Tl_{6}SI_{4} is a promising room-temperature semiconductor radiation detection material. Here, we report density functional calculations of native defects and dielectric properties of Tl_{6}SI_{4}. Formation energies and defect levels of native point defects and defect complexes are calculated. Donor-acceptor defect complexes are shown to be abundant in Tl_{6}SI_{4}. High resistivity can be obtained by Fermi level pinning by native donor and acceptor defects. Deep donors that are detrimental to electron transport are identified and methods to mitigate such problem are discussed. Furthermore, we show that mixed ionic-covalent character of Tl_{6}SI_{4} gives rise to enhanced Born effective charges and large static dielectric constant, which provides effective screening of charged defects and impurities.

## I. INTRODUCTION

Collecting radiation-generated charge carriers from a semiconductor under bias is one of the main methods of radiation detection.^{1,2} A semiconductor radiation detection material should be a dense material and contain heavy elements for efficiently absorbing high-energy radiation. High resistivity (>10^{9} Ω cm) is required to reduce dark current. Therefore, the band gap of the semiconductor should be reasonably large (>1.5 eV) to suppress thermal carriers for room temperature radiation detection. In order to detect radiation events, the radiation generated charge carriers need to diffuse through the detector material to reach electrodes. This requires excellent carrier transport efficiency and hence high-quality single-crystal materials. There is continuous interest in searching for new semiconductor radiation detection materials that meet these requirements.

Halides that contain ns^{2} ions (e.g., TlBr,^{3} CH_{3}NH_{3}PbI_{3}^{4,5}) have recently been shown to exhibit exceptional good carrier transport properties. The ns^{2} ions are the ions with outer electron configuration of ns^{2}, e.g., Tl^{+}, Pb^{2+}, and Bi^{3+}. Unlike typical compound semiconductors, which have cation-s derived conduction bands, semiconductors with ns^{2} cations have p-like conduction bands.^{6–8} The strong hybridization between the cation-p and anion-p states results in more delocalized conduction and valence bands. The occupied cation-s states also hybridize with the anion-p states, which further increases the dispersion of the valence band. As a result, the electron and hole effective masses of halides with ns^{2} ions are often small and comparable to each other.^{7,8} Furthermore, the mixed ionic-covalent character of halides with ns^{2} ions can lead to enhanced Born effective charges and large dielectric constants as reported previously.^{6,7,9,10} The resulting strong dielectric screening reduces carrier trapping and recombination at charged defects and impurities.

The mixing of halides with chalcogenides further increases covalency of the material and enables the tuning of the band gap to values appropriate for room-temperature radiation detection.^{11} Recently, a Tl-based chalcohalide, i.e., Tl_{6}SI_{4}, has been shown to exhibit promising properties for radiation detection applications.^{12} Tl_{6}SI_{4} has a tetragonal P4/mnc crystal structure and a band gap of 2.07 eV (at 24 K).^{13} The reported resistivity and electron μτ products (μ is the mobility and τ is the lifetime of the charge carriers) are ∼10^{10} Ω cm and 2.1 × 10^{−3} cm^{2}·V^{−1}, respectively. These properties are comparable to those of CdZnTe, which is the current leading room-temperature semiconductor radiation detection material.^{14,15}

Defects play important roles in carrier trapping and recombination in semiconductors. Photoluminescence (PL) measurements of Tl_{6}SI_{4} showed strong donor-acceptor recombination at low temperatures.^{13} The two observed emission energies are 1.55 eV and 1.66 eV, which are related to deep defect levels. On the other hand, the strong thermal quenching of the PL emission indicates that shallow defects are involved. The shallow Tl vacancy (*V*_{Tl}) and the deep S vacancy (*V*_{S}) and their complexes were suggested to be responsible for the observed PL emissions. The intense PL emissions suggest high defect concentrations.

In this paper, we show density functional calculations of native defects in Tl_{6}SI_{4}. The anion vacancies (*V*_{S} and *V*_{I}) are the main electron traps. *V*_{Tl} and anti-site defects (S_{I} and I_{S}) are shallow defects. These results are similar to those for Tl_{6}SeI_{4}.^{10} We further studied the defect complexes, i.e., *V*_{S}–*V*_{Tl}, *V*_{S}–2*V*_{Tl}, *V*_{S}-S_{I}, and *V*_{I}–*V*_{Tl}. $VS2+$ binds strongly with $VTl\u2212$ with binding energy of 0.56 eV. (*V*_{S}–*V*_{Tl})^{+} is the dominant donor defect in semi-insulating Tl_{6}SI_{4} under Tl-poor or S-poor conditions. The binding between the native donor and acceptor defects make the defect levels shallower and less detrimental to carrier transport. Calculations of Born effective charges and dielectric constants show that dielectric screening in Tl_{6}SI_{4} is mainly due to the lattice polarization. The relatively large static dielectric constant provides effective screening to the charged defects and impurities.

