A model description of charge carrier transport phenomena in organic molecular crystals. II. Perylene

Chemical Physics I55 ( 1991) 389-399 North-Holland

A model description of charge carrier transport phenomena in organic molecular crystals. II. Perylene * E.A. Silinsh, G.A. Shlihta and A.J. Jurgis Institute

ofPhysicalEnergetics, Latvian Academy

of Sciences, Riga 226006, Latvia, USSR

Received 13 November 1990; in final form 10 April 199 1

A model description, previously used for computer simulation studies of charge carrier transport phenomena in polyacene crystals, has been extended to a-perylene crystals. A modified Sano-Moxumder (SM) model has been used for simulation of reported experimental carrier transport characteristics as a function of electric field Band temperature T, both in the hot and in the thermalized carrier regime. The present studies demonstrate the plausibility of a nearly small molecular polaron (MP) model approach to describe phenomenologically the negafive charge carrier separation and transport phenomena in a-perylene crystals over a wide temperature (30-300 K) and electric field f 1O’- IO5V/cm) range. On the other hand, the behaviour ofpositiwcharge carrier transport characteristics in a-peryiene indicates small lattice polaron f LP) formation.

1, Introduction In our previous paper [ I] a modified [ 2 ] SanoMozumder (SM ) model [ 3 ] was used for computer simulation of reported [4-S] experimental charge carrier transport characteristics in polyacene crystals as a function of electric field 8 and temperature T. The studies of ref. [ I] demonstrated the validity of the nearly small molecular polaron (MP) model [ 9121 approach to describe phe~omenolo~c~ly the hot and thermalized charge carrier transport characteristics in polyacene crystals over a wide temperature (4.2-300 K) and electric field ( b= 103-1 O6V/cm) range. In ref. [ 1 ] the conditions for creation of hot, non-thermalized carriers and the transport mechanism at high electric field were also analyzed in some detail. Karl and co-workers [ 4,6 ] (see also refs. [ 7,131) have recently measured intrinsic time-of-flight carrier mobilities also in ultrapure a-perylene (u-PI) crystals down to z 20 K. According to Steven’s classification of molecular crystals [ 141 a-Pi, like pyrene, belongs to the less frequently encountered lattice type B - with two pairs of parallel adjacent molecules in the unit cell. Thus, * For part I, see Chem. Phys. 138 ( 1989) 347-363. 0301-0~04/91/$03.50

the role of the crystallographic molecular unit in a-P1 is played by a molecular pair consisting of a dimer structure of closely spaced molecules with a separationdistanceof3.~3~(seep.22ofref. [15]).Such a con~guration causes considerable overlap of x orbitals, and, consequently, relatively strong interaction. In other aspects the crystal structure of a-P1 is similar to that of naphthalene and anthracene: it is monoclinic, belongs to the same space group P2,/a except that there are four molecules in the unit cell (2=4). Owing to such a dimer structure a-Pl, like pyrene, exhibits strong tendency for exciton self-trapping, resulting in eximer fluorescence and other unusual optical properties. Therefore, a-perylene and pyrene have been used for decades as model compounds for exciton localization studies in organic molecular crystals (see refs. [ 12,15,16 ] and references therein ) . Karl et al. [ 4,6,7,13] observed a number of peculiarities in the behaviour of charge carrier transport cha~cte~stics, different from those of polyacene crystals, which are apparently a consequence of this unique dimer-type arrangement of molecules in the a-PI lattice. First, in a-PL crystals the usually observed characteristic temperature dependence of carrier mobility

0 1991 Elsevier Science Publishers B.V. All rights reserved.

390

p( 7-)=&r-”

E.A. Silinsh et ai. I’Charge carrier transp~ri in organic moieeuiar crystals. II.

(1)

