WO2018198121A1

WO2018198121A1 - Quantum hall edge-mode device with multiple 2-dimensional electron gasesthat are coupled electrostatically

Info

Publication number: WO2018198121A1
Application number: PCT/IL2018/050461
Authority: WO
Inventors: Mordechai HEIBLUM; Yonatan Cohen; Yuval RONEN; Vladimir Umansky; Daniel BANITT
Original assignee: Yeda Research And Development Co. Ltd.
Priority date: 2017-04-26
Filing date: 2018-04-26
Publication date: 2018-11-01

Abstract

A device for quantum computers having two or more parallel electronic layers, each containing a 2-dimensional electron gas (2DEG) in the quantum Hall effect. The layers contain different filling factors supporting different configurations of quantum Hall effect edge states. The device has multiple adjacent regions, each covered by a top gate above the layers, which allows filling factors in each region to be configured such that in the interface between two regions two edge states are juxtaposed, which counter- propagate in the juxtaposed region. This is made possible by controlling the voltage on the electrodes to increase the filling factor of one layer and decrease that of another layer without requiring separate magnetic fields for different regions. Controlling the filling factors in each region, configures the spins of counter-propagating edge states to be either opposite or aligned. Both integer, fractional, and mixed integer-fractional quantum Hall effect edge states are supported.

Description

QUANTUM HALL EDGE-MODE DEVICE WITH MULTIPLE

2-DIMENSIONAL ELECTRON GASES

BACKGROUND

[001] The present invention relates to the field of quantum computing devices. In particular, embodiments of the present invention provide an electronic device structure for counter-propagating 1 -dimensional (ID) electronic as well as anyonic states. Moreover, the spin configuration of the counter-propagating states can be chosen so that their spins are either opposite or aligned. Counter-propagating states (i.e., a pair of juxtaposed states which propagate in opposite directions in the region of their juxtaposition) having opposite spin orientations (also sometimes referred to as helical states) are desirable in quantum computing, because when they are coupled to a superconductor they form a topological superconductor with protected quantum states. In the case of electronic states, the quantum states are Majorana bound states; and in the case of counter-propagating anyonic states on the boundaries, the quantum states are parafermion states. Both Majorana bound states as well as parafermionic states are desirable in quantum computers, as they can form quantum bits (qubits) with topological protection against errors. In fact, parafermion states can exist even without a superconductor for configurations of counter-propagating anyonic states according to certain embodiments of the present invention.

[002] Consequently, efforts have been made to create device structures that exhibit counter propagating, 1 -dimensional electronic states with opposite spins. These efforts include: [003] Systems with large spin-orbit coupling (such as: InAs or InSb nanowires; 2-dimensional systems of InAs, etc.).

[004] 2-dimensional electronic systems where magnetic doping lifts the spin degeneracy, forming ferromagnetic transitions in the spectrum of a 2-dimensional electron systems.

[005] Pairs of 2-dimensional systems, one system of electrons and one system of holes.

[006] Edges of 2-dimensional topological insulators.

[007] As is well- appreciated in the field, it would be highly desirable to have devices that support counter-propagating 1 -dimensional (ID) electronic states to form topological states, for use as functional components, such as protected quantum bits in quantum computers and similar systems. This goal is attained by embodiments of the present invention.

SUMMARY

[008] Embodiments of the present invention provide a quantum Hall effect edge-state device. In one embodiment, the device forms counter-propagating one-dimensional (ID) electronic states using the edge states of the integer quantum Hall effect. In another embodiment, the device supports counter-propagating, ID anyonic states using the edge states of the fractional quantum Hall effect. In both embodiments the spin orientations of the states (opposite or aligned) can be controlled by tuning the system to the appropriate filling factors as disclosed herein. BRIEF DESCRIPTION OF THE DRAWINGS

[009] The subject matter disclosed may best be understood by reference to the following detailed description when read with the accompanying drawings in which:

[0010] Fig. 1 A conceptually illustrates a configuration of quantum Hall edge states in a double-electronic layer quantum Hall effect device according to certain embodiments of the present invention, with two edge states propagating at the edge of the lower layer.

[0011 ] Fig. IB conceptually illustrates another configuration of quantum Hall edge states in a double-layer quantum Hall effect device according to certain embodiments of the present invention, with one edge state propagating at the edge of the lower layer and one edge state propagating at the edge of the upper layer.

