Excited states

Governing equations

(1a) Electron continuity

\[ \frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \mathbf{J}_n \;-\; R_{\mathrm{free}} \;+\; G_n, \qquad \mathbf{J}_n = -\,q\mu_n n \nabla\phi \;+\; q D_n \nabla n \]

(1b) Hole continuity

\[ \frac{\partial p}{\partial t} = -\,\frac{1}{q}\nabla \cdot \mathbf{J}_p \;-\; R_{\mathrm{free}} \;+\; G_p, \qquad \mathbf{J}_p = \;q\mu_p p \nabla\phi \;-\; q D_p \nabla p \]

(1c) Free-to-free (bimolecular) recombination

\[ R_{\mathrm{free}} = k_r\,(np - n_0 p_0) \quad\text{with}\quad D_{n,p} = \frac{k_B T}{q}\,\mu_{n,p} \]

(2)

\[ \frac{dN_S}{dt} = \frac{1}{4}\gamma N_P^{2} + \frac{1}{4}\kappa_{TT}N_T^{2} - (\kappa_{\mathrm{FRET}}P_{OD} + \kappa_S + \kappa_{ISC})N_S - \Big(\tfrac{7}{4}\kappa_{SS}N_S + \kappa_{SP}N_P + \kappa_{ST}N_T\Big)N_S \]

(3)

\[ \frac{dN_T}{dt} = \frac{3}{4}\gamma N_P^{2} + \kappa_{ISC}N_S + \frac{3}{4}\kappa_{SS}N_S^{2} - (\kappa_{DEXT}P_{OD} + \kappa_T + \kappa_{TP}N_P)N_T - \frac{5}{4}\kappa_{TT}N_T^{2} \]

(4)

\[ \frac{dN_{SD}}{dt} = \kappa_{\mathrm{FRET}}P_{OD}N_S + \frac{1}{4}\kappa_{TTD}N_{TD}^{2} - (\kappa_{SD} + \kappa_{ISCD})N_{SD} - \Big(\tfrac{7}{4}\kappa_{SSD}N_{SD} + \kappa_{SPD}N_P + \kappa_{STD}N_{TD}\Big)N_{SD} - \xi P_{HO}\big(N_{SD} - WN_{OD}\big) \]

(5)

\[ \frac{dN_{TD}}{dt} = \kappa_{DEXT}P_{OD}N_T + \kappa_{ISCD}N_{SD} + \frac{3}{4}\kappa_{SSD}N_{SD}^{2} - \kappa_{TD}N_{TD} - \frac{5}{4}\kappa_{TTD}N_{TD}^{2} - \kappa_{TPD}N_{TD}N_P \]

(6)

\[ \frac{dP_{HO}}{dt} = \beta_{sp}\kappa_{SD}N_{SD} + \big(\Gamma \xi (N_{SD} - WN_{OD}) - \kappa_{CAV}\big) P_{HO} \]

(7)

\[ N_{OD} = N_{DOP} - N_{SD} - N_{TD} \]