Influence of image forces on electron transport in ferroelectric tunnel junctions

We study influence of image forces on conductance of ferroelectric tunnel junctions. We show that the influence of image forces is twofold: (i) they enhance the electroresistance effect due to polarization hysteresis in symmetric tunnel junctions at nonzero bias, and (ii) they produce the electroresistance effect due to hysteresis of dielectric permittivity of ferroelectric barrier. We study dependence of ferroelectric tunnel junction conductance on temperature and show that image forces lead to strong conductance variation with temperature.

Figure 1

Potential barrier $U\left(z\right)$ [Eq. (1)] as a function of distance for symmetric FTJ at zero bias voltage. Black like is the potential profile in the absence of surface charges effect and image forces. Dash-dotted blue line is the potential corrected due to the surface charges. Red solid line is the potential profile corrected by both the image forces and FE surface charges. Notations M and FE stand for metal and ferroelectric, respectively. FE layer thickness is $d$. $z_{1,2}$ is the position where potential $U\left(z\right)$ crosses the Fermi energy $E_{\mathrm{F}}$ (which is the zero energy). $z_{1,2}$ defines the effective barrier thickness $d_{\mathrm{eff}}=z_2-z_1;P$ denotes the FE polarization, and potentials ${\phi }_{1,2},h_{b,c}$ are introduced in the text.

Figure 2

Polarization [Eq. (3)] and dielectric constant [Eq. (4)] shown for the following parameters: $P_0=30\mu \mathrm{C}/{\mathrm{cm}}^2,V_{\mathrm{s}}=0.1$ V, $\mathrm{\Delta }\varepsilon =30$, and ${\varepsilon }_{\min }=15$. Solid lines correspond to the upper hysteresis branch. Dash-dotted lines stand for the lower hysteresis branch. The parameters correspond to ${\mathrm{Hf}}_{0.5}{\mathrm{Zr}}_{0.5}{\mathrm{O}}_2$ FE (see Ref. [39])

Figure 3

Electroresistance effect (main plot) and conductance (inset) as a function of applied voltage for the following system parameters: $d=1$ nm, $h_{\mathrm{b}}=0.5$ eV, $P_0=30\mu \mathrm{C}/{\mathrm{cm}}^2,{\delta }_1={\delta }_2=0.05$ nm, $V_{\mathrm{s}}=0.1$ V, $\mathrm{\Delta }\varepsilon =15$, and ${\varepsilon }_{\min }=30$. Solid line corresponds to the total ER effect including both mechanisms related to surface charges and image forces. Dotted line shows the ER effect due to surface charges ${\text{ER}}_{\mathrm{P}}$. Dashed-dotted line shows the ER effect due to the image forces ${\text{ER}}_{\varepsilon }$. ${\text{ER}}_{\varepsilon ,\mathrm{P}}^{\max }$ is the value of ${\text{ER}}_{\varepsilon }$ and ${\text{ER}}_{\mathrm{P}}$ at $V=-V_{\mathrm{s}}$. Inset: solid lines correspond to the upper hysteresis branch. Dash-dotted lines stand for lower hysteresis branch. Arrows show the path of hysteresis loop.

Figure 4

Maximum value of electroresistance effect ${\text{ER}}^{\max }$ as a function of (a) saturation polarization $P_0$, (b) dielectric constant variation ${\varepsilon }_{\min }$ and $\mathrm{\Delta }\varepsilon$, and (c) barrier height $h_{\mathrm{b}}$. The following parameters are used for all three plots: $d=1$ nm, ${\delta }_{1,2}=0.05$ nm, $V=-V_{\mathrm{s}}=-0.1$ V. Solid lines correspond to the total ER effect including effects of surface charges and image forces ${\text{ER}}^{\max }$. Dotted line shows the ER effect due to surface charges ${\text{ER}}_{\mathrm{P}}^{\max }$. Dashed-dotted line is the ER effect produced by the image forces $E_{\varepsilon }^{\max }$. (a) Dashed line shows ${\text{ER}}_{\varepsilon }^{\max }\times {\text{ER}}_{\mathrm{P}}^{\max }$. The barrier height $h_{\mathrm{b}}=0.5$ V, $\mathrm{\Delta }\varepsilon =15$, and ${\varepsilon }_{\min }=30$. $P_{1,2}$ denotes polarization where ${\text{ER}}_{\mathrm{P}}^{\max }$ has a derivative gap. (b) $P_0=30\mu \mathrm{C}/{\mathrm{cm}}^2,h_{\mathrm{b}}=0.5$ V, ${\varepsilon }_{\min }=50$ for the main graph and $\mathrm{\Delta }\varepsilon =0.5{\varepsilon }_{\min }$ for the inset. The ER effect due to surface charges disappears at $\varepsilon ={\varepsilon }_1$. (c) $P_0=35\mu \mathrm{C}/{\mathrm{cm}}^2,\mathrm{\Delta }\varepsilon =15$, and ${\varepsilon }_{\min }=30$.

Figure 5

Maximum value of electro-resistance effect ${\text{ER}}^{\max }$ (at $V=-V_{\mathrm{s}})$ as a function barrier thickness $d$ for the following system parameters: ${\delta }_1={\delta }_2=0.5$ nm, $P_0=30\mu \mathrm{C}/{\mathrm{cm}}^2,V_{\mathrm{s}}=0.1$ V, and $h_{\mathrm{b}}=0.5$ V, ${\varepsilon }_{\min }=50,\mathrm{\Delta }\varepsilon =50$. Solid lines correspond to the total ER effect including effects of surface charges and image forces. Dotted line shows the ER effect due to the surface charges, ${\text{ER}}_{\mathrm{P}}$. Dashed-dotted line is the ER effect produced by the image forces ${\text{ER}}_{\varepsilon }$.

Figure 6

Electroresistance effect as a function of applied bias for asymmetric FTJ with ${\delta }_1=0.5$ nm, ${\delta }_2=0.1$ nm, $d=1$ nm, $P_0=20\mu \mathrm{C}/{\mathrm{cm}}^2,V_{\mathrm{s}}=0.1$ V, and $h_{\mathrm{b}}=0.5$ V, ${\varepsilon }_{\min }=30,\mathrm{\Delta }\varepsilon =15$. Solid lines correspond to the total ER effect including effects of surface charges and image forces. Dotted line shows the ER effect due to the surface charges ${\text{ER}}_{\mathrm{P}}$. Dashed-dotted line is the ER effect produced by the image forces, ${\text{ER}}_{\varepsilon }$. The inset shows the dependence of the ER effect on the ratio ${\delta }_2/{\delta }_1$ at ${\delta }_1=0.05$ nm for the same parameters as in the main plot.

Figure 7

Temperature coefficient of resistance (TCR) as a function of temperature for FTJ with P(VDF-TrFE) barrier with thickness $d=1$ nm and barrier height $h_{\mathrm{b}}=0.5$ V. The inset shows dependence of the dielectric constant of P(VDF-TrFE) vs. temperature obtained using experimental data of Ref. [4].

Figure 8

Temperature resistance coefficient (TCR) as function of temperature for FTJ with ${\mathrm{SrCa}}_2\mathrm{Sr}\left({\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{COO}\right){}_6$(D) barrier with thickness $d=1$ nm and barrier height $h_{\mathrm{b}}=0.5$ V. The inset shows the dependence of the dielectric constant of ${\mathrm{SrCa}}_2\mathrm{Sr}\left({\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{COO}\right){}_6$(D) from Ref. [36].