# High Temperature Ferromagnetism in Cr1+xPt5-xP

Tyler J. Slade,1,2∗, Nao Furukawa,1,2 Tanner R. Smith,1,2 Juan Schmidt,1,2
Ranuri S. Dissanayaka Mudiyanselage,3 Lin-Lin Wang,1 Weiwei Xie,3,4 Sergey L.
Bud’ko,1,2
Paul C. Canfield1,2∗

###### Abstract

We present the growth and basic magnetic and transport properties of
Cr1+xPt5-xP. We show that single crystals can readily be grown from a high-
temperature solution created by adding dilute quantities of Cr to Pt-P based
melts. Like other 1-5-1 compounds, Cr1+xPt5-xP adopts a tetragonal P4/mmm
structure composed face-sharing CrPt3 like slabs that are broken up along the
c-axis by sheets of P atoms. EDS and X-ray diffraction measurements both
suggest Cr1+xPt5-xP has mixed occupancy between Cr and Pt atoms, similar to
what is found in the closely related compound CrPt3, giving real compositions
of Cr1.5Pt4.5P (x = 0.5). We report that Cr1.5Pt4.5P orders ferromagnetically
at TC = 464.5 K with a saturated moment of $\approx$ 2.1 $\mu_{\textit{B}}$/Cr
at 1.8 K. Likely owing to the strong spin-orbit coupling associated with the
large quantity of high Z, Pt atoms, Cr1.5Pt4.5P has exceptionally strong
planar anisotropy with estimated anisotropy fields of 345 kOe and 220 kOe at
1.8 K and 300 K respectively. The resistance of Cr1.5Pt4.5P has a metallic
temperature dependence with relatively weak magnetoresistance. Electronic band
structure calculations show that CrPt5P has a large peak in the density of
states near the Fermi level which is split into spin majority and minority
bands in the ferromagnetic state. Furthermore, the calculations suggest
substantial hybridization between Cr-3d and Pt-5d states near the Fermi level,
in agreement with the experimentally measured anisotropy.

1Ames National Laboratory, US DOE, Iowa State University, Ames, Iowa 50011,
USA

2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA

3Department of Chemistry and Chemical Biology, The State University of New
Jersey Rutgers, Piscataway, NJ 08854, USA

4Department of Chemistry, Michigan State University, East Lansing, MI, 48824,
USA

##  1 Introduction

From both an applied and fundamental perspective, ferromagnetism continues to
provide opportunities and challenges for new research. At a practical level,
the high cost and environmental concerns associated with the mining and
purification of rare earth elements necessitates the search for new rare earth
free permanent magnets for applications ranging from energy production, memory
storage, and high performance electric engines to emerging quantum technology
such as spintronics.[1, 2, 3, 4, 5, 6] More fundamentally, it remains
challenging to accurately predict the transition temperatures and even the
type of magnetic order observed in many materials, especially in metallic
compounds where the magnetism originates from the itinerant electrons rather
than local moments.[7, 8, 9, 10] In such itinerant ferromagnets, the physics
and possible emergent phases associated with suppressing the Curie temperature
towards 0 K using pressure, and/or chemical doping is less understood than for
antiferromagnets and remains active frontier in condensed matter research.[11,
12, 13, 14, 15, 16] These factors, amongst others, make the discovery of new,
transition metal based, ferromagnetic metals highly desirable.

Recently, several new materials in the Mn(Pt,Pd)5Pn (Pn = P, As) family,
referred to here as 1-5-1 compounds, were discovered and found to order
ferromagnetically near room temperature. Specifically, MnPt5As exhibits
ferromagnetic order below T${}_{\text{C}}$ $\approx$ 280 K and MnPd5P below
$\approx$ 295 K.[17, 18] While not ferromagnetic, the isostructural materials
MnPt5P and FePt5P also have interesting magnetic properties, as MnPt5P orders
antiferromagnetically at 192 K (with a small ferromagnetic, q = 0
component),[19, 18] and FePt5P is an itinerant antiferromagnet with three
closely spaced transitions between 70 K and 90 K.[20, 18]

The magnetic 1-5-1 compounds are part of the much larger X(Pt,Pd)5Pn family
that crystallizes in the tetragonal P4/mmm ($\\#$123) space group and where X
can be a late transition metal (Ag, Zn, Cd, Hg), a B-group element (Al–Tl), or
a 3d transition metal (Mn, Fe).[21, 22] The X(Pt,Pd)5Pn materials are
isostructural to the well-known family of heavy fermion superconductors CeTIn5
(T = Co, Rh, Ir),[23, 24, 25, 26] and similarly to how CeTIn5 can be pictured
as being derived from CeIn3, a useful way to describe the X(Pt,Pd)5Pn crystal
structure is as a quasi-two dimensional material formed by breaking up the
parent cubic compound XPt3 into X-Pt layers that are separated by sheets of P
atoms along the c-axis (see Figure 1a).

From the standpoint of materials discovery, the above picture is attractive
given that nearly all X elements that are reported to form ternary X(Pt,Pd)5Pn
compounds are also reported to form XPt3 binaries. With this in mind,
identifying known XPt3 for which a corresponding XPt5Pn is not yet reported
may be a fruitful guide to discovering new magnetic materials. Here we focus
on X = Cr. CrPt3 is a T${}_{\textit{C}}$ $\approx$ 490 K ferromagnetic with
among the highest magnetic anisotropy energies known for a cubic material.[27,
28, 29, 30, 31] A high anomalous hall effect of 1750 S$\cdot$cm-1 was recently
measured on CrPt3 films and attributed to the Berry curvature stemming from
two gaped nodal lines near the Fermi level.[32] Based on the unique properties
exhibited by CrPt3, we reasoned that a hypothetical CrPt5P may also host
interesting behavior.

Here, we report the discovery and basic physical properties of the compound
Cr1+xPt5-xP. We find that single crystals can readily be grown from solution
by adding Cr into melts based on the single phase liquid region above the Pt–P
eutectic composition.[18] Cr1+xPt5-xP is the first ternary compound in the
Cr–Pt–P phase space and adopts the same tetragonal P4/mmm crystal structure as
the other known 1–5–1 materials. Our EDS and X-ray diffraction
characterization suggests mixed occupancy between Cr and Pt atoms, giving true
compositions of Cr1+xPt5-xP with x between 0–0.5, which is similar to what is
observed in the CrPt3 parent compound. The samples characterized here have x
$\approx$ 0.5, and we show that Cr1.5Pt4.5P orders ferromagnetically below
T${}_{\text{C}}$ = 464.5 K with a saturated moment of 2.1
$\mu_{\textit{B}}$/Cr at 1.8 K. The ab-plane is the easy direction and the
magnetic anisotropy of Cr1.5Pt4.5P is exceptionally high, with an estimated
anisotropy fields of $\approx$ 345 kOe and 220 kOe at 1.8 K and 300 K
respectively. Transport measurements indicate typical metallic behavior with a
moderate negative magnetoresistance when the field is applied within the ab-
plane. Density functional theory calculations show a strong peak in the
density of states near the Fermi level which is split into majority and
minority spin bands in the magnetic state, consistent with the picture for
itinerant magnetism. Likewise, the calculations show significant hybridization
between Cr-3d and Pt-5d states, which likely underpins the strong magnetic
anisotropy.

##  2 Experimental Details

###  2.1 Crystal Growth

The staring materials were elemental Pt powder (Engelhard, 99+$\%$ purity),
red P (Alpha Aesar, 99.99$\%$) and Cr pieces (Alpha Aesar, 99.999$\%$). The
elements were weighed according to a nominal molar ratio of Cr9Pt71P20 and
contained in the bottom side of a 3-piece alumina Canfield crucible set (CCS,
sold by LSP Ceramics )[33, 34] The CCS was sealed in a fused silica ampule and
was held in place with a small amount of silica wool, which serves as
cushioning during the decanting step.[35] The ampules were evacuated three
times and back-filled with $\approx$ 1/6 atm Ar gas prior to sealing.

The crystal growth took place over three steps. First, using a box furnace,
the ampule was slowly warmed to 250°C over 6 h, then to 1180°C in an
additional 8-12 h. After dwelling at 1180°C for 6 h, the furnace was cooled to
950°C over 40 h, after which the remaining liquid phase (a Cr-depleted, Pt-P-
rich melt) was decanted by inverting the ampule into a centrifuge with
specially made metal rotor and cups.[36, 33, 35] After cooling to room
temperature, the ampule was opened and the products found to be metallic,
highly inter-grown, block-like crystals.

The remaining solidified liquid phase that was decanted and captured in the
“catch” crucible of the CCS was next reused in a second step.[18] Here, a new
3-piece CCS was assembled, using the decanted liquid in the new “growth” end.
The crucible set was sealed in an ampule as described above and heated rapidly
(in $\approx$ 2 h) to 975°C. After holding for 6 h, the furnace was slowly
cooled to 800°C over 75 h and the remaining liquid again decanted. The ampule
was opened to reveal a mixture of block-like crystals like those described
above as well as thin, nicely formed metallic plates. In the final step, the
remaining solidified liquid phase in the “catch” crucible was again reused.
After evacuating and sealing into an ampule, the furnace was heated to 825°C,
held for 6 h, and then cooled to 700°C over 60 h where the excess liquid was
decanted. The final step yielded only thin, plate-like crystals like those
also found in the second step. Pictures of several plates are shown in the
inset to Figure 1c.

Analysis with EDS suggested that both the block-like crystals and thinner
plates belonged to the same phase, with a putative chemical formula of
Cr1+xPt5-xP (see discussion section for details). This indicates that the
initial, more Cr-rich melts decanted at higher temperatures produce thicker
block-like crystals, whereas using relatively Cr-depleted melts and decanting
at lower temperatures results in the the thin plates. Both the three-
dimensional blocks and plates were malleable, and the plates could be bent and
deformed using tweezers. Furthermore, the plates could be cleaved within the
basal plane. Because the block-like crystals were generally inter-grown with
less clear orientation, the well formed, plate crystals were used in all
following measurements.

###  2.2 Crystal Structure Determination

We initially attempted to use single crystal X-ray diffraction to determine
the crystal structure of the plate-like crystals. Samples were analyzed with a
Bruker D8 Quest Eco single-crystal X-ray diffractometer equipped with a Photon
II detector and Mo radiation ($\lambda$K${}_{\text{$\alpha$}}$ = 0.71073 Å).
As noted above, the crystals are malleable and readily deformed, and as a
result, selecting a suitable piece was challenging and the reflections were
always weak and tailed (this is likely compounded by the large quantity of Pt
in the system, which is strongly absorbing). While the structure refinements
were consistent with the P4/mmm 1-5-1 arrangement[19, 17, 20] the poor data
quality rendered assessment of the structural parameters (site occupancy,
thermal parameters, etc.) impossible.

