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Manifestation of Anomalous Weak Space-Charge-Density Acentricity for a Tl4HgBr6 Single Crystal Anatoliy A. Lavrentyev,† Boris V. Gabrelian,† Tuan V. Vu,† Peter N. Shkumat,† Petro M. Fochuk,‡ Oleg V. Parasyuk,§ Iwan V. Kityk,∥ Ivan V. Luzhnyi,⊥ Oleg Y. Khyzhun,⊥ and Michal Piasecki*,# †

Department of Electrical Engineering and Electronics, Don State Technical University, Gagarin Sq. 1, 344010 Rostov-on-Don, Russian Federation ‡ Yuriy Fedkovych Chernivtsi National University, 2 Kotziubynskoho Street, UA-58012 Chernivtsi, Ukraine § Department of Solid State Physics, Lesya Ukrainka Eastern European National University, 13 Voli Avenue, UA-4025 Lutsk, Ukraine ∥ Faculty of Electrical Engineering, Częstochowa University of Technology, Armii Krajowej 17, PL-42-217 Częstochowa, Poland ⊥ Frantsevych Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, 3 Krzhyzhanivsky Street, UA-03142 Kyiv, Ukraine # Institute of Physics, J. Dlugosz University of Częstochowa, Armii Krajowej 13/15, PL-42-200 Częstochowa, Poland ABSTRACT: Density functional theory (DFT) calculations within the concept of the MBJ+U+SO (modified Becke−Johnson potential + U + spin orbit) approach were performed for a Tl4HgBr6 single crystal for the first time assuming weak noncentrosymmetry (space group P4nc). Excellent agreement was achieved between the calculated and experimental band-gap-energy magnitudes as well as the density of electronic states measured by the X-ray photoelectron spectroscopy method. It is a very principal result because usually the DFT calculations underestimate the energy-gap values. In the present study, we carry out calculations of the optical properties (absorption coefficient, real and imaginary parts of the dielectric function, electron energy-loss spectrum, refractive index, extinction coefficient, and optical reflectivity dispersions). It has been established that the principal origin of the observed weak acentricity is determined by delocalized band states at the top of the valence band originating from the p states of the Br atoms.

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centrosymmetric space group13 P4/mnc (a = 8.978 Å and c = 8.812 Å). The crystal structure of Tl4HgBr6 is shown in Figure 1a. However, recent powder X-ray diffraction refinements for the crystal structure of Tl4HgBr6 have shown14 that the crystal adopts the noncentrosymmetric structure (space group P4nc with the unit-cell lattice parameters a = 8.9539 Å and c = 8.7884 Å). The noncentrosymmetric structure for the Tl4HgBr6 compound was enforced by the existence of a mild secondharmonic-generation value (0.4−0.5 pm/V) and a relatively low piezoelectric coefficient (0.9 pm/V).14 Such controversy in the determination of the space group symmetry may be of extremely high interest for modern materials chemistry because of the possibility forming materials that may tune their symmetry by weak external fields. In the crystal under consideration, Tl and Hg atoms occupy 8c and 2a site positions, respectively, while Br atoms are situated in three nonequivalent local site positions in this compound, namely, two different 2a sites and one 8c site (Figure 1a). In the Tl4HgBr6 structure, the nearest coordination of Hg atoms includes six Br atoms (Figure 1b), while eight Br atoms are in the nearest surrounding of Tl atoms (Figure 1c). It is worth

INTRODUCTION Thallium hexabromomercurate, Tl4HgBr6, is a distinctive member among an enchanting series of thallium-bearing halides with the overall formula Tl4BX6, where B is Cd, Hg, or Pb and X is a halide anion. They attract significant interest from a scientific and applicative viewpoint because of their unusual optical and electronic features.1 As a consequence, Tl4BX6 single crystals may be counted as very attractive materials for nonlinear-optical (NLO) appliance in the middleand far-IR spectral range.1−3 Tl4BX6 halides are also expected to be up-and-coming materials for γ-ray detectors4 that might perform as well as cadmium−zinc telluride (CdZnTe or CZT), which is nowadays the best semiconducting material for ambient-temperature γ-ray spectroscopy,5−8 but be easier to manufacture in large volumes and possess lower cost. Furthermore, Tl4BX6 crystals have been found to be very useful materials for applications like solar cells,3 ion-selective electrodes,9,10 and temperature sensors.11 Among the Tl4BX6 halides, the Tl4HgBr6 crystal is of particular interest. Huart12 discovered this bromide as early as 1966, and he reported that Tl4HgBr6 melts incongruently at 575 K and crystallizes in a tetragonal structure (the unit-cell parameters were reported to be equal to a = 8.965 Å and c = 8.783 Å). Brodersen et al. have established for the first time that Tl4HgBr6 crystallizes in the © 2016 American Chemical Society

Received: July 24, 2016 Published: October 3, 2016 10547

DOI: 10.1021/acs.inorgchem.6b01389 Inorg. Chem. 2016, 55, 10547−10557

Article

Inorganic Chemistry

Figure 1. (a) Crystal structure of the Tl4HgBr6 compound and the neighbor surroundings of (b) Hg and (c) Tl atoms within the HgBr6 octahedra and TlBr8 polyhedra, respectively, in this bromide.

