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and Lifshitz Course of Theoretical Physics: Electrody- namics in Continuous Media (Butterworth-Heinemann,. 1980).  S.-Y. Lu and R.A. Chipman, Optics Comm. 146, 11. (1998).  C. Genet, E. Altewischer, M.P. van Exter, and J.P. Wo- erdman, in pre
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deformation in the OSH-PT and omnidirectional SH waves in the hosting plate. Both finite element ... Experimental testing shows that the OSH-PT exhibits good omnidirectional properties, on matter it is used as a ... monitoring (SHM). a) Author to who
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magnetoelectric coupling factor leading to a magnetic field driven gigantic change in the polarization near ... polarization cannot be reversed completely by changing the polarity of a poling electric field. (E=Â±14kV/cm). ... value of the coefficien
1International Center for Quantum Materials, School of Physics, Peking University, Beijing ... 4Department of Materials Science & Engineering, University of Utah ... [email protected]; [email protected] Abstract. The discovery of high-tempe
Jan 21, 2010 - Chia-Wei Huang and Xuedong Hu. Department of Physics, University at Buffalo, The State University of New York, Buffalo, NY 14260-1500, USA ..... cesses are the most efficient in building up NSP in the. QD, as shown in Fig. 4. ...... AC
Apr 14, 2008 - pirically observable at the epoch of the gravitational collapse of the protogalaxy. But the. Universe is a good conductor: the magnetic flux and ...
Oct 9, 2006 - netic and electric (dipolar) degrees of freedom in a class of materials known as âmultiferroicsâ1â7. ..... 3: (color online) The calculated polarization Px and Py from exact diagonalization of the full M-O-M cluster .... Research
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Aug 21, 2015 - like quantum dots (QDs) or impurity centers. A rela- tively high nuclear spin polarization can be .... leads to a shift of the RSA peaks, which reflects a change of the electron precession frequency. The left inset ... both the seleniu
D. J. Kim et al.
Polarization Relaxation Induced by Depolarization Field in Ultrathin Ferroelectric BaTiO3 Capacitors D. J. Kim,1 J. Y. Jo,1 Y. S. Kim,1 Y. J. Chang,1 J. S. Lee,1 Jong-Gul Yoon,2 T. K. Song,3,∗ and T. W. Noh1
arXiv:cond-mat/0506480v2 [cond-mat.mtrl-sci] 14 Oct 2005
ReCOE and School of Physics, Seoul National University, Seoul 151-747, Korea 2 Department of Physics, University of Suwon, Kyunggi-do 445-743, Korea 3 Department of Ceramic Science and Engineering, Changwon National University, Changwon, Kyungnam 641-773, Korea (Dated: February 2, 2008)
Time-dependent polarization relaxation behaviors induced by a depolarization field Ed were investigated on high-quality ultrathin SrRuO3 /BaTiO3 /SrRuO3 capacitors. The Ed values were determined experimentally from an applied external field to stop the net polarization relaxation. These values agree with those from the electrostatic calculations, demonstrating that a large Ed inside the ultrathin ferroelectric layer could cause severe polarization relaxation. For numerous ferroelectric devices of capacitor configuration, this effect will set a stricter size limit than the critical thickness issue. PACS numbers: 77.22.Ej, 77.22.Gm, 77.80.Dj, 77.55.+f
With recent breakthroughs in fabricating high-quality oxide films [1, 2, 3], ultrathin ferroelectric (FE) films have attracted much attention from the scientific as well as application points of view. As the FE film thickness d approaches tens of unit cell length, the FE films often show significantly different physical properties from those of bulk FE materials. Some extrinsic effects, especially coming from FE film surfaces and/or interfaces with other materials, could be very important . For some other cases, intrinsic physical quantities could play vital roles in determining the unique properties of ultrathin films. Many FE-based electronic devices have the capacitor configuration, where a FE layer is inserted between two conducting electrodes. Then, polarization bound charges will be induced at the surfaces of the FE layer, but compensated by free charge carriers in the conducting electrodes. In real conducting electrodes, however, the compensating charges will be induced with a finite extent, called the screening length λ. This will result in an incomplete compensation of the polarization charges. Such an incomplete charge compensation should induce a depolarization field Ed inside the FE layer, with a direction opposite to that of the FE polarization P . Therefore, Ed will appear in every FE capacitor, and its effects will becomes larger with the decrease of d . (For a FE film without electrodes, there is no compensation for the polarization bound charge, so the value of Ed will become even larger than that of the FE capacitor case.) Ed has been known to be important in determining the critical thickness  and domain structure of ultrathin FE films [7, 8, 9], and reliability problems of numerous FE devices [10, 11]. Recently, using a first principles calculation, Junquera and Ghosez investigated the critical thickness of BaTiO3 (BTO) layers in SrRuO3 (SRO)/BTO/SRO capacitor . For calculations, they assumed that all of the BTO and SRO layers were fully strained with the SrTiO3 substrate. By taking the real SRO/BTO interfaces into account properly, they showed that Ed could make the ferroelectricity vanish for the BTO films thinner than 6 unit cells, i.e. 2.4 nm . More recently,
