Designing Nanomagnet Arrays for Topological Nanowires in Silicon

Recent interest in topological quantum computing has driven research into topological nanowires, one-dimensional quantum wires that support topological modes, including Majorana fermions. Most topological nanowire designs rely on materials with strong spin-orbit coupling, such as $\mathrm{In}\mathrm{As}$ or $\mathrm{In}\mathrm{Sb}$, used in combination with superconductors. It would be advantageous to fabricate topological nanowires with $\mathrm{Si}$ owing to its mature technology. However, the intrinsic spin-orbit coupling in $\mathrm{Si}$ is weak. One approach that could circumvent this material deficiency is to rotate the electron spins with nanomagnets. Here we perform detailed simulations of realistic $\mathrm{Si}$/$\mathrm{Si}\mathrm{Ge}$ systems with an artificial spin-orbit gap induced by a nanomagnet array. Most of our results are generalizable to other nanomagnet-based topological nanowire designs. By studying several concrete examples, we gain insight into the effects of nanomagnet arrays, leading to design rules and guidelines. In particular, we develop a recipe for eliminating unwanted gaps that result from realistic nanomagnet designs. Finally, we present an experimentally realizable design using magnets with a single polarization.

Figure 1

Basic layout of a system with an artificial spin-orbit gap. A 1D quantum wire with weak intrinsic spin-orbit coupling (gray) is next to an array of nanomagnets with alternating polarizations. The magnets produce a rotating magnetic field in the wire that creates a spin-orbit gap in the band structure [12, 15].

Figure 2

(a) A unit cell of an example device design. The nanomagnets are 600 nm long, are 100 nm wide, have alternating polarization, and are placed on a stack consisting of a 15-nm-thick ${\mathrm{Al}}_2{0}_3$ layer (gray), 40-nm-thick $\mathrm{Si}\mathrm{Ge}$ barrier (blue), 15-nm-thick $\mathrm{Si}$ well (orange), and a thick $\mathrm{Si}\mathrm{Ge}$ buffer (blue). An accumulation gate (yellow) generates a 1D channel (wire) in the $\mathrm{Si}$ layer directly below the gate. The axis of this wire is indicated. An actual device would have many of these unit cells tiled along the wire axis. (b) The magnetic field in the plane of the quantum well. Arrows indicate the in-plane field. Colors represent the out-of-plane field. The wire axis is in blue. The magnets above the well are outlined in dashed black and gray. (c) The magnetic field along the wire axis with 1.76-T magnets. (d) The resulting band structure after the unitary transformation (Sec. 2d). The dotted light-gray curve shows the band structure resulting from a perfectly sinusoidal magnetic field [Eqs. (5) and (6)]. The light-gray curve is an excellent match except for two small gaps (inset), which are explained in Sec. 3a.

Figure 3

Band structure from the device in Fig. 2 shown in four ways: without [(a),(c)] and with [(b),(d)] the unitary transform [Eq. (4)] applied. Reduced-zone scheme [(a),(b)] and extended-zone scheme [(c),(d)] [23]. The orange curve in (c) is dashed to make the overlap with the blue curve clear. All four representations of the band structure are physically equivalent and are useful for visualizing different aspects of the band structure.

Figure 4

Results from the device in Fig. 2 where the two magnets of opposite polarization have different strengths of 1.76 and 1.07 T (appropriate for $\mathrm{Co}$ and $\mathrm{Sm}\mathrm{Co}$), respectively. (a) The magnetic field along the wire. (b) The transformed band structure in the reduced-zone scheme. Compared with Fig. 3, the blue and orange curves have moved apart due to the average magnetic field (and the same for the red and green curves). The asymmetry in the magnetic field also causes the band maxima and minima to shift slightly left or right. The inset shows that the $k$ values of orange- and blue-band extrema (marked by a vertical line) have shifted and do not occur at the same $k$ value as the other bands. (c) The band structure in the extended-zone scheme after the unitary transformation shows that the nonzero average magnetic field creates new gaps at $k=\pm (\pi /a)$. The dotted light-gray curve shows the band structure resulting from a perfectly sinusoidal magnetic field [Eqs. (5) and (6)].

Figure 5

Device modeled after the single-polarization device in Ref. [12]: an array of 600-nm-wide, 330-nm-long nanomagnets spaced 200 nm apart. The magnet is in plane with the quantum well. In Ref. [12] the magnet thickness or remanent magnetization was not stated, so we choose 60 nm and 2 T, respectively. (a) Magnetic field through the plane of the quantum well. The magnet is outlined in red. A wire located 40 nm from the magnet is shown in blue. Other colors indicate the magnitude of the magnetic field (light high, dark low). (b) The magnetic field along the wire. It rotates along the wire axis. However, the magnitudes of the $x$ and $y$ components are very different, and the average magnetic field is far from zero. (c) The resulting band structure in the reduced-zone scheme without transformation. There is no identifiable spin-orbit gap.

Figure 6

(a) A unit cell of an example single-polarization device. The material stack is the same as in Fig. 2. The axis of the wire is indicated. (b) The magnetic field in the plane of the quantum well. Arrows indicate the in-plane field. Colors represent the out-of-plane field. The wire axis is in blue. The magnets above the well are outlined in dashed black and gray. (c) The magnetic field along the wire axis with 1.76-T magnets. Despite the asymmetry of the magnet placement, the average value of each component is close to zero, and the components have similar amplitudes. (d) The resulting band structure after the unitary transformation. The dotted light-gray curve shows the band structure resulting from a perfectly sinusoidal magnetic field [Eqs. (5) and (6)]. The design shows a clear spin-orbit gap, which is more than 5 times larger than the bad gap.

Figure 7

(a) A unit cell of a vertical magnet device. The material stack is the same as in Fig. 2. (b) The magnetic field in the plane of the quantum well with alternating 1.76 and 1.07-T magnets. Arrows indicate the in-plane field. Colors represent the out-of-plane field. The wire axis is in blue. The magnets above the well are outlined in dashed black and white. (c) The magnetic field along the wire axis. (d) The resulting band structure after the unitary transformation. The dotted light-gray curve shows the band structure resulting from a perfectly sinusoidal magnetic field.

Figure 8

Transmission through two, four, six, and eight unit cells of the device shown in Fig. 2 connected to leads where $\mathbf{B}=0$. We assume that $\mathbf{B}$ in each unit cell is the same as in Fig. 2. Vertical dashed black lines show the extent of the good gap. Vertical dashed gray lines show the extent of the bad gap. The dip in transmission due to the good gap is clear even with a device that is only a few unit cells long. The dip will be even more pronounced in devices with a larger spin-orbit gap (e.g., the tower magnet design in Fig. 7).

Figure 9

Comparison of 1D (Sec. 2) and 2D (Appendix pp1) models for the device in Fig. 2. (a) The magnetic field at the center of the wire (solid lines), which is used by the 1D model, and the weighted average magnetic field across the well (dashed lines; see Appendix pp1), which is relevant to the 2D model. The curves are very similar, which suggests the magnetic field should have a similar effect in both models. (b) The band structure produced by the 1D model (solid lines) and the 2D model (dashed lines). The two models are in excellent agreement.