Localizing Holes as Polarons and Predicting Band Gaps, Defect

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX-XXX

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Localizing Holes as Polarons and Predicting Band Gaps, Defect Levels, and Delithiation Energies of Solid-State Materials with a Local Exchange-Correlation Functional Shuping Huang,*,†,‡ Pragya Verma,‡,§ and Donald G. Truhlar*,‡,§ †

College of Chemistry, Fuzhou University, Fuzhou, Fujian 350108, P. R. China Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, 207 Pleasant Street SE, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States § Nanoporous Materials Genome Center, 207 Pleasant Street SE, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States ‡

ABSTRACT: This work assesses the performance of seven exchange−correlation functionals (some with and some without a Hubbard U correction) for their ability (i) to predict band gaps of silicon, diamond, and Li-ion battery cathode materials, (ii) to localize hole polarons and predict delithiation energies in Li-ion battery cathode materials, and (iii) to predict transition levels of charge carriers of doped silicon and diamond. Both local and hybrid exchange−correlation functionals were tested. The local functionals tend to underestimate band gaps and delocalize polarons. The hybrid functionals very often give a good description of both properties, but they may not be practical for calculations involving large unit cells, large ensembles, or dynamics, and therefore a local functional with a Hubbard U correction is often used (giving the method called DFT+U), where the value of a parameter U is adjusted according to the system and the property being investigated. Keeping in mind the importance of computational cost and the undesirability of having to adjust an empirical parameter for each system or property of interest, we recently developed a local functional, namely HLE17, to try to accurately predict band gaps and excitation energies, and we validated it using mostly main-group solids and molecules. Here we test the performance of HLE17 for its ability to predict band gaps and localize polarons in other solid-state materials, and we compare its performance to that of popular local functionals (PBE, PBEsol, and TPSS), a range-separated hybrid functional with screened exchange (HSE06), and DFT+U. We find that HLE17 predicts more accurate band gaps than other local functionals and can localize holes as polarons, which other local functionals usually fail to do, and for a number of cases it is comparable in performance quality to Hubbard-corrected functionals without the need for system-specific parametrization and to hybrid functionals without the high cost. Because HLE17 does not predict accurate lattice constants, we use the single-point method of quantum chemistry, where the geometry is optimized with one functional and the band gap is calculated with HLE17, or we perform calculations with the lattice constants obtained by TPSS and both the fractional intracell coordinates and the electronic structure obtained by HLE17 (a new method denoted HLE17\\TPSS). In particular, we performed calculations by HLE17//TPSS, HLE17//HSE06, HLE17//DFT+U, and HLE17\\TPSS, and these methods usually agree well with each other and give values similar to experiment.

1. INTRODUCTION

properties, such as electron densities, their gradients, and electron kinetic energy densities. Traditional local density functionals, such a generalized gradient approximations (GGAs) that reduce to the Gàspàr−Kohn−Sham local density functional for a uniform electron gas, severely underestimate band gaps, and hence, they are also unreliable for defect levels in doped systems or operating electrodes, but knowledge of band gaps and defect levels is essential to understanding many material properties.12 To circumvent this problem, many calculations are reported in which local-functional KS-DFT is

Quantum mechanical methods are becoming more and more useful for their ability to predict properties of materials. Specifically Kohn−Sham density functional theory (KS-DFT)1 has been widely used because of it good combination of speed and accuracy.2−11 However, the accuracy and cost of Kohn− Sham calculations depend on the exchange-correlation functional, and the most successful exchange-correlation functionals are hybrid functionals that are very expensive for plane-wave calculations on extended systems.5,6 Hybrid functionals include an admixture of nonlocal Hartree−Fock exchange, which improves the accuracy for certain properties but only at a much higher cost in computation time than is required by local functionals, which are functionals that depend only on local © XXXX American Chemical Society

Received: September 9, 2017 Revised: October 6, 2017 Published: October 6, 2017 A

DOI: 10.1021/acs.jpcc.7b09000 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Just as conventional local density functionals almost always underestimate band gaps, they also tend to underestimate transition energies of defects (which are equal to the energy difference between the defect level and a band edge) in semiconductors, e.g., the transition energies of charge carriers in doped silicon and diamond,32 and here we test if these problems can be overcome with the HLE17 high-localexchange functional.

