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Hopping conduction in 3,4-cycloalkylpolypyrrole perchlorates: A model study of conductivity in polymers:
Volume 144, number 2
CHEMICAL PHYSICS LETTERS
HOPPING CONDUCTION IN 3,4XYCLOALKYLPOLYPYRROLE A MODEL STUDY OF CONDUCTIVITY IN POLYMERS:
19 February 1988
PERCHLORATES:
T.A. EZQUERRA, J. ROHE and G. WEGNER MoxPlanck-Institutfr Polymerforschung, Postfach 3148, D-6500 Maim, Federal Republic of Germany Received 15 September 1987; in final form 25 November 1987
The conductivity of 3,4-cycloalkyl-substituted polypyrrole perchlorates has been analyzed using hopping theory. The localization length of the charge carriers was found to be 0.8 nm, the actual conductivity at constant temperature being exponentially dependent on the minimum distance, R, between adjacent chains. R can be reasonably estimated from molecular models and was systematically varied between 0.38 nm (unsubstituted polypyrrole) and 1.38 nm (decamethylene chain attached in 3,4-position). The conduction is activated in the temperature range 150-300 K, the activation energy varying systematically from 0.0 12to 0.066 eV on increasing R from 0.38 to 1.38 nm.
The conduction mechanism in electronically conducting polymers and the true nature of the charge carriers remains a controversial subject [ 11. Various models have been proposed giving more-or-less precise approximation to experimental observations [ 2-51. In the case of the various salts of polypyrrole, the temperature dependence of the conductivity suggests a three-dimensional variable-range hopping of localized states: (1) However different samples give varying values of To and co indicating the existence of unknown morphological parameters, causing deviations from the variable-range hopping (VRH) model [ 61. Such modifications of the VRH model have been theoretically predicted [ 71 and have also been qualitatively observed [ 8,9], but until now a quantitative relation between the theoretical parameters of the hopping model and the experimental features is lacking. We have measured the temperature dependence of the dc conductivity of polypyrrole salts in which the 3,4 position of the pyrrole moiety was substituted by an alkyl ring. The method of synthesis of the polymers and their characterization in terms of the conductivity temperature dependence have been published elsewhere [ 10,111. In the following, a the194
oretical analysis of the data presented previously will be given in terms of a hopping model. In accordance with the theory of hopping conduction between localized states in three-dimensional disordered systems, the probability (w) for the hopping of a charge carrier between two states separated by a distance, R, is given by W-exp(-2aR-E,lkT),
(2)
a being the inverse localization length, and EA the energy barrier between the two states. Due to the lack of information concerning R, it has not been possible to distinguish between the tunneling term (2aR) and the activation energy term (EJkT). In the derivatives l-5 of polypyrrole salts investigated in the present work (see table 1) the minimum distance between polymeric chains can be controlled by changing the size of the substituent. A closer approximation than this minimum distance is only possible at the expense of an energetically unfavourable distortion of the alkyl rings. Because of the bulky substituents, adjacent pyrrole moieties of the same chain cannot be arranged in a plane, but have to be twisted as schematically depicted in fig. 1. Each individual polypyrrole backbone is cylindrically surrounded by a layer of methylene groups, the thickness of which is determined by the size of the alkyl ring fused to the pyrrole units in the 3,4 po-
0 009-2614/88/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Table 1 Formulae for the investigated polypyrrole derivatives, minimum distance R between adjacent chains as estimated with the help of molecular models, and activation energies Er, (as defined in eq. (2)) as obtained from fig. 3 ‘) Minimum distance R (nm)
EA (eV)
KJ0.T
0.38
0.012
iclo,-
0.76
0.019
Compound
1
J$ 11/p
2
Q% n -+-N w
8
N H
WO,
0.86
0.021
iClOT
1.03
0.047
+c10,
1.38
0.066
n l’l~”
4 9% N N” -ivy*
5 %
for this investigation are given in table 1. According to the hopping theory, two different temperature regimes can be distinguished. At low temperatures the charge carriers tend to hop not only to the nearest neighbour, but also to more remote sites in order to minimize the energy required for the hop. In this case the conductivity can be described by Mott’s law of variable-range hopping (eq. (1)). At higher temperatures almost exclusively nearestneighbour hops occur [ 121. In this case the conductivity can be described by a=u,exp(
T/s” 3
19 February 1988
CHEMICAL PHYSICS LETTERS
Volume 144, number 2
R
‘) See fig. 1 for the definition of R; estimated from space filing models.
sition. Thus, adjacent polymer backbones must be separated from each other in the bulk material by a distance R which can be estimated from molecular models as shown in fig. 1. The relevant distances R
1
2n+
R/2 I
2n CIO, Fig. 1. Schematic description of the polypyrrole backbone wrapped in a layer of poorly conducting methylene moieties of the alkylene chains fused to the pyrrole units in the 3,4 position. The minimum separation distance R between adjacent chants can be estimated from molecular models (compare table I ).
