Unification of the phonon mode behavior in semiconductor alloys: Theory and ab initio calculations

We demonstrate how to overcome serious problems in understanding and classification of vibration spectra in semiconductor alloys, following from traditional use of the virtual crystal approximation (VCA). We show that such different systems as InGaAs (1-$\text{bond}\to 1$-mode behavior), InGaP (modified 2-mode), and ZnTeSe (2-$\text{bond}\to 1$-mode) obey, in fact, the same phonon mode behavior—hence probably a universal one—of a percolation type (1-$\text{bond}\to 2$-mode). The change of paradigm from the “VCA insight” (an averaged microscopic one) to the “percolation insight” (a mesoscopic one) offers a promising link toward the understanding of alloy disorder. The discussion is supported by ab initio simulation of the phonon density of states at the zone center of representative supercells at intermediary composition (ZnTeSe) and at the impurity-dilute limits (all systems). In particular, we propose a simple ab initio “protocol” to estimate the basic input parameters of our semiempirical “percolation” model for the calculation of the 1-$\text{bond}\to 2$-mode vibration spectra of zinc blende alloys. With this, the model turns self-sufficient.

Figure 1

Schematic views of a $A{B}_{1-x}{C}_x$ alloy according to the VCA (a) and percolation (b) approaches. A simple gray scale code is used. In the alloy, the gray scale reinforces when the local composition becomes more like that of the corresponding pure crystal. A MREI-like correlation with the dependence of the $A\text{−}C$ TO frequency $\left(\omega \right)$ on the actual alloy composition $x$ (a) or on rescaled alloy compositions $y$ and $z$ (b), is emphasized. In case (b), critical behaviors occur at the $A\text{−}C$ $(x_C=0.19)$ and $A\text{−}B$ $(x_B=0.81)$ bond percolation thresholds. In particular, out of the percolation regime, a minor region forms a dispersion of clusters with quasistable internal structure, as evidenced by a stable gray in scheme (b). This corresponds to a stable phonon frequency. The intensity $\left(I\right)$ aspect is also indicated. For a given $A\text{−}C$ mode, this scales as the total fraction of $A\text{−}C$ bond in the alloy $\left(x\right)$ weighted by the scattering volume of the corresponding $C$-rich $\left(x\right)$ or $B$-rich $(1-x)$ host region. Subscript and superscript $B$ $\left(C\right)$ refer to the $A\text{−}B$ $\left(A\text{−}C\right)$ bond species and to the pale $B$-rich (dark $C$-rich) region, respectively. A similar frequency and/or intensity description applies to $A\text{−}B$.

Figure 2

Simplified 1-$\text{bond}\to 1$-mode TO (thick lines) MREI-VCA schemes of InGaAs (a), InGaP (b), and ZnTeSe (c). Only those TO modes actually observed in the Raman and/or IR spectra are represented. The intensity of each TO mode basically scales like the corresponding bond fraction [refer to Fig. 1a], as specified within brackets (when this is known). The optical bands used for the Elliott-CPA criterion are shown as shaded areas. The ${\omega }_{\mathrm{imp}}$ values for InGaAs, InGaP, and ZnTeSe are taken from Refs. 8, 9, 10, respectively.

Figure 3

TO (thick lines)/LO (thin lines) percolation picture for $\mathrm{Zn}{\mathrm{Te}}_{1-x}{\mathrm{Se}}_x$. This is built up from representative sets of TO (open symbols)/LO (filled symbols) frequencies taken from the literature, as indicated. The intensity of each TO mode scales as the corresponding bond fraction [see Fig. 1b], as specified within the square brackets. ZnTe:Se and ZnSe:Te refer to the impurity modes of Se and Te in ZnTe and ZnSe, respectively. The dotted lines indicate the onset of the ZnSe-like 1-$\text{bond}\to 2$-mode TO behavior just departing from the dilute limits.

Figure 4

TO (thick lines) and LO (thin lines) Raman line shapes of $\mathrm{Zn}{\mathrm{Te}}_{1-x}{\mathrm{Se}}_x$. These were calculated from the frequency and intensity TO information displayed in Fig. 3 by using Zn-Se $\left(3{\mathrm{cm}}^{-1}\right)$ and Zn-Te $\left(1{\mathrm{cm}}^{-1}\right)$ phonon dampings that scale like in the pure crystals. The bottom LO mode is divided by 2, as indicated. The ZC TO-DOS per atom for uniformly damped Zn, Se, and Te oscillators $\left(10{\mathrm{cm}}^{-1}\right)$ at $x\sim 0.5$ is shown in the inset for comparison, together with the corresponding quasirandom supercell. There, the arrows indicate a three-oscillator behavior.

Figure 5

Simplified 1-$\text{bond}\to 2$-mode TO (thick lines) percolation schemes of InGaAs (a), InGaP (b), and ZnTeSe (c), as built up from the experimental (${\omega }_{\mathrm{imp}}$, $\Delta$) values reported in Table . A direct comparison can be made with the MREI-VCA schemes of Fig. 2. The dotted lines indicate the onset of the 1-$\text{bond}\to 2$-mode TO behavior just departing from the dilute limits. The generic bond fraction corresponding to each TO branch [refer to Fig. 1b] is indicated in part (b) on the right side. The intensity of each TO mode scales accordingly. The optical bands used for the Elliott-CPA criterion are shown as shaded areas.

Figure 6

ZC TO-DOS per atom related to the dominant substituting species (shaded areas), to the isolated impurity (thin lines), and to the impurity pair (plain lines) at the Se- (upper curves) and Te-dilute (lower curves) limits of simple cubic 64-atom ZnTeSe supercells. The vibration patterns of the impurity modes are indicated with labeling (I) to (III), the impurity (Zn) species being represented by filled (open) symbols. The splitting between the mode due to the isolated impurity and the softer mode of the impurity pair provides a direct estimate of $\Delta$ (one of the two input parameters of the percolation model), as indicated by the antagonist arrows. The notation $\mathbf{q}\ne 0$ indicates a parasitical vibration mode out of the center of the Brillouin zone. In the inset, the ZC TO-DOS per atom due to the isolated Be impurity (upper curve) and to the Be impurity pair (lower curve) in ZnSe are shown, for reference purpose.