Increased localization of Majorana modes in antiferromagnetic chains on superconductors

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Increased localization of Majorana modes in antiferromagnetic chains on

superconductors

Daniel Crawford,1Eric Mascot,1Makoto Shimizu,2Roland

Wiesendanger,3Dirk K. Morr,4Harald O. Jeschke,5and Stephan Rachel1

1School of Physics, University of Melbourne, Parkville, VIC 3010, Australia

2Department of Physics, Okayama University, Okayama 700-8530, Japan

3Department of Physics, University of Hamburg, D-20355 Hamburg, Germany

4University of Illinois at Chicago, Chicago, IL 60607, USA

5Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan

Magnet-superconductor hybrid (MSH) systems are a key platform for custom-designed topological

superconductors. Ideally, the ends of a one-dimensional MSH structure will host Majorana zero-

modes (MZMs), the fundamental unit of topological quantum computing. However, some of the

experiments with ferromagnetic chains show a more complicated picture. Due to tiny gap sizes and

hence long coherence lengths MZMs might hybridize and lose their topological protection. Recent

experiments on a niobium surface have shown that both ferromagnetic and antiferromagnetic chains

may be engineered, with the magnetic order depending on the crystallographic direction of the

chain. While ferromagnetic chains are well understood, antiferromagnetic chains are less so. Here

we study two models inspired by the niobium surface: a minimal model to elucidate the general

topological properties of antiferromagnetic chains, and an extended model to more closely simulate a

real system by mimicking the proximity eﬀect. We ﬁnd that in general for antiferromagnetic chains

the topological gap is larger than for ferromagnetic ones and thus coherence lengths are shorter for

antiferromagnetic chains, yielding more pronounced localization of MZMs in these chains. While for

some parameters antiferromagnetic chains may be topologically trivial, we ﬁnd in these cases that

adding an additional adjacent chain can result in a nontrivial system, with a single MZM at each

chain end.

I. INTRODUCTION

One-dimensional (1D) topological superconductors

(TSCs) are candidates for hosting Majorana zero-modes

(MZMs) [1–3]. These quasiparticles obey non-Abelian

statistics and may be used for topological, (i.e., fault-

tolerant) quantum computing [4,5]. 1D TSCs can

be engineered — there are a myriad of proposals [6–

14] — although to date there has not been any com-

pletely unambiguous experimental realization. Magnet-

superconductor hybrid (MSH) structures constitute a

particularly promising platform for MZMs, which in-

volve depositing chains or islands of magnetic adatoms

on the surface of a superconductor by self-assembly or

single-atom manipulation using a scanning-tunneling mi-

croscope (STM) [9–11,14–16]. STM techniques allow for

both atomic-scale control of structures and also atomic-

resolution measurements such as spectroscopy [17,18],

reconstruction of density of states [19], and spin-polarized

maps [20]. A rapidly growing number of MSH systems

have been studied in recent years [15–33].

The ﬁrst MSH experiment [15] involved Fe chains

on Pb(110), with the chains grown via self-assembly.

The authors observed signature zero-energy end states,

demonstrating the viability of the platform. Subsequent

experiments replicated these results [23] while reducing

disorder [19] and increasing spectral resolution [17,22].

The state of the art progressed by transitioning from

self-assembled chains to artiﬁcially constructed Fe chains

on Re(0001) using a STM tip [16]; because these chains

are constructed atom-by-atom they are crystalline and

disorder-free. Alongside these developments, there have

also been ﬁrst attempts to engineer 2D structures involv-

ing a Pb/Co/Si(111) heterostructure [34] and Fe islands

on Re(0001)-O(2×1) [18] which showed compelling signa-

tures of chiral Majorana modes.

Because Nb is the elemental superconductor with high-

est transition temperature at ambient pressure and has

a relatively large spectral gap of 1.51 meV, it should be

an ideal MSH substrate. Only recently has it been possi-

ble to prepare a suﬃciently clean Nb(110) surface. The

ﬁrst Nb(110) experiments studied single Fe adatoms [26],

which was rapidly followed up by Mn chains [27]. In

the latter experiment there was suﬃcient spectral res-

olution to observe in-gap Yu-Shiba-Rusinov bands [35]

and to identify a signature topological band inversion.

Point-like zero-energy end states are not observed in

these Mn/Nb(110) systems but instead a periodic accu-

mulation of spectral weight along the sides of the chain,

dubbed side features; similar features were also observed

in Fe/Nb(110) systems [28]. These are identiﬁed as hy-

bridized Majorana modes, and have been proposed to

have the same origin [28] as the previously observed dou-

ble eye feature [22]. Cr chains on Nb(110) have also been

studied, with no signs of MZMs [29–32].

