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Theoretical Investigation of TaO, TaON and TaN: Electronic Band Structures and Absolute Band Edges Zhi-Hao Cui, and Hong Jiang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12370 • Publication Date (Web): 17 Jan 2017 Downloaded from http://pubs.acs.org on January 30, 2017

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The Journal of Physical Chemistry

Theoretical Investigation of Ta2O5, TaON and Ta3N5: Electronic Band Structures and Absolute Band Edges Zhi-Hao Cui and Hong Jiang∗ Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Rare Earth Material Chemistry and Application, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China E-mail: [email protected]

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January 15, 2017 Abstract Early transition metal oxides, nitrides and oxynitrides have attracted a lot of interest due to their potential application in photovoltaics and photocatalysis. In this work, a systematic investigation is conducted to the electronic band structure of Ta2 O5 polymorphs, β-Ta3 N5 and β-TaON, which are crucial for the understanding of their photocatalytic properties, based on the state-of-the-art first-principles approaches. The calculated results imply that many-body perturbation theory in the GW approximation can well overcome the severe underestimation of the band gap caused by standard density functional theory (DFT) in the local and semilocal approximations, and lead to a quantitative agreement with experiment. The effects of the electron-phonon coupling on electronic band structure are considered by the Fr¨ olich model, and especially for ǫ-Ta2 O5 , a strong electron-phonon coupling is predicted as a result of small highfrequency dielectric constant and large effective masses. Based on the analysis in terms of the phenomenological ionic model, the band gap difference between three compounds can be physically attributed to not only the well-known energy difference between O-2p and N-2p orbitals, but also the influences of the Madelung potential on the conduction band energy. By comparing the calculated absolute band edge positions to the redox potentials for water reduction and oxidation, all of the three compounds are predicted to have the potential photocatalytic property for unassisted water splitting. In addition, we have also analyzed the stability and the band gaps of different Ta2 O5 polymorphs and found that β-Ta2 O5 , the phase commonly used in theoretical studies, is actually unstable and its unusually small band gap can be attributed to strong overlapping of neighboring atomic orbitals. On the other hand, ǫ-Ta2 O5 , which is much less well studied compared to β-Ta2 O5 , leads to calculated properties that are much more consistent with experimental findings of Ta2 O5 in general. Theoretical analysis and findings presented in this work can have general implications for the understanding of electronic band structure of other early transition metal compounds.

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Introduction Since the discovery of photocatalytic water decomposition with the titanium dioxide electrode in 1972, 1 a lot of efforts have been made to find the potential semi-conductive materials for photocatalytic and photoelectrochemical applications. 2 Among all of the properties that are relevant to the photo-splitting of water, the band gap (Eg ) and the absolute band edge position (EVBM and ECBM ) with respect to the vacuum level are two crucial ones that determine whether a material is promising for unassisted water splitting. In particular, the band gap should be suitable for visible light absorption, and furthermore, ECBM should be more positive than the redox potential for water reduction (H2 /H2 O, -4.4 eV with respect to the vacuum level) while EVBM needs to be more negative than the redox potential for water oxidation (O2 /H2 O, -5.6 eV). 3–6 In the past decades, various transition metal oxide materials have been explored in order to satisfy the above two conditions. 7 However, only in the recent years, non-oxide materials, especially metal nitrides and oxynitrides, have attracted a lot of interest experimentally. 8–13 To fully realize the potential of these materials in practical applications, it is of great importance to establish a theoretical understanding of their electronic band structure, which is also necessary for rational design of new materials. First-principles electronic structure theory, at present mainly density functional theory (DFT) in the local density approximation (LDA) or various generalized gradient approximations (GGAs), has become an indispensable working tool in the interpretation and prediction of material properties. 14–16 In spite of its extensive applications, however, DFT in LDA/GGA actually faces severe problems for the theoretical description of the electronic band structure, including, in particular, the famous band gap problem, i.e. the systematically underestimation of band gaps of typical semiconductors by 30-100% . 17 The origin of this problem derives from the fact that even with the exact exchange-correlation functional (Exc [ρ]), the Kohn-Sham (KS) band gap εKS g , i.e. the energy difference between highest occupied and lowest unoccupied Kohn-Sham orbitals, often termed as the valence band maximum (VBM) and conduction band minimum (CBM) for solids, respectively, does 3



