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Ground state and lattice dynamical study of ionic conductors CaF2, SrF2 and BaF2 using density functional theory
Journal of Physics and Chemistry of Solids 72 (2011) 934–939
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ground state and lattice dynamical study of ionic conductors CaF2, SrF2 and BaF2 using density functional theory Himadri R. Soni a, Sanjeev K. Gupta a,b, Mina Talati c, Prafulla K. Jha a,n a
Department of Physics, Bhavnagar University, Bhavnagar 364022, India Dipartimento di Fisica dell’Universita degli studi di Modena e Reggio, Emilia and Centro S3, CNR-Istituto di Nanoscienze, via Campi 213/A, 41100 Modena, Italy c Institut f¨ ur Sicherheitsforschung, Forschungszentrum Rossendorf e.V.—Postfach 510119, 01314 Dresden, Germany b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 January 2011 Received in revised form 24 March 2011 Accepted 28 April 2011 Available online 12 May 2011
The present paper reports a comprehensive and complementary study on structural, electronic and phonon properties of face centered cubic fluorites, namely CaF2, BaF2 and SrF2, using first principles density functional calculations within the generalized gradient approximation. The calculated lattice constants and bulk modulus are in good agreement with available experimental data. The analysis of band structure and density of states confirms the ionic character for all the three fluorides. The phonon dispersion curves and corresponding phonon density of states obtained in the present work are consistent with the available experimental and other theoretical data. The LO–TO splitting is maximum for CaF2, which confirms that the ionicity is maximum in the case of CaF2. The phonon properties for SrF2 have been calculated for the first time. & 2011 Elsevier Ltd. All rights reserved.
Keywords: C. Ab initio calculations D. Equations-of-state D. Lattice dynamics D. Phonons
1. Introduction Ionic conductors are promising candidates for solid state electrolytes used in batteries. The fluorite CaF2 is a prototype ionic conductor showing a strong increase of conductivity with variation of external parameters such as temperature that saturates at 1420 K, where it becomes comparable to that of a matter salt [1,2]. In addition it possesses intrinsic optical properties, superior character at high temperature and is the subject of several experimental and theoretical studies in the recent past [3–20]. The alkaline-earth fluorides XF2(X¼Pb, Ca, Sr, Ba) generally crystallize to a cubic fluorite structure ðFm3mÞ consisting of a close-packed cubic alkali (X) lattice with fluorine (F ) occupying the tetrahedral sites. The ionic conduction arises in these compounds from the motional disorder in the fluorite sublattice, hopping over potential barriers and formation of various types of defects [21]. Details of ionic conduction mechanism however depend on the peculiarities of structure and dynamics as demonstrated by comparison with other ionic conductors, e.g. LaF3 [22]. Furthermore, phonon dispersion curves are an essential key ingredient for the calculation of specific heat, thermal expansion and vibrational entropy. In addition the phonons play an important role not only in the formation of various defects, hopping
n
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over potential barriers and disorder but also in several mechanical and optical properties. Therefore, a systematic characterisation of the phonon dispersion relations and phonon density of states (PDOS) of alkaline-earth fluorides is highly desirable. To the best of our knowledge there exist rich experimental and theoretical studies on phonons for one of the fluorides CaF2 [14,22–34], but for other fluorides such as SrF2 and BaF2 the situation is not so encouraging. The Raman and TO phonon modes for BaF2 have been reported in Refs. [1,3,4,27,30,32–40] and for SrF2 in Refs. [4,5,33–35]. There does not exist a single theoretical calculation for these alkaline-earth fluorides. Hence, a comprehensive theoretical study is not only essential to verify the observed experimental data but also to find a correlation between these compounds and to establish stability of the otherwise putative structure. The aim of the present work is to add to the general understanding of the ionic conduction through the knowledge of electronic and phonon properties using the ab initio pseudopotential method based on a generalized gradient approximation (GGA) of the density functional theory. The choice of GGA lies with the fact that the GGAs are intended to be an improvement on the conventional local density approximation (LDA) and indeed perform better in certain situations. As far as the success of Perdew–Burke–Ernzerhoff (PBE) exchange correlation functional for the prediction of phonon dispersion curves closure to experiments is concerned it is more or less of same quality as that of LDA [41], with exceptions in a few cases. The newly developed GW or hybrid functionals are more suitable to the electronic