-2aR)
exp( -E,IkT)
.
(3)
From this equation a linear dependence of log o on the hopping distance R should be expected at a given temperature. The logarithm of the room-temperature conductivity of the investigated polypyrrole derivatives l-5 indeed shows this linear dependence on the minimum interchain distance R as shown in fig. 2. Analyzing our results in terms of eq. (3) assumes that carriers move by hopping between regions of one chain defined by the localization length I/a! to equivalent regions of an adjacent chain. As further evaluation of the data will point out, we have no information on i&u-chain processes, but a fully consistent picture will be developed on the assumption that the controlling factor for the description of the conductivity behaviour are the inter-chain processes. If the dependence of EA on R is known, the localization length l/a can be calculated according to eq. (3). The EA values for the different polymers can be obtained from the temperature dependence of the conductivity. A linear relationship log a versus l/T has been observed between 300 and 150 K (fig. 3). The activation energy is given by the slope A log o/A l/T (table 1). A temperature dependence of a-exp( -C/,1’4) or ff wexp( -C/T1’2) cannot be ruled out based on the available data in the temperature range investigated. However, the consistency of the data analyzed in terms of eq. (3) fully substantiates the assumption. As shown in fig. 4, the activation energy can be described by the empirical formula E,-BR
(4)
with a proportionality constant B= 7.37 x lo-’ eV/nm. An additional proof for this dependence can be 195
Volume144,number2
I9 February1988
CHEMICAL PHYSICSLETTERS
0
.2
.4
.6
LO
.6
R
6
7
I3
9
10
11
12 R
12
1A
16
[nml
13
14
15
16
tnml
Fig.2. Logarithmof the roomtemperaturedc conductivityof the investigatedpolymers1 to 5 as a functionof R. (a) Room-temperature measurements; (b) measurement at (0 ) 350,(A ) 250,( + ) 200and ( X ) 150 K.
obtained by evaluating the correlation between log [T and R at different temperatures. The general relation derived from eq. (3) is =_za
&% -
6R
--- sEA 1 6R kT’
(5)
Assuming that the localization length of the charge carrier is independent of R and for that reason GalSR=O, (5) can be simplified to =_za
6E,_1_ -6RkT’
196
(6)
A plot of (6 log d6R) T versus l/T should be linear, the slope giving the value of 6E,/6 R. The data for A log a/m are obtained from fig. 2b. The predicted linearity between A log aIAR and l/T can be seen in fig. 5 from which a value for AEJAR of 7.69 X 10d2 eV/nm is calculated, which is in very good agreement with the value of J3calculated above. Introducing the proportionality between EA and R (eq. (4)) into eq. (3) and taking the logarithm, it follows log Q_ -R(2a+BIkT)
-AIkT.
(7)
19 February 1988
CHEMICAL PHYSICS LETTERS
Volume 144, number 5
3
4
6
5
1000/T
7
[K-l]
Fig. 3. Temperature dependence of the conductivity of the polymers I-5 in the range from 300 to 150 K,
The term 2a! + BlkT can be identified with the slope of the straight line in fig. 2. The value of this slope at room temperature is approximately 5-3 urn- ’ and
therefore (xs: 1.2 nm- ‘. Using this, the localization
Fig. 4. Activation energies for the investigated polymers l-5 as calculated from eq. ( 2) as a function of R (compare table I )
length of the charge carrier can be estimated as
llaw0.8 nm. Summarizing the results obtained, the conduction
Fig. 5. Slopes 6 loga/GR asobtaiued from fig. 2 versus l/T (compare eq. ( 6)).