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not correspond to the fundamental band gap Eg , due to the ignorance of the derivative discontinuity of Exc [ρ] (∆xc ). 18–20 Similar difficulties also exist for the description of the absolute band positions. 21 Currently many-body perturbation theory in the GW approximation (GWA) 17,22–24 is the state-of-the-art first-principles approach to describe the electronic band structures of semiconductors, 17,24–26 and therefore is promising for its application to solar energy conversion materials. 4,5,27–33 Moreover, the previous studies have shown the importance of GW correction to the accurate description of ionization potential with the uncertainty comparable with experiments. 21,28,34–37 In this work, we will mainly focus on three tantalum-based compounds, Ta2 O5 ,TaON and Ta3 N5 . The latter two compounds have been verified experimentally to have smaller band gaps with respect to common oxides, 3 and exhibit visible-light photocatalytic activity. 38–40 A series of studies on electronic structure of the three compounds based on DFT have been published. 41–49 Compared with pure metal oxides and nitrides, metal oxynitrides have the attractive feature that it can often combine the small band gap of nitrides with the good stability of oxides. 8–13,50,51 By considering systematically the oxide, nitride and oxynitride of the same transition metal element (i.e. Ta), our purpose in this work is to investigate the band gaps and the absolute band positions of these compounds using the state-of-the-art first-principles approaches, and further clarify the origin of the differences between them. The paper is organized as followings. The next section briefly introduces the methods to solve the band gap problem, including the hybrid functionals and the GW approach. The slab model combined with the GW corrections for absolute band position calculations is presented as well. In the third section, we present a systematic investigation of electronic band structure of Ta2 O5 , TaON and Ta3 N5 . The lattice properties after relaxation are presented and are compared with experimental studies. Then the results of band structures, absolute band positions of different methods (including electron-phonon coupling correction) are collected and the band gap difference is analyzed by the phenomenological ionic model. In addition, we also discuss the polymorphism of Ta2 O5 , and an analysis is made in order to

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interpret the gap difference of the multiple phases. The fourth section summarizes the main findings of this work.

Theory and Method The band gap problem and GW approximation As we have mentioned in the preceding section, conventional local or semi-local density functional approximations (LDA/GGA) are severely limited in accurate description of electronic band structure of materials, since the fundamental band gap does not correspond to the Kohn-Sham band gap, 18,52 Eg ≡ I − A = εKS g + ∆xc .

(1)

For the exchange-correlation energy in LDA/GGAs, which are explicit functionals of electron density, the derivative discontinuity ∆xc vanishes, which is the origin of band gap problem. In order to solve the problem, one approach is to mix a fraction of the Hartree-Fock exchange with LDA/GGAs exchange-correlation energy in the so-called hybrid functionals, which can recover the missing ∆xc to some extent. One typical hybrid functional is PBE0. 53 However, due to the long-range nature of the HF-type exchange, the computational cost of PBE0 is much larger than standard LDA/GGAs, especially for extended systems. 54 One way to improve the numerical efficiency is to use the screened hybrid functional, e.g. the HSE functional, 54,55 which has been proven to be able to describe the band gaps of many semiconducting materials very well. 56–58 Unlike density functional theory, many-body perturbation theory based on Green’s function provides a theoretically more rigorous framework for the treatment of quasi-particle excitations, as Green’s function itself describes the process of adding an electron to or removing an electron from the system intrinsically. 22,59 The electronic band structure can be

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obtained in a theoretically rigorous way by solving the quasi-particle equation, 17,26 1 [− ∇2 + Vext (x) + VH (x)]Ψn (x) + 2

Z

dx′ Σxc (x, x′ ; En )Ψn (x′ ) = En Ψn (x).

(2)

In Eq. 2, atomic unites are used, Vext (x) and VH (x) denote the external and Hartree potentials, and Σxc (x, x′ ; En ) denotes the exchange-correlation self-energy that accounts for all non-classical electron-electron interactions beyond the Hartree approximation. 17,23,26 In order to solve the quasi-particle energy En , Hedin derived a set of integrodifferential equations 22,23 and introduced the so-called GW approximation, in which Σxc in the space and time domain is approximated as the product of one-body Green’s function G and the screened Coulomb interaction W . In practice, the calculations can be further simplified by the G0 W0 approximation, 17 i Σxc (x, x ; ω) = 2π ′

Z

′

G0 (x, x′ ; ω ′ + ω)W0 (x′ , x; ω ′)eiηω dω ′

(3)

in which both G and W are calculated from Kohn-Sham orbitals in a given density functional approximation (often LDA or GGA), hence denoted as G0 and W0 , respectively. The quasiparticle energies are calculated by the first order perturbation theory based on Kohn-Sham orbital energies (ǫnk ) and wavefunctions(ψnk ),

Enk = ǫnk + Znk (ǫnk )R hψnk | Σxc (ǫnk ) − Vxc |ψnk i ,

(4)

where Znk is the quasi-particle renormalization factor, and Vxc is the Kohn-Sham exchangecorrelation potential.

Absolute band position with GW correction Absolute valence and conduction band edge positions with respect to the vacuum level, denoted as EVBM and ECBM , respectively, can be obtained from the slab model calculations. 21,60 6


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In particular, EVBM can be determined by calculating the difference between the highest occupied KS orbital and the vacuum level in a slab model calculation, (s)

(s) EVBM,s = εVBM − Vvac .

(5)

Through the paper, we use “(s)” and “(b)” to denote the quantity calculated from the slab and the bulk system, respectively. However, in the previous work, 21 we have clarified the importance of considering the bulk correction to eliminate the finite-size effects and the possible influences of surface states in the direct slab model calculations (eq. 5), which leads to (b)

(b)

(s)

(s) EVBM,b = [εVBM − εref ] − [Vvac − εref ],

(6)

where εref is the macroscopically averaged electrostatic potential in the center of the slab (s)

(b)

(εref ) or in the bulk system (εref ). 21,61 As shown in our previous study, 21 when using LDA or GGAs, the ionization potential (I = −EVBM,b) calculated in terms of Eq. 6 is systematically underestimated with respect to experiment for typical semiconductors as a result of missing the contribution from the derivative discontinuity, and it is necessary to consider the GW quasi-particle correction, which, at the G0 W0 level, can be straightforwardly obtained by considering the GW correction to (GW )

the VBM of the corresponding bulk system ∆εVBM , 21 (GW )

(KS)

(GW )

EVBM = EVBM + ∆εVBM .

