Brillouin zone unfolding method for effective phonon spectra

Thermal properties are of great interest in modern electronic devices and nanostructures. Calculating these properties is straightforward when the device is made from a pure material, but problems arise when alloys are used. Specifically, only approximate band structures can be computed for random alloys and most often the virtual crystal approximation (VCA) is used. Unfolding methods [Boykin, Kharche, Klimeck, and Korkusinski, J. Phys.: Condens. Matter 19, 036203 (2007)] have proven very useful for tight-binding calculations of alloy electronic structure without the problems in the VCA, and the mathematical analogy between tight-binding and valence-force-field approaches to the phonon problem suggests they be employed here as well. However, there are some differences in the physics of the two problems requiring modifications to the electronic-structure approach. We therefore derive a phonon alloy band-structure (vibrational-mode) approach based on our tight-binding electronic-structure method, modifying the band-determination method to accommodate the different physical situation. Using the method, we study $\mathrm{In}\left(x\right)\mathrm{Ga}\left(1-x\right)\mathrm{As}$ alloys and find very good agreement with available experiments.

Figure 1

Example of the band-determination algorithm. Red solid line: The cumulative probability, ${\mathcal{P}}_{\mathrm{cum}}$, versus energy, shows that Band 2 falls between energies $b_1 \left({\mathcal{P}}_{\mathrm{cum}}=1+\delta \right)$ and $a_2 \left({\mathcal{P}}_{\mathrm{cum}}=2-\delta \right)$, the gray shaded region. The average slope over the step in cumulative probability is $m_{\mathrm{ref}}$ (black dotted line) and the last 5% of $\left(E,{\mathcal{P}}_{\mathrm{cum}}\right)$ points in the range are fitted to a line with slope $m$ (solid black line). Because $m<m_{\mathrm{ref}}$ the step is distinct and Band 2 can be differentiated from the next-higher band(s). The probability-weighted average energy is indicated by $E_2$.

Figure 2

Random-alloy (RA) unfolded bands for InGaAs alloys. In(0.2)Ga(0.8)As along [100] (a), [110] (b), and [111] (c); In(0.5)Ga(0.5)As along [100] (d), [110] (e), and [111] (f); In(0.8)Ga(0.2)As along [100] (g), [110] (h), and [111] (i). Because the acoustic modes are qualitatively similar only the optical modes are shown for the lowest and highest mole fractions. Dots indicate average energies and dot color indicates degeneracy, $D$: red (1), blue (2), or black (3). Black lines and gray bars denote the spread in probability for the band they surround: $0.25\le {\mathcal{P}}_{\mathrm{band}}\le D-0.25$ (black) or $0.05\le {\mathcal{P}}_{\mathrm{band}}\le D-0.05$ (gray).

Figure 3

Bondlength convergence test for the [100] In(0.5)Ga(0.5)As SCs. The SCs use the special [100] PC from Appendix pp2. Three different SC sizes are examined: $101\times 4\times 2$ (black open circles), $101\times 4\times 4$ (blue solid line and blue open squares), $101\times 8\times 4$ (red solid line). The two largest cells agree well, and therefore we use the $101\times 4\times 4$ SC since it affords accuracy at less computational cost than the $101\times 8\times 4$.

Figure 4

Projection probability (a) and cumulative probability (b) for the $101\times 4\times 4$ (blue open circles and blue solid line) and $101\times 8\times 4$ (red solid circles and red dashed line) [100] SCs 75% of the way from Γ to $X$ in the PC first Brillouin zone. The set of projection probability samples is the discrete analog of the Bloch spectral density function. The cumulative probability has obviously converged, and the projection probability must be lower for the larger cell because the sum rule, Eq. (17), fixes the cumulative probability. The larger SC has twice as many samples so each must contribute less due to the fixed cumulative probability.

Figure 5

RA and VCA bands for In(0.5)Ga(0.5)As along [110]. The RA bands are represented by dots and gray bars as in Fig. 2. Dots indicate average energies and dot color indicates degeneracy, $D$: red (1), blue (2), or black (3). Gray bars denote the spread in probability for the band they surround: $0.05\le {\mathcal{P}}_{\mathrm{band}}\le D-0.05$. The VCA bands are plotted with black solid lines.

Figure 6

Sound velocity along [111] for $\mathrm{In}\left(x\right)\mathrm{Ga}\left(1-x\right)\mathrm{As}$, computed from $v_g=d\omega /dq$ at $q=0$ for the LA mode. Open blue squares and blue line (as a guide to the eye) are the RA unfolded results while red open circles are the experimental data [31]. There is very good agreement between the RA results and experiment.