ЖЭТФ, 2021, том 159, вып. 4, стр. 598-606

A DZYALOSHINSKII - MORIYA INTERACTION GUIDE

TO MAGNETS MICRO-WORLD

V. V. Mazurenkoa*, Y. O. Kvashnin^b, A. I. Lichtenstein^c,a, M. I. Katsnelson^d,a

^a Theoretical Physics and Applied Mathematics Department, Ural Federal University

620002, Ekaterinburg, Russia

^b Uppsala University, Department of Physics and Astronomy, Division of Materials Theory

SE-751 20, Uppsala, Sweden

^c I. Institut für Theoretische Physik, Universität Hamburg

D-20355, Hamburg, Germany

^d Institute for Molecules and Materials, Radboud University

NL-6525, AJ Nijmegen, The Netherlands

Received December 4, 2020,

revised version December 4, 2020

Accepted for publication December 4, 2020

Contribution for the JETP special issue in honor of I. E. Dzyaloshinskii’s 90th birthday

DOI: 10.31857/S0044451021040039

metric with respect to swapping the positions of two

spins. This was done based on a purely phenomenologi-

Abstract.

Dzyaloshinskii-Moriya interaction

cal basis. Very soon, Moriya [2] suggested the first sim-

plified microscopic explanation of these interactions, in-

(DMI) represents an antisymmetric type of magnetic

interactions that favour orthogonal orientation of

direct exchange and spin-orbit coupling (SOC) being

the key ingredients. The Hamiltonian governing these

spins and competes with Heisenberg exchange. Being

introduced to explain weak ferromagnetism in an-

interactions can be written in the following form:

∑

tiferromagnets without an inversion center between

Ĥ_DMI = D_ij[Ŝi ×

Ŝ_j],

(1)

magnetic atoms such an anisotropic interaction can be

i,j

used to analyze other non-trivial magnetic structures

where S_i is the spin moment at the site i. Nowadays

of technological importance including spin spirals and

the parameter D_ij , which is, by construction, an axial

skyrmions. Despite the fact that the corresponding

vector, is known as Dzyaloshinskii - Moriya interaction.

DMI contribution to the magnetic energy of the system

“Slow is the experience of all deep fountains: long

has a very compact form of the vector product of spins,

have they to wait until they know what has fallen into

the determination of DMI from first-principles elec-

their depths.” (F. Nietzsche).

tronic structure is a very challenging methodological

Whereas the first decades DMI were considered as

and technical problem whose solution opens a door

more or less marginal subject in magnetism (with the

into the fascinating microscopic world of complex

only exception of the phenomenon of weak ferromag-

magnetic materials. In this paper we review a few

netism) now they are the mainstream subject, of a great

such methods developed by us for calculating DMI

conceptual meaning and of a great practical importance

and their applications to study the properties of real

[3-8]. This only contribution would be sufficient to put

materials.

the name of Igor Dzyaloshinskii among the main cre-

Introduction. In a seminal paper

[1]

ators of modern physics of magnetism.

I. E. Dzyaloshinskii has introduced a novel type of

We are very thankful to the organizers for their kind

anisotropic magnetic interactions which are antisym-

invitation to participate in the special issue dedicated

to Dzyaloshinskii. In this short review we present our

^* E-mail: vmazurenko2011@gmail.com

view on the fast growing field of DMI based mostly on

598

ЖЭТФ, том 159, вып. 4, 2021

A Dzyaloshinskii - Moriya interaction guide to magnets micro-world

our own experience of calculations and analysis of DMI

D_ij = -

[Tr_σ{t_ji}Tr_σ{t_ij σ} -

parameters for specific magnetic materials.

- Tr_σ{t_ij }Tr_σ{t_jiσ}],

(4)

2. Methods for calculating the Dzyaloshin-

skii-Moriya interaction. In this section, numerical

where σ are the Pauli matrices.

Interestingly, the Moriya’s microscopic theory was

approaches for calculating DMI are discussed. We start

with a microscopic theory by Moriya [2] and show how

published in 1960, however, its first application to

quantitative analysis of the magnetic properties of real

it can be extended to analyze the dependence of DMI

sign on the occupation of the 3d shell. Then we will

materials was only done 30 years later by Coffey, Rice,

and Zhang in Ref. [14]. They have estimated D_ij

focus on a correlated band theory of the DMI, that is

free from basic limitations of the superexchange theory

for different phases of La₂CuO₄ and YBa₂Cu₃O₆ com-

and can be applied in a wide range of electronic Hamil-

pounds. It was shown that peculiarities in the crystal

structures of these systems result in different patterns

tonian parameters corresponding to insulators and me-

tals. The last subsection of the methodological part

of the DMI vectors, and as the result different ground

states with and without net magnetic moment can be

is devoted to first-principles approaches based on the

density functional theory.

realized.

