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Crystals 2013, 3(1), 14-27; doi:10.3390/cryst3010014
Abstract: We study the bound state spectrum and the conditions for entering a supercritical regime in graphene with strong intrinsic and Rashba spin-orbit interactions within the topological insulator phase. Explicit results are provided for a disk-shaped potential well and for the Coulomb center problem.
The electronic properties of graphene monolayers are presently under intense study. Previous works have already revealed many novel and fundamental insights; for reviews, see [1,2]. Following the seminal work of Kane and Mele , it may be possible to engineer a two-dimensional (2D) topological insulator (TI) phase  in graphene by enhancing the—usually very weak [5,6,7]—spin-orbit interaction (SOI) in graphene. This enhancement could, for instance, be achieved by the deposition of suitable adatoms . Remarkably, random deposition should already be sufficient to reach the TI phase [9,10,11] where the effective “intrinsic” SOI exceeds (half of) the “Rashba” SOI . So far, the only 2D TIs realized experimentally are based on the mercury telluride class. Using graphene as a TI material constitutes a very attractive option because of the ready availability of high-quality graphene samples  and the exciting prospects for stable and robust TI-based devices , see also [12,13].
In this work, we study bound-state solutions and the conditions for supercriticality in a graphene-based TI. Such questions can arise in the presence of an electrostatically generated potential well (“quantum dot”) or for a Coulomb center. The latter case can be realized by artificial alignment of Co trimers , or when defects or charged impurities reside in the graphene layer. Without SOI, the Coulomb impurity problem in graphene has been theoretically studied in depth [15,16,17,18,19,20]; for reviews, see [1,2]. Moreover, for , an additional mass term in the Hamiltonian corresponds to the intrinsic SOI (see below), and the massive Coulomb impurity problem in graphene has been analyzed in [21,22,23,24,25,26]. However, a finite Rashba SOI is inevitable in practice and has profound consequences. In particular, breaks electron-hole symmetry and modifies the structure of the vacuum. We therefore address the general case with both and finite, but within the TI phase , in this paper. Experimental progress on the observation of Dirac quasiparticles near a Coulomb impurity in graphene was also reported very recently , and we are confident that the topological version with enhanced SOI can be studied experimentally in the near future. Our work may also be helpful in the understanding of spin-orbit mediated spin relaxation in graphene .
The atomic collapse problem for Dirac fermions in an attractive Coulomb potential, , could thereby be realized in topological graphene. Here we use the dimensionless impurity strength
2. Model and Symmetries
2.1. Kane–Mele Model with Radially Symmetric Potential
We study the Kane–Mele model for a 2D graphene monolayer with both intrinsic ( ) and Rashba ( ) SOI  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
Using polar coordinates, it is now straightforward to verify (see also ) that total angular momentum, defined as
In this representation, the radial Dirac equation for total angular momentum and valley index reads
One easily checks that Equation (8) satisfies the parity symmetry relation
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 , 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 .
2.2. Zero 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
In fact, using this transformation matrix to carry out a similarity transformation, , we obtain
For parity , the block matrix in Equation (14) is formally identical to an effective problem with , fixed , and the substitutions
2.3. Solution in Region with Constant Potential
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
Using the Bessel function recurrence relation, , the set of four coupled differential Equations (7) simplifies to a set of algebraic equations
For , a solution regular at the origin is obtained by putting , which describes standing radial waves. Equation (17) then yields the unnormalized spinor
For , instead it is convenient to set in Equation (16). Using the identity , the unnormalized spinor resulting from Equation (17) then takes the form
2.4. Solution without Potential
In a free system, i.e., when for all , the only acceptable solution corresponding to a physical state is obtained when . 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)]
3. Circular Potential Well
3.1. Bound States
In this section, we study a circular potential well with radius and depth
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
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
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.
3.2. Zero Angular Momentum States
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.
3.3. Threshold for Bound States
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
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 . 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)
Likewise, for , the condition for the appearance of a new bound state is
3.4. Supercritical Behavior
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.
4. Coulomb Center
We now turn to the Coulomb potential, , generated by a positively charged impurity located at the origin, with the dimensionless coupling strength in Equation (1). We consider only the TI phase and analyze the bound-state spectrum and conditions for supercriticality. Again, without loss of generality, we focus on the point only ( ), and first summarize the known solution for [2,21,24]. In that case, is conserved, and the spin-degenerate bound-state energies are labeled by the integer angular momentum and a radial quantum number (for , is also possible)
Following the arguments in Section 2.2 for , the combination of Equation (33) with Equation (15) immediately yields the exact bound-state energy spectrum ( )
Figure 3 shows the resulting bound-state spectrum vs. for the regularized Coulomb potential. Within the energy window Equation (21), we again find that states with parity remain true bound states that dive only for , while states show supercritical diving already for . Figure 4 shows the corresponding critical couplings for , where the lowest bound state with parity turns supercritical. Note that for finite and , a unique value for is found, while for two different critical values for are found. However, this conclusion holds only for finite regularization parameter , i.e., it is non-universal. As seen in the inset of Figure 4, in the limit , both critical values for approach again, which is the value found without SOI.
Finally, for , we can then draw the same qualitative conclusions as in Section 3.4 for the potential well. In particular, we expect that all bound states turn supercritical when their energy reaches the continuum threshold at .
In this work, we have analyzed the bound-state problem for the Kane–Mele model of graphene with intrinsic ( ) and Rashba ( ) spin-orbit couplings when a radially symmetric attractive potential is present. We have focused on the most interesting “topological insulator” phase with . The Rashba term leads to a restructuring of the valence band, with a halving of the density of states in the window , where . This has spectacular consequences for total angular momentum , where the problem can be decomposed into two independent parity sectors ( ). The states remain true bound states even inside the above window and coexist with the continuum solutions as well as possible supercritical resonances in the sector. However, all bound states exhibit supercritical diving for , where the critical threshold ( or for the disk or the Coulomb problem, respectively) is lowered when the Rashba term is present. We hope that these results will soon be put to an experimental test.
This work has been supported by the DFG within the network programs SPP 1459 and SFB-TR 12.
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