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The electrical properties of amorphous polymers described by a modified variable-range hopping model
Journal of Non-Crystall£ne Sol£ds 35 & 36 (1980) 129-134 • NorCh-Hollemcl PublJ.sh£ng Company
THE ELECTRICALPROPERTIESOF AMORPHOUSPOLYMERSDESCRIBEDBY A MODIFIED VARIABLE-RANGEHOPPINGMODEL R. C01son and P. Nagels Materials Science Department S.C.K./C.E.N. B-2400 H01 Belgtum
The d.c. conductivity Od; and the thermopower S of a number of polyacene qu~none radical polymers obey.the relationship Odc = ooexp (-To/T) / a n d . S/T = a+bT-~/~, typical f o r t w o d~mensional variable-range nopping. /ne results can oe explained by HoLt's model modified, however, by allowing for a linearly varying density of states near the Fermi level, and by taking into account the form of the wave function of the localized states. Special attention is paid to the prefactor oo which contains important hopping parameters, such as a frequency factor u^, usually assumed to be not higher than a typical phonon frequency (~ 1013 s - l ) . The original calculation of M~11er and Abrahams (HA) of the, hopping rate, however, gives uo values ranging from 10~D to 10£~ s-~. For the PAQRpolymers reasonable estimates of the parameters are obtained when using I~ expressions adapted for two-dimensional systems. INTRODUCTION Polyacene quinone radical polymers (PAQR) have been extensively investigated by Pohl [1]. These amorphous materials consist of a series of fused benzeno~d rings. Their long-chain mo]ecules are reasonably planar and are probably assembled tn the form of randomly packed f l a t ribbons. In this paper we report measurements of d.c. conductivity oar and thermopower S of three PAQRmaterials. The influence of doping these polymeF~ with iodine will be demonstrated too. In order to Interpret the experimental facts use w i l l be made of a modified variable-range hopping model. EXPERIMENTAL PROCEDURESAND RESULTS Three copolymers of the PAQR-type were synthesized starting from wrene or anthraquinone as one constitutent and phthalic anhydride (PA) or pyromellitic dianhydride (PMA) as the other one. The details of the preparation and measurement techniques have been described in prevtous papers [2,3]. The measurements were carried out on highly purified materials, after prolonged outgassing in situ. The pyrenePI~ po13nner was doped by exposing part of the material to iodine vapour at ~oom temperature for 12 hours in a dessicator at a pressure of approximately 10- Torr. The experimental data of the conductivity of three undoped and one doped PAQRmat e r i a l are shown in Fig, 1. I t can be seen that the temperature dependence of Odc does not obey a relationship of the form O~c = oo exp (-A/kT). The treatment with iodine has a small but definite influence on the conductivity: Odc of the pyrenePMA po13nmr is increased by a factor of three at room temperature. The experimental results of the thermopower are plotted as a function of reciprocal temperature in Fig. 2. The absolute value of S increases with increasing temperature. Upon doping with iodine, a sign reversal of S occurs. 129
130
R. Colson, P. Nagels / A Modified Variable-Range Hopping Model
40,
o
I
I
I
~
t
I
>
-i
i..
-6O ~" -
-80 4
6 1031 T
8
10
( K -1 )
Fi gure 1 Electrical conductivity of some PAQR polymers plotted as a function of reciprocal temperature.
e
o ioyrerm PMA ,, p y r e n e FRvlA - I 2 • ;:ntl~raquinone PA clihydrOxy - ant hraq ulnone
PMA
I
I
/
I
I
i
2
3
z; 103/T
5 ( K -1 )
6
7
Figure 2 Thermopower of some PAQR polymers plotted as a function of reciprocal temperature.