extended by adding a Hubbard correction, which is a method called DFT+U.13−16 Applying the Hubbard correction often improves the accuracy over local functionals with a negligible increase in cost, but it has its own limitations, especially the need to adjust the empirical parameter U.17−23 In an earlier work,24 we showed that when DFT+U is compared to KS-DFT, the empirical correction to DFT did not improve results for all the systems and all the properties that were investigated. Also the value of U needs to be tuned not only for the system and/or the property being investigated, but also for the density functional with which it is used (the optimum value of U for different functionals may or may not be the same).24−26 In light of this situation, local density functionals that do not need any system-dependent parameters and that at the same time have accuracy similar to or better than hybrid density functionals for some of the key properties of solid-state physics and chemical physics would be highly desirable, if they can be achieved. With this motivation, the high-local-exchange functionals HLE16 and HLE1727,28 were developed recently, and they were shown to give band gaps of semiconductors similar in accuracy to those obtained with a hybrid functional. The HLE17 functional is used in the current work, and for background we note that it was obtained by raising TPSS exchange by a factor of 1.25 and simultaneously decreasing its correlation by a factor of 0.50. These factors were chosen to give improved band gaps for semiconductors while keeping reasonable values for thermochemical predictions on molecules. In the present work, the HLE17 functional is compared to previous results obtained with DFT and DFT+U methods for its ability to predict delithiation energies, band gaps, and localization of hole polarons in Li-ion battery cathode materials and to predict band gaps and defect states for doped and undoped silicon and diamond. The cathode materials considered are pure and doped lithium zirconate, and delithiation is considered for both one and two Li atoms; the Li atom is extracted from the tetrahedral site rather than the octahedral site because extraction from the former is easier.29−31 Lithium is present in the pristine cathode (Li8ZrO6) as a combination of a Li+ ion at a lattice site and an electron in a band. When we charge the battery, we move Li atoms to the anode. So, in addition to removing Li+ at a lattice site (creating a defect), this creates an unoccupied orbital in the band (i.e., a hole). But in reality, the unoccupied orbital is not spread out over the crystal as a band orbital would be (and as the orbital was in the pristine cathode); it localizes and forms a hole polaron. Conventional local functionals predict that it remains delocalized, which is wrong. Note that the missing electron in the band is a positive hole; if this hole is localized near the defect, it oxidizes a nearby O2− to make O−. Since O2− is a closed shell, it has no magnetic moment, but since O− is an open shell, it is paramagnetic. Therefore, a small magnetic moment on oxygens near the defect implies that they have mainly the character of O2−, which implies a delocalized hole. But a larger magnetic moment implies significant O− character, i.e., a hole polaron. Conduction with delocalized holes occurs by band conduction, but the creation of polarons converts this to conduction by polaron hopping. Therefore, a proper treatment of hole localization/delocalization is crucial to adequate treatment of charge transport in the battery, and we will examine magnetic moments to determine if HLE17 overcomes this deficiency of conventional local functionals.

2. SOLIDS AND PROPERTIES TESTED This section describes the solids and their properties investigated in this work; see Table 1, which shows the Table 1. Solids, the Sizes of Their Unit Cells, and the Properties Investigated in This Work solida Li8ZrO6 Li24Zr3O18 Li96Zr12O72 Li7ZrO6 Li93FeZr12O72 Li16Fe16P16O64 Si512 Si512:Ci Si512:SSi Si512:OSi Si512:InSi C512 C512:PC C512:BC

no. of atoms in unit cell 15 45 180 14 178 112 512 513 512 512 512 512 512 512

properties delithiation energy, band gap, polarons lattice constant delithiation energy, polarons delithiation energy, polarons delithiation energy delithiation energy, polarons band gap charge carrier transition levels charge carrier transition levels charge carrier transition levels charge carrier transition levels band gap charge carrier transition levels charge carrier transition levels

a

For solids indicated by Si512:ESi or C512:EC, E is the element that replaces one Si or C atom in Si512 or C512. For Si512:Ci, C is an interstitial element.

number of atoms in the unit cell of each periodic structure. We studied three Li-ion battery cathode materials, namely, Li8ZrO6 (lithium zirconate, abbreviated LZO, shown in Figure 1a), Fedoped LZO, and LiFePO4 (lithium iron phosphate, abbreviated LIP, shown in Figure 1b), and we studied diamond and silicon materials with and without doping by C, B, In, O, P, or S. We calculate lattice constants, band gaps, delithiation energies, charge carrier transition levels, and magnetic moments, which are used as an indicator of how well polarons are

Figure 1. (a) Supercell of Li8ZrO6. The zirconium ions occupy octahedral sites, and the lithium ions occupy both tetrahedral sites (labeled as Td) and octahedral sites (labeled as Oh). (b) Supercell of LiFePO4. B


