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Defect structure and electrical properties of molybdenum disulphide
ARTICLE IN PRESS
Journal of Physics and Chemistry of Solids 67 (2006) 2528–2535 www.elsevier.com/locate/jpcs
Defect structure and electrical properties of molybdenum disulphide M. Potoczek1, K. Przybylski, M. Rekas AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Al. Mickiewicza 30, 30-059 Krakow, Poland Received 23 March 2005; received in revised form 5 November 2005; accepted 7 July 2006
Abstract Electrical conductivity of molybdenum disulphide was studied in a helium–sulphur gas mixture as a function of temperature (1073–1273 K). It was found that over the whole temperature and sulphur pressure range (10–6600 Pa) studied, the material exhibits ptype conductivity. Based on literature intrinsic electronic disorder data as well as measured electrical conductivity results a defect model has been proposed. This model involves electron holes and doubly ionized interstitial sulphur ions as majority point defects as well as electrons and acceptor-type foreign ions as minority defects. r 2006 Elsevier Ltd. All rights reserved. Keywords: A. Inorganic compounds; A. Semiconductors; D. Defects; D. Electrical conductivity
1. Introduction Molybdenum disulphide, MoS2, due to its interesting semiconducting properties remains a promising electrode material for lithium batteries [1] and solar cells [2,3]. Also, an unusually low high-temperature corrosion rate of molybdenum in sulphur-containing gas atmospheres is closely related to the semiconducting properties of MoS2 [4]. The semiconducting properties of the transition metal sulphides are determined by both intrinsic and extrinsic disorder in the crystal lattice. The intrinsic disorder depends on the electronic structure, mainly the width of the forbidden energy gap (bandgap), Eg. There is a lot of controversy concerning the electronic structure of MoS2. Wilson and Yoffe [5] calculated the band structure of MoS2 using a semi-empirical tight-bonding method. They found an energy bandgap, Eg, equal to 0.2 eV. Electrical measurements (electrical conductivity and Hall effect) within the temperature range 140–670 K carried out by Fivaz and Mooser [6] and El-Mahalawy and Evans [7] gave Eg values of 0.14–0.32 and 0.38 eV, respectively. Photoemission studies [8–9] suggest that Eg is higher than 1 eV. Corresponding author. Tel.: +48 12 617 4722; fax: 48 12 634 1201.
E-mail address: [email protected] (M. Rekas). Present address: University of Technology, Faculty of Chemistry, ul. W. Pola 2, 35–959 Rzeszow, Poland. 1
0022-3697/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.07.008
The quantum-mechanic analysis revealed strong hybridization between the d z2 , dxy and d x2 y2 orbitals in MoS2 [10]. Mattheiss [11,12], using either augmented plane wave (APW) or linear combination of atomic orbitals (LCAO) method, reported Eg values equal to 1.22 [11] and 1.16 eV, respectively [12]. Different methods used for determination of the bandgap, effect of temperature on Eg as well as different polymorphic forms of used MoS2 specimens can explain the observed differences of Eg. Also, little is known about the extrinsic disorder of molybdenum disulphide. This disorder is determined by both the departure from the nonstoichiometry and by the presence of foreign ions (impurities and dopants) in the crystal lattice. In the available literature data the electrical properties of molybdenum disulphide were mostly studied with temperature below 500 K [13–17]. In such conditions no effect of sulphur vapour pressure on electrical properties and related defect structure is observed. Unlike most transition metal sulphides, molybdenum disulphide shows low departure from stoichiometric composition [18,19]. A deficit and an excess of sulphur were both observed in MoS2 [18–21]. Table 1 summarizes the available literature data concerning nonstoichiometry and related predominant defects. As can be seen, the presented literature data are scarce, fragmentary and there is no agreement between them. The literature reports concerning the high-temperature semiconducting properties and related defect structure
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Table 1 Nonstoichiometry of MoS2 Specimen
Temperature (K)
p(S2) (Pa)
Nonstoichiometry range
Pre-dominant defect
Ref
Ceramic Single crystal Ceramic
1373 1173–1373 1023–1223
Scha¨fer et al. [20] Rau [18] Suzuki et al. [19]
1073–1473
MoS2.00–MoS2.22 MoS1.9958–MoS2.0012 MoS1.983–MoS2.000 MoS1.978–MoS2.000 MoS2.025–MoS2.075
Si
Ceramic
102–105 6 103–102 105–105 8 103–105 105–105 8 103–105
of MoS2 are scarce and insufficient. Mahalawy and Evans [7] studied electrical conductivity and Hall coefficient of MoS2 single crystal within 120–1170 K at vacuum. They results may serve to determine the intrinsic electronic disorder, but cannot be used to study the ionic defect structure due to lack of the equilibrium of crystal with the surrounding gas phase (lack of dependencies of the studied parameters on sulphur vapour pressure). The purpose of the investigations, reported in this work and in the two consecutive papers (the present one and one under preparation [22], is the determination of relationships between semiconducting or transport properties and defect structure. The objective of this work is to derive defect diagrams for MoS2 and verify them on the basis of electrical conductivity data. 2. Experimental 2.1. Sample preparation The samples were prepared in the following way: Molybdenum powder of 99.97% purity, supplied by Johnson Matthey Chemical Limited, was fully sulphurized for 24 h in a mixture containing sulphur vapour and helium at the sulphur vapour partial pressure equal to 1000 Pa. The gas mixture was obtained by passing helium over the surface of the melted sulphur of 99.998% purity (POChGliwice, Poland). After refinement in an agate crucible, samples were formed from the MoS2 powder in the shape of bars under a pressure of 30 MPa. The samples, placed on a substrate of molybdenum disulphide powder, were sintered at 1323 K in pumped-out quartz ampoules, with a small weighted portion of sulphur. The mass of sulphur was selected in such a way that the pressure of sulphur vapour in the quartz ampoule at the sintering temperature was about 104 Pa. The sintering time was 30 days. The pressure of sulphur vapour was calculated on the basis of the temperature of a container with liquid sulphur, taking thermodynamic equilibrium between the liquid and the gaseous stage as an assumption. For the calculations the thermodynamic data of Barin [23] were used. The pressure of sulphur vapour applied in the above experiment, equal to 3160 Pa, corresponded to the thermodynamic equilibrium established between the sulphur vapour and a solution of liquid sulphur at the temperature of 548 K.
