Piezomagnetism

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 132-137

Section 1.5.7. Piezomagnetism

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.7. Piezomagnetism

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As we have seen, the appearance of weak ferromagnetism in antiferromagnets is closely connected with their magnetic symmetry. If the magnetic point group of the antiferromagnetic crystal contains an axis of higher than twofold symmetry, the magnetic structure is purely antiferromagnetic. By applying an external force that disturbs the symmetry of the crystal and destroys the axis of high symmetry, one may create a structure possessing weak ferromagnetism. In the previous section, we considered such reduction of the symmetry with the aid of a magnetic field applied perpendicular to the main axis of the crystal. Another possibility for symmetry reduction is to apply an external pressure and to deform the crystal. Thus, in some antiferromagnetic crystals, a ferromagnetic moment may be produced on application of external stress. This phenomenon is called piezomagnetism.

To investigate the piezomagnetic effect from the phenomenological point of view, we have to add the terms of the magnetoelastic energy in the expansion of the thermodynamic potential. The magnetoelastic terms of the least degree in the expansion of the thermodynamic potential $[\tilde{\Phi}]$ for a given stable magnetic structure will be of the type $[T_{ij}M_{k}L_{l}]$ ( $[T_{ij}]$ are the components of the elastic stress tensor $[{\bf T}]$ ). These terms must be invariant relative to the crystallographic group of the material under examination. If we consider the potential $[\Phi]$ , which is a function of T, $[{\bf T}]$ , $[\bf H]$ , the terms of the magnetoelastic energy that are responsible for piezomagnetism are of the form $[H_{i}T _{jk}]$ . Thus, for the piezomagnetic crystals the expansion of the thermodynamic potential should be expressed by $[\Phi (T, {\bf T}, {{\bf H}}) = \Phi_{0} (T, {{\bf H}}) - \textstyle\sum\limits_{i,j,k} \Lambda_{ijk} H_{i}T_{jk}. \eqno(1.5.7.1)]$ If at least one term of this expansion remains invariant under the magnetic symmetry of the given crystal, then the corresponding component $[\Lambda_{ijk}]$ will not be zero and hence $[M_{i} = -\partial \Phi/\partial H_{i} = -\partial \Phi_{0}/\partial H_{i} + \Lambda_{ijk} T_{jk}. \eqno(1.5.7.2)]$ Thus, when a stress $[T_{jk}]$ is applied, a magnetic moment is produced which is linear with the stress.

It follows from expression (1.5.7.1) that the converse of the piezomagnetic effect also exists, i.e. linear magnetostriction : $[S_{jk} = -\partial {\Phi}/\partial T_{jk} = \Lambda_{ijk} H_{i}, \eqno(1.5.7.3)]$ where $[S_{jk}]$ are the components of the deformation tensor.

1.5.7.1. Piezomagnetic effect

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The possibility of the existence of a piezomagnetic effect was first foreseen by Voigt (1928). However, he assumed that it is sufficient to consider only the crystallographic symmetry in order to predict this effect. In reality, the crystals that do not possess a magnetic structure are characterized by the transformation R being contained in the magnetic group as an independent element. The transformation R changes the sign of the magnetic vectors $[\bf H]$ , $[\bf L]$ , $[\bf M]$ . Hence, for such crystals all values of $[\Lambda_{ijk}]$ vanish and piezomagnetism is forbidden. The magnetic groups of magnetically ordered crystals (ferromagnets and antiferromagnets) contain R only in combination with other elements of symmetry, or do not contain this transformation at all. Hence the piezomagnetic effect may occur in such crystals. This statement was first made by Tavger & Zaitsev (1956). The most interesting manifestation of the piezomagnetic effect is observed in antiferromagnets, as there is no spontaneous magnetization in these materials.

From equation (1.5.7.1) it follows that $[\Lambda_{ijk}]$ is an axial tensor of third rank. Hence, apart from the restriction that piezomagnetism is forbidden for all para- and diamagnetic materials, it must be absent from the 21 magnetic point groups that contain the element $[C_{i}R = \overline{1}']$ (see Table 1.5.7.1). The stress tensor $[T_{jk}]$ is symmetrical ( $[T_{jk} = T_{kj}]$ ); see Section 1.3.2.4 . Thus the tensor $[\Lambda_{ijk}]$ is symmetrical in its last two indices. This is the reason why piezomagnetism is prohibited for three more magnetic point groups: $[{\bi O}=432]$ , $[{\bi T}_{d}=\bar{4}3m]$ and $[{\bi O}_{h}=m\bar{3}m]$ . The remaining 66 magnetic point groups were found by Tavger (1958), who also constructed the 16 corresponding forms of the piezomagnetic tensors appropriate to each point group. They are represented in Table 1.5.7.1. (See also Birss & Anderson, 1963; Birss, 1964.)