DISCUSSION
X-ray d i f f r a c t i o n measurements demonstrate the highly amorphous nature of the PAQR materials. This suggests an interpretation of t h e i r behaviour in terms of a hopping model. The ?ow value of the thermopower and i t s positive temperature coefficient, are indications of a variable-range hopping mechanism(VRH) for the charge transport. This type of conduction is characterized by a T - l / 4 dependence of the logarithm of Odc in a three-dimensional system [ 4 ] . I n o~der to look for such a dependence, the conductivitydata were plotted versus T- / 4 , but in all cases the data clearly did not obey such a relationship. To find a more suitable value of p such that Odc may be expressed as: Odc = ATb exp (-BT "p) the method of Hill 15] was adopted. The local gradient 6 = - k dlno/d(1/T) of lnOdc versus 1/T may be expressed as: .6 = bkT + BpkT(I"p) This allows p to be determined, since at s u f f i c i e n t l y low temperatures, a double logarithmic plot of 6 versus 1/T yields a straight line with slope ( l - p ) . Applying this procedure to the data of the PAQRpolymers, a straight line was always found, from which values of p between 0.36 and 0.39 were calculated. This suggests a T-1/3 behaviour, and indicates two-dimensional VRH in the large planar molecules. Fig. 3j where log Ode is plotted versus T-1/3, confirms these findings. In order to obtain an expression of Odc in the case of two-dimensional VRH, one may follow the original method of Mott [4]. The resulting formula for Odc is given by [3]:
R. Colson, P..Nagels / A Modified Variable-Range Hopping Hodel
°dc with:
= AT"2/3 exp (-BT "1/3)
A = e2(~k)'2/3 A
131
l -1 N1/3v F o
and: B = 3 [a2/(~k)] 1/3 NFI/3 0 v >,
In these formulae, I is the distance betweenothe planar molecules (= 3.5 A), ~" i s a measure of the "0 size of the localized state, Vo is § a frequency factor, and the other symbols have their conventional meanings. I t h a s to be emphasized that in this case, NF represents the number of states at the Fermi leve] per unit area and per unit energy, and has been assumed constant. Consequently, this method does not permit to derive an expression for the I I I P I I I i ,i thermopower. Therefore, a s l i g h t l y .13 .15 ,17 .19 .21 different approach based on two T J/3 ( K "113) assumptions w i l l be fol]owed. First of a l ] , the jump probability v is supposed to be given by the H i l l e r Figure 3 Abrahams (HA) expression [ ~ . Wuertz Conductiv!tv of PAORmaterials plotted and Thomas [7] extended the original versus T-I/3. HA expression to the case of sp~ orbitals for the localized wave functions. I t is easy to show that their results may be written mope general]y as: U
\
= v o exp (-2~R) exp (-A/kT) E~
: z psS 4
, e2~ ~2
ABA ( R)B
In these expressions, E1 is the defomation potential, s the velocity of sound, p the mass per unit volOme, co the vacuum permittivity and s the dielectric con~t:n~oO:n~h~a~~o.F~eSs~T:l~'a~dR2r:~rA~ot~;~:~;~i~O~n~P~p~T als' distance between two localized sites. In the case of two-dimensional systems, an analogous expression may be derived for ~0: E~ , e2= ~2 B v° : 2ps4~3 ~ ' ~ 0 ; AB (~R) Now, p is the mass per unit surface of the two-dimensional system. The other assumption o f t h e model is a linear energy dependence of the density of states N(E) near theFermi level: N(E) = NF [ I+y(E-EF)] One may proceed to derive expressions for a~c and S, using the fundamental supposition that the most probable hop is given by:
132
R. Colson, P. Nagels / A Modified Variable-Range Sopping Model
dv
~]1~ = 0 and:
10..=
t
N F, 101°eV~m "8
.SJL li,10
V l
a
o
N(E)dE = 1
c In the two-dimensional case, whereas for three-dimensional V = ~ ~R3. Hence, t t follows that for the two-dimens~onal
u £
E+A
.lO-t.
