We investigate polarization retention in 10 to 19 nm thick ferroelectric BaTiO3 (BTO) grown on Ge(001) by molecular beam epitaxy. The out-of-plane direction and reversibility of electric polarization were confirmed using piezoresponse force microscopy. After reverse-poling selected regions of the BTO films to a value P with a biased atomic-force microscope tip, we monitored relaxation of their net polarization for as long as several weeks using optical second-harmonic generation microscopy. All films retained reversed polarization throughout the observation period. 10 nm-thick BTO films relaxed monotonically to a saturation value of 0.9 P after 27 days and 19 nm films to 0.75 P after 24 h. Polarization dynamics are discussed in the context of a 1D polarization relaxation/kinetics model.
Thin ferroelectric films have potential for low-power electronic device applications. For example, films with spontaneous out-of-plane polarization could be used in ferroelectric non-volatile memory1–4 and electro-optical modulators.5–9 BaTiO3 (BTO) features a moderate remanent polarization (∼26 μC/m2) and a dielectric constant ϵ/ϵ0 of several thousands.10 In addition, thin BTO films can be integrated monolithically with common semiconductor substrates such as Si or Ge using molecular beam epitaxy (MBE), atomic layer deposition, or chemical vapor deposition, opening a plethora of exciting opportunities for hybrid devices.8,11,12
Data retention time is a key figure of merit for low-power ferroelectric non-volatile memory devices. It can be determined by measuring time-varying polarization after poling. In films thinner than 100 nm, polarization measurements using an R-C circuit13 are hampered by high leakage current and a deposited top electrode could inject charges.14 Piezoresponse force microscopy (PFM) is effective for indirectly monitoring polarization immediately after poling15,16 but is subject to tip damage over longer times. Moreover, the tip applies force to the surface, which can alter polarization kinetics. Transmission electron microscopy (TEM) measures polarization directly17 but is destructive. Here, we use optical second-harmonic generation (SHG) microscopy to monitor the break in centrosymmetry associated with ferroelectric polarization in BTO films.18 SHG is non-invasive and sensitive to small changes in polarization and can monitor polarization kinetics indefinitely.
Ga-doped Ge(001) wafers (0.019 Ω cm) were used as substrates. They were cleaned by degreasing in acetone, isopropyl alcohol, and deionized water in a sonicator for 10 min each, then exposed in-situ to oxygen plasma for 30 min, and subsequently annealed, leading to atomically flat, 2 × 1 reconstructed and contamination-free surfaces. The details of the Ge cleaning process are given elsewhere.19,20 For the thin film deposition, a customized DCA Instruments M600 MBE chamber with a base pressure of 3.0 × 10−10 Torr was used. Ba, Sr, and Ti were evaporated from Knudsen effusion cells, while the atomic flux was calibrated with a quartz crystal microbalance. First, 2 nm of SrTiO3(STO) were grown by co-deposition of Sr and Ti on the cleaned Ge(001) substrate at at 5 × 10−7 Torr molecular oxygen. This layer imposed compressive strain on BTO, enforcing out-of-plane polarization.4 Increasing the substrate temperature to with a ramp rate of crystallized the STO film. The molecular oxygen pressure was then increased to 5 × 10−6 Torr, and BTO was deposited to the desired thickness on the STO/Ge template at by shuttering the Ba and Ti effusion cells. At this temperature, BTO crystallizes as deposited. To monitor the crystal growth and surface quality, a Staib Instruments electron gun operating at 21 keV for reflection high-energy electron diffraction (RHEED) under a grazing angle of is used [see Fig. 1(a), inset]. Samples with BTO films of 10, 14, and 19 nm were prepared.
(a) scan of a 30 nm thick BaTiO3 film on a Ge(001) substrate with a 2 nm thick SrTiO3 buffer layer. The inset shows a RHEED pattern for a 30 nm thick BTO on STO/Ge, viewed along the (110) azimuth of BTO. (b) Out-of-plane PFM phase image of a 19 nm thick BTO film after electric poling. The light region is out-of-phase with the dark region indicating the opposite polarization direction. The white scale bar indicates 10 μm.