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structure calculations and determination of the band gap but not for the lattice dynamical properties. It is also expected that the phonon calculations in SrF2 and BaF2 along with CaF2 will be of immense help to understand the parameters influencing the ionic conduction, whole origin is still to be clarified. Previous theoretical work [26] has been based on phenomenological and semiempirical potential leading to somewhat quantitatively unreliable results [24]. The large LO–TO splitting observed in these compounds suggests an advanced response density functional perturbation theory that incorporates macroscopic electric fields for the lattice dynamical calculations of these compounds. In addition these compounds, particularly SrF2, are becoming important and useful for the nanoparticles dispersed medium [42,43]. The importance of different processes of interaction between phonon modes is of considerable interest, particularly during the study of phonon assisted energy transfer processes [44]. Recently, it has been observed that the Eu2 þ doped glasses and glass ceramics containing SrF2 nanocrystals exhibited a much stronger ( 20 times) broad blue emission band and longer life time of excited state than glass. The reason for this intense blue luminescence in comparison to glass is attributed to the smaller multiphonon relaxation probability due to the lower phonon energy of host (SrF2) of about 290 cm 1. The phonon modes corresponding to this energy range in the SrF2 bulk crystal are due to the ionic movement. The blue-shift in the luminescence spectrum of SrF2 nanocrystals may arise due to the phonon confinement effect as was found in the case of CaF2 by Ricci et al. [45]. Therefore, it is important and the right time to investigate the phonon dispersion curves for this group of compounds. The only reported lattice dynamical calculation using first principles method is in good agreement with the experimental phonon dispersion and density of states [22]; however there are no similar studies on any other compounds of this group. Furthermore the electronic band structure calculation performed for the three fluorides will be useful in shedding some light on the excellent transmission properties of CaF2 and to see if there is any role of electronic properties in ionic conduction. The paper is organized as follows. The computational methodology is presented in Section 2. In Section 3, we present theoretical results and discussion. The results are concluded in Section 4.
2. Computational details To determine the electronic structure and phonon properties in CaF2, SrF2 and BaF2, we performed plane-wave and pseudopotential (both norm-conserving and ultrasoft) calculations using the Quantum ESPRESSO code [47]. The generalized gradient approximation (GGA) has been used for the exchange and correlation energy density function. A fully relativistic calculation is performed for core states, whereas the valence states are treated in a scalar relativistic scheme. In order to find an appropriate energy cutoff, the total energy as a function of the energy cutoff has been calculated for 23 different energy cutoffs ranging from 10 to 125 Ry. The total energy converges near the energy cutoffs of 39, 55 and 40 Ry for CaF2, SrF2 and BaF2, respectively. In the calculation, we fixed lattice constants 5.466 A˚ for CaF2, 5.80 A˚ for SrF2 and 6.20 A˚ for BaF2 structure as an experimental values [3,4,14,34,38,48–49]. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 5 Ry. Here, for the selfconsistency, the initial potential for the next iteration is constructed using a convergence stabilization scheme. The number of sampling k-points used in the Brillouin zone (BZ) summation of the electronic density and total energy was increased till the total
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energy converged to the desirable tolerance. Thirty six special k-points in the irreducible BZ are sufficient to achieve the convergence for the total energy. To generate a (6 6 6) mesh in BZ, the scheme of Monkhorst–Pack [50] has been used. The crystal structure and associated equilibrium lattice constants for all the three fluorides have been obtained by minimizing the calculated total energies as a function of lattice constant. Furthermore, to calculate the lattice dynamics of CaF2, SrF2 and BaF2, we have used the Quantum ESPRESSO [47] code based upon density functional perturbation theory [DFPT] [51] implemented within plane wave pseudopotentials framework. In this method the dynamical matrix, which provides information on lattice dynamics of the system, can be obtained from the ground state electron charge of the nuclear geometry. The kinetic energy cutoff and numbers of k-points mentioned above are found to yield phonon frequencies converged to within 2–5 cm 1.
