Finite-size effects in cylindrical topological insulators

We present a theoretical study of a nanowire made of a three-dimensional topological insulator. The bulk topological insulator is described by a continuum-model Hamiltonian, and the cylindrical-nanowire geometry is modelled by a hard-wall boundary condition. We provide the secular equation for the eigenergies of the systems (both for bulk and surface states) and the analytical form of the energy eigenfunctions. We describe how the surface states of the cylinder are modified by finite-size effects. In particular, we provide a $1/R$ expansion for the energy of the surface states up to second order. The knowledge of the analytical form for the wavefunctions enables the computation of matrix elements of any single-particle operators. In particular, we compute the matrix elements of the optical dipole operator, which describe optical absorption and emission, treating intra- and inter-band transition on the same footing. Selection rules for optical transitions require conservation of linear momentum parallel to the nanowire axis, and a change of $0$ or $\pm 1$ in the total-angular-momentum projection parallel to the nanowire axis. The magnitude of the optical-transition matrix elements is strongly affected by the finite radius of the nanowire.

In this paper we explore the properties of a finite-radius 3D-TI cylinder, using the envelope-function description of the TI bulk band structure developed in Refs. [39,40]. Our goal is to determine the dependence of its energy spectrum and eigenfuctions on the radius R. The central point of our analysis is the analytical expression of the eigenfunctions, which allow us to express cylindrical hard-wall boundary conditions in terms of secular equations that can be approximated in the limit of large radii: we obtain approximate expressions for the eigenenergies up to second order in 1/R. The analytical functional form of the eigenfunctions, which is valid irrespective of the radius of the wire, enables the calculation of the matrix elements of any observable. As an example, we consider the dipole matrix elements for optical transitions. In particular, we find that the selection rules for absorption and emission are not modified by a finite radius, in contrast to the case of a spherical nanoparticle [36]. Numerical results are presented for two different materials, namely Bi 2 Se 3 and Bi 2 Te 3 , which show qualitatively different behaviours. We compute eigenenergies as functions of the radius R and longitudinal momentum and compare them with approximate large-radius expressions. The eigenenergies are found to be oscillating for small values of R, especially in the case of Bi 2 Te 3 . Moreover, we characterize the behaviour of eigenfunctions by plotting the average radial coordinate and the corresponding variance as a function of the radius R. As expected, the average coordinate moves towards the centre of the nanowire for small values of R, more rapidly for Bi 2 Te 3 than for Bi 2 Se 3 , while the variance increases in an oscillating fashion for increasing radii, reaching the asymptotic value more rapidly in the case of Bi 2 Se 3 with respect to Bi 2 Te 3 . Finally, we calculate numerically the dependence of the optical dipole matrix elements on the radius finding quantitative important changes with respect to the bulk situation.
The paper is organized as follows. In section 2, we present an analytic treatment for a cylindrical 3D-TI nanowire with hard-wall confinement. We conclude section 2 with a complete analytic expression for the eigenfunction of the finite-radius 3D-TI. In section 3, we study the finite size effects on the topological properties of a cylindrical 3D-TI for two different materials. Specifically, we study the eigenenergies and characterise the eigenfunctions of the system as a function of the radius of the cylinder. Finally, in section 3.3, we calculate the optical dipole matrix elements of a cylindrical TI and study their dependence on the the radius of the cylinder.

