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Phono-limited transport of charge carriers in molecular crystals
Volume 3, number 9
CHEMICAL PHYSICSLETTERS
PHONON-LIMITED
TRANSPORT
IN MOLECULAR
OF
1 Skptembor 1969
CHARGE
CRYSTALS
CARRIERS
*
R. W. MUNN** and W. SIEBRAND Division of Pure Chemistry. National Research Courcil of Canada, Ottawa, Canada
Received 10 July 1969 For strong electron-phonon interactions, the mobilities of charge carriers in molecular crystals oan be limited by phonon velocities. A model oalculation of mobilities and their temperature dependence inoiuding this limit gives results in good agreement with experiment for anthracene.
There has been considerable discussion as to whether electronic transport in molecular crystals, particularly the aromatic hydrocarbons, occurs coherently or by random,jumps between adjacent molecules (hopping) [1,2]. The answer to this question depends on the nature of the electron-phonon interaction: whether the strongest coupling is with intermolecular or intramolecular vibrations; .whether it is linear or quadratic in the phonon coordinates; anal how strong it is compared with intermolecular electronic interactions. The electron-phonon interaction has been treated by Friedman [3] for coher snt trana port, and in more detail by Gosar and Choi [4] for hopping transport; both treatments assumed a linear interaction with intermolecular vibrations. The discussions all assume that the transport is governed only by the strength of electronic interactions. However, for hopping transport the electron-phonon interaction must be strong, so that if the phonons move very slowly their speed will determine that of the electron because it is strongly coupled to them. The existence of this slow-phonon limit besides the usual slowelectron limit where the electron moves slowly was recognized by Rashba [5], who discussed the effect on crystal spectra. A measure of the relative phonon arm electron velocities is given by the ratio of their energy bandwidths. The phonon bandwidth for intramolecular vibrations is about 5 cm-I, which is two orders of magnitude smaller than calculated electron bandwidths for aromatic hydrocarbons [6], so that the siowphonon limit may well prevail in these crystals. Here we report the first calculations of hopping
transport in the slow-phonon limit, for a linearchain model of a molecular crystal. Calculations are also reported for slow-electron limit, and the results are compared with experiment for anthracene. We consider a molecular-crystal model consisting of a linear chain of diatomic molecules treated as harmonic oscillators of reduced mass m and frequency wo. The single excess electron is asarmed to interact only with intramolecular vibrations; intermolecular vibrations are suppressed by fixing the centres of mass of the oscillators at sites nd, where n i.s an integer. Three interactions are introduced: a mechanical coupling wl between adjacent oscillators, an electronic coupling J between adjacent molecules, and an electrcn-phonon interaction w2. The electron-phonon interaction is taken as the strongest of the three, giving rise to zeroth-order states in which the electron is localized at one site, and is taken quadratic in the phonon coordinates [7]. As pointed out by Rashba [5], such a quadratic interaction may involve only a small fraction of the energy of a phonon and yet still be more :mportant than a linear interaction with an energy comparable to phonon energies; furthermore, out-of-plane bending modes in aromatic hydrocarbons are known to change in frequency by about 20% on electronic excitation [8]. Except for the quadratic instead of linear electron-phonon interaction, this model is the same as discussed by Holstein [9]. The electron wavefunction is written in the tight-binding approximation as a linear superposition of localizeli electronic wal1efunctions
* Issued as NRC’ No. 10929. ** NRCC Postdoctoral Fellow 1966 to present.
