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Electronic structure and diffusion paths of Ag ions in rocksalt structured AgI
Solid State Ionics 178 (2007) 1023 – 1026 www.elsevier.com/locate/ssi
Electronic structure and diffusion paths of Ag ions in rocksalt structured AgI S. Ono a,⁎, M. Kobayashi b , S. Kashida c , T. Ohachi d a
Faculty of Engineering, Doshisha University, Kyoto 610-0321, Japan Department of Physics, Niigata University, Niigata 950-2181, Japan Department of Environmental Science, Niigata University, Niigata 950-2181, Japan Department of Electrical Engineering, Doshisha University, Kyoto 610-0321, Japan b
c d
Received 16 December 2006; received in revised form 30 March 2007; accepted 1 May 2007
Abstract The electronic structure and diffusion paths of Ag ions in the rocksalt structured phase of AgI (rs-AgI) are studied based on the full-potential linear-muffin-tin-orbital (FP-LMTO) method. Ag ions in the rs-AgI start to migrate along the 〈111〉 direction due to the low energy barrier, as is well known in the cases of AgBr and AgCl. The high ionic conductivity in rs-AgI might be expected from the viewpoint of the empirical overlap parameters between Ag 4d electrons and halogen p electrons. © 2007 Elsevier B.V. All rights reserved. Keywords: Rocksalt structured AgI; Electronic structure; Superionic conductor; Frenkel defect; FP-LMTO method
1. Introduction AgI is known as a typical superionic conductor [1]. The superionic phase of AgI appears at 420 K. The value of the ionic conductivity is as high as about 1 Ω− 1 cm− 1. This value is comparable to that of liquid electrolytes. The structure of superionic conductor α-AgI possesses a bcc lattice of iodine ions, while Ag ions are distributed over 42 crystallographic sites: 6(b) octahedral, 12(d) tetrahedral, and 24(h) trigonal. The ion diffusion paths of Ag ions between tetrahedral sites have been shown to occur in the 〈110〉 direction, i.e., via the trigonal sites. The ambient pressure data for AgI showed that the sample was initially a mixture of the wurtzite and the zincblende structured phases (AgI-II and AgI-II′, respectively) [2]. In the Phillips scale [3,4], AgI has the ionicity fi = 0.770. This is very close to the critical value fc = 0.785 which marks the idealized boundary between compounds with fourfold-coordinated structures (zincblende, wurtzite) and those with sixfold coordination (rocksalt). Furthermore, AgI might be expected to transform from the zincblende to the rocksalt under hydrostatic pressures. The structural changes of silver halides induced by hydrostatic ⁎ Corresponding author. E-mail address: [email protected] (S. Ono). 0167-2738/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2007.05.001
pressures were investigated using angle-dispersive X-ray diffraction with an image-plate device [2]. Available experimental evidences indicate that the transition from fourfold to sixfold coordination occurs. For pressures between 0.4 and 10 GPa, a rocksalt-structure phase of fcc AgI is stable at room temperature [1]. This fact is in contrast to the fact that other silver halides AgF, AgCl, and AgBr adopt the octahedrally coordinated rocksalt structure at ambient pressure and temperature. It was indicated that Frenkel defects dominated the electrical conductivities in AgCl and AgBr [5]. It is considered that Frenkel defects dominate also in rs-AgI. According to the results of the molecular dynamics (MD) simulations of AgI under hydrostatic pressures, the gradual onset of the highly conducting state is accompanied by an increasing fraction of dynamical Frenkel defects, a peak in the specific heat and anomalous behavior of the lattice expansion [6]. The measurement of the electrical conductivity of rs-AgI showed the value of the ionic conductivity was approximately 1 Ω− 1 cm− 1, i.e., only slightly less than the electrical conductivity in the α phase [1]. Thus there are two solid electrolyte phases in AgI: α phase and high temperature part of the fcc phase. The main difference between the two is that the iodine sublattice is a body-centered cubic for the α phase, while it is a face-centered cubic for the fcc phase. The conductivity is somewhat lower in the fcc phase than
