Electronic properties of semiconducting $\mathrm{Zn}(\mathrm{Si},\mathrm{Ge},\mathrm{Sn}){\mathrm{N}}_2$ alloys

We report on results of first-principles electronic structure calculations on disordered $\mathrm{Zn}(\mathrm{Si},\mathrm{Ge},\mathrm{Sn}){\mathrm{N}}_2$ alloys. These calculations on substitutional disordered alloys are carried out using the Korringa-Kohn-Rostoker Green function (KKR-GF) method in combination with the coherent potential approximation (CPA) alloy theory. The band gaps and effective masses as well as the disorder-induced finite lifetime of electronic states at the conduction band minimum and valence band maximum are evaluated by analyzing the Bloch spectral functions. Relativistic effects are found to have a small impact and in particular the influence of the spin-orbit coupling is negligible. The alloys with low Si content show band gaps and effective masses which change almost linearly with the composition. Their relatively small effective mass and long lifetime indicate a high charge carrier mobility.

Figure 1

Electronic band structures of ${\mathrm{ZnSiN}}_2$ (a), ${\mathrm{ZnGeN}}_2$ (b), and ${\mathrm{ZnSnN}}_2$ (c) represented by the BSF $A_{\mathrm{B}}(\vec{k},E)$ as a function of energy and $k$ vector. The greyscale images for the ordered compounds show an artificial finite width because of a small imaginary part of the energy (${10}^{-3}$ Ry) used for computational reasons. See the text for more information.

Figure 2

Band structure of ${\mathrm{ZnSi}}_{0.3}{\mathrm{Ge}}_{0.4}{\mathrm{Sn}}_{0.3}{\mathrm{N}}_2$ represented by the BSF $A_{\mathrm{B}}(\vec{k},E)$ calculated in a nonrelativistic (a) and fully relativistic way (b). For the fully relativistic calculations (b), the converged potential from (a) has been used. The same imaginary part of the energy as used for Fig. 1 (${10}^{-3}$ Ry) was taken. The increased width and reduced intensity of the gray lines compared to Fig. 1 reflect the smearing of the BSFs due to the disorder in the system.

Figure 3

Band gaps of the disordered alloy systems ${\mathrm{ZnSi}}_{1-x}{\mathrm{Ge}}_x{\mathrm{N}}_2$ (a), ${\mathrm{ZnGe}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$ (b), and ${\mathrm{ZnSi}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$ (c) as a function of the composition parameter $x$. The closed and open symbols indicate direct and indirect band gaps, respectively. The LDA- and QSGW-based results are taken from Ref. [1], the HSE-based results are from Refs. [9, 15], while the experimental values stem from Refs. [13, 14, 15, 17, 18, 26, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49].

Figure 4

Band gaps for the full ternary regime of $\mathrm{Zn}\left({\mathrm{Si}}_x{\mathrm{Ge}}_y{\mathrm{Sn}}_z\right){\mathrm{N}}_2$ ($x+y+z=1$) alloys obtained within the present study on the basis of the CPA. The squares show the results of the systems with indirect band gaps and the circles show the ones with direct band gaps.

Figure 5

Band gaps of ${\mathrm{ZnSiN}}_2$, ${\mathrm{ZnGeN}}_2$, and ${\mathrm{ZnSnN}}_2$ calculated for various lattice parameters. “Structure” in the legend implies that $b/a$, $c/a$, and the atomic coordinates are fixed accordingly (see the text). The vertical dotted lines indicate the original volumes of the reference systems.

Figure 6

Effective electron and hole masses, $m_{\mathrm{e}}^{*}$ (top row) and $m_{\mathrm{h}}^{*}$ (bottom row), respectively, in units of the electron rest mass $m_0$ for ${\mathrm{ZnSi}}_{1-x}{\mathrm{Ge}}_x{\mathrm{N}}_2$ (left), ${\mathrm{ZnGe}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$ (middle) and ${\mathrm{ZnSi}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$ (right). The hole masses are for the $a_1$-like subband. The QSGW-based results are from Ref. [1] and the HSE-based results are from Ref. [38].

Figure 7

Effective electron (left column) and hole (right column) masses of $\mathrm{Zn}\left({\mathrm{Si}}_x{\mathrm{Ge}}_y{\mathrm{Sn}}_z\right){\mathrm{N}}_2$ ($x+y+z=1$) for the $\vec{k}$ vector along the $k_x$ (top row), $k_y$ (middle row), and $k_z$ (bottom row) directions. The layout of the plots is the same to Fig. 4.

Figure 8

FWHM of the peaks in the Bloch spectral functions at the VBM (a) and CBM (b) of ${\mathrm{ZnSi}}_{1-x}{\mathrm{Ge}}_x{\mathrm{N}}_2$, ${\mathrm{ZnGe}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$, and ${\mathrm{ZnSi}}_{1-x}{\mathrm{Sn}}_x{\mathrm{N}}_2$. Results for ${\mathrm{Ga}}_{1-x}{\mathrm{In}}_x\mathrm{N}$ are given as a reference in addition (crosses).

Figure 9

Hole and electron lifetime due to substitutional disorder, ${\tau }_{\mathrm{h}}$ (a) and ${\tau }_{\mathrm{e}}$ (b), respectively, for the full ternary regime of $\mathrm{Zn}\left({\mathrm{Si}}_x{\mathrm{Ge}}_y{\mathrm{Sn}}_z\right){\mathrm{N}}_2$ alloys. The plot format is the same to Fig. 4. The lifetime for the pure systems, ${\mathrm{ZnSiN}}_2$, ${\mathrm{ZnGeN}}_2$, and ${\mathrm{ZnSnN}}_2$, is infinite and therefore not shown.

Figure 10

Band structures of ${\mathrm{ZnSiN}}_2$ (a) and ${\mathrm{ZnSnN}}_2$ (b) calculated with the lattice parameters of ${\mathrm{ZnGeN}}_2$.