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denoted by U in the rest of this article, which is a standard convention in the field. For lithium zirconate and doped lithium zirconate, either U = 4 or U = 6 eV is used, where the choice of 6 eV is based on our previous work,29,30 and the choice of 4 eV is examined because it is a popular choice.24,27 The core electrons of all the solids were represented by projector-augmented-wave (PAW) potentials;44,45 the PAW potentials used for Li, Zr, Fe, P, B, Si, C, In, and O are Li_sv, Zr_sv, Fe, P, B, Si, C, In_d, and O, respectively, in the notation of VASP. Plane-wave basis sets were used with a cutoff energy of 650 eV for LZO and doped LZO, 400 eV for LIP, and 450 eV for Si, C, and their doped materials. The self-consistent-field (SCF) energy convergence criterion was 10−4 eV, and the force convergence criterion was −10−3 eV/Å. A Monkhorst−Pack k-point mesh of 6 × 6 × 6, 6 × 6 × 2, and 3 × 3 × 2 was used for LixZrO6 (x = 6, 7, 8), Li24Zr3O18, and LimFenZr12O72 (m = 96, 95, n = 0; m = 93, n = 1; m = 92, n = 1), respectively. A Monkhorst−Pack k-point mesh of 2 × 2 × 2 was used for the orthorhombic (1 × 2 × 2) supercell of LIP. For LIP, only high-spin ferromagnetic states were considered. For the Si, diamond, doped Si, and doped diamond calculations, we used a unit cell with 512 (or 513 in case of Si512:Ci) atoms, and only the Γ point was calculated. Except where indicated otherwise, in the optimizations both the lattice constants and the fractional coordinates of the atoms were relaxed. We will also report some calculations where the lattice constants were fixed and only the fractional coordinates were optimized. The notation “A//B” indicates a “single-point energy calculation” with the energy calculated by method A at a geometry (lattice constants, cell shape, and fractional coordinates) optimized by method B; whereas “A” alone indicates that both the energy and the geometry were optimized by method A. The notation “A\\C” denotes a calculation with the energy and fractional coordinates optimized by method A within the fixed cell size and shape obtained by method C.

localized. All magnetic moments are given in Bohr magnetons (μB).

3. COMPUTATIONAL DETAILS The density functionals tested in this work and their types and percentages of Hartree−Fock exchange are given in Table 2. All Table 2. Methods Used in This Work method

typea

Xb

refs

PBE PBE+U PBEsol GAM+U M06-L TPSS HLE17 HSE06

GGA GGA+U GGA NGA+U meta-GGA meta-GGA meta-GGA hybrid-GGA

0 0 0 0 0 0 0 25−0

35 14, 35 36 14, 41 37 38 27 33, 34

a

GGA: generalized gradient approximation. NGA: nonseparable gradient approximation. +U: with a Hubbard correction. Meta: with kinetic energy density. Hybrid: with Hartree−Fock exchange. bX indicates the percentage of Hartree−Fock exchange; 25−0 denotes 25% Hartree−Fock exchange at short interelectronic separations and no Hartree−Fock exchange at long interelectronic separations.

exchange−correlation functionals in Table 2 except HSE0633,34 are local functionals (which means they have no Hartree−Fock exchange and no nonlocal correlation), and HSE06 is a rangeseparated hybrid functional that has 25% Hartree−Fock exchange at short interelectronic separations and no Hartree−Fock exchange at long interelectronic separations. Calculations with PBE,35 PBE+U,14,35 PBEsol,36 M06-L,37 TPSS,38 and HSE0633,34 were performed using the released version of the Vienna ab initio simulation package (VASP)39,40 and those with GAM+U14,41 and HLE1727 were performed using a locally modified version of VASP called MinnesotaVASP Functional Module (MN-VFM).42 In the Hubbard-corrected calculations, the empirical Hubbard correction was applied only to valence 3d orbitals of Fe in LiFePO4 and to the valence 2p subshells of O in lithium zirconates and doped lithium zirconates. We used the formulation of Dudarev et al.14 in which the difference between the on-site Coulomb (U) and exchange (J) integrals matters but not the actual values of U and J. The U and J values of 5.3 and 1.0 eV, respectively, are applied to Fe in LiFePO4 based on the work of Johannes et al.43 The difference U − J will be

4. RESULTS AND DISCUSSION This section is divided into three parts. In the first part, we discuss various properties of Li-ion battery cathode materials, in the second part we discuss band gaps of diamond and silicon, and in the third part we discuss transition levels of charge carriers of their doped materials. 4.1. Li-Ion Battery Cathode Materials. The experimental structure of LZO is known;29 therefore we first benchmark the

Table 3. Comparison of Unit Cell Parameters of Li24Zr3O18 Determined by Rietveld Refinement and Quantum Mechanical Calculations method

Ua (eV)

a (Å)

b (Å)

c (Å)

α (deg)

β (deg)

γ (deg)

volume (Å3)

b

− − 4 6 − 4 6 − − − −

5.49 5.51 5.48 5.47 5.45 5.50 5.49 5.46 5.47 5.23 5.46

5.49 5.51 5.48 5.47 5.45 5.50 5.49 5.46 5.47 5.23 5.46

15.47 15.58 15.49 15.46 15.39 15.56 15.52 15.46 15.47 14.78 15.39

90.00 90 90 90 90 90 90 90 90 90 90

90.00 90 90 90 90 90 90 90 90 90 90

120.0 120 120 120 120 120 120 120 120 120 120

404 410 403 400 396 408 405 399 401 350 397

expt PBE PBE+U PBE+U PBEsol GAM+U GAM+U M06-L TPSS HLE17 HSE06 a

U is applied only to oxygen atoms. bref 29. C


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Table 4. Delithiation Energy for the Reaction Li8ZrO6 → Li7ZrO6 + Li, Magnetic Moment of O in Li7ZrO6, and Band Gaps of Li7ZrO6 and Li8ZrO6 band gap (eV) of Li7ZrO6

a

method

Ua (eV)

delithiation energy (eV)

magnetic moment (μB) of O in Li7ZrO6b

spin up

spin down

band gap (eV) of Li8ZrO6

PBE PBE//PBEsol PBE//HSE06 PBE+U PBE+U PBE+U//PBE PBE+U//PBEsol PBE+U//PBEsol PBEsol GAM+U GAM+U TPSS TPSS//HSE06 HLE17 HLE17\\TPSS HLE17//PBE+U HLE17//PBEsol HLE17//GAM+U HLE17//TPSS HLE17//HSE06 HSE06

