Evolution of Berry phase and half-metallicity in ${\text{Cr}}_2{\text{Te}}_3$ in response to strain, filling, thickness, and surface termination

${\mathrm{Cr}}_2{\mathrm{Te}}_3$ is a ferromagnetic, quasi-two-dimensional layered material with perpendicular magnetic anisotropy, strong spin-orbit coupling, and nontrivial band topology. The nontrivial topology results in an intrinsic anomalous Hall conductivity (AHC) that switches sign under filling and biaxial strain. Thin films can exhibit half-metallicity. Using density-functional theory combined with maximally localized Wannier functions, we reveal the physical origins of the sensitivity of the sign of the AHC to strain and filling, and we determine the effect of surface termination on the half-metallicity. We find that thin films terminated on the Te layers are the most energetically stable, but only the thin films terminated on both sides with the partially occupied Cr layers are half metals. In bulk ${\text{Cr}}_2{\text{Te}}_3$, the sensitivity of the sign of the AHC to strain and filling results from the complex Fermi surface comprised of three bands. Filling of local minima and bands near anticrossings alters the local Berry curvature, consistent with the negative-to-positive switching of the AHC. Similarly, strain depopulates a local minimum, shifts a degenerate point closer to the Fermi energy, and causes two spin-orbit split bands to reverse their order. These findings provide a physical understanding of the evolution of the Berry phase, AHC, and half-metallicity in ${\text{Cr}}_2{\text{Te}}_3$.

Figure 1

(a) Top (001) and (b) side (210) views of bulk ${\mathrm{Cr}}_2{\mathrm{Te}}_3$. Interlayer distances are shown. The vertical and horizontal thin lines in (a), (b) show the unit cell from the (001) and (210) directions. The legend provides the labels for the three nonequivalent Cr atoms in the unit cell. The values for the $a$ and $c$ lattice constants and the interlayer distances of the fully relaxed structure are also shown. (c) The Brillouin zone and the high-symmetry lines used for plotting band structure and Berry phase.

Figure 2

Average magnetic moment per Cr atom (left axis) and magnetic anisotropy (right axis) as a function of $+U$ parameters of the Cr atom. Magnetic moments are in units of Bohr magnetons (${\mu }_B$).

Figure 3

Side views of the thin-film structures arranged from left to right in order of increasing in-plane lattice constant and decreasing stability. Interlayer distances are shown.

Figure 4

Spin-resolved density of states and electronic band structures of differently terminated ${\mathrm{Cr}}_{1+\delta }{\mathrm{Te}}_2$ few-layer films. Red (blue) indicates the majority (minority) spin channel for both the density of states and the electronic band structure ($E-k$) plots. The density of states of Te/Te-, Cr/Te-, and Cr/Cr-terminated bilayers are shown in (a)–(c), respectively, and the corresponding $E-k$ plots are shown in (d)–(f). The spin-resolved density of states of Te/Te-, Cr/Te-, and Cr/Cr-terminated monolayers are shown in (g)–(i), and the corresponding $E-k$ plots are shown in (j)–(l).

Figure 5

Anomalous Hall conductivity as a function of the Fermi energy for different biaxial strains, as indicated in the legend. Negative (positive) values are for compressive (tensile) strain. The black square is the experimental value for the AHC in a bulklike ${\text{Cr}}_2{\text{Te}}_3$ sample (24 u.c.) from Ref. [18]. The point at $E_F-E_{F,0}=10$ meV indicated by the red star is discussed in the text.

Figure 6

(a) Berry curvature calculated along the high-symmetry lines corresponding to the (b) electronic dispersion. Cross sections of the Fermi surface are shown in (c) for $k_z=0$ (through $\mathrm{\Gamma }$) and (d) for $k_z=b_3/2$ (the top of the Brillouin zone). The three bands are color coded. The numbered circled regions correspond to the same numbered circled regions in (b). (e) Cr $d$-orbital projections of the bands. (f) Te $p$-orbital projections of the bands.

Figure 7

(a), (b) Berry curvature contour plots in the two-dimensional $\mathrm{\Gamma }$-A-L-M $k$ plane for the equilibrium state ($E_F=0$) (a) and $E_F=10$ meV (b). (c) Difference of the Berry curvature plots (a) and (b), ${\mathrm{\Omega }}_z\left(k\right)[E_F$(10 meV)-$E_F$(0 meV)]. (c) Electronic band structure along the high-symmetry $k$ paths for 0% strain. The red dashed line shows the Fermi level shifted upwards by 10 meV. The orange circle labeled 4,5 corresponds to the circles labeled 4 and 5 in (c). (e) Band structure calculated along the upper solid lines in (a)–(c) at $k_z=b_3/3$. The path in $k$ space is shown in the inset. (f) Band structure calculated along the lower lines in (a)–(c) at $k_z=b_3/6$. The path in $k$ space is shown in the inset. The $x$-axis labels in (e) and (f) are in units of the reciprocal lattice vectors. (g), (h) Two-dimensional slices of the Fermi surface at a fixed $k_z$ as indicated by the labels on each plot. The labeled red circles in (g) and (h) correspond to the circles with the same labels in (e) and (f).

Figure 8

All of the results in the left panel are calculated under 1% compressive strain, and all of the results in the right panel are calculated under 1% tensile strain. (a), (e) Berry curvature along high-symmetry lines of the Brillouin zone. The vertical axes are truncated for clarity. The top insets in (a), (e) show the same plots over the full range of the Berry curvature. (b), (f) Electronic band structure plotted along the same high-symmetry lines as the Berry curvature. The bottom insets in (b) and (f) show the zoomed-in area of highlighted rectangular regions in the middle of the A-L $k$ path. Berry curvature plotted in the two-dimensional planes defined by the high-symmetry lines (c), (g) $\mathrm{\Gamma }$-A-L-M and (d), (h) $\mathrm{\Gamma }$-K-H-A. The two planes are illustrated in the inset of Fig. 7. The color bars show the signs and magnitudes of the Berry curvatures. (i), (j) Zoom-in of the two inner rings in the cross sections of the Fermi surfaces under compressive and tensile strain taken at the top of the Brillouin zone ($k_z=b_3/2$).

Figure 9

Phonon dispersions of (a) 2L ${\mathrm{Cr}}_7{\mathrm{Te}}_{12}$, (b) 2L ${\mathrm{Cr}}_9{\mathrm{Te}}_{12}$, (c) bulk ${\mathrm{Cr}}_2{\mathrm{Te}}_3$, (d) 1L ${\mathrm{CrTe}}_2$, (e) 1L ${\mathrm{Cr}}_4{\mathrm{Te}}_6$, and (f) 1L ${\mathrm{Cr}}_5{\mathrm{Te}}_6$.

Figure 10

Constant-temperature AIMD calculations for Cr/Te-terminated ${\mathrm{Cr}}_4{\mathrm{Te}}_6$ (a) and Cr/Cr-terminated ${\mathrm{Cr}}_5{\mathrm{Te}}_6$ (b). Red curve represents the temperature as a function of time. Insets show the crystal structures at 0 fs and 1 ps for each calculation.

Figure 11

Two-dimensional slices of the Fermi surface at different values of $k_z$ as shown.