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Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp

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Putting DFT to the trial: First principles pressure dependent analysis on optical properties of cubic perovskite SrZrO3

Ghazanfar Nazir a, b, *, Afaq Ahmad b, Muhammad Farooq Khan a, Saad Tariq b a b

Department of Physics and Graphene Research Institute, Sejong University, Seoul 143-747, South Korea

Centre of Excellence in Solid State Physics, University of the Punjab, Lahore, Pakistan

a r t i c l e i n f o

a b s t r a c t

Article history:

Received 8 July 2015

Received in revised form

21 July 2015

Accepted 27 July 2015

Available online 31 July 2015

Here we report optical properties for cubic phase Strontium Zirconate (SrZrO3) at different pressure values (0, 40, 100, 250 and 350) GPa under density functional theory (DFT) using Perdew-Becke-Johnson

(PBE-GGA) as exchange-correlation functional. In this article we ﬁrst time report all the optical properties for SrZrO3. The real and imaginary dielectric functions has investigated along with reﬂectivity, energy loss function, optical absorption coefﬁcient, optical conductivity, refractive index and extinction coefﬁcient under hydrostatic pressure. We demonstrated the indirect and direct bandgap behavior of SrZrO3 at

(0) GPa and (40, 100, 250 and 350) GPa respectively. In addition, static dielectric constant, Optical bandgap, Plasma frequency and Static refractive index has also been reported. We veriﬁed the Penn's model and showed the inverse relation between static dielectric constant and optical bandgap. Further, we proved the direct relation between static dielectric constant and static refractive index. Both these constants increased by increasing the pressure. Our investigation explored that the material preserve its positive value of refractive index at all pressure values and thus is not a negative index metamaterial.

Also, we measured Plasma frequency for SrZrO3 which also increase by increasing the pressure which leads to a conclusion that material is going to be destabilize.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:

First principle

Cubic phase SrZrO3

High pressure phase

Density functional theory

Optical properties

Perovskite

1. Introduction

In recent days, researchers are busy in ﬁnding materials that have potential uses like hydrogen sensors, fuel cells and data storage devices such as random access memory, high intensity violet-blue light emission, optical wave-guides, high temperature oxygen sensors and capacitors in different functional devices [1e6].

Perovskite having general formula ABO3 are very fundamental materials to be used in various functional devices. These materials due to their wonderful properties of ferroelectricity and piezoelectricity have great attraction for the researchers to investigate them with provoking details. In the recent period, the focus of experimental research is the zirconate perovskite. A very few theoretical approach has been done on these perovskite. Among zirconate perovskite, strontium zirconate (SrZrO3) is very interesting material because of its high temperature protonic conductivity [7]. Besides this, SrZrO3 also has great potential for high voltage and high capacitor reliability applications. High dielectric

* Corresponding author.

E-mail address: zafarforall2004@gmail.com (G. Nazir).

constant, large value of breakdown strength and low leakage current are some of its fundamental characteristics [8,9].

Kennedy et al. used powder neutron diffraction and Rietveld method to investigate the phase transitions in SrZrO3 [10]. These people gave the solid evidence for the pathway of SrZrO3 and said this material is ﬁrst orthorhombic (Pnma) changes to orthorhombic

(Cmcm) at about 970 K, then changes to tetragonal (14/mcm) at about 1100 K and then ﬁnally to cubic (Pm3m) at about 1400 K. It was also suggested that these materials have very high melting temperature of about 2920 K [11]. First principle method is used to study structural, electronic, optical and magnetic properties of materials [12e15]. Mete's group of researchers reported high temperature electronic properties of cubic phase SrZrO3 [16]. Terki et al. used full-potential linearized augmented plane wave (FPLAPW) method to investigate the structural, electronic and optical properties of BaTiO3 and SrZrO3 [17]. Evarestov et al. studied the density functional theory (DFT) LCAO and plane wave (PW) calculations for the known four phases of SrZrO3 [18,19].

It has been observed that previous studies on SrZrO3 mainly focused on its structural and electrical properties using experimental approach but few researchers also reported its optical

http://dx.doi.org/10.1016/j.cocom.2015.07.002

2352-2143/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

ε2 ðuÞ ¼

Z e2 h0 X jMcv ðkÞj2 d½ucv ðkÞ À ud3 k pm2 u2 v;c

BZ

where the volume integral is limited to ﬁrst Brillouin zone and dipole matrix element Mcv ðkÞ ¼ huck jeVjuvk i gives the information about direct transition between conduction band and valence band states. The relation ucv(k)¼EckÀEvk gives the account of the excitation energy used during the transition between conduction and valence band states, “e” represent polarization vector because of electric ﬁeld and “uck” give detail about periodic portion of Bloch wave function in conduction band associated with wave vector k.

