Electron-phonon deformation potential interaction in core-shell Ge-Si and Si-Ge nanowires

We settle a general expression for the Hamiltonian of the electron-zone-center optical phonon deformation potential (DP) interaction in the case of nonpolar core-shell cylindrical nanowires (NWs). On the basis of a long-range phenomenological continuum model for the optical modes and by taking into account the bulk phonon dispersions, we study the size dependence and strain-induced shift on the electron-phonon coupling strengths for Ge-Si and Si-Ge NWs. We derive analytically the DP electron-phonon Hamiltonian and report some numerical results for the frequency core modes and vibrational amplitudes. Our approach allows for the unambiguous identification of the strain and confinement effects on the optical phonons at the $\Gamma$ point. We explore the dependence of mode frequencies, phonon amplitudes, and hole-DP scattering rate on the spatial symmetry and the structural parameters of these core-shell structures, which constitute a basic tool for the characterization and device applications of these novel nanosystems.

Figure 1

Frequencies of the core modes with $n=0$ and $k_z=0$ as a function of the core radius $a$ in a core-shell system grown in the [011] direction. Left panel: Ge-Si. Right panel: Si-Ge, for fixed shell thickness $b-a=1$ nm.

Figure 2

The same as Fig. 1 for $n=1$.

Figure 3

Core phonon amplitude $U_z$ for Ge-Si and Si-Ge core-shell NWs. Left panel: $n=0$, 1, and $k_z=0$. Right panel: $n=0$ and $k_{z}a=\pi /2$. In the calculation $a=2$ nm and $b=4$ nm.

Figure 4

Same as in Fig. 3 for core phonon amplitude $U_r$.

Figure 5

Core phonon amplitude $U_{\theta }$ for Ge-Si and Si-Ge core-shell NW and $k_z=0$. Left panel: $n=0$. Right panel: $n=1$. In the calculation $a=2$ nm and $b=4$ nm.