## II. METHODS

Density functional calculations were used to calculate various properties of Tl_{6}SI_{4}. Born effective charges, dielectric constant, and defect formation energies were calculated using standard Perdew, Burke, and Ernzerhof (PBE)^{16} generalized gradient approximation (GGA). In defect calculations, the band gap was corrected using Heyd-Scuseria-Ernzerhof (HSE) hybrid functionals.^{17} In the HSE calculations, we included 25% non-local Hartree Fock exchange and set the range separation parameter at 0.2. The PBE and HSE band gaps calculated with and without spin-orbit coupling are given in Table I. The PBE calculations underestimated the band gap as expected while the HSE calculations yielded band gaps in good agreement with the experimental value. The PBE and HSE calculations were performed using projector augmented wave method (PAW)^{18} as implemented in the VASP code.^{19} Tl *d*-electrons were included as part of the valence states. The cut off energy for the plane waves was set at 259 eV and all forces were minimized to below 0.02 eV/Å in structural relaxation calculations.

We used a 2 × 2 × 2 supercell (176 atoms if defect free) and a 2 × 2 × 2 k-point mesh in the defect calculations. Experimental lattice parameters (space group P4/mnc, #128, a = b = 9.1758 Å, and c = 9.5879 Å) were used in all calculations.^{12} The defect calculations were performed without the spin-orbit interaction. Spin-orbit coupling gives a somewhat smaller band gap as shown in Table I, but would not in general be expected to significantly change the defect formation energies.^{10}

Defect formation energies are given by

where $ED$ and $Eh$ are the total energies of the defect-containing and the host (i.e., defect-free) supercells. Formation of a defect involves exchange of atoms with their respective chemical reservoirs. The second term in Eq. (1) represents the change in energy due to such exchange of atoms, where *n _{i}* is the difference in the number of atoms for the

*i*th atomic species between the defect-containing and defect-free supercells. $\mu i$ is the chemical potential for the

*i*th atomic species relative to $\mu iref$, which is the chemical potential for the elemental bulk form. The third term in Eq. (1) represents the change in energy due to exchange of electrons with its reservoir. $\epsilon VBM$ is the energy of the valence band maximum (VBM) and $\epsilon f$ is the Fermi energy relative to the VBM.

The VBM and the conduction band minimum (CBM) from the PBE calculation were corrected using the band gap from the HSE calculations. Deep defect level positions relative to band edges were obtained by calculating the defect levels using PBE functionals and then referencing them to band edges corrected by hybrid functional calculations. This method is justified by recent studies that showed that the deep defect levels calculated using local density approximation/GGA and hybrid functionals align with each other in the absolute scale.^{20–23} This approach is tested by directly calculating the defect levels for *V*_{S} using HSE calculations. The defect levels for *V*_{S} calculated using the two approaches (i.e., direct HSE calculations of defect levels vs. PBE calculated defect levels referenced to HSE calculated band edges) differ by well less than 0.1 eV. More details are given Sec. III C. Corrections to the defect formation energy due to image charges and potential alignment (between the host and a charged defect supercell)^{24,25} were applied wherever appropriate.

The shallow defect states, which have mainly the character of bulk electronic states, are spatially extended and cannot be meaningfully treated in a small supercell because the computational errors in calculating level positions could exceed ionization energies of shallow defect levels.^{26,27} In our calculations, a donor (acceptor) defect is considered shallow if adding electrons (holes) would result in electron (hole) filling of the conduction (valence) band. The positions of these shallow defect levels are not calculated. It suffices to know that these defect levels are shallow and do not significantly affect carrier transport. The focus of this work is on the deep electron traps that are most detrimental to electron transport.