VS
Ugryumova et al. [21]
Samples obtained in this way were polished on a grinding paper SiC 800, and then measuring probes, made of platinum wire of the diameter + ¼ 0.25 mm, were firmly attached to the samples. To improve the contact and to prevent shifting, small grooves were cut out on the edges of the samples before fastening the probes. Geometrical dimensions of the samples were determined by means of a micrometric screw, and the distances between the voltage probes—using Abbe’s comparator. The specimens prepared in this way were used for electrical measurements. Also, density, morphology and the impurity content were examined. The density of the samples was determined by the pycnometric method. The pycnometric density of the MoS2 sinter was equal to 4.91 g/cm3, and the apparent density was 4.46 g/cm3, which amounted to 90% of the theoretical density. From the conducted SEM morphological observations (JEOL JSM-5400) it is evident that the sintered MoS2 samples are compact and they contain very small amounts of pores, which is in good agreement with the determined apparent density. The impurity content analysis of the sintered MoS2 samples was performed by atomic absorption spectroscopy (AAS), type Philips PU 9100x. The results of analysis are listed in Table 2. 2.2. X-ray analysis The measurements were carried out by means of the Philips X-ray diffractometer, type PO 1710, applying monochromatic radiation CoKa. The X-ray phase analysis revealed that the samples were built of the MoS2 phase in which the hexagonal form (2HMoS2) prevailed over the rhombohedral one (3R-MoS2). The lattice parameters ‘‘a’’ and ‘‘c’’ of the 2H-MoS2 and 3R-MoS2 forms were determined as a function of sulphur vapour pressure, p(S2), The sulphide scales, obtained at 1273 K in the sulphur vapour pressure from 10 to 6600 Pa were used for the X-ray analysis. The details of preparation the sulphide scales are described elsewhere [4]. The lattice parameters were determined by the diffractometric method which enabled the registration of reflexes at large angles, at least to the value of 2y ¼ 1601. The strongest reflexes with simple indicates h, k, l (i.e. such that at least one from among them was equal to zero) have been selected for the determination.
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2530 Table 2 Impurity analysis Element Ionic radius
Valence (pm)a
Si
Fe
Cu
Zn
Mn
Co
Cr
Ti
Bi
Pb
Ni
+4 40
+2 61 LS 77 HS 25 6 LS 20 HS S LS–HS
+2 73
+2 74.5
+4 68
+3 96
+2 120
+2 69
6 15
4 6
4 7
4 57
o1 82
o1 8
S
S
+2 65 LS 74.5 HS 5 0 LS 15 HS S
+3 63
12 12
+2 67 LS 82 HS 5 3 LS 26 HS S LS–HS
S
S
—
—
S
Concentration (ppm) rMo4þ rMjþ 100 (%) r 4þ
30–50 38
Type of solid solution
I
Mo
LS—low spin electron configuration, HS—high spin electron configuration, S—substitutional solid solution, I—interstitial solid solution. a According to Ref. [24,25].
Table 3 Lattice parameters of 2H-MoS2 and 3R-MoS2 formed at 1273 K Sulphur vapour pressure p(S2) (Pa)
10 316 1000 6600
2H-MoS2
3R-MoS2
a (pm)
c (pm)
a (pm)
c (pm)
316.370.2 316.170.6 316.770.5 316.770.9
1206.372.2 1228.473.4 1229.775.5 1248.379.0
315.670.7 316.270.1 315.571.0 316.570.5
1834.876.8 1820.4713.1 1829.871.6 1830.670.4
Since the reflexes almost always represented a superposition of two polytypes (2H and 3R), the computer program ‘‘Fit Profile’’ from ‘‘PC APD’’ packet was used, which separated the reflexes on the basis of Fourier’s analysis (with the accuracy up to 0.0021), and then with great precision assigned the distances dhkl corresponding to the reflexes. The parameters of an elementary cell were calculated using the dependence for a hexagonal system combining the lattice constants with interplanar distances of the given family of planes. Since maximal accuracy is obtained for measurements corresponding to positions of reflexes of high 2y value, the lattice constants were determined from the positions of many reflexes and their extrapolation to 2y ¼ 1801 according to the function cos2 2y using the least-square method. As the error of the lattice constant the standard error of the free term of the extrapolation line was accepted. The results of these calculations are presented in Table 3. As can be seen from the Table 3, the lattice parameter ‘‘c’’ for 2H-MoS2 sulphide increases with p(S2). On the other hand, the value of the parameter ‘‘a’’ is independent of this pressure. In the case of 3R-MoS2 sulphide the values of the lattice constants ‘‘a’’ and ‘‘c’’ do not in fact depend on p(S2). 2.3. Electrical conductivity measurements MoS2 samples in the form of rectangular bars, with dimensions of about 20 3 2 mm, were used to measure electrical conductivity. The measurements were carried out
in the temperature range 10731273 K and sulphur vapour partial pressures of 106600 Pa, by the four-probe method, using alternate current. The measuring setup is illustrated in Fig. 1. MoS2 in the form of rectangular pellet (6) with four Pt wires wrapped around the sample was placed inside the vertical quartz tube (4). The required sulphur vapour activity in the chamber was achieved by passing the He-S2 mixture through the measuring tube. The total pressure of the gas mixture was 105 Pa. The partial pressure of sulphur is a function of temperature in the liquid sulphur container (3) and the flow rate of helium. It was determined by calibration proposed by Wakihara et al. [26]. The electrical conductivity measurements were carried out as a function of temperature (1073–1273 K) and sulphur pressure (10–6600 Pa).
3. Results Fig. 2 illustrates the electrical conductivity in the Arrhenius coordinate system. The experimental data fulfil the linear dependency, predicted by the Arrhenius equation well: E act s ¼ s0 exp , (1) kT where the s0 parameter is independent of temperature, k-Boltzmann constant.
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Fig. 1. Experimental set-up for measurements of electrical conductivity of molybdenum disulphide in sulphur vapor under the conditions of thermodynamic equilibrium. 1, 2—electrical furnaces, 3—sulphur reservoir, 4—reaction tube, 5—mixer of He and S2, 6—sample, 7—thermostating heater, 8—sulphur cold trap, 9—Dewar vessel, 10 and 11—PtRh10Pt thermocouples, 12—Pt wires sealed through the Pyrex glass.
Fig. 3. Electrical conductivity versus of sulphur pressure. Fig. 2. Electrical conductivity versus temperature in the Arrhenius coordinate system.
