Small polaron transport in amorphous V2O5 films

Journal of Non-CrystaUine Solids 124 (1990) 71-75 North-Holland

71

Small polaron transport in amorphous V205 films L. Murawski a, C. Sanchez, J. Livage and J.P. Audiere b Chimie de la Matihre Condens~e, Universit~ Paris VI, 4, place Jussieu, 75005 Paris, France a Faculty of Applied Physics and Mathematics, Technical University of Gdahsk, 80-952 Gdahsk, Poland b Laboratoire Spectrochimie des Elements de Transition, Universit~ Paris-Sua~ Batiment 420, 91405 Orsay, France Reoeived 19 April 1989 Revised manuscript received 21 February 1990

DC conductivity in amorphous V20s films exhibits T -1/4 dependence over a wide temperature range. This behavior is analysed using Triberis and Friedman percolation model of small polaron hopping. A reasonable value of density of states is obtained as a result of the application of this theory but a large pre-exponential factor that is difficult to explain is obtained. Comparison with previous calculations based on a non-percolation model of multiphonon assisted small polaron hopping is made.

1. Introduction The polaron model of electrical conductivity is generally applicable to transition metal oxide glasses and amorphous transition metal oxides [1-4]. The theoretical considerations of small polaron hopping are based on the molecular crystal model introduced by Holstein [5]. This model was extended to disordered systems by Schnakenberg [6] and Emin [7]. The main feature of the electrical conductivity in transition metal oxide glasses is that the activation energy decreases with decreasing temperature. As the temperature is lowered, the multiphonon processes are frozen out and the high temperature activation energy is expected to drop continuously from Wn + ½W D to 14"o , where WH is the polaron hopping energy and WD is a disorder term arising from the energy differences of neighboring hopping sites. In the temperature range T < 0/4, where 0 is the Debye temperature, charge carrier transport should be an acoustic phonon-assisted hopping process and in the lowtemperature limit the activation energy for a hop upward in energy is WD. In the high-temperature range (T > 0/2) a multiphonon process takes place and the activation energy for a small polaron hop upward in energy by the amount W o is Wrt + ½WD

regardless of whether the phonons involved are acoustic, optical or mixed optical and acoustic [8,9]. The experimental evidence for the above behavior has been observed by many authors in amorphous transition metal oxides and glasses [2,10-20]. Some PEOs-V2Os, P2Os-WO3 and TeO2-V205 glasses containing a large amount of transition metal oxides (> 50 mol%) exhibit In o T - 1 / 4 dependence over a relatively broad temperature range [12-17]. However, this behavior cannot be interpreted satisfactorily as Mott's variable range hopping even in the low temperature range. For example, the values of Wo calculated from Mott's formula are always much higher than the low temperature activation energy [12-17]. DC hopping conductivity in disordered systems at low temperature has been studied by many authors [21,22] using percolation-theoretic considerations. In this case the electronic transport is described as a single phonon-induced tunnelling of electrons between localised states which are randomly distributed in energy and position. It has been shown that the temperature dependence of conductivity has the Mott's law form: In o T-~/4. Recently, Triberis and Friedman have pubfished several works [23-26] on the percolation

0022-3093/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

L. Murawski et al. / Small polaron transport in a- V20~ films

72

treatment o f conductivity for small polaron hopping in disordered systems. They have also applied percolation theory to the high temperature multiphonon assisted small polaron hopping regime. The most interesting result is that T - 1 / 4 dependence of conductivity for the few phonon low-T and multiphonon high-T region should be observed. In this paper we apply Triberis and Friedman's model [24] to the experimental results of dc conductivity measurements in amorphous V205 and Li~,V205 films. We have found that the conductivities of our samples obey the T -1/4 law.

