Understanding photoluminescence of ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ doped with post-transition-metal ions using first-principles calculations

Doped lead-free halide perovskites have been widely reported for impressive photoluminescence properties. Herein we study the mechanisms of photoluminescence in ${\mathrm{Se}}^{4+}$-, ${\mathrm{Te}}^{4+}$-, ${\mathrm{Sb}}^{3+}$-, and ${\mathrm{Bi}}^{3+}$-doped ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ and also in this undoped host via first-principles calculations with hybrid density functionals. The results show that the main photoluminescence in the host, as well as isovalent and aliovalent dopants, can be attributed to highly localized self-trapped excitons composed of an electron on Zr and a $V_k$ center (molecular-like ${\mathrm{Cl}}_2^{-}$), $M{\mathrm{Cl}}_6$ ($M={\mathrm{Se}}^{4+},{\mathrm{Te}}^{4+}$), and $M{\mathrm{Cl}}_5$ ($M={\mathrm{Bi}}^{3+},{\mathrm{Sb}}^{3+}$) complexes, respectively. The systematic underestimation of the emission energies is discussed and is attributed to the over-relaxation of the excited-state geometric structures. Our results illustrate the photoluminescence processes and excited-state dynamics in host and aliovalent dopant of ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$, which may inspire further revelations of the mechanisms of photoluminescence of other materials in the tetravalent halide perovskite family.

Figure 1

The band structure (a) and the density of states (b) of ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ accompanied with (c) the square of the electric dipole transition dipole moment and (d) the absorption spectrum obtained via the calculation of the dielectric tensor. All data are achieved with the HSE06 hybrid density functional. In (a), the bands in purple, dark green, and light purple have Cl-$p$, Zr-$d$, and Cs-$s$/Cl-$p$ as the main components. In (a) and (b), the VBM is set to zero for convenience, and the HSE06 band gap is improved to 4.99 eV from the value of 3.65 eV predicted by the PBE method. In (d), the 6.23 eV absorption peak is close to the value 6.54 eV read off from the measured absorption spectrum reported in [46]. See Fig. S2 in the Supplemental Material [45] for the PBE counterpart of this figure.

Figure 2

(a) Allowed ranges of the relative chemical potentials of elements to sustain the ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ host, and (b)–(d) the formation energies calculated via the PBE method under Cl-rich (b), Cl-moderate (c), and Cl-poor (d) condition as specified in panel (a). The symbols $E_F$, CB, and VB in the legend represent the Fermi energy determined by charge neutrality and the conduction and valence bands, and the two integers such as “$0|-1$” mark the place where the valence of the defect switches from 0 to $-1$ as the Fermi energy increases. It is noted that the Fermi energy is given relative to VBM via the HSE06 method, and the VBM and CBM obtained by the PBE calculation are at 0.96 and $4.60\mathrm{eV}$ when the reference potentials of PBE and HSE06 results are aligned.

Figure 3

The two most stable interstitial candidate sites in the ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ host: (a) the ${\mathrm{Cl}}_6$ site and (b) the ${\mathrm{Cl}}_2$ site. The silver, green (large and small), and red spheres in both panels are ${\mathrm{Cs}}^{+}$, ${\mathrm{Zr}}^{4+}$, ${\mathrm{Cl}}^{-}$ and interstitial sites, respectively.

Figure 4

(a) The configuration coordinate diagram of the STE in the pristine ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ host. (b),(c) The squared orbital functions of the electron and the hole in the octahedral type STE [left one in (a)]. (d),(e) The same as (b),(c) but in the STE Zr–${\mathrm{Cl}}_2^{-}$ that contains a ${\mathrm{Zr}}^{3+}$ (an electron on ${\mathrm{Zr}}^{4+}$) and a ${\mathrm{Cl}}_2^{-}$ ($V_k$ center) formed by a hole on two combined ${\mathrm{Cl}}^{-}$. (f) The energy curve of STE along the linear interpolation between high-symmetry STE configuration $O_h$ ($C_{4v}$ for the case with a $V_{\mathrm{Cl}}^{+}$) and the STE Zr–${\mathrm{Cl}}_2^{-}$ (the case with a $V_{\mathrm{Cl}}^{+}$), showing no energy barrier along the considered relaxation routes.

