Transport properties and the semiconducting nature of TiS2

193 TRANSPORT PROPERTIES AND THE SEMICONDUCTING NATURE OF TiS2 E.M. L O G O T H E T I S , W.J. K A I S E R and Carl A. K U K K O N E N Engineering and Research Staff, Research, Ford Motor Company, Dearborn, Michigan 48121, USA

and S.P. F A I L E , R. C O L E L L A and J. G A M B O L D Department of Physics, Purdue University, W. Lafayette, Indiana 47907, USA

For a range of temperatures between 4.2 K and 700 K, we have m e a s u r e d the resistivity, Hall coefficient, thermoelectric power and infrared reflectivity of TiS2 single crystals with varying degrees of stoichiometry, and a corresponding variation in the carrier concentration by a factor of 15. T h e strong correlations established a m o n g different properties over this large range of doping provides convincing evidence that TiS2 is a semiconductor rather than a semimetal. However, our most stoichiometric samples continued to be metallic with carrier concentrations of n = 2 x 102°cm -3. T h e origin of these electrons is unknown. T h e t e m p e r a t u r e d e p e n d e n t resistivity follows approximately a power law 0 ( T ) - 0(4.2 K) cr T ,~ over the entire temperature range of the m e a s u r e m e n t s (77 I(-700 K). T h e e x p o n e n t m is not strictly 2 as would be expected for carrier-carder scattering but varies from 2.2 for o u r most stoichiometric sample to 1.85 for the least stoichiometric. W e also find that 0(300) - 0(4.2 K) varies with doping as n ~/3 rather than the n -5/3 dependence reported previously and presented as evidence for carder-carrier scattering. W e have considered optical p h o n o n scattering, but find that this m e c h a n i s m alone cannot fit o u r data. W e regard the scattering m e c h a n i s m as unknown.

1. Introduction The layered compound titanium disulphide (TiS2) has received considerable attention in recent years, to a large extent, because of its potential use as the cathode material in lithiumanode intercalation chemistry batteries [I]. For this application it is important that the electrode (cathode) material be a good conductor of both electrons and lithium ions [2, 3]. All investigators [4-9] have found that TiS2 shows metallic electronic conductivity at all temperatures and the Hall coefficient is independent of temperature. Furthermore, the a-axis electrical resistivity was reported to exhibit an unusual T2-dependence

[5, 8]. Although it is agreed that TiS2 has metallic properties, the origin of the carriers remains controversial. Early investigators considered TiS2 as a degenerate semiconductor with the extrinsic conduction electrons arising from excess titanium [4, 10]. It is well known that TiS2 tends to grow metal rich and the excess titanium atoms in nonstoichiometric Tit÷xS2 are assumed to be donors, each Ti yielding four electrons to the conduction band. Band structure calculations also find TiS2 to be a semiconductor [1l]. With improved materials preparation techniques it became possible to prepare highly stoichiometric titanium disulphide. Takeuchi and Katsuta [5] and T h o m p s o n et al. [12] have reported the

Physica 99B (1980) 193--198 © North-Holland

preparation of Til+xS2 with x-----0.001. These investigators found that even these highly stoichiometric materials exhibited metallic behavior and they concluded that this property is intrinsic and that TiS2 is a semimetal. Furthermore, T h o m p s o n [8] reported that the resistivity of TiS2 depended on temperature as p ( T ) = a + b T 2 and on carrier concentration as n -5/3. Both dependences were taken as evidence for carrier-carrier scattering. Wilson [13] reexamined the literature and claimed that all existing data with the exception of the residual carrier concentration at x ~ 0 are consistent with TiS2 being a semiconductor. H e suggested that the displacement defects inferred by Takeuchi and Katsuta [5] (vacancy in Ti site/interstitial Ti between the layers) might be donors and the source of the residual electrons. Recently Friend et al. [14] reported that the Hall coefficient of TiS2 is independent of applied pressure while that of the semimetal TiSe2 has a substantial pressure dependence. They concluded that TiS2 is a semiconductor. We have carried out extensive studies of the Hall effect, infrared reflectivity, thermoelectric power and electrical resistivity of single crystals of Ti~+xS2 with varying degree of nonstoichiometry x. These materials were grown from powders by the vapor transport technique using sulphur or iodine as the transport agent. The nonstoichiometry x was varied by changing

