Engineering Dirac electrons emergent on the surface of a topological insulator

Here, let us first focus on the cylindrical case by physically targeting a topological insulator nanowire. On the cylindrical surface of such a nanowire, equation (1) is modified as [22]

$$\begin{eqnarray}&&\begin{array}{l} {{H}_{{\rm surf}}} \\ \quad ={{v}_{F}}\left[ \begin{array}{ccccccccccccccc} 0 & -i{{p}_{z}}+\frac{\hbar }{R}\left( -i\frac{\partial }{\partial \phi }+\frac{1}{2} \right) \\ i{{p}_{z}}+\frac{\hbar }{R}\left( -i\frac{\partial }{\partial \phi }+\frac{1}{2} \right) & 0 \\ \end{array} \right], \\ \end{array}\end{eqnarray} \tag{ 2 }$$

where R is the radius of the cylinder. The additive factor, $1/2,$ that appears in the two off-diagonals of equation (2) is a spin Berry phase encoding the constraint on the direction of spin as a consequence of the spin-to-surface locking. When a Dirac electron moves around the curved, cylindrical surface of the topological insulator nanowire, the spin-to-surface locking constrains the direction of the spin to follow the change of the tangential surface. As a result, when an electron completes a $2\pi$ rotation in the configuration space, its spin also performs a $2\pi$ rotation in the spin space, while the spin is naturally double-valued. This explains the origin of the additive factor, $1/2,$ or in other words, a Berry phase, π [19, 20, 22, 23].

The orbital part of an electron state on the surface of a cylindrical topological insulator is represented as

$$\begin{eqnarray}&&\psi (\phi ,z)={{{\rm e}}^{\frac{i}{\hbar }({{L}_{z}}\phi +{{p}_{z}}z)}},\end{eqnarray} \tag{ 3 }$$

where ${{L}_{z}}={{p}_{\phi }}R$ is the orbital angular momentum in the z-direction, along the axis of the cylinder. The Berry phase, π, modifies the boundary condition with respect to the polar angle, ϕ, from periodic to anti-periodic—that is,

$$\begin{eqnarray}&&{{{\rm e}}^{\frac{i}{\hbar }{{L}_{z}}(\phi +2\pi )}}\times (-1)={{{\rm e}}^{\frac{i}{\hbar }{{L}_{z}}\phi }}.\end{eqnarray} \tag{ 4 }$$

This leads to the following quantization rule of the orbital angular momentum:

$$\begin{eqnarray}&&\frac{{{L}_{z}}}{\hbar }=\pm \frac{1}{2},\pm \frac{3}{2},\cdots .\end{eqnarray} \tag{ 5 }$$

In this half-integral quantization, L_z = 0, and therefore ${{p}_{\phi }}=0,$ is not allowed. On the other hand, the spectrum of the surface states takes the following Dirac form,

$$\begin{eqnarray}&&{{E}_{{\rm surf}}}=\pm {{v}_{F}}\sqrt{p_{\phi }^{2}+p_{z}^{2}}.\end{eqnarray} \tag{ 6 }$$

Combining equations (5) and (6), one is led to believe that the spectrum of the cylindrical topological insulator is generically gapped; see also a gapped spectrum shown in figure 1(a). The spectra shown in figure 1 are calculated using tight-binding implementation of the bulk 3D topological insulator of a rectangular prism shape [22].

Zoom In Zoom Out Reset image size

The magnitude of this energy gap does not decrease exponentially as the size of the system increases, in sharp contrast to the case of the usual hybridization gap [24]. The energy gap, $\Delta E={{v}_{F}}\hbar /R\propto {{R}^{-1}},$ decays only algebraically as a function of the size of the system. In typical topological insulators (Bi₂Se₃, Bi₂Te₃), the Fermi velocity, ${{v}_{F}}\simeq 4\times {{10}^{5}}{\rm m}\;{{{\rm s}}^{-1}}\simeq c/7500,$ and the gap is expected to be visible in an ARPES measurement for $R\lesssim {{10}^{-7}}$ m at 1 K.

