Formation and fabrication of semiconductor-superconductor nanowires and quantum devices based thereon

Abstract

The present disclosure relates to a quantum device and a method for manufacturing the same. The device includes one or more semiconductor-superconductor nanowires, each of which includes a certain length of semiconductor material and a coating of superconductor material coated on the semiconductor material. The nanowires can be formed above a substrate. In a first aspect, at least some of the nanowires are full-shell nanowires, in which the superconductor material is coated around the entire periphery of the semiconductor material along some or all of the length of the wire, wherein the device is operable to induce at least one Majorana zero-mode MZM in one or more active nanowires of the full-shell nanowires. In a second aspect, at least some of the nanowires are arranged vertically relative to the plane of the substrate in the finished device.

Description

Background Art

[0001] Quantum computing is a class of computing in which intrinsic quantum mechanical phenomena, such as superposition and entanglement of quantum states, are exploited to perform certain computations much faster than any conventional computer that might be capable of performing them. In a "topological" quantum computer, computations are performed by manipulating quasi-particles that appear in certain physical systems, called "non-abelian anyons." Anyons have unique physical characteristics that distinguish them from fermions and bosons. Non-Abelian anyons also have unique properties relative to Abelian anyons. It is these unique properties that serve as the basis for topological quantum computing, in which information is encoded as the topological properties of non-Abelian anyons; specifically, the weaving of the space-time world of non-Abelian anyons. This has some advantages over other quantum computing models. A key benefit is stability, because quantum weaving is not affected by perturbations that can cause error-induced quantum decoherence in other types of quantum computers on a large scale.

[0002] Many types of physical systems have been considered as potential hosts for non-Abelian anyons, such as the "5 / 2 fractional quantum Hall" system in condensed matter physics and topological insulator systems in contact with superconductors. Another example is a semiconductor-superconductor (SE / SU) heterostructure, such as SE / SU nanowires. Relative to these, a key advance in the field is the recognition that non-Abelian anyons in the form of "Majorana zero modes" (MZMs) can be formed in regions where a semiconductor (SE) is coupled to a superconductor (SU). Based on this phenomenon, a small network of SE / SU nanowires, each of which includes a length of semiconductor coated with a superconductor, can be used to create qubits.

[0003] A quantum bit (qubit) is an element that can undergo a measurement with two possible outcomes, but at any given time (when not being measured) can actually be in a quantum superposition of two states corresponding to different outcomes.

[0004] A "topological" qubit is a qubit implemented based on the above-mentioned technology of non-Abelian anyons in the form of MZMs. Non-Abelian anyons are a type of quasiparticle, that is, they do not refer to particles per se, but to excitations that behave at least partially like particles in an electronic liquid. Specifically, anyons are quasiparticles that appear in two-dimensional systems (two degrees of freedom in space). Majorana zero modes are specific bound states of such quasiparticles. Under certain conditions, for the purpose of quantum computing, these states can be formed close to the semiconductor / superconductor interface in the SE / SU nanowire network in a way that enables them to be manipulated as qubits. Regions or "segments" of the nanowire network between MZMs are said to be in a "topological" state.

[0005] Majorana-based qubits conventionally involve gating in order to exhibit this topological behavior. That is, an electric potential is applied to a segment of the semiconductor of one of the nanowires that form the qubit. The potential is applied via a gate terminal placed adjacent to the nanowire in a fabricated structure on a wafer. A magnetic field is also required to induce the topological state. The magnetic field is applied by an electromagnet placed outside the wafer (usually in a cold room), such as that used to induce superconductivity in superconductors.

[0006] Conventionally, building a Majorana-based topological quantum computing device involves forming superconducting islands on semiconductors. Some parts of the superconductor are topological (T), while some parts are non-topological (e.g., conventional S waves (S)). The topological segmentation supports Majorana zero modes that appear at their opposite ends. Existing techniques for realizing MZM require strong magnetic fields and electrostatic gating to drive half-shell nanowires into topological phases. MZM is induced by coupling a magnetic field to the spin component of an electron. This requires a strong magnetic field.

[0007] In some fabrication techniques, the semiconductor of the nanowires can be formed in the plane of the wafer by techniques such as selective area growth (SAG). The superconducting material can then be selectively deposited over the semiconductor or can be deposited as a uniform coating and areas that are subsequently etched away to form islands.

[0008] Another method of manufacturing a device comprising semiconductor-superconductor nanowires is disclosed by P. Krogstrup et al. in "Epitaxy of semiconductor-superconductor nanowires" in Nature Materials, pp. 400-406, January 12, 2015. The semiconductor core of the nanowire is grown vertically relative to the wafer plane, and then an angled deposition is performed to deposit a coating of the superconductor on the facets of the semiconductor core. The nanowire is then "pressed down" by ultrasonic treatment and aligned in a horizontal plane with the aid of an optical microscope. Parts of the superconductor coating are then etched away from the nanowire so that in the resulting device each nanowire is coated only with the superconducting islands mentioned above. Summary of the invention

[0009] According to a first aspect disclosed herein, coating the entire perimeter of each nanowire with a superconductor (i.e., providing a complete superconducting shell) is disclosed, and leaving the shell in place in the final device (rather than etching away portions of the superconductor or selectively depositing the portions to form superconducting islands only above the semiconductor of the wire). This full shell coating can be implemented, for example, by rotating the substrate or deposition beam during deposition of the superconductor. In the case of a full shell, the inventors have found that only a relatively weak orbiting magnetic field is required to drive the system into a topological state. The disclosed formation can also be used to alleviate other technical challenges present in earlier schemes because it avoids electrostatic tuning into the topological phase, protects the topological elements from spatial inhomogeneities, and increases the charging energy of the superconducting islands that form part of the qubit.

[0010] The disclosed formations can be used to build quantum computing structures, such as to form topological qubits, qubit systems, and quantum computers based on full-shell nanowires, including but not limited to the example designs illustrated later. Example designs include "horizontal" and "vertical" designs.

[0011] According to a second aspect, a structure is disclosed in which vertically grown nanowires remain oriented vertically relative to the wafer in the final device, rather than being flattened and oriented horizontally in the plane of the wafer as is common in the prior art. The advantage of this situation is that it allows 3D integration. The general approach requires that the nanowires be laid flat in a 2D network in the plane of the wafer, but this takes up a lot of space. Therefore, instead, the present disclosure discloses a method in which the nanowires remain vertically placed in the orientation in which they are grown and then the rest of the device is built around the nanowires.

[0012] The first aspect and the second aspect can be used together or independently. That is, in the first aspect, the nanowires can remain vertical in the plane of the wafer or be flattened and aligned horizontally, while in the second aspect, the nanowires can remain as full-shell nanowires or can be etched to remove portions of the superconductor from the wire.

[0013] In either aspect, the disclosed structures can be used to form topological qubits or topological computers or universal quantum computers comprising multiple qubits. In a second aspect, the disclosed structures can even be used to form other non-topological quantum devices.

[0014] It should be noted that "device" as referred to herein refers to a finished device, ie, a device formed in a finished die or chip. Typically, the device will also be packaged, ie, packaged in an integrated circuit package.

[0015] It should also be noted that terms such as "horizontal," "vertical," "bottom," and "top" as mentioned herein refer to the plane of the wafer or substrate. That is, horizontal refers to parallel to the plane of the substrate, and vertical refers to perpendicular to the pane of the substrate, while bottom refers to the end of the nanowire closest to the substrate, and top refers to the end farthest from the substrate. These terms do not necessarily imply anything about orientation relative to gravity.

[0016] This summary is provided to introduce in a simplified form a selection of concepts that are further described in the detailed description below. This summary is neither intended to identify key features or essential features of the claimed subject matter nor to be used to limit the scope of the claimed subject matter. The claimed subject matter is also not limited to implementations that solve any or all of the disadvantages mentioned herein. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] To assist in understanding the embodiments of the present disclosure and to show how such embodiments may be performed, reference is made, by way of example only, to the accompanying drawings, in which:

[0018] Figure 1 is a diagram of a quantum system,

[0019] Figure 1A is a diagram of a semiconductor nanowire with a complete superconducting shell.

[0020] Figure 1B is a diagram of a semiconductor nanowire with a thin shell,

[0021] Figure 2 is the topological phase diagram in the thin shell limit,

[0022] Figure 3 Part (a) is the topological phase diagram of the complete core model.

[0023] Figure 3 Part (b) shows Figure 3 The band structure at the point indicated in part (a) of

[0024] Figure 4 Part (a) shows the Majorana coherence length of the full core model,

[0025] Figure 4 Part (b) shows the Majorana wave function integrated over the radius,

[0026] Figure 5 Part (a) of FIG. 1 shows an example design of a quadruplet qubit (also referred to as a four-qubit),

[0027] Figure 5 Part (b) of shows an example design of a hexon qubit,

[0028] Figure 6 shows an example design of a network of hexon qubits,

[0029] Figure 7 Shows Figure 6 Details,

[0030] Figure 8 Another example design of a network of hexon qubits is shown,

[0031] Fig. 9 Shows Figure 8 Details,

[0032] Fig.10 shows an example design of a network of quaternion qubits,

[0033] Fig.11 Another example design of a network of quaternion qubits is shown,

[0034] Fig.12 An example fabrication process for fabricating vertical nanowires is shown.

[0035] Fig.13 An example design of the underlying controls for a vertical nanowire device is shown.

[0036] Fig.14 Part (a) shows a side view of an example design with horizontal nanowires,

[0037] Fig.14 Part (b) shows a side view of another example design with vertical nanowires,

[0038] Fig.14 Part (c) shows a side view of another example design with vertical nanowires,

[0039] Fig.15 is a sketch of the energy gap in the superconducting shell as a function of B,

[0040] Fig.16 shows the topological phase diagram of the hollow cylinder model,

[0041] Fig.17 shows the topological phase diagram of the complete cylindrical model,

[0042] Fig.18 shows the evolution of the local density of states at the end of the finite line,

[0043] Fig.19A The topological phase diagram of the disordered complete cylindrical model is shown,

[0044] Fig.19B shows the effect of the angular symmetric breaking perturbation,

[0045] Fig. 20 shows a simulation of a superconducting shell,

[0046] Fig.21 shows the topological phase diagram of the hollow cylinder model,

[0047] Fig. 22 shows the probability density of the lowest energy up and down spin modes,

[0048] Fig.23 shows the local density state in the middle of the line,

[0049] Fig.24A Phase diagrams showing different realizations of disorder, and

[0050] Fig. 24B Other phase diagrams for different disorder realizations are shown.