has been found only for negative charge carriers. In this case formula ( 1) holds for a rather wide high-T range, namely, in the c~stallo~aphic u-direction from z 100 to 300 K, in the b-direction from %60 to 300 K and in the &-direction - even from ~40 to 300 K [4,6,7]. Second, the slope values n of log v- versus log T dependences are different, viz. nao= 1.78, n&b” 1.72, the highest value being in the c’~irection with nc.c.= 2.15 [ 7 1, while in the case of Nph and ACcrystals the negative charge carriers in the high-Tregion yield a practically temperature independent ,u- ( T) curve, i.e. the slope value is approximately zero for log (P;~. ) as function of log T. In the low-temperature region below 100 K the pet-( T) dependence declines from the power law for higher electric fields and becomes electric field dependent, indicating the onset of a hot carrier transport regime. The observed field dependence of the carrier drift velocity ud( &) shows a sublinear character in this 7’ range, but it saturates ak higher field strength than in Nph crystals, namely, at &‘a (46 ) x 1O4V/cm (see figs. 1 and 6) yielding a constant mean drift velocity (u,),=const. greater than the mean thermal velocity Q,, at the given temperature [ 131. Thus, the behaviour of fi( 8, T) and ud( 8, T) dependences for negative charge carriers in U-PI are similar to those of posit& charge carriers in polyscene crystals (cf. ref. [ 1 ] ). On the other hand, the behaviour of the p+ ( 7’) dependence of positive charge carriers in a-P1 crystals differs completely from that in the polyacenes. In the a-PI case the hole mobility is thermally activated and increases with increasing temperature [ 6,7 J. The computer simulation of reported [ 4,6,7 ] negative charge carrier transport characteristics in a-P1 proves the applicability of a nearly small molecular polaron (MP) model approach both for the hot and for the thermalized carrier regimes. On the other hand, due to prevailing localization ofpositivecharge carriers in the a-P1 lattice thermally activated hopping of a small lattice polaron (LP) may be the most plausible mechanism of charge carrier transport in this case.

2. Simulation procedure In the present work we used the simulation procedure described in some details in our previous paper [ 11. The computer simulation of reported experimental data [4,6,7] was based on our modified version [ 21 of the initial Sano-Mozumder (SM) model [ 31. According to this modified approach [ 21 the charge carrier is considered as an adiabatic nearly small molecular polaron (MP ) [ 9- 12 ) produced in a photoionization process with effective mass meEand excess kinetic energy E, = mcRug/2 where u. is the radial velocity of the hot carrier at an initial position r, (with respect to the parent ion) at zero time (t = 0). There are two basic input quantities of the model which formally may be regarded as adjustable parameters, namely, the effective mass of the carrier meff and the carrier relaxation rate constant /3 [ 1,lO]. In this work, in a similar way as in ref. [ 11, the m&T) values were evaluated using two independent approaches: (a) from the reported [ 4,6,7] saturation values of the drift velocity ( u,,)~(T) dependences of hot carriers in the low-T region ( 30- 100 K); (b ) from temperature dependences of the mobilitiesp( 7”) of thermalized carriers in the high-Tregion (100-300 K). The nz,& T) dependences in the low-T region of hot carriers were estimated according to the formula [II:

hV,h meff(T)= 2[(u,),(T)]2’ where ( v~)~(8) =const. is an average field-independent saturated drift velocity of the carrier caused by cyclic repetition of carrier acceleration in the electric field b, ending in an emission of an optical lattice phonon hvph at every cycle Il,5]. In these estimates reported characteristic optical phonon frequences h vphof a-P1 crystals were used taking as limited values optical phonon energies hu,, which lie between 200 and 104 cm-’ ([17], see also p. 41 of ref. [ 121) #‘* w This excludes a number of lower-frequency modes. However, previous simulation results (see ref. [ 1] ), as well as present studies (see figs. 3 and 7) demonstrate that the inelastic scattering at higher-energy optical phonons prevails in the saturation region.

EA. Silinsh et al. /Charge carrier transport in organic molecular crystals. II.

t 0

1

2

I 3

I

L 4 E(lD’

I V cm-‘)

I 5

I

I 6

I

391

I 7

-

Fig. I. Calcuiated mean vaiues of negative charge carrier (MP- ) drift velocities v; (8) in the a-direction af an a-perylene crystal as a function of electric field 8: solid lines - computer simulation data of this work: dashed lines - approximation by a square-root ud (8) - fi dependence [ 51 according to refs. [6,7]. The experimental points of vd- (8) are those of refs. [6,7 1. m* denotes the relative effective mass of the carrier. The arrows show the critical field strength values 4 where ud G v,,,and below which the carrier thermalization sets in (where u,,,is the mean thermal velocity of the carrier).

The relaxation rate constants &( 8) in the saturation ( r&)s= const. region were evaluated, for the given rneffand 8 values, according to the following formula [II (3) Formula ( 3 ) has been derived from the expression of maximum kinetic energy meffv&,,/2 of the carrier, gained in the electric field 8’over the mean free path f, at the end of every acceleration cycle, viz. m,nvL,x/2=e~~~=ebx2(vd),t,(I), where I;(&‘) is the mean relaxation time in the saturation region ~S(&)=i/j?S(&) and VmaX=2(vd)s(see fig. 7 of ref.

