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Biased dielectric response in LuFe2O4
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Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Biased dielectric response in LuFe2 O4 Yu.B. Kudasov a,b,c,∗ , M. Markelova d , D.A. Maslov a,b,c , V.V. Platonov a,b,c , O.M. Surdin a,b,c , A. Kaul d a
Sarov Physics and Technology Institute, National Research Nuclear University “MEPhI”, Moscow, 115409, Russia National Research Nuclear University “MEPhI”, Kashirskoe shosse 31, Moscow, 115409, Russia c Russian Federal Nuclear Center – VNIIEF, pr. Mira 37, Sarov, 607188, Russia d Chemistry Department, Moscow State University, Moscow, 119991, Russia b
a r t i c l e
i n f o
Article history: Received 10 July 2016 Received in revised form 19 September 2016 Accepted 30 September 2016 Available online xxxx Communicated by M. Wu Keywords: Multiferroics Ferroelectricity Dielectric response Domain structure Depleted layer
a b s t r a c t A complex permittivity at a low level of excitation signal was measured in ceramic LuFe2 O4 . A Debye-type relaxation response with a strong temperature dependence of a characteristic frequency was observed in accordance with earlier works. A small DC bias of about 10 V/cm led to unusual changes in the dielectric response. At frequencies, which were lower than the characteristic one, the conductivity drastically increased with slight decrease of the real part of the permittivity under the bias. In the opposite case of low frequencies, there are no traces of the DC bias effect. We show that an inhomogeneous charge distribution over surface layer (domain structure) is essential for describing the biased dielectric response in LuFe2 O4 . © 2016 Elsevier B.V. All rights reserved.
1. Introduction Multiferroic materials are of a great interest due to potential applications to memory elements, filtering devices, and highperformance insulators [1–3]. Among them, LuFe2 O4 has attracted a considerable attention because of a novel mechanism of the ferroelectricity [4,5]. This substance has a hexagonal layered struc¯ Iron sites form triangular bilayers ture with the space group R 3m. (W-layers) which contain the same numbers of Fe2+ and Fe3+ ions [5]. Their arrangement, i.e. charge order (CO), varies with temperature: a two-dimensional CO exists in the range between 320 K and 500 K, a three-dimensional noncentro-symmetrical CO appears and gives rise to the ferroelectricity below 320 K [4–6]. At lower temperatures the CO is assumed to be coupled with a ferrimagnetic order [7]. Geometric frustration in both the charge and spin systems of LuFe2 O4 as well as low-dimensionality leads to complex behavior of this substance: a rich phase diagram [8], fully and partially ordered charge arrangements [6], giant magnetodielectric response [9]. AC dielectric measurements in LuFe2 O4 have revealed a Debyetype response with a temperature dependent characteristic fre-
*
Corresponding author. E-mail address: [email protected] (Yu.B. Kudasov).
http://dx.doi.org/10.1016/j.physleta.2016.09.054 0375-9601/© 2016 Elsevier B.V. All rights reserved.
quency [5,10]. A constant bias voltage led to a giant tunability effect for the real part of the dielectric permittivity [10]. The measurements in this work were performed at a high level of excitation signal as compared to a zone of linearity for the permittivity. That is why they should have been accompanied by sizable nonlinear distortions and most probably have underestimated the tunability effect. At large DC electric field intensity (above 100 V/cm) a sharp drop of resistivity, i.e. a reversible breakdown, occurred [11]. It was also mentioned that the DC bias led to increase of dielectric loss tangent at high temperatures [12]. Recent wide-band investigations of the dielectric permittivity [13–15] have cast doubt on the ferroelectric nature of the AC dielectric response in LuFe2 O4 . It was shown that an equivalent scheme with Maxwell–Wagnertype contacts and temperature dependent hopping intrinsic bulk conductivity described well the AC response without involving ferroelectricity. The key role of the surface or interface layers in the AC dielectric response was well established [13–15]. However, it should be mentioned that this model [14] contained at least 5 free parameters, some of which depended on temperature. Very recently, magnetic and electric domains on surface of LuFe2 O4 were directly observed by means of electrostatic and magnetic force microscopy (EFM and MFM) [16]. Characteristic sizes of the magnetic and electric domains strongly differed from each other. At the same time piezoresponse force microscopy (PFM) has not detected any sign of piezoelectricity. The combination of EFM
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Fig. 1. The real part of the permittivity (a) and the conductivity (b) as functions of temperature. The dash and dash–dot lines show the DC part of the intrinsic conductivity and σ v as guides for eye. A model representation of the complex conductivity is shown in the insert.
