Abstract
We obtain exact solutions to the twodimensional (2D) Dirac equation for the onedimensional PöschlTeller potential which contains an asymmetry term. The eigenfunctions are expressed in terms of Heun confluent functions, while the eigenvalues are determined via the solutions of a simple transcendental equation. For the symmetric case, the eigenfunctions of the supercritical states are expressed as spheroidal wave functions, and approximate analytical expressions are obtained for the corresponding eigenvalues. A universal condition for any square integrable symmetric potential is obtained for the minimum strength of the potential required to hold a bound state of zero energy. Applications for smooth electron waveguides in 2D DiracWeyl systems are discussed.
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Introduction
The PöschlTeller potential^{1} plays an important role in many fields of physics^{2} from modeling diatomic molecules and quantum manybody systems^{3,4,5}, to applications in astrophysics^{6, 7}, optical waveguides^{8} and quantum wells^{9, 10}, through to BoseEinstein and Fermionic condensates^{11, 12}, and supersymmetric quantum mechanics^{13}. For the onedimensional Schrödinger equation, the hyperbolic symmetric form can be solved in terms of associated Legendre polynomials and the eigenvalues are known explicitly^{1, 14}. We consider an analogous relativistic problem, that of a twodimensional Dirac particle, confined by a onedimensional PöschlTeller potential. Several solutions have been obtained for the Dirac equation with central PöschlTeller potentials^{15,16,17,18,19,20,21}, and the hyperbolicsecant potential is also known to admit analytic solutions for both the one and twobody onedimensional Dirac problems^{22,23,24}. Modified PöschlTeller potential potentials have also been employed in numerical simulations of potential barriers in bilayer graphene^{25}.
With the recent explosion of research in Dirac materials^{26} there has been a renewed interest in quasirelativistic phenomena considered in condensed matter systems of different dimensionalities. This is due to the fact that the same equations which govern Dirac fermions in relativity, map directly to the equations of motion describing the quasiparticles in systems such as graphene^{27}, carbon nanotubes^{28}, topological insulators^{29,30,31} transition metal dichalcogenides^{32} and 3D Weyl semimetals^{33}. Massless Dirac particles are notoriously difficult to confine; however, it has been demonstrated that certain types of onedimensional electrostatic waveguides in graphene, possess zero energymodes which are truly confined within the waveguide^{22, 23}, and that the number of these zero energymodes is equal to the number of supercritical states (i.e. bound states whose energy, E = −M, where M is the particle’s effective mass). Transmission resonances and supercriticality of Dirac particles through onedimensional potentials have been studied extensively^{34,35,36,37,38,39,40,41,42,43,44,45}. The majority of studies model topgated carbonbased nanostructures using abrupt potentials^{46,47,48,49,50,51,52,53,54,55,56}. However, experimental potential profiles vary smoothly over many lattice constants, with even the smallest of gate generated potentials being far from square^{57}. There is also some controversy concerning waveguides which are defined by smoothly decaying, square integrable functions, which decay at large distances as 1/x ^{n}, where n > 1. Numerical experiments imply that such waveguides always contain a zeroenergy mode^{58}, whereas analysis based upon relativistic Levinson theorem, says there is a minimum potential strength required to observe a zeroenergy mode^{45, 59,60,61,62}. Our result supports the latter, and we demonstrate this through a simple analysis of supercritical states of zero energy.
