Space Group Symmetry of Chiral Fe-deficient van der Waals Magnet $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$ Probed by Convergent Beam Electron Diffraction

Abstract

The crystal symmetry of Fe-deficient $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$ at room temperature has been investigated by a combination of selected-area electron diffraction (SAED) and convergent-beam electron diffraction (CBED). By symmetry analysis of CBED patterns along different zone axis, the space group of $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$ at room-temperature has been identified as $P6_3mc$ (No.186), which derives from the high-symmetry parent system $\text{Fe}_{\text{3}}\text{GeTe}_{\text{2}}$ ($P6_3/mmc$) by breaking the mirror symmetry along the 6-fold rotation axis. The $P3m1$ (No.156) space group previously reported for $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$ is a subgroup of $P6_3mc$ suggesting further possible symmetry breaks in this non-stoichiometric system.

Figures (5)

• Figure 1: Electron diffraction along $[0001]$ zone axis : (a) selected area diffraction pattern (scale bar is 2 nm$^{-1}$) and (b) CBED pattern with convergence angle of 3 mrad (scale bar is 5 nm$^{-1}$), indexed by using Miller-Bravais indices.
• Figure 2: Electron diffraction along $[10\bar{1}0]$ zone axis : (a) selected area diffraction pattern (scale bar is 2 nm$^{-1}$), (b) CBED pattern with convergence angle of 2 mrad (scale bar is 2 nm$^{-1}$), and (c) another CBED pattern with smaller convergence angle of 1.25 mrad recorded at a different position suggesting presence of a vertical mirror symmetry (scale bar is 2 nm$^{-1}$).
• Figure 3: Analysis of possible symmetrical differences in $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$. First row show (a) parent $P6_3/mmc$ symmetry of $\text{Fe}_{\text{3-x}}\text{GeTe}_{\text{2}}$Verchenko15, (b) possible continuous transformation into $P6_3mc$ symmetry via small shifts of top layers along c-axis, (b) transformation into $P6_3mc$ via introduction of Fe atoms inside the vdW’s gaps, and (d) proposed by Chakraborty22 reduction of symmetry down to $P3m1$ via introduction of Fe atoms inside the vdW’s gap. Second row show results of CBED simulations for $[10\bar{1}0]$ zone axis with thickness of 54 nm and convergence angle of 2 mrad for (e) original $P6_3mmc$ space group, (f) $P6_3mc$ space group through small ($O$(pm)) continuous transformation of the unit cell, and (g) $P6_3mc$ space group through occupation of different sites within the vdW’s gap by Fe atoms. Third row show results of CBED simulations for $[0001]$ zone axis with thickness of 36.5 nm and convergence angle of 2.5 mrad for (h) original $P6_3mmc$ space group, (i) $P6_3mc$ space group through small ($O$(pm)) continuous transformation of the unit cell, and (j) $P6_3mc$ space group through occupation of different sites within the vdW’s gap by Fe atoms.
• Figure 4: Symmetry analysis of $[0001]$ CBED patterns. In a first step the original CBED pattern (Fig. \ref{['fig:0001experiment']}(b) was aligned to the $xy$ coordinate system. Subsequently, the rotational and mirror symmetries have been applied, respectively. The CBED discs have been masked before computing the normalized difference between the original and transformed CBED discs. The differences are plotted in the same gray scale as the original CBED pattern to intuitively highlight the magnitude of the differences.
• Figure 5: Symmetry analysis of $[10\bar{1}0]$ CBED patterns. In a first step the original CBED pattern (Fig. \ref{['fig:10-10experiment']}(b) was aligned to the $xy$ coordinate axis. Subsequently, the rotational and mirror symmetries have been applied respectively. The CBED discs have been masked before computing the normalized (Euclidean) difference between the original and transformed CBED discs.