194 the growth t e m p e r a t u r e in the range 650°C to 900°C. By performing all four types of m e a s u r e m e n t s on each sample (or samples from the same batch) we were able to correlate one m e a s u r e m e n t with another. The strong correlations observed a m o n g different properties over a large range of carrier concentrations has allowed us to draw a n u m b e r of conclusions concerning both the electronic structure and the transport properties of titanium disulphide, In a recent publication [9] we discussed the results of the Hall effect and reflectivity measurements. W e showed there that although each type of m e a s u r e m e n t , taken separately, is consistent with either the semimetallic or the semiconducting model of TiS2, the combination of the two provides convincing evidence that TiS: is a semiconductor. In the present paper, we first review b r i e f y our Hall effect and reflectivity data and their implications for the electronic structure of TiS2. W e then proceed with the discussion of the thermoelectric power data and the information they provide concerning the source of the carriers. W e conclude with the presentation of some of our results on the resistivity of titanium disulphide and its dependence on t e m p e r a t u r e and nonstoichiometry.

2. Hall effect and reflectivity The Hall coefficient and the a-axis resistivity were measured with the van der Pauw method. Low resistance contacts to the crystals were m a d e with gold paste (Engelhard No. A1644). In the range 77-300 K, the Hall coefficient of all crystals was found to be negative (electron-like) and independent of temperature. This result is consistent with both, single-carrier and two-carrier systems [9]. On the basis of a single-carrier system, the carrier concentration, nHa, of our crystals calculated from the Hall coefficient range between 2.2× 1020 and 3.4× 102~cm 3 and are shown in table I. The reflectivity of freshly cleaved crystals was measured at r o o m t e m p e r a t u r e with a P e r k i n E l m e r infrared spectrometer. W e found that we can fit the data for each crystal with a simple single-carrier D r u d e relationship. This result alone, however, is consistent also with a twocarrier model if the two types of carriers have the same relaxation time [9]. Fig. 1 plots nHau against n,op, the carrier con-

Table I Experimental results for single crystals of titanium disulphide with varying degree of nonstoichiometry Single crystal batch n u m b e r

nHall = --1/RHe (10a° cm -3)

no~ = me~o2p/41re2 (1020 cm 3)

-S (tzV/K)

164 6 3 8 1

2.2 3.0 7.5 13.0 34.0

1.9 2.8 5.1 11.0 27.0

240 203 131 I01 56

4

I

I

I o

3

%2 =

I

o

0

I

2 ncop( I0 21 cm-3)

I

3

Fig. 1. T h e carrier concentration calculated from the Hall coefficient plotted against that calculated from the plasma frequency. T h e straight line behavior is consistent with a single-carrier model and inconsistent with a semimetal. T h e slope yields the optical mass, rnopt/rne = 1.3. In this figure and figs. 2 and 3 we include two additional data points not m e n t i o n e d elsewhere.

centration calculated from the plasma frequency

: 4ere 2 •

(1)

The fact that nHall and n~p are linearly related over a range where both vary by a factor of 15 shows total internal consistency for the singlecarrier (semiconductor) model. In this case, the slope of 1.3 can be taken as the value of the optical mass mopt since n H a l l : (moot/me)hoop.Furthermore, our analysis has shown [9] that the experimentally imposed constraint that nHa~t/n~=constant for all crystals cannot be satisfied with a two-carrier (electron-hole) system. We have thus concluded that titanium disulphide is not a semimetal but a semiconductor. In this case, a basic question to ask is what is the source of the electrons in our samples. Although

