Crystals 2013, 3(1), 14-27; doi:10.3390/cryst3010014

Article
Bound States and Supercriticality in Graphene-Based Topological Insulators
Denis Klöpfer 1, Alessandro De Martino 2 and Reinhold Egger 1,*
1
Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany; E-Mail: kloepfer@thphy.uni-duesseldorf.de
2
Department of Mathematical Science, City University London, London EC1V 0HB, UK; E-Mail: ademarti@city.ac.uk
*
Author to whom correspondence should be addressed; E-Mail: egger@thphy.uni-duesseldorf.de; Tel.: +49-211-81-14710; Fax:+49-211-81-15630.
Received: 16 November 2012; in revised form: 17 December 2012 / Accepted: 9 January 2013 /
Published: 21 January 2013

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.
Keywords:
graphene; supercriticality; spin-orbit interaction

1. Introduction

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 [3], it may be possible to engineer a two-dimensional (2D) topological insulator (TI) phase [4] 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 [8]. Remarkably, random deposition should already be sufficient to reach the TI phase [9,10,11] where the effective “intrinsic” SOI Crystals 03 00014 i001 exceeds (half of) the “Rashba” SOI Crystals 03 00014 i002. 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 [1] and the exciting prospects for stable and robust TI-based devices [4], 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 [14], 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 Crystals 03 00014 i003, an additional mass term in the Hamiltonian corresponds to the intrinsic SOI Crystals 03 00014 i001 (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 Crystals 03 00014 i002 is inevitable in practice and has profound consequences. In particular, Crystals 03 00014 i004 breaks electron-hole symmetry and modifies the structure of the vacuum. We therefore address the general case with both Crystals 03 00014 i001 and Crystals 03 00014 i002 finite, but within the TI phase Crystals 03 00014 i005, in this paper. Experimental progress on the observation of Dirac quasiparticles near a Coulomb impurity in graphene was also reported very recently [14], 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 [27].

The atomic collapse problem for Dirac fermions in an attractive Coulomb potential, Crystals 03 00014 i006, could thereby be realized in topological graphene. Here we use the dimensionless impurity strength

Crystals 03 00014 i007
where Crystals 03 00014 i008 is the number of positive charges held by the impurity; Crystals 03 00014 i009 a dielectric constant characterizing the environment; and Crystals 03 00014 i010 m Crystals 03 00014 i011s the Fermi velocity. Without SOI, the Hamiltonian is not self-adjoint for Crystals 03 00014 i012, and the potential needs short-distance regularization, e.g., by setting Crystals 03 00014 i013 with short-distance cutoff Crystals 03 00014 i014 of the order of the lattice constant of graphene [1,2]. Including a finite “mass” Crystals 03 00014 i001, i.e., the intrinsic SOI, but keeping Crystals 03 00014 i003, the critical coupling Crystals 03 00014 i015 is shifted to [24]
Crystals 03 00014 i016
approaching the value Crystals 03 00014 i017 for Crystals 03 00014 i018. In the supercritical regime Crystals 03 00014 i019, the lowest bound state “dives” into the valence band continuum (Dirac sea). It then becomes a resonance with complex energy, where the imaginary part corresponds to the finite decay rate into the continuum. Below we show that the Rashba SOI provides an interesting twist to this supercriticality story. The structure of this article is as follows. In Section 2 we introduce the model and summarize its symmetries. The case of a circular potential well is addressed in Section 3 before turning to the Coulomb center in Section 4. Some conclusions are offered in Section 5. Note that we do not include a magnetic field (see, e.g., [28,29]) and thus our model enjoys time-reversal symmetry. Below, we often use units with Crystals 03 00014 i020.

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 ( Crystals 03 00014 i001) and Rashba ( Crystals 03 00014 i002) SOI [3] in the presence of a radially symmetric scalar potential Crystals 03 00014 i021. Assuming that Crystals 03 00014 i021 is sufficiently smooth to allow for the neglect of inter-valley scattering, the low-energy Hamiltonian near the Crystals 03 00014 i022 point Crystals 03 00014 i023 is given by

Crystals 03 00014 i024
with Pauli matrices Crystals 03 00014 i025 ( Crystals 03 00014 i026) in sublattice (spin) space [1]. The Hamiltonian near the other valley ( Crystals 03 00014 i027 point) follows for Crystals 03 00014 i028 in Equation (3). We note that a sign change of the Rashba SOI, Crystals 03 00014 i029, does not affect the spectrum due to the relation Crystals 03 00014 i030. Without loss of generality, we then put Crystals 03 00014 i031 and Crystals 03 00014 i032.

