and \(\theta =\pi /2\), as well as the graphene waveguide spectra (quasi-analytically determined2) used in the mapping. In the same figure, we also plot the numerical solutions to the tilted Dirac problem obtained via a transfer matrix method (see Supplementary Material). We show in blue crosses the exact solutions given in Eq. (8) together with the complete set of mapped quasi-exact solutions given in Ref.2. It can be seen from Fig. 4 that the waveguide orientated at \(\theta =\pi /2\) contains more bound states than the waveguide orientated at \(\theta =0\). This is a result of the effective graphene potential being deeper, and thus supporting more guided modes. It should be noted that for potentials which vanish at infinity, i.e., \(V(\pm \infty )=E=0\), only the zero-energy modes are truly confined, since the density of states vanishes outside of the well. Guided modes occurring at non-zero energies can always couple to continuum states outside of the well, thus having a finite lifetime.

Figure 3
figure 3

The normalized zero-energy state (lowest positive momentum) wavefunctions of the hyperbolic secant potential, \(V(x')=-V_0/\cosh (x'/L)\), of strength \(V_0=-3.2\) and orientation: (a) \(\theta =0\) and (c) \(\theta =\pi /2\) in a tilted Dirac material defined by parameters \(v_x=0.86\,v_{\mathrm {F}}\), \(v_y=0.69\,v_{\mathrm {F}}\) and \(v_t=0.32\,v_{\mathrm {F}}\), for the \(s=1\) chirality, i.e., the K valley in the effective graphene sheet. The corresponding wavefunctions of the effective graphene waveguide are given in panels (b) and (d). The solid red and blue lines correspond to the real part of \(\psi _A\) and \(\psi _B\) respectively, while the dashed lines correspond to their imaginary parts. The grey line shows the potential as a guide to the eye.

Full size image
Figure 4
figure 4

The energy spectrum of confined states (for the \(s=1\) chirality, i.e., the K valley in the effective graphene sheet) in the hyperbolic secant potential \(V(x')=-V_0/\cosh (x'/L)\), of strength \(V_0=3.2\), as a function of dimensionless momentum along the waveguide, \(\Delta =k_y'L\), for the orientations (a) \(\theta =0\) and (c) \(\theta =\pi /2\) in a tilted Dirac material, defined by parameters \(v_x=0.86\,v_{\mathrm {F}}\), \(v_y=0.69\,v_{\mathrm {F}}\) and \(v_t=0.32\,v_{\mathrm {F}}\). The energy spectra of the effective graphene waveguide from whence they came, are given in panels (b) and (d), respectively. The black dots denote the semi-analytic eigenvalues, the blue crosses represent the quasi-exact eigenvalues, and the solid red lines show the eigenvalues numerically obtained via a transfer matrix method. The boundary at which the bound states merge with the continuum is denoted by the grey dashed lines.

Full size image

Valleytronic applications

It has been suggested that the valley quantum number can be used as a basis for carrying information in graphene-based devices95 in an analogous manner to spin in semiconductor spintronics96,97. Unlike in the case of graphene (in the conical regime), smooth electrostatic potentials in tilted Dirac materials can be utilized as a means to achieve valley polarization. The majority of studies have focused on tunneling across electrostatically-induced potential barriers, and valley filtering and beam splitting have been demonstrated43,47,52,73,86. It has also been shown that the allowed transmission angles through a potential can be controlled using magnetic barriers86. We propose a change in geometry: rather than studying chirality-dependent transmission across barriers, we shift the focus to studying guided modes along quasi-1D confining potentials. The conductance along such a channel can be measured by placing one terminal at each end. According to the Landauer formula, when the Fermi level is set to energy E (by modulating the back-gate voltage21), the conductance along the waveguide is simply \(2(n_K+n_{K'})e^{2}/h\), where \(n_K\) and \(n_{K'}\) are the number of modes belonging to the \(s=1\) and \(s=-1\) chirality, respectively (or the K and \(K'\) valley in the effective graphene sheet), at that particular energy.

