Unexpected Robustness of the Band Gaps of TiO2 under High Pressures

Titanium dioxide (TiO2) is a wide band gap semiconducting material which is promising for photocatalysis. Here we present first-principles calculations to study the pressure dependence of structural and electronic properties of two TiO2 phases: the cotunnite-type and the Fe2P-type structure. The band gaps are calculated using density functional theory (DFT) with the generalized gradient approximation (GGA), as well as the many-body perturbation theory with the GW approximation. The band gaps of both phases are found to be unexpectedly robust across a broad range pressures. The corresponding pressure coefficients are significantly smaller than that of diamond and silicon carbide (SiC), whose pressure coefficient is the smallest value ever measured by experiment. The robustness originates from the synchronous change of valence band maximum (VBM) and conduction band minimum (CBM) with nearly identical rates of changes. A step-like jump of band gaps around the phase transition pressure point is expected and understood in light of the difference in crystal structures.


Introduction
Pressure is a powerful tool for tuning the atomic and electronic structures of materials. Both inorganic and organic substances, such as Si, Ge, H 2 O, SiO 2 and C 6 H 6 , can exhibit complex condensed phases under high pressures [1][2][3][4][5]. With the application of high pressure, people are able to synthesize some novel compounds that do not exist at ambient conditions [6]. Tremendous studies have demonstrated that the electronic properties of materials can be effectively manipulated by compression, through which the metal-to-semiconductor [7,8] or insulator-to-metal transitions [9][10][11][12][13][14][15] take place. It is conjectured that insulators and semiconductors always metallize under sufficiently high external pressures [16]. The most notable examples may be the prediction of possible metallization and superconductivity in solid state hydrogen [17][18][19], and the observation of high-T c superconductivity in hydrogen sulfide under high pressures [20,21].
In principle, the electronic structures of materials under high pressures can be probed by experimental measurements on the quantities such as electrical conductivity, optical conductivity and reflectivity [22,23]. To date, however, measurements on the  [22]. There lacks the knowledge about the global behavior of electronic structures across a wide range of pressures, in particular, pressures spanning an interval with the width of ~ 100 GPa or larger.
In this work, we study the effects of pressure on titania (TiO 2 ), one of the most extensively studied wide band gap semiconductors [24][25][26], at pressures spanning over an interval of ~ 150 GPa. As an important photocatalyst for harvesting solar energy, the alignments of energy levels near the valence and conduction bands play a 3 key role [25,27]. At ambient pressure, TiO 2 commonly exist in natural minerals in three phases: rutile, anatase, and brookite [25]. All the three phases have an optical band gap of slightly higher than 3 eV [28][29][30]. At elevated pressures，TiO 2 undergoes phase transformations from the most stable rutile structure at ambient pressure to a number of high-pressure polymorphs [31,32]. For pressures above 50 GPa, TiO 2 adopts an orthorhombic cotunnite-type structure [32] until ~ 160 GPa, at which a new phase with a hexagonal Fe 2 P-type structure will prevail [33]. At ~ 650 GPa, recent theoretical studies have predicted the existence of a new tetragonal phase (space group: I4/mmm), in which the metallization of TiO 2 may occur [34,35]. The present work is focused on the pressure dependence of electronic structures of two experimentally verified high-pressure TiO 2 phases: the cotunnite-type [31,32] and the Fe 2 P-type structure [33]. At moderate or low pressures, there have been a lot of researches on the pressure dependence of the band gaps of diamond-, zinc-blende-, and wurtzite-structure semiconductors [36][37][38][39]. These works show that, the band gap (E g ) can be expressed as a function of pressure (P) [37][38][39]: where α and β are the parameters to be determined, and c p = dE g /dP is the so-called pressure coefficient [36][37][38][39], which is a key parameter describing the pressure dependence of band gaps. It is unclear whether Eq. (1) would be applicable or not to wide band gap semiconductors at P ≥ 100 GPa. From the data reported in literatures [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50], previous studies are focused on the pressure coefficients (dE g /dP) at ambient or moderate pressures, at which the values of c p can be either positive or negative, with an order of magnitude of 10 -2 to 10 -1 eV/GPa for most systems. By contrast, our studies based on first-principles calculations find that, the band gaps of the two high-pressure phases of TiO 2 are surprisingly robust to compression. Within a wide range of pressures (50 to 200 GPa for the cotunnite-type, and 110 to 260 GPa for the Fe 2 P-type), the pressure coefficients of TiO 2 are significantly smaller than that of diamond (~ 5 meV/GPa) [41,42] and silicon carbide (SiC) (~ 1.9 meV/GPa for the cubic SiC [43], and ~ 2 meV/GPa for 6H-SiC [44]), the smallest value ever measured 4 experimentally. Furthermore, we find that the pressure dependence of the band gaps of both phases is well described by a modified form of Eq. (1).

