# Integrating micromagnets and hybrid nanowires for topological quantum
computing

Malcolm J. A. Jardine  Department of Physics and Astronomy, University of
Pittsburgh, Pittsburgh, PA, 15260, USA  John P. T. Stenger  Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA
Yifan Jiang  Department of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA, 15260, USA  Eline J. de Jong  Department of Applied Physics,
Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands  Wenbo
Wang  Department of Physics, University of California, Santa Barbara CA 93106,
USA  Ania C. Bleszynski Jayich  Department of Physics, University of
California, Santa Barbara CA 93106, USA  Sergey M. Frolov  Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA

###### Abstract

Majorana zero modes are expected to arise in semiconductor-superconductor
hybrid systems, with potential topological quantum computing applications. One
limitation of this approach is the need for a relatively high external
magnetic field that should also change direction at the nanoscale. This
proposal considers devices that incorporate micromagnets to address this
challenge. We perform numerical simulations of stray magnetic fields from
different micromagnet configurations, which are then used to solve for
Majorana wavefunctions. Several devices are proposed, starting with the basic
four-magnet design to align magnetic field with the nanowire and scaling up to
nanowire T-junctions. The feasibility of the approach is assessed by
performing magnetic imaging of prototype patterns.

##  I Introduction

One-dimensional topological superconductors host Majorana zero modes (MZMs)
Oreg  _et al._ (2010); Lutchyn  _et al._ (2010). They are promising for fault-
tolerant quantum computing because of the predicted topological protection
that they facilitate Sarma  _et al._ (2015); Aasen  _et al._ (2016)). While
there has been considerable effort applied to identify MZMs in semiconductor
nanowires Mourik  _et al._ (2012); Albrecht  _et al._ (2016); Deng  _et al._
(2016), the evidence to date is not conclusive due to plausible alternative
explanations such as non-topological Andreev Bound States Lee  _et al._
(2014); Yu  _et al._ (2021a). With new developments in materials research and
experimental methods it is reasonable to expect that the present day
challenges can be overcome Frolov  _et al._ (2020). We look beyond to explore
how generating local magnetic fields using micromagnets can aid in the design
of Majorana devices.

There have been several studies on incorporating magnetic materials to induce
MZM in hybrid systems, though the questions asked were different. One class of
ideas has focused on generating synthetic spin-orbit coupling in weak spin-
orbit materials through nanomagnet patterning Klinovaja  _et al._ (2012);
Klinovaja and Loss (2013); Kjaergaard  _et al._ (2012); Desjardins  _et al._
(2019); Mohanta  _et al._ (2019); Rex  _et al._ (2020); Turcotte  _et al._
(2020); Maurer  _et al._ (2018). Another class imagines shells of magnetic
insulators on nanowires as a path to topological superconductivity through
using exchange interactions Liu  _et al._ (2020); Vaitiekėnas  _et al._
(2021). These results have sparked a debate about the feasibility and true
origin of the observed signals Pöyhönen  _et al._ (2020); Langbehn  _et al._
(2020); Khindanov  _et al._ (2020); Yu  _et al._ (2021b).

Figure 1: The Dragonfly setup with four micromagnets (blue/red) and an overlay
of the magnetic field calculated with MuMax3 (gray arrows). The nanowire runs
horizontally with the spin-orbit axis vertical, indicated by $B_{so}$.

In this work we propose how stray magnetic fields can be used to realize basic
two-Majorana building blocks, as well as four-Majorana fusion and six-Majorana
T-junction braiding devices. Locally inducing magnetic fields avoids the
reliance on a global magnetic field, and therefore fields can be oriented
differently in the nanowire set-up opening up possibilities in measuring of
complicated structures like T-junctions braiding devices and the ability to
address and manipulate individual MZMs. In the basic building block
(Dragonfly, Fig. 1), four micromagnets are arranged around the nanowire such
that field lines flow along the wire for 700 nm. We find through modular
micromagnetics and Schrödinger calculations that it should be possible to
enter the topologically superconducting state and achieve partial Majorana
separation. Higher stray fields, e.g. from stronger or thicker magnets, would
make the regime more robust. We perform magnetic force microscopy on prototype
micromagnet patterns and find that it is possible to realize the building
block Dragonfly configurations, though arranging all micromagnets in a
T-junction to the desired orientations will require very accurate control of
switching fields.

Figure 2: Magnetic field profile, as a function of position, of the four
magnet Dragonfly setup. Solid line is field amplitude and dashed line is field
angle $\theta$ relative to negative x-axis (inset). The field is averaged over
a hexagonal cross-section of the nanowire with a characteristic dimension of
100 nm. In this Figure only, the reader is invited to imagine an infinitely
long nanowire without ends and consider what field profile would be created
along such a nanowire. The horizontal dashed gray line indicates a uniform
field for entering the topological regime in an infinite nanowire.

