Magnetically induced Ferroelectricity in Bi$_2$CuO$_4$

TL;DR

This study reveals that Bi$_2$CuO$_4$ exhibits magnetically induced ferroelectricity due to spin ordering, with experimental evidence from neutron scattering and dielectric measurements showing inversion symmetry breaking and magnetoelectric effects.

Contribution

First detailed experimental investigation demonstrating magnetically induced ferroelectricity in Bi$_2$CuO$_4$, linking magnetic order to electric polarization and symmetry breaking.

Findings

Cu spins are aligned parallel to the $ab$ plane.

Electric polarization appears below antiferromagnetic transition under magnetic field.

Magnetoelectric effect explains spin-induced ferroelectricity in the material.

Abstract

The tetragonal copper oxide Bi$_2$CuO$_4$ has an unusual crystal structure with a three-dimensional network of well separated CuO$_4$ plaquettes. This material was recently predicted to host electronic excitations with an unconventional spectrum and the spin structure of its magnetically ordered state appearing at T$_N$ $\sim$43 K remains controversial. Here we present the results of detailed studies of specific heat, magnetic and dielectric properties of Bi$_2$CuO$_4$ single crystals grown by the floating zone technique, combined with the polarized neutron scattering and high-resolution X-ray measurements. Our polarized neutron scattering data show Cu spins are parallel to the $ab$ plane. Below the onset of the long range antiferromagnetic ordering we observe an electric polarization induced by an applied magnetic field, which indicates inversion symmetry breaking by the ordered state of Cu spins. For the magnetic field applied perpendicular to the tetragonal axis, the spin-induced ferroelectricity is explained in terms of the linear magnetoelectric effect that occurs in a metastable magnetic state. A relatively small electric polarization induced by the field parallel to the tetragonal axis may indicate a more complex magnetic ordering in Bi$_2$CuO$_4$.

Equations

$$f_{\mathrm{me}}=$$

$$P_c=-\partial f_{\mathrm{me}}/\partial E_c=g_1(L_a{H}_a+L_b{H}_b)+g_4{L}_c{H}_c.$$

$$f\approx \frac{a}{2}{L}^2+\frac{b_1}{4}{L}^4+\frac{b_2}{2}{L}_a^2{L}_b^2+\frac{c_1}{2}{L}^2{H}^2+\frac{c_2}{2}(\mathbf{L}\cdot \mathbf{H}{)}^2$$

$$L_a^2=\frac{\alpha (T_N(H_a)-T)}{b_1},$$

$$H_{\mathrm{cr}}^2=\frac{\alpha (T_N(0)-T)b_2}{b_1{c}_2+b_2(c_1+c_2)}.$$

$${\chi }_c={\frac{\partial P_c}{\partial E_c}|}_{E_c=0}={\chi }_L{g}_1^2{H}_a^2,$$

Full text

I Introduction

The discovery of high-temperature superconducting (HTSC) cuprates triggered an extensive research on copper-based materials, especially, low-dimensional cuprates with novel physical properties a . Among these materials, Bi2CuO4  is of special interest: its “2-1-4” chemical formula is common to the prototypical HTSC R2CuO4+δ compounds (R = rare earth etc.). However, the isovalent substitution of the rare earth $R^{3+}$ ions by Bi3+ ions of similar ionic size results in a different and unique crystal structure shown in Fig. 1. This difference might be caused by a strong covalent Bi-O bonding b . Bi2CuO4 has a tetragonal structure (space group P4/ncc), in which the Cu2+ ions exhibit a square-planar coordination by four oxygen ions. While in other cuprates the CuO4 units form infinite two-dimensional CuO2 layers (e.g., in R2CuO4+δ a ) or quasi-one-dimensional ribbons or ladders, as in LiCu2O2 c or Sr3Ca11Cu24O41 d , the CuO4 plaquettes in Bi2CuO4 are isolated from each other. As shown in Fig. 1, the CuO4 plaquettes are staggered along the $c$ direction. Each plaquette is twisted with respect to the adjacent one with a twist/rotation angle of $\sim$33*∘, thus, forming chains in the $c$-direction. These chains are separated by non-magnetic Bi3+* ions. The Cu-Cu distance along the chains is of the order of 3 Å, which is larger than in copper metal. Nuclear quadrupole resonance (NQR) experiments e indicate that the Bi3+ ions are involved in the superexchange between Cu ions, i.e. Cu2+-O2--Bi3+-O2--Cu2+ paths yield long range antiferromagnetic (AFM) ordering below the Néel temperature T_{\text{N}}$$~{}\sim 43 K.