Table 1.5.7.1 | top | pdf |
The forms of the matrix characterizing the piezomagnetic effect

Magnetic crystal class		Matrix representation $[\Lambda_{i\alpha}]$ of the piezomagnetic tensor
Schoenflies	Hermann–Mauguin
$[{\bi C}_1]$		$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} &\Lambda_{14} &\Lambda_{15} &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} &\Lambda_{24} &\Lambda_{25} &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} &\Lambda_{34} &\Lambda_{35} &\Lambda_{36} } \right]]$
$[{\bi C}_i]$	$[\bar{1}]$

$[{\bi C}_2]$	$[2\,(=121)]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 &\Lambda_{34} & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_s]$	$[m\,(=1m1)]$
$[{\bi C}_{2h}]$	$[2/m\,(=1\,2/m\,1)]$
	(unique axis y)
$[{\bi C}_2({\bi C}_1)]$	$[2'\,(=12'1)]$	$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 &\Lambda_{35} & 0 } \right]]$
$[{\bi C}_s({\bi C}_1)]$	$[m'\,(=1m'1)]$
$[{\bi C}_{2h}({\bi C}_i)]$	$[2'/m'\,(=1\,2'/m'\,1)]$
	(unique axis y)
$[{\bi D}_2]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_{2v}]$	$[mm2\,[2mm,m2m]]$
$[{\bi D}_{2h}]$
$[{\bi D}_2({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 & 0 \cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 & 0 & 0 }\right]]$
$[{\bi C}_{2v}({\bi C}_2)]$
$[{\bi C}_{2v}({\bi C}_s)]$	$[m'2'm\,[2'm'm]]$
$[{\bi D}_{2h}({\bi C}_{2h})]$
$[{\bi C}_4, \,{\bi C}_6]$	$[4,\, 6]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & \Lambda_{15}& 0 \cr 0 & 0 & 0 &\Lambda_{15} &-\Lambda_{14}& 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi S}_4,\,{\bi C}_{3h}]$	$[\bar{4},\,\bar{6}]$
$[{\bi C}_{4h},\,{\bi C}_{6h}]$	$[4/m,\,6/m]$
$[{\bi C}_4({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & \Lambda_{14}&\Lambda_{15} & 0 \cr 0 & 0 & 0 &-\Lambda_{15}&\Lambda_{14} & 0 \cr \Lambda_{31}&-\Lambda_{31} & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi S}_4({\bi C}_2)]$	$[\bar{4}']$
$[{\bi C}_{4h}({\bi C}_{2h})]$
$[{\bi D}_4,\,{\bi D}_6 ]$	$[422,\,622]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v},\,{\bi C}_{6v}]$	$[4mm,\,6mm]$
$[{\bi D}_{2d},\,{\bi D}_{3h}]$	$[\bar{4}2m\,[\bar{4}m2],\,\bar{6}m2\,[\bar{6}2m]]$
$[{\bi D}_{4h},\,{\bi D}_{6h}]$	$[4/mmm,\,6/mmm]$
$[{\bi D}_4({\bi C}_4),\,{\bi D}_6({\bi C}_6)]$	$[42'2',\,62'2']$	$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v}({\bi C}_4),\,{\bi C}_{6v}({\bi C}_6)]$	$[4m'm',\,6m'm']$
$[{\bi D}_{2d}({\bi S}_4),\,{\bi D}_{3h}({\bi C}_{3h})]$	$[\bar{4}2'm'\,[\bar{4}m'2'],\,\bar{6}m'2'\,[\bar{6}2'm']]$
$[{\bi D}_{4h}({\bi C}_{4h}),\,{\bi D}_{6h}({\bi C}_{6h})]$	$[4/mm'm',\,6/mm'm']$
$[{\bi D}_4({\bi D}_2)]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} }\right]]$
$[{\bi C}_{4v}({\bi C}_{2v})]$
$[{\bi D}_{2d}({\bi D}_2),\,{\bi D}_{2d}({\bi C}_{2v})]$	$[\bar{4}'2m',\,\bar{4}'m2']$
$[{\bi D}_{4h}({\bi D}_{2h})]$
$[{\bi C}_3]$		$[\left[\matrix{ \Lambda_{11}&-\Lambda_{11} &0 &\Lambda_{14} & \Lambda_{15} &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} &0 &\Lambda_{15} &-\Lambda_{14} &-2\Lambda_{11} \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33}& 0 & 0 & 0 } \right]]$
$[{\bi S}_6]$	$[\bar{3}]$

$[{\bi D}_3]$	$[32\,(=321)]$	$[\left[\matrix{\Lambda_{11} &-\Lambda_{11}& 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}]$	$[3m\,(=3m1)]$
$[{\bi D}_{3d}]$	$[\bar{3}m\,(=\bar{3}m1)]$
$[{\bi D}_3({\bi C}_3)]$	$[32'\,(=32'1)]$	$[\left[\matrix{ 0 & 0 & 0 & 0 & \Lambda_{15}&-2\Lambda_{22}\cr -\Lambda_{22} & \Lambda_{22} & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}({\bi C}_3)]$	$[3m'\,(=3m'1)]$
$[{\bi D}_{3d}({\bi S}_6)]$	$[\bar{3}m'\,(=\bar{3}m'1)]$
$[{\bi C}_6({\bi C}_3) ]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} & 0 & 0 & 0 &-2\Lambda_{11} \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3h}({\bi C}_3)]$	$[\bar{6}']$
$[{\bi C}_{6h}({\bi S}_6)]$
$[{\bi D}_6({\bi D}_3)]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{6v}({\bi C}_{3v})]$
$[{\bi D}_{3h}({\bi D}_3),\,{\bi D}_{3h}({\bi C}_{3v})]$	$[\bar{6}'2m',\,\bar{6}'m2']$
$[{\bi D}_{6h}({\bi D}_{3d})]$
$[{\bi T},\,{\bi T}_h ]$	$[23,\,m\bar{3} ]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{14} }\right]]$
$[{\bi O}({\bi T})]$
$[{\bi T}_d({\bi T})]$	$[\bar{4}'3m']$
$[{\bi O}_h({\bi T}_h)]$	$[m\bar{3}m']$

