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On variable range hopping in amorphous films of germanium and silicon
Thin Solid Films, 50 (1978) 151-155 © Elsevier Sequoia S.A., Lausanne--Printed in the Netherlands
151
ON VARIABLE R A N G E H O P P I N G IN A M O R P H O U S FILMS OF G E R M A N I U M A N D SILICON S. GUHA AND K. L. NARASIMHAN
Tata Institute of Fundamental Research, Bombay 400005 (India)
The experimental data on the d.c. conductivity, a.c. conductivity and thermoelectric power of evaporated or sputtered films of amorphous germanium and silicon are critically examined. It is shown that the variable range hopping theories at present available do not explain the experimental data satisfactorily. The applicability of a heterogeneous model to explain the experimental results is discussed.
The electronic properties of evaporated and sputtered films of amorphous germanium and silicon have been extensively studied in recent years. It is believed that these materials have a large number of dangling bonds and voids which give rise to states in the gap, and the electronic transport in these materials is usually considered to be due to phonon-assisted hopping in these localized states 1. For low temperatures Mott z has shown that hopping takes place between sites that are physically more distant but energetically closer, and that the conduction is essentially governed by this process of variable range hopping. Since Mott's work, more rigorous theories a have been worked out for variable range hopping and it is widely believed that the electronic properties of tetrahedrally bonded semiconductors can be understood on the basis of this model. In this paper we examine critically the experimental data on the d.c. conductivity, a.c. conductivity and thermoelectric power of these materials and show that, contrary to popular belief, the variable range hopping theory does not explain the experimental data satisfactorily. Let us first turn our attention to the d.c. conductivity. Mott 2 has shown that at low temperatures the conductivity due to electrons hopping in the localized states at the Fermi level obeys the relation tr = troeXp
{-(To~T) 1;'}
where tro and T o are related to the density of states N(EF) at the Fermi level. A T - 1/4 dependence is observed a in evaporated and sputtered films o f a - G e and a-Si and this is usually considered to be evidence for the hopping model. Stronger support comes from measurements on thin films. Theory predicts a transition from the T - 1/4 law to a T - 1,/3law as the material changes from a three-dimensional to a two-dimensional structure. This has been observed by Knotek et'al. 5 as the thickness of films of a-Ge
] 52
S. GUHA, K. L. NARASIMHAN
deposited under ultrahigh vacuum conditions was decreased below 500 A. A similar behaviour has also been reported 6 for a-Si. A closer look at the experimental results, however, shows many discrepancies between theory and experiment. The value of N(EF) obtained; from TO is about 10 is cm -3 eV-1. This is lower by a factor of 50-100 than estimates from, for example, field effect 8 or tunnelling 9 measurements. Moreover, even when the conductivity is changed by several orders of magnitude by the incorporation of hydrogen in the film during deposition ~°. there is not much change in TO but field effect results 11 indicate a considerable change in N ( E v ) . Calculations ~ based on a o, however, give physically absurd values like 10 3o e g - 1 cm 3 for N(EF). It is debatable, therefore, whether just obtaining a fit to the T ~..4law can be considered as sufficiently good evidence for the hopping model. With regard to the experimental results on thinner films, we would like to point out that it is not easy to distinguish between a T - 1/3 and T-1/4 law over a limited temperature range. This is illustrated in Fig. 1, where we show the data of Knotek e t al. 5 for a 62/~ film that has been found to obey the T - 1.,3 law. The conductivity at different temperatures was calculated by assuming a T - 1/3 law and by noting the slope of the curve and the conductivity at one particular temperature