Here R is the position coordinate of the electron, 856
Volume 3, number 9
CHEMICAL PHYSICS LETTERS
1
September 1969
and {Q) denotes dependence on the set of vibrational coordinates X, for each.molecule. The coefficients aA+) satisfy
aa,,
iAx=
where the terms in wl and J are taken as perturbations. The zeroth-order coefficients are products of oscillator functions for each oscillator, all with frequency w. except the one at the site n of the localized electron, where the frequency is (02 - w2)+. In”the glow-phonon limit, J is the larger of the two perturbations, and produces a quasicontinuous exciton band of width 4J. The perturbation $rnw~x~x,+l induces transitions in which phonons are emitted and absorbed at sites n and ?1+1, but ever. when the oscillators at these sites differ in freqnency by 20% the amplitude of multiphonon processes is negligible, so that only single-phonon processes need be considered. The rate of electron hopping between sites n and n+l is equal to the rate of phonon exchange, which is calculated by time-dependent perturbation theory. Energy conservation requires that the sites should have equal numbers of phonons before and after each exchange. The square of the matrix element for exchange of the phonon between sites IZ and ntl is then k,,n+l
. = 2(rrw~/4~~~~~~(V~+l)
6(7w~,~+l), (3)
where vlt is the number of phonons initially at site n, etc. The rate of electron hopping is the rate of exchange of all the phonons, which is given by eq. (3) divided by vrz. In the thermal average this rate is proportional to l/(l+u)2, where u = exp(- fiwo/kZ’). The complete expression for the rate requires the density of final states, which we approximate by the inverse electron bandwidth, 1/4J. The rate is then 4 7&W, W~(n--n+l) = lSJw2,(l+o)2 The rate of electron hopping from site tz to n+l and 32-1 is 2W~(n--n+l), and the diffusion coefficient D is simply 2d2’W~(n-n+l). Finally, the mobility cr is given by p = eD/kT, or 656
here-the subscript p denotes hopping in the slowphonon limit. In the siow-electron limit, y is the larger of the two perturbations, and produces a quasi-continuous phonon band of width tiwf/w,. The matrix elements of the perturbation J are simply J times an overlap between oscillator functions. This overlap is negligible unless the number of phonons at a site remains unchanged, when it is unity. Energy conservation requires that the phonons at the two sites exchange, so that the sites must start zuid finish with equal numbers of phonons. The thermal average of the squared matrix element is then J2 times the thermal probability of equal numbers of phonons at adjacent sites, or J2(1-cr)/(l+o). The density of final states is taken as the thermal average number of phonons per oscillator, u/(1-u), divided by the bandwidth. The rate of hopping from site n to n+l is then w~(‘L+z+l) The mobility
2i7J2wo
=-
gw$
[ 1u
liu
is 4s;e$ wod2 he=
(7) $04
where the subscript e denotes the slow-electron limit. In order to compare the model with experiment for anthracene, w. is chosen as the mid-point of the frequency range of out-of-plane bending vibrations, 350 cm-l. This fixes the temperature dependence of pp and peg leaving the absolute magnitudes determined by the quantities J, d, and ~1. ln fig. 1, pp calculated for a suitably-chosen numerical factor is compared with experimental data on electron mobilities in the a direction of anthracene [lo: 111. The mobility varies as l/T at low temperatures as observed by Kepler [lo]. Similar fits can be obtained to all the other mobilities of holes and electrons except for electrons in the c’ direction (I ab plane). Thus it appears that those mobilities which are decreasing functions of temperature may represent hopping transport in the slow-phonon limit, and not necessarily coherent transport as usually assumed. The electron mobility in the c’ direction shows a positive temperature dependence which does not fit eq. (5). However, the electron bandwidth in this direction is known to be much smaller than in the other directions [S], which suggests that here it may be
Volume
3. number
9
CHEMICAL
PHYSICS
5-
z-
I
j I %
L969
phonon limit is always present if there is strong electron-phonon coupling. We have also introduced electron-phonon coupling quadratic in the phonon coordinates. which prevents the neareqonential increase of mobility with temperature produced by the commonly-used Ifnear terms in the slow-electron limit. A particularly satisfying feature of the present model is that for directions in which the electronic bands are relatively wide the slow-phonon limit applies, leading to a negative temperature dependence of the mobility, whereas in directions where these bands are very narrow the slow-electron Limit takes over, leading to a positive temperature dependence. This is precisely the situation observed experimentally in anthracene.
IO-
^ ‘0 :: T > ” E 2
1 September
LETTERS
0.5 -
REFERENCES [l] S. H. Glarum. J. Phys. Chem. Solids 24 (1963) [Z] R. M. Glaeser
T(“K)
Fig.
