Ab initio calculations of the rate of carrier trapping and release at dopant sites in NaI: Tl beyond the harmonic approximation

We present ab initio calculations of electron capture and release coefficients as a function of temperature for Tl dopants in the widely used scintillator NaI, which is a soft ionic crystal. The modeled capture and release events occur by transitions mediated by multiple phonon absorption and emission between states with significantly different local geometries around the trapping site. We demonstrate that such transitions are not well-described by the normal harmonic approximation to the nuclear dynamics. We go beyond the harmonic approximation by numerically solving the vibrational Schrödinger equation for the motion along/of a single effective phonon coordinate. The localized trapped state is not correctly described by semilocal density functionals, so we employ a global hybrid functional tuned to reproduce the band gap and lattice constant of the host material. Our calculations, which combine plane-wave and embedded cluster electronic structure methods are in reasonable agreement with available experimental data for detrapping and predict an unusual temperature dependence for trapping rates.

Figure 1

Left: Band gap vs the fraction of explicit exchange included in the xc functional in our CM (blue) and PWM (yellow) calculations. Right: computed lattice constant $a$ vs band gap for the PWM calculations. The experimental band gap is taken from Ref. [79] while the lattice constant is from Refs. [79, 80]. The functionals are otherwise standard PBE for the remainder of the exchange functional and the correlation functional, except for the case $\alpha =1$, which is Hartree-Fock (HF) theory (i.e., no correlation).

Figure 2

Calculated local environment of a Tl dopant in NaI using the $\mathrm{PBE}{0}_{1/3}$ exchange-correlation approximation in plane-wave (top row) and cluster (bottom row) models without (left column) and with (right column) a trapped electron. The spin-density isosurfaces (at 0.001 electrons per cubic angstrom) associated with the trapped electron are plotted in blue. Na atoms are shown in gold, I atoms in purple, and the Tl dopant is gray.

Figure 3

Configuration coordinate diagram calculated using either cluster models (+’s) or periodic boundary conditions (×’s) and HF (purple), HSE06 (blue), or $\mathrm{PBE}{0}_{1/3}$ (yellow and green). The origin of the energy scale is the energy of a free electron in the ground-state $\mathrm{T}{\mathrm{l}}^{+}$ structure, and the abscissa is Q, the effective 1D phonon coordinate. Solid lines are quadratic fits near the appropriate minimum. See the text for details.

Figure 4

Final configuration-coordinate diagram used for calculation of the vibrational structure. The ×’s mark explicitly calculated points (and agree with the green ×’s in Fig. 3), the curved solid lines are the interpolated/extrapolated adiabatic potential energy surfaces, and the horizontal solid lines mark every 20th vibrational level (excluding the $v=0$ levels for clarity), drawn between the classical turning points in the harmonic approximation.

Figure 5

Electron-phonon matrix elements and density of states from CM calculations and density of states from PWM calculations near the conduction-band minimum.

Figure 6

Capture coefficient (solid lines) and release rate (broken lines) as a function of temperature. The calculated values, shown in both panels, are in black. In the left panel, the theoretical release rate (dotted purple) is compared to Arrhenius curves from thermally stimulated luminescence measurements by Dietrich et al. [5] and Gridin et al. [51] as well as an Arrhenius curve resulting from analysis of scintillation decay curves by Kerisit et al. [33]. In the right panel, our predictions (purple) are compared to values that we would obtain if the trapped state was shifted by $\delta E=0.1\mathrm{eV}$ (green) or $\delta E=-0.1\mathrm{eV}$ (blue) in energy. The capture rate for $\delta E=-0.1\mathrm{eV}$ is below the bottom of the graph.

Figure 7

Arrhenius fit (yellow line) to the calculated capture coefficient (decorated black curve—same data as the solid purple curve in Fig. 6).

Figure 8

Heat map of the squared matrix elements ${\left|\langle \mathrm{I}\left|Q\right|F\rangle \right|}^2$ between different vibrational states of the upper (horizontal axis) and lower (vertical axis) PES's. The green +’s mark transitions that are allowed by energy conservation. Since none of the plotted quantities involves the thermal occupation factors, this figure describes both trapping and detrapping probabilities.