Quantum theory of magnetoelectric properties of rare-earth alumoborates: holmium alumoborate

N.V. Kostyuchenko a,1,2 , A.I. Popov1,3, A.K. Zvezdin1,2 1 A. M. Prokhorov General Physics Institute of Russian Academy of Sciences, Moscow, 38 Vavilov

Str., 119991, Russia;

2 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, 9 Institutsky Per., 141700, Russia

3 National Research University of Electronic Technology, Zelenograd, Moscow region, 5 Pas. 4806, 124498, Russia

ae-mail: nvkost@gmail.com

Keywords: magnetoelectric materials, rare-earth ions, crystal-field parameters, aluminum borates

Abstract. The magnetization processes of HoAl3(BO3)4 rare-earth aluminum borates have been

studied theoretically. Magnetic properties of the crystals were examined. The dependencies of the

magnetic susceptibility on the magnitude and direction of magnetic field were calculated. Study of a

magnetoelectric effect was performed and the dependencies of the polarization on the strength and

orientation of a magnetic field and temperature were obtained. A comparison of the theoretical and

experimental data was performed, their consistency has been ascertained.

Introduction

In recent years, a new class of magnetoelectric materials, namely rare-earth aluminum borates

RAl3(BO3)4, where R = La-Lu, has been intensively studied both theoretically and experimentally

[1,2,3]. These compounds are convenient for marking out the rare-earth ion contribution to the

magnetoelectricity of rare-earth multiferroics (the absence of iron ions eliminates the problem of

determining an effective magnetic field, which influences rare-earth ions). Magnetoelectricityin

aluminum borates is owed to the presence of rare-earth f-ions, which is also typical for rare-earth

ferroborates [4,5,6]. Weak exchange interactions between rare-earth f-ions (compared with d-d and

f-d exchange), large orbital moments of rare-earth ions and weak orbital moment quenching by a

low-symmetry crystal field result in domination of single-ion mechanisms of magnetoelectric effect

over two-ion ones, which are typical for transition metals with d -electrons.

In this study, the quantum theory of magnetoelectric effect in rare-earth aluminum borates has

been developed by an example of holmium aluminum borate. It is currently impossible to derive the

theory from the first principles, that’s why these calculations are based on the usage of crystal-field

parameters that are obtained from spectroscopic experiments. One of the goals of this study is

approbation of the crystal-field parameters, obtained from optical experiments for the description of

magnetic and magnetoelectric phenomena. The level of consistency, achieved between the

theoretical and experimental results [7] counts in favor of the theory that has been developed.

Magnetization and susceptibility

The environment symmetry of rare earth ions in aluminum borates is described by point

symmetry group D3. In aluminum borates, rare earth ions interact with the crystal field and the

external field ˆCF J B

H H g JHµ= + .

2 (2) 4 (4) 6 (6) 4 (4) (4) 6 (6) (6) 6 (6) (6)

0 0 0 0 0 0 3 3 3 3 3 3 6 6 6[ ] [ ] [ ]CF

H B C B C B C iB C C iB C C B C C− − − − −= + + + + + + + + (1)

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In this equation ( ) ( ) ( )k k

q q

i

C C i=∑ , where ( ) ( )k

qC i are single electron irreducible tensor operators,

defined by the reduced matrix elements 0

0 0( ) 2 1k l

l kl C l l C

′′ || || = + . The crystal-field parameters for

Ho3+ allowing to describe the location of a large number of energy levels, that are obtained from

spectroscopic data, are [1]: B20=491cm-1, B4

0=-1150cm-1, B60=327cm-1, B3

4=-797cm-1, B36=-62cm-1,

B66=-162cm-1. In present work the energy levels and wave functions of the holmium ions have been

numerically determined at the different values of H at H||a and H||b.

The magnetization and magnetic susceptibility are determined by:

J B

MNg J

H

αα α αβ

β

µ χ∂

= − ⋅ , =∂

M (2)

where N is the number of Ho 3+ ions per volume unit. The symbol ... means a thermodynamic

averaging over wave functions of a rare earth ion.

A comparison of the numerically calculated magnetic susceptibility along (χz) and perpendicular

(χx) C 3 axis with corresponding experimental data is shown in the Fig. 1. The theory is in a good

agreement with the experiment [7,3].

Fig. 1. The temperature dependences of the magnetic susceptibility along the c - and a -

crystallographic axes. Solid lines – theory, dashed lines – experiment. The experimental data are

taken from [7].