In 2D Dirac materials, the lowenergy spectrum of the charge carriers can be described by a Dirac Hamiltonian^{63} of the form
where \({\hat{k}}_{x}=i\frac{\partial }{\partial x}\), \({\hat{k}}_{y}=i\frac{\partial }{\partial y}\), σ _{x,y,z} are the Pauli spin matrices. v plays the role of the speed of light and k _{z} is proportional to the particle inplane effective mass. In graphene, the charge carriers are massless, k _{z} = 0, and the dispersion is linear, v = v _{F} ≈ 10^{6} m/s is the Fermi velocity and s _{K} has the value of ±1 for the K and K′ valley respectively^{64}. For narrowgap nanotubes and certain types of graphene nanoribbons^{65, 66} the operator \({\hat{k}}_{y}\) can by replaced by the number k _{y} = E _{g}/(2ħv _{F}), which in the absence of applied field is fixed by geometry, where E _{g} is the value of the bandgap. For nanotubes, E _{g} can be controlled by applying a magnetic field along the nanotube axis^{24, 67,68,69,70}. When k _{z} is finite, Eq. (1) can be used as a simple model for silicene^{71} or Weyl semimetal^{26, 72}. It has also been proposed that when graphene is subjected to a periodic potential on the lattice scale, for example, graphene on top of a latticematched hexagonal boron nitride^{73} the Dirac fermions acquire mass with E _{g} = 2ħv _{F}k _{z} being of the order of 53 meV.
In what follows we shall consider a particle described by the Hamiltonian (1) subjected to a onedimensional potential U(x), which varies on a scale much larger than the lattice constant of the corresponding Dirac material, therefore allowing us to neglect intervalley scattering for the case of graphene. We shall also set s _{K} = 1, and note that the other valley’s wave function can be obtained by exchanging k _{y} → −k _{y}. The Hamiltonian acts on the twocomponent Dirac wavefunction
to yield the coupled firstorder differential equations
and
where \(\tilde{x}=x/L\) and L is a constant. V = UL/ħv _{F} and the charge carrier energy, ε, have been scaled such that E = εL/ħv _{F}. The charge carriers propagate along the ydirection with wave vector k _{y} = Δ/L, which is measured relative to the Dirac point, Ψ_{A}(x) and Ψ_{B}(x) are the wavefunctions associated with the A and B sublattices of graphene and finally M = k _{z}L represents an effective mass in dimensionless units.
In what follows we consider the relativistic quasionedimensional PöschlTeller potential problem which can be applied to describe e.g. graphene waveguides. We obtain the exact energy eigenfunctions for this potential and formulate a method for calculating the eigenvalues of the bound states. We then analyze the energyspectrum of the symmetric PöschlTeller potential and obtain expressions for the eigenvalues of the supercritical states. By analyzing the zeroenergy supercritical states we obtain a universal threshold condition for the minimum potential strength required for a potential to possess a zeroenergy mode, for any squareintegrable potential. We also show that the eigenfunctions in the nonrelativistic limit restore the onedimensional Schrödinger equation solutions. Finally, we analyze the eigenvalue spectrum for the modified PöschlTeller potential which includes an asymmetry term.
Relativistic onedimensional PöschlTeller problem
In this section we consider the potential
which is a linear combination of the symmetric PöschlTeller potential with an additional term which enables the introduction of asymmetry^{74}. This potential belongs to the class of quantum models, which are quasiexactly solvable^{23, 75,76,77,78,79,80,81}, where only some of the eigenfunctions and eigenvalues are found explicitly. The depth of the well is given by – (a − b)^{2}/4a, and the potential width is characterized by the parameter L, which was introduced after Eq. (4). For the case of b = 0, the potential transforms into the symmetric PöschlTeller potential, while if a = 0, the potential is a smooth potential step, which can be used to model a pn junction^{82, 83}. The symmetric and asymmetric forms of the potential are plotted in Fig. 1.
