Calculation of electronic density of states induced by impurities in $\mathrm{Ti}{\mathrm{O}}_2$ quantum dots

We have theoretically calculated and analyzed the electronic density of states (DOS) in crystalline $\mathrm{Ti}{\mathrm{O}}_2$ nanoparticles in air (or vacuum) affected by the presence of impurities near the particle surface. The effect of temperature has been incorporated by changing the permittivity value of the quantum dot, in accordance with reported dependences. Our model is described by a truly three-dimensional effective mass Hamiltonian with a realistic finite steplike confining potential and the Coulomb terms, including dielectric mismatch effects. The presence of impurities in nanometer systems naturally induces band tailing with details depending on the dielectric response of involved media. Our calculations predict a dependence of the DOS on temperature. Moreover, a high degree of electronic density delocalization within the quantum dots is found, which has important implications on developing transport models.

Figure 1

One-electron energy levels of a 10 nm radius spherical $\mathrm{Ti}{\mathrm{O}}_2$ nanocrystal with a hydrogenic donor impurity located at 9 nm from the QD center, as a function of the QD dielectric constant. Insets show the common logarithmic representation of the density of states in the vicinity of the conduction band edge of a statistical distribution of doped $\mathrm{Ti}{\mathrm{O}}_2$ QDs (with an average 10 nm radius and a dispersion of 10%), for two different values of the QD dielectric constant. In all cases, the outer medium is air or vacuum.