− − − 4 6 6 4 6 − 4 6 − − − − 6 − 4 − − −

3.71 3.72 3.79 3.38 3.12 3.31 3.58 3.42 3.87 2.95 2.70 4.28 3.83 7.41 4.69 4.49 4.68 4.44 4.98 4.49 4.04

0.31 0.30 0.54 0.67 0.72 0.63 0.47 0.57 0.28 0.75 0.79 0.29 0.58 0.66 0.71 0.72 0.46 0.71 0.48 0.71 0.70

5.65 5.66 5.66 5.29 5.36 5.65 5.67 5.67 5.38 5.55 5.43 5.50 5.55 6.29 6.42 5.72 5.96 5.96 5.98 5.95 7.09

metallic metallic 0.45 0.90 1.59 0.90 metallic 0.67 metallic 1.33 1.83 metallic 0.38 0.86 1.19 1.19 metallic 1.18 0.48 1.19 2.05

4.93 5.14 5.11 5.07 5.14 5.37 5.37 5.39 4.94 5.16 5.14 5.15 5.46 6.05 5.94 5.80 5.82 5.65 5.58 5.82 6.78

U is applied only to oxygen atoms. bThe magnetic moment of O4 in Li7ZrO6.

The band gaps calculated for LZO are given in Table 4 where they may be compared to the experimental value of 5.75 eV.29 We see that HSE06 overestimates the band gap by ∼1 eV. In comparison, HLE17 at any of the three realistic geometries predicts a much more accurate band gap that is within 0.2 eV of the experimental value. Table 4 also reports the calculated magnetic moment on one of the oxygen atoms of Li7ZrO6. In particular, we created Li7ZrO6 by removing Li1 of Li8ZrO6 in Figure 8a of ref 31, and we report the magnetic moment of O4, which is one of the oxygens close to the Li vacancy. The magnetic moment is a measure of how well the hole polarons are localized on the oxygen atom, where a higher value is indicative of conversion of O2− to O− and hence of greater hole localization. The hybrid functional in Table 4, namely HSE06, correctly predicts the formation of hole polarons. The local functionals PBE, PBEsol, and TPSS do not predict localized oxygen hole polarons in Li7ZrO6, while the magnetic moments for HLE17, for most of the Hubbard-corrected methods, and for HSE06 do indicate localized hole polarons in this material. Although HLE17 localizes the hole polaron even at the incorrect geometry, single-point calculations at better geometries result in better localization of the polaron, with a magnetic moment on oxygen within 2% of the value predicted by HSE06. Also reported in Table 4 are the band gaps of Li7ZrO6. The up-spin and down-spin band gaps in Table 4 correspond respectively to the gap between the valence band and the conduction band and the gap between the valence band and the midgap polaron. When the hole is not localized in the downspin valence band, the band structure is half-metallic, i.e., metallic for one spin but with a gap for the other. We see a rough correlation between the magnetic moment on O and the position of the polaron in the gap. For example, with only one exception, if the magnetic moment on O is below 0.5, the

methods shown in Table 2 for their ability to correctly predict the lattice constants of LZO. The results are summarized in Table 3. The best geometries (lattice constants and volumes) are obtained with PBE+U (U = 4 eV) and GAM+U (U = 6.0 eV) and the next best are obtained with TPSS. After DFT+U and TPSS, the next best geometries are by M06-L; the geometries given by HSE06, PBE, and PBEsol are less accurate than these. As was pointed out in our earlier work,28 HLE17 is not good for geometries, and when compared to experiments it was shown to underestimate both bond lengths of molecules and lattice constants of solids. The results shown in Table 3 for LZO agree with what we had observed in ref 28. for solids other than LZO. Therefore, in the rest of the tables, with HLE17, we report single-point calculations on structures given by methods with smaller geometrical errors in Table 3 or by HLE17\\TPSS calculations. Even though the DFT+U geometries are slightly better than the TPSS geometries, TPSS is preferred for such geometrical calculations for the simple reason that, unlike DFT +U, it does not depend on the empirical parameter U. Table 4 presents the delithiation energy for the solid-state reaction Li8ZrO6 → Li7ZrO6 + Li as obtained by various quantum mechanical methods. The hybrid functional in Table 4, namely HSE06, gives a reasonable delithiation energy. Optimization of LZO by HLE17 leads to a very large delithiation energy (7.41 eV). Such a large delithiation energy can be attributed to the underestimation of lattice constants by HLE17 corresponding to the atoms in the unit cell being closer to each other, which makes it difficult to remove a Li atom from LZO. Table 4 shows that single-point calculations with HLE17 on geometries predicted by methods that give better lattice constants than HLE17, result in a reasonable delithiation energy, only 10−15% higher than the experimental value29 of about 4.0 eV. D