The activation energy, Eact, of electrical conductivity determined from the slope of the straight-line (the slope equal to ðE act log eÞ=kT in Fig. 2) was (0.63570.003) eV. The results of electrical conductivity measurements as a function of sulphur pressure are presented in Fig. 3. According to Fig. 3, the dependence of electrical conductivity as a function of sulphur pressure p(S2) can be approximated by s ¼ A pðS2 Þ1=m ,
(2)
where A is the parameter independent of p(S2), m is the reciprocal power dependence exponent. The parameter m
determined from the slope of the straight-lines (the slope equal to 1/m in Fig. 3), assumes the value of 19.470.9. 4. Discussion 4.1. Defect equlibria According to the nonstoichiometry data collected in Table 1, MoS2 exhibits either an excess or a deficit of sulphur in the crystal lattice. Disorder occurring only in the sulphur sublattice is assumed in both mentioned cases [18–21]. The electrical properties presented in this paper indicate a p-type behaviour of MoS2 within the experimental
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range of temperature and p(S2). Hence, below we will discuss defect structure in the p-type regime corresponding to sulphur excess. Like the cited authors [18–21], we assume that only the sulphur sublattice is disordered. The simplest defect model which explains sulphur excess involves interstitial sulphur ions, Sij which are formed according to the reaction: 1 S 2 3Sj (3) i þ jh . 2 The mass action law applied to equilibrium (3) results in the following equation: j K j ¼ ½S j i p PS 2 , 1=2
(4)
where j ¼ 0,1,2; p ¼ [h ]. The sum of concentrations of the sulphur interstitial defects (expressed in molar ratio) results in sulphur nonstoichiometry, x in MoS2+x: 0 00 ½SX i þ ½S i þ ½S i ¼ x.
(5)
Since at high temperatures (as is the case in our investigation) the point defects are usually completely ionised, so we assume that j ¼ 2. Then, for the lattice to be electroneutral the following has to be true: 2½S00i
þ n þ A ¼ p,
(6)
where n denotes the concentration of electrons: n ¼ [e0 ], effective concentration of either acceptor-type (A40) or donor-type (Ao0) foreign ions. Taking into consideration the intrinsic electronic disorder: 0
nil3e þ h
(7)
and its equilibrium constant, Ki: K i ¼ np from Eqs. (4), (6) and (8) we obtain 2 p 2 ðp Ap K i Þ . PS2 ¼ 2K 2
(8)
(9)
The absolute concentrations, cdef , of defects expressed in cm3 can be recalculated into molar ratios, [def], using the following relationship: N Ad ¼ 1:877 1022 ½def, (11) 2M where d is the density of the material (4.99 g cm3 [27], NA the Avogadro’s number (NA ¼ 6.022 1023 mol1], M the molar weight of MoS2 (M ¼ 160.07 g mol1). From Eqs. (10) and (11) we have ð1:16 0:01Þ eV K i ¼ ð1:20 0:01Þ 103 exp . (12) kT cdef ½cm3 ¼ ½def
Fig. 4 (right axis) illustrates the temperature dependence of Ki. The numerical data of Ki determined by Eqs. (8) and (10) are first determined for MoS2. 4.3. Equilibrium constant K2 The equilibrium constant K2 is defined by K2 ¼
½S00i p2 pðS2 Þ1=2
.
Also, this equilibrium constant was known. We determined this value from the electrical conductivity data presented in Figs. 2 and 3. Electrical conductivity of solids can be expressed by the following relationship: s ¼ sion þ eðmn n þ mp pÞ,
(14)
where sion is ionic conductivity, e is elementary charge (e ¼ 1.602 1019 C); mn and mp are the mobilities of electrons and holes, respectively. As our experiments consisted of passing DC current through the sample in 1273 K for long periods of time without observing changes of electrical conductivity, we can assume that ionic conductivity in MoS2 is negligibly low. This result agrees with the results of Wilson and Yoffe [5]. Also, as mentioned in the Introduction, the very low
The Eq. (9) enables us to determine defect diagrams if both K2 and Ki are known. 4.2. Equilibrium constant Ki El-Mahalawy and Evans [7], basing on electrical conductivity and Hall effect studies, determined the concentration of electronic defects (n, p) in a single crystal of MoS2 within 140–820 K. Taking into account their results (concentrations of n and p), as well as Eg ¼ 1.16 eV reported by Mattheiss [12]. We get: Eg K i ½cm6 ¼ np ¼ K 0 exp kT ð1:16 0:01ÞeV 41 ¼ ð4:23 0:03Þ 10 exp . ð10Þ kT
(13)
Fig. 4. Van’t Hoff plots of Ki (right axis) and K2 (left axis).
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high-temperature corrosion rate of Mo confirms this assumption. The Hall effect studies, by Fivaz and Moser [6] determined both mobilities as mn ¼ 2:766 108 T2:6 cm2 V1 s1
(15)
and mp ¼ 10 cm2 V1 s1
(16)
From the combination of Eqs. (8) and (13) we have p¼
s þ ðs2 4mn mp e2 K i Þ1=2 . 2mp e
(17)
The experimental data on electrical conductivity presented in Fig. 3 and Eqs. (10), (15) and (16) are used to determine the concentrations of electron holes from Eq. (17). Then, taking into consideration the following relationship (resulting from the combination of Eqs. (13), (6) and (8)): 2K 2 pðS 2 Þ1=2 ¼ pðp2 Ap K i Þ.
(18)
Both K2 and A were determined from the Eq. (18). K2 (expressed in cm9 Pa1/2) is the following function of temperature: K 2 ½cm9 Pa1=2 ¼ ð6:76 0:70Þ 1062 ð2:095 0:018Þ eV exp kT
ð19Þ
or for concentration of point defects expressed in molar ratio: K 2 ½Pa1=2 ¼ ð1:02 0:11Þ 104 ð2:095 0:018Þ eV exp . kT
ð20Þ
Fig. 4 (left axis) illustrates the temperature dependence of K2 .The linear dependence of log K2 versus 1/T is observed (as predicted by the van’t Hoff equation). The effective concentration of acceptor-type foreign ions was found as A ¼ ð3:63 0:79Þ 1017 cm3 ¼ ð19:3 4:2Þ ppm:
(21)
This value is related with the impurity content in the studied MoS2 sample, given in Table 2. The listed elements, M, can dissolve in MoS2 crystal lattice forming either substitutional Ms Mo or interstitial Mij+ point defects. The total effective concentration of impurity-type defects, A, is then: X X A¼ s½Ms j½Mjþ (22) i Mo s
j
where j is typical valence of element M and s ¼ 4j (relative valence of Mj+ against to the lattice ions Mo4+). According to the Hume–Rothery rules [29], small differences between the ionic radius of foreign, Mj+, and native ions, Mo4+, (jðrMo4þ rMjþ Þ=rMo4þ j100p15%, rMo4þ ¼ 65 pm [24,25]) are required to form substitutional
Fig. 5. Comparison of experimental results (points) and theoretically predicted dependences (lines) of electrical conductivity as a function of sulphur pressure.