3. Results and discussion The theoretical ground state constants such as lattice parameters and bulk modulus of the three fluorides CaF2, SrF2 and BaF2 are obtained using the method discussed in the above section and listed in Table 1, which also includes the available experimental and other theoretical data for comparison. It is clear that the present theoretical values agree reasonably well with the experimental data [4,14,38,48,49,52–59]. It is generally observed that GGA yields higher lattice constant. The total energy versus volume curve is fitted to the Murnaghan equation of state [60] to obtain the bulk modulus value as given below: " # 0 B0 V0 B0 V ðV0 =VÞB0 EðVÞ ¼ E0 0 þ 0 þ 1 ð1Þ B0 B0 1 B00 1 where B0 and B0 0 are the bulk modulus and its derivative, respectively, E0 is the ground state total energy and V0 is the volume. The bulk modulus so obtained is presented in Table 1 and compared with the available experimental and other theoretical data. There is an excellent agreement between the present and experimental data, while there is significant variation within the previous theoretical data [61–65]. The self-consistent band structures along with the electronic density of states (DOS) for CaF2, SrF2 and BaF2 are presented in Fig. 1. The DOS reflects all features of the band structure. The calculated band gap for CaF2, SrF2 and BaF2 is 7.85, 7.07 and 7.12 eV, respectively, in agreement with earlier calculations [9,10,13,61,62,64–68]. However the present band gap values are lower than the experimental values [15,38], which may be due to the use of GGA, which normally underestimates the band gap by a factor of about two. This discrepancy is due to the fact that DFT Kohn–Sham states do not correctly take into account the quasiparticle self-energy in our calculations [69]. We have also calculated the charge density distributions for CaF2, SrF2 and BaF2 (presented only for CaF2 in Fig. 2), which show that these compounds are ionic in nature and there exists an ionic bond along the direction of alkali and florine ions. In order to get an idea about the behavior of phonons and the role played by them in ionic conductivity in these fluorides, the phonon dispersion curves (PDC) along with phonon density of states (DOS) for CaF2, SrF2 and BaF2 in the cubic fluorite structures ðFm3mÞ are calculated using the method discussed above and presented in Fig. 3. The shape of the phonon spectrum for all three materials is similar. The unit cell of a cubic structure with space group Fm3m presents nine degrees of freedom, six of them corresponding to optical modes and three others to acoustic modes. Fig. 3 reveals that all calculated phonon branches in the whole Brillouin zone yield positive frequencies for all three
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Table 1 Calculated equilibrium lattice constants, bulk modulus and band gap for CaF2, BaF2 and SrF2. Parameters CaF2
SrF2
BaF2
Bulk modulus (GPa)
Present work
Experimental
Other work
77.2
87 7 5a, 82.07 0.7b, 82.0c, 81.0 71.2f,81.1g, 84.7h, 81.0i
79.54d, 84.7e, 82.7e, 103j,k, 77k, 91l, 82.14m, 85l 5.4872c, 5.563e,
˚ Lattice constant (A)
5.501
5.466a, 5.4630n
Band gap (eV)
7.85
11.8p
Bulk modulus (GPa)
64.3
69.0s, 74.53t
5.444e, 5.33o, 5.50l 7.39d, 7.24j, 7.27k, 7.01k, 7.7q, 6.85o, 11.5o, 11.38r, 10.87l 66.2u, 68.8v, 117.54w, 77.11w, 79.52w, 77.35w 5.846u, 5.856z, 5.685w,
˚ Lattice constant (A)
5.817
5.80x, 5.799y
Band gap (eV)
7.07
9.70aa, 11.25p
5.845w, 5.880w, 5.895w 6.90u, 7.55v, 11.306w, 8.806w, 11.185w, 20.941w
57ab 6.20ab,ac, 6.184s
61l 6.35ad, 6.26l, 6.05ae,
11.0p
6.094ae, 6.279ae, 5.990ae, 6.251ae, 6.265ae 11.30l, 7.49m
Bulk modulus (GPa) ˚ Lattice constant (A)
Band gap (eV)
53 6.14
7.12
a
Ref. [48] Ref. [53] c Ref. [54] d Ref. [62] e Ref. [63] f Ref. [55] g Ref. [56] h Ref. [57] i Ref. [58] j Ref. [9] k Ref. [13] l Ref. [64] m Ref. [12] n Ref. [34] o Ref. [67] p Ref. [52] q Ref. [66] r Ref. [68] s Ref. [3] t Ref. [59] u Ref. [61] v Ref. [10] w Ref. [65] x Ref. [14] y Ref. [38] z Ref. [7] aa Ref. [15] ab Ref. [4] ac Ref. [49] ad Ref. [46] ae Ref. [39] b
compounds, indicating the dynamical stability of the structure. A critical assessment of PDC and phonon DOS reveals that there are two regions in which phonon modes are distributed. The top region, which is due to the optical phonons, is mainly due to fluorine ion (F ) in all the three compounds. The dispersive longitudinal phonon mode is quite above and well separated from the other modes. The second region is mixed with both optical and acoustical phonon branches arising from the vibrations of alkali atoms mixed with fluorine atomic vibrations. The LO–TO splitting is in general large for all the three compounds; however this is maximum for CaF2. The mass ratio of anion to cation as well as the trend of mass for anion are clearly seen in the phonon dispersion curves. The phonon density of states, which is vital as it requires the computation of phonon modes in the entire Brillouin zone,
demonstrates all general features of PDC and a clear phonon gap can be easily seen in the phonon DOS plot. The gap increases from CaF2 to SrF2 to BaF2. This is due to the fact that the gap between LO and TO increases and hence the ionicity becomes stronger. The sharp peaks in the phonon DOS correspond to the flat modes of the phonon dispersion curves belonging to both optical and acoustical branches. The zone center phonon modes are of special importance since they can be measured by various techniques. The optical phonons at G-point belong to the following irreducible representations: 2Fu þ1Tg. Tg and Fu are the Raman and infrared active modes, respectively. We present in Table 2 the calculated frequencies for the longitudinal optical (LO), transverse optical (TO) and Raman mode along with available experimental and other theoretical calculated values. Table 2 reveals that the present values are
H.R. Soni et al. / Journal of Physics and Chemistry of Solids 72 (2011) 934–939
500
Frequency (cm-1)
Energy (eV)
10 0 -10 -20 Γ
X
L
X
W
DOS
400 300 200 100 0
10
Γ
X
W
Γ
L
PDOS (arb. units)
400
0
Frequency (cm-1)
Energy (eV)
937
-10
-20 Γ
X
L
X
W
300 200 100
DOS
0
10
Γ
X
W
Γ
L
PDOS (arb. units)
Γ
X
W
Γ
L
PDOS (arb. units)
5
400
-5 -10 -15 -20 -25
Γ
X
L
X
W
DOS
Fig. 1. Electronic band structure and corresponding total density of states for CaF2, SrF2 and BaF2.