[II). In the sublinear region of the vd( 8) dependence (see figs. 1 and 6) hot charge carriers, accelerated in the electric field B, are presumably scattered at acoustic lattice phonons [ I,5 1. In this case the fielddependent relaxation rate constant /3=(8) may be calculated according to an equation obtained in terms of the acoustic deformation potental model which gives a pd( 8) -,,/% dependence (See eq. (9) of ref. [ t ] and eq. ( 5 ) of ref. [ 5 ] ). However, for a-perylene

crystals this approach, as can be seen from figs. 1 and 6, does not provide satisfactory agreement with the experimental tid( 8) curves. It has been shown [ I] that the effective scattering rate COIIStSIIt 8. ( 8) in the sublinear vd( 8) regiOtI may be estimated according to the following formula, analogous to formula (3 ): (4) Calculated according to eq. (4) /I.( 8) values, used in the simulation procedure as the input parameters, provide a good agreement with the experimental vd( 8) curves (see figs. 1 and 6). sincevd(B)<(vd)s,Ba(~)>a($h,),where

ehris

the threshold field strength above which generation of optical phonons becomes dominant and the sublinear vd( &) dependence passes over to the saturation region of (u&=const. In the high-temperature range (100-300 K) of thermalized carriers the m&T) values were estimated from reported experimental ,u( T) dependences [4,6,7] according to the following formula [ 1,101:

E.A. Silinsh et al. /Charge carrier transport m organic molecular crystals. II.

392

%ff( T) =

3e2102

[P(T) 12kT’

which results from relating the expressions of the mean thermal velocities: uth= p(T) kT/e10 and uth = JmW, respectively. In general, formula (5 ) is an isotropic one. However, if the experimental components of the mobility tensor are used one obtains the corresponding m,ff tensor values. The T-values were estimated according to the following formula [ 1 ] :

B(T)=

I*(TPT

-jgr’

0

(6)

from &h, using ~,~=f~/~=f~~. These estimates were carried out under the assumption of a constant mean free path lo of the carrier, equal to the lattice constant in the corresponding direction of carrier transport in a-P1 (cf. ref. [ 11). In the simulation procedure mean charge carrier trajectories r(t) and velocities u(t) were also determined as a function of temperature T, electric field d and angle 6 between the initial radial velocity v, of the carrier and the external electric field B, as well as the corresponding dispersion parameters of r and v, viz. a, and cr,,, caused by stochastic force effects of carrier motion. The initial charge carrier-parent ion separation length r,, from which thermalization is supposed to start, has been taken equal to the nearest-neighbour distance along the corresponding crystal axis of a-Pl, viz. rO=ao and r, =cb (cf. refs. [ 1,2] ).

d > 6 x 1O4V/cm than in naphthalene crystals where saturation begins already at ~92 1 x 1O4V/cm for both types of carriers (cf. ref. [ 1] ). In the sublinear region of vd( 8) the corresponding parameter pa( 8) was evaluated according to formula (4). As can be seen from fig. 1, the simulation curves provide good agreement with the experimental points of the vd( b) dependences. On the other hand, the analytically derived formula vd ( 6) - fi (see section 2 and refs. [ 1,5 ] ) gives less satisfactory agreement with experimental data. The arrows in fig. 1 show the critical field &Cvalues at which thermalization of charge carriers takes place. As may be seen from fig. 1, the & values, at which V*SZ&h, are temperature dependent. It may be shown that the temperature dependence of the critical field &,( T), above which hot carriers can be created, follows an empirical formula, proposed earlier in ref. [ 101: &‘=(T) -exp(aT”*)

(7)

.

Fig. 2 shows the G,( T) dependence, indicated by the arrows in fig. 1, in a log ~8~versus T’/’ plot. As can be seen from fig. 2, the proposed empirical formula (7) may be regarded as valid for the description of the ~9~ ( T) dependence. On the whole, the vd( 8) dependences in fig. 1. demonstrate that in the electric field range from 8,

3. Simulation results and their interpretation 3.1. Negative charge carrier (MP-) transport in the a-direction of an a-perylene crystal Fig. 1. shows the computer simulation results of drift velocity v; (8) dependences of the negative charge carriers in the hot carrier transport regime at low T ( T< 100 K) in an a-P1 crystal as reported in refs. [ 4,6,7]. The input parameters m,& T) and B( T) of the simulation model were determined according to section 2. As may be seen from fig. 1, the saturation regime of vd( ~Y)-+(v~)~=const., at which the scattering at optical lattice phonons prevails, can be reached only at considerably higher field strength

1.0

1’1 30

I

40

I

I

60 80 ~~2 (K%) -

J

Fig. 2. Estimated temperature-dependent critical electrical field &(T) values at which hot, non-thermalized charge carriers can be created plotted as log ECversus T’12. The S; values, indicated by the arrows in fig. I have been used.

EA. Silinsh et al. /Charge carrier transport in organic molecular crystais. il.