solid solutions. As shown in Table 2, most of the listed ions conform to this rule. Only Si4+ (too small) and Pb4+, Bi3+ (both too large) are the exception to this rule. Among the elements listed in Table 2, titanium (Ti4+) ions do not have any effect on electrical properties of MoS2 due to identical valence (+4) with Mo4+ ions. So, according to Vervey’s controlled valence, Ti4+ ions form neither acceptor- nor donor- type canters when forming the substitutional solid solution. Small silicon ions can incorporate into interstitial positions of the crystal lattice but usually it forms a silica phase as a precipitate at grain boundaries of ionic crystals [28]. According to Eq. (19) and the concentration of impurities (Table 2), the estimated maximum value of the parameter Amax ¼ 112 ppm if we assume that silicon does not form solid solutions. Otherwise, A is within 8 and 86 ppm range if we assume that silicon incorporates interstitially in crystal lattice. The determined parameter A (Eq. (21)) is positive and does not surpass Amax, which remains in full agreement with the first case. Fig. 5 illustrates the electrical conductivity on partial pressure of sulphur determined experimentally (points) and theoretically (solid lines) based on the numerical values given by Eqs. (10), (19) and (21). As can be seen, the agreement between both sets of data is satisfactory at lower p(S2) but less so at higher levels of sulphur concentrations. The observed discrepancies at high p(S2) may indicate that assumptions about the ideal behaviour of point defects are not valid at higher p(S2) (in Eq. (13)). Instead, of the concentrations, the activities should be taken into account. 4.4. Defect diagrams The dependence of electron hole concentration on p(S2) was determined from relationship (9). Then, using Eqs. (8) and (13) the concentration of electrons (n) and ½S00i as a function of sulphur partial pressure were determined. The obtained relationships allow the plotting of the defect
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Fig. 6. Defect concentrations in MoS2+x containing A ¼ 19.3 ppm at 1073 K.
Fig. 8. Departure from stoichiometry, x, of MoS2+x vs. sulphur pressure.
At low p(S2), nx assumes a value close to 2.5. This value is close to the theoretical value nx ¼ 2, resulting from Eq. (13) assuming roughly the approximation that p is independent of sulphur pressure. However, at high p(S2) nx assumes a value of ca. 5. It indicates that this approximation is not valid at higher p(S2).
5. Conclusions
Fig. 7. Defect concentrations in MoS2+x containing A ¼ 19.3 ppm at 1273 K.
diagrams of MoS2+x. Figs. 6 and 7 show defect diagrams for MoS2+x containing effective concentration of acceptortype impurities A ¼ 19.3 ppm at 1073 and 1273 K, respectively. Similar dependencies (as between those for 1073 and 1273 K) have been observed at intermediate temperatures. As shown in Figs. 6 and 7, the predominant electronic defects within the entire range of p(S2) are holes. At low p(S2), holes are mainly compensated by electrons (intrinsic semiconductor). On the other hand, at high p(S2) extrinsic defects (sulphur interstitial ions) participate in charge compensation. As follows from experimental data, the effective concentration of acceptor-type impurities (A) plays a minor role in the semiconducting properties of the studied material. The effect of impurities becomes less important with increasing temperature. Fig. 8 illustrates the departure from stoichiometry composition (excess of sulphur) x as a function of p(S2) determined from Eq. (5). Excess of sulphur x increases with p(S2). This increase may be characterized by the parameter nx: q log x 1 nx ¼ . (23) q log pðS 2 Þ T
The electrical conductivity measurements revealed that MoS2 exhibits a p-type conductivity at temperatures of 1073–1273 K and sulphur pressures of 10–6600 Pa. Defect chemistry of molybdenum disulphide was discussed in terms of point defects involving:
electron holes as majority electronic defects; electrons as minority electronic defects; sulphur interstitial doubly-ionized atoms; and foreign ions which can provide either acceptor- or donor-type centers in the crystal lattice.
The derived defect diagrams indicate that MoS2 exhibits p-type properties within entire range of T and p(S2). The defect diagrams of MoS2 may be considered within two p(S2) regimes.
1. At low p(S2) the intrinsic electronic disorder predominates, there is a departure from stoichiometry, x, changing with p(S2) as: xp(S2)1/2.5. 2. At high p(S2) (above 103 Pa), the departure from stoichiometry plays an important role in the overall defect concentration and related electrical properties of MoS2.
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Acknowledgements Financial support from the Polish State Committee for Scientific Research (KBN). Projects no. 4 T08D 007 24 and no. 3 0881 9101 and is gratefully acknowledged. References [1] G. Pistoia (Ed.), Lithium Batteries-New Materials, Developmments and Perspectives, Industrial Chemistry Library, vol. 5, Elsevier, Amsterdam, 1994. [2] S. Chandra, Photoelectrochemical Solar Cells, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, 1985. [3] Ya.V. Pleskov, Ya.Yu. Gurevich, Semiconductor Photoelectrochemistry, Consultants Bureau, New York, London, 1986. [4] K. Przybylski, M. Potoczek, Trans. Mat. Res. Soc. Jpn 14A (1994) 169. [5] J.A. Wilson, A.D. Yoffe, Adv. Phys. 18 (1969) 193. [6] R. Fivaz, E. Mooser, Phys. Rev. A 163 (1967) 743. [7] S.H. El-Mahalawy, B.L. Evans, Phys. Stat. Sol. (b) 79 (1977) 713. [8] P.M. Wiliams, R.F. Shepherd, J. Phys. C 6 (1973) L36. [9] J.C. McMenamin, W.E. Spicer, Phys. Rev. Lett. 29 (1972) 1501. [10] C. Julien, T. Sekine, M. Balkanski, Solid State Ionics 48 (1991) 22. [11] L.F. Mattheiss, Phys. Rev. Lett. 30 (1973) 784. [12] L.F. Mattheiss, Phys. Rev. B 8 (1973) 3719. [13] J. Molenda, K. Krzywanek, Bull. Pol. Acad. Sci. 43 (1995) 209.