The XPS and XES measurements14 have revealed that the principal manifestation of the Br 4p states takes place in the upper valence sub-band, with also their essential contributions in the remaining valence-band regions. The XPS data have established low hygroscopicity of the bromide Tl4HgBr6, a property that is very definable for use of this material in optoelectronic apparatuses, exploiting ambient conditions.14 However, to the best of our knowledge, theoretical bandstructure calculations of peculiarities for the electronic state energy distribution of the atoms constituting the Tl4HgBr6 compound, crystallizing in the noncentrosymmetric space group P4nc, are absent. The unique band-structure calculations, to the best of our enlightenment, have been made by Brik et al.17 for Tl4HgBr6 assuming that this crystal is crystallized in the centrosymmetric space group P4/mnc. To fill this shortcoming, in the present contribution, we will do first-principles bandstructure calculations of Tl4HgBr6 based on density functional theory (DFT), as implemented within the augmented plane wave + local orbitals (APW+LO) method, adopting the WIEN2k package18 to explore the total densities of states (DOSs), partial densities of states (PDOSs), and electronic bands through selected symmetry paths over the first Brillouin zone (BZ). Furthermore, we report on calculations of the principal optical characteristics of the Tl4HgBr6 compound, crystallizing in the noncentrosymmetric P4nc structure, namely, the absorption coefficient, dielectric function (its real and imaginary parts), electron energy-loss spectrum, refractive index, extinction coefficient, and optical reflectivity spectra. Tl4HgBr6 is a representative of a quite large group of A4BX6 compounds19,20 that have been actively investigated in recent years. Although information on Tl4HgBr6 is sparse compared to others, we have recently established14 that it crystallizes in a noncentrosymmetrical structure (space group P4nc), which was the impetus for a detailed study of its NLO properties. The high average atomic number (Z) of Tl4HgBr6 constituent elements is a good reason to test their use as detectors of ionizing radiation. In 2012, Bolotnikov21 graded detector materials, identifying four groups according to their viability and degree of investigation. The first group includes CdTe and CdZnTe, which are currently the most used. The second group includes also classic CdMnTe as well as HgI2 and TlBr. The third group is the alloys of CdTe and CdSe, GaAs, and a-Se. Finally, the fourth-largest group consists of materials that in the near future should attract the greatest attention of researchers. These include, along with a number of others, thallium- and mercury-containing compounds such as TlPbI3, Tl4HgI6, Tl3As2Se3, TlGaSe2, and HgxBr1−xI2, which by their parameters fit well to such uses. Later, Wang et al.22 determined that the mobility−lifetime product (μτ) of Tl4CdI6 is on the order of 10−4 cm2/V for both electrons and holes, which compares well to the CZT parameters. From this, we can conclude that

mentioning that the Hg−Br chemical bond distances within the HgBr6 octahedra are substantially shorter than the Tl−Br ones within the TlBr8 polyhedra (see Figure 1b,c). The HgBr6 octahedra are stacked along the c crystallographic direction, while the Tl atoms accommodate the channels formed by these octahedra (Figure 2).

Figure 2. Stacking of the HgBr6 octahedra in relation to the principal crystallographic axes in Tl4HgBr6. (Note: the presentation of the atoms is the same as that in Figure 1.)

The ionic-transport properties of Tl4HgBr6 have been explored in ref 15 based on the measurements of the 205Tl spin−lattice relaxation time, T1. Nagase et al.15 have established that, with increasing temperature, the T1 value gradually decreases up to about 400 K and then rather steeply decreases at temperatures above 400 K. The motion activated at high temperatures was attributed to the translational self-diffusion of the Tl+ ions.15 Moreover, measurements of the temperature dependence of the electrical conductivity, σ, in Tl4HgBr6 have revealed15 that, upon heating, the σT value increases exponentially up to about 480 K, yielding a conductivity activation energy of 47 kJ·mol−1, followed by a change in the slope of the log(σT) versus 1/T curve. Nagase et al.15 have attributed the above behavior of the Tl4HgBr6 compound to a transition from extrinsic to intrinsic conductivity. Studies of the absorption coefficient in Tl4HgBr6 single crystals indicate14 that the bromide is a semiconducting material with a band-gap-energy value equal to 2.43 eV at room temperature. The band-gap energy of Tl4HgBr6 enlarges to 2.48 eV with decreasing temperature to 100 K.14 Because of the fact that the insight of the electronic structure of solids carries out a decisive role in the apprehension of their physical properties and creation of the chemical bonding within them,16 X-ray photoelectron spectroscopy (XPS) and X-ray emission spectroscopy (XES) were applied in ref 14 to measure the binding energies of the core-level electrons and the valence-band spectra as well as the XES Br Kβ2 band, giving information about peculiarities of the energy distribution of the Br 4p states. 10548


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The iteration process has been verified by taking into consideration alterations in the integral charge difference q = ∫ |ρn − ρn−1| dr, where ρn−1(r) and ρn(r) are the input and output charge densities, respectively. The calculations were discontinued when the case of q ≤ 0.0001 was reached. It is worth emphasizing that the complex dielectric function ε(ω) = ε1(ω) + iε2(ω) is directly affiliated with the electron energy band structure and the function is the most considerable for calculations of the optical response of solids regarding electromagnetic interaction. It is known that ε2(ω) should contain interband and intraband transitions; however, the latter ones are important only for metals.29 Furthermore, interband transitions contain direct and indirect transitions. Phonon contributions included in the indirect interband transitions are ignored because of heavy atoms, and the direct transitions between the occupied and unoccupied (valence− conduction band) states are recognized in the present calculations of the optical characteristics of Tl4HgBr6. It is necessary to emphasize that, when calculating the ε(ω) dispersion, we require precise energy eigenvalues and electronic wave functions.30 The symmetry of the Tl4HgBr6 structure yields only two principal diagonal nonzero components of the second-rank dielectric tensor, namely, εxx(ω) and εzz(ω), through the a and c crystallographic axes, respectively. These components of the dielectric tensor ε2(ω) are needed for the whole depiction of the linear optical susceptibility of the compound. We were able to retrieve the mentioned diagonal nonzero components by applying the expression31

