Origin of temperature-independent electron mobilities in organic molecular crystals

0038—1098/78/1022—0309 $02.00/0

Solid State Communications, Vol.28, pp.3O9312. ~ Pergamon Press Ltd. 1978. Printed in Great Britain.

ORIGIN OF TEMPERATURE—INDEPENDENT ELECTRON MOBILITIES IN ORGARIC MOLECULAR CRYSTALS H. Sumi Electrotechnical Laboratory, Tanashi, Tokyo 188, Japan

(Received. 13 September 1978 by Y. Toyozawa) Recently the almost—temperature—independent mobility was observed for electrons in anthracene, naphthalene, and As 2 33 in the direction with a narrow bandwidth. It is attributed to the phonon-induced electron hopping between planes along which the electron motion is coherent in a two dimensions.]. energy band. The mobility increases abruptly below about the Debye temperature due to the increasing contribution of the electron transfer without phonon participation, as observed in naphthalene. It has been known that the electron mobility

electron mobility in the c’ direction is not changed by the hydrostatic pressure which does 1 not break the symmetry although that along the (a,b)The plane phonon—induced increases linearly electron with hopping pressure. (PIER) itself was first investigated by Gosar & Choi.9 They assumed, however, that the electron motion is incoherent in all directions due to strong electron—phonon interactions, and obtained that the PIER gives a mobility component proportional to T”2 which cannot explain the experiments. Nadhukar & Post10 remarked that the mobility component given by the PIER is independent of temperature in the Haken & Reineker model11 which regards the transfer integral of electron as flactuating very rapidly due to lattice vibra— tions. This model can be justified only when the energy dispersion of lattice vibrations is much larger than the total conduction-band width.12 In most organic molecular crystals,

in the c’ direction of anthracene is almost in— dependent of temperature CT) .~ The electron no—

bility hand, decreases in the a with and b increasing directions,temperature on the other and it is this type of temperature dependence that is usually observed in organic molecular crys— tels.2 Therefore one of the central problems of organic molecular crystals has been how to cx— plain the abnormal temperature dependence of the electron mobility in the c’ direction of anthra— cene and the anisotroF between the c’ and the two other directions. Recently mobility meas— urements were performed by Schein over a wide temperature range from about 100 K to about 500 K.’ He showed that the almost—tenxperature—inde— pendant mobility (ATIM) persisting through the temperature range above can be explained neither by the band-conduction model nor by the hoppingconduction one presented so far. The ATIM was also observed in As 2 63 •2 Furthermore Schein et al. found very recently a remarkable fact that 5 turns to the Al’fl4 observed also in naphthalene Anthracene, naphthalene, increase abruptly below aboutand 100 As K. 2 33 have a com~ncharacter: The conduction—band width in one di.rection is very narrow (on the order of 10 “-.20 cm~in anthracene and naphthalene) compared It is in with that in two other2’68 directions (onthis the narrow order band direction the ATIM was observed. We of 1000 cm~in that them). treat explicitly only anthracene and naphthalene hereafter, but the analogous idea can easily be applied to As 2S3 too. The narrow bandwidth is given by the nearly complete cancellation of 6’ two Then is reasonable to assume that the sign. trana— large itinteratomic integrals of opposite fer integral of electron in the c’ direction is changed very much by a small angle of molecular rotation which breaks the sy~maetryof the molec— ular arrangement. The aim of the present brief conmiunication is to show that the ATIM in the c’ direction can be obtained when the electron no— tion in the c’ direction is dominated by the in— coherent electron hopping induced by rotational vibrations under the condition that the electron motion along the (a,b) plane perpendicular to

however, the former is at most 100 cm~,~ while the latter is on the order of 1000 cm~~ So present work, on the model other hand, starts from the the Haken & Reineker is inapplicable. The conditions that the electron motion is incoher— cut only in one direction with a narrow band— width and that lattice—vibration energies are much smaller than the total conduction—band width. electron the sum the of the term Jintegral of the rigid Let usas express transfer of lattice and the term induced by rotational vi— brations, as ~m ~ (f~mw2) ~°m~°m n0t~ (i) represents the annihilation operator for the rotaional vibration at the lattice site m and its energy is denoted by ~ neglecting the energy dispersion. y represents the cou— pling energy. The sites m and n are a pair of nearest neighbors in the c’ direction. The hopping process due to J in (1) conserves the no— mentum component q along the (a,b) plane and the transition probability is given by the golden rule as ..BE 2 2 C Pq( E) Here 0m

hi 1 ,,.2L~2

the c’ direction is coherent in the two—dinen— sional Bloch—band states. This interpretation

