Electron $g$-factor engineering for nonreciprocal spin photonics

We study the interplay of electron and photon spin in nonreciprocal materials. Traditionally, the primary mechanism to design nonreciprocal photonic devices has been magnetic fields in conjunction with magnetic oxides, such as iron garnets. In this work, we present an alternative paradigm that allows tunability and reconfigurability of the nonreciprocity through spintronic approaches. The proposed design uses the high spin-orbit coupling (SOC) of a narrow-band-gap semiconductor (InSb) with ferromagnetic dopants. A combination of the intrinsic SOC and a gate-applied electric field gives rise to a strong external Rashba spin-orbit coupling (RSOC) in a magnetically doped InSb film. The RSOC which is gate alterable is shown to adjust the magnetic permeability tensor via the electron $g$ factor of the medium. We use electronic band structure calculations $(\mathrm{k}·\mathrm{p}$ theory) to show that the gate-adjustable RSOC manifest itself in the nonreciprocal coefficient of photon fields via shifts in the Kerr and Faraday rotations. In addition, we show that photon spin properties of dipolar emitters placed in the vicinity of a nonreciprocal electromagnetic environment are distinct from reciprocal counterparts. The Purcell factor $\left(F_p\right)$ of a spin-polarized emitter (right-handed circular dipole) is significantly enhanced due to a larger $g$ factor while a left-handed dipole remains essentially unaffected. Our search for novel nonreciprocal material platforms can lead to electron-spin-controlled reconfigurable photonic devices.

Figure 1

The schematic represents the arrangement considered in this work. The left panel (a) shows a unit cell of ferromagnetically doped InSb (red atom denotes In while blue stands for Sb) irradiated with a beam of light (wavy line) that traverses its body and emerges on the opposite side. The passage of the light beam is governed by the constitutive parameters, ${\epsilon }_1$ and ${\mu }_1$, of InSb, which is gyrotropic with an inherent magnetization. Note that for gyrotropy to be observed, an out-of-plane magnetic field (Hz) is applied to the device. The permeability tensor in this case is significantly modified by the external Rashba spin-orbit coupling (RSOC) that exists on the InSb slab. The amplitude of transmission of an incident beam through the slab, marked as an angled wavy blue line in the middle panel (b), is therefore linked to the strength of the RSOC. The RSOC in (b) is identified by its characteristic spin-momentum locking, where the tangential green lines indicate the spin-polarization vectors. The right panel (c) is a possible realization of a gyrotropic and nonreciprocal optical device. It is fitted with a metal gate that allows a dynamic tuning of RSOC, leading to the necessary modulation of the light beam. We elucidate here, via demonstration of such optical control, an indirect but robust connection between the electron spin and diverse photonic applications.

Figure 2

The Landau dispersion for the conduction electrons of a $15.0\mathrm{nm}$ InSb slab for several values of an external $z$-axis directed magnetic field is shown here. The left panel (a) was prepared by diagonalizing the Hamiltonian; the desired InSb band parameters such as the effective mass and the fundamental band gap were obtained from an $8\times 8k·p$ Hamiltonian adapted for slablike structures. A note about the band structure calculations and their numerical implementation can be found in the Appendix and Ref. [16]. The upper (lower) set of curves in red (blue) denotes the dispersion of the spin-up (spin-down) conduction electrons. The right panel (b) is the effective $g$ factors of the conduction electrons computed directly from the Landau dispersion curves. They are shown for two values of the Rashba parameter, a dynamically tunable quantity, an attribute which we harness to describe the coupling between electron spin and optical nonreciprocity in this paper.

Figure 3

The twin optical phenomena of Kerr and Faraday rotation is shown here. The solid lines contained within the ellipses represent the polarization axes which suffer rotation (drawn separately with respective angles marked as ${\theta }_K$ and ${\theta }_F)$ as an incident light beam on the InSb slab is partly reflected and transmitted. Note that this configuration describes the polar magneto-optical Kerr effect (PMOKE) where the magnetization $\left(\mathbf{M}\right)$ is oriented normal to the plane.

