Electronic structure of $\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ as observed by angle-resolved photoemission spectroscopy

We investigate the electronic band structure and Fermi surfaces (FSs) of $\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ single crystals with angle-resolved photoemission spectroscopy. We evidence hole bands centered around the $M$ points and electron bands centered around the $\mathrm{\Gamma }$ point in reciprocal space. Electron and hole bands exhibit an open, tubular structure along the $c$ axis, confirming the quasi-two-dimensional character of this highly anisotropic, nanolamellar compound. Dependence of the photoionization cross sections on beam light polarization and orientation allows us to assess the orbital character of each observed band locally. Despite some differences, density functional theory calculations show a good agreement with experiment.

Figure 1

(a) Schematics of the ARPES experimental setup configuration in the S-pol mode and (b) configuration in the P-pol mode. In both configurations the light beam and polarization stay horizontal, and the $x, y$, and $z$ axis are attached to the sample; $z$ is always normal to the sample. In (a) the $y$ axis stays vertical and the sample is rotated around it, so that the $x$ and $z$ axis move with respect to the light beam and the detector slit. The detector slit is vertical. ${\theta }_y$ is the angle between the line defined by the emitted electron beam detected in the middle of the detector slit (“mid-position”) and the $z$ axis. ${\theta }_x$ is the angle between the same line and an electron beam line detected in another, given part of the slit. ${\theta }_x$ and ${\theta }_y$ define the value of $k_x$ and $k_y$ in reciprocal space. In (b) the sample is rotated around the $x$ axis so that both the $y$ and $z$ axis move with respect to the light beam and slit, as explained in the figure. The detector slit is horizontal. ${\theta }_x$ is now describing the sample rotation and ${\theta }_y$ the angle between the mid-position line and an arbitrary one in the detector slit. ${\theta }_k$ is the angle between the emitted electron beam and the normal to the sample ($z$ axis): it is the physical angle which determines the photoionization cross section of a given orbital and is the variable appearing in Eq. (1). (c) A schematic of the $\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ unit cell.

Figure 2

$\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ band structure calculated from the wien2k software. $E_F$ is chosen as the energy origin. The main orbital character of each band is indicated by the color code defined in the figure.

Figure 3

$\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ Fermi surfaces, as predicted from wien2k DFT calculations and xcrysden software.

Figure 4

(a) Predicted cross section of the FS in the $\mathrm{\Gamma }$-plane $(k_z=0)$. The yellow halo schematically indicates which bands should give a substantial ARPES signal in the S-pol mode if the $\mathrm{\Gamma }$ point roughly corresponds to the angle values ${\theta }_x={\theta }_y=0$. The halo is obtained by comparing the orbital characters indicated in Fig. 2 with the photoionization cross sections estimated in Fig. 5; we make it appear whenever there is a marked $d{z}^2$ character, or when there is a $d_{xz}+d_{yz}$ character and substantial angle values. (b) FS cross section in the A-plane and P-pol mode. It is worth noting that a general asymmetry in the ARPES signal should be obtained along the wave vector direction parallel to ${\theta }_y$.

Figure 5

Orbital component of the photoionization differential cross section as a function of the angles, calculated in the P-pol configuration and the simplest level approximation.

Figure 6

Orbital component of the photoionization differential cross section as a function of the angles, calculated in the S-pol configuration and the simplest level approximation.

Figure 7

Orbital component of the photoionization differential cross section as a function of the angles, calculated in the S-pol configuration and using the closed-form expression of Ref. [13].

Figure 8

Fermi-surface images of $\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ obtained by ARPES with S-pol configuration and $h\nu =100\mathrm{eV}$ photons (a) and with P-pol configuration and $h\nu =50\mathrm{eV}$ photons (b), respectively. Polarization directions are indicated with arrows. Yellow solid and dashed lines are results of DFT calculations in $\mathrm{\Gamma }\mathrm{KM}$ and AHL plane, respectively. ARPES measurement lines in Figs. 9, 10, and 11 are indicated with indices ($hv$-dep ($S/P$-pol), 1–6). Experimentally observed Fermi surfaces in the S (P)-pol modes have been indicated as $\alpha$ (${\alpha }^{\prime }$), $\beta$ (${\beta }^{\prime }$), and $\gamma$ (${\gamma }^{\prime }$).

Figure 9

$k_z$ dependence of MDC spectra at $E_F$ (a) and Fermi-surface image (b) obtained by photon-energy-dependent ARPES performed at the slices indicated in Fig. 8. (c) Interplane Fermi surface calculated by DFT along $\mathrm{\Gamma }\mathrm{K}$-LH (gray solid lines) and $\mathrm{\Gamma }\mathrm{M}$-HA (gray dashed lines), respectively. Q2D topology of the calculated Fermi surface has been expected as shown with yellow shaded lines. Though the observed Fermi surfaces show little $k_z$ dependence at the $k_{\parallel }$ position, we have estimated the inner potential as $V_0=19\mathrm{eV}$ in Eq. (2) from the systematic intensity variations.

Figure 10

Band structure images of $\mathrm{C}{\mathrm{r}}_2\mathrm{AlC}$ obtained by ARPES along no. 1, $\mathrm{\Gamma }\mathrm{K}$ (a); no. 2, $\mathrm{\Gamma }\mathrm{M}$ (b); no. 3, $\mathrm{\Gamma }\mathrm{K}$(AH) (c); and no. 4, $\mathrm{\Gamma }\mathrm{M}$(AL) (d) slices indicated in Fig. 8, respectively. MDCs at $E_F$ are indicated at the top of each image to estimate the $k_F$ points. White solid and dashed lines are results of DFT calculation along $\mathrm{\Gamma }\mathrm{K}/\mathrm{\Gamma }\mathrm{M}$ and AH/AL lines, respectively.

Figure 11

(a–c) Detailed band dispersions at the holelike FS around the M point (a), the large (b), and small (c) electron-like FSs around the $\mathrm{\Gamma }$ point, respectively. Red solid and black dashed lines are the peak positions of MDC spectra and the fitting curves based on a standard free-electron model. (d) Comparison of the width of the MDC spectra obtained by the fitting with Lorentz functions, which should correspond to the imaginary part of the self-energy, e.g., the lifetime of the quasiparticles. Yellow shaded energy ranges around 30 and 80 meV correspond to intervals where the calculated phonon density of state is predicted to be high (from Ref. [18]).