In a 2D Dirac material subject to a quasi-1D potential, the introduction of the tilt parameter breaks the \(E(\Delta )=E(-\Delta )\) symmetry for a given valley. Indeed, for a given valley, the additional tilt term increases the particle velocity along one direction of the barrier and decreases it along the opposite direction; and vice versa for the other valley. For the case of type-I Dirac materials, i.e., \(t<T\), it can be seen from Fig. 5 that for a given sign of \(\Delta\), the eigenvalues of the critical solutions (sometimes referred to as the zero-momentum solutions: i.e., bound states with energy \(|E|=|\Delta |\)) belonging to the two valleys are different. Thus, providing that \(t\ne 0\) and \(\cos {\theta }\ne 0\), there exists a range of energies for which there will be more bound states propagating along a particular direction belonging to one valley than the other, i.e., valley polarization. The degree of valley polarization can be controlled by varying the strength of the electrostatic potential and by changing the position of the Fermi level, which in practical devices is achieved by modulating the back-gate voltage21. Full valley polarization can be achieved for energies less than the lowest-lying supercritical state (defined as a bound state with energy \(E = -\Delta\)) belonging to the valley where the tilt term enhances the particle’s velocity, indicated by the shaded region in Fig. 5. However, although full valley polarization along a particular direction can be realized, to generate a valley-polarized current one must lift the \(y^{\prime }=-y^{\prime }\) symmetry. This can be achieved by applying an in-plane electric field along the waveguide.

For type-III tilted Dirac materials, i.e., \(t=T\), full valley polarization occurs for all energies and all orientations of the waveguide. For such materials the infinite number of positive-energy critical solutions of graphene map onto the zero-energy modes of a type-III Dirac material. Consequently, the infinitely many zero-energy modes will give rise to a sharp peak in the conductance along the channel when the Fermi energy is in the proximity of \(E=0\). This is in stark contrast to the case of graphene, where the creation of an infinite number of zero-energy states requires an infinitely deep and wide potential4,98,99,100,101. Since the potential possesses infinitely many bound states, which infinitely accumulate at \(E=0\), a type-III Dirac material could be used as a THz emitter; namely, the Fermi level could be set below \(E=0\), and optical photons can be absorbed from low-lying energy levels to \(E=0\). Then the photo-excited carriers can relax back down to the Fermi level via the emission of THz photons through the closely spaced energy levels in the proximity of zero energy.

Lastly, for type-II tilted Dirac materials, i.e., \(t>T\), full valley polarization occurs for all energies; however, bound states occur only for orientations within the range \(-1/\sqrt{t^{2}-T^{2}}<\tan \left( \theta +n\pi \right) <1/\sqrt{t^{2}-T^{2}}\). In the limit at which \(\theta\) becomes imaginary, i.e., the boundary at which the equi-energy surfaces become unbounded, the effective potential required for mapping diverges.

Figure 5
figure 5

The energy spectrum of confined states in the hyperbolic secant potential \(V(x')=-V_0/\cosh (x')\), of strength \(V_0=3.2\), as a function of \(\Delta\) for the \(s=1\) and \(s=-1\) chirality (i.e., the K and \(K'\) valley in the effective graphene sheet), depicted by the red and blue lines, respectively. The waveguide is orientated at \(\theta =0\) and the tilted Dirac material is defined by parameters \(v_x=0.86\,v_{\mathrm {F}}\), \(v_y=0.69\,v_{\mathrm {F}}\) and \(v_t=0.32\,v_{\mathrm {F}}\). The boundary at which the bound states merge with the continuum is denoted by the grey dashed lines. The shaded area represents the energy range for which full valley polarization can be achieved for a given value of \(\Delta\).

Full size image

Conclusion

We have shown that if the eigenvalues and eigenfunctions of a quasi-1D potential in graphene are known, then they can be used to obtain the corresponding results for the same potential (with modified strength, orientated at arbitrary angle), for a 2D Dirac material hosting anisotropic, tilted Dirac cones. Therefore, all the rich physics associated with guiding potentials in graphene, e.g., THz pseudogaps in bipolar waveguides, can be revisited in the context of tilted Dirac materials, but with the distinct advantage of knowing the eigenvalues and eigenfunctions. We have also shown that in stark contrast to smooth electron waveguides in graphene, valley degeneracy can be broken in tilted 2D Dirac materials for a broad range of waveguide orientations, anisotropy and tilt parameters. The degree of valley polarization along the waveguide can be controlled by varying the potential strength of the top gate, and also by changing the back-gate voltage. Tilted 2D Dirac materials, such as 8-Pmmn borophene, are therefore promising building blocks for tunable valleytronic devices.

Methods

The Supplementary Information contains a full description of the transfer matrix method used to calculate the band structure of guiding potentials in 2D Dirac materials with tilted Dirac cones.