[0051] Glossary of Mathematical Symbols

[0052] The following symbols are mentioned in this article:

[0053] Direction of nanowires

[0054] The radial direction of the nanowire

[0055] R1: (optional) insulation core radius

[0056] R2: Semiconductor radius

[0057] R3: outer radius of the superconducting shell

[0058] R: Shell radius

[0059] Magnetic field applied to the nanowire

[0060] Electromagnetic vector potential

[0061] H0: Hamiltonian of the semiconductor core

[0062] e: the charge of an electron

[0063] m*: effective mass

[0064] μ: Chemical potential

[0065] α r or α: the strength of the spin-orbit coupling

[0066] σ i : spin 1 / 2 Pauli matrix

[0067] △0: s-wave superconducting pairing potential

[0068] ψ: Nambu basis

[0069] H BdG : Hamiltonian close to the nanowire

[0070] momentum

[0071] d: thickness of the superconductor shell

[0072] λ L :London penetration depth

[0073] Φ: Magnetic flux

[0074] Φ0: magnetic flux quantum

[0075] Or φ: angular coordinate

[0076] Number of windings

[0077] n eff : Prefactor of the effective Zeeman term

[0078] Ψ: Pauli matrix representing the particle / hole degrees of freedom in Nambu space

[0079] m J or m j : Angular quantum number

[0080] U: Uniform transformation

[0081] E gap : Overall energy gap

[0082] Vz: Zeeman Energy

[0083] Ec : Charging energy

[0084] J z : Angular momentum operator

[0085] L z : orbital angular momentum

[0086] Ψ: Majorana wave function

[0087] ξ: frequency of the coherence length

[0088] m e : The mass of an electron DETAILED DESCRIPTION

[0089] This disclosure introduces calculations and simulations showing that over a wide range of values ​​of B-field, spin-orbit coupling, chemical potential, radius, and temperature, semiconductor wires with a full shell of superconductors can enter a topological phase that supports MZMs, a key element in topological approaches to quantum computing.

[0090] This led the inventors to a series of other theoretical activities, accumulating in terms of solid analytical verification of topological phases, extensive numerical simulations to determine the parameters of entering the phases, and the actual characteristics of the phases (coherence length and gap). Then, with the full-shell path to MZM in hand, the inventors turned to the following question: "Can this new full-shell technology be utilized in the design of qubits, arrays of qubits, and all the ways of scalable architecture for topological quantum computers?" The inventors found that, in fact, the answer to all these questions is "yes". The present disclosure includes theoretical work that supports the discovery of scalable design of topological quantum computers based on full-shell lines.

[0091] In fact, there are at least two aspects to the present disclosure: I) the work that can be done with the now known topological phases of full-shell wires; and II) the work that can be done with vertical "forests" of vapor-liquid-solid (VLS) grown nanowires as quantum computing elements.

[0092] The connection between I) and II) is that, in developing scalable technology for full-shell wires, the inventors realized that it would be a pity to have to mess up these wires because the wires grow (VLS) into beautiful ordered vertical arrays. Therefore, it is disclosed to leave the forest in place and build the rest of the topological quantum computer around it. (Although horizontal designs are also possible in the first aspect.) Once along the road to achieving 3D integration of quantum computing elements through vertical forests of VLS wires, the inventors realized that there are fruitful applications even outside the topological field. Conventional condensed matter qubits (such as transporters and gaters) will have limited fidelity and will have to be produced in the millions for quantum computers based on these qubits that can support the necessary error correction layer. It is recognized herein that vertical forests of semiconductor wires have the potential to be used to form dense arrays of Majorana-based qubits or conventional qubits.

[0093] Figure 1 A schematic illustration of an example system 1 according to an embodiment of the present disclosure is given. For example, the system 1 can be a quantum computer, such as a topological quantum computer or a universal quantum computer. The system 1 includes a device 2 in the form of a bare die (chip) disposed in a cold storage chamber 3. The bare die can be packaged in an integrated circuit (IC) package (not shown) and can be connected to external equipment (also not shown) via pins of the package, and the external equipment is arranged to control the device and / or measure from the device via contacts between the inside and the outside of the device 2. The system 1 also includes an electromagnet 4 that is typically disposed within the cold storage chamber 3.

[0094] The device 2 has been formed from a wafer using a layered manufacturing process. The device 2 includes a substrate 5 defining a plane. The device 2 also includes one or more layers 6 formed over the substrate 5. These layers may include, for example, a dielectric layer formed over the substrate 5, a semiconductor material layer formed over the dielectric layer, a semiconductor material layer formed over the semiconductor layer, and a filler material layer formed over the superconductor layer.

[0095] The device 2 comprises one or more semiconductor-superconductor nanowires 7, preferably a plurality of such wires. The wires 7 are embedded in one or more of the layers 6. Each of the nanowires 7 comprises a length of semiconductor material 9 (the core of the nanowire 7) defining an axis. The cross section of the core may be circular, elliptical or polygonal (i.e., a shape in a channel perpendicular to the axis of the wire). Each nanowire 7 also comprises a coating of superconducting material 10 formed over at least a portion of the outer portion of the length of the nanowire. According to the first aspect disclosed herein, the superconductor material 10 forms a full path in the plane of the axis (i.e., in cross section) along a portion or the entire length of the nanowire (a portion of the full path along the axis in a direction parallel to the axis) around the periphery (e.g., circumference) of the semiconductor core 9. This is what is meant by a full-shell nanowire. One, some or all of the nanowires 7 in the device 2 may be in the form of a full-shell nanowire.

[0096] To make such a formation, the semiconductor core 9 of the nanowire 7 can be grown vertically, for example, using a growth method previously disclosed by Krogstrup et al. as cited in the background section. Vertical here means perpendicular to the plane of the substrate 5. Then, the superconductor material 10 can be deposited on the core 9 by means of an angled beam deposition technique. In order to form a full shell coating, either the wafer or the beam can be rotated during the deposition of the superconductor 10. The semiconductor material 9 is preferably a material with high spin-orbit coupling, such as InAs (indium arsenide) or InSb (indium antimonide). The superconductor material 10 is preferably an s-wave superconductor, such as Al (aluminum) or Nb (niobium).

[0097] According to the second aspect disclosed herein, the vertical nanowires 7 can be left in a vertical position in the finished device 2. In order to physically support the vertical nanowires, the deposited layer 6 includes one or more layers of filler material, which can be, for example, plastic or wax. For example, the filler can be a plastic PMMA.

[0098] Alternatively, some or all of the nanowires 7 (eg, full-shell nanowires) may be depressed and their axes aligned in a horizontal plane (parallel to the substrate 5) according to a previously known approach disclosed by Krogstrup et al.

[0099] In the final system 1, then, one, some or all of the nanowires 7a (e.g., some or all of the vertical nanowires) are used as active nanowires, i.e., operable quantum components of a quantum structure (such as a qubit). For example, one, some or all of the nanowires 7 can be used to induce Majorana zero modes within the nanowires 7, which can be used to form a structure that acts as a Majorana-based qubit. As will be discussed in more detail later, in embodiments, bundles of vertical nanowires 7a can be used to form qubit structures, such as four qubits (quadrupole qubits) or hexon qubits. In other embodiments, the nanowires 7 can be used to form other kinds of qubits, such as qubits based on transporters or gaters.

[0100] In an embodiment, all nanowires 7 or at least all active nanowires 7a may be aligned with their axes in the same direction, ie parallel to each other.

[0101] Formed in one or more of the layers 6 are one or more layers of circuitry 8, which may, for example, include one or more gates, interconnects, semiconductor networks, and / or other electrical contacts 8. In an embodiment, some or all of any semiconductors used in the circuitry 8 may be formed of the same semiconductor material 9 as the core of the nanowire 7. In an embodiment, some or all of any conductors used in the circuitry 8 may be formed of the same superconductor material 10 as the coating of the nanowire 7. The layer(s) of the circuitry 8 may, for example, be arranged to enable any or all of the following operations: connecting the nanowires 7 together into a quantum structure (such as a qubit); controlling the nanowires 7 or a quantum structure formed therefrom; and / or taking a reading from the nanowires or a quantum structure formed therefrom. For example, a gate may be arranged to tune a semiconductor in an area without a superconductor (to form a point, tunnel junction, or considerable electron density in the semiconductor). For example, a gate may be arranged to tune a tunnel junction in a semiconductor network of the circuitry 8.

[0102] Some of the nanowires 7x do not necessarily serve as operational quantum components, but instead serve only as vertical electrical contacts (i.e., vias) between layers of circuitry 8 in structure 6 and / or to make external contact beyond the top of device 2, e.g., to connect to pins of a package and ultimately to external equipment. Alternatively or additionally, a plurality of conventional vias may be used.

[0103] Figure 1The diagram shows, for example, a bottom layer 8b of the circuit arrangement, which may be formed from a base layer of semiconductor and / or superconductor, formed directly above the substrate 5 or having only a dielectric layer before being connected to the substrate 5. The conductor of this layer 8b may, for example, be formed from the same superconductor as the islands of superconductor connecting the nanowires of the qubits together and using a stitching location step (see below). Figure 1 Also shown is an example top layer or circuitry 8t formed on top of the filler layer. This layer may, for example, include contacts to the outside of the device, such as for connection to external equipment (not shown). Figure 1 Also illustrated are one or more inactive vertical nanowires 7x which, due to their properties, are not used as quantum components but rather as vias between upper and lower layers 8b, 8t of the circuit device. Similar techniques can be used to connect partially through layer 6 of the device to intermediate layers of the circuit device.

[0104] Note: In some embodiments, control lines or other such circuit devices 8 may be formed in a different layer, since each qubit requires a large number of gates and space becomes limited very quickly.

[0105] It should be understood that Figure 1 The layers of the circuit device 8 and indeed the arrangement of the nanowires 7 shown in FIG. 8 are highly schematic and are intended to illustrate the principles rather than actual circuits or quantum structures. It should also be noted that Figure 1 It is not intended to show that the nanowires 7 are necessarily arranged directly on the substrate 5. For example, there may be one or more layers between the substrate 5 and the nanowires 7, such as a dielectric layer.

[0106] The device 2 of the form described above can be used to implement a topological quantum computer based on Majorana zero mode (MZM). In operation, the chamber 2 is cooled to a temperature at which the superconductor material 10 in the active nanowire 7a exhibits superconductivity, and the electromagnet 4 is operated to apply a magnetic field B in a direction parallel to the axis of the nanowire 7a, or at least has a substantial component in a direction parallel to the axis of the active nanowire 7a. This induces one or more MZMs in each of the active nanowires 7a, so that it can play its role, for example, as a part of a network of MZM-based qubits or such qubits. In an embodiment, a pair of MZMs is induced, one MZM is induced at each end of the nanowire. In an embodiment using a vertical nanowire 7, the MZM that appears at the bottom end (closest to the substrate 5) can be used to provide the required quantum operation, for example, as part of a qubit.

[0107] It should be noted that in the disclosed full-shell arrangement, then, unlike conventional nanowires, since the superconducting shell 10 completely shields any magnetic field to the interior, it is not necessary to tune the full-shell wire into or out of the topological phase by electrostatic potential. Instead, the active full-shell wire 7a to be topological has a reproducible growth so that the full-shell wire can enter the topological phase. In view of the findings disclosed herein that the full-shell nanowire supports MZMs, suitable values ​​of B field, spin-orbit coupling, chemical potential, radius and temperature can be selected analytically or experimentally from a wide range of values ​​so that these parameters are inside the topological phase (see, for example, Figures 1 to 4 as well as Figures 16 to 19A and associated work). The inventors have performed simulations which demonstrate that topological phases can be obtained over a wide range of such parameters (see also Fig.19B and Fig. 24B ). Further, the topological phase can be induced using only a much weaker magnetic field B than required for previous devices. The field is applied along the wire direction. The minimum value depends on the radius of the wire, but is typically around 100 mT. This is different from half-shell nanowires, in which part of the nanowire is not covered by the superconductor shell and the electric field penetrates into the semiconductor core.

