We consider a hybrid heterostructure containing an inorganic quantum well in close proximity with organic material as overlayer. The resonant optical pumping of Frenkel exciton can lead to an efficient indirect pumping of Wannier excitons. As organic material in such a hybrid structure , we consider crystalline tetracene. In tetracene , the singlet exciton energy is close to twice the one of triplet exciton state and singlet exciton fission into two triplets can be efficient. This process in tetracene is thermally activated and we investigate here how the temperature - dependent exciton energy transfer affects the functional properties of hybrid organic-inorganic nanostructures. We have obtained the exact analytical solution of diffusion equation for organics at different temperatures defining different diffusion lengths of excitons. The effectiveness of energy transfer in hybrid with tetracene was calculated by definite method for two selected temperatures that open possibility to operate in full region of temperatures. Temperature dependence of energy transfer opens a new possibility to turn on and off the indirect pumping due to energy transfer from the organic subsystem to the inorganic subsystem.
Optical properties of hybrid organic—nonorganic nanostructures for different applications have attracted interest in theory and experiments [
QW exciton transition. As this probe has a frequency lower than the Frenkel exciton energy in the organic overlayer, it would be not concentrated in the organic part of the hybrid structure. Variations in the pumping intensity of the organic component would produce, via non radiating energy transfer to the inorganic QW, variations in the concentration of Wannier excitons in the QW, and thus optical control of the QW nonlinearity can be achieved.
We calculate here the efficiency of such indirect optical pumping of tetracene. The parameters describing tetracene and its particular role in the thermally activated singlet exciton fission into two triplets, are taken from the experimental study by Laubel and Baessler [
As organic subsystem in the hybrid structure, we consider an overlayer of crystalline tetracene. The numerical data for tetracene were taken from [
The diffusion kinetic equation for organics in range 0 < x < L has the following form in accordance with [
L D 2 d 2 ρ d x 2 = ρ − I 0 α τ e − α x (1)
In Equation (1), I 0 is the external radiation flux from region x < 0 . In order to solve the Equation (1) of second order, the corresponding boundary condition on the boundary x = 0 is necessary. The boundary condition of continuity for energy flow in the organic matter and the external flow on the inlet border is the kinetic representation for the mechanism of conversion for a flow of light in region x < 0 to flow of energy, i.e. flow of excitons in organics. This boundary equation, corresponding in dimension to the Equation (1) has the following form
− 1 α 2 τ ( d ρ ( x ) d x ) x = 0 = I 0 (2)
The solution of the linear Equation (1) should include the spatial dependence determined both by the external spatially attenuated “influence” in the right part (1), and the spatial dependence corresponding to the “own” solution at zero external influence. In this case, the proper “own” solution should be also spatially attenuated. These two solutions are presented in general form with factors A and B as follows
ρ ( x ) = A e − α x + B e − x L D (3)
Substitution (3) in (1) due to the corresponding cancellation of the “own” solution leads to the following definition for factor A
A = I 0 α τ 1 1 − ( α L D ) 2 (4)
Substituting the ratio (4) for A in border Equation (2), we obtain the following definition of the factor B
B = − I 0 α τ ( α L D ) 3 1 − ( α L D ) 2 (5)
As a result, substituting (4) and (5) in (3), we obtain solution in the region 0 < x < L and ρ ( 0 ) in the following form
ρ ( x ) = I 0 α τ 1 [ 1 − ( α L D ) 2 ] [ e − α x − ( α L D ) 3 e − x L D ] (6a)
ρ ( 0 ) = I 0 α τ 1 [ 1 + ( α L D ) ] [ 1 + ( α L D ) + ( α L D ) 2 ] (6b)
Diffusion coefficient of tetracene D = L D 2 / τ = 3.3 × 10 − 3 cm 2 / sec and α = 1 / L p , L p = 0.5 × 10 − 5 cm practically do not depend on temperature in region α L D < 1 [
α τ = ( 1 v ) ( L D L p ) 2 , v = D L p = 6.6 × 10 3 cm / sec (7)
Then substituting (7) in (6) gives complete solution in the region 0 < x < L and ρ ( 0 ) in the following form
ρ ( x ) = I 0 1 v ( α L D ) 2 1 [ 1 − ( α L D ) 2 ] [ e − α x − ( α L D ) 3 e − x L D ] (8a)
ρ ( 0 ) = I 0 1 v ( α L D ) 2 1 [ 1 + ( α L D ) ] [ 1 + ( α L D ) + ( α L D ) 2 ] (8b)
Using the same as in (2) definition of flows I ( x ) inside organics at 0 < x < L and on outlet border flux I L coming out of the organic area to region x > L we can define them in the following forms
I ( x ) = − 1 α 2 τ ( d ρ ( x ) d x ) = − v α ( L P / L D ) 2 ( d ρ d x ) , 0 < x < L (9a)
I L = − 1 α 2 τ ( d ρ ( x ) d x ) x = L = − v α ( L P / L D ) 2 ( d ρ d x ) x = L (9b)
Substituting the dependence (8), we obtain after differentiation the following definition of flux inside organics at 0 < x < L and the light flux departing from the organics to region x > L
