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Analytical modelling for p-i-n structured semiconductor devices

Cite as: AIP Advances 9, 025102 (2019); https://doi.org/10.1063/1.5045090 Submitted: 16 June 2018 . Accepted: 02 January 2019 . Published Online: 04 February 2019

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AIP Advances 9, 025102 (2019); https://doi.org/10.1063/1.5045090

9, 025102

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Analytical modelling for p-i-n structured semiconductor devices

Cite as: AIP Advances 9, 025102 (2019); doi: 10.1063/1.5045090 Submitted: 16 June 2018 Accepted: 2 January 2019 Published Online: 4 February 2019

Stephanie Reidt 1,a) and Bart E. Pieters 2

AFFILIATIONS

1 University of Siegen, Department of Electrical Engineering and Computer Science, Hölderlinstr. 3, 57076 Siegen, Germany 2 IEK5-Photovoltaik, Forschungszentrum Jülich, D-52425 Jülich, Germany

ABSTRACT

An analytical model for illuminated p-i-n structures such as Solar Cells and related devices has been developed. Starting from the semiconductor equations in their most general form, and introducing assumptions for the recombination and electrical ﬁeld functions, it is aimed at modelling not only the collection of photogenerated carriers but all transport mechanisms such as carrier drift and diffusion. Using this model, the behaviour of thin ﬁlm cells which is observed in operation is described in detail. This includes the dependencies of I/V curves on temperature, insolation and spectral effects, and electronic materials properties. The modeling of recombination effects results in a voltage dependent photocurrent where the carrier collection efﬁciency depends on the bias voltage. This model allows for relatively fast calculation of I/V curves as it is computationally less extensive than full device numerical simulations. Therefore it may be applicable for compact models i.e. in solar cell performance modelling.

I. INTRODUCTION

Analytical Modelling of semiconductor devices often pro- vides highly useful insights into the underlying physics. This is even more the case if the modelling results in a compact or closed form analytical description that shows fundamen- tal tendencies, and at the same time exhibits how the devices’ behaviour corresponds to its internal (material or construc- tion) parameters, as well to the conditions of its operation. As the semiconductor equations in their general form are complex, they are solvable in closed form only when simpliﬁ- cations are introduced into the equation system. These need to be selected so that they are well in accordance with the properties of the materials described. For amorphous and/or thin ﬁlm materials, this is challenging as all transport and recombination mechanisms play a role throughout the whole structure, making it difﬁcult to ﬁnd effective simpliﬁcations. Existing models for semiconductor devices mostly focus on certain predominant mechanisms. For example, the well- known Shockley diode I/V characteristics 1 is derived from minority carrier diffusion transport and recombination as mechanisms that prevail in (mono-)crystalline materials. The

solution of the diffusion equation shows ostensively how a diode is in principle behaving (forward exponential function, reverse saturation current) and how the behaviour is generally determined by the diode’s parameters and operating condi- tions such as temperature and external/terminal voltage. 2 On the other hand, deviations from the ideal I/V curve that result from ohmic resistance, space charges, recombination or high injection, are not visible. Being focused on minority carriers, the Shockley model does not couple the transport equations of majority and minority carriers and thus it cannot properly describe recom- bination in the space charge region where concentrations of both carrier types are in the same order of magnitude. As this applies in the same manner for the most part of the i-layer, the Shockley diode equation is not particularly well suited for describing p-i-n devices in general. In contrast, models for amorphous silicon solar cells such as 35 neglect diffusion transport completely and solve the remaining semi- conductor differential equation system which is then reduced by one order for each charge carrier type, greatly facilitat- ing the mathematical handling especially if recombination is accounted for. The generation rate is assumed as constant

throughout the structure, therefore, no spectral effects appear. Not many material parameters are included in these models so that their calculation of the open circuit voltage, for instance, is not very reliable. A reverse saturation current is not present in the models so that basic “diode” characteristics are missing. All of the simplifying assumptions outlined above are generally not applicable when it comes to p-i-n structured devices, even more so if the employed material is micro- or non-crystalline. This work goes beyond these assumptions and most of the limitations above, only slightly simplifying the primary semiconductor equations, accounting for drift and diffusion transport mechanisms. Using the practical example of a p-i-n structured a-Si:H Solar Cell, this work describes the develop- ment of an analytical model. We used device simulations with Advanced Semiconductor Analysis (ASA) to verify the appro- priateness of our assumptions throughout the derivation. In section II we introduce the geometrical settings and subsequently start with the standard drift/diffusion semicon- ductor equations. A suitable function for the recombination rate in n, p is then chosen. After this we apply a series of transformations in order to facilitate the further handling. The resulting differential equations are solved for sev- eral assumed electrical ﬁeld functions. The differential equa- tions are used partly in a simpliﬁed form (version 2) which enables compact solutions for any given electrical ﬁeld func- tion. For clarity, this is demonstrated for some E(x) functions that allow for closed form solutions of the occurrent integrals. For all variants of the system of differential equations under consideration, a certain constant appears (K A ) that is pro- portionally related to the total electrical current through the device. The I/V characteristics of the solar cell are simulated in section III. The model is parametrized and the calculation of the carrier concentration proﬁles by means of auxiliary functions is explained. We then demonstrate the behaviour of the model and how the simulated characteristics respond to environmental inﬂuences such as insolation and tempera- ture. By comparing the model with results obtained with ASA, we show that the model is well in accord with more detailed models. Due to the detailed approach, temperature, insolation intensity and spectral effects are all accounted for. The model calculates the dominant transport mechanisms, depending on the depth in the i-layer. It shows a diode-like behaviour including a reverse saturation current as much as devia- tions from the “ideal” diode I/V characteristics, and a voltage dependent photocurrent.