and PFM suggested the non-displacive, electronic ferroelectricity in LuFe2 O4 . However, both types of the force microscopy are sensitive to surface states only, and the nature of the bulk dielectric response remains still ambiguous. In the present Letter we perform measurements of complex permittivity of ceramic LuFe2 O4 at a very low level of excitation signal to avoid the nonlinear distortions. We also propose a model of the dielectric response in this compound. 2. Experimental technique The ceramic samples were prepared using chemical homogenization method. Ash-free paper filters were soaked with a mixed solution of lutetium and iron nitrates taken in the molar ratio 1:2, then dried and burned. The resulted ash was annealed at 600 °C during 1 h in air for total carbon elimination. The residual oxide powder was pelletized and the pellets were sintered during 30 h at 1000 °C in a sealed quartz ampoule containing oxygen getter Fe/FeO. The proper oxygen stoichiometry was formed after final thermal annealing of the samples, also 30 h at 1000 °C in the sealed ampoule, with oxygen getter FeO/Fe3 O4 . According to our results obtained by Ce-redox titration, the oxygen stoichiometry of the obtained samples correspond to the formula LuFe2 O4.005 , what is very close to the data of Ref. [17]. After surface finish the samples were discs of 5 mm in diameter and about 1.5 mm in thickness. As shown earlier [13] a contacts fabrication technique strongly influenced the measurement results. We used thin indium foils attached to the sample by silver paste to each face. During all the measurements the sample was under constant pressure at contacts of about 5 N. The dielectric response was investigated by means of HP4263B LCR meter at 120 Hz, 1 kHz, 10 kHz, and 100 kHz. The excitation electric field was typically 0.7 V/cm. To be convinced of absence of the nonlinear distortions we lowered it to about 0.07 V/cm. A built-in voltage source was used to generate the DC bias. Measurements of frequency dependence of the biased dielectric response at fixed temperature were performed by means of GW Instek 78110G LCR meter with the same excitation electric field. The temperature of the sample was determined by a calibrated high-sensitive thermocouple of the L type with an absolute error less than 2 K in the whole temperature range. A relative temperature error between the curves at the same frequency and different DC biases was within 1 K. The real part of the complex permittivity (ε ) and conductivity derived from the imaginary part of the permittivity (σ = ωε0 ε ) at the zero bias are plotted in Fig. 1. They are in a good agreement with the previous measurements [5,10], that is, the Debye-type relaxation [18] response with strong temperature dependence was observed. An insignificant discrepancy in the absolute values of the
permittivity with that measured previously [10] stemmed from a slight difference in ceramics density. The temperature dependence of the characteristic relaxation time corresponded to that established earlier [5,10]. Application of the weak DC bias caused a peculiar effect on the AC conductivity at all the frequencies as shown in Fig. 2. The conductivity increased drastically at frequencies below the characteristic one ( f < f C ) or temperatures above that corresponded to the characteristic frequency (T > T C ). The real part of the permittivity decreased slightly at the same conditions. In the opposite case ( f > f C or T < T C ) no changes in the permittivity or conductivity were observed. It should be pointed out that the transition between the two regimes was very sharp (Fig. 2). The frequency dependence of the biased dielectric response at fixed temperature is shown in Fig. 3. The permittivity and conductivity are normalized to unit magnitude, that is, σ0 and ε0 are the maximum values of the conductivity and permittivity, correspondingly. The Debye-type response [18] is clearly seen in this figure. The vertical dash line is plotted for the sake of convince to display the transition from one regime to another. 3. Model of biased dielectric response The basic wide-band model of dielectric response in LuFe2 O4 involves two relaxation times [13–15]. One of the corresponding characteristic frequencies for ceramic samples varies from 200 Hz to 20 kHz depending on temperature [15]. The other is well above 1 MHz and therefore lies far from the frequency range used for the measurements. That is why we apply a simplified model with a single relaxation time which is related to surface layer. Firstly let us consider a depleted layer at surface of LuFe2 O4 in terms of Schottky barrier [19]. Its capacitance depends on the DC bias U DC as C ∝ (U DC − U 0 )−1/2 where U 0 is a constant and q0 U 0 >> k B T , q0 is the unit charge. At the same time, the conductance of the barrier is G ∝ exp (q0 U DC /k B T ). As usual we assume q0 U DC >> k B T . Then the characteristic relaxation frequency f C = G /C should have a strong dependence on the DC bias. This obviously contradicts to the experimental data shown in Fig. 2. To overcome the problem we take into account the inhomogeneous charge distribution or domain structure in the surface layer that has been observed in Ref. [16]. The complex conductivity can ∗ (ω) where the star be represented as a sum σ ∗ (ω) = σi (ω) + σdw denotes complex values, σi (ω) = σ DC + σh (ω) is the intrinsic DC ∗ (ω) is the and frequency-dependent hopping conductivity [14], σdw complex conductivity corresponding to electric polarization of the surface layer.