195 we have not a t t e m p t e d to determine the nonstoichiometry of our crystals, some information m a y be obtained from the m e a s u r e m e n t of the thermoelectric power. This work is discussed in the next section. 3. Thermoelectric power For the m e a s u r e m e n t of the thermoelectric power, the two ends of a given crystal were pressed against two copper blocks. A series of t e m p e r a t u r e differences A T were established between the two blocks and the thermoelectric power S was determined from the slope of the thermovoltage vs A T line after correcting for the thermoelectric power of copper. T h e values of S at 300 K are shown in table I and plotted against na~, in fig. 2. T h e sign of S was negative indicating that the majority carriers are electrons. For a single-carrier free electron model, the thermoelectric power is given by [15] S =

Acr2k2T 3eEF '

(2)

where EF = h E(3~2n)2/3/2m * is the Fermi energy and m* the effective mass. T h e value of A depends on the scattering mechanism: A = 1 for impurity scattering and A = 3 for p h o n o n scattering at high temperatures. In this model, S is expected to be proportional to n -2/3. T h e solid line in fig. 2 is a plot of S calculated from eq. (2) with A = 3 and m* = mr. The Fermi energy was 10a

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i

,

,

calculated from the electron concentration using the relationship k 3 = (3"n'2nHa,)[3. (The factor of 1/3 arises because we assume three equivalent spheres of electrons centered on the faces of the hexagonal Brillouin zone of TiS2). We see in fig. 3 that the experimental results for S are in reasonable agreement with the single-carrier free electron theory. This, in turn, implies that m * is of the order of unity, in agreement with our earlier determination that mopt/me= 1.3. The agreement between experiment and the simple theory is best for the nonstoichiometric samples. For our most stoichiometric sample the measured S is about 2/3 of the calculated value. The reason for this variation is not known. The thermoelectric power data may be used to derive some information concerning the nonstoichiometry of our crystals and the source of the carriers in titanium disulphide. This can be done by comparing our data with those of T h o m p s o n et al. [12], who reported the dependence of the thermoelectric power of pressed powders of Ti~+xS2 on nonstoichiometry x. (The carrier concentrations in these materials are not known since Hall effect m e a s u r e m e n t s could not be made in pressed powders.) For the purpose of making a comparison, we first assume that all electrons arise from excess titanium and cap JO 3

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Fig. 2. The measured thermoelectric power at 300 K plotted against the carder concentration derived from the Hall coefficient. The solid line is the thermoelectric power calculated from a single-carder free electron model with m* = me (see text).

Fig. 3. The measured thermoelectric power at 300 K plotted against the nonstoichiometry x of Til+xS2 crystals. The values of x were calculated from the carder concentration nHa. assuming that the carders arise from excess titanium and each titanium atom contributes four electrons. The solid line reproduces the data of T h o m p s o n et al. [12] who measured S and x of Til+xS2 powders. The dotted line represents the S m n -2/3 or x -2~3 predictions of a single-carrier free electron model.

196 culate the nonstoichiometry x of our samples from x = nHa,/6.98 × 1022. The deduced values of x range from 0.003 for sample #164 to 0.049 for sample #1. In fig. 3, we plot the thermoelectric power of our samples as a function of the deduced x together with the data of T h o m p s o n et al. [12]. W e find an excellent agreement between the two sets of data for large deviation from stoichiometry, x >-0.01, where S varies as x -2/3. This strongly suggests that the electrons in these samples originate from excess Ti atoms which donate four electrons each to the conduction band. At low x, however, there is a definite discrepancy between our data and those of T h o m p s o n et al. [12]. For our most stoichiometric sample with S = - 2 4 0 / z V / K our assumption yields x-~ 0.003 whereas a powder sample of T h o m p s o n et al. [12] with the same value of S would have x ~ 0 . 0 0 1 . If this discrepancy is not due to errors in the measurements, the question is what is its origin. We note that we could bring the two sets of data at low x in agreement if we assume that, at low x, each excess Ti a t o m contributes m o r e than four electrons. This possibility has been raised by Wilson [13]. An alternative approach is to assume that the samples contain about 1.5x 102o electrons/cm 3 that do not derive from excess titanium. In that case, the carrier concentration would be given by n = 1.5 x 1020 + 7.0 x 1022. X ( c m - 3 ) .

