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Electric field independent mobilities in molecular crystals
Volume 62. aumber 2
CHEMICAL PHYSICS L?Z-ITFZRS
1 April 1979
ELECTRIC FIELD INDEPENDENT MOBILITIES IN MOLECULAR CRYSTALS L.B. SCHEIN Xerox Corporation _ WI I-I, Websrer Research Center. Rockesfer. New York I4644. USA
and A_R_ McGHIE
Received 6 December 1975
It Lsexperimentally demonstrated that the electron drift mobility ;tlong the c’ direction of anthmcene and nnphthalene is independent of the electric field to high electric fields (~17 V/pm) at low temperatures (=I00 K). These data protide zddirionrrl evidence qdnsi the appIicabiiity of smti polaron modeIs to molecular crystals and provide a challenge to recent& proposed hopping models of chage transport xihich can account for the temperature independence of the drift mobility.
Recent observations [I] of a band-hopping transition in naphthalene have provided evidence that the appro,x.imateIy temperature independent drift mobility JL observed in several molecular cqstals over wide temperature ranges [2] is due to the hopping of Iocalized carriers_ However, a detailed understanding of the Iocaiization mechanism and the mechanism of transport remains to be determined_ Small polaron hopping mod&, which have been worked out in several appro-ximations (adiabatic znd nonadiabatic, high and low temperature, strong and weak coupling) appear urrable :o account for the observed weak temperature dependence ofp [?I_ More recent hoppin:: models of charge transport have been able to account for the weak temperature dependence of p by including the effect of the temperature dependence of the electron overlap integral (off-diagonal disorder) [3--51 and by considering anharmonic effects oft he site energy [6] _ It is the purpose of this paper to present measurements of the electric tieId dependence of the electron drift mobility to both high electric fields u~xi Iow temperatures, thereby providing an additional test of proposed transport models_ We beiieve that these data provide additional evidence against small polaron modeIs and provide a challenge to the newer models of 356
charge transport for which the electric Reid dependence remains to be calculated_ Our data are shown in fig. 1 and 2_ Data are shown for electrons in the c’ direction of anthracene (fig. 1) and naphthalene (fig. 2) In figs. I and 2 the inverse transit time 7- t is plotted against the appIied voltage (the eIectric field is given by the appIied voltage divided by the crystal thickness); an observed linear dependence demonstrates that the mobility is field independent since --I I
= p(E)E/L ,
(1)
where L is the crystal thickness. These measurements were performed in the hopping temperature range, above the band-hopping transition [ 1] where mobility is appro.ximateIy temperature independent [ 1 ,?I. The measurements on anthracene were made at 100 K to electric fields of 16.1 V/m; the measurements on naphthalene were made at 160 K to electric fields of 173 V/m_ Both measurements indicate that the mobility is electric field independent to within ~352, which is the maximum departure of the data points from the leas; squared fitted straight line_ Similar data were obtained on many samples prepared from our crystals and crystaB supplied by other sources
. ANTHf?ACENE ELECTRONS. C’ DIRECTION T=IOOK 62,un
0
200
1 April 1979
CHEMICAL PHYSICS LElTERS
Volume 62, number 2
400 ”
600 tvoLTs1
/
800
1000
Fig_ I_ The inverse electron transit time versus the applied voltage for a 62 km thick (determined from the transit time) anthracene crystal polished, as described in ref. [ I], along the c’ direction_ The smail intercept m&yindicate a small amount of spxe charge in the sample or mn> be an artifact of the least square iitting procedure_ The obserred linearity demonstrates ttit the mobdity is electric field independent to 16-I V/pm (the maximum applied voltage divided by the crystal thickness) at 100 K.