First studies of the magnetic properties of Bi2CuO4 suggested a quasi-1D spin-1/2 chain model with possible spin dimerizaton f . However, neutron diffraction measurements g ; h ; i ; j ; k revealed a long range three-dimensional (3D) AFM order of C-type: in the magnetic unit cell, which coincides with the chemical one, spins in chains are parallel, while spins in neighboring chains are antiparallel. However, the orientation of Cu spins was not unambiguously determined and two magnetic structures were proposed – one with spins parallel to the $c$ axis g ; j and one with spins within the $ab$ plane k . Optical spectroscopy experiments including polarized far-infrared and Raman spectra on single crystals suggest the $c$ axis orientation of copper spins l ; m ; n ; o . Specific heat and thermal expansion studies of the critical behavior around $T_{\text{N}}$ are consistent with the 3D Ising universality class, thus, also suggesting an easy $c$ axis anisotropy p . However, other experimental studies, including the high field anisotropic antiferromagnetic resonance (AFMR) q ; r ; s and torque magnetometry measurements t support an easy plane anisotropy in Bi2CuO4. Theoretical band structure calculations provided conflicting predictions u ; v . Despite many efforts, this issue remains controversial.

Spin excitations in Bi2CuO4  have been investigated by inelastic neutron scattering w ; x . The magnon dispersion spectrum can be fitted using the spin wave theory with up to four AFM exchange terms. In Ref. [x, ] both intrachain and interchain exchange interactions were found to be antiferromagnetic, which indicates magnetic frustration in Bi2CuO4.

Previous analysis of the photoelectron spectra shows that Bi2CuO4 is a charge-transfer insulator with a large band gap (around 2 eV) y . The high resistivity prevents a transport study on Bi2CuO4 at low temperature. Dc conductivity measurements are only feasible at temperatures far above $T_{\text{N}}$ z . Hence, dielectric or ac conductivity measurements seem to be better suited to study Bi2CuO4. The temperature and frequency dependent dielectric response may yield further information of microscopic physical properties of Bi2CuO4. However, so far, there exists only one report of the dielectric properties of Bi2CuO4 in the temperature range from 100 K to 300 K which is well above $T_{\text{N}}$ aa .

The discovery of giant magnetoelectric effects in multiferroic manganites with coupled magnetic and electric dipoles ab ; ac , attracted a lot of interest in multiferroic and magnetoelectric materials. The spontaneous electric polarization observed in these frustrated magnets is induced by complex spiral spin structures which break time reversal and inversion symmetries. These unusual magnetic states are highly susceptible to applied magnetic fields, allowing for the magnetic control of electric polarization and, in some cases, for the electric control of magnetization, which is of great importance both for fundamental physics and technological applications ad ; ae . A similar control can be achieved in materials showing a linear magnetoelectric effect, such as Cr2O3, where an electric polarization is induced by an applied magnetic field and vice versa xa ; xb ; xc . The linear magnetoelectric effect also requires simultaneous breaking of time-reversal and inversion symmetry, but is usually observed in magnets with k = 0 magnetic states.

Copper oxides provide a rich playground for studies of magnetoelectric phenomena. The ferroelectricity induced by spin spirals was observed in the multiferroic Cu-chain compounds, LiCu2O2 and LiCuVO4 al ; xd ; xe as well as in the three-dimensional CuO with a high spin ordering temperature xf . Magnetically-induced ferroelectricity and the linear magnetic response were recently observed in the spiral and skyrmion phases of the chiral cubic magnet Cu2OSeO3 xg ; xh .

Here, we systematically investigate anisotropic physical properties of floating-zone grown Bi2CuO4 single crystals including magnetic susceptibility, magnetization, specific heat, polarized neutron, dielectric constant and pyroelectric measurements. Although the k = 0 AFM spin structure in Bi2CuO4 has been established decades ago, it was not noticed before that it breaks inversion symmetry and thus can give rise to magnetoelectric phenomena. We report the observation of the field-induced ferroelectricity in Bi2CuO4, which we associate with the linear magnetoelectric effect in a metastable state. Moreover, we

II Experimental

The growth of Bi2CuO4  single crystals was carried out in an optical floating zone furnace equipped with four elliptical mirrors (Crystal Systems Corp.). The polycrystalline rods were prepared from appropriate mixtures of Bi2O3 and CuO with high purities. The mixed starting materials were sintered in air at 725*∘C for 72 hours with several intermediate grindings. Finally, the obtained powder was pressed into a rod with $\sim$7 mm in diameter and $\sim$12 cm in length under a hydrostatic pressure of $\sim$60 MPa. Afterwards, the rod was sintered at 750∘C for 24 hours. During the floating zone growth the crystal was grown in a pure oxygen atmosphere. Due to the low surface tension, the molten zone of the Bi2*CuO4 is difficult to control. This might be the reason why in earlier studies a fast growth rate was reported. Such a high growth rate yields products that easily crack and the quality of these crystals is not good. Therefore, we lowered the low growth speed to 3 mm/hour while both feed and seed rods were counter-rotated at the rates of 20-40 rpm. Finally, we obtained boules with typically 5 cm length. The corresponding growth direction changed to [1 0 2] compared to the [0 0 1]-direction in single crystals grown with faster growth rate. Our as-grown single crystals were oriented by Laue diffraction, and cut into different rectangular plates of 1 mm thickness and with edges parallel to the crystallographic principal axes.