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Fig. 2. The effect of DC bias on the conductivity and permittivity.
Fig. 3. The normalized values of the conductivity (the upper panel) and permittivity (the lower panel) as functions of the frequency at fixed temperature and different values of the DC bias.
Since we assume the polarization of the surface layer to be inhomogeneous it can be described in terms of electric domain motion [20]
mx¨ + η x˙ + kx = α E
(1)
where x is the domain wall displacement, m and η are the specific effective mass and viscosity of the domain wall, k is the stiffness coefficient for pinning center, and α E is the driving force under the electric field intensity E which varies with time. The effective mass of the domain wall in ferroelectrics is usually associated with a striction effect at the domain boundaries [20,21]. However there was no piezoresponse observed in LuFe2 O4 [16] and the first term in Eq. (1) can be omitted (m → 0). A free domain wall moves under the driving and viscosity forces only. In this case its motion gives ∗ = σ . It is worth noticing rise to the real part of conductivity σdw v that the viscosity can be represented as η ∝ 2 /rc where is the
order parameter and rc is the domain wall thickness [20,21]. That is why η decreases drastically with increase of temperature approaching zero at the critical temperature ( → 0 and rc → ∞ at T → T C ). The conductivity corresponding to the viscous flow decreases with temperature reduction since σ v ∝ /η . A schematic behavior σ v is shown in Fig. 1. A total “freezing” of the domain structure occur at about 100 K. The pinning force describing by the third term in Eq. (1) together with viscosity leads to the Debye-type relaxation response [18]. The stiffness coefficient also decreases with the temperature rise but this dependence should be much weaker as compared to the viscosity. That is why the critical frequency f c = k/η is a function of temperature as shown in Fig. 1 by the arrows. At low frequency or high temperature the pinning force determines the domain wall motion and causes the high almost nondispersive real part of permittivity Fig. 1. In the opposite case the viscous flow of the domain walls gives rise to the conductivity. The weak DC bias applied to the electric domain wall causes its partial detachment from pinning centers [20]. If a domain wall is in the viscous flow regime the DC bias has no effect on the dielectric response since the pinning centers plays no role in this case. In the pinning regime some fraction of the domain walls leaves the pinning centers under the DC bias and goes into the viscous regime. Then the complex conductivity can be written as ∗ = (1 − v )σ ∗ + v σ where v is the fraction of the domain walls σdw v p which are defined under the DC bias, σ p∗ is the conductivity in the pinning regime, which is mainly imaginary (the real part of the permittivity). Thus, we observed a sizable increase of conductivity at temperatures higher than the characteristic one and slight decrease of the real part of the permittivity with the characteristic frequency being independent on the DC bias. A detailed discussion of the model will be presented elsewhere. 4. Conclusion The biased dielectric response presented in Fig. 2 and Fig. 3 can not be explained by variation of the capacity and conductivity of
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homogeneous depleted layer under the bias since this should cause a sizable shift of the characteristic frequency. Instead we have observed a rise of the conductivity and decrease of the permittivity at f < f C with constant f C . That is why we have to extend the basic model of the dielectric response in LuFe2 O4 taking into account the inhomogeneous charge distribution or domain structure of the depleted surface layer which was directly observed by EFM [16]. Despite the fact that the response was discussed in terms of domain structure the results obtained do not necessarily imply bulk ferroelectricity of LuFe2 O4 because the low-frequency behavior is apparently caused by surface layer. It is also worth mentioning that in contrast to traditional ferroelectrics the domain wall pinning in LuFe2 O4 is very weak that leads to drastic changes in the response at very small values of the bias. The work was partially supported by Russian Foundation for Basic Research (Projects No. 13-02-01194 and No. 13-03-01249). References [1] [2] [3] [4] [5]
M. Bibes, A. Barthélémy, IEEE Trans. Electron Devices 54 (2007) 1003. J. van den Brink, D.I. Khomskii, J. Phys. Condens. Matter 20 (2008) 434217. A.P. Pyatakov, A.K. Zvezdin, Phys. Usp. 182 (2012) 593. N. Ikeda, K. Saito, H. Kito, J. Akimitsu, K. Siratori, Ferroelectrics 161 (1994) 111. N. Ikeda, H. Ohsumi, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe, H. Kito, Nature 436 (2005) 1136.
