Automated calculations of exchange magnetostriction

2025-05-02 0 0 2.5MB 49 页 10玖币

侵权投诉

P. Nievesa,∗, S. Arapana, S. H. Zhangb,c, A. P. K ˛adzielawaa, R. F. Zhangb,c,

D. Leguta

aIT4Innovations, VŠB - Technical University of Ostrava, 17. listopadu 2172/15, 70800

Ostrava-Poruba, Czech Republic

bSchool of Materials Science and Engineering, Beihang University, Beijing 100191, PR China

cCenter for Integrated Computational Materials Engineering, International Research Institute

for Multidisciplinary Science, Beihang University, Beijing 100191, PR China

Abstract

We present a methodology based on deformations of the unit cell that allows

to compute the isotropic magnetoelastic constants, isotropic magnetostrictive co-

efﬁcients and spontaneous volume magnetostriction associated to the exchange

magnetostriction. This method is implemented in the python package MAELAS

(v3.0), so that it can be used to obtain these quantities by ﬁrst-principles calcu-

lations and classical spin-lattice models in an automated way. We show that the

required reference state to obtain the spontaneous volume magnetostriction com-

bines the equilibrium volume of the paramagnetic state and magnetic order of the

ground state. We identify an error in the theoretical expression of the isotropic

magnetostrictive coefﬁcient λα1,0for uniaxial crystals given in previous publica-

tions, which is corrected in this work. The presented computational tool may be

helpful to provide a better understanding and characterization of the relationship

between the exchange interaction and magnetoelasticity.

Keywords: Magnetostriction, Magnetoelasticity, Exchange interaction,

High-throughput computation, First-principles calculations

1. Introduction

The isotropic magnetoelastic constants (biso) give a contribution to the magne-

toelastic energy that depends on the strain but not on the magnetization direction

∗Corresponding author.

E-mail address: pablo.nieves.cordones@vsb.cz

Preprint submitted to Elsevier February 16, 2023

arXiv:2210.00791v3 [cond-mat.mtrl-sci] 15 Feb 2023

at saturated state. They are mainly originated by the volume dependence of both

atomic magnetic moments and exchange interaction [1], so that are responsible for

the so-called exchange magnetostriction[1–3]. Other less signiﬁcant contributions

to biso are the isotropic contribution of the dipolar magnetostriction (form effect)

and crystal effect [1]. From the minimization of the elastic energy and isotropic

contribution of magnetoelastic energy with respect to strain, one can derive the

isotropic contribution to the fractional change in length ([l−l0]/l0) which is char-

acterized by the isotropic magnetostrictive coefﬁcients λiso. These quantities lead

to the isotropic contribution to the spontaneous volume magnetostriction (ωs)[1–

3], deﬁned as the fractional volume change between magnetically ordered and

paramagnetic (PM) state (ωs= [VFM −VPM]/VPM) [4]. This property is responsi-

ble for interesting features like anomalies in the thermal expansion coefﬁcient in

magnetic materials[2, 3]. For example, Invar alloy (Fe65Ni35) exhibits a very large

ωs(∼10−2) that cancels the normal thermal expansion, leading to nearly zero

net thermal expansion over a broad range of temperatures [2]. Nowadays, Invar

alloys and their extensions (Super-Invar, Stainless-Invar, Elinvar, Super-Elinvar,

Co-Elinvar)[2] are widely used in many commercial applications such as preci-

sion machine tools, precision pendulums, precision capacitors, precision moulds,

transistor bases, lead frames for integrated circuits, thermostats, bending meters,

gravity meters, ﬂow meters, astronomical telescopes, seismographic devices, mi-

crowave guides, resonant cavities, laser light sources and radar echo boxes [2].

Exchange magnetostriction might also play an important role in novel magnetic

phenomena like laser-induced ultrafast magnetism [5].

At an arbitrary temperature T, we have ωs(T)=(VFM(T)−VPM(T))/VPM(T).