Tl4HgBr6 is a promising material for nonlinear optics and optoelectronics. In particular, the present materials are promising for ambient and low-temperature nonlinear optics because they have a higher potential of the NLO susceptibilities χ(2) due to the existence of the heavy-atom cationic subsystem, which influences the photoinduced NLO properties, and χ(2) factor, which is limited for the ZnTe-like materials. Additionally, varying even slightly the cationic subsystem using technological processes, one can achieve the desired optoelectronic parameters. The main advantage of the present materials with respect to the well-known ZnTe crystals is their less expensive technology for obtaining of quality crystals (up to several times). It is also necessary to add that the thermomobility of Tl atoms is prevailingly localized in the voids, and therefore Tl4HgBr6 is not as much a polluting or hazardous substance as one might think after its constituents.

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METHOD OF CALCULATIONS

The DFT band-structure calculations of Tl4HgBr6 are made within a framework of the APW+LO method, as implemented in the WIEN2k package.18 We apply the full potential by relying on the orbital momentum spreading inside and outside the atomic spheres and the unit-cell lattice parameters and atomic locations established for the Tl4HgBr6 compound (Table 1).14 In the present calculations, the

ε2ij(ω) =

Table 1. Atomic Coordinates in the Tl4HgBr6 Structure, as Used in the Present APW+LO Calculations (Space Group P4nc; a = 8.9539 Å and c = 8.7884 Å14) atom

Wyckoff site

X

Y

Z

Tl Hg Br1 Br2 Br3

8c 2a 2a 2a 8c

0.1450 0 0 0 0.1388

0.6495 0 0 0 0.3228

0.270 0.017 0.2923 0.7211 0.023

4π 2e 2 ∑ ⟨knσ |pi |kn′σ ⟩⟨kn′σ |pj |knσ ⟩ Vm2 ϖ2 nn ′ σ × fkn (1 − fkn ′ ) δ(Ekn ′ − Ekn − ℏω)

(1)

where m and e are the mass and charge of electrons, respectively, ω is the angular frequency of electromagnetic radiation, V is the unit cell volume, p stands for the momentum operator, |knσ⟩ is the wave function of the compound possessing the momentum (wave vector) k, and σ is the spin that corresponds to the energy eigenvalue, Ekn. The Fermi distribution function, f kn, commits a definite score of transitions from the occupied band to unoccupied band in solids, while the term δ(Ekn′ − Ekn − ℏω) specifies a term for preservation of the total energy, emerging from the summation of the integrated DOSs. Finestructure peculiarities in the absorbing part of the dielectric function define the allowed electric-dipole transitions between the valence and conduction bands. With the aim of identifying the peculiarities, we need to compare values of the optical matrix elements. Determined fine-structure peculiarities are suitable for the transitions with significant values of optical-dipole matrix elements. The real part of the dielectric function ε1(ω) can be gained from the imaginary part of the dielectric function ε2(ω) employing Kramers− Kronig’s relationship32

MT principal RMT minkmax parameter [Rmin infers the smallest muffin-tin (MT) sphere radius, and kmax determines the value of the largest k vector in the plane-wave space] is admitted to be 7.0. In the potential dissolution, the valence wave functions inside the MT spheres are extended up to lmax = 10 and beyond the MT spheres up to lmax = 4. The charge density is Fourier spread up to the value Gmax = 12 au−1, where 1 atomic unit (au) = 0.529177 Å. The MT sphere radii of the constituting atoms in the calculations are applied as 2.5, 2.39, and 2.16 au for the Tl, Hg, and Br atoms, respectively. In the present calculations, the basis function is composed of the atomic orbitals of Tl, Hg, and Br, as tabulated in Table 2. The entire number of semicore and valence electrons (besides core electrons, which are also considered in the calculations) per unit cell is equal to 392. We applied the generalized gradient approximation (GGA), as developed by Perdew, Burke, and Ernzerhof (PBE),23 and the modified Becke− Johnson (MBJ) potential24 to account for the exchange-correlation potential. In addition, the PBE+U25,26 and MBJ+U24,27 models were employed. Further, we use the tetrahedron method28 in order to integrate over the BZ. The BZ sampling is made using 1000 k points within the irreducible wedge of the BZ for the present DFT calculations.

ε1(ω) = 1 +

2 P π

∫0

∞

ω′ε2(ω′) d ω′ ω′2 − ω2

(2)

where P corresponds to the principal value of the integral. The principal optical dispersion spectra of solids such as the absorption coefficient α(ω), refractive index n(ω), extinction coefficient k(ω), optical reflectivity coefficient R(ω), and electron energy-loss spectrum L(ω) are extracted from the ε1(ω) and ε2(ω) dispersion functions using the equations reported in refs 25, 33, and 34:

Table 2. Atomic Orbitals Adopted in the Present APW+LO Calculations of the Electronic Structure of Tl4HgBr6 atom

core electrons

semicore electrons

valence electrons

number of electrons involved in the APW+LO calculations

Tl Hg Br

1s22s22p63s23p63d104s24p64d104f145s2 1s22s22p63s23p63d104s24p64d104f145s2 1s22s22p63s23p6