V = ~R2, systems, Immediately case:
~R2 [ N(E) +~TNFA ] = 1 and: A = 2kTz/[2+~R2N(E)kTz]
u
The symbol z is a notation for the quant i t y 2~R-B. These equations allow the iterative determination of the hopping parameters R and A. The conductivity and thermopower follow from the Kubo-Greenwood formulae [8]:
tl;
0
~24
.26
.28 .30 T -l/d; ( K "1#')
.3~
Figure 4 Temperature dependence of the conductivity of a three-dimensional system, as predicted by the modified VRH model. The resulting formulae are therefore:
f af Odc = - J o(E) ~-EdE o(E) = e2v(E)R2(E)N(E) 2 k2T dlno(E)
S = "3"-
e
dE
EF
2 2 Odc = e VoFRFNF exp (-2~RF) exp (-AF/kT) k2T
S--T~-
1 aR [~Fa-E
(2_ZF) +
+y] EF
The energy derivatives are given by:
[
( I+YAF) kTzF y [ .2AF(I+YAF/Z)+ZFk T + 1] 1 aR = (3(~RF-B)kT ]~F ~-E EF {(I+'YAF/2)+(I+'yA F) [(XRF+ 2AF(I+YAF/Z)+ZFkT] }
FA1
I
[ ykTZF (3~RF'f3)kT 2 aR + ~AF(I+YAF/Z)+ZFk T aA EF = [ aRF + ZAF(I+YAF/Z)+ZFkT ] ~FF ~-E EF a-Z"
Similar expressions are obtai ned for three-dimensional systems. In order to ]ook for the dependence of Oac and S on the different parameters of this.model, one may plot some numerical results as a function of temperature. This
R. Co2son, P. Nagels / A Nodified Variable-Range Hopping Model
~
1
I~3
N~: lOmeV'~3 a :5~ =10
• 3
[o I-
-JO IB,IO 4
6
103/T ( K"1 )
8
.24
~
Figure 5 Temperature dependence of the thermopower of a three-dimensional system as predicted by the modified VRH model.
I
I,
.26
I
I
I
I
I
28 .30 T "1/4 { K"l/g*)
I
32
Figure 6 Demonstration of the law S/T = a+bT'l/4 for a three-dimensional system.
has been done for three-dimens|onal systems in Figs. 4-6. From these figures one can draw some deftnite conclusioos. First of 811, even i f B > 2 and 7 # O, the log.odc data s t i l l obey the T-1/4 (or T-1/3) law. A positive value of 7 implies a nigner conouc¢ivicy rot cne same values of the remaining parameters. Furthermore, the absolute value of the thermopower is an increasing function of tem~ereture. Horeo~e~, the quantity S/T decreases with temperature nearly as a+bT-174 (or a+bT" [ in the case of two-dimensiona] systems). The thermopower and the quantity 7 have opposite sigp ~. TABLE 1 Parameters for variable-range hopping conductivity Hateri al
7(eV "1)
-0.30 wrene-PHA 0.45 wrene-Pt4A-I 2 anthraquinqne-PA 0.30 dihydroxy-anthr.-PHA 0.02
NF(eV'lcm "2)
0
~o(S"1) ~oeXp(-2aRF) RF(A) AF(eV) (T=150K) (T=150K) CT=150K) :T=150K)
17oxlo13
21sxlo
1.71x1013 5.80x1012 7.90x1012
4.73x1016 2.78x109 2.40x1017 1.54x107 2.63x1017, 1.35x108
8.3oxlo 8
42 41 58 53
0.!1 0.11 0.16 0.14
These conclusions show that the proposed modif4cattons of the origtnal VRH model predict a behaviour very similar to that observed on the PAQRpolymers and qther well-known inorganic amorphous materials. Both log Odc and S/T show the T'z/~ dependence (Fig. 3 and F~g. 7).Furthermore, this modeT-demonstrates the sign reversa1 of the thermopower and the slight increase of Odc upon doping with iodine to be caused by a sign reversal of 7. Detailed data for the various parameters of
|34
R. Colson, P. Nagels / A Modified Variable-Range Hopping Model
.20 .16 A
.12
>= I--
.08 .04
L
0
8 o
-D4 -.08
.c I-
-.12
-.16 -.20
ii
IIl
~ II I I I I'
I
12
.14 .16 T "1/3 ( K -V3)
.~B
Figure 7 The quantity S/T of the PAQR polymers plotted versus T-1/3.