frequency (cm-1)
Energy (eV)
0
300 200 100 0
Fig. 3. Phonon dispersion curves and phonon density of states for (a) CaF2, (b) SrF2 and (c) BaF2.
Fig. 2. Total charge density distribution in the (1 1 1) plane for CaF2.
generally in good agreement with the available experimental [22,23,27–30,33,35,36,39,40] and theoretical [22,24,26,31–33,37–39] data for CaF2, BaF2 and SrF2 compounds. The slight discrepancy in phonon frequencies with available experimental data in these compounds can be attributed to two facts. First there is a slight overestimation of lattice constant, resulting in underestimation of frequencies, and second is the temperature effect in the case of experimental data as it is obtained at room temperature and thus has certain anharmonic contribution in contrast to the theoretical results,
which are strictly harmonic [22]. The largest discrepancy with the experimental results is observed for the transverse optical (TO) branch. The frequency of TO phonons is grossly underestimated. This may be attributed to decreased electronic screening in the case of GGA, leading to higher inter atomic force constants or softness of the responsible bond. In particular the theoretical lattice constant is larger than the experimental one, resulting in lower theoretical frequencies.
4. Conclusion In conclusion, we have presented the structural, electronic and phonon properties of three fluorides, namely CaF2, SrF2 and BaF2, using the generalized gradient approximation of the density functional theory and obtained an overall consistent description of the phonon properties in these compounds. Our calculated results for the lattice constants, bulk modulus and electronic properties agree reasonably well with the earlier theoretical and
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Table 2 Calculated frequency of the phonon modes at some main symmetry points of the Brillouin zone for CaF2, SrF2 and BaF2. Compound
Parameters
Present calculation
Other calculation
Experimental
CaF2
LO (cm 1)
453.82
Raman (cm 1)
309.56
473.01b, 466.35d 319.37b, 322.21e, 322.17g, 327.46h
TO (cm 1)
225.96
472.22a, 486.56b, 473.13c, 447.75c 272.03a, 335.98b, 309.85b,327f 318.66c, 298.87c 253.12a, 279.61b, 241.87b,289i,j 286.95c, 227.08c
LO (cm 1) Raman (cm 1) TO (cm 1)
361.97 283.28 194.09
LO (cm 1) Raman (cm 1) TO (cm 1)
301.42 216.18 129.62
SrF2
BaF2
260.34b, 257.18h, 257.16e, 269.98k
285l 251m,j 351.46n,349o 247f,252.03n,257o 213q,j,199.34n,199.34o
344k,326d,p,330h, 241h,242p 187.5g,184d,189h
a
Ref. [26] Ref. [22] c Ref. [24] d Ref. [27] e Ref. [23] f Ref. [32] g Ref. [28] h Ref. [29] i Ref. [31] j Ref. [33] k Ref. [30] l Ref. [36] m Ref. [35] n Ref. [38] o Ref. [39] p Ref. [40] q Ref. [37] b
available experimental results. The computed band structure and electronic density of states are in agreement with earlier theoretical data for all the three compounds using our calculated lattice constants and electronic structures. Lattice dynamics of the three fluorides has been studied by employing a linear response approach based on the density functional perturbation theory. The phonon properties for these compounds also agree reasonably well with the available experimental data. The phonon properties of SrF2 are reported for the first time. Both the electronic and phonon properties calculations suggest that the compounds are ionic in nature and the ionic bonding is in the direction of X and F ion. The LO–TO splitting is maximum for CaF2, which suggests that CaF2 is the most ionic. These compounds have an indirect band gap.