To better determine the phase and assess the purity of the crystals, samples
were analyzed with powder X-ray diffraction (PXRD). A representative selection
of $\approx$5-10 plate-like crystals was ground to a fine powder, sifted
through a 33 micron mesh sieve, and analyzed using a Rigaku Miniflex-II
instrument operating with Cu-K$\alpha$ radiation with $\lambda$ = 1.5406 Å
(K$\alpha_{1}$) and 1.5443 (K$\alpha_{2}$) Å at 30 kV and 15 mA. The powder
patterns were refined using GSAS-II software.[37]

###  2.3 Chemical Composition

The chemical composition was determined by energy dispersive X-ray
spectroscopy (EDS) quantitative chemical analysis using an EDS detector
(Thermo NORAN Microanalysis System, model C10001) attached to a JEOL scanning-
electron microscope (SEM). The measurements were performed at three different
positions on each crystal’s face (perpendicular to c-axis), revealing good
homogeneity in each crystal. An acceleration voltage of 16 kV, working
distance of 10 mm and take off angle of 35∘ were used for measuring all
standards and crystals with unknown composition. A MnPt5P single crystal was
used as a standard for Pt and P quantification, and a LaCrGe3 single crystal
was used as a standard for Cr. The spectra were fitted using NIST-DTSA II
Microscopium software.[38] The average compositions and error bars were
obtained from these data, accounting for both inhomogeneity and goodness of
fit of each spectra.

###  2.4 Physical Property Measurements

Magnetization measurements were performed in a Quantum Design Magnetic
Property Measurement System (MPMS-3) SQUID magnetometer operating in the VSM
mode. For the measurements at 300–600 K, the sample was mounted on a heater
stick with alumina cement, and radiation shielding temperature homogeneity
across the sample was ensured by a Cu foil. For the measurements conducted at
1.8–300 K, the sample was mounted on a quartz sample holder with GE 7031
varnish. The temperature dependent measurements were carried out using a 0.5
K/min warming/cooling rate.

Resistance measurements were performed using a Quantum Design Physical
Property Measurement System (PPMS) in AC transport mode. The samples were
prepared by cutting the crystals into rectangular bars, and the contacts were
made by spot welding 25 $\mu$m thick annealed Pt wire onto the samples in
standard four point geometry so that the current was applied within the ab-
plane. To ensure good mechanical strength, a small portion of silver epoxy was
painted over the spot-welded contacts, and typical contact resistances were
$\approx$ 1 $\Omega$. The transverse magnetoresistance was measured up to 90
kOe with the field applied perpendicular to the direction of current flow.

###  2.5 Electronic Band Structure Calculations

Band structure, density of states, and total energy for CrPt5P were calculated
in density functional theory[39, 40] (DFT) using the PBEsol[41] exchange-
correlation functional with spin-orbit coupling (SOC) included. All DFT
calculations were performed in VASP[42, 43] with a plane-wave basis set and
projector augmented wave method.[44] We used the unit cell of 7 atoms with a
$\Gamma$-centered Monkhorst-Pack[45] (10 $\times$ 10 $\times$ 6) k-point mesh
with a Gaussian smearing of 0.05 eV. The kinetic energy cutoff was 319 eV. The
ferromagnetic moment direction was changed along different axis to find the
preferred direction.

##  3 Results and Discussion

###  3.1 Phase Determination and Crystal Structure of
Cr${}_{1+\text{x}}$Pt${}_{5-\text{x}}$P

Table 1: Atomic Coordinates and Isotropic Atomic Displacement Parameters for
Cr1+xPt5-xP determined by Rietveld refinement of the PXRD pattern in Figure
1b. See the text for information on refinement of the Pt site occupancy. The
refinement statistics are R${}_{\text{wp}}$ = 11.5 and GOF = 3.2.

Atoms | Wycoff | Occ. | x | y | z |  U${}_{\text{iso}}$ (Å2)
---|---|---|---|---|---|---
Pt1 |  1a | 1 | 0 | 0 | 0 | 0.006(1)
Pt2 |  4i | 1 | 0 | 1/2 | 0.2905(1) | 0.006(2)
Cr3 |  1c | 1 | 1/2 | 1/2 | 0 | 0.01
P4 |  1b | 1 | 0 | 0 | 1/2 | 0.01

Figure 1: (a) Crystal structure of (a) CrPt5P and (b) CrPt3. The color code is
blue = Cr, grey = Pt, and red = P. (c) Pictures of several CrPt5P single
crystals on a mm grid. (d) Powder X-ray diffraction pattern and Le Bail
refinement for CrPt5P.

Cr1+xPt5-xP (discussion of x in following paragraphs) is a previously unknown
1-5-1 compound that adopts a layered tetragonal crystal structure with the
space group P4/mmm ($\\#$123), like the previously reported XPt5P
analogues.[21, 22, 17, 19, 20] The structure is illustrated in Figure 1a and
consists of slabs of face sharing CrPt12 polyhedra that span the ab-plane and
which are separated along the c axis by sheets of P atoms. As noted in the
introduction, CrPt5P can also be visualized as a composite built from single
layers of CrPt3 (crystal structure shown in Figure 1b) in the Cu3Au structure,
which are separated along the c-axis by Pt–P–Pt slabs. This second viewpoint
provides a useful lens to view the magnetic properties.

Figure 1d shows a powder X-ray diffraction pattern collected from several
plate-like crystals. The Bragg peaks are somewhat broad compared to those
collected on other 1-5-1 compounds,[18] which is likely because the
Cr1+xPt5-xP plates are malleable and challenging to grind into a fine powder.
Nevertheless, the experimental pattern agrees well with the reflections
expected from the P4/mmm structure, and there are no significant peaks from
secondary phases, indicating good sample purity. The lattice parameters
determined from Le Bail refinements of the powder data are a = 3.8922(2) Å and
c = 6.8565(1) Å which are very similar to those reported for MnPt5P and
FePt5P.[19, 20] The Le Bail refinement statistics are R${}_{\text{wp}}$ = 8.66
and GOF = 2.39, indicating a reasonable refinement.

Figure 2: (a) Field dependent magnetization isotherms at temperatures between
1.8–300 K. The solid points are the data and the dashed lines are
extrapolations to estimate the anisotropy fields. (b) Full magnetization
isotherms in the easy direction (H $\perp$ $\textit{c})$ at 1.8 K and 400 K
showing hysteresis when raising/lowering the field. Figure 3: (a) Temperature
dependence of M/H (left axis) and M/H-1 (right axis). The light blue line
marks the Curie-Weiss fit to the high temperature data. The inset shows the
temperature dependence of the spontaneous moment M${}_{\text{s}}$ at select
temperatures between 1.8-470 K, and the dashed line is an order-parameter fit
(see text for more details). (b) Arrot plots between 460–470 K. The inset
shows a closeup of the data between 464–466 K.

Elemental analysis with EDS indicated that the samples are enriched with Cr
and deficient in Pt compared to the expected stoicheometry of CrPt5P, with
true composition Cr1.53(4)Pt4.5(1)P0.94(3). The fact that the excess Cr is
offset by a corresponding Pt deficiency, i.e.
Cr${}_{1+\text{x}}$Pt${}_{5-\text{x}}$P, suggests there is mixed occupancy
between Cr and Pt atoms where the Cr site is fully occupied and the extra Cr
sits on the Pt sites. Considering that the parent compound CrPt3 is known to
form a solid solution with mixed occupancy between Cr and Pt atoms, where the
Cu3Au structure is maintained for compositions $\approx$20–40 $\%$ Cr,[28, 31]
it is reasonable to believe that Cr${}_{1+\text{x}}$Pt${}_{5-\text{x}}$P can
also accommodate excess Cr atoms on the Pt sites. Unfortunately, because the
samples are malleable and easily deformed, we were unable to select a
suitable, strain-free, piece for single crystal X-ray refinement (see
experimental section), and the Bragg peaks were always smeared/broadened.
Consequentially, we were not able to satisfactorily assess the site occupancy
using single crystal refinements. We next attempted Rietveld refinements of
our PXRD data using the known atomic positions of the structural analogue
MnPt5P as starting parameters.[19] Refining the Pt occupancy f gives fPt
$\approx$ 0.9 for both Pt sites, which is consistent with partial replacement
of the Pt atoms with substantially lower Z Cr, in qualitative agreement with
the Cr1.5Pt4.5P composition implied by the EDS analysis. We note that the
thermal displacement parameters U for Cr and P were fixed to 0.01 in order to
ensure a stable refinement. This is likely necessary given that the large and
heavy Pt atoms contribute more than 70$\%$ atomic weight in Cr1+xPt5-xP and
can significantly absorb X-rays. The refinements are otherwise consistent with
those in other XPt5P systems.[19, 20] Given the limitations of our powder
data, we were unable to obtain a stable refinement when explicitly attempting
to model the disorder by introducing Cr onto the two Pt sites, and therefore
cannot currently determine whether the excess Cr preferentially occupies one
of the two Pt sites. Ultimately, our PXRD and EDS results both provide clear
evidence for mixed occupancy between Cr and Pt atoms on the Pt-sublattice;
however, quality single crystal diffraction data or high resolution
synchrotron PXRD is needed for fully quantitative structural characterization.

###  3.2 Magnetic Properties of Cr${}_{1+\text{x}}$Pt${}_{5-\text{x}}$P

Figure 2a shows magnetization isotherms temperatures collected from 0–70 kOe
and at temperatures between 1.8–300 K. Measurements were conducted on a plate-
like crystal with EDS composition of Cr1.5Pt4.5P, and the field was applied
both along the c-axis and within the ab-plane (H $\perp$ c). The M(H) results
indicate Cr1.5Pt4.5P is ferromagnetic below at least 300 K with strong easy-
plane anisotropy. As will be shown below, the Curie temperature is $\approx$
465 K. Full M(H) loops obtained at 1.8 K and 400 K are displayed in Figure 2b
and show clear hysteresis between increasing and decreasing field sweeps,
confirming the ferromagnetic nature of Cr1.5Pt4.5P. The coercive field is
$\approx$ 0.2 kOe, indicating the ferromagnetism is rather soft, similar to
what is observed in other 1-5-1 ferromagnets MnPd5P and MnPt5As, which both
have small coercive fields under 50 Oe.[18, 17, 17] In the H $\perp$ c
orientation (i.e. with the field applied in the basal plane), the
magnetization increases in a nearly step-like fashion at low fields, quickly
reaching saturation by $\approx$ 4 kOe, and the saturated moment falls
monotonically with increasing temperature, from 2.1 $\mu_{\textit{B}}$/Cr at
1.8 K to 1.8 $\mu_{\textit{B}}$/Cr at 300 K. At 1.8 K, the saturated moment of
$\approx$ 2.1 $\mu_{\text{B}}$ is very close to the expected 2
$\mu_{\text{B}}$ for S = 3/2 Cr3+.