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in the bcc phase and, in fact, most solid electrolytes that have the fcc structure have lower conductivities than those that have the bcc structure. Kleppmann et al. [7,8] introduced the quadrupolar deformabilities of Ag ions in order to elucidate the mechanism of the ionic conductivities and the phonon dispersion relations in silver halides. Also, they estimated the decrease of the activation energies owing to the Ag deformabilities, which are ΔE (AgBr) ∼ − 0.37 eV and ΔE(AgCl) ∼ − 0.22 eV. These results compare favorably with the differences of the activation energy between alkali halides and silver halides of about 0.3 ∼ 0.4 eV. They suggested that the quadrupolar deformabilities of Ag ions led to the decrease of the activation energies for ionic conductions in silver halides. In this paper, we investigate the mechanism of ionic conductivity in rs-AgI on the basis of the electronic structure theory. We study the electronic structure and the variations of the total energies along ion diffusion paths using the FP-LMTO method. We will discuss the origin of the superionicity by comparing the results conducted by making use of the empirical overlap parameters between Ag 4d electrons and halogen p electrons.
Fig. 2. Total energy variations for AgCl, AgBr and rs-AgI along the bold arrow shown in Fig. 1.
exchange energy. The convergence is assumed if the selfconsistent total energy difference between subsequent iterations is less than 10− 6 Ry. The density of states (DOS) is calculated using the tetrahedron method, where 6 × 6 × 6 division in the kspace is used. Details of the calculation methods are described in Ref. [9]. 3. Results and discussion
2. Calculation method Band calculations are carried out using the full-potential linear muffin-tin-orbital program LMTART, where the local density approximation (LDA) is used [9]. In this calculation, the space is divided into muffin-tin spheres and the interstitial region. Within the muffin-tin spheres, the charge density and potential are expanded using spherical harmonics, and in the interstitial region they are expanded in plane waves. The initial charge density is taken as a superposition of the neutral atomic charge densities. The total energy is estimated as the sum of the following terms: Etot = Tval + Tcor + Eel + Exc where Tval and Tcor are the kinetic energies of the valence and core electrons and Eel is the electrostatic energy including electron–electron, electron–nucleus and nucleus–nucleus interactions and Exc is the
Fig. 1. Schematic view of ion diffusion paths. Two arrows show ion diffusion paths from the octahedral site to the cation vacancy. The bold arrow shows a Ag ion starts to migrate along the 〈111〉 direction toward the cation vacancy via the center of the tetrahedron, while the thin arrow shows a Ag ion starts to migrate directly toward the cation vacancy.
Recently, many studies along the spirit of this study have been done to elucidate the mechanism of the ionic conduction. For example, the electronic structure of the ternary silver compound Ag3SI was studied in order to clarify the microscopic origin of the structural phase transition and the fast ionic conduction [10]. The electronic structure and the Li diffusion paths in the lithium doped lanthanum titanate were also studied [11]. At first, we evaluate the energy required for the phase transition from the zincblende structure to the rocksalt-structure phase fcc AgI. We estimate the value varying the lattice constant of the rocksalt-structure phase fcc AgI until the total energy difference becomes zero. The ambient pressure data for AgI showed that the sample was initially a mixture of the wurtzite and the zincblende structured phases (AgI-II and AgI-II′, respectively) [2]. However, we adopt the zincblende structure when we estimate the value of the total energy. If we take a lattice constant of the rocksalt-structure phase fcc AgI as about a = 6.32 Å, the energy difference becomes zero between that of the rocksalt-structure phase fcc AgI and that of the zincblende
Fig. 3. Total energy variations for AgCl, AgBr and rs-AgI along the 〈110〉-axis.