These data were obtained by standard transient photoconductivity techniques discussed previously [ 1, 71. The transit time, which equals the crystal thickness divided by the product of the mobihty and the applied electric field, of a photogenerated sheet of charge carriers is measured_ The experiments were performed with a quadrupled (2650 A light) Nd I glass laser (Holobeam model 331) which provides a highly absorbed (approximately 0.1 m) laser flash of sufficient energy (10 mJ) and of short enough duration (20 ns) to avoid carrier trapping at the low temperatures of these measurements. These measurements were made in a Statham SD14 liquid nitrogen fed test chamber which maintained the sample temperatures to within AO.1 K. The highest attainabie electric fields were determined by dielectric breakdown; this field was madmized by (1) cooling the sample slowly and (2) by using appropriate pulse circuitry to limit the spplicalion time of the electric field to several ms. A measure of the expected sensitivity of the mobility to an electric field E is determined by the ratio of enE/2kT where a is an intermolecular spacing, k is Boltzmann’s constant and T is the absolute temperature_ Physicahy, the reason large values of e&$X-T are crucial for testing the electric field dependence of fl is straightforward: e&+/2 is the energy shift of an energy level across a molecular spacing CIdue to the presence OI an applied eIectric field. If the electron is localized at a site due to the presence of an energy well of depth IV, this depth is decreased in the presence of an applied field by an amount IV - eaE/2 and transition rates, which are determined by exp(--lV/kT), become esp [-(W-
e&/2)/X-T]
= exp(-W/H)
exp(e&/ZkT)
_
This result is borne out by detailed cakulations polaron literature is described by 1-10~x-1
in the [S] in which the field dependence
sinh x exp(-x2kTJ2
IV) ,
0)
where x = e&‘/l)kT Fig. 2. SimiIar data as shown in fig. 1 for a 69.3 Drnthick naphthalene crystal. The observed linearity demonstrates that the mobilit> is electric field independent to 17.3 V/Mm at 160 I;.
and Jt’is the polaron binding energy. The sinh x term an exponential dependence for large argument and the additional term in eq_ (3) esp(-x2X-T/ ZIV), results from the necessity within the polaron modapproaches
357
Vo!ume 62. number 2
CHEMICAL
PHYSICS
Fw 3. Predicted electric field dependence of the mobility from small pokuon theory 181 wiirh the small pohron binding energy Was the parameter_ For constant W, the predicted field dependence increases with incresin~ x = eaE/zlllI
el for the occurrence of an ener,T coincidence be-
tween hopping sites before a hop can occur. This function is pIotted in fig. 3 with IVas a parameter at T= 100 K: a value of one indicates a field independent mobility. The figure indicates that as x increases the departure of D from fieId independence increases for constant W_Due to a canceIIation of fieId dependence of x-* sinh x and exp(-x%T/2IV), there e_xistsa band of IV’s for which no field dependence is predicted. The boundaries of this band W, and W_ are given approximate& by
1 April 1979
LETTERS
for small x where p is the experimental error (0.03 in these experiments). IV+- iv_ is decreased by both lowering the temperature and increasingx (Le., the electric field)_ Table 1 compares the value of eaE/2kT of these measurements with previous measurements. A “typical” room temperature mobility measurement can usuaUy be made to 10 V/w before dieIectric breakdown destroys the crystal. For anthracene, where the intermoIecular distance along c’ is 9.2 A, this gives eQE/ 2kTof 0.18. Kepler and Hoesterey [9] reported high field room measurements in anthracene to 16 V/m, giving eaEj2kTof 0.29 and Schein et al. [I] reported measurements at 134 K to S-5 V/m in r.aphthalene (a = 7.3 A) giviztgeaEj2kT of O-27_ The values of eaE/ 2kT for the data presented in figs. 1 and 3, (0.86 and 0.46, respectively) cIearly exceed these values of eaEj XT by Iarge factors thereby providing a more sensitive test of the electric field dependence of the mobility. One learns from fig. 3 and eq_ (3), which describes the eIectric field dependence predicted for small poIaron hopping models, that at the value of eaE/2kTobtained from fig. I, 0.86, the field dependence of p is predicted to exceed 3% for aII values of the polaron binding energy except a narrow band of IV’s,21-35 meV. For the naphthalene experiment (fig- 2), this band is X-266 meV, corresponding to eaE/ZT of 0.46 and T = 160 K. Such large values of IVwould predict activated temperature dependence for JLabove 100 K which is not observed [1,2]. It is concluded that the electric field independence of the mobility evident in figs. 1 and 2 is inconsistent with the predictions of small polaron theory. This supports previous work [2] in which it is argued that small polaron theory
Table 1 Comparison of high tieid measurements _-w-__ Material
-__--_ --__--
-__-tvpicll
-------_
---__L-
Max E
eaE
(V/m)
(meV)
T(K)
X-T
eaE/2X-T
CmeV)
anthracene
10.0
9.3
300
25.9
0.1%
Kepler and Hsesrerep [9] Schein el aI_ [I]
anthracene
16.0
14.7
300
25.9
0.29
8.5
6.2
134
11.6
0.27
fix-I
anthrxene ruphthaiena
16.1 17.3
14.8 12.6
100 160
8.6 13.8
0.86 0.46
me3surement.s
fig-2
I__-_1_---
mphthdene
VoIume 62. number 2
1 April 1979
CHEMICAL PHYSICS LETTERS
cannot describe the approximately temperature independent mobilities observed over wide temperature ranges in anthracene and several other molecular crystals. For those hopping models which predict cc 0
exp(e&QkT) the fieId dependence will cause a deviation from linearity of the inverse transit versus applied voltage. For the maximum electric fields used in figs. 1 and 2, the deviation should be 128% (e”-86 - 1) and 5770, respectively_ Clearly such an electric field dependence of the mobility is not observable in these data; the maximum deviation of the data points from the fitted least squared straight line is zW?L The electric field dependence of the mobility predicted by recent hppping models which embody offdiagonal disorder [3-S] and anharmonic effects on the site energy [6] have not yet been determined. However, the relatively small departure of these models conceptually from standard hopping models leads to the expectation that they may predict an electric field dependence similar to the predictions of small polaron theory or eq. (2)_ In either case, these data would remain unaccounted for.