In the magnetic hard direction (H $\parallel$ c), the magnetization shows a
small increase at the lowest fields and settles into a linear field dependence
as H approaches 70 kOe, reaching a maximum of $\approx$ 0.7 $\mu_{\text{B}}$
at 1.8 K, which is only a third of the saturated 2.1 $\mu_{\text{B}}$ measured
when the field is applied in the ab-plane. To determined the anisotropy fields
HA, we extrapolated the tangents of the H $\parallel$ c and H $\perp$ c curves
to locate the field in which the respective lines meet, as shown in Figure 2a,
giving an estimated HA of $\approx$ 345 kOe at 1.8 K that falls monotonically
with increasing temperature to $\approx$ 220 kOe at 300 K. Note that all of
our M(H) measurements in the hard direction (H $\parallel$ c) show a small
low-field saturation below $\approx$ 2 kOe, followed by more linear behavior
at high fields. This is most likely from to a small misorientation of the
sample (or part of the sample, due to malleability) such that the field is not
applied exactly along the c-axis; however, at this time we cannot fully rule
out the possibility of a small axial component to the ferromagnetism. Assuming
the low-field saturation is from imperfect sample orientation, the anisotropy
fields discussed above will slightly underestimate the true values.

Figure 3a shows the temperature dependence of the magnetization (M/H) measured
in a H = 1 kOe applied field for Cr1.5Pt4.5P between 300–600 K. On cooling,
M/H increases rapidly beginning near 470 K, characteristic of the onset of
ferromagnetic order. The right axis of Figure 3a shows the inverse (M/H)-1 of
the temperature dependent M/H data above 300 K. (M/H)-1 has a nearly linear,
Curie-Weiss like temperature dependence above $\approx$ 500 K. Curie-Weiss
fits to the H/M-1 data above 500 K give an estimated Weiss temperature
$\Theta$ = 475 K, where the positive value is consistent with ferromagnetic
ordering, and an effective moment $\mu_{\text{eff}}$ = 3.5
$\mu_{\text{B}}$/Cr, which like the 1.8 K saturated moment discussed above, is
in reasonably agreement with the 3.87 $\mu_{\text{B}}$ anticipated for
paramagnetic Cr3+ moments.

Because the applied field will broaden and enhance a ferromagnetic transition,
the Curie temperature is poorly defined in temperature dependent magnetization
data. To more accurately determine TC, we also collected magnetization
isotherms at temperatures above 400 K and assembled Arrot plots.[46] Figure 3c
and its inset show the data collected every 1 K between 460 K and 470 K. Here,
the plot of M2 against H/M should run through the origin at the Curie
temperature, and our results indicate TC $\approx$ 464.5 K.

The inset to Figure 3a shows the magnitude of the spontaneous magnetization
(M${}_{\text{s}}$), estimated by extrapolating the tangents of the high field
M(H) data to zero field, increases in an order-parameter like manner as the
temperature is lowered, approaching 2.1 $\mu_{\text{B}}$/Cr at 1.8 K. Linear
fits to a plot of log(M${}_{\text{s}}$) vs. log(TC–T) give a slope $\beta$
$\approx$ 0.23. The dashed line in the inset to Figure 3a shows the
$(T_{C}-T)^{\beta}$ captures the experimental data well, and the estimated
$\beta$ $\approx$ 0.23 suggests the magnetism in CrPt5P is not mean-field like
(where $\beta$ = 0.5).

###  3.3 Transport properties of Cr${}_{1+\text{x}}$Pt${}_{5-\text{x}}$P

Figure 4: (a) Temperature dependent resistance of CrPt5P between 1.8–375 K.
(b) Transverse magnetoresistance at 2 K.

Figure 4a shows the temperature dependent resistivity $\rho$ between 1.8–375
K. As expected, the behavior is characteristic of a metal, decreasing
monotonically with cooling and approaching saturation as the lowest
temperature is reached. No transitions are observed below 375 K, which is
consistent with the single TC = 464.5 K transition detected in the magnetic
data. The residual resistance ratio RRR = $\rho$(300 K)/$\rho$(1.8 K) is 16,
indicating reasonably high crystal quality. Considering the mixed occupancy
between Cr and Pt atoms inferred from the EDS and X-ray diffraction analysis,
16 may appear a high RRR for a compound with such a substantial degree of
crystallograrphic disorder ($\approx$ 10 $\%$ of the Pt sites are occupied by
excess Cr); however, the RRR of 16 is considerably lower than the RRR
$\approx$ 110 reported for single crystals of the structural analogue
MnPt5P.[18] Using a similar effective temperature below the Neel temperature
still gives RRR $\approx$ 70 for MnPt5P, suggesting the atomic disorder in
Cr1.5Pt4.5P raises the residual resistance compared to what would be expected
in an ideal, disorder-free crystal.

Figure 4b shows the transverse magnetoresistance, defined as
[R(H)–R(0)]/R(0)), collected at 2 K with the field applied both along the
c-axis and within the ab-plane. In both cases, the current flow was within the
plane and the field perpendicular to the direction of the current. At 2 K and
for H $\parallel$ c, CrPt5P has a relatively small positive magnetoresistance
that reaches 20 $\%$ at 9 T. When the field is applied in the ab-plane (the
magnetic easy direction), the 2 K magnetoresistance first has a negative field
dependence at low H that crosses over to a positive slope near $\approx$ 15
kOe and then increases with a similar field dependence as the H $\parallel$ c
data over the remaining range of H. The negative slope at low fields is
typical of ferromagnets and most likely comes from the decrease in spin
disorder scattering as the field aligns the magnetic domains before crossing
over to positive behavior at higher H.

###  3.4 Discussion

Whereas the high Pt content and planar anisotropy likely limit the feasibility
of CrPt5P for practical applications, the 220 kOe (at 300 K) and 345 kOe (at
1.8 K) anisotropy fields are exceptionally high. For comparison, Nd2Fe14B,
which is among the most widely used permanent magnets, has an anisotropy field
of 82 kOe near room temperature.[47] Other examples include MnBi (Ha = 50
kOe),[48] HfMnP (Ha = 100 kOe),[49] FePt (Ha = 100 kOe),[50] and CoPt (Ha =
140 kOe).[51] Clearly the anisotropy of CrPt5P is substantially stronger. As
magnetic anisotropy arises from spin orbit coupling, it is very likely the
substantial quantity of large-Z Pt atoms in CrPt5P underpins the high
anisotropy.

The 464.5 K Curie temperature of Cr1.5Pt4.5P is a high ordering temperature
for a compound with only 1/7 moment bearing Cr atoms per formula unit. This
likely indicates that the Cr atoms do not behave as local moments and that
CrPt5P is better described as an itinerant ferromagnet in which the magnetism
arises from the exchange-splitting of the conduction electrons.[7, 8, 9] A
simple experimental metric for evaluating the degree of itinerancy in a
metallic compound is the Rhodes-Wohlfarth ratio qc/qs,[52] which is calculated
from the high temperature effective moment and base temperature saturated
moment using the following expressions:

$\mu_{eff}^{2}=q_{c}(q_{c}+2)^{2}\mu_{B}$

$\mu_{sat}=q_{s}\mu_{B}$

$q_{c}/q_{s}=(-1+\sqrt{1+(\mu_{eff}/\mu_{B})^{2}})/(\mu_{sat}/\mu_{B})$

For compounds with relatively low magnetic ordering temperatures, high ratios
suggest itinerant behavior, and qc/qs should be close to unity when the
magnetism is local moment like (i.e. in rare-earth containing materials). The
Rhodes-Wohlfarth ratio also converges towards 1 in itinerant magnets when the
ordering temperatures is high ($>$ 300 K).[10] For Cr1.5Pt4.5P, we find qc/qs
= 1.3, which is consistent with the expectation for an itinerant magnetic with
a relatively high, 465 K, Curie temperature. Because qc/qs of 1 are also
expected for local moment systems, which may be consistent with the trivalent
oxidation state of Cr implied by the magnetic measurements, this result is
ultimately somewhat ambiguous; however, the band structure calculations
discussed below also support itinerant, Stoner-like magnetism.

Figure 5: (a) Electronic band structure for CrPt5P in the ferromagnetic phase
(with moments along the a-axis. (b) Density of states in the non-magnetic
phase. (c) Density of states in the ferromagnetic phase. The calculations
assume the ideal 1-5-1 composition (no Cr-Pt site disorder)

To provide greater insight into the magnetism and strong anisotropy measured
in CrPt5P, we used density functional theory (DFT) to calculate the electronic
band structure, which is displayed in Figure 5a. Our calculations considered
the ideal 1-5-1 composition, i.e. without excess Cr and mixed occupancy
between Cr and Pt atoms. Consistent with the experimental results, the
calculations show CrPt5P to be a metal with many well dispersed bands crossing
the Fermi level (EF). The orbital projected density of states (PDOS) for
nonmagnetic and ferromagnetic (with the moment along the a-axis) phases of
CrPt5P are respectively given in Figures 5b and 5c. In both cases, the
majority of the states near EF are derived from Cr-3d and Pt-5d orbitals. The
main contributions from P orbitals are significantly deeper in energy, with a
band of P-3p based states from -10 to -5 eV and the P-3s states spanning -14
to -13 eV.

