Supporting Information: Controlling mobility in. perovskite oxides by ferroelectric modulation of. atomic-scale interface structure

Transcription

1 Supporting Information: Controlling mobility in perovskite oxides by ferroelectric modulation of atomic-scale interface structure Andrei Malashevich,, Matthew S. J. Marshall,, Cristina Visani,, Ankit S. Disa, Haichao Xu,,, Frederick J. Walker,, Charles H. Ahn,,,, and Sohrab Ismail-Beigi,,,, Center for Research on Interface Structures and Phenomena (CRISP), Yale University, New Haven, Connecticut 06520, USA Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA Advanced Materials Laboratory, Fudan University, Shanghai , People s Republic of China Department of Mechanical Engineering and Materials Science, Yale University, New Haven, Connecticut 06520, USA Department of Physics, Yale University, New Haven, Connecticut 06520, USA sohrab.ismail-beigi@yale.edu Computational details The density functional theory (DFT) calculations are performed with the Perdew-Zunger parameterization 1 of the local density approximation (LDA) for exchange and correlation as implemented in the Quantum ESPRESSO software. 2 The energy cutoff for the planewave basis is chosen to be 35 Ry with the corresponding charge density cutoff of 280 Ry. 1

2 The Brillouin zone integrations are performed on a regular 6 6 mesh of k points using the Gaussian smearing technique with a smearing width of 5 mry. The atoms are described by Vanderbilt ultrasoft pseudopotentials. 3 For Ti 3d orbitals we use the LDA+U method 4 with the onsite Coulomb repulsion U Ti = 8 ev to increase the band gap of PbTiO 3 and better align the Ti and Ni d states. Since we use periodic boundary conditions, we separate the slabs by 10 Å from their (001) images. The dipole correction technique 5 is used to remove any spurious electric field in the vacuum region. The polar state of ferroelectric PbTiO 3 is enforced by fixing the atomic positions in a 1 u.c.-thick layer of PbTiO 3 near the Pt side of the slab to theoretical values of bulk PbTiO 3 strained to either LaAlO 3 or SrTiO 3. All structural relaxations are performed until the Cartesian components of forces on atoms are below the threshold of 30 mev/å. The Wannier analyses 6,7 of the electronic band structures are performed with the Wannier90 code. 8 As described in the main text, the majority of the calculations are carried out with Pt atoms placed on top of PbTiO 3 to simulate the top electrode and provide a reservoir of free charge carriers. There is an alternative approach in which no metal electrode is placed on top of a ferroelectric, but rather a bound charge is introduced on its surface. 9 Specifically, one can substitute a cation on the ferroelectric surface with a virtual atom having a predefined formal valence. By performing a calculation with the dipole correction, 5 one guarantees to have zero electric field in the vacuum region and, by virtue of the polarization surface theorem, 10 constant polarization inside the ferroelectric. The advantages of this method are the following: 1) There is no need to place additional metal atoms on top of the ferroelectric; 2) The transport calculations (see below) are simplified, as there is no need to disentangle metallic states of Ni and the states coming from the top electrode at the Fermi level; 3) One can perform full structural relaxation of the interface without the need to fix any part of the ferroelectric by hand to a proper polar state; 4) By changing the value of the bound charge of the virtual atom, one can set the polarization inside the ferroelectric to an arbitrary (within reason) value. For this method to work properly, however, one needs to make sure 2