[0108] In the case of a full shell, it may still involve controlling the charging energy of the Majorana qubit and electrostatically changing the induced charge on the island. For this reason, only a weak electric field is required, while in order to change the band structure, one needs to apply a very large electric field. This refers to the gating of the superconducting island (e.g., by one or two electrons) connecting the nanowire 7, which may require controlling the charge of the island to protect against the influence of quasiparticle poisoning and to achieve readout (as in the case of a partial shell, this is easier than gating to change the chemical potential of the semiconductor to induce a topological phase, which is impossible in any case in the case of a full shell, because the semiconductor is completely shielded by the superconductor). The island in question can be formed, for example, in the lower layer 8b of the circuit device, wherein each island connects a group of active nanowires 7a together to the qubit (see below). The gating can be performed at any position on the island, for example, by gating the full shell wire and / or the superconducting structure (e.g., spiral) connecting the full shell wire. The electric field can be applied by a gate close to the full shell wire. The exact position of the gate is not important. Preferably, the gate should be close to only a small area of ​​the full shell surface (although a larger area would increase coupling to the gate, thereby reducing the required voltage change).

[0109] Figure 1AA semiconductor nanowire core 9 with a complete superconducting shell 10 is shown, which is confined by a weak axial magnetic field B. The central region indicates that an optional insulating (sub) core 11 may be present within the semiconductor core 9 (axially with the semiconductor core 9 along its central axis). R1 ​​is the radius of the outer limit of the insulating sub-core 11 (if present) from the central axis of the nanowire 7 (and the inner limit of the semiconductor core 9). R1 ​​can be equal to 0. R2 is the radius of the outer limit of the core of the semiconductor 9 from the central axis of the nanowire 7 (and the inner limit of the semiconductor coating 10). R3 is the radius of the outer limit of the semiconductor coating 10 (and of the nanowire itself).

[0110] The Hamiltonian of this full-shell nanowire with radial spin-orbit coupling can be modeled as follows.

[0111]

[0112] Item 1 represents kinetic energy. represents the radial spin-orbit field. The last term represents a pairing with winding number n.

[0113] The system has rotational symmetry: The above BdG (Bogoliubov de Genes) Hamiltonian is commuted with the following generalized angular momentum operator:

[0114] [J z ,H BdG ]=0

[0115] Where L z is the orbital angular momentum. Therefore, one obtains the quantum number notation for the angular momentum state:

[0116]

[0117] To obtain an angle-independent Hamiltonian, the angular dependence can be removed by performing the following uniform transformation.

[0118]

[0119] In each m J In the sector, the Hamiltonian takes the following form:

[0120]

[0121] It should be noted that the Rashba term representing the strength of Rashba coupling will not be averaged to zero. In other words, the spin-orbit coupling) will not cancel. Another way to detect this is to study the angular dependence of the eigenvectors in the original basis. Due to the non-trivial winding with φ, the average of the spin-orbit (SO) term is non-zero.

[0122] Figure 1B The thin shell limit is shown in the figure, where R1→R2=R. j = 0 sector and n = 1 gives the solution. Considering m j = 0 and n = 1, the thin shell with radius R is composed of a hollow semiconductor 9 and a superconducting shell 10, such as Figure 1B As shown in .

[0123] The Hamiltonian becomes simplified:

[0124]

[0125] in and μ(R) is the renormalized chemical potential, and Vz(R) is the effective Zeeman term, where The penultimate term α r p r σ y τ z is a valid RashbaSO term that does not exchange with the Zeeman term. The last term Δ0(r)τ x It is s wave pairing.

[0126] This Hamiltonian has all the elements of the standard 1D model. This gives the possibility of topological phases.

[0127] Note: Majorana solutions belong to m J = 0 sector in order to be particle-hole symmetric. This is only possible if n is an odd number.

[0128] Figure 2 The simulated topological phase diagram in the shell limit of the solution for n = 1 as the winding number is shown. The left panel shows the overall energy gap E as a function of μ and α gap (The global energy gap is the energy difference between the ground state and the first excited state of a system without boundaries (such as an infinitely long line). According to the above equation, the dashed line represents the energy difference between the ground state and the first excited state of a system with no boundaries (such as an infinitely long line). J =0 is the boundary of the topological phase in the sector. Here, n eff =1 / 2, R / R0=1 / 2, and As a reference, using the actual parameters m* = 0.026 me and Δ0 = 0.2 meV, we obtain α0 ≈ 17 meV nm and R0 ≈ 85 nm. The right panel shows the overall energy gap and n at fixed μ / Δ0 = 2 and α / α0 = 1.eff and R. The circles in the left panel mark the μ and α values ​​used in the right panel, and the circles in the right panel mark the R and φ values ​​used in the left panel.

[0129] The dotted line indicates The phase diagram shows that there is a stable topological region, which is marked with reference numeral 12 in the figure.

[0130] Now consider the complete semiconductor core limit R1 = 0. The Hamiltonian can be modeled again as follows:

[0131]

[0132] The first of these Represents kinetic energy, the middle two terms represents the radial spin-orbit field, and the last term represents a pairing with winding number n.

[0133] Figure 3 Part (a) shows the simulated topological phase diagram of the complete core model with R2 = 100 nm, R3 = 3000 nm, m* = 0.026 me and Δ0 = 0.2 meV. gap is plotted as a function of μ and α. The dashed line represents the J = p in sector 0 z = 0 to find the boundary of the topological phase obtained by finding the zero energy crossing. Figure 3 Part (b) shows Figure 3 The band structure at the points indicated by circles in part (a): Figure 3 The upper circle in part (a) is Figure 3 The lower plot in part (b) corresponds to, Figure 3 The middle circle in part (a) is Figure 3 corresponds to the middle plot in part (b) of , and Figure 3 The lower circle in part (a) is Figure 3 The upper plot in part (b) corresponds to Figure 3 In part (b) of the diagram, the shading of the bands indicates to which m the bands belong. J Sector. Figure 3 In each plot in part (b) of FIG. 1 , the band above E=0 is associated with m increasing with increasing darkness. J The band below E = 0 corresponds to positive values ​​of , while the band below E = 0 corresponds to negative values ​​(increasing amplitude) that decrease with increasing darkness. The parameters from top to bottom are μ = 1 meV and α = {2, 20, 35) meV nm.

[0134] Figure 4 The simulated Majorana splitting energy and coherence length are shown. Figure 4 Part (a) shows the Majorana coherence length of a full-core model with the same parameters as in Figure 3 . The coherence length is obtained by fitting the exponential decay of the Majorana wave function in a wire of length 3 μm. The dashed line represents the boundary of the topological phase in the m J = 0 sector, and the shaded region is gapless due to the higher m J region. The histogram indicates the frequency ξ of the coherence length in the gapped topological phase. Figure 4 Part (b) shows the Majorana wave function Ψ integrated by radio in the case of μ = 1 meV and α = 30 meV nm. The solid line is obtained by fitting the envelope of the wave function to a double-exponential in the interval 0.5 μm < z < 2.5 μm.

[0135] As shown above, full-shell nanowires support Majorana zero modes in a wide range of parameters, so one can use these systems for quantum computing purposes.

[0136] In a conventional nanowire without a full coating of superconductor, the induction of MZM is due to the spin effect of the magnetic field B. However, in the full-shell nanowire 7, the induction of MZM is due to the coupling between the magnetic field B and the winding of the superconducting phase by means of the orbital effect of the magnetic field B. As a result, MZM can be induced at a significantly lower magnitude of the magnetic field. Thus, this allows quantum devices or computers to work with a lower magnetic field.

[0137] The idea outlined by Karzig et al in (Phys. Rev. B 95, 235305 (2017) “Scalable designs for quasiparticle-poisoning-protected topological quantum computation

[0138] with Majorana zero modes” is also applied in this case. In fact, by combining topological (T) and non-topological (e.g., conventional s-wave (S)) segments, one can construct a superconducting island with finite charging energy and Majorana-induced ground-state degeneracy. In addition to the flat design elucidated by Karzig et al (2017), full-shell nanowires allow a vertical design. The following figures illustrate several arrangements incorporating a vertical “forest” of topological wires into qubit and computer designs. Then, the following is a description of example fabrication methods and routes that can be followed to implement these designs as devices.

[0139] The vertical semiconductor nanowires 7 can be used for a variety of quantum computing applications. For example, vertical nanowires 7 with full or partial shells can host Majorana zero modes (MZMs), which when combined in superconducting islands become qubits (quads, 4MZMs), hexons (6MZMs), or larger quantum Hilbert spaces.

[0140] Vertical nanowires 7 with or without shells can also act as connectors between different planes (e.g., semiconductor, contact and / or gate planes), providing a valuable tool for scalable 3D integration solutions to layout and spatial problems in the quantum sector of a quantum computer.

[0141] As another example, vertical semiconductor nanowires 7 with full or partial shells can also act as conventional qubits, such as Gatemons, and in this capacity contribute to solving the 3D integration problem.

[0142] Figure 5 Part (a) of FIG. 5 illustrates an example design of a quaternion qubit. Figure 5 Part (b) of illustrates an example design of a hexon, which is a qubit plus an auxiliary qubit. The design adopts a vertical geometry. In either case, each quadrupole or hexon includes a portion of a semiconductor network 14 that can be formed in the plane of the substrate 5. The semiconductor network 14 can be formed in part or in whole of the same semiconductor material as the nanowire 9 (e.g., InAs or InSb). Each quadrupole or hexon also includes a superconducting island 15, which can also be formed in the plane of the substrate 5. The superconducting island can be formed in part or in whole of the same superconductor material (e.g., Al or Nb) as the material used to form the nanowire coating 10. Each quadrupole or hexon also includes a plurality of vertical nanowires 7a, which are arranged to be called into a topological phase (active nanowire) and formed vertically to the plane of the substrate 5. Each quadrupole or hexon can also optionally include one of the plurality of full-shell nanowires 7x, which is disconnected from the semiconductor network 14 but connected to a voltage source and acts as an electric gate.

[0143] The quaternion includes four vertical active nanowires 7a. The hexon includes six active topological nanowires 7a. In either case, each nanowire has one end connected to the superconducting island 15. In an embodiment, the island 15 includes multiple "arms" of superconductor material, which are connected to each other at a common point (e.g., the center of the island 15), but are otherwise separated from each other. For example, in an embodiment, the arm can take the form of a concentric spiral arm. Each active nanowire 7a is connected to the distal end of a different corresponding arm in the arm of the superconductor island 15 at its lower end (the end closest to the substrate 5), such as one end of a corresponding spiral arm in the spiral arm. The semiconductor network 14 is connected to the superconductor island 15 at one or more points (e.g., at one or more points (e.g., two, more, or all such points) where the active nanowire 7a meets the superconductor island 15). In an embodiment, this is at the distal end of one, two, more, or all arms in the superconductor arm.