Comparing the density of states calculations for nonmagnetic and ferromagnetic
CrPt5P, we observe a narrow peak in the DOS around the Fermi level in the
paramagnetic phase. The spike in DOS is composed primarily of Cr-3d orbital
states with a smaller but still substantial contribution from Pt-5d orbitals,
suggesting the hybridization between Cr-3d and Pt-5d orbitals in CrPt5P is
strong. As magnetic anisotropy arises from spin orbit coupling, the
substantial hybridization between Cr-3d and the high Z Pt-5d states at EF is
consistent with the strong magnetic anisotropy observed experimentally. Once
in the ferromagnetic state, the peak at EF is split, where the lower spin
majority band is approximately centered at EF and the higher spin minority
band at 2 eV. Both upper and lower peaks retain significant contributions from
both Cr-3d and Pt-5d orbital states. The calculations shown in Figures 5b and
5c match what is anticipated for an itinerant ferromagnet, where a locally
high density of states near EF is split into upper and lower, majority and
minority, spin bands by the on-site Coulomb repulsion, giving rise to a net
magnetization associated with the conduction electrons.[9, 7] Our calculations
therefore support CrPt5P as an itinerant ferromagnet, which is consistent with
its Rhodes-Wohlfarth ratio and high Curie temperature.

Integrating the contributions of the spin up and down electrons, we calculated
a magnetic moment of 3.2 $\mu_{\text{B}}$/Cr, which is in good agreement with
the experimental $\mu_{\text{eff}}$ $\approx$ 3.5 $\mu_{\text{B}}$/Cr inferred
from the Curie-Weiss fits. Moreover, the calculations indicate the most
favorable ferromagnetic configuration is with the ordered moment within the
ab-plane, consistent with the experimental observations, and we calculate a
magnetic anisotropy energy -1.261 meV/f.u. Although this suggests weaker
anisotropy than the experimental results, the DFT calculations overall
satisfactorily capture the magnetism experimentally observed in CrPt5P.

Because the Cr1+xPt5-xP studied here deviate from the ideal stoicheometry,
with x $\approx$ 0.5, we also constructed a (2x2x1) supercell with two Pt
sites, one at 4i and the other at 1a, substituted by Cr, in order to
preliminarily study the effect of excess Cr on the electronic structure. The
calculations indicate that excess Cr placed on either Pt sites prefers to
couple antiferromagneticallly with the host Cr; however, the additional Cr
changes the DOS away from E${}_{\text{F}}$ at +2.0 eV and -1.0 eV, and we find
that the DOS near E${}_{\text{F}}$ is not strongly altered by the presence of
excess Cr. Consequentially, it is unclear from these calculations how strongly
the excess Cr will alter the magnetism. In our experimental M(H) data, we
observe no evidence for a ferrimagnetic state associated with AFM coupling
between Cr moments associated with the different crystallographic positions,
and the saturated moments are in reasonable agreement with the expectations
for trivalent Cr, both suggesting Cr1.5Pt4.5P shows simple ferromagnetic
order. A more detailed study on the effect of mixed occupancy requires
configurational thermodynamics using cluster expansion method or coherent
potential approximation, ideally in tandem with measurements on crystals with
different compositions (x).

As described in the discussion of the crystal structure, Cr1+xCr5-xP can be
envisioned as single layers of CrPt3 that span the ab-plane and which are
divided along the c-axis by planes of P atoms. From this perspective, it is
interesting to compare the magnetic properties of Cr1+xCr5-xP with those of
the parent compound CrPt3. In CrPt3, the Cr atoms also order ferromagnetically
with a Curie temperature of $\approx$ 490 K.[27, 28, 29, 30] In Cr1+xCr5-xP,
the insertion of the layer of P atoms nearly doubles the Cr-Cr distance along
the c-axis from 3.874 Å in CrPt3 to 6.857 Å in CrPt5P, whereas the in-plane
Cr-Cr distances (the a-lattice parameters) are approximately the same in each
compound. Considering that the salient structural feature of face sharing Cr-
Pt polyhedra is conserved in both compounds, and that the electronic states
near the Fermi level in CrPt5P are essentially derived only from Cr and Pt
based orbital states, the similar magnetic properties suggests that the
underlying physics governing the magnetism in each compound may be similar and
that the addition of the P-layer in Cr1+xCr5-xP serves to weaken the exchange
interaction between magnetic moments and slightly lower the ordering
temperature. As Cr1+xCr5-xP and CrPt3 share similar crystallographic and
magnetic properties, investigation of topological transport properties, like
the high anomalous hall effect found in CrPt3, may be an attractive future
direction for work on Cr1+xCr5-xP.

##  4 Summary and Conclusions

We grew single crystals of Cr1+xPt5-xP (x = 0.5) and found that it is a
remarkably anisotropic ferromagnet with a TC = 464.5 K and a 1.8 K anisotropic
field of 345 kOe. We show that single crystals can readily be grown out
ternary Cr-Pt-P solutions. Like all other members of the X(Pt,Pd)5P family,
Cr1+xPt5-xP crystallizes in the P4/mmm space group and is comprised of
CrPt3-like layers spaced out by sheets of P atoms along the c-axis. Our EDS
and XRD analysis indicate Cr-enrichment, likely pointing toward site occupancy
disorder and true compositions of Cr1.5Pt4.5P, which is similar to the
disorder found in the parent compound CrPt3. Likely owing to the high Pt
content, CrPt5P has very strong magnetic anisotropy in which the ab-plane is
the easy direction. The estimated anisotropy fields are $\approx$ 345 kOe and
220 kOe respectively at 1.8 K and 300 K. Cr1.5Pt4.5P exhibits metallic
transport behavior and has relatively weak transverse magnetoresistance.
Electron band structure calculations are consistent with itinerant magnetism,
showing the CrPt5P has a local peak in the density of states near the Fermi
level that is split into upper and lower bands in the ferromagnetic state.
Likewise, our calculations suggest strong hybridization between Cr-3d and
Pt-5d states, in agreement with the experimentally observed strong magnetic
anisotropy.

Acknowledgements

Work at Ames National Laboratory was supported by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences, Materials Sciences and
Engineering Division. Ames National Laboratory is operated for the U.S.
Department of Energy by Iowa State University under Contract No. DE-
AC02-07CH11358. TJS, PCC, and LLW were supported by the Center for Advancement
of Topological Semimetals (CATS), an Energy Frontier Research Center funded by
the U.S. Department of Energy Office of Science, Office of Basic Energy
Sciences, through Ames National Laboratory under its Contract No. DE-
AC02-07CH11358 with Iowa State University. W.X. is supported by NSF-
DMR-2053287.

Conflicts of Interest

The authors have no conflicts of interest to declare.

*corresponding authors’ email<EMAIL_ADDRESS><EMAIL_ADDRESS>
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# Anisotropic transport properties in prismatic topological insulator
nanowires

Hallmann Ó. Gestsson  Dept. of Engineering, Reykjavik University, Menntavegi
1, IS-102 Reykjavik, Iceland  Dept. of Physics and Astronomy, University
College London, Gower Street, London WC1E 6BT  Andrei Manolescu  Dept. of
Engineering, Reykjavik University, Menntavegi 1, IS-102 Reykjavik, Iceland
Jens H. Bardarson  Dept. of Physics, KTH Royal Institute of Technology,
Stockholm, SE-106 91 Sweden  Sigurdur I. Erlingsson  Dept. of Engineering,
Reykjavik University, Menntavegi 1, IS-102 Reykjavik, Iceland

###### Abstract

The surface of a three dimensional topological insulator (TI) hosts surface
states whose properties are determined by a Dirac-like equation. The
electronic system on the surface of TI nanowires with polygonal cross-
sectional shape adopts the corresponding polygonal shape. In a constant
transverse magnetic field, such an electronic system exhibits rich properties
as different facets of the polygon experience different values of the magnetic
field due to the changing magnetic field projection between facets. We
investigate the energy spectrum and transport properties of nanowires where we
consider three different polygonal shapes, all showing distinct properties
visible in the energy spectrum and transport properties. Here we propose that
the wire conductance can be used to differentiate between cross-sectional
shapes of the nanowire by rotating the magnetic field around the wire.
Distinguishing between the different shapes also works in the presence of
impurities as long as conductance steps are discernible, thus revealing the
sub-band structure.

##  I Introduction

Properties of charged particles moving in two dimensions (2D) in a constant
magnetic field are described by Landau levels, and this holds both for
particles governed by the Schrödinger equation and the Dirac equation. The
corresponding energy spectra do not depend on the direction of the magnetic
field, only on the magnitude of the perpendicular projection of the field, in
the absence of Zeeman coupling. In systems where the magnetic field projection
normal to the surface changes sign, interesting physics can occur, e.g., the
presence of states with trajectories along the spatial lines of zero normal
magnetic field. Such so-called snake states have been studied in
nanostructures, first for semiconductors in the integer quantum Hall regime
Müller (1992); Reijniers and Peeters (2000); Zwerschke et al. (1999), and
later in graphene Oroszlány et al. (2008).

Snake states can also occur in a constant magnetic field if the relative
orientation of the electronic system changes with respect to the direction of
the field, which is the situation for electrons localized on the surface of a
nanowire Tserkovnyak and Halperin (2006); Ferrari et al. (2008); Manolescu et
al. (2013); Rosdahl et al. (2015); Manolescu et al. (2016); Chang and Ortix
(2017). For semiconducting nanowires the electronic system may be engineered
to follow the surface using Fermi level pinning Heedt et al. (2016), or a
core-shell heterostructure with an insulating core and a conducting shell
Rieger et al. (2012); Fan et al. (2006); Heurlin et al. (2015).

Figure 1: A nanowire assuming a hexagonal cross section. We take the $x$\- and
$s$-coordinates to parameterize the longitudinal direction of the wire and its
perimeter, respectively. The vectors $\vec{B}$ and $\vec{n}$ represent the
magnetic field and surface normal, respectively, and $\Phi$ is the angle
between them. Points $s_{\pm}$ are midpoints on corresponding facets with
fixed normal $\vec{n}_{\pm}$. Facet length $L$ is fixed.

An interesting feature of the semiconductor nanowires is that the cross
section typically assumes a polygonal shape, e.g., hexagonal. Notably, if the
electrons are constrained to such a narrow shell, the lowest energy states
become localized at the corners of the polygonal cross section for
Schrödinger-like systems Ballester et al. (2012); Royo et al. (2014); Sitek et
al. (2015), while in a magnetic field the electronic states show properties
that can change from Landau levels to snaking states depending on which facet
the electrons are located Ferrari et al. (2009). Such nanowires can be
considered tubular conductors, with a prismatic geometry, and their electrical
conductance in the presence of a transverse magnetic field is expected to
depend on the relative angles between the magnetic field and the facets of the
nanowire surface Urbaneja Torres et al. (2018). However, to the best of our
knowledge, this kind of anisotropic conductivity has not been reported in
experimental studies.

