Supplementary Material A: Boundary conditions and photogeneration
The concept of surface recombination velocity for electrons and holes at electronic contacts
along with the work function of the electrodes have been widely utilized as in device simulations
as Neumann boundary conditions for continuity equations [Eq. (1) and Eq. (2)]. This model
relates the electron and hole concentrations to their corresponding electric currents at the vicinity
of the contact,
J n  qSn  n  n0  S ,
(A1a)
J p  qS p  p  p0  S .
(A1b)
Where Sn and Sp are electron and hole surface recombination velocity values, respectively.
Since current at the contact can be calculated via two formulations [Eq. (C3) and Eq. (A1)], for
each charge carrying particle these two currents, current at the contact and the vicinity of the
contact, must be equal. Therefore, two boundary conditions at each contact can be obtained for
electron and hole continuity equations [Eq. (2) and Eq. (3)].
Similar to electrons and holes, excitons can also interact with metal contacts and get
recombined. Therefore, the exciton recombination process at the contacts can be characterized by
surface recombination velocity. Exciton density in a metal and exciton injection rate into a metal
are both zero. Substituting Ki and X2 with zero in Eq. (9) and using the flux formulation by the
surface recombination velocity leads to the boundary conditions for singlet and triplet excitons at
the contacts.
D
dX
 SX X ,
dx
(A2a)
1
S X  kr  kd .
(A2b)
Where SX is exciton surface recombination. As a boundary condition for Poisson equation,
back contact potential can be assumed zero. This boundary condition along with the potential
difference between front and back contacts (|x=0 - |x=l) leads to two Dirichlet boundary
conditions for the Poisson equation [Eq. (1)].
Photogeneration of energy carrying particles
Light absorption and its distribution inside a solar cell determines the photo-excitation rate
(generation of free electron/hole pairs or excitons) inside the cell and directly transform into the
short circuit current of the solar cell. Light absorption (A) inside the device is equal to energy
loss of the light. The loss of electromagnetic field is equals to the divergence of energy flux,
Poynting vector (S), based on definition. It must be noted that this absorption can be inter band
which lead to photo generation or intra band which is heat up the device. Intra band absorption is
due to free carrier absorption or plasmonic effect. In simple organic devices without plasmonic
structure density of charge carriers is low and the ratio between intra band transitions over total
absorption energy, (), is almost one. Since light is an electromagnetic wave its energy flux is
obtained by cross product of Electric field(E) and complex conjugate of magnetic field(H).
GOp ,np 
GOp ,np 
 

. Re E ( )  H ( )*
hc
 ( ) ( ),
(A3a)

 
. Re E     H   
*
hc

    1     ,


(A3b)
2
Where  , h, c and  are ratio between exciton generation rate and total generation rate,
Planck constant , speed of the light and light wavelength respectively.
The intensity of Electric and magnetic fields are determined by solving the Maxwell equations.
In complex structures, such as nano structures, Maxwell equations (ME) do not have an
analytical solution and they should solve numerically. Finite difference time domain for three
dimensional structure (3D-FDTD) and finite element method (FEM) are common algorithm to
solve these equations. Based on the result of ME equations GOp,np and GOp,X can be determined
[Eq. (A3)] and replace in Eq. (2) up to (4).
Photogeneration rate of electron-hole pairs and excitons in general can be obtained by solving
the Maxwell’s equations, using often EM-wave simulators in 2 and 3 dimensions and transfer
matrix method in 1 dimension, for the given structure and applying the Eq. (A1). In a one
dimensional light absorber when penetration depth of light is smaller than the thickness of the
absorber, the interference of EM-wave between the incident and reflected light can be neglected,
and the Beer-Lambert law can be used to calculate photogeneration rate of excitons and electronhole pairs as,
GOp ,np  x       1       0    e     x ,
(A4a)
GOp , X  x           0    e    x ,
(A4b)
0     i    1  R     ,
(A4c)
i    

Re E     H   
*
hc
 .S .
(A4d)

