Percolation versus cluster models for multimode vibration spectra of mixed crystals: GaAsP as a case study

We reexamine the phonon mode behavior of GaAsP, the leading system among those alloys that exhibit “anomalous” vibration spectra of the one-$\text{bond}\to \text{multimode}$ type, the sign of a far-from-random substitution if we refer to the 1-$\text{bond}\to 4$-mode cluster model (microscopic insight), which has been accepted through use. In fact, we show that the phonon behavior of GaAsP obeys a basic version of the 1-$\text{bond}\to 2$-mode percolation model (mesoscopic insight), just as the phonon behaviors of InGaAs, InGaP, and ZnTeSe [O. Pagès et al., Phys. Rev. B 77, 125208 (2008)], the leading systems representing the three admitted variants of the “nominal” 1-$\text{bond}\to 1$-mode type for random alloys as covered by the modified random element isodisplacement model (macroscopic insight). With this, GaAsP and its like are rehabilitated as random alloys in principle, and further, the percolation paradigm generalizes to all types and subtypes of the traditional classification of phonon mode behavior of semiconductor alloys, based on the virtual crystal approximation. The discussion is supported by phenomenological modeling of representative Raman and infrared-reflectivity spectra of GaAsP taken from the literature via the percolation model, ab initio insight into the phonon mode behavior of basic impurity motifs, and existing experimental/ab initio bond length data.

Figure 1

Cluster type description of a random $A{\mathrm{B}}_{1-x}{\mathrm{C}}_x$ crystal in terms of five possible $A$-centered cluster units labelled $i=0$–to–4 according to the number of C atoms at the corners (lower panel). The VCA-type host medium is schematically represented by a grayscale. The corresponding 1-$\text{bond}\to 4$-TO mode behavior is shown in the upper panel. The generic forms of the TO-(frequency $\omega$, intensity $I$) $x$ dependencies for each relevant cluster type (superscript $i$) are summarized above the crystal drawing by putting an emphasis on the short $A$-C bond (subscript C). The native bond lengths are schematically indicated at $x\sim 0,1$. For a given bond the TO frequencies (thin lines) are ranked from top to bottom in the sense of the increasing fraction of the short bond in the clusters. The intensity of an individual TO mode scales as the total fraction of like bonds, i.e., as $x$ for $A$-C and $(1-x)$ for $A$-B, weighted by both the fraction $f_i\left(x\right)$ of the host cluster in the crystal and the fraction of like bonds in that cluster, i.e., $i∕4$ for $A$-C and $(4-i)∕4$ for $A$-B. The corresponding MREI dependencies ($\omega$: thick lines, $I$: within brackets) are also shown (upper panel), for comparison. For sake of completeness the TO-LO bands used for the Elliott’s criterion are indicated (shaded areas). Counterparts of the lower and upper panels for the percolation model are shown in Figs. 1(b) and 5(a) of Ref. 4.

Figure 2

(Color online) TO (thick-red lines)/LO (coupled–thin lines, uncoupled–dotted lines) percolation scheme of $\mathrm{Ga}{\mathrm{As}}_{1-x}{\mathrm{P}}_x$ as built up from representative frequency sets related to pure-TO (open symbols), pure-LO (filled symbols), as well as undetermined-symmetry (crossed symbols) modes in the literature [circles: VB–Ref. 9, stars: Strahm and McWorter–Ref. 24, triangles: Pistol and Liu–Ref. 25]. The data from Ref. 25 were uniformly blueshifted by $2.7{\mathrm{cm}}^{-1}$ so that the TO frequency in the pure GaP crystal, chosen as an arbitrary reference, matches $365{\mathrm{cm}}^{-1}$ for all three data sets. In the Ga-As spectral range the TO frequency vs $x$ dependency is better described by a straight line (thick line) than by the proper MREI curve (dashed line). The alloy-related LO-(uncoupled—“unc,” coupled) curves were subsequently calculated on this simplified TO basis. The fraction of oscillator per TO branch, which fixes the intensity of each individual TO mode, is indicated within square brackets. The circle marks a strong $\vec{E}$-coupling regime in the Ga-P spectral range. In the labeling of the modes, the (Ga-As, Ga-P) subscripts and (As,P) superscripts refer to the bond species and to the host region, respectively. Superscripts “−” and “+” refer to coupled modes.

Figure 3

Ab initio ZC TO-DOS per impurity atom of the relevant Ga-centered VB clusters at the $P$-and As-dilute limits. ${\omega }_{\mathrm{imp}}$ is indicated in each case. The symbols (∥ chain) and (⊥ chain) refer to out-of-phase cation and anion vibrations along and perpendicular to (in-plane and out-of-plane) the chain of impurity bonds forming the 2—imp. motif (a pair of neighboring substitutional impurities, see text). A finite $\Delta$ value is identified for the $P$ impurity only (refer to the double arrow).

Figure 4

(Color online) Reference percolation-type TO (thick-red curves), LO-coupled (thin curves), and LO-uncoupled (dotted curves) Raman line shapes of $\mathrm{Ga}{\mathrm{As}}_{1-x}{\mathrm{P}}_x$ as derived via Eq. (3) by using the TO-frequency (thick-red curves) and TO-intensity (refer to square brackets) information given per oscillator in Fig. 2. A small (TO, LO) individual phonon damping of $\sim 1{\mathrm{cm}}^{-1}$ was chosen for a clear overview of the whole collection of oscillators at each $x$ value.

Figure 5

Comparison of theoretical LO-coupled Raman line shapes for random $\mathrm{Ga}{\mathrm{As}}_{1-x}{\mathrm{P}}_x$ (thin lines) calculated by injecting in Eq. (3) the three-oscillator [Ga-As, ${\left(\mathrm{Ga}\text{−}\mathrm{P}\right)}^{\mathrm{P}}$, ${\left(\mathrm{Ga}\text{−}\mathrm{P}\right)}^{\mathrm{As}}$] version of the percolation model displayed in Fig. 2, with corresponding raw LO-pure Raman spectra taken from the literature (symbols, digitized from Fig. 2 of Pistol and Liu–Ref. 25, and uniformly blueshifted by $2.7{\mathrm{cm}}^{-1}$ for sake of consistency with Fig. 2).

Figure 6

(Color online) Original series of raw IR reflectivity spectra of $\mathrm{Ga}{\mathrm{As}}_{1-x}{\mathrm{P}}_x$ taken by VB (symbols, reproduced from Fig. 1–pure crystals–and Figs. 9 to 13–alloys–of Ref. 9) compared with two series of theoretical curves: cluster model [thin curves, recomputed from Eq. (4) by using the TO-(frequency, oscillator strength, phonon damping) input parameters per oscillator adjusted by VB–see Ref. 9] and percolation model [thick-red curves, calculated by injecting in Eq. (4) the corresponding information as available in Fig. 2, a slight adjustment being allowed for the phonon damping–see text].

Figure 7

Schematic view of the 1-$\text{bond}\to 2$-TO (thick lines) percolation scheme derived for GaAsP. The fractions of individual TO oscillators are indicated within brackets. The oblique rescaled-MREI segments are simplified to straight lines, for clarity. Shaded areas represent the TO-LO bands used for the Elliott’s criterion. A direct comparison can be made with the percolation schemes of InGaAs, ZnTeSe, and InGaP, the systems representing the three admitted variants of the MREI type, as displayed in Fig. 5 of Ref. 4.