The real part of dielectric function ε(u) can be calculated from imaginary part using well known relation of Kramers-Kronig given by: Z∞

0

u0 ε2 ðu0 Þ du0 u02 À u2

The symbol P in the about relation tell us about principal value of integral. By knowing both the real and imaginary values of dielectric functions, different important optical properties can be determined. The refractive index of the material can be determined using the following relation:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ31 2

2

ε2 ðuÞ þ ε2 ðuÞ ε1 ðuÞ

1

2

5

þ nðuÞ ¼ 4

2

2

Whereas at low frequency i.e. at (u¼0), we have the following relation: 1

nð0Þ ¼ ε 2 ð0Þ

The above relation is called static refractive coefﬁcient. The extinction coefﬁcient can be computed by using following relation:

2

Àε ðuÞ þ kðuÞ ¼ 4 1

2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ31 2 ε2 ðuÞ þ ε2 ðuÞ

1

2

5

2

=

Basic calculations of our research is done using full potential linearized augmented plane wave (FP-LAPW) based on density functional theory (DFT) as implemented in Wien2k. In addition to this, Kohn-Sham equations are calculated by applying full potential-linearized augmented plane wave (FP-LAPW) using selfconsistent method [23e28]. We performed pressure dependent characteristics instead of temperature dependent in order to study the optical response of the material under high pressure by using standard DFT at 0 K temperature. In this well-known FP-LAPW method, Slater's theory of Mufﬁn-Tin radii has been used in order to divide this space into two regions. Close to the atoms, all interesting quantities are expanded in terms of spherical harmonics and as expansion of plane waves with wave vector having cut-off value of

KMAX in the interstitial region. The earlier kind of expansion is stated within a well-known mufﬁn-tin sphere of radius RMT around every nucleus. The choice of sphere radii predicts the rate of convergence of these expansions, but this affects only the speed of the calculation. We choose RMT x KMAX ¼ 7 as convergence parameters for the matrix element that represent number of fundamental functions, in which KMAX is associated with plane wave cutoff in momentum space (k-space) and RMT represent the smallest radius of Mufﬁn-Tin cube among all the atomic sphere radii. The value of Mufﬁn-Tin radii for “Barium”, “Zirconium” and “Oxygen” are 2.5, 1.98 and 1.79 au respectively. Energy separation is considered to be À6 Ry. We used generalized gradient approximation

(GGA-PBE) given by Perdew et al.for the calculation of exchangeecorrelation potential [29,30]. We did our calculation without including spin-orbit effects. We choose maximum value of l to be lmax ¼ 10 for wave function expansion inside the atomic spheres.

Outside the mufﬁn-tin spheres, the expansion of charge density is done by Fourier series. For reciprocal space integration under different pressure values in the irreducible Brillouin zone, 35 kpoints in grid of 10 Â 10 Â 10 meshes which is equal to 1000-k points in the complete Brillouin zone scheme has been used to achieve self-consistency. Optical characteristics for SrZrO3 are studied by dielectric function ε(u)¼ε1(u)þε2(u) consist of real and imaginary dielectric parameters. Following equations formulated by Ehrenreich and Cohen give brief description about the real and imaginary parts of dielectric function [29,31].

2

P

p

=

2. Computational details

ε1 ðuÞ ¼ 1 þ

=

properties. Ref. [17,20,21] reported the existence of large conductivity spectra at room temperature shown by SrZrO3. In many reports the calculation for band structure of cubic phase of SrZrO3

(which is stable above 1400 K) [16,20,22]. In this paper, we investigate all the possible optical properties of cubic phase SrZrO3 under different values of pressure (GPa).

33

Reﬂectivity of the mentioned material is calculated by:

R¼

ðn À 1Þ2 þ k2 ðn þ 1Þ2 þ k2

In the same way, energy loss function L(u), absorption coefﬁcient a(u) and optical conductivity s(u) are determined using the following relation [32].

LðuÞ ¼ Imð À 1=e ðuÞÞ ε aðuÞ ¼ 4pkðuÞ=l sðuÞ ¼ ð2Wcv h0 uÞ=Eo

Where “Wcv” in the above relation represent transition probability per unit time.