II. METHOD AND THEORY

A. Geometry

Solar Cells are a widely used application for p-i-n devices, especially when non-crystalline materials are employed. This work focuses on the simulation of a p-i-n solar cell structure of amorphous silicon which is a commonly used and techni- cally optimized structure for solar cells. Fig. 1 illustrates the

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FIG. 1. Cross Section of p-i-n structured Solar Cell under investigation.

structure of the considered device. The top layer consists of p-doped a-Si:H and usually has a thickness of approx. 10 nm. Then follows an intrinsic layer of 300-400 nm thick- ness, where most of the absorption takes place, and the bottom layer of the cell which consists of n-doped a-Si:H. Within this structure, space charge regions exist almost throughout the whole depth of the p and n layers which are very thin compared to the i layer, and therefore absorption in the top and the bottom layer contributes very little to the pho- tocurrent and the solar cell performance. In addition, there are space charge regions in the i-layer adjacent to the p-i and i-n interfaces. All space charges evoke an electrical ﬁeld inside the device that is typically higher at the interfaces as trapping in Dangling Bonds distorts the electrical ﬁeld. 6 The electrical ﬁeld as a function of position is shown in Fig. 2 where curve (A) depicts a spatial function of the elec- trical ﬁeld as obtained from theoretical considerations and numerical simulations from other authors 4,7,8 and our own simulations carried out with the device simulator program ASA (Advanced Semiconductor Analysis). 9 The electrical ﬁeld has

FIG. 2. Spatial functions of electrical ﬁeld E(x) (absolute value) and generation rate G(x).

a negative direction relative to the x axis. Curve (B) shows a uniform electrical ﬁeld which has often been used in analyti- cal modelling such as in Refs. 3 and 10, as a constant function greatly simpliﬁes the handling of the semiconductor equa- tions. It will also be employed for some simulations in this work. Curve (C) depicts an approximation of (A). Both the uni- form as well as the hyperbolic approximation will be used in this work. The area below the respective curve equals the voltage drop across the i-layer, V i . For V i we write

V i = V bi V,

where V is terminal voltage of the cell, and V bi is the built-in potential. In this work we will solve the semiconductor transport equations within the i-layer. The adjacent p and n-layers are accounted for through boundary conditions for the charge carrier concentrations at the edges of the i-layer. Also in this ﬁgure, the generation function G(x) is shown. G(x) is derived from the Lambert-Beer attenuation law and therefore is represented by a decaying exponential function.

B. Semiconductor equations

A model that takes into account both diffusion and drift transport mechanisms can be formulated for the collective charge carrier concentrations, i.e., not only for excess carriers or photogenerated carriers. The current densities of electrons and holes in the one- dimensional case are described as: 11

and

j n = qµ n nE + qD n

j p = qµ p pE qD p

Continuity equations:

dn

dx

dp

dx

and

1 dj n

q dx

+ G R = 0

dj p

1

q dx

+ G R = 0

(1a)

(1b)

(2a)

(2b)

Substituting Eqs. (1a), (1b) into (2a), (2b), respectively they become

d

2

d

2 (D n · n(x)) + µ n dx (E(x) · n(x)) R + G = 0

dx

d

2

2 D p · p(x) µ p dx (E(x)

dx

d

· p(x)) R + G = 0

(3a)

(3b)

where β = kT and D n = µ n kT = µ n , D p = m p kT = m p (Einstein

Relations). Equations (3a) and (3b) are coupled via the recombina- tion rate R that depends on carrier concentrations as well as via E(x). G(x) also occurs in both equations, as it effects elec- trons and holes in the same manner but irrespective of carrier concentrations:

G(x) = G 0 e αx ,G 0 = G 0 (α), α = α(λ)

q

q

β

q

β

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where G 0 = Generation rate at the surface of the cell. G 0 is proportional to the insolation intensity and to the absorption coefﬁcient α.

