Electronic and optical properties of SrTiO3 under pressure effect: Ab initio study

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Electronic and optical properties of SrTiO3 under pressure effect: Ab initio study L. Khaber a, A. Beniaiche a, A. Hachemi b,n a b

Institute of Optics and Mechanics, University of Setif 1, 19000 Setif, Algeria Laboratory of Physics Quantic and Dynamic Systems, Department of Physics, Faculty of Sciences, University of Setif 1, 19000 Setif, Algeria

art ic l e i nf o

a b s t r a c t

Article history: Received 27 November 2013 Received in revised form 19 March 2014 Accepted 25 March 2014 by F. Peeters

We present an investigation on hydrostatic pressure dependence of electronic and optical properties of the perovskite SrTiO3 in cubic and tetragonal structures, including band structure, density of states, dielectric function, refractive index, absorption coefficient, reflectivity and electron energy-loss function. Calculations are performed using ab initio pseudopotential density functional method with the generalized gradient approximation. The cubic structure is optimized at zero pressure while the tetragonal structure is optimized under its structural phase transition pressure, which is previously calculated to be 6.0 GPa. Comparison between electronic and optical properties of the considered structures has been done. We find that the electronic and optical properties of the cubic phase are quite different from those of the tetragonal one. & 2014 Published by Elsevier Ltd.

Keywords: D. Electronic properties D. High pressure D. Optical properties D. Phase transition

1. Introduction The high pressure behavior of perovskites has been extensively examined in recent years as a model system for understanding minerals under compression [1–5]. Although structural phase transitions in the solid state have been intensively studied, ferroelectric titanate perovskite is still of great interest to condensedmatter physics. In a previous work [4], we considered the structural cubic– tetragonal phase transition in strontium titanate, SrTiO3. The material's wide range of physical properties, such as semiconductivity, superconductivity, incipient ferroelectricity, and catalytic activity [5–10], make it an important material with the potential to be used in numerous high-tech applications. Strontium titanate is cubic Pm3mðO1h Þ and paraelectric at room temperature. Below 105 K, it undergoes a structural phase transition from cubic to an antiferrodistortive (AFD) tetragonal structure I4=mcmðD18 4h Þ, which is associated with zone-boundary phonon condensation at the R point [1–3,6–17]. This structural phase transition involves the rotation of the TiO6 octahedra about the [0 0 1] axis. The rotation angle is less than 21 [4]. At temperatures between 50 and 105 K in the tetragonal phase, the electric susceptibility behaves according to the Curie– Weiss law and the temperature transition to a ferroelectric state is

n

Corresponding author. Tel.: þ 213 36 62 01 36. E-mail address: [email protected] (A. Hachemi).

extrapolated to occur at approximately 36 K [1]. Strontium titanate also exhibits a very large dielectric constant (3 0 0) at ambient conditions. At low temperatures, the crystal remains paraelectric and the dielectric constant increases significantly, deviating from the Curie–Weiss law and saturating at a value of 2  104 at temperatures under 10 K [18–20]. In this paper, we investigate the electronic and optical properties of the cubic and tetragonal structures of the perovskite SrTiO3. First, we calculate the electronic structures, because the optical properties depend on the electronic transitions between occupied bands and empty bands. Second, we calculate the optical properties, including the dielectric function, absorption, refractive index, extinction coefficient, reflectivity and electron energy loss function for both structures and compare between them.

2. Calculation method The first-principles calculations were performed by employing the pseudo-potential plane waves (PP-PW) approach based on DFT and implemented using the CASTEP (Cambridge Serial Total Energy Package) code [21,22]. The major advantages of this approach are as follows: (1) the ease of computing the forces and stresses, (2) good convergence control with respect to all computational parameters, (3) favorable scaling with the number of atoms in the system, and (4) the ability to simplify calculations by neglecting core electrons. For both the cubic and tetragonal phases, we treated exchange and correlation effects using

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the generalized gradient approximation (GGA) potential described in Perdew et al. (PW91) [23]. Convergence was determined based on Brillouin zone (BZ) sampling and the size of the basis set. Convergence was achieved with 6  6  6 and 5  5  3K-points for the cubic and tetragonal phases, respectively. A special k-point mesh [24,25] and the size of the basis set were determined using a cut-off energy equal to 660 eV. The valence electron orbitals used for the calculation of the band structures and the DOSs are as follows: (1) O: 2s and 2p; (2) Ti: 3s, 3p, 3d, and 4s; and (3) Sr: 4s, 4p, and 5s.