Model
We consider an infinitely long cylinder of TI of radius R, whose axis is in the z-direction. The bulk TI is described by the Hamiltonian [39,40] where p = (p x , p y , p z ) is the momentum operator, m(p) = m 0 + m 1 p 2 z + m 2 (p 2 x + p 2 y ) is the mass term and p ± = p x ± ip y . The Hamiltonian (1) is written in the basis When the sign of m 0 /m 2 is negative, the material is in the topological insulating phase, causing isolated boundaries to host surfaces states represented by gapless Dirac cones. The coefficients m 0 , m 1 and m 2 , as well as the coefficients A and B of the linear-momentum terms depend on the material [41]. The values of the parameters for the most common TIs are reported in Table 1 where and with the mass terms given by the expressions The Hamiltonian H 0 commutes both with p z and with the z-component of the total angular momentum (L z + 2 σ z ) ⊗ τ 0 , where τ 0 is the identity matrix in the orbital pseudo-spin subspace. In the following, to avoid cluttering the notation, we set = 1. The commutation relations of H 0 discussed above suggest the following Ansätz for the wave function: where k z is the eigenvalue of p z and j (half integer) the eigenvalue of the z component of the total angular momentum. Solving the eigensystem requires applying the Hamiltonian Eq. (1) to the wavefunciton in Eq. (6). The calculation is detailed in Appendix A. In order to solve the radial part of the eigensystem, we make further Ansätze for the Φ i (ρ) and rewrite Eq. (6) as where J n (z) is a Bessel function of the first kind and κ and the coefficients c 1 , . . . , c 4 need to be determined. In order for the Ansätz of Eq. (7) to be an eigenfunction of H 0 with energy E, the parameter κ needs to take one of the following two values

Finite-size effects in cylindrical topological insulators
For the coefficients (c 1 , c 2 , c 3 , c 4 ) T there are four independent solutions (two for κ + and two for κ − ) given by The general solution for the wavefunction with quantum numbers k z , j and E is a linear combination of the four independent solutions obtained above: We can now solve the confinement problem by assuming a hard-wall cylindrical confinement potential of radius R. We need to impose the boundary condition Ψ(R, ϕ, z) = 0. This leads to as system of equations for the coefficients α s and β s which has non-trivial solutions for energies obeying the secular equation where we have defined the function T j (z) = J j+1/2 (z) . A detailed derivation of the secular equation is provided in Appendix A. In the case k z = 0, the problem decouples in two 2 × 2 problems and we have two independent secular equations which are analogous to Eq. (28) of Ref. [36]. The k z = 0 energy eigenstates associated with solutions of Eq. (12a) have β s = 0 and therefore their only nonvanishing spinor components are the first and the fourth. Conversely, the eigenstates corresponding to solutions of Eq. (12b) have α s = 0 and therefore their only nonvanishing spinor components are the second and the third. Taking into account the transformation properties of the basis states under spatial inversion, it is straightforward to show that eigenstates associated with energy eigenvalues arising from the secular equation (12a) [(12b)] are also parity eigenstates with eigenvalue (−1) j− 1 2 [(−1) j+ 1 2 ]. Even for finite k z , the spinors multiplied by α s [β s ] in the Ansätz (10) remain parity eigenstates with eigenvalue (−1) j− 1 2 [(−1) j+ 1 2 ]. However, as the energy eigenstates for nonzero k z are superpositions of these opposite-parity spinors, they are not eigenstates of parity.
Once we fix the quantum number j and k z and solve the secular equation (11) we obtain a series of solutions both with positive and negative energies. Of these, we will only consider the two, one positive and one negative, with the smallest absolute value of the energy. We will indicate the positive(negative)-energy solution with s = +(−).
‡ Furthermore, we will restrict our analysis to energies that lie within the bulk gap.
The quantum numbers that we will use to label the states are s = ±, j, k z . The secular problem yields the full knowledge of the eigenfunctions. In order to simplify the notation, in the following we rewrite the eigenfunction Eq. (10) as where the wavefunction obeys the normalisation condition 4

Results
In order to understand the effect of a finite radius of the cylinder and how it affects the topologically protected surface states, we start from the large-radius limit.

Large-radius expansion
A natural length scale in this context is the effective Compton length R 0 = A m 0 . In the following we perform an expansion in R 0 /R and find corrections to the asymptotic (large R) results obtained by Imura et al. [42]. To this aim, we make use of Hankel's asymptotic expansion for the Bessel function [43] The functions P (n, z) and Q(n, z) are power serieses of 1/z.

