Controlling the spin-torque efficiency with ferroelectric barriers

Nonequilibrium spin-dependent transport in magnetic tunnel junctions comprising a ferroelectric barrier is theoretically investigated. The exact solutions of the free electron Schrödinger equation for electron tunneling in the presence of interfacial screening are obtained by combining Bessel and Airy functions. We demonstrate that the spin transfer torque efficiency, and more generally the bias dependence of tunneling magneto- and electroresistance, can be controlled by switching the ferroelectric polarization of the barrier. In particular, the critical voltage at which the in-plane torque changes sign can be strongly enhanced or reduced depending on the direction of the ferroelectric polarization of the barrier. This effect provides a supplementary way to electrically control the current-driven dynamic states of the magnetization and related magnetic noise in spin transfer devices.

Figure 1

(a) Schematics of a magnetic tunnel junction comprising two ferromagnets and a ferroelectric insulator; (b) Potential profile of the junction with positive polarization (screening) and applied voltage $V_a$.

Figure 2

Voltage dependence of (a) TMR for zero (black symbols), positive (red symbols), and negative (blue symbols) FEP and (b) TER for P (black symbols) and AP (red symbols) configurations. The parameters are the same as in the text, with $U_{\mathrm{B}}=2$ eV, $P=0,\pm 60\mu \mathrm{C}/{\mathrm{cm}}^2$, and $d=1$ nm (open symbols) or 1.5 nm (filled symbols), corresponding to $V_c^{\mathrm{L},\mathrm{R}}\approx 0.35$ V and 0.5 V, respectively.

Figure 3

Voltage dependence of the (a,c) in-plane and (b,d) out-of-plane torques exerted on the right layer for $P=0$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ (black symbols), $P=\pm 20$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ (green symbols), $P=\pm 40$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ (blue symbols), and $P=\pm 60$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ (red symbols). The parameters are $U_{\mathrm{B}}=2$ eV (a,b) $d=1$ nm and (c,d) $d=1.5$ nm, with $\theta =\pi /2$. The filled and open symbols refer to positive and negative FEP, respectively.

Figure 4

(a) Dependence of the critical voltage at which the in-plane torque changes sign as a function of the barrier thickness for $P\in [-60,+60]$ $\mu \mathrm{C}/{\mathrm{cm}}^2$; (b) dependence of the critical voltage at which the out-of-plane torque reaches its extremum as a function of the barrier thickness for $P=40$ and $60\mu \mathrm{C}/{\mathrm{cm}}^2$. The barrier height is $U_{\mathrm{B}}=2$ eV.

Figure 5

Bias dependence of the efficiency of the in-plane torque, $T_{\parallel }/j_e$, for a barrier thickness (a) $d=1$ nm and (b) $d=1.5$ nm. The filled symbols represent the efficiency calculated with $P=\pm 40$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ (black and red symbols, respectively), corresponding to (a) $V_c^{\mathrm{L},\mathrm{R}}\approx 0.23$ V and (b) 0.34 V, as explained in the text. The open symbols represent the efficiency calculated with a normal insulating barrier with asymmetric potential, i.e. in the absence of the interfacial depletion/accumulation region. The average barrier height is $U_{\mathrm{B}}=2$ eV in all the cases.

Figure 6

Spatial dependence of the spin density (a,b,c) $S_x$ and (d,e,f) $S_y$ in the right electrode calculated at the Fermi energy and perpendicular incidence $k=0$. The parameters are (a,d) $P=0$ $\mu \mathrm{C}/{\mathrm{cm}}^2$, (b,e) $P=40$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ and (c,f) $P=-40$ $\mu \mathrm{C}/{\mathrm{cm}}^2$ and $V_a=\pm 0.5$ V (black and red, respectively). The units are given in inverse volume of the unit cell ${\Omega }^{-1}$.