Eg=1,862eV

SrTiO3 (Cubic 0GPa)

10 5 0

EF

-5 -10

3. Results and discussion -15

3.1. SrTiO3 under pressure R

X

The phase transition in perovskites as well as in other materials is accompanied by anomalies of different physical quantities. In a previous study [4], we have associated the first phase transition from the cubic phase to the tetragonal phase, at instabilities in evolution of elastics parameters under hydrostatic pressure near 6 GPa. Under hydrostatic pressure, the transition pressure at 6 GPa was observed by several authors such as, Bonello et al. [26], Ishidate et al. [27,28] and Grzechnik et al. [29]. The results of Fischer et al. [31] show that a phase transition at about 6 GPa and at room temperature, is identified as the one that occurs at atmospheric pressure and 105 K. The transition pressures at room temperature reported in the literature differ significantly from one author to another, in the range from 6 GPa [26–30] to 9.6 GPa [5,31]. Guennou et al. [5] discuss some possible causes for these scattered results like the effect of non-hydrostatic pressure conditions and the crystal defects. We think that the effect of anisotropic stress can be the most likely reason. In a theoretical prediction [4] near 14 GPa, We have showed that a second phase transition takes place from a tetragonal to an orthorhombic phase. Lyttle [12] found anomalies in the spectra of X-ray diffraction around 60 K and 30 K which let suggest a transition to a phase with undetermined symmetry. Grzechnik et al. [29], also, observed changes in the intensity of Raman scattering spectra of the second order around 15 GPa and they attributed that to a transition from the tetragonal phase to a phase with undetermined symmetry. Only Cabaret et al. [32] considered this new phase at 14 GPa with the CaIrO3 orthorhombic perovskite structure with space group Cmcm (D17 2h ).This phase transition is not confirmed by others authors like Guennou et al. [5] and stays unexplored. 3.2. Electronic analysis under pressure 3.2.1. Cubic phase Previous experimental data describing the electronic properties show an indirect band gap of approximately 3.25 eV along the R-Γ direction [33]. However, theoretical studies reveal an energy band gap around 1.9 eV using the LDA or GGA approximations [34], with an error of approximately 40% when compared to the experimental results. The accuracy of the theoretical results is improved when using the B3LYP method, which yields a deviation of less than 10% between the experimental and theoretical results. Our results were collected within the GGA-PW91 framework. The calculated energy band gap at 0 GPa is 1.862 eV along the R-Γ direction (see Fig. 1), which is 42% lower than that observed in the experimental data. This deviation is related to the use of the GGA approximations, which are known to underestimate the energy band gap values [35]. The energy band gap increases

Γ

M

R

Fig. 1. (Color online) Calculated band structure of the cubic phase of STO at 0 GPa.

1.92

SrTiO3 - Cubic phase 1.91

Gap energy (eV)

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Energy (eV)

2

1.90 1.89 1.88 1.87 1.86 0

1

2

3

4

5

6

Pressure (GPa) Fig. 2. (Color online) Energy gap versus increasing pressure for the cubic phase of STO.

linearly with increasing pressure and continues along the R-Γ direction as shown in Figs. 2 and 3, respectively. At a pressure of 6 GPa, the energy band gap increases by approximately 2.7%. Based on the partial density of state (PDOS), as presented in Figs. 4 and 5, the valence band structure of the STO is composed of two parts that are separated by a pseudo gap of approximately 9.6 eV. The lower part occurs at approximately  15 eV and contains Sr-p, O-s, and a few electrons from the Ti-(p,d) orbitals. The upper part occurs from  5 eV up to the Fermi level and contains O-p, Ti-d, and a few Ti-p electrons. The Ti-d and Sr-p states dominate the conduction band. Increasing the pressure to 6 GPa for cubic phase STO has a weak effect on the band structure and the PDOS curves.

3.2.2. Tetragonal phase The tetragonal phase of the STO occurs above pressures of 6 GPa. Our results, as presented in Fig. 6, show a direct band gap energy of 2.06 eV along the Γ-Γ direction. Similar results were reported earlier, but the value of the energy band gab reported here is approximately 60% less than the experimentally determined one [36]. Moreover, the use of the B3PW hybrid exchange correlation functionally overestimates the energy band gap by 8% compared to the experimental results [37]. Fig. 7 presents the effect of pressure on the energy band gap. The energy band gap increases linearly at pressures from 6 to 12 GPa. Moreover, no

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SrTiO3 (Cubic 6GPa)

SrTiO3 Tetra 6GPa

Eg=1,914eV

Eg=2,068eV

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Fig. 3. (Color online) Calculated band structure of the cubic phase of STO at 6 GPa.

SrTiO3 (Cubic) 0 GPa

Fig. 6. (Color online) Calculated band structure of the tetragonal phase of STO at 6 GPa.

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Pressure (GPa) Fig. 4. (Color online) Calculated total and partial densities of states of the cubic phase of STO at 0 GPa.

SrTiO3 Cubic 6GPa

6

Fig. 7. (Color online) Energy gap versus increasing pressure for the tetragonal phase of STO.