Using polar coordinates, it is now straightforward to verify (see also [21]) that total angular momentum, defined as

Crystals 03 00014 i033
is conserved and has integer eigenvalues Crystals 03 00014 i034. For given Crystals 03 00014 i034, eigenfunctions of Crystals 03 00014 i035 must then be of the form
Crystals 03 00014 i036
Next we combine the radial functions to (normalized) four-spinors
Crystals 03 00014 i037

In this representation, the radial Dirac equation for total angular momentum Crystals 03 00014 i034 and valley index Crystals 03 00014 i038 reads

Crystals 03 00014 i039
with Hermitian matrix operators (note that Crystals 03 00014 i001 denotes the intrinsic SOI and not the Laplacian)
Crystals 03 00014 i040
Crystals 03 00014 i041
where we use the notation
Crystals 03 00014 i042

One easily checks that Equation (8) satisfies the parity symmetry relation

Crystals 03 00014 i043
Note that this “parity” operation for the radial Hamiltonian is non-standard in the sense that the valley is not changed by the transformation Crystals 03 00014 i044, spin and sublattice are flipped simultaneously, and only the Crystals 03 00014 i045-coordinate is reversed. (We will nonetheless refer to Crystals 03 00014 i044 as parity transformation below.) A second symmetry relation connects both valleys,
Crystals 03 00014 i046

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 Crystals 03 00014 i034, the current density operator has angular component Crystals 03 00014 i047 and radial component Crystals 03 00014 i048 for arbitrary Crystals 03 00014 i034. When real-valued entries can be chosen in Crystals 03 00014 i049, the radial current density thus vanishes separately in each valley. We define the (angular) spin current density as Crystals 03 00014 i050. 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 Crystals 03 00014 i022 point ( Crystals 03 00014 i051) and omit the Crystals 03 00014 i052-index henceforth; the degenerate Crystals 03 00014 i053 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 Crystals 03 00014 i054.

2.2. Zero Total Angular Momentum

For arbitrary Crystals 03 00014 i021, we now show that a drastic simplification is possible for total angular momentum Crystals 03 00014 i055, 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 Crystals 03 00014 i056 sector, exact statements about what happens for Crystals 03 00014 i055 are valuable and can be explored along the route sketched here.

The reason why Crystals 03 00014 i055 is special can be seen from the parity symmetry relation in Equation (10). The parity transformation Crystals 03 00014 i044 connects the Crystals 03 00014 i057 sectors, but represents a discrete symmetry of the Crystals 03 00014 i055 radial Hamiltonian Crystals 03 00014 i058 [see Equation (8)] acting on the four-spinors in Equation (6). Therefore, the Crystals 03 00014 i055 subspace can be decomposed into two orthogonal subspaces corresponding to the two distinct eigenvalues of the Hermitian operator Crystals 03 00014 i044. This operator is diagonalized by the matrix

Crystals 03 00014 i059
Crystals 03 00014 i060

In fact, using this transformation matrix to carry out a similarity transformation, Crystals 03 00014 i061, we obtain

Crystals 03 00014 i062
For Crystals 03 00014 i055, the upper and lower Crystals 03 00014 i063 blocks decouple. Each block has the signature (“parity”) Crystals 03 00014 i064 corresponding to the eigenvalues in Equation (13), and represents a mixed sublattice-spin state, see Equations (6) and (12).

For parity Crystals 03 00014 i064, the Crystals 03 00014 i063 block matrix in Equation (14) is formally identical to an effective Crystals 03 00014 i003 problem with Crystals 03 00014 i055, fixed Crystals 03 00014 i065, and the substitutions

Crystals 03 00014 i066
This implies that for Crystals 03 00014 i055 and arbitrary Crystals 03 00014 i021, the complete spectral information for the full Kane–Mele problem (with Crystals 03 00014 i004) directly follows from the Crystals 03 00014 i003 solution.