An important feature of the one-band considera-

2.1. Microscopic theory of DMI. The first mic-

tion of the DMI is that the Moriya’s results were ob-

roscopic theory of the antisymmetric anisotropic ex-

tained by using an assumption of the constant U value

change interaction was developed by Moriya in 1960

without orbital dependence as well as by neglecting the

and presented in Ref. [2]. It is based on the Anderson’s

intra-atomic (Hund’s) exchange contribution. Further

idea on superexchange interaction [9] and formulated

development of the microscopic theory of the antisym-

on the basis of the simplest electronic model account-

metric anisotropic interaction was mainly related to its

ing the on-site Coulomb interaction and the spin-orbit

generalization to multi-orbital electronic Hamiltonians.

coupling on the level of the hopping integrals. Such an

As was shown in Ref. [13] inter-orbital Coulomb and

electronic model can be written in the following form

intra-atomic exchange interactions play an important

role in formation of the DMI.

∑

1^∑

Ĥ₌

Another important peculiarity of the one-band con-

â^†iσâ_jσ′ +

Uâ^†iσâ^†iσ′â_iσ′â_iσ,

(2)

sideration of the DMI was demonstrated in Ref. [15].

ij,σσ^′

i,σσ^′

It was shown that the resulting spin model, Eq. (3) is

characterized by a specific symmetry of the symmet-

where â^†iσ(a_iσ) are the creation (annihilation) opera-

↔

ric anisotropic exchange interaction tensor,

Γ_ij whose

is the element

tors. U is local Coulomb interaction, t

principal axis coincides with DMI for each bond. It

of the spin-resolved hopping matrix. Formally, Eq. (2)

means that the state of a system with weak ferromag-

is nothing but the Hubbard model [10-12] that was offi-

netism is higher in energy than the pure (compensated)

cially introduced three years later in 1963. In the limit

antiferromagnetic state.

when the on-site Coulomb interaction is much larger

Despite of the above-mentioned and other limita-

than the hopping integrals such a Hubbard model can

tions of the one-band approach for calculating magnetic

be reduced to the spin model

interaction parameters, it provides a very simple and

∑

transparent way to analyze the properties of the inter-

Ĥspin₌

J_ij Ŝ_i Ŝ_j +

D_ij[Ŝi ×

Ŝ_j] +

actions. For instance, it can be used for analysis of the

dependence of the DMI sign on the occupation of the 3d

∑

Γ_ij Ŝ_j,

(3)

shell experimentally observed in the series of isostruc-

tural weak ferromagnets, MnCO₃, FeBO₃, CoCO₃, and

NiCO₃ as it was done by us in Ref. [16]. We consider

↔

the case of a transition metal oxide for which the crys-

where

Ŝ is the spin operator, J_ij , D_ij and

Γ_ij

tal field splitting is much larger than the spin-orbit cou-

are the isotropic exchange interaction, antisymmet-

pling, the latter can be treated as a perturbation. The

ric anisotropic (Dzyaloshinskii-Moriya) and symmet-

corresponding expression for Dzyaloshinskii-Moriya in-

ric anisotropic interactions, respectively. The summa-

teraction can be presented in the following form

tion runs twice over all pairs. In terms of the Hub-

bard Hamiltonian parameters the resulting expression

′

D^nnij

Cn′nji - C

bn′nji],

(5)

for the DMI has the following form [2, 13]:

599

V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, M. I. Katsnelson

ЖЭТФ, том 159, вып. 4, 2021

The difference between DMIs obtained for a system

with two and six electrons is related to the difference

between Cn′ n₂₁ and Cm′ m₂₁,

(

)

Cn′ nji = -Cm′ mji = -λLmn

bm′ nji - bn′ mji

(7)

2ΔE

where ΔE = ϵⁿⁱ - ϵ^mi.

(for the system with two elec-

It means that D

trons) and D

(with six electrons) are of different

Fig. 1. (Color online) a) Minimal tight-binding model used for

signs. Thus, on the level of Moriya’s approach, the

explaining the DMI sign change at variation of the occupation.

sign of the DMI depends on the occupation of the ex-

The horizontal lines represent the electron levels and hoppings

cited states. Depending on the symmetry and occu-

are shown with arrows. b, c) Two antiferromagnetic ground

pation, each pair of 3d orbitals can result in positive

states corresponding to the S = 1/2 case, obtained in the

or negative contribution to the total DMI between two

model for different orbital fillings: N = 2 (b) and N = 6 (c)

atoms. It should be noted that similar dependence of

the DMI sign on the occupation of the 3d shell can be

also found in some series of metallic systems. In this

is the (unperturbed) hopping integral be-

where b

sense interesting methodological results were obtained

tween n-th ground orbital state of i-th atom and n^′-th

in Refs. [17-19].

