First principle study of the effect of pressure on the optical properties of cubic-LaAlO3 compound

Computational Materials Science 84 (2014) 360–364

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First principle study of the effect of pressure on the optical properties of cubic-LaAlO3 compound M.R. Benam ⇑, N. Abdoshahi, M. Majidiyan Sarmazdeh Department of Physics, Payame Noor University, P.O. BOX 19395-3697, Iran

a r t i c l e

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Article history: Received 27 September 2013 Received in revised form 2 December 2013 Accepted 16 December 2013 Available online 9 January 2014 Keywords: Perovskite LaAlO3 Density functional theory Optical properties Pressure

a b s t r a c t In this paper, some optical properties of cubic-LaAlO3 have been calculated at various pressures. These properties are dielectric function, energy loss function, optical conductivity and refractive index. In our calculation we have used density function theory (DFT) and generalized gradient approximation (GGA). The results showed that with increasing pressure the dielectric function and static refraction index are decreased at low energies. The analysis of energy loss function in terms of pressure confirmed that below 41.1 GPa, the plasmon energy is increased and then decreased. Furthermore, with increasing pressure the optical band gap was increased and reached to 5.91 eV. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Lanthanum Aluminates, LaAlO3 (LAO) with perovskite structure, has attracted the interest of the scientific society due to its excellent individual properties. The most important properties are high dielectric constant (between 23 and 27) [1], wide band gap (5.5–6.5 eV) [2–4] and thermal stability in contact with Si [3,5–7]. Recently, epitaxial growth of LAO on Si has attracted a huge interest in electronic industry due to its excellent lattice matching of cubic-LAO on Si substrates [3,7–11]. LAO is a high quality insulating buffer and therefore can be used as a substrate for high-temperature superconducting layers [12]. LAO has a perovskite ABO3 (A = La and B = Al) structure, which is a combination of rocksalt-LaO with rutile-AlO2 structure. LAO has two different phases. At the ambient condition it crystallizes to rhombohedral perovskite structure with space group R-3c. Above its critical temperature, Tc = 850 K, it changes to cubic structure with space group Pm-3m [13–15]. Experimental investigations, using powder synchrotron X-ray diffraction (XRD) and Raman spectroscopy at high-pressure, show that there is a rhombohedral to cubic phase transition at pressures around 14.0 GPa [16,17]. Recent Photoemission study of epitaxial LAO films, grown on SrTiO3 (0 0 1) substrates, has shown that LAO has also a tetragonal lattice which is a slightly distorted cubic lattice [18]. Recently the cubic phase of LAO has been synthesized and studied experimentally [13,19]. Yuan et al. [20] have shown that if the diameter of LAO nano-crystal be smaller than 100 nm, it will have a ⇑ Corresponding author. Tel.: +98 5118433638; fax: +98 5118683001. E-mail addresses: [email protected], [email protected] (M.R. Benam). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.12.034

stable cubic structure at room temperature with indirect band gap. But when the LAO nano-crystals are embedded in an amorphous Lu2O3 matrix, the nano-crystals experience a net strain and pressure from Lu2O3 matrix, which causes a distortion of the AlO6 octahedral and leads to the growth of direct band gap LAO nano-crystals with rhombohedral structure. These findings suggest that the change of LAO crystal parameters with pressure is an important idea for engineering and controlling the physical properties of LAO. Therefore, theoretical studying of the effect of pressure on the physical properties of LAO would be an effective way for predicting the change of its properties and improving its applications in industry, which is the main goal of this research.

2. Computational methods We have performed full potential linear augmented plane waves (FP-LAPW) method as implemented in WIEN2k package [21]. In our calculations we have used density function theory [22] and generalized gradient approximation (GGA) [23] for exchange–correlation potential. The radius of Muffin-Tin sphere for La, Al and O atoms were selected to be 2.26, 1.51 and 1.51 a.u., respectively. The positions of La, Al and O atoms in unit cell were chosen to be (0.0, 0.0, 0.0), (0.5, 0.5, 0.5) and (0.5, 0.5, 0.0), respectively [24]. The space group of cubic-LAO is Pm-3m. The separation energy between core and valance states was set to 8.0 Ry. The parameter RMTKMax (number of basis functions) was set to 7.0 where RMT is the smallest radius of the Muffin-Tin spheres and KMax is the cut-off wave vector. We considered 4000 k-points in the first Brillouin

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zone which is equal to 120 k-point in reduced Brillouin zone. The convergence criterion for self-consistent fields was taken the difference between incoming and outcoming charge densities to be less than 0.0001e. In our optical calculations, we have considered the interband transition, dipole and long wavelength approximations. Based on these simplifications the following Kramerz-Kronig transformations were used for obtaining the optical properties [25].