The previously mentioned examples referred to systems described by the
Schrödinger equation. In more recent years, topological insulator (TI) Kane
and Mele (2005); Hasan and Kane (2010); Qi and Zhang (2011); Ando (2013) have
come to the forefront as materials that host surface states in nanowires
Dufouleur et al. (2013); Cho et al. (2015); Jauregui et al. (2016); Dufouleur
et al. (2017); Münning et al. (2021); Roessler et al. (2023). This is due to
the fact that TIs are characterized by an insulating bulk (either 3D or 2D),
but conducting surface states (2D or 1D, respectively) whose low energy
properties are described by the Dirac Hamiltonian, with a linear energy-
momentum relation Qi and Zhang (2011); Liu et al. (2010). These nanowires show
rich Aharanov-Bohm related phenomena Bardarson et al. (2010); Zhang and
Vishwanath (2010a); Rosenberg et al. (2010); Bardarson and Moore (2013);
Dufouleur et al. (2018) for magnetic fields parallel to the nanowire axis
Dufouleur et al. (2013); Cho et al. (2015); Jauregui et al. (2016); Dufouleur
et al. (2017); Münning et al. (2021); Roessler et al. (2023), or Landau level
and snake-like states for transverse fields Zhang et al. (2011); Shevtsov et
al. (2012); Tang and Fu (2014); Vafek (2011); Brey and Fertig (2014); Ilan et
al. (2015); Xypakis and Bardarson (2017); Erlingsson et al. (2017) that can
give rise to rich transport phenomena, e.g., reversal of thermoelectric
currents Erlingsson et al. (2017). Effective 2D models for TI nanowires have
been used to investigate, e.g., Andreev reflection in T-junctions Fuchs et al.
(2021), or cone shaped nanowires Kozlovsky et al. (2020). All the 2D models
assume a fixed Fermi velocity, but for example in topological crystalline
insulators different terminations of the crystalline lattice can host
different velocities leading to hinge and corner states Nguyen et al. (2022);
Skiff et al. (2023).

In this paper we consider the electronic properties of TI nanowires of
polygonal shape in a constant transverse magnetic field. Different projections
of the magnetic field along different polygon facets lead to energy spectra
showing unique characteristics for each structure, e.g., two separate sets of
Landau levels on different facets. We propose that the conductance of the
wire, as a function of magnetic field orientation, can be used to
differentiate between cross-sectional shapes of the nanowires. To model the
transport we introduce a modified Wilson mass term which faithfully reproduces
the correct Landau levels on different facets. We use the recursive Green’s
function method to calculate the transmission function and show that for low
enough impurity densities the anisotropic conductance reveals the underlying
nanowire shape.

The structure of the paper is as follows: Following the Introduction, the
model system and Hamiltonian is discussed in Sec. II. We also outline the
Landau level approximation for general facets of the polygon in Sec. II.1. The
modification to the Wilson mass term for transport calculation is outlined in
Sec. III along with transmission function calculations. The conductance
anisotropy is discussed in Sec. IV, and finally we discuss the results in Sec.
V.

##  II Model Hamiltonian and facet Landau levels

We consider a nanowire composed of a TI material such that the wire surface
can support edge modes, i.e., quasi-particles which propagate along the wire
surface whilst the bulk remains insulating. Furthermore, we assume that there
is a constant magnetic field perpendicular to the length of the wire. The wire
surface is taken to be parameterized by $x$ and $s$, where $x$ spans the
entire length of the wire and $s$ is the arc-length variable for the cross
section perimeter of the wire, as shown in Fig. 1. The effective Hamiltonian
describing the surface modes of such a system is Ostrovsky et al. (2010);
Zhang and Vishwanath (2010b); de Juan et al. (2019)

$H_{\mathrm{surf}}(x,s)=v_{F}\left\\{\sigma_{x}\left[p_{x}+eA_{x}(s)\right]+\sigma_{y}p_{s}\right\\},$
(1)

where $p_{s}=-i\hbar\partial_{s}$ is the momentum operator of the perimeter
variable $s$. Here $v_{F}$ is the Fermi velocity, $e$ is the electronic
charge, $\sigma_{x}$ and $\sigma_{y}$ are the Pauli matrices, $p_{x}$ is the
momentum operator along the length of the wire, and $A_{x}$ is the
$x$-component of the vector field. We choose the gauge such that
$A_{y}=A_{z}=0$, and

$A_{x}(y,z)=B[\sin(\phi)z-\cos(\phi)y],$ (2)

where $B=|\vec{B}|$ and $\phi$ is the angle between $\vec{B}$ and the
$z$-axis. For the linear-in-momentum surface Hamiltonian the polygon corners
have no effect on the surface states Lee (2009); Vafek (2011); Brey and Fertig
(2014); Messias de Resende et al. (2017) and the influence of the polygonal
shape only enters via discontinuities in the slope of $A_{x}(s)$.

In the absence of a magnetic field $H_{\mathrm{surf}}$ admits a simple
analytical spectrum. Squaring the Hamiltonian in Eq. (1) and using
$\\{\sigma_{x},\sigma_{y}\\}=0$ leads to

$H_{\mathrm{surf,0}}^{2}=\hbar^{2}v_{F}^{2}\left(p_{x}^{2}+p_{s}^{2}\right)\sigma_{0},$
(3)

with $\sigma_{0}$ being the $2\times 2$ identity matrix. Applying the plane
wave ansatz $\psi(x,s)=\frac{1}{\sqrt{LP}}e^{i(k_{x}x+k_{s}s)}$, where $L$ is
the length of the wire and $P$ is the cross section perimeter length, yields
eigenvalues of the form $E_{\pm}=\pm\hbar v_{F}\sqrt{k_{x}^{2}+k_{s}^{2}}$. A
nontrivial spin connection has been gauged away from the Hamiltonian and into
the boundary condition $\psi(x,s+P)=-\psi(x,s)$, leading to
$k_{s}=\frac{2\pi}{P}\left(n+\frac{1}{2}\right)$ with integer $n$ Bardarson et
al. (2010); Zhang and Vishwanath (2010a); Rosenberg et al. (2010); Bardarson
and Ilan (2018).

###  II.1 Facet Landau levels

In the presence of a perpendicular magnetic field the vector potential
$A_{x}(s)$ changes according to the form of the prismatic nanowire. For
translationally invariant wires where $[p_{x},H_{\mathrm{surf}}]=0$, $p_{x}$
can be replaced by its eigenvalue $\hbar k_{x}$ in Eq. (1).

Figure 2: Normalized vector potential, $A_{x}(s)/LB$, as a function of the
perimeter variable $s$ for two different orientations of the magnetic field
for different cross sections and orientation: a) triangle for $\phi=0$, and
$\phi=\pi/2$, b) square $\phi=0$, $\phi=\pi/4$, and c) hexagon for $\phi=0$
and $\phi=\pi/6$.

Depending on the orientation of the magnetic field, and the local normal
$\vec{n}(s)$, Landau levels can form at different facets, at different values
of $k_{x}$. To illustrate how this occurs we consider the hexagonal nanowire
with a magnetic field tilted from the $z$-axis by an angle $\phi$, and $\Phi$
is the angle between $\vec{B}$ and $\vec{n}$. Focusing on the tilted facets
characterized by the two angles $\Phi_{+}$ and $\Phi_{-}$, see Fig. 1, the
vector potential can be linearized around the center on each facet, indicated
by $s_{\pm}$, resulting in

$\displaystyle A^{\pm}_{x}(s)$ $\displaystyle=$
$\displaystyle-B\cos(\Phi_{\pm})(s-s_{\pm})+A_{x}(s_{\pm}).$ (4)

The resulting linearized version of Eq. (1) is then

$\displaystyle H^{[\pm]}_{\mathrm{surf}}$ $\displaystyle=$ $\displaystyle
v_{F}\left\\{\sigma_{x}\left[-eB^{\pm}(s-s_{\pm})+\hbar
k_{x}+eA_{x}(s_{\pm})\right]\right.$ (5)
$\displaystyle\left.\quad+\sigma_{y}p_{s}\right\\},$

where $B^{\pm}=B{\cos(\Phi_{\pm})}$. Landau levels centered on $s_{\pm}$ will
form at $k_{x}$-values that satisfy $\hbar k_{x}+A_{x}(s_{\pm})=0$, see App.
A. Moving away from these $k_{x}$-values by $\Delta k_{x}$ shifts the Landau
level center coordinate by $\Delta s_{\pm}=\frac{\hbar}{eB^{\pm}}\Delta
k_{x}$.

The properties of $A_{x}$ as a function of the perimeter variable $s$ are
shown in Fig. 2, for a) a triangle, b) a square, and c) a hexagon. The
coordinate system is chosen such that the $x$-axis (the longitudinal
direction) intersects the centroid of the nanowire, and $s=0$ is placed at the
top left corner of each shape, and increases in the clockwise direction. The
slope of each piece-wise linear part gives the normal projection of the
magnetic field on the corresponding facet. Two different orientations of
magnetic fields are shown for each shape. The blue and red dots indicate the
position of the $s$-points (for $k_{x}=0$), where Landau levels will be
centered, which depends on the cross section shape and magnetic field
orientation. Focusing on one such $k_{x}$ value, e.g., for $s_{+}$ the
Hamiltonian can be written as

$H^{[+]}_{\mathrm{surf}}=\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{+})|}}{\sqrt{2}\ell_{c}}\left(a_{[+]}\sigma_{+}+a_{[+]}^{\dagger}\sigma_{-}\right),$
(6)

where $\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$, $\ell_{c}^{2}=\hbar/eB$ is the
magnetic length, and the ladder operator for the $\vec{n}_{+}$ facet is
defined in in Eq. (25) in App. A. Being able to focus on a given facet and
defining the facet Hamiltonian in Eq. (6) relies on the wave function being
nonzero only on that facet (up to exponentially small corrections). In terms
of the magnetic length $\ell_{c}$ and side length $L$ this condition
translates to $\ell_{c}/L\ll\sqrt{|\cos(\Phi_{+})|}$. The Hamiltonian in Eq.
(6) can be diagonalized using basis states
$\\{|n-1,\uparrow\rangle,|n,\downarrow\rangle\\}$ for $n=1,2,3,\dots$
resulting in a pair of eigenenergies (negative and positive)

$\varepsilon^{[+]}_{n,\pm}=\pm\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{+})}|}{\ell_{c}}\sqrt{2n},$ (7)

along with a single state $|0,\downarrow\rangle$ with eigenenergy
$\varepsilon^{[+]}_{0}=0$. In a similar way the Hamiltonian for the
$\vec{n}_{-}$ segment can be written as

$H^{[-]}_{\mathrm{surf}}=\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{-})}|}{\sqrt{2}\ell_{c}}\left(a_{[-]}^{\dagger}\sigma_{+}+a_{[-]}\sigma_{-}\right),$
(8)

where the corresponding ladder operator is defined in Eq. (27) in App. A. The
Hamiltonian in Eq. (8) can be diagonalized with the basis states
$\\{|n-1,\downarrow\rangle,|n,\uparrow\rangle\\}$ for $n=1,2,3,\dots$, where
the spin indices have been switched. The resulting pair of eigenenergies
(negative and positive)

$\varepsilon^{[-]}_{n,\pm}=\pm\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{-})}|}{\ell_{c}}\sqrt{2n},$ (9)

along with a single state $|0,\uparrow\rangle$ with eigenenergy
$\varepsilon^{[-]}_{0}=0$.