Since the stress tensor $[T_{jk}]$ is symmetrical, it has only six independent components. Therefore the notation of its components can be replaced by a matrix notation (Voigt's notation, see Section 1.3.2.5 ) in the following manner: $[\matrix{\hbox{Tensor notation}\hfill&\hbox{Matrix notation}\hfill\cr T_{11}\hfill & T_{1}\hfill\cr T_{22}\hfill & T_{2}\hfill\cr T_{33}\hfill & T_{3}\hfill\cr T_{23}, T_{32}\hfill & T_{4}\hfill\cr T_{31}, T_{13}\hfill & T_{5}\hfill\cr T_{12}, T_{21}\hfill & T_{6}\hfill\cr&\cr}]$ In matrix notation, equation (1.5.7.2) may be written in the form $[M_{i} = \Lambda_{i\alpha}T_{\alpha}, \eqno(1.5.7.4)]$ where [i = 1, 2, 3] and $[\alpha = 1, 2, 3, 4, 5, 6]$ . These notations are used in Table 1.5.7.1. Notice that $[\Lambda_{ij}=\Lambda_{ijj}]$ for [j=1,2,3] , $[\Lambda_{i4}=2\Lambda_{i23}]$ , $[\Lambda_{i5}=2\Lambda_{i31}]$ , and $[\Lambda_{i6}=2\Lambda_{i12}]$ .

The form of the matrix $[\Lambda_{i\alpha}]$ depends on the orientation of the axes of the Cartesian coordinate system (CCS) with respect to the symmetry axes of the point group of the crystal under consideration. These symmetry axes may be rotation axes, rotoinversion axes or mirror-plane normals, all possibly combined with time reversal. The usual orientations of the CCS with respect to the symmetry axes can be expressed by the order of the entries in the Hermann–Mauguin symbol. An entry consists (apart from possible primes and bars) of a number [N=1] , 2, 3, 4 or 6 or the letter m or [N/m] ( $[={\textstyle{N \over m}}]$ ). The conventional rules will be followed: in the monoclinic and orthorhombic crystal systems the x, y and z axes of the CCS are parallel to the symmetry axes given in the first, second and third entries, respectively. In the monoclinic system, there is only one symmetry axis, which is usually chosen parallel to the y axis, and a short Hermann–Mauguin symbol with only one entry is usually used, e.g. [2/m] instead of $[1\,2/m\,1]$ . In the trigonal and hexagonal systems, the z, x and y axes are parallel to the symmetry axes given in the first, second and third entries, respectively. In the tetragonal system, the z axis is parallel to the symmetry axis given in the first entry, and the x and y axes are parallel to the symmetry axes given in the second entry, which appear in two mutually perpendicular directions. In the cubic system, the symmetry axes given in the first entry appear in three mutually perpendicular directions; the x, y and z axes of the CCS are chosen parallel to these directions. Alternative orientations of the same point group that give rise to the same form of $[\Lambda_{i\alpha}]$ have been added between square brackets [] in Table 1.5.7.1. Notice that the Schoenflies notation does not allow us to distinguish different orientations of the CCS with respect to the symmetry axes.

The forms of $[\Lambda_{i\alpha}]$ for frequently encountered orientations of the CCS other than those given in Table 1.5.7.1 are

(1) $[112, 11m, 11\,2/m]$ (unique axis z): $[\left[\matrix{0 & 0 & 0 &\Lambda_{14} &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} &\Lambda_{25} & 0 \cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 & 0 &\Lambda_{36}}\right]\semi]$
(2) $[112', 11m', 11\,2'/m']$ (unique axis z): $[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} & 0 & 0 &\Lambda_{16} \cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 & 0 &\Lambda_{26} \cr 0 & 0 & 0 &\Lambda_{34} &\Lambda_{35} & 0}\right]\semi]$
(3) $[22'2', 2m'm', mm'2'\,[m2'm'], mm'm']$ : $[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} &0 &0 & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{26} \cr 0 & 0 & 0 & 0 &\Lambda_{35} & 0 }\right]\semi]$
(4) $[2'22', m'2m', 2'mm'\,[m'm2'], m'mm']$ : $[\left[\matrix{0 & 0 & 0 & 0 & 0 &\Lambda_{16} \cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 & 0 & 0 \cr 0 & 0 & 0 &\Lambda_{34} & 0 & 0 }\right]\semi]$
(5) $[4'2'2, 4'm'm, \overline{4}\,'m'2, \overline{4}\,'2'm, 4'/mm'm]$ : $[\left[\matrix{0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &-\Lambda_{15}& 0 & 0 \cr \Lambda_{31} &-\Lambda_{31}& 0 & 0 & 0 & 0}\right]\semi]$
(6) $[6'2'2, 6'm'm, \overline{6}\,'m'2, \overline{6}\,'2'm, 6'/m'm'm]$ : $[\left[\matrix{0 & 0 & 0 & 0 & 0 &-2\Lambda_{22}\cr -\Lambda_{22} &\Lambda_{22} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 }\right]\semi]$
(7) $[312, 31m, \overline{3}1m]$ : $[\left[\matrix{0 & 0 & 0 &\Lambda_{14} & 0 &-2\Lambda_{22}\cr -\Lambda_{22} &\Lambda_{22} & 0 & 0 &-\Lambda_{14}& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 }\right]\semi]$
(8) $[312', 31m', \overline{3}1m']$ : $[\left[\matrix{\Lambda_{11} &-\Lambda_{11}& 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{15} & 0 &-2\Lambda_{11}\cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0}\right].]$