6 x 104 V/cm the hot carrier scattering on the acoustic lattice phonons prevails. The possible range of meff values in the low-T region ( TG 100 K) was estimated by formula (2 ) using reported experimental ( vd)$( T) data at c%=6 x 1O4 V/cm (see fig. 1) and presumed limiting values of opticai phonon energies ktrph (see section 2). The co~esponding RZ& T) ranges in the low-T hot carrier region ( Tf 100 K) are shown by vertical bars in fig. 3. In the high-temperature region (see fig. 3) the rn& T) values were estimated from reported experimental ,u( 7’) dependences [4,6,7] according to formula (5) under the assumption of a constant mean free path &of the carrier, equal to the lattice constant in the direction of the a-axis, viz. I,= 1.14 nm. As may be seen from fig 3, the calculated meff value of a thermalized carrier at T= 100 K gets on a straight line in the semi-log plot which coincides with the upper end of the vertical bars of the possible m&T) range of hot carriers. First, these results show that in this T range (30 G Tq 100 K) the me& T) dependence follows the exponential law [ 1 ] : up to I=

(8) Second, the coincidence of the line with the upper end of the vertical bars indicates that the scattering most probably occurs at the higher-energy optical phonons around h &,h= 200 cm- ’ rather than those at lower energies, It should be emphasized that the coincidence at T= 100 K of the m* value, estimated by formula ( 5 ) with the upper end of the vertical bar of the m *range, calculated according to formula ( 2 ) , may serve as an additional indication of the validity of both approaches. The extrapolated meffvalue at zero temperature rn& is rather small - practically equal to two free electron masses mt; sz2m,. The parameter T, in the approximation formula (8) equals 32.3 K (-22 cm-‘). As may be seen from fig. 3, at higher temperatures, above 100K, the me& T) curve declines from the exponential dependence (8) and shows a tendency towards saturation. This behaviour of the rn& T) dependence is reminiscent of similar character of the rn& T) curve in the case of negutiw charge carrier transport in the c’-direction of a naphthalene crystal (seeref. [i],fig. 11). For the high-T region, where the mobility p= T--n formula ( 1) holds, it is possible to derive the rn& 7”) dependence from eq. ( 5 ) . Thus, inserting formula ( 1) in (5) one obtains the following equation: m&T)=

1

L. 0

a

1 *

50

I

I

I

1

I

I

100

150

200

250

T(K)

300

-

Fig. 3. Estimated values of relative effective masses m*f T) of a negative charge carrier (MP-) and mean values of carrier free paths &( T) in the n-direction of a-perylene as a function of temperature T. The vertical bars indicate the possible m* range in the hot carrier regime (formula (2) ), the filled circles - m* values in the temperature range of the~aiized carriers, evaluated according to formula (5 ).

393

3e=i2 + T2"-' .

Hence, in the high-T region one may expect a power-law dependence of m& 7")instead of an exponential one. Indeed, it may be shown that in the temperature range 100 < T< 300 K the m&T) dependence (see fig= 3) can be linearized in a log rneff versus 1ogT plot with the slope value equal to (2n,,-- 1) ~2.56 for no,.,=1.78 [7]. However, this approximation does not fit for the low-T region at TclOOK. Fig. 3 also demonst~tes the extremely small mean free path &,of the carrier, equal to the lattice constant in the a-direction, viz. ao= 1.14 nm, which remains so small down to very low temperatures. Only below T= 80 K the lo values begin to increase. It ought to be reminded that in naphthalene crystals this increase

394

E.A. Silinsh et al. / Chnrge carrier transport tn organic tno~ecul~rtrystals. II.

of I0 begins already at = 150 IS (see figs. 4 and 5 of ref. [ I] ). These data confirm the feasibility of using this lo value (&=a,) in formula (5) for evaluating the effective mass temperature dependence nz*( T) in the high-Tregion. Fig. 4 shows a plot of computer-simulated mean trajectories of negative charge carriers in the u&plane of a-PI under the influence of Coulombic and external electric fields as a function of the angle 8 between the initial velocity vector 4 of the ejected carrier and the external field & (for a detailed description of the simulation model see refs. [ 1,2 ] ) . The simulation has been performed using the following input parameters: T=60 K, m*=11.4m, (see fig. I), j?=j.&=6.7~ 10” s-’ and &=6X lo4 V/cm (&]]a). The initial kinetic energy Ek of the ejected electron is assumed to be E k = m,&/2 = 1.3 eV and the corresponding value of vo= 2.1 X 1O7cm/s. Fig. 4 demonstrates that there emerge two main types of mean carrier trajectories as a function of the angle 0. In the range between 8= 0’ and 8= 135’ hat carriers are created from the very beginning. Carriers moving along the direction of the field d at &LO” (trajectory 1) do not thermalize but leave the Coulombic capture radius r, and obtain a constant mean drift velocity, (Q)$= 1.34~ lo6 cm/s, as reported in refs. [4,6 1, which is greater than the mean thermal velocity ( vth= 8.8 X 1O5cm/s). Carriers ejected from the parent ion at angles 0” < 8~ 135 * are deflected by the external field and also obtain the reported ( va),

m'I_ll.I,rn,

T=60K &=6x10'