[6] Y.B. Kudasov, D.A. Maslov, Phys. Rev. B 86 (2012) 214427. [7] W. Wu, V. Kiryukhin, H.-J. Noh, K.-T. Ko, J.-H. Park, W. Ratcliff II, P.A. Sharma, N. Harrison, Y.J. Choi, Y. Horibe, et al., Phys. Rev. Lett. 101 (2008) 137203. [8] X.S. Xu, M. Angst, T. Brinzari, R.P. Hermann, J.L. Musfeldt, A.D. Christianson, D. Mandrus, B.C. Sales, S. McGill, J.-W. Kim, et al., Phys. Rev. Lett. 101 (2008) 227602. [9] M.A. Subramanian, T. He, J. Chen, N.S. Rogado, T.G. Calvarese, A.W. Sleight, Adv. Mater. 18 (2006) 1737. [10] C.-H. Li, X.-Q. Zhang, Z.-H. Cheng, Y. Sun, Appl. Phys. Lett. 92 (2008) 182903. [11] C. Li, X. Zhang, Z. Cheng, Y. Sun, Appl. Phys. Lett. 93 (2008) 152103. [12] Y. Liu, C.-H. Li, X.-Q. Zhang, Z.-H. Cheng, Y. Sun, J. Appl. Phys. 104 (2008) 104112. [13] A. Ruff, S. Krohns, F. Schrettle, F. Schrettle, P. Lunkenheimer, A. Loidl, Eur. Phys. J. B 85 (2012) 290. [14] D. Niermann, F. Waschkowski, J. de Groot, M. Angst, J. Hemberger, Phys. Rev. Lett. 109 (2012) 016405. [15] S. Lafuerza, J. García, G. Subías, J. Blasco, K. Conder, E. Pomjakushina, Phys. Rev. B 88 (2013) 085130. [16] I.K. Yang, J. Kim, S.H. Lee, S.-W. Cheong, Y.H. Jeong, Appl. Phys. Lett. 106 (2015) 152902. [17] T. Sekine, T. Katsura, J. Solid State Chem. 17 (1976) 49. [18] S. Kasap, P. Capper (Eds.), Springer Handbook of Electronic and Photonic Materials, Springer Science+Business Media, Inc., New York, USA, 2006. [19] S.M. Sze, Physics of Semiconductor Device, Wiley-Interscience, New York, 1981. [20] B.A. Strukov, A.P. Levanyuk, Ferroelectric Phenomena in Crystals: Physical Foundations, Springer-Verlag, Berlin Heidelberg, 1998. [21] A.S. Sidorkin, Domain Structure in Ferroelectrics and Related Materials, Cambridge International Science Publishing, 2006.