.

fdE E e~’p (E) q q

is consistent with the observation that the 309

(2)

310

ORIGIN OF ELECTRON MOBILITIES Vol. 28, No.4 1 with kB representing the n= 8iry/A, c~~(2/v)(~/S)(J/~u , 2 (i2) Here 8 equals (kBTYand p (E) denotes the density Boltzriann constant 1) of electron states with %nergy E and momentum ~ 2~u2d , (13) component q. In the hopping process due to the phonon—induced term in (1) electrons can hop to and every momentum state uniformly with the compo— = ~ <[fdE”f(E’) p(E+E’) ]~> . (i14) nent of the site vector along the (a,b) plane being conser~redand the transition probability is Here represents the thermal average de— given by fined by 21T fdEfdE~?e_BEp(E)g(E1)p(E÷E?) fdEe~p(E)A(E) f~e~~p(E) (3) = f~e~p(E) . (i5)

w2

with p(E) ~iç~

~PqC5’)~

(~)

Here N1 is the total number of lattice sites in a single (a,b) plane and g(E) is the spectral function of the rotational vibrations written as

Now we treat, contrary to Madhukar & Post, the case where the phonon spectral functions g(E) and. f(E) are both much narrower than the electron one p(E) for~1<<~and~A~<>~lul andnkBT>>~!w and 2( 2 and we get

with n(E) representing the Bose—Einstein distri— bution function. p CE) and p(E) are normalized as fp (E)dE = ~ = 1. The electron mobili— ty ~ with a representing the lattice constant in the c’ direction, by 2/kBT)(Wl+Wz). (6) ~i= (ea We denotes the conduction—band energy for momentum component q by Ea neglecting the energy dispersion in the c’ diredtion. Then Pq(E’) can be written as 2+ r(E)2], (7) Pq(E’)’r(E)/[(E~q) with the use of the energy—dependent width r(E).

1 kBT (16) Taking into account the step—like van—Hove sin— gularity at the edges of the two—dimensional en— ergy band, we can approximate p(E) by ~ constant approximat,~d by 0.5 of and respectiv,çely. (~~‘) independent E. ~(*.wl/kBT)2 Then and l1~ can be If r~l,c~ can be neglected compared with in (11) at high temperatures and the nobility ii is almost independent of temperature there. We consider that this is actually the case in anthracene, naphthalene, and As 2 S3~ In numerical calculations we set p(E) to be equal exactly to d’ for 0
In the present work we assume, contrary to Gosar & Choi, that r(E) is much smaller than the total width d of the conduction band {E }. Then we can regard p(E) of (14) as the den~ity—of—state function for the conduction band, and also we can get 2= p(E)/(2Trr(E)). (8) Ni~’~Pq(E) to the first order in the scattering of electron with (acoustic and/or non—polar optical) lattice vibrations within the (a,b) plane r(E) ~ written as r(E)=i~sh~ 1fdE’f(E’)p(E+E’). (9)

Up

Here 5 and ~1w1represent respectively the inter— action energy with lattice vibrations and the maximum the spectral cut—off function energyof of lattice them. vibrations f(E) represents de— fined by f(E) =N’E (wk/Cu1) [n(flL~)d(E_?iwk)

(10)

number sites inrespectively the whole crystal and Here N of andlattice ?lwk represent the total the energy of the lattice vibration with threed.inensional momentum k. Then substituting (8) and (9) into (2) and (3) we can rewrite the no— bility p of (6) as ii= with

n(ea2/~)[?~÷

ii~]

(11)

liz

P2

f<~p(E)>, and

~=

1(1)2<

~

~(~wz/kBT)n(~wz),

(17)

which is independent of t~. c~i~ is important only at low temperatures of kBT ~ that is, of k,..T <0 and 0 for x <0. The dependence of and the kBT/~lw2 dependence of are showo in Fig.1, where was calculated with the use of the Debye and the Einstein mod— in the Debye els for the phonon energy ~lWk. 5For kBT<<~wj model is and proportional to (~lu to (~kil/kBT) 1/k~T)exp(~w1/k,,T) in the Em— stein one, but the ifferences ~etween the two models are not appreciable for k~T’~~~ZW1. Figure 2 shows the kBT/~U3I dependence of ?~+Eli1 for