The magnetic properties of our samples were studied using a SQUID magnetometer (MPMS-5XL, Quantum Design Inc.). The measurements of specific heat were carried out using a standard thermal relaxation calorimetric method in a commercial Physical Property Measurement System (PPMS, Quantum Design Inc.).

Polarized neutron scattering experiments were performed at the IN12 triple-axis spectrometer at Institut Laue-Langevin and diffuse neutron scattering spectrometer DNS at the Heinz Maier-Leibnitz Zentrum dnsA ; dnsB . The measurements at IN12 have been carried out using CryoPad setup for spherical neutron polarimetry. The flipping ratio was 21. The measurements at DNS we have done with two different setups. In the standard setup for xyz-polarization analysis we have studied the scattering in spin-flip and non-spin-flip channels for three different neutron polarizations. In this case the magnetic field at the sample was limited to the instrument guide field (around 10 G). The flipping ratio was above 26. Additionally, we have measured the sample mounted between two permanent magnets providing the field the sample roughly around 2.5 kG. In this case due to the depolarization of the neutron beam by magnets, we performed no polarization analysis. The incoming wavelength for all measurements at DNS was $\lambda$ $=$ 4.2Å.

Temperature dependent powder x-ray diffraction measurements have been performed on a powder x-ray diffractomter (Bruker D8 Discover A25) which is equipped with a Johansson monochromator for Cu-K$\alpha$ radiation and which was optimized for low background and high resolution. The measurements of our (crushed) single crystals confirm the absence of impurity phases. The temperature dependence of the lattice constants was measured using a He cryostat (Oxford Phenix).

For dielectric measurements, the plate-shaped samples were further polished thinner to a thickness of 0.1-0.3 mm. We applied silver paint to both sides as electrodes to form a parallel-plate-capacitor. The samples were glued on the cryogenic stage of the customized probe that was inserted into a cryostat. A high-precision capacitance bridge (AH2700a, Andeen-Hagerling Inc.) or a LCR meter (E4980AL Keysight Technologies) was used fore the dielectric measurements. The main sources of errors such as residual impedance in the whole circuit have been carefully compensated. In our measurements, various excitation levels (from 100 meV to 10 V) were used and no apparent difference was found, thus, confirming the intrinsic nature of our observations.

To verify the ferroelectric nature of the possible multiferroic transitions, the spontaneous electric polarization (P) has been measured via the integration of the corresponding pyroelectric current. Firstly, we fully polarized the specimens with a static electric field of 1000 kV/m during the cooling process from temperatures above $T_{\text{N}}$, then removed the electric field and short-circuited the sample at lowest temperature in order to remove any possible trapped charge carriers. The pyroelectric current was, then, measured during the warming process at different heating rates (2-4 K/min).

III Experimental Results

III.1 Magnetic properties

We measured the temperature dependence of the magnetic susceptibility $\chi$ of a Bi2CuO4  single crystal in different magnetic fields and field-directions (from 0.1 T to 5 T and with H parallel to $c$- or $a$- axis). In the zero-field-cooling (ZFC) and the field-cooling (FC) $\chi$(T) curves no essential difference was observable.

As shown in Fig. 2(a), $\chi$(T) increases on cooling, reaches its maximum within a broad hump around $\sim$50 K, drops abruptly around $\sim$43.5 K and decreases further until an upturn below $\sim$20 K occurs. The broad hump in $\chi$(T) is indicative for frustration or short-range fluctuations - a typical behavior observed in low dimensional antiferromagnetic systems. The drop around $\sim$43.5 K indicates the emergence of long-range AFM order (i.e. $T_{\text{N}}$); see also d$\chi$/dT(T) in the inset of Fig. 2(b) which exhibits a sharp peak at T_{\text{N}}$$\sim43.5 K. A strong anisotropy of $\chi$(T) can be even observed at high temperature. We plotted the temperature dependent inverse susceptibility ($\chi^{-1}$) in Fig. 2b. A strong deviation from linear behavior can be observed especially for H $\|$ a even at high temperatures above $T_{\text{N}}$. Apparently, a simple Curie-Weiss law is not applicable to Bi2CuO4. Note, that also ESR experiments q report an anisotropic g factor for Bi2CuO4.

