Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 1 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 2 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 3 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 4 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 ) 5 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 6 (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 7 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 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 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 9 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 ) 10 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 11 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 12 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 13 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 14 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 15 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 16