High-ﬁeld Studies on Layered Magnetic and Polar Dirac Metals Novel Quantum Transport Phenomena Coupled with Spin-valley Degrees of Freedom

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High-ﬁeld Studies on Layered Magnetic and Polar Dirac Metals:

Novel Quantum Transport Phenomena Coupled with Spin-valley Degrees of

Freedom

Hideaki Sakai∗

Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

Abstract

Recently, the interplay between the Dirac/Weyl fermion and various bulk properties, such as magnetism,

has attracted considerable attention, since unconventional transport and optical phenomena were discov-

ered. However, the design principles for such materials have not been established well. Here, we propose

that the layered material AMnX2(A: alkaline and rare-earth ions, X: Sb, Bi) is a promising platform for

systematically exploring strongly correlated Dirac metals, which consists of the alternative stack of the X−

square net layer hosting a 2D Dirac fermion and the A2+-Mn2+-X3−magnetic block layer. In this article,

we shall review recent high-ﬁeld studies on this series of materials to demonstrate that various types of

Dirac fermions are realized by designing the block layer. First, we give an overview of the Dirac fermion

coupled with the magnetic order in EuMnBi2(A=Eu). This material exhibits large magnetoresistance by

the ﬁeld-induced change in the magnetic order of Eu layers, which is associated with the strong exchange

interaction between the Dirac fermion and the local Eu moment. Second, we review the Dirac fermion cou-

pled with the lattice polarization in BaMnX2(A=Ba). There, spin-valley coupling manifests itself owing to

the Zeeman-type spin–orbit interaction, which is experimentally evidenced by the bulk quantum Hall eﬀect

observed at high ﬁelds.

arXiv:2210.01717v1 [cond-mat.str-el] 4 Oct 2022

I. INTRODUCTION

The Dirac/Weyl fermion in solids is a quasi-particle state described by a relativistic Dirac/Weyl

equation. Bismuth has been long known as a typical three-dimensional Dirac system1–3, whereas

graphene4and surface states of topological insulators5,6 have been recently extensively investi-

gated as two-dimensional ones. Their unique physical properties, such as ultrahigh mobility be-

yond conventional semiconductors, have attracted much attention in terms of potential application

as well as fundamental physics7. In particular, reﬂecting the topologically nontrivial band struc-

ture, the relativistic quantum Hall eﬀect (QHE) manifests itself in graphene8,9 and topological

insulator ﬁlms10. There, the half-integer quantization of the Hall plateaus and the zero-energy

Landau level at the charge neutral Dirac point were experimentally discovered8,9, which are as-

sociated with the Berry phase of the Dirac fermion and hence have no analog in conventional 2D

electron gas.

More recently, the interplay between relativistic quasi-particles and various physical proper-

ties, such as magnetism, polarity (ferroelectricity), and electron correlation, has been of particular

interest. For instance, in magnetic topological insulator thin ﬁlms, quantum Hall phenomena

coupled with magnetic order were experimentally observed. Typical examples are the quantum

anomalous Hall eﬀect showing a Hall plateau at zero ﬁeld11,12 and an axion insulating state show-

ing a zero Hall plateau13. In addition to 2D thin ﬁlms, 3D bulk materials, so called Dirac/Weyl

semimetals14,15, have also attracted signiﬁcant interest for the coupling with a wide range of quan-

tum phenomena. In Weyl semimetals, the peculiar magnetic order and (polar) lattice structure give

rise to a Weyl point (i.e., nondegenerate linearly crossing bands), leading to many unprecedented

bulk properties, such as large anomalous Hall/Nernst eﬀects16–21, magnetoresistance eﬀects (chi-

ral anomalies)22–25, and magnetooptical responses26,27. In perovskite iridates, the highest mobility

among the oxides was reported, which likely originates from the Dirac point formed in the vicinity

of the Mott insulating (antiferromagnetic) state28.