The spin Berry phase, π, can be interpreted in such a way that a fictitious magnetic flux is induced and pierces the cylinder. Indeed, the additive factor, $1/2,$ which appears in the two off-diagonals of equation (2) can be regarded as a vector potential

$$\begin{eqnarray}&&{\boldsymbol{A}} =\frac{\Phi }{r}\hat{\phi },\end{eqnarray} \tag{ 7 }$$

created around a flux tube of strength Φ, identical to half of a flux quantum, ${{\Phi }_{0}}=h/e$. This corresponds to the Berry phase,

$$\begin{eqnarray}&&\alpha =2\pi \frac{\Phi }{{{\Phi }_{0}}}=\pi .\end{eqnarray} \tag{ 8 }$$

An electron on the cylindrical surface does not touch the fictitious flux itself, since the fictitious flux tube is deep inside the cylinder. Yet quantum mechanically, the spectrum of the surface state is still influenced by this effective flux; the electron feels a vector potential, and its spectrum is affected by the vector potential, even when there is no magnetic field at any position where the electron is allowed to exist (Aharonov–Bohm effect). Here, the effective flux is induced by a constraint on the spin of the surface state. Thus, the electron on the cylindrical surface induces an effective flux, while its quantum mechanical motion is influenced by the flux created by the electron itself; therefore, we call it the intrinsic Aharonov–Bohm effect. To confirm this scenario, we repeated the same numerical simulation in the presence of a real external magnetic flux designed to cancel the fictitious flux, as shown in equation (8). As expected, the spectrum of the surface state becomes gapless once the Berry phase, π, is cancelled by the external flux, as seen in figure 1(b). Related experimental results in Bi₂Se₃ nanowires have been reported in [25, 26].

Magnetic monopoles do not exist in nature. This is what we learn in courses on elementary electromagnetism, and the statement is, of course, true at the microscopic level. In matter, however, Maxwell equations are modified, or at least it is convenient to replace them with effective equations. Then, depending on the nature of the effective media, analogues of a magnetic monopole can appear. This indeed happens in the case of the spherical topological insulator.

Similarly to the cylindrical case (equation (2)), on a spherical surface of a topological insulator, equation (1) is modified to [27]

$$\begin{eqnarray}&&\begin{array}{l} {{H}_{{\rm surf}}} \\ \quad =\frac{v}{R}\left[ \begin{array}{ccccccccccccccc} 0 & -{{\partial }_{\theta }}+\frac{i{{\partial }_{\phi }}}{{\rm sin} \theta }-\frac{1}{2}{\rm cot} \frac{\theta }{2} \\ {{\partial }_{\theta }}+\frac{i{{\partial }_{\phi }}}{{\rm sin} \theta }-\frac{1}{2}{\rm tan} \frac{\theta }{2} & 0 \\ \end{array} \right]. \\ \end{array}\end{eqnarray} \tag{ 9 }$$

There are corrections due to the spin Berry phase in the off diagonals of equation (9). Here, in the spherical case, they are understood as vector potentials associated with a magnetic monopole. The explicit form of the vector potential differs depending on the choice of the gauge in such a way that

$$\begin{eqnarray}&&{{{\boldsymbol{A}} }_{{\rm I}}}=\frac{g}{4\pi r}{\rm tan} \frac{\theta }{2}\ \hat{\phi }\end{eqnarray} \tag{ 10 }$$

when the singularity (the Dirac string) is chosen on the $-z$ axis, while

$$\begin{eqnarray}&&{{{\boldsymbol{A}} }_{{\rm II}}}=\frac{-g}{4\pi r}{\rm cot} \frac{\theta }{2}\ \hat{\phi }\end{eqnarray} \tag{ 11 }$$

when the Dirac string is on the $+z$ axis. Consistent with the Dirac quantization condition, equation (9) corresponds to the case of an effective magnetic monopole of strength $g=\pm 2\pi ,$ which is the smallest value compatible with the Dirac quantization condition. The electron on the surface of a spherical topological insulator behaves as if there is an effective magnetic monopole at the center of the sphere.