Whereas the susceptibility of Bi2CuO4 is field independent above $T_{\text{N}}$ it exhibits an unusual field dependence below $T_{\text{N}}$ (for H $\|$ a) - up to $\sim$1 T $\chi$ is increasing with increasing size of H. Moreover, we measured the field-dependent magnetization and susceptibility at different temperatures. As shown in Fig. 3(a,b) the M-H curve is non-linear below 0.5-1 T for H $\|$ a and for temperatures below $T_{\text{N}}$. This behaviour might be indicative for a metamagnetic transition with a kind of spin-reorientation. In contrast to that we observe only linear M-H relations and field-independent susceptibilities for H $\bot$ a. Our results for Bi2CuO4 are consistent with older studies in literature k ; t .

III.2 Polarized neutron scattering

To resolve the long-standing controversy regarding the orientation of ordered Cu spinsg ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; t , we studied the magnetic structure of Bi2CuO4 by polarized neutron diffraction. Figure 4 shows the basically magnetic neutron scattering intensities in the $\sigma_{zz}$ and $\sigma_{yy}$ spin-flip channels. No intensity in the $\sigma_{yy}$ channel is observed. Hence, the magnetic moments in Bi2CuO4 have no projection on the $c$ axis and are entirely aligned within the $ab$-plane. Thus, the $c$-axis alignment of the copper spins as suggested in Refs. g ; j can be discarded and the alignment within the $ab$-plane k is corroborated. We obtained the same result in a polarized neutron measurement at 3.5 K using the DNS spectrometer at the FRM II, see Fig. 5.

Magnetoelectrically induced ferroelectricity (see Sec. III.7) and competing exchange interactions x prompted us to look for signs of incommensurate spin ordering in Bi2CuO4. In fields up to $\sim 0.25$ T we could not observe any incommensurability within the experimental resolution of our measurements at the DNS spectrometer (see Fig. 5). Further high-resolution experiments in higher magnetic fields would be desirable.

III.3 Lattice parameters

We have measured Bi2CuO4 by means of powder x-ray diffraction on a powder x-ray diffractometer (Cu-K$\alpha$ radiation) that we optimized for high resolution. Within these measurements we were not able to find any indications for the occurrence of peak splittings or superstructure reflections below 300 K. However, our high resolution measurements are able to resolve an anomalous increase of the $a$- and an anomalous decrease of the $c$-lattice constant on cooling through $T_{\text{N}}$, see Fig. 6. This is indicative for magneto elastic coupling as was also observed in other transition metal compounds magcoup ; magcoupB . Our observed anomalous increase of the lattice constants is in agreement with reported thermal expansion measurements p .

III.4 Heat capacity

Further evidence for long-range antiferromagnetic order in Bi2CuO4 is revealed from our specific heat (C**p) measurements. As shown in Fig. 7, C**p(T) shows a $\lambda$-shaped anomaly at $T_{\text{N}}$. We have measured the specific heat also on heating and on cooling around $T_{\text{N}}$, see the upper left inset of Fig. 7. The absence of any hysteresis indicates a second order phase transition. This is also consistent with the absence of any structural transition at $T_{\text{N}}$ observed in our powder x-ray diffraction measurements.

Both, lattice vibrations (phonons) and magnetic excitations (magnons) contribute to the specific heat $C_{p}$. At low temperatures the phonon contribution of $C_{p}$ obeys the Debye T3 power law. As can be seen in the lower right inset in Fig. 7, the low temperature part of $C_{p}$(T) can be fitted quite well with a function proportional to T3, thus, suggesting that also the magnetic contribution of $C_{p}$ obeys a T3 law. The T3 dependence - rather than T3/2 behaviour - further confirms long-range AFM ordering in Bi2CuO4. However, it remains difficult to extract the pure magnetic part of $C_{p}$(T) accurately due to the absence of nonmagnetic isostructural materials in order to estimate the contribution of the lattice vibrations to the total specific heat. To get a rough estimate of the entropy removed by the magnetic ordering, which is remarkably indicated by the peak of Cp(T) around $T_{\text{N}}$, we use a polynomial fit to the high-temperature ($>$60 K) and low-temperature ($<$25 K) regimes for the background (the red solid line shown in main panel of Fig. 7). The entropy obtained from integrating the peak in Cp(T)/T (for 25 K $<$ T $<$ 60 K) amounts to 1.82 J mol*-1* K*-1* which is only 32%  of the full magnetic entropy associated with R$\cdot$ln(2S+1) $\thickapprox$ 5.76 J mol*-1* K*-1* as expected for an S $=$ 1/2 spin system. Most of the remaining magnetic entropy is considered to be removed by the short-range ordering well above $T_{\text{N}}$, as usually observed in the low-dimensional frustrated spin systems.

III.5 Anisotropic dielectric properties as zero field

In view of the tetragonal symmetry of Bi2CuO4 we measured the dielectric constant for an electric field $E$ applied either within the $ab$ plane, $\varepsilon$ab, or along the $c$-axis, $\varepsilon$c. Two thin plate samples that were cut from our floating-zone-grown single crystal were used to investigate the anisotropic dielectric properties.