3 that the surface of the ferroelectric remains insulating, which sometimes may not be trivial to achieve. We use this method for some of our interfaces (one example is discussed in the next section) and find good agreement with the calculations with Pt electrodes (in terms of both interfacial crystal structure and conductivity). It is worth mentioning that some types of interfaces are particularly difficult to model. Specifically, the LaO/TiO 2 interface in the depletion state is not trivial because the combined effect of the ferroelectric polar field and the polar field from the LaO-terminated LaNiO 3 film results in significant accumulation of electron charge density at the interface. This raises the Fermi level and, due to a (well known) DFT band gap underestimation, renders the entire ferroelectric layer metallic. To avoid this unphysical behavior we model a very thick layer of PbTiO 3 of at least 8 unit cells. We also find that in this case the calculations with bound charges introduced by virtual atoms are more stable compared to the calculations with a metallic electrode on top of the ferroelectric. Choice of LDA and U parameter As stated above, we have chosen to use LDA with U = 0 for LaNiO 3 based on prior literature showing this specific choice is best for describing bulk LaNiO 3. For example, when comparing LDA, GGA, LDA+U, GGA+U and hybrid functionals, LDA with U = 0 provides the best description of lattice modes and electronic properties for LaNiO Our own calculations agree with that work: a reasonable value of U 4 ev for Ni in LDA+Uor GGA+U for LaNiO 3 leads to significant problems since the material becomes magnetic, half-metallic or even insulating and charged ordered for large U, none of which agrees with the experimental facts that LaNiO 3 is metallic, paramagnetic and has no charge order at all temperatures. Separately, using LDA+U =0 is known to provide a correct quantitative description of the interfacial structure (at picometer resolution) and orbital properties of LaNiO 3 -based heterostructures Furthermore, prior work 16 has shown that modest values of U 3 ev do 3

4 not lead to any significant changes in the properties of LNO at an interface. In addition, regarding the interfacial conductivity change and using U on Ni, modest values of U on Ni do not change the theoretical conclusions. 16 Of course, larger values of U on Ni will change the results, but given the failure of LDA+U for LaNiO 3 for large U, it is unclear whether one should trust such computations. Hence, we have decided to use U = 0 for Ni in LaNiO 3. To improve beyond the LDA, most likely one must employ an electronic structure method beyond band theory to properly include correlation effects in LaNiO 3. Regarding U on Ti for PbTiO 3, we are not using LDA+U to model localized electronic correlations since Ti 4+ is already fully ionized. Instead, we use it as a simple lever to increase the band gap of PbTiO 3 away from the too small LDA prediction and closer to experiment in order to improve the band alignment across the interface. Prior work has shown that values between 4 and 8 ev all act similarly. 14,15 Transport calculations Figure S1: (Color online) Electronic band structure of the 4 u.c. LaNiO 3 /PbTiO 3 interface with LaO/TiO 2 interface in the accumulation (left panel) and depletion (right panel) states. The LaAlO 3 strain is imposed. The bands are projected (red) onto Ni e g maximally localized Wannier functions. The Fermi level is at zero ev. The remaining bands at the Fermi level come from the Pt states on the surface of PbTiO 3. In this work, we perform conductivity calculations within the relaxation time approximation by computing the band-averaged products of group velocities multiplied by the scattering time τ. We assume a constant τ that is independent of the polarization state: 4

5 when computing the conductivity on/off ratio, the precise value of τ is irrelevant as it drops out from the ratio. Admittedly, a constant scattering time approximation is crude but it (i) allows to make qualitative predictions of the transport properties of the interface in a computationally feasible manner, and (ii) permits clear separation of changes of band properties (our primary focus) from scattering mechanisms. In principle, an improved approach would involve calculating τ for some scattering mechanism, e.g., electron-phonon scattering. In practice, this approach is not easy to implement or interpret. In the case of phonon scattering, computing the electron-phonon interactions for such an interfacial system ( 100 atoms) from first principles is extremely expensive computationally given present day computer time budgets and capabilities. Separately, phonons are not expected to play a major role in the conductivity modulation in our system because the measured on/off ratio is greater at lower temperatures (see Figure 3 of main text) where phonons are less operative. Our low temperature data (Figure S10 below) indicates that the dominant scattering mechanism in our devices stems from electron-electron interactions. While one could attempt to compute τ due to electron-electron interactions, the interpretation is difficult. Electron-electron interactions lead to scattering but also to band narrowing (especially in electronically correlated transition metal oxides such as nickelates). One simple choice would be to absorb this band narrowing (i.e., increase of effective mass) empirically into τ. However, one could also decide to narrow the bands used in the Fermi-surface averaging while including in τ only the pure electron-electron scattering component (which must be separated out and defined in some way). In brief, attempting to compute τ accurately and sensibly from first principles is a challenge for the future. From the technical viewpoint, the transport calculations are performed in two steps. First, we calculate the electronic band structure near the Fermi level and disentangle the metallic Ni e g states from other spurious states associated with the Pt electrode. To disentangle the states, we calculate the maximally localized Wannier functions corresponding to the Ni 3d, O 2p, and Pt 5d orbitals and project the electronic band structure on these 5

