Optical Signatures of Periodic Magnetization: The Moiré Zeeman Effect

Detecting magnetic order at the nanoscale is of central interest for the study of quantum magnetism in general, and the emerging field of moiré magnets in particular. Here, we analyze the exciton band structure that arises from a periodic modulation of the valley Zeeman effect. Despite long-range electron-hole exchange interactions, we find a sizable splitting in the energy of the bright circularly polarized exciton Umklapp resonances, which serves as a direct optical probe of magnetic order. We first analyze quantum moiré magnets realized by periodic ordering of electron spins in Mott-Wigner states of transition metal dichalcogenide monolayers or twisted bilayers: we show that spin valley–dependent exciton-electron interactions allow for probing the spin-valley order of electrons and demonstrate that it is possible to observe unique signatures of ferromagnetic order in a triangular lattice and both ferromagnetic and Néel order in a honeycomb lattice. We then focus on semiclassical moiré magnets realized in twisted bilayers of ferromagnetic materials: we propose a detection scheme for moiré magnetism that is based on interlayer exchange coupling between spins in a moiré magnet and excitons in a transition metal dichalcogenide monolayer.

Figure 1

(a) Momentum space representation of the periodic potential problem, where the area in light gray is the first Brillouin zone. The green dot represents the zero-momentum state, which is connected through the set of reciprocal lattice vectors $\mathcal{G}$ to the first Umklapp states, in orange. (b) Schematic representation of the explored optical resonances. The solid blue (dashed gray) lines correspond to the dispersion of the transversely (linearly) polarized excitonic eigenmode. The red (blue) wavy arrows correspond to resonances belonging to the ${\sigma }^{-}$ (${\sigma }^{+}$) subspace.

Figure 2

(a) Energy spectrum of the two circularly polarized subspaces as a function of the periodic magnetic field amplitude $\mathrm{\Delta }=g{\mu }_B|B_{\mathbf{G}}|$, assuming vanishing average field. The energy of the states belonging to the ${\sigma }^{-}$ (${\sigma }^{+}$) subspace is shown in solid red (dashed blue) lines. As $\mathrm{\Delta }$ increases, the Umklapp states notably split, while the zero-momentum excitons remain quasidegenerate. (b),(c) Exciton band structure for a periodic magnetic field with vanishingly small (b) and finite strength yielding $\mathrm{\Delta }=3\mathrm{meV}$ (c). We denote the high-symmetry points of the superlattice with $m$, $\gamma$, and $\kappa$. The degeneracy of the lowest energy Umklapp scattered bands in (b) is lifted for $\mathrm{\Delta }\ne 0$ in (c). To the right, the corresponding zero-momentum spectral functions for ${\sigma }^{+}$ and ${\sigma }^{-}$ subspaces in dashed blue and solid red, respectively. For $\mathrm{\Delta }=0$, the spectral functions present a single degenerate peak for each zero-momentum exciton. For $\mathrm{\Delta }\ne 0$, the transverse Umklapp resonances in each subspace become bright, resulting in additional peaks in $\mathcal{A}(E)$. The remarkable splitting of the Umklapp peaks is a unique signature of the moiré Zeeman effect. (d) Oscillator strengths of the finite quasimomentum states as a function of $\mathrm{\Delta }$, with $a=10\mathrm{nm}$. In solid red lines (dashed blue lines), the results corresponding to the ${\sigma }^{-}$ (${\sigma }^{+}$) subspace. For $\mathrm{\Delta }\gtrsim 1.5\mathrm{meV}$, the two resonances are bright enough to act as a direct optical probe of the periodic magnetic field.

Figure 3

(a) Schematic of the exciton energy shift as a function of electron density. The difference between the ${\sigma }^{+}$ and ${\sigma }^{-}$ exciton is proportional to the effective constant magnetic field due to valley Zeeman effect. (b) Schematic of the exciton band structures around the $\gamma$ point of the ferromagnetic triangular lattice for different values of constant external magnetic field $B_e$ and $\lambda \ne \mu$. The robustness of the Umklapp peaks against constant magnetic fields allows for direct manipulation of the zero momentum resonances. (c) Spectral functions for the ferromagnetic (FM) and antiferromagnetic (AFM) honeycomb lattice, where the solid green (dotted navy blue) lines correspond to the ${\sigma }^{-}$ $({\sigma }^{+})$ subspace. We consider a moiré lattice constant of $a=15\mathrm{nm}$ and the experimentally relevant parameters $n_e\lambda =2\mathrm{meV}$ and $n_e\mu =4\mathrm{meV}$. The net magnetization in the FM leads to a large splitting of the main exciton resonances, while ${\sigma }^{+}$ and ${\sigma }^{-}$ excitons remain degenerate in the AFM. Moreover, although both scenarios present energetically split Umklapp peaks, the signatures are notably distinct. The AFM presents two Umklapp peaks for each subspace, which are subspace degenerate. On the other hand, the FM has single Umklapp peaks, which are not subspace degenerate.

Figure 4

Schematic representation of a ${\mathrm{CrI}}_3$ twisted bilayer with a generic spin distribution in the top layer. On top of the bilayer, the probe-TMD monolayer in green.