RKKY interactions mediated by topological states in transition metal doped bismuthene Emmanuel V. C. Lopes1aE. Vernek2 1and Tome M. Schmidt1b

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RKKY interactions mediated by topological states in transition metal

doped bismuthene

Emmanuel V. C. Lopes,1, a) E. Vernek,2, 1 and Tome M. Schmidt1, b)

1)Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais 38400-902,

Brazil

2)Nanoscale and Quantum Phenomena Institute, and Department of Physics & Astronomy, Ohio University, Athens, Ohio 45701,

USA

(Dated: 23 March 2023)

We have investigated magnetic interactions between transition metal ions in bismuthene topological insulators with

protected edge states. We ﬁnd that these topological states have a crucial role in the magnetic interactions in 2D

topological insulators. Using ﬁrst-principles and model Hamiltonian, we make a comparative study of transition metal

doped bulk and nanoribbon bismuthene. While a direct overlap between the transition metal prevails in gapped bulk

bismuthene, at the borders of nanoribbons, a long-range magnetism is present. The exchange interactions are well

described by a Ruderman-Kittel-Kasuya-Yosida-like Hamiltonian mediated by massive and topological states. Our

results show a dominance of antiferromagnetism promoted by the topological states, preserving the spin-locked Dirac

crossing states due to a global time-reversal symmetry preservation. This extended magnetic interactions mediated

mainly by massless electrons can increase the spin diffusion length being promising for fast dissipationless spintronic

devices.

I. INTRODUCTION

Since graphene experimental synthesis in 2004,1the class

of two-dimensional (2D) materials became an important sub-

ﬁeld in condensed matter physics. A great deal of effort

has been dedicated to predict and synthesize those materi-

als within potential technology applications, such as quantum

computing, spintronics, high speed and storage devices.2–5

One special subclass of 2D materials is the quantum spin Hall

(QSH) insulator, largely studied motivated by the presence of

strong spin-orbit coupling (SOC), that leads to band inversion

in bulk with spin-polarized edge channels protected by time-

reversal symmetry. Since 2006, many of these new materials

were predicted and synthesized. Examples encompass HgTe,

bismuthene, silicene, stanene6–13 and many others predicted,

including topological crystalline insulators.14–20

Much of the interest in these 2D materials relies on the

possibility of their application for practical purposes. There-

fore, ﬁne control of their intrinsic electronic and magnetic

properties is pivotal. Fortunately, doping provides promis-

ing ways to manipulate electronic properties in 2D materials.

More interestingly, it has been showed that magnetic prop-

erties can be modiﬁed by replacement of atoms in monolay-

ers like graphene by cobalt or nitrogen,21–23 for instance. As

compared to normal 2D materials, QSH insulators have an ad-

vantage for applications since massless spin-locked states are

present at the edges or at the interfaces with trivial materials.

As such, control over the spins of the conducting electrons has

a consequence on the currents. Therefore, a detailed compre-

hension of interaction of electrons in these topological states

with magnetic impurities is crucial to design devices for prac-

tical applications.

In this context, bismuthene is a promising QSH system,

thanks to its large bandgap as compared to other 2D topo-

a)Electronic mail: emmanuelvictor96@gmail.com

b)Electronic mail: tschmidt@ufu.br

logical insulators (TIs).24,25 Also the search for high speed

dissipationless devices using bismuthene has become more

intense after its synthesis on distinct surfaces using differ-

ent techniques.9,10,15,26–29 Moreover, bulk bismuthene doped

with transition metals (TMs) is predicted to rich magnetic

properties.30–33 The TMs provide the bismuthene with mag-

netic impurities which, at a low density, can interact with

each other leading to stable magnetic phases. What so far

is poorly understood is the role played by topological states

on the magnetic stability and, consequently, in the 1D spin

current. The spin diffusion length must be greater than the de-

vice length scale to preserve the electron’s spin.34 In this way,

it is expected that long-range exchange interactions mediated

by edge states may stabilize a magnetic order in the system.35

Indeed, long-range Ruderman-Kittel-Kasuya-Yosida (RKKY)

interactions36–38 have been predicted in 3D topological crys-

talline insulator surfaces39 and in doped MoS2ﬂakes.40,41

Motivated by the development progress of the 2D TI bis-

muthene, in this work, we used it as a platform to investigate

the role played by topological states on the magnetic stability,

by a direct substitution of bismuth atoms by TM ions. A com-

parative study of the magnetic correlations is performed in the

bulk as well at the edges, where the topological phase has a di-

rect impact on the exchange energy between impurities. With

these results, we propose a low-energy effective Hamiltonian

that accounts for the topological edge states and is coupling

to the magnetic impurities. With this, upon integrating out the

electrons’ degrees of freedom, we derive an effective coupling

between the impurities. The remarkable agreement between

the RKKY-like couplings obtained from the effective model

and the ab initio calculations conﬁrms the role of the topolog-

ical edge states in the inter-impurity exchange interaction.

II. METHODOLOGY

We combine ﬁrst-principles calculations and a model

Hamiltonian to elucidate the effect of the topological states

on the magnetic exchange interactions. The ab initio elec-

arXiv:2210.14658v3 [cond-mat.mtrl-sci] 22 Mar 2023

tronic structures were computed within density functional the-

ory (DFT) using the projector augmented waves implemented

in the VASP code.42,43 A cut-off energy of 420 eV of the

kinetic energy for the plane wave basis set has been used.