Many connections between the different forms of $[\Lambda_{i\alpha}]$ given above and in Table 1.5.7.1 have been derived by Kopský (1979a,b) and Grimmer (1991). These connections between the forms that the matrix can assume for the various magnetic or crystallographic point groups hold for all matrices and tensors that describe properties of materials, not just for the special case of piezomagnetism.

Dzyaloshinskii (1957b) pointed out a number of antiferromagnets that may display the piezomagnetic effect. These include the fluorides of the transition metals, in which the piezomagnetic effect was first observed experimentally (see Fig. 1.5.7.1) (Borovik-Romanov, 1959b). Below we shall discuss the origin of the piezomagnetic effect in fluorides in more detail.

Figure 1.5.7.1 | top | pdf |

The dependence of the magnetic moment of CoF₂ on the magnetic field. (1) Without stress; (2) under the stress T_xz = 33.3 MPa (Borovik-Romanov, 1960).

The fluorides of transition metals MnF₂, CoF₂ and FeF₂ are tetragonal easy-axis antiferromagnets (see Fig. 1.5.5.3). It is easy to check that the expansion of the thermodynamic potential $[\tilde{\Phi}]$ up to terms that are linear in stress $[T_{ij}]$ and invariant relative to the transformations of the crystallographic space group $[{\bi D}^{14}_{4h} =]$ $[ P4_{2}/mnm]$ is represented by $[\eqalignno{\tilde{\Phi} &= \tilde{\Phi}_{0} + (A/2){\bf L}^{2} + (a/2)(L^{2}_{x} + L^{2}_{y}) &\cr&\quad+ (B/2){\bf M}^{2} + (b/2)(M^{2}_{x} + M^{2}_{y}) &\cr&\quad + d(L_{x}M_{y} + L_{y}M_{x}) &\cr&\quad + 2\lambda_{1}(M_{x}T_{yz} + M_{y}T_{xz})L_{z}&\cr&\quad + 2\eta _{1}(L_{y}T_{yz} + L_{x}T_{xz})L_{z} &\cr&\quad + 2\lambda_{2}M_{z}L_{z}T_{xy} + 2\eta_{2}L_{x}L_{y} T_{xy} - {\bf MH}. &(1.5.7.5)}]$ In this expression, the sums ( $[T_{ij} + T_{ji}]$ ) that appear in the magnetoelastic terms have been replaced by $[2T_{ij}]$ , as $[T_{ij} \equiv T_{ji}]$ .

The analysis of expression (1.5.7.5) in the absence of stresses proves that fluorides may possess weak ferromagnetism provided that $[a \,\lt\, 0]$ ( $[L_{z} = 0]$ ) (see Section 1.5.5.1). Here we shall discuss the easy-axis structure of the fluorides MnF₂, CoF₂, FeF₂ (see Fig. 1.5.5.3b). In the absence of magnetic fields and stresses only $[L_{z} \neq 0]$ for this structure. All other components of the vector $[\bf L]$ and the magnetization vector $[\bf M]$ are equal to zero. The magnetic point group is $[{\bi D}_{4h}({\bi D}_{2h})= 4^{\prime}/mmm^{\prime}]$ .

To transform the potential $[\tilde{\Phi}(L_{i}, M_{j}, T_{k\ell})]$ [(1.5.7.5)] into the form $[\Phi(T, {\bf T}, {\bf H})]$ [(1.5.7.1)], one has to insert into the magnetoelastic terms the dependence of the components of $[\bf L]$ and $[\bf M]$ on the magnetic field. The corresponding relations, obtained by minimization of (1.5.7.5) without the magnetoelastic terms, are $[\matrix{M_{x} = \displaystyle{{a}\over{a(B + b) - d^{2}}} H_{x}\semi \hfill& L_{x} = -\displaystyle{{d}\over{a(B + b) - d^{2}}} H_{y}\semi\hfill \cr M_{y} = \displaystyle{{a}\over{a(B + b) - d^{2}}} H_{y}\semi \hfill& L_{y} = -\displaystyle{{d}\over{a(B + b) - d^{2}}} H_{x}\semi\hfill\cr M_{z} = \displaystyle{{1}\over{B}} H_{z}\semi\hfill&L_{z} \simeq {\rm constant}.\hfill} \eqno(1.5.7.6)]$ To a first approximation, the component $[L_{z}]$ does not depend on the magnetic field.

