Acoustic phonons and strain in core/shell nanowires

We study theoretically the low-energy phonons and the static strain in cylindrical core/shell nanowires (NWs). Assuming pseudomorphic growth, isotropic media, and a force-free wire surface, we derive algebraic expressions for the dispersion relations, the displacement fields, and the stress and strain components from linear elasticity theory. Our results apply to NWs with arbitrary radii and arbitrary elastic constants for both core and shell. The expressions for the static strain are consistent with experiments, simulations, and previous analytical investigations; those for phonons are consistent with known results for homogeneous NWs. Among other things, we show that the dispersion relations of the torsional, longitudinal, and flexural modes change differently with the relative shell thickness, and we identify new terms in the corresponding strain tensors that are absent for uncapped NWs. We illustrate our results via the example of Ge/Si core/shell NWs and demonstrate that shell-induced strain has large effects on the hole spectrum of these systems.

Figure 1

Static core strain and its effect on the hole spectrum of Ge/Si core/shell NWs as a function of the relative shell thickness $\gamma$. In the top figure, the nonzero components ${\epsilon }_{\perp }^c$ and ${\epsilon }_{zz}^c$ of Eqs. (39) and (40) are plotted for the parameters in the text. The resulting splitting ${\Delta }_{\mathrm{BP}}$ [Eq. (47), bottom figure] can be as large as $\sim 30$ meV and strongly affects the low-energetic hole states, as explained in detail in Ref. [16] [wherein $\delta \left(\gamma \right)={\Delta }_{\mathrm{BP}}\left(\gamma \right)$].

Figure 2

Dispersion relation of gapless phonon modes in Ge/Si core/shell NWs. (Top) The thin red lines are numerically calculated roots of the determinant that comprises the nine boundary conditions described in Sec. 5a. For the parameters in Appendix pp1 and an assumed core (shell) radius of $R_c=\text{10}\text{nm}$ $\left(R_s=\text{14}\text{nm}\right)$, phonons with gapped spectra were found at $\omega >5\times {10}^{11}{\text{s}}^{-1}$, i.e., $\hslash \omega >0.3\text{meV}$. The dashed black lines correspond to $\omega =v_{t,0}\left|q_z\right|,\omega =v_{l,0}\left|q_z\right|$, and $\omega ={\zeta }_{f,0}{q}_z^2$, respectively, and agree well with the exact result even at relatively large $q_z$. (Bottom) The mode velocities $v_{t,0}$ [torsional, Eq. (110)], $v_{l,0}$ [longitudinal, Eq. (116)], and ${\zeta }_{f,0}/R_s$ [flexural, Eq. (138)] are plotted as a function of the relative shell thickness $\gamma$.

Figure 3

Radial dependence of the displacement field (top) and core strain (bottom) in cylindrical coordinates due to the torsional phonon mode in a Ge/Si NW with core radius $R_c=\text{10}\text{nm}$ and shell radius $R_s=\text{14}\text{nm}$. Thin red lines correspond to the exact, numerical solution of the ansatz described in Sec. 5a, while dashed black lines are calculated with the algebraic expressions listed in Sec. 5b. Besides the parameters for Ge and Si in the text (see also Appendix pp1), we use $c_{q_{z}t}=0.01$ and $q_z=0.2/R_s$. The global phase factor $exp\left[i(q_{z}z-{\omega }_{q_{z}t}\tau )\right]$ is set to 1. Excellent agreement between the exact and approximate solutions is found even when $q_z$ is relatively large. We note that $0=u_r=u_z$ and $0={\epsilon }_{rr}^c={\epsilon }_{\varphi \varphi }^c={\epsilon }_{zz}^c={\epsilon }_{rz}^c$.

Figure 4

Comparison between exact results and derived formulas for the longitudinal phonon mode, analogous to Fig. 3. Good agreement is found for all components at $q_z=0.2/R_s$ assumed here, and the quality of the approximation increases with decreasing $|q_z|$. The longitudinal mode features $u_{\varphi }=0$ and $0={\epsilon }_{r\varphi }^c={\epsilon }_{\varphi z}^c$. For details, see Sec. 5c.

Figure 5

Comparison between exact and approximate solutions for a flexural phonon mode $\left(n=1\right)$, analogous to Figs. 3 and 4. Assuming again $q_z=0.2/R_s$, the top and middle figures exhibit very good agreement for the displacement and the diagonal strain components, respectively. In the bottom figure, ${\epsilon }_{rz}^c$ and ${\epsilon }_{\varphi z}^c$ are well approximated by Eqs. (157) and (158) (dashed black lines, $\tilde{\gamma }\to \infty )$. Equations (151) and (152) (not plotted, $\tilde{\gamma }\ll 1)$ yield a slightly worse but still good approximation. We note that the quantitative agreement can be improved by taking $\tilde{\gamma }\sim 1$ of the studied NW fully into account in the terms of order $c_{q_z{f}_{\pm }}\mathcal{O}\left({\delta }^3\right)$ (thin dotted lines for ${\epsilon }_{rz}^c$ and ${\epsilon }_{\varphi z}^c$, expressions too lengthy for text). For ${\epsilon }_{r\varphi }^c$, the observed deviation between Eq. (150) (dashed black line) and the exact result decreases rapidly with decreasing $|q_z|$, as expected. We illustrate that corrections of order $c_{q_z{f}_{\pm }}\mathcal{O}\left({\delta }^4\right)$ become important in the considered example by taking them into account (thin dotted line for ${\epsilon }_{r\varphi }^c$, expressions too long for text). For details, see Sec. 5d.