We study the Kane–Mele model for a 2D graphene monolayer with both intrinsic () and Rashba () SOI [3] in the presence of a radially symmetric scalar potential . Assuming that is sufficiently smooth to allow for the neglect of inter-valley scattering, the low-energy Hamiltonian near the point is given by (3) with Pauli matrices () in sublattice (spin) space [1]. The Hamiltonian near the other valley ( point) follows for in Equation (3). We note that a sign change of the Rashba SOI, , does not affect the spectrum due to the relation . Without loss of generality, we then put and .

Using polar coordinates, it is now straightforward to verify (see also [21]) that total angular momentum, defined as (4) is conserved and has integer eigenvalues . For given , eigenfunctions of must then be of the form (5) Next we combine the radial functions to (normalized) four-spinors (6)

In this representation, the radial Dirac equation for total angular momentum and valley index reads (7) with Hermitian matrix operators (note that denotes the intrinsic SOI and not the Laplacian) (8) where we use the notation (9)

One easily checks that Equation (8) satisfies the parity symmetry relation (10) Note that this “parity” operation for the radial Hamiltonian is non-standard in the sense that the valley is not changed by the transformation , spin and sublattice are flipped simultaneously, and only the -coordinate is reversed. (We will nonetheless refer to as parity transformation below.) A second symmetry relation connects both valleys, (11)

Using Equation (10), this relation can be traced back to a time-reversal operation. Equations (10) and (11) suggest that eigenenergies typically are four-fold degenerate.

When projected to the subspace of fixed (integer) total angular momentum , the current density operator has angular component and radial component for arbitrary . When real-valued entries can be chosen in , the radial current density thus vanishes separately in each valley. We define the (angular) spin current density as . Remarkably, the transformation defined in Equation (11) conserves both (total and spin) angular currents, while the transformation in Equation (10) reverses the total current but conserves the spin current. Therefore, at any energy, eigenstates supporting spin-filtered counterpropagating currents are possible. However, in contrast to the edge states found in a ribbon geometry [3], these spin-filtered states do not necessarily have a topological origin.

We focus on one point () and omit the -index henceforth; the degenerate Kramers partner easily follows using Equation (11). In addition, using the symmetry (10), it is sufficient to study the model for fixed total angular momentum .

For arbitrary , we now show that a drastic simplification is possible for total angular momentum , which can even allow for an exact solution. Although the lowest-lying bound states for the potentials in Section 3 and Section 4 are found in the sector, exact statements about what happens for are valuable and can be explored along the route sketched here.

The reason why is special can be seen from the parity symmetry relation in Equation (10). The parity transformation connects the sectors, but represents a discrete symmetry of the radial Hamiltonian [see Equation (8)] acting on the four-spinors in Equation (6). Therefore, the subspace can be decomposed into two orthogonal subspaces corresponding to the two distinct eigenvalues of the Hermitian operator . This operator is diagonalized by the matrix (12) (13)

In fact, using this transformation matrix to carry out a similarity transformation, , we obtain (14) For , the upper and lower blocks decouple. Each block has the signature (“parity”) corresponding to the eigenvalues in Equation (13), and represents a mixed sublattice-spin state, see Equations (6) and (12).

For parity , the block matrix in Equation (14) is formally identical to an effective problem with , fixed , and the substitutions (15) This implies that for and arbitrary , the complete spectral information for the full Kane–Mele problem (with ) directly follows from the solution.

We start our analysis of the Hamiltonian (3) with the general solution of Equation (7) for a region of constant potential. Here, it suffices to study , since and enter only through the combination in Equation (8). In Section 3, we will use this solution to solve the case of a step potential.

The general solution to Equation (7) follows from the Ansatz (16) where the are real coefficients; is one of the cylinder (Bessel) functions; or ; and denotes a real spectral parameter. In particular, is a generalized radial wavenumber. We here assume true bound-state solutions with real-valued energy. However, for quasi-stationary resonance states with complex energy, and the may be complex as well.

Using the Bessel function recurrence relation, , the set of four coupled differential Equations (7) simplifies to a set of algebraic equations (17) Notably, does not appear here, and therefore the spectral parameter depends only on the energy . The condition of vanishing determinant then yields a quadratic equation for , with the two solutions (18) Which Bessel function is chosen in Equation (16) now depends on the sign of and on the imposed regularity conditions for and/or .

For , a solution regular at the origin is obtained by putting , which describes standing radial waves. Equation (17) then yields the unnormalized spinor (19)

For , instead it is convenient to set in Equation (16). Using the identity , the unnormalized spinor resulting from Equation (17) then takes the form (20) where the modified Bessel function describes evanescent modes, exponentially decaying at infinity.

In a free system, i.e., when for all , the only acceptable solution corresponding to a physical state is obtained when [30]. For , at least one in Equation (18) for all , and the system is gapless. However, the TI phase defined by has a gap as we show now.