C. The recombination function

Next, for recombination a suitable function of the carrier concentrations needs to be chosen that represents the recom- bination mechanisms well. Furthermore, it has to be consid- ered that the chosen function will inﬂuence the form of the resulting differential equations, and possibly, their solvability. Fundamental considerations lead to a Shockley-Read type of recombination function: 12,13

R = R(x) =

n(x) · p(x) n i 2

τ p · n(x) + τ n · p(x)

where τ n , τ p are the (average) carrier lifetimes. The complexity of this function however would compli- cate the solution of the resulting differential equations. A lin- ear law has been proposed by Refs. 14 and 15, relating to the recombination behavior of dangling bonds:

R = R(x) = n(x)

τ

n

+ p(x)

τ

p

(4)

This R(x) function is linear in both n and p and not overly complex. Yet the authors of Ref. 12 point out that in a strict sense this relation is a good approximation when the ﬁeld is uniform, as can be assumed, for example, for reverse bias. So the question remains if it is a good approximation under

positive terminal voltage, i.e. normal solar cell operation con-

ditions. For this reason, we carried out device simulations using the ASA program, 9 which was designed to model in detail

all processes occurring inside the semiconductor materials,

including recombination rates. Figure 3 depicts the spatial distribution of the generation rate R(x) as calculated by ASA for a p-i-n-structured device of

FIG. 3. Recombination rate proﬁle according to the ASA simulations compared to linear recombination law, at a forward bias of 0.5V and under STC illumination (1000 W/m 2 , AM1,5).

amorphous silicon at a forward bias of 0.5V, which is near the maximum power point of the considered cell. For the simula- tion parameters we took a parameter set established by Liang et al. 16 We modiﬁed the parameter set slightly as we used the defect-pool model of Powel and Deane from 1996 17 to describe the spatial distribution of defects within the intrinsic layer. The simulated recombination rates with ASA include the effect of the energy and spatial variation in defect density according to the used defect pool-model. The recombination rate proﬁle is high at the p-i and i-n interfaces and drops in between to an almost constant level that is one to two orders of magnitude smaller than at the interfaces. The qualitative behavior of recombination rate proﬁle is reproduced by the linear recombination model in Eq. (4). Note, however, that without the impact of this spatial variation in defect density according to the defect-pool model, Eq. (4) would be a rather poor approximation of the recombination rate distribution. We ﬁt the carrier lifetimes τ n and τ p to reproduce the recombination rate in the center of the i-layer by least squares optimization. Fig. 4 depicts the ﬁtted parameters τ n , τ p , in a cell under STC conditions as a function of the applied termi- nal voltage. We can observe the carrier lifetime is not con- stant, however, the variation is within one order of magnitude, i.e. the variation is small compared to the variation in carrier concentrations as a function of the applied voltage. The carrier lifetimes are related to the quality of the a-Si:H layers. 18 With decreasing τ values, the total recombi- nation increases, reﬂecting deterioration of the semiconduc- tor material which is represented by an increased density of defect states, e.g. resulting from the Staebler Wronski effect. Hence, charge carriers are increasingly likely to be trapped and to recombine. The ﬁtted carrier lifetime parameters’ mag- nitudes are well in accordance with measurements in undoped a-Si:H. 19 Substituting the recombination rate from Eq. (4) into Eqs. (3a) and (3b), they read as

FIG. 4. Fitted charge carrier lifetimes as functions of the terminal voltage.

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d

2

d

2 n(x) + β dx (E(x) · n(x))

dx

d

2

d

2 p(x) β dx (E(x) · p(x))

dx

β τ n n(x)

µ

n

β τ p p(x)

µ

p

β τ p p(x) = G 0 β

µ

n

µ

n

β τ n n(x) = G 0 β

µ

p

µ

p

e

αx

(5a)

e

αx

(5b)

Due to the choice for the R function both electrons and holes

appear in both equations and they cannot be solved separately.

In the following these equations will be transformed in sev-

eral steps, such as scaling and linear combinations in order to obtain a more compact form. The mathematical transformations used for this purpose and are detailed in Appendix A. The system of Eqs. (5a) and (5b) leads to a third order dif- ferential equation which is solvable under simplifying assump- tions. Several solutions for special cases are deduced and discussed. The investigated scenarios are:

Solution of the 3rd order differential equation for uni- form electrical ﬁeld, see Appendix A 2 a;

Simpliﬁcation of the third-order ODE to a second- order ODE, and its Solutions, see Appendix A 2 b

Furthermore, an integration constant is identiﬁed that is a direct representation of the current density inside the cell and therefore used for calculations of I/V characteristics in the following chapters (see Appendix A 2 c).

III. PARAMETERS, RESULTS AND DISCUSSION

In the following we use a p-i-n structured amorphous

hydrogenated silicon solar cell as an example to demonstrate how charge carrier concentrations and current densities can

be calculated by the method developed above.