[0012] Fig. 1C conceptually illustrates counter propagating edge states with opposite spins (i.e., helical states) in a double-layer quantum Hall effect device according to certain embodiments of the present invention, with the left half-plane of the two electronic layers having the filling factor shown in Fig. 1A, and the right-half plane having the filling factor of Fig. IB. Combining these two configurations as shown leads to counter-propagating edge states with opposite spins (namely, helical states) at the interface region between the two half-planes.

[0013] Fig. 2 A is a schematic side view of a GaAs/AlGaAs heterostructure of a device according to an embodiment of the present invention, with a double quantum well having two GaAs quantum wells separated by a barrier.

[0014] Fig. 2B is an energy-level diagram of the conduction band in the region of the double quantum well of the device of Fig. 2A and the wave functions of the 1^st and 2^nd electronic sub-bands of the double quantum well. These two sub-bands implement the two quantum Hall layers of the embodiment described above. [0015] Fig. 2C illustrates the Landau level energies of the two sub-bands of the device of Fig. 2A as a function of magnetic field. As shown, the energies of the 2^nd sub-band's

Landau levels are shifted upwards by the sub-bands' energy difference.

[0016] Fig. 3A conceptually illustrates Landau level positions and the corresponding edge states for the double-layer quantum Hall configuration of Fig. 1 A.

[0017] Fig. 3B conceptually illustrates Landau level positions and the corresponding edge states for the double-layer quantum Hall configuration of Fig. IB.

[0018] Fig. 3C schematically illustrates a device according to an embodiment of the present invention showing two separate and adjacent regions - one region in the configuration of Fig. 3A, and one region in the configuration of Fig. 3B - and the transition between them with counter-propagating 1 -dimensional edge states along the interface.

[0019] Fig. 3D conceptually illustrates Landau level positions and the corresponding edge states for the device schematic of Fig. 3C.

[0020] Fig. 4 is a top view of the physical layout of a device according to an embodiment of the invention, corresponding to the schematic representation of Fig. 3C.

[0021 ] Fig. 5 A is a schematic side view of a device as shown in top view in Fig. 4, showing an arrangement of metallic gates according to an embodiment of the present invention.

[0022] Fig. 5B is a schematic side view of a device as shown in top view in Fig. 4, showing an arrangement of metallic gates according to another embodiment of the present invention.

[0023] For simplicity and clarity of illustration, elements shown in the figures are not necessarily drawn to scale, and the dimensions of some elements may be exaggerated relative to other elements. In addition, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.

DETAILED DESCRIPTION

[0024] Embodiments of the present invention feature a configuration of two parallel 2- dimensional electron gases (2DEGs), each having its own quantum Hall effect states and its own filling factor (under an applied magnetic field) and together forming counter-propagating integer and fractional quantum Hall edge states. The configuration is described in detail below, including the edge states and Landau level structure. When two regions having different filling factors are juxtaposed as in the present embodiments, counter-propagating edge states are formed at the interface region. Moreover, the spin configuration of the counter-propagating states is controllable so that their spins can be chosen to be either opposite or aligned, depending on the chosen filling factors of the adjacent regions.

[0025] When a device featuring counter-propagating integer quantum Hall edge states with opposite spins, according to certain embodiments of the present invention, is coupled to a conventional superconducting contact and a superconducting gap is induced, a topological superconductor forms hosting Majorana bound states at its boundaries (at the two ends of the region of the counter-propagating edge states where superconductivity is induced). To localize the Majorana bound states a small gap must be opened in the region of the counter-propagating states where there is no induced superconductivity. This is an advantage of the present invention: proper gating and choice of distance between the quantum Hall layers controls the coupling between the counter-propagating edge states. [0026] In a similar manner, when a device featuring counter-propagating fractional quantum Hall edge states with opposite spins, according to other embodiments of the present invention, is coupled to a conventional superconducting contact and a superconducting gap is induced, a topological superconductor is formed, hosting parafermion states at its boundaries. Certain additional embodiments of the present invention feature parafermion states without the need for a superconductor, by providing configurations of counter-propagating anyonic states. In further embodiments, combinations of fractional and integer Hall effect states are used.

[0027] Fig. 1A - Fig. 1C conceptually illustrate edge state configurations in a double- layer quantum Hall effect device according to various embodiments of the present invention. A lower layer 2-dimensional electron gas (2DEG) is symbolically represented as L_\, and an upper layer 2D EG layer is symbolically represented as Li.