is the corresponding

orbital state of j-th atom, C

It is important to discuss the limits of the Moriya’s

hopping renormalized by SOC and U is the on-site

theory of DMI from the point of view of its using to

Coulomb interaction. Thus, Cn′nji is given by

study real physical systems. In its original formula-

[

]

tion it is limited to the systems with the spin state

(Lm′n^′

)^∗

L^mni

Cn′nji = -λ

bm′nji +

bn′m

(6)

of S = 1/2. Real transition metal compounds and

ϵ^mi - ϵⁿ

ϵm′j-ϵj′

nanosystems are of multi-orbital nature. In this case,

the main question is how to define the numerous hop-

where λ is the spin-orbit coupling constant, L^mni is the

ping and Coulomb interaction parameters of the Hub-

matrix element of the orbital angular momentum be-

bard model. In principle, one can use approximations

tween the single m-th excited state and the n-th ground

of different types to define the parameters [20, 21] by

state Wannier functions which are centered at i-th ion,

using available experimental data. Another approach is

while ϵⁿⁱ represents the energy of the n-th Wannier or-

based on performing density functional theory (DFT)

bital at the i-th ion.

calculations and their parametrization using wannier-

Tight-binding model we considered contains two

ization procedure developed in Refs. [22, 23] to con-

atoms having non-degenerate (n and n^′) and high-ener-

struct the Wannier functions [24]. Then, on this basis

gy (m and m^′) levels. The schematic visualization of

the electronic model parameters are calculated. The

the model with the allowed hopping paths is presented

most accurate numerical scheme to estimate local (U)

in Fig. 1. In the simplest case one can assume that the

and non-local Coulomb interaction parameters taking

same hopping integrals between high-energy (m and

screening effects into account is based on the con-

m^′) and low-energy (n and n^′) levels, b

= b

strained random phase approximation [25].

The hoppings between orbitals of different symmetry

The situation becomes even more complicated if

require more detail analysis, since they define the DMI

one simulates a compound with a strong spin-orbit

in the system in question. We assume that the geom-

coupling. For this case effective numerical schemes

etry of the model system is fixed, which means that

based on the superexchange theory can be found in

hopping integrals do not change with variation of the

Refs. [26, 27].

occupation.

Our tight-binding model has two ground states with

2.2. Correlated band theory for DMI. We

different occupations N that correspond to the S = 1/2

start with correlated band theory of DMI developed

case: N = 2 and N = 6 (Fig. 1). In the case N = 2,

by us in Ref. [28]. It is based on the consideration of

the ground state magnetic orbital is of symmetry n(n^′),

the general Hamiltonian of interacting electrons in a

while for N = 6 it is m(m^′). Another difference be-

crystal:

tween these configurations is the different occupation

∑

1^∑

Ĥ₌

of the excited states: they are empty and fully occu-

c†1t₁₂c₂ +

c†1c†2U₁₂₃₄c₃c₄,

(8)

pied for N = 2 and N = 6, respectively.

1234

600

ЖЭТФ, том 159, вып. 4, 2021

A Dzyaloshinskii - Moriya interaction guide to magnets micro-world

where 1 = (i₁, m₁, σ₁) is the set of site (i₁), orbital (m₁)

where

and spin (σ₁) quantum numbers and t₁₂ are hopping

∫

integrals that contain the spin-orbit coupling. These

N_ji = 〈c^†ic_j〉 = -¹

Im G_ji(E) dE

transfer couplings can be found by the Wannier-para-

meterization of the first-principle band structure with

-∞

the spin-orbit coupling.

is the inter-site occupation matrix and

Ĝ is the Green

We will take into account only the local Hubbard-li-

function of the system, E_F is the Fermi energy. The

ke interactions, keeping in

Ĥ_u only terms with i₁ =

occupation matrix can be calculated by using a static

= i₂ = i₃ = i₄. This assumption corresponds to the

(such as DFT+U [29]) or a dynamic mean-field ap-

DFT+U Hamiltonian [29] that is also a starting point

proach (DFT+DMFT [30-32]).

for the DFT+DMFT (Dynamical Mean-Field Theory)

Note that the occupation matrix is calculated in the

[30-32]. It is crucially important for the later consid-

corresponding collinear states, which strictly speaking

eration that the interaction term

Ĥ_u is supposed to be

can be done self-consistently only within constrained

rotationally invariant.

calculations [34]. Using the decomposition of the to-

We start with a collinear magnetic configuration,

tal moment

Ĵ into orbital and spin moments, we have

for instance an antiferromagnetic state, which is close

a natural representation of the Dzyaloshinskii-Moriya

to the real ground state (weak ferromagnet), but does

vector (13) as a sum of the orbital and spin contribu-

not coincide with it due to the DMI. Let us re-define

tions which are related with the rotations in orbital and

the DM Hamiltonian (Eq. (1)) in a slightly different

spin space, respectively.

way:

∑

The resulting expression Eq. (13) is of general na-

H_DMI = D^′ij[e_i × e_j],

(9)

ture and its spin part can be also derived in the case of

the metallic systems as it was shown in [35].

