Comprehensive study on ferroelectricity induced by a proper-screw-type magnetic ordering in multiferroic ${\text{CuFeO}}_2$: Nonmagnetic impurity effect on magnetic and ferroelectric order

We refined the magnetic structure of a ferroelectric (FE) phase of multiferroic ${\text{CuFe}}_{1-x}{\text{Ga}}_x{\text{O}}_2$ with $x=0.035$ by complementary use of spherical neutron polarimetry and a four-circle neutron-diffraction measurement, revealing that the proper-screw-type magnetic structure in the ferroelectric phase has a finite ellipticity of $\sim 0.9$. By means of polarized neutron-diffraction and in-situ pyroelectric measurements, we also investigated the quantitative relationship between the macroscopic ferroelectric polarization $\left(P\right)$ and the asymmetry in volume fractions with left-handed and right-handed helical magnetic order in ${\text{CuFe}}_{1-x}{\text{Al}}_x{\text{O}}_2$ with $x=0.0155$ and ${\text{CuFe}}_{1-x}{\text{Ga}}_x{\text{O}}_2$ with $x=0.035$. These measurements revealed that the substitution of a small amount of nonmagnetic ${\text{Ga}}^{3+}$ or ${\text{Al}}^{3+}$ ions does not significantly change the magnitude of the local ferroelectric polarization but does reduce the sensitivity of $P$ to the poling electric field $\left(E_p\right)$. This implies that the mobility of the magnetic domain walls, which is sensitive to magnetic defects due to nonmagnetic substitution, determines the sensitivity of $P$ to $E_p$ because of a one-to-one correspondence between the magnetic and ferroelectric domains.

Figure 1

(Color online) (a) Crystal structure of CFO with the hexagonal basis. (b) The magnetic structure in the FE-ICM phase in $a_m\times b_m\times 2c_m$ cell with monoclinic basis. (c) The definition of the relative phase shift $\delta$. [(d)–(f)] The definition of (d) (110), (e) $\left(1\overline{2}0\right)$, and (f) $\left(\overline{2}10\right)$ domains and the $c$ projection of the magnetic propagation wave vectors in each domain.

Figure 2

(Color online) [(a)–(d)] $\omega$ dependence of the polarization matrix terms of $P_{y^{\prime },y^{\prime }}$, $P_{y^{\prime },z^{\prime }}$, $P_{z^{\prime },y^{\prime }}$, and $P_{z^{\prime },z^{\prime }}$ at $T=1.4\text{K}$. The imperfection of the instrumental beam polarization was corrected by the observed $P_{x^{\prime },x^{\prime }}$ term, which should be $-1$ for a completely polarized neutron beam. The open circles denote the observed values of the polarization matrix terms. The solid and dotted lines denote the calculation for an elliptic helical magnetic structure with ${\mu }_x/{\mu }_z=0.895$ and a proper helical magnetic structure $\left({\mu }_x/{\mu }_z=1.0\right)$, respectively. The insets of (b) and (c) show the magnifications near $\omega =10°$ and the calculations (colored solid lines) for helical magnetic structures whose minor axis cants from the $x$ axis toward the $z$ axis by 15, 0, or $-15°$. (e) A schematic of the $\left(H,H,L\right)$ scattering plane and the definition of $\omega$. The black arrows denote the position of the magnetic reflections measured in the present work. (f) The definition of Cartesian coordinates for the magnetic moments at ${\text{Fe}}^{3+}$ sites $\left(x,y,z\right)$ and the polarization direction of the incident and scattered neutrons $\left(x^{\prime },y^{\prime },z^{\prime }\right)$. (g) Temperature variation in ${\mu }_x/{\mu }_z$ deduced from the results of the present measurement. The solid line is a guide to the eyes.

Figure 3

$\left|F_{\text{cal}}\right|$ vs $\left|F_{\text{obs}}\right|$ plots obtained from the magnetic structure analysis for $\text{CFGO}\left(x=0.035\right)$ at (a) $T=2.8\text{K}$ and (b) $T=6.5\text{K}$.

Figure 4

(Color online) (a) The schematic of the present experimental setup and the definition of the “positive” poling electric field. (b) The definition of the hexagonal basis in the crystal structure of CFO. The [110] direction differs in the positions of ${\text{O}}^{2-}$ ions from the $\left[\overline{1}\overline{1}0\right]$ direction. (c) The schematic of the reciprocal lattice map of CFGO and the directions of the polarization vector of the incident neutrons. (d) and (e) The diffraction profiles of $\left(H,H,\frac{3}{2}\right)$, reciprocal lattice scans for the $\left(q,q,\frac{3}{2}\right)$, and $\left(\frac{1}{2}-q,\frac{1}{2}-q,\frac{3}{2}\right)$ magnetic Bragg reflections at $T=2.0\text{K}$ in the FE-ICM phase. [(f)–(g)] The relationship between the spin helicity and the direction of the induced ferroelectric polarization.