Zero axial momentum
We start by considering the case of zero axial momentum (k z = 0), with the goal to understand the j-dependence of the surface states. We will consider only one of the two secular equations, namely Eq. (12a) which can be recast as For realistic materials, see Table 1, and small values of energies E |m 0 |, κ ± = k ± iq with q > 0. In the large-radius limit qR 1, we keep only the terms proportional to ‡ In principle, we could introduce another integer quantum number to label the different solutions as in the case of a particle in a box. exp(qR) in Eq. (14). The secular equation reduces to Taking the zeroth order of the Hankel's expansion [i. e. P (n, z) = 1 and Q(n, z) = 0], the secular equation becomes This equation has a zero-energy solution if m 0 /m 2 < 0, i. e. when the system is in the topological phase. Next, we consider the next two terms in the Hankel's expansion, that is P (n, z) = 1 − (4n 2 − 1)(4n 2 − 9)/(128z 2 ) and Q(n, z) = (4n 2 − 1)/(8z), and insert them into Eq. (16). After some tedious but otherwise standard algebra, we obtain the eigenenergies up to second-order in R 0 /R The first term is in agreement with Ref. [42], the second term gives the first correction to the asymptotic result. The other solution, with the opposite sign, arises from solving Eq. (12b).

Finite axial momentum
In this section we assume that k z R 1. Proceeding in the same way as for case k z = 0, in zeroth-order in R 0 /R the secular equation for the case of non-zero axial momentum reduces to This equation has the solutions which represents the linear dispersion of the surface modes.
Considering the Hankel's expansion up to terms in 1/z 2 , that is P (n, z) = 1 − (4n 2 − 1)(4n 2 − 9)/(128z 2 ) and Q(n, z) = (4n 2 − 1)/(8z), we obtain the eigenenergies up to second order in R 0 /R E = ± Bk z + 1 2 which corresponds to the Taylor expansion in second order in 1/(k z R) of the result by Imura et al. [42], E = ± B 2 k 2 z + A 2 j 2 /R 2 . Notice that we are not allowed to take the k z → 0 limit, as this result has been derived assuming k z 1/R. and is indicated by a black solid line.

Numerical results
In this section we present numerical results for two different materials, namely Bi 2 Se 3 and Bi 2 Te 3 , using the parameters of Table 1. We use the following units for length and momentum, respectively, where R 0 = 1.49 nm for Bi 2 Se 3 and 1.35 nm for Bi 2 Te 3 . Figure 1 shows how the eigenenergies in units of E R = A/R depend on the radius of the cylinder for the two materials and for three different values of j. Here we show only the positive energies, that is s = +. Solid curves refer to the exact result obtained by solving Eqs. (12), while the dashed curves refer to the large-radius analytic expression Eq. (18). We observe that the latter solutions approximate well the numerical results when R 6R 0 for Bi 2 Se 3 , and when R 20R 0 for Bi 2 Te 3 , respectively. For Bi 2 Se 3 it is worthwhile noticing that at R = 6R 0 , especially for j = 3/2 and 5/2, the normalized eigenenergies have not yet reached the asymptotic (R R 0 ) value [represented by the thin solid lines, see Eq. (18)]. On the other hand, when the radius of the cylinder is small, Fig. 1 shows an oscillatory behaviour, especially in the case of Bi 2 Te 3 , that is more pronounced for smaller values of j, similarly to a spherical nanoparticle [36]. For Bi 2 Te 3 , the effect of these oscillations are so large that, for some values of the radius, the surface-state energy goes to zero. This oscillatory behaviour is a consequence of the  fact that the wavefunction is no longer localized on the surface of the cylinder. The oscillations are consistent with the results of Ref. [29] (see also Appendix B).
In Fig. 2 we show the positive eigeneregies, divided by the asymptotic value E R,j,kz = B 2 k 2 z + A 2 j 2 /R 2 , as a function of wavevector k z . Finite-size effects appear in this plot as deviations from unity of the normalized eigenenergies and are more pronounced form small values of k z .
Since we have the full knowledge of the eigenfunctions, we can calculate the expectation values of any single-particle operator. The average of the radial coordinate in the state Ψ s,j,kz is simply given by and its variance by Figure 3 (top panels) shows that the average of the radial coordinate, ρ s,j,kz , approaches R for large values of the radius as expected for topologically-protected surface states. The average of the radial position for both materials increases monotonically with the radius of the cylinder, showing weak oscillations only for the case of Bi 2 Te 3 . As shown in Fig. 3 (bottom panels), the variance in itself approaches, in an oscillatory fashion, a constant value of the order of R 0 for large values of radius (the variance varies very little for R 8R 0 for Bi 2 Se 3 and R 24R 0 for Bi 2 Te 3 ). Since the value of R 0 is similar for the two materials (R 0 = 1.5 nm for Bi 2 Se 3 and R 0 = 1.35 nm for Bi 2 Te 3 ), we can conclude that in Bi 2 Se 3 the asymptotic form of the surface states is reached for smaller values of the radius compared to Bi 2 Te 3 .