The authors would Eke to acknowledge support from and discussions with C.B. Duke. This wcrk was supported in part by a National Science Foundation Materials Research Laboratory Program under Grant Xo. DMR 76-80994. Additional samples of anthracene and naphthalene were supplied by 1. ZschokkeGranacher.
References 111 L.B. Schein. C.B. Duhe and A-R. McChie, Ph>s. Rev. Letters 40 (1978)
197.
(21 L-B. Schein, Chem. Phls. Letters 48 (1977) 571. [31 H. H&en and P. Reineker, 2. Physik 249 (1972) 253.
141 A. Miadhukar and W’_Post, Qh>c. Reb. Letters 39 (1977) 1424; 40 (1978)
70(E).
[51 IL Sumi, Solid State Commun. 28 (1978) I61 [71 ISI PI
309; 3. Chem. Phbs., submitted for publication. S. Efrima and H. Meriu, J. Chem. Phgs. 69 (1978) 5113; Chem. Phys. Letters 60 (1979) 226. L-B. Schein. Phls. Rer. BI5 (1977) 1024. LG. Austin, J. Phys. C5 (1972) 1687. R.G. Kepler kd D.C. Hoesterey, Phys. Rev_ B9 (1974) 2743.
359
CHEMICAL PHYSICS L?Z-ITFZRS
1 April 1979
ELECTRIC FIELD INDEPENDENT MOBILITIES IN MOLECULAR CRYSTALS L.B. SCHEIN Xerox Corporation _ WI I-I, Websrer Research Center. Rockesfer. New York I4644. USA
and A_R_ McGHIE
Received 6 December 1975
It Lsexperimentally demonstrated that the electron drift mobility ;tlong the c’ direction of anthmcene and nnphthalene is independent of the electric field to high electric fields (~17 V/pm) at low temperatures (=I00 K). These data protide zddirionrrl evidence qdnsi the appIicabiiity of smti polaron modeIs to molecular crystals and provide a challenge to recent& proposed hopping models of chage transport xihich can account for the temperature independence of the drift mobility.
Recent observations [I] of a band-hopping transition in naphthalene have provided evidence that the appro,x.imateIy temperature independent drift mobility JL observed in several molecular cqstals over wide temperature ranges [2] is due to the hopping of Iocalized carriers_ However, a detailed understanding of the Iocaiization mechanism and the mechanism of transport remains to be determined_ Small polaron hopping mod&, which have been worked out in several appro-ximations (adiabatic znd nonadiabatic, high and low temperature, strong and weak coupling) appear urrable :o account for the observed weak temperature dependence ofp [?I_ More recent hoppin:: models of charge transport have been able to account for the weak temperature dependence of p by including the effect of the temperature dependence of the electron overlap integral (off-diagonal disorder) [3--51 and by considering anharmonic effects oft he site energy [6] _ It is the purpose of this paper to present measurements of the electric tieId dependence of the electron drift mobility to both high electric fields u~xi Iow temperatures, thereby providing an additional test of proposed transport models_ We beiieve that these data provide additional evidence against small polaron modeIs and provide a challenge to the newer models of 356
charge transport for which the electric Reid dependence remains to be calculated_ Our data are shown in fig. 1 and 2_ Data are shown for electrons in the c’ direction of anthracene (fig. 1) and naphthalene (fig. 2) In figs. I and 2 the inverse transit time 7- t is plotted against the appIied voltage (the eIectric field is given by the appIied voltage divided by the crystal thickness); an observed linear dependence demonstrates that the mobility is field independent since --I I