5p65d10 5p65d10 3d104s2

6s26p1 6s2 4p5

19 18 17

10549


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Inorganic Chemistry α ij(ω) =

2ωkij(ω) c

(3)

nij(ω) =

1 [ ε1ij(ω)2 + ε2ij(ω)2 + ε1ij(ω)]1/2 2

(4)

kij(ω) =

1 [ ε1ij(ω)2 + ε2ij(ω)2 − ε1ij(ω)]1/2 2

(5)

(nij − 1)2 + kij2 R (ω) = ij = (n + 1)2 + kij2 ij

Lij(ω) = − Im(ε−1)ij =

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ε1ij + iε2ij − 1 ε1ij + iε2ij + 1

ε2ij(ω) ij ε1 (ω)2 + ε2ij(ω)2

2

(6)

(7)

RESULTS AND DISCUSSION To begin with, we remind everyone about what was just mentioned earlier, that we chose the mBJ+U+SO approach as the principal method available in the WIEN2k program. The primary reason was that the general opinion, presented in many papers and confirmed by our earlier experience, was that the MBJ approach is the best for estimation of the Eg value and is particularly suitable for halides. Spin−orbit coupling (SOC) is also very useful because it gives the possibility of receiving SOC effects for the heavy ions like the experiments give. Previously, in some of our contributions, we included the SOC effects in band-structure calculations. However, it leads to a significant increase of the calculation time, and it demands a big space of a calculating cluster. The SOC did not influence much the energy distribution of the electronic states over the valence- and conduction-band areas; however, it influences the p and d semicore and core electron levels, giving two levels (as in experiments) instead of one in the band-structure calculations ignoring SOC. Such features were just shown in the TlI crystals.35 Data of our APW+LO band-structure calculations of curves of total DOSs performed for Tl4HgBr6 within the PBE, PBE+U, MBJ, and MBJ+U+SO approaches are presented in Figure 3. In this figure, for comparison, the XPS valence-band spectrum of Tl4HgBr6 is also shown, as measured in ref 14. From Figure 3, it is evident that the spectral curve of the total DOS of Tl4HgBr6 calculated using the PBE approximation23 exhibits significant energy underestimation of the positions for the Tl 5d and Hg 5d PDOSs, which are detected to be spectrally shifted by about 1.0−1.2 eV toward the Fermi energy, with respect to the positions of the fine-structure features F and D, respectively, of the XPS valence-band spectrum of Tl4HgBr6. In our opinion, the nonconformity of the theoretical total DOS curve and the experimental XPS valence-band spectrum of Tl 4 HgBr 6 appeared as a result of skipping of strongly correlated d electrons in GGA, as elaborated on by Perdew et al.23 This problem can be resolved within the PBE+U model25,26 by taking into consideration the correction parameter U, which is added exclusively for strongly correlated electrons: the U value is used as a fitting parameter to adjust to the experimental data. In our DFT calculations made within the PBE+U model,25,26 we adopt the values of U = 0.368 Ry for Tl 5d and U = 0.257 Ry for Hg 5d. The correct U parameter is difficult to obtain theoretically, and it is usually considered as an adjustable parameter. In the present study, the U parameters were used for semicore electrons, and their use cannot significantly disturb the optical parameters because they are very sensitive to the Eg

Figure 3. Curves of total DOSs calculated for the Tl4HgBr6 compound within the PBE, PBE+U, MBJ, and MBJ+U+SO approaches compared on a common energy scale with the XPS valence-band spectrum of this compound recorded in ref 14.

values and energy positions of the PDOSs in the valence- and conduction-band regions. Following our calculations, inclusion of the U parameter did not cause any substantive changes in the total DOS and PDOS distributions within the valence and conduction-band regions. It shifts somewhat the positions of the Tl 5d and Hg 5d semicore electrons. The application of the PBE+U model25,26 in the present APW+LO calculations of Tl4HgBr6 causes a spectral shift of the Tl 5d and Hg 5d PDOSs away from the Fermi energy, terminating in better coincidence with the finestructure peculiarities of the theoretical curves for the total DOSs and the experimental XPS spectrum of the valence band. Furthermore, it is well-known that the band-gap values, Eg, evaluated by calculations within the PBE approximation23 are, as a rule, underestimated in the semiconductors and insulators.24,36 As a result of this shortage, the present DFT band-structure calculations of Tl4HgBr6 carried out using the PBE approximation23 reveal the value of Eg = 1.834 eV, which is underestimated with respect to Eg = 2.43 eV determined experimentally for the compound under consideration at ambient conditions.14 The use of the PBE+U approximation25,26 results in an increase in the band-gap value of Tl4HgBr6 up to 1.875 eV. Nevertheless, we have derived Eg = 2.455 eV, when using the MBJ potential24 in our band-structure calculations of Tl4HgBr6. This theoretical Eg value almost coincides with that determined experimentally for Tl4HgBr6 in ref 14. It should be emphasized that Tl and Hg atoms are considered to be heavy elements.37,38 Therefore, in the present DFT calculations within the MBJ+U potential,24,27 SOC39−41 has been accounted for as well (we refer to these calculations, for clarity, as MBJ+U+SO). It is reported that consideration of the SOC approach for the first-principles band-structure calculations of heavy-element-bearing compounds (e.g., U, Np, Pu, Am, Th, etc.) alters a number of parameters, in 10550