For this purpose, we assume the geometric and mate- rial properties of a real single cell operating under realistic surrounding conditions in a close to realistic operation mode (generator quadrant or dark forward operation, resp.).

A. Parametrization

For calculating I/V characteristics the model needs to be parametrized. The required parameters are listed in Table I. The electrical ﬁeld function in the i-layer is a function of the voltage over the i-layer V i . In case of the uniform ﬁeld the

electric ﬁeld is simply E i = V i /d (d: thickness of the i layer),

 in case the electric ﬁeld is non-uniform, the function needs to be scaled such that that its integral over the i-layer equals

V i , see for example, Appendix A 2 b with a hyperbolic ﬁeld function. The built-in Voltage usually has a value of approx. 1.0-1.2V, depending on the material properties. 20

1. Boundary conditions

Whereas the above parameters are inserted into the dif- ferential equations prior to their solving, the following four parameters are boundary values that adjust the charge carrier concentrations of majority and minority carriers at the edges

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TABLE I. Parameter list for the analytical p-i-n model and the corresponding values we use.

Parameter

Value

Remarks

Built-in Voltage (V bi ) thickness of p-layer thickness of i-layer thickness of n-layer Mobility electrons (i-layer):

1.05 V 10 nm 330 nm 20 nm 2 Vs/cm 2 0.3 Vs/cm 2 4.9e-6 s 1.1e-8 s

V bi (real cells) = 1.0-1.2V. 20 not directly relevant for simulation

not directly relevant for simulation

16

values taken from curve ﬁtting results of section II C, Fig. 4

of the i-layer, and therefore serve to determine the integration constants in the solutions for the auxiliary functions q and m. Reasonable assumptions need to be made. The carrier concen- trations at the boundaries may either be directly obtained by ﬁtting experimental results or we may use values from our ASA simulations. We use results from the ASA simulations as these are readily available to us. Table II lists the effective doping and carrier concentrations at the p-i and i-n interfaces from the ASA simulations. Since amorphous silicon material is considered here, these effective doping values correlate with free charge car- rier concentrations that are several orders of magnitude lower. Hydrogenated amorphous silicon material possesses high densities of localized states in the mobility gap. 21 Large amounts of charge carriers are trapped in localized states in the band gap and thus cannot contribute to current transport. Under non-equilibrium conditions, semiconductor theory indicates that the majority concentrations almost stay at their equilibrium levels, which is the case in dark forward opera- tion mode, but also under realistic illumination levels up to 1000 W/m 2 . Likewise ASA predicts only a very slight increase compared with the dark case. We use the majority carrier concentrations given in Table II for all simulations. For the minority carriers, the situation is fundamentally different. In dark forward operation of the solar cell diode, basic semiconductor theory indicates that these boundary val-

ues are proportional to exp qV , V being the terminal volt-

age, which is well in accordance with ASA results so that this relationship will be used for our simulations.

kT

TABLE II. Doping and equilibrium carrier concentrations at the p-i and i-n interfaces according to the ASA simulations

Parameter

Value

effective doping concentration, p-layer effective doping concentration, n-layer

p-i interface hole concentration (majority carriers) electron concentration (minority carriers)

i-n interface hole concentration (minority carriers) electron concentration (majority carriers)

1.19e25 m -3 1.1e24 m -3

1.37e19 m -3 6.7e3 m -3

52 m -3 1.8e21 m -3

Under illumination, the minority carrier concentrations signiﬁcantly increase as is expected, for example under

1000W/m 2 AM1.5 spectrum insolation at a terminal voltage of

0V:

electron concentration (minority carriers) at p-i interface n(x pi ) = 1e17m -3 = n

hole concentration (minority carriers) at i-n interface p(x in ) = 2.4e17m -3 = p

For our simulations, we interpolate these values linearly with the insolation power from their respective equilibrium values up to the ASA values at 1000 W/m 2 , as real cells exhibit an almost linear dependence of photocurrent on insolation. Semiconductor theory predicts that under illumination, a sec- ond term needs to be superimposed that is proportional to

0

pi

0

in

exp 2kT , which is expectable for a high injection situation,

and has a prefactor of n˜ i = 1e11m -3 . These results are consistent with ASA ﬁndings. So we get for the boundary values:

qV

n(x pi ) = n

0

pi ·

G

G AM1.5

+

n˜ i ·

G

G AM1.5 ·

exp

2kT

qV

p(x in ) = p

0

in ·

G

G AM1.5 + n˜ i ·

G AM1.5 · exp

G

2kT

qV

B. I/V characteristics

Appendix B details the calculation procedure for a speci- ﬁed working point. On the basis of this procedure, the follow- ing simulations have been carried out:

1. Simulations with uniform electrical ﬁeld

For the calculation of I/V characteristics two types of

electrical ﬁeld function have been chosen for investigation:

A uniform ﬁeld for the third-order differential equation and

a uniform or a hyperbolic function for the second-order

differential equation. The insolation is accounted for in the form of the

Lambert-Beer attenuation law (Appendix A 1) or, alternatively

if it simpliﬁes the mathematical evaluation, as a converging

Taylor series expansion of the exponential function. Losses and gains resulting from refraction and reﬂection at the front surface of the cell or at the back contact are not considered. Fig. 5 shows I/V characteristics of the illuminated cell for temperature variation:

FIG. 5. Variation of temperature in illuminated I/V characteristics, 2nd or 3rd order DE, uniform electrical ﬁeld.