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Journal of Physics and Chemistry of Solids 67 (2006) 2528–2535 www.elsevier.com/locate/jpcs
Defect structure and electrical properties of molybdenum disulphide M. Potoczek1, K. Przybylski, M. Rekas AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Al. Mickiewicza 30, 30-059 Krakow, Poland Received 23 March 2005; received in revised form 5 November 2005; accepted 7 July 2006
Abstract Electrical conductivity of molybdenum disulphide was studied in a helium–sulphur gas mixture as a function of temperature (1073–1273 K). It was found that over the whole temperature and sulphur pressure range (10–6600 Pa) studied, the material exhibits ptype conductivity. Based on literature intrinsic electronic disorder data as well as measured electrical conductivity results a defect model has been proposed. This model involves electron holes and doubly ionized interstitial sulphur ions as majority point defects as well as electrons and acceptor-type foreign ions as minority defects. r 2006 Elsevier Ltd. All rights reserved. Keywords: A. Inorganic compounds; A. Semiconductors; D. Defects; D. Electrical conductivity
1. Introduction Molybdenum disulphide, MoS2, due to its interesting semiconducting properties remains a promising electrode material for lithium batteries [1] and solar cells [2,3]. Also, an unusually low high-temperature corrosion rate of molybdenum in sulphur-containing gas atmospheres is closely related to the semiconducting properties of MoS2 [4]. The semiconducting properties of the transition metal sulphides are determined by both intrinsic and extrinsic disorder in the crystal lattice. The intrinsic disorder depends on the electronic structure, mainly the width of the forbidden energy gap (bandgap), Eg. There is a lot of controversy concerning the electronic structure of MoS2. Wilson and Yoffe [5] calculated the band structure of MoS2 using a semi-empirical tight-bonding method. They found an energy bandgap, Eg, equal to 0.2 eV. Electrical measurements (electrical conductivity and Hall effect) within the temperature range 140–670 K carried out by Fivaz and Mooser [6] and El-Mahalawy and Evans [7] gave Eg values of 0.14–0.32 and 0.38 eV, respectively. Photoemission studies [8–9] suggest that Eg is higher than 1 eV. Corresponding author. Tel.: +48 12 617 4722; fax: 48 12 634 1201.
E-mail address: [email protected] (M. Rekas). Present address: University of Technology, Faculty of Chemistry, ul. W. Pola 2, 35–959 Rzeszow, Poland. 1
0022-3697/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.07.008
The quantum-mechanic analysis revealed strong hybridization between the d z2 , dxy and d x2 y2 orbitals in MoS2 [10]. Mattheiss [11,12], using either augmented plane wave (APW) or linear combination of atomic orbitals (LCAO) method, reported Eg values equal to 1.22 [11] and 1.16 eV, respectively [12]. Different methods used for determination of the bandgap, effect of temperature on Eg as well as different polymorphic forms of used MoS2 specimens can explain the observed differences of Eg. Also, little is known about the extrinsic disorder of molybdenum disulphide. This disorder is determined by both the departure from the nonstoichiometry and by the presence of foreign ions (impurities and dopants) in the crystal lattice. In the available literature data the electrical properties of molybdenum disulphide were mostly studied with temperature below 500 K [13–17]. In such conditions no effect of sulphur vapour pressure on electrical properties and related defect structure is observed. Unlike most transition metal sulphides, molybdenum disulphide shows low departure from stoichiometric composition [18,19]. A deficit and an excess of sulphur were both observed in MoS2 [18–21]. Table 1 summarizes the available literature data concerning nonstoichiometry and related predominant defects. As can be seen, the presented literature data are scarce, fragmentary and there is no agreement between them. The literature reports concerning the high-temperature semiconducting properties and related defect structure
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Table 1 Nonstoichiometry of MoS2 Specimen
Temperature (K)
p(S2) (Pa)
Nonstoichiometry range
Pre-dominant defect
Ref
Ceramic Single crystal Ceramic
1373 1173–1373 1023–1223
Scha¨fer et al. [20] Rau [18] Suzuki et al. [19]
1073–1473
MoS2.00–MoS2.22 MoS1.9958–MoS2.0012 MoS1.983–MoS2.000 MoS1.978–MoS2.000 MoS2.025–MoS2.075
Si
Ceramic
102–105 6 103–102 105–105 8 103–105 105–105 8 103–105
of MoS2 are scarce and insufficient. Mahalawy and Evans [7] studied electrical conductivity and Hall coefficient of MoS2 single crystal within 120–1170 K at vacuum. They results may serve to determine the intrinsic electronic disorder, but cannot be used to study the ionic defect structure due to lack of the equilibrium of crystal with the surrounding gas phase (lack of dependencies of the studied parameters on sulphur vapour pressure). The purpose of the investigations, reported in this work and in the two consecutive papers (the present one and one under preparation [22], is the determination of relationships between semiconducting or transport properties and defect structure. The objective of this work is to derive defect diagrams for MoS2 and verify them on the basis of electrical conductivity data. 2. Experimental 2.1. Sample preparation The samples were prepared in the following way: Molybdenum powder of 99.97% purity, supplied by Johnson Matthey Chemical Limited, was fully sulphurized for 24 h in a mixture containing sulphur vapour and helium at the sulphur vapour partial pressure equal to 1000 Pa. The gas mixture was obtained by passing helium over the surface of the melted sulphur of 99.998% purity (POChGliwice, Poland). After refinement in an agate crucible, samples were formed from the MoS2 powder in the shape of bars under a pressure of 30 MPa. The samples, placed on a substrate of molybdenum disulphide powder, were sintered at 1323 K in pumped-out quartz ampoules, with a small weighted portion of sulphur. The mass of sulphur was selected in such a way that the pressure of sulphur vapour in the quartz ampoule at the sintering temperature was about 104 Pa. The sintering time was 30 days. The pressure of sulphur vapour was calculated on the basis of the temperature of a container with liquid sulphur, taking thermodynamic equilibrium between the liquid and the gaseous stage as an assumption. For the calculations the thermodynamic data of Barin [23] were used. The pressure of sulphur vapour applied in the above experiment, equal to 3160 Pa, corresponded to the thermodynamic equilibrium established between the sulphur vapour and a solution of liquid sulphur at the temperature of 548 K.
