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The microscopic model of the electron transfer in disordered solid matrices
M. V. Basilevsky A. V. Odinokov Photochemistry Center, RAS S. V. Titov Karpov Institute of Physical Chemistry

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1. Charge carrier mobilities in OLED materials
OLED = Organic Light Emitting Diode Active ET centers of OLED systems are dimers (M±)M appearing in applied electric fields Typical monomer (M) molecules

AlQ3 Dielectric permittivity for AlQ3: Static: s = 2.84 (calculated) or 3 ± 0.3 (experiment) Optical: 2 < < 3
*)

CBP Conclusion: the ET in OLED active centers is mainly associated with local molecular modes, rather than with medium polarization modes, as in usual ET theories
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*) V. Ruhle et al, JCTC 7 (2011) 3335-3345


2. The ET without medium polarization
x ­ essentially quantum coordinate (two states 1 and 2) X(Xk) ­ local molecular modes (n or n' states: 1n and 2n') Q(Qv) ­ medium modes with continuous frequency spectrum (the bath) Both polarization and aqoustic phonon modes may be included
The model MLD (spin-boson) DJ (generalized spinboson) Non-spin-boson (the present work)

MLD ­ Marcus-Levich-Dogonadze (traditional ET) DJ ­ Dogonadze-Jortner (including local modes)
Comment Strong x-Q, Q - polarization Strong x-Q and x-X, Q ­ polarization. The same bath Q for all transitions 1n2n' Strong x-X and weak X-Q, Q ­ aqoustic phonons, n - dependent

The interaction scheme x Q x Q x Q X X X

The connection to the continuum bath Q is obligatory in order to dissipate the 3 energy misfit of a ET reaction. This assures the convergence of rate integrals.


3. Active ET local motions: Reorganization mode X (intramolecular)
Transfer integral:
n

J

nn

^ = J 0 n ( X ) J X n ( X )

d ^ J X = exp - (shift operator ) dX

(X) are oscillator functions:

J

nn

= J 0 n ( X ) | n ( X + )

2 m0 2 Er = 2 (reorganization energy)

J0 and Er (or ) are the basic parameters Marcus (1956; M) Levich, Dogonadze (1959; LD) MLD mechanism of ET is the shift of the equilibrium position X is the reaction energy change
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0


4. Active ET local motions: promotion mode X (intermolecular)
Transfer integral:
J
nn

^ ^ = 1 ( x)n ( X ) J 2 ( x)n ( X ) = J 0 n ( X ) J X n ( X )

2 (x) are electron (or H) ^ J X = exp - ( X - X 0 ) - ( X - X 0 ) functions

(

)

n

(X) are oscillator functions

J0,

and

are the basic parameters

Miller, Abrahams (1960; MAET) Trakhtenberg, Klochikhin, Pshezhetski (1982; T H transfer) MAT mechanism of ET or HT

mAmB sin - = 2 ( mA + mH ) ( mB + mH ) = 80 (H transfer)

1/ 2

= 0.6

0

( ET )

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5. ET dynamics/kinetics
Rate constant is determined by the dissipation of the energy misfit
0

K (T ) = const cos t exp ( - (t ) ) dt (1) 0 -
is the reaction energy change

+

is the frequency;

(t) is extremely complicated in the full theory (LD, 1959; the reorganization mode X with frequency 0) Marcus (1956; the reorganization mode X):
K (T ) = J
2 0

Invoking the promotion mode X is quite unusual in the ET theory (i.e. the MA mechanism is usually disregarded)

( + Er )2 в ex p - E r k BT 4 k BT E r



(2)

Eq. (2) is the asymptotic limit of (1), purely classical, i.e. 0/kT<<1

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6. The ET kinetics (Levich, Dogonadze, 1959)
K (T ) = 2 C (T ) Z (T ) (1)

Eq. (1) is derived from the Fermi "Golden Rule"
E 2 J nn n ( E ) n ( E ) the reaction probability flux k BT E - n ( E ) the partition function k BT -

The present work follows our approach suggested in 2006:
C (T ) = Z (T ) =


n ,n 0




n 0



dE exp dE exp

The energy distributions n(E) are the basic quantities. The level broadening appears owing to the interaction of the mode X with medium modes Q :