real materials are obtained by comparing the experimental results ofode and S with the theoretical expressionS. In case of the PAQRpolylners, the best fi~s were obtained with 8 = 2 and ~-z = 5A. The results of such a procedure are collected in Table I. The high vo values mayappear surprising. In n~arly all previous interpretations on amorphous m t e r i a l s , i t has been customary tocompare ~o directly with the ~ximpm phonon frequency ~Dh (~ 10 s'Z). However, referring to the original derivations of Vo by NA [6] and by Mort [8], the factor Vo exp (-2~R) is expected to have a value not higher than Vph.Therefore, the experimental vo values are quite reasonable. In fact, they seem to be too low, since for normal values of the material constants (s = lO00ms "1, = 10, E1 = 10 eV, p/1 = 1000, kg m-3) one finds a value of about 1020 s-I for Vo. However, extremely high values of the dielectric constant (up to 105) have been observed in PAQR polymers [1]. This confims the low ~o values we derived in this work.
REFERENCES [1] Pohl, H.A., J. Biological Phys. 2 (1974) 113-172. [2] Colson, R., Nagels, P. and Speeckaert, R. in: Proc. 7th Int. Conference on Amorphous and Liquid Semiconductors, Spear, W.E. (ed.) (G.G. Stevenson, Dundee, 1977) 775-779. [3] Colson R. and Nagels, P., Phil. Hag. 38 (1978) 503-514. [4] Mott, N.F., Phil. Hag. 12 (1969) 835-852. [5] H i l l , R.M., Phys.stat.sol. (a) 35 (1976) K29. [6] M i l l e r , A. and Abrahams, E.A., Phys. Rev. 120 (1960) 745-755. [7] Wuertz, D. and Thomas, P., Phys.stat.sol. (b) 88 (1978) K73. [8] Mort, N.F. and Davis, E.A., Electronic Processes in Non-crystalline solids, (Oxford U.P., Oxford 1971).
THE ELECTRICALPROPERTIESOF AMORPHOUSPOLYMERSDESCRIBEDBY A MODIFIED VARIABLE-RANGEHOPPINGMODEL R. C01son and P. Nagels Materials Science Department S.C.K./C.E.N. B-2400 H01 Belgtum
The d.c. conductivity Od; and the thermopower S of a number of polyacene qu~none radical polymers obey.the relationship Odc = ooexp (-To/T) / a n d . S/T = a+bT-~/~, typical f o r t w o d~mensional variable-range nopping. /ne results can oe explained by HoLt's model modified, however, by allowing for a linearly varying density of states near the Fermi level, and by taking into account the form of the wave function of the localized states. Special attention is paid to the prefactor oo which contains important hopping parameters, such as a frequency factor u^, usually assumed to be not higher than a typical phonon frequency (~ 1013 s - l ) . The original calculation of M~11er and Abrahams (HA) of the, hopping rate, however, gives uo values ranging from 10~D to 10£~ s-~. For the PAQRpolymers reasonable estimates of the parameters are obtained when using I~ expressions adapted for two-dimensional systems. INTRODUCTION Polyacene quinone radical polymers (PAQR) have been extensively investigated by Pohl [1]. These amorphous materials consist of a series of fused benzeno~d rings. Their long-chain mo]ecules are reasonably planar and are probably assembled tn the form of randomly packed f l a t ribbons. In this paper we report measurements of d.c. conductivity oar and thermopower S of three PAQRmaterials. The influence of doping these polymeF~ with iodine will be demonstrated too. In order to Interpret the experimental facts use w i l l be made of a modified variable-range hopping model. EXPERIMENTAL PROCEDURESAND RESULTS Three copolymers of the PAQR-type were synthesized starting from wrene or anthraquinone as one constitutent and phthalic anhydride (PA) or pyromellitic dianhydride (PMA) as the other one. The details of the preparation and measurement techniques have been described in prevtous papers [2,3]. The measurements were carried out on highly purified materials, after prolonged outgassing in situ. The pyrenePI~ po13nner was doped by exposing part of the material to iodine vapour at ~oom temperature for 12 hours in a dessicator at a pressure of approximately 10- Torr. The experimental data of the conductivity of three undoped and one doped PAQRmat e r i a l are shown in Fig, 1. I t can be seen that the temperature dependence of Odc does not obey a relationship of the form O~c = oo exp (-A/kT). The treatment with iodine has a small but definite influence on the conductivity: Odc of the pyrenePMA po13nmr is increased by a factor of three at room temperature. The experimental results of the thermopower are plotted as a function of reciprocal temperature in Fig. 2. The absolute value of S increases with increasing temperature. Upon doping with iodine, a sign reversal of S occurs. 129
130
R. Colson, P. Nagels / A Modified Variable-Range Hopping Model
40,
o
I
I
I
~
t
I
>
-i
i..