2. Sample preparation and results of previous investigations i19,201 Details of sample preparation have been previously published [19,20]; thus we briefly describe experimental techniques and results o f dc conductivity measurements of V205 and LixV20s films. We have applied two methods of evaporation: vapor deposition [19] and flash evaporation [20]. The temperature of the substrate was changed from 77 to 673 K in the case of vapor deposited samples. The vacuum and temperature of the molten oxide were carefully controlled and constant during the evaporation process ( P - 10 -6 Torr, T = 1113 ___5 K). The only variable in our experiment was the temperature of the substrate during deposition. Thus, we expected that C = V4+/Vtot (i.e. the ratio of the concentration of V4+ ions to the total concentration of vanadium ions) was constant for all samples. This was confirmed by ESR and chemical analysis from which we have obtained the value C ~- 1.4%. Conductivity depends strongly on the quenching rate that is the difference of temperature between molten oxide and the substrate. The conductivity decreases and the activation energy increases with an increase of the quenching rate. We used the following expression [1,6] in order to fit our experimental data: tl

=

oo

e x p / - --~-~o 4WH tanh 4ht°° ~ ) exp(

-

WD/2kT),

(1) where ht00 = kO.

A computer analysis of the conductivity curves showed that the disorder term WD in activation energy increases from 0.05 to 0.22 eV when the substrate temperature decreases from 673 to 393 K [19]. However, these films were not very homogeneous [27,28]. Differential thermal analysis (in situ) [28] showed several relaxations below the glass transition temperature. These processes observed for highly quenched films are attributed to relaxations processes from highly metastable state towards more stable states. More homogeneous films were obtained using the flash evaporation method [20]. In this case the flash evaporated particles are more uniform and therefore more homogeneous films were obtained that did not exhibit relaxation processes below Tg. The optimum evaporation temperature was - 1350-1400 K while that of the substrate temperature was between 77 and 300 K. The conductivity was higher than in V205 vapor deposition films obtained at similar quenching rates. Analysis of the conductivity curves showed that the disorder term WD is much lower for flash evaporated films [20].

3. Percolation treatment of small polaron hopping in V20s amorphous films The percolation approach to hopping conduction in disordered systems is based on the assumption that electron motion between localised states, which are randomly distributed in energy and position, can be treated as a problem of current flow through a network of impedances Zij which connect the different lattice sites. The values of these impedances depend on the site energies Ei, Ej and the distance R~j between the two sites. The impedances can vary by many orders of magnitude because of the exponential dependence of Z~j on Rij and E i, Ej. The conductivity of the material is characterised by the critical impedance Zjj = Z c which interconnect small clusters of low impedances through percolation paths in the sample. When the site energies are not the same the effect of correlation between neighboring impedances should be taken into account because the energy site affects the incoming as well as the outgoing Z.

73

h Murawski et al. / Smallpolaron transport in a-VgO5 films

The percolation treatment of dc conductivity for the high-temperature and low-temperature small polaron hopping regime presented by triberis and Friedman [24,25] leads to the expression for a critical impedance which, considering the effect of correlation, has the form

"7

0~ kO

Z¢ = Z o exp( To/T ) 1/4, where TO= 12.5 ot3//Nk, a -1 is the spatial extent of the electronic wave function and N is the density of states qr

Zo = kT/e2%;

~1/2.

"Yo= ( J Z / h )( 4 W ~ k T }

'

J is the electron transfer integral. In order to demonstrate that our experimental results obey the above formula, we plot the dc conductivity versus T a/4 for several V205 and LixV205 films. Figures 1 and 2 show a linear behavior for vapor deposited V205 and flash evaporated V205 and LixV205 films, respectively. The T -1/4 dependence is observed over a broad temperature range. The linearity has been verified by computer calculation in order to find the best fit to the experimental results. We found that the temperature exponent is 0.25 + 0.02. Table 1 presents the values of TOobtained from the slope of T -a/4 dependences of In o for vapor deposited V205 and flash evaporated V205 and LixV205 films. The densities of states were evaluated from the Triberis-Friedman formula assuming a = 1 A-1. This value of a seems to be reasonable if compared with that of V2Os-P205 glasses [2,12,13]. On the other hand, the basic feature of the small polaron should be satisfied namely, that a - l < ro < R. In a previous paper [19], we have estimated the values of small polaron radius in V20 % films depending on the quenching rate as: 1.6-3 A. The average V - V distance can be evaluated from the density measurements leads to the value of R = 3.84 A [19]. Knowing the V 4÷ ions concentration C = 0.014, we can find the number of charge carriers: n = C R - 3 = 2.5 X 1 0 20 c m - 3 . T h i s value is comparable