Substituting Ψ_{A} = (Ψ_{1} + Ψ_{2})/2 and Ψ_{B} = (Ψ_{1} − Ψ_{2})/2 allows Eqs (3,4) to be reduced to a single secondorder differential equation in Ψ_{1} (Ψ_{2})
where s = −(−1)^{j} and j = 1, 2 correspond to the spinor components Ψ_{1} and Ψ_{2} respectively. By making the natural change of variable \(Z=[1+\,\tanh (\tilde{x})]\mathrm{/2}\) and using the wave function of the form \({{\rm{\Psi }}}_{j}=\exp (pZ){Z}^{n}{(1Z)}^{m}{\psi }_{j}(Z)\) allows Eq. (6) to be reduced to the Heun confluent equation^{84} in variable Z:
where n = β/2, m = γ/2, P = α/2, \({\mathop{{\rm{\Delta }}}\limits^{ \sim }}^{2}={M}^{2}+{{\rm{\Delta }}}^{2}\), s _{α,β,γ} = ±1 and the parameters α, β, γ, δ _{s} and η _{s} are: α = ias _{α}, \(\beta ={s}_{\beta }\sqrt{{\mathop{{\rm{\Delta }}}\limits^{ \sim }}^{2}{E}^{2}}\), \(\gamma ={s}_{\gamma }\sqrt{{\mathop{{\rm{\Delta }}}\limits^{ \sim }}^{2}{(Eb)}^{2}}\), \({\delta }_{s}=a(\frac{1}{2}bis)\) and η _{s} = β ^{2}/2 − (a − b)(E − is)/2. This same method of reducing a system of coupled firstorder differential equations to the Heun confluent equation has been exploited to solve various generalisations of the quantum Rabi model^{85}, and notably the quasiexact solutions of the PöschlTeller family potentials and Rabi systems are closely related^{86}. In some instances, the resulting Heun confluent functions can be terminated as a finite polynomial^{23, 87}, allowing particular eigenvalues to be obtained exactly, providing the parameters obey special relations. When this is not the case, the energy spectrum can be obtained fully via the Wronskian method^{23, 85, 88, 89}, which is the method we shall utilize.
Equation (7) has regular singularities at Z = 0 and 1, and an irregular one at Z = ∞ which is outside the domain of \(\tilde{x}\). The solutions to Eq. (7) are given by
where \({A}_{j,{s}_{\alpha },{s}_{\beta },{s}_{\gamma }}\) and \({B}_{j,{s}_{\alpha },{s}_{\beta },{s}_{\gamma }}\) are constants, and H is the Heun confluent function^{90}, which has a value of 1 at the origin. For Z < 1, H(α, β, γ, δ _{s}, η _{s}, Z) = (1 − Z)^{−γ}H(α,β, −γ,δ _{s},η _{s},Z) and \(H(\alpha ,\,\beta ,\,\gamma ,\,{\delta }_{s}\,,{\eta }_{s},\,Z)=\) \(\exp (\alpha Z)H(\alpha,\,\beta,\,\gamma,\, {\delta }_{s},\, {\eta }_{s},\,Z)\), therefore, as expected from the general theory of second order differential equations, the full solution is given by a linear combination of just two linearly independent functions
where β is not an integer. It should be noted that when a = 0 and α = δ = 0, the Heun confluent functions appearing in Eq. (9) reduces to a Gauss hypergeometric functions and for the case of a massless particle Eq. (9) reduce down to the solutions obtained in ref. 82. If, however, β is an integer then H(α, −β, γ, δ, η _{s}, Z) is divergent and B _{i} has to be set to zero, and the second linearly independent solution can be constructed from a series expansion about the point 1 − Z. The solutions to the Heun confluent equation thus far have been given as power series expansions about the point Z = 0. However, these power series rapidly diverge as Z approaches the second singularity; therefore, at Z = 1 we must construct solutions as power series expansions in variable 1 − Z:
The constants C _{i} and D _{i} are found by matching the two power series expansions and their derivatives at Z _{0} where 0 < Z _{0} < 1.
For bound states, we require that \({\mathop{{\rm{\Delta }}}\limits^{ \sim }}^{2} > {E}^{2}\) and \({\mathop{{\rm{\Delta }}}\limits^{ \sim }}^{2} > {(Eb)}^{2}\). These conditions ensure that the bound states are inside the effective bandgap (which accounts for the motion along the yaxis). As x → −∞, Z → 0 and as x → ∞, Z → 1, therefore we may write the asymptotic expressions of Ψ_{j} as
and
Therefore, for bound states, B _{i} (A _{i}) is zero for s _{β} = 1 (s _{β} = −1) and D _{i} (C _{i}) is zero for s _{γ} = 1 (s _{γ} = −1). Clearly the choice of s _{β} and s _{γ} is arbitrary, therefore, from hereon in we set both to 1 unless otherwise stated. In this instance, the energy eigenvalues are found from the condition:
where 0 < Z _{0} < 1.