The I/V curves in Figs. 5 and 6 show the basically expected behavior when varying the cell temperature or the illumination power, as is observed in practice. The ﬁll factors of the 2nd order DE characteristics are rel- atively high due to the fact that the 2nd order DE neglects a part of the recombination processes in the cell, as the terms comprising the factors a and b in Eqs. (A4a) or (A5b), resp. are neglected. On these grounds the curves closely resem- ble the Shockley exponential diode function, and therefore the behavior of solar cells of low defect materials which possess a smaller density of states in the bandgap that can be active as recombination centers. I/V characteristics calculated by means of the 3rd order DE also show the expected behavior concerning cell temper- ature and insolation power (Figs. 5 and 6). Compared to the characteristics calculated from the 2nd order DE, it is obvious

FIG. 6. Variation of insolation in I/V characteristics, 2nd and 3rd order DE, uniform electrical ﬁeld.

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FIG. 7. Variation of parameter κ 1 proportional to the holes’ lifetime under constant illumination - 3rd order differential equation compared to 2nd order DE.

that the short circuit current and the ﬁll factor of the curves are lower, due to the comprehensive recombination inﬂu- ences that the 3rd order equations include. Appendix C details simulations with non-uniform electri- cal ﬁeld and spectral effects on the shape of simulated I/V curves.

The 2nd order DE (see Appendix A) does not account for recombination in the same way as the 3rd order DE does (as the inﬂuence of the parameters a, b or the coupling κ, see Appendix A 1 and A 2 respectively has been neglected). Inves- tigating more closely the effect of the coupling κ via the 3rd order DE leads to the result that its value has a strong impact on the ﬁll factor, and at the same time, on short circuit current and on the open circuit voltage as shown in Fig. 7. The basic relation of carrier lifetimes to the ﬁll factor has been shown before. 15 , 22 An increasing value for κ or a decreasing κ 1 = µ n τ n means that the carrier mobility and lifetime product becomes more asymmetrical with the holes being the carriers with shorter lifetime. An identical reduction of both lifetimes, that of electrons and holes, leads to a less lowered ﬁll factor but to a signiﬁ- cant reduction of both short circuit current and open circuit voltage.

p τ p

µ

C. Comparison with ASA calculation results and speed

Finally we compare the results with ASA simulations. To do so we take the irradiation dependent JV curves as calcu- lated with ASA at 300K and determine effective parameters for the analytical model assuming a uniform electric ﬁeld. The boundary conditions at the p-i and i-n interfaces are as before in Section III A. We varied the total generation rate under AM1.5, carrier lifetimes, mobilities, and the built-in potential.

FIG. 8. Variation of illumination from 0 to 1000W/m 2 , this model.

TABLE III. Effective parameter values obtained from ﬁtting the model to ASA simulations.

Parameter

Value

Hole mobility

Built in potential

2.36e27 m -3 s -1 3.9e-6 s 2.9e-5 s 8.06e-6 m 2 V -1 s -1 5.98e-6 m 2 V -1 s - 0.984 V

In Fig. 8 we compare the results from the analytical model

(lines) and ASA (dots). We achieve a good agreement of the analytical model with deviations less than 3Am -2 . The used effective parameters are listed in Table III. As one of the reasons for using simpliﬁed models is a reduced computational effort it is interesting to compare the performance of the analytical model with ASA. It must be noted that such a comparison is rather complicated as one can only compare implementations and thus the results depend on the quality of the code. In this case we compare the analytical model, implemented in Matlab, with ASA, which is compiled C code. Our Matlab implementation outperforms ASA by about a factor 5.

