### 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 HoMnO_{3} and
in RNiO_{3} (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 Cu_{2}MnSnS_{4}.
*This compound crystallizes in the space group I -42m (n** ^{◦}*121) where only
the manganese atoms carry a spin since the copper atoms here are Cu

^{+}ions.

The Mn^{2+} 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 T* _{N}*=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 Cu*_{2}MnSnS_{4}
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 Cu*_{2}*MnSnS*_{4} *in the (a,c) plane. Arrows *
*indi-cate the magnetic moments on the manganese atoms.*

*F = F*_{0}+*a*

2*L*^{2}+ *b*

4*L*^{4}+ *c*

2*M*^{2}+ *d*
4*M*^{4}
+1

2 X

*i=x,y,z*

*(ν*_{i}*L*^{2}_{i}*+ β*_{i}*M*_{i}^{2})
*+ δL*_{x}*L*_{z}*+ γM*_{x}*M*_{z}

*+ σ*_{1}*L*_{x}*M*_{y}*+ σ*_{2}*L*_{z}*M*_{y}*+ σ*_{3}*L*_{y}*M*_{z}*+ σ*_{5}*L*_{y}*M*_{x}

(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 (L* _{x}* and L

*). The mag-netic ordering is described by the magmag-netic space group P21.*

_{z}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 L*x*L*z*P*y*. 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*
h_{1} h_{2} Rh_{1} Rh_{2}

L* _{x}* 1 1 -1 -1

L*y* 1 -1 -1 1

L* _{z}* 1 1 -1 -1

M* _{x}* 1 -1 -1 1

M*y* 1 1 -1 -1

M* _{z}* 1 -1 -1 1

P* _{x}* 1 -1 1 -1

P* _{y}* 1 1 1 1

P*z* 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 =F*_{0}+ *a*

2*L*^{2}+ *b*

4*L*^{4}+*α*
2*P*^{2}
*+ βL*_{x}*L*_{z}*P** _{y}*+ 1

2 X

*i=x,y,z*

*ν*_{i}*L*^{2}* _{i}* (6.25)

If we minimize 6.25, we find:

*∂F*

*∂L*_{x}*= aL**x**+ bL*^{3}_{x}*+ βL**z**P**y**+ ν**x**L**x*

*∂F*

*∂L*_{y}*= aL*_{y}*+ bL*^{3}_{y}*+ ν*_{y}*L*_{y}

*∂F*

*∂L*_{z}*= aL*_{z}*+ bL*^{3}_{z}*+ βL*_{x}*P*_{y}*+ ν*_{z}*L*_{z}

*∂F*

*∂P*_{x}*= αP**x*

*∂F*

*∂P*_{y}*= αP*_{y}*+ βL*_{x}*L*_{z}

*∂F*

*∂P*_{z}*= αP*_{z}

(6.26)

From 6.26, we can find two series of equalities:

*P*_{x}*= P** _{z}* = 0

*P*

_{y}*= −β*

*αL*_{x}*L*_{z}

*L*_{x}*= 0 or L** _{x}* =

*−(a + ν*

*+*

_{x}

^{β}

_{α}^{2}

*L*

^{2}

*)*

_{z}*b*

*L*_{y}*= 0 or L** _{y}* =

*−(a + ν*

*)*

_{y}*b*

*L**z* *= 0 or L**z* = *−(a + ν** _{z}*+

^{β}

_{α}^{2}

*L*

^{2}

*)*

_{x}*b*

(6.27)

Consequently, we have different magnetic ordering patterns possible.

However, from experiment we know that we have the situation (L* _{x}*,L

_{z}*)6=(0,0).*

Above T* _{N}*, we have L

*= L*

_{x}*= 0 and consequently P*

_{z}*= 0. However, below T*

_{y}*N*, we have (replacing the expression of L

*x*in L

*z*and vice-versa):

*L** _{x}*=

*−α(abα*

^{2}

*+ aβ*

^{2}

*+ β*

^{2}

*ν*

_{z}*+ αβν*

*)*

_{x}*α*

^{2}

*b*

^{2}

*− β*

^{4}

*L** _{z}* =

*−α(abα*

^{2}

*+ aβ*

^{2}

*+ β*

^{2}

*ν*

_{x}*+ αβν*

*)*

_{z}*α*

^{2}

*b*

^{2}

*− β*

^{4}

(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:

*P*_{y}*= −* *β*
*αL*_{x}*L*_{z}

*= −* *β*

*α* *×−α(abα*^{2}*+ aβ*^{2}*+ β*^{2}*ν*_{z}*+ αβν** _{x}*)

*α*

^{2}

*b*

^{2}

*− β*

^{4}

*×−α(abα*^{2} *+ aβ*^{2}*+ β*^{2}*ν*_{x}*+ αβν** _{z}*)

*α*

^{2}

*b*

^{2}

*− β*

^{4}

(6.29)

*If we assume that the relativistic components ν** _{j}* have a negligible effect
on the value of the spontaneous polarization, we can rewrite 6.29 as:

*P*_{y}*' −a*^{2}*αβ* *(bα*^{2}*+ β*^{2})^{2}

*(α*^{2}*b*^{2}*− β*^{4})^{2} (6.30)
*From 6.30, we see that the spontaneous polarization appearing below*
T* _{N}* is proportional to a

^{2}. 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:

*P**x* = *−κ*_{xx}*λ*_{1}*L*_{x}*1 − κ*_{xx}*χ*_{xx}*λ*^{2}_{1}*L*^{2}_{x}*H**x*

*P** _{y}* =

*−κ*

_{yy}*λ*

_{2}

*L*

_{x}*1 − κ*

_{yy}*χ*

_{yy}*λ*

^{2}

_{2}

*L*

^{2}

_{x}*H*

_{y}*P*

*=*

_{z}*−κ*

*zz*

*λ*3

*L*

*z*

*1 − κ*_{zz}*χ*_{zz}*λ*^{2}_{3}*L*^{2}_{z}*H*_{z}

(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 T* _{N}*. Consequently, we will
have an increase of this polarization under magnetic field by an amount of:

*∆P*_{y}*= −β*

*α* *×* *−α(abα*^{2}*+ aβ*^{2}*+ β*^{2}*ν*_{z}*+ αβν** _{x}*)

*α*

^{2}

*b*

^{2}

*− β*

^{4}

*×* *−α(abα*^{2}*+ aβ*^{2}*+ β*^{2}*ν*_{x}*+ αβν** _{z}*)

*α*

^{2}

*b*

^{2}

*− β*

^{4}

*−* *−κ**yy**λ*2*L**x*

*1 − κ*_{yy}*χ*_{yy}*λ*^{2}_{2}*L*^{2}_{x}*H**y*

*' −a*^{2}*αβ* *(bα*^{2}*+ β*^{2})^{2}

*(α*^{2}*b*^{2}*− β*^{4})^{2} *−* *−κ*_{yy}*λ*_{2}*L*_{x}*1 − κ**yy**χ**yy**λ*^{2}_{2}*L*^{2}_{x}*H*_{y}

(6.32)

We show using Landau and group theory that Cu_{2}MnSnS_{4} 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 HoMnO_{3} and in RNiO_{3} [10].

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

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