= p(E)E/L ,
(1)
where L is the crystal thickness. These measurements were performed in the hopping temperature range, above the band-hopping transition [ 1] where mobility is appro.ximateIy temperature independent [ 1 ,?I. The measurements on anthracene were made at 100 K to electric fields of 16.1 V/m; the measurements on naphthalene were made at 160 K to electric fields of 173 V/m_ Both measurements indicate that the mobility is electric field independent to within ~352, which is the maximum departure of the data points from the leas; squared fitted straight line_ Similar data were obtained on many samples prepared from our crystals and crystaB supplied by other sources
. ANTHf?ACENE ELECTRONS. C’ DIRECTION T=IOOK 62,un
0
200
1 April 1979
CHEMICAL PHYSICS LElTERS
Volume 62, number 2
400 ”
600 tvoLTs1
/
800
1000
Fig_ I_ The inverse electron transit time versus the applied voltage for a 62 km thick (determined from the transit time) anthracene crystal polished, as described in ref. [ I], along the c’ direction_ The smail intercept m&yindicate a small amount of spxe charge in the sample or mn> be an artifact of the least square iitting procedure_ The obserred linearity demonstrates ttit the mobdity is electric field independent to 16-I V/pm (the maximum applied voltage divided by the crystal thickness) at 100 K.
These data were obtained by standard transient photoconductivity techniques discussed previously [ 1, 71. The transit time, which equals the crystal thickness divided by the product of the mobihty and the applied electric field, of a photogenerated sheet of charge carriers is measured_ The experiments were performed with a quadrupled (2650 A light) Nd I glass laser (Holobeam model 331) which provides a highly absorbed (approximately 0.1 m) laser flash of sufficient energy (10 mJ) and of short enough duration (20 ns) to avoid carrier trapping at the low temperatures of these measurements. These measurements were made in a Statham SD14 liquid nitrogen fed test chamber which maintained the sample temperatures to within AO.1 K. The highest attainabie electric fields were determined by dielectric breakdown; this field was madmized by (1) cooling the sample slowly and (2) by using appropriate pulse circuitry to limit the spplicalion time of the electric field to several ms. A measure of the expected sensitivity of the mobility to an electric field E is determined by the ratio of enE/2kT where a is an intermolecular spacing, k is Boltzmann’s constant and T is the absolute temperature_ Physicahy, the reason large values of e&$X-T are crucial for testing the electric field dependence of fl is straightforward: e&+/2 is the energy shift of an energy level across a molecular spacing CIdue to the presence OI an applied eIectric field. If the electron is localized at a site due to the presence of an energy well of depth IV, this depth is decreased in the presence of an applied field by an amount IV - eaE/2 and transition rates, which are determined by exp(--lV/kT), become esp [-(W-
e&/2)/X-T]
= exp(-W/H)
exp(e&/ZkT)
_
This result is borne out by detailed cakulations polaron literature is described by 1-10~x-1
in the [S] in which the field dependence
sinh x exp(-x2kTJ2
IV) ,
0)
where x = e&‘/l)kT Fig. 2. SimiIar data as shown in fig. 1 for a 69.3 Drnthick naphthalene crystal. The observed linearity demonstrates that the mobilit> is electric field independent to 17.3 V/Mm at 160 I;.