Figure 3: Energy spectra for the triangle for the two magnetic field
orientations shown in Fig. 2a). The magnetic field value corresponds to
$\ell_{c}/L=0.125$. The two dashed lines indicate energies $E=0.97$ and $1.92$
that are relevant for subsequent transport calculations. The vertical dash-
dotted line indicates the $k_{x}L$-value corresponding to the center of the
facet where the zero-magnetic field states occur, see text for details. The
zero energy Landau levels are doubly degenerate.

The energy spectrum, as a function of $k_{x}$, for a triangular nanowire is
shown in Fig. 3 for the two different magnetic field orientations shown in
Fig. 2a). Only the positive energy is shown since the negative part is simply
mirrored due to an electron-hole symmetry, i.e., the anticommutation relation
$\\{H_{\mathrm{surf}},\sigma_{z}\\}=0$ results in any state $|\psi\rangle$
with energy $\varepsilon\geq 0$ having a corresponding state
$\sigma_{z}|\psi\rangle$ with negative energy $-\varepsilon$. For magnetic
field orientation $\phi=0$ the tilted facets of the triangle have
$\cos(\Phi_{+})=1/2$ and on the bottom facet $|\cos(\Phi_{-})|=1$. The
spectrum shows an interesting feature of every other LL being nondegenerate at
energies $\sqrt{1},\sqrt{3},\sqrt{5},\dots$ (in units of $\hbar
v_{F}/\ell_{c}$) while LL at energies $\sqrt{0},\sqrt{2},\sqrt{4},\dots$ are
doubly degenerate. The LL energies, given by Eqs. (7) and (9), coincide if
$|\cos(\Phi_{+})|m_{+}=|\cos(\Phi_{-})|m_{-}$ where $m_{\pm}\in\mathbb{N}$ are
the LL indices for the $\Phi_{\pm}$ facets (the $\Phi_{+}$ facet consists of
the top left and right facets). This happens for the even $m_{+}$ states on
the $\Phi_{+}$ facet which will hybridize with the state on the opposing
facet, into symmetric and anti-symmetric states, as they evolve into linear-in
$k_{x}$ states for $|k_{x}|\gg L/\ell_{c}^{2}$. The odd $m_{+}$ states on the
$\Phi_{+}$ facet never coincide with the eigenenergies of states on opposite
facets. As $k_{x}$ is increased these states will be coupled to states with
different energies (2nd order perturbation) without hybridizing, as they
evolve into linear-in $k_{x}$ states for $|k_{x}|\gg L/\ell_{c}^{2}$.

In the $\phi=\pi/2$ orientation the magnetic field is parallel to the bottom
side, leading to a zero-magnetic field projection on that facet. The two
tilted facets will have opposite projection and will hybridize at the apex for
positive value of $k_{x}$. For negative values of $k_{x}$ the wave function is
pushed toward the $\vec{\bm{B}}\cdot\vec{\bm{n}}=0$ facet. This corresponds to
the red curve in Fig. 2a, where one can see that $A_{x}(s)$ is constant for
$s\in[L,2L]$. Setting $s=3L/2$, corresponding to the center of the constant
region, results in a value of $k_{x}L$ that is determined by

$\hbar k_{x}+eA_{x}(s)\bigr{|}_{s=3L/2}=0.$ (10)

Note the coordinate system is placed in the centroid of the triangle which, in
our chosen gauge, results in $A_{x}(s)=-\frac{\sqrt{3}}{4}BL$ in the constant
region. The resulting value $k_{x}L=\frac{\sqrt{3}L^{2}}{4\ell_{c}^{2}}$ is
indicated by a vertical line for $\phi=\pi/2$. A set of states will form there
which, to lowest order, have energies $\sim\hbar
v_{F}\frac{\pi}{L}\times\mathrm{integer}$.

Figure 4: Energy spectra for the square for the two magnetic field
orientations shown in Fig. 2b). The magnetic field value corresponds to
$\ell_{c}/L=0.125$. The two dashed lines indicate energies $E=0.97$ and $1.92$
that are relevant for subsequent transport calculations. The vertical dash-
dotted lines indicates the $k_{x}L$-values where the zero-magnetic field
states occur, see text for details. The zero energy Landau levels are doubly
degenerate.

Figure 4 shows the square nanowire spectra for two magnetic field
orientations. For $\phi=0$ the two side facets have zero projection of the
magnetic field so that for $|k_{x}L|\sim L^{2}/2\ell_{c}^{2}$ the Landau
levels evolve into corresponding quantized zero-magnetic field states. The
$s$-values at the center of the flat sections of $A_{x}(s)$ in Fig. 2b are
$s=3L/2$ and $7L/2$. In this case, the condition in Eq. (10) results in
$|k_{x}L|=L^{2}/2\ell_{c}^{2}$, indicated by the vertical dashed line. Note
that each Landau level evolves into two states on the side facet, the same
applies to the zero Landau level which is doubly degenerate and splits into a
positive energy state and a negative energy state (not shown). For the
$\phi=\pi/4$ orientation the magnetic field projection is never zero and takes
constant value $\cos(\Phi_{+})=1/\sqrt{2}$ for the two facets with positive
projection, but $\cos(\Phi_{-})=-1/\sqrt{2}$ for the negative projection
facets. For $|k_{x}L|\sim L^{2}/2\ell_{c}^{2}$ the Landau levels hybridize and
evolve towards a linear in $k_{x}$ dependence when $|k_{x}L|\gg
L^{2}/\ell_{c}^{2}$. The dispersing states occur in the two corners where
$\vec{\bm{B}}\cdot\vec{\bm{n}}$ changes sign.

The hexagon is the final shape we consider. The energy spectra for the two
orientations in Fig. 2c are shown in Fig. 5. For the $\phi=\pi/6$ orientation
the spectrum is similar to the $\phi=0$ spectrum for the square in Fig. 4
since on two facets $\vec{\bm{B}}\cdot\vec{\bm{n}}=0$, and on the remaining
facets $\cos(\Phi_{+})=|\cos(\Phi_{-})|=\sqrt{3}/2$. The zero-magnetic field
states, where $\vec{\bm{B}}$ is parallel to the facets (red curve in Fig. 2c),
are determined by Eq. (10) at $s=5L/2$ and $11L/2$. These value of $s$ on the
hexagon result in $k_{x}L=\pm\sqrt{3}L/2\ell_{c}^{2}$, indicated by the
vertical lines. The $\phi=0$ orientation for the hexagon shows that Landau
levels can form on two facets with the same sign of $\cos(\Phi_{+})$, but
different values reflecting the different facet orientation. On the top
surface $\cos(\Phi_{+})=1$ but on the adjacent facet $\cos(\Phi_{+})=1/2$
resulting in $\varepsilon_{1,+}=1$, in units of $\frac{\hbar
v_{F}}{\ell_{c}}$.

Figure 5: Energy spectra for the hexagon for the two magnetic field
orientations shown in Fig. 2c). The magnetic field value corresponds to
$\ell_{c}/L=0.125$. The two dashed lines indicate energies $E=0.97$ and $1.92$
that are relevant for subsequent transport calculations. The vertical dash-
dotted lines indicates the $k_{x}L$-values where the zero-magnetic field
states occur, see text for details. The zero energy Landau levels are doubly
degenerate.

The effectively smaller magnetic field on the tilted facets translates to a
smaller width of the Landau levels, i.e., smaller $k_{x}$ interval, before
they hybridizes with the states on the facets with the opposite projection.

The energy spectra in Figs. 3-5 are obtained by direct numerical
diagonalization of Eq. (1) written in a basis of $76$ plane wave basis states,
see below Eq. (3), along with the two spin states. The number of basis states
is chosen such that the first 10 Landau levels (flat) conincide with the
analytical energy levels for the facets given in Eqs. (7) and (9).

##  III Lattice model in the presence of impurities

In the previous section we analyzed the Landau levels forming on different
facets assuming translational invariance along the wire. In the presence of
impurities $k_{x}$ is no longer a good quantum number and we need to solve the
Dirac eigenvalue problem in terms of both the surface variables $(x,s)$. To
model the effect of randomly distributed point impurities on the transport
properties of the wire we discretize the longitudinal ($x$) variable of
$H_{\mathrm{surf}}(x,s)$ using a finite difference approach with a lattice
constant $a$. This entails approximating the action of $\partial_{x}$ onto the
spatial component $\psi(x,s)$ of the wave function as

$\partial_{x}\psi(x,s)\approx\frac{1}{2a}\left[\psi(x+a,s)-\psi(x-a,s)\right],$
(11)

such that the Hamiltonian is approximated as

$\displaystyle H_{\mathrm{surf}}\psi(x,s\\!\\!\\!\\!$
$\displaystyle\\!\\!)\\!\\!$ $\displaystyle\\!\\!\approx
v_{F}\left[e\sigma_{x}A_{x}(s)-i\hbar\sigma_{y}\partial_{s}\right]\psi(x,s)$
(12) $\displaystyle-$ $\displaystyle\frac{i\hbar
v_{F}\sigma_{x}}{2a}\left[\psi(x+a,s)-\psi(x-a,s)\right].$

Discretizing the linear in momentum Hamiltonian in Eq. (1) will lead to
Fermion doubling.Nielsen and Ninomiya (1981); Stacey (1982) For two-
dimensional linearly dispersing Hamiltonians an extra quadratic term can be
added to solve the Fermion doubling Masum Habib et al. (2016); Zhou et al.
(2017)

$H_{W}=\frac{v_{F}\beta}{\hbar}\biggl{\\{}\sigma_{z}\left[(p_{x}+eA_{x}(s))^{2}+p_{s}^{2}\right]\biggr{\\}},$
(13)

where $\beta$ is a parameter with dimension of length. The value of
$\beta=a/\sqrt{3}$ gives a linear in momentum dependence of $k_{x}$ over a
large part of the Brillouin zone, but still lifts the Fermion doubling
Erlingsson et al. (2018).