Thus far most theoretical and experimental work has

focused on chains with ferromagnetic (FM) order. Most

simulations of 1D MSH systems are also usually based on

simple models which couple magnetic and superconduct-

ing orbitals, and only for one-dimensional structures. Re-

alistic systems of course consist of many atoms, involving

s-, p-, and d-orbitals, and are constructed from a large 3D

arXiv:2210.11587v2 [cond-mat.supr-con] 13 Feb 2023

FIG. 1. Geometry of magnetic chains (M=Mn, Fe, Cr)

on Nb(110) surfaces. Distances between magnetic ions are

determined by the substrate as (a) d[1¯

11]

M−M= 2.86˚

A, (b)

d[001]

M−M= 3.30˚

A, and (c) d[1¯

10]

M−M= 4.51˚

superconducting substrate with a (short) chain deposited

somewhere on the surface. Realistic conventional super-

conductors also have small spectral gaps (typically <2

meV), while toy models consider gap sizes of hundreds of

meV. Some experiments [15] appear to be consistent with

these very simple models featuring point-like MZMs, but

later experiments revealed a more complex spatial struc-

ture of the low-energy modes [22,27]. It seems that more

realistic models, possibly based on ab initio methods [28],

are necessary to capture all relevant details of MSH sys-

tems.

Ferromagnetism is not essential for realizing MZMs

in MSH chains. Simple single-band models of antiferro-

magnetic (AFM) chains show MZMs [36,37], as do anti-

ferromagnetic nanowires [38] and superconducting helical

magnets [13]. Not only is the presence of MZMs invari-

ant to the speciﬁc magnetic ground state, but for instance

classical Monte-Carlo methods show that FM, AFM, and

spin spiral ground states exist in MSH chains [37,39].

Inspired by these results we present density functional

theory (DFT) calculations for Mn, Fe, and Cr chains

on Nb(110), oriented along three crystalline directions.

We ﬁnd that exchange couplings between the magnetic

moments may be FM or AFM depending on the type of

adatom and on the chain direction. Indeed, recent exper-

iments [20,40,41] have found that Mn atoms deposited

on the surface of Nb(110) are ferromagnetic along the

[001] direction and antiferromagnetic along the [1¯

11] di-

rection. In these experiments spin-polarized STM is used

to measure the diﬀerential conductance of Mn ultrathin

ﬁlms and chains. Applying a soft out-of-plane magnetic

ﬁeld to the magnetic sample reveals FM or AFM order

in diﬀerent directions.

In this work we investigate the spectral and topological

properties of chains of magnetic adatoms on the surface

of a conventional superconductor motivated by the recent

experimental developments in Nb-based MSH structures.

Depending on the crystalline direction we simulate either

ferromagnetic or antiferromagnetic chains. We typically

include the substrate in the simulations, and also focus

on small gaps so as to account for more realistic scenarios.

In Sec. II we present magnetic couplings of adatom chains

on Nb(110) derived from DFT. In Sec. III we introduce

two tight-binding models inspired by the Nb(110) sur-

face. In sections Sec. IV we study their topological phase

diagrams. In sections Sec. V, Sec. VI, and Sec. VII we

study diﬀerent geometries of these models in real space,

including searching for side features. In Sec. VIII we dis-

cuss our results and summarize the work in Sec. IX.

II. DFT MODELING

We employ DFT calculations based on the full po-

tential local orbital (FPLO) basis set [42] and general-

ized gradient approximation (GGA) exchange correlation

functional [43] to investigate the magnetic interactions of

transition metal (TM) chains on the Nb(110) surface.

For this purpose, we construct three diﬀerent supercells

with two symmetry inequivalent TM sites for the three

directions [1¯

11], [001], and [1¯

10] on the Nb(110) sur-

face (Fig. 1). These correspond to nearest, next-nearest,

and third-nearest neighbor distances for the TM adatoms

(with exchange couplings J1,J2,J3, respectively); equiv-

alently, these correspond to TM-TM distances 2.86 ˚

3.30 ˚

A, and 4.51 ˚

A, respectively. This has been found to

be the equilibrium position on the Nb(110) surface both

theoretically and experimentally. We use the projector

augmented wave basis as implemented in VASP [44,45]

to relax the relevant supercells for each TM considered.

We then extract the Heisenberg exchange interaction by

a simple version of DFT energy mapping which is very

successful for insulating quantum magnets [46] but has

been shown to work for metallic systems as well [47]. For

this purpose, we calculate the energies of FM and AFM

states with high precision. Note that due to the metal-

lic nature of the TM chain on Nb(110) systems, the en-

ergy mapping is not as precise as for insulating magnets.

Metallicity has the consequence that magnetic moments

can diﬀer between FM and AFM spin conﬁgurations as

well as for diﬀerent TM distances. Nevertheless, the ap-

proach can give robust information about the sign of the

TABLE I. Magnetic exchange energies (JiM2) and total mag-

netic moment per transition metal adatom (MT M ), calculated

within GGA and at least 6 ×6×6kpoints (the kmesh was

not reduced in the slab direction kz). Positive (negative) mag-

netic exchange energies indicates AFM (FM) order.

transition metal direction JiM2(meVµ2

B)MTM (µB)