##  II Numerical Results

##### Brief Methods

The micromagnetic simulation software MuMax3 Vansteenkiste  _et al._ (2014) is
used to simulate realistic single domain-sized micromagnets, including
hysteretic magnetization and stray fields. Hysteresis simulations are
performed to obtain the required magnetization state. The parameters for
cobalt were used, details can be found in section V.4. The stray magnetic
fields are integrated over an imagined hexagonal nanowire cross-section to
obtain a one-dimensional field profile which can in principle extend to
infinity. The field profile is used as input into a basic one-dimensional
Majorana nanowire model Stenger  _et al._ (2018) to obtain the energy spectrum
and calculate Majorana wavefunctions $\gamma_{1}$ and $\gamma_{2}$. For this
step a finite nanowire length is chosen over which the field profile can be
utilized for a given set-up.

##### Dragonfly setup

The setup presented in Fig. 1 is our basic configuration. The four
micromagnets, that resemble a dragonfly, induce field lines along the
nanowire, coming in at the right end, and flowing out of the left end. The
field does not deviate by more than $30^{\circ}$ over a 700 nm long nanowire
segment, see Fig. 2, and maintains a relatively uniform amplitude.
Furthermore, for a given hexagonal cross-section the standard deviation is of
order 5% which justifies the procedure of producing a one-dimensional field
profile by averaging over the cross-section, at least for the central segment
of the nanowire length, see Fig. S10. The nearly parallel field is required
for MZMs formation, since such field is mostly orthogonal to the effective
spin-orbit field, $B_{so}$ Lutchyn  _et al._ (2010); Oreg  _et al._ (2010);
Osca  _et al._ (2014); Rex and Sudbø (2014), hence why we use angle relative
to the nanowire when presenting the stray field profile. Four, rather than
two, magnets are required to cancel y-fields and enhance x-fields, with no
external field required. Micromagnets placed parallel to the nanowire, e.g. as
a shell around the nanowire, produce largely y-fields that are highly non-
uniform, concentrated outside the poles of the magnets (see Fig. S8 Liu  _et
al._ (2020)).

Figure 3: (a) Probability distributions for two Majorana wavefunctions
$\gamma_{1}$ and $\gamma_{2}$. The first excited state (dashed line) is a bulk
nanowire state. (b) Magnetic field profile reproduced from Fig. 2 over a
smaller range. Hard boundaries are introduced at $\pm 350$ nm to calculate the
wavefunctions. There is no applied external field.

The micromagnet widths are different to ensure different switching fields and
to aid in the preparation of a desired mutual magnetization pattern Jiang  _et
al._ (2020). The magnets are rotated at 45 degrees to aid magnetization
through the hysteresis simulation. Magnets on opposite sides of the nanowire
are perpendicular, and therefore the external field used for magnetization is
also perpendicular for the pairs of magnets. The out of plane direction
(z-component) of the simulated field is insignificant due to the symmetry of
the micromagnet arrangement (see supplementary information S6).

Given the simulated magnetic field profile along the 700 nm long nanowire,
there is clear Majorana polarization of the two MZMs, $\gamma_{1}$ and
$\gamma_{2}$, shown in Fig. 3. Though we assume a relatively low magnetic
field for the topological transition, 50 mT. The two Majorana wavefunctions
are highly overlapping. The lowest (partially separated Majorana) and the
first excited energy levels have energies $E_{0}/\Delta=0.28$ and
$E_{1}/\Delta=0.82$, where $\Delta$ is the superconducting gap. We note that a
small global external field can be applied to boost Zeeman energy.

##### Double Dragonfly

The next device (inset Fig. 4) combines two Dragonfly set-ups and an
additional vertical magnet (dashed border), elongating the topological
nanowire region. The concept can be repeated multiple times to further extend
Majorana separation or create multiple pairs of MZM, for instance in Majorana
fusion experiments Aasen  _et al._ (2016). The right Dragonfly unit is
reflected: this means magnetic fields point opposite in the left and right
nanowire segments. Naively, MZM generated by opposite magnetic fields cannot
couple as they possess opposite spin Jiang  _et al._ (2013). Nevertheless,
left and right MZM form a pair due to field rotation provided by the central
magnet. Using the same topological criterion we now find a more substantial
MZM polarization of the ground state $E_{0}$ (red/blue) with more noticeable
tailing off of the wavefunctions. This is because the total length of the
device is increased and two MZMs in the middle (dashed line in Fig. 4a)
hybridize and reduce their overlap with end MZMs. Application of an external
field (10’s of mT) in the positive y-direction in this case aligns the total
field closer with the nanowire. Supplementary information presents other
versions of the double Dragonfly.

Figure 4: (a) Probability distributions of two lowest energy states for double
Dragonfly setup with gate open (shown in inset). MZM (red/blue) with
$E_{0}/\Delta=5.1\times 10^{-4}$ . The first excited state (grey dashed) with
$E_{1}/\Delta=1.6\times 10^{-1}$. External field of 40mT is applied in
positive y-direction. Inset: two dragonfly configurations, with zigzag
indicating an electrostatic gate capable of dividing the nanowire in two
parts. (b) Magnetic field profile along the wire, showing amplitude (solid)
and angle (dashed).

