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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 [

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 [

The atomic collapse problem for Dirac fermions in an attractive Coulomb potential,

We study the Kane–Mele model for a 2D graphene monolayer with both intrinsic (

Using polar coordinates, it is now straightforward to verify (see also [

In this representation, the radial Dirac equation for total angular momentum

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

We focus on one

For arbitrary

The reason why

In fact, using this transformation matrix to carry out a similarity transformation,

For parity

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

The general solution to Equation (7) follows from the Ansatz

Using the Bessel function recurrence relation,

For

For

In a free system,

For

In this section, we study a circular potential well with radius

We always stay within the TI phase

The continuity condition for the four-spinor at the potential step,

Numerical solution of Equation (26) yields the bound-state spectrum shown in

Bound-state spectrum (

Surprisingly, for

Therefore an intriguing physical situation arises for

Returning to arbitrary total angular momentum

The threshold values

From the above equations, we can then infer the threshold values

Likewise, for

As can be seen in

Critical potential depth

Since the parity decoupling in

We now turn to the Coulomb potential,

Following the arguments in

Finally, for

Bound state energies with angular momentum

Main panel: Critical Coulomb impurity strength

In this work, we have analyzed the bound-state problem for the Kane–Mele model of graphene with intrinsic (

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