(7)

The energy of the conduction band edge with respect to the vacuum level (ECBM ) can then be easily calculated by adding the GW band gap of the bulk system to EVBM , (GW )

(GW )

ECBM = EVBM + Eg(GW ) .

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(8)


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Computational details All DFT and G0 W0 calculations are performed by using the Vienna ab-initio simulation package (VASP) 62,63 with the projector augmented wave (PAW) method. 64 For DFT calculations, both GGA and the hybrid functional are considered. The GGA functional in the PBEsol 65 parameterization is used for structural optimization since its good performance in the description of structural parameters and surface properties. 65 And the hybrid functional in the HSE06 form 54,55 is used for electronic band structure calculations to partly overcome the underestimation of band gaps by LDA/GGA functionals. 56,57 The crystal structures of all systems concerned are relaxed by using a Γ-centered k-mesh determined by requiring the number of k points × the length of lattice vectors ≈ 30 ˚ A and a large kinetic energy cutoff of 550 eV for plane waves. The latter is necessary to eliminate the possible influence of the Pulay stress on the accuracy of optimized lattice constants. 66 The energy convergence A. After the relaxation, criterion is 10−6 eV and the force convergence criterion is 10−4 eV/ ˚ the electronic structures are calculated with a kinetic energy cutoff of 450 eV. Ab initio molecular dynamics (AIMD) simulations within the canonical ensemble at 50 K, 200 K, 500 K and 1000 K are performed to check the stability of β-Ta2 O5 , in which the time step is chosen to be 0.25 fs, and the total simulation time is about 10 ps. For G0 W0 calculations in this work, we mainly use the KS orbital from PBEsol as the input, hence labeled as G0 W0 @PBEsol or simply G0 W0 for abbreviation. The band gap convergences with respect to the number of unoccupied states and k points have been carefully tested. The band structure diagrams of HSE06 and G0 W0 along high-symmetry k-path are obtained by using the Wannier interpolations. 67 The effects of the electron-phonon coupling (EPC) on electronic band structure are estimated based on the Frölich model, 68,69 from which the renormalization energy of the band edges can be evaluated as, 70 ∆E ≈ −ωLO (α + 0.0159α2 + 0.000806α3),

8


(9)

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where α is the coupling constant,

α≡

e2 2m∗ ωLO 1 1 1 ( )2 ( − ). 2~ωLO ~ ε∞ ε0

(10)

We calculated the longitudinal optical phonon frequency ωLO at the Γ point (LO-TO splitting is considered, see Supporting Information for details), the high-frequency (i.e. electroniconly) dielectric constant ε∞ , the static dielectric constant ε0 by the density functional perturbation theory (DFPT) method, 71,72 and the effective mass m∗ at the VBM and CBM by the finite difference approach. For the slab model calculations of surface energies and absolute band positions, the thickness of the slab is as large as 13-25 ˚ A to ensure the center of the slab can be regarded as the bulk phase. We have also made a convergence test of surface energy with respect to slab thickness, see Table S1 in Supporting Information. The thickness of the vacuum is fixed as 15 ˚ A to eliminate the fictitious interaction between mirrors. The macroscopic electrostatic potential is calculated on a 4-times more dense Fourier grid, and then interpolated in spline form on a 10-times more dense grid to ensure the numerical accuracy.

Results and Discussions Crystal structures Ta2 O5 exists in various polymorphs, including the most commonly studied β-Ta2 O5 (space group Pmmm), 41,44,45,73–75 which is believed to be stable at low temperature and undergo a phase transition to α-Ta2 O5 at rather high temperature (∼1630 K). 73 Another typical phase is labeled by ǫ-Ta2 O5 (space group C2/c), 76 which can be prepared by both the hydrothermal approach 76 and high-pressure synthesis. 77 Besides, the X-ray powder diffraction measurements also indicate the existence of the polymorph with large number of oxygen vacancies, 78 hexagonal phase, 79,80 high-pressure prepared phase, 77 as well as the amorphous phase. 81 In

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Figure 1: Crystal structures of Ta2 O5 , TaON and Ta3 N5 . this work, we mainly focus on the first two representative phases, i.e. β-Ta2 O5 and ǫ-Ta2 O5 , since they are prepared in relatively mild conditions and are commonly considered to be stable enough for practical applications, especially for the β phase. We note in passing that our consideration of different polymorphic structures of Ta2 O5 is by no means exhaustive, and our purpose here is mainly to investigate the effects of changing crystal structures on electronic band structure. In contrast to the complicated polymorphism of Ta2 O5 , βTaON (space group P21 /c) and β-Ta3 N5 (space group Cmcm), in spite of the presence of other phases, 42,82 are considered to be the most important polymorphs in practice. All crystal structures considered in this work are illustrated in Figure 1. The lattice parameters optimized by PBEsol and those measured by experiment are collected in Table 1. For β-Ta2 O5 and ǫ-Ta2 O5 , the optimized lattice constants obtained from PBSsol calculations agree very well with experiment, which is consistent with the general good performance of the PBEsol GGA functional in predicting crystal structural