##### T-junction braiding device

The T-junction set-up, shown in Fig. 5, explores how micromagnets could be
used to realize a MZM braiding setup. The horizontal left and right arm
sections are made by chaining together Dragonflies, and a perpendicular leg
section in the middle has a micromagnetic configuration similar to a single
Dragonfly. Adding a second dragonfly to the leg does not substantially improve
MZM separation due to an external field in the y-direction (see supplementary
information). This allows the field to be parallel to the nanowire in
perpendicular sections of the wire, something that is not possible in a
uniform external field.

The simulated stray field (supplementary Fig. S7a), with an additional
y-direction external magnetic field of 40 mT, allowed MZMs to separate or
couple across any two sections of the T-junction nanowire, seen in Fig. 6,
where each arm of the junction has an electrostatic gate, $G_{1}$, $G_{2}$ and
$G_{3}$, to control which sections are connected and disconnected.

With all three gates pinching-off the nanowire we see that the three lowest
energy ground states are pairs of MZMs (red, green and blue) confined to
separate wire segments (Fig. 6a). With gate $G_{3}$ cutting off the
perpendicular leg section, Fig. 6b, the lowest energy state (red) is localized
at the ends of the top wire, demonstrating that the micromagnets allow this to
act as a single, long topological superconductor. The second lowest energy
state of the entire system is MZMs isolated in the perpendicular nanowire
(blue), and the third lowest state (green) has weight mostly at the junction,
representing two nearby MZM. We note that the fourth energy state is larger in
energy with more weight in the bulk of the nanowire, suggesting it is not a
partially hybridized MZM pair.

By pinching off gates $G_{1}$ or $G_{2}$, Figs. 6c and 6d, we cut off either
the left or the right section of nanowire. MZM states couple and decouple
across the T-junction, In panel (d) the blue wavefunction extends over entire
left and leg segments of the T-junction, making the red wavefunction appear
purple. These results suggest that a repeated Dragonfly setup could be used to
realize a braiding setup, in principle, although with relatively complex
micromagnet configurations.

Figure 5: T-Junction setup. The top two sections of nanowire are 5000 nm in
length in total and the perpendicular section is 1100 nm. Zigzag lines
indicate electrostatic gates. Circles indicate desired positions of 6 MZM with
all gates on.

(a)

(b)

(c)

(d)

Figure 6:  Probability distributions (color) of the ground, second and third
lowest energy with (a) all gates (b) gate $G_{3}$, (b) gate $G_{1}$, (c) gate
$G_{2}$ activated. Colors chosen so that red wavefunction always has a weight
on the left end. The energies for the pairs of Majorana states are given in
the table below.

| Energies of
---
setup $/\Delta$
(a) | (b) | (c) | (d)
$E_{0}$ |  | $1.6\times 10^{-3}$
---
| $1.6\times 10^{-5}$
---
| $3.5\times 10^{-4}$
---
| $7.2\times 10^{-4}$
---
$E_{1}$ |  | $2.3\times 10^{-3}$
---
| $9.4\times 10^{-3}$
---
| $1.6\times 10^{-3}$
---
| $2.3\times 10^{-3}$
---
$E_{2}$ |  | $9.4\times 10^{-3}$
---
| $4.2\times 10^{-2}$
---
| $4.3\times 10^{-3}$
---
| $3.1\times 10^{-3}$
---
$E_{3}$ (not shown) |  | $1.5\times 10^{-1}$
---
| $1.1\times 10^{-1}$
---
| $1.3\times 10^{-1}$
---
| $1.2\times 10^{-1}$
---

##  III Magnetic Force Microscopy

We take the first step to evaluate these device concepts experimentally, by
studying magnetization patterns of micromagnets. Ferromagnetic micro-strips
are fabricated in a design that represents a simplified version of the
braiding T-junction setup (Fig. 7). The design features three Dragonflies
arranged in a T with the same dimensions as Fig. 1 but a shorter distance
between magnets, and an extra vertical magnet. The strips are written by
electron beam lithography (EBL) and the metal is deposited by electron beam
evaporation from a CoFeB source (atomic ratio 30/55/15 before deposition) to a
thickness of 20 nm. This is thinner than typical nanowires, and was done to
reduce magnetic signal and permit higher resolution imaging. It is believed
that the final strips are mostly CoFe without boron Bureau-Oxton (2014).

Magnetic force microscopy (MFM) and atomic force microscopy (AFM) are
performed on triple Dragonflies (Fig. 7). An attempt is made to take advantage
of hysteretic magnetization and prepare magnets preferentially in the desired
Dragonfly configuration so that stray magnetic fields are along the imaginary
nanowire in between the micromagnets. The range of field where wide and narrow
micromagnets are antiparallel is determined from separate SQUID measurements
to be between 15 and 20 mT, in agreement with Ref. Jiang  _et al._ (2020),
when reduced CoFe thickness is accounted for.

Twenty-four T-junctions with three Dragonflies each are imaged, and six of 72
total Dragonflies are magnetized as required for MZM generation. Fig. 7(b)
shows one such section with the magnetization marked by arrows (supplementary
Fig. S11 shows all data). The occurrence of four micromagnets in the right
orientation is consistent with random magnetization. While in our MuMax3
simulations it is possible to run through the hysteretic magnetization cycle
and prepare micromagnets in the desired configuration, experimental variations
in switching fields highlight a challenge. Optimized micromagnet fabrication
will yield sharper switching and more reproducible switching fields in the
future.