197
Volume 144, number 2
CHEMICAL PHYSICS LETTERS
can be understood as a hopping of localized charge carriers between adjacent chains separated by poorly conducting (“insulating”) regions of methylene groups. The probability for the hopping process is controlled by the substituent size which represents the minimum interchain distance. It is also worth noting that the value of the localization length found here for the substituted polypyrrole perchlorates is in very good agreement with the value estimated for the conducting salts of unsubstituted polypyrrole with polystyrenesulfonate as counterion [ 71. Furthermore, the present results and their analysis emphasize the overwhelming importance of interchain charge-carrier hopping processes for the macroscopic conduction behaviour of polymers. If intrachain processes were the controlling factor, the results obtained with the cycloalkylpolypyrroles could not be easily understood, because the conformational twist angle between adjacent rings in the backbone is practically independent of the ring size, as is immediately seen from space filling molecular models. The present work was supported in part by the BMFT (Project No. 03C156A 3). We wish to thank
198
19 February 1988
the CSIC (Spain) for additional support (for TAE) and the Stiftung Volkswagenwerk for a scholarship for JR.
References [] T.A. Skotheim, ed., Handbook of conducting polymers, Vol. 2 (Dekker, New York, 1986). [2] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. B22 (1980) 2209. [ 31 Y. Tomkiewicz, T.D. Schultz, H.D. Broom, T.C. Clarke and W.B. Street, Phys. Rev. Letters 43 (1979) 1532. [ 41 S. Kivelson and A.J. Epstein, Phys. Rev. B29 (1984) 336. [ 51J.C. Scott, P. PIIuger, M. Krombi and G.B. Street, Phys. Rev. B28 (1983) 40. [6] D.T. Glatzhofer, J. Ulanski and G. Wegner, Polymer 28 (1987) 449. [ 71 P. Sheng and J. Klafter, Phys. Rev. B27 (1983) 2583. [ 81 J. Ulanski, D.T. Glatzhofer, M. Przybylski, F. Kremer and G. Wegner, Polymer 28 (1987) 859. [9] E.K. Sichel, M. Knowles, M. Rubner and J. Georger Jr., Phys. Rev. B25 (1982) 5574. [lo] J. Riihe, Ch. Kriihnke, T.A. Ezquerra and G. Wegner, Springer series on solid state science ( 1987), to be published. [ 111 J. Rilhe, Ch. Krohnke, T.A. Ezquerra, F. Kremer and G. Wegner, Ber. Bunsenges. Physik. Chem. 9 (1987) 885. [ 121 N.E Mott and E.A. Davis, Electronic processes in non-crystalline materials (Clarendon Press, Oxford, 1979).
CHEMICAL PHYSICS LETTERS
HOPPING CONDUCTION IN 3,4XYCLOALKYLPOLYPYRROLE A MODEL STUDY OF CONDUCTIVITY IN POLYMERS:
19 February 1988
PERCHLORATES:
T.A. EZQUERRA, J. ROHE and G. WEGNER MoxPlanck-Institutfr Polymerforschung, Postfach 3148, D-6500 Maim, Federal Republic of Germany Received 15 September 1987; in final form 25 November 1987
The conductivity of 3,4-cycloalkyl-substituted polypyrrole perchlorates has been analyzed using hopping theory. The localization length of the charge carriers was found to be 0.8 nm, the actual conductivity at constant temperature being exponentially dependent on the minimum distance, R, between adjacent chains. R can be reasonably estimated from molecular models and was systematically varied between 0.38 nm (unsubstituted polypyrrole) and 1.38 nm (decamethylene chain attached in 3,4-position). The conduction is activated in the temperature range 150-300 K, the activation energy varying systematically from 0.0 12to 0.066 eV on increasing R from 0.38 to 1.38 nm.