Above the Curie or Néel temperature (T>Tc), the spontaneous exchange magne-

tostriction is zero ωs(T) = 0. Below the Curie or Néel temperature (T<Tc), one

needs to know the equilibrium volume at hypothetical PM-like state VPM(T)at

T<Tc, which is difﬁcult to characterize experimentally and theoretically [2–4].

In experiment, such equilibrium volume is typically estimated by an extrapola-

tion using the Debye theory and the Grüneisen relation from above the Curie or

Néel temperature[2, 3]. On the other hand, the PM state at zero-temperature can

be studied with ﬁrst-principles calculations through the Stoner model[6, 7], dis-

ordered local moment (DLM) approach[4, 8–10] and special-quasirandom struc-

tures (SQS)[11]. For example, ﬁrst principles calculations of ωsshowed that the

Stoner model might overestimate it in body-centered cubic (bcc) Fe [4, 6], while a

better quantitative agreement with experiment is achieved through the disordered

local moment (DLM) approach[4].

In practice, the theoretical study of the PM state at zero-temperature and ωs

with current available methods is possible but is not easy. So in this sense, it

would be desirable to ﬁnd alternative ways to compute these quantities which

could be easily automatized[12–14], making it more suitable for high-throughput

screening. For example, such kind of strategy might be helpful to further improve

Invar alloys or to discover new families of magnetic materials with very large

ωs(>10−2). In previous versions of the python package MAELAS[13, 14], it

was possible to compute anisotropic magnetoelastic constants (bani) and magne-

tostrictive coefﬁcients (λani) in automated way but not the isotropic ones. Here,

we release a new version of MAELAS (v3.0) where we extend its capabilities by

implementing a methodology to compute biso,λiso and ωs. To do so, we make use

of the universal notation proposed by E. du Tremolet de Lacheisserie [1] which,

thanks to its rigorous theoretical derivation based on the framework of group the-

ory, naturally decomposes the deﬁnition of the magnetoelastic constants arising

from isotropic and anisotropic magnetic interactions. The source ﬁles of this new

version are available in GitHub repository [15].

The paper is organized as follows. In Section 2, we describe the methodol-

ogy to compute biso,λiso and ωs, while Section 3 is devoted to technical details

about its implementation in MAELAS. This method is benchmarked in Section

4. In Section 5, we discuss about some limitations of this version and future per-

spectives. The paper ends with a summary of the main conclusions and future

perspectives (Section 6).

2. Methodology

We consider two cases depending on whether the anisotropic magnetic inter-

actions are included or not.

2.1. Including only isotropic magnetic interactions

In magnetic materials where the main source of magnetic anisotropy arises

from spin-orbit coupling (SOC), one can easily compute the total energy with-

out anisotropic magnetic interactions through ﬁrst-principles calculations by just

switching off the SOC. In this case bani =0 and the magnetoelastic energy con-

tains only the isotropic term. This approach also makes it possible to speed up

this type of task, since including SOC is more computationally demanding.

The method to compute biso,λiso and ωs, including only isotropic magnetic

interactions, is derived from the total spin-polarized energy (E) including elastic

(Eel) and isotropic magnetoelastic (Eiso

me ), that is,

E(ε

ε) = Eel(ε

ε) + Eiso

me (ε

ε),(1)

where ε

εis the strain tensor. Here, the elastic energy is considered up to second

order in the strain, while the isotropic magnetoelastic energy contains only linear

terms in the strain and no dependence on magnetization direction α

α. This means

that we assume that the total energy of the system is invariant under rotations

of the magnetization at the saturated state. Hence, the anisotropic magnetoe-

lastic must be negligible (Eani

me Eiso

me ). For example, this assumption could be

accomplished by not including the SOC in the calculation of the energy for mag-

netic materials where the main source of magnetic anisotropy arises from SOC.