[0144] By means of such an arrangement or a similar one, then, under conditions of a magnetic field component B perpendicular to the active nanowire 7a, a pair of MZMs 13 can be invoked in each such topological nanowire 7a, one at each end.

[0145] Arranging the superconducting islands 15 into thin arms (e.g. spiral arms) is useful (but not necessary) because it helps prevent the formation of eddy currents due to quasiparticle poisoning, i.e., circulating currents that can drive the superconductor 15 into a positive phase and thereby hinder the formation of an MZM.

[0146] Figure 6 A top view of an example wafer in the case of a quantum device comprising a hexon network is shown. Superconductor (e.g. Al) islands 15 link six MZMs 13 on the wafer to form an "insect" (with six legs - in other words, six MZMs connected by a central body). The additional six MZMs 13 are located vertically above the top of the full shell line 7a. The latter are decoupled. The spiral minimizes MZM hybridization, making the charging energy E at the limit of the dense spiral c maximizing, and allowing the superconductor material 15 to be vortex-free in a weak vertical B field. Vertically located above each X in the figure is a topological full-shell nanowire 7a. Figure 7 Further details of an example of how other full shell wires 7x can be used as electrical gates 16 to open or close a tunnel junction are shown. The full shell 10 is connected to the superconductor spiral 15 of the superconductor island. The details show where the superconductor network 14 joins the adjacent hexons and where in each hexagon the semiconductor network 14 is connected to one of the active vertical nanowires 7a (where the MZM 13 appears). The gate 16 is formed by one of the non-topological state nanowires 7x.

[0147] Note: In the case where B is vertical and weak, many of the constraints on the "comb design" are relaxed, allowing hexagonal symmetry. It should also be noted that due to the lattice structure of the superconductor, such as aluminum, the spiral form of the superconductor island 15 may not necessarily be continuously curved, but have a series of straight edges. The term "spiral" in this article does not necessarily imply that it is preferably continuously curved.

[0148] Figure 8 A top view of a wafer in another example design for a quantum device including a hexon network is shown. Here, the arms of the superconductor islands 15 are not in the form of spirals, but in the form of radial arms, but their design is similar to that of the conventional superconductor islands 15. Figure 6 The radial arms act like a spiral because they minimize the MZM hybridization, making the charging energy E c maximized, and allows the superconductor 15 to be vortex-free in a weak vertical B field. Fig. 9 Further details of an example of how other full shell wires 7x can be used as electrical gates 16 to open or close the tunnel junction are shown. The full shell 10 is connected to the radial superconductor arms of the superconductor island 15. Again, in the case where B is vertical and weak, many of the constraints on the "comb design" are relaxed, allowing hexagonal symmetry

[0149] Fig.10 Shown with Figure 6 and Figure 7 1, but the quadrupole comprises only four active vertical nanowires per semiconductor island 15 instead of six, and the quadrupole is linked by square or octagonal cells of semiconductor 14 instead of hexagonal combs. Each semiconductor island (e.g., Al) 15 supports four plus four MZMs (one at each end of each active vertical nanowire 7a). The tunnel junctions in the semiconductor network 14 can be formed by Figure 7 and Fig. 9 The other vertical full-shell nanowires 7x shown in detail are gate controlled. Fig.10 The advantage of the design is that the superconductor island 15 is slightly smaller and the charging energy E c Higher.

[0150] Fig.11 Another top view of the square layout of the quadruplets is shown, which is similar to Fig.10 The quadruple arrangement is similar, but as Figure 8 In the embodiment of FIG. 1 , radial arms are provided instead of spiral arms 15. Similarly, the tunnel junction in the semiconductor network 14 can be formed by Figure 7 and Fig. 9 The other vertical full-shell nanowires 7x shown in detail are gate controlled. Fig.10The advantage of the design is that the superconductor island 15 is slightly smaller and the charging energy E c Higher.

[0151] Fig.12 Some example manufacturing steps are shown that can be used to form devices having structures disclosed herein, such as Figures 1 to 1B or Figures 5 to 11 equipment. Fig.12 Fabrication steps are included to produce selective area growth (SAG) or vapor-liquid-solid (VLS) superconductors on ridges or cliffs of the disclosed designs.

[0152] Step i) starts with a substrate 5 and a dielectric layer 17 formed above the substrate 5, where etching windows 22 are formed through the dielectric 17 for selective area growth (SAG). These windows 22 are placed at locations where vertical nanowires 7 are to be grown (one window 22 is placed for each such nanowire 7). The dielectric 17 can be any suitable dielectric insulator. The etching of the windows can be performed using any suitable known etching process.

[0153] From step i), the method proceeds to step ii) or ii'). In either case, in this next step, the substrate 5 is etched with the trenches 23. Any suitable known etching process can be used. In variant ii), a further layer of dielectric 17 is applied, which covers the inner surface of the trenches 23. In variant ii)', a further layer of dielectric 17 can be applied, but if so, it does not cover the trenches 23. Any suitable selective deposition technique can be used to apply the further layer of dielectric, for example using a mask. The dielectric material 17 applied in step ii) or ii') can be the same as the dielectric material applied in step i).

[0154] The method then proceeds from step ii) to step iii) or from step ii') to step iii'). In either variant, here, a portion of the semiconductor material 18 is formed in the window 22, which window is formed for each vertical nanowire 7 in step i). The semiconductor 18 is deposited using selective area growth (SAG). In variant iii'), the semiconductor material 18 can be deposited in the inner surface of the groove 23. The same semiconductor material 18 as the semiconductor material used to fill the window can be used, and SAG can also be used. The portions of the semiconductor 18 in the window 22 and the groove 23 can be formed in the same SAG step.

[0155] The method then proceeds from step iii) to step iv) or from step iii') to step iv'). In either variant, this step comprises growing the semiconductor core 9 of the nanowire 7 from a semiconductor material 18 by growing the core 9 of the wire vertically under SAG conditions using VLS. The semiconductor material 18 used to form the nanowire core 9 in this way may be the same semiconductor material (e.g. InSb or InAs) used in step iii) or iii'), respectively.

[0156] The method then proceeds from step iv) to step v) or from step iv') to step v'). In either variant, this step comprises depositing a superconductor material 19 on the dielectric 17 and the core 9 of the vertical nanowire 7. The superconductor material 19 may be, for example, Al or Nb. The deposition may be performed by angular deposition of the semiconductor material 19 with a grazing angle. Fig.12 v') shows a beam flux of superconductor material. The beam flux can also be used in v), but is not shown for the sake of simplifying the illustration. In order to completely cover the wire, the beam or wafer can be rotated (while maintaining a grazing angle so as not to fill the grooves). Depending on the angle, the superconductor 19 may or may not be in contact with the semiconductor 18 if present in the groove 23. The superconductor material 19 formed on the core 9 of the nanowire 7 forms a coating 10. Either the wafer or the beam can be rotated to achieve a full shell 10 around the vertical nanowire 7. Areas of superconductor material 19 formed in the plane of the substrate 5 can be used to form superconducting islands 15.

[0157] Depending on the application, it may be desirable to grow semiconductor 18 in trenches 23 (e.g. to protect quantum dots from superconductor deposition) or to keep trenches 23 free of semiconductor growth (e.g. to provide better electrical insulation). These are the unfilled (ii-v) and filled (ii'-v') cases, respectively.

[0158] Semiconductors 18 and superconductors 19 other than the semiconductors and superconductors used to form the core 9 and coating 10 of the vertical nanowires 7, respectively, may be used to form part or all of one of the layers of the circuit device 8, such as the bottom layer 8b of the circuit device. As mentioned, this may include superconducting islands 15, quantum dots in the semiconductor 18, other parts of the semiconductor network 14, gates 16, contacts for gating, and / or other contacts (e.g., external contacts). Alternatively or additionally, some or all of these components and / or other components may be formed in one or more subsequent steps in one or more other layers 6 formed above the substrate 5.

[0159] Fig.13An isometric view of an example design for all underlying controls is shown. Here, any arbitrary pattern can be etched in the superconductor 19 in the bottom plane of the SAG. The dashed circle shows a side view of the semiconductor layer 18 between the substrate 5 and the superconductor layer 19. The nanowire core 9 can be formed using VLS. Other areas of the semiconductor 18 in the plane of the substrate 5 can be formed using SAG. Fig.13 The dielectric layer 17 is not shown in FIG. 1 , but it should be understood that the dielectric layer 17 may be present in embodiments.

[0160] Fig.14 Some other design features that may be employed in embodiments are illustrated. Fig.14 Part (a) of shows a side view of the wafer in which a filler material 20 has been deposited over the semiconductor layer 18, the superconductor layer 19, and any other portion of the bottom layer 8b of the circuit device. The filler material 20 may be, for example, wax or a plastic, such as PMMA. The filler material is used to mechanically support the vertical nanowires 7 in the final finished device 2. The bottom layer 8b of the circuit device may include one or more portions (e.g., semiconductors) that are suitable for forming one or more external contacts 21b with external equipment (not shown). A top layer 8t of the circuit device may be formed over the filler material 20. The top layer 18t of the circuit device may include one or more portions (e.g., superconductors 19) that are suitable for forming one or more external contacts 2lt with external equipment. Fig.14 Part (b) of shows a similar arrangement, alternatively or additionally, the arrangement comprises at least one intermediate layer 8i of the circuit device, with a filler 20 formed below and above the intermediate layer. The intermediate layer 8i of the circuit device may include one or more parts (e.g., semiconductor 19) for forming one or more external contacts 21i with external equipment. The external contacts 21 formed by the bottom layer, the intermediate layer and / or the top layer of the circuit device 8 may, for example, be used to control the qubits (such as for gating) and / or to make measurements and / or to supply power to power the device 2. For example, in an embodiment, the intermediate layer of the circuit device 8i may include a gating plane. Fig.14 Part (c) of FIG. 8 shows an isometric view of another example, in which a contact 24 of the gate 16 is formed in the top layer of the circuit device 8t above the filler 20. This contact 24 can be used to apply a potential controlled by external equipment for gating purposes. Alternatively, the gate contact 24 can be formed between fillers in the middle layer 8i or bottom layer 8b of the circuit device 8.

[0161] Other Discussions

[0162] As described above, the present disclosure describes a novel model system that supports Majorana zero modes based on semiconductor nanowires with complete superconducting shells. It is shown that in the presence of a radial spin-orbit field, the winding of the superconducting order parameter is sufficient to drive the system into a topological phase, significantly reducing the magnitude of the required magnetic field. The resulting topological phase persists over a large range of chemical potentials, opening a path to achieving Majorana zero modes in an experimental system that is easy to manufacture and has reproducible material quality.

[0163] Majorana zero modes (MZMs) are expected to revolutionize quantum computing through topological quantum information processing. In the past decade, the study of MZMs has evolved into one of the most active fields in modern condensed matter physics, with amazing progress in its experimental realization and theoretical understanding. This activity has been greatly promoted by proposing simple, experimentally feasible systems. Particularly promising routes involve approaching semiconductor nanowires (e.g., see Lutchyn et al., "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures (Majorana Fermions and Topological Phase Transition in Semiconductor-Superconductor Heterostructures)" published in Physical Review Letters 105, 077001 (2010) on August 13, 2010). The basic elements in these schemes are very simple: spin-orbit coupling, Zeeman field, and induced superconductivity. Nonetheless, the required coexistence of large (about 1 T) magnetic fields with superconductivity and the need for careful control of the chemical potential in the semiconductor poses significant challenges to the consistent realization of MZMs in nanofabricated devices, requiring ongoing experimental improvements.