Z

4pe2 X m2 x2 c;v

Im

eij ðxÞ ¼

Re

eij ðxÞ ¼ rij þ p p

2

Z

1

0

dkhC k jpi jV k ihV k jpj jC k idðeck  ev k  xÞ

ð1Þ

x0 Im eij ðx0 Þ 0 dx x02  x2

where jck i and jv k i are the conduction and valence states of an electron with wavevector k. 3. Optical properties 3.1. Dielectric function The interaction of incident photons or electrons with a solid can be described with a complex dielectric function as:

eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ

ð2Þ

where e1(x) is the real part and e2(x) is the imaginary part of dielectric function. The real part of dielectric function is obtained from its imaginary part, using Kramerz-Kronig transformations of Eq. (1). Fig. 1a shows the changes of e1(x) in terms of incident photon energy at different pressures. In order to see the variations clearly, we have displayed its low energy spectrum from 0 to 9 eV at Fig. 1b. We see that with increasing pressure, the static dielectric function, e1(0), is decreased but the maximum peak of e1(x) is increased and shifts toward the higher energies. The roots of the dielectric function show the energies in which the maximum absorption is occurred. The first root, last root and the static dielectric function have been listed in Table 1. It is seen that with increasing pressure the static dielectric function, e1(0), is decreased and reached to 4.08 at 80.9 GPa. It was found that with increasing pressure the first root and the last root are increased up to 49.5 GPa and 33.6 GPa, receptively and then they are decreased. The remarkable change of the last root around 33.6 GPa maybe due to the transition of indirect to direct band gap at these pressures [26]. Fig. 2 shows the change of the band structure at different pressures. This figure clearly illustrates the change of the character of the band gap from indirect to direct at pressures more than 26 GPa. Finally, we see that the pressure apparently has not any clear effect on the dielectric function for energies higher than 30 eV. 3.2. Energy-loss function The energy-loss function, L(x), represents the amount of energy that a photon losses during the inelastic interaction with solid. In free-electron approximation, the energy-loss function is given by following equation [27]:

    1 e2 xCx2P ¼ 2 ¼ Im 2 2 2 eðxÞ e1 þ e2 ðxCÞ þ ðx2  x2 Þ

ð3Þ

P

where C is the inverse of the relaxation time and xP is the plasmon frequency. At x = xP the denominator of L(x) is minimized and the energy-loss function will have a peak. In fact, the photons with this energy will excite the plasmons of the solid. Plasmon frequency is related to the free electron density, n, by [27]:

Fig. 1. The variation of e1(x) in terms of incident photon energy and pressure (a) from 0 to 35 eV, (b) from 0 to 9 eV.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pne2 xP ¼ m

ð4Þ

Although the above equation is for free electrons, it can be generalized to photons at long wavelength approximation [27]. The energy loss function in terms of the incident photon energy and pressure have been shown in Fig. 3. We see that the L(x) is negligible below 9 eV and the first peak occurres at the first root of e1(x) (about 9.14–10.5 eV). These peaks shift toward higher energies with increasing pressure. The plasmon energy is occurred at the position of the largest peak of L(x). Plasmon energy values at various pressures have been summarized at Table 1. It is seen that by increasing pressure up to 41.1 GPa, the plasmon energy increases and then decreases.

3.3. Optical conductivity The electrons of valance band (VB) transit to conduction band (CB) by absorbing the incident photons and contribute to electric conductivity. This increase in electric conductivity is called optical conductivity. Fig. 4a shows the change of optical conductivity of LAO in terms of incident photon energy for interband transitions. For better analyzing, we have divided it into three main regions. In the first region (0–10 eV), there are some remarkable peaks

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Table 1 Static dielectric function, first root and last root of dielectric function, plasmon energy and optical conductivity threshold (OCT) at different pressures. Pressure (GPa)

0.0

0.7

5.7

11.4

17.9

26.8

33.6

41.1

49.5

58.9

63.9

80.9

e1(0)