Symmetric PöschlTeller potential solutions
In general, relating Ψ_{1} to Ψ_{2} is nontrivial, since neither a known expression exists which connects Heun confluent functions about two different singular points for arbitrary parameters, nor is there a general expression relating the derivative of the confluent Heun function to other confluent Heun functions, though particular instances have been obtained^{91, 92}. However, for the symmetric PöschlTeller potential (i.e. b = 0) one can obtain the relation:
Therefore, \({A}_{2}={{\rm{\Omega }}}_{{s}_{\beta }}{A}_{1}\) and \({B}_{2}={{\rm{\Omega }}}_{{s}_{\beta }}{B}_{1}\), where \({{\rm{\Omega }}}_{{s}_{\beta }}=(E+i\beta )/(M+i{\rm{\Delta }})\).
In pristine graphene, only the symmetric form of Eq. (5) will contain nonleaky modes at zero energy. Nonzeroenergy modes will have a finite lifetime since they can always couple to continuum states outside of the waveguide, whereas zeroenergy modes are fully confined since the density of states vanishes at zero energy outside of the well. Asymmetric forms of Eq. (5) never contain truly bound modes since the density of states cannot vanish on both sides of the potential simultaneously. Notably, this is somewhat counterintuitive as for the Schrödinger problem a symmetric potential always contains a bound state, which can be removed by asymmetry. The emergence of bound states for a relativistic problem with an infinitely wide barrier is a manifestation of the Klein tunneling phenomenon^{22, 23}.
We shall now consider the symmetric form of Eq. (5) for massless particles. Accordingly, we set b = 0 and M = 0, and in this instance the symmetrized real functions^{22, 23}, are given by Ψ_{I} = Ψ_{A} + iΨ_{B} and Ψ_{II} = Ψ_{A} − iΨ_{B}, where
and
where Φ = H(α, β, γ, δ _{1}, η _{1}, Z) and \(\tan \,\theta =E/\beta \). By employing the identity Eq. (14), the derivatives appearing in Eq. (13) can be expressed in terms of Heun confluent functions. It immediately follows that at the origin \({{\rm{\Psi }}}_{{\rm{I}}}^{\ast }{{\rm{\Psi }}}_{{\rm{I}}{\rm{I}}}+{{\rm{\Psi }}}_{{\rm{I}}}{{\rm{\Psi }}}_{{\rm{I}}{\rm{I}}}^{\ast }=0\), which in terms of the functions Ψ_{A} and Ψ_{B} yields:
where Z = 1/2 corresponds to x = 0. Substituting Eq. (15) and Eq. (16) into Eq. (17) results in the condition
From Eq. (15) and Eq. (16), \({A}_{1}{{\rm{\Psi }}}_{A}^{\ast }={A}_{1}^{\ast }{{\rm{\Psi }}}_{A}\,\exp (i\theta )\) and \({A}_{1}{{\rm{\Psi }}}_{B}^{\ast }={A}_{1}^{\ast }{{\rm{\Psi }}}_{B}\,\exp (i\theta )\). Therefore, the condition from which the eigenvalues of the spectrum are determined, Eq. (17), can be written as \({({{\rm{\Psi }}}_{A}+i{{\rm{\Psi }}}_{B})}_{Z=\frac{1}{2}}\) \({({{\rm{\Psi }}}_{A}i{{\rm{\Psi }}}_{B})}_{Z=\frac{1}{2}}=0\). This condition can be understood in terms of parity. In principle, one can construct from these functions odd and even solutions. However, since the even modes of Ψ_{I} occur at the same energies as the odd modes of Ψ_{II} and vice versa, one can obtain the eigenvalues when the symmetrized functions Ψ_{I} or Ψ_{II} are zero at the origin. The functions Ψ_{I} and Ψ_{II} are related to the earlier introduced functions Ψ_{1} and Ψ_{2} by
and
Using Eqs (19, 20) together with Eqs (9),(10) allows Ψ_{I} and Ψ_{II} to be expressed explicitly in terms of Heun functions.
