Inversion center breaking due to antiferromag- antiferromag-netic ordering

In document University of Groningen Orbital ordering and multiferroics Nenert, Gwilherm (Page 22-29)

6.4.1 Introduction

Various groups looked for magnetically induced ferroelectrics. The main idea is that antiferromagnetic ordering may break the inversion center of a given structure [13]. Until very recently the magnetically induced ferro-electrics exhibited low polarization values. It has been proposed that the ferroelectric displacements should not rely on the presence of spin orbit cou-pling alone in order to have a significant polarization [10]. We treat here the case where the inversion center is broken by the antiferromagnetic or-dering. We predict a new antiferromagnetic induced ferroelectric due to a non E type antiferromagnetism contrary to the orthorhombic HoMnO3 and in RNiO3 (R = rare-earth) [10].

6.4.2 Cu

2

MnSnS

4

Introduction

The title compound has been studied in the search for new dilute magnetic semiconductors (DMS’s). Most of the new designed DMS have been based on Mn-II-VI compositions with a zinc-blende or wurtzite crystal structure.

However, the main issue in these materials in the light of application is the difficulty to align the Mn spins even with a high magnetic field. In this type of materials, it has been demonstrated that the necessity for a high magnetic field to align the spins is directly related to the amount of Mn in the structure. In order to investigate more in details the relationship be-tween the critical field to align the spins and the structure of recent DMS’s, T. Fries and coworkers investigated the magnetic structure of Cu2MnSnS4. This compound crystallizes in the space group I -42m (n121) where only the manganese atoms carry a spin since the copper atoms here are Cu+ions.

The Mn2+ ions occupy the Wyckoff position 2a. This compound presents an antiferromagnetic structure characterized by a propagation wave-vector −→ k

= (1/2,0,1/2) below TN=8.8K. A representation of the magnetic structure is presented in figure 6.7.

Magnetic ordering The little group of −→

k describing the magnetic structure of Cu2MnSnS4 is identical to the one describing the magnetic structure of LiFeP2O7 (see section 6.3.2). Consequently, we can write a similar expression for the free-energy:

a C

b

Figure 6.7: Magnetic structure of Cu2MnSnS4 in the (a,c) plane. Arrows indi-cate the magnetic moments on the manganese atoms.

F = F0+a

2L2+ b

4L4+ c

2M2+ d 4M4 +1

2 X

i=x,y,z

iL2i + βiMi2) + δLxLz+ γMxMz

+ σ1LxMy + σ2LzMy+ σ3LyMz+ σ5LyMx

(6.24)

We will not discuss here the free-energy and the different magnetic sym-metries since there are the same as for LiFeP2O7. Experimentally, it has been observed that the structure orders magnetically in the magnetic struc-ture described by 2 antiferromagnetic components (Lx and Lz). The mag-netic ordering is described by the magmag-netic space group P21.

Magnetic ordering induces polarization

We will have a look at the possible breaking of the inversion center. In other words, can we have appearance of spontaneous polarization while the compound orders magnetically? In order to determine the extra terms susceptible to appear, we investigated the transformation properties of the different magnetic components. We present the results in table 6.10.

From table 6.10, we notice an interesting term LxLzPy. Consequently, we can write a simplified expression for the free energy including this new term:

6.4. Inversion center breaking due to antiferromagnetic ordering 147 h1 h2 Rh1 Rh2

Lx 1 1 -1 -1

Ly 1 -1 -1 1

Lz 1 1 -1 -1

Mx 1 -1 -1 1

My 1 1 -1 -1

Mz 1 -1 -1 1

Px 1 -1 1 -1

Py 1 1 1 1

Pz 1 -1 1 -1

Table 6.10: Transformation properties of the components of the different vectors of interest for the magnetic space group I-42m associated to k=(1/2, 0, 1/2).