[0028] The terms "upper" and "lower" herein refer to the position of the center of the 2DEG along the z-axis normal to the plane of the 2DEG. The term "center" in this context is defined by the center of the probability density of the corresponding wave function - each 2DEG is associated with a single wave function along the z-axis (by definition) and thus has some spread in the z-direction. This spread leads to some overlap of the wave functions, such that some of the density of the upper 2DEG wave function can lie below some of the density of the lower 2DEG wave function, and some of the density of the lower 2DEG wave function can lie above some of the density of the upper 2DEG wave function. However, by defining the "center" z-position according to the centers of the respective wave functions, it is possible to refer to the "upper" and "lower" 2DEGs without confusion. [0029] Each of L_\ and Li has its own filling factor, symbolically represented as Vi and v₂, respectively, with the generalized double-layer filling symbolically represented as v = (vi,v₂), which characterizes the state of the structure. In the cases presented in Fig. 1A - Fig. 1C, edge states propagate in different configurations. In these non-limiting examples, all edge states propagate counterclockwise {i. e. a counterclockwise chirality) when seen from above, but some edge states have a spin-up orientation, while others have a spin-down orientation (the spin of each edge state is according to the underlying Landau level' s spin).

[0030] Fig. 1 A illustrates a double-layer quantum Hall system 101 having a generalized filling factor 101F, namely v = (2,0). A lower layer

and thus has a path for an edge state 101C with a spin-up orientation and an edge state 101D with a spin-down orientation. An upper layer (L ) 101B is in filling factor v₂=0 and thus has no edge states. A magnetic field source 110 provides a Hall effect magnetic field.

[0031 ] Fig. I B illustrates a double-layer quantum Hall system 102 having a generalized filling factor 102F where v = (1 , 1). A lower layer (L_\) 102A is in filling factor \_\=\ and thus has a path for a single edge state 102C with a spin-up orientation and an upper layer (La) 102B is in filling factor v₂=l and thus has a single edge state 102D also with a spin-up orientation. A magnetic field source 120 provides a Hall effect magnetic field.

[0032] Fig. 1 C illustrates a double-layer quantum Hall system 103 with two regions, a left half 105 and a right half 106, each in a different generalized filling factor. Left half 105 has a generalized filling factor 103G, namely v_g = (2,0), while right half 106 has a generalized filling factor 103H, namely v_¾ = (1 ,1). A lower layer (L_\) 103A has a path for a spin-up orientation edge state 103C which extends over both left half 105 and right half 106. Lower layer (L_\) 103A also has path for a spin-down orientation edge state 103D which extends only over left half 105. Upper layer (La) 103B has a path for a spin-up orientation edge state 103E which extends only over right half 106. Inset 104 shows a close-up view of the interface region between left half 105 and right half 106, where spin-down orientation edge state 103D and spin-up orientation edge state 103E counter-propagate. A magnetic field source 130 provides a Hall effect magnetic field.

[0033] In this disclosure, two parallel 2DEG layers are used as a non-limiting example for purposes of illustration and discussion. However, other multiple 2DEG layers are also possible within the scope of the present invention, with the generalized filling factor extended to cover higher multiples of layers. In addition, within the two parallel 2DEG layers, a non-limiting example of v_g = (2,0) and v_¾ = (1 ,1) is used for purposes of illustration and discussion. However, other filling factors are also possible within the scope of the present invention. Furthermore, a non-limiting example of two counter- propagating states having opposite spins is used for purposes of illustration and discussion. However, it is also possible to have two counter-propagating states having the same spin orientation within the scope of the present invention. In a non-limiting example, a Vg = (3,0) and v_¾ = (2,l) configuration (in the left and right halves, respectively) results in counter-propagating edge states with the same spin orientation, whereas a v_g = (4,0) and v_¾ = (3,1) configuration results in counter-propagating edge states with opposites spins (such as in the v_g = (2,0) and v_¾ = (1 , 1) configuration). To create counter-propagating fractional quantum Hall edge states, fractional filling factors are used. For example when the left and right halves are tuned to filling factors (4/3,0) and (1 ,1/3), respectively, counter-propagating 1/3 fractional quantum Hall edge states are formed. [0034] To create two phases with different filling factors such that counter-propagating states form at the interface (e.g. (2,0) and (1 ,1); (4,0) and (3,1); (4/3,0) and (1 ,1/3); etc.), it is normally necessary to use a different magnetic field for each phase. To configure two such filling factors juxtaposed over a micrometer scale (or smaller) requires spatial magnetic field variations which are extremely difficult to produce in practice. The present invention overcomes this restriction by providing a device in which the required filling factors coexist in the same magnetic field. A double-layer quantum Hall system according to the present embodiment allows a change in the voltage on a top gate electrode located above the quantum Hall system to increase the density of one 2DEG and therefore increase its filling factor, while decreasing the density of the other 2DEG and therefore decreasing its filling factor. This feature of the present invention allows tuning two adjacent regions of the device to the required filling factors (e.g. (2,0) and (1 ,1); (4,0) and (3,1); (4/3,0) and (1 ,1/3); etc.) by applying two different voltages to two different top gate electrodes, each located above one of the regions, all of which are within the same magnetic field.