The conduction mechanism in electronically conducting polymers and the true nature of the charge carriers remains a controversial subject [ 11. Various models have been proposed giving more-or-less precise approximation to experimental observations [ 2-51. In the case of the various salts of polypyrrole, the temperature dependence of the conductivity suggests a three-dimensional variable-range hopping of localized states: (1) However different samples give varying values of To and co indicating the existence of unknown morphological parameters, causing deviations from the variable-range hopping (VRH) model [ 61. Such modifications of the VRH model have been theoretically predicted [ 71 and have also been qualitatively observed [ 8,9], but until now a quantitative relation between the theoretical parameters of the hopping model and the experimental features is lacking. We have measured the temperature dependence of the dc conductivity of polypyrrole salts in which the 3,4 position of the pyrrole moiety was substituted by an alkyl ring. The method of synthesis of the polymers and their characterization in terms of the conductivity temperature dependence have been published elsewhere [ 10,111. In the following, a the194
oretical analysis of the data presented previously will be given in terms of a hopping model. In accordance with the theory of hopping conduction between localized states in three-dimensional disordered systems, the probability (w) for the hopping of a charge carrier between two states separated by a distance, R, is given by W-exp(-2aR-E,lkT),
(2)
a being the inverse localization length, and EA the energy barrier between the two states. Due to the lack of information concerning R, it has not been possible to distinguish between the tunneling term (2aR) and the activation energy term (EJkT). In the derivatives l-5 of polypyrrole salts investigated in the present work (see table 1) the minimum distance between polymeric chains can be controlled by changing the size of the substituent. A closer approximation than this minimum distance is only possible at the expense of an energetically unfavourable distortion of the alkyl rings. Because of the bulky substituents, adjacent pyrrole moieties of the same chain cannot be arranged in a plane, but have to be twisted as schematically depicted in fig. 1. Each individual polypyrrole backbone is cylindrically surrounded by a layer of methylene groups, the thickness of which is determined by the size of the alkyl ring fused to the pyrrole units in the 3,4 po-
0 009-2614/88/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Table 1 Formulae for the investigated polypyrrole derivatives, minimum distance R between adjacent chains as estimated with the help of molecular models, and activation energies Er, (as defined in eq. (2)) as obtained from fig. 3 ‘) Minimum distance R (nm)
EA (eV)
KJ0.T
0.38
0.012
iclo,-
0.76
0.019
Compound
1
J$ 11/p
2
Q% n -+-N w
8
N H
WO,
0.86
0.021
iClOT
1.03
0.047
+c10,
1.38
0.066
n l’l~”
4 9% N N” -ivy*
5 %
for this investigation are given in table 1. According to the hopping theory, two different temperature regimes can be distinguished. At low temperatures the charge carriers tend to hop not only to the nearest neighbour, but also to more remote sites in order to minimize the energy required for the hop. In this case the conductivity can be described by Mott’s law of variable-range hopping (eq. (1)). At higher temperatures almost exclusively nearestneighbour hops occur [ 121. In this case the conductivity can be described by a=u,exp(
T/s” 3
19 February 1988
CHEMICAL PHYSICS LETTERS
Volume 144, number 2
R
‘) See fig. 1 for the definition of R; estimated from space filing models.
sition. Thus, adjacent polymer backbones must be separated from each other in the bulk material by a distance R which can be estimated from molecular models as shown in fig. 1. The relevant distances R
1
2n+
R/2 I
2n CIO, Fig. 1. Schematic description of the polypyrrole backbone wrapped in a layer of poorly conducting methylene moieties of the alkylene chains fused to the pyrrole units in the 3,4 position. The minimum separation distance R between adjacent chants can be estimated from molecular models (compare table I ).
-2aR)
exp( -E,IkT)
.
(3)
From this equation a linear dependence of log o on the hopping distance R should be expected at a given temperature. The logarithm of the room-temperature conductivity of the investigated polypyrrole derivatives l-5 indeed shows this linear dependence on the minimum interchain distance R as shown in fig. 2. Analyzing our results in terms of eq. (3) assumes that carriers move by hopping between regions of one chain defined by the localization length I/a! to equivalent regions of an adjacent chain. As further evaluation of the data will point out, we have no information on i&u-chain processes, but a fully consistent picture will be developed on the assumption that the controlling factor for the description of the conductivity behaviour are the inter-chain processes. If the dependence of EA on R is known, the localization length l/a can be calculated according to eq. (3). The EA values for the different polymers can be obtained from the temperature dependence of the conductivity. A linear relationship log a versus l/T has been observed between 300 and 150 K (fig. 3). The activation energy is given by the slope A log o/A l/T (table 1). A temperature dependence of a-exp( -C/,1’4) or ff wexp( -C/T1’2) cannot be ruled out based on the available data in the temperature range investigated. However, the consistency of the data analyzed in terms of eq. (3) fully substantiates the assumption. As shown in fig. 4, the activation energy can be described by the empirical formula E,-BR
(4)
with a proportionality constant B= 7.37 x lo-’ eV/nm. An additional proof for this dependence can be 195
Volume144,number2
I9 February1988
CHEMICAL PHYSICSLETTERS
0
.2
.4
.6
LO
.6
R
6
7
I3
9
10
11
12 R
12
1A
16
[nml
13
14
15
16
tnml
Fig.2. Logarithmof the roomtemperaturedc conductivityof the investigatedpolymers1 to 5 as a functionof R. (a) Room-temperature measurements; (b) measurement at (0 ) 350,(A ) 250,( + ) 200and ( X ) 150 K.