Figure 5

Formation energy line segments for defects involving Sb doping under a Cl-moderate circumstance. The upper bound of the chemical potential Sb under this condition is determined by the coexistence of ${\mathrm{Cs}}_3{\mathrm{Sb}}_2{\mathrm{Cl}}_9$, and the appropriate value is adopted. Refer to Fig. 2 for other notation.

Figure 6

Vacuum referred bind energies of CTLs and polaron levels in the band gap calculated with the HSE06 method. Note: the e-polaron is an extra electron localized on the ${\mathrm{Zr}}^{4+}$ ion, and the h-polaron is a hole on two adjacent ${\mathrm{Cl}}^{-}$ ions forming ${\mathrm{Cl}}_2^{-}$ after relaxation. The $+{\mathrm{v}}_{\mathrm{Cl}}$ to the right of (Sb/${\mathrm{Bi})}_{\mathrm{Zr}}$ is the compounded defect [(Sb/${\mathrm{Bi})}_{\mathrm{Zr}}+V_{\mathrm{Cl}}$]. The electronic configurations for the CTLs of defects (excluding the polarons) above the dashed line are ${\left(ns\right)}^{2}np/{\left(ns\right)}^2$ and those underneath are ${\left(ns\right)}^2/\left(ns\right)$, except that (0/1) of (Sb/${\mathrm{Bi})}_{\mathrm{Zr}}$ is ${\left(ns\right)}^1/{\left(ns\right)}^0$.

Figure 7

The configuration coordinate diagram of ${\mathrm{Te}}_{\mathrm{Zr}}^0$ (${\mathrm{Te}}^{4+}$) in ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$ (a) and the squared orbital functions of the excited state at configurations $ez$ (b) and $cz$ (c), which represent elongated and compressed ${\mathrm{TeCl}}_6^{2-}$ octahedra, respectively. $\mathrm{\Delta }$(Te-Cl) in panel (a) is the difference of bond length of the axial (Te-Cl) bond from the average.

Figure 8

The configuration coordinate diagram of ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$:${\mathrm{Sb}}^{3+}$ along three linearly interpolated routes grd–MMCT, MMCT–sp, and sp–grd. Here the structures “grd,” “MMCT,” and “sp” represent the equilibrium geometric configuration of the ground state, (${\mathrm{Sb}}_{\mathrm{Zr}}+{\mathrm{Zr}}_{\mathrm{Zr}}^{-}$), and the ${}^3{P}_1$ excited state, respectively. The curves “grd,” “MMCT,” and “p1” and “p3” plot the energies of the ground state, MMCT, and the ${}^3{P}_1$ and ${}^1{P}_1$ states, respectively. NR is short for nonradiative relaxation and denotes two important relaxation processes. It is noted that under (near) highly symmetric structures “grd” and “MMCT,” the ${}^1{P}_1$ and ${}^3{P}_1$ are the states of ${\mathrm{Sb}}^{3+}$ with one electron excited from Sb-$s$ to $j=1/2$ (the lowest two) and $j=3/2$ (the other four) Sb-$p$ orbitals split dominantly by SOC. Under other distorted structures, the $sp$ electronic configuration splits to many more states due to the combination of SOC, ligand-field, and Coulomb interactions. Here the energies of the ${}^1{P}_1$ and ${}^3{P}_1$ states are estimated by the single electron excitations from Sb-$s$ to the lowest and the third Sb-$p$ throughout.

Figure 9

(a) The configuration coordinate diagram of ${\left[{\mathrm{SbCl}}_5\right]}^{2-}$ complex in ${\mathrm{Cs}}_2{\mathrm{ZrCl}}_6$, and (b),(c) the square of the orbital functions nominally named Sb-$5p$ and Sb-$5s$ in the relaxed electronic excited states. $\mathrm{\Delta }$(Sb-Cl) in panel (a) means the bond length difference of the $C_4$ axial (Sb-Cl) bond with respect to the average of the five bonds. The system at ${}^3{P}_1$ or ${}^1{P}_1$ state will relax nonradiatively to the luminescent state to emit a photon.