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Figure 4. PDOS curves for the (a) Tl s and Tl p, (b) Hg s and Hg p, (c) Tl d and Hg d, and (d) Br s and Br p originated bands of Tl4HgBr6 (calculated with MBJ+U+SO).

in Figure 4. From this figure, it is obvious that the Br 4p states are the principal contributors to the valence-band region of the Tl4HgBr6 compound. They contribute principally to the upper and central portions of the valence band (sub-bands A1 and B in Figure 4) with somewhat smaller contributions at the top (sub-band A) and bottom (sub-band C) of the band in almost equal proportions. The above theoretical data are in remarkable agreement with recent experimental measurements of the energy distribution of the valence Br p states in Tl4HgBr6.14 It should be pointed out that Br 4p state contribution over the valence band is a general feature of bromide crystals, as shown in several studies.44−47 Also, there is a significant difference between the present calculations using the meta-GGA Becke−Johnson + U + SO exchange-correlation functional with the crystal structure of the P4nc space group and the prior calculations of Brik et al.17 using the PBE exchange-correlation functional and performed on the relaxed crystal structure of the P4/mnc space group. One must note that the agreement with the absorption experiments seems better in this last case than with the present new calculations. This may be a consequence of scissor-factor inclusions in ref 17, which is fitted by an optical energy band gap. However, it is clear that the present calculations are in much better coincidence with the XPS experiments and for the energy band gap than Brik’s calculations. Additionally, it is necessary to

particular, the magnetic moments, degree of localization of the 5f states, chemisorption energy of light atoms on the surfaces of heavy atoms, elastic modulus, etc.42,43 As can be seen from the data of the present APW+LO calculations made within the MBJ +U+SO approximation, the main fine-structure peculiarities of the curve for the total DOS spectrum coincide well with those obtained for the Tl4HgBr6 compound employing MBJ20 and PBE+U25,26 approximations. Nevertheless, it is apparent (see Figure 3) that the theoretical total DOS curve calculated by adopting the MBJ+U+SO approximation is better fitted to the experimental XPS valence-band spectrum of the compound under consideration. In addition, our DFT calculations made within the MBJ+U+SO approximation reveal the value of spin− orbit XPS splitting for Pb 5d5/2,3/2 electrons equal to 2.5 eV. This theoretical value correlates well with the experimental Pb 5d5/2,3/2 spin−orbit splitting found for this compound experimentally in ref 14. For comparison, curves of the total DOSs and principal PDOSs obtained in calculations within the MBJ+U+SO approximation for the Tl4HgBr6 compound are plotted together with the XPS valence-band spectrum of the studied bromide in Figure 3. However, detailed curves of the total DOSs and PDOSs for the Tl, Hg, and Br atoms are shown in Figure 4. Our DFT data show that the primary part of the valence band of the Tl4HgBr6 compound consists of four spectrally isolated sub-bands that are marked as A, A1, B, and C 10551


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contributions of the Br 6p states at the bottom of the valence band for the bromide under study, while in accordance with the data by Brik et al.,17 the Hg 5d and Tl 6s states overload the lower portion of the valence band and form two nonseparating sub-bands in Tl4HgBr6. Our DFT calculations indicate that the sub-bands formed by the Hg 5d states are well separated by the energy from the bottom of the valence band of Tl4HgBr6. It should be pointed out that the principle features of the occupation of the valence band of Tl4HgBr6 seem to be similar to those established for other Tl-containing ternary bromides, in particular for Tl3PbBr550 and TlPb2Br551. Like in the case of Tl4HgBr6, the DFT calculations made in refs 50 and 51 indicate that the valence bands of Tl3PbBr5 and TlPb2Br5 are formed mainly by the Br 4p states contributing predominantly in the upper part of the valence band, while the Tl 6s states are the principle contributors to its bottom. Band dispersions for the Tl4HgBr6 compound are presented in Figure 5 for special symmetry directions of the tetragonal BZ

