Intersubband plasmon-phonon modes in quantum wires at high temperatures

Coupled intersubband plasmon-phonon modes are studied in a multisubband parabolic quantum wire at room temperatures. These modes are found by calculating the spectral weight function which is related to the inelastic Raman spectra. We use a 13 subband model. The plasmon-phonon coupling strongly modifies the dispersion relation of the intersubband modes in the vicinity of the optical phonon frequency ${\omega }_{LO}$. Extra modes show up as a result of the electron-phonon interaction. We carefully study the density and temperature dependence of these extra modes. We also show that coupled intersubband plasmon-phonon modes should be observed for temperatures as high as 300 K.

Figure 1

(Color online) The plasmon dispersion relations obtained from the peaks in (a) $S^{\text{even}}(q,\omega )$ and (b) $S^{\text{odd}}(q,\omega )$ for $T=50\mathrm{K}$ and $N_e={10}^6{\mathrm{cm}}^{-1}$. Thin solid lines show the dispersion relations without an electron-phonon interaction. The horizontal dashed lines indicate the frequencies of LO and TO phonons, ${\omega }_{LO}=5.17{\omega }_0$ and ${\omega }_{TO}=4.75{\omega }_0$, respectively. The inset shows the vicinity of $\omega ={\omega }_{LO}$. The intensity scale there is changed according to the change in the $\omega$-axis scale.

Figure 2

(Color online) The plasmon dispersion relations at zero temperature (solid lines) obtained from the zeros of (a) $\det \mid {\epsilon }_{\alpha \beta }^{\text{even}}(q,\omega )\mid =0$ and (b) $\det \mid {\epsilon }_{\alpha \beta }^{\text{odd}}(q,\omega )\mid =0$. Open dots represent the plasmon dispersion relations obtained from the peaks in (a) $S^{\text{even}}(q,\omega )$ and (b) $S^{\text{odd}}(q,\omega )$. The electron density is the same as in Fig. 1. The horizontal dashed lines indicate the frequencies of LO and TO phonons, ${\omega }_{LO}=5.17{\omega }_0$ and ${\omega }_{TO}=4.75{\omega }_0$, respectively. The inset shows the vicinity of $\omega ={\omega }_{LO}$. The green (dark) and yellow (bright) regions describe the SPE continua related to the first and second subbands, respectively.

Figure 3

The spectral weight (a) $S^{\text{even}}(q,\omega )$ and (b) $S^{\text{odd}}(q,\omega )$ for $q=0.05k_0$ and $T=50\mathrm{K}$. The electron density varies from $N_e=1.0\times {10}^6{\mathrm{cm}}^{-1}$ (the lowest curve) to $N_e=2.35\times {10}^6{\mathrm{cm}}^{-1}$ (the top curve) with a difference of $0.15\times {10}^6{\mathrm{cm}}^{-1}$. The insets indicate the position of the peaks as a function of $N_e$. Vertical and horizontal dashed lines indicate the frequencies of TO and LO phonons.

Figure 4

(Color online) The temperature dependence of the peaks in (a) $S^{\text{even}}(q,\omega )$ and (b) $S^{\text{odd}}(q,\omega )$ for $q=0.05k_0$ and $N_e=1.0\times {10}^6{\mathrm{cm}}^{-1}$. The colors and the horizontal dashed lines indicate the same as in Fig. 1. The insets show the vicinity of the frequency $\omega ={\omega }_{LO}$.

Figure 5

(Color online) Intersubband coupled modes $(1,n)$ obtained by mapping the spectral weight function [(a) and (c)] $S^{\text{even}}(q,\omega )$ and [(b) and (d)] $S^{\text{odd}}(q,\omega )$ at $T=300\mathrm{K}$. The electron density in (a) and (b) is $N_e={10}^6{\mathrm{cm}}^{-1}$, while we take $N_e=2.22\times {10}^6{\mathrm{cm}}^{-1}$ in (c) and (d). Horizontal dashed lines indicate the frequencies of TO and LO phonons.