Magnetoelectric effect in HoAl3(BO3)4

The mechanism of the magnetoelectric effect has already been described in [6,8,9]. We adapt it

to HoAl3(BO3)4 here and find the expression for the magnetoelectric operator of rare earth

aluminum borate [6]:

96 Trends in Magnetism: Nanomagnetism (EASTMAG-2013)

meH = −ED (3)

where Dα ( x y zα = , , ) are the operators for the components of the effective electric dipole

moment of a rare earth ion [6,9]:

2 2 2 4 4 4 6 6 6 4 4 4 6 6 6

2 2 2 2 2 2 2 2 2 4 4 4 4 4 4

2 2 2 4 4 4 6 6 6 4 4 4 6 6 6

2 2 2 2 2 2 2 2 2 4 4 4 4 4 4

4 4 4 6 6 6

3 3 3 3 3 3

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

x

y

z

D b C C b C C b C C b C C b C C

D b i C C b i C C b i C C b i C C b i C C

D b C C b C C

− − − − −

− − − − −

− −

= + + + + + + + + +

= − + − + − + − + −

= − + − . (4)

Constants bpq contain contributions of the electronic bp

q(e) and ionic bpq(i) mechanisms to the

polarization [6]. Values of bpq(e) are expressed in the terms of odd crystal field parameters [6] that

are not known with a sufficient degree of precision. We are aware of only one paper [10] where the

parameters of the odd crystal field has been calculated for Pr3+

in praseodymium iron borate in the

frame of the point-charge model. Generally, the bpq values are treated as phenomenological

parameters in the Hme Hamiltonian and can be determined from the fitting of the theoretical results

to the experimental ones.

Holmium aluminum borate is a good example because it reveals a large magnitude of the

magnetoelectric effect. The magnetoelectric effect has been studied in: [7] and [3]. The data they

present differ significantly. The study presented below shows that the data from [7] are preferable. It

is the data we use to compare the theoretical calculations with.

Using Eq. 4 the polarization along x -axis a

P can be expressed in the form:

2 2 2 4 4 4 6 6 6 4 4 4 6 6 6

2 2 2 2 2 2 2 2 2 4 4 4 4 4 4a xP N D d C C d C C d C C d C C d C C− − − − −= = + + + + + + + + + , (5)

where k k

q qd Nb= . Here k

qd are empirical constants determined from a comparison between the

theoretical and experimental data.

In the weak fields H << ∆/µ =20 kOe (∆ is the distance to the overlying levels, µ≈10 µB – the

magnetic moment of a holmium ion at 0T = K), the magnetoelectric energy is a product of the

components of the electric field and the basis functions of even powers of H . This expansion is

given by:

}

2 2 1 2 2 2 2 2

2 4

2 2 2 2 2

4

2 ( ) 4

4 ( ) 3

me x x y y x y x x y x y

y x y x y z z x x y

F N C E H H E H H C E H H H H

E H H H H C E H H H H

= − = − − ⋅ + − − +

+ ⋅ − + − + ...

E D (6)

Eq. 6 determines the behavior of the electric polarization on the magnitude and the direction of

an external magnetic field. In weak fields when polarization linearly depends on 2H the condition

( ) ( )a a a b

P H P H= − is performed.

In the case of strong fields (H≥20kOe) we should calculate the terms <Ckq+Ck

-q>. These averages

are the functions of the external field and temperature. We calculate them for the cases H||a and

H||b. The dkq values are required for a quantitative description. The values for terbium aluminum

borate d22=-1.25·105 µC/m2, d4

2=-2.12·105 µC/m2 [11] were taken as the first approximation. The

values of the parameters obtained from comparison between theory and experiment are: dkq:

d22=-1.12·105 µC/m2, d4

2=-2.00·105 µC/m2, d44=-1.18·104 µC/m2, d6

2=-7.44·104 µC/m2, d64=-1.77·104

µC/m2. Though we cannot claim the unambiguity of the dkq parameters, a good concordance between

the theory and the experiment demonstrates the reasonableness of such a choice.

The dependencies of the polarization are presented in Fig. 2. A good agreement with

experimental data [7] has been demonstrated.

Solid State Phenomena Vol. 215 97

Fig. 2. The dependences of the polarization of HoAl3(BO3)4 along the a axis at the different

temperatures in the cases of H a|| and H b|| . The polarization along the a axis at 5T K= is

shown in the inset. Solid lines – theory, dashed lines – experiment. The experimental data are taken

from [7].

Conclusions

The dependencies of the magnetic susceptibility and electric polarization of HoAl3(BO3)4 on the

magnitude and direction of an external field and temperature have been calculated in this work with

use of the crystal-field parameters Bkn from [1]. A comparison of theoretical results with

experimental data given in [7] has been performed, showing a good agreement.

An analysis of magnetoelectric properties for this compound has also been performed. The

peculiarities of the electric polarization induced by the external magnetic field have been studied.

Polarization anisotropy ( ( ) ( )a a a b

P H P H| |≠| | ) is shown on Fig. 2 (see inset). The comparison of the

98 Trends in Magnetism: Nanomagnetism (EASTMAG-2013)

theoretical result, acquired in this study, with the complex of experimental data demonstrates their

quantitative consistency and indicates that the parameters of crystal field obtained from

spectroscopic experiments enable a quite precise description of the magnetic and magnetoelectric

phenomena.

Acknowledgements

We are very grateful to B. Z. Malkin who has shown us the article [1] where crystal field

parameters for HoAl3(BO3)4 that we were using in our study were presented.

This work is supported by the Russian Foundation for Basic Research (projects 13-02-01093 and

12-02-01261).

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