The transition level of a defect, $\epsilon (q/q\u2032)$, corresponding to a change in its charge state between $q$ and $q\u2032$, is given by the Fermi level, at which the formation energies, $\Delta H(q)$ and $\Delta H(q\u2032)$, for charge states $q$ and $q\u2032$ are equal to each other

## III. RESULTS AND DISCUSSION

### A. Born effective charges and dielectric properties

Halides are typically ionic compounds. The mixing of a halide and a chalcogenide in Tl_{6}SI_{4} reduces ionicity. Tl has low electronegativity. Tl-6p-derived conduction band is spatially extended and has strong hybridization with the valence band states as demonstrated by the calculated density of states.^{12} These all lead to enhanced covalency. The mixed ionic and covalent character gives rise to enhanced Born effective charges and large static dielectric constant in Tl_{6}SI_{4}. The calculated Born charges are larger than their respective nominal ionic charges by about a factor of two, as shown in Table II. The enhanced Born effective charges indicate strong lattice polarization. As a result, the lattice contribution to the static dielectric constant are significant, i.e., $\epsilon ionXX=11.60$ and $\epsilon ionZZ=10.43$. The electronic contribution to the static dielectric constant are calculated to be $\epsilon optXX=7.54$ and $\epsilon optZZ=8.11$, respectively. Combining the electronic and lattice parts, the static dielectric constant $\epsilon st$ is calculated to be $\epsilon stXX=19.14$ and $\epsilon stZZ=18.54$. Note that the electronic contribution to $\epsilon st$ is likely overestimated somewhat due to the smaller band gap obtained in PBE. The screening is mainly provided by the lattice polarization rather than the electronic polarization. The calculated static dielectric constant is large for a compound of over 2 eV band gap. Such a large $\epsilon st$ may provide effective screening of the charged defects and impurities and therefore may reduce the carrier scattering and trapping.

Z^{*}
. | Nominal ionic charge . | a
. | b
. | c
. |
---|---|---|---|---|

Tl1 | 1 | 1.94 | 1.94 | 1.96 |

Tl2 | 1 | 1.97 | 1.97 | 2.09 |

S | −2 | −3.17 | −3.17 | −4.07 |

I | −1 | −2.13 | −2.13 | −2.04 |

Z^{*}
. | Nominal ionic charge . | a
. | b
. | c
. |
---|---|---|---|---|

Tl1 | 1 | 1.94 | 1.94 | 1.96 |

Tl2 | 1 | 1.97 | 1.97 | 2.09 |

S | −2 | −3.17 | −3.17 | −4.07 |

I | −1 | −2.13 | −2.13 | −2.04 |

The dielectric properties of Tl_{6}SI_{4} are similar to those of Tl_{6}SeI_{4}, which is also a radiation detection material. The calculated Born effective charges and the lattice contributions to the dielectric constants for these two compounds are very close to each other.^{10} The electronic contribution to the dielectric constant in Tl_{6}SI_{4} is slightly smaller than that in Tl_{6}SeI_{4}^{10} due to the larger band gap of Tl_{6}SI_{4} (2.07 eV for Tl_{6}SI_{4}^{13} vs. 1.86 eV for Tl_{6}SeI_{4}^{11}). Therefore, the static dielectric constant of Tl_{6}SI_{4} (∼19) is slightly smaller than that of Tl_{6}SeI_{4} (∼20) calculated at the PBE level.^{10}

### B. Phase diagram of Tl_{6}SI_{4}

To form single-phase Tl_{6}SI_{4} and avoid competing binary phases (e.g., TlI, TlI_{3}, Tl_{2}S, Tl_{4}S_{3}, TlS, and Tl_{2}S_{5}), the following conditions should be met:

where Δ*H*(Tl_{6}SI_{4}), Δ*H*(TlI), Δ*H*(TlI_{3}), Δ*H*(Tl_{2}S), Δ*H*(Tl_{4}S_{3}), Δ*H*(TlS), and Δ*H*(Tl_{2}S_{5}) are the heats of formation (per formula unit) of Tl_{6}SI_{4}, TlI, TlI_{3}, Tl_{2}S, Tl_{4}S_{3}, TlS, and Tl_{2}S_{5}, respectively. The chemical potential ranges for single-phase Tl_{6}SI_{4} are shown in Fig. 1 along with those for other competing phases.

### C. Native defects in Tl_{6}SI_{4}

Figure 2 shows the native defect formation energies calculated using chemical potentials corresponding to points A, B, C, and D in Figure 1. The defects that have been studied include I vacancy (*V*_{I}), S vacancy (*V*_{S}), Tl vacancy (*V*_{Tl}), and two antisites (S_{I} and I_{S}). These defects are studied because the vacancies and the antisites on anion sites are the dominant native defects in a similar compound, Tl_{6}SeI_{4}.^{10} We have further studied several defect complexes (*V*_{S}–*V*_{Tl}, *V*_{S}-S_{I}, *V*_{S}–2*V*_{Tl}, and *V*_{I}–*V*_{Tl}). The binding between $VTl\u2212$ and $VS2+$ ($VI+$) lowers the total energy by 0.56 eV (0.11 eV). $VTl\u2212$ is more strongly bound to $VS2+$ than to $VI+$ due to stronger Coulomb binding and strain relaxation. $(VS\u2212VTl)+$ can bind a second $VTl\u2212$ to form $(VS\u22122VTl)0$ with additional binding energy of 0.47 eV. These large binding energies make these $VS\u2212VTl$ defect complexes important defects as shown in Fig. 2. $(VS\u2212VTl)+$ is the dominant native donor defect in semi-insulating Tl_{6}SI_{4} under Tl-poor or S-poor conditions as shown in Figures 2(a) and 2(b). $(VS\u2212SI)+$ has binding energy of 0.26 eV, which is smaller than that of $(VS\u2212VTl)+$. This is because $SI\u2212$ is the second nearest neighbor while $VTl\u2212$ is the nearest neighbor to $VS2+$.