4.62 9.18 28.12 28.20 4.44

4.61 9.18 28.12 28.20 4.44

4.58 9.42 28.36 28.45 4.61

4.53 9.59 28.53 28.61 4.77

4.49 9.75 28.77 28.86 4.93

4.42 10.00 29.10 29.18 5.18

4.38 10.24 29.26 29.35 5.24

4.33 10.32 26.90 29.43 5.31

4.20 10.49 26.65 28.20 5.42

4.14 9.67 27.14 27.39 5.59

4.13 9.67 26.98 27.47 5.77

4.08 10.00 26.90 26.98 5.91

First root (eV) Last root (eV)  hxP (eV) OCT (eV)

Fig. 2. The change of the band structure of LAO compound with pressure. The band gap shifts from indirect to direct at pressures more than 26 GPa.

which are due to the electron transition from O-2p states at VB to La-4f states at CB of the t-DOS (see Fig. 5). The change of the optical conductivity threshold (OCT) with pressure has been shown in Fig. 4b and Table 1. We see that with increasing pressure, OCT increases and reaches from 4.45 eV at zero pressure, to 5.91 eV at 80.0 GPa. Second region is expanded from 10 to 20 eV where the number of peaks are large and their intensity are decreased with pressure. These peaks mainly are related to optical transition from O-2p, Al-3s, and Al-3p states of the VB to the unoccupied La-4f, La-5d, Al-3s, Al-3p, and O-2p states of the CB [26] (see Fig. 5). In the third region (20–35 eV) there is a remarkable peak at high energies for every pressure. Considering the calculated t-DOS of LAO (Fig. 5) and according to the transition laws, this

peak is related to the electron transition from La-5p deep states of VB to La-5d state at CB. 3.4. Refraction index Fig. 6 shows the change of refraction index, n(x), in terms of incident photon energy and pressure. The changes are similar to the variations of the real part of dielectric function. We see that with increasing pressure the refraction index shifts to the higher energies. Static refraction index, n(0), is an important physical quantity which can be obtained experimentally. We have calculated n(0), and n(x) in the visible region of electromagnetic spectrum, at various pressures (see Table 2).

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Fig. 3. The energy loss function in terms of incident photon energy and pressure.

363

Fig. 5. The contribution of atomic orbitals to t-DOS of LAO at zero pressure. Fermi energy has been set to zero.

Fig. 6. The change of refraction index in terms of incident photon energy and pressure. The inset clearly shows decreasing of the refractive index with increasing pressure in the range of 0–7 eV.

Fig. 7 shows the change of refraction index in terms of wavelength at visible spectrum. It is seen that n(x) decreases with increasing wavelength at constant pressure, and decreases with increasing pressure at constant wavelength.

4. Conclusion

Fig. 4. The change of (a) optical conductivity and (b) optical conductivity threshold of LAO in terms of incident photon energy and pressure.

According to these data, the n(0) and n(x) are reduced with increasing wavelength and pressure.

In conclusion, dielectric function, energy loss function, optical conductivity and refractive index of cubic-LAO at various pressures were calculated using full potential linear augmented plane waves (FP-LAPW). We showed that the pressure has a remarkable effect on these properties, especially in low energy range. It was shown that with increasing pressure the dielectric function and static refraction index at low energies decreases. We also showed that the plasmon energy increases below 41.1 GPa and then decreases with pressure. The optical band gap was investigated in terms of pressure and we showed that the optical band gap increased with pressure and reached from 4.45 eV at zero pressure, to 5.91 eV at

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Table 2 Static refraction index and refraction index values in the visible region of electromagnetic spectrum, at various pressures. Pressure (GPa)

0.0

0.7

5.7

11.4

17.9

26.8

33.6

41.1

49.5

58.9

63.9

80.9

n(0) n(400.0 nm) n(646.8 nm) n(741.0 nm)

2.14 2.32 2.20 2.19

2.14 2.32 2.20 2.19

2.14 2.30 2.19 2.18

2.12 2.28 2.17 2.16

2.11 2.26 2.16 2.15

2.10 2.23 2.14 2.13

2.09 2.22 2.13 2.12

2.07 2.20 2.12 2.11

2.05 2.16 2.09 2.08

2.03 2.14 2.07 2.06

2.03 2.14 2.06 2.06

2.02 2.12 2.05 2.04

Fig. 7. The change of refraction index in terms of wave length at visible region of electromagnetic spectrum.

80.0 GPa. The refraction index at the visible region of electromagnetic spectrum, was decreased by increasing pressure and wavelength. Acknowledgments This work has been done in Nanophysics lab of Payame-Noor University. The authors want to thank from WIEN2K community for their helpful comments. References [1] B.E. Park, H. Ishiwara, Appl. Phys. Lett. 82 (2003) 1197.

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