IV. CONCLUSIONS AND FUTURE PROSPECTS

A new model for the description of p-i-n structures has

been presented that is focusing on the charge carrier densi- ties and transport processes within the intrinsic layer. Even though the calculations do not lead to a closed form I/V char- acteristics equation, the description is thoroughly analytical and the tendencies the model shows are very well consistent with the behavior of real cells. The model presented above is capable of calculat- ing charge carrier concentration proﬁles and current den- sity components for both drift and diffusion transport

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mechanisms. Temperature, insolation intensity and spectral effects are demonstrated. A set of parameters is included that directly or indirectly reﬂect the quality of the semiconduc- tor material and can be adapted accordingly. Hence, it qual- iﬁes for detailed calculations of the device in thermodynamic equilibrium as well as for biased and/or illuminated operating conditions. The presented model was compared to device simulations where we demonstrate we can accurately match the device simulation results. Furthermore, we could show the model can be efﬁciently evaluated with little computational effort. The model can be applied to other materials than amor- phous silicon partially in analogy with the presented example. To achieve this, the electrical ﬁeld function, the recombination function, the material parameters and boundary conditions for carrier concentrations need to be reviewed with respect to the intended purpose, and revised according to speciﬁc needs. Future investigation of the model’s parameters’ behavior when changing the operating conditions such as temperature and illumination may show if it may be suited for quantitatively accurate simulations, for example, curve correction proce- dures 23 with a similar accuracy as was shown in Ref. 24 for application in crystalline cells.

APPENDIX A 1. Transformations

a. 1st transformation

In a ﬁrst step we rewrite the continuity equations (5a, b) by uniting the constant β with the electrical ﬁeld and the carrier mobilities so that

β · E = E 1 ,

and the system spells as

d 2 n

2 +

d

dx

dx (E 1 · n)

β

µ n

β

µ p

=

=

1

µ n1 ,

1

µ p1

1

µ

n1 τ n

n

1

µ

n1 τ p

p = G 0

d 2 p

d

dx

2

dx (E 1 · p)

1

µ

p1 τ n

n

b. 2nd transformation

1

µ

p1 τ p

p = G 0

1

µ

n1

e

αx

1

µ

p1

e αx .

This transformation introduces a scaled length coordi-

nate by multiplying with

1

µ n1 τ n :

x

n1 τ n = x 1 dx = µ n1 τ n dx 1 ,

µ

E 1 µ n1 τ n = E 2 ,

α µ n1 τ n = α 1

Thus the factor

differential equations

1

µ n1 τ n appears in almost all terms in the

1 d 2 n

+

1

d

µ n1 τ n

dx

1 2

µ n1 τ n

dx

1 (E 2 · n)

1 d 2 p

1

d

µ n1 τ n

dx

1 2

µ n1 τ n

dx

1 (E 2 · p)

1

n

µ n1 τ n

1

n

µ p1 τ n

1

1

µ˜ n

1

µ n1 τ p

1

µ p1 τ p

p = G 0

p = G 0

µ

p1

e α 1 x 1

e α 1 x 1

This form suggests to expand the system by µ n1 τ n . The differ- ential equations then become

d 2 n

dx

d

1 2 + dx

1 (E 2 · n) n τ n p = G 0 τ n e α 1 x 1

τ

p

d 2 p

1 (E 2 · p) µ n1 n µ n1 τ n p = G 0 µ n1 τ n

µ

p1

µ p1 τ p

µ

p1

d

dx

1 2 dx

e α 1 x 1 .

Next, the charge carrier concentrations are scaled with the lifetimes τ n , τ p .

c. 3rd transformation

τ

p

τ n

d 2 n 1

d

1 2 + τ n dx 1 (E 2 ·

dx

n = n 1 · τ n ,

p = p 1 · τ p

n 1 ) τ n n 1 τ n τ p p 1 = G 0 τ n e α 1 x 1

τ

p

d 2 p 1

dx 1 2 τ p dx 1 (E 2 · p 1 ) τ n

d

µ µ n 1 τ p

n1

p1

µ n1 τ n

µ p1 τ p

p 1 = G 0 µ n1 τ n

µ

p1

e α 1 x 1

Division by τ n , τ p , respectively yields

d 2 n 1

d

dx 1 2

+

dx 1 (E 2 · n 1 ) n 1 p 1 = G 0 e α 1 x 1

1 2 dx 1 (E 2 · p 1 ) µ n1 τ n n 1 µ n1 τ n p 1 = G 0 µ n1 τ n

dx

µ p1 τ p

µ p1 τ p

µ p1 τ p

d 2 p 1

d

e α 1 x 1

When introducing the coupling κ = µ n1 τ n and writing the dif-

µ p1 τ p

ferential operator as the system of differential equations is written in the more compact form:

(A1a)

n

1

+

(E 2 n 1 ) (n 1 + p 1 ) = G 0 e α 1 x 1

(A1b)

As both equations have a similar form, linear combina- tions of (A1a) and (A1b) appear to be advantageous and lead to the 4th transformation.

d. 4th transformation

We now compose a new set of differential equations by forming both the sum and the difference of equations (A1a) and

(A1b):

p

1

(E 2 p 1 ) κ(n 1 + p 1 ) = κ G 0 e α 1 x 1

κ · (5a) (5b) :

κ · n

1

p

1

+ κ(E 2 n 1 ) + (E 2 p 1 ) = 0

or

κ · (5a) + (5b) :

κ · n

1

+ p

1

+ κ(E 2 n 1 ) (E 2 p 1 ) 2κ(n 1 + p 1 )

= 2κ G 0 e α 1 x 1

(κ · n 1 p 1 ) + (E 2 (κ n 1 + p 1 )) = 0

ARTICLE

(κ · n 1 + p 1 ) + (E 2 (κ n 1 p 1 )) 2κ(n 1 + p 1 ) = 2κ G 0 e α 1 x 1

and the appearance of linear combinations of n1, p1 suggests

that renaming them is of avail:

κn 1 p 1 = m,

and

κn 1 + p 1 = q

a = 1 κ,

b = 1 + κ.