Figure 4

We numerically calculate the Kerr (a) and Faraday (b) rotation which arises from reflected and transmitted rays for two gate fields and several incoming frequencies. The incident light is assumed to make an angle of $\pi /4$ with the normal to the plane of incidence. A higher electric field (which augments the $g$ factor) widens the Kerr rotation angle and also pushes the peak past the one obtained for a lower bias. In addition, the Kerr angle is negative in the same frequency range for which the permeability plots dip below the zero mark (see Fig. 5). The inset in (a) quantitatively assesses the ellipticity of the reflected beam and a profile in agreement with that of the Kerr rotation. The Faraday rotation in (b) which quantifies the plane of rotation of electric field for transmitted waves exhibits a similar behavior for a higher gate bias and records a minimum at the same frequency as noted for its Kerr counterpart. Note that the Kerr and Faraday rotation and the measure of ellipticity are evaluated using the transmission formalism whose governing equations are summarized in Eq. (14) in the main text. The material system used in these calculations is a 30.0 nm wide InSb well under an external magnetic field of 0.8 T and intrinsic magnetization of 0.3 T. The Gilbert damping constant, as usual, is set to 0.04.

Figure 5

The permeability dispersions for two different values of the $g$ factor, where we made use of Eq. (4) and set the external $z$-axis directed magnetic field to 0.8 T, are shown in the above plots. The dispersion curves that use a $g$-factor value of 22 (25) are depicted by a dotted (solid) set of lines. Additionally, the intrinsic magnetization (parallel to the external magnetic field) and the dimensionless Gilbert damping constant were assumed to be 0.3 T and 0.04, respectively. The dispersion on the left (a) shows the real and imaginary components of the diagonal elements of the permeability tensor while the right panel (b) furnishes the corresponding curves for the off-diagonal entries. Note that the dispersions for both the diagonal and off-diagonal components besides displaying a functional dependence on the $g$ factor also peak at a resonant frequency. A switch of signs is also observed for a frequency range in both cases.

Figure 6

The numerically determined Purcell factor $\left(F_p\right)$ for two sets of circularly polarized dipoles of opposite handedness $\left(d_{1,2}=1/\sqrt{2}\left[x\pm iy\right]\right)$ placed at a certain distance from a $30.0\mathrm{nm}$ InSb slab with (assumed) intrinsic magnetization is shown here. We select two values of the $g$ factor for this calculation, where the lower (higher) number corresponds to an electric field of $8\left(5\right)\times {10}^6\mathrm{V}/\mathrm{m}$. For the case of the $d_1$ dipole, a large enhancement in the Purcell factor is observed, which correlates with an increase in the electromagnetic density of states arising from a constructive interference of electric field in the vicinity of the dipole, $d_1$. The placement of the second dipole $\left(d_2\right)$, however, in contrast, leads to no significant uptick in the electric field and the $F_p$ remains close to unity. The $F_p$ also in the case of the $d_1$ dipole reaches a higher value for a $g$ factor pushed upward through a stronger gate bias. For smaller distances $\left(z_0\right)$ from the slab, a more intense electric field operates that gives rise to a more robust $F_p$ uptick; this trend falls off for larger $z_0$ values in agreement with the usual inverse square law for electric fields. Note that the arrow curving around the dipole $d_1$ in panel (a) represents the emission of a right circularly polarized light; for $d_2$, the sense of polarization of emitted light is the exact opposite.

Figure 7

The Rashba spin-orbit coupling (RSOC) leads to two nondegenerate Fermi concentric energy contours for the spin-up and spin-down ensemble (a). The right panel (b) shows the band structure of conduction electrons of a $6.0\mathrm{nm}$ InSb quantum well obtained from a $k·p$ calculation. The two “winged profiles” in the right panel (b) denote the energy contours for the spin-up (higher energy) and spin-down electrons. Notice that InSb is an ideal candidate material to observe RSOC as it satisfies the twin criteria of a large intrinsic spin-orbit coupling (0.78 eV) and a small band gap (0.43 eV at Brillouin zone center). In the present case, the Rashba coupling parameter was artificially enhanced to $4.0\mathrm{eV}\AA$ for a more vivid portrayal of the spin splitting.