Inserting the relations (1.5.7.6) for $[M_{i}]$ and $[L_{i}]$ into the magnetoelastic terms of (1.5.7.5), one gets the following expression for the corresponding terms in $[\Phi(T, H_{i}, T_{jk})]$ : $[\eqalignno{\Phi(T, H_{i}, T_{jk}) &= \Phi_{0}(T, H_i) + 2L_{z}{{a\lambda_{1} - d\eta_{1}}\over{a(B + b) - d^{2}}}T_{yz}H_{x} &\cr&\quad +2L_{z}{{a\lambda_{1} - d\eta_{1}}\over{a(B + b) - d^{2}}}T_{xz}H_{y} + 2L_{z}{{\lambda_{2}}\over{B}}T_{xy}H_{z}. &\cr&&(1.5.7.7)}]$ In this case, the expression for the magnetoelastic energy contains only three components of the stress tensor: $[T_{yz}]$ , $[T_{xz}]$ and $[T_{xy}]$ . Using (1.5.7.2), we get formulas for the three main components of the piezomagnetic effect: $[\eqalignno{M_{x} &= 2L_{z} {{d\eta_{1} - a\lambda_{1}}\over{a(B + b) - d^{2}}}T_{yz} = 2\Lambda_{xyz}T_{yz} = \Lambda_{14}T_{4}, &(1.5.7.8)\cr M_{y} &= 2L_{z} {{d\eta_{1} - a\lambda_{1}}\over{a(B + b) - d^{2}}}T_{xz} = 2\Lambda_{yxz}T_{xz} = \Lambda_{25}T_{5}, &(1.5.7.9)\cr M_{z} &= -2L_{z} {{\lambda_{2}}\over{B}}T_{xy} = 2\Lambda_{zxy}T_{xy} = \Lambda_{36}T_{6}. &(1.5.7.10)\cr}%fd1.5.7.10]$ In all three cases, the piezomagnetic moment is produced in the direction perpendicular to the shear plane. Comparing (1.5.7.8) and (1.5.7.9), we see that $[\Lambda_{25} = \Lambda_{14}]$ . This is in agreement with the equivalence of the axes x and y in the tetragonal crystals. If the stress is applied in the plane xz (or yz), the vector $[\bf L]$ turns in the shear plane and a component $[L_{x}]$ (or $[L_{y}]$ ) is produced: $[L_{x} = 2L_{z}{{\eta_{1}(B+b) - d\lambda_{1}}\over{a(B+b) - d^{2}}}T_{xz}. \eqno(1.5.7.11)]$ For $[T _{xy}]$ stress, no rotation of the vector $[\bf L]$ occurs.

Formulas (1.5.7.8)–(1.5.7.10) show that in accordance with Table 1.5.7.1 the form of the matrix $[\Lambda _{i\alpha}]$ for the magnetic point group $[{\bi D}_{4h}({\bi D}_{2h})=4^{\prime}/mmm^{\prime}]$ is $[\boldLambda_{i\alpha} = \left[\matrix{0 & 0 & 0 & \Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 & \Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 & \Lambda_{36}} \right]. \eqno(1.5.7.12)]$

The relations (1.5.7.8)–(1.5.7.10) show that the components of the piezomagnetic tensor $[\Lambda_{ijk}]$ are proportional to the components of the antiferromagnetic vector $[\bf L]$ . Thus the sign of the piezomagnetic moment depends on the sign of the vector $[\bf L]$ and the value of the piezomagnetic effect depends on the domain structure of the sample (we are referring to S-domains ). The piezomagnetic moment may become equal to zero in a polydomain sample. On the other hand, piezomagnetism may be used to obtain single-domain antiferromagnetic samples by cooling them from the paramagnetic state in a magnetic field under suitably oriented external pressure.

There are relatively few publications devoted to experimental investigations of the piezomagnetic effect. As mentioned above, the first measurements of the values of the components of the tensor $[\Lambda_{ijk}]$ were performed on crystals of MnF₂ and CoF₂ (Borovik-Romanov, 1960). In agreement with theoretical prediction, three components were observed: $[\Lambda_{xyz} = \Lambda_{yxz}]$ and $[\Lambda_{zxy}]$ . The largest value obtained for these components was Λ₁₄ = 21 × 10⁻¹⁰ Oe⁻¹. The piezomagnetic effect was also observed for two modifications of α-Fe₂O₃ (Andratskii & Borovik-Romanov, 1966). The magnetic point group of the low-temperature modification of this compound is $[{\bi D}_{3d} = \bar{3}m]$ . In accordance with form (7) given above, the following nonzero components $[\Lambda_{ijk}]$ were found for the low-temperature state: $[\eqalignno{\Lambda_{xyz} &= - \Lambda_{yxz}, &(1.5.7.13)\cr \Lambda_{yyy} &= - \Lambda_{yxx} = -\Lambda_{xxy}. &(1.5.7.14)}%fd1.5.7.14]$ The values of these components are one order of magnitude smaller than for CoF₂.

The temperature dependence of the components is similar for the piezomagnetic tensor and the sublattice magnetization. This means that the magnetoelastic constants $[\lambda_{1}]$ and $[\lambda_{2}]$ (as well as the constants B and d) in the relations (1.5.7.7) and (1.5.7.8) depend only slightly on temperature.

