Band alignment at metal/ferroelectric interfaces: Insights and artifacts from first principles

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Band alignment at metal/ferroelectric interfaces: Insights and artifacts from first principles

Massimiliano Stengel, Pablo Aguado-Puente, Nicola A. Spaldin, and Javier Junquera

Phys. Rev. B 83, 235112 – Published 7 June 2011

Abstract

Based on recent advances in first-principles theory, we develop a general model of the band offset at metal/ferroelectric interfaces. We show that, depending on the polarization of the film, a pathological regime might occur where the metallic carriers populate the energy bands of the insulator, making it metallic. As the most common approximations of density functional theory are affected by a systematic underestimation of the fundamental band gap of insulators, this scenario is likely to be an artifact of the simulation. We provide a number of rigorous criteria, together with extensive practical examples, to systematically identify this problematic situation in the calculated electronic and structural properties of ferroelectric systems. We discuss our findings in the context of earlier literature studies, where the issues described in this work have often been overlooked. We also discuss formal analogies to the physics of polarity compensation at LaAlO $_{3}$ /SrTiO $_{3}$ interfaces, and suggest promising avenues for future research.

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Received 2 March 2011

DOI:https://doi.org/10.1103/PhysRevB.83.235112

Authors & Affiliations

Massimiliano Stengel¹, Pablo Aguado-Puente², Nicola A. Spaldin³, and Javier Junquera²

¹Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, E-08193 Bellaterra, Spain
²Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Avda. de los Castros s/n, E-39005 Santander, Spain
³Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland

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Vol. 83, Iss. 23 — 15 June 2011