Figure 7: (a) Atomic force microscopy (AFM) of a T-junction setup with three
Dragonfly magnet configurations. Magnetic film thickness is 20 nm. (b)
Magnetic force microscopy (MFM) data on a different T-Junction of the same
design. Arrows indicate magnetization direction, white dashed lines are
example of magnet dimensions. Blue dashed line is where the Majorana nanowire
is envisioned. The magnetic history was a magnetizing field from 100 mT to -16
mT to 0 mT applied at 45 degree direction, then 100 mT to -16 mT to 0 mT at
135 degrees.

##  IV Conclusions, limitations

We consider device concepts in which micromagnets generate stray field
patterns suitable for the generation of Majorana zero modes. Our approach
assumes micromagnets placed next to semiconductor nanowires that possess
strong spin-orbit coupling, and are coated with superconducting shells. The
requirements on the stray magnetic fields are that they are of sufficient
strength to drive a topological transition, and should be oriented as much as
possible along the nanowire. The building block of our magnetic design is a
Dragonfly configuration in which four micromagnets are magnetized such that
the magnetic field lines flow out of one pair of micromagnets, along the
nanowire, and into the other pair (Fig. 1). By repeating the Dragonfly pattern
along the nanowire, we can extend the length of the topological segment with
addition of coupling magnets. The approach can also be applied to T-junctions
required for Majorana braiding experiments, in which case magnetic field turns
into the T-junction leg that is perpendicular to the junction top.

Among the limitations is the still limited strength of stray fields from
micromagnets. In previous experiments the typical stray fields are in the
range of tens of milliTesla Jiang  _et al._ (2020). This is in principle
sufficient to enter the topological regime in large g-factor semiconductor
such as InSb nanowires, but it limits the parameter space for Majorana
separation and manipulation. Stronger magnetic materials, or thicker
micromagnets can help. Additionally, the Meissner effect from the
superconducting layer on top of the nanowires was not taken into account,
however the effect on the magnetic field is assumed to be small as magnetic
fields penetrate through such thin superconducting films. Disorder in real
nanowire devices will put limitations on the achievable MZM separation
distance.

A challenge that became apparent from magnetic imaging of prototype structures
is how to prepare all micromagnets in the appropriate relative magnetic
orientation. This becomes harder when the configurations increase in
complexity such as for T-junctions. Though basic two-Majorana experiments
should be possible already now, in the future better control over coercive
fields, further pattern optimization, and direct magnetic writing can be
deployed.

Among future ideas, a promising path is using a Y-junction instead of a
T-junction Harper  _et al._ (2019), an approach that may require fewer
micromagnets as some can be shared between the branches. Rather than gate-
controlled MZM coupling, magnetic field mediated coupling can be investigated,
by flipping micromagnets at the junctions. The implementation of the Poisson-
Schrödinger equation to model MZM in the 3D geometry of a single nanowire
could be integrated with 3D stray field profiles rather than simplified one-
dimensional profiles integrated over the nanowire cross-section, which will
provide information on the effect of the non-uniformity of the field combined
with the effect of electrostatic confinement and disorder.

To summarize, nanoscale control over the field magnitude and direction is
advantageous, particularly for advanced geometries where some nanowire
segments may run perpendicular to others. Micromagnets could offer novel ways
to control Majorana-based devices, for instance by switching magnetization in
order to reposition or couple MZMs.

Acknowledgements and funding. We thank S. Meynell and D. Yang for helpful
discussions. Work in Pittsburgh (numerics and nanofabrication) supported by
the Department of Energy DE-SC-0019274. A.B.J. acknowledges the support of the
NSF Quantum Foundry through Q-AMASE-i program award DMR-1906325.

Code and data availability. Example MuMax3 scripts, numerical results data,
Majorana simulation Mathematica notebooks are available on Zenodo Jardine  _et
al._ (2021), or GitHub via https://github.com/frolovgroup.

Author Contributions. M.J. and E.J. performed MuMax3 simulations. M.J., E.J.
and J.S. performed Majorana simulations. Y.J. fabricated micromagnet devices.
W.W. and A.B.J. performed MFM/AFM imaging. M.J. and S.F. wrote the manuscript
with input from co-authors.

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##  V Supplementary Information

###  V.1 Background on Majorana nanowire model

MZMs are non-Abelian in nature and are their own antiparticle. They appear as
topological, zero-energy states localized at the ends of a 1D system. They
were first theoretically investigated in condensed matter systems by Kitaev in
one-dimensional, spinless (p-wave), topological superconductors Kitaev (2001).
However, p-wave superconductors have yet to be realized experimentally, and,
therefore, models that are experimentally accessible have been utilized in
recent times. The prospect for MZMs that this work is utilizes is using
semiconductor-nanowire, superconductor hybrid systems under external magnetic
fields Oreg  _et al._ (2010); Lutchyn  _et al._ (2010). These hybrid systems
utilize the nanowire’s inherently strong spin-orbit coupling; proximity-
induced s-wave superconductivity via contact with conventional
superconductors; and the magnetic field to act as effective p-wave
superconductors.