I ( x ) = I 0 [ e − α x − ( α L D ) 2 e − ( x / L D ) ] 1 − ( α L D ) 2 (10a)
I L = I 0 [ e − α L − ( α L D ) 2 e − ( L / L D ) ] 1 − ( α L D ) 2 (10b)
dash curve and for lower temperature T = 184 K by the solid curve. The vertical dash and solid line are fixing diffusion length for these temperatures. According to
The previous part gives in kinetic approach the light flux from organic that irradiate semiconductor QW but this approach does not include distance between organic and QW, corresponding interaction between them and flux in semiconductor. It is needed to include in consideration these interactions and flux for completeness of kinetic approach. Diffusion equation includes two parameters τ and L_{D} that define the transport of energy in hybrid as whole combined structure. But time of life τ in micro dynamic of excitons for crystalline systems of interacting dipoles is parameter of second order in comparison with optical molecular frequencies, energy of dipoles interaction, etc. It is defined actually by proper nonlinearities of high orders and is introduced usually as fixed phenomenological parameter. In the same time diffusion Equation (1) includes only one parameter-diffusion length L_{D} that is directly defined in micro dynamics by frequencies and dipole-dipole interactions. So it is possible to investigate energy transfer in hybrid by including in consideration spatially dependent diffusion length L D ( v ) ( x ) . The region of border between organics and QW may be simulated in diffusion equation by decreasing of diffusion length in organics on this border and subsequent increasing in direction to semiconductor. It is known that diffusion length is defined by forth of dipole oscillators and distance between them. If these distances are increasing then diffusion length decreases. The border between organics and semiconductor is actually natural place for distance increasing and diffusion length decreasing. We have used for combined diffusion length L D ( v ) ( x ) the following spatially dependence
L D ( v ) ( x ) = L D − f ( x ) + f ( 0 ) (11)
f ( x ) = h [ ( L C ) 2 ( x − σ L D ) 2 + ( L C ) 2 − ( L C ) 2 ( σ L D ) 2 + ( L C ) 2 ] .
This dependence distracts from diffusion length of organics L_{D} additional Lorenz function with maximum at x = σ L D . Then spatial dependence L D ( v ) ( x ) has minimum at this coordinate between left organics and right semiconductor. The length L_{C} defines width of Lorenz minimum and parameter h defines depth of this minimum. The additional constant term in (11) is using to conserve identity L D ( v ) ( 0 ) = L D . These few parameters h , σ , L C were used for adjusting to hybrid structure. Small additional corrections to (11) may be used for taking into account exact tail of dependence in region of semiconductor. The solution of diffusion equation with dependence (11) will give in different regions flux between organics and semiconductor, flux in semiconductor and transmitted light flux from hybrid. Diffusion equation has now the following form
d 2 ρ d x 2 = 1 L D ( v ) ( x ) ( ρ − I 0 α τ e − α x ) (12)
The analytical solution of diffuse equation is cumbersome and may be completely impossible, but it is no problem with computer solution for this smooth function (11) without any dangerous for computer cracks in L D ( v ) ( x ) . This function (11) with L D = 400 Å , L c = 100 Å and h = 104 , σ = 1.2 is demonstrated in
Procedure of computer solution for diffuse equation with dependence (11) by iteration method is shortly demonstrating in Appendix 1.
The function (11) with other parameters L D = 300 Å , L c = 104 Å and h = 54 , σ = 1.2 is demonstrated in
The chosen set of parameters in (11) may be used for adjusting computer calculation to composed structure of real hybrids. These results demonstrate that energy transfer in whole hybrid construction may be investigated in frame of kinetic approach with diffusion equation at varying diffusion length. Estimations of energy transfer effectiveness by ratio outlet flow to inlet flow or relation exciton densities on outlet border to density on inlet border may be easily found
from received results. It is evident that dependences D ( x ) and F ( x ) in
It will be shown in next part that micro dynamical approach gives with some important additions results that are close to that found above in kinetic approach with diffusion equation.
Micro dynamic approach includes many dynamical parameters—frequency and
intensity of external inlet flow, own frequencies of organics and QW, Coulomb and radiation interactions inside organic, inside QW and between them. This approach is additional to kinetic approach that includes these parameters in only two kinetic parameters—external intensity and diffusion length. It is needed to demonstrate shortly more common micro dynamical process of energy transfer in hybrid organics/semiconductor that includes frequencies and interactions in composed hybrid.