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Figure 1
Schematic representation of the band offset at a metal/insulator junction, illustrating the main quantities discussed in the text.Reuse & Permissions
Figure 2
Schematic representation of a symmetric short-circuited ferroelectric capacitor in a polarized configuration within the imperfect-screening model. $t$ is the thickness of the ferroelectric film. M and FE represent, respectively, the metal electrode and the ferroelectric film. Both materials are assumed to be separated by a vacuum layer of thickness $λ_{eff}$ . The thick solid line indicates the opposite of the electrostatic potential, $- V (z)$ .Reuse & Permissions
Figure 3
Schematic representation of the effect of free-charge redistribution onto the band diagram of a paraelectric capacitor with a negative $ϕ_{n}$ . (a) Band alignment under perfect interface screening (i.e., when $ρ_{free}$ vanishes), and (b) after charge spillout and electrostatic reequilibration. The corresponding profiles of the electric displacement field within the ferroelectric films are displayed in (c) and (d), obtained after integrating Eq. (13).Reuse & Permissions
Figure 4
(a) Paraelectric capacitor with a Schottky-like band alignment in the paraelectric structure. (b) When the polar instability sets in, the band alignment becomes pathological, the conduction band is locally populated (red shaded area), and the film becomes partially metallic (light shaded area bounded by the dashed line).Reuse & Permissions
Figure 5
Schematic representation of the effects of dielectric nonlinearity on the band diagram of a centrosymmetric capacitor. The effective potential felt by the conduction electrons is $- {\overset{̲}{\overset{̲}{V}}}_{H} (z)$ (see text). (a) Ferroelectric material. (b) Paraelectric material.Reuse & Permissions
Figure 6
(a) Schematic representation of $ϕ_{n}$ and $ϕ_{p}$ in an unpolarized SrRuO $_{3}$ /PbTiO $_{3}$ /SrRuO $_{3}$ capacitor. $E_{V}$ , $E_{C}$ , $E_{F}$ , and $Δ 〈 V 〉$ were defined in Sec. 2a. The calculated values are also indicated in the figure. The black solid line represents $- {\overset{̲}{\overset{̲}{V}}}_{H} (z)$ . The dashed line represents the hypothetical position of the CBM if $E_{C}$ were shifted to reproduce the experimental band gap. (b) Profile of ${ρ̄}_{free}$ as defined in Eq. (26). (c) Profile of the layer-by-layer polarization $P_{j}^{Z}$ . The size of the capacitor corresponds to $n$ $=$ 5.5 unit cells of SrRuO $_{3}$ and $m$ $=$ 12.5 unit cells of PbTiO $_{3}$ . Only the top half of the symmetric supercell is shown.Reuse & Permissions
Figure 7
PDOS of the inequivalent TiO $_{2}$ layers in the unpolarized PbTiO $_{3}$ /SrRuO $_{3}$ capacitor (solid curves with gray shading). The bottom curve lies next to the electrode, and the top one lies in the center of the PbTiO $_{3}$ film. Only the PDOS on half of the symmetric supercell are shown. The bulk PDOS curves (red dashed) are aligned to match the Ti(3 $s$ ) peak at $E ~ - 57$ eV. The Fermi level is located at zero energy.Reuse & Permissions
Figure 8
LDOS integrated over the NbO $_{2}$ layers of the KNbO $_{3}$ /SrRuO $_{3}$ heterostructure (solid curves with gray shade). The bottom curve lies next to the electrode, and the top one lies in the middle of the KNbO $_{3}$ film. Only the PDOS on half of the symmetric supercell are shown. The bulk LDOS (red dashed curves) are aligned to match the O( $2 s$ )-derived peaks. The Fermi level is located at zero energy.Reuse & Permissions
Figure 9
Calculated free charge for paraelectric SrRuO $_{3}$ /KNbO $_{3}$ /SrRuO $_{3}$ heterostructure. Black curve: Planar-averaged ${\overset{̲}{ρ}}_{free}$ . Red dashed: ${\overset{̲}{\overset{̲}{ρ}}}_{free}$ , nanosmoothed using a Gaussian filter. Blue symbols: Finite differences of the local $P_{j}$ (shown as a black curve in Fig. 10), calculated using the Wannier-based layer polarization described in Sec. 3b2.Reuse & Permissions
Figure 10
Local polarization profile in the SrRuO $_{3}$ /KNbO $_{3}$ /SrRuO $_{3}$ capacitor. Black circles: Polarization from Wannier-based layer polarizations. Red squares: Approximate polarization from “renormalized” Born effective charges (see Sec. 3b3). Analogous results for a paraelectric SrRuO $_{3}$ /BaTiO $_{3}$ /SrRuO $_{3}$ capacitor are shown for comparison (blue diamonds).Reuse & Permissions
Figure 11
Layer rumplings (cation-oxygen vertical relaxations) in the centrosymmetric KNbO $_{3}$ /SrRuO $_{3}$ (black line, empty circles) and PbTiO $_{3}$ /SrRuO $_{3}$ (red line, filled circles) capacitors. Dashed vertical lines indicate the location of the BO $_{2}$ planes. The shaded areas correspond to the SrRuO $_{3}$ electrode region.Reuse & Permissions
Figure 12
(a) Calculated free charge and (b) local polarization profile for a paraelectric Pt/BaTiO $_{3}$ /Pt capacitor with TiO $_{2}$ -type interfaces. All symbols have the same meaning as in Figs. 9 and 10.Reuse & Permissions