from Knotek's data. Since we used the T-1/3 law for obtaining the conductivity, any error that might have arisen from reading the conductivity at various temperatures directly from the In o v e r s u s T - 13 plot of Knotek e t al. was avoided. We see from Fig. 1 that the fit to a T - 14 law is also extremely good, i.e. the same data points give a good fit to both T 13 and T - 1/4 over the limited range of temperature between 80 and 250 K (which is the temperature range in which the conductivity of thinner films had been measured). The evidence for a transition from the T 1/4 law to a T-1.3 law is therefore not unambiguous and we should look for other independent evidence for the hopping model. Measurement of the a.c. conductivity is considered to be useful for this purpose. According to the Austin-Mott formula 12 for hopping in localized states, the a.c. conductivity a~c is related to frequency by the relation O'ac OC N ( E v ) 2 0 9 '
where for frequencies below 10s Hz s = 0.8. Such an uY law has been observed in various amorphous materials 13 and is usually explained in terms of the AustinMott formula. Estimates ~4 of N(EF) from a.c. conductivity data have been made for a-Ge and the values are found to be an order of magnitude higher than those obtained from T Ofor the same films. The magnitude of s, however, has been found ~5 to be dependent on temperature; s decreases as the temperature is lowered. It has been shown 15 that the observed decrease in s with temperature cannot be explained by the hopping model; in fact the theoretical models predict a temperature dependence which is just opposite to the observed behaviour. We now consider the results for the thermoelectric power. It has been shown 16 that if hopping takes place at the Fermi level the thermopower will be very small. A very small thermoelectric power is observed at low temperatures in a-Ge and a-Si and this is usually considered 4 to be evidence for the hopping model. It has recently been pointed out by Emin 17 that, when the density of states is symmetrical about the Fermi level, variable range hopping predicts the thermoelectric power to be
VARIABLE RANGE H O P P I N G IN
Ge
AND Si FILMS
153
10-7
10-e
10-~
1
0
-
~
~
0.2.50 0.260 0.270 0.280 0290 0300 0310 0.32,0 T-V~
Fig. 1. Plot of the data of Knotek et al.S for a 62 ,~ film as a function of T - ~/'*.
proportional to T 1/2. Lewis 18 has measured the thermoelectric power o f a - G e and aSi over the temperature range in which variable range hopping is believed to take place, and he finds that for a-Ge the thermoelectric power is temperature independent between 70 and 300 K. Similar results are observed for a-Si between 200 and 300 K. This is obviously not compatible with the predictions from the variable range hopping theory. We thus find that the variable range theories that are available at present do not explain the experimental results satisfactorily. It should be mentioned, however, that most of these theories assume N ( E r ) to be independent of energy, whereas in reality field effect experiments 8 indicate that this is incorrect. Efforts have been made 19 to calculate the d.c. conductivity by considering various shapes of energy distribution near the Fermi level. However, a satisfactory answer to the problem of proper estimates of N(EF) from tro and T O is yet to emerge. Calculations of the thermoelectric power have also been made ~8 using three different density of states models at the Fermi level. None of these models is found to explain the experimental data (namely, the temperature-independent thermoelectric power) satisfactorily. It may be argued, however, that, in view of the presence of a large number of voids 2° with diameters from that o f a divacancy to about 20/~, any theoretical work that considers the medium to be homogeneous cannot be very successful. It has been shown that heterogeneities can cause band bending 2~ and charge and potential fluctuations 22. Transport in semiconductors with potential fluctuations has been studied 23 on the basis of a heterogeneous model. Adler et al. 24 have shown that it is possible to obtain a T-1/4 law within the framework of this model. The model also