emphasize that the two crystal structures are very close to each other, and it may be an additional factor of the observed differences. So, our calculations without the scissor factor may be a huge reason for using calculations of such kinds of compounds. It is also important that some differences in the structure manifesting a weak acentricity may be crucial for future deeper studies of the such kinds of materials, which will be a separate fundamental study in the future. Further, as can be seen from Figure 4, the Tl 6s and Hg 6s states are among other substantive contributors to the valenceband bottom, while the Tl 6s states contribute also to the top of the valence band of Tl4HgBr6. Small admixtures of the Hg 6p and Tl 6p states in the central part of the valence band of Tl4HgBr6 are also detected in our band-structure calculations plotted in Figure 4. The bottom of sub-band C is energy-detached from the top of sub-band D, which is formed from the contributions of the prevailing Hg 5d5/2 states (see Figures 3 and 4), by a gap of about 2 eV. Further, sub-band C is detached from the top of sub-band D, which is formed from the contributions of the prevailing Hg 5d5/2 states (Figures 3 and 4), with an energy gap of about 2 eV. Further, sub-bands F and G of the experimental XPS valence-band spectrum of Tl4HgBr6 are formed mainly by the contributions of the Tl 5d3/2 and Tl 5d5/2 states, respectively, and sub-band E originated from the Hg 5d3/2 states. It should be mentioned that significant contributions of the Br 4s states in the energy region corresponding to sub-band G and the Br 4p states in sub-bands E and D are also found in the present DFT calculations. Our calculations manifest that the Br 4p states are highly hybridized with the Hg 6p states in the energy region, which corresponds to the position of the central sub-band B of the valence band of Tl4HgBr6 (see Figure 4), while high hybridization of the Br 4p states with the Tl 6s and Hg 6s states at the valence-band bottom is also typical for the electronic structure of the bromide under study. The presence of the above-mentioned hybridization of the electronic Tl 6s, Hg 6s, and Hg 6p states with the Br 4p states indicates that a significant input of the covalent component to the chemical bonding takes place in addition to the ionic component in the Tl4HgBr6 compound. With respect to the occupancy of the conduction band of Tl4HgBr6, the present DFT calculations display that its bottom is dominated by Hg 6s* state contributions, with somewhat smaller contributions of the Br 4p* states as well (see Figures 3 and 4). It is necessary to underline that the data of the present APW +LO band-structure calculations carried out for Tl4HgBr6 crystallizing in the noncentrosymmetric space group P4nc resemble those made in ref 17, assuming that the title bromide crystallizes in the centrosymmetric space group P4/mnc. Like in the present work, data by Brik et al.17 pointed out that the Tl4HgBr6 compound is a direct-band-gap semiconductor with Eg = 1.779 eV and Eg = 1.324 eV when the exchange-correlation effects are treated in the calculations with the PBE19 and Ceperley−Alder−Perdew−Zunger48,49 functionals, respectively. In addition, the results by Brik et al.17 indicate also that the valence band of Tl4HgBr6 predominantly originates from the contributions of the Br 4p states, while the bottom of the conduction band is composed especially by the contributions of the unoccupied Hg 6s states. The main difference of the results of our APW+LO band-structure calculations from those derived in ref 17 is that the present data confirm the prevailing contributions of the Tl 6s and Hg 6s states with slightly smaller

Figure 5. Electronic bands at broad selected symmetry paths within the first BZ of Tl4HgBr6.

(Figure 6). The diagram of the tetragonal BZ used in the present DFT calculations is similar to that reported previously in a review.52 The coordinates of the BZ k points, within the restricted region of the BZ studied for the band dispersions presented in Figure 6, are as follows: X (0.5, 0.0, 0.0), A (0.5, 0.5, 0.5), R (0.0, 0.5, 0.5), Γ (0.0, 0.0, 0.0), X (0.5, 0.0, 0.0), R (0.5, 0.0, 0.5), Z (0.0, 0.0, 0.5), and Γ (0.0, 0.0, 0.0). It is worth indicating that the value of Eg (2.456 eV) found in the present DFT calculations of Tl4HgBr6 adopting the MBJ+U+SO approximation is close to that determined experimentally in ref 14. The dispersions of the curves located in the vicinity of the valence-band maxima (VBM) and conduction-band minima (CBM) are rather flat in k space, as shown in Figure 5. This may indicate high effective mass and low electron mobility. The present APW+LO calculations allow one to conclude that 10552


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Inorganic Chemistry

there is no need to perform the scissor-factor-correction technique, and we report here the results of DFT calculations of the main optical parameters that were obtained using the theoretically derived Eg value for Tl4HgBr6. As one can observe from the data plotted in Figure 7, the absorption coefficient dispersion α(ω) for the Tl4HgBr6 compound increases strongly for photon energies above 3.5 eV. So, the spectral region of strong absorption in the bromide under consideration spreads from about 3.5 eV to about 27 eV. In the above-mentioned energy region, the absorption coefficient α(ω) for Tl4HgBr6 possesses a number of spectral peaks that originate from different electronic transitions. The relative peak intensities alternate with changes in the photon energy within the calculated absorption region, indicating that the Tl4HgBr6 compound is a rather perspective material for its use in optoelectronic devices. The calculated real part of the dielectric function ε1(ω) dispersion for Tl4HgBr6 is plotted in Figure 8. From this figure,

Figure 6. Diagram of the BZ for a tetragonal structure that is a characteristic of the Tl4HgBr6 compound.

valence- and conduction-band extrema in the Tl 4HgBr6 compound are located at the BZ Γ point (0.0, 0.0, 0.0). Therefore, the Tl4HgBr6 bromide is a direct-gap material. The results of our DFT calculations, performed using the MBJ+U +SO approximation, for the main optical characteristics of the Tl4HgBr6 compound, namely, the absorption coefficient α(ω), dielectric functions ε1(ω) and ε2(ω), refractive index n(ω), extinction coefficient k(ω), optical reflectivity coefficient R(ω), and electron energy-loss spectrum L(ω) dispersion, are presented in Figures 7−13. It is worth to remind everyone

Figure 8. Real part of the dielectric function ε1(ω) of Tl4HgBr6 (calculations within the MBJ+U+SO approach).

one can see that the ε1(ω) curve of the Tl4HgBr6 bromide is described by the occurrence of at least five well-resolved spectral fine-structure peculiarities, namely, A (∼4.5 eV), B (∼5.5 eV), C (∼7 eV), D (∼10.5 eV), and E (∼17.5 eV). The energy positions of the above-mentioned features for the ε1(ω) curve of Tl4HgBr6, as is apparent from a comparison of Figures 8 and 9, correspond well to those of the respective peculiarities of the imaginary part of the dielectric function ε2(ω). As Figure 8 reveals, the calculated real part of the dielectric function ε1(ω) decreases beginning from about 5.5 eV up to about 8 eV and then the ε1(ω) function enhances with increasing photon

Figure 7. Absorption coefficient dispersion α(ω) of Tl4HgBr6 (calculations within the MBJ+U+SO approach).