(a) scan of a 30 nm thick BaTiO3 film on a Ge(001) substrate with a 2 nm thick SrTiO3 buffer layer. The inset shows a RHEED pattern for a 30 nm thick BTO on STO/Ge, viewed along the (110) azimuth of BTO. (b) Out-of-plane PFM phase image of a 19 nm thick BTO film after electric poling. The light region is out-of-phase with the dark region indicating the opposite polarization direction. The white scale bar indicates 10 μm.
Immediately after deposition, all samples were transferred to an in-situ x-ray photoelectron spectroscopy analysis chamber equipped with a VG Scienta R3000 electron analyzer and a monochromatic Al Kα radiation (hν = 1486.6 eV) source with the x-rays incident at from the normal to check the stoichiometry of the grown films. To determine the out-of-plane lattice constant of the samples, ex-situ x-ray diffraction (XRD) measurements of the BTO films were carried out using a Rigaku Ultima IV diffractometer. Only highly stoichiometric films that showed good crystallinity were considered for subsequent SHG characterization.
Figure 1(a) shows a symmetric θ − 2θ scan of a 30 nm-thick BTO film grown on Ge, with a 2 nm-thick STO buffer layer. Only peaks from the Ge substrate and single orientation BTO were observed. High-resolution scans around the BTO(002) peak yielded an out-of-plane lattice constant of 4.03–4.06 Å, indicating the out-of-plane polarization of BTO films, which is consistent with the reciprocal space map on the 16 nm-thick BTO film.4 Rocking curve scans around the BTO(002) peak show a full-width at half-maximum of . Similar features were observed on the thinner films studied here.
To pole the BTO films, we used a Park XE70 atomic force microscope (AFM) with a commercially available Pt-coated silicon tip (AppNano ANSCM-PC). The AFM tip acts as a top electrode and Ge as a bottom electrode. 8–10 V was applied to the tip to flip the electric polarization of the BTO films. PFM measurements were carried out at 2–3 V with a frequency of 270 kHz at a scan rate of 4–8 μm/s.
To study the retention time of the reversely poled BTO films, a rectangular polarization pattern as shown in Fig. 1(b) was created. It consisted of an outer reversely poled rectangle (up-polarization) of 20 × 40 μm2 and an inner 8 × 12 μm2 rectangle that was reverse-poled second time, yielding down-polarization, like the surrounding as-grown film. The relative phase of the PFM image between the outer and the inner rectangle is , indicating the opposite direction of out-of-plane polarization (see Fig. S1 in the supplementary material for the PFM amplitude image).
For SHG microscopy, laser pulses (780 nm wavelength, 76 MHz repetition rate, and 150 fs duration) from a Ti:Sapphire laser were focused to a diameter of 2.2 μm at an angle of incidence of onto the BTO films. The film's bulk polarization P(z, t), its surface, and its buried interface all break local centrosymmetry and are thus all SHG sources.24 Since the bulk P is the subject of interest, we used s-polarized fundamental light and detected p-polarized SH (Second Harmonic) light through filters with a photo-multiplier tube. We found this polarization configuration to be relatively most sensitive to changes in P and least sensitive to the film's surface and interface. SHG raster scans (Fig. 2) were performed with a step size of 2 μm. Half of the pattern was scanned every 10 min and the entire pattern every 2 h to ensure that laser irradiation did not cumulatively alter the polarization patterns. SHG intensities were normalized to those from the as-grown part of the sample. Since the coherence length of the SHG process is less than a few hundreds of nanometers in BTO,21,22 while the absorption depth is ∼10 μm,23 the coherence length determines the probe depth of SHG. The SHG includes contribution from all three layers. To see the effect of the underlying structure, we have performed test measurements on the Ge substrate-only and the 2 nm-STO layer on Ge without BTO films. They show that its contribution to SHG is small and the same for all three samples. Therefore, BTO is the dominant source of the collected SHG intensity, which can be written as25