VS
Ugryumova et al. [21]
Samples obtained in this way were polished on a grinding paper SiC 800, and then measuring probes, made of platinum wire of the diameter + ¼ 0.25 mm, were firmly attached to the samples. To improve the contact and to prevent shifting, small grooves were cut out on the edges of the samples before fastening the probes. Geometrical dimensions of the samples were determined by means of a micrometric screw, and the distances between the voltage probes—using Abbe’s comparator. The specimens prepared in this way were used for electrical measurements. Also, density, morphology and the impurity content were examined. The density of the samples was determined by the pycnometric method. The pycnometric density of the MoS2 sinter was equal to 4.91 g/cm3, and the apparent density was 4.46 g/cm3, which amounted to 90% of the theoretical density. From the conducted SEM morphological observations (JEOL JSM-5400) it is evident that the sintered MoS2 samples are compact and they contain very small amounts of pores, which is in good agreement with the determined apparent density. The impurity content analysis of the sintered MoS2 samples was performed by atomic absorption spectroscopy (AAS), type Philips PU 9100x. The results of analysis are listed in Table 2. 2.2. X-ray analysis The measurements were carried out by means of the Philips X-ray diffractometer, type PO 1710, applying monochromatic radiation CoKa. The X-ray phase analysis revealed that the samples were built of the MoS2 phase in which the hexagonal form (2HMoS2) prevailed over the rhombohedral one (3R-MoS2). The lattice parameters ‘‘a’’ and ‘‘c’’ of the 2H-MoS2 and 3R-MoS2 forms were determined as a function of sulphur vapour pressure, p(S2), The sulphide scales, obtained at 1273 K in the sulphur vapour pressure from 10 to 6600 Pa were used for the X-ray analysis. The details of preparation the sulphide scales are described elsewhere [4]. The lattice parameters were determined by the diffractometric method which enabled the registration of reflexes at large angles, at least to the value of 2y ¼ 1601. The strongest reflexes with simple indicates h, k, l (i.e. such that at least one from among them was equal to zero) have been selected for the determination.
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2530 Table 2 Impurity analysis Element Ionic radius
Valence (pm)a
Si
Fe
Cu
Zn
Mn
Co
Cr
Ti
Bi
Pb
Ni
+4 40
+2 61 LS 77 HS 25 6 LS 20 HS S LS–HS
+2 73
+2 74.5
+4 68
+3 96
+2 120
+2 69
6 15
4 6
4 7
4 57
o1 82
o1 8
S
S
+2 65 LS 74.5 HS 5 0 LS 15 HS S
+3 63
12 12
+2 67 LS 82 HS 5 3 LS 26 HS S LS–HS
S
S
—
—
S
Concentration (ppm) rMo4þ rMjþ 100 (%) r 4þ
30–50 38
Type of solid solution
I
Mo
LS—low spin electron configuration, HS—high spin electron configuration, S—substitutional solid solution, I—interstitial solid solution. a According to Ref. [24,25].
Table 3 Lattice parameters of 2H-MoS2 and 3R-MoS2 formed at 1273 K Sulphur vapour pressure p(S2) (Pa)
10 316 1000 6600
2H-MoS2
3R-MoS2
a (pm)
c (pm)
a (pm)
c (pm)
316.370.2 316.170.6 316.770.5 316.770.9
1206.372.2 1228.473.4 1229.775.5 1248.379.0
315.670.7 316.270.1 315.571.0 316.570.5
1834.876.8 1820.4713.1 1829.871.6 1830.670.4
Since the reflexes almost always represented a superposition of two polytypes (2H and 3R), the computer program ‘‘Fit Profile’’ from ‘‘PC APD’’ packet was used, which separated the reflexes on the basis of Fourier’s analysis (with the accuracy up to 0.0021), and then with great precision assigned the distances dhkl corresponding to the reflexes. The parameters of an elementary cell were calculated using the dependence for a hexagonal system combining the lattice constants with interplanar distances of the given family of planes. Since maximal accuracy is obtained for measurements corresponding to positions of reflexes of high 2y value, the lattice constants were determined from the positions of many reflexes and their extrapolation to 2y ¼ 1801 according to the function cos2 2y using the least-square method. As the error of the lattice constant the standard error of the free term of the extrapolation line was accepted. The results of these calculations are presented in Table 3. As can be seen from the Table 3, the lattice parameter ‘‘c’’ for 2H-MoS2 sulphide increases with p(S2). On the other hand, the value of the parameter ‘‘a’’ is independent of this pressure. In the case of 3R-MoS2 sulphide the values of the lattice constants ‘‘a’’ and ‘‘c’’ do not in fact depend on p(S2). 2.3. Electrical conductivity measurements MoS2 samples in the form of rectangular bars, with dimensions of about 20 3 2 mm, were used to measure electrical conductivity. The measurements were carried out
in the temperature range 10731273 K and sulphur vapour partial pressures of 106600 Pa, by the four-probe method, using alternate current. The measuring setup is illustrated in Fig. 1. MoS2 in the form of rectangular pellet (6) with four Pt wires wrapped around the sample was placed inside the vertical quartz tube (4). The required sulphur vapour activity in the chamber was achieved by passing the He-S2 mixture through the measuring tube. The total pressure of the gas mixture was 105 Pa. The partial pressure of sulphur is a function of temperature in the liquid sulphur container (3) and the flow rate of helium. It was determined by calibration proposed by Wakihara et al. [26]. The electrical conductivity measurements were carried out as a function of temperature (1073–1273 K) and sulphur pressure (10–6600 Pa).
3. Results Fig. 2 illustrates the electrical conductivity in the Arrhenius coordinate system. The experimental data fulfil the linear dependency, predicted by the Arrhenius equation well: E act s ¼ s0 exp , (1) kT where the s0 parameter is independent of temperature, k-Boltzmann constant.
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Fig. 1. Experimental set-up for measurements of electrical conductivity of molybdenum disulphide in sulphur vapor under the conditions of thermodynamic equilibrium. 1, 2—electrical furnaces, 3—sulphur reservoir, 4—reaction tube, 5—mixer of He and S2, 6—sample, 7—thermostating heater, 8—sulphur cold trap, 9—Dewar vessel, 10 and 11—PtRh10Pt thermocouples, 12—Pt wires sealed through the Pyrex glass.
Fig. 3. Electrical conductivity versus of sulphur pressure. Fig. 2. Electrical conductivity versus temperature in the Arrhenius coordinate system.