that principal input to the optical spectra of solids is due to transitions occurring from the top of the valence band to the bottom of the conduction band. Figure 7 indicates that the optical absorption edge, the so-called first critical point, appears in Tl4HgBr6 at about 2.7 eV. As already mentioned above, the band-gap value, E g , derived in the present APW+LO calculations, employing the MBJ+U+SO approximation, is equal to 2.456 eV, being in sufficiently good coincidence with the experimental Eg value derived for Tl4HgBr6 at 300 K, namely, Eg = 2.43 eV.14 It is well-known that DFT band-structure calculations usually bring underestimated values of the energy gaps in semiconductors and insulators. Therefore, because the calculated optical properties are extremely susceptible to the amounts of the interband energy interval, a scissor-factor-corrected Eg value is generally used when calculating the optical parameters. The above-mentioned scissor values are obtained as a difference between the calculated and experimentally measured Eg values for semiconductors/insulators.53 However, because of the fact that the theoretically and experimentally derived band-gap values of the Tl4HgBr6 compound coincide with each other,

Figure 9. Imaginary part of the dielectric function ε2(ω) of Tl4HgBr6 (calculations within the MBJ+U+SO approach). 10553


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Inorganic Chemistry energy up to about 10 eV, followed by its decrease to about 12.5 eV. At energies higher than 17.5 eV, the ε1(ω) curve decreases slowly to about 18.5 eV and then it increases monotonously up to about 27 eV. As can be seen from Figure 8, static dielectric constants of Tl4HgBr6 at zero frequency are as follows: ε1xx(0) = 7.918 and ε1zz(0) = 8.520. In addition, the calculated imaginary part of the dielectric function ε2(ω) reduces beginning from about 5.5 eV, revealing several spectral peaks marked as C, D, and E in Figure 9, and tends to about zero at energies of about 22−27 eV. Following the results of our DFT band-structure calculations for total DOSs and PDOSs of Tl4HgBr6 shown in Figures 3 and 4, one can conclude that the spectral peaks resolvable on the ε2(ω) curve (Figure 9) are composed because of the electronic transitions as follows: peaks A (∼4 eV) and B (∼5 eV) originate as a result of transitions from the upper valence Tl s states to the unoccupied Tl p states at the bottom of the conduction band, peak C (∼6 eV) originates as a result of transitions from Tl s to Tl p states and from Hg p to Hg s states, peak D (∼10.5 eV) is due to transitions from Hg s to Hg p states as well as from Tl s to Tl p states, and finally peak E (∼17 eV) is formed as a result of transitions from Br s to Br p states as well as from Hg d and Tl d to Tl p states. It is crucial that the above-presented interpretation of the origin of the peaks observed on the ε2(ω) curve of Tl4HgBr 6 includes only interband transitions accounting for the transition-dipole selection rules for the matrix elements of the electronic transition probabilities because in our calculations of the imaginary part of the dielectric function ε2ij(ω) the joint DOSs have been taken into account as follows:28 ρ(ω) =

Figure 10. Refractive index n(ω) dispersion of Tl4HgBr6 (calculations within the MBJ+U+SO approach).

The calculated extinction coefficient k(ω) dispersion for the Tl4HgBr6 compound is presented in Figure 11. From a

∑ ∫ dk ⃗ δ[εc(k ⃗) − εv(k ⃗) − ℏω] (8)

c,v

where the matrix element of the momentum generates the transition probability from the valence band (v) to the conduction band (c) and depicts the selection rules that do not allow the possibility of electronic transitions involving alterations of the orbital quantum number Δl = 0: Im ε2ij(ω , 0) α

1 ω2

Figure 11. Extinction coefficient k(ω) dispersion of Tl4HgBr6 (calculations within the MBJ+U+SO approach).

∑ ∫ dk ⃗ δ(εc(k ⃗) − εv(k ⃗) − ℏω)

comparison of Figures 9 and 11, one can conclude that the k(ω) curve is similar to the ε2(ω) curve in Tl4HgBr6. It is worth indicating that some visible deviations of the shapes for the k(ω) and ε2(ω) curves from each other can be explained by the fact that the MBJ+U+SO potential adopted in our DFT calculations of Tl4HgBr6 is not acquitted for the case of a medium with a nonzero absorption coefficient. Maximum values of the theoretically achieved extinction coefficient k(ω) are positioned within the energy region of about 5−20 eV, with several fine-structure peculiarities. Figure 12 shows the calculated optical reflectivity coefficient R(ω) of Tl4HgBr6. Two nonzero components of the R(ω) curve are equal to 22.62% and 23.98%, and they correspond to reflections extrapolated to 0 eV. From a comparison of Figures 10 and 11, one can see that the shape of the R(ω) curve closely follows that of the k(ω) curve for energies ranging from about 3 to 12 eV, while the energy positions of the fine-structure peculiarities observed on the R(ω) curve coincide well with those of the curves of the extinction coefficient k(ω) and of the imaginary part of the dielectric function ε2(ω) of Tl4HgBr6 (cf. Figures 9, 11, and 12). Furthermore, Figure 12 shows that sharp minima at energies of about 10, 17, and 25.5 eV are characteristic of the optical reflectivity coefficient R(ω) of Tl4HgBr6, and their

c,v

⟨ck |⃗ pi ̂ |vk ⃗⟩⟨vk |⃗ pĵ |ck ⃗⟩

(9)