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Table 1: Results of lattice parameters and relative energies of conventional unit cells of Ta2 O5 , TaON and Ta3 N5 from PBEsol. The quantities in the parentheses are obtained from the experimental literature cited in the Ref. column. ∆E denotes the relative energy with respect to β-Ta2 O5 . Species β-Ta2 O5 β ′ -Ta2 O5 ∗ ǫ-Ta2 O5 β-TaON β-Ta3 N5

a [˚ A] b [˚ A] c [˚ A] β[◦ ] ∆E [eV/f.u.] 6.43 (6.22) 3.67 (3.68) 7.72 (7.79) 0.00 6.36 3.75 7.60 -1.82 12.83 (12.79) 4.87 (4.85) 5.52 (5.53) 103.8 (104.3) -2.37 4.95 (4.96) 5.02 (5.03) 5.17 (5.18) 99.8 (99.6) 3.88 (3.89) 10.22 (10.21) 10.24 (10.26)

Ref. 74 77 83 84

The structure is optimized after a small displacement along the maximum imaginary phonon mode. ∗

properties, 65 and therefore the PBEsol optimized structure can be used as a good starting point for the following electronic structure calculations. Table 1 also collects the relative energy per formula unit (f.u.) of different polymorphic phases of Ta2 O5 , taking the energy of β-Ta2 O5 as zero. Surprisingly, β-Ta2 O5 is much less stable than ǫ-Ta2 O5 by about 2.37 eV per formula unit, which is a rather large energy difference, and the consideration of vibration contributions (zero-point energy and finite temperature effects) or high-order correlation effects beyond GGA (e.g. those accounted for by using adiabatic connection fluctuationdissipation theorem in the random phase approximation 85 ) is unlikely to change the relative stability of the two phases. The poor stability of β-Ta2 O5 compared to ǫ-Ta2 O5 is surprising since β-Ta2 O5 is often regarded as the stable phase at the ambient condition, 73,74 and is often used in theoretical study of Ta2 O5 . 44,45,75 It is therefore worth to have a closer look at the stability of βTa2 O5 . The phonon analysis indicates a large imaginary frequency (∼394 cm−1 ) of βTa2 O5 at the Γ point, consistent with the previous report. 41,75 In contrast, ǫ-Ta2 O5 shows no imaginary frequency from the DFPT phonon calculations at the Γ point. Furthermore, the AIMD simulations indicate that even at a low temperature of 50 K, the β phase would undergo a strong distortion after about 230 fs, which follows the vibration mode of the maximum imaginary frequency. By relaxing the symmetry of the β phase and slightly 11



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distorting the structure in terms of the vibration mode of the maximum imaginary frequency, or taking a low-energy structure from the MD simulation trajectories, we can obtain a crystal structure that is similar to β phase, but without imaginary phonon frequencies, denoted as β ′Ta2 O5 henceforth (see Figure 1 (b)). Structurally β-Ta2 O5 and β ′ -Ta2 O5 differ mainly in two aspects: (i) the TaO6 octahedra in β ′ -Ta2 O5 are strongly distorted, and (ii) the connection between TaO6 octahedra also change significantly, which are clearly illustrated in Figure 2. We can therefore come to the conclusion that β-Ta2 O5 with the Pmmm symmetry is indeed unstable at low temperature and ambient pressure. In the next Section, we will further show that regarding electronic band structures, the results for β-Ta2 O5 are also inconsistent with experimental findings, which implies that β-Ta2 O5 may not be a realistic model to simulate Ta2 O5 commonly prepared experimentally.

Figure 2: (a) The top view from c direction of β-Ta2 O5 , and the maximum imaginary phonon mode is also labeled. (b) The top view from c direction of β ′ -Ta2 O5 . (c)The side view from b direction of β-Ta2 O5 . (d) The side view from b direction of β ′ -Ta2 O5 .

Electronic band structure In this section, we will present the results on the electronic band structure of the three compounds. The band structure diagrams from both PBEsol and G0 W0 @PBEsol are collected in Figure 3. It is obvious that G0 W0 predicts a larger band gap by pushing the VBM to lower energy and CBM to the opposite, with almost unchanged dispersion of band energies. For 12


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c)

6

6

b)

5

4

4

3

3

energy [eV]

5

2 1

2 1

0

0

-1

-1

-2 S 4

Γ

R

S

Γ

Y

X

Z

Γ

U

Z

d)

-2 Γ 4

3

3

2

2 energy [eV]

energy [eV]

a)

energy [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

0

-1

-1

Γ

Y

A

B

D

E

C

L

Z

Γ

X

Y

1

0

-2 Z

Y

-2 Γ

Y

T

Z

A

R

S

Γ

Figure 3: Band structures of different compounds: a) β-Ta2 O5 , b) ǫ-Ta2 O5 , c) β-TaON , d) β-Ta3 N5 . The black lines denote the results from PBEsol and the red dots denote the results from G0 W0 @PBEsol. In each case, the valence band maximum (VBM) from PBEsol is taken as the common energy zero (Fermi level) for both PBEsol and G0 W0 band structures.