EF

SrTiO3 Tetra 12GPa

Eg=2,153eV

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Energy (eV) Fig. 5. (Color online) Calculated total and partial densities of states of the cubic phase of STO at 6 GPa.

significant change in the curve dispersions for the band structures (Figs. 6 and 8) or the density of states (Figs. 9 and 10) for pressures of 6 and 12 GPa are observed for the tetragonal phase of the STO. According to Figs. 9 and 10, the lower part of the valence band (around  15 eV) is composed of Sr-(s, p), O-s, and a few electrons from the Ti-(p, d) orbitals. From 5 eV up to the Fermi level, O-p and Ti-(p, d) electrons dominate and Sr-s electrons are in the

-15 Z

X

P

N

Fig. 8. (Color online) Calculated band structure of the tetragonal phase of STO at 12 GPa.

minority. The conduction band above the Fermi level contains electrons from the Ti-d and Sr-p orbitals. We conclude this part by the main modifications raised by the phase transition cubic–tetragonal of STO: – The band gap energy change from indirect gap for the cubic to direct band gap for the tetragonal structure.

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SrTiO3 cubic (0GPa)

SrTiO3 6GPa (Tetra)

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Fig. 11. (Color online) Calculated real and imaginary parts of dielectric function for the cubic phase at 0 GPa.

SrTiO3 cubic (0GPa)

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Absorption (Cm-1)

TDOS (States/ eV. Unit cell)

Fig. 9. (Color online) Calculated total and partial densities of states of the tetragonal phase of STO at 6 GPa. 15

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Os Op

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Fig. 10. (Color online) Calculated total and partial densities of states of the tetragonal phase of STO at 12 GPa.

– The relatively increase of the band gap energy from cubic to tetragonal. – The upper curves of the BV (the bottom BC curves) turn to be more flat along ΓX direction for the tetragonal phase. – The band gap energy increases rapidly with pressure for the tetragonal phase than for the cubic one. Q3– Sr(s, p) and O(s, p) states move slightly to higher energy level from the cubic to tetragonal structure. 3.3. Optical analysis under pressure 3.3.1. Cubic phase The calculated optical properties for the cubic phase of the STO are summarized in Figs. 11–14. Fig. 11 shows the real and imaginary part of the dielectric function for the energy range between 0 and 45 eV. To account for the structures observed in the optical spectra, it is customary to consider transfers from occupied to unoccupied bands in the electronic energy band structure, especially at high symmetry points in the Brillouin zone [38]. Fig. 11 shows the real and imaginary parts of the dielectric function for the STO. Based on the electronic structure, the imaginary part ε2 ðωÞ is directly connected with the electron transfers in the band structure of the STO. The curve of the imaginary part ε2 ðωÞ contains six peaks in the energy range from 0 to 40 eV. The imaginary part of the dielectric function ε2(ω) shows six important structures that are labeled A (4.38 eV), B (7.95 eV), C (11.21 eV), D (19.91 eV), E (22.85 eV), and F (36.31 eV) in

0

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frequency (eV) Fig. 12. Calculated absorption coefficient for the cubic phase at 0 GPa.

SrTiO3 cubic (0GPa) 3.0 n k

2.5

Refractive Index

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1.5

1.0

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Energy (eV) Fig. 13. (Color online) Calculated refractive index for the cubic phase at 0 GPa.

Fig. 11. The first peak (A) in the region at 4.38 eV originates from the transfer of the extreme valence electrons (O-p states) to the first conduction band (Ti-d states) along the (Γ–Γ) direction. This peak is a result of the fundamental gap in the energy (direct gap). Peaks B and C correspond to transfers from the Ti-p þd and O-p valence bands to the Ti-d and O-p conduction bands. It is noted that a peak in ε2 ðωÞ does not correspond to a single inter-band

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A1 6

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R( )

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Energy (eV) Fig. 15. (Color online) Calculated real and imaginary parts of dielectric function for the tetragonal phase at 6 GPa.

Fig. 14. Calculated loss function and reflectivity for the cubic phase at 0 GPa.