The spherical topological insulator naturally models a topological insulator nanoparticle. Taking it, therefore, as an artificial atom, let us focus on its low-lying electronic levels. The angular part of the wave function is described by an anti-periodic analogue of the spherical harmonics. A few examples are shown in figure 2, where ${\boldsymbol{ \alpha }} (\theta ,\phi )={{{\rm e}}^{{\rm i}m\phi }}{{{\boldsymbol{ \alpha }} }_{nm}}(\theta )$ represents the angular dependence of a surface eigenfunction, specified by quantum numbers n, m. An analogue of the s-orbital is

$$\begin{eqnarray}&&{{{\boldsymbol{ \alpha }} }_{0\frac{1}{2}}}=\left[ \begin{array}{ccccccccccccccc} {{\alpha }_{0\frac{1}{2}+}} \\ {{\alpha }_{0\frac{1}{2}-}} \\ \end{array} \right]=\frac{1}{\sqrt{4\pi }}\left[ \begin{array}{ccccccccccccccc} {\rm cos} \frac{\theta }{2} \\ -{\rm sin} \frac{\theta }{2} \\ \end{array} \right],\end{eqnarray} \tag{ 12 }$$

as seen in figure 2(a), while

$$\begin{eqnarray}&&{{{\boldsymbol{ \alpha }} }_{0\frac{3}{2}}}=\sqrt{\frac{3}{2\pi }}\left[ \begin{array}{ccccccccccccccc} {\rm sin} \frac{\theta }{2}\ {{{\rm cos} }^{2}}\frac{\theta }{2} \\ -{{{\rm sin} }^{2}}\frac{\theta }{2}\ {\rm cos} \frac{\theta }{2} \\ \end{array} \right]\end{eqnarray} \tag{ 13 }$$

can be regarded as an anti-periodic version of the p-orbital, as seen in figure 2(b). Further details on such 'monopole harmonics' [28] and the spectrum of the artificial atom are given in [27].

Zoom In Zoom Out Reset image size

To highlight the even/odd feature, we first consider the spectrum of a WTI sample with top, bottom, and side surfaces. The top and bottom surfaces are oriented normal to the z-axis. To realize a typical situation, we assume that the top and bottom surfaces are gapped surfaces (no Dirac cone). We then consider electronic states on a side surface, here chosen to be on the zx-plane. The two Dirac cones are typically located at k_z = 0 and at ${{k}_{z}}=\pi$. Low-energy electron states at or in the vicinity of the Dirac points may be represented by a plane wave:

$$\begin{eqnarray}\begin{array}{rcl} {{\psi }_{1}} & = & {{{\rm e}}^{{\rm i}({{k}_{x}}x+{{q}_{1}}z)}}, \\ {{\psi }_{2}} & = & {{{\rm e}}^{{\rm i}[{{k}_{x}}x+(\pi +{{q}_{2}})z]}}. \\ \end{array}\end{eqnarray} \tag{ 14 }$$

Here, the crystal momentum, q₁ or q₂, of the electron is measured from the corresponding Dirac cone. Now, if we consider the presence of top and bottom surfaces, located respectively at $z={{N}_{z}}$ and z = 1 for example, we need to confine the above electron in the region of $1\leqslant z\leqslant {{N}_{z}}.$ This means that we impose the boundary condition such that the wave function of the surface state, ψ, which may be expressed as a linear combination of ${{\psi }_{1}}$ and ${{\psi }_{2}},$ satisfies

$$\begin{eqnarray}\begin{array}{rcl} \psi (z) & = & {{c}_{1}}{{\psi }_{1}}+{{c}_{2}}{{\psi }_{2}}, \\ \psi (z=0) & = & 0,\quad \psi (z={{N}_{z}}+1)=0. \\ \end{array}\end{eqnarray} \tag{ 15 }$$

To cope with the boundary condition at z = 0, we choose the constants c₁ and c₂ such that ${{c}_{2}}=-{{c}_{1}}=1$. Also, at a given energy, E, one can set q₁ and q₂ such that ${{q}_{2}}=-{{q}_{1}}\equiv q$. We are then left with

$$\begin{eqnarray}&&\psi (z)={{{\rm e}}^{{\rm i}{{k}_{x}}x}}\left[ {{{\rm e}}^{{\rm i}qz}}-{{(-1)}^{z}}{{{\rm e}}^{-{\rm i}qz}} \right],\end{eqnarray} \tag{ 16 }$$

and this must vanish at $z={{N}_{z}}+1$. Therefore, for N_z odd,

$$\begin{eqnarray}&&q=\pm \frac{\pi }{2({{N}_{z}}+1)}2n\end{eqnarray} \tag{ 17 }$$

where $n=0,\pm 1,\pm 2,\cdots$ can be an arbitrary integer. Similarly, for N_z even, the vanishing of equation (16) at $z={{N}_{z}}+1$ implies

$$\begin{eqnarray}&&q=\pm \frac{\pi }{2({{N}_{z}}+1)}(2m-1)\end{eqnarray} \tag{ 18 }$$