6 Wannier functions. 6 8 The result of this procedure for the LaO/TiO 2 interface with LaAlO 3 strain is shown in Fig. S1. One can verify that the Wannier projection is reasonable near the Fermi level by comparing it to the projection on Löwdin atomic orbitals. As an example, we show the projection for the accumulation state in Fig. S2. Figure S3 shows an example of an analogous calculation when the virtual-atom method is used. The figure shows that in this case, there are no spurious states at the Fermi level, and the electronic structure due to Ni e g states is the same as in the Pt electrode case shown in Fig. S1 (top panel). Second, in the basis of the above Wannier functions, we construct a tight-binding model and calculate the conductivity semiclassically within the constant scattering time approximation via 17 σ αβ (n, k) = e 2 τv α (n, k)v β (n, k), (1) where τ is the scattering time, n is the band label, k is the wave vector in the Brillouin zone, α and β are Cartesian directions, and the group velocity for bands ɛ n,k is defined as v α (n, k) = 1 h ɛ n,k. (2) k α From these equations one can determine conductivity at temperature T as σ αβ (T, E F ) = 1 1 Ω N [ σ αβ (n, k) n,k ] f(t, ɛ), (3) ɛ ɛ=ef where Ω is the unit cell volume, N is the number of k points in the Brillouin zone, and f(t, ɛ) is the Fermi-Dirac distribution. An additional advantage of using a Wannier-based tightbinding description of the electronic structure for the purposes of conductivity calculations (other than the possibility to disentangle the bands of interest) is efficiency. In particular, in our calculations, we use a k-point mesh to ensure proper convergence with respect to the Brillouin zone sampling. The results of conductivity calculations are reported for T = 300 K in the main text. 6

7 Figure S2: (Color online) Electronic band structure of the 4 u.c. LaNiO 3 /PbTiO 3 interface with LaO/TiO 2 interface in the accumulation state projected onto the Ni d, Pt d, and Pb s Löwdin atomic orbitals. The LaAlO 3 strain is imposed. The Fermi level is at zero ev. Figure S3: (Color online) Electronic band structure of the 4 u.c. LaNiO 3 /PbTiO 3 interface with LaO/TiO 2 interface in the accumulation state. The SrTiO 3 strain is imposed. The bands are projected (red) onto Ni e g maximally localized Wannier functions. The Fermi level is at zero ev. Ni e g are the only states at the Fermi level when virtual atoms are used instead of metallic Pt. 7

8 Role of octahedral rotations Figure S4: (Color online) Simulated structure of the LaNiO 3 /PbTiO 3 (001) interface with LaO-terminated LaNiO 3 in accumulation and depletion states. The simulation is performed in the 1 1 supercell. The strain corresponds to the SrTiO 3 substrate. The differences in vertical coordinates of Ni and O z, as well as Ni O Ni bond angles θ in the interfacial NiO 2 layer, are indicated. The majority of the calculations in this work are performed on 1 1 supercells, which do not allow NiO 6 octahedral rotations present in LaNiO 3. To understand the role of octahedral rotations, we perform several calculations of the LNO/PTO interfaces using c(2 2) supercells. Compared to 1 1 supercells, the corresponding c(2 2) supercells have twice 8

9 Figure S5: (Color online) Simulated structure of the LaNiO 3 /PbTiO 3 (001) interface with LaO-terminated LaNiO 3 in accumulation and depletion states. The simulation is performed in a c(2 2) supercell. The strain corresponds to the SrTiO 3 substrate. The average differences in vertical coordinates of Ni and O z, as well as Ni O Ni bond angles θ 1 and θ 2 in the interfacial NiO 2 layer, are indicated. 9