The exchange-correlation was described by the generalized

gradient approximation within the Perdiew-Burke-Ernzehof’s

functional,44 including fully relativistic pseudopotentials. To

get a better description of the electronic correlation of the

localized 3dTM orbitals, we use the DFT+U45 approxima-

tion. The onsite optimized effective Hubbard U parameter of

3.25 eV has been obtained for dorbitals.46,47 The topologi-

cal invariants have been computed using the Wannier charge

center proposed by Soluyanov et al.48,49 The inter-impurity

exchange interaction was studied by appropriate large unit

cells, with up to 308 atoms in bulk and slab systems. The

coupling between the TM impurities has been also interpreted

by using model Hamiltonian50 and compared with DFT re-

sults. These low-lying energy bands take into account the in-

teraction between the TM impurities, including the topologi-

cal edge states, through second order perturbation theory, as

will be discussed in Sec. IV.

III. RESULTS AND DISCUSSION

We will use a vanadium ion as a prototype of 3dTM doped

bismuthene to investigate the effects of the topological states

on the magnetic interactions. The results can be extended to

other 3das well as to 4dTM ions. Vanadium doped bulk bis-

muthene forms covalent bonds with Bi host atoms, inducing

a small inward relaxation of the Bi nearest neighbors. The

bond-lengths of the Bi neighbors to the impurity is 2.81 Å,

which is shorter than the pristine Bi-Bi distance of 3.12 Å, in

agreement with previous results and similar to other TMs.31–33

Defects can modify the electronic properties of topologi-

cal insulators.30,51 As we can see in Fig. 1, a magnetic im-

purity induces spin splitting which is enhanced by the strong

SOC interactions. The impurity introduces new energy lev-

els inside the bulk bandgap, with strong p-d hybridization

moving the Fermi level toward the valence band maximum

(VBM) [Fig. 1(b)]. The correction in the Coulomb interac-

tions by introducing the onsite effective U in the TM dor-

bitals opens up the bandgap, while keeping the p-type charac-

ter [see Fig. 1(c)]. The topological non-trivial phase character-

ized by an inverted bandgap [showed for pristine bismuthene

in Fig. 1(a)] is also better described with the introduction of

the effective U interaction (compare Figs. 1(b) and 1(c)). The

topological gap is now between the impurity state and the split

conduction band. The band structure shown in Fig. 1is for 2%

of V doping concentration. The topological gap is maintained

by increasing the TM doping (checked up to 5%), but it re-

duces the global bulk energy bandgap.31

A. Magnetic interactions mediated by TM doped gapped

bulk bismuthene

As the TM impurity introduces new energy levels around

the Fermi level (as shown in Fig. 1), it brings a new possi-

MΓK

(c)

MΓK

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

E - EF (eV)

(a)

MΓK

(b)

FIG. 1. Band structure with atomic orbitals projected on pxy (red)

and pz(blue) Bi ions, and transition metal d(green) orbitals for: pris-

tine bismuth monolayer (a), vanadium doped without (b) and with (c)

effective U interaction. The Femi level (horizontal dashed line) is at

zero energy.

bility for magnetic research in bismuthene, once this substi-

tution allows not only magnetic interaction between the ions,

but also conduction electrons can maximize orbital overlap.52

Our calculations show that non-interacting TM ion doped bulk

bismuthene has an out-of-plane magnetization of 3 µBper

ion. By increasing the number of impurities in an enlarged

unit cell, we can investigate the magnetic impurity interac-

tions. We ﬁrst focus on the bulk doping, where V impurities

have been diluted doped at Bi sites in a 2D periodic Bloch

supercell. Figure 2(a) shows the total energy difference be-

tween adjacent TM ions aligned ferromagnetically (FM) and

antiferromagnetically (AFM) as a function of the distance be-

tween the impurities. The exchange interactions support a FM

phase for most of the V-V distance in bulk bismuthene, with a

small oscillation between the two phases. The most stable FM

conﬁguration occurs for a short-range exchange interaction

for a V-V distance around 9 Å with the FM phase 120 meV

more stable than the AFM one. This scenario is consistent

with a predominant direct exchange interaction, however, the

magnetic exchange energy is nonvanishing even for distances

around 15 Å [Fig. 2(a)], indicating some indirect magnetic

exchange contribution that can be mediated by acceptor states

[see Fig. 1(c)].

B. Magnetic interactions mediated by topological states

The effect of massless spin-locked electrons on the mag-

netic interactions has been investigated by promoting TM

doped at the borders of a bismuthene ribbon, where the topo-

logical edge states are. In contrast to TM doped bulk, the

interaction between the impurities and topological states turns

the magnetic phase strongly dependent on the TM distances,

with a strong oscillation, as shown in Fig. 2(b). The pres-

ence of topological states induces mostly antiferromagnetism,

contrary to the gapped bulk system. The oscillation on the

magnetic exchange interaction as a function of inter-impurity

distance is one of the main characteristics of RKKY interac-

tion, as predicted in doped bulk 2D materials.52–55 The mag-

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RKKYinteractionsmediatedbytopologicalstatesintransitionmetaldopedbismutheneEmmanuelV.C.Lopes,1,a)E.Vernek,2,1andTomeM.Schmidt1,b)1)InstitutodeFísica,UniversidadeFederaldeUberlândia,Uberlândia,MinasGerais38400-902,Brazil2)NanoscaleandQuantumPhenomenaInstitute,andDepartmentofPhysics&Astronomy,OhioUniv...

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RKKY interactions mediated by topological states in transition metal doped bismuthene Emmanuel V. C. Lopes1aE. Vernek2 1and Tome M. Schmidt1b

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