The Kitaev chain model is described by this Hamiltonian Kitaev (2001),

$\hat{H}_{p}=-\mu\sum_{i}\left[c_{i}^{\dagger}c_{i}\right]-\frac{1}{2}\sum_{i}\left[tc_{i}^{\dagger}c_{i+1}+\Delta
e^{i\phi}c_{i}c_{i+1}+H.c.\right].$ (1)

where $c_{i}$ destroys a spinless fermion at site i, $c_{i}^{\dagger}$ creates
a spinless fermion, $\mu$ is the chemical potential, $t>0$ is the site hopping
energy and $\Delta e^{i\phi}$ is the p-wave superconducting pairing term. When
this model is written in terms of Majorana fermion operators $\gamma_{1}$ and
$\gamma_{2}$ Kitaev showed that this describes a system with Majorana fermions
that have on site and nearest neighbour interactions. The key insight was that
in the topological regime, defined by $\left|\mu\right|<t$, unpaired Majorana
fermions are left at the ends of the 1D chain which could be potentially
detected and manipulated.

To be able to effectively describe p-wave superconductors in a real system the
semiconductor nanowire model is used, with Hamiltonian:

$\begin{array}[]{ll}\hat{H}_{NW}=&\left(\frac{p^{2}}{2m^{*}}-\mu+\alpha_{R}\left(\boldsymbol{\sigma}\times\mathbf{p}\right)\cdot\mathbf{\hat{z}}\right)\tau_{z}\\\
&+V_{z}\left(\cos\theta\sigma_{x}+\sin\theta\sigma_{y}\right)+\Delta_{s}\tau_{x}\sigma_{y}\end{array}$
(2)

which acts on the Nambu spinor basis $\hat{\psi}_{x}=\left(c_{\uparrow
x},c_{\downarrow x},c_{\uparrow x}^{\dagger},-c_{\downarrow
x}^{\dagger}\right)$. The momentum $\mathbf{p}$ points in the direction of a
section of nanowire, $\Delta_{s}$ is the s-wave superconducting pairing term,
$V_{z}=\frac{1}{2}g_{eff}\mu_{B}B$ is the Zeeman energy, $\theta$ is the angle
of the field relative to the positive x-axis, $\tau$ and $\sigma$ are Pauli
matrices representing the particle-hole and spin spaces respectively, and
$\alpha_{R}$ is the Rashba spin orbit coupling term. The SOC term has
$\cdot\mathbf{\hat{z}}$ as this is the direction of the internal electric
field which induces the Rashba SOC, this direction arises because of the
crystal asymmetry from the nanowire growing on the substrate, and the nanowire
coating can affect this as well Manchon  _et al._ (2015). These parameters are
taken constant because the nanowire is considered to be made of a homogeneous
material. To be in the topological phase the following condition must be met

$V_{z}>\sqrt{\mu^{2}+\Delta_{s}^{2}}$ (3)

When this system is deep in the topological phase it was shown that this
reduces to a p-wave topological system described by Eq.(1). To simulate the
model in a finite one-dimensional system the continuum model is discretized on
a 1D lattice in the electronic state basis, giving Eq. (4).

###  V.2 Further Reading

For basic introductory review of Majorana nanowire topic consider reading
Leijnse and Flensberg (2012). A perspective giving an explanation on Rashba
spin-orbit coupling one can read Manchon  _et al._ (2015). Other relevant work
on nanowires that deal with areas such as surface effects, effects of the
contacts, electric field considerations, and try to comprehensively implement
the model Winkler  _et al._ (2019); Stenger  _et al._ (2018). For work that
utilizes the Poisson Schrödinger equation to model MZMs in the 3D geometry of
a single nanowire Stenger  _et al._ (2018); Winkler  _et al._ (2019); Woods
_et al._ (2018), where the use of 3D field profiles could give a batter
picture of systems in future studies. Further work on implementing
micromagnets for use in Majorana based devices Jiang  _et al._ (2020). There
are several different proposed braiding schemes using nanowire-network
junction devices Stenger  _et al._ (2018); Alicea  _et al._ (2011), with
options such as using local chemical potential changes Harper  _et al._ (2019)
or flux gates Stenger  _et al._ (2019).

(a)

(b)

Figure S1: This compares the a rotated and un-rotated Dragonfly configuration.
The solid line is the rotated set-up and the dashed line is the straight set-
up. Rotating the magnets makes them easier to magnetize through hysteresis.
This figure is for a 1000 nm nanowire segment.