Here, B is the background SH field, A includes the geometric Fresnel factors and the nonlinear susceptibility of BTO, l is the thickness of the film, and is the out-of-plane polarization within the BTO film. The term “spot” denotes the area of the focused laser spot on the sample, and and denote the wave vectors at fundamental (ω) and SH (2ω) frequencies inside the BTO film. Neighboring +P(up) and −P(down) regions generate SHG fields with the same amplitudes but shifted in phase by π. Lateral integration accounts for the possibility that as the initially up-poled film relaxes, unresolved lateral distribution of +P and −P domains (∼20 nm on 19 nm film26) could develop within the laser spot,15,28 reducing the effective polarization and the SHG signal associated with the center location (x0, y0) due to destructive interference. Upon integrating over l, Eq. (1) can be written more simply as
where is A multiplied by an additional factor resulting from the integration and Peff (t) is the time-varying effective polarization at each spot center location of the BTO film. Vertically separated domains with polarizations and of opposite sign and similar magnitude also generate SH fields that interfere destructively. Since BTO dominates the SHG signal, we anticipate that , except when .
Time evolution of SHG micrographs on (a)10, (b)14, and (c)19 nm BTO samples. Numbers on the top and inside of micrographs indicate time passed after poling. The scale bar is 10 μm.
Time evolution of SHG micrographs on (a)10, (b)14, and (c)19 nm BTO samples. Numbers on the top and inside of micrographs indicate time passed after poling. The scale bar is 10 μm.
We monitored SHG intensity from all samples for several weeks. It took 2 h to finish poling, measure PFM, and transfer the sample into the SHG setup. Thus, the first SHG scans were recorded at t = 2 h. On all samples, we observed the same poling pattern as PFM. The singly reversed polarization region consistently yielded higher SHG intensities than the as-grown part of the sample, indicating that the SH field from the reverse-poled BTO film interfered constructively with the background SH field B. This enhanced SHG was clearly distinguishable at t = 2 h on all samples (see Fig. 2, left column). SHG from the poled regions was non-uniform: bright structures ranging from 2 to 10 in diameter were observed. One possibility is that this structure reflects the development of transversely structured domains. To test this possibility, we carried out time-dependent PFM imaging with a resolution of nm (see supplementary material for the results). The results showed a transversely uniform PFM phase throughout the poled region for up to 24 h after poling (Fig. S2), ruling out the presence of polarization-reversed domains larger than nm. Thus, we tentatively attribute the micron-scale SHG structure to irregular poling due to tip damage or to surface particles. Nevertheless, this non-uniformity did not detract from observations of relaxation of the single-reversed region, which is the main subject of interest. The SHG intensity from the reversely polarized domain remained visible on the 10(14) nm sample after 532(365) h [see Figs. 2(a) and 2(b)]. However, it disappeared after 100 h on the 19 nm thick BTO film [see Fig. 2(c)].
Figures 3(a)–3(c) were obtained from Fig. 2 using Eq. (2). We normalized Peff (t) to the value of P = 1 in the fully reversely polarized (up) domain and to P = –1 in the original state (down). To extract Peff (t) from SHG data, we needed a SHG intensity at t = 0. This value was estimated by a stretched-exponential decay, describing a random-walk process.27 This model has been shown to fit short-time (up to a few hours) polarization kinetics. Short time data in Fig. 2 were indeed described accurately, as in the previous literature,15,28,29 but not long time saturation, because this model overlooked different decay rates of different unit cells. For comparison, the black data points show the constant P0(t), extracted from the as-grown region. The statistical uncertainties in Peff (t)/P0, which average less than 7% of Peff (t)/P0 [< 4% in Fig. 3(b)], originate mainly from laser fluctuations and small surface inhomogeneities due to adsorbed impurities. These uncertainties were less than 70% of its reported change due to polarization dynamics in Fig. 3(a), less than 20% in (b), and less than 30% in (c). After poling, Peff (t) on the reversely poled region (the red data points) of each film decreased monotonically by 10–25% and most rapidly on the thickest film.