#

-

v cm“ Ella

s-1 -30

-20

-10

0

10

20 Z(“rn

vafue for given parameters T, d and m* (trajectories 2-4). On the other hand, carriers ejected at angles 180” >, 6% 135 o30’ thermalize within the Coulombie radius r, forming bound charge pair (CP) states which need additional thermal activation to become free carriers according to the Onsager mechanism (see ref. [2] and references therein) (trajectories 5-7). On the whole the mean carrier trajectories in fig. 4 are rather similar to those of a naphthalene crystal (cf.fig.2ofref. fl]). However, there is a minor difference. In the case of a naphthalene crystal there appears a relatively wide range of angles ( 130” < 8< 140” ) at which a rather peculiar type of mean trajectories emerge [ I]. In the case of a-P1 this inte~ediate range of transition from hot to thermalized carriers is very narrow, of the order of 30’ (0.5’ 1; viz. the transition occurs at 8=135”15’~15’. The calculated mean values of carrier drift velocities Q = vd( t ) as a function of time, as well as the dist~bution curves of carrier coordinates a(r) and velocities q(u), and the corresponding dispersion parameters cr, and o;, are aiso similar to those of the naphthalene crystal (see figs. 3,7 and 8 of ref. [ I] ). Fig. 5 shows the calculated field dependences of the carrier mean free path 1,( 8) and 1,( R) for scattering at optical, respectively, acoustic lattice phonons. These dependenees have been obtained for negative charge carrier transpo~ in the a-direction of the a-PI crystal at T= 80 K with m*= 23.2 m, (see fig. 1). The I, ( 6) dependence was evaluated according to formula [ 11:

30

LO

1 -

Fig. 4. Calculated mean (r> values of charge carrier trajectories r( f?)= {x( t ), z( t ) } in the &-plan of an a-perylene crystal for different values of the angle 6 between the initial radial velocity v0 of the carrier and the external electric field 8= -E,; ( 1) 8= 0” , (2) 8=90”, (3) 8=120”, (4) 8=135”, (5) 8=t40”, (6) f&160”, (7) &=180”.

Thus, the I,( 6) value decreases reciprocally with increasing field strength I (see fig. 5 ). The /,( B) value was evaluated in the following way. The mean scattering time at acoustic phonons r, ( 8) in the sublinear region of vd( 8) is related to the relaxation rate constant as r, ( b) = 1I/3, ( &). The value of a, ( 8) can be determined according to formula ( 4 ) . Consequently, the mean free path i,(d) equals

Thus, the i, ( b) dependence may be obtained directly

E.A. Silinsh et al. /Charge carrier transport in organic molecular crystals. II.

395

3.2. Negative charge carrier (MP-) transport in the L-direction of an cu-perylene crystal

--

6 t

Fig. 5. Calculated curves of carrier mean free path I.( 8) and I,( 8) for the case of scattering at acoustic and optical lattice phonons, respectively, as a function of the electric field 8 for carriers moving in the u-direction of an a-perylene crystal.

from the experimental sublinear v, ( 8) dependences (fig. 1) for given d and T values. As can be seen from fig. 5,1. ( 8) decreases with decreasing field d and approaches the lattice constant value uo= 1.14 nm at I= 1 x 1O4V/cm. Thus, at this low field limit 1, becomes equal to the mean thermal scattering length lo= u. (see fig. 3 ). By the way, the lo value may be evaluated independently from other simulation data, viz. lo=?,vth, where rr is the carrier relaxation time 7,= 1/p= r1h/2 (see fig. 3 of ref. [ 1 ] ) and Vu, is the mean thermal velocity of the carrier. For thermalized carriers at T=80Kina-P17,=2.13x10-‘3sand~,h=7.0X105 cm/s. Consequently, lo= 1.49 nm. As we see, this rather rough evaluation result is pretty close to the lattice constant value a, = 1.14 nm and the 1, value at b= 1 x lo4 V/cm (see fig. 5). Returning once more to fig. 5, one can see that at field values above d= lo6 V/cm the inequality l,( 8’) < I,( 8) determines the transition from dominant carrier scattering at acoustic phonons to scattering at optical phonons, corresponding to the transition from the sublinear dependence of Q,(B) to the saturation region of carrier drift velocity ( u~)~= const. (see fig. 1).