obtained by evaluating the correlation between log [T and R at different temperatures. The general relation derived from eq. (3) is =_za
&% -
6R
--- sEA 1 6R kT’
(5)
Assuming that the localization length of the charge carrier is independent of R and for that reason GalSR=O, (5) can be simplified to =_za
6E,_1_ -6RkT’
196
(6)
A plot of (6 log d6R) T versus l/T should be linear, the slope giving the value of 6E,/6 R. The data for A log a/m are obtained from fig. 2b. The predicted linearity between A log aIAR and l/T can be seen in fig. 5 from which a value for AEJAR of 7.69 X 10d2 eV/nm is calculated, which is in very good agreement with the value of J3calculated above. Introducing the proportionality between EA and R (eq. (4)) into eq. (3) and taking the logarithm, it follows log Q_ -R(2a+BIkT)
-AIkT.
(7)
19 February 1988
CHEMICAL PHYSICS LETTERS
Volume 144, number 5
3
4
6
5
1000/T
7
[K-l]
Fig. 3. Temperature dependence of the conductivity of the polymers I-5 in the range from 300 to 150 K,
The term 2a! + BlkT can be identified with the slope of the straight line in fig. 2. The value of this slope at room temperature is approximately 5-3 urn- ’ and
therefore (xs: 1.2 nm- ‘. Using this, the localization
Fig. 4. Activation energies for the investigated polymers l-5 as calculated from eq. ( 2) as a function of R (compare table I )
length of the charge carrier can be estimated as
llaw0.8 nm. Summarizing the results obtained, the conduction
Fig. 5. Slopes 6 loga/GR asobtaiued from fig. 2 versus l/T (compare eq. ( 6)).
197
Volume 144, number 2
CHEMICAL PHYSICS LETTERS
can be understood as a hopping of localized charge carriers between adjacent chains separated by poorly conducting (“insulating”) regions of methylene groups. The probability for the hopping process is controlled by the substituent size which represents the minimum interchain distance. It is also worth noting that the value of the localization length found here for the substituted polypyrrole perchlorates is in very good agreement with the value estimated for the conducting salts of unsubstituted polypyrrole with polystyrenesulfonate as counterion [ 71. Furthermore, the present results and their analysis emphasize the overwhelming importance of interchain charge-carrier hopping processes for the macroscopic conduction behaviour of polymers. If intrachain processes were the controlling factor, the results obtained with the cycloalkylpolypyrroles could not be easily understood, because the conformational twist angle between adjacent rings in the backbone is practically independent of the ring size, as is immediately seen from space filling molecular models. The present work was supported in part by the BMFT (Project No. 03C156A 3). We wish to thank
198
19 February 1988
the CSIC (Spain) for additional support (for TAE) and the Stiftung Volkswagenwerk for a scholarship for JR.
References [] T.A. Skotheim, ed., Handbook of conducting polymers, Vol. 2 (Dekker, New York, 1986). [2] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. B22 (1980) 2209. [ 31 Y. Tomkiewicz, T.D. Schultz, H.D. Broom, T.C. Clarke and W.B. Street, Phys. Rev. Letters 43 (1979) 1532. [ 41 S. Kivelson and A.J. Epstein, Phys. Rev. B29 (1984) 336. [ 51J.C. Scott, P. PIIuger, M. Krombi and G.B. Street, Phys. Rev. B28 (1983) 40. [6] D.T. Glatzhofer, J. Ulanski and G. Wegner, Polymer 28 (1987) 449. [ 71 P. Sheng and J. Klafter, Phys. Rev. B27 (1983) 2583. [ 81 J. Ulanski, D.T. Glatzhofer, M. Przybylski, F. Kremer and G. Wegner, Polymer 28 (1987) 859. [9] E.K. Sichel, M. Knowles, M. Rubner and J. Georger Jr., Phys. Rev. B25 (1982) 5574. [lo] J. Riihe, Ch. Kriihnke, T.A. Ezquerra and G. Wegner, Springer series on solid state science ( 1987), to be published. [ 111 J. Rilhe, Ch. Krohnke, T.A. Ezquerra, F. Kremer and G. Wegner, Ber. Bunsenges. Physik. Chem. 9 (1987) 885. [ 121 N.E Mott and E.A. Davis, Electronic processes in non-crystalline materials (Clarendon Press, Oxford, 1979).