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Table 5. Comparison of Unit Cell Parameters of the Delithiated Structures, Li7ZrO6 and Li6ZrO6, from Quantum Mechanical Calculations

a

method

Ua (eV)

a (Å)

b (Å)

PBE PBE+U PBE+U PBEsol GAM+U GAM+U TPSS HLE17 HSE06

− 4 6 − 4 6 − − −

6.02 5.99 5.99 5.99 6.03 6.01 6.00 5.72 5.96

5.92 5.86 5.85 5.88 5.87 5.86 5.92 5.60 5.82

PBE PBE+U TPSS HLE17 HSE06

− 6 − − −

5.97 5.84 5.89 5.58 5.81

5.31 6.31 6.28 5.97 6.28

c (Å) Li7ZrO6 6.14 6.10 6.09 6.07 6.12 6.12 6.11 5.82 6.06 Li6ZrO6 5.95 5.89 5.93 5.66 5.87

α (deg)

β (deg)

γ (deg)

54.1 55.0 55.0 53.8 54.9 55.0 53.8 54.9 55.1

55.4 55.9 55.8 54.5 55.7 55.9 54.9 55.6 55.9

55.1 55.8 55.8 55.0 55.9 56.0 54.8 55.9 55.8

136 136 135 131 137 137 134 118 133

52.2 53.2 52.4 53.0 53.3

50.8 53.6 51.6 51.9 53.5

58.1 58.3 57.4 58.7 58.2

133.5 135 132 115 133

volume (Å3)

U is applied only to oxygen atoms.

Table 6. Delithiation Energy (eV) for the Reaction Li96Zr12O72 → Li95Zr12O72 + Li and Magnetic Moment of O9 in Li95Zr12O72 delithiation energy magnetic moment (μB)b

TPSS

HSE06

PBE+Ua

HLE17//HSE06

HLE17//TPSS

HLE17\\TPSS

4.24 0.02

4.33 0.68

3.28 0.70

4.78 0.68

5.17 0.03

5.07 0.02 (0.28, 0.25)c

a U = 6 eV is applied only to oxygen atoms. bMagnetic moment on one O in Li95Zr12O72. cThe magnetic moments of O17 and O68 which are close to the Li vacancy.

Table 7. Delithiation Energy (eV) for the Reaction Li7ZrO6 → Li6ZrO6 + Li and Magnetic Moment of the Six O Atoms in Li6ZrO6 delithiation energy magnetic moment (μB)b

TPSS

HSE06

PBE+Ua

HLE17

HLE17//HSE06

HLE17//TPSS

4.22 0.480 0.353 0.163 0.109 0.380 0.250

3.83 0.692 0.143 0.083 0.033 0.705 0.032

2.93 0.715 0.149 0.084 0.038 0.727 0.034

7.37 0.680 0.174 0.093 0.051 0.689 0.065

4.40 0.710 0.145 0.085 0.031 0.723 0.035

4.56 0.607 0.285 0.089 0.058 0.541 0.243

a U = 6 eV is applied only to oxygen atoms. bThe magnetic moment is listed for each of the six O atoms in Li6ZrO6, with the largest values in bold font.

HLE17//HSE06. The HLE17//GAM+U and HLE17\\TPSS results (also HLE17//PBE+U) are very encouraging from a cost standpoint since they do not require any calculations with Hartree−Fock exchange. Table 5 compares the calculated lattice parameters of delithiated structures, Li7ZrO6 and Li6ZrO6, by different methods. Similar to what was shown in Table 3 for Li24Zr3O18, we find that as compared to other methods, the HLE17 functional underestimates the lattice constants (a, b, and c) of Li7ZrO6 and Li6ZrO6. Table 6 gives results for delithiation energy and magnetic moment of oxygen for a unit cell that is 12 times larger than what is presented in Table 4. This corresponds to a smaller percentage delithiation, i.e., the cell energy at a point in time closer to the start of delithiation. For the methods that are common to Tables 4 and 6, all of them except for TPSS predict larger delithiation energy for Li96Zr12O72 than Li8ZrO6, and all of them give a smaller magnetic moment of oxygen in Li95Zr12O72 than in Li7ZrO6. The large difference in magnetic