Theoretical methods
The first-principles calculations were carried out by the Vienna ab initio simulation package (VASP) [51,52], which is based on density functional theory (DFT). A plane wave basis set and the projector-augmented-wave (PAW) potentials [53,54] are employed to describe the electron-ion interactions. The exchange-correlation interactions of electrons are described by the generalized gradient approximation (GGA) within the PBE formalism [55]. The energy cutoff for plane waves is 600 eV. For the structural optimization and total energy calculations, a 6×10×5 and 6×6×10 Monkhorst-Pack k-mesh [56] is generated for sampling the Brillouin zone (BZ) of the cotunnite-type and Fe 2 P-type structure, respectively. These set of parameters yield convergence of total energies to within a level of 10 -2 meV/Ta 2 O 5 unit. Since DFT is a ground state theory and not suitable to estimate the band gap values, we have employed the GW method [57,58], which explicitly includes the many-body effects (exchange and correlation) of electrons to calculated the energy levels of low-lying excited states and consequently the band gaps of TiO 2 under high pressures. Specially, we take the one-shot GW approach implemented in the VASP code [59] to calculate the energy spectrum. The quasiparticle energies and wave functions are obtained by solving a Schrödinger-type equation [58]: where T is the kinetic energy operator of electrons, is the external potential due to ions, V H is the electrostatic Hartree potential, Σ is the electron self-energy operator, and and ( ⃗) are the quasiparticle energies and wave functions, respectively.
Within the GW approximation proposed by Hedin [57], the term Σ can be calculated as follows: where G is the Green's function, W is the dynamically screened Coulomb interaction, 5 and δ is a positive infinitesimal. In the GW calculations on the cotunnite-type and Fe 2 P-type TiO 2 , a 4×6×3, 4×4×6 k-mesh is used to sample the BZ, respectively. The number of energy bands involved in the GW calculations is 192 for both systems.

Lattice Parameters and Enthalpy as a Function of Pressure
The dependence of the lattice parameters with pressure is shown in previous experimental work has shown that the cotunnite-type structure preserves at ambient pressure upon rapid decompression and quenched in liquid nitrogen (77 K) [62]. Moreover, the coexistence of different TiO 2 phases below 60 GPa has also been experimentally demonstrated by compression-decompression method using the diamond-anvil cell (DAC) technique [32]. In this context, the coexistence of the two high-pressure phases studied here is naturally expected for P > 100 GPa. 6

Band Gaps as a Function of Pressure
The calculated band gaps of the cotunnite-type and Fe 2 P-type TiO 2 are shown in where α and β have the same meanings as in Eq. (1); E g (P 0 ) is the band gap at the starting point of pressure range under investigation. The value of P 0 is 50 GPa and 110 GPa, respectively, for the cotunnite-type and Fe 2 P-type TiO 2 . The fitting parameters are summarized in Table I. Consequently, the pressure coefficient is calculated as the first derivative of E g with respect to P: This is a linear function of pressure P. Specially, c p = α when P = P 0 . The pressure variation of c p is shown in Figures 2(c) and 2(d), for the two phases. In the case of GGA calculations, the value of α is 0.117 meV/GPa and -0.813 meV/GPa, for the cotunnite-type and Fe 2 P-type TiO 2 , respectively. For the more accurate GW method, α is 1.831 meV/GPa for the cotunnite-type and 0.489 meV/GPa for the Fe 2 P-type structure. As seen from Table I, all the values of β are the order of magnitude of -10 -3 meV/GPa 2 , which result in the linear decrease of c p with pressure. Meanwhile, the point c p = 0 gives the pressure P m = P 0 -α/(2β), at which the band gap E g reaches its maximum. For instance, from the parameters describing the GW band gaps (Table I), P m is calculated to be ~ 180.1 GPa and ~ 140.0 GPa, for the cotunnite-type and Fe 2 P-type structure, respectively. Compared to the pressure coefficient of diamond (~ 5 meV/GPa) [41,42] and that of SiC (~ 1.9 meV/GPa for cubic SiC and ~ 2 meV/GPa for 6H-SiC) [43,44], which is the smallest value ever measured by experiments, the pressure coefficients of the two high-pressure TiO 2 phases found in our work are still significantly smaller. In particular, the region in which | | ≤ 0.5 meV/GPa, i.e., one 7 order of magnitude smaller than that of diamond, is highlighted in Figures 2(c)  ].
To make a comparison, the variation of the band gaps of rutile TiO 2 under pressure is also studied (Supplemental Material, Figure S1). The GW band gap (2.92 eV) of rutile at P = 0 compare well with the experimental value (~ 3 eV) at ambient pressure [28][29][30], and verifies the reliability of GW calculations. The fitting parameters are listed in Table I. From the data given by both GGA and GW calculations, the pressure coefficient of rutile turns out to be significantly larger than that of the cotunnite-type and the Fe 2 P-type TiO 2 . A simple comparison between the starting and ending pressure points gives that, the volume of crystal unit cell is contracted by ~ 18.3% and ~ 14.2% for the cotunnite-type and Fe 2 P-type, with the GW band gaps changed by ~ +0.14 eV and ~ −0.14 eV, respectively. As a contrast, at a volume contraction of ~ 15%, the GW band gap of rutile TiO 2 is increased by ~ 0.59 eV ( Figure S1). The plateau-like behavior (Figures 2(a), 2(b)) suggests that a step-like jump of the band gaps of TiO 2 can be observed when the applied pressure is increased gradually from 50 to 260 GPa, across the phase transition pressure point (~ 115 GPa).
On the other hand, the GW band gaps (2.989 eV and 1.876 eV) of the cotunnite-type and Fe 2 P-type at 160 GPa are comparable with the values (3.0 eV and 1.9 eV) estimated by previous work [33], and rationalize the empirical corrections therein.