F =F0+ a

2L2+ b

4L4+α 2P2 + βLxLzPy+ 1

2 X

i=x,y,z

νiL2i (6.25)

If we minimize 6.25, we find:

∂F

∂Lx = aLx+ bL3x+ βLzPy+ νxLx

∂F

∂Ly = aLy+ bL3y+ νyLy

∂F

∂Lz = aLz+ bL3z + βLxPy + νzLz

∂F

∂Px = αPx

∂F

∂Py = αPy + βLxLz

∂F

∂Pz = αPz

(6.26)

From 6.26, we can find two series of equalities:

Px= Pz = 0 Py = −β

αLxLz

Lx = 0 or Lx = −(a + νx+βα2L2z) b

Ly = 0 or Ly = −(a + νy) b

Lz = 0 or Lz = −(a + νz+βα2L2x) b

(6.27)

Consequently, we have different magnetic ordering patterns possible.

However, from experiment we know that we have the situation (Lx,Lz)6=(0,0).

Above TN, we have Lx = Lz = 0 and consequently Py = 0. However, below TN, we have (replacing the expression of Lx in Lz and vice-versa):

Lx= −α(abα2+ aβ2+ β2νz+ αβνx) α2b2− β4

Lz = −α(abα2+ aβ2+ β2νx+ αβνz) α2b2− β4

(6.28)

Using 6.28 to put in 6.27, we find that there is a spontaneous polarization possible while going in the magnetic ordered state:

Py = − β αLxLz

= − β

α ×−α(abα2+ aβ2+ β2νz+ αβνx) α2b2− β4

×−α(abα2 + aβ2+ β2νx+ αβνz) α2b2− β4

(6.29)

If we assume that the relativistic components νj have a negligible effect on the value of the spontaneous polarization, we can rewrite 6.29 as:

Py ' −a2αβ (bα2+ β2)2

2b2− β4)2 (6.30) From 6.30, we see that the spontaneous polarization appearing below TN is proportional to a2. This is the first time to our knowledge that a non-oxide material exhibits such property.

6.4. Inversion center breaking due to antiferromagnetic ordering 149

Magnetoelectric properties

We have been describing in the previous section, the spontaneous polar-ization susceptible to arise going trough the antiferromagnetic ordering.

However, this is not the only polarization possible. We have to deal here with the same point group than in the case of LiFeP2O7 (see section 6.5).

Consequently, under the application of a magnetic field, we will have the following polarization components which will appear:

Px = −κxxλ1Lx 1 − κxxχxxλ21L2xHx

Py = −κyyλ2Lx 1 − κyyχyyλ22L2xHy Pz = −κzzλ3Lz

1 − κzzχzzλ23L2zHz

(6.31)

From 6.31, we can predict the appearance of some polarization along the three direction of the crystal under the application of a magnetic field.

This induced polarization by a magnetic field is the characteristic of mag-netoelectricity. However, here we have to deal with a structure giving rise to some spontaneous polarization along y under TN. Consequently, we will have an increase of this polarization under magnetic field by an amount of:

∆Py = −β

α × −α(abα2+ aβ2+ β2νz+ αβνx) α2b2 − β4

× −α(abα2+ aβ2+ β2νx+ αβνz) α2b2− β4

−κyyλ2Lx

1 − κyyχyyλ22L2xHy

' −a2αβ (bα2+ β2)2

2b2− β4)2 −κyyλ2Lx 1 − κyyχyyλ22L2xHy

(6.32)

We show using Landau and group theory that Cu2MnSnS4 can be con-sidered as a new magnetically induced ferroelectric. In addition to a sponta-neous polarization, we show that a linear magnetoelectric effect is allowed.

Moreover, the ferroelectric displacements are not expected to rely on the presence of spin-orbit coupling. Consequently, the polarization should be quite strong as in the predicted orthorhombic HoMnO3 and in RNiO3 [10].

Contrary to these compounds, the polarization should be tunable by electric and magnetic fields.

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In document University of Groningen Orbital ordering and multiferroics Nenert, Gwilherm (Page 22-29)

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