and Jt’is the polaron binding energy. The sinh x term an exponential dependence for large argument and the additional term in eq_ (3) esp(-x2X-T/ ZIV), results from the necessity within the polaron modapproaches
357
Vo!ume 62. number 2
CHEMICAL
PHYSICS
Fw 3. Predicted electric field dependence of the mobility from small pokuon theory 181 wiirh the small pohron binding energy Was the parameter_ For constant W, the predicted field dependence increases with incresin~ x = eaE/zlllI
el for the occurrence of an ener,T coincidence be-
tween hopping sites before a hop can occur. This function is pIotted in fig. 3 with IVas a parameter at T= 100 K: a value of one indicates a field independent mobility. The figure indicates that as x increases the departure of D from fieId independence increases for constant W_Due to a canceIIation of fieId dependence of x-* sinh x and exp(-x%T/2IV), there e_xistsa band of IV’s for which no field dependence is predicted. The boundaries of this band W, and W_ are given approximate& by
1 April 1979
LETTERS
for small x where p is the experimental error (0.03 in these experiments). IV+- iv_ is decreased by both lowering the temperature and increasingx (Le., the electric field)_ Table 1 compares the value of eaE/2kT of these measurements with previous measurements. A “typical” room temperature mobility measurement can usuaUy be made to 10 V/w before dieIectric breakdown destroys the crystal. For anthracene, where the intermoIecular distance along c’ is 9.2 A, this gives eQE/ 2kTof 0.18. Kepler and Hoesterey [9] reported high field room measurements in anthracene to 16 V/m, giving eaEj2kTof 0.29 and Schein et al. [I] reported measurements at 134 K to S-5 V/m in r.aphthalene (a = 7.3 A) giviztgeaEj2kT of O-27_ The values of eaE/ 2kT for the data presented in figs. 1 and 3, (0.86 and 0.46, respectively) cIearly exceed these values of eaEj XT by Iarge factors thereby providing a more sensitive test of the electric field dependence of the mobility. One learns from fig. 3 and eq_ (3), which describes the eIectric field dependence predicted for small poIaron hopping models, that at the value of eaE/2kTobtained from fig. I, 0.86, the field dependence of p is predicted to exceed 3% for aII values of the polaron binding energy except a narrow band of IV’s,21-35 meV. For the naphthalene experiment (fig- 2), this band is X-266 meV, corresponding to eaE/ZT of 0.46 and T = 160 K. Such large values of IVwould predict activated temperature dependence for JLabove 100 K which is not observed [1,2]. It is concluded that the electric field independence of the mobility evident in figs. 1 and 2 is inconsistent with the predictions of small polaron theory. This supports previous work [2] in which it is argued that small polaron theory
Table 1 Comparison of high tieid measurements _-w-__ Material
-__--_ --__--
-__-tvpicll
-------_
---__L-
Max E
eaE
(V/m)
(meV)
T(K)
X-T
eaE/2X-T
CmeV)
anthracene
10.0
9.3
300
25.9
0.1%
Kepler and Hsesrerep [9] Schein el aI_ [I]
anthracene
16.0
14.7
300
25.9
0.29
8.5
6.2
134
11.6
0.27
fix-I
anthrxene ruphthaiena
16.1 17.3
14.8 12.6
100 160
8.6 13.8
0.86 0.46
me3surement.s
fig-2
I__-_1_---
mphthdene
VoIume 62. number 2
1 April 1979
CHEMICAL PHYSICS LETTERS
cannot describe the approximately temperature independent mobilities observed over wide temperature ranges in anthracene and several other molecular crystals. For those hopping models which predict cc 0
exp(e&QkT) the fieId dependence will cause a deviation from linearity of the inverse transit versus applied voltage. For the maximum electric fields used in figs. 1 and 2, the deviation should be 128% (e”-86 - 1) and 5770, respectively_ Clearly such an electric field dependence of the mobility is not observable in these data; the maximum deviation of the data points from the fitted least squared straight line is zW?L The electric field dependence of the mobility predicted by recent hppping models which embody offdiagonal disorder [3-S] and anharmonic effects on the site energy [6] have not yet been determined. However, the relatively small departure of these models conceptually from standard hopping models leads to the expectation that they may predict an electric field dependence similar to the predictions of small polaron theory or eq. (2)_ In either case, these data would remain unaccounted for.
The authors would Eke to acknowledge support from and discussions with C.B. Duke. This wcrk was supported in part by a National Science Foundation Materials Research Laboratory Program under Grant Xo. DMR 76-80994. Additional samples of anthracene and naphthalene were supplied by 1. ZschokkeGranacher.
References 111 L.B. Schein. C.B. Duhe and A-R. McChie, Ph>s. Rev. Letters 40 (1978)
197.
(21 L-B. Schein, Chem. Phls. Letters 48 (1977) 571. [31 H. H&en and P. Reineker, 2. Physik 249 (1972) 253.
141 A. Miadhukar and W’_Post, Qh>c. Reb. Letters 39 (1977) 1424; 40 (1978)
70(E).
[51 IL Sumi, Solid State Commun. 28 (1978) I61 [71 ISI PI
309; 3. Chem. Phbs., submitted for publication. S. Efrima and H. Meriu, J. Chem. Phgs. 69 (1978) 5113; Chem. Phys. Letters 60 (1979) 226. L-B. Schein. Phls. Rer. BI5 (1977) 1024. LG. Austin, J. Phys. C5 (1972) 1687. R.G. Kepler kd D.C. Hoesterey, Phys. Rev_ B9 (1974) 2743.
359