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by the experimental results for E(AgCl) = 0.28 eV [7], E(AgBr) = 0.33 eV [7] and E(rs-AgI) = 0.3 eV [1], respectively. If we take an ion diffusion path directly from one site surrounded by the octahedron of anions to a vacant site, the energy barriers for mobile ions are larger than those along the bold arrow shown in Fig. 1. These features are shown in Fig. 3. The energy maximum at the position 0.5 corresponds to the face-centered position of the base, while the position 1.0 corresponds to the octahedral site. The values of energy barrier per one mobile ion are as high as about 1.2∼ 1.4 eV. The direct path is too high in energy. The low energy path through F (face of the octahedron) and T (center of the tetrahedron) is favor for mobile ions. We can also confirm the low minimum site energies at the tetrahedral site from Fig. 2. These results are reasonable for AgCl, AgBr and rs-AgI with many Frenkel defects. Our results suggest that the tetrahedral sites are enough low energy sites for mobile Ag ions and regarded as the stable sites for Frenkel defect formation. Figs. 4 and 5 show the band structure and the density of states (DOS) of rs-AgI, respectively. The band around − 13 eV is derived from I 5s state. The bands around from − 4.5 eV to 0 eV are derived from Ag 4d and I 5p states. The band structure of rsAgI is similar to that obtained by the full-potential linearized augmented plane-wave (FP-LAPW) method [13].
Fig. 4. Band structure of rs-AgI. The solid line denotes the Fermi energy.
structure. The value of the lattice constant is close to that obtained experimentally [12]. This result indicates that the phase transition from the zincblende structured phases to the rocksalt-structure phase fcc AgI is relatively easy to occur under hydrostatic pressures. Next, we investigate diffusion paths of Ag ions. Fig. 1 shows the schematic view of ion diffusion paths. Two arrows show ion diffusion paths from the octahedral site to the cation vacancy. The bold arrow shows a Ag ion starts to migrate along the 〈111〉 direction toward the cation vacancy via the center of the tetrahedron, while the thin arrow shows a Ag ion starts to migrate directly toward the cation vacancy. Fig. 2 shows the total energy variations for AgCl, AgBr and rs-AgI along the bold arrow shown in Fig. 1. The position 0.5 corresponds to the center of the tetrahedron, while the position 1.0 corresponds to the octahedral site. The results indicate the total energy maximum appears at the point r = (a/2)(1/3,1/3,1/3) for all compounds. This position corresponds to the center of the numbered anion triangle from 1 to 3, where the closest approach occurs. The another energy maximum at the position about 0.67 corresponds to the center of the numbered anion triangle from 2 to 4. It is also suggested the values of energy barrier per one mobile ion are as high as about 0.3 eV for all compounds. These results can compare well with the activation energies obtained
Fig. 5. Density of states of rs-AgI. The solid line denotes the Fermi energy.