where e_i is a unit vector in the direction of the i-th site

2.3. DFT-based methods. In this section we

magnetic moment and D^′ij is the Dzyaloshinskii-Mo-

discuss mean-field approaches for calculating the DMI

riya vector. We analyze the magnetic configuration

that are realized on the basis of the numerical methods

that is slightly deviated from the collinear state,

of the density functional theory. The net DMI can be

assessed by calculating the DFT total energies for the

e_i = η_ie₀ + [δφ_i × η_ie₀],

(10)

two sets of spin spiral states having opposite helicities

where η_i = ±1, e₀ is the unit vector along the vector

[36-38] or by using Berry phase theory [39]. In order

of antiferromagnetism, and δφ_i are the vectors of small

to calculate the individual pair-wise DMI, one can em-

angular rotations.

ploy the magnetic force theorem [40]. According to this

Substituting Eq. (10) into Eq. (9) one finds for the

theorem, the variation of the total energy of the system

variation of the magnetic energy:

due to a magnetic excitation can be expressed through

∑

the variation of the single-particle energy.

δE = D^′ij(δφ_i - δφ_j).

(11)

In 3d systems, the spin-orbit coupling in itself can

be also considered as a perturbation [41,42]. One can

Now we should calculate the same variation for the

consider a mixed perturbation scheme with respect to

microscopic Hamiltonian (8). Similar to the procedure

the rotation and spin-orbit coupling, which leads to the

used in Ref. [33] to derive exchange interactions for the

antisymmetric anisotropic DMI [43]

LDA+DMFT approach, we consider the effect of the

local rotations

∫

D^zij = -

dϵ ×

R_i = eiδϕi Ĵⁱ ,

(12)

8πS_iS_j

-∞

∑

on the total energy; here

Ĵ_i

= L_i +

Ŝ_i is the total

× Tr_m(Δ_iG^↓ikH^sok↓↓G^↓kj Δ_j G^↑ji -

moment operator,

Li and

Ŝ_i are the orbital and spin

moments, respectively.

- Δ_iG^↑ikHsok ↑↑G^↑kjΔ_jG^↓ji+Δ_iG^↓ijΔ_jG^↑jkHsok ↑↑G↑ki -

The resulting DMI is given by anticommutator of

- Δ_iG^↑ijΔ_jG^↓jkHsok ↓↓G^↓ki).

(14)

and t_ij :

Other components of the Dzyaloshinskii-Moriya vector

D^′ij = -

Tr_m,σN_ji[Ĵ, t_ij ]₊,

(13)

for particular bond can be obtained from the z ones by

601

V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, M. I. Katsnelson

ЖЭТФ, том 159, вып. 4, 2021

rotation of the coordinate system. Similar expression

for DMI was obtained by Solovyev et al. [44].

We have applied the developed method for cal-

culating the DMI to give a microscopic explanation

to the scanning tunneling microcopy experiments per-

formed for chains of manganese atoms on CuN sur-

face [43]. Weak ferromagnetism due to the DMI be-

tween neighbouring manganese atoms was predicted.

Another important example is a first-principles study

of the molecular nanomagnet Mn₁₂ for which most the-

oretical works on molecular magnets are mainly relied

on the so-called rigid-spin model. Within such a model

a complex system of interacting spins is replaced by

Fig. 2. (Color online) Local atomic and magnetic orders in the

just one big spin, with some magnetic anisotropy being

weak ferromagnets. The ions of the two magnetic sublattices

introduced artificially. However, such a description is

are represented by blue (site 1) and red (site 2) spheres, with

black arrows denoting the direction of their spins. Oxygen

rather simplistic and largely ignores intermolecular in-

atoms between the two adjacent transition metal layers are

teractions. Previously, it was predicted that the DMI

represented as yellow spheres. The dotted circles highlight the

plays a crucial role in the physics of molecular mag-

twist of the oxygen layer. The bottom panel shows the occu-

nets [45] and in particular magnetic tunneling effects in

pation of the 3d level of a magnetic ion. The left and right

Mn₁₂ [46]. In our work [47] we have demonstrated that

panels show the two possible magnetic configurations which

the account of the inter-atomic anisotropic exchange

stabilize depending on the 3d occupation and, therefore, the

interactions in Mn₁₂ gives opportunity to reproduce ex-

sign of the DMI, for a net ferromagnetic moment pointing

citation energies observed in the inelastic neutron scat-

along the magnetic field H. SAF M denotes the direction of

tering experiments for this system.

the antiferromagnetic spin structure. This figure is reproduced

In the case of the systems with strong spin-orbit

with permission from Ref. [16]

coupling one could still use similar Green’s function

approach within the magnetic force theorem [48]. The

expressions for DMI, which do not rely on the small-

rection of the weak ferromagnetic moment with respect

ness of spin-orbit coupling constant have been derived

to the crystallographic axes and hence the “sign” of DM

independently by several groups [49-53].