Origin of the Robustness of Band Gaps upon Compression
To understand the robustness of the band gaps upon compression, we studied considerable difference, which leads to the monotonic increase of band gap ( Figure   S1). As seen from Figures 2(e) and 2(f), the difference between the VBM given by GGA and GW calculations is very small for both phases, and the GW corrections on band gaps come mainly from the upshift of CBM.

Comparison of the Structural and Electronic Properties of the Two Phases
We have further studied the atomic and electronic properties of the two high-pressure phases. The local atomic bonding structures of the cotunnite-type and  atoms are assigned to one Ti atom: (1/4) × n 4c + (1/5) × n 5c = 2 together with the condition n 4c + n 5c = 9, one has n 4c = 4 and n 5c = 5, which is respectively the number of four-and five-coordinated O around each Ti. In the meantime, the total number of O 4c and O 5c is equal in the crystal unit cell, which is 4 in cotunnite-type and 3 in the Fe 2 P-type TiO 2 . The nine-coordination is achieved via the periodic extension of unit cells, as indicated in Figures 3(a) and 3(c).
The sharp and discrete RDFs peaks shown in Figures 3(b) [33].
The electron densities associated with the wave function of VBM and CBM, i.e., | | 2 and | | 2 , are displayed in Figure 4, for the two high-pressure phases of

Understanding the Difference of Band Gaps
As mentioned above, at pressure P ~ 115 GPa, the two high-pressure phases of TiO 2 have equal enthalpy, the same coordination numbers of Ti and O in the first coordination shells, and similar averaged value of Ti-O bond lengths (Table II).
Besides, the volume per formula unit is also very close, which is ~ 20.253 Å 3 /TiO 2 for the cotunnite-type and ~ 20.197 Å 3 /TiO 2 for the Fe 2 P-type. Despite these similarities, their band gaps differ by ~ 1.06 eV. Such difference should originate from the different crystal structures (Supplemental Table I (Table III). To the first-order perturbation of correction [58], the quasiparticle energies within the GW method is given by:

Conclusions and outlook
In conclusion, we have studied the large-pressure-scale behavior of the structural and electronic properties of TiO 2 , a wide band gap semiconductor. Calculation of enthalpies suggests the coexistence of two high-pressure phases across a wide range of pressures. For pressures which spanning over an interval of ~ 150 GPa, the band gaps (E g ) of both phases are rather robust to compression. The related pressure coefficients are found to be significantly smaller than that of diamond and SiC, which has the smallest pressure coefficient ever reported by experiments. For both phases of TiO 2 , the variation of E g with pressure is well described by a quadratic polynomial.
The robustness of E g is due to the nearly identical rates of changes of the VBM and CBM with pressure. Such unusual properties may have potential applications in optical devices at extreme conditions such as high pressure. Detailed analysis on crystal structures and the wave function characteristics of VBM and CBM helps to understand the difference in E g of the two structures. Finally, the pressure dependence of the optical properties, and the expected gap closure and metallization with further increasing pressures, will be the subject of a forthcoming work.

Supplementary data
The supplementary data related to this article can be found on the website.  The left panels (a, c) are for the cotunnite-type and the right panels (b, d) are for the Fe 2 P-type. 22