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 Crystals 03 00014 i067, since Crystals 03 00014 i068 and Crystals 03 00014 i069 enter only through the combination Crystals 03 00014 i070 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

Crystals 03 00014 i071
where the Crystals 03 00014 i072 are real coefficients; Crystals 03 00014 i073 is one of the cylinder (Bessel) functions; Crystals 03 00014 i074 or Crystals 03 00014 i075; and Crystals 03 00014 i076 denotes a real spectral parameter. In particular, Crystals 03 00014 i077 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, Crystals 03 00014 i076 and the Crystals 03 00014 i072 may be complex as well.

Using the Bessel function recurrence relation, Crystals 03 00014 i078, the set of four coupled differential Equations (7) simplifies to a set of algebraic equations

Crystals 03 00014 i079
Notably, Crystals 03 00014 i034 does not appear here, and therefore the spectral parameter Crystals 03 00014 i076 depends only on the energy Crystals 03 00014 i068. The condition of vanishing determinant then yields a quadratic equation for Crystals 03 00014 i076, with the two solutions
Crystals 03 00014 i080
Which Bessel function is chosen in Equation (16) now depends on the sign of Crystals 03 00014 i081 and on the imposed regularity conditions for Crystals 03 00014 i082 and/or Crystals 03 00014 i083.

For Crystals 03 00014 i084, a solution regular at the origin is obtained by putting Crystals 03 00014 i074, which describes standing radial waves. Equation (17) then yields the unnormalized spinor

Crystals 03 00014 i085

For Crystals 03 00014 i086, instead it is convenient to set Crystals 03 00014 i075 in Equation (16). Using the identity Crystals 03 00014 i087, the unnormalized spinor resulting from Equation (17) then takes the form

Crystals 03 00014 i088
where the modified Bessel function Crystals 03 00014 i089 describes evanescent modes, exponentially decaying at infinity.

2.4. Solution without Potential

In a free system, i.e., when Crystals 03 00014 i067 for all Crystals 03 00014 i090, the only acceptable solution corresponding to a physical state is obtained when Crystals 03 00014 i084 [30]. For Crystals 03 00014 i091, at least one Crystals 03 00014 i084 in Equation (18) for all Crystals 03 00014 i068, and the system is gapless. However, the TI phase defined by Crystals 03 00014 i005 has a gap as we show now.

For Crystals 03 00014 i005, Equation (18) tells us that for Crystals 03 00014 i092 and for Crystals 03 00014 i093, both solutions Crystals 03 00014 i081 are positive and hence (for given Crystals 03 00014 i034 and Crystals 03 00014 i052) there are two eigenstates Crystals 03 00014 i094 for given energy Crystals 03 00014 i068. However, within the energy window [with Crystals 03 00014 i095 in Equation (18)]

Crystals 03 00014 i096
we have Crystals 03 00014 i097 and Crystals 03 00014 i098, i.e., only the eigenstate Crystals 03 00014 i099 represents a physical solution. Both Crystals 03 00014 i081 are negative when Crystals 03 00014 i100, 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 Crystals 03 00014 i095, with halved density of states in the energy window (21). One may then ask at which energy ( Crystals 03 00014 i101 or Crystals 03 00014 i102) the supercritical diving of a bound state level in an impurity potential takes place.

3. Circular Potential Well

3.1. Bound States

In this section, we study a circular potential well with radius Crystals 03 00014 i014 and depth Crystals 03 00014 i103

Crystals 03 00014 i104

We always stay within the TI phase Crystals 03 00014 i005, where bound states are expected for energies Crystals 03 00014 i105 in the window Crystals 03 00014 i106. For Crystals 03 00014 i107, the corresponding radial eigenspinor [see Equation (6)] is written with arbitrary prefactors Crystals 03 00014 i108 in the form

Crystals 03 00014 i109
with Equation (19) for Crystals 03 00014 i110. Here, the Crystals 03 00014 i111 follow from Equation (18) by including the potential shift,
Crystals 03 00014 i112
For Crystals 03 00014 i113, the general solution is again written as
Crystals 03 00014 i114
However, now Crystals 03 00014 i094 is given by Equation (20), since Crystals 03 00014 i086 for true bound states with only evanescent states outside the potential well.

