Download The microscopic model of electron transfer reaction in disordered

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
CBP
AlQ3
Dielectric permittivity for AlQ3:
Static: εs = 2.84 (calculated)
or 3 ± 0.3 (experiment)*)
Optical: 2 < ε < 3
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
*) V. Ruhle et al, JCTC 7 (2011) 3335-3345
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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)
MLD – Marcus-Levich-Dogonadze (traditional ET)
DJ – Dogonadze-Jortner (including local modes)
The interaction
scheme
x
X
Strong x-Q,
Q - polarization
X
Strong x-Q and x-X, Q – polarization. The
same bath Q for all transitions 1n→2n’
X
Strong x-X and weak X-Q,
Q – aqoustic phonons, n - dependent
Q
DJ (generalized spinboson)
x
Non-spin-boson (the
present work)
x
Q
Q
Comment
The connection to the continuum bath Q is obligatory in order to dissipate the
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energy misfit Δ of a ET reaction. This assures the convergence of rate integrals.
3. Active ET local motions: Reorganization
mode X (intramolecular)
Transfer
integral:
n(X)
J nn′ = J 0 ϕ n ( X ) Jˆ X ϕ n ( X ′)
are oscillator functions:
d ⎞
⎛
Jˆ X = exp ⎜ −δ
⎟ (shift operator )
dX ⎠
⎝
J nn′ = J 0 ϕ n ( X ) | ϕ n ( X + δ )
mω02 2
Er =
δ
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 X0
is the reaction energy change
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4. Active ET local motions: promotion
mode X (intermolecular)
Transfer integral:
J nn′ = χ1 ( x)ϕn ( X ) Jˆ χ 2 ( x)ϕn′ ( X ) = J 0 ϕn ( X ) Jˆ X ϕn′ ( X )
(
2
(x) are electron (or H) ˆ
J X = exp − μ ( X − X 0 ) −ν ( X − X 0 )
functions
n(X)
J0,
)
are oscillator functions
and
are the basic parameters
Miller, Abrahams (1960; MA→ET)
Trakhtenberg, Klochikhin,
Pshezhetski (1982; T → H transfer)
MAT mechanism of ET or HT
1/2
⎤
mAmB
⎛π
⎞ ⎡
sin ⎜ − θ ⎟ = ⎢
⎥
⎝2
⎠ ⎣ ( mA + mH )( mB + mH ) ⎦
θ = 80 (H transfer)
θ = 0.60 ( ET )
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5. ET dynamics/kinetics
+∞
⎛ Δ ⎞
K (T ) = const ∫ cos ⎜
t ⎟ exp ( −Φ (t ) ) dt (1)
⎝ ω0 ⎠
−∞
Rate constant is determined by the dissipation of the energy misfit Δ
0
is the frequency;
is the reaction energy change
(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 ) =
Invoking the promotion mode X is
quite unusual in the ET theory
(i.e. the MA mechanism is usually
disregarded)
J 02
⎡ (Δ + Er )2 ⎤
× exp ⎢ −
⎥
E r k BT
⎣ 4 k B TE r ⎦
π
Eq. (2) is the asymptotic limit of (1),
purely classical, i.e. ħ 0/kT<<1
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(2 )
6. The ET kinetics (Levich, Dogonadze, 1959)
K (T ) =
2π C (T )
Z (T )
(1)
Eq. (1) is derived from the Fermi “Golden
Rule”
The present work follows our approach suggested in 2006:
∞
⎛ E ⎞ 2
C (T ) = ∑ ∫ dE exp ⎜ −
⎟ J nn′ ρ n ( E ) ρ n′ ( E ) → the reaction probability flux
k
T
n ,n ′ 0
⎝ B ⎠
∞
⎛ E ⎞
Z (T ) = ∑ ∫ dE exp ⎜ −
⎟ ρ n ( E ) → the partition function
n 0
⎝ k BT ⎠
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 , Qν ) = X ∑ Cν Qν
ν
The interaction strength is
1⎞
⎛
ε n = ⎜ n + ⎟ ω0 ;
2⎠
⎝
⎡
ω0 ⎤
Γ n = ⎢ (2 n + 1) coth
− 1⎥ γ
k
T
B
⎣
⎦
n-dependent continuous
frequency spectrum
destroys the spin-boson
model
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7. The spectral functions
ρ n ( E ) = ρ n (ω ) =
+∞
∫
n(t)
ρ n (t ) exp(iωt )dt ; ω =
and f(t)
E − εn
−∞
ω0
(the energy level distribution)
⎡ Γn
⎤
1
ρ n (t ) =
exp ⎢ −
