Paramagnetic electronic structure of CrSBr: Comparison between ab initio $GW$ theory and angle-resolved photoemission spectroscopy

We explore the electronic structure of paramagnetic CrSBr by comparative first-principles calculations and angle-resolved photoemission spectroscopy. We theoretically approximate the paramagnetic phase using a supercell hosting spin configurations with broken long-range order and applying quasiparticle self-consistent $GW$ theory, without and with the inclusion of excitonic vertex corrections to the screened Coulomb interaction ($\mathrm{QS}GW$ and $\mathrm{QS}G\widehat{W}$, respectively). Comparing the quasiparticle band-structure calculations to angle-resolved photoemission data collected at 200 K results in excellent agreement. This allows us to qualitatively explain the significant broadening of some bands as arising from the broken magnetic long-range order and/or electronic dispersion perpendicular to the quasi-two-dimensional layers of the crystal structure. The experimental band gap at 200 K is found to be at least 1.51 eV at 200 K. At lower temperature, no photoemission data can be collected as a result of charging effects, pointing towards a significantly larger gap, which is consistent with the calculated band gap of approximately 2.1 eV.

Figure 1

(a) Structure of the CrSBr unit cell (Cr: blue, S: yellow, Br: brown). (b) Bulk (black) and (001) surface (cyan) Brillouin zones (BZs) for CrSBr with high-symmetry points. Solid red lines indicate the paths along which bands have been calculated.

Figure 2

Orbital-resolved quasiparticle band structures for AFM-$b$ CrSBr within DFT, $\mathrm{QS}GW$, and $\mathrm{QS}G\widehat{W}$. The individual (colored) bandwidths indicate Cr $d$ orbital (blue) and Br $p$ orbital (red) contribution to each state.

Figure 3

$\mathrm{QS}G\widehat{W}$ band structures for different magnetic easy axes.

Figure 4

$\mathrm{QS}GW$ band structures for supercells in FM-$c$ and AFM-$c$ together with the paramagnetic approximation.

Figure 5

Illustration of charging at low temperature. (a) Photoemission intensity as a function of energy and temperature at ${\overline{\mathrm{\Gamma }}}_{10}$ for $h\nu =100$ eV (for the definition of the high-symmetry points, see Fig. 6). (b) Photoemission intensity at $k_x=0{\AA }^{-1}$ for three selected temperatures. Strong charging leads to an energy shift of the spectra below $T\approx 170$ K.

Figure 6

Electronic structure of CrSBr determined by ARPES at a photon energy of $h\nu =100$ eV. (a) Photoemission intensity integrated in a 30 meV window around the VBM. (b) Photoemission intensity in the $k_x$ direction along ${\overline{\mathrm{\Gamma }}}_{10}-{\overline{\mathrm{\Gamma }}}_{11}-{\overline{\mathrm{\Gamma }}}_{12}$ and ${\overline{\mathrm{\Gamma }}}_{00}-{\overline{\mathrm{\Gamma }}}_{01}-{\overline{\mathrm{\Gamma }}}_{02}$. (c) Corresponding cuts in the $k_y$ direction along the ${\overline{\mathrm{\Gamma }}}_{10}-{\overline{\mathrm{\Gamma }}}_{00}, {\overline{\mathrm{\Gamma }}}_{11}-{\overline{\mathrm{\Gamma }}}_{01}$, and ${\overline{\mathrm{\Gamma }}}_{12}-{\overline{\mathrm{\Gamma }}}_{02}$ directions. The red arrows mark the point at which the VBM is observed. (d) Magnification of the VBM region in panels (b) and (c). The red curve in the upper panel is the photoemission intensity at ${\overline{\mathrm{\Gamma }}}_{10}$ as a function of energy.

Figure 7

Photoemission intensity and AFM-$b \mathrm{QS}G\widehat{W}$ calculation ($h\nu =100$ eV) for two equivalent but different paths in the extended zone scheme (indicated on the right-hand side).

Figure 8

Dispersion along $k_z$. (a) Constant energy surface in a range of 30 meV around the VBM as a function of $k_z$ and $k_y$. Consecutive $\mathrm{\Gamma }$ points are marked in blue, while the BZ is shown in green. (b),(c) Energy dispersion as a function of $k_z$ at $k_y=0 {\AA }^{-1}$ (i.e., ${\overline{\mathrm{\Gamma }}}_{00}$) and $k_y=1.33 {\AA }^{-1}$ (i.e., ${\overline{\mathrm{\Gamma }}}_{10}$), respectively.