3. Result and discussion

To discuss the internal structure of any material, the optical properties play a very supportive role. The Optical Properties of materials suggest the feasibility and suitability as industrial point of view especially in opto-electronics. Fig. 1 show theoretical cubic perovskite structure of compound SrZrO3. Atomic arrangements of

Sr, Zr and O atoms on different sites of cubic perovskite are: Sr at 8 corners, Zr at the center and O atoms on the 6 faces.

Figs. 2e10 show meaningful study of different optical parameters for cubic phase of SrZrO3 at different values of pressure at (0,

40, 100, 250 and 350) GPa calculated for energy range upto 45 eV.

We veriﬁed Penn's model by knowing the inverse relation between optical bandgap and static dielectric constant. Fig. 2 shows the variation of optical bandgap and static dielectric constant at different pressure values from (0, 40, 100, 250 and 350) GPa. We can say that optical bandgap and static dielectric constant satisfy Penn's model. This inverse relation between ε1(0) and bandgap can be explained by penn model mathematically given by [33,34].

34

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 1. The compound SrZrO3 has an ideal cubic perovskite structure. The Strontium,

Oxygen and Zirconium atoms preferably reside at corners, on the faces and at the center respectively of single cubic unit cell.

À

Á2 ε1 ð0Þz1 þ ħup Eg

The relation given by Penn's model can be used to determine Eg by knowing the values of ε1(0) and plasma energy “ħup ”. The measured values of static dielectric constant ε1(0) at different pressures (0, 40, 100, 250 and 350) GPa along with their bandgap values and plasma frequencies are given in Table 1.

The real and imaginary parts of dielectric function determined at different values of pressure are shown in Fig. 3. The real part of dielectric function ε1(u) gives information about how much a

Fig. 2. Static dielectric constant and optical bandgap for cubic phase SrZrO3 at (0, 40,

100, 250 and 350) GPa pressure values.

Fig. 3. Frequency dependent dielectric functions of cubic phase SrZrO3 at (0, 40, 100,

250 and 350) GPa pressure values (real part: black lines, imaginary part: red lines). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

material is polarized. It is observed from the Fig. 3 that the measured values of static dielectric constantε1(0), low energy limit ofε1(u), depends upon the bandgap of the material. It can be observe from the Fig. 3 that the real part of dielectric function start increasing from its static value ε1(0) given in Table 1 and reaches a highest peak at about 4.784 eV, 5.497 eV, 6.026 eV, 7.292 eV and

6.887 eV for cubic phase of SrZrO3 at different values of pressure from (0, 40, 100, 250 and 350) GPa. After reaching this peak value, real dielectric function start decreasing with some low peaks exist at different higher energy values and becomes zero at certain value of energy of about 9.740 eV, 11.117 eV, 12.150 eV, 13.110 eV and

13.270 eV at (0, 40, 100, 250 and 350) GPa respectively. On further increase in energy, it can be seen that dielectric function becomes negative which means that in these regions of energy, the medium will totally reﬂect all the incident electromagnetic waves and thus exhibiting metallic nature of the material. The graph also reveals that hump is shifted toward the higher energies by increasing the value of pressure upto 350 GPa.

It has been understood that imaginary part of dielectric function ε2(u) play very impressive role in the optical properties for any material. It has deep effect on the absorption of the medium. Absorption in the material is large for large value of imaginary part of dielectric function ε2(u) Fig. 3 also shows the behavior of imaginary part of dielectric function versus energy. It is clear that absorption starts at about 3.172 eV, 3.738 eV, 3.664 eV, 2.828 eV and 2.213 eV for cubic phase of SrZrO3 at (0, 40, 100, 250 and 350) GPa respectively. These points are related to minimum direct bandgap transition between maxima of the valence band to the minima of the conduction band. Fig. 3 also represents the strong absorbing nature of the material in these energy regions from 3.172 eV to 37.965 eV at

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 4. Frequency dependent reﬂectivity of cubic phase SrZrO3 at (0, 40, 100, 250 and

350) GPa pressure values.

Fig. 5. Frequency dependent energy loss function of cubic phase SrZrO3 at different values at (0, 40, 100, 250 and 350) GPa pressure values.

35

Fig. 6. Frequency dependent Absorption coefﬁcient of cubic phase SrZrO3 at (0, 40,

100, 250 and 350) GPa pressure values.

0 GPa, 3.738 eVe40.031 eV at 40 GPa, 3.664 eVe43.241 eV at

100 GPa, 2.828 eVe40.720 eV at 250 GPa and 2.123 eVe41.175 eV.

These regions consist of different peaks which exist as a result of interband transition between valence band and conduction band.