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It has been recently pointed out that the p–d hybridization can lead to a local lattice instability with double well (DW) formation [14]. It seems to be possible to conclude that the presence of p−d electronic state hybridization in the center of Brillouin-zone is an essential prerequisite to the formation of a local DW for some part of crystal constituent ions. The proper distortion of symmetry can be reached by applying an external pressure. The neutron diffraction measurements show the structural evidence for a fast-ion transition in the high pressure rocksalt phase of AgI being observed at the temperatures high enough to provide the effective mixing of p−d states which would be forbidden by symmetry selection rules at lower temperature [12]. AgBr and AgCl can be considered as ambient pressure counterparts of rs-AgI. They both possess the rocksalt structure and show an anomalous increase in conductivity as the temperature is raised. It seems they both are apt to become the superionic conductors. However, there is no evidence for a fastion transition in theses materials and it has been proposed that the transition would exist in AgBr if the anion sublattice did not melt before the fast-ion phase was established. We have studied the relationship between the difference of p−d hybridizations and the activation energies in noble metal halides [15,16]. We have shown that the strength of p−d hybridization for AgX (X = halogen) is smaller than that for CuX. Harrison [17] gave the overlap parameters Vpdσ,π(ι) between Ag 4d electrons and halogen p electrons as follows: 3=2
Vpdr;p ðiÞ ¼ gpdr;p
ћ2 r d ; me i7=2
where ι is the bonding length, ηpdσ = − 2.95, ηpdπ = 1.36, rd is 0.89 Å for Ag, me is the electron mass and ћ is the Planck constant. If we follow the expressions of the overlap parameters given by Harrison, p−d hybridizations in rs-AgI are smaller than those in AgCl and AgBr because of the lattice constants of AgCl (a = 5.54 Å), AgBr (a = 5.77 Å) and rs-AgI (a = 6.1596 Å) [12]. The high ionic conductivity in rs-AgI is also expected on the basis of the empirical parameters. 4. Conclusion The electronic structure and ion diffusion paths in rs-AgI have been investigated using the FP-LMTO method. The low energy path through F (face of the octahedron) and T (center of the tetrahedron) is favor for mobile ions. This is consistent with
the results which were well known in the cases of AgBr and AgCl. We have also confirmed the low minimum site energies at the tetrahedral site. These results are reasonable for AgCl, AgBr and rs-AgI with many Frenkel defects. Our results suggest that the tetrahedral sites are enough low energy sites for mobile Ag ions and regarded as the stable sites for Frenkel defect formation. The band structure has been also calculated. The bands around from − 4.5 eV to 0 eV have been derived from Ag 4d and I 5p states. These electronic states have been hybridized and have played important roles to elucidate the mechanism of the ionic conductivity in rs-AgI. If we follow the expressions of the overlap parameters between Ag 4d electrons and halogen p electrons given by Harrison, p−d hybridizations in rs-AgI are smaller than those in AgCl and AgBr because of the long lattice constant for rs-AgI, compared with that for AgCl and AgBr. The strength of p−d hybridization was discussed in relation to the activation energy for the ionic conduction in our recent studies [15,16]. The high ionic conductivity in rs-AgI is also expected as shown in the cases of AgBr and AgCl on the basis of the empirical parameters. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
B.E. Mellander, Phys. Rev., B 26 (1982) 5886. S. Hull, D.A. Keen, Phys. Rev., B 59 (1999) 750. J.C. Phillips, Rev. Mod. Phys. 42 (1970) 317. J.C. Phillips, Bonds and Bands in Semiconductors, Academic, New York, 1973. J.K. Aboagye, R.J. Friauf, Phys. Rev., B 11 (1975) 1654. D.A. Keen, S. Hull, A.C. Barnes, P. Berastegui, W.A. Chichton, P.A. Madden, M.G. Tucker, M. Wilson, Phys. Rev., B 68 (2003) 14117. W.G. Kleppmann, H. Bilz, Commun. Phys. B 1 (1976) 105. W.G. Kleppmann, W. Weber, Phys. Rev., B 20 (1979) 1669. S.Y. Savrasov, Phys. Rev., B 54 (1996) 16470. W. Shimosaka, S. Kashida, M. Kobayashi, Solid State Ionics 176 (2005) 349. S. Ono, Y. Seki, S. Kashida, M. Kobayashi, Solid State Ionics 177 (2006) 1145. D.A. Keen, S. Hull, W. Hayes, N.J.G. Gardner, Phys. Rev. Lett. 77 (1996) 4914. C.M.I. Okoye, Phys. Status Solidi 234 (2002) 580. A. Rakitin, M. Kobayashi, Phys. Rev., B 53 (1996) 3088. S. Ono, M. Kobayashi, H. Iyetomi, T. Tomoyose, Solid State Ionics 139 (2001) 249. M. Kobayashi, H. Iyetomi, S. Ono, T. Tomoyose, Int. J. Mod. Phys. B 15 (2001) 678. W.A. Harrison, Electronic Structure and Properties of Solids, Freeman, San Francisco, 1980.