10 as many atoms and may become computationally expensive, especially in problematic cases such as the depletion state of LaO/TiO 2 interface described in the Computational Details section, for which one needs to model a thick PbTiO 3 layer. We also notice that in the c(2 2) simulations of the depletion state of the LaO/TiO 2 interface, the PbTiO 3 layers may become prone to instabilities associated with rotations of TiO 6 octahedra. We believe that this unusual behavior is due to the thinness of the PbTiO 3 film and should disappear for thicker simulated films. However, given that the computations are already quite expensive and difficult, increasing the PbTiO 3 thickness is not possible at this time. To prevent these rotations in our present system, we simply freeze out in-plane motions of atoms in PbTiO 3 and allow only polar atomic displacements during structural relaxation. We emphasize that this should not have any significant effect on the behavior of LNO, but future computations on even larger systems (with more powerful computers) can clarify this issue. Figures S4 and S5 show the structure of the LaO-terminated LNO/PTO interface in the accumulation and depletion states with the entire system strained to the (calculated) SrTiO 3 lattice constant. One can see that structural distortions at the interface are more complicated in the c(2 2) case and can be described as a combination of NiO 6 octahedral rotations and polar displacements discussed in the main text. In the c(2 2) case, the polar distortions in the depletion state are larger than in the accumulation state, as is the case in the 1 1 calculation. In the accumulation state, the average oxygen-cation distance at the NiO 2 layers is close to zero, while in the depletion state the oxygens are slightly above Ni, consistent with polar displacements shown in Fig. 1 of the main text. We have also computed the semiclassical conductivities for the two structures shown in Fig. S5. We find for accumulation and depletion σ acc /τ = Ω 1 m 1 s 1 and σ dpl /τ = Ω 1 m 1 s 1, respectively. Due to NiO 6 octahedral rotations, the conductivities go down compared to the corresponding 1 1 calculations. Mainly, the depletion state is affected. The conductivity on/off ratio increases slightly, from 1.25 in 1 1 calculations to 1.4 in c(2 2) calculations. 10

11 Substrate modeling In our work, the effect of the substrate is indirect: we compute the lattice parameters of bulk LaAlO 3 and SrTiO 3 (STO) and use these lattice constants for the supercells describing LNO/PTO interface. The substrates are not included explicitly in the calculations. To be more precise, the bottom LNO interface is modeled as a LaO-terminated surface exposed to a vacuum region. Therefore, the only effect from the substrate considered in this work is strain. The inclusion of the substrate explicitly in the calculation is computationally demanding. Since we are interested in the LNO/PTO interface in two different polar states of PTO, it is expected that the substrate/lno interface will not play a major role on the ferroelectric switching effect of conductivity in LNO film. In other words, the substrate/lno interface remains the same regardless of the state of the ferroelectric layer. The ferroelectric switching affects only the top portion of the LNO film of 1-2 unit cell thickness. However, the conductivity of the bottom portion of the LNO film, while not affected by the ferroelectric, could in principle be affected by the substrate. This means that both σ acc and σ dpl may be affected by the substrate (in the same way), and the σ acc /σ dpl ratio may depend on the explicit inclusion of the substrate in the calculations. For this reason, we perform an additional calculation of the LaO-terminated LNO/PTO interface with the STO substrate included explicitly. We find in this case σ acc /τ = Ω 1 m 1 s 1 and σ dpl /τ = Ω 1 m 1 s 1 (compared to σ acc /τ = Ω 1 m 1 s 1 and σ dpl /τ = Ω 1 m 1 s 1 computed without the explicit substrate). Thus, we conclude that explicit inclusion of the substrate in calculations is not crucial. Experimental details Heterostructures of epitaxial PbZr 0.2 Ti 0.8 O 3 (PZT) on LaNiO 3 are experimentally realized using a combination of molecular beam epitaxy (MBE) and off-axis RF magnetron sputtering. First, 3.5 or 4 unit cells of LaNiO 3 are grown on TiO 2 -terminated SrTiO 3 (001) using 11