[0035] Fig. 2 A - Fig. 2C illustrate aspects of a double quantum well Hall effect device according to various embodiments of the present invention.

[0036] Fig. 2 A is a schematic side view of a GaAs/AlGaAs heterostructure 200 of a device according to an embodiment of the present invention, with a GaAs upper quantum well 204 and a GaAs lower quantum well 206 separated by an AlAs barrier 205 as seen more clearly in a magnified side view inset 201. Other features of heterostructure 200 include a GaAs cap 202, AlGaAs spacers 203 and 207, an AlGaAs/GaAs superlattice 208, and a GaAs substrate 209. [0037] In various embodiments of the present invention, on top of cap 202 a thin insulating layer 210 is deposited and on top of insulating layer 210 a thin metal conducting layer 211 is deposited to form a metallic gate (described herein below).

[0038] Fig. 2B illustrates an energy- level diagram of the conduction band and the wave functions of the 1^st and 2^nd electronic sub-bands of the double quantum well in the region of the double quantum well of the device of Fig. 2A at zero magnetic field for a particular value of the top gate voltage. A region 214 corresponds to upper quantum well 204 and a region 216 corresponds to lower quantum well 206. A region 215 represents barrier 205. In this embodiment of the invention, the double quantum well is approximately 40 nm thick, with a barrier approximately 3 nm thick. Each sub-band of the double quantum well thus forms a 2DEG. In this non-limiting example, the sub- bands appear to reside in different quantum wells, but in general this is not the case. Thus, it is important to stress that it is the two sub-bands, not the two quantum wells, which constitute the two 2DEGs. According to other embodiments of the present invention, the coupling between the two 2DEGs and thus the coupling between the counter-propagating edge states can be controlled by the dimensions of the barrier between the two quantum wells. The ability to use a semiconducting heterostructure such as a GaAs/AlGaAs heterostucture is a benefit of the present invention, because such heterostructures can be designed in a highly controllable fashion (for example by growing them in a Molecular Beam Epitaxy system) and because device fabrication methods are well-established for such structures, allowing for versatile device architecture, reproducibility and scalability.

[0039] Fig. 2C is an ideal energy fan diagram showing the Landau levels of the two sub-bands of the device of Fig. 2A as a function of magnetic field, showing the level crossings. An E₂ energy 224 corresponds to the energy of the 2ⁿ sub-band at zero magnetic field and an E_\ energy 226 corresponds to the energy of the 1 ^st sub-band at zero magnetic field. If the magnetic field is tuned to values near some Landau level crossing regions, the top gate can induce the required filling factors transitions (e.g. from (2,0) to (1 , 1); from (4,0) to (3 ,1); etc.) in the following way: As the top gate voltage increases, the energy of the upper 2DEG (2^nd sub-band) goes down with respect to the energy of the lower 2DEG (1^st sub-band), i.e. Z¾ - Ei decreases. This change in gate voltage changes the shape of the double quantum well, which decreases the energy difference between the sub-bands (2DEGs), and in turn induces Landau level crossings at a fixed magnetic field, i. e. a Landau level of the upper 2DEG (2^nd sub-band) goes below a Landau level of the lower 2DEG (1 ^st sub -band) due solely to the change in the top gate voltage, with no change in magnetic field.

