Piezomagnetism and magnetoelastic memory in uranium dioxide

The thermal and magnetic properties of uranium dioxide, a prime nuclear fuel and thoroughly studied actinide material, remain a long standing puzzle, a result of strong coupling between magnetism and lattice vibrations. The magnetic state of this cubic material is characterized by a 3-k non-collinear antiferromagnetic structure and multidomain Jahn-Teller distortions, likely related to its anisotropic thermal properties. Here we show that single crystals of uranium dioxide subjected to strong magnetic fields along threefold axes in the magnetic state exhibit the abrupt appearance of positive linear magnetostriction, leading to a trigonal distortion. Upon reversal of the field the linear term also reverses sign, a hallmark of piezomagnetism. A switching phenomenon occurs at ±18 T, which persists during subsequent field reversals, demonstrating a robust magneto-elastic memory that makes uranium dioxide the hardest piezomagnet known. A model including a strong magnetic anisotropy, elastic, Zeeman, Heisenberg exchange, and magnetoelastic contributions to the total energy is proposed.

pressure-induced weak ferromagnetism [2]. Measurements in a field H = 5 kOe, on the other hand, reproduce ambient pressure results [2]. Supplementary Figures 1a and c show the axial and transverse magnetostriction ε a (T ), and ε t (T ), close to the AFM transition.
The drop at T N supports reduction in the volume of the unit cell in agreement with earlier conclusions [3,4]. Also shown are the coefficients of thermal expansion α T a (T ), and α T t (T ) which are truly first-order like at T N .
The observed linear magnetostriction (LMS) in the ordered state, precluded on the basis of time reversal symmetry considerations in most AFM materials, is strong in UO 2 . The converse of the LMS phenomena is the piezomagnetic (PZM) effect. Borovik-Romanov [5] has considered the piezomagnetism from a phenomenological point of view, adding bilinear terms of magnetoelastic energy to the expansion of the thermodynamic potential per unit volume: where σ jk are the components of the elastic strain tensor. If at least one term of this expansion remains invariant under the magnetic symmetry of the crystal, then the corresponding component axial Λ ijk is not zero and hence the magnetization is given by Thus, when a stress σ jk is applied, a magnetic moment M i linear in the stress is produced.
It follows from Equation (1) that there exists also the LMS effect: where jk are components of the deformation tensor.
The possibility of existence of LMS/PZM in a system is thus related with a non zero tensor The subset of 66 magnetic point groups that can indeed display PZM/LMS were listed by Tavger [6] and Briss [7]; the space group P a3 (m3), used to describe the AFM state in UO 2 , is among them (see Supplementary Ref. [8]).
The magnetostriction tensor for the magnetic space group Pa3, point group m3 of UO 2 in Voigt's notation is: The dependence of the strain tensor with the applied magnetic field in Supplementary Equation (3) can be explicitly written as: For an applied field H along the [111] direction, the strain tensor can be diagonalized to Where the first two directions are perpendicular and the third parallel to the [111] direction.

Supplementary Note 2: The model Hamiltonian
We consider a classical Hamiltonian where the free degrees of freedom are the orientations of the magnetic moments of the four U atoms at the 4a positions in the Pa3 unit cell.
Interaction with the magnetic field The Zeeman term gives the following contribution to the total energy: If the magnetic field is applied along the (111) direction, one has:

Magnetic anisotropy
The Jahn-Teller distortion stabilizes the 3-k magnetic order at T < T N producing a local anisotropy that can be written as: This term is not enough to establish the relative orientation of the magnetic moments. A Heisenberg like interaction compatible with the symmetry operations of the magnetic group have to be included [10][11][12].

Heisenberg interaction
The Heisenberg contribution to the total energy, can be written as: where the S iα (v i ) are the three components of the magnetic moments with the z component alongv i .

Elastic energy
The elastic energy for a cubic crystal is: where c 11 = 389 GPa, c 12 = 119 GPa, c 44 = 60 GPa, B = 1/3 (c 11 + 2c 12 ) 207 GPa, and a = 5.47Å. Only shear components of the strain can be kept in the expression above because they are the only components appearing in the magnetoelastic term (see below) :

Magnetoelastic energy
To couple the elastic deformation to the magnetic order we include the magnetoelastic contribution (Supplementary eq. (1)) to the total energy : where V = a 3 , Λ is given by equation (4), and the magnetoelastic contribution now reads : where Λ 14 is proportional to the staggered magnetization : The minimization of the elastic and magnetoelastic contributions to the total energy gives : The strain tensor for pure shear is = 0 xy /2 xz /2 In the case of a magnetic field along (111). xy = xz = yz , and the diagonalization of the strain tensor gives: Thus :  T/s (red). This shows that the switching can be partial, allowing for tuning of dε/dH. These characteristics make the gradual reorientation of magnetic moments a peculiar memory effect in UO 2 .