In the absence of high concentrations of impurities, the Fermi level should be pinned at $\epsilon fpin$, at which the formation energy lines of lowest-energy native donor and acceptor defects cross, as shown in Figure 2. The lowest energy donor and acceptor defects vary with the chemical potentials of involved elements. $\epsilon fpin$ generally moves higher with Tl chemical potential, since Tl-rich conditions favor the formation of native donor defects such as anion vacancies and suppress the formation of native acceptor defects such as Tl vacancies. $\epsilon fpin$ calculated for points A, B, C, and D, which are four corners of the stable region of Tl_{6}SI_{4} in the phase diagram, stays in the middle range of the band gap. This suggests that high resistivity is robust even with variation of growth conditions. This is consistent with the experimentally observed high resistivity of Tl_{6}SI_{4} (∼10^{10} Ω cm).

Among native defects shown in Fig. 2, I_{S} is a shallow donor, while *V*_{Tl} and S_{I} are shallow acceptors. *V*_{S}, *V*_{I}, and complexes *V*_{S}–*V*_{Tl} and *V*_{S}-S_{I} are capable of localizing electrons at deep defect levels. Since the deep defects are most detrimental to carrier transport, the charge transition levels of these deep defects are calculated and shown in Fig. 3. (The shallow levels are not calculated as discussed in Sec. II.) When the Fermi level is near CBM, $VS0$, $VI\u2212$, $(VS\u2212VTl)\u2212$, and $(VS\u2212SI)\u2212$ are stable and each of them induces a filled single-particle defect level deep inside the band gap. When the Fermi level is near midgap as required for a semiconductor radiation detector, these native deep donor defects are stable as $VS2+$, $VI+$, $(VS\u2212VTl)+$, and $(VS\u2212SI)+$. The trapping of the first electron on each defect level is most relevant to the low-photon-flux conditions. These electron trapping levels are calculated to be *E*_{c} − 0.32 eV [(2+/+) level for *V*_{S}], *E*_{c} − 0.22 eV [(+/0) level for *V*_{I}], *E*_{c} − 0.16 eV [(+/0) level for *V*_{S}–*V*_{Tl}], and *E*_{c} − 0.20 eV [(+/0) level for *V*_{S}-S_{I}], as shown in Fig. 3. The electron trapping levels for the donor-acceptor complexes are shallower than their corresponding uncomplexed donors. This is because the coupling of the donor and acceptor levels pushes up the donor level and pushes down the acceptor level as schematically shown in Fig. 4. The stronger the coupling between the donor and the acceptor levels, the higher the donor level would be. This explains why the electron trapping level for *V*_{S}–*V*_{Tl} is the shallowest. Figure 3 shows that *V*_{S}, *V*_{S}–*V*_{Tl} and *V*_{S}-S_{I} are negative-*U* centers,^{28} at which the trapping of the second electron on the defect level lowers the total energy more than that of the first electron [e.g., the (+/0) level is lower than the (+2/+) level for *V*_{S}.]. This is due to the strong structural relaxation of *V*_{S} upon electron trapping.

The defect levels shown in Fig. 3 are obtained by referencing the PBE calculated defect levels to HSE calculated band edges. This approach has been shown to be adequate in previous calculations (see Sec. II for details). To further test the reliability of this approach, we calculated the defect levels for *V*_{S} directly using HSE functionals. Since HSE calculations are very time-consuming, we used only Г point in the HSE calculations. (Note that results in Figs. 2 and 3 were obtained using a 2 × 2 × 2 k-point mesh.) For a fair comparison, we also performed PBE calculations using only Г point. The results are summarized in Table III. The defect levels for *V*_{S} calculated using the two approaches (i.e., direct HSE calculations of defect levels vs. PBE calculated defect levels referenced to HSE calculated band edges) differ by well less than 0.1 eV. These results show that the approach used for calculating defect formation energies and defect levels in Figs. 2 and 3 are sufficiently accurate, and confirm that *V*_{S} is a deep donor.

