Theoretical Approach to Electroresistance in Ferroelectric Tunnel Junctions

In this paper, a theoretical approach comprising the nonequilibrium Green’s function method for electronic transport and the Landau-Khalatnikov equation for electric polarization dynamics is presented to describe polarization-dependent tunneling electroresistance (TER) in ferroelectric tunnel junctions. Using appropriate contact, interface, and ferroelectric parameters, the measured current-voltage characteristic curves in both inorganic ($\mathrm{Co}/\mathrm{BaTi}{\mathrm{O}}_3/{\mathrm{La}}_{0.67}{\mathrm{Sr}}_{0.33}{\mathrm{MnO}}_3$) and organic ($\mathrm{Au}/\mathrm{PVDF}/\mathrm{W}$) ferroelectric tunnel junctions can be well described by the proposed approach. Furthermore, under this theoretical framework, the controversy of opposite TER signs observed experimentally by different groups in $\mathrm{Co}/\mathrm{BaTi}{\mathrm{O}}_3/{\mathrm{La}}_{0.67}{\mathrm{Sr}}_{0.33}{\mathrm{MnO}}_3$ systems is addressed by considering the interface termination effects using the effective contact ratio defined through the effective screening length and dielectric response at the metal-ferroelectric interfaces. Finally, our approach is extended to investigate the role of a ${\mathrm{CoO}}_x$ buffer layer at the $\mathrm{Co}/\mathrm{BaTi}{\mathrm{O}}_3$ interface in a ferroelectric tunnel memristor. It is shown that in order to have a significant memristor behavior not only the interface oxygen vacancies but also the ${\mathrm{CoO}}_x$ layer thickness may vary with the applied bias.

Figure 1

Schematics of FTJs in the (a) absence and (b) presence of a nonpolar DE layer between the FE and metallic electrode. $M_1$ and $M_2$ are top and bottom metallic electrodes, respectively.

Figure 2

Schematics of electrostatic potential profiles due to (a) applied electric field, (b) built-in field, and (c) depolarization field for FTJs with (bottom panel) and without (top panel) a nonpolar DE layer between the FE and top metallic electrode.

Figure 3

Energy band diagrams at a bias voltage $V_a$ satisfying ${\mu }_2-{\mu }_1=e{V}_a$ for FTJs (a) without and (b) with a nonpolar DE layer between the FE and metallic electrode. Arrows in the FE represent the direction of the electric polarization.

Figure 4

(a) Comparison between FTJ ($\mathrm{Co}/\mathrm{BTO}/\mathrm{LSMO}$) experimental data ($\text{diameter}=700\mathrm{nm}$) [7] and simulation results using the following band diagram parameters: $t_{\mathrm{FE}}=2\mathrm{nm}$, ${\varphi }_1={\varphi }_2=7.15\mathrm{eV}$, $E_{F1}=E_{F2}=6.5\mathrm{eV}$, ${\epsilon }_1=2.4$, ${\epsilon }_2=9.6$, ${\lambda }_1=0.5\times {10}^{-10}\mathrm{m}$ [42], ${\lambda }_2=1\times {10}^{-10}\mathrm{m}$ [42], $m^{*}=0.8m_0$. (b) Simulated FE hysteresis loop for FTJ ($\mathrm{Co}/\mathrm{BTO}/\mathrm{LSMO}$) experiments [7] ($V_c\sim \pm 3\mathrm{V}$, ${\epsilon }_{\mathrm{FE}}\sim 11$, and $P_r\sim 0.3\mathrm{C}/{\mathrm{m}}^2$) with the following LK parameters: $\gamma =1.8\times {10}^{-2}\mathrm{msec}/\mathrm{F}$, ${\alpha }_1=-2.77\times {10}^7\mathrm{m}/\mathrm{F}$, ${\alpha }_{11}=-5.35\times {10}^8{\mathrm{m}}^5/{\mathrm{C}}^2\mathrm{F}$, and ${\alpha }_{111}=2\times {10}^{10}{\mathrm{m}}^9/{\mathrm{C}}^4\mathrm{F}$.

Figure 5

(a) Comparison between FTJ ($\mathrm{Au}/\mathrm{PVDF}/\mathrm{W}$) experimental data [9] and simulation results using the following band diagram parameters: $t_{\mathrm{FE}}=2\mathrm{nm}$, ${\varphi }_1=6.76\mathrm{eV}$, ${\varphi }_2=6.7\mathrm{eV}$, $E_{F1}=E_{F2}=6.5\mathrm{eV}$, ${\epsilon }_1=6.5$, ${\epsilon }_2=20$, ${\lambda }_1=0.75\times {10}^{-10}\mathrm{m}$ [45], ${\lambda }_2=0.45\times {10}^{-10}\mathrm{m}$ [46], $m^{*}=0.1m_0$. (b) Simulated FE hysteresis loop for FTJ ($\mathrm{Au}/\mathrm{PVDF}/\mathrm{W}$) experiments [9] ($V_c\sim \pm 1\mathrm{V}$, ${\epsilon }_{\mathrm{FE}}\sim 4.4$, and $P_r\sim 0.18\mathrm{C}/{\mathrm{m}}^2$) with the following LK parameters: $\gamma =1.5\times {10}^{-3}\mathrm{m}\sec /\mathrm{F}$, ${\alpha }_1=-1.38\times {10}^9\mathrm{m}/\mathrm{F}$, ${\alpha }_{11}=-2.67\times {10}^{10}{\mathrm{m}}^5/{\mathrm{C}}^2\mathrm{F}$, and ${\alpha }_{111}=8\times {10}^{11}{\mathrm{m}}^9/{\mathrm{C}}^4\mathrm{F}$.

Figure 6

Energy band diagrams at 0.1 V for both polarization states with two different effective contact ratios: (a) 1.98 and (b) 0.49. The dark blue and the green correspond to the polarization states pointing to the top and bottom contacts, respectively. Red dashed lines represent chemical potentials at both contacts.

Figure 7

TER at $V_a=0.1\mathrm{V}$ and effective contact ratio versus top contact (a) dielectric constant and (b) screening length. TER and effective contact ratios are defined as $I_{\uparrow }/I_{\downarrow }$ and $[{\lambda }_1{\epsilon }_2/({\lambda }_2{\epsilon }_1)]$, respectively.

Figure 8

(a) Energy band diagrams at 0.1 V for high- and low-resistance states in a FTJ with a ${\mathrm{CoO}}_x$ buffer layer at the interface. ${\varphi }_c$ for high- and low-resistance states are 6.6 and 0.1 eV, respectively. (b) Comparison with experimental data [22] using various ${\varphi }_c$ for high- and low-resistance states and different writing voltages. In addition to $t_{\mathrm{DE}}=0.6\mathrm{nm}$, $t_{\mathrm{FE}}=1\mathrm{nm}$, ${\varphi }_1$, and ${\varphi }_c$, the simulation parameters are the same as those in Fig. 4. (c) Tunneling current versus applied voltage under different ${\mathrm{CoO}}_x$ thicknesses with ${\varphi }_c$ being 3.3 eV.

Figure 9

Schematics illustrating the electron wave vector in the spherical coordinate and the NEGF approach to FTJs without and with a nonpolar DE layer between the FE and metallic electrode.