Optical transitions in cylindrical topological insulators
The Hamiltonian in Eq. (1) can be written as: where σ i and τ i are Pauli matrices in spin and orbital-pseudo-spin space, respectively. The velocity operator v = ∂H 0 /∂k| k=0 for the the Hamiltonian (24) and the optical dipole operator d = er are connected by the fundamental relation [44]: where d τ σ,τ σ = τ σ |er|τ σ and v τ σ,τ σ = τ σ |v|τ σ . Here |τ σ represents the basis functions in the orbital and spin space of the Hamiltonian H 0 defined in Eq. (1), and τ is the eigenvalue of τ z associated with the eigenstate |τ σ . Following the procedure of Ref. [36], we straightforwardly obtain for the optical-dipole operator in envelope function representation. The expression given in Eq. (26) accounts on the same footing for both envelope-function-mediated (sometimes called intraband ) transitions, which are associated with the first term on the r.h.s., and basis-function-mediated (interband ) transitions, which are subsumed in the remaining three terms. The optical dipole matrix elements are given by Using Eq. (10) and performing the integrals over ϕ and z, we obtain and where we have defined the overlap integrals and the matrix elements of radial position (R mn ) s j ,k z s,j,kz = R 0 dρ ρ 2 Φ * m,s ,j ,k z (ρ)Φ n,s,j,kz (ρ).
For circular polarization in the plane perpendicular to the nanowire axis, we find the conventional selection rule j = j ± 1, which is mandated by the conservation of totalangular-momentum projection (including the photon's) parallel to the nanowire axis. In addition, linear momentum k z parallel to the nanowire axis is conserved in any optical transition. The energy threshold for absorption is associated with transitions between (s = +, j = ±1/2, k z = 0) and (s = −, j ∓ 1/2, k z = 0). At the subband edge (k z = 0 and k z = 0) for d x + id y only the overlap integral (S 14 )  Fig. 6. Again, the selection rules for optical transitions are consistent with the basic symmetries associated with a cylindrical-nanowire geometry, and finite-size effects are manifested as significant quantitative changes in the magnitude of dipole matrix elements.

Conclusions
In this paper we have studied a nanowire made of TI. In particular, we have provided the analytical form of the energy eigenfuctions, which is central to the derivation of an analytical secular equation for the eigenenergies. This secular equation, on one hand, enables an analytical expansion for large radii and, on the other hand, is amenable to straightforward numerical solution. We study the dependence of the eigenenergies on the radius of the wire and we find oscillations as a function of the radius, which are very pronounced for Bi 2 Te 3 . The analytical form of the energy eigenfuctions enables the computation of the matrix elements of any single-particle operator. We have considered the optical dipole matrix elements. While we find the usual selection rules for absorption/emission, the value of the matrix elements is strongly dependent on the radius of the cylinder.

Appendix A. Secular equation for confined states
In this Appendix we provide the detailed derivation of the secular equation for the state of the TI cylinder. Acting with the Hamiltonian (2) on the wave function Eq. (6) and looking for eigenfunctions with energy E, we obtain    where J n (z) is a Bessel function of the first kind and κ and the coefficients c 1 , . . . , c 4 need to be determined. Substituting the Ansätz Eq. (A.2) in (A.1), we obtain the following equation for the coefficients  where we have introduced the following abbreviation ∆ ± = m 2 κ 2 ± + m 1 k 2 z + m 0 − E. The general solution with quantum numbers k z , j and E can therefore be written as  Figure B1. Eigenenergies with j = 1 2 and s = + for a cylinder of Bi 2 Te 3 as a function of radius for k z = 0. The solid red curve is the exact numerical solution of Eq. (11) while the dashed black curves is obtained by means of the Hankel's expansion at first order in 1/z.