154
S. GUHA, K. L. NARASIMHAN
predicts 4 a difference in the activation energy of the conductivity and the slope of the thermoelectric power v e r s u s temperature plot, as has been observed experimentally. Recently 25 Guha and Narasimhan have calculated the a.c. conductivity of a heterogeneous medium using the effective medium theory 26. They have found that it is possible to obtain a temperature dependence of s similar to the observed behaviour. The results obtained from the heterogeneous models, however, are critically dependent on the parameters that define the heterogeneity. In the absence o f any data on these parameters, attempts to analyse the experimental data quantitatively are not very meaningful. In conclusion, we have shown that the theories available at present that are based on variable range hopping do not explain the transport in evaporated or sputtered films o f a - G e or a-Si quantitatively. Whilst it is possible to explain the data on the basis o f a heterogeneous model, the results obtained from this model are very sensitive to the parameters that define the heterogeneity and no quantitative comparison can therefore be made between theory and experiment. REFERENCES 1 For a recent review, see W. Paul, Thin Solid Films, 33 ( t 976) 381. 2 N . F . Mort, Philos. Mag., 19(1969) 835. 3 V. Ambegaonkar, B. I. Halperin and J. S. Langer, Phys. Rev., Sect. B, 4 ( 1971 ) 2612. W. Brenig, G. H. Dohler and P. Wolfle, Z. Phys., 246 (1971 ) 1 ; 258 (1973) 381. M. Pollak, J. Non-cryst. Solids, 11 (1972) 1. S. Kirkpatrick, in J. Stuke and W. Brenig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 183. 4 See, for example, H. Mell, in J. Stuke and W. Brenig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 203. 5 M . L . Knotek, M. Pollak, T. M. D o n o v a n and H. K u n t z m a n , Phys. Rev. Lett., 30 (1973) 853. 6 M . L . Knotek, SolidState Commun., 17(1975) 1431. 7 D . K . Paul and S. S. Mitra, Phys. Rev. Lett., 31 (1973) 1000. M. H. Brodsky and R. J. Gambino, J. Non-cryst. Solids, 8 10 (1972) 739. 8 A. Madan, P. G. LeComber and W. E. Spear, J. Non-cryst. Solids, 29 (1976) 239. M. Hirose, M. Taniguchi and Y. Osaka, Jpn. J. Appl. Phys., 15 (1976) 175. A. K. Malhotra and G. W. Neudeck, J. Appl. Phys., 46 (1975) 2690. 9 J . W . O s m u n and H. Fritzsche, AlP Conf. Proc., 20 (1974) 333. J. J. Hauser, AlP Conf. Proc., 20 (1974) 338. 10 A.J. Lewis, Phys. Rev., Sect. B, 14 (1976) 658. 11 A . K . Malhotra and G. W. Neudeck, Appl. Phys. Lett., 28 (1976) 47. 12 I.G. Austin and N. F. Mott, Adv. Phys., 18 (1969) 41. 13 For a recent review, see M. Pollak, in G. Kolomiets (ed.), Proc. 6th Int. ConL on Amorphous and Liquid Semiconductors, Leningrad, 1975, Nauka, Leningrad, 1976. 14 S.C. Agarwal, S. G u h a and K. L. Narasimhan, J. Non-tryst. Solids, 18 (1975) 429. 15 K . L . Narasimhan, S. G u h a and S. C. Agarwal, Solid State Commun., 20 (1976) 573. 16 N . F . Mott and E. A. Davis, Electronic Processes in Non-crystalline Materials, Clarendon Press, Oxford, 1971. 17 D. Emin, Phys. Rev. Lett.,35(1975)882. 18 A . J . Lewis, Phys. Rev., Sect. B, 13 (1976) 2565. 19 M. Pollak, J. Non-cryst. Solids, 11 (1972) 1. M. Pollak, M. L. Knotek, H. K u r t z m a n and H. Glick, Phys. Rev. Left., 30 (1973) 856. K. M aschke, H. Overhof and P. Thomas, Phys. Status Solidi B, 62 (1974) 113. 20 S.C. MossandJ. F. Graczyk, Phys. Rev. Lett.,23(1969) l167. M. H. Brodskv and R. S. Title, Phys. Rev. Lett., 23 (1969) 581.
VARIABLE RANGE HOPPING IN
21 22 23 24 25 26
Ge AND Si FILMS
155
W. Paul, G. A. N. Connell and R. J. Temkin, Adv. Phys., 22 (1973) 529. D. "1".Pierce and W. E. Spicer, Phys. Rev. Lett., 27 (1971) 1217. H. Fritzsche, in J. Tauc (ed.), Amorphous and Liquid Semiconductors, Plenum Press, New York, 1973. Chap. 5. M . H . Cohen, J. Non-cryst. Solids, 14 (1970) 391. R. Zallen and H. Scher, Phys. Rev., Sect. B, 4 (1971) 4471. D. Adler, L. P. Flora and D. Senturia, Solid State Commun., 12 (19733 9. S. Guha and K. L. Narasimhan, Phys. Rev., Sect. B, (1978) to be published. B.E. Springett, Phys. Rev. Lett., 31 (19733 1463.