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Ground state and lattice dynamical study of ionic conductors CaF2, SrF2 and BaF2 using density functional theory Himadri R. Soni a, Sanjeev K. Gupta a,b, Mina Talati c, Prafulla K. Jha a,n a
Department of Physics, Bhavnagar University, Bhavnagar 364022, India Dipartimento di Fisica dell’Universita degli studi di Modena e Reggio, Emilia and Centro S3, CNR-Istituto di Nanoscienze, via Campi 213/A, 41100 Modena, Italy c Institut f¨ ur Sicherheitsforschung, Forschungszentrum Rossendorf e.V.—Postfach 510119, 01314 Dresden, Germany b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 January 2011 Received in revised form 24 March 2011 Accepted 28 April 2011 Available online 12 May 2011
The present paper reports a comprehensive and complementary study on structural, electronic and phonon properties of face centered cubic fluorites, namely CaF2, BaF2 and SrF2, using first principles density functional calculations within the generalized gradient approximation. The calculated lattice constants and bulk modulus are in good agreement with available experimental data. The analysis of band structure and density of states confirms the ionic character for all the three fluorides. The phonon dispersion curves and corresponding phonon density of states obtained in the present work are consistent with the available experimental and other theoretical data. The LO–TO splitting is maximum for CaF2, which confirms that the ionicity is maximum in the case of CaF2. The phonon properties for SrF2 have been calculated for the first time. & 2011 Elsevier Ltd. All rights reserved.
Keywords: C. Ab initio calculations D. Equations-of-state D. Lattice dynamics D. Phonons
1. Introduction Ionic conductors are promising candidates for solid state electrolytes used in batteries. The fluorite CaF2 is a prototype ionic conductor showing a strong increase of conductivity with variation of external parameters such as temperature that saturates at 1420 K, where it becomes comparable to that of a matter salt [1,2]. In addition it possesses intrinsic optical properties, superior character at high temperature and is the subject of several experimental and theoretical studies in the recent past [3–20]. The alkaline-earth fluorides XF2(X¼Pb, Ca, Sr, Ba) generally crystallize to a cubic fluorite structure ðFm3mÞ consisting of a close-packed cubic alkali (X) lattice with fluorine (F ) occupying the tetrahedral sites. The ionic conduction arises in these compounds from the motional disorder in the fluorite sublattice, hopping over potential barriers and formation of various types of defects [21]. Details of ionic conduction mechanism however depend on the peculiarities of structure and dynamics as demonstrated by comparison with other ionic conductors, e.g. LaF3 [22]. Furthermore, phonon dispersion curves are an essential key ingredient for the calculation of specific heat, thermal expansion and vibrational entropy. In addition the phonons play an important role not only in the formation of various defects, hopping
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over potential barriers and disorder but also in several mechanical and optical properties. Therefore, a systematic characterisation of the phonon dispersion relations and phonon density of states (PDOS) of alkaline-earth fluorides is highly desirable. To the best of our knowledge there exist rich experimental and theoretical studies on phonons for one of the fluorides CaF2 [14,22–34], but for other fluorides such as SrF2 and BaF2 the situation is not so encouraging. The Raman and TO phonon modes for BaF2 have been reported in Refs. [1,3,4,27,30,32–40] and for SrF2 in Refs. [4,5,33–35]. There does not exist a single theoretical calculation for these alkaline-earth fluorides. Hence, a comprehensive theoretical study is not only essential to verify the observed experimental data but also to find a correlation between these compounds and to establish stability of the otherwise putative structure. The aim of the present work is to add to the general understanding of the ionic conduction through the knowledge of electronic and phonon properties using the ab initio pseudopotential method based on a generalized gradient approximation (GGA) of the density functional theory. The choice of GGA lies with the fact that the GGAs are intended to be an improvement on the conventional local density approximation (LDA) and indeed perform better in certain situations. As far as the success of Perdew–Burke–Ernzerhoff (PBE) exchange correlation functional for the prediction of phonon dispersion curves closure to experiments is concerned it is more or less of same quality as that of LDA [41], with exceptions in a few cases. The newly developed GW or hybrid functionals are more suitable to the electronic