It is thus important to systematically explore the Dirac/Weyl semimetals to expand their variety

as well as establish the underlying physics, which has never been demonstrated thus far. This is be-

cause each Dirac/Weyl semimetal has its own peculiar magnetic and/or lattice structure and hence

each material was discovered or predicted independently. How can we explore the Dirac/Weyl

semimetals systematically? To address this issue, the layer structure consisting of the insulat-

ing block and conducting Dirac/Weyl fermion layers should be promising, since the Dirac/Weyl

fermion layer can be controlled and modiﬁed by the coupling with the physical properties of the

block layer, such as magnetic order, lattice distortion, and charge imbalance. This concept of ma-

terial design (known as the block-layer concept or nanoblock integration) was ﬁrst proposed for

exploring high-Tccuprate superconductors29, followed by the application to a variety of strongly

correlated systems (e.g., high-performance thermoelectric materials30 and colossal magnetoresis-

tive materials31). Here we propose that the layered material AMnX2(A: alkaline and rare-earth

ions, X: Sb, Bi) can realize the block-layer concept for the Dirac/Weyl semimetals, since AMnX2

consists of the alternative stack of the X−square net layer hosting quasi 2D Dirac fermions and

the Mott insulating A2+-Mn2+-X3−block layer [Fig. 1(a)].

In this review, we give an overview of the novel physics of Dirac fermions coupled with the

magnetic order and lattice polarization in this series of compounds. Owing to the high-mobility

features of the Dirac fermion layer spatially separated from the block layer, high-ﬁeld measure-

ments using a pulsed magnet enable us to observe quantum oscillations (and even quantum Hall

eﬀects) with high accuracy. This plays a vital role in revealing the electronic structure as well as

transport properties of the correlated Dirac fermion in AMnX2.

II. MEASUREMENTS AT HIGH FIELDS

We performed transport and magnetization measurements up to ∼55 T using a non-destructive

mid-pulse magnet equipped with a 900 kJ capacitor bank at the International MegaGauss Science

Laboratory at the Institute for Solid State Physics, The University of Tokyo. All the measure-

ments were performed on single crystals of AMnX2grown by a high-temperature solution growth

technique32. The measurement temperature range was 1.4–150 K. The resistivity was measured by

a lock-in technique at 100 kHz with a typical AC excitation of 1–10 mA, where the ﬁeld direction

was controllable by using a sample probe equipped with a rotating stage. The tilt angle of the ﬁeld

was determined by two pick-up coils ﬁxed parallel and perpendicular to the sample stage. As de-

tailed in Sects. IV C and V C, the measurement of the angular dependence of quantum oscillation

is essential for unveiling the microscopic properties of the Dirac fermions. The magnetization at

high ﬁelds was measured by the induction method, using coaxial pickup coils.

FIG. 1: (Color online) (a) Lattice structure of AMnX2(A: alkaline and rare-earth ions, X: Sb, Bi). (b) Band

structure for A=Sr, a basic material of AMnX2. Upper panel: 2D band structure calculated by a tight-binding

method for the X−square net coordinated with Sr (without the SOC). Lower panel: schematic Dirac band

on the Γ-M line obtained by a full ﬁrst-principles calculation. The energy gap is opened at the Dirac point

by the SOC while keeping the spin-degenerate state. (c) Square-net X−layer coordinated with magnetic Eu

ions (A=Eu), where the Dirac band exhibits spin splitting due to the exchange interaction with local Eu2+

spins (S=7/2). (d) Distorted (polar) X−layer coordinated with Ba ions (A=Ba), where the Dirac band

exhibits valley-contrasting spin splitting due to the SOC with broken inversion symmetry (nonzero electric

by the author.