-6O ~" -
-80 4
6 1031 T
8
10
( K -1 )
Fi gure 1 Electrical conductivity of some PAQR polymers plotted as a function of reciprocal temperature.
e
o ioyrerm PMA ,, p y r e n e FRvlA - I 2 • ;:ntl~raquinone PA clihydrOxy - ant hraq ulnone
PMA
I
I
/
I
I
i
2
3
z; 103/T
5 ( K -1 )
6
7
Figure 2 Thermopower of some PAQR polymers plotted as a function of reciprocal temperature.
DISCUSSION
X-ray d i f f r a c t i o n measurements demonstrate the highly amorphous nature of the PAQR materials. This suggests an interpretation of t h e i r behaviour in terms of a hopping model. The ?ow value of the thermopower and i t s positive temperature coefficient, are indications of a variable-range hopping mechanism(VRH) for the charge transport. This type of conduction is characterized by a T - l / 4 dependence of the logarithm of Odc in a three-dimensional system [ 4 ] . I n o~der to look for such a dependence, the conductivitydata were plotted versus T- / 4 , but in all cases the data clearly did not obey such a relationship. To find a more suitable value of p such that Odc may be expressed as: Odc = ATb exp (-BT "p) the method of Hill 15] was adopted. The local gradient 6 = - k dlno/d(1/T) of lnOdc versus 1/T may be expressed as: .6 = bkT + BpkT(I"p) This allows p to be determined, since at s u f f i c i e n t l y low temperatures, a double logarithmic plot of 6 versus 1/T yields a straight line with slope ( l - p ) . Applying this procedure to the data of the PAQRpolymers, a straight line was always found, from which values of p between 0.36 and 0.39 were calculated. This suggests a T-1/3 behaviour, and indicates two-dimensional VRH in the large planar molecules. Fig. 3j where log Ode is plotted versus T-1/3, confirms these findings. In order to obtain an expression of Odc in the case of two-dimensional VRH, one may follow the original method of Mott [4]. The resulting formula for Odc is given by [3]:
R. Colson, P..Nagels / A Modified Variable-Range Hopping Hodel
°dc with:
= AT"2/3 exp (-BT "1/3)
A = e2(~k)'2/3 A
131
l -1 N1/3v F o
and: B = 3 [a2/(~k)] 1/3 NFI/3 0 v >,
In these formulae, I is the distance betweenothe planar molecules (= 3.5 A), ~" i s a measure of the "0 size of the localized state, Vo is § a frequency factor, and the other symbols have their conventional meanings. I t h a s to be emphasized that in this case, NF represents the number of states at the Fermi leve] per unit area and per unit energy, and has been assumed constant. Consequently, this method does not permit to derive an expression for the I I I P I I I i ,i thermopower. Therefore, a s l i g h t l y .13 .15 ,17 .19 .21 different approach based on two T J/3 ( K "113) assumptions w i l l be fol]owed. First of a l ] , the jump probability v is supposed to be given by the H i l l e r Figure 3 Abrahams (HA) expression [ ~ . Wuertz Conductiv!tv of PAORmaterials plotted and Thomas [7] extended the original versus T-I/3. HA expression to the case of sp~ orbitals for the localized wave functions. I t is easy to show that their results may be written mope general]y as: U