The computer simulation results of the reported [ 67 ] drift velocity v; ( 8) dependences of the negative charge carriers in the hot carrier transport regime at low temperatures ( TG 80 K) are shown in fig. 6. As can be seen from fig. 6 the calculated vd( &‘) curves are in good agreement with experimental data both in the sublinear and in the saturation region. On the other hand, approximation by a square root vd ( E) N &? dependence [ 5 1, according to ref. [ 61 gives rather poor agreement with experimental data and, obviously, is not suitable for describing the saturation region. In the c’-direction the (ud)s values are smaller than in the u-direction by a factor of two. However, the saturation region starts at lower field strength (cf. figs. 1 and 6). Also, in this case the critical field values G, for hot carrier creation (shown by arrows in fig. 6) are strongly temperature dependent and can again be satisfactorily approximated by formula (7 ). The effective mass T-dependence (fig. 7) was estimated according to the procedure described in sections 2 and 3.1. As can be seen, for the negative charge carrier transport in the c’-direction, the exponential law (8) for rn& T) holds both in the hot and partially in the thermalized carrier regions up to 200 K. Only a higher temperatures, viz. at T>200 K, the m,,(T) curve declines from the exponential dependence (8). It is interesting to notice that in this case the straight line in the semi-log plot crosses the middle part of the vertical bars of the possible m*(T) range. This may indicate that the scattering of the carriers takes place at middle range optical lattice phonons, with frequencies around 130-l 60 cm-’ (cf. ref. [17] andp.41 ofref. [12]). The extrapolated rn: at T= 0 in this case equals 10. The parameter To in eq. (8) equals 43.1 K (30 cm-‘). Also in this case the rn& T) dependence in fig. 7 can be linearized in the high-T region ( 100 < T< 300 K) in the log meff versus log T plot according to formula (9). Since in the c’-direction nC.C. = 2.15, the corresponding (2n,,,, - 1) value in the power-law (9) equals 3.3. This explains why the exponential m,,(T) dependence extends up to x 200 K in fig. 7. It may be shown that the exponential function of m,,( T) (eq. (8 ) ) and the power-law

E.A. Silinsh et al. /Charge carrier transport in orgamc molecular cr?,stals. II

396

Fig. 6. Calculated mean values of negative charge carrier (MP- ) drift velocities v,- ( 6) in the c’-direction of an a-perylene crystal as a function of the electric field &: the solid lines represent computer simulation data of this work; the dashed lines an approximation by the square-root type vd ( 6) _ $ dependence [ 5 ] according to ref. [ 6,7]. The experimental points of vd- ( 8) are those of refs. [ 6,7]. The arrows show the critical-field 8, at which vd=vlh’f10 is the low-field Ohmic mobility.

function of m,& T) - T 2n- ’ ( eq. ( 9 ) ) practically coincide for 2n - 1z 3 in the 1OO- 150 K range. Thus, the deviation of the exponential m,& T) dependence at higher temperatures seems to be common for negative charge carriers in u-P1 (see figs. 3 and 7 ) as well as in naphthalene crystals (cf. fig. 11 ofref. [l]). As can be seen from fig. 7, also in the cl-direction the carrier mean free path 1, remains constant lo % CA= 1.1 nm over a wide T range down to 80 K. Only below 80 K one observes an increase in the f. value. Thus, a small value of lo, practically equal to the lattice constant, is a general property of polyacenes (see refs. [ 1,2] ) and of a-P1 over a wide temperature range. This characteristic feature should be taken into account in new trials of developing more realistic transport models of charge carriers in organic molecular crystals.

r

3.3. Positive charge carrier transport in an a-perylene crystal 111111 50

I

100

150

200

250 T(K)

3

-

Fig. 7. Estimated values of relative effective mass m*(T) of a negative charge carrier (MP- ) and the mean values of carrier free paths 10( T) in the c’-direction of a-perylene as a function of temperature T. The vertical bars indicate the possible range of m* in the hot carrier regime (formula (2) ), the filled circles m* values of thermalized carriers (formula (5) ).