Table 2: Results of the band gap (in eV) from different theoretical approaches and experiment. Note that the phase of Ta2 O5 used to measure the experimental band gap is undetermined. Species Ta2 O5 β-Ta2 O5 β ′ -Ta2 O5 ǫ-Ta2 O5 β-TaON β-Ta3 N5 a

Ref. 3;

b

PBEsol

HSE06 G0 W0

scGW

G0 W0 with EPC

Expt. 3.9 , 4.2b , 4.5c a

0.31 2.59 3.14 1.86 1.20

Ref. 86;

c

1.52 4.08 4.67 3.02 2.16

1.81 4.25 4.81 2.95 2.20

d

Ref. 39;

Ref. 87;

2.42 5.24 5.76 3.25 2.38

13

3.71 3.96 2.75 2.07


2.4 a , 2.8 2.1a

d


4.80

1.88

4.78

1.84 ε-Ta2O5

4.76 Eg [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4.74 100 2.96

200

β-Ta2O5

1.80 1.76 100

300

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200

300

2.19

2.94

2.16 2.92

β-TaON

2.13

2.90 100 200 300 400 100 Emax [eV]

β-Ta3N5

200

300

Figure 4: The calculated G0 W0 band gaps fitted as a function of highest unoccupied state energy considered. clarity, we summarize the electronic band structure information from different approaches in Table 2. To obtain numerically converged G0 W0 band gaps, we fit the calculated band gap by (see Figure 4) 88 Eg (x) =

a + Eg∞ , x − ǫ0

(11)

where x denotes the energy of highest unoccupied state considered in GW calculations, a, ǫ0 and Eg∞ (i.e. the extrapolated band gap) are fitting parameters. It is interesting to note that for these materials with a 5d0 electronic configuration, the calculated G0 W0 band gap converges quite quickly with respect to the number of unoccupied states, in contrast to the situations observed in the systems with semicore d-states like ZnO. 88 In Table 2, we also present the band gaps of different compounds obtained from the quasi-particle self-consistent GW (scGW) scheme. 89,90 We can see that the band gaps from scGW are systematically larger than those from PBEsol-based G0 W0 , which is consistent in previous findings that the scGW approach without considering vertex corrections tend to significantly overestimate the band gaps of common semiconductors. 90,91 At the level of the GW approximation to the self-energy, the “best” results (i.e. in terms of the agreement with experiment) for normal systems with moderate electron correlation can often be obtained by G0 W0 or partially (energy-only) self-consistent GW0 with standard LDA or GGA Kohn-

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Sham orbital energies and wave functions as the input. 33,88,90,92,93 Thus we mainly consider the G0 W0 @PBEsol results in this work. From Table 2, we can see that PBEsol-GGA severely underestimates the band gaps of all these compounds, and both the hybrid functional (HSE06) and G0 W0 @PBEsol increase the band gaps dramatically. It is noteworthy that the band gap of β-Ta2 O5 is unexpectedly small, much smaller than that of ǫ-Ta2 O5 and the experimental values for Ta2 O5 , and it is even smaller than that of TaON and Ta3 N5 calculated at the same level, which is contrary to the general trend that the band gap usually follows the order of oxide > oxynitride > nitride for compounds of the same metal. β−Ta2O5

Ta(6s) Ta(5d) O (2p)

β’-Ta2O5

ε−Ta2O5

0 -8

-6

-4

-2

0 2 4 energy [eV]

6

8

10

Figure 5: DOS of ǫ-Ta2 O5 , β ′ -Ta2 O5 and β-Ta2 O5 by PBEsol. All three DOS are shifted so that the Ta’s inner core 5s states are aligned. The dashed line indicates the VBM of ǫ-Ta2 O5 . The dramatic difference between the band gaps of β-Ta2 O5 and ǫ-Ta2 O5 is remarkable, and calls for an in-depth understanding of its physical origin. To this end, we use the phenomenological ionic model, 31,94,95 in which the band gap of a typical ionic insulating system can be decomposed as 1 a c Eg = (Aa − I c ) + e(|VMa | + |VMc |) − (∆Epol + ∆Epol ) − (WVB + WCB ), 2

15


(12)