The width of the absorption region ﬁrst increase upto 100 GPa and then decrease with the increase in pressure. Maximum absorption takes place in energy range of 3.664 eVe43.241 eV with energy width equal to 39.577 eV at 100 GPa and minimum at 0 GPa with energy width equal to 34.793 eV. We can also see that the absorption width shift toward the lower value of energy as the pressure exceeds 100 GPa.

To investigate deeply the surface behavior of the material, its reﬂectivity is measured which is equal to the ratio of incident power to reﬂected power. Fig. 4 represent reﬂectivity versus energy for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa. Fig. 4 shows that the zero frequency limit of reﬂectivity for cubic phase SrZrO3 is equal to 0.129, 0.127, 0.128, 0.130, 0.131 at (0, 40, 100, 250 and 350)

GPa respectively. It then increase from its zero limit as with the increase in pressure and reached at the highest peak of reﬂectivity for cubic phase SrZrO3 at 24.424 eV, 24.375 eV, 26.453 eV,

27.130 eV, 27.560 eV for (0, 40, 100, 250 and 350) GPa respectively.

These peaks produced as a result of interband transition between valence band and conduction band. On the other hand, the minimum value of reﬂectivity occurs in energy range from 10 eV to

40 eV as a result of collective plasma resonance. Imaginary part of dielectric function can be used to measure the depth of plasma resonance [34]. We can see very interesting behavior shown by the peak value of reﬂectivity. It shifts toward higher energy value as the pressure exceeds 40 GPa consistent with the imaginary part of dielectric function [35,36].

The energy loss function L(u) is an important parameter which describes the energy loss for electron moving in a material. Fig. 5

36

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 7. Optical bandgap of cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values.

show energy loss function L(u) versus at (0, 40, 100, 250 and 350)

GPa. The peaks in energy loss function L(u) gives us brief detail about characteristics related to plasma resonance and hence the associated frequency known as plasma frequency. No energy loss occur for photons having energy less than 6.186 eV, 7.096 eV,

7.637 eV, 8.449 eV, 8.670 eV at (0, 40, 100, 250 and 350) GPa respectively but as soon as the photons exceed these amount of energy, energy loss will start increasing and get the maximum peak at 26.035 eV, 32.246 eV, 35.124 eV, 36.969 eV, 37.571 eV for (0, 40,

100, 250 and 350) GPa respectively. It can be seen that highest peak

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Fig. 8. Frequency dependent optical conductivity of cubic phase SrZrO3 at (0, 40, 100,

250 and 350) GPa pressure values (real part: black lines, imaginary part: red lines). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

shifts toward higher value of energy with the increase in pressure.

The highest peak has value 5.709 eV occur at 36.969 eV because of pressure value equal to 250 GPa. The observed peak in the energy loss function associated with the plasma frequency and can be treated as an interface between metallic and dielectric behavior.

Absorption coefﬁcient is very important factor that gives information about the decay of light intensity per unit distance in medium. The frequency dependent absorption coefﬁcient for cubic phase SrZrO3 at different pressure is shown in Fig. 6. The change in frequency is described in terms of energy with range from 0 upto

45 eV. The highest peaks occurred at 23.613 eV, 24.191 eV,

24.769 eV, 25.605 eV, 26.035 eV for (0, 40, 100, 250 and 350) GPa respectively. We can see largest value of peak equal to 382.809 eV exist at pressure value of 100 GPa. So we can say that by increasing pressure, the absorption peak shift toward higher value of energy upto 350 GPa. It has also been observed that there is no absorption for photon energy less than 4.033 eV, 4.525 eV, 5.288 eV, 5.755 eV,

5.989 eV for (0, 40, 100, 250 and 350) GPa respectively. However with photon energy greater than these values, absorption coefﬁcient start increasing, which is associated with the direct bandgap values 3.307 eV, 3.175 eV, 3.023 eV, 2.605 eV, 2.156 eV calculated for cubic phase SrZrO3 at different high pressures, respectively.

We can see many peaks within the studied energy range, and the structures of these peaks can be understood from the interband transitions between valence band and conduction band. Fig. 7 shows optical bandgap values determined by measuring the direct and indirect behavior of the cubic phase SrZrO3 at different pressure values by using theoretical approach of square of absorption. The real and imaginary part of optical conductivity is also

37

Fig. 9. Frequency dependent refractive indices of cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values (refractive index: black lines, extinction coefﬁcient: red lines). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

studied for cubic phase SrZrO3 as shown in Fig. 8. It is observed that the real part of conductivity remained at zero and starts to originate from 3.66, 4.16, 4.31, 5.54 and 5.76 eV for different pressure values at (0, 40, 100, 250 and 350) GPa respectively. This is in good agreement with the bandgap of the material at different pressure.