Figure 13
$n$ -type Schottky barrier as a function of interface doping in KNbO $_{3}$ / $A$ O-terminated SrRuO $_{3}$ , where $A$ is a fictitious atom with atomic number $Z = 37 + x$ . Only the Sr atoms at the interfacial layer are replaced by this fictitious atom. The dashed line is a linear regression of the data between $x = 0$ and $0.3$ , where the interface is nonpathological from the band-alignment point of view. Blue and red empty symbols represent, respectively, the results for $x = 0.5$ and 1.0, where the interface is already pathological. All values were obtained from Eq. (24), using either the Nb(4 $s$ ) (squares) or the O(2 $s$ ) (circles) semicore peaks of the central NbO $_{2}$ layer as a reference.Reuse & Permissions
Figure 14
(a) Conduction electrons and (b) local (Wannier-based) polarization profiles extracted from the calculations with $x = 0.0$ , 0.1, 0.2, and 0.3 (filled circles, thin black curves), 0.5 (empty blue circles, dashed blue curve), and 1.0 (empty red circles, solid red curve). In (a) only half of the KNbO $_{3}$ film is shown.Reuse & Permissions
Figure 15
Results for the polarized PbTiO $_{3}$ /SrRuO $_{3}$ interface for increasing polarization of the film. (a) planar averaged ${\overset{̲}{ρ}}_{free}$ . Black, red, green, and blue curves correspond to the results for $d = 0.20$ , 0.40, 0.60, and 0.74 $e$ , respectively. The sharp peaks in ${\overset{̲}{ρ}}_{free}$ correspond to the Ti ions in the PbTiO $_{3}$ film. (b) layer polarizations from the Wannier-based analysis. Same color code as in (a).Reuse & Permissions
Figure 16
Calculated results for the fully polarized PbTiO $_{3}$ /SrRuO $_{3}$ interface at $d = 0.74$ . (a) Local polarization from the Wannier-based layer polarizations, and (b) planar-averaged ${\overset{̲}{ρ}}_{free}$ (black curve), macroscopically averaged ${\overset{̲}{\overset{̲}{ρ}}}_{free}$ (red dashed curve), and finite differences of the polarization shown in (a) (blue squares) for an $m = 8$ unit-cell-thick PbTiO $_{3}$ film. (c) and (d) are the corresponding figures for an $m = 5$ unit-cell-thick PbTiO $_{3}$ film. The sharp peaks in ${\overset{̲}{ρ}}_{free}$ correspond to the Ti ions in the PbTiO $_{3}$ film.Reuse & Permissions
Figure 17
(a) Profile of the layer-by-layer polarization ${P̃}_{j}^{Z}$ , defined in Eq. (33), in the relaxed polar configuration of a short-circuited SrRuO $_{3}$ /PbTiO $_{3}$ /SrRuO $_{3}$ capacitor. The dashed line represents the bulk spontaneous polarization under the same in-plane strain as in the capacitor. (b) ${ρ̄}_{free} (z)$ as defined in Eq. (26) (black solid line), and its nanosmoothed average ${\overset{̲}{\overset{̲}{ρ}}}_{free} (z)$ (red dashed line). The blue line represents the profile of the bound charge, computed as a finite-difference derivative of ${P̃}_{j}^{Z}$ .Reuse & Permissions
Figure 18
Layer-by-layer PDOS on the TiO $_{2}$ layers of the polar SrRuO $_{3}$ /PbTiO $_{3}$ /SrRuO $_{3}$ ferroelectric capacitor. Meaning of the lines as in Fig. 7, but now the PDOS on all the TiO $_{2}$ layers are plotted (there is no longer a mirror-symmetry plane). The squares represent the position of the local band edges, computed following the recipe of Sec. 3a3. The dashed lines are a linear interpolation of the calculated local band edges.Reuse & Permissions
Figure 19
Schematic representation of the impact of the charge spillage on the ferroelectric stability. M are the metal electrodes, and FE is the ferroelectric film. The polarization points to the right-hand side.Reuse & Permissions
Figure 20
LDOS of selected TiO $_{2}$ layers in the electrostatically doped LaAlO $_{3}$ /SrTiO $_{3}$ system. The insets are a blow up of the regions indicated by the small squares. The energy scale of the insets comrpises between $- 0.25$ and 0.25 eV.Reuse & Permissions
Figure 21
Polarization $P$ in a BaTiO $_{3}$ film and PbTiO $_{3}$ bulk as a function of the reduced electric displacement field $d = DS$ . Data are taken from Ref. 20 (see Section IIIC.1) and Ref. 32.Reuse & Permissions
Figure 22
(a) Gaussian ( $σ = 0.15$ eV) and Fermi-Dirac ( $σ = 0.075$ eV) occupation functions. (b) Kernel of the occupation functions as defined in the text. (c), (d) Fourier transform of the smearing kernels $g$ , assuming an energy window of $[- 1, 1]$ .Reuse & Permissions
Figure 23
Left-hand side: Fermi-Dirac occupation function, identical to that of Fig. 22a (solid curve); hypothetical orbital located at an energy of 0.15 eV above the Fermi level (dashed line); the thermal occupation of this state yields a total charge of 0.119 electrons (red dot). Right-hand side: DOS corresponding to the single isolated orbital at an energy of 0.15 eV above the Fermi level, smeared by using the $g_{FD}$ kernel of Fig. 22b; the integral of the DOS up to the Fermi level (shaded area) yields the exact same charge of 0.119 electrons.Reuse & Permissions

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