Exploring non-trivial band structure and spin polarizations in $d$-wave altermagnets tailored by anisotropic optical fields

Abstract

The subject of the present paper is a detailed theoretical investigation of the energy spectrum and bandgaps, as well as collective properties and linear response, in $d$-wave altermagnets in the presence of an off-resonance optical dressing field. We consider the altermagnets with both $d_{x^2-y^2}$ and $d_{xy}$ pairing symmetries and focus on anisotropic dressing fields applied to an anisotropic and non-linear electron Hamiltonian. We have uncovered several crucial properties of the resulting electron-dressed state; specifically, we found that a finite bandgap is opened by linearly polarized irradiation, a phenomenon not observed in Dirac materials. A number of crucial properties of the electron dressed states in the presence of the linearly polarized light can be uncovered only in the second-order perturbation expansion, which is often omitted. We demonstrate that introducing an anisotropic driving field leads to several subtle yet important changes in the Edelstein susceptibilities of altermagents, enabling the fine-tuning of their spin polarizations. All these results must be in high demand due to the rapidly developing fields of spintronics and device physics.

Figures (12)

• Figure 1: (Color online) The schematics for a setup of a two-dimensional altermagnetic material (d-wave type) in the presence of an off-resonance dressing field with different polarizations (generally elliptical; circular, and linear, as limiting cases). The electrostatic gating provided by the two electrodes enables Rashba spin-orbit coupling in this material.
• Figure 2: (Color online). Energy spectrum of a $d$-wave altramagnet with $d_{x^2-y^2}$ pairing symmetry in the absence of the spin-orbit coupling ($\rho=0$). Panels $(a)$ and $(b)$ demonstrate the spin-resolved ($s=\pm1$) anisotropic energy dispersions $\epsilon_s(\vec{\bf k})$ obtained from Eq. \ref{['genen1']} as the functions of the $k_x$- and $k_y$- components of the wave vector $\vec{\bf k}$. The black and red lines correspond to the values of spin index $s=\pm 1$. Plots $(c)$ and $(d)$ represent the constant-energy cuts corresponding to $\epsilon_1 = 2.0\,E_{(0)}$ and $\epsilon_1 = 1.0\, E_{(0)}$. Here, the red and black curves are related to the positive and negative out-of-plane spin polarizations $\text{sign}[ \langle \hat{S}_z \rangle ] = \text{sign}[ \langle \Psi_s(\vec{\bf k}) \vert \hat{\Sigma}^{(2)}_z \vert \Psi_s(\vec{\bf k}) \rangle \,] = \pm 1$, which are in principle not equivalent to the values of the spin index $s=\pm 1$.
• Figure 3: (Color online) Energy spectrum of a $d$-wave altramagnet with $d_{x^2-y^2}$ pairing symmetry in the presence of the spin-orbit coupling ($\rho=0.53$). Panels $(a)$ and $(b)$ demonstrate the spin-resolved ($s=\pm1$) anisotropic energy dispersions $\epsilon_s(\vec{\bf k})$ obtained from Eq. \ref{['genen1']} as the functions of the $k_x$- and $k_y$- components of the wave vector $\vec{\bf k}$. The black and red lines correspond to the values of spin index $s=\pm 1$. Plots $(c)$ and $(d)$ represent the constant-energy cuts corresponding to $\epsilon_1 = 2.0\,E_{(0)}$ and $\epsilon_1 = 1.0\, E_{(0)}$. Here, the red and black curves are related to the positive and negative out-of-plane spin polarizations $\text{sign}[ \langle \hat{S}_z \rangle ] = \text{sign}[ \langle \Psi_s(\vec{\bf k}) \vert \hat{\Sigma}^{(2)}_z \vert \Psi_s(\vec{\bf k}) \rangle \,] = \pm 1$, which are in principle not equivalent to the values of the spin index $s=\pm 1$.
• Figure 4: (Color online) Energy spectrum of a $d$-wave altramagnet with $d_{x^2-y^2}$ pairing symmetry in the presence of the spin-orbit coupling ($\rho=0.53$) and an off-resonance dressing field with effective coupling parameter $\mathcal{K}_\omega = 0.6\,k_{(0)}$ and $\beta = 0.9$ (nearly circular polarization). Panels $(a)$ and $(b)$ demonstrate the spin-resolved ($s=\pm1$) anisotropic energy dispersions $\epsilon_s(\vec{\bf k})$ obtained from Eq. \ref{['genen1']} as the functions of the $k_x$- and $k_y$- components of the wave vector $\vec{\bf k}$. The black and red lines correspond to the values of spin index $s=\pm 1$. Plots $(c)$ and $(d)$ represent the constant-energy cuts corresponding to $\epsilon_1 = 2.0\,E_{(0)}$ and $\epsilon_1 = 1.0\, E_{(0)}$. Here, the red and black curves are related to the positive and negative out-of-plane spin polarizations $\text{sign}[ \langle \hat{S}_z \rangle ] = \text{sign}[ \langle \Psi_s(\vec{\bf k}) \vert \hat{\Sigma}^{(2)}_z \vert \Psi_s(\vec{\bf k}) \rangle \,] = \pm 1$, which are in principle not equivalent to the values of the spin index $s=\pm 1$.
• Figure 5: (Color online) Energy spectrum of a $d$-wave altramagnet with $d_{x^2-y^2}$ pairing symmetry in the absence of the spin-orbit coupling ($\rho=0$) but exposed to an off-resonance dressing field with effective coupling parameter $\mathcal{K}_\omega = 0.6\,k_{(0)}$ and $\beta = 0.9$ (nearly circular polarization). Panels $(a)$ and $(b)$ demonstrate the spin-resolved ($s=\pm1$) anisotropic energy dispersions $\epsilon_s(\vec{\bf k})$ obtained from Eq. \ref{['genen1']} as the functions of the $k_x$- and $k_y$- components of the wave vector $\vec{\bf k}$. The black and red lines correspond to the values of spin index $s=\pm 1$. Plots $(c)$ and $(d)$ represent the constant-energy cuts corresponding to $\epsilon_1 = 2.0\,E_{(0)}$ and $\epsilon_1 = 1.0\, E_{(0)}$. Here, the red and black curves are related to the positive and negative out-of-plane spin polarizations $\text{sign}[ \langle \hat{S}_z \rangle ] = \text{sign}[ \langle \Psi_s(\vec{\bf k}) \vert \hat{\Sigma}^{(2)}_z \vert \Psi_s(\vec{\bf k}) \rangle \,] = \pm 1$, which are in principle not equivalent to the values of the spin index $s=\pm 1$.