[m5G; v1.190; Prn:7/10/2016; 12:53] P.1 (1-4)
Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Biased dielectric response in LuFe2 O4 Yu.B. Kudasov a,b,c,∗ , M. Markelova d , D.A. Maslov a,b,c , V.V. Platonov a,b,c , O.M. Surdin a,b,c , A. Kaul d a
Sarov Physics and Technology Institute, National Research Nuclear University “MEPhI”, Moscow, 115409, Russia National Research Nuclear University “MEPhI”, Kashirskoe shosse 31, Moscow, 115409, Russia c Russian Federal Nuclear Center – VNIIEF, pr. Mira 37, Sarov, 607188, Russia d Chemistry Department, Moscow State University, Moscow, 119991, Russia b
a r t i c l e
i n f o
Article history: Received 10 July 2016 Received in revised form 19 September 2016 Accepted 30 September 2016 Available online xxxx Communicated by M. Wu Keywords: Multiferroics Ferroelectricity Dielectric response Domain structure Depleted layer
a b s t r a c t A complex permittivity at a low level of excitation signal was measured in ceramic LuFe2 O4 . A Debye-type relaxation response with a strong temperature dependence of a characteristic frequency was observed in accordance with earlier works. A small DC bias of about 10 V/cm led to unusual changes in the dielectric response. At frequencies, which were lower than the characteristic one, the conductivity drastically increased with slight decrease of the real part of the permittivity under the bias. In the opposite case of low frequencies, there are no traces of the DC bias effect. We show that an inhomogeneous charge distribution over surface layer (domain structure) is essential for describing the biased dielectric response in LuFe2 O4 . © 2016 Elsevier B.V. All rights reserved.
1. Introduction Multiferroic materials are of a great interest due to potential applications to memory elements, filtering devices, and highperformance insulators [1–3]. Among them, LuFe2 O4 has attracted a considerable attention because of a novel mechanism of the ferroelectricity [4,5]. This substance has a hexagonal layered struc¯ Iron sites form triangular bilayers ture with the space group R 3m. (W-layers) which contain the same numbers of Fe2+ and Fe3+ ions [5]. Their arrangement, i.e. charge order (CO), varies with temperature: a two-dimensional CO exists in the range between 320 K and 500 K, a three-dimensional noncentro-symmetrical CO appears and gives rise to the ferroelectricity below 320 K [4–6]. At lower temperatures the CO is assumed to be coupled with a ferrimagnetic order [7]. Geometric frustration in both the charge and spin systems of LuFe2 O4 as well as low-dimensionality leads to complex behavior of this substance: a rich phase diagram [8], fully and partially ordered charge arrangements [6], giant magnetodielectric response [9]. AC dielectric measurements in LuFe2 O4 have revealed a Debyetype response with a temperature dependent characteristic fre-
*
Corresponding author. E-mail address: [email protected] (Yu.B. Kudasov).
http://dx.doi.org/10.1016/j.physleta.2016.09.054 0375-9601/© 2016 Elsevier B.V. All rights reserved.
quency [5,10]. A constant bias voltage led to a giant tunability effect for the real part of the dielectric permittivity [10]. The measurements in this work were performed at a high level of excitation signal as compared to a zone of linearity for the permittivity. That is why they should have been accompanied by sizable nonlinear distortions and most probably have underestimated the tunability effect. At large DC electric field intensity (above 100 V/cm) a sharp drop of resistivity, i.e. a reversible breakdown, occurred [11]. It was also mentioned that the DC bias led to increase of dielectric loss tangent at high temperatures [12]. Recent wide-band investigations of the dielectric permittivity [13–15] have cast doubt on the ferroelectric nature of the AC dielectric response in LuFe2 O4 . It was shown that an equivalent scheme with Maxwell–Wagnertype contacts and temperature dependent hopping intrinsic bulk conductivity described well the AC response without involving ferroelectricity. The key role of the surface or interface layers in the AC dielectric response was well established [13–15]. However, it should be mentioned that this model [14] contained at least 5 free parameters, some of which depended on temperature. Very recently, magnetic and electric domains on surface of LuFe2 O4 were directly observed by means of electrostatic and magnetic force microscopy (EFM and MFM) [16]. Characteristic sizes of the magnetic and electric domains strongly differed from each other. At the same time piezoresponse force microscopy (PFM) has not detected any sign of piezoelectricity. The combination of EFM
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Fig. 1. The real part of the permittivity (a) and the conductivity (b) as functions of temperature. The dash and dash–dot lines show the DC part of the intrinsic conductivity and σ v as guides for eye. A model representation of the complex conductivity is shown in the insert.