Change in Peff (t) on the singly poled area (red dots), compared to P0(t), the effective polarization on the as-grown part of the sample (black dots), for BTO film thicknesses of (a) 10, (b) 14, and (c) 19 nm. (d) plots Eq. (2) as a function of time-dependent polarization on the BTO film. The red circle indicates the SHG intensity when the BTO film is fully reversely polarized (Pup) and the blue circle in the original state (Pdown).
Change in Peff (t) on the singly poled area (red dots), compared to P0(t), the effective polarization on the as-grown part of the sample (black dots), for BTO film thicknesses of (a) 10, (b) 14, and (c) 19 nm. (d) plots Eq. (2) as a function of time-dependent polarization on the BTO film. The red circle indicates the SHG intensity when the BTO film is fully reversely polarized (Pup) and the blue circle in the original state (Pdown).
The solid black curve in Fig. 3(d) shows a relation between SHG intensity and the effective polarization Peff (t), calculated from Eq. (2). Immediately after poling, Peff (t) on the reversely poled region has a net positive value, which is confirmed using PFM. If the polarization saturated to a negative value, SHG intensity, following the parabolic curve in Fig. 3(d), would have previously passed through zero when the destructive interference condition was satisfied. However, no such zero, or even minimum, in SHG intensity was observed. Instead, SHG intensity decreased monotonically on all samples. We can thus infer that the reversed domain did not return to the original state but remained in a net positive polarization state. In particular, the observed relaxation of SHG intensity from the 19 nm sample to the value of the as-grown sample must then be interpreted as relaxation of polarization NOT to the original “down” state [blue circle in Fig. 3(d)] but rather to a net “up” state [green arrow in Fig. 3(d)] that coincidentally yielded the same SHG intensity [horizontal dashed line in Fig. 3(d)] as the original state, which is consistent with the time-dependent PFM measurement (see Fig. S1 in the supplementary material).
To gain insight into polarization evolution on a long time scale, we used the Landau-Khalatnikov (LK) theory.30 In the absence of the external field, the LK equation reduces to
where F is the free energy density of the ferroelectric, P the polarization per unit volume, and ρ the resistivity which ranges from 107 to 1010 Ω m in BTO.31 Free energy below the Curie temperature can be written as a series expansion in the Landau-Devonshire form.32 In general, interactions of each dipole with neighbors in all three dimensions contribute to free energy and can lead to both lateral and vertical restructuring of P(x, y, z). However, we have no evidence of lateral domain structures larger than our PFM resolution (∼50 nm), even though they would form energetically more favorable (neutral) domain walls than a vertically restructured P. Moreover, we lack quantitative information on defects (e.g., vacancies, dislocations, and grain boundaries) needed to model the seeding of restructuring P. Thus, here, as a first step, we present a simplified LK model based on nearest-neighbor interactions of each dipole in the vertical direction only, in order to exhibit both its strengths and weaknesses in explaining observed polarization dynamics. Accordingly, we represent a vertical cross-section as a one-dimensional ferroelectric chain comprised of N cells (layers). In this case
Here, the polarization is treated as a scalar quantity with a positive (up polarization) or negative (down polarization) sign. The expansion coefficients can be extracted from the experimental data. The value of α is , where is the dielectric constant for a BTO thin film. β is ,33 where Ps is the homogeneous spontaneous polarization (0.20 C/m2).34 As there are no experimental data available to estimate the coupling constant k, representing the coupling effect between the neighboring layers (cells), it was left as a free parameter for analyzing the data. Substituting Eq. (4) into Eq. (3) and introducing generalized time t = ρ τ
In this 1D model, every cell, except boundaries, was initially set to 0.20 C/m2, positive Ps. For the boundary conditions, the first cell in the chain of N had a constant polarization of zero, indicating clamping to the substrate, and the last cell to –0.10 C/m2, representing the titanium atom at the TiO2 surface being shifted downwards from the high symmetry position.35,36 Excluding the boundary cells, integration of the system of differential Eqs. (5) using 4th order Runge-Kutta returns polarization of each cell as a function of time.