\
= v o exp (-2~R) exp (-A/kT) E~
: z psS 4
, e2~ ~2
ABA ( R)B
In these expressions, E1 is the defomation potential, s the velocity of sound, p the mass per unit volOme, co the vacuum permittivity and s the dielectric con~t:n~oO:n~h~a~~o.F~eSs~T:l~'a~dR2r:~rA~ot~;~:~;~i~O~n~P~p~T als' distance between two localized sites. In the case of two-dimensional systems, an analogous expression may be derived for ~0: E~ , e2= ~2 B v° : 2ps4~3 ~ ' ~ 0 ; AB (~R) Now, p is the mass per unit surface of the two-dimensional system. The other assumption o f t h e model is a linear energy dependence of the density of states N(E) near theFermi level: N(E) = NF [ I+y(E-EF)] One may proceed to derive expressions for a~c and S, using the fundamental supposition that the most probable hop is given by:
132
R. Colson, P. Nagels / A Modified Variable-Range Sopping Model
dv
~]1~ = 0 and:
10..=
t
N F, 101°eV~m "8
.SJL li,10
V l
a
o
N(E)dE = 1
c In the two-dimensional case, whereas for three-dimensional V = ~ ~R3. Hence, t t follows that for the two-dimens~onal
u £
E+A
.lO-t.
V = ~R2, systems, Immediately case:
~R2 [ N(E) +~TNFA ] = 1 and: A = 2kTz/[2+~R2N(E)kTz]
u
The symbol z is a notation for the quant i t y 2~R-B. These equations allow the iterative determination of the hopping parameters R and A. The conductivity and thermopower follow from the Kubo-Greenwood formulae [8]:
tl;
0
~24
.26
.28 .30 T -l/d; ( K "1#')
.3~
Figure 4 Temperature dependence of the conductivity of a three-dimensional system, as predicted by the modified VRH model. The resulting formulae are therefore:
f af Odc = - J o(E) ~-EdE o(E) = e2v(E)R2(E)N(E) 2 k2T dlno(E)
S = "3"-
e
dE
EF
2 2 Odc = e VoFRFNF exp (-2~RF) exp (-AF/kT) k2T
S--T~-
1 aR [~Fa-E
(2_ZF) +
+y] EF
The energy derivatives are given by:
[
( I+YAF) kTzF y [ .2AF(I+YAF/Z)+ZFk T + 1] 1 aR = (3(~RF-B)kT ]~F ~-E EF {(I+'YAF/2)+(I+'yA F) [(XRF+ 2AF(I+YAF/Z)+ZFkT] }
FA1
I
[ ykTZF (3~RF'f3)kT 2 aR + ~AF(I+YAF/Z)+ZFk T aA EF = [ aRF + ZAF(I+YAF/Z)+ZFkT ] ~FF ~-E EF a-Z"
Similar expressions are obtai ned for three-dimensional systems. In order to ]ook for the dependence of Oac and S on the different parameters of this.model, one may plot some numerical results as a function of temperature. This
R. Co2son, P. Nagels / A Nodified Variable-Range Hopping Model
~
1
I~3
N~: lOmeV'~3 a :5~ =10
• 3
[o I-
-JO IB,IO 4
6
103/T ( K"1 )
8
.24
~
Figure 5 Temperature dependence of the thermopower of a three-dimensional system as predicted by the modified VRH model.
