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mKdV equation approach to zero energy states of graphene

We utilize the relation between soliton solutions of the mKdV and the combined mKdV-KdV equation and the Dirac equation to construct electrostatic fields which yield exact zero energy states of graphene.

Introduction.-The relation between solutions of nonlinear evolution equations and nonrelativistic Schrödinger equation is well known. For example, the soliton solutions of the KdV equation u t +u xxx +6uu x = 0 can be used to construct reflectionless potentials of the one dimensional Schrödinger equation [1][2][3][4]. There is also a similar relation between the solition solutions of the mKdV and combined mKdV-KdV equation and the Dirac equation [2,[5][6][7][8][9]. However this relation has been exploited to a lesser extent to find out electrostatic fields admitting exact solutions of Dirac equation [10].
It may be recalled that the dynamics of electrons in graphene is governed by (2 + 1) dimensional massless Dirac equation with the exception that the velocity of light c is replaced by the Fermi velocity (v F = c/300) [11]. In graphene one of the important problems is to confine or control the motion of the electrons. Such confinement can be achieved, for example, by using various types of magnetic fields [12]. On the other hand it is generally believed that because of Klein tunneling electrostatic confinement is relatively more difficult compared to magnetic confinement. However it has been shown recently that certain types of electrostatic fields can indeed produce confinement [13] and zero energy states which were earlier found using various magnetic fields [17] can also be found using electrostatic fields [14][15][16].
In this Letter it will be shown that the Dirac equation of the graphene model is closely related to nonlinear evolution equations, namely the mKdV equation and the combined KdV-mKdV equation [7]. To be more specific, the soliton solutions of the above mentioned equations actually act as electrostatic potentials of the Dirac equation for the charge carriers in graphene. Here our objective is to use this correspondence to obtain several electrostatic field configurations which admit exact zero energy solutions of the graphene system.
Zero energy states in graphene.-The motion of electrons in graphene in the presence of an electrostatic field or potential is governed by the equation where v F is the Fermi velocity, σ x,y are the Pauli spin matrices and U (x, y) is the potential. Here we would consider a potential depending only on the x coordinate and consequently the wavefunction can be taken as Then from Eq.(1) we obtain where V (x) = U (x)/hv F and ǫ = E/hv F . It is interesting to note that Eqs. (3) and (4) are invariant under the following transformations: This means that if the spinor ψ = e iky y (ψ A , ψ B ) t is a solution for k y , then ψ = e −ikyy (ψ B , ψ A ) t is a solution for −k y (here "t" means transpose). That is, eignestates with opposite signs of k y are spin-flipped.
Here we shall consider k y = 0, since for k y = 0 the wave function ψ(x) is normalizable only in a finite region if V (x) is real.
Defining ψ 1,2 = (ψ A ± ψ B ) we obtain from Eqs. (3) and (4) The above equations can be easily decoupled and the equations satisfied by the components ψ 1,2 can be obtained as Note that Eqs. (8) and (9) can be interpreted as a pair of Schrödinger equations with energy dependent potentials which for ǫ = 0 read It is also interesting to note that when V (x) is an even function the potentials in (26) or the ones appearing in Eqs. (8) or (9) are PT symmetric [18]. It will now be shown that the equations for graphene's zero energy states are related to solutions of nonlinear evolution equations. mKdV equation.-The mKdV equation for a real field u(x, t) is given by where the subscript t and x indicate the time and space derivatives, respectively. This equation is the compatibility condition of a set of linear equations [2,7] for certain functions v 1,2 . It is seen that the transformations map Eqs. (3) and (4) into (16) and (17), and vice versa for ǫ = 0. Accordingly, the transformation changes Eq. (13) into Comparing the two sets of Eqs. (16), (17) and (8)  The soliton solutions of the mKdV equation can be obtained by the inverse scattering method. For the boundary conditions u(x, t) → 0 as |x| → ∞, the N -soliton solution is given by [6,7] u(x, t) = −2 ∂ ∂x tan −1 Im det(I + A) Re det(I + A) . (18) where I is the N × N unit matrix, and A denotes the N × N matrix with elements m, n = 1, 2, . . . , N, ζ n = iη n , η n > 0, (19) d n (t) = d n (0) exp(8iζ 3 n t).
One-and two-soliton potentials.-We shall present some examples of solutions of the mKdV equation that provide exactly solvable potentials for the graphene's problem. Only solutions u(x) which correspond to confining potential V (x) = u(x) are presented.
By taking ζ = iη, η > 0, one obtains a one-soliton solution of the mKdV equation This gives an exactly solvable potential V (x) = u(x) admitting one zero energy bound state with k 2 y = η 2 . The case η = 1/2 correspond to the reflectionless potential with one bound state, which is a special case of the example considered in [14,16].
Next, starting with two eigenvalues One obtains a suitable solution with two bound states for ǫ = 0 and k 2 y = η 2 1 and η 2 2 : where ǫ 1 , ǫ 2 are constants. This expression provides infinitely many potentials for the graphene system with two bound states. In Fig 1 we present plots of the potential (21) for different values of the parameters and it is seen that depending on the parameter values the potential can be is a single well or a double well potential. The special choice ǫ 1 = ǫ 2 = 1, η 1 = 1/2 and η 2 = 3/2 gives the reflectionless potential This corresponds to the reflectionless potential with two bound states, which is a special case of the example in [14,16]. In general, continuing the process, Eqs. (18) with (19) allow one to construct potentials with N -bound states for the graphene system. Periodic solutions.-The mKdV equation also admits periodic solutions [7,[19][20][21]. A periodic solution corresponding to one-gap state is given in [7] u where sn(η x, m) is the Jacobi elliptic function with the modulus m(0 ≤ m ≤ 1) and the scale Here a > b > c are constants. This solution of mKdV equation thus furnishes a periodic potential of the graphene system that admits a zero energy bound state with k 2 y = η 2 . In the limit m → 1(b = c = 0) , sn → tanh and η → a/2, and u(x) → −a sech ax, which is just the 1solution case in Eq. (20).
It turns out that the solution u(x) of this equation is connected to the potential of the Dirac equation by The N -soliton solutions u(x) of Eq.(27) are known [7]. For illustration we present the one-soliton solution Summary.-In this Letter we have utilized the relation between the solutions of nonlinear evolution equations, namely, the mKdV equation and combined mKdV and KdV equation and the Dirac equation to find a large number of potentials admitting exactly known zero energy solutions. There are of course many other solutions of the aforementioned equations [22] and depending on the specific need of the problem or experiment some of these solution can play an important role. -