1.5.7.2. Linear magnetostriction

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From expression (1.5.7.3), it follows that a deformation of the sample may occur in a magnetic field. This deformation is linear with respect to the field. By its linear dependence, this effect differs essentially from ordinary magnetostriction, which is quadratic in the magnetic field. Most substances display such quadratic magnetostriction. The linear magnetostriction may be observed only in those ordered magnetics that belong to one of the 66 magnetic point groups that allow piezomagnetism and are listed in Table 1.5.7.1. The distinctive feature of linear magnetostriction is the dependence of its sign on the sign of the magnetic field and on the sign of the antiferromagnetic vector $[\bf L]$ . The sign of $[\bf L]$ characterizes the domain state of the specimen. Thus, observation of linear magnetostriction gives information about the domain state. In some materials, it has been observed that a sudden transition from one domain state to the opposite may occur in strong magnetic fields.

Linear magnetostriction (LM) was observed in CoF₂ by Borovik-Romanov & Yavelov (1963) in a magnetic field applied parallel to the fourfold axis. The relations for the LM in CoF₂ can be obtained by differentiating the expression of the thermodynamic potential $[\Phi]$ [(1.5.7.7)]. If the magnetic field is applied along the y axis, a deformation $[S_{xz}]$ appears: $[S_{xz} = -\partial\Phi / \partial T_{xz} = 2L_{z}{{d\eta_{1} - a\lambda_{1}}\over {a(B+b) - d^{2}}}H_{y} = 2\Lambda_{yxz} H_{y} = \Lambda_{25}H_{2}. \eqno(1.5.7.15)]$ A similar formula holds for $[S_{yz}]$ if the magnetic field is applied parallel to the x axis (with $[\Lambda_{14}]$ , which is equal to $[\Lambda_{25}]$ ).

If the magnetic field is applied parallel to the fourfold axis, the $[S_{xy}]$ component of the deformation appears: $[\eqalignno{S_{xy} &= -\partial\Phi / \partial T_{xy} = -2L_{z}(\lambda_{2}/B) H_{z} = -2L_{z}\lambda_{2}\chi_{ \parallel } H_{z} &\cr&\quad = 2\Lambda_{zxy}H_{z} = \Lambda_{36}H_{3}. &(1.5.7.16)}]$

If the relations (1.5.7.15) and (1.5.7.16) are compared with (1.5.7.8)–(1.5.7.10), it is apparent that in accordance with theory the components of the tensors of the piezomagnetic effect (PM) and LM are identical.

Prokhorov & Rudashevskii (1969, 1975) extended the investigation of LM in CoF₂. They discovered that if the applied field becomes larger than 20 kOe, a jump in the magnetostriction occurs and it changes its sign (see Fig. 1.5.7.2). This jump is the result of a transition of the magnetic structure from one domain state (L₊) into the opposite state ( $[\bf L]$ ₋). To explain such a transition, one has to take into account the term of third power in the expansion of the magnetic energy (Scott & Anderson, 1966), $[U_{m} = A_{i}H_{i} + {\textstyle{1 \over 2}}\chi _{ij}H_{i}H_{j} + C_{ijk}H_{i}H_{j}H_{k}. \eqno(1.5.7.17)]$ $[C_{ijk}]$ is an axial time-antisymmetric tensor , the sign of which depends on the sign of the domain. This term defines the dependence of the magnetic energy on the sign of the antiferromagnetic domain .

Figure 1.5.7.2 | top | pdf |

Linear magnetostriction of CoF₂ (Prokhorov & Rudashevskii, 1975).

To date, CoF₂ and MnF₂ are unique in that LM and PM occur without rotating the antiferromagnetic vector $[\bf L]$ if the magnetic field is applied along the fourfold axis (or pressure along a $[\langle 110\rangle]$ axis). In all other cases, these effects are accompanied by a rotation of $[\bf L]$ and, as a result, the creation of new components $[L_{i}]$ . To the latter belongs the LM in the low-temperature modification of $[\alpha]$ -Fe₂O₃, which was observed by Anderson et al. (1964) (see also Scott & Anderson, 1966; Levitin & Shchurov, 1973). This compound displays PM, therefore it is obvious that LM will also occur (see Table 1.5.7.1).

LM has been observed in some orthoferrites. One of the orthoferrites, DyFeO₃ at low temperatures, is a pure antiferromagnet, the vector $[\bf L]$ of which is aligned along the y axis. Its magnetic point group ( $[{\bi D}_{2h}=mmm]$ ) allows PM and LM. The latter was observed when a magnetic field was applied parallel to the z axis by Zvezdin et al. (1985). There it was shown that $[\Lambda_{zxy} \neq 0]$ if $[0 \,\lt\, H \,\lt\, H_{c}]$ . At H_c ≃ 4 kOe, a first-order phase transition into a weakly ferromagnetic state with magnetic point group $[{\bi D}_{2h}({\bi C}_{2h})=m'm'm]$ $[({\bf L} \parallel Ox, {\bf M}_{D} \parallel Oz)]$ occurs.