Eq. (18) is formally the same as Eq. (13) but computationally faster. In Fig. 2 we plot the numerically obtained solutions of Eq. (18) for the potential defined by a = 24 and b = 0. The dashed lines represent the boundary at which the bound states merge with the continuum which occurs at the energies E = ±Δ and E + a/4 = Δ. For the potential of strength a = 24 we find that there are four zeroenergy bound modes, occurring at Δ = 0.597, 2.276, 3.817 and 5.282. Their normalized wavefunctions are shown in Fig. 3.
As mentioned previously, the number of zeroenergy modes is equal to the number of supercritical states (neglecting the spin and valley degrees of freedom). For the symmetric PöschlTeller potential, the eigenvalues of these supercritical states can be determined approximately, via simple analytic expressions. Moving to the symmetric basis, (Ψ_{I}, Ψ_{II})^{T}, allows the pair of coupled first order differential equations, Eq. (3) and Eq. (4) to be reduced to a single second order differential equation in Ψ_{II}:
For supercritical states, E = −Δ, Eq. (21) transforms into the differential equation for the angular prolate spheroidal wave functions^{93}:
where \(\eta =\,\tanh (z)\), c = ±V _{0}, \({{\rm{\Psi }}}_{{\rm{II}}}=\sqrt{1{h}^{2}}{S}_{1N}\), and S _{1N} are the spheroidal wave functions. λ _{1N} = a(a − 8Δ)/16, where the permissible values of λ _{1N} must be determined to assure that S _{1N}(c, η) are finite at η = ±1. The permissible λ _{1N} can be obtained via the asymptotic expansion
where N = 1, 2, 3, … and q = 2N − 1^{93}. Keeping only the terms of expansion shown in Eq. (23) yields the following eigenvalues:
where N is restricted to ensure that E is negative. The resulting approximate eigenvalues for the symmetric PöschlTeller potential of strength a = 24 are E = −2.475, −1.555 and −0.734 respectively, and are indicated as black crosses in Fig. 2. The approximate eigenvalues deviate increasingly from the numerically exact results, E = −2.473, −1.542 and −0.682, with decreasing y. It should be noted that a refinement of the approximate values of λ _{1N} can be found in ref. 93.
For the hyperbolic secant potential, \(V={V}_{0}/\cosh (\tilde{x})\), it was found that there was a minimum potential strength of V _{0} = 1/2, required to observe a zeroenergy mode^{22}. According to the Landauer formula, the conductance along the waveguide when the Fermi level is set to the Dirac point is 4ne ^{2}/h, where n is the number of zeroenergy modes. The existence of a threshold in the potential strength needed for the waveguide to contain a zeroenergy mode allowed us to suggest that such waveguides could be used as switchable devices. However, later numerical calculations utilizing a variable phase method implied that powerdecaying potentials always possess a bound mode^{58}. This result cast serious doubt in the validity of employing exponentially decaying potentials as a suitable model for graphene waveguides, since realistic potential profiles decay a power of distance rather than exponentially. Notably, the threshold potential strength at which the first zeroenergy mode appears can be obtained from the condition of the first bound state coinciding with the first supercritical state, i.e. E = −Δ = 0. In this instance, Eq. (21) can be solved exactly:
For even modes of Ψ_{I}, C _{2} = 0, whereas, for odd modes of Ψ_{I}, C _{1} = 0. In the absence of the potential, when E = Δ = 0 the two first order differential equations in Ψ_{I} and Ψ_{II} decouple, and Eq. (21) reduces to a first order differential equation. As E = −Δ → 0 and x → ±∞ (where the potential is zero), Eq. (25) and Eq. (26) are required to be linearly dependent^{94} and the Wronskian of the solutions Ψ_{I} and Ψ_{II} is zero^{95}. Consequently, the bound modes satisfy the condition: Ψ_{I}(±∞)^{2} = Ψ_{II}(±∞)^{2}. Therefore, the threshold potential strength at which the first zeroenergy mode appears is found by the condition
Therefore, for any squareintegrable potential, the threshold for the appearance of the first bound state of zero energy is only a function of the integrated potential. Notably, this is the same result obtained as relativistic Levinson theorem^{45, 59,60,61,62}. For the PöschlTeller potential, Eq. (27) yields a = π, which agrees with Eq. (18). For \(V={V}_{0}/\cosh (\tilde{x})\), Eq. (27) yields V _{0} = 1/2 which restores the result of ref. 22. This result implies that squareintegrable power decaying potentials do indeed have a threshold, in contrast to the numerically predicted result of ref. 58. In this respect, exponentially decaying potentials are not that different from powerdecaying and are perfectly suitable for the modeling of topgated Weyl semimetals.
