CBS of bulk PbTiO₃

In order to face the difficulty of the problem step by step, we have proceed first with a centrosymmetric tetragonal phase (P4/mmm space group), assuming the in-plane theoretical lattice constant, a, of an hypothetically thick SrTiO₃ substrate and only allowing the out-of-plane lattice constant, c, to relax. This is the structure that PbTiO₃ displays at the center of the unpolarized capacitor.

In essence a CBS, calculation consists in sampling the two-dimensional Brillouin zone (2DBZ) parallel to the interface, and for every k_// search for complex k _⊥ associated with real eigenvalues of the system. Imaginary bands always connect extrema of other bands (either real or complex),^[Reference Chang^22] and it is often assumed that complex bands with the shortest imaginary wave vector q are those connecting the edges of the valence and conduction bands. The analysis of the real band structure (see Suppl. Fig. 2 in supplemental data) reveals that, in the non-polar P4/mmm phase, PbTiO₃ possesses a direct gap at X, (π/a, 0, 0) and a slightly larger one at Z, (0, 0, π/c).

Complex bands at such high-symmetry points have an associated k with the form (π/a, 0, iq) and (0, 0, π/c + iq), respectively, which correspond to complex bands at ${\bi k}_{//} = \bar{X}$ and $\bar{\it\Gamma}$ in the 2DBZ, shown in Figs. 1(a) and 1(b). We observe that the states with the shortest q (largest penetration lengths) consist of a Δ₁ singlet in the lower energy part of the band gap and a Δ₅ doublet for energies close to the conduction band edge. The smaller decay factor q takes values from 0.10 to 0.15 in units of (2π/c) at the center of the gap and tends to zero as the energy approaches the band edges. In Fig. 1(c), we plot together all the energy bands with associated complex values of k = (k_//, k _z + iq), with ${\bi k}_{//} = \bar{\it\Gamma}$ or $\bar{X}$. Nevertheless, we have to keep in mind that the CBS at $\bar{\it\Gamma}$ and $\bar{X}$ might not be representative of the whole CBS of the system, since they are only two points of special symmetry of the total 2DBZ. For a comprehensive analysis, we should gather into a plot like Fig. 1(c), the imaginary part of the complex bands coming from all k_//. Instead of this, that would result in some of the infinite numbers of complex bands obscuring others, we plot the “density of complex bands” in Fig. 1(d). In this 2D-histogram, for every value of the energy and imaginary part of the wave vector q, we represent in a red scale the number of complex bands contributing to that q(E) point. The analysis of Fig. 1(d) reveals that the total CBS is clearly dominated by bands at q ≈ 0.2 and q ≈ 0.3. To identify the k_// that contributes to those clusters of bands we resolve the structure and, for every k_//, we plot the smallest q at a given energy [see Suppl. Fig. 3(a) in the supplemental data]. We find that from energies between the middle of the gap and the bottom of the conduction band, the most penetrating bands come from k_// lying in a narrow rectangle centered along the $\bar{\it\Gamma} - \bar{X}$ path, while from the top of the valence band to the center of the gap, the bands with smaller q are due to states in small circular (around $\bar{\it \Gamma} $) or square (around $\bar{X}$) regions centered at the high-symmetry points. The much larger area of the 2DBZ outside those sections contributes to the darker cluster of imaginary bands at q ≈ 0.2. Similar results have been previously reported for other perovskites.^[Reference Velev, Belashchenko, Stewart, van Schilfgaarde, Jaswal and Tsymbal^11, Reference Velev, Duan, Belashchenko, Jaswal and Tsymbal^17, Reference Hinsche, Fechner, Bose, Ostanin, Henk, Mertig and Zahn^23, Reference Wortmann and Blügel^24] The main messages that can be drawn might be summarized as follows: (i) the most penetrating bands (the most important for tunneling, especially in thick films) come from high-symmetry points or lines in the 2DBZ whose relative weight is small. That is the reason why those bands might be totally inappreciable in the 2D-histogram of Fig. 1(d) (see the region marked with a square, where the bands with q ≈ 0.1 at $\bar{X}$ and $\bar{\it \Gamma} $ previously discussed were expected); and (ii) for ultra-thin dielectric layers or for the computation of integrated quantities (such as the interfacial dipole) it is important to consider higher-order imaginary bands and non-high-symmetry points in the 2DBZ, since associated phenomena might be dominated by them.^[Reference Mavropoulos, Papanikolau and Dederichs^25, Reference Butler, Zhang, Schulthess and MacLaren^26]

Figure 1.

