Simulation of hydrogenated amorphous and microcrystalline silicon optoelectronic devices

Simulation of hydrogenated amorphous and microcrystallinesilicon optoelectronic devices

Alessandro Fantonia,*, Manuela Vieirab, Rodrigo Martinsa

aUninova/FCT-UNL, Quinta da Torre, Monte-de-Caparica 2825, PortugalbISEL, R. Conselheiro EmõÂdio Navarro, Lisboa P-1900, Portugal

Received 7 December 1998; accepted 28 April 1999

Abstract

This paper is concerned with the modelling and simulation of amorphous and microcrystalline silicon optoelectronic

devices. The physical model and its mathematical formulation are extensively described. Its numerical reduction is also

discussed together with the presentation of a computer program dedicated to the simulation of the electrical behaviour of such

devices. This computer program, called ASCA (Amorphous Silicon Solar Cells Analysis), is capable of simulating, on one-

and two-dimensional domains, the internal electrical behaviour of multi-layer structures, homojunctions and heterojunctions

under simple or complex spectra illumination and externally applied biases.

The applications of the simulator presented in this work are the analysis of �c/a-Si:H p-i-n photovoltaic cell in thermal

equilibrium and illuminated by monochromatic light and the AM1.5 solar spectrum, with and without polarisation. We also

study the appearance within the device of lateral components of the electric field and current density vectors when the

illumination is not uniform. # 1999 IMACS/Elsevier Science B.V. All rights reserved.

Keywords: Optoelectronics; Semiconductor device simulation; Amorphous; Microcrystalline silicon p-i-n junctions; Solar

cells

1. Introduction

Amorphous silicon (a-Si:H) for low power applications appeared on the scene in 1975. Spear and LeComber [43] produced, for the first time, amorphous silicon doped with boron and phosphorus, andCarlson and Wronski [9] obtained the first a-Si:H junction. Since the beginning, this material attracted agreat deal of interest because it can be deposited on large areas at very low costs as compared withcrystalline silicon. As a matter of fact, one of the most interesting aspects of a-Si:H is its versatility andthe fact that it may be used, besides for photovoltaic applications, in a wide variety of devices,including thin film transistors, image sensors, light emitting diodes and random access memories.

Mathematics and Computers in Simulation 49 (1999) 381±401

ÐÐÐÐ

* Corresponding author. Tel.: +351-1-2954464; fax: +351-1-2941365; e-mail: [email protected]

0378-4754/99/$20.00 # 1999 IMACS/Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 8 - 4 7 5 4 ( 9 9 ) 0 0 0 5 5 - 5

The use of amorphous silicon for photovoltaic applications is limited by the light induceddegradation of the material [44]. Called the Staebler±Wronski effect, this effect is basically an intrinsicproperty of the material which shifts its defect equilibrium according to the operating conditions. It canresult in up to 20±40% of efficiency degradation, unless checked by design modifications such asthinner intrinsic layers and the use of multi-junctions. Minimisation of this effect has been one of themain problems in amorphous silicon physics and technology, and this is probably the main reason why,in the last few years, the amorphous silicon community has been focusing on microcrystalline silicon(mc-Si:H) rather than on amorphous silicon itself.

The first successful deposition of mc-Si:H (a two-phase material composed by grains of crystallinesilicon imbedded in an amorphous silicon tissue) was made in 1968 [48]. The interest in producingmicrocrystalline silicon p-i-n solar cells has recently been on the increase, as they are more stable thantheir a-Si:H counterparts. Microcrystalline based solar cells have demonstrated remarkable efficiencyon entirely mc-Si:H p-i-n cells [27] and they have been successfully incorporated in tandem structures[34]. In any case, the understanding of the mc-Si:H based devices is still in an early stage and somequestions about their properties are currently a major subject of debate [53].

Computer simulation supports research in fundamental areas of materials science that have an impacton the development and processing of new materials. In this field (including semiconductor and solidstate physics), a computer simulation deals with the interpretation of experimental observations andlinks theory and experiment. It is a way to approach a matter where the theory is not yet able to fullyexplain the experimental results obtained or where the calculations that the theory suggests are toocomplicated to achieve a solution without making recourse to drastic and unrealistic simplifications.This is the case of amorphous semiconductor physics and devices: the equations describing phenomenaand device behaviour cannot be completely solved without recourse to numerical techniques, and in thelast instance, to a computer simulation.

The first full numerical modelling of a semiconductor device based on partial differential equationswhich describe the device as a whole in one unified manner was suggested by Gummel [14] in 1964for the one-dimensional bipolar transistor. Such an approach was further developed and applied byDe Mari [11] to the pn junction theory and by Sharfetter and Gummel [37] to a silicon read diodeoscillator. These papers remain important references for developers of semiconductor device simulationprograms.

In 1982 Swartz [45] published the first work describing a computer model of an a-Si:H p-i-n solarcell in which the transport equation was self-consistently solved on a domain defined on the entiredevice. However in this model the trapped charge in the intrinsic layer was ignored and it used a singlelevel recombination model with constant lifetimes. Shortly later, Okamoto [31] and Sichanugrist [40]developed models based on the assumption that the electric field profile and the charge distribution aretotally controlled by the intrinsic layer, excluding the p and n regions from their analysis.

In 1983 Hack and Shur [16] developed a comprehensive model based on the complete self-consistentsolution of the transport equations for the entire p-i-n diode in steady state condition. This modelincludes a Shockley±Read±Hall recombination model [38] for a continuous distribution of localisedstates in the mobility gap based on the Taylor and Simmons statistics [41]. Such a model has beensuccessively employed for the analysis of amorphous silicon alloy p-i-n solar cells [17] and, in its two-dimensional formulation, for numerical simulations of a-Si:H thin film transistors [18].