Figure 4(a) shows the relaxation time (TR; time to reach a saturated polarization), as a function of chain length (N) for a coupling constant . TR first increases with N, reaching a peak at critical thickness Nc. Below Nc, the coupling allows the boundary conditions to dominate [see Fig. 4(b), upper panel], and it takes longer to reach saturation for thicker films. Above Nc, TR decreases with increasing N and then approaches a constant value. Experimentally, the relaxation time decreases with the thickness, suggesting that we are above the critical thickness. Nc monotonically increases with k, as shown in the inset of (a). Polarization evolution on each cell of the 1D chain is shown in Fig. 4(c) for N = 175. Polarization on the 2nd and the (N–2)th cells, closest cells to the boundaries, decays fast to saturation, approximately to the boundary values within 100 s after poling. However, cells in the middle relax slowly and after a certain delay. For example, for i = 88, the cell maintained the reversed polarization for 50 s before decay and took about 800 s to reach the saturation. Therefore, we conclude that the rate of polarization decay is controlled mainly by the bulk of the film. We calculated normalized Peff at each time point for four different chain lengths above the critical thickness [see Fig. 4(d)]. Polarization on every cell, except boundaries, was integrated to get Peff. In Fig. 4(d), the shorter chain, analogous to the thinner film in the experiment, takes longer time to reach saturation and is qualitatively consistent with Fig. 3. However, relaxation time estimated in the model is much shorter than what we observed in Fig. 3. For more accurate modeling and unit conversion of the chain length (N) to length dimension, quantitative information on 3D polarization dynamics, defects, boundary conditions, and the coupling parameter k will be needed.
(a) Relaxation time vs. length of the 1D chain when the coupling constant k = 1011. The inset shows the shift in critical thickness Nc. (b) shows polarization distribution across the chain length when saturated. The upper panel of (b) is N = 125, which is below Nc, and the lower panel is above Nc, N = 175. Polarization evolution on the selected cells of the 1D chain is shown in (c) up to 1500 s. (d) is time-dependent Peff for N = 145, 155, 165, and 175.
(a) Relaxation time vs. length of the 1D chain when the coupling constant k = 1011. The inset shows the shift in critical thickness Nc. (b) shows polarization distribution across the chain length when saturated. The upper panel of (b) is N = 125, which is below Nc, and the lower panel is above Nc, N = 175. Polarization evolution on the selected cells of the 1D chain is shown in (c) up to 1500 s. (d) is time-dependent Peff for N = 145, 155, 165, and 175.
In conclusion, using optical SHG microscopy, we studied polarization retention kinetics of BaTiO3 thin films integrated on Ge (001) by MBE. We find that thinner films have longer retention times, which is consistent with a 1D Landau-Khalatnikov model. This suggests that the critical thickness, when the behavior of the film is no longer dominated by the boundaries, is below 10 nm. While for films with a thickness of 19 nm, the saturation value reached 0.75 P after a day, the films with a thickness of 10 nm took a month to reach 0.9 P. In addition, above the coercive thickness, polarization P is saturated to a positive value which agreed with the experimental data. Annealing or chemical doping could affect the retention time by controlling the density of impurities or induced strain on the film, respectively.
See supplementary material for PFM phase and amplitude images after poling and time-dependent high resolution PFM phase images.
This research was supported by the Robert Welch Foundation (F-1038) and by the Air Force Office of Scientific Research under Grant Nos. FA9550-14-1-0090 and FA9550-12-10494. B.H. acknowledges the Moncrief Undergraduate Summer Internship for making his stay at UT Austin possible. The poling and PFM work (L.Z. and K.L.) was supported by the NSF Division of Materials Research Award No. 1707372.