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structure calculations and determination of the band gap but not for the lattice dynamical properties. It is also expected that the phonon calculations in SrF2 and BaF2 along with CaF2 will be of immense help to understand the parameters influencing the ionic conduction, whole origin is still to be clarified. Previous theoretical work [26] has been based on phenomenological and semiempirical potential leading to somewhat quantitatively unreliable results [24]. The large LO–TO splitting observed in these compounds suggests an advanced response density functional perturbation theory that incorporates macroscopic electric fields for the lattice dynamical calculations of these compounds. In addition these compounds, particularly SrF2, are becoming important and useful for the nanoparticles dispersed medium [42,43]. The importance of different processes of interaction between phonon modes is of considerable interest, particularly during the study of phonon assisted energy transfer processes [44]. Recently, it has been observed that the Eu2 þ doped glasses and glass ceramics containing SrF2 nanocrystals exhibited a much stronger ( 20 times) broad blue emission band and longer life time of excited state than glass. The reason for this intense blue luminescence in comparison to glass is attributed to the smaller multiphonon relaxation probability due to the lower phonon energy of host (SrF2) of about 290 cm 1. The phonon modes corresponding to this energy range in the SrF2 bulk crystal are due to the ionic movement. The blue-shift in the luminescence spectrum of SrF2 nanocrystals may arise due to the phonon confinement effect as was found in the case of CaF2 by Ricci et al. [45]. Therefore, it is important and the right time to investigate the phonon dispersion curves for this group of compounds. The only reported lattice dynamical calculation using first principles method is in good agreement with the experimental phonon dispersion and density of states [22]; however there are no similar studies on any other compounds of this group. Furthermore the electronic band structure calculation performed for the three fluorides will be useful in shedding some light on the excellent transmission properties of CaF2 and to see if there is any role of electronic properties in ionic conduction. The paper is organized as follows. The computational methodology is presented in Section 2. In Section 3, we present theoretical results and discussion. The results are concluded in Section 4.
2. Computational details To determine the electronic structure and phonon properties in CaF2, SrF2 and BaF2, we performed plane-wave and pseudopotential (both norm-conserving and ultrasoft) calculations using the Quantum ESPRESSO code [47]. The generalized gradient approximation (GGA) has been used for the exchange and correlation energy density function. A fully relativistic calculation is performed for core states, whereas the valence states are treated in a scalar relativistic scheme. In order to find an appropriate energy cutoff, the total energy as a function of the energy cutoff has been calculated for 23 different energy cutoffs ranging from 10 to 125 Ry. The total energy converges near the energy cutoffs of 39, 55 and 40 Ry for CaF2, SrF2 and BaF2, respectively. In the calculation, we fixed lattice constants 5.466 A˚ for CaF2, 5.80 A˚ for SrF2 and 6.20 A˚ for BaF2 structure as an experimental values [3,4,14,34,38,48–49]. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 5 Ry. Here, for the selfconsistency, the initial potential for the next iteration is constructed using a convergence stabilization scheme. The number of sampling k-points used in the Brillouin zone (BZ) summation of the electronic density and total energy was increased till the total
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energy converged to the desirable tolerance. Thirty six special k-points in the irreducible BZ are sufficient to achieve the convergence for the total energy. To generate a (6 6 6) mesh in BZ, the scheme of Monkhorst–Pack [50] has been used. The crystal structure and associated equilibrium lattice constants for all the three fluorides have been obtained by minimizing the calculated total energies as a function of lattice constant. Furthermore, to calculate the lattice dynamics of CaF2, SrF2 and BaF2, we have used the Quantum ESPRESSO [47] code based upon density functional perturbation theory [DFPT] [51] implemented within plane wave pseudopotentials framework. In this method the dynamical matrix, which provides information on lattice dynamics of the system, can be obtained from the ground state electron charge of the nuclear geometry. The kinetic energy cutoff and numbers of k-points mentioned above are found to yield phonon frequencies converged to within 2–5 cm 1.