For , Equation (18) tells us that for and for , both solutions are positive and hence (for given and ) there are two eigenstates for given energy . However, within the energy window [with in Equation (18)] (21) we have and , i.e., only the eigenstate represents a physical solution. Both are negative when , and no physical state exists at all. This precisely corresponds to the topological gap in the TI phase [3]. Note that due to the Rashba SOI, the valence band edge is characterized by the two energies , with halved density of states in the energy window (21). One may then ask at which energy ( or ) the supercritical diving of a bound state level in an impurity potential takes place.

In this section, we study a circular potential well with radius and depth (22)

We always stay within the TI phase , where bound states are expected for energies in the window . For , the corresponding radial eigenspinor [see Equation (6)] is written with arbitrary prefactors in the form (23) with Equation (19) for . Here, the follow from Equation (18) by including the potential shift, (24) For , the general solution is again written as (25) However, now is given by Equation (20), since for true bound states with only evanescent states outside the potential well.

The continuity condition for the four-spinor at the potential step, then yields a homogeneous linear system of equations for the four parameters (). A nontrivial solution is only possible when the determinant of the corresponding matrix (which is too lengthy to be given here but follows directly from the above expressions) vanishes (26) Solving the energy quantization condition (26) then yields the discrete bound-state spectrum (). It is then straightforward to determine the corresponding spinor wavefunctions.

Numerical solution of Equation (26) yields the bound-state spectrum shown in Figure 1. When exceeds a (-dependent) “threshold” value, , a bound state splits off the conduction band edge. When increasing further, this bound-state energy level moves down almost linearly, cf. inset of Figure 1, and finally reaches the valence band edge at some “critical” value . (For , we will see below that this definition needs some revision.) Increasing even further, the bound state is then expected to dive into the valence band and become a finite-width supercritical resonance, i.e., the energy would then acquire an imaginary part.

Surprisingly, for , we find a different scenario where supercritical diving, with finite lifetime of the resonance, happens only for half of the bound states entering the energy window (21). Noting that states with different parity do not mix, see Section 2.2, we observe that all bound states enter the valence band as true bound states (no imaginary part) throughout the energy window (21) while the valence band continuum is spanned by the states. We then define for bound states as the true supercritical threshold where . However, the bound states become supercritical already when reaching .

Therefore an intriguing physical situation arises for in the energy window (21). While states are true bound states (no lifetime broadening), they coexist with states which span the valence band continuum or possibly form supercritical resonances. For , however, all bound states dive, become finite-width resonances, and eventually become dissolved in the continuum.

Returning to arbitrary total angular momentum , we observe that whenever hits a possible threshold value , a new bound state is generated, which then dives into the valence band at another potential depth (and so on). Analytical results for all possible threshold values follow by expanding Equation (26) for weak dimensionless binding energy For and , Equation (26) yields after some algebra (27) where is the Euler constant and . The binding energy approaches zero for , where Equation (27) simplifies to (28) For vanishing Rashba SOI , this reproduces known results [25]. For any , we observe that the bound state in Equation (28) exists for arbitrarily shallow potential depth .

The threshold values for higher-lying bound states also follow from the binding energy (27), since vanishes for and for . When one of these two conditions is fulfilled at some , a new bound state appears for potential depth above . This statement is in fact quite general: By similar reasoning, we find that the threshold values for follow by counting the zeroes of . Without SOI, this has also been discussed in [31]. Note that this argument immediately implies that no bound state with exists for .

From the above equations, we can then infer the threshold values for all bound states with or in analytical form. These are labeled by and (for , corresponds to parity) (29) where is the th zero of the Bessel function.

Likewise, for , the condition for the appearance of a new bound state is (30) Close examination of this condition shows that no bound states with exist for . We conclude that bound states in a very weak potential well exist only for .

As can be seen in Figure 1, the lowest bound state is also the first to enter the valence band continuum for . For , the critical value is known to be [25] (31) with . The energy of the resonant state acquires an imaginary part for [25]. For , we have obtained implicit expressions for , plotted in Figure 2. Note that these results reproduce Equation (31) for . The almost linear decrease of with increasing , see Figure 2, can be rationalized by noting that the valence band edge is located at . Thereby supercritical resonances could be reached already for lower potential depth by increasing the Rashba SOI. Similarly, with increasing disk radius , the critical value decreases, see the inset of Figure 2. For the lowest bound state, the critical value in fact follows in analytical form, (32) where .

Since the parity decoupling in Section 2.2 only holds for , it is natural to expect that all bound states turn into finite-width resonances when . This expectation is confirmed by an explicit calculation as follows. Within in the window , a true bound state should not receive a contribution from for , but instead has to be obtained by matching an Ansatz as in Equation (23) for the spinor state inside the disk () to an evanescent spinor state . However, the matching condition is then found to have no real solution , i.e., there are no true bound states with in the energy window (21). We therefore conclude that all bound states turn supercritical when . Note that this statement includes the lowest-lying bound state (which has ). This implies that a finite Rashba SOI can considerably lower the potential depth required for entering the supercritical regime.