1i~

u~

= ~ where the Debye model was used for 1i1.ex— Takin~into account the Reman—scattering 15 we can energy reasonably estimate periments’~and the calculated dispersion of lattice vibrations ~1 and to be both about 70 cm~ (~l0OkB). ~~antit3’ in (11) has a dimension of mobility and ~a/~! has a magnitude about 10 cm2! V.sec for a~9”~8 appropriate to anthracene and naphtha.lene. The observed magnitude of the ATIM is about O.1V~0.5cm2/V.sec, so fl in (11) should be about 0.1. The observed temperature dependence of the mobility is well approximated by the

X

Vol. 28, No.4

ORIGIN OF ELECTRON NOBILITIES

1.5

311

—Debye Model

-

—

Einstein Model

)J.2

0.5-

_____________

0

1

3 kT/’hW1

or

5

kT/-~&2

Fig.l. kBT/*t~ dependence of ~, and k~Tf~lw1 dependence of p1 calculated with the use of the Debye and the Einstein models for the phonon spectrum.

I

=)~~+8)~~ w2)

~

kT/f~U)1 Fig.2. kBT/~ dependence of = 0.1, 0.2, 0.5, and 1.0 when model and. W1 are assumed.

=

~1

+ E~

for

of the Debye

312

ORIGIN OF ELECTRON MOBILITIES

curve for E=0.2 in Fig.2, which is in fact al— most independent of temperature forlOOK T’~l00%5O0K and increases abruptly below about when ~l!kB~lO0K. It is reasonable from the band— structure calculation to estimate J to be about 14 cmi. Then we get S/El~0.lfrom (12). Equa— tion (9) gives that r(EYis about 2vkBTS/El (~o.6 kBT) for ~ Then the initial assumption o±~ r(E) <<~ (in which El is on the order of 1000 cm1 i~1500kB) for the coherent electron motion along the (a,b) plane is satisfied fairly well for T~500IC. The bandwidth in the c’ direction is 14j (~20kB) in the present model and it is considerably smaller than r(E) for T~l00K. Therefore the initial assumption of the incoher— ent electron motion in the c’ direction is jus— tified. We can get from (1) that the square average of the phonon—induced term of the transfer inte— gral is about 2ykBT for kBT~1u 2. Two molecules molecule has inthree vibration participate the rotational transfer integral andmodes each with energies about 140%l20 cm1in anthracene and naphthalene.”15 Assuming that these three modes contribute equally to induce the transfer integral at high temperatures, we can estimate the average amplitude of the transfer integral induced by one rotational vibration mode of one molecule to be about (*IkBT)”2 S (flElkBT/(2llir))hhi which is about 17 cm~at 300K for ~ . The average rotational angle of each molecule has

Vol. 28, No.4

been measured by X-ray diffraction studies to be6 about ~3%14°for each rotational vibrationthat mode.’ Then corresponds to the condition rotations with angle 3%14°of one molecule in— crease on the average the transfer integral by about 17 cm’. Taking into account the fact that the transfer integral of electron along the(a,b) plane and that of hole in all directions are on the order of 100 cn°, we understand that the condition mentioned above for the coupling gy between electrons and rotational vibrations is quite acceptable. With the use of S/El ~ 0.1 we can also reproduce satisfactorily the observed magnitude (~i.O%i.6 cn2/V.sec at 300K in anthracene) and temperature dependence of the mobility along the (a,b) plane, as will be sho~melsewhere. Thus the PIEH in the c’ direction resolves also the mystery3 for a long time that the observed electron 2/V.sec nobility at 300 in K) thehas c’ adirection magnitude(~ compara— 0.14% ble 0.5 to cn that along the (a,b) plane while the cal— culated conduction—band width in the c’ direc— tion is very small on the order of about one hundredth to about one fiftieth of that along the (a,b) plane. Acknowledgements — The author wishes to thank Professor Y. Toyozawa of the University of Tokyo for valuable discussions.

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