4.2 K values were obtained by immersing the samples into a liquid H e reservoir. For T > 300 K, the samples were m o u n t e d in a quartz cell that was placed in a furnace. Oxidation of the crystals was avoided by continuously flowing through the cell helium gas that was first passed through a copper oxygen-getter kept at 500°C. Fig. 4 shows some typical results on the temperature dependence of the resistivity p for a n u m b e r of crystals with varying degree of nonstoichiometry. In this figure, we have plotted the logarithm of [ p ( T ) - p ( 4 . 2 K)], the resistivity at t e m p e r a t u r e T minus that at 4.2 K, as a function of the logarithm of the temperature. The solid lines represent least-square fits of the data to a power-law dependence over the whole temperature range from 77 K to 700 K; the accompanying numbers are the corresponding slopes. W e find that the resistivity of Til+~S2 varies approximately as p ( T ) - p(4.2 K) = a . T m,

where the value of m ranges between 2.2 for our most stoichiometric crystals and 1.85 for the least stoichiometric. We do not view this power law t e m p e r a t u r e dependence as fundamental, but the fact that the exponent is much larger than unity tends to rule out s i m p l e acoustic p h o n o n scattering for which p goes as T for T >, TDebye (~235 K in TiS2). The variability in the value of the

(3)

At small x, the 1.5 x 102o residual electrons per cm 3 are dominant, but above x = 0.01 they are a negligible fraction of the total. This argument provides no evidence for the existence of extra residual carriers, but our data and those of T h o m p s o n et al. can be viewed consistently with this picture. T h e origin of these residual carriers, if they indeed exist, is not clear. O n e possibility is that they arise f r o m the displacement defects [5, 13]. It appears very desirable to carry out the S vs. x determination on single crystals where the carrier concentration can also be obtained from Hall effect measurements.

(4)

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4. Electrical resistivity T h e m e a s u r e m e n t of the a-axis electrical resistivity of the TiS2 crystals below r o o m temperature was m a d e in a liquid N2 cryostat. T h e

Fig. 4. T h e logarithm of the difference between the resistivities at t e m p e r a t u r e T and at 4.2 K plotted against the logarithm of T for a n u m b e r of single crystals. T h e solid lines are power-law fits of the data over the entire t e m p e r a t u r e range 77-700 K. T h e n u m b e r s are the values of the exponent m in eq. (4).

197 104

I

I

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+

the resistivity phonons.

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103 Peon-0'73

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the

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5. C o n c l u s i o n s

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I

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CARRIER CONCENTRATION RHALL(Cm-3)

Fig. 5. The difference between the 300 K and 4.2 K resistivities plotted against the carrier concentration nHa,. A leastsquares fit yields p ~ n -073. For comparison, the /9 ~ n 5/3 dependence obtained by Thompson [8] is also indicated.