[0164] In the present disclosure, semiconductor nanowires completely covered by superconducting shells are described as alternative structures for realizing MZMs. Although having similar simplicity and practical feasibility to the original nanowire proposals (e.g., Lutchyn et al.), full-shell nanowires can be used to provide any one or more of several key advantages. First, the topological transformations in the full-shell wires are driven by field-induced windings of superconducting order parameters (rather than by the Zeeman effect), so the required magnetic field may be very low (about 0.1 T). Moreover, the full shell naturally protects the semiconductor from impurities and random surface doping, thereby achieving a reproducible way to grow many wires with substantially the same properties. Although the full-shell wire does not allow direct electrostatic gating of the electron density in the semiconductor core, it is demonstrated below that by carefully designing the wire properties (e.g., by selecting the correct radius), it is possible to obtain wires that naturally contain MZMs under a certain magnetic field.

[0165] It is known that time reversal symmetry can be broken and MZMs can be localized in topological insulators and semiconductor heterostructures using carefully chosen superconducting phase differences or vortices, but the corresponding implementation usually requires careful tuning of the flux, which complicates the scalable approach with multiple MZMs (see the above references by Karzig et al.). Here, it is shown that the quantized number of superconducting windings in a full-shell wire is a natural and more robust implementation of the required phase difference, resulting in a considerable region of the topological phase space. Below, these ideas are first demonstrated in a simple model of a hollow wire, where an analytical mapping of the standard model of topological superconductors is possible. These results are then supplemented with numerical and analytical studies of topological phases in the opposite state, where the electron density is dispersed over the entire semiconductor core.

[0166] Theoretical model: A nanowire consisting of a semiconductor (SM) core and a complete superconducting (SC) shell is considered, e.g. Figure 1A This illustrates a semiconductor nanowire core 9 with a complete superconducting shell 10, which is confined by a weak axial magnetic field B at the lower left. In the cross-sectional detail, the region of r<R1 indicates the possible presence of an insulating core 11 in the semiconductor. Fig.15 is a sketch of the energy gap in the superconducting shell as it varies with the magnetic field, showing typical Little-Parks lobes. The different lobes correspond to different numbers of windings n of the superconducting order parameter around the wire. The period is B0≈4Φ0 / π(R2+R3)2 with Φ0=h / 2e superconducting flux quanta.

[0167] It is assumed that the semiconductor (e.g., InAs) has a large Rashba spin-orbit coupling due to the intrinsic electric field at the semiconductor-superconductor interface. The system undergoes The direction of the magnetic field (i.e., ) limit. For electromagnetic vector potential Using cylindrical coordinates and a symmetry gauge, the effective Hamiltonian for the semiconductor core can be written as (where h = l):

[0168]

[0169] Here, is the electron momentum, e>0 is the charge, m* is the effective mass, μ is the chemical potential, α is the strength of the radial spin-orbit coupling, and finally σi is the spin-1 / 2 Pauli matrix. Note that both μ and α can depend on the radial coordinate r. The vector potential A φ =Φ(r) / 2πr (where Φ(r) = πBr 2 ) is the flux through the cross section of radius r. For simplicity, we neglect the Zeeman term due to the small magnetic fields required in these devices.

[0170] The shell (e.g. made of Al) induces superconducting correlations in the nanowire due to the Andreev process at the semiconductor-superconductor (SM-SC) interface. If the coupling to the superconductor is weak, the induced pairing in the nanowire can be expressed as a local potential See Appendix B. Nambu Foundation The Bogoliubov-de-Gennes (BdG) Hamiltonian for the approaching nanowire is then given by:

[0171]

[0172] Assume that the thickness of the SC shell is less than the London penetration depth: R3→R2<<λ L Therefore, the magnetic flux through the SC shell is not quantified. However, the magnetic field induces windings of the superconducting phase, i.e., with Angular coordinates and Order parameter of winding number In practice, the winding number n adjusts itself to the value of the external magnetic field so that the free energy of the superconducting shell is minimized. This is the familiar Little-Parks effect: the variation of the winding number produces periodic lobes in the energy spectrum of the superconducting shell, see Fig.15 and Appendix A.

[0173] Note the following rotational symmetry of the BdG Hamiltonian:

[0174] [J z ,H BdG ]=0, where

[0175] Among them, the τi matrix that works in the Nambu space is introduced. BdG The eigenstates of can therefore be determined by the conserved quantum number m J mark:

[0176]

[0177] The wave function must be single-valued, which is true for m J The following constraints are imposed:

[0178]

[0179] Now, through the uniform transformation Remove the angular dependence of HBdG, that is, in:

[0180]

[0181] Note that, naively, one would want to average out the spin-orbit coupling. However, mJ The nontrivial structure of the eigenvectors gives rise to a finite number of matrix elements proportional to the Rashba spin-orbit coupling.

[0182] We now show that the above BdG Hamiltonian supports topological superconductivity and Majorana zero modes. First, note that the particle-hole symmetry involves having opposite energy and angular quantum numbers m. J state, that is, P = τ y σ y K (where K represents the complex conjugate) PΨ E，mJ =Ψ -E，-mJ Therefore, non-degenerate Majorana zero-energy solutions can only appear in m J = 0 sector. This in turn implies that Majorana zero modes can only appear when the number of windings n is an odd number.

[0183] Hollow Cylinder Model: We now focus on the limit of semiconductors forming thin-walled hollow cylinders (i.e. Figure 1A and Fig.15 R1≈R2 in (R1≈R2). This approximation is motivated by the fact that in some semiconductor-superconductor heterostructures (such as InAs / Al) there is an accumulation layer so that the electron density is concentrated within the screening length (about 20 nm) from the interface. In this case, one can consider only the lowest energy radial mode in equation (6). This allows an analytical solution of the model. If one considers the sector m for n=1 J = 0, then this yields the simplified Hamiltonian:

[0184]

[0185] Here, Δ = Δ(R2) and the effective chemical potential and Zeeman energy are defined as:

[0186]

[0187] Note that when the core is penetrated by one flux quantum (i.e., Φ(R2) = Φ0), V z = 0. This state corresponds to trivial (i.e., s-wave) superconductivity. However, small deviations in the magnetic field can drive the system into a topological phase (note that the magnetic flux through a finite-thickness superconducting shell can be significantly the same as the magnetic flux through a semiconducting shell). In fact, the Zeeman and spin-orbit terms in equation (7) do not commute, so V z In p z = 0 opens a gap in the spectrum.

[0188] One can map equation (7) to a Majorana nanowire model (e.g., the model of Lutchyn et al. cited above) and determine the topological phase diagram by calculating the topological index Q. The topological quantum phase transition between a trivial (Q=1) superconducting phase and a nontrivial (Q=-1) superconducting phase occurs when:

[0189]

[0190] exist Fig.16 The resulting phase diagram is shown in FIG, where the dashed line indicates the phase at m J = 0 when the topological transformation closes the gap. Fig.16 The topological phase diagram of the hollow cylinder model is shown. Panel (a) shows the overall energy gap E as a function of chemical potential and spin-orbit coupling. gap . Multiply the energy gap by the topological index Q = ±1. The dotted line represents m J =0 sector, as expressed using written labels and according to equation (9). Here, As indicated by the black asterisk in panel (b). and As a reference, using the actual parameters m* = 0.026me and Δ = 0.2meV, α0 ≈ 17meV nm and R0 ≈ 85nm are obtained. Right: The overall energy gap as a function of flux and R at fixed μ / Δ = 2 and α / α0 = 1, as indicated by the black asterisks in (a). Panels (c to e) show the band structure at the points indicated in (a) using black triangles, which illustrate m J = Closing and reopening of the body gap in sector 0.

[0191] Close to the transformation, the quasiparticle spectrum is changed from m J = 0 sector is determined and given by:

[0192]

[0193] One can estimate the quasiparticle velocity v using Eq. 10 to find v = αΔ√(Δ 2 +μ0 2 ) and the corresponding coherence length ξ~v / E gap , where E gap is m J = 0 sector.

[0194] Well-defined topological phases require that the quasiparticle gap is finite in all angular momentum channels, and not just for m. J = 0. Therefore, Fig.16 In the figure, it is shown that by considering all m JThis limits the range of topological phases, since the energy gap is larger at high m under large chemical potentials and / or spin-orbit coupling. J disappears from the sector (see Appendix C). Fig.16 We demonstrate that in the hollow cylinder model, there exists a topological phase with voids within a finite range of all model parameters, where the optimal quasiparticle gap is comparable in magnitude to △ and the corresponding coherence length ξ is about 100 nm.

[0195] Full Cylindrical Model: Now consider the case where the electron density is uniform in the semiconductor core (i.e., Figure 1A and Fig.15 R1=0). People can J The quantum numbers are numerically solved for the radial modes in the core. The superconductor is effectively treated as a boundary condition where r = R2, thereby neglecting the effects of the magnetic field penetrating the shell, so that Φ(R3) ≈ Φ(R2). This treatment of proximity effects is reasonable for thin superconducting shells in the dirty limit (see Appendices A to D for technical details).

[0196] Fig.17 The topological phase diagram of the complete cylindrical model with parameters suitable for InAs-Al hybrid semi-superconductor nanowires is shown. Panel (a) shows the topological phase diagram of the complete cylindrical model with R2 = 100nm, m* = 0.026me, Δ = 0.2meV and Φ(R2) = Φ0 / 2. Panel (b) shows the same situation if Φ(R2) = Φ0. The overall energy gap E gap multiplied by the topological index Q is plotted as a function of μ and α. The dashed line represents the J = p in sector 0 z = 0. Panels (c to f) show the band structure at the points indicated in (b). The parameters from left to right are μ = 1 meV and α = {2, 20, 30, 35) meVnm.

[0197] As in the hollow cylinder model, a stable topological phase is found that extends over a fairly large range of chemical potentials and has a maximum topological gap of about 30 μeV. As in the previous case, due to m J ≠6, so most of the topological phases are gapless. The topological gap is smaller in the full cylindrical model due to the large expansion of the radial wave function in the semiconductor core. Moreover, in this case, the cancellation of the superconducting windings by orbital effects is not exact, so that for suitable parameters, the topological phase also appears at Φ(R2)=Φ0, see also Appendix E. Fig.17 (b to e) show different m JThe momentum dispersion of the sectors, which illustrates the topological transformation inside and outside the gapped topological phase. As can be seen in Fig.17 (b), the bands formed in the wire core have a significantly flat dispersion, reminiscent of the Caroli - de Gennes - Matricon vortex state.