interaction remained unknown. For the first time, this

was unambiguously identified for FeBO₃ using resonant

3. Applications.

x-ray diffraction in Ref. [62]. We have performed ab

initio calculations and extracted the DM vectors using

3.1. Weak ferromagnetism in antiferromag-

nets. Discovery of the weak ferromagnetism in iron

Eq. (13). It was found that the theoretical calculations

hematite, Fe₂O₃ (Ref. [54, 55]) was the starting point

do not only reproduce the correct sign of DM vectors,

but also give a very good estimate of the canting angle.

for development of the DMI theory. More specifically,

Fe₂O₃ is pure antiferromagnet with magnetic moments

Next, we addressed the series of isostructural cal-

parallel to the trigonal axis c at T < 260 K. In the tem-

cite oxides, namely: MnCO₃, FeBO₃, CoCO₃, NiCO₃.

perature range between 260 and 950 K, the magnetic

They all exhibit weak ferromagnetism and experiments

moments are in-plane and a small canting of the mag-

based on the technique developed in Ref. [62], have re-

netic moment exists. As the result of this canting there

vealed the change of canting angle sign across the series

is net magnetic moment in the antiferromagnetic sys-

[16]. More specifically, the compounds MnCO₃ and

tem. The weak ferromagnetism in Fe₂O₃ was explored

FeBO₃ are characterized by the rotation sense which

with first-principles DFT calculations in Ref. [56] and

differs from that for the CoCO₃ and NiCO₃ systems.

with Green’s function approach based on the magnetic

It is schematically shown in Fig. 2. Taking into account

force theorem in Ref. [57].

that these compounds have the same crystal structure

A more interesting situation concerning weak ferro-

(and the same crystallographic chirality), such a sign

magnetism in antiferromagnets is observed in transition

change can be attributed to the difference in the occu-

metal oxides having calcite structure. Previous magne-

pation of the 3d shell.

tization measurements [58-61] have confirmed the exis-

To provide a theoretical support to these experi-

tence of the non-compensate in-plane magnetization in

ments in Ref. [16] we have performed first-principles

several materials of this kind. However, the precise di-

calculations within local density approximation taking

602

ЖЭТФ, том 159, вып. 4, 2021

A Dzyaloshinskii - Moriya interaction guide to magnets micro-world

into account the on-site Coulomb interaction U and

spin-orbit coupling (DFT+U+SO). The resulting mag-

netic configuration is antiferromagnetic one character-

ized by a canting of the magnetic moments. Such a

canted state is the lowest-energy state for all the sys-

tems under consideration. The calculated magnitudes

and signs of the canting angles are in good agreement

with experimental data. However, the full calculation

does not provide a truly microscopic understanding of

the DMI sign change phenomena. For that a minimal

tight-binding model based on the Moriya’s theory as

Fig. 3. (Color online) Skyrmionic magnetic structure obtained

described in the methodological part of this paper be-

from the Monte Carlo simulations for C2F system. This figure

comes extremely useful.

is adopted from Ref. [74]

3.2. Magnetic skyrmions. Investigation of

skyrmions is a widely studied topic in the modern ma-

and nanosystems with heavy adatoms Sn/Si(111),

terial science [63-71]. Previous experimental and theo-

Pb/Si(111) and Sn/SiC(0001). The key quantity here

retical studies on the topologically-protected magnetic

allowing the formation of the topologically protected

skyrmion excitations were fully focused on the tran-

magnetic structures is the anisotropic Dzyaloshin-

sition metal crystals and nanosystems. This seems

skii-Moriya interaction.

to be natural, since these materials are characterised

by well-localised magnetic moments originated from

Perspectives. Despite there were several

the partially-filled 3d states and magnetic anisotropy

decades of intensive investigations on DMI and related

[72,73] that facilitates an experimental detection of the

phenomena we think that this is still young and very

distinct magnetic textures. In works [74-76] a new class

promising research field within which one could focus

of materials, surface nanostructures with sp element re-

on the following directions for future investigations.

vealing skyrmion excitations at experimentally achiev-

From the very beginning DMI is considered as a rep-

able magnetic fields and temperatures was introduced.

resentative of one of the smallest energy scales in the

The non-trivial result, that such sp-electron systems

magnetic Hamiltonians of the strongly correlated sys-

are, in principle, characterized by a magnetic state, was

tems, |D_ij | ≪ J_ij . It is due to the DMI always contains

experimentally confirmed in Refs. [77-79].