and PFM suggested the non-displacive, electronic ferroelectricity in LuFe2 O4 . However, both types of the force microscopy are sensitive to surface states only, and the nature of the bulk dielectric response remains still ambiguous. In the present Letter we perform measurements of complex permittivity of ceramic LuFe2 O4 at a very low level of excitation signal to avoid the nonlinear distortions. We also propose a model of the dielectric response in this compound. 2. Experimental technique The ceramic samples were prepared using chemical homogenization method. Ash-free paper filters were soaked with a mixed solution of lutetium and iron nitrates taken in the molar ratio 1:2, then dried and burned. The resulted ash was annealed at 600 °C during 1 h in air for total carbon elimination. The residual oxide powder was pelletized and the pellets were sintered during 30 h at 1000 °C in a sealed quartz ampoule containing oxygen getter Fe/FeO. The proper oxygen stoichiometry was formed after final thermal annealing of the samples, also 30 h at 1000 °C in the sealed ampoule, with oxygen getter FeO/Fe3 O4 . According to our results obtained by Ce-redox titration, the oxygen stoichiometry of the obtained samples correspond to the formula LuFe2 O4.005 , what is very close to the data of Ref. [17]. After surface finish the samples were discs of 5 mm in diameter and about 1.5 mm in thickness. As shown earlier [13] a contacts fabrication technique strongly influenced the measurement results. We used thin indium foils attached to the sample by silver paste to each face. During all the measurements the sample was under constant pressure at contacts of about 5 N. The dielectric response was investigated by means of HP4263B LCR meter at 120 Hz, 1 kHz, 10 kHz, and 100 kHz. The excitation electric field was typically 0.7 V/cm. To be convinced of absence of the nonlinear distortions we lowered it to about 0.07 V/cm. A built-in voltage source was used to generate the DC bias. Measurements of frequency dependence of the biased dielectric response at fixed temperature were performed by means of GW Instek 78110G LCR meter with the same excitation electric field. The temperature of the sample was determined by a calibrated high-sensitive thermocouple of the L type with an absolute error less than 2 K in the whole temperature range. A relative temperature error between the curves at the same frequency and different DC biases was within 1 K. The real part of the complex permittivity (ε ) and conductivity derived from the imaginary part of the permittivity (σ = ωε0 ε ) at the zero bias are plotted in Fig. 1. They are in a good agreement with the previous measurements [5,10], that is, the Debye-type relaxation [18] response with strong temperature dependence was observed. An insignificant discrepancy in the absolute values of the
permittivity with that measured previously [10] stemmed from a slight difference in ceramics density. The temperature dependence of the characteristic relaxation time corresponded to that established earlier [5,10]. Application of the weak DC bias caused a peculiar effect on the AC conductivity at all the frequencies as shown in Fig. 2. The conductivity increased drastically at frequencies below the characteristic one ( f < f C ) or temperatures above that corresponded to the characteristic frequency (T > T C ). The real part of the permittivity decreased slightly at the same conditions. In the opposite case ( f > f C or T < T C ) no changes in the permittivity or conductivity were observed. It should be pointed out that the transition between the two regimes was very sharp (Fig. 2). The frequency dependence of the biased dielectric response at fixed temperature is shown in Fig. 3. The permittivity and conductivity are normalized to unit magnitude, that is, σ0 and ε0 are the maximum values of the conductivity and permittivity, correspondingly. The Debye-type response [18] is clearly seen in this figure. The vertical dash line is plotted for the sake of convince to display the transition from one regime to another. 3. Model of biased dielectric response The basic wide-band model of dielectric response in LuFe2 O4 involves two relaxation times [13–15]. One of the corresponding characteristic frequencies for ceramic samples varies from 200 Hz to 20 kHz depending on temperature [15]. The other is well above 1 MHz and therefore lies far from the frequency range used for the measurements. That is why we apply a simplified model with a single relaxation time which is related to surface layer. Firstly let us consider a depleted layer at surface of LuFe2 O4 in terms of Schottky barrier [19]. Its capacitance depends on the DC bias U DC as C ∝ (U DC − U 0 )−1/2 where U 0 is a constant and q0 U 0 >> k B T , q0 is the unit charge. At the same time, the conductance of the barrier is G ∝ exp (q0 U DC /k B T ). As usual we assume q0 U DC >> k B T . Then the characteristic relaxation frequency f C = G /C should have a strong dependence on the DC bias. This obviously contradicts to the experimental data shown in Fig. 2. To overcome the problem we take into account the inhomogeneous charge distribution or domain structure in the surface layer that has been observed in Ref. [16]. The complex conductivity can ∗ (ω) where the star be represented as a sum σ ∗ (ω) = σi (ω) + σdw denotes complex values, σi (ω) = σ DC + σh (ω) is the intrinsic DC ∗ (ω) is the and frequency-dependent hopping conductivity [14], σdw complex conductivity corresponding to electric polarization of the surface layer.