using pulsed laser deposition with a reflection high energy electron diffraction monitoring system, we fabricated high-quality fully-strained SRO/BTO/SRO capacitors on SrTiO3 substrates with d between 5 and 30 nm [2, 12]. With a very low leakage current, we could directly measure their P -E hysteresis loops . In this letter, we report the time-dependent polarization changes of the ultrathin BTO films. We find that the net P of the ultrathin BTO films decreases quite rapidly in time. We will show that the P relaxation should originate from Ed . By compensating for Ed with an external potential, we can determine the Ed values of the SRO/BTO/SRO capacitors experimentally. These measured Ed values agree with the values from the electrostatic calculations. Finally, we will discuss the effect of the P relaxation on a practical size limitation imposed on actual FE devices. In our earlier report , we obtained the thicknessdependent remnant polarization Pr values from the P E hysteresis loops, measured at 2 kHz in ultrathin FE films as thin as 5 ∼ 30 nm. With further studies on the frequency dependence of the Pr values in P -E hysteresis loops, as shown in Fig. 1(a) for a 15 nm thick BTO capacitor, we found differences in the Pr values when the measuring frequency is varied. These results suggest that the FE domain dynamics should play an important role for ultrathin FE films, where the FE domain wall motion is known to be strongly suppressed . Note that the first principles calculation (FPC) and the Landau-Devonshire calculation (LDC) do not consider the domain dynamics, so their predicted polarization values should be called as spontaneous polarization Ps . Since the P value significantly affects the subsequent analysis of P relaxation, precise determination of Ps values is necessary. To determine the precise values of Ps , we applied pulse trains, which are schematically shown in the inset of Fig. 1(a) . The interval between write and read pulses was set to 1 µs to minimize the effects of the P relaxation, and the current responses under the read pulse were measured. The total amount of charge is obtained by integrating the current responses in time. The read pulses with different heights were used to obtain the linear part of the polarization under an external electric
2 field The Ps values can be obtained by extrapolating the linear part of the polarization to zero electric field. The triangles (black) and circles (green) in Fig. 1(b) show the Pr values measured at 2 and 100 kHz, respectively. Also, the squares (red) show the Ps values from the pulse test. The solid (green) and dashed (blue) curves show the theoretical predictions from the FPC  and the LDC , respectively, which take account of Ed . Note that neither of these theories can explain the thickness-dependence of Ps quantitatively. However, it is known that the FPC predicts systematically somewhat lower bulk lattice constants compared to real values, so the compressive stress predicted by the FPC could be smaller than that in the fully strained sample, resulting in a smaller Ps . To avoid this systematic error, we normalized the polarization values to those of a 30 nm thick BTO capacitor. We found that the thickness-dependent scaling of Ps also follows the FPC predictions quite well, as shown in Fig. 1(c). The large difference in P values between the 2 and the 100 kHz tests indicates that there should be a strong change in the net P between 10 and 500 µs. Time-
FIG. 2: (color-online) (a) Polarization relaxation in the 15 nm thick BTO capacitor. The inset shows the schematics of measuring relaxation behaviors under an external voltage. (b) Slowing down of the relaxation behavior under external voltages in the 15 nm thickness BTO capacitor. (c) Thicknessdependent Ed in the ultrathin BTO capacitors. The Ed values obtained from relaxation behaviors experimentally (red solid circles) and those from electrostatic calculations with the parameters determined and measured Ps in this work (open squares). The (green) line indicates the Ed from electrostatic calculations with the polarization values obtained from first principles calculation.
