Resonant tunnel magnetoresistance in double-barrier planar magnetic tunnel junctions

We present a theoretical approach to calculate the spin-dependent current and tunnel magnetoresistance (TMR) in a double-barrier magnetic tunnel junction (DMTJ), in which the magnetization of the middle ferromagnetic metal layer can be aligned parallel or antiparallel in relation to the fixed magnetizations of the left and right ferromagnetic electrodes. The electron transport through the DMTJ is considered as a three-dimensional problem, taking into account all transmitting electron trajectories as well as the spin-dependent momentum conservation law. The dependence of the transmission coefficient and spin-polarized currents on the applied voltage is derived as an exact solution to the quantum-mechanical problem for the spin-polarized transport. In the range of the developed physical model, the resonant tunneling, nonresonant tunneling, and enhanced spin filtering can be explained; the simulation results are in good agreement with experimental data.

Figure 1

Schematic view of the planar DMTJ. Thick red and blue arrows show possible magnetization directions of the ferromagnetic layers. $L_{1\left(2\right)}$ are the insulator thicknesses, and $L_W$ is the width of the middle ferromagnetic layer ${\mathit{FM}}^{\mathrm{W}}$. The black arrow inside the cone within the solid angle ${\Omega }_L$ indicates the direction of the electron trajectory, which has the trajectory angle ${\theta }_L$ with the $z$ axis.

Figure 2

Schematic energy diagram for the potential profile of the ${\mathit{FM}}^{\mathrm{L}}{/\mathrm{I}}_1{/\mathit{FM}}^{\mathrm{W}}{/\mathrm{I}}_2{/\mathit{FM}}^{\mathrm{R}}$ DMTJ with applied bias $V_a$, where $V$ are voltage drops on the first and second barriers ($V_a=2V$). The $U_{B1\left(2\right)}$ are the heights of the barriers above $E_{\mathrm{F}}$. The enhanced parabolic curves present dispersion relations for spin-up electrons ($\uparrow$ arrows) and correspond to the spin-up majority conductance subbands for the parallel (P) case. The small parabolic curves belong to the spin-down minority subbands with spin-down electrons ($\downarrow$ arrows). The arrows inside the brackets of the middle layer show the electron spin direction for the antiparallel (AP) case. The electron spin-conduction channels passing through minority or majority subbands are determined by the electron-tunneling trajectory with conserved spin direction. They are shown at the bottoms of the subbands as blue dash-dot-dotted lines and red dashed lines for the P case and as dash-dotted and solid lines for the AP case.

Figure 3

Transmission coefficients versus $V_a$ for four configurations of the conduction channels at each of the P and AP magnetic moment alignments and with two fixed trajectory angles: ${\theta }_{L,s}=0.0{}^{\circ }$ (solid lines) and ${\theta }_{L,s}=10{}^{\circ }$ (dashed lines). The parameters of the conduction bands’ spin polarizations are ${\delta }_{L\left(R\right)}=0.38$, ${\delta }_W=0.96$. Other parameters are as follows: $L_{1\left(2\right)}=12\AA$, $L_W=13.6\AA$, $U_{B1\left(2\right)}=1.8\mathit{eV}$, $V_{01\left(2\right)}=0.1\mathit{eV}$, and $m_{1\left(2\right)}=0.4m_e$.

Figure 4

TMR as a function of the middle ferromagnetic layer ${\mathit{FM}}^{\mathrm{W}}$ thickness $L_W$, calculated with the same parameters as in Fig. 3. The four curves correspond to different electron effective masses $m_{1\left(2\right)}$ of the insulating barriers.

Figure 5

Resonant, flat-dome TMR behavior for different ${\delta }_{L\left(R\right)}$ (${\delta }_R={\delta }_L$) and fixed ${\delta }_W=0.96$, $m_{1\left(2\right)}/m_e=0.4$. The other parameters are the same as in Fig. 3.

Figure 6

Bias voltage dependence of the normalized TMR of DMTJs in the case of coherent tunneling. $L_{1\left(2\right)}=25\AA$, $L_W=10\AA ,$ $m_{L,W,R}=0.8m_e$, $m_{1,2}=0.4m_e$.

Figure 7

TMR versus voltage for single and double barrier magnetic tunnel junctions in the case of consecutive tunneling (SMTJs connected in series). Numerical parameters for SMTJ: $U_B=2.8$ eV, $L=23\AA ;$ for DMTJ: SMTJ-1(2): $U_{B1\left(2\right)}=$ 2.7 eV, $L=25\AA ,$ $m_{1\left(2\right)}=0.46m_e$, $m_{L\left(R\right)}=0.76m_e$.