Electronic structure and diffusion paths of Ag ions in rocksalt structured AgI S. Ono a,⁎, M. Kobayashi b , S. Kashida c , T. Ohachi d a
Faculty of Engineering, Doshisha University, Kyoto 610-0321, Japan Department of Physics, Niigata University, Niigata 950-2181, Japan Department of Environmental Science, Niigata University, Niigata 950-2181, Japan Department of Electrical Engineering, Doshisha University, Kyoto 610-0321, Japan b
c d
Received 16 December 2006; received in revised form 30 March 2007; accepted 1 May 2007
Abstract The electronic structure and diffusion paths of Ag ions in the rocksalt structured phase of AgI (rs-AgI) are studied based on the full-potential linear-muffin-tin-orbital (FP-LMTO) method. Ag ions in the rs-AgI start to migrate along the 〈111〉 direction due to the low energy barrier, as is well known in the cases of AgBr and AgCl. The high ionic conductivity in rs-AgI might be expected from the viewpoint of the empirical overlap parameters between Ag 4d electrons and halogen p electrons. © 2007 Elsevier B.V. All rights reserved. Keywords: Rocksalt structured AgI; Electronic structure; Superionic conductor; Frenkel defect; FP-LMTO method
1. Introduction AgI is known as a typical superionic conductor [1]. The superionic phase of AgI appears at 420 K. The value of the ionic conductivity is as high as about 1 Ω− 1 cm− 1. This value is comparable to that of liquid electrolytes. The structure of superionic conductor α-AgI possesses a bcc lattice of iodine ions, while Ag ions are distributed over 42 crystallographic sites: 6(b) octahedral, 12(d) tetrahedral, and 24(h) trigonal. The ion diffusion paths of Ag ions between tetrahedral sites have been shown to occur in the 〈110〉 direction, i.e., via the trigonal sites. The ambient pressure data for AgI showed that the sample was initially a mixture of the wurtzite and the zincblende structured phases (AgI-II and AgI-II′, respectively) [2]. In the Phillips scale [3,4], AgI has the ionicity fi = 0.770. This is very close to the critical value fc = 0.785 which marks the idealized boundary between compounds with fourfold-coordinated structures (zincblende, wurtzite) and those with sixfold coordination (rocksalt). Furthermore, AgI might be expected to transform from the zincblende to the rocksalt under hydrostatic pressures. The structural changes of silver halides induced by hydrostatic ⁎ Corresponding author. E-mail address: [email protected] (S. Ono). 0167-2738/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2007.05.001
pressures were investigated using angle-dispersive X-ray diffraction with an image-plate device [2]. Available experimental evidences indicate that the transition from fourfold to sixfold coordination occurs. For pressures between 0.4 and 10 GPa, a rocksalt-structure phase of fcc AgI is stable at room temperature [1]. This fact is in contrast to the fact that other silver halides AgF, AgCl, and AgBr adopt the octahedrally coordinated rocksalt structure at ambient pressure and temperature. It was indicated that Frenkel defects dominated the electrical conductivities in AgCl and AgBr [5]. It is considered that Frenkel defects dominate also in rs-AgI. According to the results of the molecular dynamics (MD) simulations of AgI under hydrostatic pressures, the gradual onset of the highly conducting state is accompanied by an increasing fraction of dynamical Frenkel defects, a peak in the specific heat and anomalous behavior of the lattice expansion [6]. The measurement of the electrical conductivity of rs-AgI showed the value of the ionic conductivity was approximately 1 Ω− 1 cm− 1, i.e., only slightly less than the electrical conductivity in the α phase [1]. Thus there are two solid electrolyte phases in AgI: α phase and high temperature part of the fcc phase. The main difference between the two is that the iodine sublattice is a body-centered cubic for the α phase, while it is a face-centered cubic for the fcc phase. The conductivity is somewhat lower in the fcc phase than
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S. Ono et al. / Solid State Ionics 178 (2007) 1023–1026