022

0.2/,

0.26

0.28

030 0.32 T-v4 [ K- 114]

Fig. 1. Plot of dc conductivity versus T -1/4 for tion (1113 K) of amorphous V205 films at the strate temperatures: 2, 293 K; 4, 458 K; 5, 523 2(f), flash evaporated sample 1373/77

vapor deposidifferent subK; 6, 573 K; K.

to the density of states obtained from the Triberis-Friedman formula. Therefore, we can presume that the percolation approach to small polaron hopping satisfactorily explains our data. Large discrepancies exist if one considers the preexponential factor %. If one calculates o0 from the absolute experimental value of conductivity, this value appears to be unreasonably high (1015-1018 ) S / c m . The Triberis-Friedman model does not consider the pre-exponential factor. It is possible to obtain the expression for o0 if the correlation length L 0 of the critical network is determined. According to Shklovskii and Efros [29], O"0 = ( L o Z o )

-1

74

L Murawski et a L / Small polaron transport in a-V205 films

'oE10-3 t_o

10.4

10-5

10-6

10.7

lOat

V205 f flosh

10.9

I

0.22

t

I

0.24

i

0.26

i

0.28

i

0. 0

T-114 [ K -114]

Fig. 2. The dependence of T -1/4 for flash evaporated (1373 K) amorphous V20s and LixV20s films. The substrate temperatures: 1, 77 K; 2, 287 K; 3 287 K (Lio.esV2Os); 4, 287 K (Lio.lo4V2Os).

silicon and germanium [30]. Ortuno and PoUak [31] have shown that this problem can be eliminated if we assume that the density of localised states near the Fermi level is an exponential function of energy. They have obtained, from the slope of T -1/4 dependence as well as from the value of o0, a reasonable density of states. It remains a question if the same approach could be applied to the small polaron hopping transport in a narrow bandwidth of localised states as in the case of amorphous V205films. In the interpretation put forward b y Emin [8,9] the T -1/4 behavior of the conductivity is explained by the temperature dependence of individual transition rates, assisted b y multiphonon interactions. The weak dependence of the conductivity on temperature can be attributed to freezing out of multiphonon process as T decreases. Emin [8] has pointed out that the observed non-activated temperature dependence of dc conductivity below the material's Debye temperature (440-580 K in V205 films [19,20]) can be a manifestation of multiphonon hopping between well-located states. In contrast to the T r i b e r i s - F r i e d m a n percolation model, Emin's theory predicts some deviation from T -1/4 in the lowest and the highest ( > 0) temperature range, so that the plot of In e vs. T -1/4 is 'S' shaped. This means that one does not expect to observe a large discrepancy between the measured o0 and that predicted by this theory. This m a y be also confirmed by our calculations [19,20] using

Table 1 (1) Vapor deposited films (1113 K)

It leads to the following formula for the pre-exponential factor [21]:

For example, in the case of sample no. 6, we obtained Z o = 5 x 10 .5 S if one assumes J = 0.02 e ¥ and W n = 0.2 eV. Taking T0 = 8 X 108 K, a = 1 .~-1 and T = 300 K, the pre-exponential factor o0 is - 1 S / c m . An unreasonably large pre-exponential factor for conductivity is an old problem with interpretations of hopping transport data on amorphous

Sample Substrate temperature (K) TO (K) N ( c m - 3 e V -1)

2

4

5

6

293 458 523 573 3.85 X 109 2.2 x 109 1.5 X 109 8.05 X 108 4.2x10 ]9 6.6x1019 9.85x1019 1.8X102°

(2) Flash evaporated films (1373 K) Sample Substrate temperature (K) To (K) N ( c m - 3 e V -1)

1

2

Lio.esV205 LioaoaV205

77 287 287 287 2.3 x 109 1.75 X 109 1.55 x 109 1.1 × 109 6.3x1019 8.3x1019 9.4X1019 1.3X10 z°

L. Murawski et al. / Small polaron transport in a-V205 films

formula (1) which was derived by Schnakenberg [6] for non-adiabatic phonon-assisted small polaron hopping in the case when the electron interacts only with optical phonon modes. This expression is a simplified form of Emin's more general theory [9]. We have not observed a discrepancy between theoretical and experimental values of o0 in a computer fit of our experimental data to eq. (1).

