Ground state and optical conductivity of interacting polarons in a quantum dot

The ground-state energy, addition energies, and optical absorption spectra are derived for interacting polarons in parabolic quantum dots in three and two dimensions. A path integral formalism for identical particles is used in order to take into account the fermion statistics. The approach is applied to both closed-shell and open-shell systems of interacting polarons. Using a generalization of the Jensen-Feynman variational principle, the ground-state energy of a confined $N$-polaron system is analyzed as a function of $N$ and of the electron-phonon coupling constant $\alpha$. In contrast to few-electron systems without the electron-phonon interaction, three types of spin polarization are possible for the ground state of the few-polaron systems: (i) a spin-polarized state, (ii) a state where the spin is determined by Hund’s rule, and (iii) a state with the minimal possible spin. A transition from a state fulfilling Hund’s rule to a spin-polarized state occurs when the electron density is decreased. In the strong-coupling limit, the system of interacting polarons turns into a state with the minimal possible spin. These transitions should be experimentally observable in the optical absorption spectra of quantum dots.

Figure 1

Total spin of a system of interacting polarons in a parabolic quantum dot as a function of the number of electrons for $\hslash {\Omega }_0=0.5H^{*}$ (a) and for $\hslash {\Omega }_0=0.1H^{*}$ (b).

Figure 2

Addition energy of a system of interacting polarons in a parabolic quantum dot as a function of the number of electrons for $\hslash {\Omega }_0=0.5H^{*}$ (a) and for $\hslash {\Omega }_0=0.1H^{*}$ (b).

Figure 3

The total spin (a) and the addition energy (b) of a system of interacting polarons in a 2D parabolic GaAs quantum dot as a function of the number of electrons for $\hslash {\Omega }_0=0.5$ $H^{*}$. Inset: the experimentally observed addition energy vs number of electrons in a cylindrical GaAs quantum dot for two values of the diameter$D=0.5\mu \mathrm{m}$ and $D=0.44\mu \mathrm{m}$.[43]

Figure 4

Optical conductivity spectra of $N=20$ interacting polarons in CdSe quantum dots with $\alpha =0.46$, $\eta =0.656$ for different confinement energies close to the transition from a spin-polarized ground state to a ground state obeying Hund’s rule. Inset: the first frequency moment $⟨\omega ⟩$ of the optical conductivity as a function of the confinement energy.

Figure 5

The first frequency moment $⟨\omega ⟩$ of the optical conductivity as a function of the number of electrons for systems of interacting polarons in CdSe quantum dots with $\alpha =0.46$, $\eta =0.656$, and $0.143{\omega }_{\text{LO}}\left(\hslash {\Omega }_0\approx 0.04H^{*}\right)$. Open squares denote the spin-polarized ground state; full dots denote the ground state, obeying Hund’s rule; open triangles denote the ground state of the third type, with more than one partly filled shells, which is not totally spin-polarized. Inset: the total spin of the system of interacting polarons as a function of $N$.

Figure 6

Optical conductivity spectra of $N=10$ interacting polarons in a quantum dot with $\alpha =2$, $\eta =0.6$ for several values of the confinement frequency from ${\Omega }_0=0.6{\omega }_{\text{LO}}$ to ${\Omega }_0=1.4{\omega }_{\text{LO}}$. The spectrum for ${\Omega }_0={\omega }_{\text{LO}}$ corresponds to the confinement-phonon resonance.

Figure 7

The function $\Theta \left(N\right)$ and the addition energy $\Delta \left(N\right)$ for systems of interacting polarons in CdSe quantum dots with $\alpha =0.46$, $\eta =0.656$ for $\hslash {\Omega }_0=0.1H^{*}$ (panels a, b) and for ${\Omega }_0=0.04H^{*}$ (panels c, d). Open squares denote the spin-polarized ground state; full dots denote the ground state, obeying Hund’s rule; open triangles denote the ground state of the third type, with more than one partly filled shells, which is not totally spin-polarized.

Figure 8

The function $\Theta \left(N\right)$ and the addition energy $\Delta \left(N\right)$ for systems of interacting polarons in quantum dots with $\alpha =3,$ $\eta =0.25$, and ${\Omega }_0={\omega }_{\text{LO}}$ (panels a, b) and with $\alpha =3,$ $\eta =0.3$, and ${\Omega }_0=0.5{\omega }_{\text{LO}}$ (panels c, d).

Figure 9

Real (a) and imaginary (b) parts of the memory function $\chi \left(\omega \right)∕\omega$ for a system of interacting polarons in a quantum dot for $N=10,$ $\alpha =2,$ $\eta =0.6,$ and ${\Omega }_0=0.6{\omega }_{\text{LO}}$. The dashed line in panel a represents $\left({\omega }^2-{\Omega }_0^2\right)∕\omega .$ The vertical arrows in panel a indicate the roots of Eq. (44). The height of peaks in panel b represents the relative intensity of the $\delta$-like peaks of $\left[-\mathrm{Im}\chi \left(\omega \right)∕\omega \right]$.