down-spin manifold is metallic, and if it is above 0.5, it is semiconducting. In most cases the conclusion from the downspin band structure about whether the hole is localized agrees with the conclusion from the oxygen magnetic moments, but for HLE17//PBEsol it does not. The HSE06 geometry has a localized polaron, and there will be a geometry distortion ready to accommodate a polaron in the PBE, TPSS, and HLE17 single-point energy calculations with this geometry. But the PBEsol geometry does not have that local distortion, so there is no distorted site at which the polaron can be localized. As expected from the kind of argument in the previous paragraph, the HLE17 calculation with lattice constants from TPSS but fractional coordinates optimized by HLE17 (HLE17\ \TPSS) gives results closer to HSE06 than to HLE17 and HLE17//TPSS in the delithiation energy of Li8ZrO6 → Li7ZrO6 + Li, the band gap of Li7ZrO6, and the magnetic moment of O in Li7ZrO6. All things considered, the best results in Table 4 are for HLE17//GAM+U, HLE17\\TPSS, HLE17//PBE+U, and E


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The Journal of Physical Chemistry C Table 8. Delithiation Energy (eV) for the Reaction Li93FeZr12O72 → Li92FeZr12O72 + Li delithiation energy a

TPSS

HSE06

PBE+Ua

HLE17//HSE06

HLE17//TPSS

3.75

4.13

3.24

6.02

4.77

U = 6 eV is applied only to oxygen atoms.

Table 9. Delithiation Energy (eV) for the Reaction Li16Fe16P16O64 → Li15Fe16P16O64 + Li and Magnetic Moments of Fe in Li16Fe16P16O64 and Li15Fe16P16O64 delithiation energy magnetic moment (μB) of Fe in Li16Fe16P16O64b magnetic moment (μB) of 16 Fe atoms in Li15Fe16P16O64

a

TPSS

HSE06

PBE+Ua

HLE17//PBE+U

HLE17//TPSS

3.06 3.595 3.627 3.668 3.645 3.623 3.642 3.624 3.630 3.687 3.624 3.637 3.648 3.624 3.646 3.624 3.629 3.643

4.00 3.703 3.701 3.708 3.710 3.706 3.701 3.704 3.708 4.254 3.704 3.700 3.707 3.700 3.702 3.701 3.706 3.700

4.82 3.78 3.778 3.783 3.785 3.781 3.778 3.780 3.784 4.322 3.780 3.778 3.782 3.778 3.778 3.778 3.782 3.778

3.45 3.652 3.663 3.703 3.688 3.662 3.681 3.666 3.680 3.868 3.661 3.676 3.693 3.665 3.679 3.664 3.677 3.686

3.62 3.647 3.671 3.717 3.687 3.668 3.684 3.670 3.674 3.740 3.668 3.684 3.696 3.670 3.694 3.671 3.674 3.689

U = 4.3 eV is applied only to iron atoms. bThe magnetic moment for all 16 Fe atoms is the same.

of values, even when using only realistic geometries. Experimentally, the delithiation energy seems to be about the same after doping as before, within a few tenths of an electonvolt.30 Comparison to Table 6 shows that HSE06 and especially PBE+U give reasonable results in this respect, whereas TPSS and HLE17//HSE06 show large differences, and HLE17//TPSS is intermediate. Table 9 gives delithiation energies and magnetic moments for another Li-ion battery cathode material, namely LiFePO4. For delithiation energies, the two single-point HLE17 values are closer to HSE06 than are TPSS or PBE+U. For magnetic moments of the 16 Fe atoms in the fully lithiated material, Li16Fe16P16O64, all methods give similar values (3.6−3.8 μB). In the partially delithiated material, Li15Fe16P16O64, the magnetic moments of the 16 iron atoms are not the same. Table 9 shows that HSE06 and PBE+U do predict a localized hole polaron on one of the irons in the partially delithiated material, while the two HLE17 methods and TPSS have delocalized holes. 4.2. Silicon and Diamond. Our calculated PBE band gaps for Si (0.65 eV) and C (4.13 eV) are close to the ones (0.61 eV for Si and 4.21 eV for C) calculated by Deak et al.32 The calculated gaps by HLE17//PBE and HSE06//PBE for Si are respectively 1.65 and 1.23 eV, and those for diamond are respectively 5.02 and 5.30 eV. Thus, the mean absolute deviation of the PBE band gaps from the HSE06 ones is 0.88 eV, and this is lowered to 0.35 eV by HLE17. The calculated band gaps of Si and diamond by HLE17//TPSS are 1.63 and 5.02 eV, respectively. 4.3. Doped Silicon and Diamond. In Si512:Ci, the carbon atom is a split interstitial in Si. The C and a Si atom close to it share a lattice site in a dumbbell configuration. Both atoms are three-coordinated, having a p orbital orthogonal to the three sp2 bonds. That p orbital of the C is doubly occupied, giving