[0198] Fig.18 Part (a) of shows the evolution of the local density of states (DOS) varying with the magnetic field at the end of a finite wire. Panel (a) shows the DOS at the end of the wire varying with the flux for μ = 1.1 meV, α = 30 meV nm, and the same parameters as in Fig.17 part (b) of (Φ(R2)=Φ0, the Majorana coherence length of the complete cylindrical model with the same parameters as in Fig.17 part (b) of. The coherence length is obtained by fitting the exponential decay of the Majorana wave function to a 3 - μm - long wire. The dashed line represents the boundary of the topological phase in the m J = 0 sector, and due to the higher m J sectors, the red - shaded region is gapless. The inset shows the Majorana wave function integrated within a radius range of μ = 1 meV and α = 30 meV nm (this point is also indicated by a black dot). The red solid line is obtained by fitting the envelope of the wave function to a double - exponential in the interval 0.5 μm < z < 2.5 μm. The histogram indicates the frequency of finding a given coherence length inside the gapped topological phase. Panel (b) shows the DOS varying with the flux at the end of the wire for μ = 1.1 meV and α = 30 meV nm.

[0199] As the flux Φ(R2) increases, the winding number n changes by one in half - integer multiples of Φ0. The change in the winding number leads to a discontinuous jump in the density of states. At energies comparable to the pairing gap, the DOS regenerates the periodic Little - Parks lobes already plotted in Fig.15 . However, at lower energies, the DOS reveals a different structure of the sub - gap spectrum in the semiconductor core. Peaks in the DOS are visible at zero energy in the n = 1 and n = 3 lobes, but not in the n = 0 and n = 2 lobes, which is consistent with the fact that Majorana zero modes should appear only for odd values of n. Inside the odd - numbered lobes, one can see a characteristic asymmetry of the sub - gap spectrum with respect to the center of the lobe, which is caused by the difference in the magnetic flux penetrating the wire core and the superconducting shell. Since both μ0 and V z depend on the magnetic field, the disappearance of the Majorana zero - energy states inside the odd - numbered lobes occurs, see equation (8), which leads to a topological phase transition at a certain point. When the DOS is plotted in the middle of the wire, the overall gap closing is clearly visible, see Appendix F.

[0200] In Fig.18 In part (b) of Figure 1, Majorana hybridization due to finite nanowire length (L 3 μm) is investigated, and the coherence length is extracted by fitting the Majorana wave function, see inset. Fig.17 The topological gap is relatively small, but people still find short Majorana coherence lengths, with the minimum coherence length being about 250 nm, which is consistent with Fig.17 The small group velocity correlation of the overall state of part (c).

[0201] Now consider a perturbation that breaks the angular symmetry discussed above. Such perturbations (e.g., disorder or other defects in the superconducting shell) are ubiquitous in real devices and will couple different m J This can actually have a beneficial effect on the stability of the topological phase, due to the coupling of different m J The sectors can be Fig.17 The gap is induced in the gapless region that appears with large μ and α. Therefore, such perturbations can lead to significantly larger topological phase space volumes. The inventors have considered the effects of disorder in superconductors and have confirmed this speculation, see Fig.19A .

[0202] Fig.19A Panel (a) shows the Fig.17 Topological phase diagram of the disordered complete cylindrical model with the same parameters as in . To implement the cylindrical symmetry breaking perturbation, the Hamiltonian (2) is discretized on a square lattice of a = 10 nm in a two-dimensional cross section. The disorder potential δU on each lattice site in the superconductor is uniformly distributed δU∈[-U,U], where U = 2meV. The gap multiplied by the topological index is averaged over the twelve disordered configurations. The red dashed line indicates the phase boundary where there is no disorder. Panel (b) shows the band structure at α = 35eVnm and μ = 1.5meV, which is used to increase the disorder strength from top to bottom, U = {0,2,4}meV. A single disordered configuration is shown.

[0203] In fact, the inventors have found that the phase space of the topological phases in full-shell nanowires is very large, thereby supporting MZM over a wide range of parameters (B-field, spin-orbit coupling, chemical potential, radius and temperature). This is Fig.19B It has been proved in.

[0204] Fig.19B Results showing the effect of angular symmetry-breaking perturbations on the phase diagram. Such perturbations (e.g., shape deformations or disorder in superconducting shells) are ubiquitous in real devices and will couple different m J eigenstates, which can have a beneficial effect on the stability of the topological phase. JThe perturbation of each sector can open the gap in the areas of large μ and |α|. The shadowed area is topological. Panel (a) shows a topological phase diagram of a hollow cylindrical model with angular symmetry of fracture. By introducing in the z direction, it depends on The anisotropic effective mass of is used to break the symmetry: The results are shown for q=1, Φ(R2) / Φ0=1 / 2, and R / R0=1 / 2.

[0205] Fig.19B Panel (b) shows the use of Φ(R2) / Φ0=1 with Figure 3 Topological phase diagram of the disordered complete cylindrical model with the same parameters in part (a). To achieve cylindrical symmetric rupture disturbance, the Hamiltonian is discretized on a square lattice with a=10 nm in a two-dimensional cross-section. The disordered potential ΔU at each lattice site in the superconductor is uniformly distributed ΔU∈[-U,U], where U=2meV. Fig.19B The scenario considers rotational symmetry rupture disorder, which translates unchanged in the z-direction. In the twelve implementations, multiply the gap by the topological index to average. The dotted line indicates that there is no disordered phase boundary. The energy band structures at α=35 eV nm and μ=1.5 meV are shown below, which are used to increase the disorder intensity U={0,2,4} meV from left to right. A single unordered configuration is shown.

[0206] In summary, we have investigated novel physical systems that support Majorana zero modes based on semiconductor nanowires covered by superconducting shells. Using a combination of analytical and numerical methods, we have calculated topological phase diagrams and showed that the model supports robust topological superconductivity in a fairly large parameter space. The topological phase can be characterized by calculating the quasiparticle gap and the effective coherence length. The existence of easily accessible robust topological phases in all-shell nanowires opens the way to realizing topological quantum computing devices.

[0207] Appendix A: Model of a disordered superconducting shell .

[0208] In this section, a disordered superconducting shell (e.g., Al shell) is considered, with inner and outer radii R2 and R3, respectively, see Figure 1A and Fig.15 Assume that the shell thickness d≡R3-R2<<λ L , where λ L is the London penetration length in the bulk superconductor. In this case, the shielding of the magnetic field by the superconductor is weak and can be neglected. The effective Hamiltonian of the SC shell in cylindrical coordinates can be expressed as:

[0209]

[0210] Here, is the electron momentum operator, e>0 is the charge, m is the electron mass in the SC, μ (s) is the chemical potential in the SC, τ i It is the Pauli matrix representing the Nambu space, △0 is the overall SC gap when B=0, n is the number of windings of the SC phase, and Vimp represents the short-distance impurity scattering potential. Performing a gauge transformation that produces actual order parameters is inspiring, i.e. Δ0[cos(nφ)τ x +sin(nφ)τ y ]→Δ0τ x ). Gauge transformation introduces effective vector potential in:

[0211]

[0212] where Φ(r) = πBr2 and Φ0 = h / 2e. From this argument it follows that the solution to equation (A1) should be periodic with respect to Φ0, see Fig. 20 That is, the winding number adjusts itself to the magnetic field value so that the energy of the superconductor is minimized. Specifically, for each winding number n, the maximum value of the quasiparticle gap occurs at:

[0213]

[0214] Since typical magnetic fields of interest are less than 100 mT (for which Zeeman splitting is negligible), the Zeeman contribution is negligible.

[0215] Fig. 20 A simulation of a superconducting shell with R1 = R2 = 60 nm and R3 = 70 nm without a semiconductor core is shown. Actual parameters corresponding to Al are used: m* = me, μ = 10 eV and Δ0 = 0.34 meV

[32] . The Hamiltonian equation (A1) is discretized on a square lattice of a = 0.1 nm using the Kwant package. In (a), we show the clean case, where the superconductivity is almost immediately destroyed by the magnetic field. In (b), we show the disordered case using an on-site disorder potential δU, which is randomly sampled from δU∈[-U, U] with U = 2 eV. To model the oxidized Al2O3 layer, disorder is imposed in the 5 nm thick outer layer.

[0216] To understand the magnetic field dependence of the quasiparticle gap, one needs to calculate the randomness of the disordered SC shell. The Green function of the disordered superconductor is e and the corresponding diffuse coherence length (where ξ0=V F / △ is the coherence length within the overall clean limit (V F is the Fermi velocity in the SC). For simplicity, we assume that the thickness of the superconducting shell is d&ξ d , so that the characteristics of the system can be obtained by calculating the Green function of the disordered overall superconductor in magnetic fields B and n=0. Larkin considers this problem, Larkin shows that within the self-consistent Bom approximation, the normal and exceptional Matsubara Green functions are given by the following equation:

[0217]

[0218] Where H = eB / 4m and m J is the eigenvalue of angular momentum and is an eigenvalue of the Hamiltonian:

[0219] in

[0220] function and Determined by the following equation:

[0221]

[0222] where τ is the elastic scattering time and is the angular momentum cutoff. In the limit H→0, due to the linear term in m J vanishes when averaged over, so the first-order correction to the above equation appears to be quadratic. In fact, one can show that the self-consistent solution for τ→0 is given by:

[0223]

[0224] in is the characteristic scale of the magnetic field effect in the problem. Here, The correction to the pairing gap is therefore controlled by a small parameter k << 1. In terms of flux quanta, this condition reads Note that disorder suppresses the orbital effect of the magnetic field and results in a weak dependence of the pairing gap on the magnetic field (ie, quadratic relative to linearity). In other words, disordered superconductors can withstand much higher magnetic fields compared to clean superconductors, see Fig. 20 Finally, the above analysis can be extended to n≠6. After some manipulation, one finds that:

[0225]

[0226] This estimate is consistent with numerical calculations, see Fig. 20 .

[0227] Appendix B: Derivation of the effective Hamiltonian .

[0228] In the previous section, we derived the Green function for a disordered superconducting ring. We can now use these results to study the proximity effect of the SC ring on the semiconductor core. Here we consider the SC shell to be thin d~l e so that In this case, one can neglect the magnetic field dependence of the self-energy of the entire lobe. (Alternatively, when ξ d When R3, people can Therefore, one can use the zero-field Green function for disordered superconductors to study the proximity effect, which is obtained by using the clean Green function replace and And obtained.

[0229] One can now integrate the superconducting degrees of freedom and calculate the effective self-energy due to tunneling between the semiconductor and the superconductor. Using the gauge convention when ∆0 is real-valued, the tunneling Hamiltonian between the SM and the SC is given by:

[0230]

[0231] where r and r' refer to the SM and SC domains, respectively. T(r,r') is the tunneling matrix element between the two subsystems, and Ψ and is the fermion annihilation and creation operator in the corresponding subsystem. One can calculate the SC self-energy due to tunneling and obtain:

[0232]

[0233] where Γ(r) is a rapidly decaying function far from r = R2, which describes the tunneling between the two subsystems. Note that the SC self-energy in this approximation is the same as for a clean superconductor, because The ratio is independent of τ.

[0234] The Green function of a semiconductor can be expressed as:

[0235] G -1 (ω n )=-iω n -H SM -∑ (SC) (r,ω n ) (B3)

[0236] To calculate the quasiparticle energy spectrum, one must find the pole above the Green function.