The continuity condition for the four-spinor at the potential step, Crystals 03 00014 i115 then yields a homogeneous linear system of equations for the four parameters ( Crystals 03 00014 i116). A nontrivial solution is only possible when the determinant of the corresponding Crystals 03 00014 i117 matrix Crystals 03 00014 i118 (which is too lengthy to be given here but follows directly from the above expressions) vanishes

Crystals 03 00014 i119
Solving the energy quantization condition (26) then yields the discrete bound-state spectrum ( Crystals 03 00014 i120). 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 Crystals 03 00014 i121 exceeds a ( Crystals 03 00014 i034-dependent) “threshold” value, Crystals 03 00014 i122, a bound state splits off the conduction band edge. When increasing Crystals 03 00014 i121 further, this bound-state energy level moves down almost linearly, cf. inset of Figure 1, and finally reaches the valence band edge Crystals 03 00014 i123 at some “critical” value Crystals 03 00014 i124. (For Crystals 03 00014 i055, we will see below that this definition needs some revision.) Increasing Crystals 03 00014 i121 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.

Crystals 03 00014 g001 200
Figure 1. Bound-state spectrum ( Crystals 03 00014 i120) vs. Rashba SOI ( Crystals 03 00014 i002) for a circular potential well with depth Crystals 03 00014 i125 and radius Crystals 03 00014 i126. Only the lowest-energy states with Crystals 03 00014 i127 are shown. The red dotted line indicates Crystals 03 00014 i123. The left panel shows Crystals 03 00014 i055 bound states with parity Crystals 03 00014 i064. The right panel shows Crystals 03 00014 i128 bound states. The inset displays the Crystals 03 00014 i056 bound-state energies vs. potential depth Crystals 03 00014 i121 for Crystals 03 00014 i129. At some threshold value Crystals 03 00014 i130 (where Crystals 03 00014 i131 for the lowest state shown), a new bound state emerges from the conduction band. This state dives into the valence band for some critical value Crystals 03 00014 i132, where the valence band edge is at energy Crystals 03 00014 i133. For the second bound state in the inset, Crystals 03 00014 i122 ( Crystals 03 00014 i134) is shown as red (blue) triangle.

Click here to enlarge figure

Figure 1. Bound-state spectrum ( Crystals 03 00014 i120) vs. Rashba SOI ( Crystals 03 00014 i002) for a circular potential well with depth Crystals 03 00014 i125 and radius Crystals 03 00014 i126. Only the lowest-energy states with Crystals 03 00014 i127 are shown. The red dotted line indicates Crystals 03 00014 i123. The left panel shows Crystals 03 00014 i055 bound states with parity Crystals 03 00014 i064. The right panel shows Crystals 03 00014 i128 bound states. The inset displays the Crystals 03 00014 i056 bound-state energies vs. potential depth Crystals 03 00014 i121 for Crystals 03 00014 i129. At some threshold value Crystals 03 00014 i130 (where Crystals 03 00014 i131 for the lowest state shown), a new bound state emerges from the conduction band. This state dives into the valence band for some critical value Crystals 03 00014 i132, where the valence band edge is at energy Crystals 03 00014 i133. For the second bound state in the inset, Crystals 03 00014 i122 ( Crystals 03 00014 i134) is shown as red (blue) triangle.
Crystals 03 00014 g001 1024

3.2. Zero Angular Momentum States

Surprisingly, for Crystals 03 00014 i055, 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 Crystals 03 00014 i064 do not mix, see Section 2.2, we observe that all Crystals 03 00014 i136 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 Crystals 03 00014 i137 states. We then define Crystals 03 00014 i134 for Crystals 03 00014 i138 bound states as the true supercritical threshold where Crystals 03 00014 i139. However, the Crystals 03 00014 i140 bound states become supercritical already when reaching Crystals 03 00014 i123.