in the bcc phase and, in fact, most solid electrolytes that have the fcc structure have lower conductivities than those that have the bcc structure. Kleppmann et al. [7,8] introduced the quadrupolar deformabilities of Ag ions in order to elucidate the mechanism of the ionic conductivities and the phonon dispersion relations in silver halides. Also, they estimated the decrease of the activation energies owing to the Ag deformabilities, which are ΔE (AgBr) ∼ − 0.37 eV and ΔE(AgCl) ∼ − 0.22 eV. These results compare favorably with the differences of the activation energy between alkali halides and silver halides of about 0.3 ∼ 0.4 eV. They suggested that the quadrupolar deformabilities of Ag ions led to the decrease of the activation energies for ionic conductions in silver halides. In this paper, we investigate the mechanism of ionic conductivity in rs-AgI on the basis of the electronic structure theory. We study the electronic structure and the variations of the total energies along ion diffusion paths using the FP-LMTO method. We will discuss the origin of the superionicity by comparing the results conducted by making use of the empirical overlap parameters between Ag 4d electrons and halogen p electrons.
Fig. 2. Total energy variations for AgCl, AgBr and rs-AgI along the bold arrow shown in Fig. 1.
exchange energy. The convergence is assumed if the selfconsistent total energy difference between subsequent iterations is less than 10− 6 Ry. The density of states (DOS) is calculated using the tetrahedron method, where 6 × 6 × 6 division in the kspace is used. Details of the calculation methods are described in Ref. [9]. 3. Results and discussion
2. Calculation method Band calculations are carried out using the full-potential linear muffin-tin-orbital program LMTART, where the local density approximation (LDA) is used [9]. In this calculation, the space is divided into muffin-tin spheres and the interstitial region. Within the muffin-tin spheres, the charge density and potential are expanded using spherical harmonics, and in the interstitial region they are expanded in plane waves. The initial charge density is taken as a superposition of the neutral atomic charge densities. The total energy is estimated as the sum of the following terms: Etot = Tval + Tcor + Eel + Exc where Tval and Tcor are the kinetic energies of the valence and core electrons and Eel is the electrostatic energy including electron–electron, electron–nucleus and nucleus–nucleus interactions and Exc is the
Fig. 1. Schematic view of ion diffusion paths. Two arrows show ion diffusion paths from the octahedral site to the cation vacancy. The bold arrow shows a Ag ion starts to migrate along the 〈111〉 direction toward the cation vacancy via the center of the tetrahedron, while the thin arrow shows a Ag ion starts to migrate directly toward the cation vacancy.
Recently, many studies along the spirit of this study have been done to elucidate the mechanism of the ionic conduction. For example, the electronic structure of the ternary silver compound Ag3SI was studied in order to clarify the microscopic origin of the structural phase transition and the fast ionic conduction [10]. The electronic structure and the Li diffusion paths in the lithium doped lanthanum titanate were also studied [11]. At first, we evaluate the energy required for the phase transition from the zincblende structure to the rocksalt-structure phase fcc AgI. We estimate the value varying the lattice constant of the rocksalt-structure phase fcc AgI until the total energy difference becomes zero. The ambient pressure data for AgI showed that the sample was initially a mixture of the wurtzite and the zincblende structured phases (AgI-II and AgI-II′, respectively) [2]. However, we adopt the zincblende structure when we estimate the value of the total energy. If we take a lattice constant of the rocksalt-structure phase fcc AgI as about a = 6.32 Å, the energy difference becomes zero between that of the rocksalt-structure phase fcc AgI and that of the zincblende
Fig. 3. Total energy variations for AgCl, AgBr and rs-AgI along the 〈110〉-axis.