Many orthoferrites and orthochromites that possess weak ferromagnetism belong to the same point group, which possesses an ordinary centre of symmetry. Thus PM and LM are allowed for these phases of orthoferrites. If the magnetic field is applied parallel to [Ox] , they undergo a reorientation transition at which both vectors, $[{\bf L}]$ and $[{\bf M}_D]$ , being orthogonal, rotate in the xz plane. These intermediate angular phases belong to the magnetic point group $[{\bi C}_{2h}({\bi C}_i)=2'/m']$ .

LM was observed by Kadomtseva and coworkers (Kadomtseva, Agafonov, Lukina et al., 1981; Kadomtseva, Agafonov, Milov et al., 1981) in two such compounds, YFeO₃ and YCrO₃. The $[\Lambda_{xxz}]$ components of the LM tensor were measured, which are allowed for the $[{\bi D}_{2h}({\bi C}_{2h})=m'm'm]$ state.

The experimental data obtained to date for PM and LM are summarized in Table 1.5.7.2. The values of the components $[\Lambda_{i\alpha}]$ can be converted to SI units using 1 Oe⁻¹ = 4π × 10⁻³ m A⁻¹ = 4π × 10⁻³ T Pa⁻¹.

Table 1.5.7.2 | top | pdf |
Experimental data for the piezomagnetic effect (PM) and for linear magnetostriction (LM)

a : antiferromagnetic phase; w: weak ferromagnetic phase.

Compound	$[\Lambda_{i\alpha}\times 10^{10}]$ (Oe⁻¹)	T (K)	PM or LM	Reference^†
MnF₂	$[\Lambda_{14} \simeq 0.2]$	20	PM	(1)
CoF₂	$[\Lambda_{14} = 21]$	20	PM	(1)
	$[\Lambda_{36} = 8.2]$	20	PM	(1)
	$[\Lambda_{36} = 9.8]$	4	LM	(3)
DyFeO₃	$[\Lambda_{36} = 6.0]$	6	LM	(8)
YFeO₃	$[\Lambda_{15} = 1.7]$	6	LM	(6)
YCrO₃	$[\Lambda_{15} \simeq 1]$	6	LM	(7)
α-Fe₂O₃ (a)	$[\Lambda_{22} = 1.9]$	78	LM	(4)
	$[\Lambda_{22} = 3.2]$	77	PM	(2)
	$[\Lambda_{22} = 1.3]$	100	LM	(5)
	$[\Lambda_{14} = 0.3]$	78	LM	(4)
	$[\Lambda_{14} = 1.7]$	77	PM	(2)
	$[\Lambda_{14} = 0.9]$	100	LM	(5)
α-Fe₂O₃ (w)	$[\Lambda_{23} = 2.5]$	292	PM	(2)

^† References: (1) Borovik-Romanov (1959b

, 1960

); (2) Andratskii & Borovik-Romanov (1966

); (3) Prokhorov & Rudashevskii (1969

, 1975

); (4) Anderson et al. (1964

); (5) Levitin & Shchurov (1973

); (6) Kadomtseva, Agafonov, Milov et al. (1981

); (7) Kadomtseva, Agafonov, Lukina et al. (1981

); (8) Zvezdin et al. (1985

1.5.7.3. Linear magnetic birefringence

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The magnetic contribution to the component of the dielectric permittivity $[ \delta\varepsilon_{ij}]$ can be represented as a series in the powers of the components of the magnetization and the antiferromagnetic vector. The magnetic birefringence (also called the Cotton–Mouton or Voigt effect) is described by the real symmetrical part of the tensor $[\delta\varepsilon_{ij}]$ . In paramagnetic crystals, the magnetization $[{\bf M}]$ is proportional to the applied magnetic field $[{\bf H}]$ , and the series has the form $[\delta\varepsilon_{ij} = Q^{MM}_{ijk\ell} M_k M_\ell = Q^{MM}_{ijk\ell} \chi^M_{kr} \chi^M_{\ell s} H_r H_s = \Gamma_{ijrs} H_r H_s.\eqno(1.5.7.18)]$ The tensor $[\Gamma_{ijrs}]$ is symmetric with respect to both the first and the second pair of indices. The symmetry of this tensor implies that the diagonal components of the permittivity tensor include magnetic corrections. The modification of the diagonal components gives rise to birefringence in cubic crystals and to a change $[\Delta n^{\rm pm}]$ of the birefringence in uniaxial and lower-symmetry crystals. It follows from (1.5.7.18) that this birefringence is bilinear in the applied field. Bilinear magnetic birefringence can be observed in uniaxial crystals if the magnetic field is applied along the x axis perpendicular to the principal z axis. In the simplest case, a difference in the refractive indices [n_x] and [n_y] arises: $[{\Delta}n^{\rm pm} = n_x - n_y = {{1}\over{2n_0}}(\delta\varepsilon_{xx} - \delta\varepsilon_{yy}) = {{1}\over{2n_0}}(\Gamma_{xxxx}-\Gamma_{yyxx})H_x^2, \eqno(1.5.7.19)]$ where [n_0] is the refractive index for the ordinary beam.