Therefore an intriguing physical situation arises for Crystals 03 00014 i055 in the energy window (21). While Crystals 03 00014 i136 states are true bound states (no lifetime broadening), they coexist with Crystals 03 00014 i137 states which span the valence band continuum or possibly form supercritical resonances. For Crystals 03 00014 i093, 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 Crystals 03 00014 i034, we observe that whenever Crystals 03 00014 i121 hits a possible threshold value Crystals 03 00014 i122, a new bound state is generated, which then dives into the valence band at another potential depth Crystals 03 00014 i124 (and so on). Analytical results for all possible threshold values Crystals 03 00014 i122 follow by expanding Equation (26) for weak dimensionless binding energy Crystals 03 00014 i141 For Crystals 03 00014 i142 and Crystals 03 00014 i056, Equation (26) yields after some algebra

Crystals 03 00014 i143
Crystals 03 00014 i144
where Crystals 03 00014 i145 is the Euler constant and Crystals 03 00014 i146. The binding energy approaches zero for Crystals 03 00014 i147, where Equation (27) simplifies to
Crystals 03 00014 i148
For vanishing Rashba SOI Crystals 03 00014 i003, this reproduces known results [25]. For any Crystals 03 00014 i149, we observe that the Crystals 03 00014 i056 bound state in Equation (28) exists for arbitrarily shallow potential depth Crystals 03 00014 i121.

The threshold values Crystals 03 00014 i122 for higher-lying Crystals 03 00014 i056 bound states also follow from the binding energy (27), since Crystals 03 00014 i150 vanishes for Crystals 03 00014 i151 and for Crystals 03 00014 i152. When one of these two conditions is fulfilled at some Crystals 03 00014 i130, a new bound state appears for potential depth above Crystals 03 00014 i122. This statement is in fact quite general: By similar reasoning, we find that the threshold values Crystals 03 00014 i122 for Crystals 03 00014 i055 follow by counting the zeroes of Crystals 03 00014 i153. Without SOI, this has also been discussed in [31]. Note that this argument immediately implies that no bound state with Crystals 03 00014 i055 exists for Crystals 03 00014 i147.

From the above equations, we can then infer the threshold values Crystals 03 00014 i122 for all bound states with Crystals 03 00014 i055 or Crystals 03 00014 i056 in analytical form. These are labeled by Crystals 03 00014 i154 and Crystals 03 00014 i064 (for Crystals 03 00014 i055, Crystals 03 00014 i155 corresponds to parity)

Crystals 03 00014 i156
where Crystals 03 00014 i157 is the Crystals 03 00014 i158th zero of the Crystals 03 00014 i159 Bessel function.

Likewise, for Crystals 03 00014 i160, the condition for the appearance of a new bound state is

Crystals 03 00014 i161
Close examination of this condition shows that no bound states with Crystals 03 00014 i160 exist for Crystals 03 00014 i147. We conclude that bound states in a very weak potential well exist only for Crystals 03 00014 i056.

3.4. Supercritical Behavior

As can be seen in Figure 1, the lowest Crystals 03 00014 i056 bound state is also the first to enter the valence band continuum for Crystals 03 00014 i124. For Crystals 03 00014 i003, the critical value is known to be [25]

Crystals 03 00014 i162
with Crystals 03 00014 i163. The energy of the resonant state acquires an imaginary part for Crystals 03 00014 i164 [25]. For Crystals 03 00014 i165, we have obtained implicit expressions for Crystals 03 00014 i134, plotted in Figure 2. Note that these results reproduce Equation (31) for Crystals 03 00014 i166. The almost linear decrease of Crystals 03 00014 i134 with increasing Crystals 03 00014 i002, see Figure 2, can be rationalized by noting that the valence band edge is located at Crystals 03 00014 i123. Thereby supercritical resonances could be reached already for lower potential depth by increasing the Rashba SOI. Similarly, with increasing disk radius Crystals 03 00014 i014, the critical value Crystals 03 00014 i134 decreases, see the inset of Figure 2. For the lowest Crystals 03 00014 i167 bound state, the critical value in fact follows in analytical form,
Crystals 03 00014 i168
where Crystals 03 00014 i169.

Crystals 03 00014 g002 200
Figure 2. Critical potential depth Crystals 03 00014 i134 for the lowest Crystals 03 00014 i056 bound state level in a disk with Crystals 03 00014 i170. The obtained Crystals 03 00014 i003 value matches the analytical prediction Crystals 03 00014 i171 from Equation (31), while Crystals 03 00014 i172 near the border of the TI phase ( Crystals 03 00014 i173). Inset: Crystals 03 00014 i134 vs. radius Crystals 03 00014 i014 with several values of Crystals 03 00014 i002 (given in units of Crystals 03 00014 i001) for the lowest bound state.