CBS of centrosymmetric PbTiO₃ at (a) ${\bi k}_{//}=\bar{\it\Gamma} $and (b) ${\bi k}_{//}=\bar{X}$. Black lines at central panels correspond to bands with a real value of k _⊥. At left and right panels, complex bands with the form k _⊥ = iq and k _⊥ = π/c + iq, respectively, are plotted as black lines. In red we plot complex bands with k _z ≠ 0 or π/c, with the projections on the real plane plotted on the central panel and their projections on the imaginary plane on the side panels (following Ref. Reference Chang22). Horizontal dashed lines delimit the band gap. In (c), we plot together the imaginary part of bands with complex values of k, with ${\bi k}_{//}=\bar{\it\Gamma} $ or $\bar{X}$. (d) DOS (in arbitrary units) with respect to the energy and the imaginary wave vector q. The density of complex bands is plotted in a red scale, where darker regions represent a larger density of bands. The black solid line represents the effective penetration length.

Figure 2.

(a) Schematic representation of the interface region of the simulated (SrRuO₃)_9.5/(PbTiO₃)_8.5 superlattice in an unpolarized configuration (Ti atoms in green, Ru in magenta, O in blue, Sr in yellow, and Pb in gray). Only one of the two symmetric interfaces is shown. (b) Layer-by-layer PDOS on the atoms at the different layers, as denoted by the layer labeling. The plot at the top corresponds to the central atomic plane of the PbTiO₃ film. Dashed lines delimit one of the energy windows used to perform the integrations in Eq. (2).

Figure 3.

(a) Decay of the MIGS for the energy window inside the gap of the SrRuO₃/PbTiO₃ capacitor shown in Figure 2. (a). The solid curve is a fit to a hyperbolic cosine, from which the effective penetration δ(E) is obtained. (b) Effective imaginary wave vector q _eff(E) obtained from the CBS [using only bands at $\bar{\it\Gamma} $ (blue dotted line), $\bar{\it\Gamma} $ and $\bar{X}$ (green dotted), and the full 2DBZ (red line)] and from the fit of the decay of the PDOS inside the insulating material for SrRuO₃/PbTiO₃ (solid black line) and Pt/PbTiO₃ capacitors (dashed black line). (c) Total DOS of MIGS in the SrRuO₃/PbTiO₃ (solid black line) and Pt/PbTiO₃ capacitors (dashed black line) obtained integrating the layer-by-layer PDOS from the interface to the center of the PbTiO₃ film. In (b) and (c), the zero of energies is set to the top of the valence band and the horizontal lines indicate the Fermi level of the corresponding capacitor.

It is sensible to think that the charge transferred from the metal to the insulator through the MIGS reflects the collective contribution of all complex bands in the gap of the insulator. Here we also give a recipe to infer from the CBS an effective value of the imaginary wave vector for a given energy, $q_{\rm eff}^{\rm CBS}$. This can be estimated by fitting

(1)$$\sum\limits_{{\bi k}_{//}} \sum\limits_n e^{ - 2q_n \left(E\comma \, {\bi k}_{//} \right)z} \approx e^{ - 2q_{\rm eff}^{\rm CBS} \left(E \right)z}\comma$$

which comparison with the q _eff obtained from the realistic simulation of the interface is discussed below.

MIGS in first-principles simulations of capacitors

The explicit influence of the metal/ferroelectric interface in the decay of the MIGS is captured in the simulation of realistic capacitors from first-principles. Here, we have simulated short-circuited (SrRuO₃)_m/(PbTiO₃)_n and (Pt)_l/(PbTiO₃)_p capacitors as model systems, with m = 9.5, n = 8.5, l = 9.5 and p = 6.5 unit cells, respectively. In both cases the PbTiO₃ films are terminated in a TiO₂ atomic layer, and we have checked that the band-alignment in the most stable paraelectric configuration is non-pathological.^[Reference Stengel, Aguado-Puente, Spaldin and Junquera^27]

The analysis of MIGS in full capacitor simulations requires working with a spatially resolved density of states (DOS). Here, we have chosen the layer by layer projected density of states (PDOS), as defined in Eq. (21) of Ref. Reference Stengel, Aguado-Puente, Spaldin and Junquera27. The results for the SrRuO₃/PbTiO₃ capacitor in the unpolarized configuration are plotted in Fig. 2(b) (similar results are obtained for the Pt/PbTiO₃ interface). The decay of the evanescent tails of the MIGS can be clearly observed, with the PDOS showing a clean gap for atomic layer further than two unit cells from the interface.