###  V.3 Majorana Model Used

The implemented one-dimensional semiconductor-superconductor hybrid model is
given by a discretized Hamiltonian,

$\begin{array}[]{ll}\hat{H}=&\sum_{n}\left[\left(-\mu_{n}+\varepsilon_{0}\right)\left|n\right\rangle\left\langle
n\right|\tau_{z}\otimes\sigma_{0}\right.\\\ \\\
&+t\left(a\right)\left(\left|n+1\right\rangle\left\langle
n\right|+H.c.\right)\tau_{z}\otimes\sigma_{0}\\\ \\\
&+i\alpha\left(a\right)\left(\left|n+1\right\rangle\left\langle
n\right|-H.c.\right)\tau_{z}\otimes\left(\boldsymbol{\sigma}\times\mathbf{x}\right)\cdot\mathbf{\hat{z}}\\\
\\\ &+\tau_{z}\otimes
V_{z,n}\left(\cos\theta_{n}\sigma_{x}+\sin\theta_{n}\sigma_{y}\right)\left|n\right\rangle\left\langle
n\right|\\\ \\\
&\left.-\Delta_{s}\tau_{x}\otimes\sigma_{x}\left|n\right\rangle\left\langle
n\right|\right],\end{array}$ (4)

where

$\begin{array}[]{ccc}t\left(a\right)=\frac{\hbar^{2}}{2m^{*}a^{2}}&&\alpha\left(a\right)=\frac{\alpha_{R}}{2a}\end{array}.$
(5)

The meaning and values used for the parameters are as follows: $n$ labels the
lattice site, $a$ is the lattice constant and
$\varepsilon_{0}=\cos\left(ka\right)$ with $k$ the crystal momentum and this
is an offset to bring the Fermi energy to the bottom of the band. To deal with
nanowires in different directions $\mathbf{x}$ is a unit vector pointing along
a particular section of nanowire. The s-wave superconducting pairing term is
$\Delta_{s}=0.08$ meV, the Rashba spin orbit coupling term $\alpha_{R}=0.2$
eVÅ, effective electron mass $m^{*}=0.04m_{e}$, and $g_{eff}=50$ such as in
InSb wires. Lastly, the Zeeman energy is
$V_{z,n}=\frac{1}{2}g_{eff}\mu_{B}B_{n}$, with the magnetic field, $B_{n}$,
and relative angle of the field to the x-axis, $\theta_{n}$, being both site
dependent and in general have a complicated structure due to the use of the
micromagnets, a notable difference to many other works. The simulated magnetic
field from MuMax3 is inputted into the 1D nanowire-superconductor model via
$V_{z,n}$, and the spectrum and eigenstates calculated via exact
diagonalization. The effects of the non-uniform structure of the Zeeman energy
were utilized and investigated by constructing longer devices with many
micromagnets, and a T-junction set-up where perpendicular wires had different
field directions.

Interpreting Majorana results. To investigate viability of the proposed
devices, the MZMs signatures of state polarization and zero-energy pinning are
required. A single electronic Majorana state, with energy $E_{0}$, can be
written in terms of the two Majorana wavefunctions via
$\gamma_{1}=c+c^{\dagger}$ and $\gamma_{2}=-i\left(c-c^{\dagger}\right)$ and,
due to the topological nature of MZMs, polarization of the two MZM will be
apparent, seen as the localisation of two Majoranas at opposite ends of a
single topological region of nanowire. In short length systems the MZMs can be
overlapping, and also could have a complicated structure due to the variation
of the magnetic field or nanowire length. The lowest eigenstates of the system
are single quantum states that are pairs of MZMs. The first energy/excited
state, denoted $E_{1}$ if there is a single pair of MZMs present, should have
weight predominately in the bulk, with little overlap with the MZMs. The
second signature is the near-zero energy of the MZM state, with good energy
separation from the first energy state.

###  V.4 Micromagnetic Simulation Details

##### MuMax3

Vansteenkiste  _et al._ (2014) is a GPU-accelerated micromagnetic simulation
program that uses finite difference discretization methods, accurate sized
magnetic domains within a single magnet for micro- to nano- scale system
simulations, and was implemented to obtain the magnetic field used in the
hybrid nanowire model. The micromagnetic structure is first constructed for
each set-up in MuMax3, the requisite magnetization states achieved through
hysteresis, if possible, and then the stray field is averaged over the
hexagonal nanowire cross-section (such as for InSb wires) with diagonal width
of 100 nm, typical of experiments. Upon averaging, an effective one-
dimensional magnetic field profile is obtained. Note that the magnetic field
can be in principle calculated for an arbitrary coordinate, meaning that the
MuMax calculation does not need to encode a fixed nanowire length. This one-
dimensional field profile is then used in the Majorana model to calculate
several lowest energy energies and wavefunctions. At this step a finite
nanowire length is chosen. The code used for the calculations can be found on
Zenodo Jardine  _et al._ (2021).

##### Hysteresis.

To obtain the required magnetization state through hysteresis the micromagnets
start with many randomly orientated domains and then an external magnetizing
field is applied and the system was relaxed at regular time intervals while
ramping up and down the field. It was found that angling the micromagnets at
45 degrees made obtaining the desired magnetization directions easier, as
magnets on the opposite side of the nanowires were then perpendicular to each
other. This allows the micromagnets to be closer together, however angling the
magnets reduces the magnetic field, see Fig. S1, so these effects have to be
balanced to obtain the strongest magnetic field possible. The hysteresis
process takes into account the material of magnets, which include saturation
magnetization, anisotropy constants and exchange stiffness. One limitation
present is the material of the nanowire, leads and other components were not
taken into account in the simulation, but the main magnetic effects are
presumed to come from the magnets themselves.

(a)

(b)

(c)

Figure S2: Comparing the double Dragonfly with different external fields. (a)
40 mT in the positive y direction (b) no field and (c) 40 mT in the positive x
direction.

