Enhanced spin-current generation in Dirac altermagnets through Klein tunneling

Abstract

Altermagnets have recently emerged as a new platform for spintronics applications, offering spin-split electronic bands despite vanishing net magnetization. Here, we investigate spin-current generation in Dirac altermagnets and identify Klein tunneling as an efficient mechanism for enhancing spin transport. Using a low-energy Dirac model combined with scattering theory, we demonstrate that Klein tunneling in altermagnets is strongly spin-dependent and can be used to effectively control the electronic spin-current polarization by, for instance, adjusting the height, width and orientation of the potential barrier. Finally, we explore how the l-wave symmetry of the Dirac altermagnet shapes the spin-current polarization and transmission, focusing especially on the d- and g-wave cases. Particularly promising results are obtained for the g-wave Dirac altermagnet, as it is found that the presence of a potential barrier can significantly boost the spin-current polarization, even when the intrinsic polarization due to the spin-split band structure is vanishingly small. For a barrier implemented via electrostatic gating, such a mechanism would in turn allow the spin-current polarization to be switched on and off via a gate voltage.

Figures (6)

• Figure 1: Schematic of Klein tunneling in a Dirac altermagnet. (a) Orientation of the barrier with respect to the altermagnet dispersion. The coordinate system is always chosen in such a way that the barrier is along the $y$-direction. (b) Sketch of the proposed tunnel junction. An $\ell$-wave Dirac altermagnet (here, $\ell = d$) is subjected to a potential barrier of height $V_0$ between $x=0$ and $x=W$, with $W$ denoting the barrier width. The spin-split electronic bands of the altermagnet are illustrated, with red corresponding to the spin-up (w.r.t. the $z$-axis) band and blue corresponding to the spin-down band. In this case, the angle $\theta$ between the altermagnet and the barrier is chosen to be $0$. For $|V_0| \gg |E|$, where $E$ is the electron energy, the system is in the Klein tunneling regime. Applying a voltage difference $V$ across the altermagnet induces a spin-polarized current, whose polarization is strongly affected by Klein tunneling.
• Figure 2: Transmission of a $d$-wave Dirac altermagnet. (a) Low-energy structure of a $d$-wave Dirac altermagnet. (b) Spin- and angle-resolved transmission, as a function of the group velocity angle $\varphi=\arctan(v^g_{y,\sigma}/v^g_{x,\sigma})$ of the incident electron, for an angle $\theta=0^\circ$ and (c) an angle $\theta = 18^\circ$ between the barrier and the altermagnet. Red (blue) denotes spin-up (spin-down) transmission. (d) Spin- and angle-resolved transmission as a function of the barrier width $W$; the upper (lower) part of the panel corresponds to spin-up (spin-down). (e) Transmission as a function of the velocity anisotropy ratio $v_x/v_y$. (f) Spin-up and (g) spin-down transmission as a function of the barrier-region orientation angle $\theta$. Dashed lines in (d) and (e) indicate the parameters for which (b) and (c) are obtained. In all plots, results are shown for $V_0=10$ (arb. units), $E=0.4 \, V_0$, $W=7$ (arb. units), $v_F=1$ (arb. units), $\xi=0.25\, v_F$, and $\theta = 0^{\circ}$, unless stated otherwise.
• Figure 3: Transmission of a $g$-wave Dirac altermagnet. (a) Low-energy structure of a $g$-wave Dirac altermagnet. Spin- and angle-resolved transmission, as a function of the group velocity angle $\varphi=\arctan(v^g_{y,\sigma}/v^g_{x,\sigma})$ of the incident electron, for an angle $\theta=0^\circ$ and (c) an angle $\theta = 5.8^\circ$ between the barrier and the altermagnet. Red (blue) denotes spin-up (spin-down) transmission. (d) Spin- and angle-resolved transmission as a function of the barrier width $W$; the upper (lower) part of the panel corresponds to spin-up (spin-down). (e) Transmission as a function of the form factor constant $\xi$. (f) Spin-up and (g) spin-down transmission as a function of the barrier-region orientation angle $\theta$. Dashed lines in (d) and (e) indicate the parameters for which (b) and (c) are obtained. In all plots, results are shown for $V_0=5$ (arb. units), $E=0.2 \, V_0$, $W=5$ (arb. units), $v_F=1$ (arb. units), $\xi=0.05\, v_F$, and $\theta = 0^{\circ}$, unless stated otherwise.
• Figure 4: Spin-current polarization $\mathcal{P}$ in a $d$-wave Dirac altermagnet. Yellow curves show the spin-current polarization $\mathcal{P}$ for different barrier parameters: (a) $V_0 = 20$ (arb. units), $eV = 0.05 \,V_0$, $W = 0.1$ (arb. units); (b) $V_0 = 20$ (arb. units), $eV = 0.05 \,V_0$, $W = 0.5$ (arb. units); (c) $V_0 = 50$ (arb. units), $eV = 0.02 \,V_0$, $W = 0.1$ (arb. units); (d) $V_0 = 50$ (arb. units), $eV = 0.02 \,V_0$, $W = 0.5$ (arb. units), as a function of the velocity anisotropy ratio $v_x/v_y$. The gray line indicates the spin-current polarization in the absence of a barrier, i.e., the intrinsic polarization of the Dirac altermagnet. Insets show the spin-resolved current ratio between the barrier ($J_{x,\sigma}$) and no-barrier ($J^{(0)}_{x,\sigma}$) cases, with red for spin-up electrons and blue for spin-down electrons, and we have used barrier angle $\theta = 0^{\circ}$. (e) Spin-current polarization corresponding to the parameters of panel (c), shown as a function of the barrier angle $\theta$ ranging from $0^\circ$ to $45^\circ$. For clarity, the curves for different $\theta$ values are vertically offset. The spacing between the black horizontal lines corresponds to $\Delta \mathcal{P} = 0.25$, and each successive curve represents an increase of $5^\circ$ in $\theta$. In all plots, results are shown for $v_F=1$ (arb. units) and $T = 0$.
• Figure 5: Spin-current polarization $\mathcal{P}$ in a $g$-wave Dirac altermagnet. Yellow curves show the spin-current polarization $\mathcal{P}$ for different barrier parameters: (a) $V_0 = 5$ (arb. units), $eV = 0.2 \,V_0$, $W = 2.5$ (arb. units); (b) $V_0 = 5$ (arb. units), $eV = 0.2 \,V_0$, $W = 5$ (arb. units); (c) $V_0 = 10$ (arb. units), $eV = 0.1 \,V_0$, $W = 2.5$ (arb. units); (d) $V_0 = 10$ (arb. units), $eV = 0.1 \,V_0$, $W = 5$ (arb. units), as a function of the form factor constant $\xi$. The gray line indicates the spin-current polarization in the absence of a barrier, i.e., the intrinsic polarization of the Dirac altermagnet. Insets show the spin-resolved current ratio between the barrier ($J_{x,\sigma}$) and no-barrier ($J^{(0)}_{x,\sigma}$) cases, with red for spin-up electrons and blue for spin-down electrons, and we have used barrier angle $\theta = 0^{\circ}$. (e) Spin-current polarization corresponding to the parameters of panel (c), shown as a function of the barrier angle $\theta$ ranging from $0^\circ$ to $45^\circ$. For clarity, the curves for different $\theta$ values are vertically offset. The spacing between the black horizontal lines corresponds to $\Delta \mathcal{P} = 0.25$, and each successive curve represents an increase of $2.5^\circ$ in $\theta$. In all plots, results are shown for $v_F=1$ (arb. units) and $T = 0$.