Fig. 8 shows the temperature-dependent $\varepsilon$c and $\varepsilon$ab measured in zero field for different frequencies. We observed a strong anisotropy in permittivity: \varepsilon$${}_{ab}\sim 18 is much lower than \varepsilon$${}_{c}\sim 30, which resembles the anisotropy in magnetic susceptibility discussed above.

Both $\varepsilon$c and $\varepsilon$ab slightly decrease with decreasing temperature and saturate at lowest temperature. $\varepsilon$ab and $\varepsilon$c also exhibit a strong frequency dependence characteristic of a Debye-type dipolar relaxation-like behavior. For analysis of the frequency dispersion, the corresponding dielectric loss (tan$\delta$) curves are also shown. At each measuring frequency the loss exhibits a peak corresponding to the knee-like change of $\varepsilon$. As frequency increases, the peak shifts to higher temperatures with a concomitant decrease of magnitude. The peak temperature and the frequency can be fitted with the Arrhenius law, $f=f_{0}\exp(-Q/kT_{peak})$, where $f$ is the characteristic frequency, $k$ is the Boltzmann constant, Tpeak is the peak temperature, $f_{0}$ is a pre-factor and $Q$ is the activation energy. Fitting the two sets of data from Fig. 8(a) and (b) yields nearly the same activation energy of $\sim$22 meV. We note that in Raman scattering experiments on Bi2CuO4 a featured mode around 150 cm*-1* was observed at low temperature, which results from a two-magnon scattering process. The corresponding energy ($\sim 20$ meV) is close to $Q$ derived from dielectric dispersion analysis, which suggests that the observed dielectric relaxation at low temperatures can result from the coupling of electric dipoles to magnons.

Across the magnetic ordering temperature $T_{\text{N}}$, a weak ‘kink’ appears in $\varepsilon$c(T) at all frequencies. However, no corresponding anomalous behavior was found in the dielectric loss indicating the absence of ferroelectric transition in zero field. This conclusion is corraborated by our pyroelectric measurements discussed below. No anomaly is observed in $\varepsilon$ab(T).

III.6 In-plane dielectric response in an applied magnetic field

Next we studied the in-plane dielectric properties under applied magnetic fields. We present the data measured at 1 kHz because of the relatively large separation of the dispersion peak from $T_{\text{N}}$ and a better signal-to-noise ratio at low frequencies.

We applied $H$ parallel and perpendicular to the measuring electric field $E$. Up to 9 T no change can be observed in $\varepsilon$ab and tan$\delta_{ab}$, as shown in Fig. 9. The corresponding pyroelectric measurements also show no electric polarization (not shown). We also investigated the in-plane magnetodielectric effect at temperatures below and above $T_{\text{N}}$ for several values of the applied field $H$. As shown in Fig. 9(c) for $H\perp E$ no magnetodielectric effect is observed. Similar results were found for $H\parallel E$.

III.7 Out-of-plane dielectric properties in an applied field and magnetically-induced ferroelectricity

In this section we describe the effect of magnetic field on the dielectric permitivity along the tetragonal $c$ axis, $\varepsilon$c. We show that under an applied magnetic field Bi2CuO4 becomes ferroelectric with the field-dependent electric polarization, $P_{c}$, parallel to the $c$ axis.

III.7.1 Out-of-plane dielectric response in $H\parallel c$

We performed dielectric measurements for a variety of magnetic fields applied along the $c$-direction. Figure 10 shows the detailed results for $H$ = 9 T. The most remarkable observation is the sharp $\lambda$-shaped peak observed in both $\varepsilon$c(T) and tan$\delta_{c}$ for all frequencies, which is indicative of a ferroelectric transition. The peak position is independent of frequency and coincides with $T_{\text{N}}$ determined from magnetic susceptibility and specific heat measurements. Compared with the zero field data (Fig. 8(b)) shows that the frequency-dependent dielectric relaxation behavior is essentially unaffected by the external magnetic field.

The direct evidence for the ferroelectric nature of the transition at $T_{\text{N}}$ is given by our pyroelectric current measurements shown in Fig. 10(c). A small but very sharp peak emerges as temperature approaches $T_{\text{N}}$. The pyroelectric current drops to zero above $T_{\text{N}}$. The corresponding $c$-axis electric polarization, $P_{c}$, obtained by integrating the pyroelectric current, appears below $T_{\text{N}}$ and reaches its (rather small) saturation value of about 0.18$\mu$C/m2. The reversal of the poling electric field changes sign of $P_{c}$. This proves the emergence of ferroelectricity in Bi2CuO4under the applied magnetic field.