additional relativistic small parameter, the ratio of elec-

Our first-principles calculations [74-76] have con-

tron velocity in atoms to the velocity of light. However,

firmed a long-range character of the magnetic states in

such a dominance of the Heisenberg exchange interac-

graphene derivatives C₂H and C₂F as well as in sur-

tion can be overcome by different means. For instance,

face nanostructures Si(111):{C, Si, Sn, Pb} and in Sn

manipulation of magnetic interactions via a strong pe-

on SiC(0001). The corresponding Wannier functions in

riodic in time electromagnetic field [80] (“Floquet engi-

these systems demonstrate that substantial amount of

neering”) suggests that using real nanosystems and real

the electron density is concentrated in the interstitial

values of the laser fields one can reach the regime when

region. In all the cases we have found that the values of

the Heisenberg exchange J_ij is arbitrarily small, or even

the calculated hopping integrals are much smaller than

equal to zero, whereas the Dzyaloshinskii-Moriya pa-

that of the Coulomb interactions, which gives us oppor-

rameter D_ij remains constant. As an interesting exam-

tunity to construct a Heisenberg-type Hamiltonian for

ple of such a situation, a new class of two-dimensional

the localized spins S = 1/2 within the superexchange

Heisenberg-exchange-free materials where a completely

theory.

new type of skyrmions (Fig. 4) that emerge as the re-

The constructed spin models for graphene deriva-

sult of the competition between the DMI and uniform

tives, Si(111):{C, Si, Sn, Pb} and Sn monolayer on

magnetic field has been introduced [81].

SiC(0001) surface were solved by means of the Monte

Another fascinating research field is related to a

Carlo methods, which gives us opportunity to de-

quantum skyrmions that in contrast to the classical

fine the magnetic phases of these materials depend-

counterpart are practically unexplored. The main

ing on the external magnetic field and temperature. It

methodological problem here is how to characterize the

was found that one can stabilize skyrmionic solutions

topology of quantum system with a three-dimensional

in the case of the semifluorinated graphene (Fig. 3)

magnetic structure when the orientation of a spin is

603

V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, M. I. Katsnelson

ЖЭТФ, том 159, вып. 4, 2021

S. W. Cheong and M. Mostovoy, Nature Mater. 6, 13

(2007).

M. Bode, M. Heide, K. von Bergmann, P. Ferriani,

S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch,

S. Blügel, and R. Wiesendanger, Nature 447, 190

(2007).

M. Heide, G. Bihlmayer, and S. Blügel, Phys. Rev.

B 78, 140403(R) (2008).

Fig. 4. (Color online) Abbreviation of Dzyaloshinskii-Moriya

S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422

interaction written with nanoskyrmions. This is a result

(2013).

of Monte Carlo simulations of the Heisenberg-exchange-free

model on the square lattice with non-regular site occupation.

P. W. Anderson, Phys. Rev. 115, 2 (1959).

Arrows and colors depict the in-plane and out-of-plane spin

projections, respectively

10.

J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963).

11.

M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).

ill-defined. One of the possible solutions was recently

12.

J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).

proposed by some of us in Ref. [82] where scalar chiral-

ity operator was introduced to define a quantum analog

13.

T. Yildirim, A. B. Harris, A. Aharony, and O. En-

of the topological charge.

tin-Wohlman, Phys. Rev. B 52, 10239 (1995).

The other direction we consider to be very at-

tractive is DMI applications in quantum computing.

14.

D. Coffey, T. M. Rice, and F. C. Zhang, Phys. Rev.

Existence of the anisotropic exchange interactions

B 44, 10112 (1991).

between qubits can be used for preparing highly

15.

L. Shekhtman, O. Entin-Wohlman, and A. Aharony,

entangled quantum states that play an important role

Phys. Rev. Lett. 69, 836 (1992).

in quantum information processing [83,84]. The future

of DMI looks bright and promising.

16.

G. Beutier, S. P. Collins, O. V. Dimitrova,

V. E. Dmitrienko, M. I. Katsnelson, Y. O. Kvashnin,

Funding. The work of A. I. L. and M. I. K. is

A. I. Lichtenstein, V. V. Mazurenko, A. G. A. Nisbet,

E. N. Ovchinnikova, and D. Pincini, Phys. Rev. Lett.

supported by European Research Council via Synergy

119, 167201 (2017).

Grant 854843-FASTCORR. Y. O. K. acknowledges the

financial support from the Swedish Research Council

17.

A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538

(VR) under the project No. 2019-03569. The work of

(1980).

V. V. M. was supported by Act 211 Government of

the Russian Federation, contract 02.A03.21.0006.

18.

V. Kashid, T. Schena, B. Zimmermann, Y. Mokrou-

sov, S. Blügel, V. Shah, and H. G. Salunke, Phys.

Rev. B 90, 054412 (2014).

The full text of this paper is published in the English

version of JETP.

19.

A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Blügel,

and A. Manchon, Phys. Rev. Lett. 117,

247202

(2016).

REFERENCES

20.

A. S. Moskvin, M. A. Vigura, and A. P. Agafonov,

1. I. Dzyaloshinskii, J. Phys. Chem. Sol. 4, 241 (1958).

Sov. Phys. Solid State 28, 1631 (1986).

2. T. Moriya, Phys. Rev. 120, 91 (1960); T. Moriya, in

21.

A. Moskvin, J. Magn. Magn. Mater. 400, 117 (2016),

Magnetism, Vol. I, ed. by G. T. Rado and H. Suhl,

Proc. of the 20th Int. Conf. on Magnetism, Barcelona

Acad. Press, New York (1963), p. 86.