##### The magnetic material parameters

of Cobalt are used due to several factors, first is that the constructed
micromagnet devices are thought to be mainly composed of CoFe. Additionally,
Cobalt is a common material used in the fabrication of micromagnets, and it
allows for a relatively high saturation magnetization, and thus a stronger
stray field, while also allowing for a reasonable coercive field due to
Cobalt’s exchange stiffness and cubic anisotropy constant. This allows for an
appropriate demagnetization/stray field in the nanowire region, while also
reliably obtaining the magnetization direction required for the setup, and
additionally allowing for a small external field to be applied without
accidentally flipping the magnets. The parameters used are a saturation
magnetisation of $1.44\times 10^{6}$ A/m, exchange stiffness $3.0\times
10^{-11}$ J/m, anisotropy constant $K_{1}$ $4.5\times 10^{5}$, anisotropy
constant $K_{2}$ $1.5\times 10^{5}$ and a Landau-Lifshitz damping constant of
0.01.

Micromagnets of two different widths are reliable and sufficient enough for
hysteresis and field production purposes, the dimensions of all magnets in
this work are either that of the thin magnet, with dimensions $130\times
1000\times 100$ nm, or the thick magnet with dimensions $250\times 1000\times
100$ nm, as the different sized magnets flip at different coercive fields. The
nanowire used in the MuMax simulations has a hexagonal cross-section with a
long diagonal length of 100 nm.

###  V.5 Supplementary Results

####  V.5.1 Dragonfly

For the Dragonfly setup, the positions and dimensions of the set-up are as
follows. The micromagnets used were two different widths, with thin magnet
dimensions $130\times 1000\times 100$ nm or thick magnet with dimensions
$250\times 1000\times 100$ nm, so the different sized magnets flip at
different coercive fields. The magnets are rotated at 45 degrees to aid the
reliability of magnetization directions. The nanowire has a hexagonal cross-
section, with a long diagonal length of 100 nm, and length of 700 nm so it
spanned the region of parallel field from the magnets. The centers of the
micromagnets are displaced from the middle of the nanowire to the left and
right by 250 nm, with magnets of the different widths opposite each other. The
y-displacement of the micromagnets from the nanowire is chosen such that the
nearest corner of each magnet is 40 nm from the nanowire. These positions are
one workable configuration, and changing the positions, dimensions or magnet
material parameters will change the magnetic field profile, but this is the
optimal set-up found.

Due to the magnetic field being calculated through simulated hysteresis the
magnets are not perfect single domains, in particular at the edges of the
micromagnets the boundary causes non-uniformities. However, the net effect of
this is small, but it explains why some of the state’s energies and positions
are not perfectly symmetric. Lastly, we note the difficulty to obtain a field
larger than $\sim$ 150 mT in the current set-up, additionally the
superconducting gap is not necessarily fixed experimentally, which will change
the field requirements.

Figure S3:  (a) Probability distributions for two lowest energy states, this
system shows two overlapping and degenerate states with no clear Majorana
polarization and no higher energy state being gapped out. The energies of both
states $E_{1}/\Delta=3.9\times 10^{-2}$ and $E_{2}/\Delta=4.6\times 10^{-2}$.
(b) Field profile along wire. External field is 40mT in positive y-direction

####  V.5.2 Double Dragonfly without extra magnet

This double Dragonfly device (inset Fig. S3(a)), is different from the one
shown in the main text as it only combines two individual Dragonfly set-ups
(without the middle 9th magnet). The magnetic field has a large region of low
magnitude field in the middle section, see bottom panel of Fig. S3(b), and
there is an external field of 40mT applied in positive y-direction. The
eigenstate profiles and energy spectrum suggest there are two pairs of
uncoupled MZM on each side of the wire.

####  V.5.3 Double Dragonfly with extra Magnet

Figure S4: The double Dragonfly setup with the magnet in the middle and a
potential gate, dimensions given. Toggling this gate allows the left and right
side of the wire to be coupled or uncoupled.

Full dimensions of the double Dragonfly with the extra magnet are given in
Fig.S4. A gate is turned on in the middle region of the nanowire (red zig-zag
line), uncoupling the ends of the nanowire and causing two pairs of MZMs to
appear. This is implemented as a large potential barrier spanning around 10
sites, and would be implemented experimentally by pinching off the nanowire.
These two pairs of MZMs live in the parallel region of field, similar to the
single Dragonfly set-up, with energies of $E_{0}/\Delta=3.1\times 10^{-2}$ and
$E_{1}/\Delta=3.2\times 10^{-2}$, and the first excited state at a larger
energy of $E_{2}/\Delta=8.9\times 10^{-1}$.

The states in the double Dragonfly device under different external magnetic
fields are shown in S2. Panel (a) is the same figure in the main text with an
external field in the y-direction which aids coupling. Compare to panel (b)
without the external field where the MZM states are less clear with less
polarization and larger energies. Panel (c) shows how applying a Bx field with
the double Dragonfly set-up gives preference to one side over the other. This
is what would happen if a double Dragonfly was used in the leg section of the
T-junction.

