Dielectric function of ${\mathrm{CuBr}}_x{\mathrm{I}}_{1-x}$ alloy thin films

We study the dielectric function of ${\mathrm{CuBr}}_x{\mathrm{I}}_{1-x}$ thin film alloys using spectroscopic ellipsometry in the spectral range between 0.7 eV and 6.4 eV, in combination with first-principles calculations based on density functional theory. Through the comparison of theory and experiment, we attribute features in the dielectric function to electronic transitions at specific $\mathbf{k}$-points in the Brillouin zone. The observed band gap bowing as a function of alloy composition is discussed in terms of different physical and chemical contributions. The band splitting at the top of the valence band due to spin-orbit coupling is found to decrease with increasing Br-concentration, from a value of 660 meV for CuI to 150 meV for CuBr. This result can be understood considering the contribution of copper $d$ orbitals to the valence band maximum as a function of the alloy composition.

Figure 1

Experimental (black squares) and calculated (solid red lines) spectra of the pseudodielectric functions $\langle {\varepsilon }_1\rangle$ (left column) and $\langle {\varepsilon }_2\rangle$ (right column) by means of the transfer matrix technique shown to be exemplary for angles of incidence (AOI) of ${45}^{\circ }$ and ${65}^{\circ }$ of ${\mathrm{CuBr}}_{0.5}{\mathrm{I}}_{0.5}$ thin film.

Figure 2

Experimental spectra of the real (${\varepsilon }_1$) and imaginary (${\varepsilon }_2$) parts of the dielectric function of ${\mathrm{CuBr}}_x{\mathrm{I}}_{1-x}$ thin films as a function of the alloy composition for $0\le x\le 1$. The main peaks are marked with vertical arrows and labeled with $E_0$, $E_0+{\mathrm{\Delta }}_0$, $E_1$, and $E_{\text{0}}^{\prime }$. The spectra have been offset vertically for the sake of clarity. The inset shows a zoom of ${\varepsilon }_2$ in the vicinity of the excitonic resonances at the $\mathrm{\Gamma }$-point. The dashed red arrows highlight the bowing of the excitonic resonance $E_0$ and the monotonic shift of the $E_0+{\mathrm{\Delta }}_0$ transition, respectively.

Figure 3

Top: Band structures of CuI (a) and CuBr (b) calculated using PBE0 including spin-orbit coupling. The main electronic transitions around the symmetry point in the Brillouin zone are indicated by vertical black arrows and labeled consistently with the notation of Fig. 2. The colors distinguish groups of bands. Bottom: Calculated absorption spectra (i.e., imaginary part of the dielectric function) of CuI. The contributions of all bands at selected symmetry points and contributions of transitions involving different groups of bands over the entire Brillouin zone are shown in panels (c) and (d), respectively. The definition of how the region around the symmetry points was set is given in the computational details. Analogous calculated spectra for CuBr can be found in Sec. III of the Supplemental Material [57].

Figure 4

Energy difference $\mathrm{\Delta }{E}_{\text{0}}\left(x\right)=E_{\text{0}}\left(x\right)-E_{\text{lin}}\left(x\right)$ between the measured transition energy $E_0$ and the weighted average of the ternary transition energies $E_{\text{lin}}$ as a function of alloy composition. The black symbols represent the experimental values at 300 K, while the red symbols represent the values resulting from the calculated band structures for the corresponding alloy compositions for the lowest energy configurations. The solid lines represent the model fit using $bx(1-x)$.

Figure 5

(a) Experimentally observed (black symbols) as well as theoretically calculated values (red symbols) for the spin-orbit splitting at the $\mathrm{\Gamma }$-point ${\mathrm{\Delta }}_0$ as a function of the alloy composition. The dashed gray line indicates the almost linear decrease of the spin-orbit splitting with increasing Br content. (b) The effect of Br content on the contribution of copper $d$ orbitals to the VBM at the $\mathrm{\Gamma }$ point. Again, the experimental data are indicated by black symbols, while the values obtained from the DFT calculations are shown in red. Fat band structures of CuI (c) and CuBr (d) highlighting the contribution of copper to the corresponding bands. The figures were made using pyprocar [91].