Although the Hamiltonian in Eq. (13) solves the Fermion doubling it will lead
to a shift of the Landau levels, which is in opposite direction depending on
the projection of $\vec{B}$ on the local normal $\vec{n}(s)$. To illustrate
this energy shift, and also show how it can remedied, we solve the facet
Landau levels in the presence of $H_{W}$ in Eq. (13) for the homogeneous wire.
Starting with the definition of the ladder operators for the positive
projection in Eq. (25) and applying it to $H^{[+]}_{\mathrm{surf}}+H_{W}$
results in

$\displaystyle H^{[+]}_{\mathrm{surf},W}$ $\displaystyle=$
$\displaystyle\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{+})|}}{\sqrt{2}\ell_{c}}\Biggl{[}a_{[+]}\sigma_{+}+a_{[+]}^{\dagger}\sigma_{-}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\\!\sigma_{z}2\sqrt{2}\sqrt{|\cos(\Phi_{+})|}\frac{\beta}{\ell_{c}}\left(a_{[+]}^{\dagger}a_{[+]}+\frac{1}{2}\right)\Biggr{]}\\!,$
(14)

which has shifted the previously zero energy eigenstate $|0,\downarrow\rangle$
and the other eigenenergies are found by diagonalization in the
basis$\\{|n-1,\uparrow\rangle,|n,\downarrow\rangle\\}$, $n=1,2,\dots$. The
resulting eigenenergies are

$\displaystyle\varepsilon^{[+]}_{0}$ $\displaystyle=$
$\displaystyle-\frac{\hbar
v_{F}}{\ell_{c}}\frac{\beta}{\ell_{c}}|\cos(\Phi_{+})|$ (15)
$\displaystyle\varepsilon^{[+]}_{n,\pm}$ $\displaystyle=$
$\displaystyle\frac{\hbar
v_{F}}{\ell_{c}}\left(-\frac{\beta}{\ell_{c}}|\cos(\Phi_{+})|\right.$ (16)
$\displaystyle\pm$
$\displaystyle\left.\frac{\sqrt{|\cos(\Phi_{+})|}}{\sqrt{2}}\sqrt{4n+8|\cos(\Phi_{+})|\frac{\beta^{2}}{\ell_{c}^{2}}n^{2}}\right).$

For the negative projection Landau levels, the same procedure starting from
Eq. (8) and the definition of ladder operators in Eq. (27) results in the
following Hamiltonian

$\displaystyle H^{[-]}_{\mathrm{surf},W}$ $\displaystyle=$
$\displaystyle\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{-})|}}{\sqrt{2}\ell_{c}}\Biggl{[}a_{[-]}^{\dagger}\sigma_{+}+a_{[-]}\sigma_{-}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\\!\sigma_{z}2\sqrt{2}\sqrt{|\cos(\Phi_{-})|}\frac{\beta}{\ell_{c}}\left(a_{[-]}^{\dagger}a_{[-]}+\frac{1}{2}\right)\Biggr{]}\\!.$
(17)

Now, the shifted zero energy state is $|0,\uparrow\rangle$ and other
eigenergies are obtained by diagonalizing in the basis
$\\{|n-,\downarrow\rangle,|n,\uparrow\rangle\\}$, $n=1,2,\dots$ The resulting
eigenenergies are

$\displaystyle\varepsilon^{[-]}_{0}$ $\displaystyle=$
$\displaystyle\frac{\hbar
v_{F}}{\ell_{c}}\frac{\beta}{\ell_{c}}|\cos(\Phi_{-})|$ (18)
$\displaystyle\varepsilon^{[-]}_{n+1,+}$ $\displaystyle=$
$\displaystyle\frac{\hbar
v_{F}}{\ell_{c}}\left(\frac{\beta}{\ell_{c}}|\cos(\Phi_{-})|\right.$
$\displaystyle\pm$
$\displaystyle\\!\\!\\!\\!\left.\frac{\sqrt{|\cos(\Phi_{-})|}}{\sqrt{2}}\sqrt{4n+8|\cos(\Phi_{-})|\frac{\beta^{2}}{\ell_{c}^{2}}n^{2}}\right)\\!\\!.$
(19)

On inspection of Eqs. (15)-(16) and (18)-(19), one can see that the shift that
breaks the Landau level degeneracy can be removed by adding a term $v_{F}\beta
eB\cos(\Phi)=v_{F}\beta e\vec{B}\cdot\vec{n}(s)$. The amended Wilson mass term
accounting for different orientation of surface normals with respect to the
applied magnetic field is thus

$\displaystyle H_{W,B}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{v_{F}\beta}{\hbar}\biggl{\\{}\sigma_{z}\left[(p_{x}+eA_{x}(s))^{2}+p_{s}^{2}\right]+\sigma_{0}\frac{e}{\hbar}\vec{B}\cdot{\vec{n}}(s)\biggr{\\}}.$

Figure 6:  The transmission function $T(E)$ for a) triangle, b) square, and c)
hexagon. The curves corresponding to different impurity densities are offset
for clarity. For comparison, the gray lines show $T(E)$ for the clean wire
shifted by $\Delta E=\langle V\rangle$. The impurity strength is
$V_{\mathrm{imp}}=0.45\frac{\hbar v_{F}}{\ell_{c}}$, $a=0.0025L$,
$N_{x}=2400$, and $l_{c}/L=0.20$. The arrows indicate prevalent step sizes
$\Delta T$.

We can now discretize $H_{\mathrm{surf}}+H_{W,B}$, given by Eqs. (1) and
(LABEL:eq:H_wilsonB), respectively, in the $x$-variable (see App. B for
details) and model randomly placed short range impurities with

$V(x,s)=\sum_{n}V_{\mathrm{imp}}\delta(s-s_{n})\delta(x-x_{n}),$ (21)

where $V_{\mathrm{imp}}$ is the impurity strength. The density of impurities
is controlled by the parameter $a_{\mathrm{imp}}$, which gives the average
distance between impurities in units of $a$. We then use standard transport
methods based on the recursive Green’s functions method Ferry and Goodnick
(1997); Datta (1995). This is outlined in the case of circular nanowires in
Ref. Erlingsson et al. (2017), which can be directly applied here by replacing
the cylindrical perimeter $R\varphi$ with the equivalent polygonal perimeter
variable $s$. From the nanowire Green’s function one obtains the transmission
function $T(E)$.Ferry and Goodnick (1997)

In Fig. 6 we show the transmission function for the three cross-sectional
shapes, a) triangle, b) square, and c) hexagon. The transmission calculations
are done for a single impurity configuration, i.e. there is no averaging over
impurity configurations. The different curves, which are offset for clarity,
correspond to increased impurity concentrations, parameterized by
$\frac{a_{\mathrm{imp}}}{a}$. Narrow features in $T(E)$ tend to get washed out
by increased impurity concentration, but for all cases considered here the
quantized steps are quite evident. Note that in Fig. 6b (square) and Fig. 6c
(hexagon), the transmission steps are of size $\Delta T=2$, but for the
triangle in Fig. 6a the step size is predominantly of size $\Delta T=1$. This
is due to the nondegenerate Landau levels and asymmetric zero-magnetic field
states in the triangle shown in Fig. 3 which contribute only one conducting
channel.

Note that the curves corresponding to clean systems show no spurious levels
splitting due to the Wilson mass term. Experimentally there are already
transport measurements that show evidence of quantized TI nanowire states
Münning et al. (2021); Roessler et al. (2023). For such samples the nanowire
cross-sectional shape can be identified from transport properties, as we will
discuss in the next section.

##  IV Conductance and magnetic field orientation

Having established an accurate method of calculating transmission function in
TI prismatic nanowires we can make predictions regarding the two terminal
conductance, and how it behaves as a function of magnetic field orientation.
In Fig. 7 we plot the conductance for a) triangle, b) square, and c) hexagon,
as a function of magnetic field orientation angle $\phi$.

Figure 7: Conductance $G_{c}$ as a function of angle $\phi$ for a) triangle,
b) square, and c) hexagon. The conductance is calculated at two energies for
each shape, corresponding to the vertical lines in Figs. 3-5, with
$l_{B}/L=0.125$. The arrows denote the size of the conductance jumps $\Delta
G_{c}$.

There are two features that allow us to identify the shape of the nanowire
from this: $(i)$ the number of times repeated features appear in $G_{c}(\phi)$
for $\phi\in[-\pi/2,\pi/2]$, and $(ii)$ how large the conductance steps are.
In Fig. 7b) the features is $G_{c}(\phi)$ are repeated twice, for both
energies, indicating an underlying square cross-section. The number of
repeating patterns in Figs. 7a) and 7c) is three, reflecting the underlying
three-fold symmetry of both triangle and hexagon. However, the step sizes are
different: $\Delta G_{c}=1$ for the triangle and $\Delta G_{c}=2$ for the
hexagon. The origin of the difference can be understood by comparing Fig. 3
for $\phi=\pi/2$ (triangle), and Fig. 5 for $\phi=\pi/3$ (hexagon). For the
triangle only one facet can be parallel to the applied magnetic field. This
leads to the observed asymmetry in the energy spectrum. As the magnetic field
rotates the zero-magnetic field states move up or down relative to $E$ leading
to jumps of $\Delta G_{c}=1$. In the case of the hexagon, two facets can be
parallel to the applied $B$-field. This leads to two sets of such zero-
magnetic field states which results in $\Delta G_{c}=2$, as is observed in
Fig. 7c).

As can be seen in Fig. 6a (triangle) steps of $\Delta T=2$ are possible for
certain energies (see $E=1.33$ for the clean wire). But if the $G_{c}(\phi)$
can be measured for different energies, then it should possible to establish
whether $\Delta T=2$ is predominant (hexagon) or $\Delta T=1$ is predominant
(triangle).

##  V Conclusions

We have studied surface electrons in nanowires made of a TI material, in the
presence of an external magnetic field perpendicular to the longitudinal axis
of the nanowire. The cross section of the nanowire is considered polygonal, as
in most of the experimental realizations. In the present work we have
discussed the triangular, square, and hexagonal cases. The component of the
magnetic field normal to the electrons’ trajectories depends on the facet of
the prism, and consequently peculiar energy spectra are obtained, combining
local Landau levels with free motion. We have shown that the transmission
function depends on the cross sectional geometry. And, naturally, for each
geometry the conductance is anisotropic, i.e., it changes with the angular
direction of the magnetic field.