FIG. 1: (color-online) (a) Upper halves of hysteresis loops for 15 nm thick BTO capacitor at the measurement frequencies 2 and 100 kHz. The values of spontaneous polarization Ps were obtained from linear extrapolation of pulse measurements. (b) The polarization values obtained from pulse measurements, first principles calculation (FPC), and LandauDevonshire calculation (LDC). (c) Normalized behaviors of LDC, FPC, and Ps to the values of BTO capacitors with 30 nm thickness.
dependent P changes were investigated by applying two kinds of pulse trains, as shown in the inset of Fig. 2(a). For the write and the read pulses with the same (opposite) polarities, the amount of nonswitching (switching) P can be determined . The difference ∆P , between the switching and the nonswitching P should be twice as large as the net P . As shown in Fig. 2(a), ∆P decreases quite rapidly for the film with d = 15 nm; ∆P falls to less than 10% of the Ps value within a relaxation time trelax of 1000 s. As shown with the solid squares (black) in Fig. 2(b), ∆P decay follows a power-law dependence on trelax . Similar power-law decays of ∆P were observed for all the BTO films in the thickness range of 5 ∼ 30 nm. Note that such a strong polarization relaxation could pose a serious problem in capacitor-type ultrathin FE devices. What is the origin of such strong polarization relaxations? We thought that they could be closely related
3 to large Ed induced inside the BTO films. To verify this idea, we slowed down the relaxation phenomena by applying an external voltage, as shown in the inset of Fig. 2(a). The values of the applied external electric field Eext were obtained by dividing the applied external voltage by the corresponding film thickness. When the external field is applied in the opposite direction of Ed , the potential gradient inside the FE layer will decrease. Figure 2(b) shows that the slope of the power-law decay becomes smaller, as Eext increases. Assuming that the depth of the double-well potential for BTO ferroelectricity can be considered negligible compared to the effect of Ed , we approximately determined experimental Ed values from the applied electric field under which the slope becomes zero. Since Ed is proportional to P , the Ed value should increase slightly on application of Eext . After correcting this minor contribution, we could determine the Ed values, which are plotted as solid circles (red) in Fig. 2(c). From electrostatic calculations on the capacitor geometry, Mehta et al. showed that 2ǫF /d P , (1) Ed = − ǫ0 ǫF 2ǫF /d + ǫe /λ where d is the thickness of the FE layer, and ǫF and ǫe are the relative dielectric constants of the FE layer and the electrode, respectively . To obtain theoretical Ed values for our SRO/BTO/SRO capacitors, we have to know accurate values of ǫe , λ, and ǫF . Unfortunately, the reported physical parameter values in the literature vary [5, 6, 16, 17]. Also, we could not find any definite experimental study on ǫe . To obtain the value of ǫe for an SRO electrode, we used optical spectroscopy. We measured the optical reflectivity spectra of epitaxial SrRuO3 films (thickness: about 0.5 µm) in a wide frequency region between 5
FIG. 3: (color-online) (a) Frequency-dependent dielectric functions of SRO. (b) Capacitance-electric field curve of the SRO/BTO/SRO capacitor with 5 nm thick BTO.