moments yielded by HLE17//HSE06 and HLE17//TPSS in Table 6 cannot be explained by the small difference in lattice constants because HSE06 and TPSS give similar lattice constants for both fully lithiated and partially delithiated LZO (See Tables 3 and 5). Therefore, we attribute it to a difference in fractional coordinates in the cell. This comparison shows that when one judges the quality of solid-state geometry predictions, it is not sufficient to simply consider the lattice constants. The HLE17\\TPSS method does not work as well for Li95Zr12O72 as does HLE17//HSE06; the locations of the holes are different in the two cases (see Table 6). The delithiation energy and magnetic moment on oxygen corresponding to the removal of second Li atom from LZO are shown in Table 7. Again, similarly to what was shown in Table 4, the delithiation energy by HLE17 is much higher than those by TPSS, HSE06, and PBE+U, but the delithiation energy calculated using HLE17 energies at the HSE06 or TPSS geometry (HLE17//HSE06 or HLE17//TPSS) is more reasonable. The values of magnetic moments on the first and fifth oxygen atoms of Li6ZrO6 show that all methods except TPSS predict two localized oxygen hole polarons. The HLE17\ \TPSS calculation for Li6ZrO6 led to an unreasonable structure (some of the atoms are too close) and failed (although the HLE17\\HSE06 calculation works well), so HLE17\\TPSS is not shown in Table 7. This is explained by the different character of the TPSS lattice. In the PBE+U and HSE06 optimized geometries in Table 5, the angle α is close to β. But in the TPSS optimized geometry, the angle α is about 2 deg larger than β. Table 8 shows the delithiation energy corresponding to the removal of a Li atom from iron-doped Li96Zr12O72, where an iron atom has replaced three octahedral Li atoms to give the doped material, Li93FeZr12O72. Here there is a wide distribution F


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In other doped materials, one silicon or carbon atom in silicon or diamond solid is replaced by another element (In, B, P, S, or O). The B and In atoms have less valence electrons than C and Si atoms, and doping with them gives holes, while S and P have more valence electrons than Si and doping with them gives negative charge carriers. The sulfur has a covalent radius comparable to silicon, and that enables it to be coerced into sp3 hybridization and 4-fold coordination. There are no localized states left for the fifth and sixth valence electrons of S, so they have energies close to the conduction band edge. The O and S are in the same column in the periodic table and have the same number of valence electrons. The covalent radius of O (60 pm) is much smaller than that of Si (110 pm), and the oxygen optimizes to an off-center position where it bonds to only two Si neighbors and forms two lone pairs. The other two silicon atoms (the distance between the two silicon atoms is 3.39 Å when optimized with the PBE functional) have dangling bonds, forming acceptor states. The PBE, HLE17\\TPSS, and HSE06//PBE calculations yield respectively 0.64, 1.66, and 1.87 eV for the energy by which the configuration with the O off-center is lower than that of the one with the O on-center. It is encouraging that this HLE17 energy calculation agrees so well with this much more expensive HSE06 energy calculation. The charge transition levels of donors, E(+/0), with respect to the conduction band minimum (CBM) and of acceptors, E(0/−), with respect to the valence band maximum (VBM) for doped Si and C are shown in Table 10. The charge transition levels are calculated by the ΔKS method,46 and the average potentials between the perfect crystal and the defective supercell are aligned by using the method suggested in ref 47. The calculated acceptor level for Si 512 :In Si by HLE17\\TPSS are very close to the experimental value. Both HSE06 and HLE17 slightly underestimate the acceptor levels for C512:BC, and slightly overestimate the acceptor levels for Si512:Ci and Si512:SSi. The transition levels predicted by PBE and TPSS are underestimated compared to both HLE17 and HSE06. We find that HLE17\\TPSS performs better than HLE17//PBE, except for Si512:SSi.

rise to a double donor level in the lower half of the band gap, while the p orbital of Si is empty, corresponding to a double acceptor level in the upper part. Figures 2a,b and 3a,b show that

Figure 2. (a) Partial charge density by HLE17\\TPSS for the donor state of Si512:Ci and (b) partial charge density by HLE17\\TPSS for the acceptor state of Si512:Ci. The blue and brown balls represent silicon and carbon, respectively. The isosurface level is set to be 0.02.

Figure 3. (a) Partial charge density by HLE17//PBE for the donor state of Si512:Ci and (b) partial charge density by HLE17//PBE for the acceptor state of Si512:Ci. The blue and brown balls represent silicon and carbon, respectively. The isosurface level is set to be 0.0002.

5. CONCLUDING REMARKS The performance of popular density functionals widely used for solid-state calculations has been tested for various properties of solids and compared to the newly developed local functional, HLE17. For lithium zirconate, delithiated lithium zirconate, doped lithium zirconate, and lithium iron phosphate, we examined delithiation energies, band gaps, and localization of hole polarons (iron hole polarons in one case and oxygen hole

the HLE17\\TPSS method can describe the localized defect states correctly, but HLE17//PBE cannot. The HLE17\\TPSS scheme was motivated by the fact that TPSS gives (by design) reasonable lattice constants, HLE17 gives (by design) reasonable band gaps, and the “\\” scheme allows HLE17 to optimize the local surroundings (fractional coordinates) at the defect to create a polaron, and this example shows that the strategy performs well.