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Fig. 2. The effect of DC bias on the conductivity and permittivity.
Fig. 3. The normalized values of the conductivity (the upper panel) and permittivity (the lower panel) as functions of the frequency at fixed temperature and different values of the DC bias.
Since we assume the polarization of the surface layer to be inhomogeneous it can be described in terms of electric domain motion [20]
mx¨ + η x˙ + kx = α E
(1)
where x is the domain wall displacement, m and η are the specific effective mass and viscosity of the domain wall, k is the stiffness coefficient for pinning center, and α E is the driving force under the electric field intensity E which varies with time. The effective mass of the domain wall in ferroelectrics is usually associated with a striction effect at the domain boundaries [20,21]. However there was no piezoresponse observed in LuFe2 O4 [16] and the first term in Eq. (1) can be omitted (m → 0). A free domain wall moves under the driving and viscosity forces only. In this case its motion gives ∗ = σ . It is worth noticing rise to the real part of conductivity σdw v that the viscosity can be represented as η ∝ 2 /rc where is the
order parameter and rc is the domain wall thickness [20,21]. That is why η decreases drastically with increase of temperature approaching zero at the critical temperature ( → 0 and rc → ∞ at T → T C ). The conductivity corresponding to the viscous flow decreases with temperature reduction since σ v ∝ /η . A schematic behavior σ v is shown in Fig. 1. A total “freezing” of the domain structure occur at about 100 K. The pinning force describing by the third term in Eq. (1) together with viscosity leads to the Debye-type relaxation response [18]. The stiffness coefficient also decreases with the temperature rise but this dependence should be much weaker as compared to the viscosity. That is why the critical frequency f c = k/η is a function of temperature as shown in Fig. 1 by the arrows. At low frequency or high temperature the pinning force determines the domain wall motion and causes the high almost nondispersive real part of permittivity Fig. 1. In the opposite case the viscous flow of the domain walls gives rise to the conductivity. The weak DC bias applied to the electric domain wall causes its partial detachment from pinning centers [20]. If a domain wall is in the viscous flow regime the DC bias has no effect on the dielectric response since the pinning centers plays no role in this case. In the pinning regime some fraction of the domain walls leaves the pinning centers under the DC bias and goes into the viscous regime. Then the complex conductivity can be written as ∗ = (1 − v )σ ∗ + v σ where v is the fraction of the domain walls σdw v p which are defined under the DC bias, σ p∗ is the conductivity in the pinning regime, which is mainly imaginary (the real part of the permittivity). Thus, we observed a sizable increase of conductivity at temperatures higher than the characteristic one and slight decrease of the real part of the permittivity with the characteristic frequency being independent on the DC bias. A detailed discussion of the model will be presented elsewhere. 4. Conclusion The biased dielectric response presented in Fig. 2 and Fig. 3 can not be explained by variation of the capacity and conductivity of
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homogeneous depleted layer under the bias since this should cause a sizable shift of the characteristic frequency. Instead we have observed a rise of the conductivity and decrease of the permittivity at f < f C with constant f C . That is why we have to extend the basic model of the dielectric response in LuFe2 O4 taking into account the inhomogeneous charge distribution or domain structure of the depleted surface layer which was directly observed by EFM [16]. Despite the fact that the response was discussed in terms of domain structure the results obtained do not necessarily imply bulk ferroelectricity of LuFe2 O4 because the low-frequency behavior is apparently caused by surface layer. It is also worth mentioning that in contrast to traditional ferroelectrics the domain wall pinning in LuFe2 O4 is very weak that leads to drastic changes in the response at very small values of the bias. The work was partially supported by Russian Foundation for Basic Research (Projects No. 13-02-01194 and No. 13-03-01249). References [1] [2] [3] [4] [5]
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