1. Electron
mobilities
in anthracene.
lines are the theoretical curves. p
The solid
for the a direction
The b&ken lines are the and 12, for the c’ direction. e.xperimental data from A. ref. [lo]; B. ref. [Ill; C. ref. (121: and D. ref. 1131.
smaller than the phonon bandwidth, so that (7) rather than eq. (5) would apply. This is posed by fig. 1 where p,(c’). calculated for sonable values of J, d and ~1, is compared
eq. suprea-
with experimental data for anthracene [lo-131. The agreement between theory and experiment is again very satisfactory. In summary, we have extended the theory of carrier transport in molecular crystals by considering the regime where carrier mobilities are governed ty phonon velocities. Such a slow-
and R. S. Berry.
ZGPi.
J. Chem. Phys. 4%
(1966) 3797. [3] L.Friedman. Phys.Rev. 140 (1965) Al649. [4] P. Gosar and S.-I. Choi. Phys. Rev. 150 (1966) 529. [S] E. I. Rashba. Zh. Eksperim. i l’eor. Fizz. 50 (1966) 1064, English transl. Soviet Phys. JETP 23 (1966) 708. [S] R. Silbey. J. Jortner, S. A. Rice and BI.T.Vala. 5. Chem. Phvs. 42 (1965) 733; 13 (1965) 2925. [7] R. W. hlunn and iV. Siebrand, J: Chem. Phys.. to be published. [S] A. P. Best, F. RI. Garforth. C. K. Ingoold. H. G. Poole and C. L. Wilson. J. Chem. Sot. (1918) 406 (part I) - 516 (part XII). [9] T. Holstein. Ann. Phya. 8 (1959) 325. 343. [lo] R. G. Kepler. Phonons and phonon interactions (W. A. Benjamin_ Xe\v York, 1964) p_ 578. [ll] G. T. Pott and D. F.Williams, J.Chem. Phys., to be published. [12] I. Nakada and Y. Ishihars. J. Phvs. Sot. Japan 19 (1964) 695. 1131J. Fourny and G. Delacate. J. Chem. Phys.50 (1969) 1028.
651
CHEMICAL PHYSICSLETTERS
PHONON-LIMITED
TRANSPORT
IN MOLECULAR
OF
1 Skptembor 1969
CHARGE
CRYSTALS
CARRIERS
*
R. W. MUNN** and W. SIEBRAND Division of Pure Chemistry. National Research Courcil of Canada, Ottawa, Canada
Received 10 July 1969 For strong electron-phonon interactions, the mobilities of charge carriers in molecular crystals oan be limited by phonon velocities. A model oalculation of mobilities and their temperature dependence inoiuding this limit gives results in good agreement with experiment for anthracene.
There has been considerable discussion as to whether electronic transport in molecular crystals, particularly the aromatic hydrocarbons, occurs coherently or by random,jumps between adjacent molecules (hopping) [1,2]. The answer to this question depends on the nature of the electron-phonon interaction: whether the strongest coupling is with intermolecular or intramolecular vibrations; .whether it is linear or quadratic in the phonon coordinates; anal how strong it is compared with intermolecular electronic interactions. The electron-phonon interaction has been treated by Friedman [3] for coher snt trana port, and in more detail by Gosar and Choi [4] for hopping transport; both treatments assumed a linear interaction with intermolecular vibrations. The discussions all assume that the transport is governed only by the strength of electronic interactions. However, for hopping transport the electron-phonon interaction must be strong, so that if the phonons move very slowly their speed will determine that of the electron because it is strongly coupled to them. The existence of this slow-phonon limit besides the usual slowelectron limit where the electron moves slowly was recognized by Rashba [5], who discussed the effect on crystal spectra. A measure of the relative phonon arm electron velocities is given by the ratio of their energy bandwidths. The phonon bandwidth for intramolecular vibrations is about 5 cm-I, which is two orders of magnitude smaller than calculated electron bandwidths for aromatic hydrocarbons [6], so that the siowphonon limit may well prevail in these crystals. Here we report the first calculations of hopping