We can obtain a measure of the spatial distribution of the probability density at a given energy integrating the PDOS in small windows centered at different energies inside the band gap,

(2)$$Q_{{\rm PDOS}} \left({E_i\comma \; z_j } \right)=\vint_{E_i - {{\Delta E} \over 2}}^{E_i+{{\Delta E} \over 2}} {\rho \left({\,j\comma \; E} \right)dE}\comma$$

where ρ(j, E) denotes the PDOS on all the atomic orbitals of a given layer j, located at z(j). The decay of this quantity when we move away from the interface provides a direct way to obtain the effective decay length of the PDOS, δ(E). Since δ refers to the decay of the density, and given the exponential behavior of the tails of the MIGS, we can relate δ with an effective decay of the wave functions as $q_{\rm eff}^{\rm MIGS} \left(E \right)= 1/\left[2\delta \left(E \right)\right]$. Numerically, δ(E) is estimated first integrating the PDOS after Eq. (2) in energy windows of width ΔE = 0.02 eV. Then, for every energy sampled, we plot the exponentially decaying Q _PDOS curve as a function of the distance to the interface [Fig. 3(a)], and fit it to a function $Q_{\rm PDOS} \left(E\comma \; z \right)\approx \cosh \left[z/\delta \left(E \right)\right]$. A hyperbolic cosine was used to account for the presence of two interfaces. Repeating this procedure for energy windows covering the whole gap we obtain the energy dependence of the effective imaginary wave vector, $q_{\rm eff}^{\rm MIGS} \left(E \right)$, that is plotted in Fig. 3(b) for the two different capacitors.

Comparison of decay factors obtained from bulk and interface simulations

In Fig. 3(b), we plot the effective values of the decay factors obtained from a bulk CBS calculation [using Eq. (1) and scaling to the band gap obtained with Siesta to account for the larger gap obtained with Quantum-Espresso], and from the evanescent behavior of the MIGS at the interface for PbTiO₃-based capacitors with SrRuO₃ and Pt electrodes. When the full 2DBZ is used for the calculation of $q_{\rm eff}^{\rm CBS} \left(E \right)$ its agreement with $q_{\rm eff}^{\rm MIGS} \left(E \right)$ is very good, supporting the idea that the penetration of the MIGS is mostly determined by the bulk properties of the insulator. The correspondence is remarkably good for the capacitor with Pt electrodes, for which a featureless dependency of $q_{\rm eff}^{\rm MIGS} \left(E \right)$ with the energy is observed. For the metal oxide SrRuO₃ a more structured shape of the decay factor is noticed. In this case, to understand the peaks we have to consider interfacial effects like the symmetry filtering^[Reference Velev, Belashchenko, Stewart, van Schilfgaarde, Jaswal and Tsymbal^11, Reference Bibes, Villegas and Barthélémy^15]: the wave functions in the electrodes should match evanescent states in the insulator that are compatible by symmetry. The inspection of the bulk band structure of SrRuO₃ (see Suppl. Fig. 4 in the supplemental data) reveals the presence of a band below the Fermi energy with Δ₅ symmetry along the Γ − X line that nicely matches the complex bands of PbTiO₃ of the same symmetry, with k_// along the $\bar{\it\Gamma} - \bar{X}$ line of the 2DBZ. Above the Fermi energy there are no bands with the appropriate symmetry to link the complex bands with minimal q. The matching has to be done with higher-order complex bands with larger q, explaining the sharp increase in $q_{\rm eff}^{\rm MIGS} $. In the case of Pt, there is a band with the appropriate symmetry that crosses completely the band gap of PbTiO₃. In Fig. 3(b), we also plot $q_{\rm eff}^{\rm CBS} $ obtained taking into account only complex bands from $\bar{\it\Gamma} $ and $\bar{X}$. As anticipated, it can be clearly noted that only considering high-symmetry points of the 2DBZ results in a severe underestimation of the effective decay factor of the MIGS.

