Strong influence of complex band structure on tunneling electroresistance: A combined model and ab initio study

The tunneling electroresistance (TER) for ferroelectric tunnel junctions (FTJs) with ${\text{BaTiO}}_3$ and ${\text{PbTiO}}_3$ barriers is calculated by combining the microscopic electronic structure of the barrier material with a macroscopic model for the electrostatic potential, which is caused by the ferroelectric polarization. The TER ratio is investigated in dependence on the intrinsic polarization, the chemical potential, and the screening properties of the electrodes. A change in sign in the TER ratio is obtained for both barrier materials in dependence on the chemical potential. The inverse imaginary Fermi velocity describes the microscopic origin of this effect; it qualitatively reflects the variation and the sign reversal of the TER. The quantity of the imaginary Fermi velocity allows to obtain detailed information on the transport properties of FTJs by analyzing the complex band structure of the barrier material.

Figure 1

(Color online) Schematics of the tunnel barrier potential along the current direction for the ferroelectric insulator with the barriers polarization pointing to the right [(a), $P_{+}$] or to left [(b), $P_{-}$] interface. The potential is a superposition of the electrostatic potential (colored line), the electronic potential, which determines the bottom of the bands in the two electrodes and the potential barrier created by the ferroelectric insulator. $\Lambda$ and $\lambda$ are the screening lengths of the electrodes, [cf. Eq. (1)].

Figure 2

(Color online) Symmetry-resolved complex band structures for ferroelectric (a) ${\text{BaTiO}}_3$ and (b) ${\text{PbTiO}}_3$ at $\overline{\Gamma }$ in the 2D Brillouin zone $\left({\mathit{k}}_{\parallel }=0\right)$. Middle panels show the conventional bands structure $\left(\text{Im}{k}_z=0\right)$ along the $\Gamma \text{-Z}$ line. Left and right panels display imaginary bands of the first $\left(k_z=0\right)$ and the second kind $\left(k_z=\frac{\pi }{c}\right)$, respectively.

Figure 3

(Color online) Smallest imaginary part of the complex wave vector of ${\text{BaTiO}}_3$ [upper row, (a)–(c)] and ${\text{PbTiO}}_3$ [lower row, (d)–(f)] in the 2D Brillouin zone (edges of the Brillouin zone at $\pm \pi /a$). The imaginary parts are shown for different energies relative to the valence-band maximum: (a) 0.77 eV, (b) 1.30eV, (c) 1.77eV, (d) 1.10eV, (e) 1.40eV, and (f) 2.10 eV.

Figure 4

(Color online) Conductances and TERs for the two ferroelectric configurations of a FTJ with ${\text{BaTiO}}_3$ [upper row, (a)–(c)] and ${\text{PbTiO}}_3$ [lower row, (d)–(f)] as a ferroelectric barrier. The barrier thickness was fixed to four unit cells $\left(\approx 1.6\text{nm}\right)$. Both, conductance and TER, are shown in dependence on the absolute value of the electrical polarization as well as for different chemical potentials with respect to the valence-band maximum: (a) 0.77 eV, (b) 1.30 eV, (c) 1.77 eV, (d) 1.10 eV, (e) 1.40 eV, and (f) 2.10 eV (that is, as in Fig. 3). Blue (full) and red (open) circles denote the dependence of the conductance for the polarization pointing to the left and right interface. Black dashed vertical lines indicate the value for the bulk polarization. The inset in (b) and (e) depicts the conductances in a smaller scale, showing a small but pronounced variation. $\Lambda$ and $\lambda$ are the screening lengths in the left and right electrode, respectively, with fixed ratio $\Lambda /\lambda =9$.

Figure 5

(Color online) TER versus chemical potential for BTO (a) and PTO (c) relative to the valence-band maximum of the ferroelectric. The screening lengths’ ratio is fixed to 9. In the lower part the corresponding inverse imaginary Fermi velocity for (b) BTO and (d) PTO is depicted. For further details see text.

Figure 6

(Color online) TER dependence on the polarization strength and the position of the chemical potential for FTJ with (a) BTO or (b) PTO barriers. The screening lengths’ ratio was fixed to $\Lambda /\lambda =9$.

Figure 7

(Color online) The conductance for the two ferroelectric polarization states and the resulting TER of a FTJ with ${\text{PbTiO}}_3$ as a ferroelectric barrier material is shown. Blue (full) and red (open) circles denote the dependence of the conductance for the polarization pointing to the left and right interface. The absolute polarization is fixed to ${\mathit{P}}_{\text{PTO}}=70\mu \text{C}/{\text{cm}}^2$. Both, conductance and TER, are shown in dependence on the ratio $\frac{\Lambda }{\lambda }$ of the screening lengths of the electrodes material and for different chemical potentials with respect to the valence band: (a) 1.10 eV and (b) 1.40 eV.