[0237] In the hollow cylindrical limit, Γ(r = R2) is constant and one can find the low energy spectrum analytically. In fact, in expanding equation (B3) to small ω n Afterwards, the quasiparticle poles are determined from the spectrum of the following effective Hamiltonian:

[0238]

[0239] By comparing with equation (2), one can establish the correspondence between the renormalized parameters and bare parameters of the semiconductor and the proximity-induced gap △=Δ0Γ / (Δ0+Γ).

[0240] Appendix C: Higher m j The influence of state on the gap

[0241] As demonstrated above, with larger m j The states with ≠6 may close the gap, thus limiting the range of topological phases. Here, we provide a novel approach for the hollow cylinder model in parameter space due to the high m j First, consider the finite m of the BdG Hamiltonian (7) j Extensions:

[0242]

[0243] in:

[0244]

[0245] Among them, it is defined

[0246] From equation (C1), we can observe that for different m j The pairing gap in the sector does not necessarily open around zero energy, but is shifted by τ z , independent term A mj +Vzσ z , which acts as a Pauli limiting field and leads to a pair breaking effect. In other words, corresponding to m J The ≠6 particle and hole bands have different Fermi momenta, which prevents opening of a pairing gap in the spectrum.

[0247] When there is no pairing term Δτ x and the spin-orbit term αp z σ y τ zWhen considering the Hamiltonian (C1) with αp , one can obtain an illustrative analytical estimate of the conditions when a gapless state exists. It should be noted that neglecting the spin mixing term slightly overestimates the existence of a gapless state, due to the fact that the gapless state opens a gap when particle and hole bands of different σz components mix. In fact, αp z σ y τ z For the estimated m j = 0 sectors is crucial. However, for m j ≠6, the misalignment of the band at the Fermi level becomes the dominant effect.

[0248] For the spin eigenvalue σ z = ±1 for any m j ≠6 When both of the following two conditions are satisfied, the existence of a gapless state can be estimated.

[0249]

[0250]

[0251] where β1, β2 are numerical factors in the interval [0,1], and condition (C6) determines whether the particle and hole band crossing is farther from the chemical potential than β1△. It is assumed that the most efficient pairing for the gap ~△ can be captured by setting β1=1. Condition (C7) involves a specific m j and the position of the bottom of the band of spin states. For Δ = 0, the band is at least partially occupied when (C7) holds β2 = 0. In the presence of pairing, the particle band and the hole band repel each other, which shifts the bottom of the particle band to higher energy compared to the case of Δ = 0. This effect can be captured qualitatively by choosing a finite β2 < 1.

[0252] One can observe that for sufficiently large m j or a sufficiently large α (when m j When m is kept constant, condition (C6) is satisfied to understand the generally limited range of the gapped region in the α-μ plane. At the same time, large m j The state usually violates condition (C7) because the bottom of the band is shifted upward by ∝m 2 j , which needs to be compensated by a sufficiently large μ. Therefore, it is desirable to find a solution that satisfies condition (C6) while still being compatible with condition (C7) for large μ (which achieves large m j ) or a gapless state of very large α.

[0253] These considerations Fig.21 It has been verified in Fig.21 The bounds predicted by equations (C6) and (C7) are shown. Fig.16 The good agreement between the numerical simulations (using β1=1 and β2=1 / 2) of the gap reported in . In particular, the figure reveals that m J = ±2 sectors are those that maximally restrict the range of the topological phase at large μ and α. Note that Fig.21 The slope of the blue line in is independent of β1 and β2.

[0254] Fig.21 The phase diagram of the hollow cylinder model as a function of μ and α is shown, as in Fig.16 The left panel of the diagram shows the contours of the analytical estimate of the boundaries of the gapped region of the phase diagram. The horizontal and sloping dashed lines at the bottom of the diagram are given by conditions (C6) and (C7) for m j = ±1 to determine, and the upper dotted line is J = corresponds to condition (C7) in the ±2 sectors.

[0255] Appendix D: Details on the numerical simulation of the clean system .

[0256] From a numerical point of view, the solution of equation (B3) for the poles is not optimal, given that one has to solve nonlinear equations for ω. Therefore, one can take an alternative approach to coupling the SM to an artificial clean superconductor. One can use parameters for the superconductor and the tunneling Hamiltonian so that they are reproduced after integrating out the SC degrees of freedom in the final equation (B3).

[0257] In order to obtain the correct self-energy equation (B2), it is necessary to make the thickness of the simulated clean superconductor significantly larger than the coherence length. This is done by using R3 = 3 μm. Fig.17 and Fig.18 All parameters are chosen independently of r, except:

[0258]

[0259] Here, Φ(R2) corresponds to the flux penetrating the semiconductor core. Based on the above arguments, we simulated the semiconductor without a magnetic field. Equation (6) can be solved via the finite difference method using a discrete length of 5 nm.

[0260] Appendix E: Complete cylindrical semiconductor model in the small radius limit .

[0261] In this section, the complete cylindrical limit discussed previously is considered ( Figure 1A and Fig.15 Using the effective model, we analytically prove that a topological phase exists when exactly one superconducting flux quantum penetrates the core. The results in this section complement the numerical calculations in the main text above. The effective Hamiltonian of the model is given by:

[0262]

[0263] Moreover, unlike the hollow cylinder limit, one must solve the radial part of equation (E1). The dimensionless variable b = eBR2 is introduced 2 =πBR2 2 / Φ0. The proximity-induced gap △(r) must disappear in the middle of the core, lim r→0 Δ(r) = 0. We consider below the case where Δ(r) = Δr / R2. Although the specific choice of the radial dependence of Δ(r) is not important for demonstrating the existence of a topological phase.

[0264] Consider the limit When n = 1, the inner limit is m J = 0 sector. In this case, the current problem can be simplified because the Hamiltonian (7) becomes separable when a→0 and the spin-orbit effect can be included perturbatively. In the limit a→0, the electron spin is conserved and the Bogoliubov transform of the diagonalized Hamiltonian (El) can be expressed as:

[0265]

[0266] Among them, the transformation coefficient U λ，pz，σ (r) and V λ，pz，σ (r) is given by the solution of equation (E1). Ignoring the spatial dependence of △(r), the function U λ，pz，σ (r) and V λ，pz，σ (r) can be approximately expressed as:

[0267] Uλ, pz, σ(r)=uλ, σ(pz)fλ, σ(r) (E3)

[0268] Vλ, pz, σ(r) = vλ, σ(pz)fλ, σ(r) (E4)

[0269] The single particle wave function f λ,σ (r) is defined by the following radial Schrodinger equation:

[0270]

[0271] The linear term in b represents a constant energy shift:

[0272]

[0273] After introducing the dimensionless coordinate x = r / R2 and the dimensionless energy κ λ，σ =2m*R2 2 (ε λ，σ +δ σ) after that, the above equation becomes:

[0274]

[0275] The normalized eigenstate of this equation that satisfies the boundary condition fλ,σ(x=1)=0 is:

[0276]

[0277] Here, 1F1 is the Kummer confluent hypergeometric function, and the coefficient C λσ Determined by the normalization condition:

[0278]

[0279] The corresponding eigenvalues ​​are:

[0280]

[0281] Where K λ,σ is the zero of the appropriate Kummer convergence hypergeometric function of the two spins. Taking all things into consideration, for b = 1 and n = 1, the lowest eigenvalue of equation (E5) is:

[0282]

[0283] It should be noted that different values ​​of b will affect the numerical coefficients reported above.

[0284] At the Limit Within , one can project the system onto the lowest energy manifold (ie, λ = 1) and integrate over the radial coordinates. After some algebraic manipulation, the effective Hamiltonian takes the simple form (at most constants):

[0285]

[0286] The effective parameters are given by:

[0287]

[0288] where the numerical constant A is given by the overlap integral σ , B σ , C, D:

[0289]

[0290] C=∫0 1 ∫ 1.↑ (x)f 1.↓ (x)dx=0.93...,

[0291] D-∫0 1 x 2 f1.↑ (x)f 1.↓ (x)dx-0.465....(E18, E19, E20, E21)

[0292] One can note that, in contrast to the hollow cylinder model, the Zeeman term remains finite at b = 1 (i.e., one flux quantum). As mentioned in the main text, this is because the semiconductor state is distributed through the semiconductor core, rather than being confined to r = R2, so that the flux cannot completely cancel the effect of the winding number.

[0293] Thus, it has been shown that the full cylinder model also maps onto the Majorana nanowire model (e.g. of Lutchyn et al.) and supports a topological superconducting phase. The topological quantum phase transition from the topological trivial (i.e., s-wave) to the non-trivial (i.e., p-wave) phase occurs at:

[0294]

[0295] It should be noted that this only takes into account m J = 0 sector. In order to study other m J sectors and ensuring that the quasiparticle gap does not close within the topological phase can be done numerically as shown previously in the main text.

[0296] Appendix F: Topological Phase Transitions .

[0297] Fig. 22 The probability densities for the lowest energy spin-up (dashed lines) and spin-down (solid) modes are shown.

[0298] Fig.23 It shows (a): for Fig.18 Same parameters as in part (b), flux-dependent DOS in the middle of the line, and (b): flux-dependent DOS at the end of the line at μ = 1.5 meV and α = 40 meVnm.

[0299] In order to understand the topological phase transition within the same lobe as a function of the magnetic flux, it is useful to study the calculated overall DOS (e.g. in the middle of the line). Fig.23 The n=1 lobe in part (a) of , where the topological phase transition manifests itself by closing the whole. Fig.18 It is also instructive to compare the global and local DOS at the ends of the lines shown in . One can note the asymmetry with respect to the center of the lobe, which is due to the different dependencies of the semiconducting and superconducting states arising there.

[0300] The asymmetry depends on the parameters and Fig.23The boundary DOS of different sets are shown in part (b) of , where the zero bias peak extends throughout the n=1 and n=3 lobes. However, it should be noted that according to Fig.17 In part (a) of , the system is gapless for this parameter at Φ(R2) = Φ0. However, as Fig.19A As discussed in part (a) of the paper, a symmetry-breaking perturbation (e.g., disorder) can lead to m J A gap of ≠6 states opens, thus stabilizing the topological phase.

[0301] Appendix G: Details on the numerical simulations of disordered systems.

[0302] for Figures 19A to 19B For the disorder simulations in , the Kwant package has been used to discretize the Hamiltonian equation (2) on a 2D square lattice using a lattice spacing of 10 nm. The system is assumed to be translationally invariant along the z direction, where the disordered configuration repeats along the z axis. This trick is required because a full 3D simulation would be computationally prohibitive. To accommodate the higher computational cost, we used a smaller R3 of 1.5 μm in these simulations. Fig.24A Shown is the method for obtaining Fig.19A Phase diagram for different disorder realizations with average values ​​in . Fig. 24B Shown is the method for obtaining Fig.19B Phase diagram for different disorder realizations with average values ​​in .

[0303] in conclusion

[0304] It should be understood that the above embodiments are described by way of example only.