12 (a) Intensity (a.u.) (b) Time(s) Figure S6: (Color online) (a) RHEED image along the [100] direction taken in situ following growth of 3.5 u.c. LaNiO 3 on TiO 2 -terminated SrTiO 3 (001). Sharp RHEED spots indicate an atomically abrupt surface. (b) RHEED oscillations of the 3 uc LaNiO 3 layer and the subsequent deposition of 1 ML of LaO, after a wait time of 3 mins. Each oscillation signals the growth of 1 uc of LaNiO 3 (or 1 ML of LaO). The growth rate is 1 uc/min, in agreement with the QCM calibration. 12

13 1 µm 4 Height (nm) 3 2 ~3.5-4 Å Distance (µm) Figure S7: (Color online) (Top) Ex situ atomic force micrograph of 3.5 u.c. LaNiO 3 grown on SrTiO 3 (001) using oxygen plasma assisted MBE. (Bottom) Line cut perpendicular to the step edges showing approximately unit cell step height, indicating a single termination and an atomically abrupt interface. 13

14 oxygen plasma-assisted molecular beam epitaxy. The substrate temperature is 590 C during deposition of LaNiO 3 at an oxygen partial pressure of approximately Torr. In order to grow a 3.5 unit cell thick film that exhibits a single termination of LaO, 3 full unit cells are co-deposited on TiO 2 -terminated SrTiO 3 (001). This produces an NiO 2 -terminated surface of LaNiO 3. The Ni shutter is then closed and a single unit cell of LaO is deposited. More details on the film growth can be found in Refs. 12,18 The film thickness is precisely controlled by measuring the deposition rate with a quartz crystal monitor (QCM), as well as by measuring the oscillations of the reflection high-energy electron diffraction (RHEED) specular spot intensity during growth (Fig. S6(b)). The post-growth RHEED image shown in Fig. S6(a) exhibits sharp spots, indicative of an atomically abrupt surface. Similarly, the atomic force micrograph (AFM) taken following growth of a typical 3.5 u.c. thick LaO-terminated LaNiO 3 film exhibits well defined atomic terraces, indicating an atomically abrupt surface termination (Fig. S7). The well-defined surface enables atomically defined interfaces with PZT to be formed, which is crucial when studying sensitive interfacial phenomena. Following the deposition of LaNiO 3, a nm thick layer of PZT is grown by using off-axis RF magnetron sputtering at 520 C in 225 mtorr of Ar:O 2 (1:3 ratio), as described elsewhere. 16 Gold contacts ( nm thick) are used for the top gate and channel electrodes. Device Characteristics Hall measurements and carrier density We study the electric transport behavior of the PZT/LNO ferroelectric devices to extract and separate the contribution of carrier density changes and mobility changes that combine to describe the electrical conductivity modulation upon ferroelectric switching. The room temperature Hall resistance versus magnetic field and the sheet resistance are measured on devices with both NiO 2 /PbO and LaO/TiO 2 interfacial terminations for both depletion and accumulation states and are shown in Fig. S8(a)-(d). We note that these data are from a 14

15 Figure S8: (Color online) Caption: (a)-(b) Room temperature Hall resistance (R xy ) as a function of magnetic field H measured on a PZT/LNO device with a NiO 2 /PbO interfacial termination, in depletion (a) and accumulation (b) states, respectively. (c)-(d) Same as (a)- (b) but from a PZT/LNO device with a LaO/TiO 2 interfacial termination. The thick red lines are linear fits for the R xy v.s. H data. The slope values from the linear fits and the resulting sheet resistances are indicated in each panel. (e) Sheet carrier density and carrier mobility of the devices in depletion and accumulation states for both interfaces at room temperature. 15