The activation energy, Eact, of electrical conductivity determined from the slope of the straight-line (the slope equal to ðE act log eÞ=kT in Fig. 2) was (0.63570.003) eV. The results of electrical conductivity measurements as a function of sulphur pressure are presented in Fig. 3. According to Fig. 3, the dependence of electrical conductivity as a function of sulphur pressure p(S2) can be approximated by s ¼ A pðS2 Þ1=m ,
(2)
where A is the parameter independent of p(S2), m is the reciprocal power dependence exponent. The parameter m
determined from the slope of the straight-lines (the slope equal to 1/m in Fig. 3), assumes the value of 19.470.9. 4. Discussion 4.1. Defect equlibria According to the nonstoichiometry data collected in Table 1, MoS2 exhibits either an excess or a deficit of sulphur in the crystal lattice. Disorder occurring only in the sulphur sublattice is assumed in both mentioned cases [18–21]. The electrical properties presented in this paper indicate a p-type behaviour of MoS2 within the experimental
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range of temperature and p(S2). Hence, below we will discuss defect structure in the p-type regime corresponding to sulphur excess. Like the cited authors [18–21], we assume that only the sulphur sublattice is disordered. The simplest defect model which explains sulphur excess involves interstitial sulphur ions, Sij which are formed according to the reaction: 1 S 2 3Sj (3) i þ jh . 2 The mass action law applied to equilibrium (3) results in the following equation: j K j ¼ ½S j i p PS 2 , 1=2
(4)
where j ¼ 0,1,2; p ¼ [h ]. The sum of concentrations of the sulphur interstitial defects (expressed in molar ratio) results in sulphur nonstoichiometry, x in MoS2+x: 0 00 ½SX i þ ½S i þ ½S i ¼ x.
(5)
Since at high temperatures (as is the case in our investigation) the point defects are usually completely ionised, so we assume that j ¼ 2. Then, for the lattice to be electroneutral the following has to be true: 2½S00i
þ n þ A ¼ p,
(6)
where n denotes the concentration of electrons: n ¼ [e0 ], effective concentration of either acceptor-type (A40) or donor-type (Ao0) foreign ions. Taking into consideration the intrinsic electronic disorder: 0
nil3e þ h
(7)
and its equilibrium constant, Ki: K i ¼ np from Eqs. (4), (6) and (8) we obtain 2 p 2 ðp Ap K i Þ . PS2 ¼ 2K 2
(8)
(9)
The absolute concentrations, cdef , of defects expressed in cm3 can be recalculated into molar ratios, [def], using the following relationship: N Ad ¼ 1:877 1022 ½def, (11) 2M where d is the density of the material (4.99 g cm3 [27], NA the Avogadro’s number (NA ¼ 6.022 1023 mol1], M the molar weight of MoS2 (M ¼ 160.07 g mol1). From Eqs. (10) and (11) we have ð1:16 0:01Þ eV K i ¼ ð1:20 0:01Þ 103 exp . (12) kT cdef ½cm3 ¼ ½def
Fig. 4 (right axis) illustrates the temperature dependence of Ki. The numerical data of Ki determined by Eqs. (8) and (10) are first determined for MoS2. 4.3. Equilibrium constant K2 The equilibrium constant K2 is defined by K2 ¼
½S00i p2 pðS2 Þ1=2
.
Also, this equilibrium constant was known. We determined this value from the electrical conductivity data presented in Figs. 2 and 3. Electrical conductivity of solids can be expressed by the following relationship: s ¼ sion þ eðmn n þ mp pÞ,
(14)
where sion is ionic conductivity, e is elementary charge (e ¼ 1.602 1019 C); mn and mp are the mobilities of electrons and holes, respectively. As our experiments consisted of passing DC current through the sample in 1273 K for long periods of time without observing changes of electrical conductivity, we can assume that ionic conductivity in MoS2 is negligibly low. This result agrees with the results of Wilson and Yoffe [5]. Also, as mentioned in the Introduction, the very low
The Eq. (9) enables us to determine defect diagrams if both K2 and Ki are known. 4.2. Equilibrium constant Ki El-Mahalawy and Evans [7], basing on electrical conductivity and Hall effect studies, determined the concentration of electronic defects (n, p) in a single crystal of MoS2 within 140–820 K. Taking into account their results (concentrations of n and p), as well as Eg ¼ 1.16 eV reported by Mattheiss [12]. We get: Eg K i ½cm6 ¼ np ¼ K 0 exp kT ð1:16 0:01ÞeV 41 ¼ ð4:23 0:03Þ 10 exp . ð10Þ kT
(13)
Fig. 4. Van’t Hoff plots of Ki (right axis) and K2 (left axis).
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high-temperature corrosion rate of Mo confirms this assumption. The Hall effect studies, by Fivaz and Moser [6] determined both mobilities as mn ¼ 2:766 108 T2:6 cm2 V1 s1
(15)
and mp ¼ 10 cm2 V1 s1
(16)
From the combination of Eqs. (8) and (13) we have p¼
s þ ðs2 4mn mp e2 K i Þ1=2 . 2mp e
(17)
The experimental data on electrical conductivity presented in Fig. 3 and Eqs. (10), (15) and (16) are used to determine the concentrations of electron holes from Eq. (17). Then, taking into consideration the following relationship (resulting from the combination of Eqs. (13), (6) and (8)): 2K 2 pðS 2 Þ1=2 ¼ pðp2 Ap K i Þ.
(18)
Both K2 and A were determined from the Eq. (18). K2 (expressed in cm9 Pa1/2) is the following function of temperature: K 2 ½cm9 Pa1=2 ¼ ð6:76 0:70Þ 1062 ð2:095 0:018Þ eV exp kT
ð19Þ
or for concentration of point defects expressed in molar ratio: K 2 ½Pa1=2 ¼ ð1:02 0:11Þ 104 ð2:095 0:018Þ eV exp . kT
ð20Þ
Fig. 4 (left axis) illustrates the temperature dependence of K2 .The linear dependence of log K2 versus 1/T is observed (as predicted by the van’t Hoff equation). The effective concentration of acceptor-type foreign ions was found as A ¼ ð3:63 0:79Þ 1017 cm3 ¼ ð19:3 4:2Þ ppm:
(21)
This value is related with the impurity content in the studied MoS2 sample, given in Table 2. The listed elements, M, can dissolve in MoS2 crystal lattice forming either substitutional Ms Mo or interstitial Mij+ point defects. The total effective concentration of impurity-type defects, A, is then: X X A¼ s½Ms j½Mjþ (22) i Mo s
j
where j is typical valence of element M and s ¼ 4j (relative valence of Mj+ against to the lattice ions Mo4+). According to the Hume–Rothery rules [29], small differences between the ionic radius of foreign, Mj+, and native ions, Mo4+, (jðrMo4þ rMjþ Þ=rMo4þ j100p15%, rMo4þ ¼ 65 pm [24,25]) are required to form substitutional
Fig. 5. Comparison of experimental results (points) and theoretically predicted dependences (lines) of electrical conductivity as a function of sulphur pressure.