151
ON VARIABLE R A N G E H O P P I N G IN A M O R P H O U S FILMS OF G E R M A N I U M A N D SILICON S. GUHA AND K. L. NARASIMHAN
Tata Institute of Fundamental Research, Bombay 400005 (India)
The experimental data on the d.c. conductivity, a.c. conductivity and thermoelectric power of evaporated or sputtered films of amorphous germanium and silicon are critically examined. It is shown that the variable range hopping theories at present available do not explain the experimental data satisfactorily. The applicability of a heterogeneous model to explain the experimental results is discussed.
The electronic properties of evaporated and sputtered films of amorphous germanium and silicon have been extensively studied in recent years. It is believed that these materials have a large number of dangling bonds and voids which give rise to states in the gap, and the electronic transport in these materials is usually considered to be due to phonon-assisted hopping in these localized states 1. For low temperatures Mott z has shown that hopping takes place between sites that are physically more distant but energetically closer, and that the conduction is essentially governed by this process of variable range hopping. Since Mott's work, more rigorous theories a have been worked out for variable range hopping and it is widely believed that the electronic properties of tetrahedrally bonded semiconductors can be understood on the basis of this model. In this paper we examine critically the experimental data on the d.c. conductivity, a.c. conductivity and thermoelectric power of these materials and show that, contrary to popular belief, the variable range hopping theory does not explain the experimental data satisfactorily. Let us first turn our attention to the d.c. conductivity. Mott 2 has shown that at low temperatures the conductivity due to electrons hopping in the localized states at the Fermi level obeys the relation tr = troeXp
{-(To~T) 1;'}
where tro and T o are related to the density of states N(EF) at the Fermi level. A T - 1/4 dependence is observed a in evaporated and sputtered films o f a - G e and a-Si and this is usually considered to be evidence for the hopping model. Stronger support comes from measurements on thin films. Theory predicts a transition from the T - 1/4 law to a T - 1,/3law as the material changes from a three-dimensional to a two-dimensional structure. This has been observed by Knotek et'al. 5 as the thickness of films of a-Ge
] 52
S. GUHA, K. L. NARASIMHAN
deposited under ultrahigh vacuum conditions was decreased below 500 A. A similar behaviour has also been reported 6 for a-Si. A closer look at the experimental results, however, shows many discrepancies between theory and experiment. The value of N(EF) obtained; from TO is about 10 is cm -3 eV-1. This is lower by a factor of 50-100 than estimates from, for example, field effect 8 or tunnelling 9 measurements. Moreover, even when the conductivity is changed by several orders of magnitude by the incorporation of hydrogen in the film during deposition ~°. there is not much change in TO but field effect results 11 indicate a considerable change in N ( E v ) . Calculations ~ based on a o, however, give physically absurd values like 10 3o e g - 1 cm 3 for N(EF). It is debatable, therefore, whether just obtaining a fit to the T ~..4law can be considered as sufficiently good evidence for the hopping model. With regard to the experimental results on thinner films, we would like to point out that it is not easy to distinguish between a T - 1/3 and T-1/4 law over a limited temperature range. This is illustrated in Fig. 1, where we show the data of Knotek e t al. 5 for a 62/~ film that has been found to obey the T - 1.,3 law. The conductivity at different temperatures was calculated by assuming a T - 1/3 law and by noting the slope of the curve and the conductivity at one particular temperature from Knotek's data. Since we used the T-1/3 law for obtaining the conductivity, any error that might have arisen from reading the conductivity at