Figure S5: (a) Probability distributions for two lowest energy states for
double Dragonfly setup with the 9th magnet and the gate closed (inset). The
probability distributions shows two separate pairs of MZMs in each section of
the wire, the left is shown in the Majorana basis with energy
$E_{1}/\Delta=3.1\times 10^{-2}$ (red and cyan), the other pair in the right
section is in the electronic state basis with energy $E_{2}/\Delta=3.1\times
10^{-2}$ (black- dashed line). The third state is well separated at
$E_{3}/\Delta=8.9\times 10^{-1}$. (b) Field profile along wire. External field
is 40mT in positive y-direction Figure S6: $B_{z}$ field component along the
wire for the double Dragonfly with the middle ninth magnet. The $B_{z}$ data
for other set-ups available on Zenodo Jardine  _et al._ (2021).

####  V.5.4 T-junction magnetic fields

(a)

(b)

Figure S7: (a) Magnetic field of top wire for T-Junction. (b) Magnetic field
in the T-Junction leg (vertical segment of nanowire). There is an external
field of 40mT applied in positive y-direction.

The magnetic field profile input into the nanowire model was made by taking
the simulated central three single Dragonfly set-ups (with the middle magnet)
from MuMax3, and then the top field profile was elongated by attaching two
flipped copies of the top field profile and and connecting these on the ends.
This had to done as it wasn’t possible to accurately simulate all the magnets
in the final T-junction set-up, however due to the modular nature of the
Dragonfly set-ups this is reasonable. The magnetic field in the top nanowire
rotates smoothly across the length of the nanowire (see Fig. S7a), and the
whole nanowire is one, mostly continuous topological region. The magnetic
field in the perpendicular section of nanowire is similar to that of the
single Dragonfly set-up, with a parallel region of around 700 nm, note this is
very uniformly pointing up (positive y) due to magnets on opposite sides of
the nanowire having the same widths, see Fig. S7b. The applied external
magnetic field, of 40 mT, was added in the positive y-direction to facilitate
maximum coupling.

####  V.5.5 Field uniformity

To show the uniformity of the field near and inside the nanowire the following
is considered, with all data on Zenodo. The single micromagnet in Fig. S8
shows the field is only highly non-uniform very near the micromagnet poles,
and drops off rapidly away from the micromagnet. Note that the field is very
weak at the magnet sides, hence why a multiple magnet design is required to
obtain strong enough stray fields. The most inhomogeneous region is therefore
where the magnets come closest to the nanowire Fig. S9 which shows the field
magnitude in the area between two micromagnets in a 5 nm thick layer. In the
region of the nanowire (marked by red lines) the field variation is
significant near the tips of the magnets, but the field is much more uniform
away from the tips on the left. Fig. S10 plots the standard deviation of the
field magnitude over the nanowire cross-section, note this is for a finer
resolution than Fig 3. The standard deviation is relatively low compared to
the mean for the nanowire segment in between the four magnets, away from the
magnet tips. Over the majority of the nanowire the standard deviation is less
than 10% of the mean. Slices inside the hexagonal nanowire region show the
same. Full 3D data are available online.

Figure S8: A heat map for magnetic field magnitude for a micromagnet of
dimensions 230X1000X100 nm. This is shown for a 2D X-Y plane slice through the
middle of the micromagnet (50nm), the magnet is shown as grey. Note the field
is very weak away from the micromagnet ends.  Figure S9: A X-Y plane heat map
for magnetic field magnitude in a small region near the nanowire, the magnet
areas have been set to 0 field (black triangular regions). This is for a 5 nm
thick slice in the middle where the nanowire is (marked by red lines). A
Mathematica script is available online to look at all the different X-Y slices
at different z-positions.  Figure S10: Shows the magnetic field magnitude’s
standard deviation (dashed) across the hexagonal cross-section and mean value
over the nanowire region for the single Dragonfly set-up. The insets are field
magnitude cross sections of the nanowire at different positions. Note the mean
value is the same data that is plotted in Fig. 3 (b). A Mathematica script is
available on Zenodo to look at all Y-Z slices along the nanowire.

Figure S11:  Magnetic force microscopy (MFM) data on a 24 T-junction set-ups
using three Dragonfly’s. The magnet dimensions are the same as in Fig. 1 but
the distance between them differs. Top left: Atomic force microscopy (AFM) of
one T-junction setup with three dragonfly magnet configurations. Magnetic film
height 20 nm. Dashed ovals point out magnetizations favorable for generating
MZM. The magnetic history was a magnetizing field from 100 mT to -16 mT to 0
mT applied at 45 degree direction, then 100 mT to -16 mT to 0 mT at 135
degrees.

Figure S12:  Study design, Volume and Duration of study. The bulk of the this
work was carried out over nine months to find an appropriate micromagnet and
nanowire set-up that demonstrated MZMs in the hybrid model. Approximately 5
different single Dragonfly configurations were tested and simulated, which was
then used to construct the double Dragonfly and the T-junction. There were
around 10 different designs tested for the double Dragonfly and T-junction.
The experimental data was collected on a single chip where 24 devices had AFM
and MFM measurements taken after an hysteresis cycle was applied to magnetize
the configurations.