where joint DOSs are multiplied by the matrix element of the probability of electronic transitions. The calculated refractive index dispersion n(ω) of the Tl4HgBr6 compound is presented in Figure 10. From a comparison of Figures 8 and 10, it is clear that the shapes and energy positions of features for the ε1(ω) and n(ω) curves are similar to each other. The calculated refractive index n(ω) of Tl4HgBr6 at zero frequency is close to nxx(0) = 2.814 and nzz(0) = 2.919. Maximum values of the refractive index n(ω) of the Tl4HgBr6 compound, as Figure 10 presents, are located in the energy range 2−6 eV, with a few small peaks emerging at other specific energies. As can be seen from Figure 10, the refractive index n(ω) of the Tl4HgBr6 compound gets very close to zero at high energies. The origin of peaks A (∼4 eV), B (∼5 eV), C (∼6−7 eV), D (∼10.5 eV), and E (∼17.5 eV) detected in the present DFT calculations for the refractive index n(ω) is explained by the above-reported interband electronic transitions corresponding to PDOSs for the Tl4HgBr6 compound, as plotted in Figures 3 and 4. 10554


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al.17 was found a peak at about 5 eV in the absorption, which is significantly lower than in the present calculations. This may be a consequence of the difference in the structural type used in ref 17 with respect to the present work. It may be useful also for understanding of the cationic substitutions.

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CONCLUSIONS In the present study, we have carried out DFT calculations for the total DOSs and PDOSs for Tl4HgBr6 single crystals using the APW+LO method, as implemented in the WIEN2k package. Contrary to previous electronic band calculations based on the P4mnc structure,17 we determined the band-gap magnitude adopting the noncentrosymmetric space group P4nc, yielding a very good agreement with the experiment.14 The results are very important because usually the DFT calculations underestimate these magnitudes. The theoretical results have been compared with XPS measurements. The excellent coincidence of the experimental and theoretical data confirms the existence of the P4nc noncentrosymmetric crystal structure. The DFT calculations indicate that the Br 4p states give the principal contributions to the valence band of Tl4HgBr6 crystallizing in the noncentrosymmetric space group P4nc. Moreover, they define the weak degree of the space-chargedensity acentricity. It was established that this acentricity is formed prevailingly by central and upper valence sub-bands. Among other substantive contributors are the valence cationic states associated with Tl and Hg atoms. The Tl 6s states contribute prevailingly at the top and bottom of the valence band, while the principal contributions of the Hg 6s states occur at its bottom. Regarding the occupation of the conduction band of Tl4HgBr6, our APW+LO calculations manifest that its bottom mainly consisted of the unoccupied Hg 6s states, with slightly smaller contributions of the unoccupied Br 4p states as well. Finally, using the MBJ+U+SO approximation in the DFT calculations of Tl4HgBr6, we obtained excellent agreement with the measured XPS valence-band spectrum of this compound.14 The calculations reveal that the Tl4HgBr6 compound is a direct-gap material: the VBM and CBM are positioned at the Γ point. The present calculations indicate that Tl4HgBr6 is a very prospective material for its use in optoelectronic devices. We have established that the shapes and energy positions of finestructure peculiarities of the curves for the real part of the dielectric function ε1(ω) and refractive index n(ω) of Tl4HgBr6 are similar to each other and the values of the calculated refractive index at zero frequency are close to nxx(0) = 2.814 and nzz(0) = 2.919. In addition, the present calculations of the imaginary part of the dielectric function ε2(ω) and extinction coefficient k(ω) indicate that the k(ω) curve closely follows the ε2(ω) curve in Tl4HgBr6. Two nonzero components of the calculated optical reflectivity coefficient R(ω) of Tl4HgBr6 are equal to 22.62% and 23.98%, and they correspond to reflections at 0 eV. Furthermore, the DFT calculations allow one to conclude that the anisotropy of the electron energy-loss spectrum L(ω) for two different components of the tensor (xx and zz) of the Tl4HgBr6 compound is rather insignificant at energies ranging from about 3 to 27 eV.

Figure 12. Optical reflectivity R(ω) of Tl4HgBr6 (calculations within the MBJ+U+SO approach).

energy positions coincide with those of similar minima detected on the curve of extinction coefficient k(ω) (see Figure 11). The presence of the minima for energies at about 10 and 17 eV is also typical for the curve of the imaginary part of the dielectric function ε2(ω) of Tl4HgBr6 (Figure 9). Figure 11 shows that maximum values of the reflectivity coefficient R(ω) of the Tl4HgBr6 compound are detected at energies ranging from about 18 to 32 eV, with some smaller peaks appearing at other specific energies. Figure 13 presents the electron energy-loss spectrum L(ω) of Tl4HgBr6. From this figure, one can see that the L(ω) curve

Figure 13. Electron energy-loss spectrum L(ω) of Tl 4 HgBr 6 (calculations within the MBJ+U+SO approach).

increases quietly beginning from about 3.5 eV and yields two small maxima at about 9.5 and 17 eV, followed by a monotonous sharp increase up to about 25 eV. This case results in the appearance of a maximum positioned at this energy value. A comparison of Figures 11 and 13 allows to one conclude that the spectral maximum at about 25 eV on the L(ω) curve corresponds to the plasma frequency of Tl4HgBr6. Furthermore, as presented in Figure 13, the anisotropy of the L(ω) function for two different components of the tensor (xx and zz) of the Tl4HgBr6 compound is rather insignificant at energies ranging from about 3 to 27 eV. It is necessary to emphasize that experimentally in the 4−5.2 eV range by Brik et

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 10555


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Inorganic Chemistry Notes

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The authors declare no competing financial interest.

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