Finally, not only the decay rate of MIGS is relevant but the interfacial DOS of MIGS is also important. We find that in this respect the two electrodes behave differently as well [see Fig. 3(c)], with Pt/PbTiO₃ capacitor showing a much larger surface density of MIGS than the SrRuO₃/PbTiO₃ one. This is consistent with the much larger DOS in the vicinity of the Fermi level of bulk Pt with respect to SrRuO₃.

Influence of the ferroelectric polarization

The influence of the ferroelectric distortion in the spatial and energetic distribution of the evanescent states is known to play a mayor role on the electro^[Reference Velev, Duan, Belashchenko, Jaswal and Tsymbal^17, Reference Hinsche, Fechner, Bose, Ostanin, Henk, Mertig and Zahn^23, Reference Caffrey, Archer, Rungger and Sanvito^28] and magnetoresistance^[Reference Velev, Belashchenko, Stewart, van Schilfgaarde, Jaswal and Tsymbal^11, Reference Caffrey, Archer, Rungger and Sanvito^13, Reference Velev, Duan, Burton, Smogunov, Niranjan, Tosatti, Jaswal and Tsymbal^29] of ferroelectric tunnel junctions. Unfortunately, a thorough analysis of such effect using the arguments developed in this work is limited by band alignment issues^[Reference Stengel, Aguado-Puente, Spaldin and Junquera^27] that restrict the possible electrodes and the magnitude of the polarization that can be explored. In our case, only the ferroelectric capacitor with SrRuO₃ electrodes and n = 8.5 is non-pathological,^[Reference Stengel, Aguado-Puente, Spaldin and Junquera^27] with a polarization within PbTiO₃ of 24 µC/cm². Again, the agreement between $q_{\rm eff}^{\rm MIGS} $ (see Supplemental data for details about how to compute $q_{\rm eff}^{\rm MIGS} $ in the polar state) and $q_{\rm eff}^{\rm CBS} $ (computed from the CBS of bulk PbTiO₃ at the same polarization as in the capacitor) is very good (see Suppl. Fig. 5 in the supplemental data). A general increase of the effective decay factor (i.e., shorter penetration lengths) with the polarization is observed. This growth of q is due to two different factors. First, the polarization opens the gap. Second, it pushes up the flat band along the Γ − X line at the bottom of the conduction band in the paraelectric configuration (see Suppl. Fig. 6 in the supplemental data). Both facts increase the imaginary part of those bands that link to the bottom of the conduction band with k_// along the $\bar{\it\Gamma} - \bar{X}$ line. This effect has already been reported for the complex bands at $\bar{\it\Gamma} $ in BaTiO₃,^[Reference Velev, Duan, Belashchenko, Jaswal and Tsymbal^17, Reference Velev, Duan, Burton, Smogunov, Niranjan, Tosatti, Jaswal and Tsymbal^29] and was invoked to explain the impact of ferroelectricity in the tunnel conductivity of ferroelectric junctions.

Summary

The main conclusions that can be drawn from this work are: (i) although the most penetrating bands are located at high-symmetry points, for the study of tunneling across ultra-thin layers or integrated quantities (where many k_// contribute) it is important to consider complex bands in the whole 2DBZ; (ii) we have given a general recipe to estimate from the CBS of the bulk insulator the effective decay factor (and its dependence with the energy within the gap) of the MIGS in realistic interfaces, showing a very good agreement with the results obtained from first-principles simulations of capacitors; (iii) to explain the fine details, especially when metal-oxide electrodes are used, a symmetry filtering analysis has to be performed; and (iv) the ferroelectric polarization can be used as a knob to increase the decay factor of the CBS. The accuracy of the model used to predict the effective decay length of MIGS would allow one to screen materials to be used as barriers in tunnel electro- and magnetoresistance junctions based on knowledge. We hope this work will encourage experiments and set the ground for future theoretical studies of MIGS, tunneling and related phenomena.