Identification of microscopic spin-polarization coupling in the ferroelectric phase of magnetoelectric multiferroic ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$

We have performed synchrotron radiation x-ray and neutron diffraction measurements on magnetoelectric multiferroic ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$ $\left(x=0.0155\right)$, which has a proper helical magnetic structure with incommensurate propagation wave vector in the ferroelectric phase. The present measurements revealed that the ferroelectric phase is accompanied by lattice modulation with a wave number $2q$, where $q$ is the magnetic modulation wave number. We have calculated the Fourier spectrum of the spatial modulations in the local electric polarization using a microscopic model proposed by T. Arima [J. Phys. Soc. Jpn. 76, 073702 (2007)]. Comparing the experimental results with the calculation results, we found that the origin of the $2q$-lattice modulation is not the conventional magnetostriction but the variation in the metal-ligand hybridization between the magnetic ${\text{Fe}}^{3+}$ ions and ligand ${\text{O}}^{2-}$ ions. Combining the present results with the results of a previous polarized neutron diffraction study [Nakajima et al., Phys. Rev. B 77, 052401 (2008)], we conclude that the microscopic origin of the ferroelectricity in ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$ is the variation in the metal-ligand hybridization with spin-orbit coupling.

Figure 1

(Color online) (a) Crystal structure of ${\text{CuFeO}}_2$ with the hexagonal basis $\left({\mathit{a}}_h,{\mathit{b}}_h,{\mathit{c}}_h\right)$. (b)-(c) Relationship between the hexagonal basis and the monoclinic basis $\left(\mathit{a},\mathit{b},\mathit{c}\right)$. (d) Schematic drawing of the proper helical magnetic structure in ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$.

Figure 2

(Color online) Schematic drawing of $x\text{−}H\text{−}T$ phase diagram of ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$.

Figure 3

(Color online) The x-ray diffraction intensity maps obtained by the ${\left(H,K,0\right)}_h$ reciprocal lattice scans around the reciprocal lattice position of ${\left(2,2,0\right)}_h$ at (a) $T=5.5\text{K}$, (b) $T=10.5\text{K}$, (c) $T=11.0\text{K}$, and (d) $T=19.2\text{K}$. These intensity maps were measured in a warming process. (e) Temperature variations of the lattice constants $a$ and $b$ in the monoclinic notation, which were measured on heating. (f) Temperature variations of the intensities of (0, 3, 2) and $\left(0,2+2q^{\text{PD}},2\right)$ reflections. The inset shows the diffraction profiles of the $\left(0,2+2q^{\text{PD}},2\right)$ reflections at typical temperatures. The data of $\left(0,2+2q^{\text{PD}},2\right)$ are magnified by a factor of 20.

Figure 4

(Color online) (a) Illustration of the reciprocal lattice $\left(0KL\right)$ zone. [(b)-(c)] Diffraction profiles of the (b) $\left(0K0\right)$ and (c) $\left(0K\frac{1}{2}\right)$ reciprocal lattice scans in the FEIC phase. Contamination peaks from the Cu sample holder are masked by the shaded areas. The inset of (b) shows the magnification around the reciprocal lattice position of $\left(0,3-2q^{\text{FEIC}},0\right)$. [(d)–(e)] The diffraction profiles of (d) $\left(0,3+2q^{\text{FEIC}},0\right)$ and (e) $\left(0,5-2q^{\text{FEIC}},0\right)$ reflections. (f) Temperature dependences of the integrated intensity of the $\left(0,3-2q^{\text{FEIC}},0\right)$ reflection and the neutron diffraction intensity of the magnetic Bragg reflection corresponding to the FEIC magnetic ordering (taken from Ref. 26), under zero field.

Figure 5

(Color online) (a)The $H_{\parallel b}$ dependence of the integrated intensity of $\left(0,3-2q^{\text{FEIC}},0\right)$ reflection observed in the x-ray diffraction measurements. (b) The $H_{\parallel b}$-dependence of ${\left(\frac{1}{2}-q_h^{\text{FEIC}},\frac{1}{2}-q_h^{\text{FEIC}},\frac{3}{2}\right)}_h$ magnetic Bragg reflection observed in the neutron diffraction measurements at $T=4.5\text{K}$. (d) The field dependence of the integrated intensity of $\left(0,3-2q^{\text{FEIC}},0\right)$ reflection normalized by the data of the neutron diffraction measurements. (d) Schematic drawings of a conical structure with uniform magnetization component along the helical axis. [(e)–(f)] Schematic drawings of the fractions of three magnetic domains (e) under zero field, and (f) under applied field along the $b$ axis (${\left[110\right]}_h$ axis). The directions of the filled arrows denote the ${\left[001\right]}_h$ projections of the propagation wave vectors. The sizes of arrows qualitatively show the fractions of each domain.

Figure 6

(Color online) (a) A ${\text{Fe}}_4{\text{O}}_2$ cluster used for the calculation of the local electric dipole moment in Ref. 18. (b) The magnetic structure in the FEIC phase with the $a\times b\times 2c$ cell. (c) The phase differences in the magnetic modulations $\left(\Delta {\Phi }_{i,j}^{\text{mag}}\right)$ between the ${\text{Fe}}^{3+}$ sites.

Figure 7

(Color online) [(a)–(b)] ${\delta }_{\text{mag}}$ dependence of the difference between the phases of (a) $2q$ and (b) $4q$ modulations. [(c)–(d)] The phase differences of the (c) $2q$ and (d) $4q$ components in the spatial modulations in the local electric polarization.

Figure 8

(Color online) Fourier spectrum mappings of the spatial modulations in the local polarization, which are calculated using (a) the $d\text{−}p$ hybridization model with ${\delta }_{\text{mag}}=0,\pi$, (b) that with ${\delta }_{\text{mag}}=76°$, and (c) the magnetostriction model with finite ellipticity. The size of a symbol, except for ${\mathit{\tau }}_{\text{even}}$ and ${\mathit{\tau }}_{\text{odd}}$, qualitatively shows the intensity of the spectrum ${\left|P\left(\mathit{Q}\right)\right|}^2$, which is normalized by the intensity of the strongest peak in the spectrum. The black arrows denote the position of the satellite reflections observed in the zero-field FEIC phase.

Figure 9

(Color online) (a) $\theta$ dependence of the intensities of the Fourier spectra for the spatial modulations in the local polarization. To enhance the visibility, the data of $3q$ and $4q$ are magnified by a factor of 200. (b) Fourier spectrum mapping of the spatial modulations in the local polarization, which are calculated with the $d\text{−}p$ hybridization model with ${\delta }_{\text{mag}}=76°$, $\alpha =0.1$, and $\theta =10°$. The size of the symbols, except for ${\mathit{\tau }}_{\text{even}}$ and ${\mathit{\tau }}_{\text{odd}}$, corresponds to the intensity of the spectrum ${\left|P\left(\mathit{Q}\right)\right|}^2$, which are normalized using the intensity of the most intense peak in the spectrum. The black arrows denote the positions of the satellite reflections observed in the present measurements.