When the photon energy exceed these particular values, the real

Fig. 10. Static Refractive indices for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa pressure values.

38

G. Nazir et al. / Computational Condensed Matter 4 (2015) 32e39

Table 1

Static Dielectric Constant ε1(0), Optical Band-gap(Eg), Plasma Frequency up and Static Refractive Index n0 at (0, 40, 100, 250 and 350) GPa pressure values.

Pressure (GPa)

Static dielectric constant ε1(0)

Band-gap(Eg)

Plasma frequency (up)

Static refractive Index (n0)

0

40

100

250

350

4.526

4.452

4.496

4.555

4.570

3.307

3.157

3.023

2.605

2.156

0.2714

0.3188

1.0127

1.0514

1.0386

2.122

2.113

2.122

2.132

2.136

part of conductivity start increasing and attains the maximum peak values at 23.231 eV, 23.539 eV, 23.797 eV, 12.618 eV, 13.011 eV for pressure at (0, 40, 100, 250 and 350) GPa respectively. It is observed that as the pressure is increased, the maximum real conductivity peak shifted toward higher value of energy for (0, 40 and 100) GPa.

With the further increase in pressure, the highest peak shift toward the lower value. of energy as seen for 250 GPa and then again it goes toward higher value of energy for 350 GPa. After reaching at highest peak, it starts decreasing with the further increase in photon energy

(eV).

The refractive index and the extinction coefﬁcient for cubic phase SrZrO3 can be determined by the relation:

n2 À k2 ¼ ε1 And 2nk ¼ ε2

Fig. 9 shows the broad spectrum of refractive index and extinction coefﬁcient over wide range of energy. The Figure shows that the spectrum of n(u) nearly follow ε1(u) pattern [37]. It can be seen from the Figure that static refractive index n0 values are 2.122,

2.113, 2.122, 2.132, 2.136 for cubic phase SrZrO3 at (0, 40, 100, 250 and 350) GPa values of pressure which is in good agreement with static dielectric function values with the formula (n2 ¼ ε1 ð0Þ). We

0

can see that the peak in the refractive index shift toward higher value of energy as the pressure increase from (0, 40, 100, 250 and

350) GPa. In the middle of the graph, we can see different humps which vanish at higher values of photon energy. It is due to the fact that beyond certain value of energy, the material will no longer remain transparent and start absorbing high energy photons. At certain value of energy, the value of refractive index becomes less than unity as can be seen from Fig. 9.

Refractive index with value less than unity (vg ¼ c/n) describes that group velocity of the falling radiation becomes greater than speed of light. This means that group velocity move toward the negative domain and nature of the medium will no longer remain linear. In other words, we can say that the material converts to superluminal medium for high energy photons [38,39]. The extinction coefﬁcient is also shown in the Fig. 9. The extinction coefﬁcient gives information about the absorption of light, and at the same time, absorption characteristics at the band edges. The peaks in refractive index as well as in the extinction coefﬁcient are due to the interband transition of electrons from valence band to conduction band. The response of k(u) is closely related with the response of ε2(u) at different values of pressure [40e42]. Fig. 10 shows the behavior of static refractive index with different values of pressure (GPa) which increased as the value of pressure is increased from 0 GPa to 350 GPa. To the best of our knowledge, there is lack of data available on pressure dependent optical properties of cubic phase SrZrO3. We therefore hope that our work will motivate researcher to do theoretical studies in this direction using different exchange-correlation functional, so we can compare our results with them to get better understanding about the material. 4. Conclusion

In summary, we performed ﬁrst principle computational analysis on cubic phase SrZrO3 to measure the optical properties. We have seen that at 0 GPa, the SrZrO3 has indirect nature and become direct as the pressure equals to 40 GPa, after that it remained direct with further increase in pressure upto 350 GPa as conﬁrmed by the optical bandgap measurements. Further, static dielectric constant, plasma frequency, static refractive indices are also measured which increase as the pressure is increased. However, the optical bandgap is decreased with the increased in pressure consistent with Penn's model. Also material showed positive refractive index value at all pressure which leads to conclusion that our material did not change into negative index metamaterial under pressure measurement.

Plasma frequency is also increased with pressure increase exhibiting de-stability of the material which is conﬁrmed by increment in the optical bandgap values. We try our best to discuss all the optical properties in detail and show path to the researcher to do experimental study for comparison.

Acknowledgment

We have no ﬁnancial support for this work.

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