Influence of Magnetic Field and LO Phonon Effects on the Spin Polarization State Energy of Strong-Coupling Bipolaron in a Quantum Dot

On the basis of Lee–Low–Pines unitary transformation, the influence of magnetic field and LO phonon effects on the energy of spin polarization states of strong-coupling bipolarons in a quantum dot (QD) is studied by using the variational method of Pekar type. The variations of the ground state energy $E_0$ and the first excited state the energy $E_1$ of bipolarons in a two-dimensional QD with the confinement strength of QDs $\omega _0$ , dielectric constant ratio $\eta $ , electron–phonon coupling strength $\alpha $ and cyclotron resonance frequency of the magnetic field $\omega _{c}$ are derived when the influence of the spin and external magnetic field is taken into account. The results show that both energies of the ground and first excited states ( $E_0$ and $E_1)$ consist of four parts: the single-particle energy of electrons $E_\mathrm{e}$ , Coulomb interaction energy between two electrons $E_\mathrm{c}$ , interaction energy between the electron spin and magnetic field $E_\mathrm{S}$ and interaction energy between the electron and phonon $E_{\mathrm{e-ph}}$ ; the energy level of the first excited state $E_1$ splits into two lines as $E_1^{(1+1)}$ and $E_1^{(1-1)}$ due to the interaction between the single-particle “orbital” motion and magnetic field, and each energy level of the ground and first excited states splits into three “fine structures” caused by the interaction between the electron spin and magnetic field; the value of $E_{\mathrm{e-ph}}$ is always less than zero and its absolute value increases with increasing $\omega _0$ , $\alpha $ and $\omega _c$ ; the effect of the interaction between the electron and phonon is favorable to forming the binding bipolaron, but the existence of the confinement potential and Coulomb repulsive energy between electrons goes against that; the bipolaron with energy $E_1^{(1-1)}$ is easier and more stable in the binding state than that with $E_1^{(1+1)}$ .