3. Results and discussion The theoretical ground state constants such as lattice parameters and bulk modulus of the three fluorides CaF2, SrF2 and BaF2 are obtained using the method discussed in the above section and listed in Table 1, which also includes the available experimental and other theoretical data for comparison. It is clear that the present theoretical values agree reasonably well with the experimental data [4,14,38,48,49,52–59]. It is generally observed that GGA yields higher lattice constant. The total energy versus volume curve is fitted to the Murnaghan equation of state [60] to obtain the bulk modulus value as given below: " # 0 B0 V0 B0 V ðV0 =VÞB0 EðVÞ ¼ E0 0 þ 0 þ 1 ð1Þ B0 B0 1 B00 1 where B0 and B0 0 are the bulk modulus and its derivative, respectively, E0 is the ground state total energy and V0 is the volume. The bulk modulus so obtained is presented in Table 1 and compared with the available experimental and other theoretical data. There is an excellent agreement between the present and experimental data, while there is significant variation within the previous theoretical data [61–65]. The self-consistent band structures along with the electronic density of states (DOS) for CaF2, SrF2 and BaF2 are presented in Fig. 1. The DOS reflects all features of the band structure. The calculated band gap for CaF2, SrF2 and BaF2 is 7.85, 7.07 and 7.12 eV, respectively, in agreement with earlier calculations [9,10,13,61,62,64–68]. However the present band gap values are lower than the experimental values [15,38], which may be due to the use of GGA, which normally underestimates the band gap by a factor of about two. This discrepancy is due to the fact that DFT Kohn–Sham states do not correctly take into account the quasiparticle self-energy in our calculations [69]. We have also calculated the charge density distributions for CaF2, SrF2 and BaF2 (presented only for CaF2 in Fig. 2), which show that these compounds are ionic in nature and there exists an ionic bond along the direction of alkali and florine ions. In order to get an idea about the behavior of phonons and the role played by them in ionic conductivity in these fluorides, the phonon dispersion curves (PDC) along with phonon density of states (DOS) for CaF2, SrF2 and BaF2 in the cubic fluorite structures ðFm3mÞ are calculated using the method discussed above and presented in Fig. 3. The shape of the phonon spectrum for all three materials is similar. The unit cell of a cubic structure with space group Fm3m presents nine degrees of freedom, six of them corresponding to optical modes and three others to acoustic modes. Fig. 3 reveals that all calculated phonon branches in the whole Brillouin zone yield positive frequencies for all three
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Table 1 Calculated equilibrium lattice constants, bulk modulus and band gap for CaF2, BaF2 and SrF2. Parameters CaF2
SrF2
BaF2
Bulk modulus (GPa)
Present work
Experimental
Other work
77.2
87 7 5a, 82.07 0.7b, 82.0c, 81.0 71.2f,81.1g, 84.7h, 81.0i
79.54d, 84.7e, 82.7e, 103j,k, 77k, 91l, 82.14m, 85l 5.4872c, 5.563e,
˚ Lattice constant (A)
5.501
5.466a, 5.4630n
Band gap (eV)
7.85
11.8p
Bulk modulus (GPa)
64.3
69.0s, 74.53t
5.444e, 5.33o, 5.50l 7.39d, 7.24j, 7.27k, 7.01k, 7.7q, 6.85o, 11.5o, 11.38r, 10.87l 66.2u, 68.8v, 117.54w, 77.11w, 79.52w, 77.35w 5.846u, 5.856z, 5.685w,
˚ Lattice constant (A)
5.817
5.80x, 5.799y
Band gap (eV)
7.07
9.70aa, 11.25p
5.845w, 5.880w, 5.895w 6.90u, 7.55v, 11.306w, 8.806w, 11.185w, 20.941w
57ab 6.20ab,ac, 6.184s
61l 6.35ad, 6.26l, 6.05ae,
11.0p
6.094ae, 6.279ae, 5.990ae, 6.251ae, 6.265ae 11.30l, 7.49m
Bulk modulus (GPa) ˚ Lattice constant (A)
Band gap (eV)
53 6.14
7.12
a
Ref. [48] Ref. [53] c Ref. [54] d Ref. [62] e Ref. [63] f Ref. [55] g Ref. [56] h Ref. [57] i Ref. [58] j Ref. [9] k Ref. [13] l Ref. [64] m Ref. [12] n Ref. [34] o Ref. [67] p Ref. [52] q Ref. [66] r Ref. [68] s Ref. [3] t Ref. [59] u Ref. [61] v Ref. [10] w Ref. [65] x Ref. [14] y Ref. [38] z Ref. [7] aa Ref. [15] ab Ref. [4] ac Ref. [49] ad Ref. [46] ae Ref. [39] b
compounds, indicating the dynamical stability of the structure. A critical assessment of PDC and phonon DOS reveals that there are two regions in which phonon modes are distributed. The top region, which is due to the optical phonons, is mainly due to fluorine ion (F ) in all the three compounds. The dispersive longitudinal phonon mode is quite above and well separated from the other modes. The second region is mixed with both optical and acoustical phonon branches arising from the vibrations of alkali atoms mixed with fluorine atomic vibrations. The LO–TO splitting is in general large for all the three compounds; however this is maximum for CaF2. The mass ratio of anion to cation as well as the trend of mass for anion are clearly seen in the phonon dispersion curves. The phonon density of states, which is vital as it requires the computation of phonon modes in the entire Brillouin zone,
demonstrates all general features of PDC and a clear phonon gap can be easily seen in the phonon DOS plot. The gap increases from CaF2 to SrF2 to BaF2. This is due to the fact that the gap between LO and TO increases and hence the ionicity becomes stronger. The sharp peaks in the phonon DOS correspond to the flat modes of the phonon dispersion curves belonging to both optical and acoustical branches. The zone center phonon modes are of special importance since they can be measured by various techniques. The optical phonons at G-point belong to the following irreducible representations: 2Fu þ1Tg. Tg and Fu are the Raman and infrared active modes, respectively. We present in Table 2 the calculated frequencies for the longitudinal optical (LO), transverse optical (TO) and Raman mode along with available experimental and other theoretical calculated values. Table 2 reveals that the present values are
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500
Frequency (cm-1)
Energy (eV)
10 0 -10 -20 Γ
X
L
X
W
DOS
400 300 200 100 0
10
Γ
X
W
Γ
L
PDOS (arb. units)
400
0
Frequency (cm-1)
Energy (eV)
937
-10
-20 Γ
X
L
X
W
300 200 100
DOS
0
10
Γ
X
W
Γ
L
PDOS (arb. units)
Γ
X
W
Γ
L
PDOS (arb. units)
5
400
-5 -10 -15 -20 -25
Γ
X
L
X
W
DOS
Fig. 1. Electronic band structure and corresponding total density of states for CaF2, SrF2 and BaF2.