transport in the slow-phonon limit, for a linearchain model of a molecular crystal. Calculations are also reported for slow-electron limit, and the results are compared with experiment for anthracene. We consider a molecular-crystal model consisting of a linear chain of diatomic molecules treated as harmonic oscillators of reduced mass m and frequency wo. The single excess electron is asarmed to interact only with intramolecular vibrations; intermolecular vibrations are suppressed by fixing the centres of mass of the oscillators at sites nd, where n i.s an integer. Three interactions are introduced: a mechanical coupling wl between adjacent oscillators, an electronic coupling J between adjacent molecules, and an electrcn-phonon interaction w2. The electron-phonon interaction is taken as the strongest of the three, giving rise to zeroth-order states in which the electron is localized at one site, and is taken quadratic in the phonon coordinates [7]. As pointed out by Rashba [5], such a quadratic interaction may involve only a small fraction of the energy of a phonon and yet still be more :mportant than a linear interaction with an energy comparable to phonon energies; furthermore, out-of-plane bending modes in aromatic hydrocarbons are known to change in frequency by about 20% on electronic excitation [8]. Except for the quadratic instead of linear electron-phonon interaction, this model is the same as discussed by Holstein [9]. The electron wavefunction is written in the tight-binding approximation as a linear superposition of localizeli electronic wal1efunctions
* Issued as NRC’ No. 10929. ** NRCC Postdoctoral Fellow 1966 to present.
Here R is the position coordinate of the electron, 856
Volume 3, number 9
CHEMICAL PHYSICS LETTERS
1
September 1969
and {Q) denotes dependence on the set of vibrational coordinates X, for each.molecule. The coefficients aA+) satisfy
aa,,
iAx=
where the terms in wl and J are taken as perturbations. The zeroth-order coefficients are products of oscillator functions for each oscillator, all with frequency w. except the one at the site n of the localized electron, where the frequency is (02 - w2)+. In”the glow-phonon limit, J is the larger of the two perturbations, and produces a quasicontinuous exciton band of width 4J. The perturbation $rnw~x~x,+l induces transitions in which phonons are emitted and absorbed at sites n and ?1+1, but ever. when the oscillators at these sites differ in freqnency by 20% the amplitude of multiphonon processes is negligible, so that only single-phonon processes need be considered. The rate of electron hopping between sites n and n+l is equal to the rate of phonon exchange, which is calculated by time-dependent perturbation theory. Energy conservation requires that the sites should have equal numbers of phonons before and after each exchange. The square of the matrix element for exchange of the phonon between sites IZ and ntl is then k,,n+l
. = 2(rrw~/4~~~~~~(V~+l)
6(7w~,~+l), (3)
where vlt is the number of phonons initially at site n, etc. The rate of electron hopping is the rate of exchange of all the phonons, which is given by eq. (3) divided by vrz. In the thermal average this rate is proportional to l/(l+u)2, where u = exp(- fiwo/kZ’). The complete expression for the rate requires the density of final states, which we approximate by the inverse electron bandwidth, 1/4J. The rate is then 4 7&W, W~(n--n+l) = lSJw2,(l+o)2 The rate of electron hopping from site tz to n+l and 32-1 is 2W~(n--n+l), and the diffusion coefficient D is simply 2d2’W~(n-n+l). Finally, the mobility cr is given by p = eD/kT, or 656
here-the subscript p denotes hopping in the slowphonon limit. In the siow-electron limit, y is the larger of the two perturbations, and produces a quasi-continuous phonon band of width tiwf/w,. The matrix elements of the perturbation J are simply J times an overlap between oscillator functions. This overlap is negligible unless the number of phonons at a site remains unchanged, when it is unity. Energy conservation requires that the phonons at the two sites exchange, so that the sites must start zuid finish with equal numbers of phonons. The thermal average of the squared matrix element is then J2 times the thermal probability of equal numbers of phonons at adjacent sites, or J2(1-cr)/(l+o). The density of final states is taken as the thermal average number of phonons per oscillator, u/(1-u), divided by the bandwidth. The rate of hopping from site n to n+l is then w~(‘L+z+l) The mobility