For the simplest 1D crystal organic chain with own energy of excitons E 0 and energy of interaction between the nearest neighbors, V quantum Hamilton operator of energy has the following form
H ^ = E 0 ∑ n | n 〉 〈 n | + V ∑ n ( | n 〉 〈 n + 1 | + | n + 1 〉 〈 n | ) (12)
In relation (12) index n numerate organic molecules of chain. Solution of Heisenberg equation i h ( d / d t ) ψ = H ^ ψ may be found in the following form
| ψ ( t ) 〉 = ∑ n p n ( t ) | n 〉 (13)
The amplitudes p n are solution of following system of equations
i h d d t p n ( t ) = E 0 p n ( t ) + V [ p n + 1 ( t ) + p n − 1 ( t ) ] (14)
We transform (14) for hybrid to following system of equations for standard dipole oscillators p n ( t ) = p n exp ( − i ω t ) . Then at external excitation of border dipole n = 0 with intensity I 0 and frequency ω system of equations has the following form
ω p 0 = ω 0 p 0 + V 0 p 1 + I 0 ω p 1 = ω 0 p 1 + V 0 p 0 + V 1 p 2 ω p 2 = ω 0 p 2 + V 1 p 1 + V 2 p 3 ⋮ ω p n = ω 0 p n + V n − 1 p n − 1 + V n p n + 1 , 3 ≤ n ≤ N − 1 ⋮ ω p N = ω 0 p N + V N − 1 p N − 1 + V N ( W ) p N + 1 ( W ) , (15)
ω p N + 1 ( W ) = ω 0 ( W ) p N + 1 ( W ) + V N ( W ) p N
In (15) ω 0 is own frequency of organics, V n is interaction between n and n + 1 dipoles, frequency, dipole and interactions with QW are noted by upper index W. We shall use new original recurrent procedure of solution for equation system (15). In opposite to usual procedure, we shall suppose that amplitude of vibration of last QW amplitude p N + 1 ( W ) is known. Then by “revert” step by step we define all other amplitudes up to the second formula in (15)
p N = ( 1 / V N ( W ) ) ( ω − ω 0 ( W ) ) p N ( W ) p N − 1 = ( 1 / V N − 1 ) [ ( ω − ω 0 ) p N − V N ( W ) p N + 1 ( W ) ] p N - − 2 = ( 1 / V N − 2 ) [ ( ω − ω 0 ) p N − 1 − V N − 1 p N ] p N − n = ( 1 / V N − n ) [ ( ω − ω 0 ) p N − n + 1 − V N − n + 1 p N − n + 2 ] , 3 ≤ n ≤ N − 1 p 1 = ( 1 / V 1 ) [ ( ω − ω 0 ) p 2 − V 2 p 3 ] p 0 = ( 1 / V 0 ) [ ( ω − ω 0 ) p 1 − V 1 p 2 ] (16)
Using of the first relation in (15) gives only “definition” of I 0 , that is unimportant for system of linear equations. Dividing on “found” I 0 will give only unimportant rate setting solution.
V n = V [ 1 − 50 ( n − 60 ) 2 + 100 ] (17)
Peculiarities of dynamic in energy transfer were also investigated by classic micro dynamic of classical system that is analog to hybrid. It was considered at
impulse excitation of 1D system with 15 identical harmonic oscillators in crystal chain with own frequency ω 1 = 1 and interaction force f 1 = 0.1 . Last oscillator of another type has different own low frequency ω 2 = 0.95 and different interaction f 2 = 0.2 with last organic oscillator. Impulse of vibration was generated on opposite border of crystal at start. Evolution of system that was simulated on computer is solution for system of classical differential equations of second order. It is demonstrated by
strongly depressed in comparison with neighbor dipoles. This behavior has analogy with behavior of flow in hybrid on
The authors declare no conflicts of interest regarding the publication of this paper.
Dubovskiy, O.A. and Agranovich, V.M. (2019) Exciton Energy Transfer in Hybrid Organics—Semiconductor Nanostructure. Soft Nanoscience Letters, 9, 17-33. https://doi.org/10.4236/snl.2019.92002
The relation (10) was used for definition ρ ( x 0 ) in initial point x 0 and ρ ( x 0 + d x ) in next point x 0 + d x at small step d x . The iteration method use the following relation for the second derivative
d 2 ρ ( x ) d x 2 = 1 d x 2 [ ρ ( x + d x ) + ρ ( x − d x ) − 2 ρ ( x ) ] (A1)
This relation is reverting to the form where ρ ( x + d x ) is presenting as function of second derivative, ρ ( x ) and ρ ( x − d x )
ρ ( x + d x ) = d x 2 ( d 2 ρ ( x ) d x 2 ) x + 2 ρ ( x ) − ρ ( x − δ ) (A2)
So from two initial function value ρ ( d x ) and ρ ( 0 ) at known second derivative we have next value ρ ( 2 d x ) . Then from ρ ( d x ) and ρ ( 2 d x ) we have ρ ( 3d x ) etc. This procedure is useful only for smooth dependences defining second derivative.