Thermooptical evidence of carrier-stabilized ferroelectricity in ultrathin electrodeless films

Ferroelectric films may lose polarization as their thicknesses decrease to a few nanometers because of the depolarizing field that opposes the polarization therein. The depolarizing field is minimized when electrons or ions in the electrodes or the surface/interface layers screen the polarization charge or when peculiar domain configuration is formed. Here, we demonstrate ferroelectric phase transitions using thermooptical studies in ∼5-nm-thick epitaxial Pb0.5Sr0.5TiO3 films grown on different insulating substrates. By comparing theoretical modeling and experimental observations, we show that ferroelectricity is stabilized through redistribution of charge carriers (electrons or holes) inside ultrathin films. The related high-density of screening carriers is confined within a few-nanometers-thick layer in the vicinity of the insulator, thus resembling a two-dimensional carrier gas.


S1. Crystal structure
PSTO ceramics possess the room-temperature tetragonal crystal structure and lattice parameters a = b  0.392 nm and c  0.396 nm [1]. Compared to a cubic bulk PSTO cell (lattice parameter a0 = (a 2 c) 1/3  0.393 nm), a biaxial in-plane misfit strain [sa = as/a0 -1] is expected in a cube-on-cube-type epitaxial film of PSTO, coherently grown onto a cubic substrate with the lattice parameter as. The film-substrate mismatch between coefficients of thermal expansion can lead to an additional strain arising on cooling from the high temperature of deposition to room temperature. Here, this thermal strain is omitted for simplicity. The in-plane misfit strain sa, in-plane and out-of-plane lattice parameters a and c, respectively, tetragonality (c/a-1), and unit-cell volume V = a 2 c are estimated for the films coherent to the square surface cells of LSAT(001), STO(001), and DSO(011) substrates (Table S1). Table S1. Theoretical in-plane misfit strain sa, lattice parameters a and c, tetragonality (c/a-1), and unit cell volume in epitaxial PSTO films on different substrates compared to those in bulk PSTO.   The grown films are cube-on-cube-type epitaxial, with the in-plane lattice parameters equal to those on the substrates surfaces. The out-of-plane lattice parameters of the films on STO are similar, indicating the same in-plane compressive strain in these films. The larger in-plane compression and the in-plane tension are detected in the films on LSAT and DSO, correspondingly. The strains in the films are close to the theoretical estimations (Table S1). The optical properties of the thin films were probed using variable-angle spectroscopic ellipsometry (VASE). This is a rapidly advancing method experiencing remarkable progress in both the equipment and the methodological aspects. The method is superior in terms of accuracy compared to reflection and/or transmission analysis. State-of-the-art VASE allows for high-precision studies of optical properties of films with a thickness of only a few atomic layers and of separate macromolecules on top of arbitrary substrates [2]. Accurate analyses of ellipsometric data for wide bandgap materials are only possible if the spectral range is sufficiently expanded into the high-energy (UV) region. For this reason, an J. A. Woollam ellipsometer with an extended photon energy range is the most suitable choice.
Ellipsometric data were collected using a variable-angle rotating-analyzer spectroscopic ellipsometer over a spectral range from 0.74 eV to 9.0 eV. Each sample was measured at two angles of incidence (Θ = 65 o , 70 o ). This allowed us to obtain spectra of ellipsometric angles (∆, ψ) with an excellent accuracy of 0.2 deg for ∆ and 0.04 deg for ψ [3]. VASE data analysis was based on numerical inversion and minimization of possible artifacts. The analysis was performed using the WVASE32 software package [4]. The complex refractive index as a function of photon energy (here -optical properties for brevity), the thickness of the films, and the surface roughness were obtained from the analysis.
For data processing, the experimental ellipsometric spectra were fitted using a model considering a stack of a semi-infinite substrate, a film, a surface roughness layer, and ambient air. The parameterization of the initial dielectric functions of the films was based on the multi-oscillator model. The optical properties of the surface roughness layer were represented by a Bruggeman effective medium approximation [5]. The initial optical spectra and thickness of the film and surface roughness layer were extracted using a least-square regression analysis. After a refinement of the initial optical properties and layer thickness, the thickness was fixed and numerical inversion was used to extract optical properties from the measured spectra of ellipsometric angles ∆ and ψ. The dielectric functions and the optical properties of the substrates were determined from separate measurements. The data analysis accounted for the surface roughness layer. Several substrates of each type from MTI Corp. were studied. The optical properties of the substrates were used in the VASE data processing.
The mean square error (MSE) for thin-film samples was in the range of 0.1 -0.25. This excellent accuracy indicates a high reliability of the obtained results [6]. A typical example of a VASE data fits is shown in Fig. S4.
The absorption coefficient α was obtained using the relationship  = 4k/, where k is the extinction coefficient (the imaginary part of the complex index of refraction) and  is the wavelength of the light.