Click here to enlarge figure

Figure 2. Critical potential depth Crystals 03 00014 i134 for the lowest Crystals 03 00014 i056 bound state level in a disk with Crystals 03 00014 i170. The obtained Crystals 03 00014 i003 value matches the analytical prediction Crystals 03 00014 i171 from Equation (31), while Crystals 03 00014 i172 near the border of the TI phase ( Crystals 03 00014 i173). Inset: Crystals 03 00014 i134 vs. radius Crystals 03 00014 i014 with several values of Crystals 03 00014 i002 (given in units of Crystals 03 00014 i001) for the lowest bound state.
Crystals 03 00014 g002 1024

Since the parity decoupling in Section 2.2 only holds for Crystals 03 00014 i055, it is natural to expect that all Crystals 03 00014 i174 bound states turn into finite-width resonances when Crystals 03 00014 i175. This expectation is confirmed by an explicit calculation as follows. Within in the window Crystals 03 00014 i176, a true bound state should not receive a contribution from Crystals 03 00014 i177 for Crystals 03 00014 i113, but instead has to be obtained by matching an Ansatz as in Equation (23) for the spinor state inside the disk ( Crystals 03 00014 i107) to an evanescent spinor state Crystals 03 00014 i178. However, the matching condition is then found to have no real solution Crystals 03 00014 i120, i.e., there are no true bound states with Crystals 03 00014 i174 in the energy window (21). We therefore conclude that all Crystals 03 00014 i174 bound states turn supercritical when Crystals 03 00014 i175. Note that this statement includes the lowest-lying bound state (which has Crystals 03 00014 i056). This implies that a finite Rashba SOI can considerably lower the potential depth Crystals 03 00014 i134 required for entering the supercritical regime.

4. Coulomb Center

We now turn to the Coulomb potential, Crystals 03 00014 i179, generated by a positively charged impurity located at the origin, with the dimensionless coupling strength Crystals 03 00014 i180 in Equation (1). We consider only the TI phase Crystals 03 00014 i005 and analyze the bound-state spectrum and conditions for supercriticality. Again, without loss of generality, we focus on the Crystals 03 00014 i022 point only ( Crystals 03 00014 i051), and first summarize the known solution for Crystals 03 00014 i003 [2,21,24]. In that case, Crystals 03 00014 i181 is conserved, and the spin-degenerate bound-state energies are labeled by the integer angular momentum Crystals 03 00014 i034 and a radial quantum number Crystals 03 00014 i182 (for Crystals 03 00014 i183, Crystals 03 00014 i184 is also possible)

Crystals 03 00014 i185
The corresponding eigenstates then follow in terms of hypergeometric functions. The lowest bound state is Crystals 03 00014 i186, which dives when Crystals 03 00014 i187; note that Crystals 03 00014 i015 precisely corresponds to Crystals 03 00014 i134 in Section 3. In particular, for ( Crystals 03 00014 i188 states we define Crystals 03 00014 i015 in the same manner. Next we discuss how this picture is modified when the Rashba coupling Crystals 03 00014 i002 is included.

Following the arguments in Section 2.2 for Crystals 03 00014 i055, the combination of Equation (33) with Equation (15) immediately yields the exact bound-state energy spectrum ( Crystals 03 00014 i182)

Crystals 03 00014 i189
The corresponding eigenstates then also follow from [21,24]. The very same reasoning also applies to a regularized Crystals 03 00014 i190 potential [23,24], where Crystals 03 00014 i191 is replaced by the constant value Crystals 03 00014 i192. Here, Crystals 03 00014 i014 is a short-distance cutoff scale of the order of the lattice spacing. The solution of the bound-state problem then requires a wavefunction matching procedure, which has been carried out in [24]. Thereby we can already infer all bound states for Crystals 03 00014 i055.