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by the experimental results for E(AgCl) = 0.28 eV [7], E(AgBr) = 0.33 eV [7] and E(rs-AgI) = 0.3 eV [1], respectively. If we take an ion diffusion path directly from one site surrounded by the octahedron of anions to a vacant site, the energy barriers for mobile ions are larger than those along the bold arrow shown in Fig. 1. These features are shown in Fig. 3. The energy maximum at the position 0.5 corresponds to the face-centered position of the base, while the position 1.0 corresponds to the octahedral site. The values of energy barrier per one mobile ion are as high as about 1.2∼ 1.4 eV. The direct path is too high in energy. The low energy path through F (face of the octahedron) and T (center of the tetrahedron) is favor for mobile ions. We can also confirm the low minimum site energies at the tetrahedral site from Fig. 2. These results are reasonable for AgCl, AgBr and rs-AgI with many Frenkel defects. Our results suggest that the tetrahedral sites are enough low energy sites for mobile Ag ions and regarded as the stable sites for Frenkel defect formation. Figs. 4 and 5 show the band structure and the density of states (DOS) of rs-AgI, respectively. The band around − 13 eV is derived from I 5s state. The bands around from − 4.5 eV to 0 eV are derived from Ag 4d and I 5p states. The band structure of rsAgI is similar to that obtained by the full-potential linearized augmented plane-wave (FP-LAPW) method [13].
Fig. 4. Band structure of rs-AgI. The solid line denotes the Fermi energy.
structure. The value of the lattice constant is close to that obtained experimentally [12]. This result indicates that the phase transition from the zincblende structured phases to the rocksalt-structure phase fcc AgI is relatively easy to occur under hydrostatic pressures. Next, we investigate diffusion paths of Ag ions. Fig. 1 shows the schematic view of ion diffusion paths. Two arrows show ion diffusion paths from the octahedral site to the cation vacancy. The bold arrow shows a Ag ion starts to migrate along the 〈111〉 direction toward the cation vacancy via the center of the tetrahedron, while the thin arrow shows a Ag ion starts to migrate directly toward the cation vacancy. Fig. 2 shows the total energy variations for AgCl, AgBr and rs-AgI along the bold arrow shown in Fig. 1. The position 0.5 corresponds to the center of the tetrahedron, while the position 1.0 corresponds to the octahedral site. The results indicate the total energy maximum appears at the point r = (a/2)(1/3,1/3,1/3) for all compounds. This position corresponds to the center of the numbered anion triangle from 1 to 3, where the closest approach occurs. The another energy maximum at the position about 0.67 corresponds to the center of the numbered anion triangle from 2 to 4. It is also suggested the values of energy barrier per one mobile ion are as high as about 0.3 eV for all compounds. These results can compare well with the activation energies obtained
Fig. 5. Density of states of rs-AgI. The solid line denotes the Fermi energy.
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It has been recently pointed out that the p–d hybridization can lead to a local lattice instability with double well (DW) formation [14]. It seems to be possible to conclude that the presence of p−d electronic state hybridization in the center of Brillouin-zone is an essential prerequisite to the formation of a local DW for some part of crystal constituent ions. The proper distortion of symmetry can be reached by applying an external pressure. The neutron diffraction measurements show the structural evidence for a fast-ion transition in the high pressure rocksalt phase of AgI being observed at the temperatures high enough to provide the effective mixing of p−d states which would be forbidden by symmetry selection rules at lower temperature [12]. AgBr and AgCl can be considered as ambient pressure counterparts