2i7J2wo
=-
gw$
[ 1u
liu
is 4s;e$ wod2 he=
(7) $04
where the subscript e denotes the slow-electron limit. In order to compare the model with experiment for anthracene, w. is chosen as the mid-point of the frequency range of out-of-plane bending vibrations, 350 cm-l. This fixes the temperature dependence of pp and peg leaving the absolute magnitudes determined by the quantities J, d, and ~1. ln fig. 1, pp calculated for a suitably-chosen numerical factor is compared with experimental data on electron mobilities in the a direction of anthracene [lo: 111. The mobility varies as l/T at low temperatures as observed by Kepler [lo]. Similar fits can be obtained to all the other mobilities of holes and electrons except for electrons in the c’ direction (I ab plane). Thus it appears that those mobilities which are decreasing functions of temperature may represent hopping transport in the slow-phonon limit, and not necessarily coherent transport as usually assumed. The electron mobility in the c’ direction shows a positive temperature dependence which does not fit eq. (5). However, the electron bandwidth in this direction is known to be much smaller than in the other directions [S], which suggests that here it may be
Volume
3. number
9
CHEMICAL
PHYSICS
5-
z-
I
j I %
L969
phonon limit is always present if there is strong electron-phonon coupling. We have also introduced electron-phonon coupling quadratic in the phonon coordinates. which prevents the neareqonential increase of mobility with temperature produced by the commonly-used Ifnear terms in the slow-electron limit. A particularly satisfying feature of the present model is that for directions in which the electronic bands are relatively wide the slow-phonon limit applies, leading to a negative temperature dependence of the mobility, whereas in directions where these bands are very narrow the slow-electron Limit takes over, leading to a positive temperature dependence. This is precisely the situation observed experimentally in anthracene.
IO-
^ ‘0 :: T > ” E 2
1 September
LETTERS
0.5 -
REFERENCES [l] S. H. Glarum. J. Phys. Chem. Solids 24 (1963) [Z] R. M. Glaeser
T(“K)
Fig.
1. Electron
mobilities
in anthracene.
lines are the theoretical curves. p
The solid
for the a direction
The b&ken lines are the and 12, for the c’ direction. e.xperimental data from A. ref. [lo]; B. ref. [Ill; C. ref. (121: and D. ref. 1131.
smaller than the phonon bandwidth, so that (7) rather than eq. (5) would apply. This is posed by fig. 1 where p,(c’). calculated for sonable values of J, d and ~1, is compared
eq. suprea-
with experimental data for anthracene [lo-131. The agreement between theory and experiment is again very satisfactory. In summary, we have extended the theory of carrier transport in molecular crystals by considering the regime where carrier mobilities are governed ty phonon velocities. Such a slow-
and R. S. Berry.
ZGPi.
J. Chem. Phys. 4%
(1966) 3797. [3] L.Friedman. Phys.Rev. 140 (1965) Al649. [4] P. Gosar and S.-I. Choi. Phys. Rev. 150 (1966) 529. [S] E. I. Rashba. Zh. Eksperim. i l’eor. Fizz. 50 (1966) 1064, English transl. Soviet Phys. JETP 23 (1966) 708. [S] R. Silbey. J. Jortner, S. A. Rice and BI.T.Vala. 5. Chem. Phvs. 42 (1965) 733; 13 (1965) 2925. [7] R. W. hlunn and iV. Siebrand, J: Chem. Phys.. to be published. [S] A. P. Best, F. RI. Garforth. C. K. Ingoold. H. G. Poole and C. L. Wilson. J. Chem. Sot. (1918) 406 (part I) - 516 (part XII). [9] T. Holstein. Ann. Phya. 8 (1959) 325. 343. [lo] R. G. Kepler. Phonons and phonon interactions (W. A. Benjamin_ Xe\v York, 1964) p_ 578. [ll] G. T. Pott and D. F.Williams, J.Chem. Phys., to be published. [12] I. Nakada and Y. Ishihars. J. Phvs. Sot. Japan 19 (1964) 695. 1131J. Fourny and G. Delacate. J. Chem. Phys.50 (1969) 1028.
651