[0305] More generally, according to one definition of the first aspect of the present disclosure, a device can be provided, comprising: one or more semiconductor-superconductor nanowires, each of the one or more semiconductor-superconductor nanowires comprising a certain length of semiconductor material and a coating of superconductor material coated on the semiconductor material; each nanowire in one, some or all of the nanowires is a full-shell nanowire, and the superconductor material is coated around the entire periphery of the semiconductor material along some or all of the length of the semiconductor material; and wherein the device is operable to induce at least one Majorana zero-mode MZM in one or more active nanowires in the nanowires by applying a magnetic field component parallel to the active nanowire, the one or more active nanowires in the nanowires including at least one or more full-shell nanowires in the full-shell nanowires.

[0306] In an embodiment, the device is operable to induce a MZM at least by applying a magnetic field component from outside the device.

[0307] The device may include a substrate and one or more layers formed over the substrate, wherein the nanowires may be formed in one or more of the layers.

[0308] In embodiments, one, some or all of the full-shell nanowires may be vertical relative to the substrate.The device may include at least one layer of filler material disposed between the vertical nanowires to mechanically support the vertical nanowires.

[0309] In embodiments, one, some, or all of the full-shell nanowires may be horizontal in the plane of the substrate.

[0310] In an embodiment, the device may also include one or more layers of circuit devices formed in one or more of the layers, and the one or more layers of circuit devices are used to connect the nanowires together into the quantum structure, control the nanowires or quantum structure and / or measure from the nanowires or quantum structure.

[0311] In an embodiment, one or more of the nanowires other than the active nanowires may be arranged as a conductive via between layers of the circuitry or between a layer of the circuitry and an exterior of the device.

[0312] In an embodiment, one or more layers of the circuit device may include a semiconductor network connecting active nanowires for opening or closing a tunnel junction in the semiconductor network.

[0313] In embodiments, the device may include one or more qubits, each qubit comprising a plurality of nanowires in an active nanowire, wherein one, some, or all of the plurality of nanowires in each qubit are full-shell nanowires. In some such embodiments, each qubit in the qubits may be any of: a quadrupole qubit where the plurality is four; or a hexon qubit where the plurality is six.

[0314] In an embodiment, each qubit may include a horizontal superconducting island formed in a plane parallel to the substrate, wherein the superconducting island is divided into arms that respectively join the lower ends of corresponding vertical nanowires in a plurality of vertical nanowires in the qubit to a common point of the superconducting island. In some such embodiments, the arms may take the form of concentric spiral arms.

[0315] In an embodiment, the superconducting islands of each qubit may be formed from the same superconductor material as the coating of the nanowires.

[0316] According to another definition of the first aspect, there may be provided a method of operating the disclosed apparatus as defined, the method comprising: applying a magnetic field component parallel to the active nanowires so as to induce at least one MZM in each of the active nanowires; wherein the inducing comprises coupling of windings of a superconducting phase of a superconductor material introduced by magnetic flux of the magnetic field component through the active nanowires to the magnetic field component so as to induce a topological phase by means of an orbital effect of the magnetic field component. The coupling acts as if a strong Zeeman field is present.

[0317] The method may also include cooling the device to induce superconductivity in the superconductor material.

[0318] According to one definition of the second aspect, a device can be provided, comprising: a substrate defining a plane; one or more layers formed above the substrate; and one or more semiconductor-superconductor nanowires formed in one or more of the layers; wherein each of the nanowires comprises a certain length of semiconductor material and a coating of superconductor material coated on at least a portion of the semiconductor material; and wherein each of one, some or all of the nanowires is vertical relative to the plane of the substrate.

[0319] In embodiments, each of one, some or all of the nanowires may be a full shell nanowire, with the superconductor material coated around the entire perimeter of the semiconductor material along some or all of its length.

[0320] In an embodiment, the apparatus is operable to induce at least one MZM in one or more of the active nanowires by applying a magnetic field component parallel to the active nanowires.

[0321] In an embodiment, a MZM may be formed at each end of each active nanowire.

[0322] In an embodiment, the active nanowires may include one or more of the full-shell nanowires.

[0323] In an embodiment, the device may include one or more qubits, each qubit comprising a plurality of active nanowires.

[0324] In embodiments, one, some, or all of the plurality of nanowires in each qubit may be full-shell nanowires.

[0325] In embodiments, one, some, or all of the qubits may be MZM-based qubits. In some such embodiments, each of the one, some, or all of the MZM-based qubits may be any of: the plurality of four quadrupon qubits; or the plurality of six hexon qubits.

[0326] In an embodiment, each qubit may include a horizontal superconducting island formed in a plane parallel to the substrate, wherein the superconducting island is divided into arms that respectively join the lower ends of corresponding vertical nanowires in a plurality of vertical nanowires in the qubit to a common point of the superconducting island. In some such embodiments, the arms may take the form of concentric spiral arms.

[0327] In an embodiment, the device may also include one or more layers of circuit devices formed in one or more of the layers, and the one or more layers of circuit devices are used to connect the nanowires together into the quantum structure, control the nanowires or quantum structure and / or measure from the nanowires or quantum structure.

[0328] In an embodiment, one or more of the non-active ones of the nanowires may be arranged as a conductive via between layers of the circuitry or between a layer of the circuitry and an exterior of the device.

[0329] In an embodiment, one, some or all of the qubits may be transporter or gater based qubits.

[0330] In an embodiment, the layer of the wafer may include at least one layer of filler material disposed between the vertical nanowires to mechanically support the vertical nanowires.

[0331] According to another definition of the second aspect, a method for manufacturing a device can be provided, the method comprising: setting a substrate; growing vertical lengths of semiconductor material perpendicular to the substrate; subsequently coating at least a portion of each of at least some of the vertical lengths of semiconductor material with a superconductor material to form vertical semiconductor-superconductor nanowires; and after the device is completed, maintaining the vertical nanowires in a vertical orientation relative to the substrate.

[0332] In an embodiment, the method may include, after said growing, forming circuitry around the vertical nanowires to form one or more qubits, each qubit comprising a plurality of vertical nanowires.

[0333] In an embodiment, the completion of the device may include any one, multiple or all of the following: forming a gate or contact at the top of each vertical nanowire in some or all of the vertical nanowires; depositing filler material between the vertical nanowires; and / or packaging the device in an integrated circuit package.

[0334] In an embodiment, the growth may be performed using a vapor-liquid-solid (VLS) growth process.

[0335] In an embodiment, the method may be used to manufacture a device according to any embodiment of the first aspect or the second aspect disclosed herein.

[0336] In any of the first or second aspects, by way of example implementation, the semiconductor material may be InAS or InSb. The semiconductor material may be Al or Nb. The superconducting islands of the nanowires connecting each qubit may be formed of the same superconductor material as the coating of the nanowires. The filler material is plastic or wax. The gates, contacts and / or circuit devices between the nanowires may be formed of metals or semiconductors. The at least one MZM induced in each active nanowire may be a pair of MZMs. In the case of a vertical qubit design, the lower MZM in the pair may be the effective MZM of the qubit. The device may be packaged in an integrated circuit package.

[0337] In an embodiment, a quantum computer may be provided, comprising: a device disclosed in any embodiment of the first aspect or the second aspect; and an electromagnet arranged to apply a magnetic field component parallel to the active nanowire. The quantum computer may include a cold storage chamber, and the device is placed in the cold storage chamber to induce superconductivity in the superconductor.

[0338] Other applications or variations of the disclosed techniques or structures may become apparent to those skilled in the art once the disclosure herein is given.The scope of the present disclosure is not limited by the embodiments described above.

Claims

1. A semiconductor-superconductor nanowire-based device comprising: substrate; one or more layers formed over the substrate; as well as one or more semiconductor-superconductor nanowires formed in one or more of the layers, each of the one or more semiconductor-superconductor nanowires comprising a length of semiconductor material and a coating of superconductor material coated on the semiconductor material; each of one, some or all of said nanowires is a full shell nanowire, said superconductor material being coated around the entire perimeter of said semiconductor material along some or all of said length of said semiconductor material; wherein one, some or all of the full-shell nanowires are vertical relative to the substrate; as well as The device is operable to induce at least one Majorana zero-mode MZM in one or more of the active nanowires by applying a magnetic field component parallel to the active nanowires, wherein the one or more active nanowires in the nanowires include at least one or more of the full-shell nanowires.

2. The apparatus of claim 1, wherein the apparatus is operable to induce the MZM at least by applying the magnetic field component from outside the apparatus.

3. The device of claim 1, further comprising at least one layer of filler material disposed between the vertical nanowires to mechanically support the vertical nanowires.

4. The device of claim 1, wherein one or some of the full-shell nanowires are horizontal in the plane of the substrate.

5. The device of claim 1 further comprising one or more layers of circuitry formed in one or more of the layers, the one or more layers of circuitry being used to connect the nanowires together into a quantum structure, control the nanowires or quantum structure, and / or perform measurements from the nanowires or quantum structure.

6. The device of claim 5, wherein one or more of the nanowires other than the active nanowires are arranged as conductive vias between layers of the circuit device or between a layer of the circuit device and an exterior of the device.

7. The apparatus of claim 5, wherein the one or more layers of circuitry include a semiconductor network connecting the active nanowires for opening or closing a tunnel junction in the semiconductor network.

8. The device of claim 1, wherein the device comprises one or more qubits, each qubit comprising a plurality of nanowires in the active nanowires, wherein one, some or all of the plurality of nanowires in each qubit are full-shell nanowires.

9. The apparatus of claim 8, wherein each of the qubits is any of: a quadrupon qubit of the plurality of four; or a hexon qubit of the plurality of six.

10. The device of claim 1 , wherein the device comprises one or more qubits, each qubit comprising a plurality of nanowires in the active nanowires, wherein one, some or all of the plurality of nanowires in each qubit are full-shell nanowires; and each qubit comprises a horizontal superconducting island formed in a plane parallel to the substrate, wherein the superconducting island is divided into arms, each of the arms joining a lower end of a corresponding vertical nanowire in a plurality of vertical nanowires in the qubit to a common point of the superconducting island.

11. The apparatus of claim 10, wherein the arms are in the form of concentric spiral arms.

12. The apparatus of claim 10, wherein the superconducting island of each qubit is formed of the same superconductor material as the coating of the nanowire.

13. The apparatus of claim 1, wherein the apparatus is packaged in an integrated circuit package.

14. The apparatus of any one of claims 1-13, wherein one or more of the following are true: - the semiconductor material is InAs or InSb, and / or - The superconductor material is Al or Nb.

15. The device of claim 3, wherein the filler material comprises plastic or wax.

16. A quantum computer comprising: The device according to claim 1; as well as An electromagnet is arranged to apply a component of the magnetic field parallel to the active nanowire.

17. A quantum computer according to claim 16, comprising a cold room in which the device is placed for inducing superconductivity in the superconductor.

18. A method of operating the apparatus according to claim 1, the method comprising: applying a component of the magnetic field parallel to the active nanowires to induce the at least one MZM in each of the active nanowires; The inducing comprises coupling a winding of a superconducting phase of the superconductor material introduced by a magnetic flux of the magnetic field component passing through the active nanowire to the magnetic field component to induce a topological phase by means of an orbital effect of the magnetic field component.

19. The method of claim 18, comprising refrigerating the apparatus to induce superconductivity in the superconductor.

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