3
Where i() and 0() are the photon fluxes [per unit area (cm2) per wavelength (µm-1) per
second (s-1)] associated with the incident light and the light that enter the top layer of the cell
excluding the reflection from the top surface, R(). For a standard illumination condition, φi can
be replaced by the photon flux of AM1.5 spectrum. α() represents the absorption coefficient of
the medium and Ŝ represents the unit- vector normal to the top surface of the cell. () represents
what percentage of the absorbed light leads to exciton generation. Assuming that the all the light
absorption in the absorption spectrum of the solar cell is due to the inter-band transitions, the
percentage of the absorbed light that leads to free electron/hole pair generation would be equal to
1-().
4
Supplementary material B: Numerical method for solving the unified charge transport
model
In this equation set, the density of free electrons and holes can be formulated by the quasi-Fermi
energy levels and the intrinsic Fermi level which in turn is the function of electric potential,
electron affinity, and a reference potential level.
n  NC e
E fn bl  XeBC  Xe  q
p  NV e
KT
,
(B1a)
bl  XeBC  XeBC  Eg  q  E fp
KT
.
(B1b)
Where K, T, Nv, NC, bl, e, eBC, Eg, Efn and Efp are Boltzmann constant, absolute
temperature, effective density of states in LUMO or conduction band and HOMO or valence
band, the reference potential (difference between work function of metal and the electron affinity
of the semiconductor at the back contact), electron affinity of the semiconductor, electron
affinity of the semiconductor at the back contact, band gap, and quasi-Fermi levels for electrons
and holes, respectively. Solution of the cross-coupled Poisson-continuity equations leads to
distribution of (x) throughout the device. Knowing (x) and material parameters the LUMO(x)
and HOMO(x) can be obtained by LUMO(x)=bl+eBC-e(x)-q(x) and HOMO(x)=bl+eBC-
e(x)-q-Eg(x), respectively. In the bulk of material, total current JT is the sum of electron and
hole currents, where electron and hole currents (Jn and Jp) are given by,
J n  qn n  qDnn,
(B2a)
J p  q p p  qD pp,
(B2b)
JT  J n  J p .
(B2c)
5
The formulation for the unified charge transport model, described by Eqs (1) through (6), can
be implemented in 3, 2 or 1 dimensions by expanding the Laplacian and gradient operators (2
and ) in 3, 2, or 1 dimensions, respectively.
The continuity equations of charge carriers, Eq. (2) and Eq. (3), are coupled together via
recombination rates and Poisson equation, along with the contact boundary conditions of Eq.
(A1). It should be noted that electron-hole generation due to exciton dissociation at the
donor/acceptor interface are included in Eq. (2) and Eq. (3) through the generation rates
described in Eq. (A4). This system is a nonlinear coupled PDE and there is no known analytical
solution for this system. Consequently, quasi-Fermi levels for electron and hole and the electric
potential are considered as the system variables in equation set of (1) through (3) and the
Newton-Raphson and Gummel iterative methods are used to obtain the numerical solution.
The PDEs are discretized using a finite difference scheme with smaller mesh sizes near the
interface. Furthermore, to improve convergence properties the material parameters are allowed to
vary smoothly across thin interfaces. This assumption appears to be more realistic than the
assumption of a discontinuity in the parameters due to intermixing of materials near the
interface. Since the Newton-Raphson method is sensitive to initial values the algorithm is
initiated with the Gummel method which can provide a better convergence in the case of a naive
initial guess. After several iterations of the Gummel method, the Newton-Raphson method is
initiated with the terminal value of Gummel method as the initial value. The initial value for
Gummel method is derived from the thermal equilibrium. The residual errors of the discretized
equation are computed after every iteration by plugging the variables in the equation sets of Eqs.
(1), (2) and (3) in all mesh points and obtaining the associated Euclidean norms over the entire
6
PDE system. If the error is sufficiently small the Newton iterations are terminated. Otherwise,
the variable values are updated and the next iteration is performed. The descritized form of these
equations are
F1  2
k x  h   h  x  k   h  k   x 

khk  h 
  x  E fpq  x  blq   e BCq   e  x  E g  x 
E fnq  x   x  blq   e BCq   e  x 


q
KTq
KTq


NV e
 NC e
 N D  N A  p T  nT   0




,
F2  2
(B3a)
kE fnq  x  h   hE fnq  x  k   h  k E fnq  x 

khk  h 
1 E fnq  x  h   E fnq  x  k   n  x  h    n  x  k 
.

 n ( x)
kh
kh
E fnq  x  h   E fnq  x  k 
kh
.
1
.
KTq
 E fnq  x  h   E fnq  x  k    x  h     x  k   eq  x  h    eq  x  k  




kh
kh
kh


2
ni
Gn
px 


0
E fnq ( x ) blq   e BCq   eq ( x )  ( x )
E fnq ( x ) blq   e BCq   eq ( x )  ( x )
 n ( x)
KTq
KTq
 n ( x) N C e
 n ( x) N C e
,
F3  2
1
 p ( x)
(B3b)
kE fpq x  h   hE fpq x  k   h  k E fpq x 

khk  h 
 p x  h    p x  k  E fpq x  h   E fpq x  k 
kh
kh

1 E fpq x  h   E fpq x  k 
.
KTq
kh
 x  h   x  k  E fpq x  h   E fpq x  k   eq x  h    eq x  k  Egq x  h   Egq x  k 





kh
kh
k h
k h


2
Gp
 n x 
ni


0
 blq  XeBCq   eq ( x )  E gq ( x )  ( x )  E fpq ( x )
 blq  XeBCq   eq ( x )  E gq ( x )  ( x )  E fpq ( x )