exponent rn with stoichiometry is not consistent with simple electron-electron scattering for which we expect a pure T 2 dependence. Fig. 5 shows the dependence of p ( 3 0 0 K ) p(4.2K) on the carrier concentration nHa,. We find that O ~//-0.73. For comparison, we also show in fig. 5 the p ~ n -5t3 relationship obtained by T h o m p s o n [8] under the assumption that the carrier concentration in his crystals can be calculated from the excess Ti, x, and that each Ti atom contributes four electrons. The two results are in clear disagreement. The reason cannot be T h o m p s o n ' s assumptions in calculating n from x, since our thermoelectric power data have shown these assumptions to be valid, at least for x > 0.01 (n larger than a few times 1020 carriers per cm3). Our findings that p ( 3 0 0 K ) - p ( 4 . 2 K ) is proportional to //-0.73 rather than to n -5/3 and the absence of a clear T2-dependence of the resistivity raises strong doubts concerning the previous interpretation [8] of electron--electron scattering as the dominant scattering process in (stoichiometric) TiS2. W e have attempted but have been unable so far to determine the nature of the scattering process that give rise to the dependences described above. Wilson [13] has suggested Fivaz-mode optical p h o n o n scattering, but we find that this model cannot fit our data over a substantial temperature range. On the other hand, the approximate n -2/3 dependence of

With our extensive studies of the reflectivity, Hall effect, thermoelectric power and resistivity of titanium disulphide and their dependences on temperature and nonstoichiometry, we have succeeded in clarifying the electronic structure of this material and obtaining new information concerning the transport properties. The main conclusions are as follows: (1) TiS2 is not a semimetal but a semiconductor. (2) Our most stoichiometric samples were found to still be metallic with an electron concentration of about 2 × 1020cm -3. The origin of these carriers is not known. (3) In highly nonstoichiometric samples, x ~0.01, the conduction electrons appear to arise from excess Ti atoms, each contributing four electrons to the conduction band. (4) The temperature dependence of the resistivity is not strictly T 2, but approximately a power law with an exponent that ranges between 2.2 and 1.8 decreasing with increasing nonstoichiometry. (5) The dependence of the resistivity on carrier concentration is not the previously reported n -513 but rather n -°'73. (6) We find no compelling evidence for electron-electron scattering and optical p h o n o n scattering cannot explain the data. We consider the scattering mechanism as unknown. References

[1] M.S. Whittingham, Science 192 (1976) 1126; and in: Progress in Solid State Chemistry 12 (1978) p. 1. [2] B.C.H. Steele, in: Superionic Conductors, G.D. Mahan and W.L. Roth, eds. (Plenum Press, New York, 1976) p. 47. [3] D.W. Murphy, F.J. DiSalvo, J.N. Carides and J.W. Waszczak, Mat. Res. Bull. 13 (1978) 1395. [4] F.K. McTaggart and A.B. Wadsley, Aust. J. Chem. 11 (1958) 445. [5] S. Takeuchi and H. Katsuta, J. Jap. Inst. Metals 34 (1970a) 758; and J. Jap Inst. Metals 34 (1970b) 764. [6] A.H. Thompson, K.R. Pisharody and R.F. Koehler, Jr., Phys. Rev. Lett. 29 (1972) 163. [7] J.A. Benda, Phys. Rev. B10 (1974) 1409. [8] A.H. Thompson, Phys. Rev. Lett. 35 (1975) 1786. [9] E.M. Logothetis, W.J. Kaiser, Carl A. Kukkonen, S.P. Faile, R. Colella and J. Gambold, J. Phys. C 12 (1979) L521. [10] D.L. Greenway and R. Nitsche, J. Phys. Chem. Solids 26 (1%5) 1445.

198 [11] H.W. Myron and A.J. Freeman, Phys. Rev. B9 (1974) 481; A. Zunger and A.J. Freeman, Phys. Rev. B16 (1977) 906. [12] A.H. Thompson, F.R. Gamble and C.R. Symon, Mat. Res. Bull. 10 (1975) 915. [13] J.A. Wilson, Phys. Stat. Sol. (b) 86 (1978) 11.

[14] R.H. Friend, D. Jerome, W.Y. Liang, J.C. Mikkelsen and A.D. Yoffe, J. Phys. C: Solid St. Phys. 10 (1977) L705. [15] See for example J.M. Ziman, Electrons and Phonons (Oxford Univ. Press, London, 1960) p. 396.