transfer because many direct or indirect transfers may be found in the band structure with an energy corresponding to the same peak [39,40]. Peaks D, E, and F are associated with the transfer of deep electron excitations from Sr-p, Ti-p þd, and O-s to the conduction bands. The static dielectric constant ε1 ð0Þ calculated at the equilibrium lattice constant is approximately 6.45. The calculated linear absorption spectrum αðωÞ is shown in Fig. 12. The absorption edge starts from approximately 1.41 eV, corresponding to the energy gap ΓValence-ΓConduction. This absorption edge originates from electron transitions from the O-p electron states located at the top of the valence bands to the empty Ti-d electron states that dominate the bottom of the conduction bands. The refractive index and the extinction coefficient are shown in Fig. 13. The static refractive index nð0Þ is found to have a value of 2.53. The nðωÞ increases with increasing photon energy to reach its maximum value of about 2.99 in the ultraviolet energy range, and then it decreases to minimum value of about 0.38. The local maximum value of the extinction coefficient kðωÞ corresponds to the parameter ε1 ðωÞ being equal to zero. The origin of the structures in the imaginary part of the dielectric function also explains the structures observed in the refractive index. The electron energy loss function LðωÞ is an important factor describing the energy loss of a fast electron traversing through a material. The apparent peaks in the LðωÞ spectra represent a property associated with the plasma resonance (a collective oscillation of the valence electrons) and the corresponding frequency, which is referred to as the plasma frequency ωP [41]. The peaks of LðωÞ correspond to trailing edges in the reflection spectra. For instance, the prominent peaks of LðωÞ (as observed in Fig. 14), are situated at energies that correspond to abrupt reductions in RðωÞ (see Fig. 10(b)). The calculated reflectivity has a maximum value of approximately 46.45% at 24.64 eV.

3.3.2. Tetragonal phase In this section, we discuss the calculation of the real (ε1) and imaginary (ε2) parts of the dielectric function as a function of the energy over the range of 0–45 eV for the tetragonal phase of SrTiO3. The results are presented in Fig. 15. Based on the electronic structure, the imaginary part ε2 ðωÞ is directly connected with the electron transfers in the band structure of the STO. The curve in the imaginary part ε2 ðωÞ exhibits four peaks in the energy range from 0 to 45 eV. The imaginary part of the dielectric function ε2(ω)

350000 300000

Absorption(cm-1)

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 61 62 63 64 65 66

5

250000 200000 150000 100000 50000 0 0

10

20

30

40

Energy(eV) Fig. 16. Calculated absorption coefficient for the tetragonal phase at 6 GPa.

shows four important structures that are labeled as A1 (4.59 eV), B1 (7.86 eV), C1 (22.21 eV), and D1 (36.27 eV), as observed in Fig. 15. This peak is a result of the fundamental energy gap (direct gap). Peaks B1 and C1 are equivalent to the transfers from the Ti-pþ d and O-p valence bands to the Ti-d and O-p conduction bands. It is noted that a peak in ε2 ðωÞ does not correspond to a single inter-band transfer because many direct or indirect transfers may be observed in the band structure for an energy corresponding to the same peak [39,40]. Peak D1 is attributed to the transfer of deep electrons from the Sr-p, Ti-p þd, and O-s orbitals to the conduction bands. The static dielectric constant ε1 ð0Þ calculated from the equilibrium lattice constant is approximately 4.91. The calculated linear absorption spectrum αðωÞ is shown in Fig. 16. The absorption edge starts at approximately 1.91 eV, which corresponds to the energy gap ΓValence-ΓConduction. This absorption edge originates from an electron transfer from the O-p electron state located at the top of the valence band to the empty Ti-d electron state that dominates the bottom of the conduction bands. The refractive index and the extinction coefficient are shown in Fig. 17. The static refractive index nð0Þ has a value of 2.21. The nðωÞ increases with increasing photon energy to reach its maximum value of about 2.72 in the ultraviolet energy range, then it decreases to minimum value of about 0.044. The local maximum value of the extinction coefficient kðωÞ corresponds to a ε1 ðωÞ value of zero. The origin of the structures in the imaginary part of

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References

3.0

[1] [2] [3] [4]

2.5 n k

Refractive Index

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

2.0

[5] [6]

1.5

[7] 1.0

[8] [9] [10] [11]

0.5

0.0 0

10

20

30

40

Energy (eV) Fig. 17. (Color online) Calculated refractive index for the tetragonal phase at 6 GPa.

the dielectric function also explains the structures observed in the refractive index. The principal modifications of the optical properties throughout the phase transition cubic–tetragonal of STO are: – The phase transition cubic–tetragonal weaken the peak intensity of the dielectric functions, and the C and D peaks in the cubic phase disappear. – The cubic–tetragonal transition reduces the values of the static dielectric constant ε1 ð0Þand the static refractive index n(0).

4. Conclusion In summary, we have investigated the electronic and optical properties of the perovskite SrTiO3 in the cubic and tetragonal structures under pressure effect using ab initio pseudopotential density functional method. The calculated band structure and density of states spectra suggest the semiconducting character of the considered material in its both structural phases which is in agreement with previous literature. The cubic structure has an indirect energy band gap whereas the tetragonal one has a direct energy band gap. The dielectric function and some optical constants such as absorption coefficient, reflectivity, refractive index, extinction coefficient and electron energy-loss function were presented for a wide energy range from 0 to 45 eV. Some differences between the electronic and optical properties of the cubic and tetragonal structures were reported.

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Please cite this article as: L. Khaber, et al., Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2014.03.018i

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