Table 10. Charge Transition Levels of Donors, E(+/0), with Respect to the CBM and of Acceptors, E(0/−), with Respect to the VBM in (eV)a

C512:PC Si512:SSi Si512:Ci C512:BC Si512:Ci Si512:OSi Si512:InSi a

PBE

HSE06//PBE

−0.25 −0.22 −0.55

−0.45 −0.34 −1.04

0.21 0.47 0.51 0.08

0.28 1.02 0.97 0.11

HLE17//PBE donor −0.27 −0.41 −1.03 acceptor 0.18 1.19 1.22 0.10

TPSS

HLE17\\TPSS

exptb

−0.25 −0.26 −0.59

−0.45 −0.48 −1.21

−0.58 −0.32 −0.89

0.21 0.52 0.57 0.09

0.24 1.22 1.00 0.14

0.37 1.05 0.95 0.15

CBM denotes conduction band minimum; VBM denotes valence band maximum. bThe experimental data are from ref 48 and references therein. G


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Although the focus of this work is on testing and validating HLE17, we note that the HLE17 functional is interesting not only in its own right but also for suggesting broader possibilities for future functional development by not restricting the local exchange.

polarons in others). For silicon and diamond, we examined band gaps and for doped silicon and doped diamond, we examined transition levels of charge carriers. These tests are particularly interesting because the systems and problems considered here (except for pure diamond and pure silicon) are quite different from those on which HLE17 was previously validated and quite different from those that were considered when the parameters were defined. For LZO and doped-LZO, the delithiation energy calculated using HLE17 for both band structure and geometry optimization is much higher than the values calculated by PBE+U and HSE06; this is due to the fact that HLE17 does not give good lattice constants, so we recommend and further tested composite calculations, in particular, HLE17//TPSS, HLE17//HSE06, and HLE17//DFT+U in which HLE17 single-point calculations were performed at geometries optimized by TPSS, and HLE17\\TPSS, in which HLE17 band structures and fractional coordinates were obtained with lattice constants obtained by other methods, and these composite calculations gave much more reasonable results. In the summary below we discuss only the single-point HLE17 calculations and HLE17\\TPSS results. As is well-known, the popular PBE local functional underestimates band gaps, while the hybrid functional (HSE06) and local functionals with a Hubbard U correction can give reasonably good estimates (although one obtains comparable accuracy by PBE+U only when the parameter U is readjusted for each case). It is encouraging that single-point calculations with the local HLE17 functional, without needing a Hubbard correction or system-specific parameters, also give realistic band gaps in most cases. For the band gap of LZO, the single-point HLE17 calculations agree much better with experiments than PBEsol, TPSS, PBE+U, and HSE06. In Si512, C512, and their doped materials, PBE and TPSS were found to underestimate band gap states compared to HSE06 and single-point HLE17 calculations, and HSE06 and singlepoint HLE17 calculations were found to agree within 0.32 eV. In partially delithiated lithium zirconate, local density functionals were shown to delocalize the holes, but the holes are localized (thereby creating polarons) by single-point HLE17 and by HSE06 and PBE+U. In lithium iron phosphate, HSE06 and PBE+U predict localized iron hole polarons, whereas single-point HLE17 cannot. In the doped silicon and diamond, the HLE17\\TPSS method can describe the localized defect states correctly. In view of HSE06 being a computationally expensive method because it is a hybrid functional and DFT+U being dependent on the parameter U, the conclusion from the present work is that HLE17\\TPSS is the preferred method for calculating band gaps and polaronic structures where conventional local functionals fail and hybrid calculations are too expensive. This combination takes advantage of the better aspects of both the functionals, namely that TPSSunlike HLE17gives reasonably good lattice constants, and HLE17unlike TPSS predicts realistic band gaps and lattice distortions in the vicinity of defects. The present work on defect systems (delithiated cathodes and doped materials) shows that HLE17, which was shown in previous work to predict realistic band gaps for main-group semiconductors, can also predict realistic electronic and polaronic structures for defect systems, including some systems containing transition metals.

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AUTHOR INFORMATION

Corresponding Authors

*(S.H.) E-mail: [email protected]. *(D.G.T.) E-mail: [email protected]. ORCID

Shuping Huang: 0000-0003-4815-1863 Pragya Verma: 0000-0002-5722-0894 Donald G. Truhlar: 0000-0002-7742-7294 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Peter Deák for suggestions on the doped silicon calculations. S.H. acknowledges financial support from the National Natural Science Foundation of China (No. 21703036). As part of Nanoporous Materials Genome Center, this work was partially supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under Award DEFG02-12ER16362. Computations were performed using resources of (1) the Molecular Science Computing Facility in the William R. Wiley Environmental Molecular Sciences Laboratory of Pacific Northwest National Laboratory sponsored by the U.S. Department of Energy, (2) Minnesota Supercomputing Institute, and (3) the National Energy Research Scientific Computing Center.

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REFERENCES

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Localizing Holes as Polarons and Predicting Band Gaps, Defect

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