Magnetic structure and magnetoelectric properties of the spin-flop phase in ${\mathrm{LiFePO}}_4$

We investigate the magnetic structure and magnetoelectric(ME) effect in the high-field phase of the antiferromagnet ${\mathrm{LiFePO}}_4$ above the critical field of $31\mathrm{T}$. A neutron diffraction study in pulsed magnetic fields reveals the propagation vector to be $\mathbf{q}=0$ for the high-field magnetic structure. Pulsed-field electric polarization measurements show that, at the critical field, the low-field off-diagonal ME coupling ${\alpha }_{ab}$ is partially suppressed, and the diagonal element ${\alpha }_{bb}$ emerges. These results are consistent with a spin-flop transition where the spin direction changes from primarily being along the easy $b$ axis below the transition to being along $a$ above. The persistence of off-diagonal ME tensor elements above the critical field suggests a lowering of the magnetic point-group symmetry and hence a more complex magnetic structure in the high-field phase. In addition, neutron diffraction measurements in low magnetic fields show no observable field-induced spin canting, which indicates a negligible Dzyaloshinskii-Moriya interaction. The observed spin-flop field supports the Hamiltonian recently deduced from inelastic neutron studies and indicates that the system is less frustrated and with a larger single-ion anisotropy than originally thought. Our results demonstrate the effectiveness of combining pulsed-field neutron diffraction and electric polarization measurements to elucidate the magnetic structures and symmetries at the highest attainable field strengths.

Figure 1

(a) The basis vectors of $\mathrm{Li}M{\mathrm{PO}}_4$ for $\mathbf{q}=0$ structures in the $Pnma$ space group. Only the lattice of magnetic ions is shown. The arrows represent the spin orientation on the four magnetic sites labeled $1–4$ as shown. (b) Projections onto the $(b,c)$ and $(a,b)$ planes for the zero-field magnetic structure of ${\mathrm{LiFePO}}_4, [C_y+C_x+A_z]$. Canting angles are exaggerated for clarity.

Figure 2

(a) Schematic of the neutron Laue diffraction backscattering setup at NOBORU. The incoming beam is polychromatic with wave vector $\mathbf{k}$, scattered wave vector ${\mathbf{k}}^{\prime }$, and scattering vector $\mathbf{Q}=\mathbf{k}-{\mathbf{k}}^{\prime }$. The pulsed magnet coils (gray) allow a maximum $2\theta ={42}^{\circ }$. The magnetic field is parallel to the $b$ axis of the sample. (b) Position-sensitive detector image for a pulsed-field measurement. The detector has 16 vertical tubes, each with 60 pixels. The color represents the neutron intensity integrated over 148 pulses and all neutron flight times. The high-intensity pixel enclosed in the black rectangle corresponds to neutron momentum transfers along $\mathbf{Q}=\left(0K0\right)$. (c) Neutron intensity vs time of flight for the high-intensity pixel shown in (b). The top axis shows the corresponding value of $K$ along $\mathbf{Q}=\left(0K0\right)$. The magnetic field strength (gray line) is indicated on the right-hand axis. The time delay between the neutron and magnet pulse has been adjusted to reach the maximum field strength of $34\mathrm{T}$ at $\mathbf{Q}=\left(030\right)$.

Figure 3

Neutron diffraction measurement of ${\mathrm{LiFePO}}_4$ with pulsed magnetic fields. (a) Diffraction intensity as a function of $K$ along $\mathbf{Q}=\left(0K0\right)$ in zero applied magnetic field. Green lines indicate nuclear Bragg reflections. Inset focuses on the $\left(020\right)$ reflection. (b) Diffraction intensity along $\mathbf{Q}=\left(0K0\right)$ in pulsed magnetic fields. Zero-field data from (a) has been scaled to use as background. A Gaussian (yellow) line has been fitted to the pulsed-field data as a guide to the eye of the $\left(030\right)$ reflection. The strength of the magnetic field pulse is shown on the right-handside in gray.

Figure 4

Electric polarization $P_i$ in applied pulsed magnetic fields $H_j$. Background measurements obtained at 60 K have been subtracted from all data, except $P_b{H}_c$, as explained in the text. (a) Overviewof all nine combinations of $P_i$ and $H_j$ for various temperatures. (b) Electric polarizations $P_a$ and $P_b$ as a function of magnetic fields applied along $b$ clearly show the phase transition. The critical field $B_{\mathrm{SF}}=31\mathrm{T}$ is indicated in by the gray vertical line. The dot-dashed lines represent linear fits to the polarization as a function of field to extract ${\alpha }_{ij}$. (c) Temperature dependence of ${\alpha }_{ij}$. Notice that the values obtained are merely proportional to ${\alpha }_{ij}$. Our values are compared with scaled results from Refs. [19, 22].

Figure 5

Sample rotation $\omega$ scan of different Bragg peaks at low and high magnetic fields. All scans were performed at 2 K. Scans (a)–(d) are with magnetic fields along $a$, while scan (e) is with field along $c$.

Figure 6

Magnetic structures of ${\mathrm{LiFePO}}_4$ in (a) the spin-flop and (b) the low-field phase. (c) Experimental phase diagram of ${\mathrm{LiFePO}}_4$ for magnetic fields along $b$. Figure includes results from our neutron diffraction and electric polarization measurements as well as magnetization data from Refs. [26, 28]. (d) and (e) Phase diagrams as determined from mean-field calculations with two different sets of parameters [8, 23].

Figure 7

Identification of powder lines arising from the sample environment (cryostat wall and magnet coils) and nuclear Bragg peaks from the sample. (a) Experimental geometry showing the position of the cryostat wall, magnet coils, and sample. Note that the sketch is not to scale. (b) Neutron diffraction data collected at zero field as a function of pixel number and time of flight(ToF) for detector tube No. 9. Color contours represent the neutron counts. Nuclear Bragg peaks $\left(020\right), \left(040\right)$, and $\left(060\right)$ are observed as high-intensity spots around pixel No. 20. Powder lines have intensity spread out homogeneously over all pixels. (c) Neutron counts as a function of ToF collected at zero field and $4.5\mathrm{K}$ in detector tube No. 9 and pixel No. 20. Nuclear Bragg peak positions from the sample are marked with red lines. Powder lines from the cryostat wall (aluminum) and magnet coils (copper) are marked with, respectively, gray and blue lines.

Figure 8

(a) Experimental setup for polarization measurement. (b) Two consecutive magnetic field pulse of duration 6.1 ms. (c) The measured displacement current during the time of the pulse. (d) The deduced electric polarization. As evident, the electric polarization along $a$ has a plateau in the spin flop phase, for field applied along the $b$ axis, as discussed in the main text.