. | PBE (2 × 2 × 2) . | PBE (Г-only) . | HSE (Г-only) . |
---|---|---|---|

(2+/+) | E_{C} − 0.32 eV | E_{C} − 0.40 eV | E_{C} − 0.35 eV |

(+/0) | E_{C} − 0.70 eV | E_{C} − 0.71 eV | E_{C} − 0.69 eV |

(2+/0) | E_{C} − 0.51 eV | E_{C} − 0.55 eV | E_{C} − 0.52 eV |

. | PBE (2 × 2 × 2) . | PBE (Г-only) . | HSE (Г-only) . |
---|---|---|---|

(2+/+) | E_{C} − 0.32 eV | E_{C} − 0.40 eV | E_{C} − 0.35 eV |

(+/0) | E_{C} − 0.70 eV | E_{C} − 0.71 eV | E_{C} − 0.69 eV |

(2+/0) | E_{C} − 0.51 eV | E_{C} − 0.55 eV | E_{C} − 0.52 eV |

To reduce the concentration of anion vacancies, especially *V*_{S}, which induces a relatively deep (+2/+) electron trapping level in the band gap (*E*_{c} − 0.32 eV) as shown in Fig. 3, Tl poor conditions and relatively high Fermi level are desirable. However, the Tl-poor conditions promote the formation of Tl vacancies (which are acceptor defects) and suppress the formation of anion vacancies (which are donor defects); therefore result in low Fermi level. This is evident in Figures 2(a) and 2(d), where low $\epsilon fpin$ correlates with low Tl chemical potential. To solve this problem, one may dope Tl_{6}SI_{4} with shallow donors under a Tl-poor growth condition. The resulting high S and I chemical potentials and relatively high Fermi level should suppress the formation of anion vacancies. Note that the Fermi level should be carefully controlled to maintain high resistivity and the introduced donor impurities should be shallow to avoid excessive electron trapping.

Although the (2+/+) level for *V*_{S}, the (+/0) level for *V*_{I}, the (+/0) level for *V*_{S}–*V*_{Tl}, and the (+/0) level for *V*_{S}-S_{I} are all below the CBM, the empty single-particle levels of $VS2+$, $VI+$, $(VS\u2212VTl)+$, and $(VS\u2212SI)+$ are above the CBM. These single-particle levels would be below the CBM if they are occupied, because electron occupation of these levels provides energy incentive for Tl ions around the anion vacancy to move close to each other, which enhances the hybridization of the Tl orbitals and lower the single-particle defect level. Before electron trapping, lattice vibration is needed to move the Tl ions towards the center of the anion vacancy and consequently moves the single-particle defect level below the CBM. Therefore, although electron trapping at $VS2+$, $VI+$, $(VS\u2212VTl)+$, or $(VS\u2212SI)+$ is exothermic, it involves a kinetic barrier, which should reduce the trapping cross-section.

## IV. CONCLUSIONS

Density functional calculations are performed to study native point defects and dielectric properties of Tl_{6}SI_{4}, a promising room-temperature semiconductor radiation detector material. The mixed ionic-covalent character of Tl_{6}SI_{4} leads to enhanced Born effective charges and consequently large static dielectric constant, which increases screening of charged defects and impurities. The Fermi level can be pinned by native donors and acceptors deep inside the band gap, leading to high resistivity. *V*_{S} and *V*_{I} are deep donors with electron trapping levels calculated at *E*_{c} − 0.32 eV (+2/+ level) and *E*_{c} − 0.22 eV (+/0 level), respectively. $VS2+$ can bind with $VTl\u2212$ and $SI\u2212$, forming electrically active low-energy donor complexes, $(VS\u2212VTl)+$ and $(VS\u2212SI)+$. Due to donor-acceptor coupling, the (+/0) electron trapping levels of $(VS\u2212VTl)+$ and $(VS\u2212SI)+$ are shallower, located at 0.16 eV and 0.20 eV below CBM. Therefore, $VS2+$ should be most detrimental to electron transport. A Tl-poor growth condition combined with shallow donor doping should reduce the concentration of $VS2+$ in Tl_{6}SI_{4}.

## ACKNOWLEDGMENTS

The authors are grateful for stimulating discussion with Csaba Szeles, Bruce W. Wessels, and Mercouri G. Kanatzidis. This work was supported by the Department of Homeland Security, Domestic Nuclear Detection Office (Grant No. HSHQDC-14-R-B0009).