various temperatures directly from the In o v e r s u s T - 13 plot of Knotek e t al. was avoided. We see from Fig. 1 that the fit to a T - 14 law is also extremely good, i.e. the same data points give a good fit to both T 13 and T - 1/4 over the limited range of temperature between 80 and 250 K (which is the temperature range in which the conductivity of thinner films had been measured). The evidence for a transition from the T 1/4 law to a T-1.3 law is therefore not unambiguous and we should look for other independent evidence for the hopping model. Measurement of the a.c. conductivity is considered to be useful for this purpose. According to the Austin-Mott formula 12 for hopping in localized states, the a.c. conductivity a~c is related to frequency by the relation O'ac OC N ( E v ) 2 0 9 '
where for frequencies below 10s Hz s = 0.8. Such an uY law has been observed in various amorphous materials 13 and is usually explained in terms of the AustinMott formula. Estimates ~4 of N(EF) from a.c. conductivity data have been made for a-Ge and the values are found to be an order of magnitude higher than those obtained from T Ofor the same films. The magnitude of s, however, has been found ~5 to be dependent on temperature; s decreases as the temperature is lowered. It has been shown 15 that the observed decrease in s with temperature cannot be explained by the hopping model; in fact the theoretical models predict a temperature dependence which is just opposite to the observed behaviour. We now consider the results for the thermoelectric power. It has been shown 16 that if hopping takes place at the Fermi level the thermopower will be very small. A very small thermoelectric power is observed at low temperatures in a-Ge and a-Si and this is usually considered 4 to be evidence for the hopping model. It has recently been pointed out by Emin 17 that, when the density of states is symmetrical about the Fermi level, variable range hopping predicts the thermoelectric power to be
VARIABLE RANGE H O P P I N G IN
Ge
AND Si FILMS
153
10-7
10-e
10-~
1
0
-
~
~
0.2.50 0.260 0.270 0.280 0290 0300 0310 0.32,0 T-V~
Fig. 1. Plot of the data of Knotek et al.S for a 62 ,~ film as a function of T - ~/'*.
proportional to T 1/2. Lewis 18 has measured the thermoelectric power o f a - G e and aSi over the temperature range in which variable range hopping is believed to take place, and he finds that for a-Ge the thermoelectric power is temperature independent between 70 and 300 K. Similar results are observed for a-Si between 200 and 300 K. This is obviously not compatible with the predictions from the variable range hopping theory. We thus find that the variable range theories that are available at present do not explain the experimental results satisfactorily. It should be mentioned, however, that most of these theories assume N ( E r ) to be independent of energy, whereas in reality field effect experiments 8 indicate that this is incorrect. Efforts have been made 19 to calculate the d.c. conductivity by considering various shapes of energy distribution near the Fermi level. However, a satisfactory answer to the problem of proper estimates of N(EF) from tro and T O is yet to emerge. Calculations of the thermoelectric power have also been made ~8 using three different density of states models at the Fermi level. None of these models is found to explain the experimental data (namely, the temperature-independent thermoelectric power) satisfactorily. It may be argued, however, that, in view of the presence of a large number of voids 2° with diameters from that o f a divacancy to about 20/~, any theoretical work that considers the medium to be homogeneous cannot be very successful. It has been shown that heterogeneities can cause band bending 2~ and charge and potential fluctuations 22. Transport in semiconductors with potential fluctuations has been studied 23 on the basis of a heterogeneous model. Adler et al. 24 have shown that it is possible to obtain a T-1/4 law within the framework of this model. The model also
154