frequency (cm-1)
Energy (eV)
0
300 200 100 0
Fig. 3. Phonon dispersion curves and phonon density of states for (a) CaF2, (b) SrF2 and (c) BaF2.
Fig. 2. Total charge density distribution in the (1 1 1) plane for CaF2.
generally in good agreement with the available experimental [22,23,27–30,33,35,36,39,40] and theoretical [22,24,26,31–33,37–39] data for CaF2, BaF2 and SrF2 compounds. The slight discrepancy in phonon frequencies with available experimental data in these compounds can be attributed to two facts. First there is a slight overestimation of lattice constant, resulting in underestimation of frequencies, and second is the temperature effect in the case of experimental data as it is obtained at room temperature and thus has certain anharmonic contribution in contrast to the theoretical results,
which are strictly harmonic [22]. The largest discrepancy with the experimental results is observed for the transverse optical (TO) branch. The frequency of TO phonons is grossly underestimated. This may be attributed to decreased electronic screening in the case of GGA, leading to higher inter atomic force constants or softness of the responsible bond. In particular the theoretical lattice constant is larger than the experimental one, resulting in lower theoretical frequencies.
4. Conclusion In conclusion, we have presented the structural, electronic and phonon properties of three fluorides, namely CaF2, SrF2 and BaF2, using the generalized gradient approximation of the density functional theory and obtained an overall consistent description of the phonon properties in these compounds. Our calculated results for the lattice constants, bulk modulus and electronic properties agree reasonably well with the earlier theoretical and
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Table 2 Calculated frequency of the phonon modes at some main symmetry points of the Brillouin zone for CaF2, SrF2 and BaF2. Compound
Parameters
Present calculation
Other calculation
Experimental
CaF2
LO (cm 1)
453.82
Raman (cm 1)
309.56
473.01b, 466.35d 319.37b, 322.21e, 322.17g, 327.46h
TO (cm 1)
225.96
472.22a, 486.56b, 473.13c, 447.75c 272.03a, 335.98b, 309.85b,327f 318.66c, 298.87c 253.12a, 279.61b, 241.87b,289i,j 286.95c, 227.08c
LO (cm 1) Raman (cm 1) TO (cm 1)
361.97 283.28 194.09
LO (cm 1) Raman (cm 1) TO (cm 1)
301.42 216.18 129.62
SrF2
BaF2
260.34b, 257.18h, 257.16e, 269.98k
285l 251m,j 351.46n,349o 247f,252.03n,257o 213q,j,199.34n,199.34o
344k,326d,p,330h, 241h,242p 187.5g,184d,189h
a
Ref. [26] Ref. [22] c Ref. [24] d Ref. [27] e Ref. [23] f Ref. [32] g Ref. [28] h Ref. [29] i Ref. [31] j Ref. [33] k Ref. [30] l Ref. [36] m Ref. [35] n Ref. [38] o Ref. [39] p Ref. [40] q Ref. [37] b
available experimental results. The computed band structure and electronic density of states are in agreement with earlier theoretical data for all the three compounds using our calculated lattice constants and electronic structures. Lattice dynamics of the three fluorides has been studied by employing a linear response approach based on the density functional perturbation theory. The phonon properties for these compounds also agree reasonably well with the available experimental data. The phonon properties of SrF2 are reported for the first time. Both the electronic and phonon properties calculations suggest that the compounds are ionic in nature and the ionic bonding is in the direction of X and F ion. The LO–TO splitting is maximum for CaF2, which suggests that CaF2 is the most ionic. These compounds have an indirect band gap.
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