f (t ) ⎥ ( the spectral function )
(1)
2π
⎣ 2 ω0
⎦
1 − b|t|
f (t ) = t + ( e − 1) (Kubo, Toda, Hashitsume, 1978, 1986)
b
⎡
⎤
1⎞
ω0
Γ
⎛
(2)
Γ n = ⎢(2n + 1) coth
− 1⎥ γ ; ε n = ⎜ n + ⎟ ω0 − 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 , Qν ) = X ∑ Cν Qν = XQ, Q = ∑ Cν Qν
ν
(3)
ν
Parameters b (Eq. (1)) and (Eq. (2)) can be extracted from the correlation
function C(t)=<Q(t=0)*Q(t)>, where Q(t) (Eq. (3)) is the collective medium
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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
Er
ω0
⎛ Δ ⎞
⎟
= E2 – E1 – the reaction energy change ⎜
ω
⎝ 0⎠
,
– the parameters of the transfer
integral J = J0 exp (- X- X2)
}
}
for the
reorganization
mode X
for the
promotion
mode X
⎛ Δ ⎞
= E2 – E1 – the reaction energy change ⎜
⎟
ω
⎝ 0⎠
-the strength of the mode/medium interaction
1
f
(
t
)
=
t
+
exp( − b t ) − 1)
(
b – the parameter of Kubo function
b
ω
The important parameter ξ = 0
kBT
determines the kinetic regime in the whole temperature range
}
specific
for the
present
approach
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9. What is the target of computation
K (T ) =
2π C (T ) ← the reaction probability flux
Z (T )
← the partition function
+∞
⎛ Δ ⎞
⎡ γ
⎤
⎛
J 02
Δ ⎞
dv
exp ⎜ −
C (T ) =
z
×
exp
×
exp
Re
(
)
i
v
f
⎜
⎟
⎟ ∫
⎢
⎥ × I ( z , z ′)
2π
⎝ 2k BT ⎠ −∞ ω0
⎝ ω0 ⎠
⎣ ω0
⎦
iξ
iξ
ω
z = − + v; z ′ = − − v; ξ = 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 ⎡
⎛ γ
⎞
⎤ ⎫⎪
γ ⎛
ξ⎞
1
=
ξ
Z (T ) = exp ⎜
f ( z ) ⎟ 2sinh ⎨ ⎢iz +
coth
f
(
z
)
;
z
⎬
⎜
⎟
⎥
ω0 ⎝
2⎠
i
⎪⎩ 2 ⎣
⎝ 2 ω0
⎠
⎦ ⎪⎭
For the Kubo function the analytical continuation in the complex time plane
z = v – iu is required:
+∞
⎡
⎤
1 ρ1 (v)
γ
dv, where ρ1 (v) = exp ⎢ −
ρ ( z) = ∫
f (v ) ⎥
π i −∞ v − z
⎣ 2 ω0
⎦
1
f (v) = v + ⎣⎡exp(−b v ) − 1⎦⎤
b
( the Kubo function )
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10. Reminder
ω0
ω0
ξ=
=
;
k BT ωT
ωT =
k BT
This important parameter essentially determines the kinetic regime, in
which the ET process proceeds
ξ << 1 for a classical regime
ξ >> 1 for a quantum regime
So, ξ < 1 for:
ω0 , 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
Er = 3 ω0
Rate,
105s-1
Marcus
Δ = 0.3 ω0
ω0 = 120cm−1
b=0.1
ξ=
full computation b=1
J 0 = 10 −5 eV
γ = 0.2 ω0
ω0
kBT
ξ
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12. The ET kinetics for typical
liquid phase polar environment
Rate,
Er = 10 ω0
10 s
Δ = 3.5 ω0
4 -1
Marcus
ω0 = 120cm−1
full computation b=0.1
b=1
−5
J 0 = 10 eV
γ = 0.2 ω0
ξ=
ω0
kBT
ξ
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13. The ET kinetics in the hightemperature regime: ξ<1. Typical
OLED systems
Rate,
105s-1
Marcus
b=0.1
b=0.3
b=1
b=3
Er = 3 ω0
J 0 = 10 −5 eV
γ = 0.2 ω0
ξ=
ω0
kBT
Δ = 0.3 ω0
ω0 = 120cm −1
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ξ
14. The ET kinetics for promotion
mode
Rate, s-1
b=0.1
b=0.3
μ ⋅ l0 = 2.1
Δ = 0.06 ω0
ω0 = 120cm
−1
ξ=
b=1
ω0
kBT
b=3
ξ
J 0 = 10 −5 eV
γ = 0.2 ω0
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Колебательный анализ
локального центра ПЗ
(выявление активных
локальных мод)
Интеграл переноса,
зависимость
от расстояния
Энергии
реорганизации
Er(in) и Er(out);
тепловой
эффект
Расчёт константы
скорости ПЗ
(а) Одна локальная мода
(b) Несколько локальных
мод
Плотности колебательных
уровней активных
локальных мод,
распределения
тепловых эффектов
Параметры
фононного
спектра (расчёт
кор. функций
активных мод)
Расчёт
подвижности
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