S3. Modeling
A single-domain ferroelectric film with the out-of-plane (normal to substrate surface) polarization P3(z) was considered. Here z is the out-of-plane coordinate across the film from its bottom to the top. The polarization was calculated using the Landau-Ginzburg-Devonshire (LGD) theory with the Euler-Lagrange equation and boundary conditions given as follows: Here, ,  , and  are the expansion coefficients of the LGD potential, g = g33 is the tensor component of gradient energy coefficients,  is the extrapolation length [7], and L is the thickness of the ferroelectric film. The electric potential  was found from the Poisson equation: ( The calculations were performed for three types of electrostatic boundary conditions: the film is (a) short-circuited between perfect conducting electrodes; (b) open-circuited and sandwiched between two insulators; and (c) sandwiched between an insulator and conducting electrode. The electric potential is zero at the short-circuited interfaces, i.e., for the film with one top electrode, and for the film-insulator boundary. The conditions (6) assume clean non-conducting film-insulator interfaces, where free surface charge is absent and the electrical displacement is continuous [9][10][11][12][13]. The electric field outside the ferroelectric film was set to zero to prevent the energy of the system reaching infinity.
The density of free charge carriers in the ferroelectric film was calculated considering the ferroelectric as a donor-doped semiconductor with a thin donor level, whose activation energy is Ed [14][15][16][17]. The donor density is [18]: where 0 d N is the density of donor centers, ( ) is the Fermi-Dirac distribution function, kB = 1.380710 −23 J/K, T is the absolute temperature, and F E is the Fermi energy. The conduction band (CB) electron density is [14,19]: where C E is the bottom of the CB. For an effective-mass density of states ( ) where is a polylogarithmic function. The Fermi level is found from the condition of electro-neutrality: where ( ) ( ) The critical thickness for ferroelectricity was considered as that corresponding to a second order ferroelectric-to-paraelectric transition. Because the spontaneous polarization and depolarizing field are very small in the immediate vicinity of the transition, a Debye approximation for the charge density can be used. For self-screening by free carriers with , the Poisson equation acquires the form: where ( ) is the screening radius. The critical thickness was calculated for the above described three types of electrostatic boundary conditions. Numerical simulations of the polarization and screening charge density were performed using the following parameters: T = 0.753×10 6 m/(F K), TC = 700 K,  = 1.12×10 9 J m 5 /C 4 ,  = 10 9 J m 9 /C 6 , Q12 = -0.02 m 4 /C 2 , s11 = 7.0×10 -12 Pa -1 , s12 = -2.0×10 -12 Pa -1 , g33 = 5.0×10 -10 m 3 /F, 33 b = 10, 33 a = 300, and the intrinsic density of charge carriers in the film n0 = 10 25 m -3 . The extrapolation length  is related to the properties of interfaces. The length is  = 0 for the polarization P3 = 0 at the interface, and it tends to infinity (  ) for the boundary condition 0 3    z P . In the absence of accurate physical model for the length , rather small lengths are assumed for the film-insulator interfaces and larger lengths are assumed for the film-electrode interfaces, as specified below. The three types of stacks were analyzed: electrode-film-electrode, insulator-film-insulator, and insulator-film-electrode. The critical thickness, polarization, and density of screening charge were calculated.
The critical thickness Lcr (a) for ferroelectricity in the film sandwiched between two perfect conducting electrodes is equal to where is the coefficient  renormalized by misfit strain um (if any) and is the correlation length [12]. This critical thickness exists for 0  a , and it depends on temperature. Often, the correlation length  is small; therefore, , and the expression (12) is approximated by As shown in the expressions (12) and (13), the low-temperature (T < TC) critical thickness is sensitive to the properties of the film-electrode interfaces, described by the extrapolation lengths 1 and 2. For the selected parameters and similar film-electrode interfaces (1 = 2 = 50 nm), the calculated roomtemperature critical thickness is very small:  ) (a cr L 0.4 nm. We note that the critical thickness can be additionally affected by such phenomena as interfacial band bending, formation of the Schottky barrier, and interface capacitance, which are ignored here for simplicity. When both electrodes are absent and the film is sandwiched between two perfect insulators, the critical thickness Lcr (b) is equal to Here ( ) is the effective screening length and    is the dimensionless coefficient [13]. Typically, the effective screening length is much larger than the correlation length:    . The critical thickness is then approximately For the selected parameters and the two similar 100-nm-thick insulating layers, the roomtemperature single-domain ferroelectric state is found to be stable if the thickness of the film is at least 12 nm.
The calculations do not account for the materials' bandgaps. Qualitatively, considering the bandgaps of ≥ 3 eV in ferroelectrics and those of > 3 eV in most insulating substrates, the interfacial band alignment is expected to either have no effect or to prevent charge from leaking out of the film.
Because the substrates are usually low-permittivity dielectrics, the assumed absence of an electric field and, consequently, polarization outside the films is justified.
The stack insulator-film-electrode represents a case with one short-circuited interface and another open-circuited interface, where the boundary conditions (5) and (6) are valid. An analytical expression for the critical thickness takes the form (16): For the screening length larger than the correlation length (    ), expression (16)  .
For the selected parameters, for a 100-nm-thick bottom insulator and perfect top electrode, the calculated critical thickness is approximately 6 nm.