16 separate batch of devices than those shown in the main text, and thus there are inevitable differences in the measured conductivities. The device with LaO termination shows a somewhat larger change in sheet resistance than the one with NiO 2 termination. From the sheet resistance and the linear slope of Hall resistance versus magnetic field, we extract the sheet carrier density and mobility, as summarized in Fig. S8(e). For both terminations, the ferroelectric switching barely changes the sheet carrier density (from cm 2 in depletion to cm 2 in accumulation state for the NiO 2 terminated device, and from cm 2 to cm 2 for LaO terminated device, respectively). On the other hand, the carrier mobility changes significantly from depletion to accumulation state (from cm 2 /Vs to cm 2 /Vs for the NiO 2 terminated device, and from cm 2 /Vs to cm 2 /Vs for the LaO terminated device, respectively). The carrier mobility accounts for almost all the change in resistance, indicating that the changes of bond angles and bond lengths at the interface play the major role in the ferroelectric switching of electric conductivity for these PZT/LNO interfaces. Hysteresis measurements and built-in fields Ferroelectric hysteresis loops were measured using a ferroelectric test module (TF Analyzer 2000) at a frequency of 5 Hz. We observe large values of the remanent polarization ( 55 µc/cm 2 ) and similar switching fields ( 138 kv/cm) for both LaO and NiO 2 interface terminations, as shown in Fig. S9(a) and (b), respectively. According to the structural model depicted in Fig. 1 (main text), a polar field builds up at the LNO/PZT interface due to the alternation of charged atomic planes in the LaNiO 3 layer. In principle, this polar field is expected to either oppose or favor the switching of the polarization in the PZT layer resulting in a shift of the ferroelectric hysteresis towards negative voltage for LaO termination and positive voltage for NiO 2 termination of the LaNiO 3 layer. 19 However, we estimate the built-in field arising from the polar film/ferroelectric interface to be relatively small ( kv/cm, see below). In our measurements, no shift is visible in the hysteresis loop for the 16

17 Figure S9: (Color online) Polarization vs. electric field hysteresis loops measured for PZT-LNO ferroelectric field effect devices fabricated on SrTiO 3 substrates with (a) 3.5 uc (LaO/TiO 2 interface) and (b) 4 uc (NiO 2 /PbO interface) LaNiO 3 channels. NiO 2 termination, while a small shift of +2.5 kv/cm is observed for the LaO-terminated sample. We believe that this shift is primarily due to the different nature of the electrodes at each side of the PZT capacitor (LaNiO 3 and gold) and the fact that the resistance of the LaNiO 3 electrode changes significantly according to the PZT polarization. A number of mechanisms other than the interface polarization have been shown to shift the hysteresis loop, including fatigue, domain-wall pinning effects, and point defects which may play a role in shaping our hysteresis loops. A rough estimate of the expected magnitude of the built-in field at the LNO/PZT interface can be obtained by using an electrostatic model of the LaNiO 3 layers along the (001) direction as alternately charged conducting sheets with areal charge density σ = ±e/a where A is the area of the interfacial unit cell. If unscreened, such a charge distribution creates a diverging potential, but it is compensated by a surface charge of magnitude σ s = ±e/(2a). This charge is spread over the screening length (λ) of the material and leads to a surface po- 17

18 tential V s = σ s λ/(4ɛ), where ɛ is the dielectric constant. Using a typical value for perovskite oxides of ɛ = 25 and assuming a screening length of 2-3 unit cells (see Ref. 13 ), this model provides an approximate upper bound for the built-in voltage of V b < V. The sign of this voltage depends upon charge of the terminating surface, as expected. Applied to the device, the voltage would correspond to a relatively small shift in the PZT/LNO hysteresis loop of kv/cm. Low-temperature transport 10-4 Sheet conductance (S) p= ln(t) Figure S10: (Color online) Low-temperature transport measurements of the 3.5 uc (LaO/TiO 2 interface) LNO/PZT device in the accumulation state. The orange line shows the result of Eq. 4 with p = 2, indicating the dominance of electron-electron scattering. To gain insight into the dominant scattering mechanisms and the role of disorder in our LNO/PZT devices, we consider their low-temperature transport properties. Metallic LaNiO 3 thin films and heterostructures have previously been shown to display weak localization when confined to 2D. 12,23 The behavior is characterized by a minimum in the resistivity, below which the sheet conductance follows a universal logarithmic temperature dependence, ( ) σ(t ) = σ 0 + pe2 T πh ln. (4) T 0 The parameter p, derives from the temperature dependence of the inelastic scattering rate and depends on the predominant scattering mechanism in the material. 18