(2015).

3. H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys.

22.

N. Marzari and D. Vanderbilt, Phys. Rev. B 56,

Rev. Lett. 95, 057205 (2005).

12847 (1997).

4. I. A. Sergienko and E. Dagotto, Phys. Rev. B 73,

23.

N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and

094434 (2006).

D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).

604

ЖЭТФ, том 159, вып. 4, 2021

A Dzyaloshinskii - Moriya interaction guide to magnets micro-world

24.

A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-Su. Lee,

43.

A. N. Rudenko, V. V. Mazurenko, V. I. Anisimov,

I. Souza, D. Vanderbilt, and N. Marzari, Comput.

and A. I. Lichtenstein, Phys. Rev. B 79,

144418

Phys. Commun. 185, 2309 (2014).

(2009).

25.

F. Aryasetiawan, K. Karlsson, O. Jepsen, and

44.

I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev.

U. Schönberger, Phys. Rev. B 74, 125106 (2006).

Lett. 76, 4825 (1996).

26.

I. V. Solovyev, New J. Phys. 11, 093003 (2009).

45.

M. I. Katsnelson, V. V. Dobrovitski, and B. N. Har-

mon, Phys. Rev. B 59, 6919 (1999).

27.

I. V. Solovyev, V. V. Mazurenko, and A. A. Katanin,

Phys. Rev. B 92, 235109 (2015).

46.

H. A. De Raedt, A. H. Hams, V. V. Dobrovitski,

M. Al-Saqer, M. I. Katsnelson, and B. N. Harmon,

28.

M. I. Katsnelson, Y. O. Kvashnin, V. V. Mazurenko,

J. Magn. Magn. Mater. 246, 392 (2002).

and A. I. Lichtenstein, Phys. Rev. B 82, 100403(R)

47.

V. V. Mazurenko, Y. O. Kvashnin, Fengping Jin,

(2010).

H. A. De Raedt, A. I. Lichtenstein, and M. I. Kat-

29.

V. I. Anisimov, F. Aryasetiawan, and A. I. Lichten-

snelson, Phys. Rev. B 89, 214422 (2014).

stein, J. Phys.: Condens. Matter 9, 767 (1997).

48.

M. I. Katsnelson and A. I. Lichtenstein, Phys. Rev.

30.

A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev.

B 61, 8906 (2000).

B 57, 6884 (1998).

49.

L. Udvardi, L. Szunyogh, K. Palotás, and P. Wein-

31.

G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudo-

berger, Phys. Rev. B 68, 104436 (2003).

venko, O. Parcollet, and C. A. Marianetti, Rev. Mod.

50.

H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209

Phys. 78, 865 (2006).

(2009).

32.

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin,

51.

A. Secchi, A. Lichtenstein, and M. Katsnelson, Ann.

and G. Kotliar, J. Phys.: Condens. Matter 9, 7359

Phys. 360, 61 (2015).

(1997).

52.

S. Mankovsky and H. Ebert, Phys. Rev. B 96, 104416

33.

M. I. Katsnelson and A. I. Lichtenstein, Eur. Phys.

(2017).

J. B 30, 9 (2002).

53.

Y. O. Kvashnin, A. Bergman, A. I. Lichtenstein, and

34.

G. M. Stocks, B. Ujfalussy, X. D. Wang, D. M. C. Ni-

M. I. Katsnelson, Phys. Rev. B 102, 115162 (2020).

cholson, W. A. Shelton, Y. Wang, A. Canning, and

B. L. Gyorffy, Phil. Mag. B 78, 665 (1998).

54.

T. Smith, Phys. Rev. 8, 721 (1916); L. Neel, Rev.

Mod. Phys. 25, 58 (1953).

35.

T. Kikuchi, T. Koretsune, R. Arita, and G. Tatara,

Phys. Rev. Lett. 116, 247201 (2016).

55.

S. V. Vonsovsky, Magnetism, Vol. 2, Wiley, New York

(1974).

36.

H. Yang, A. Thiaville, S. Rohart, A. Fert, and

M. Chshiev, Phys. Rev. Lett. 115, 267210 (2015).

56.

L. M. Sandratskii, M. Uhl, and J. Kübler, J. Phys.:

Condens. Matter 8, 983 (1996); L. M. Sandratskii and

37.

B. Zimmermann, G. Bihlmayer, M. Böttcher,

J. Kübler, Europhys. Lett. 33, 447 (1996).

M. Bouhassoune, S. Lounis, J. Sinova, S. Heinze,

S. Blügel, and B. Dupé, Phys. Rev. B 99, 214426

57.

V. V. Mazurenko and V. I. Anisimov, Phys. Rev.

(2019).

B 71, 184434 (2005).

38.

L. M. Sandratskii, Phys. Rev. B 96, 024450 (2017).

58.