of rs-AgI. They both possess the rocksalt structure and show an anomalous increase in conductivity as the temperature is raised. It seems they both are apt to become the superionic conductors. However, there is no evidence for a fastion transition in theses materials and it has been proposed that the transition would exist in AgBr if the anion sublattice did not melt before the fast-ion phase was established. We have studied the relationship between the difference of p−d hybridizations and the activation energies in noble metal halides [15,16]. We have shown that the strength of p−d hybridization for AgX (X = halogen) is smaller than that for CuX. Harrison [17] gave the overlap parameters Vpdσ,π(ι) between Ag 4d electrons and halogen p electrons as follows: 3=2
Vpdr;p ðiÞ ¼ gpdr;p
ћ2 r d ; me i7=2
where ι is the bonding length, ηpdσ = − 2.95, ηpdπ = 1.36, rd is 0.89 Å for Ag, me is the electron mass and ћ is the Planck constant. If we follow the expressions of the overlap parameters given by Harrison, p−d hybridizations in rs-AgI are smaller than those in AgCl and AgBr because of the lattice constants of AgCl (a = 5.54 Å), AgBr (a = 5.77 Å) and rs-AgI (a = 6.1596 Å) [12]. The high ionic conductivity in rs-AgI is also expected on the basis of the empirical parameters. 4. Conclusion The electronic structure and ion diffusion paths in rs-AgI have been investigated using the FP-LMTO method. The low energy path through F (face of the octahedron) and T (center of the tetrahedron) is favor for mobile ions. This is consistent with
the results which were well known in the cases of AgBr and AgCl. We have also confirmed the low minimum site energies at the tetrahedral site. These results are reasonable for AgCl, AgBr and rs-AgI with many Frenkel defects. Our results suggest that the tetrahedral sites are enough low energy sites for mobile Ag ions and regarded as the stable sites for Frenkel defect formation. The band structure has been also calculated. The bands around from − 4.5 eV to 0 eV have been derived from Ag 4d and I 5p states. These electronic states have been hybridized and have played important roles to elucidate the mechanism of the ionic conductivity in rs-AgI. If we follow the expressions of the overlap parameters between Ag 4d electrons and halogen p electrons given by Harrison, p−d hybridizations in rs-AgI are smaller than those in AgCl and AgBr because of the long lattice constant for rs-AgI, compared with that for AgCl and AgBr. The strength of p−d hybridization was discussed in relation to the activation energy for the ionic conduction in our recent studies [15,16]. The high ionic conductivity in rs-AgI is also expected as shown in the cases of AgBr and AgCl on the basis of the empirical parameters. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
B.E. Mellander, Phys. Rev., B 26 (1982) 5886. S. Hull, D.A. Keen, Phys. Rev., B 59 (1999) 750. J.C. Phillips, Rev. Mod. Phys. 42 (1970) 317. J.C. Phillips, Bonds and Bands in Semiconductors, Academic, New York, 1973. J.K. Aboagye, R.J. Friauf, Phys. Rev., B 11 (1975) 1654. D.A. Keen, S. Hull, A.C. Barnes, P. Berastegui, W.A. Chichton, P.A. Madden, M.G. Tucker, M. Wilson, Phys. Rev., B 68 (2003) 14117. W.G. Kleppmann, H. Bilz, Commun. Phys. B 1 (1976) 105. W.G. Kleppmann, W. Weber, Phys. Rev., B 20 (1979) 1669. S.Y. Savrasov, Phys. Rev., B 54 (1996) 16470. W. Shimosaka, S. Kashida, M. Kobayashi, Solid State Ionics 176 (2005) 349. S. Ono, Y. Seki, S. Kashida, M. Kobayashi, Solid State Ionics 177 (2006) 1145. D.A. Keen, S. Hull, W. Hayes, N.J.G. Gardner, Phys. Rev. Lett. 77 (1996) 4914. C.M.I. Okoye, Phys. Status Solidi 234 (2002) 580. A. Rakitin, M. Kobayashi, Phys. Rev., B 53 (1996) 3088. S. Ono, M. Kobayashi, H. Iyetomi, T. Tomoyose, Solid State Ionics 139 (2001) 249. M. Kobayashi, H. Iyetomi, S. Ono, T. Tomoyose, Int. J. Mod. Phys. B 15 (2001) 678. W.A. Harrison, Electronic Structure and Properties of Solids, Freeman, San Francisco, 1980.