(
x
)
p
KTq
KTq
 p ( x) NV e
 p ( x) NV e
,
(B3c)
where k and h show the forward and backward spacing between adjacent slices, respectively.
7
It is important to note that Newton iterations are performed with fixed coupling coefficients
Upon termination, the coupling coefficients are refined and the Newton iterations are restarted
with the previous solution as the initial guess. This process is continued until the residual terms
for all equations are sufficiently small. The PDE are discretized using a finite difference scheme.
In this case three point central difference for 2nd derivatives and central difference for 1st
derivative is used. Subsequently the PDE is reduced to a non-linear system of algebraic
equations.
Numerical methods can be used to approximate the solution of this system. Newton method is
one of the fastest and efficient numerical method for solving nonlinear PDE problem, however
the performance is sensitive to the initial conditions . Due to the complexity of the Electronic
system Eq. (1) up to Eq. (3), good convergence can be achieved when the difference between the
initial guess and final solution is relatively small, thus finding a good initial guess is the essential
step in order to solve the problem. Another alternative for Newton method is Gummel method in
which Eq. (1) up to Eq. (3) are decoupled. With the assumption of other variables are constant,
each equation is solved seperately and in sequence. Since based on this assumption, each
equation is a simple PDE system, convergence is possible for a wider range of intitial conditions.
However, the convergence rate is slow due to decoupled approach. In this research the numerical
algorithm is started with Gummel method due to its better stability and after obtaining a refined
initial estimate of values, the Newton method is started.
The initial value of the Gummel method is provided based on thermal equilibrium. Under
thermal equilibrium condition Efn and EFp are equal and constant thus this complex system [Eq.
(1) up to Eq. (3)] can be expressed by single Poisson equation, which reduces to linear system
8
when discretized. Result of thermal equilibrium is a good initial guess for the Gummel method
loop. After several iterations of the Gummel method, the values of the variables are relatively
close to the final values hence Newton-Raphson method is initiated with results of Gummel
method as the initial values.
IV curves for the device can be achieved by sweeping the voltage and determining the
current under certain condition (illumination, parameters, etc). In this mode in first step Gummel
method is employed. However in other steps the result of previous condition is used as initial
guess of Newton method. The variation of the variables in each step must be small enough to
maintain the closeness the initial values to the final results.
Initially the coupling coefficients are assumed to be zero. If these coupling coefficients, given in
the matrix C, are not zero by assuming C=0, all variables can be determined with the
aforementioned approach and based on electron and hole distribution, the exciton generation rate
due to free electron hole trapping can be determined and added to the exciton generation rate in
exciton continuity equation and new exciton distribution can be calculated. After several
refinement cycles, density of carriers in coupled system is determined. Based on this coupling
electron and hole density can decrease due to transformation to exciton and increase due to
exciton dissociation. These two different effects have the potential to destabilize the coupling
coefficient refinement by introducing cycling behavior. However, increasing in the electron and
hole density is based on two phenomena. First, the exciton dissociation rate in the bulk is small
(<10-5) relative to electron and hole density due to nature of organic materials. Consequently the
exciton dissociation rate does not have a major effect on the stability of this algorithm. Second,
JXd at the interfaces and Its sensitivity to electron and hole density is less than one which
maintains the stability of refinement cycle. After a sufficient number of iterations, all variables
9
are then simultaneously updated via the Newton-Raphson algorithm. The main algorithm is
continued until residual errors or relative change in state variables is sufficiently small. This
method is one of many possible methods, including time marching methods such as the method
of lines.
Both electric potential at the back contact and electron quasi-Fermi level are defined as the
zero potential. Hence, the LUMO level at the back contact is obtained by
10
Supplementary material C: Material parameters used in the case study
Material parameters associated with the acceptor and donor media
Parameters
PCBM
P3HT
*
References
Hole mobility
h (cm2/ V s)
1.5  10-3
1.6  10-4
A1, A4, A6
Electron mobility
e (cm2/ V s)
3  10-3
4  10-5
A1, A2, A3, A4, A5, A6,
A7, A8, A9
Conduction band
(LUMO)*
Valence band
(HOMO) *
Exciton diffusion
coefficient
Exciton lifetime
EC (e V)
4.2
3.2
A10, A11, A12, A13,
A14, A15, A16, A17
EV (e V)
6
5.1
A10, A11, A12, A13,
A14, A15, A16, A17
DX (cm2/s)
9  10-5
9  10-5
A18, A19, A20
X (ns)
36
36
Assumption
Thickness
L (nm)
100
100
Device parameter
Recombination
constant
Barrier energy at top
contact
Barrier energy at back
contact
Effective density of
state at the LUMO
 (cm3/s)
1  10-9
1  10-9
A21, A22
b0 (e V)
0
-
Device parameter
bL (e V)
-
0
Device parameter
NC (cm-3)
5  1019
5  1019
A1, A2, A7, A8, A9, A23
Effective density of
state at the HOMO
Exciton surface
recombination
Exciton
velocity surface
dissociation velocity
Exciton interface
injection velocity
Electron (hole)
surface recombination
velocity
NV (cm-3)
5  1019
5  1019
A1, A2, A7, A8, A9, A23
Kr (cm.s-1)
3 105
3  105
Assumption
Kd(cm.s-1)
7  105
7  105
Assumption
Ki(cm.s-1)
3  10-2
3  10-2
Assumption
3  104
3  104
A24, A25, 4, 26
Sn (Sp) (cm.s-1)
In reference to the vacuum level
11
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