The system of equations may now be rewritten as

 m + (E 2 q) = 0 (A2a) q + (E 2 m) − a · m − b · q = −G 1 e −α 1 x 1 , (A2b)

where 2κ G 0 = G 1 . The next step will be made with the objective of merg-

ing the above system of equations into only one differential equation with only one variable. Equation (A2a) can be formally integrated and solved for m‘ and m, respectively:

 m + E 2 q = K A , K A : integration constant m = K A − E 2 q (A3a) and m = K A · x 1 − E 2 qdx 1 + K B (A3b)

after second integration. Substituting m and m into equation (A2b) yields

q E 2 2 + b q (E 2 a) E 2 q

dx

1

a)(K A x 1 + K B ) (A4a)

which is an integro-differential equation corresponding to the 3rd order differential equation

q E 2 2 + b q + a · E 2 q = a · K A + G 1 α 1 e α 1 x 1 (A4b)

= G 1 e α 1 x 1 K A E 2 (E 2

Note that so far no assumption for the electric ﬁeld func- tion has yet been made and therefore, the above equation describes the general, one-dimensional case, independent of space charges.

2. Solutions of the differential equation using simplifying assumptions

a. Version 1: Solution of the 3rd order differential equation for uniform electrical ﬁeld E 2 (x) = const.

The assumption E 2 = const. leads to a special case which greatly simpliﬁes the form of the differential equations and

their solutions. As discussed in section II A, the electrical ﬁeld func- tion inside a real cell is not uniform but exhibits maxima at the interfaces on both sides of the i-layer. In the mid- dle of the i-layer, though, there is a spacious region where

the electrical ﬁeld varies only slightly and can therefore be assumed as almost constant. From this point of view, it makes sense to investigate the solutions of the problem with uniform electrical ﬁeld in the ﬁrst step. With E 2 = const. equations (A2a) and (A2b) become

m + E 2 q = 0

q + E 2 m a · m b · q = G 1 e α 1 x 1

(A5a)

(A5b)

By one-time differentiation of (A5b) and subsequent substitu- tion of equations (A2a) solved for m and (A3a) solved for m into it the differentiated equation (A5b) it becomes

q E 2 2 q b ·

q a · (K A E 2 q) = G 1 α 1 e α 1 x 1

or

= a · K A + G 1 α 1 e α 1 x 1 , (A5c)

resp., with all source terms on the right side. Thus, assuming an uniform electrical ﬁeld, a linear 3rd order differential equation with constant coefﬁcients is obtained. This is a solvable problem. The homogeneous equa- tion is

q E 2 2 + b q + a · E 2 q and it has the complementary functions

q E 2 2 + b q + a · E 2 q

= 0

q

n (x 1 ) = C · e λ n x 1 ,

n = 1, 2, 3

where the parameters λ n are the roots of the characteristic polynomial

λ 3 E 2 2 + b λ + a · E 2 = 0, and therefore the general solution of the homogeneous differ- ential equation is

q H (x 1 ) = C 1 e λ 1 x 1 + C 2 e λ 2 x 1 + C 3 e λ 3 x 1

where λ 1 , λ 2 , λ 3 are roots of the cubic equation above which can be calculated by means of the Cardanic Formulae. 25 The present equation is already in the form of a depressed cubic as a term with the factor z 2 is missing:

The discriminant

z 3 + Pz + Q = 0,

P

= E 2 2 + b ,

Q = aE 2 .

= Q

2

2 + P

3

3

for which parameters we have is always negative and there- fore, three roots can be calculated which all are real:

λ

1 = 4P · cos 3 · arccos

3

1

2

27

P

3

Q

+ π

3

λ

2 = + 4P · cos 3 · arccos

3

1

27

Q

2

P

3

λ

ARTICLE

3 = 4P · cos 3 · arccos

3

1

2

27

P

3

Q

π

3

Therefore the solution of the homogeneous differential equa- tion

q H (x 1 ) = C 1 e λ 1 x 1 + C 2 e λ 2 x 1 + C 3 e λ 3 x 1 is inﬂuenced by the parameters E 2 and κ and depends on the actual operating point/cell voltage. Next, the inhomogeneous differential equation needs to be solved:

q (E 2 2 + b)q + a · E 2 q = a · K A + G 1 α 1 e α 1 x 1

The particular integral can be found by using an ansatz cor- responding to the functional type of the source term which consists of a constant in addition to a term proportional to

e α 1 x 1 :

q p = K 1 + K 2 e α 1 x 1 q p = α 1 K 2 e α 1 x 1 ,

q

p

=

α 1 3 K 2 e α 1 x 1

Substituting the derivatives into the inhomogeneous equation yields

α 1 3 K 2 e α 1 x 1 (E 2 2 + b)(α 1 )K 2 e α 1 x 1 + aE 2 (K 1 + K 2 e α 1 x 1 )