Consider now a magnetically ordered crystal which can be characterized by an antiferromagnetic vector $[{\bf L}_0]$ and a magnetization vector $[{\bf M}_0]$ in the absence of a magnetic field. Applying a magnetic field with components [H_r] , we change the direction and size of $[{\bf L}_0]$ and $[{\bf M}_0]$ , getting additional components $[L^H_k = \chi^L_{kr} H_r]$ and $[M^H_k = \chi^M_{kr} H_r]$ . This is illustrated by the relations (1.5.7.6). Instead of (1.5.7.18) we get $[\eqalignno{\delta\varepsilon_{ij} &= Q^{LL}_{ijk\ell} L_k L_\ell + Q^{ML}_{ijk\ell} M_k L_\ell + Q^{MM}_{ijk\ell} M_k M_\ell &\cr&= Q^{LL}_{ijk\ell} L_{0k} L_{0\ell} + Q^{ML}_{ijk\ell} M_{0k} L_{0\ell} + Q^{MM}_{ijk\ell} M_{0k} M_{0\ell} &\cr&\quad +[2Q^{LL}_{ijk\ell} \chi^L_{kr} L_{0\ell} + Q^{ML}_{ijk\ell} (\chi^M_{kr} L_{0\ell} + M_{0k} \chi^L_{\ell r}) + 2Q^{MM}_{ijk\ell} \chi^M_{kr} M_{0\ell}] H_r. &\cr&&(1.5.7.20)}]$ The terms in the middle line of (1.5.7.20) show that in an ordered state a change in the refractive indices occurs that is proportional to [L_0^2] in antiferromagnets and to [M_0^2] in ferromagnets. The terms in square brackets show that a linear magnetic birefringence may exist. In the special case of a tetragonal antiferromagnet belonging to the space group $[{\bi D}^{14}_{4h}=P4_2/mnm]$ with $[{\bf L}_0]$ parallel to the principal axis z, the linear birefringence occurs in the xy plane if the magnetic field is applied along the z axis (see Fig. 1.5.5.3). In this case, $[{\bf M}_0=0]$ , $[\chi^L_{kz}=0]$ for all k, $[\chi^M_{xz} = \chi^M_{yz} = 0]$ and $[\chi^M_{zz} = 1/B]$ [see (1.5.7.6)]. Therefore the terms in square brackets in (1.5.7.20) differ from zero only for one component of $[\delta {\varepsilon}_{ij}]$ , $[\delta {\varepsilon}_{ij} = Q^{ML}_{xyzz}L_{0z}H_z/B = q_{zxy}H_z{\rm sign}(L_{0z}). \eqno(1.5.7.21)]$ As a result, $[{\Delta}n^{\rm af} = n_{x^{\prime}} - n_{y^{\prime}} = {{1}\over{2n_{0}}} \delta {\varepsilon}_{xy} = {{1}\over{2n_{0}}}q_{zxy}H_{z}{\rm sign}(L_{0z}), \eqno(1.5.7.22)]$ where $[x^{\prime}, y^{\prime}]$ are the optic axes, which in these tetragonal crystals are rotated by $[{\pi}/4]$ relative to the crystallographic axes.

Comparing relation (1.5.7.22) with (1.5.7.3), one can see that like LM, there may be linear magnetic birefringence. The forms of the tensors that describe the two effects are the same.

Linear magnetic birefringence has been observed in the uniaxial antiferromagnetic low-temperature α-Fe₂O₃ when the magnetic field was applied perpendicular to the threefold axis (Le Gall et al., 1977; Merkulov et al., 1981). The most impressive effect was observed in CoF₂ when the magnetic field was applied along the fourfold axis. The crystal ceased to be optically uniaxial and a difference $[(n_{x^{\prime}} - n_{y^{\prime}}) \propto H_{z}]$ was observed in accordance with (1.5.7.22). Such linear magnetic birefringence does not exist in the paramagnetic state. Linear birefringence has also been observed in CoCO₃ and DyFeO₃. For details of these experiments, see Eremenko et al. (1989). These authors also used linear birefringence to make the antiferromagnetic domains visible. A further review of linear magnetic birefringence has been given by Ferré & Gehring (1984).

Piezomagnetism, linear magnetostriction and linear birefringence in fluorides can be clearly demonstrated qualitatively for one particular geometry. As shown in Fig. 1.5.7.3, the crystallographically equivalent points 1 and 2 are no longer equivalent after a shear deformation applied in the plane xy. During such a deformation, the distances from the magnetic ions to the nearest fluoride ions increase in points 1 and decrease in points 2. As a result, the values of the g-factors for the ions change. Evidently, the changes of the values of the g-factors for different sublattices are opposite in sign. Thus the sublattice magnetizations are no longer equal, and a magnetic moment arises along the direction of sublattice magnetization. On the other hand, if we increase the magnetization of one sublattice and decrease the magnetization of the other by applying a magnetic field parallel to the z axis, the interactions with the neighbouring fluoride ions also undergo changes with opposite signs. This gives rise to the magnetostriction. These considerations can be applied only to antiferromagnets with the fluoride structure. In these structures, single-ion anisotropy is responsible for the weak ferromagnetism , not the antisymmetric exchange interaction of the form $[{\bf d}[{\bf S}_{i} \times {\bf S}_{k}]]$ .

Figure 1.5.7.3 | top | pdf |

Variation of symmetry of the crystal field in the presence of the piezomagnetic effect in CoF₂. The unshaded atoms lie at height [c/2] above the xy plane (see Fig. 1.5.5.3).

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