Band offsets in InP/ZnSe nanocrystals evaluated using two-photon transitions analysis

Abstract

We present a semi-analytical theoretical kp-study of the energy structure and optical transitions in spherical core-shell InP/ZnSe nanocrystals. We use the eight-band Kane model and the six-band Luttinger Hamiltonian in the spherical approximation to calculate the electron and hole energy spectra, respectively. The influence of the Coulomb interaction is considered perturbatively. The one- and two-photon absorption spectra are calculated as functions of the band offsets between the InP core and ZnSe shell. Exciton states responsible for the main features in the two-photon absorption spectra of InP/ZnSe nanocrystals are identified and the spectral dependence of the linear-circular dichroism signal is predicted. We show that in the presence of inhomogeneous broadening, the transition to the ground two-photon-active exciton state can be hidden behind intense transitions to higher-lying states. A comparison of the calculated one- and two-photon absorption spectra with the available experimental data shows that, depending on the lattice strain in the InP core, the range of possible valence band offsets is 0.85-1 eV. The determined range exceeds the natural valence band offset of 0.57 eV and indicates the presence of electric dipoles formed by the preferential Zn-P bonds at the InP/ZnSe heterointerface.

Figures (5)

• Figure 1: (a) Schematic of an InP/ZnSe NC with core radius $a_c$ and the total NC radius $a$. (b) and (c) show the energy potential profiles for electrons with barrier (band offset) $U^e_B=E_c^{\rm ZnSe} -E_c^{\rm InP}$ and hole with barrier $U_B^h = E_v^{\rm InP} -E_v^{\rm ZnSe}$ corresponding to the case of preferential P-Zn (b) or In-Se (c) bonds at the InP/ZnSe heterointerface, respectively. Gradient arrows show the direction of the sum interface dipole. The dotted lines show the natural band offsets defined as the differences of the bulk energy band extrema with respect to the vacuum level.
• Figure 2: (a) Electron and hole energy levels given in the electron representation for a InP/ZnSe NC with $a_c=1.45$ nm and $a=4.2$ nm as functions of the band offsets $U_B^e$ and $U_B^h$ with fixed $U_B^e+U_B^h=1.4$ eV. Electron and hole energies are counted from $E_{\rm c}^{\rm InP}$ and $E_{\rm v}^{\rm InP}$, respectively. Hole states with energies above the dashed line $-U_B^h$ and electron states with energies below the dashed line $U_B^e$ are classified as localized in the InP core. Electron and hole energy levels dependencies on the total NC radius $a$ with fixed core radius $a_c=1.45$ nm for (b) the natural valence band offset $U_B^h=0.57$ eV Wei1998 and (c) valence band offset $U_B^h=0.85$ eV determined from modeling of the one- and two-photon absorption data.
• Figure 3: The heat maps of the one-photon absorption intensity ${\cal I}^0_{\rm 1PA}(\hbar\omega)$ for different valence band offset values calculated for the inhomogeneous broadening of each exciton transition (a) $\Gamma=10$ meV and (c) $\Gamma=50$ meV, respectively. Red lines in panels (b,d) show slices of the spectra ${\cal I}^0_{\rm 1PA}(\hbar\omega)$ from panels (a,c) for $U_B^h=0.85$ eV (horizontal red line ). Vertical red ($\rm I_{\rm 1PLE},\rm II_{\rm 1PLE}$) and black lines ($\rm I_{\rm TDA},\rm II_{\rm TDA}, \rm III_{\rm TDA}$) correspond to the one-photon transition energies determined in Ref. Respekta2025 from 1PLE and differential absorption spectra, respectively. Black lines in panels (b,d) show the calculated differential absorption spectra $\Delta{\cal I}_{\rm 1PA}(\hbar\omega)$. (e,f) Schematic representation of one-photon transitions to electron states of the $S$- and $P$-symmetry, respectively.
• Figure 4: The heat map of the linearly-polarized two-photon absorption intensity ${\cal I}_{\rm 2PA}^{\rm lin}(2\hbar\omega)$ for different valence band offset values, calculated for the inhomogeneous broadening of each exciton transition (a) $\Gamma=10$ meV and (c) $\Gamma=50$ meV, respectively. Solid and dashed white lines correspond to $nP_{3/2}1S_e$ and $nP_{1/2}1S_e$ exciton energies, respectively. Solid and dashed black lines correspond to $nS_{3/2}1P_e$ and $nS_{1/2}1P_e$ exciton energies, respectively. Panels (b,c) show cutoffs of the heat maps from panels (a,b) at the level $U_B^h=0.85$ eV (horizontal red line). Vertical dashed lines correspond to the two-photon transition energies ($\rm I_{\rm 2PLE},\rm II_{\rm 2PLE},\rm III_{\rm 2PLE}$) observed in Ref. Respekta2025. (e,f) Schematic representation of two-photon transitions to electron states of the $S$- and $P$-symmetry, respectively.
• Figure 5: The two-photon absorption spectrum in the case of linear ${\cal I}_{\rm 2PA}^{\rm lin}(2\hbar\omega)$ (blue) and circular ${\cal I}_{\rm 2PA}^{\rm circ} (2\hbar\omega)$ (black) polarization of photons, calculated for the inhomogeneous broadening of each exciton transition (a) $\Gamma=10$ meV and (b) $\Gamma=50$ meV, respectively. (c) Spectral dependence of the linear-circular dichroism signal for absorption spectra from panel (b) and the ${\cal I}_{\rm 2PA}^{\rm lin}(2\hbar\omega)$ spectrum calculated for the inhomogeneous broadening of each exciton transition $\Gamma=10$ meV.