meV and 30 eV and performed a Kramers-Kronig analysis to obtain the frequency-dependent dielectric function, ǫ(ω) [=ǫ′ (ω) + iǫ′′ (ω)]. The details of these measurements and analysis were published elsewhere [18, 19]. The open squares in Fig. 3(a) and the inset show experimental values of ǫ′ (ω) and ǫ′′ (ω), respectively. Note that ǫe in Eq. (1) represents the dielectric response from the bound charges, namely bound electrons and phonons. Since SRO is metallic, there should be a large contribution from the free Drude carriers, which masks the dielectric response from the bound charges. To obtain ǫe , we decompose ǫ(ω) into a free carrier contribution ǫcoherent (ω) and a bound electron contribution ǫbound (ω) by fitting the experimental ǫ(ω) with a series of Lorentz oscillators, which are displayed as the dotted (blue) lines in the inset of Fig. 3(a). The dash-dotted (blue) lines indicate the bound electron contribution. From the dc limit of ǫbound (ω), we could estimate that the bound electron contribution to ǫe is about 8.17. The phonon contribution to ǫe was evaluated in a similar way by analyzing the phonon spectra and found to be about 0.28 . Consequently, ǫe is determined to be about 8.45. Using the carrier density n0 ∼ = 1.2 × 1022 /cm3 of SRO , the experimental value of ǫe , and the effective mass of an electron mef f ∼ = 7me , where me is the mass of a free electron [21, 22], we applied the free electron model and obtained λ = 0.8 ± 0.1 ˚ A [5, 23]. We also measured ǫF from the capacitance-electric field C-E curves of BTO capacitors. Figure 3(b) shows the C-E curve for the 5 nm BTO capacitor. The C-E curve has the hysteretic behavior typical for a FE capacitor. The BTO capacitors with 5 ∼ 30 nm thickness show almost the same ǫF -E curves. The ǫF values can vary from 70 to 230 depending on the applied E. Since most of our experiments were performed under a finite applied field, which corresponds to a value between 1 and 2 MV/cm, the ǫF were estimated to be about 80 . With the measured values of ǫe , λ, and ǫF , we could estimate the theoretical Ed values from Eq. (1) with the Ps values obtained from the pulse test. The open squares in Fig. 2(c) are the theoretical Ed values. The solid (green) line shows the theoretical Ed values with the Ps values, obtained from the FPC. These theoretical Ed values from the electrostatic model agree quite well with the experimental Ed values, determined from the polarization relaxation. It should be noted that the Ed values are comparable with or even larger than the measured coercive fields (in our samples, 300 ∼ 400 kV/cm). These large Ed values can cause P reversal and FE domain formation, which will result in a reduction of the net P value as time elapses. The fact that two independent determinations provided nearly the same Ed values demonstrates that the polarization relaxation behavior should be dominated by Ed inside the FE layer. Note that the Ed -induced ∆P decay comes intrinsically from the incomplete compensation of the P charges (due to the finite screening length of the electrodes) in real conducting electrode, so that it will inevitably pose a fundamental limit for most FE device applications using the capacitor configuration. This limitation should
4 be much more severe than that due to the critical thickness of the FE ultrathin films . Even if the FE film is thicker than the critical thickness, it is feasible that the Ed -induced ∆P decay is large enough to make the net P decrease significantly, resulting in retention failures for numerous FE devices. As d decreases, Ed increases significantly. With the current miniaturization trends in some FE devices, the large value of Ed should play a very important role in determining the ultimate size limits of FE devices. In order to reduce device failure due to the polarization relaxation, we can try to select better electrode and FE materials. Noble metals, such as Pt, have been considered better electrodes because they have high carrier density (resulting in λ values smaller than that of SRO). However, the ǫe values of typical noble metals are much smaller than that of SRO, i.e. 8.45 , so Ed in capacitors with noble metal electrodes can be large. For example, Ed in the range of 500 ∼ 900 kV/cm is expected for a 15 nm thick BTO film with noble metal electrodes (typically, λ = 0.4 ∼ 0.5 ˚ A, ǫe = 2 ∼ 4). Thus, the Ed induced P relaxation for the ultrathin BTO capacitors with the noble metal electrodes could be at least equal to or worse than that with SRO electrodes. Proper FE material selection can be another option. Since PbTiO3 is known to have a much deeper double-well potential than that of BTO [15, 26], the P relaxation should occur at a much lower rate even with the same value of Ed . Optimization of FE materials should be of great importance for the improvement of ultrathin film nanoscale FE device performances. In summary, we demonstrated that the depolarization field inside the ferroelectric film could cause a severe polarization relaxation. By slowing down the relaxation under an external field, we could determine the depolarization field in a real capacitor of ultrathin SrRuO3 /BaTiO3/SrRuO3 experimentally, which result is in good agreement with electrostatic calculations. Our investigation demonstrates that the depolarization field originates from intrinsic properties of electrode material such as the finite screening length and that the depolarization field should play an important role in domain dynamics in ultrathin FE films. The polarization relaxation due to the depolarization field could pose a serious size limitation for ultrathin ferroelectric devices. The authors thank Prof. Sug-Bong Choe in Seoul National University for valuable discussions. This work was financially supported by the Korean Ministry of Science and Technology through the Creative Research Initiative program and by KOSEF through CSCMR.
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