These results can be useful to characterize the real TI nanowires fabricated
in the labs Münning et al. (2021); Roessler et al. (2023). The presence of the
conduction electrons on the surface and the polygonal cross section (often
hexagonal) should lead to conductance features that we described, which should
be robust in the presence of a modest amount of disorder. The sample quality
can therefore be tested, for example by rotating the sample or the magnetic
field (in a vector magnetic system), and comparing the conductance with the
external geometry of the nanowire.

###### Acknowledgements.

This research was supported by the Icelandic Research Fund, Grant 195943, the
Knut and Alice Wallenberg Foundation (KAW) via the project Dynamic Quantum
Matter (2019.0068), and from the Swedish Research Council (VR) through Grants
No. 2019-04736 and No. 2020-00214.

##  Appendix A Facet orientation and ladder operators

We start with the linearized surface Hamiltonian in Eq. (5). Intoducing the
$k_{x}$-dependent center coordinate

$s_{\pm}(k_{x})\equiv s_{\pm}+\frac{eA_{x}(s_{\pm})+\hbar k_{x}}{eB^{\pm}},$
(22)

Eq. (5) can be written as

$\displaystyle H^{[\pm]}_{\mathrm{surf}}$ $\displaystyle=$ $\displaystyle
v_{F}\left\\{\sigma_{x}\left[-eB_{\pm}(s-s_{\pm}(k_{x}))\right]+\sigma_{y}p_{s}\right\\},$
(23)

where $B^{\pm}=B\cos(\Phi_{\pm})$. Rewriting Eq. (23) in terms of
$\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$ results in

$\displaystyle H^{[\pm]}_{\mathrm{surf}}=v_{F}$
$\displaystyle\left\\{\frac{\sigma_{+}}{2}\left[-eB_{\pm}(s-s_{\pm}(k_{x}))-ip_{s}\right]\right.$
(24)
$\displaystyle\left.+\frac{\sigma_{-}}{2}\left[-eB_{\pm}(s-s_{\pm}(k_{x}))+ip_{s}\right]\right\\}.$

The sign of $\cos(\Phi_{\pm})$ will determine the how the ladder operators are
defined in terms of $s$ and $p_{s}$, as we show below. For concreteness we now
assume $\cos(\Phi_{+})>0$. The requirement $[a,a^{\dagger}]=1$, fixes the
definition of the ladder operator on that facet, i.e.

$a_{[+]}=\frac{\ell_{c}}{\sqrt{2}\hbar\sqrt{|\cos(\Phi_{+})|}}\left[-\frac{\hbar}{\ell_{c}^{2}}\cos(\Phi_{+})(s-s_{+})+ip_{s}\right].$
(25)

In terms of these ladder operators the surface Hamiltonian of the facet is

$H^{[+]}_{\mathrm{surf}}=\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{+})|}}{\sqrt{2}\ell_{c}}\left(a_{[+]}\sigma_{+}+a_{[+]}^{\dagger}\sigma_{-}\right).$
(26)

The zero energy state of the above Hamiltonian is
$|0\rangle|\\!\\!\downarrow\rangle$ since
$a_{[+]}\sigma_{+}|0\rangle|\\!\\!\downarrow\rangle=0$, and
$a_{[+]}^{\dagger}\sigma_{-}|0\rangle|\\!\\!\downarrow\rangle=0$.

Repeating the same proceedure for the other facet results in the ladder
operator

$a_{[-]}=\frac{\ell_{c}}{\sqrt{2}\hbar\sqrt{|\cos(\Phi_{-})}|}\left[\frac{\hbar}{\ell_{c}^{2}}\cos(\Phi_{-})(s-s_{+})-ip_{s}\right],$
(27)

and the corresponding surface Hamiltonian of the facet

$H^{[-]}_{\mathrm{surf}}=\frac{\hbar
v_{F}\sqrt{|\cos(\Phi_{-})|}}{\sqrt{2}\ell_{c}}\left(a_{[-]}^{\dagger}\sigma_{+}+a_{[-]}\sigma_{-}\right).$
(28)

Again, the zero energy state of the above Hamiltonian is
$|0\rangle|\\!\\!\uparrow\rangle$, since
$a_{[-]}^{\dagger}\sigma_{+}|0\rangle|\\!\\!\uparrow\rangle=0$, and
$a_{[-]}\sigma_{-}|0\rangle|\\!\\!\uparrow\rangle=0$.

##  Appendix B Discretization with quadratic term

Starting from Eq. (12), it can be written as

$H_{\mathrm{surf}}\psi(x,s)=H_{0}\psi(x,s)+V\psi(x+a,s)+V^{\dagger}\psi(x-a,s),$
(29)

where
$H_{0}=v_{F}\left(e\sigma_{x}\hat{A}_{x}(s)-i\hbar\sigma_{y}\partial_{s}\right)$
acts as an on-site potential of the lattice points and $V=-\frac{i\hbar
v_{F}}{2a}\sigma_{x}$ acts as a nearest neighbour coupling term.

Just as for the continuous model, the discrete model admits a simple
analytical expression for its energy spectrum when the magnetic field is
absent. Applying a Bloch ansatz $\psi(x,s)=e^{ik_{x}x}\phi_{k_{x}}(s)$ results
in

$\displaystyle H_{\mathrm{surf}}\psi(x,s)$ $\displaystyle=$
$\displaystyle\left(\hat{H}_{0}+\hat{V}e^{ik_{x}a}+\hat{V}^{\dagger}e^{-ik_{x}a}\right)e^{ik_{x}x}\phi_{k_{x}}(s)$
$\displaystyle=$ $\displaystyle\hbar
v_{F}\left(\frac{\sin(k_{x}a)}{a}\sigma_{x}+k_{s}\sigma_{y}\right)e^{ik_{x}x}\phi_{k_{x}}(s),$

which leads to eigenvalues

$E_{\pm}=\pm\hbar v_{F}\sqrt{\frac{\sin^{2}(k_{x}a)}{a^{2}}+k_{s}^{2}},$ (30)

which will converge to the spectrum of the continuous model in the limit
$a\rightarrow 0$. The anti-periodic boundary conditions is inherited by
$\phi_{k_{x}}$, i.e., $\phi_{k_{x}}(s)=-\phi_{k_{x}}(s+P)$, which results in
$k_{s}=\frac{2\pi}{P}(n+\frac{1}{2})$ with integer $n$.

There is however a serious issue with the discrete spectrum solution which is
that it will close around the edges of the Brillouin zone ($k_{x}=\pm\pi/a$),
effectively doubling the number of modes. This is the well known Fermion
doubling problem. As it is an unwanted consequence of projecting our model
onto a lattice, we remedy the problem by adding a Wilson mass term [Eq. (13)]
to our surface model. The finite difference version of the quadratic term
yields

$\displaystyle\partial_{x}^{2}\psi(x,s)\approx\frac{\left[\psi(x+a,s)-2\psi(x,s)+\psi(x-a,s)\right]}{a^{2}},$

such that the on-site potential and nearest neighour terms become

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle
v_{F}\left[e\sigma_{x}\hat{A}_{x}(s)-i\hbar\sigma_{y}\partial_{s}+\frac{\beta}{\hbar}\sigma_{z}\left(\frac{2\hbar^{2}}{a^{2}}+e^{2}\hat{A}_{x}^{2}(s)\right)\right],$
$\displaystyle V$ $\displaystyle=$ $\displaystyle
v_{F}\left[-\frac{i\hbar}{2a}\sigma_{x}+\frac{\beta}{\hbar}\sigma_{z}\left(-\frac{\hbar^{2}}{a^{2}}-\frac{i\hbar
e\hat{A}_{x}(s)}{a}\right)\right].$ (33)

For $B=0$, the plane wave solution now results in

$\displaystyle H_{\mathrm{surf}}\psi(x,s)=\hbar v_{F}$
$\displaystyle\biggl{[}\frac{\sin(k_{x}a)}{a}\sigma_{x}+4\beta\frac{\sin^{2}(k_{x}a/2)}{a^{2}}\sigma_{z}$
(34) $\displaystyle+k_{s}^{2}\sigma_{y}\biggr{]}e^{ik_{x}x}\phi_{k_{x}}(s),$

and corresponding eigenvalues

$E_{\pm}=\pm\hbar
v_{F}\sqrt{\frac{\sin^{2}(k_{x}a)}{a^{2}}+16\beta^{2}\frac{\sin^{4}(k_{x}a/2)}{a^{4}}+k_{s}^{2}}.$
(35)

Taylor expanding the sinusoidal terms involving $k_{x}$ gives

$\frac{\sin^{2}(k_{x}a)}{a^{2}}+16\beta^{2}\frac{\sin^{4}(k_{x}a/2)}{a^{4}}\approx
k_{x}^{2}+\left(\beta^{2}-\frac{a^{2}}{3}\right)k_{x}^{4},$ (36)

which implies that the linear behaviour of the energy spectrum is best
maintained by setting $\beta=\frac{a}{\sqrt{3}}$ Erlingsson et al. (2018). In
addition to maintaining the structure of the Dirac cone around the origin, the
Wilson mass term succeeds in ”lifting away” the spurious Dirac cone at the
edges of the Brillouin zone. We can see this by considering the lowest energy
modes at the edges where we have $E_{\pm}(k_{x}=\pm\pi/a,k_{s}=0)=\pm\hbar
v_{F}\frac{4}{a\sqrt{3}}$ such that the Dirac cone can always be lifted
further away by decreasing the lattice constant $a$. Note also that in the
limit of $a\rightarrow 0$ that the energy spectrum converges to the one of the
continuum model.

Having $H_{0}$ and $V$, along with an appropriate impurity potential
$V_{\mathrm{imp}}$, defined they can then be written as matrices, indicated by
$[\cdot]$, in a truncated basis of plane waves along the perimeter, and the
spin basis. The Green’s function lattice equation then becomes

$\displaystyle(E[\mathbb{I}]-[H])G_{i,j}-[V]G_{i+1,j}-[V^{\dagger}]G_{i-1,j}=[\mathbb{I}]\delta_{i,j},$

where $[H]=[H_{0}]+[V_{\mathrm{imp}}]$ and $i,j$ are lattice positions along
the wire. The above equation can then be implemented using standard recursive
GF method Ferry and Goodnick (1997); Datta (1995); Erlingsson et al. (2017).

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