solid solutions. As shown in Table 2, most of the listed ions conform to this rule. Only Si4+ (too small) and Pb4+, Bi3+ (both too large) are the exception to this rule. Among the elements listed in Table 2, titanium (Ti4+) ions do not have any effect on electrical properties of MoS2 due to identical valence (+4) with Mo4+ ions. So, according to Vervey’s controlled valence, Ti4+ ions form neither acceptor- nor donor- type canters when forming the substitutional solid solution. Small silicon ions can incorporate into interstitial positions of the crystal lattice but usually it forms a silica phase as a precipitate at grain boundaries of ionic crystals [28]. According to Eq. (19) and the concentration of impurities (Table 2), the estimated maximum value of the parameter Amax ¼ 112 ppm if we assume that silicon does not form solid solutions. Otherwise, A is within 8 and 86 ppm range if we assume that silicon incorporates interstitially in crystal lattice. The determined parameter A (Eq. (21)) is positive and does not surpass Amax, which remains in full agreement with the first case. Fig. 5 illustrates the electrical conductivity on partial pressure of sulphur determined experimentally (points) and theoretically (solid lines) based on the numerical values given by Eqs. (10), (19) and (21). As can be seen, the agreement between both sets of data is satisfactory at lower p(S2) but less so at higher levels of sulphur concentrations. The observed discrepancies at high p(S2) may indicate that assumptions about the ideal behaviour of point defects are not valid at higher p(S2) (in Eq. (13)). Instead, of the concentrations, the activities should be taken into account. 4.4. Defect diagrams The dependence of electron hole concentration on p(S2) was determined from relationship (9). Then, using Eqs. (8) and (13) the concentration of electrons (n) and ½S00i as a function of sulphur partial pressure were determined. The obtained relationships allow the plotting of the defect
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Fig. 6. Defect concentrations in MoS2+x containing A ¼ 19.3 ppm at 1073 K.
Fig. 8. Departure from stoichiometry, x, of MoS2+x vs. sulphur pressure.
At low p(S2), nx assumes a value close to 2.5. This value is close to the theoretical value nx ¼ 2, resulting from Eq. (13) assuming roughly the approximation that p is independent of sulphur pressure. However, at high p(S2) nx assumes a value of ca. 5. It indicates that this approximation is not valid at higher p(S2).
5. Conclusions
Fig. 7. Defect concentrations in MoS2+x containing A ¼ 19.3 ppm at 1273 K.
diagrams of MoS2+x. Figs. 6 and 7 show defect diagrams for MoS2+x containing effective concentration of acceptortype impurities A ¼ 19.3 ppm at 1073 and 1273 K, respectively. Similar dependencies (as between those for 1073 and 1273 K) have been observed at intermediate temperatures. As shown in Figs. 6 and 7, the predominant electronic defects within the entire range of p(S2) are holes. At low p(S2), holes are mainly compensated by electrons (intrinsic semiconductor). On the other hand, at high p(S2) extrinsic defects (sulphur interstitial ions) participate in charge compensation. As follows from experimental data, the effective concentration of acceptor-type impurities (A) plays a minor role in the semiconducting properties of the studied material. The effect of impurities becomes less important with increasing temperature. Fig. 8 illustrates the departure from stoichiometry composition (excess of sulphur) x as a function of p(S2) determined from Eq. (5). Excess of sulphur x increases with p(S2). This increase may be characterized by the parameter nx: q log x 1 nx ¼ . (23) q log pðS 2 Þ T
The electrical conductivity measurements revealed that MoS2 exhibits a p-type conductivity at temperatures of 1073–1273 K and sulphur pressures of 10–6600 Pa. Defect chemistry of molybdenum disulphide was discussed in terms of point defects involving:
electron holes as majority electronic defects; electrons as minority electronic defects; sulphur interstitial doubly-ionized atoms; and foreign ions which can provide either acceptor- or donor-type centers in the crystal lattice.
The derived defect diagrams indicate that MoS2 exhibits p-type properties within entire range of T and p(S2). The defect diagrams of MoS2 may be considered within two p(S2) regimes.
1. At low p(S2) the intrinsic electronic disorder predominates, there is a departure from stoichiometry, x, changing with p(S2) as: xp(S2)1/2.5. 2. At high p(S2) (above 103 Pa), the departure from stoichiometry plays an important role in the overall defect concentration and related electrical properties of MoS2.
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Acknowledgements Financial support from the Polish State Committee for Scientific Research (KBN). Projects no. 4 T08D 007 24 and no. 3 0881 9101 and is gratefully acknowledged. References [1] G. Pistoia (Ed.), Lithium Batteries-New Materials, Developmments and Perspectives, Industrial Chemistry Library, vol. 5, Elsevier, Amsterdam, 1994. [2] S. Chandra, Photoelectrochemical Solar Cells, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, 1985. [3] Ya.V. Pleskov, Ya.Yu. Gurevich, Semiconductor Photoelectrochemistry, Consultants Bureau, New York, London, 1986. [4] K. Przybylski, M. Potoczek, Trans. Mat. Res. Soc. Jpn 14A (1994) 169. [5] J.A. Wilson, A.D. Yoffe, Adv. Phys. 18 (1969) 193. [6] R. Fivaz, E. Mooser, Phys. Rev. A 163 (1967) 743. [7] S.H. El-Mahalawy, B.L. Evans, Phys. Stat. Sol. (b) 79 (1977) 713. [8] P.M. Wiliams, R.F. Shepherd, J. Phys. C 6 (1973) L36. [9] J.C. McMenamin, W.E. Spicer, Phys. Rev. Lett. 29 (1972) 1501. [10] C. Julien, T. Sekine, M. Balkanski, Solid State Ionics 48 (1991) 22. [11] L.F. Mattheiss, Phys. Rev. Lett. 30 (1973) 784. [12] L.F. Mattheiss, Phys. Rev. B 8 (1973) 3719. [13] J. Molenda, K. Krzywanek, Bull. Pol. Acad. Sci. 43 (1995) 209.
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