= a · K A + G 1 α 1 e α 1 x 1

then by comparison of coefﬁcients follows

and

aE 2 · K 1 = a · K A thus K 1 = K A

E

2

α 1 3 K 2

+ α 1 E 2 2 + b K 2 + aE 2 K 2 = G 1 α 1

Subsequent solving for K2 this results in

K 2 (α 1 3 + α 1 (E 2 2 + b) + aE 2 ) = G 1 α 1

or

K 2 =

G 1 α 1

α 1 3 + α 1 (E 2 2 + b) + aE 2

.

Therefore the general solution to Eq. (A5c) is

q A (x 1 ) = q H (x 1 ) + q P (x 1 ) =

C 1 e λ 1 x 1 + C 2 e λ 2 x 1

+C 3 e λ 3 x 1 + K A + K 2 e α 1 x 1

E

2

(A6)

From the result for q(x) the function m(x) can be calculated:

m (x 1 ) = K A E 2 q(x 1 )

m (x 1 ) = E 2 C 1 e λ 1 x 1 E 2 C 2 e λ 2 x 1 E 2 C 3 e λ 3 x 1 E 2 K 2 e α 1 x 1

then we integrate with respect to x 1 :

m(x 1 ) = E 2 C 1 e λ 1 x 1 E 2 C 2 e λ 2 x 1 E 2 C 3 e λ 3 x 1 + E 2

λ

1

λ

2

λ

3

α

1

K 2 e α 1 x 1 + K B

The constant K B can be determined by substituting the solu-

tion functions q(x 1 ), m(x 1 ) into differential equation (A5b)

b K A aK B = 0

2

E

so the solution for m(x 1 ) is

and hence

K B =

aE b 2 K A ,

m(x 1 ) = E 2 C 1 e λ 1 x 1 E 2 C 2 e λ 2 x 1 E 2 C 3 e λ 3 x 1 + E 2

α 1

λ

1

λ

2

λ

3

K 2 e α 1 x 1

b

aE 2 K A

(A7)

Boundary conditions are the effective (modiﬁed) doping con- centrations at x=0 and x=D which need to be transformed to q and m values (see transformations 1-4 in chapter one). With the aid of four boundary conditions for the charge carrier con- centrations n and p, the four constants C 1 , C 2 , C 3 and K A can be calculated from a system of linear equations:

q(0) = C 1 + C 2 + C 3 + K A + K 2

E

2

q(D 1 ) = C 1 e λ 1 D 1 + C 2 e λ 2 D 1 + C 3 e λ 3 D 1 + K A

E

2

+ K 2 e α 1 D 1

m(0) = E 2 C 1 E 2 C 2 E 2 C 3 + E 2 K 2

λ

1

λ

2

λ

3

α 1

b

aE 2 K A

m(D 1 ) = E 2 C 1 e λ 1 D 1 E 2 C 2 e λ 2 D 1 E 2 C 3 e λ 3 D 1 + E 2

λ

1

λ

2

λ

3

α

1

K 2 e α 1 D 1

b

aE 2 K A

where D1 is the thickness of the i-layer in transformed dimen- sions, or in a matrix notation:

1

e λ 1 D 1

1

e λ 2 D 1

1

e λ 3 D 1

E 2

E 2

λ

1

λ

1

e λ 1 D 1

E 2

λ

2

E 2 e λ 2 D 1

λ

2

E 2

λ

3

E 2 e λ 3 D 1

λ

3

=

q(0) K 2

q(D 1 ) K 2 e α 1 D 1

m(0) E 2 α 1

K

2

m(D 1 ) E 2 α 1

K 2 e α 1 D 1

 

aE 2

1

E

1

2

E

2

b

aE 2

b

C 1

C

2

C 3

K

A

(A8)

b. Version 2: Simpliﬁcation of the third-order ODE to a second-order ODE, and its solutions

In case the terms with the factors a, b in equation (A5b) are small compared to the rest of the terms this means that recombination can be neglected and that equation (A5b) can be simpliﬁed to

q + E 2 m = G 1 e α 1 x 1 Using this result we can formally integrate eqs. (A5a) and (A5b) and we obtain

m + E 2 q = K A

(A9a)

re: Eq. (A3a) alternatively: (which is) mathematically equivalent

to eq. (A3a)

q + E 2
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