Figure 11 shows $\varepsilon$c(T) measured at the frequency 1 kHz for different applied magnetic fields ranging from 0 to 9 T. For magnetic fields below \sim$$\sim 0.5 T the small kink around $T_{\text{N}}$ observed in zero field remains unchanged. As the magnetic field is increased further, a dielectric peak appears (see e.g. the blue curve measured at 1.5 T in the inset of Fig. 11(a)). This dielectric peak develops gradually with increasing $H$ and no saturation is observed up to our instrumental limit of 9 T. The transition temperature $T_{\text{N}}$ at which the peak is observed is almost unaffected by $H$. The same behavior is observed in the corresponding loss curves shown in Fig. 11(b). Hence, the dielectric anomaly associated with the onset of ferroelectricity in Bi2CuO4 appears for applied magnetic fields (>$$\sim 0.5 T) and becomes more pronounced as the magnetic field increases. The results for field and temperature dependence of $P_{c}$ (Fig. 11(c)) clearly show that Bi2CuO4 becomes ferroelectric under an applied magnetic field $H\|c$.

We also studied the magnetodielectric coupling for $H\|c$, i.e. the dependence of the dielectric constant on an applied magnetic field. Figure 11(d) shows the relative change of the permitivity, $\frac{\Delta\varepsilon}{\varepsilon}=\frac{\varepsilon_{c}(H)-\varepsilon_{c}(0)}{\varepsilon_{c}(0)}$, for several temperatures below and above the transition temperature, $T_{\text{N}}$. A prominent magnetodielectric effect is only observed for $T$ close to $T_{\text{N}}$. At $T=43$ K, the value of $\frac{\Delta\varepsilon}{\varepsilon}$ grows with increasing field $H$ ($>$1 T) reaching $\sim 0.06$% at 9 T.

III.7.2 Out-of-plane dielectric response in $H\parallel a$

We performed similar measurements for magnetic fields applied along the $a$ direction ($H\perp E$). The results are shown in Fig. 12. In contrast to the behavior observed for $H\|c$, the dielectric peak in $\varepsilon$c(T) at $T_{\text{N}}$ is discernable already at very low magnetic fields. At about $\sim 0.5$ T the peak reaches its maximum size and becomes smaller as $H$ is increased further (see the inset in Fig. 12(a)). Concomitantly with the peak in $\varepsilon_{c}$, a sharp peak in the dielectric loss, tan $\delta_{c}$, appears and disappears (see Fig. 12(b)).

Fig. 12(c) shows the electric polarization $P_{c}$ measured in $H\|a$. A low field of just 0.2 T induces ferroelectricity in Bi2CuO4 with a spontaneous polarization of about 0.14$\mu$C/m2 at 4 K. After $P_{c}$ reaches its maximum value, it decreases upon further increase of the magnetic field and vanishes around 6 T.

Finally, we also studied the magnetodielectric effect for $H\|a$. As for $H\|c$, there is no discernable magnetodielectric effect at temperatures far from $T_{\text{N}}$. Only around the dielectric peak region we are able to observe a prominent magnetodielectric effect in low fields. At 43 K, $\frac{\Delta\varepsilon}{\varepsilon}$ rises sharply with increasing field up to its maximum value at about $\sim 0.5$ T, and then decreases upon further increase of $H$.

IV Discussion

In zero field, previous neutron diffraction measurements on Bi2CuO4 reveal a k = 0 collinear spin structure with parallel spins in the chains running along the $c$ and antiparallel spins in neighboring chains. However, the orientation of ordered magnetic moments is poorly understood and remains controversial. In other magnetic Cu2+ (S = 1/2) compounds with the CuL4 (L=O, Cl, Br) square-planar coordination (e.g. in CuCl${}_{2}\cdot$2H2O, CuCl2, CuBr2 or LiCuVO4 ) an easy-plane anisotropy is observed. Our polarized neutron data suggest that this is also the case for Bi2CuO4.

Our observation of a metamagnetic behavior in low in-plane fields (of about $\sim 0.5$ T at 2 K) is consistent with previous findings t ; k . The non-linearity of the $M(H)$ curve likely originates from the field-dependent volume fraction of the four antiferromagnetic domains, in which spins are (anti)parallel to the $a$ or $b$ axis. We cannot exclude, however, a more complex spin-reorientation transition taking place at $H_{a}\sim 0.5$ T.

Concomitant with the emergence of a long range magnetic ordering at $T_{\text{N}}$, we observe a transition into a ferroelectric state with the polarization along the $c$-direction, which appears under an applied magnetic field. This transition is manifest in both the dielectric constant, \varepsilon$${}_{c}(T,H), and polarization, $P_{c}(T,H)$, measurements. The transition occurs for magnetic fields applied either in the $ab$-plane or along the $c$ direction. The electric polarization $P_{c}$ as a function of the magnetic field along the $a$ and $c$ axes at 35 K (at which the polarization nearly saturates) is shown in Fig. 13. As we discuss below, the magnetoelectric response provides an important additional information on magnetic ordering in Bi2CuO4.