4. Conclusions Comparison of the above theories indicates that the non-percolation approach of Schnakenberg [6] and Emin [7] is more appropriate for the interpretation of the temperature dependence of dc conductivity in V205 and LixV205 films. As was previously shown [19,20] this theory explains the change of conductivity with quenching rate. It has been pointed out that the disorder term WD in the activation energy increases with quenching rate. In terms of the Triberis-Friedman percolation model the change of the conductivity is attributed to the change of the density of states. There is a agreement with the Triberis-Friedman model but it is difficult to explain the large value of pre-exponential factor.

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[5] [6] [7] [8] [9]

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T. Holstein, Ann. Phys. (NY) 8 (1959) 325. J. Schnakenberg, Phys. Status Solidi 28 (1968) 623. D. Emin, Adv. Phys. 24 (1975) 305. D. Emin, Phys. Rev. Left. 32 (1974) 303. E. Gorham-Bergeron and D. Emin, Phys. Rev. B15 (1977) 3667. [10] M. Sayer and A. Mansingh, Phys. Rev. B6 (1972) 4629. [11] J. Livage, J. Phys. (Paris) 42 (1980) C4-981. [12] G.N. Greaves, J. Non-Cryst. Solids 11 (1973) 427. [13] LG. Austin and E.S. Garbett in: Electronic and Structural Properties of Amorphous Semiconductors, eds. P.G. Le Comber and J. Mort (Academic Press, New York, 1973) p. 393. [14] A. Mansingh, A. Dhawan, R.P. Tandon and J.K. Vald, J. Non-Cryst. Solids 27 (1978) 309. [15] A. Mansingh, A. Dhawan and M. Sayer, J. Non-Cryst. Solids 33 (1979) 351. [16] V.K. Dhawan, A. Mansingh and M. Sayer, J. Non-Cryst. Solids 51 (1982) 87. [17] F.P. Koffyberg, J. Non-Cryst. Solids 28 (1978) 231. [18] C. Sanchez, R. Morineau and J. Livage, Phys. Status Solidi (a)76 (1983) 661. [19] C. Sanchez, J. Livage, J.P. Audiere and A. Madi, J. Non-Cryst. Solids 65 (1984) 285. [20] L. Murawski, C. Gledel, C. Sanchez, J. Livage and J.P. Audiere, J. Non-Cryst. Solids 89 (1987) 98. [21] H. BiSttger and V.V. Bryksin, in: Hopping Conduction in Solids (Akademie, Berlin, 1985) p. 113. [22] M. Pollak, J. Non-Cryst. Solids 11 (1972) 1. [23] G.P. Triberis and L.R. Friedman, J. Phys. C 14 (1981) 4631. [24] G.P. Triberis and L.R. Friedman, J. Phys. C 18 (1985) 2281. [25] G.P. Triberis, J. Non-Cryst. Solids 74 (1985) 1. [26] G.P. Triberis and L.R. Friedman, J. Non-Cryst. Solids 79 (1986) 29. [27] J.P. Audiere, A. Madi and J.C. Grenet, J. Mater. Sci. 17 (1982) 2973. [28] J.P. Audiere and A. Madi, Thin Solid Films 101 (1983) L29. [29] B.I. Shldovskii and A.L. Efros, Usp. Fiz. Nauk 117 (1975) 401. [30] A.M. Szpilka and P. Vi~d~or, Philos. Mag. B45 (1982) 485. [31] M. Ortuno and M. Pollak, Mag. 1347 (1983) L93.