III. BASIC ELECTRONIC AND LATTICE STRUCTURES OF AMNX2

The X−layer in AMnX2works as a conducting Dirac fermion layer and dominates the transport

properties [shaded layer in Fig. 1(a)]. From the tight-binding calculation, the band structure for

the X−square net coordinated with the Asite ions exhibits the spin-degenerate 2D linear bands,

which cross on the Γ-M line and around the X point [upper panel of Fig. 1(b)]. On the other hand,

the A2+-Mn2+-X3−layer works as a block layer, where the Mn layer is a robust Mott insulator with

the antiferromagnetic (AFM) order around room temperature irrespective of the Aand Xspecies.

The ﬁrst-principles calculation for AMnX2indeed predicted that quasi-2D massive Dirac bands

are formed as a bulk band33,34, where the energy gap at the charge neutral Dirac point originates

from the spin-orbit coupling (SOC) of the X−ion [lower panel of Fig. 1(b)]. Among the AMnX2

materials, SrMnBi2(A=Sr, X=Bi) ﬁrst drew attention in 201135–37. This material is regarded as a

basic material of AMnX2, since it exhibits simple spin-degenerate quasi-2D Dirac bands, as clearly

observed along the Γ-M line by angle-resolved photoemission spectroscopy (ARPES)35,38,39.

In SrMnBi2, the AFM order of the Mn layer apparently has little inﬂuence on the Dirac

fermion40; no resistive anomalies were observed at the magnetic transition temperature35–37. In

order to enhance the impact of the block layer on the Dirac fermion, the substitution of the Asites

would be most eﬀective, since the Asite located adjacent to the X−layer can aﬀect its electronic

and lattice structures directly. In Sect. IV, we ﬁrst review the Dirac fermion coupled with the

magnetic order in EuMnBi2[Fig. 1(c)], where nonmagnetic Sr in the Asite is substituted with

magnetic Eu hosting local spin S=7/2. High-ﬁeld transport measurements have revealed that the

AFM order of Eu layers strongly aﬀects not only the quantum transport but also the Dirac band

via the exchange interaction. Next, in Sect. V, we review the Dirac fermion coupled with lattice

polarization Pin BaMnX2[Fig. 1(d)], where Sr is substituted with Ba hosting the larger ionic ra-

dius to induce polar lattice distortion in the X−layer. The SOC together with the broken inversion

symmetry gives rise to the spin polarization dependent on Dirac valleys, which is clearly reﬂected

in the bulk half-integer QHE observed at high ﬁelds. In Sect. VI, we will give a summary of this

article.

Note here that AMnX2has another crystal structure with the staggered arrangement of Asites

below and above the X−layer, as is the case for CaMnX241,42, YbMnX243–45,AMnSb2(A=Sr,

Eu)46–50, and so on. For this layer structure, the nonsymmorphic symmetry inherent in the space

group P4/nmm or Pnma results in not a simple 2D Dirac band, but complex bands hosting a large

nodal line51, which is beyond the scope of this review.

IV. DIRAC FERMION COUPLED WITH MAGNETIC ORDER IN EUMNBI2

A. Large magnetoresistance eﬀect of high-mobility Dirac fermions

For EuMnBi2, Eu layers exhibit the AFM order below TN=22Kat0T52,53, where Eu moments

order ferromagnetically within the ab plane and align along the c-axis in the sequence of up-

up-down-down [Fig. 2(a)]53–55. The Bi square net intervenes between Eu layers with magnetic

moments up and down, reminiscent of a natural spin-valve-like structure. Importantly, the AFM

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High-eldStudiesonLayeredMagneticandPolarDiracMetals:NovelQuantumTransportPhenomenaCoupledwithSpin-valleyDegreesofFreedomHideakiSakaiDepartmentofPhysics,OsakaUniversity,Toyonaka,Osaka560-0043,JapanAbstractRecently,theinterplaybetweentheDirac/Weylfermionandvariousbulkproperties,suchasmagnetism,hasat...

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High-ﬁeld Studies on Layered Magnetic and Polar Dirac Metals Novel Quantum Transport Phenomena Coupled with Spin-valley Degrees of Freedom

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