Although the magnetic field dependence of the electric polarization is nonlinear, the most likely source of the field-induced ferroelectricity is the linear magnetoelectric effect xa ; xb ; xc . We assume that the antiferromagnetic order parameter in Bi2CuO4 is, $\mathbf{L}=\mathbf{M}_{1}-\mathbf{M}_{2}+\mathbf{M}_{3}-\mathbf{M}_{4}$, where $\mathbf{M}_{i}$, $(i=1,2,3,4)$ is the sublattice magnetization labeled by the four independent Cu sites occupying the 4c Wykoff positions in the crystallographic unit cell with the coordinates $\mathbf{r}_{1}=(1/4,1/4,z)$, $\mathbf{r}_{2}=(3/4,3/4,1/2-z)$, $\mathbf{r}_{3}=(1/4,1/4,1/2+z)$ and $\mathbf{r}_{4}=(3/4,3/4,-z)$. Spatial inversion, $\mathbf{r}\rightarrow-\mathbf{r}$ interchanges the sites 1 and 4 and the sites 2 and 3, so that under inversion $\mathbf{L}$ changes sign. The inversion symmetry breaking by this antiferromagnetic ordering allows for a linear magnetoelectric effect. We note that the non-centrosymmetric antiferromagnetic order is incompatible with the interpretation of the metamagnetic behavior in terms of a weak ferromagnetic moment yy .

The free energy density describing the linear mangetoelectric coupling contains 4 terms allowed by P4/ncc symmetry of the crystal lattice,

[TABLE]

where $g_{\alpha}$ $(\alpha=1,2,3,4)$ are the coupling constants. The $c$-component of the electric polarization induced by an applied magnetic field is then

[TABLE]

Similar expressions can be obtained for $P_{a}$ and $P_{b}$, but since no significant in-plane electric polarization was observed, the coupling constants $g_{2}$ and $g_{3}$ must be relatively small.

We first discuss the magnetoelectric behavior of Bi2CuO4 in H $\|$ a (see Figs. 12(c) and 13(a)). Our polarized neutron scattering data show that in zero field the antiferromagnetic vector $\mathbf{L}$ lies in the $ab$ plane. It can be oriented either along the crystallographic axes (the $\pm[100]$ and $\pm[010]$ directions) or along the diagonals (the $\pm[110]$ and $\pm[1\bar{1}0]$ directions). In the latter case, the magnetic field parallel to the $a$ axis would result in a gradual rotation of $\mathbf{L}$ toward the $[100]$ plane. Instead, we observe a metamagnetic transition at $H_{a}\sim 0.5$ T (see Fig. 3), which indicates that in zero field Bi2CuO4  is divided into domains with $L\|a$ and $L\|b$. The zero-field magnetic susceptibility is then the average of the longitudinal ($L\parallel H$) and transverse ($L\perp H$) magnetic susceptibilities. Above the metamagnetic transition the $L_{a}$-domains disappear and the magnetic susceptibility becomes purely transversal, which explains why above $\sim 0.5$ T it increases approximately by factor of $2$ (see Fig. 3). The peculiar dependence of $P_{c}$ on $H_{a}$ shown in Fig.  13(a) is consistent with this scenario. As follows from Eq.(2), the linear magnetoelectric coefficient $\alpha_{ca}=\frac{\partial P_{c}}{\partial H_{a}}=g_{1}L_{a}$, which is why $P_{c}$ grows linearly with the field below $\sim 0.5$ T, when the $L_{a}$-domains are present, and decreases above $\sim 0.5$ T, when these domains disappear.

Below we discuss the spin-reorientation transition in more detail. Because of tetragonal symmetry of Bi2CuO4, the $L\|b$ state has a lower free energy than the $L\|a$ for an arbitrarily small $H_{a}$. However, in low applied magnetic fields the motion of domain walls separating the $L\|a$ and $L\|b$ domains can be hindered by pinning and the ferroelectric state with $L_{a}\neq 0$ can survive as a metastable state. There is, however, an upper critical field, $H_{\rm cr}$, above which the ferroelectric state becomes locally unstable and disappears. This can explain the nature of the metamagnetic transition observed at $\sim 0.5$ T.

This transition as well as the observed magnetoelectric response can be described using the phenomenological Landau theory. Assuming that both the vector order parameter and the applied magnetic field are confined to the $ab$ plane, while the electric field is parallel to the $c$ axis, we can write the free energy density of Bi2CuO4 in the form,

[TABLE]

where $L^{2}=L_{a}^{2}+L_{b}^{2}$, $H^{2}=H_{a}^{2}+H_{b}^{2}$, $(\mathbf{L}\cdot\mathbf{H})=L_{a}H_{a}+L_{b}H_{b}$, $\chi_{0}$ is the background magnetic susceptibilty and $a=\alpha(T-T_{N}(0))$, $T_{N}(0)$ being the transition temperature in zero magnetic field. The quartic anisotropy coefficient, $b_{2}>0$, favors $\mathbf{L}$ (anti)parallel to the $a$ and $b$ axes, while $c_{2}>0$ favors $\mathbf{L}\perp\mathbf{H}$.