where $m=1,2,3,\cdots ,$ and $\pm (2m-1)$ is an arbitrary odd integer. Since the spectrum of the surface state is given as [31]

$$\begin{eqnarray}&&E=\pm {{v}_{F}}\sqrt{k_{x}^{2}+{\rm sin} \;{{q}^{2}}},\end{eqnarray} \tag{ 19 }$$

equations (17) and (18) signify that the surface spectrum is gapless when N_z is odd, while it is gapped when N_z is even [30, 31]. Also, replacing N_z with N_h, one can equally apply equations (17) and (18) for characterizing the 1D helical modes that appear along a step formed on the surface of a WTI [18] (cf figure 3 and discussion given in the next section).

Zoom In Zoom Out Reset image size

We have so far argued that WTIs are susceptible to a specific type of size effect that is strongly dependent on the parity of the number of layers stacked in the direction of ${\boldsymbol{ \nu }} =({{\nu }_{1}},{{\nu }_{2}},{{\nu }_{3}})$. Here, we demonstrate that this even/odd feature indeed makes a WTI more useful than an STI. We consider the following Wilson–Dirac-type effective Hamiltonian on the cubic lattice,

$$\begin{eqnarray}\begin{array}{rcl} {{H}_{{\rm bulk}}} & = & {{v}_{F}}{\sum }_{\mu =x,y,z}{\rm sin} {{k}_{\mu }}\ {{\tau }_{x}} \otimes {{\sigma }_{\mu }}+m({\boldsymbol{k}} )\ {{\tau }_{z}} \otimes {{1}_{2}}, \\ m({\boldsymbol{k}} ) & = & {{m}_{0}}+{\sum }_{\mu =x,y,z}2{{m}_{2\mu }}(1-{\rm cos} {{k}_{\mu }}), \\ \end{array}\end{eqnarray} \tag{ 20 }$$

where two types of Pauli matrices, σ and τ, represent real and orbital spins, respectively, and ${{1}_{2}}$ is a 2 × 2 identity matrix. We have chosen the parameters ${{m}_{0}}=-{{v}_{F}},$ ${{m}_{2x}}={{m}_{2y}}{\mkern 1mu} =0.5{{v}_{F}},$ ${{m}_{2z}}=0.1{{v}_{F}},$ and v_F = 2 so that a WTI with weak indices, ${\boldsymbol{ \nu }} =(0,0,1),$ is realized [30]. By simply patterning the surface of a WTI, as seen in figure 3, one can realize a nanocircuit of a perfectly conducting 1D channel.

The even/odd feature on the spectrum of a WTI sample with side surfaces is equally applied to a system with atomic scale steps on the gapped surfaces [18]. In figure 4, we show the spatial profile of the wave function of some low-lying states in a geometry with an atomic scale step. In figure 4(a), the height of the step N_h is 1, while in figure 4(b), the step is 2 atomic scale high. As shown in the two panels, the wave function has strong amplitude along the step when N_h is odd, while it has visibly no weight in the step region when N_h is even.

Zoom In Zoom Out Reset image size

These 1D channels that appear when N_h is odd are also shown to be helical and robust against non-magnetic disorder [18, 32, 33]. In the supplemental material³ , we demonstrate the robustness of such an odd number of channels against disorder. Related to this, the robustness of the underlying WTI has also been studied [34, 35] (see also section 4). In any case, such a nanocircuit that appears on a patterned surface of a WTI is perfectly conducting, not only in the clean limit, but also in the presence of disorder. Therefore, a perfectly conducting 1D channel is stable, at least within the phase coherence length; to estimate the order of its magnitude, note that this is typically 200 nm for Bi₂Se₃ at 5 K [36].

Recently, a WTI was discovered experimentally in a bismuth-based layered bulk material, Bi₁₄Rh₃I₉, built from grapheme-like Bi-Rh sheets [37]. Other candidates for WTIs are layered semiconductors [38], superlattices of topological and normal insulators [39–41]. The superlattice has also been considered in different contexts including as a protocol for realizing a Weyl semimetal [42], and it is indeed fabricated experimentally [43]. Still another possibility to realize a WTI-like situation is to use topological crystal insulators [44–47]. These are close relatives of the standard topological insulator, protected by crystalline symmetries instead of the time reversal symmetry.