[0040] According to various embodiments of the present invention, charge is allowed to transfer between the sub-bands (2DEGs). In a non-limiting example, charge can flow from the lower 2DEG (1 ^st sub-band) to the upper 2DEG (2^nd sub-band) due to the gate- induced Landau level crossing. This inter-sub-band charge transfer decreases the density in the lower 2DEG and increases the density of the upper 2DEG, which leads to the required filling factor transition. However, two factors can affect this:

[0041] First, an increase in gate voltage also increases the total density. For this change in total density not to cause a transition away from the desired quantum Hall plateau, sufficient disorder must be present to create enough localized bulk states to widen the quantum Hall plateau.

[0042] Second, inter-sub-band charge transfer introduces charging energy associated with the mutual capacitance between the two sub-bands, which resists the inter- sub -band charge transfer itself. To minimize this effect the heterostructure must be designed so that the two quantum wells and thus the two sub-bands are close enough to increase the mutual capacitance between the sub-bands and thereby reduce the charging energy. The 40nm GaAs double quantum well with a 3nm AlAs barrier in its center, as in the embodiment shown in Fig. 2A, is a non-limiting example of a heterostructure design having a suitably-small charging energy associated with the inter-sub- band charge transfer. A substantially larger double quantum well, for example, might not function properly.

[0043] Fig. 3 A - Fig. 3D illustrate counter-propagating 1 -dimensional edge states in the quantum Hall effect according to various embodiments of the present invention.

[0044] Fig. 3A shows Landau level positions and the corresponding edge states 101C and 101D with filling factors (2,0) for the double-layer layer shown in Fig. 1A - having lower level L_\ (101 A) and upper level La (101B). A Landau level 301 C corresponds to lower-level (L_\) spin-up orientation edge state 101 C, and a Landau level 301D corresponds to lower-level (L_\) spin-down orientation edge state 101D. Also shown is a Landau level 301E corresponding to upper-level (La) spin-up orientation edge state 102D (Fig. IB).

[0045] Fig. 3B shows Landau level positions and the corresponding helical edge states 102C and 102D with filling factors (1 , 1) for the double-layer layer shown in Fig. IB - having lower level L_\ (102A) and upper level L₂ (102B). A Landau level 302C corresponds to lower-level (L_\) spin-up orientation edge state 102C, and a Landau level 302D corresponds to upper-level (L ) spin-up orientation edge state 102D. Also shown is a Landau level 302E corresponding to lower-level (L ) spin-down orientation edge state 101D (Fig. 1A).

[0046] Fig. 3C is a schematic top view of a device 330 according to various embodiments of the present invention, which corresponds to the configuration of Fig. 1C. A left side 331 has filling factor (2,0), and a right side 332 has filling factor (1 ,1). Spin-up orientation edge state 103C, spin-down orientation edge state 103D, and spin- up orientation edge state 103E are the edge states also shown in Fig. 1C and previously discussed herein. According to these embodiments, region 333 is where counter- propagating edge states 103D and 103E are juxtaposed. Within region 333, at the transition between left side 331 and right side 332, there are two helical states - spin- down electrons (or anyons in the case of fractional quantum Hall edge states) travel upward, and spin-up electrons (or anyons in the case of fractional quantum Hall edge states) travel downward.

[0047] Fig. 3D illustrates the Landau levels and the corresponding edge states of the region depicted in Fig. 3C. A Landau level 334C corresponds to lower-level (Li) spin- up orientation edge state 103C, a Landau level 334D corresponds to lower-level (Li) spin-down orientation edge state 103D, and a Landau level 334E corresponds to upper- level (L₂) spin-up orientation edge state 103E. It is noted that at the interface region (corresponding to region 333 in Fig. 3C), Landau levels 334D and 334E cross.

[0048] At the crossings of Landau levels of different 2DEGs, an increase in the top gate voltage causes an increase of the filling factor of one 2DEG and a decrease of the filling factor of the other 2DEG, as noted previously.

[0049] Fig. 4 is a top view of the physical layout of a device 401 according to an embodiment of the invention, corresponding to the schematic representation of Fig. 3C. Device 401 is fabricated on a Hall bar 402 with a double quantum well heterostructure as illustrated in side view in Fig. 2A and described previously. A metallic gate 403 covers the left-half side of Hall bar 402, and a metallic gate 404 covers the right-half side of Hall bar 402. Ohmic connections to device 401 are typified by an ohmic connection 415. An inset view 410 shows a magnified portion of the interface region between the left-half side and the right-half side of device 401. A spin-down edge state 405 with upward-moving electrons counter-propagates with a spin-up edge state 406 with downward-moving electrons.