19 Figure S10 shows the low temperature sheet conductance as function of ln(t ) for the LNO/PZT device with a 3.5 uc LaNiO 3 channel (LaO/TiO 2 interface) in accumulation mode. We find that Eq. (4) describes the data well with p = 2. This value of p corresponds to inelastic scattering dominated by electron-electron interactions in the clean limit 24 (electronphonon scattering results in p = 3), which agrees with the previous reports on LaNiO 3 thin films and heterostructures. It should also be noted that these localization effects will not be prevalent near room temperature. Thus, as emphasized in the main text, we consider the effects of strong electronic correlations to be the primary source of discrepancy between the observed and theoretically predicted on/off ratios in our devices. References (1) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, (2) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Dal Corso, A.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M. J. Phys.: Condens. Matter 2009, 21, (19pp). (3) Vanderbilt, D. Phys. Rev. B 1990, 41, (4) Cococcioni, M.; de Gironcoli, S. Phys. Rev. B 2005, 71, (5) Bengtsson, L. Phys. Rev. B 1999, 59, (6) Marzari, N.; Vanderbilt, D. Phys. Rev. B 1997, 56, (7) Souza, I.; Marzari, N.; Vanderbilt, D. Phys. Rev. B 2001, 65,

20 (8) Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Computer Physics Communications 2008, 178, (9) Stengel, M.; Aguado-Puente, P.; Spaldin, N. A.; Junquera, J. Phys. Rev. B 2011, 83, (10) Vanderbilt, D.; King-Smith, R. D. Phys. Rev. B 1993, 48, (11) Gou, G.; Grinberg, I.; Rappe, A. M.; Rondinelli, J. M. Phys. Rev. B 2011, 84, (12) Kumah, D. P.; Disa, A. S.; Ngai, J. H.; Chen, H.; Malashevich, A.; Reiner, J. W.; Ismail-Beigi, S.; Walker, F. J.; Ahn, C. H. Advanced Materials 2014, 26, (13) Kumah, D. P.; Malashevich, A.; Disa, A. S.; Arena, D. A.; Walker, F. J.; Ismail-Beigi, S.; Ahn, C. H. Phys. Rev. Applied 2014, 2, (14) Chen, H.; Kumah, D. P.; Disa, A. S.; Walker, F. J.; Ahn, C. H.; Ismail-Beigi, S. Phys. Rev. Lett. 2013, 110, (15) Disa, A. S.; Kumah, D. P.; Malashevich, A.; Chen, H.; Arena, D. A.; Specht, E. D.; Ismail-Beigi, S.; Walker, F.; Ahn, C. H. Physical Review Letters 2015, 114, (16) Marshall, M. S. J.; Malashevich, A.; Disa, A. S.; Han, M.-G.; Chen, H.; Zhu, Y.; Ismail-Beigi, S.; Walker, F. J.; Ahn, C. H. Phys. Rev. Applied 2014, 2, (17) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Harcourt College Publishers: New York, (18) Disa, A. S.; Kumah, D. P.; Ngai, J. H.; Specht, E. D.; Arena, D. A.; Walker, F. J.; Ahn, C. H. APL Materials 2013, 1, (19) Lichtensteiger, C.; Weymann, C.; Fernandez-Pena, S.; Paruch, P.; Triscone, J.-M. New Journal of Physics 2016, 18,

21 (20) Koval, V.; Viola, G.; Tan, Y. In Ferroelectric Materials - Synthesis and Characterization; Pelaiz-Barranco, D. A., Ed.; InTech, (21) Hong, S.; Nakhmanson, S. M.; Fong, D. D. Reports on Progress in Physics 2016, 79, (22) Pike, G. E.; Warren, W. L.; Dimos, D.; Tuttle, B. A.; Ramesh, R.; Lee, J.; Keramidas, V. G.; Evans, J. T. Applied Physics Letters 1995, 66, (23) Scherwitzl, R.; Gariglio, S.; Gabay, M.; Zubko, P.; Gibert, M.; Triscone, J.-M. Phys. Rev. Lett. 2011, 106, (24) Abrahams, E.; Anderson, P.; Lee, P.; Ramakrishnan, T. Phys. Rev. B 1981, 24,