Note that Eiso

me must be purely originated by isotropic magnetic interactions like

isotropic exchange. Unfortunately, the deﬁnition of magnetoelastic constants in

some conventions mixes the contribution of both isotropic and anisotropic mag-

netic interactions, so that it is not possible to write Eiso

me in terms of these magnetoe-

lastic constants. This problem can be overcome by using the universal deﬁnition

of the magnetoelastic constants proposed by E. du Tremolet de Lacheisserie on

the basis of group theory[1], which fully decouples the contribution of isotropic

and anisotropic magnetic interactions in the deﬁnition of the magnetoelastic con-

stants. The explicit form of Eiso

me , based on Lacheisserie convention [1], for each

supported crystal symmetry in MAELAS is shown in Appendix A. In Section 5,

we discuss about possible limitations of the method presented here.

The basic idea of this new method is to compute the total spin-polarized en-

ergy (Eq.1) for a set of deformed unit cells in such a way that we can get the

i-th isotropic magnetoelastic constant biso

ifrom a polynomial ﬁtting of the energy

versus strain data

V0E(ε

εi(s)) = Φi(Ci j)s2+Γibiso

is+1

V0E0,(2)

where V0is the equilibrium volume of the reference unstrained state (s=0), Γi

is a real number, Φidepends on the elastic constants Ci j,E0is the energy at the

unstrained state, and sis the parameter used to parameterize the strain tensor ε

εi(s).

In practice, Eq. 2 is ﬁtted to a third order polynomial

f(s) = As3+Bs2+Cs+D,(3)

where A,B,Cand Dare ﬁtting parameters, so that the i-th isotropic magnetoe-

lastic constant biso

iis given by

biso

i=C

Γi.(4)

Additionally, we note that the ﬁtting parameter B=Φi(Ci j)contains informa-

tion about the elastic constants, which is the basis of the program AELAS to

Figure 1: Schematic diagram showing the reference state (unstrained unit cell s=0) required by

the -mode 3 of MAELAS to generate the deformed unit cells, and derive the isotropic magne-

toelastic constants biso. This reference state is constructed by combining the equilibrium lattice

parameters of the PM state and the magnetic order of the GS. In this ﬁgure we assume that ωs<0

(VGS <VPM), and the GS of the material is FM (VGS =VFM). If the GS is AFM, then the FM

conﬁgurations in these unit cells should be changed to AFM.

derive Ci j[12]. The computed elastic constants Ci j with AELAS (energy-strain

method) may include high order magnetic corrections from the second order in

strain isotropic magnetoelastic constants[1, 16, 17]. In Section 4.1, we show that

a third order polynomial ﬁt (Eq.3) is more accurate than second order to compute

biso

iand ωsin a broad range of applied strains. In Table 1 we show the selected

set of biso

i, parameterized strain tensor ε

εi(s)and the corresponding value of Γithat

fulﬁls Eq. 2. If the elastic constants are also computed[12], then one can also

calculate the isotropic magnetostrictive coefﬁcients (λiso) and isotropic contribu-

tion to spontaneous volume magnetostriction ωiso

susing the theoretical equations

derived from magnetoelasticity (λiso(biso

k,Ci j)and ωiso

s(λiso)), see Appendix A.

We implemented this method in the python package MAELAS for all supported

crystal symmetries in previous version of MAELAS [13, 14]. This new method is

executed in MAELAS by using tag -mode 3 in the command line, see more details

in Section 3. The calculated biso by this approach depends strongly on the refer-

ence state (unstrained unit cell s=0) used to generate the deformed unit cells,

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AutomatedcalculationsofexchangemagnetostrictionP.Nievesa,,S.Arapana,S.H.Zhangb,c,A.P.Kadzielawaa,R.F.Zhangb,c,D.LegutaaIT4Innovations,VB-TechnicalUniversityofOstrava,17.listopadu2172/15,70800Ostrava-Poruba,CzechRepublicbSchoolofMaterialsScienceandEngineering,BeihangUniversity,Beijing100191,PRChin...

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Automated calculations of exchange magnetostriction

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