[0050] Regarding the physical size of a device illustrated in Fig. 4, as a non-limiting example, the total length (vertical direction) is approximately 800 μπι, the total width (horizontal direction) is approximately 200 μπι and the length of the interface between the top gates is approximately 7 μπι to 20 μπι.

[0051 ] Fig. 5 A is a schematic side view of a device as shown in top view in Fig. 4, showing an arrangement of metallic gates 501 A and 501 B according to an embodiment of the present invention. Gates 501A and 501B are deposited on top of insulating layer 210, which is directly upon cap 202 of the heterostructure shown in Fig. 2A. Gate 501A is electrically isolated from gate 501B by a gap 502. In some embodiments of the present invention the gap size can be approximately 80nm wide.

[0052] Fig. 5B is a schematic side view of a device as shown in top view in Fig. 4, showing an arrangement of metallic gates 511A and 511 B according to another embodiment of the present invention. Gate 511A is deposited on top of insulating layer 201, and gate 501B are deposited on top of an additional insulating layer 512, which covers both gate 511A and insulating layer 210. Gate 511A is electrically isolated from gate 511B by insulating layer 512, and therefore can overlap as shown. [0053] In a non- limiting example, insulating layers 210 and 512 are Hf0₂, approximately 10 nm thick. Gates 511A and 511B are gold, approximately 20 nm thick, on titanium, approximately 5 nm thick.

[0054] According to other embodiments of the present invention, different combinations of counter-propagating edge states can be formed by employing different gating of the two 2DEGs.

[0055] While embodiments of the present invention have been disclosed in the non- limiting context of a GaAs double quantum well, other embodiments include multiple parallel 2DEGs and other configurations. Additional non-limiting examples include: a double quantum well of InAs or other semiconducting heterostructures; bi-layer graphene systems; and a wide GaAs quantum well with one or more occupied sub- bands.

Claims

CLAIMS What is claimed is:

1. A device for use in a quantum computer, the device comprising:

a magnetic field source to provide a quantum Hall effect magnetic field;

at least two parallel electronic layers, each layer of the at least two parallel electronic layers being operative to contain a two- dimensional electron gas with a specified filling factor, wherein the device includes at least two adjacent regions with different filling factors, at least one of which has a path for a propagating quantum Hall effect edge state; and

an interface region having two juxtaposed paths respectively

associated with two different two-dimensional electron gases, for propagating quantum Hall effect edge states juxtaposed over a portion of the respective paths,

wherein the two juxtaposed quantum Hall effect edge states

at the interface region are counter-propagating.

2. The device of claim 1 , wherein the filling factors of the two adjacent regions are chosen so that the counter-propagating edge states have opposite spin orientations.

3. The device of claim 1, wherein the interface region has two counter-propagating paths and is a region of an S-wave superconductor.

4. The device of claim 1 , wherein a layer is a sub-band of a quantum well in a direction normal to the 2-dimentional layer plane.

5. The device of claim 4, wherein the at least two layers are two sub-bands of two quantum wells separated by a barrier.

6. The device of claim 4, wherein the at least two layers are two sub-bands of a single quantum well.

7. The device of claim 1 , wherein two layers are in bilayer graphene.

8. The device of claim 2, wherein the two counter-propagating edge states have the same spin.

9. The device of claim 2, wherein the two counter-propagating edge states have opposite spins.

10. The device of claim 1 , wherein a filling factor corresponds to an integer quantum Hall effect state.

11. The device of claim 1, wherein a filling factor corresponds to a fractional quantum Hall effect state.

12. The device of claim 1, wherein at least one filling factor corresponds to an integer quantum Hall effect state and at least one other filling factor corresponds to a fractional quantum Hall effect state.

13. The device of claim 1 , wherein the edge states are electronic states.

14. The device of claim 1 , wherein the edge states are anyonic states.

15. The device of claim 1 , wherein the device forms a topological superconductor that hosts Majorana bound states.

16. The device of claim 1 , wherein the device forms a topological superconductor that hosts parafermions.

17. The device of claim 1, wherein the magnetic field source provides a single magnetic field of substantially the same field strength for all regions and layers.

18. The device of claim 17, wherein the interface region has juxtaposed paths associated with two different two-dimensional electron gases having different filling factors.

ABSTRACT