Mn [1¯

11] 27 2.7

Mn [001] -29 2.6

Mn [1¯

10] -9 2.0

Fe [1¯

11] -10 2.0

Fe [001] -26 1.8

Fe [1¯

10] 23 1.7

Cr [1¯

11] 10 2.6

Cr [001] -41 2.7

Cr [1¯

10] 10 2.4

FIG. 2. Schematics of magnetic adatoms on an Nb-inspired

superconducting substrate. We investigate the following se-

tups: (a) Shiba chain with FM order; (b) Shiba chain with

AFM order; (c) T-junction comprised of an AFM chain and

a FM chain; and (d) AFM two-leg ladder.

exchange and about the relative size of the exchanges in

chains running along diﬀerent directions on the Nb(110)

surface. The results of our calculations are summarized

in Table I. While Mn chains along [001] and [1¯

10] are

FM, along the [1¯

11] direction they are AFM, in agree-

ment with Ref. [20,27,41]. We predict Fe chains to have

AFM order along [1¯

10] and FM along [1¯

11] and [001],

and Cr chains to have AFM order along [1¯

10] and [1¯

11],

and FM along [001]. Thus we can choose FM or AFM

coupled chains by choosing an appropriate TM adatom

and crystal direction.

III. MODELS AND METHOD

We work with two models inspired by the Nb(110) sur-

face:

•the minimal model, in which the unit cell has only

one orbital on a square lattice. Both superconduc-

tivity and magnetic couplings are associated with

this orbital. In one direction the magnetic cou-

plings are AFM and in the other they are FM.

Magnetism is restricted to chains on a subset of

the lattice.

•the extended model, in which the unit cell has sepa-

rate superconducting and magnetic sites. Because

the substrate is included in the unit cell we work

with this model only in a 1D geometry. By compar-

ing this model to the minimal model we can test for

trends in the topological physics of AFM chains.

For both models we assume a constant order parame-

ter for simplicity; while we can treat the order parameter

self-consistently this does not change any of the impor-

tant physics [48]. We also only consider disorder-free

surfaces because FM chains are robust against disorder

[48].

As seen in Fig. 1, the [001] and [1¯

11] directions form

a 60°angle with respect to each other. In the minimal

model we place the AFM and FM chains at right angles

to each other; this model is concerned only with the es-

sential physics and so the choice of the relative angles

has been neglected in the following. The model could be

extended to the full bcc(110) surface, but this is left for

future work.

The 1D FM Bogoliubov–de Gennes (BdG) Hamiltoni-

ans we consider possess only particle-hole symmetry so

they belong to the D symmetry class in the periodic table

of topological classes [49–51]. Thus the relevant topologi-

cal invariant is a Z2index, Kitaev’s Majorana number [1],

M= sign(Pf[i˜

H(0)]Pf[i˜

H(π)]),(1)

where ˜

H(k) is the Hamiltonian in the Majorana basis

at momentum k. In contrast, the 1D AFM BdG Hamilto-

nians we consider possess particle-hole, chiral, and time-

reversal symmetries so they belong to the BDI symmetry

class. In this case the relevant topological invariant is a

Zindex, the winding number [52],

W=1

2πi Zπ

−π

dk ∂kdet[Vk]

det[Vk],(2)

where Vkis the oﬀ-diagonal block of the Hamiltonian

in the Majorana basis at momentum k. Only when W

is odd do we get unpaired MZMs at chain ends, because

pairs of MZMs annihilate; in fact, we can relate the two

invariants by M=Wmod 2 [38]. Nontrivial phases

W 6= 0,M=−1 are only valid when there is a bulk spec-

tral gap, and phase transitions, i.e., changes of W,M

are associated with a gap closing. Thus by computing

the spectral gap in periodic boundary conditions (PBC)

we can conﬁrm a topological phase via gap closings, and

in open boundary conditions (OBC) we can identify po-

tential topological phases by identifying extended regions

with zero-energy states in parameter space (i.e., via the

bulk-boundary correspondence). When we include the

substrate we cannot compute the invariant because the

system becomes inhomogenous. In these cases we rely on

the bulk-boundary correspondence to identify topological

phases.

A. Minimal model

We start with the prototypical FM Shiba lattice

model [21,53,54]. There is a single orbital per unit cell

(with spin degree of freedom) which captures both su-

perconductivity and magnetism. The superconductor is

modeled as a two-dimensional square lattice Λ spanned

by ˆ

e1and ˆ

e2with magnetic adatoms occupying a subset

Λ∗⊆Λ. We extend this model to support AFM order by

doubling the unit cell such that there is AFM order in the

xdirection and FM order in the y(i.e., row-wise AFM

order on a square lattice). We emphasize that though

the magnetic unit cell is doubled, all other parameters

remain identical on both sublattices. For simplicity we

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IncreasedlocalizationofMajoranamodesinantiferromagneticchainsonsuperconductorsDanielCrawford,1EricMascot,1MakotoShimizu,2RolandWiesendanger,3DirkK.Morr,4HaraldO.Jeschke,5andStephanRachel11SchoolofPhysics,UniversityofMelbourne,Parkville,VIC3010,Australia2DepartmentofPhysics,OkayamaUniversity,Okayama7...

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Increased localization of Majorana modes in antiferromagnetic chains on superconductors

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