In contrast to the activationless mobility temperature dependence of the type fi”- - T -“, characteristic for negative charge carriers, the positive carriers in aPl exhibit thermally activated mobility p+(T) dependences. Karl and co-workers [ 6 ] have found the p+(T) values. (The direction of the electric field parallel to the vertical on the crystal plate, the L-di-

EA. Silinsh PI al. /Charge

carrier transport

rection, forms the following angles with the crystallographic axes: zi L, a= 54.5”, & L, b=90” and 4 L, c’ = 35.3’ ) . However, the p+ ( T) dependence in this case is not a simple exponential p+ ( T) N exp( - EJkT). The activation energy E, actually decreases with increasing temperature and practically reaches the value zero at 300 K. The mobilityp+ is rather small, of the order of 0.1 to 0.3 cm2/ Vs in the 120 to 300 K region [ 6,7 1. Very often such a thermally activated mobility in organic crystals is caused by multiple shallow trapping due to the presence of chemical impurities or structural defects [ 16,181. In such a case, instead of the real microscopic mobility, one observes a small effective trap limited mobility. However, the a-P1 crystals, used in p+ ( T) measurements of ref. [ 61, were obtained after very careful purification and the observed mobility should be regarded as a microscopic one. Karl and co-workers [6] have found similar thermally activated mobility dependences also for the principal tensor components ,ucC:and ~2 in the case of positive charge carrier transport in 2,3-dimethyl-naphthalene and for the mobilitypu,,! of negativecarriers in the c’direction of a pyrene crystal [ 61. In all these cases the mobility value is small, below 0.4 cm’/Vs. This means that pronounced localization of charge carriers prevails and, consequently, the mean residence time of a carrier on a particular molecule may be long enough to form a lattice polaron [ 6,12 1. Since, however, the mobility activation energy E, changes with temperature, a simple small lattice polaron model may not be valid and one should try to search for a modified, more universal polaron-type hopping transfer mechanism (cf. ref. [ 201).

4. Discussion Let us now summarize the main results obtained in this paper and in a previous one [ 1 ] on charge carrier transport phenomena in a-perylene and polyscene crystals. It has been demonstrated that in the low-T region the exponential rn& T) dependence (8) is valid both for the positive and the negative charge carriers in naphthalene crystals (see ref. [ 11, figs. 4,5 and 11) and for the negative charge carriers in a-PI crystals (see figs. 3 and 7 in this paper ) .

in organic molecular crystals. II.

397

In the early stages of the development of small lattice polaron theories, an expression of the rneffdependence upon temperature, equivalent with eq. ( 8 ) was proposed by Jamashita and Kurosawa [ 2 1 ] and Sewell [22] (see also ref. [23] ) which was assumed to be valid for a polaron band approach description of charge carrier transport at low temperatures. Actually, in our case, a band model description of charge carrier transport may be physically reasonable only at the lowest temperatures where the mean free paths lo of the carrier is of the order of at least several lattice constants lo> a, (see figs. 3 and 7, and ref. [ 131). As the temperature increases the effective width 6E, of the polaron band drastically decreases, inversely to the exponentially growing effective mass of the carrier, viz. 6Ep= 1/rnefi. Consequently, the charge carrier becomes strongly localized and the band model Bloch wave approach ought to be replaced by a classical particle representation. As a matter of fact, the power-type rn& T) dependence (9) has been derived in terms of such an approach. We have shown that in the high-Tregion eq. (9) is valid for the description of the rn,& T) dependences of negative charge carriers in naphthalene (see fig. 11 of ref. [ 1 ] ) and a-P1 crystals (see figs. 3 and 7). However, for the positive charge carriers in the a and c’-directions of naphthalene crystals the exponential rn.& T) dependence (8) holds for the whole temperature range from 10 up to 300 K (see figs 4 and 5 of ref. [ 1 ] ). The cause of this apparent contradiction might be the following. It may be shown that in the high-T region ( 150 < T,< 300 K) these m,,( T) dependences can be linearized also in a log meff versus log T plot according to eq. (9). In this case the slope n values of the dependence p(T) _ T-’ are large, viz. naa= 2.9 and n,,., = 2.8 [ 7 1. As a result, the corresponding exponents (2n - 1) of the power-type rn,@(T) dependence (9) equal (2n,,- 1) ~4.8 and (2n,,,. - 1) ~4.6, respectively. It may be shown that for these (2n - 1) values the power-type (9) and exponential (8) dependences practically coincide in the 150 < T< 300 K range. It should be emphasized once more that the most characteristic feature ofboth thepositiveand the negative charge carrier mobility in polyacenes is the lack of thermal activation yielding a decreasing p( T) dependence with increasing temperature according to a

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E.A. Silinsh et al. /Charge carrier transport in organtc molecular crystals. II.