S. GUHA, K. L. NARASIMHAN
predicts 4 a difference in the activation energy of the conductivity and the slope of the thermoelectric power v e r s u s temperature plot, as has been observed experimentally. Recently 25 Guha and Narasimhan have calculated the a.c. conductivity of a heterogeneous medium using the effective medium theory 26. They have found that it is possible to obtain a temperature dependence of s similar to the observed behaviour. The results obtained from the heterogeneous models, however, are critically dependent on the parameters that define the heterogeneity. In the absence o f any data on these parameters, attempts to analyse the experimental data quantitatively are not very meaningful. In conclusion, we have shown that the theories available at present that are based on variable range hopping do not explain the transport in evaporated or sputtered films o f a - G e or a-Si quantitatively. Whilst it is possible to explain the data on the basis o f a heterogeneous model, the results obtained from this model are very sensitive to the parameters that define the heterogeneity and no quantitative comparison can therefore be made between theory and experiment. REFERENCES 1 For a recent review, see W. Paul, Thin Solid Films, 33 ( t 976) 381. 2 N . F . Mort, Philos. Mag., 19(1969) 835. 3 V. Ambegaonkar, B. I. Halperin and J. S. Langer, Phys. Rev., Sect. B, 4 ( 1971 ) 2612. W. Brenig, G. H. Dohler and P. Wolfle, Z. Phys., 246 (1971 ) 1 ; 258 (1973) 381. M. Pollak, J. Non-cryst. Solids, 11 (1972) 1. S. Kirkpatrick, in J. Stuke and W. Brenig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 183. 4 See, for example, H. Mell, in J. Stuke and W. Brenig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 203. 5 M . L . Knotek, M. Pollak, T. M. D o n o v a n and H. K u n t z m a n , Phys. Rev. Lett., 30 (1973) 853. 6 M . L . Knotek, SolidState Commun., 17(1975) 1431. 7 D . K . Paul and S. S. Mitra, Phys. Rev. Lett., 31 (1973) 1000. M. H. Brodsky and R. J. Gambino, J. Non-cryst. Solids, 8 10 (1972) 739. 8 A. Madan, P. G. LeComber and W. E. Spear, J. Non-cryst. Solids, 29 (1976) 239. M. Hirose, M. Taniguchi and Y. Osaka, Jpn. J. Appl. Phys., 15 (1976) 175. A. K. Malhotra and G. W. Neudeck, J. Appl. Phys., 46 (1975) 2690. 9 J . W . O s m u n and H. Fritzsche, AlP Conf. Proc., 20 (1974) 333. J. J. Hauser, AlP Conf. Proc., 20 (1974) 338. 10 A.J. Lewis, Phys. Rev., Sect. B, 14 (1976) 658. 11 A . K . Malhotra and G. W. Neudeck, Appl. Phys. Lett., 28 (1976) 47. 12 I.G. Austin and N. F. Mott, Adv. Phys., 18 (1969) 41. 13 For a recent review, see M. Pollak, in G. Kolomiets (ed.), Proc. 6th Int. ConL on Amorphous and Liquid Semiconductors, Leningrad, 1975, Nauka, Leningrad, 1976. 14 S.C. Agarwal, S. G u h a and K. L. Narasimhan, J. Non-tryst. Solids, 18 (1975) 429. 15 K . L . Narasimhan, S. G u h a and S. C. Agarwal, Solid State Commun., 20 (1976) 573. 16 N . F . Mott and E. A. Davis, Electronic Processes in Non-crystalline Materials, Clarendon Press, Oxford, 1971. 17 D. Emin, Phys. Rev. Lett.,35(1975)882. 18 A . J . Lewis, Phys. Rev., Sect. B, 13 (1976) 2565. 19 M. Pollak, J. Non-cryst. Solids, 11 (1972) 1. M. Pollak, M. L. Knotek, H. K u r t z m a n and H. Glick, Phys. Rev. Left., 30 (1973) 856. K. M aschke, H. Overhof and P. Thomas, Phys. Status Solidi B, 62 (1974) 113. 20 S.C. MossandJ. F. Graczyk, Phys. Rev. Lett.,23(1969) l167. M. H. Brodskv and R. S. Title, Phys. Rev. Lett., 23 (1969) 581.
VARIABLE RANGE HOPPING IN
21 22 23 24 25 26
Ge AND Si FILMS
155
W. Paul, G. A. N. Connell and R. J. Temkin, Adv. Phys., 22 (1973) 529. D. "1".Pierce and W. E. Spicer, Phys. Rev. Lett., 27 (1971) 1217. H. Fritzsche, in J. Tauc (ed.), Amorphous and Liquid Semiconductors, Plenum Press, New York, 1973. Chap. 5. M . H . Cohen, J. Non-cryst. Solids, 14 (1970) 391. R. Zallen and H. Scher, Phys. Rev., Sect. B, 4 (1971) 4471. D. Adler, L. P. Flora and D. Senturia, Solid State Commun., 12 (19733 9. S. Guha and K. L. Narasimhan, Phys. Rev., Sect. B, (1978) to be published. B.E. Springett, Phys. Rev. Lett., 31 (19733 1463.