Figure 3 shows the resulting Crystals 03 00014 i055 bound-state spectrum vs. Crystals 03 00014 i180 for the regularized Coulomb potential. Within the energy window Equation (21), we again find that states with parity Crystals 03 00014 i136 remain true bound states that dive only for Crystals 03 00014 i193, while Crystals 03 00014 i137 states show supercritical diving already for Crystals 03 00014 i175. Figure 4 shows the corresponding critical couplings Crystals 03 00014 i015 for Crystals 03 00014 i064, where the lowest Crystals 03 00014 i055 bound state with parity Crystals 03 00014 i155 turns supercritical. Note that for finite Crystals 03 00014 i014 and Crystals 03 00014 i166, a unique value for Crystals 03 00014 i015 is found, while for Crystals 03 00014 i004 two different critical values for Crystals 03 00014 i015 are found. However, this conclusion holds only for finite regularization parameter Crystals 03 00014 i014, i.e., it is non-universal. As seen in the inset of Figure 4, in the limit Crystals 03 00014 i018, both critical values for Crystals 03 00014 i015 approach Crystals 03 00014 i017 again, which is the value found without SOI.

Finally, for Crystals 03 00014 i174, we can then draw the same qualitative conclusions as in Section 3.4 for the potential well. In particular, we expect that all Crystals 03 00014 i174 bound states turn supercritical when their energy Crystals 03 00014 i120 reaches the continuum threshold at Crystals 03 00014 i194.

Crystals 03 00014 g003 200
Figure 3. Bound state energies with angular momentum Crystals 03 00014 i055 ( Crystals 03 00014 i120 in units of Crystals 03 00014 i001) vs. dimensionless impurity strength Crystals 03 00014 i180 for the Coulomb problem with regularization parameter Crystals 03 00014 i195 and Rashba SOI Crystals 03 00014 i129. Solid black (dashed blue) curves correspond to parity Crystals 03 00014 i136 ( Crystals 03 00014 i137). Results for radial number Crystals 03 00014 i196 (with increasing energy) are shown. Red dotted lines denote Crystals 03 00014 i197.

Click here to enlarge figure

Figure 3. Bound state energies with angular momentum Crystals 03 00014 i055 ( Crystals 03 00014 i120 in units of Crystals 03 00014 i001) vs. dimensionless impurity strength Crystals 03 00014 i180 for the Coulomb problem with regularization parameter Crystals 03 00014 i195 and Rashba SOI Crystals 03 00014 i129. Solid black (dashed blue) curves correspond to parity Crystals 03 00014 i136 ( Crystals 03 00014 i137). Results for radial number Crystals 03 00014 i196 (with increasing energy) are shown. Red dotted lines denote Crystals 03 00014 i197.
Crystals 03 00014 g003 1024
Crystals 03 00014 g004 200
Figure 4. Main panel: Critical Coulomb impurity strength Crystals 03 00014 i015 vs. Rashba SOI Crystals 03 00014 i002 for Crystals 03 00014 i195 and the lowest Crystals 03 00014 i167 bound states. Inset: Crystals 03 00014 i015 vs. cutoff scale Crystals 03 00014 i014 for Crystals 03 00014 i129.

Click here to enlarge figure

Figure 4. Main panel: Critical Coulomb impurity strength Crystals 03 00014 i015 vs. Rashba SOI Crystals 03 00014 i002 for Crystals 03 00014 i195 and the lowest Crystals 03 00014 i167 bound states. Inset: Crystals 03 00014 i015 vs. cutoff scale Crystals 03 00014 i014 for Crystals 03 00014 i129.
Crystals 03 00014 g004 1024

5. Conclusions

In this work, we have analyzed the bound-state problem for the Kane–Mele model of graphene with intrinsic ( Crystals 03 00014 i001) and Rashba ( Crystals 03 00014 i002) spin-orbit couplings when a radially symmetric attractive potential Crystals 03 00014 i021 is present. We have focused on the most interesting “topological insulator” phase with Crystals 03 00014 i005. The Rashba term Crystals 03 00014 i002 leads to a restructuring of the valence band, with a halving of the density of states in the window Crystals 03 00014 i200, where Crystals 03 00014 i201. This has spectacular consequences for total angular momentum Crystals 03 00014 i055, where the problem can be decomposed into two independent parity sectors ( Crystals 03 00014 i064). The Crystals 03 00014 i136 states remain true bound states even inside the above window and coexist with the continuum solutions as well as possible supercritical resonances in the Crystals 03 00014 i137 sector. However, all Crystals 03 00014 i174 bound states exhibit supercritical diving for Crystals 03 00014 i202, where the critical threshold ( Crystals 03 00014 i134 or Crystals 03 00014 i015 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.

Acknowledgements

This work has been supported by the DFG within the network programs SPP 1459 and SFB-TR 12.

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