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where “a” denotes the anionic species whose states dominate the VBM, and “c” indicates the cationic species whose states dominate the CBM. In the first term (free ions contribution), A and I are the atomic electron affinity and ionization energy, respectively, which are certainly the same for different polymorphic structures of Ta2 O5 . The difference of the third term (the polarization contribution) between different compounds is expected to be small because of their qualitatively similar chemical bonding. We have also confirmed that the contribution of the second term (the Madelung potential VM ), calculated in terms of Bader effective charges, is nearly identical in different phases of Ta2 O5 . Thus the only remaining factor is the band broadening, which can be clearly illustrated by the density of states (DOS) shown in Figure 5. It is clear that both the valence band and conduction band are much wider in β-Ta2 O5 than in β ′ -Ta2 O5 and ǫ-Ta2 O5 , which is therefore the main origin of the smaller band gap in the β phase. The strong band broadening can be further attributed to the crystal structure features of the β phase. From Figure 1, we can see that the TaO6 octahedra in the β phase are very regular with only little distortion, compared with those in the β ′ or ǫ phase. This feature will significantly enhance the overlap between neighboring atomic orbitals and therefore lead to a large band broadening. In addition, the Ta-O bond length in the β phase is slightly shorter (about 1.9 ˚ A), which can also enhance the effective overlap between atomic orbitals on the ligand O atoms and the central Ta atom. From that point of view, one can also understand naturally that the structurally more strongly distorted β ′ -Ta2 O5 has a significantly larger band gap due to relatively weaker inter-atomic orbital overlap (see also the band structure of β ′-Ta2 O5 in Figure S1 in Supporting Information). Another evidence of strong band broadening is the crystal-field splitting of Ta-5d states. In all of the three phases, Ta and O form the octahedral coordination (although the β ′ phase is distorted to a large extent). But from Figure 5, only ǫ-Ta2 O5 and β ′ -Ta2 O5 show obvious splitting between t2g and eg band, and β-Ta2 O5 , with the almost regular octahedral coordination, shows only a flat, continuous d band. The phenomenon derives from the strong band broadening in the β phase, which makes the gap between t2g and eg “squeezed out”. To summarize, the

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reason why the β phase has such a small band gap is the strong overlap between orbitals derived from the nearly regular octahedral coordination. Together with the instability of the β phase discussed in the previous section, we can plausibly argue that β-Ta2 O5 with its currently accepted structure can not be regarded as a realistic representation of Ta2 O5 that is used in practical applications. Considering its remarkable stability compared to β-Ta2 O5 and β ′ -Ta2 O5 , we think ǫ-Ta2 O5 is more suitable to serve as the representative phase of Ta2 O5 in the ambient condition, or at least as the main component of amorphous Ta2 O5 samples that are usually investigated in experiment. In the following, we will mainly consider theoretical results from ǫ-Ta2 O5 unless stated otherwise. Ta(6s) Ta(5d) O(2p) N(2p)

Ta3N5

PDOS [arb. unit]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TaON

Ta2O5

-8

-6

-4

-2

0

2

energy [eV]

4

6

8

10

Figure 6: DOS of ǫ-Ta2 O5 , β-TaON and β-Ta3 N5 by PBEsol. All three DOS are shifted so that the Ta’s inner core 5s states are aligned. The dashed line indicates the VBM of ǫ-Ta2 O5 . Another obvious feature from Table 2 is the trend that the band gap decreases in the order of oxide > oxynitride > nitride, which is consistent with experimental findings. 3 To obtain an in-depth understanding of underlying physics, we plot the DOS of three compounds, shown in Figure 6. When aligned in term of the semicore Ta-5s states that are present in all three compounds, the VBM of Ta3 N5 and TaON, which are both dominated by N-2p characters, is about 0.5 eV higher than that of ǫ-Ta2 O5 , which is consistent with the energetic order of O- and N-2p atomic orbital. Another obvious feature is that conduction bands of TaON and 17



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Ta3 N5 move towards lower energy. This can be attributed to the difference of the electrostatic potential at the Ta site, since the CBM is mainly of Ta-5d characters. In particular, the calculated Madelung potential at the position of Ta, decreases from oxide to nitride (29.77 V in Ta2 O5 , 28.80 V in TaON , and 27.11 V in Ta3 N5 ), which indicates that the CBM state is more stable in Ta3 N5 than in Ta2 O5 from a purely electrostatic point of view. Figure 6 also shows that the band widths of the three compounds are similar, which is consistent with nearly same Ta-O(N) bond lengths (all being about 2.0 ˚ A) in their structures. We can therefore conclude that both the orbital characters and the electrostatic factor are essential to understand the band gap difference between the three compounds: higher-energy of N2p orbital pushes up the VBM while the less repulsive electrostatic potential experienced by electrons occupying CBM pushes downs the conduction band energy in oxynitride and nitride. Table 3: Results of high-frequency and static dielectric constant, longitudinal optical phonon frequency at Γ point (in 2π·THz), effective mass at VBM and CBM (in m0 ), the corresponding coupling constant and the renormalization energy (in eV) by PBEsol. The dielectric constant and the effective mass are arithmetic trace-averaged and geometric trace-averaged respectively. Species β ′ -Ta2 O5 ǫ-Ta2 O5 β-TaON β-Ta3 N5

ε∞ 5.49 5.68 8.89 10.97

ε0 32.67 32.45 26.09 42.85

ωLO m∗VBM 29.48 3.26 40.87 5.64 32.13 1.22 30.24 0.52

m∗CBM 0.71 1.71 0.86 0.64

αVBM 2.89 3.09 0.83 0.51

αCBM 1.35 1.70 0.70 0.73

∆EVBM +0.37 +0.55 +0.11 +0.06

∆ECBM -0.17 -0.30 -0.09 -0.07

We further compare our theoretically calculated band gaps to the experiment values. Since the results from HSE06 and G0 W0 are close to each other, and the latter is in general more accurate than the former, we will mainly consider the G0 W0 results in the following discussion. It should be noted that experimental band gaps often suffer from several uncertainty factors including the sample quality, the particular experimental techniques, and the way to extract the band gap from experiment, to name a few only. 88 As a result, the experimental values for the band gap of a particular compound usually exhibit significant scattering. It is therefore necessary to exercise some caution when comparing theory and experiment as far 18