typical power law fi( T) - T -“, where n may assume a rather wide range of values ( n = O-2.9 ), different for negative and positive charge carriers and for different directions of movement in the lattice. Similar ,U( T) dependences have been observed for a number of other aromatic and heterocyclic organic molecular crystals (see ref. [ 71 and references therein). It has sometimes been argued that such a “negative” .LL(T) dependence speaks in favour of a band-type, coherent carrier motion, similar in nature to transport in conventional inorganic semiconductors. However, such a simple assumption is in contradiction to other important transport features in organic molecular crystals. First, the mean free path lo of carriers over a wide T range ( 150-300 K in polyacenes [ 11 and 80-300 K in a-Pl), where the ,u( T) - T -’ dependence holds, is of the order of one lattice constant (/,=O.S-1.0 nm). This means that the carrier is scattered practically at every lattice site and one should use some kind of hopping mode1 for an adequate description of such type of motion. The controversial character of this problem is also reflected in the recent review by Movaghar on conduction in molecular solids [ 19,201. The author emphasizes that in this case the band-type transport approach may be valid only at low temperatures. However, the typical mobility dependence p( T) - T -’ at higher temperatures “cannot be said to be completely understood” and “a unique and genera1 description for the behavior... is at present lacking” [ 201. In order to give a plausible physical interpretation of experimental results of recent carrier transport studies [ 4-8,13,18] and their computer simulation in refs. [ 1,2] and in the present paper we have proposed to apply, in a phenomenological approach, a mode1 of a nearly small molecular polaron [ 2,9,1 O121. In our opinion, the present successful simulation of reported experimental mobility data [48,13,18 1, demonstrates the plausibility of the molecular polaron (MP) mode1 approach for a phenomenological description of the charge carrier separation and transport phenomena in polyacene and a-P1 crystals over a wide temperature and electric field range. In this approach a charge carrier emerges as a heavy quasi-particle with an exponential (8) or power-law type (9 ) temperature-dependent effective

mass (which depends on the carrier sign and the direction of transport in the crystal lattice). Thus, in our opinion, the nearly small molecular polaron approach reconciles the low and high temperature transport regimes. It has been shown [ 9,10,12] that a nearly small molecular polaron may be formed as the result of the interaction of a charge carrier with intramolecular vibrations of the molecule, on which it is localized during the residence time, and also with polar, IR modes of the nearest-neighbour molecules. The nearly small molecular polaron may be visualized as a slightly delocalized ionic state in a neutral molecular crystal which moves by hopping via tunneling from one crystal site to another [ 9,10,12 1. The molecular polaron approach is also consistent with energy data of ionized states in polyacene crystals, since it has been shown that the conductivity levels of thermalized carriers in these crystals are not those of free electrons but are relaxed molecular polaron states separated by an adiabatic energy gap Ead [ 2,9,11,12]. The results presented above may serve as a phenomenological and conceptual basis for the development of a comprehensive and self-consistent nearly small molecular polaron theory - a challenge to theoreticians working in the polaron field. The specific cases of thermally activated carrier transport, discussed in section 3.3, apparently require a novel small lattice polaron mode1 approach, taking into account the observed temperature dependence of the activation energy E,=E,( T) for polaron hopping. We should briefly discuss also the phenomenon of anomalous temperature-independent mobility of negative charge carriers over a wide Trange ( T= lOO300 K) in the L-direction of naphthalene and anthracene crystals (see ref. [ 6,7 ] ). A number of authors have argued about the possible physical reasons for such anomalous mobility behaviour (for references and details see p. 37 of ref. [ 151 and p. 360 of ref. [ 161). Recently, Kenkre et al. [24] have returned to the problem and demonstrated that the specific behaviour of injected negative charge carriers in a naphthalene crystal can be treated in terms of the polaron theory. On the other hand, Karl [ 61 has proposed a simple empirical approach, according to which the effective nearly T-independent carrier mobility in the c’-di-

E.A. Silinsh et al. /Charge carrier transport in organic molecular crystals. II.

rection of naphthalene may be presented as a superposition of the c’-components of two independent but differently temperature dependent p ( T) tensors: ,u,=aT-‘.48 and p2=bexp( -EJkT). We have extended this approach in terms of polaron models assuming that these two mobility p(T) components may be regarded as a manifestation of two different, coexisting transport mechanisms: a nearly small molecular polaron, which moves by hopping via tunneling without activation energy, displaying a typical pu(T) dependence, ,uMp=aTen, and a small latticepolaron which moves by thermally activated hopping and thus exhibits a typical exponential mobility dependence (see ref. [ 11). However, also in this case a more detailed and comprehensive polaron theory is required. Finally, we should emphasize, that our studies demonstrate also the plausibility of the molecular polaron approach for a description of hot carrier creation and transport phenomena. Both experimental [ 4-81 and our simulation data show that there exists a strongly temperature-dependent critical electric field value ~9~at which hot carriers can be created. However, the possible theoretical implications of the proposed empirical formula (7) of the observed &( T) dependence are not yet clear and require further studies.

Acknowledgement We are greatly indebted to Professor N. Karl for stimulating and highly relevant comments, which helped us to improve the final version of the paper.

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