Supplemental information for “Colloidal Type-II CdS/ZnSe NCs : Synthesis, Electronic Structures, and Spectroscopic Properties”
S.A. Ivanov, A. Piryatinski, J. Nanda, S. Tretiak, D. Werder, V. I. Klimov
Theoretical model.
To describe quantum states in a spherically symmetric Type-II core/shell nanocrystals (NC) characterized by the core radius R and the shell width H, we use effective mass approximation. We assume the existence of single energy bands for both electrons and holes. This assumption represents simplification compared to multi-band models that explicitly take into account mixing between, e.g., different valence sub-bands.1 However, this simplified approach still allows us to capture essential trends of core- and shell-size dependences …show more content…
(6)
The remaining contribution to the net Coulomb energy comes from the self-interaction of each charge with its own image, which is the so-called “dielectric solvation energy”
()()aaaaHRaaarVrrdrVˆ02ρ∫+=. (7)
In above expressions, ()()2rraaℜ=ρ is the electron or hole density, and a = e, h; ()herrW,ˆ is a spherically symmetric component of the point charges direct Coulomb interaction operator, is a spherically symmetric component of the polarization energy operator, and ( herrU,ˆ ()rVˆ is the dielectric solvation energy operator.
To obtain the explicit form of these operators we solve analytically the Poisson equation for a potential due to a point charge in the core/shell structure with the standard boundary conditions that account for discontinuity of the dielectric constant at hetero-interfaces.3 As a result, we obtain the following solutions:
3
()()()()()()[](hecescheheherrqRrRrrRrRrrW,max2,ˆ2εεεθθθθ−+−+−−−= (8) ()()()()()(HRqRqrRrRrrUsesecscehehe+−−−−−−=εεεεεεθθ11,ˆ22 (9) where cε , sε and eε are the dielectric constants of the core, the shell, and the environment, respectively, and ()xθ is the Heaviside step function. The expression for the dielectric solvation energy is more complex and can be represented as a sum of two …show more content…
Taking into account all relevant Coulomb terms, we can write the following expression for the energy of the 1S single-exciton state that comprises the 1S electron and the 1S hole heehehhegXVVUWEEEE++++++=12, (13) where Eg12 is the indirect energy gap of the hetero-NCs. The latter expression was used to calculate emission energies as a function of R and H. First three terms represent the band gap and kinetic energies of electron and hole, respectively, and the last four terms are components of total Coulomb correction: Veh = Weh + Ueh + Ve + Vh.
References:
1. Efros, A. L.; Rosen, M. Annu. Rev. Mater. Sci. 2000, 30, 475.
4
2. Haus, J. W.; Zhou, H. S.; Honma, I.; Komiyama, H. Phys. Rev. B 1993, 47(3), 1359.
3. Jackson, J. D.; Classical Electrodynamics, John Wiley, New York, 1975.
S.A. Ivanov, A. Piryatinski, J. Nanda, S. Tretiak, D. Werder, V. I. Klimov
Theoretical model.
To describe quantum states in a spherically symmetric Type-II core/shell nanocrystals (NC) characterized by the core radius R and the shell width H, we use effective mass approximation. We assume the existence of single energy bands for both electrons and holes. This assumption represents simplification compared to multi-band models that explicitly take into account mixing between, e.g., different valence sub-bands.1 However, this simplified approach still allows us to capture essential trends of core- and shell-size dependences …show more content…
(6)
The remaining contribution to the net Coulomb energy comes from the self-interaction of each charge with its own image, which is the so-called “dielectric solvation energy”
()()aaaaHRaaarVrrdrVˆ02ρ∫+=. (7)
In above expressions, ()()2rraaℜ=ρ is the electron or hole density, and a = e, h; ()herrW,ˆ is a spherically symmetric component of the point charges direct Coulomb interaction operator, is a spherically symmetric component of the polarization energy operator, and ( herrU,ˆ ()rVˆ is the dielectric solvation energy operator.
To obtain the explicit form of these operators we solve analytically the Poisson equation for a potential due to a point charge in the core/shell structure with the standard boundary conditions that account for discontinuity of the dielectric constant at hetero-interfaces.3 As a result, we obtain the following solutions:
3
()()()()()()[](hecescheheherrqRrRrrRrRrrW,max2,ˆ2εεεθθθθ−+−+−−−= (8) ()()()()()(HRqRqrRrRrrUsesecscehehe+−−−−−−=εεεεεεθθ11,ˆ22 (9) where cε , sε and eε are the dielectric constants of the core, the shell, and the environment, respectively, and ()xθ is the Heaviside step function. The expression for the dielectric solvation energy is more complex and can be represented as a sum of two …show more content…
Taking into account all relevant Coulomb terms, we can write the following expression for the energy of the 1S single-exciton state that comprises the 1S electron and the 1S hole heehehhegXVVUWEEEE++++++=12, (13) where Eg12 is the indirect energy gap of the hetero-NCs. The latter expression was used to calculate emission energies as a function of R and H. First three terms represent the band gap and kinetic energies of electron and hole, respectively, and the last four terms are components of total Coulomb correction: Veh = Weh + Ueh + Ve + Vh.
References:
1. Efros, A. L.; Rosen, M. Annu. Rev. Mater. Sci. 2000, 30, 475.
4
2. Haus, J. W.; Zhou, H. S.; Honma, I.; Komiyama, H. Phys. Rev. B 1993, 47(3), 1359.
3. Jackson, J. D.; Classical Electrodynamics, John Wiley, New York, 1975.