Figure 5

(Color online) The $E_p$ dependences of (a)$D_{\left(110\right)}\left(E_p\right)$ in the (110) domain, (b) the sum of $I_{\text{ON}}+I_{\text{OFF}}$, and (c) the observed and calculated values of $P$ in $\text{CFGO}\left(x=0.035\right)$ at $T=2.0\text{K}$. The solid lines are guides to the eyes, drawn so as to be symmetric.

Figure 6

(Color online) (a) A schematic of the distribution of the RH (open arrows) and LH (filled arrows) helical magnetic orderings in the (110), $\left(1\overline{2}0\right)$, and $\left(\overline{2}10\right)$ domains when the macroscopic $P$ emerges along the [110] axis. The size of the arrows shows the fractions of the LH (or RH) helical magnetic ordering. (b) The relationship between the direction of $E_p$ and the local ferroelectric polarization in (110) domain, and (c) that in $\left(1\overline{2}0\right)$ and $\left(\overline{2}10\right)$ domains. $E_{\text{eff}}$ is the projection of $E_p$ onto the direction of the ferroelectric polarization in each domain. (d) and (e) The schematic drawing of $E_p$ dependence of (d) $D_{\left(110\right)}\left(E_p\right)$ and (e) electric polarization projected on the [110] axis in each magnetic domain. (f) $V_{\left(110\right)}$ dependence of $P$ in the case of $D_{\left(110\right)}=D_{\left(1\overline{2}0\right)}=D_{\left(\overline{2}10\right)}=1$.

Figure 7

(Color online) (a) and (b) Diffraction profiles of $\left(H,H,\frac{3}{2}\right)$, reciprocal lattice scans for the $\left(q,q,\frac{3}{2}\right)$, and $\left(\frac{1}{2}-q,\frac{1}{2}-q,\frac{3}{2}\right)$ magnetic Bragg reflections at $T=2.0\text{K}$ in the FE-ICM phase of $\text{CFAO}\left(x=0.015\right)$. (c) A comparison between the profiles of $I_{\text{ON}}-I_{\text{OFF}}$ (open circles) and $I_{\text{ON}}+I_{\text{OFF}}$ (filled circles) for the $\left(q,q,\frac{3}{2}\right)$ and $\left(\frac{1}{2}-q,\frac{1}{2}-q,\frac{3}{2}\right)$ magnetic Bragg reflections at $T=2.0\text{K}$.

Figure 8

Temperature variations in the diffraction profiles of ${\left(q,q,\frac{3}{2}\right)}_h$ and $\left(\frac{1}{2}-q,\frac{1}{2}-q,\frac{3}{2}\right)$ magnetic Bragg reflections in the cooling process with $E_p$ of $+1\text{MV}/\text{m}$. The open and filled symbols denote the diffraction intensities from the sum of the collinear and helical magnetic orderings $\left(I_{\text{ON}}+I_{\text{OFF}}\right)$ and the reflection only from the helical magnetic ordering $\left(I_{\text{ON}}-I_{\text{OFF}}\right)$, respectively. The data of $I_{\text{ON}}-I_{\text{OFF}}$ were multiplied by $A{\left(\kappa \right)}^{-1}{D}_{\left(110\right)}{\left(E_p\right)}^{-1}$, where $D_{\left(110\right)}{\left(E_p\right)}^{-1}$ was deduced from the intensities for $\left(\frac{1}{2}-q,\frac{1}{2}-q,\frac{3}{2}\right)$ reflections at $T=5.14\text{K}$ in order to directly compare with $I_{\text{ON}}+I_{\text{OFF}}$. The horizontal bars represent the experimental resolution.

Figure 9

(Color online) The $E_p$ dependences of (a) $D_{\left(110\right)}\left(E_p\right)$ in the (110) domain, (b) the sum of $I_{\text{ON}}+I_{\text{OFF}}$, and (c) the observed and calculated values of $P$ in $\text{CFAO}\left(x=0.015\right)$ at $T=2.0\text{K}$. The solid lines are guides to the eyes, drawn so as to be symmetric.

Figure 10

(Color online) Temperature variations in $P$ under an applied magnetic field of 12 T along the $c$ axis measured after cooling processes with several $E_p$. The inset shows the $E_p$ dependence of $P$ at $T=2.0\text{K}$ and $H=12\text{T}$. The solid line is a guide to the eyes.

Figure 11

The $E_p$ dependences of $D_{\left(110\right)}\left(E_p\right)$ in $\text{CFGO}\left(x=0.035\right)$, $\text{CFAO}\left(x=0.015\right)$, and $\text{CFAO}\left(x=0.02\right)$ (taken from Ref. 17) at $T=2\text{K}$. The solid and dashed lines are guides to the eyes.