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as the band gap is concerned. For example, the band gap of Ta2 O5 , measured by several groups, 3,86,87 often without characterizing the crystal structure simultaneously, scatter in the range of 3.9-4.5 eV. With these uncertainties taken into account, we can see that for Ta3 N5 the G0 W0 band gap agrees very well with experiment, but for TaON and Ta2 O5 (using the theoretical result for ǫ-Ta2 O5 ), the G0 W0 band gaps show a certain overestimation, especially for Ta2 O5 . Such over-estimation of the band gap by LDA or GGA-based G0 W0 has also been observed in other early transition metal oxides with apparent d0 electronic configuration such as TiO2 , 28,29,96 SrTiO3 , 96,97 ATaO3 (A=Li, Na, K), 31,97 LiNbO3 , 98 and WO3 , 69 to name a few typical cases only. For such compounds, there is increasing evidence indicating that the electron-phonon coupling can play a significant role in reducing the band gap. 28,69,98 Following Refs. 69, we estimate the effects of EPC on electronic band structure by calculating the valence and conduction band renormalization energy ∆E based on the simplified Frölich model (see eq. 9), and the related quantities for β ′ -Ta2 O5 , ǫ-Ta2 O5 , βTa3 N5 , and β-TaON are collected in Table 3. We can see that compared with Ta-oxynitride and nitride, ǫ-Ta2 O5 shows a significantly smaller electronic dielectric constant (i.e. ε∞ ) and larger effective mass (m∗ ), which can be attributed to its larger band gap and flatter valence and conduction band edges, as shown in Figure 3. As a result, the consideration of EPC reduces the band gap of ǫ-Ta2 O5 by as much as 0.85 eV. A slightly weaker decreasing is also observed for β ′ -Ta2 O5 . For Ta3 N5 and TaON , the effects of EPC are much weaker, and the band gap decreases only by 0.1-0.2 eV. We note in passing that our result for Ta3 N5 is in agreement with the previous calculations. 99 Having considered the EPC effects, the G0 W0 band gaps of all three compounds show a rather good agreement with experiment. Since experimental band gaps of the compounds considered in this work are all obtained from optical absorption, excitonic effects that describe electron-hole interaction may also need to be considered. However, in 3D bulk materials, the excitonic effects on the band gap are expected to be weak. 100 In particular, Morbec et al 99 reported, based on the GW plus Bethe-Salpeter equation (BSE) approach, that the exciton binding energy in Ta3 N5 is only

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about 0.05 eV. We can therefore expect that considering excitonic effects plays a marginal role for the band gaps of Ta-oxides and oxynitrides as well.

Absolute band edge positions We further consider the energy of VBM and CBM with respect to the vacuum level, the socalled absolute band edge positions, using the approach described in the preceding section. Since the absolute band energies is highly surface dependent, it is important to search for the most stable exposed surface of concerned systems. The relative stability of different exposed surfaces of β-TaON 101 and β-Ta3 N5 102 has been theoretically investigated previously. In contrast, the surface stability of Ta2 O5 has been rarely studied previously to our best knowledge, probably due to the complex polymorphism of Ta2 O5 . It was found that in both β-TaONand β-Ta3 N5 , the (100) surface is the most stable, while in TaON, the (111) surface is only slightly less stable than the (100) one. 101 Based on the previous study, for TaON we consider only its (100) and (111) surfaces with either O- or N-termination, for βTa3 N5 we consider its (100) surface, and for the surfaces of Ta2 O5 , we have considered several low-index surfaces of ǫ-Ta2 O5 to the find the most stable exposed surface. The calculated surface energies are collected in Table 4. The detailed surface structures and corresponding macroscopic electrostatic potentials can be found in Figure S2 in Supporting Information. Among the low-index surfaces of ǫ-Ta2 O5 we have considered, the (100) surface has the lowest surface energy, and therefore will be used for the calculation of absolute band positions. For TaON, we find that on both (100) and (111) surfaces, the O-termination is more stable than the N-termination. The calculated surface energies are quite close to those reported in Ref. 101, except that the difference between (100) and (111) surfaces are slightly larger. Both (100) and (111) surfaces can be important in reality as a result of the closeness of their surface energies. For β-Ta3 N5 , our calculated surface energy, 1.41 J/m2 , for the (100) surface is larger than that in Ref. 102 by about 0.3 J/m2 , possibly due to the use of different computational approaches. 20


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Table 4: Results of surface stability. The value in parentheses are from the previous work indicated in Ref. column. System ǫ-Ta2 O5

β-TaON

β-Ta3 N5 a

from Ref. 101;

b

surface slab thickness [˚ A] (100) 23.8 (010)-O 14.1 (010)-Ta 14.5 (001) 13.8 (100)-O 18.9 (100)-N 18.9 (111)-O 16.0 (111)-N 16.2 (100) 13.6

Esurf [J/m2 ] 0.92 1.13 2.28 1.71 1.44 (1.65a ) 3.47 1.53 (1.66a ) 2.32 1.41 (1.1b)

from Ref. 102.