I
I,
.26
I
I
I
I
I
28 .30 T "1/4 { K"l/g*)
I
32
Figure 6 Demonstration of the law S/T = a+bT'l/4 for a three-dimensional system.
has been done for three-dimens|onal systems in Figs. 4-6. From these figures one can draw some deftnite conclusioos. First of 811, even i f B > 2 and 7 # O, the log.odc data s t i l l obey the T-1/4 (or T-1/3) law. A positive value of 7 implies a nigner conouc¢ivicy rot cne same values of the remaining parameters. Furthermore, the absolute value of the thermopower is an increasing function of tem~ereture. Horeo~e~, the quantity S/T decreases with temperature nearly as a+bT-174 (or a+bT" [ in the case of two-dimensiona] systems). The thermopower and the quantity 7 have opposite sigp ~. TABLE 1 Parameters for variable-range hopping conductivity Hateri al
7(eV "1)
-0.30 wrene-PHA 0.45 wrene-Pt4A-I 2 anthraquinqne-PA 0.30 dihydroxy-anthr.-PHA 0.02
NF(eV'lcm "2)
0
~o(S"1) ~oeXp(-2aRF) RF(A) AF(eV) (T=150K) (T=150K) CT=150K) :T=150K)
17oxlo13
21sxlo
1.71x1013 5.80x1012 7.90x1012
4.73x1016 2.78x109 2.40x1017 1.54x107 2.63x1017, 1.35x108
8.3oxlo 8
42 41 58 53
0.!1 0.11 0.16 0.14
These conclusions show that the proposed modif4cattons of the origtnal VRH model predict a behaviour very similar to that observed on the PAQRpolymers and qther well-known inorganic amorphous materials. Both log Odc and S/T show the T'z/~ dependence (Fig. 3 and F~g. 7).Furthermore, this modeT-demonstrates the sign reversa1 of the thermopower and the slight increase of Odc upon doping with iodine to be caused by a sign reversal of 7. Detailed data for the various parameters of
|34
R. Colson, P. Nagels / A Modified Variable-Range Hopping Model
.20 .16 A
.12
>= I--
.08 .04
L
0
8 o
-D4 -.08
.c I-
-.12
-.16 -.20
ii
IIl
~ II I I I I'
I
12
.14 .16 T "1/3 ( K -V3)
.~B
Figure 7 The quantity S/T of the PAQR polymers plotted versus T-1/3.
real materials are obtained by comparing the experimental results ofode and S with the theoretical expressionS. In case of the PAQRpolylners, the best fi~s were obtained with 8 = 2 and ~-z = 5A. The results of such a procedure are collected in Table I. The high vo values mayappear surprising. In n~arly all previous interpretations on amorphous m t e r i a l s , i t has been customary tocompare ~o directly with the ~ximpm phonon frequency ~Dh (~ 10 s'Z). However, referring to the original derivations of Vo by NA [6] and by Mort [8], the factor Vo exp (-2~R) is expected to have a value not higher than Vph.Therefore, the experimental vo values are quite reasonable. In fact, they seem to be too low, since for normal values of the material constants (s = lO00ms "1, = 10, E1 = 10 eV, p/1 = 1000, kg m-3) one finds a value of about 1020 s-I for Vo. However, extremely high values of the dielectric constant (up to 105) have been observed in PAQR polymers [1]. This confims the low ~o values we derived in this work.
REFERENCES [1] Pohl, H.A., J. Biological Phys. 2 (1974) 113-172. [2] Colson, R., Nagels, P. and Speeckaert, R. in: Proc. 7th Int. Conference on Amorphous and Liquid Semiconductors, Spear, W.E. (ed.) (G.G. Stevenson, Dundee, 1977) 775-779. [3] Colson R. and Nagels, P., Phil. Hag. 38 (1978) 503-514. [4] Mott, N.F., Phil. Hag. 12 (1969) 835-852. [5] H i l l , R.M., Phys.stat.sol. (a) 35 (1976) K29. [6] M i l l e r , A. and Abrahams, E.A., Phys. Rev. 120 (1960) 745-755. [7] Wuertz, D. and Thomas, P., Phys.stat.sol. (b) 88 (1978) K73. [8] Mort, N.F. and Davis, E.A., Electronic Processes in Non-crystalline solids, (Oxford U.P., Oxford 1971).