Minimizing the free energy Eq.(3) with respect to $L_{a}$ for $L_{b}=H_{b}=0$, we obtain the order parameter in the metastable state,

[TABLE]

where $T_{N}(H_{a})=T_{N}(0)-\frac{(c_{1}+c_{2})}{\alpha}H_{a}^{2}$ is the transition temperature in the applied field. The (temperature-dependent) critical field, $H_{\rm cr}$, above which this state is unstable, is given by

[TABLE]

The electric polarization, given by Eq.(2), is approximately propoportional to $H_{a}$, for $H_{a}<H_{\rm cr}$ (the magnetic field dependence of $L_{a}$ (see Eq.(4)) is weak, since our experiment shows that the transition temperature is practically field independent). Finally, the dielectric susceptibility associated with the antiferromagnetic order is given by

[TABLE]

where $\chi_{L}$ is the antiferromagnetic susceptibility: $\chi_{L}=\left(\frac{\partial^{2}f}{\partial L_{a}^{2}}\right)^{-1}\propto|T-T_{N}(H_{a})|^{-1}$ near the transition (see Ref. [ya, ] , where also the effect of spin fluctuations on the critical behavior of the dielectric susceptibility of a linear magnetoelectric material is discussed). The dielectric susceptibility diverges because in the applied magnetic field a linear magnetoelectric material becomes ferroelectric, since for $H_{a}\neq 0$ the antiferromagnetic order, $L_{a}$ is linearly coupled to the electric field, $E_{c}$. Equation (6) is consistent with our observations (see Sec. III.7.2): the magnetic field dependence of $\varepsilon_{c}$ is only strong close to $T_{N}$, where the divergence of $\chi_{c}$ compensates for the weakness of the magnetoelectric response.

The explanation of the magnetoelectric response in H $\|$ c (see Figs. 11(c) and 13(b)) is less straightforward. According to Eq.(2), the linear magnetoelectric coefficient $\alpha_{cc}=\frac{\partial P_{c}}{\partial H_{c}}=g_{4}L_{c}$ is only nonzero if $L_{c}\neq 0$ . However, $L_{c}$ is zero in zero field and H $\|$ c is unlikely to result in a transition that rotates $\mathbf{L}$ out of the $ab$ plane. Also the linear dependence of $M_{c}$ on $H_{c}$ (Figs. 3(a) and (c)) shows the absence of any transition. One possible explanation is a small in-plane magnetic field component resulting from a small misalignment of the sample or demagnetizing fields. Note that $\alpha_{cc}$ is $\sim 15$ times smaller than $\alpha_{ca}$ for $H_{a}<0.5$ T.

Another possible explanation is that the actual symmetry of Bi2CuO4 is lower than tetragonal. We did not, however, find any indication for a symmetry lowering, even though our high-resolution powder X-ray diffraction measurements allow us to resolve tiny thermal expansion effects (see Fig. 6). Ferroelectricity in Bi2CuO4 can, in principle, be induced by a complex magnetic ordering, such as an incommensurate spin spiral with a small wave vector. The rotation of the spiral plane in an applied magnetic field, which transforms a non-ferroelectric helical spiral into a ferroelectric cycloidal spiral, can mimic linear magnetoelectric response, as was observed in the CoCr2Se4 spinel yb ; yc . However, also in the spiral state scenario it is difficult to explain why the out-of-plane electric polarization is induced by both in-plane and out-of-plane magnetic fields.

V Conclusions

A combination of several experimental techniques was applied to clarify the nature of magnetic ordering in floating zone grown Bi2CuO4 single crystals. Polarized neutron scattering measurements clearly show that ordered magnetic moments of Cu ions are confined to the $ab$ plane, which resolves the long standing controversy. In addition, we observe a ferroelectric polarization parallel to the $c$ axis induced by applied magnetic fields, which is consistent with inversion symmetry breaking by the C-type antiferromagnetic ordering. We argue that the electric polarization in the magnetic field parallel to the $a$ axis originates from the linear magnetoelectric effect in a metastable state, which disappears above a critical magnetic field. The relatively weak response observed in the magnetic field parallel to the $c$ axis does not seem to be compatible with symmetry of Bi2CuO4 and remains a puzzle. Further studies of the magnetic states of this material are necessary.

Acknowledgements.

The authors would like to thank L. H. Tjeng and O. Stockert for fruitful discussions and Ch. Becker, T. Mende and S. Wirth for their support on the experimental construction of the dielectric measurement system.