Finally, it should be noted that in the nonrelativistic limit, Eq. (10) restores the well known results^{14} for the bound state functions of the Schrödinger equation for the PöschlTeller potential. In the limit that α and δ are much smaller than β and γ (i.e. large Δ):
where Q = \((1+\gamma \beta \pm \sqrt{1+{\gamma }^{2}+{\beta }^{2}4({\eta }_{s}+{\delta }_{s})})/2\) and _{2}F _{1} is the Gauss hypergeometric function. Substituting E = E _{SE} + Δ, b = 0 and s _{β} = −s _{γ} = 1 into Eq. (10), results in the nonrelativistic bound state functions
where \(\varepsilon ={s}_{\beta }\sqrt{2{E}_{{\rm{SE}}}{\rm{\Delta }}}\) and \(T=(1+\sqrt{1+2a{\rm{\Delta }}})\mathrm{/2}\). For the solutions to be finite at Z = 0, we require that ε − T = −N where N = 0, 1, 2, …. When this criteria is met, the Gauss hypergeometric function is a polynomial of degree N and the energy levels are given by
Modified PöschlTeller potential solutions
We shall now consider the case of finite b, which represents a smooth asymmetric waveguide. Previously considered asymmetric waveguides varied abruptly on the same scale as the graphene lattice constant^{54, 96,97,98,99}. We shall now consider more realistic smooth asymmetric waveguides, which fit closer to experimentally achievable potential profiles. In Fig. 4, we plot the energy spectrum for the potential defined by the parameters a = 24 and b = 2. The introduction of the asymmetry term b reduces the number of modes at E = 0, which are now quasibound modes for the massless case (Fig. 4a), since they can couple to continuum states outside of the waveguide. Naturally, for massive Dirac fermions full confinement is possible across a range of energies. In Fig. 5, we show a schematic diagram of the dispersion of a gapped Dirac material, subjected to the modified PöschlTeller potential defined by a = 24 and b = 2. For a particle of mass M = 2, it can be seen that for the energy range E = 0 to E = 2 there are no continuum states outside of the well, therefore in that range all bound solutions will be nonleaky. The corresponding energy spectrum of confined states is shown in Fig. 4b.
Conclusions
We have analyzed the behavior of quasirelativistic twodimensional particles subjected to a modified PöschlTeller potential. Our results have direct applications to electronic waveguides in 2D Dirac materials. For the symmetric PöschlTeller potential, explicit forms were obtained for the eigenvalues of supercritical states. A universal expression, for any symmetric potential, was obtained for the critical potential strength required to observe the first zeroenergy state. The well known results for the PöschlTeller potential are recovered in the nonrelativistic limit.
Change history
04 March 2020
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
We are grateful to Charles Downing for the critical reading of the manuscript. This work was supported by the EU H2020 RISE project CoExAN (Grant No. H2020644076), EU FP7 ITN NOTEDEV (Grant No. FP7607521), FP7 IRSES projects CANTOR (Grant No. FP7612285), QOCaN (Grant No. FP7316432), and InterNoM (Grant No. FP7612624). R.R.H. acknowledges financial support from URCO (Project No. 09F U 1TAY151TAY16) and Research Links Travel Grant by the British Council Newton Fund.
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Hartmann, R.R., Portnoi, M.E. Twodimensional Dirac particles in a PöschlTeller waveguide. Sci Rep 7, 11599 (2017). https://doi.org/10.1038/s4159801711411w
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DOI: https://doi.org/10.1038/s4159801711411w
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