A. Kosterov, T. Frederichs, and T. von Dobeneck,

Phys. Earth Planet. Interiors 154, 234 (2006), De-

39.

F. Freimuth, S. Blügel, and Y. Mokrousov, J. Phys.:

velopments in Techniques and Methods Related to

Condens. Matter 26, 104202 (2014).

Rock Magnetism.

40.

A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov,

59.

M. P. Petrov, G. A. Smolensky, A. P. Paugurt, and

and V. A. Gubanov, J. Magn. Magn. Mater. 67, 65

S. A. Kizhaev, AIP Conf. Proc. 5, 379 (1972).

(1987).

60.

A. S. Borovik-Romanov and V. I. Ozhogin, Sov. Phys.

41.

P. Bruno, Phys. Rev. B 39, 865 (1989).

JETP 12, 18 (1961).

42.

I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys.

61.

N. M. Kreines and T. A. Shal’nikova, Sov. Phys.

Rev. B 52, 13419 (1995).

JETP 31, 280 (1970).

605

V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, M. I. Katsnelson

ЖЭТФ, том 159, вып. 4, 2021

62.

V. E. Dmitrienko et al., Nature Phys. 10, 202 (2014).

73.

E. Torun, H. Sahin, C. Bacaksiz, R. T. Senger, and

F. M. Peeters, Phys. Rev. B 92, 104407 (2015).

63.

A. N. Bogdanov and D. A. Yablonskii, Sov. Phys.

JETP 68, 101 (1989).

74.

V. V. Mazurenko, A. N. Rudenko, S. A. Nikolaev,

D. S. Medvedeva, A. I. Lichtenstein, and M. I. Kat-

64.

S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer,

snelson, Phys. Rev. B 94, 214411 (2016).

A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Sci-

ence 323, 915 (2009).

75.

D. I. Badrtdinov, S. A. Nikolaev, M. I. Katsnel-

son, and V. V. Mazurenko, Phys. Rev. B 94, 224418

65.

A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch,

(2016).

R. Ritz, P. G. Niklowitz, and P. Böni, Phys. Rev.

Lett. 102, 186602 (2009).

76.

D. I. Badrtdinov, S. A. Nikolaev, A. N. Rudenko,

M. I. Katsnelson, and V. V. Mazurenko, Phys. Rev.

66.

W. Münzer, A. Neubauer, T. Adams, S. Mühlbauer,

B 98, 184425 (2018).

C. Franz, F. Jonietz, R. Georgii, P. Böni, B. Peder-

sen, M. Schmidt, A. Rosch, and C. Pfleiderer, Phys.

77.

S. Glass, G. Li, F. Adler, J. Aulbach, A. Fleszar,

Rev. B 81, 041203 (2010).

R. Thomale, W. Hanke, R. Claessen, and J. Schäfer,

Phys. Rev. Lett. 114, 247602 (2015).

67.

X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park,

78.

G. Li, P. Höpfner, J. Schäfer, C. Blumenstein,

J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura,

S. Meyer, A. Bostwick, E. Rotenberg, R. Claessen,

Nature 465, 901 (2010).

and W. Hanke, Nature Commun. 4, 1620 (2013).

68.

N. Nagaosa and Y. Tokura, Nature Nanotechnol. 8,

79.

H. González-Herrero, J. M. Gómez-Rodríguez,

899 (2013).

P. Mallet, M. Moaied, J. José Palacios, C. Salgado,

M. M. Ugeda, J.-Y. Veuillen, F. Yndurain, and I. Bri-

69.

O. Janson, I. Rousochatzakis, A. A. Tsirlin, M. Be-

huega, Science 352, 437 (2016).

lesi, A. A. Leonov, U. K. Rößler, J. van den Brink,

and H. Rosner, Nature Commun. 5, 5376 (2014).

80.

E. A. Stepanov, C. Dutreix, and M. I. Katsnelson,

Phys. Rev. Lett. 118, 157201 (2017).

70.

S. Heinze, K. von Bergmann, M. Menzel, J. Brede,

A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and

81.

E. A. Stepanov, S. A. Nikolaev, C. Dutreix,

S. Blügel, Nature Phys. 7, 713 (2011).

M. I. Katsnelson, and V. V. Mazurenko, J. Phys.:

Condens. Matter 31, 17LT0 (2019).

71.

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel,

82.

O. M. Sotnikov, V. V. Mazurenko, J. Colbois, F. Mi-

B. Wolter, K. von Bergmann, A. Kubetzka, and

R. Wiesendanger, Science 341, 636 (2013).

la, M. I. Katsnelson, and E. A. Stepanov, arXiv:

2004.13526.

72.

I. Beljakov, V. Meded, F. Symalla, K. Fink, S. Shall-

83.

L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).

cross, M. Ruben, and W. Wenzel, Nano Lett. 14,

3364 (2014).

84.

P. W. Shor, SIAM J. Comput. 26, 1484 (1997).

606