Table 5: Results of absolute band positions from PBEsol and G0 W0 for the surfaces of Ta2 O5 , TaON , and Ta3 N5 compared to available experimental values. Surface PBEsol ǫ-Ta2 O5 (100) (010)-O β-TaON (100)-O (111)-O β-Ta3 N5 (100) a b

-7.22 -7.66 -7.04 -6.52 -5.42

EVBM [eV] G0 W0 Expt. -7.83a, -7.93b -7.94 -8.37 -6.6b -7.67 -7.15 -6.02a,b -6.04

PBEsol -4.90 -5.34 -5.38 -4.86 -4.38

ECBM [eV] G0 W0 Expt. -3.93a , -4.03b -3.98 -4.41 -4.1b -4.92 -4.40 -4.14a , -3.92b -3.99

by electrochemical measurement from Ref. 3; by ultra-violet photoemission spectroscopy measurement from Ref. 3

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We calculated absolute band positions with or without the GW correction and the results are collected in Table 5.

The related experimental results obtained from photo-

electrochemical (PEC) measurement or ultra-violet photoemission spectroscopy (UPS) are also collected for comparison. We can see the GW correction is pretty significant to push EVBM to lower energy and ECBM to higher energy. The G0 W0 results for ǫ-Ta2 O5 obtained from the (100) surface are nearly identical to experimentally measured values, and those from the (010)-O terminated surface show a systematic shift towards lower energy by about 0.4 eV. For β-TaON, the calculated G0 W0 VBM and CBM energies from the (100)-O surfaces are systematically lower by nearly 1 eV than experimental values, but the discrepancy is significantly reduced when considering the results obtained from the (111)-O surface, which could imply that the (111)-O surface of β-TaON may play a more important role than (100) surface in determining the absolute band positions of this compound. For β-Ta3 N5 , the G0 W0 results again agree very well with experimental values. Considering the significant error bar in experimental values, the overall agreement between the G0 W0 prediction and experiment is remarkable. Our investigations based on the G0 W0 approach have therefore confirmed, from a theoretical perspective, that in all of the three compounds, EVBM is more negative than -5.6 eV, and ECBM is more positive than -4.4 eV, which indicates their potential application for unassisted water splitting, and visible light photocatalytic activity for TaON and Ta3 N5 , in particular. It is important to note that our discussion on the application of these materials in photo-splitting of water is based on the results obtained from vacuum calculations, similar to a lot of previous studies. 28,32,33,37,69 However, an explicit consideration of the interface with water can be important for a more realistic and quantitative treatment of the issue, and has been addressed by several recent studies. 103–105

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Conclusions To summarize, we have investigated electronic band structures and absolute band positions of Ta2 O5 , TaON and Ta3 N5 by different first-principles approaches, including GGA (PBEsol), the hybrid functional (HSE06) and the GW approach (G0 W0 @PBEsol). Both the hybrid functional and the GW approach can well reproduce experimental band gaps, especially after the effects of the electron-phonon coupling on electronic band structures have been taken into account based on the simplified Frölich model. The latter are particularly significant for ǫ-Ta2 O5 as a result of its small high-frequency dielectric constant and large effective mass. We also analyzed the origin of the band gap difference among the oxide, oxynitride and nitride, and we found that both the energy difference between oxygen and nitrogen 2p-orbitals, and the difference in the electrostatic (Madelung) potential in these compounds play important roles in determining their electronic band structure. We have investigated the relative stability and electronic band structures of several polymorphic phases of Ta2 O5 , and we found that the commonly studied β-Ta2 O5 is structurally unstable, and leads to a significantly distorted structure once the crystal symmetry is relaxed; ǫ-Ta2 O5 is energetically much more stable, and its band gap obtained from the G0 W0 approach with the EPC correction agrees very well with commonly measured experimental values, implying that ǫ-Ta2 O5 is more suitable than β-Ta2 O5 as a realistic representation of Ta2 O5 that is measured experimentally. We further show that electronic band structure of Ta2 O5 in different polymorphic structures is strongly affected by different inter-atomic orbital overlapping, which can be further related to inter- and intra-TaO6 octahedra structural distortion. We have also evaluated the absolute valence and conduction band positions with respect to the vacuum level in Ta2 O5 , TaON and Ta3 N5 by combining the PBEsol slab model calculations with the G0 W0 correction to the bulk valence band maximum, which can well reproduce experimental findings, and has confirmed that TaON and Ta3 N5 are visible light active and have appropriate band edge positions for photo-catalytic splitting of water to hydrogen and oxygen. 23



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Acknowledgments This work is supported by National Natural Science Foundation of China (Project Numbers 21373017, 21673005 and 21321001). ZHC acknowledges the support of the Hui-Chun Chin and Tsung-Dao Lee Chinese Undergraduate Research Endowment (CURE).

Supporting Information Information on the details for phonon calculations, the band structure of β ′ -Ta2 O5 , surface structures and the corresponding macroscopic electrostatic potentials for different surfaces considered in this work, and the convergence of the calculated surface energies with respect to the slab thickness. This material is available free of charge via the Internet at http://pubs.acs.org.

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Electronic Band Structures and Absolute Band Edges - ACS Publications

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