W ( X , Q ) = X

The interaction strength is n-dependent continuous frequency spectrum destroys the spin-boson model



C Q

1 n = n + 0 ; 2

0 n = ( 2 n + 1) co th - 1 k BT

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7. The spectral functions
n ( E ) = n ( ) =
+ -

n

(t) and f(t)
E -
n



n (t ) exp(it )dt ; =

0

(the energy level distribution)
n 1 n (t ) = exp - f (t ) ( the spectral function ) (1) 2 2 0 1 - bt| f (t ) = t + ( e | - 1) (Kubo, Toda, Hashitsume, 1978, 1986) b 1 0 (2) n = (2n + 1) coth - 1 ; n = n + 0 - i n 2k BT 2 2 Eq. (2) is derived based on the quantum relaxation equation (Bloch, Redfield); is the strength of X/medium interaction:
W ( X , Q ) = X



C Q = XQ, Q =



C Q

(3)

Parameters b (Eq. (1)) and (Eq. (2)) can be extracted from the correlation function C(t)=, where Q(t) (Eq. (3)) is the collective medium 8 coordinate (the medium induced random force).


8. The basic parameters
0

­ the frequency of the local mode X

J0 ­ the transfer integral Er ­ the reorganization energy

E

r 0

= E2 ­ E1 ­ the reaction energy change 0
­ the parameters of the transfer




,

= E2 ­ E1 ­ the reaction energy change 0 -the strength of the mode/medium interaction 1 f (t ) = t + ( b ­ the parameter of Kubo function b The important parameter = 0
kBT

integral J = J0 exp (- X- X2)

}

}

for the reorganization mode X for the promotion mode X

exp( - b t ) - 1

)

}

specific for the present approach
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determines the kinetic regime in the whole temperature range


9. What is the target of computation
K (T ) = 2 C (T ) the reaction probability flux Z (T ) the partition function Re f ( z ) в I ( z , z v в exp 0 J 02 exp C (T ) = 2
+ dv - в exp i 2k BT - 0 0 i i z =- + v; z =- - v; = 0 2 2 k BT

)

I(z,z') is the two-point integral kernel (the known analytical expression). It is specific for reorganization and promotion modes. Includes the Kubo function f(z). 1 1 Z (T ) = exp f ( z ) 2 sinh iz + coth f ( z ) ; z = 0 2 0 2 i 2

For the Kubo function the analytical continuation in the complex time plane z = v ­ iu is required: + 1 1 (v) dv, where 1 (v) = exp - ( z) = f (v ) i - v - z 2 0 10 1 ( the Kubo function ) f (v) = v + exp(-b v ) - 1 b


10. Reminder
0 = = ; k BT T
0



T =

k BT

This important parameter essentially determines the kinetic regime, in which the ET process proceeds

<< 1 for a classical regime >> 1 for a quantum regime So, < 1 for: 0 , cm -1 : 400
T, K : 200 130 100 65 30 10 > 600 > 300 > 200 > 150 > 100 > 45 > 15
The present code covers this range
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11. The ET kinetics in typical OLED systems
Rate, 105s-1 b=0.1 Marcus

Er = 3 0

= 0.3 0

0 = 120cm-1

=
full computation b=1

0
kBT


J 0 = 10 -5 eV

= 0.2 0

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12. The ET kinetics for typical liquid phase polar environment Rate, Er = 10 0 10 s = 3.5 0
4 -1

Marcus

0 = 120cm-1
= 0
kBT


full computation b=0.1

b=1

J 0 = 10 eV

-5

= 0.2

0

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13. The ET kinetics in the hightemperature regime: <1. Typical OLED systems
Rate, 105s-1 Marcus

b=0.1 b=0.3 b=1 b=3

Er = 3
J 0 = 10 -5 eV

=

0
kBT

0 0 -1

= 0.3


= 0.2 0

0 = 120cm

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14. The ET kinetics for promotion mode
Rate, s
-1

b=0.1 b=0.3

l0 = 2.1 = 0.06

0 -1

0 = 120cm
b=1 b=3

=

0
kBT



J 0 = 10 -5 eV

= 0.2 0

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( ) , ( . )

Er(in) Er(out);

() (b)

,


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