Remarkable work was carried out during the 1990s on a-Si:H solar cells characteristics andperformance in steady state condition [3,7,13,21,42,50] and in transient conditions [4,19,33,36].

382 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

The work presented here is dedicated to the modelling and simulation of amorphous andmicrocrystalline silicon photodevices. The physical model and its mathematical formulation areextensively described. The numerical reduction is also discussed together with the presentation of acomputer program dedicated to the simulation of the electrical behaviour of such devices, whichrepresents the natural application of these models. This computer program, called ASCA, is capable ofsimulating the internal electric behaviour of multi-layer structures, homojunctions and heterojunctionsunder simple or complex spectra illumination and externally applied bias. This simulator permits theanalysis of structures defined by one- or two-dimensional geometries.

2. The physical model

2.1. Density of states and occupation probability in a-Si:H

The electrical and optical properties of a solid material can be deducted from the distributionof the electronic states throughout its atomic structure. This leads to a function, called the densityof states (DOS), which defines the number of allowed electronic energetic states per unit of volumeper unit energy. Such a function depends only on the microscopic structure of the material underconsideration.

As, by definition, in amorphous materials the lattice periodicity cannot be assumed, the wave vectoris no longer a good parameter to describe the allowed electronic states. However the short range orderin amorphous material structures is the same as in the corresponding crystalline one, and the density ofstates distribution of an amorphous solid should not be very different from that of its correspondingcrystalline one [1].

In the absence of long range order, the singularities in the density of states distribution at the bordersof the bands are expected to be replaced by more gradually decreasing distributions, entering into theenergy gaps [22]. These regions are called band tails and the extent of such tails can be considered ameasure of the degree of disorder in the atomic structure. Roughly speaking, we can say that the higherthe degree of disorder in the atomic structure is, higher will be the density of allowed electronic statesinside the gap between the conduction and valence bands. The main characteristic of such states is that,because of the lack of long range periodicity, they are not extended along the atomic lattice, butlocalised. The carriers that fall in such localised states become trapped and do not contribute to anytransport phenomena, except through tunnelling from an occupied to an unoccupied state at equivalentenergies. Moreover, besides the tail states, one expects to observe localised states inside the gaporiginating in the presence of defects in the atomic structure [26].

In amorphous silicon it can be assumed that the trap centres are subdivided into two species: one inwhich a trap is charged for electrons and neutral for holes (donor-like), the other where it is charged forholes and neutral for electrons (acceptor-like). The model of the DOS, as a function of the energy level(E), utilised in this study consists of two exponential distributions of tails states (ge(E)) in superpositionwith two Gaussian shaped DOS distribution (gg(E)) representing the dangling bond states [35]. Both ofthese distributions are subdivided into donor-like states (gD(E)) and acceptor-like states (gA(E)). Thismodel leads to an expression for the DOS g(E), defined as:

g�E� � geD�E� � ge

A�E� � ggD�E� � g

gA�E�: (1)

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 383

The two exponential tails are defined as:

geD�E� �

g0

2exp ÿE ÿ Em

ED

� �(2)

geA�E� �

g0

2exp

E ÿ Em

EA

� �(3)

where Em is the energy where the tail states reach the minimum value g0, while EA and ED are thecharacteristic energy slopes, respectively, for the acceptor-like and donor-like states.

The two Gaussian distributions are defined as:

ggD�E� �

NgD

�D

exp ÿ�E ÿ EgD�2

2�D

" #(4)

ggA�E� �

NgA

�A

exp ÿ�E ÿ EgA�2

2�A

" #(5)

where EgD and E

gA are the energy positions for the two Gaussian peaks, �D and �A are the standard

deviations while NgD and N

gA are the total number of states in the Gaussian distributions.

The same model can be applied to doped a-Si:H, but it should be considered a direct dependence ofthe exponential tails minimum value on the doping density [52]. The effect of the density of dopant onthe DOS profile must be taken into account also in the deep defect states distribution. As the Fermilevel (EF) shifts, charged defects are more readily formed, giving a higher defect density further awayfrom EF. If the Fermi energy shifts towards the conduction band (resp. valence band) band for n-type(resp. p-type) material, the donor-like (resp. acceptor-like) defect distribution is enhanced, maintainingits position unchanged [6].

Once the density of states function is defined, it is possible to produce explicit expressions for theprobability occupation function of both the a-Si trap species. Considering C the ratio of the section of acharged trap (�C) and of a neutral one (�N):

C � �C=�N; (6)

one obtains two different occupation functions [15]. For acceptor-like states (defined by �C��p,�N��n) the probability of occupation fA(E) is:

fA�E� � n� CNv exp��Ev ÿ E�=kT �n� Cp� Nc exp��E ÿ Ec�=kT � � CNv exp��Ev ÿ E�=kT � (7)

where Nc and Nv are the densities of states in the conduction and the valence bands, Ec and Ev thebottom and top limits of the conduction and the valence bands, k is the Boltzmann constant, T theabsolute temperature, and n and p the concentration of free electrons and holes, respectively. For donor-like states (defined by �C��n, �N��p), the occupation probability fD(E) is:

fD�E� � nC � Nv exp��Ev ÿ E�=kT �Cn� p� CNc exp��E ÿ Ec�=kT � � Nv exp��Ev ÿ E�=kT � : (8)

384 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

2.2. Recombination and trapping

Once defined the density of states profile, the Taylors±Simmons statistics can be applied to theShockley±Read±Hall model to construct a model, specific for a-Si:H, able to describe the distributionof trapped charge and the recombination rate of the carriers by means of localised energy states. Acarrier that becomes trapped does not contribute to transport phenomena except through its contributionto the charge density distribution, and thereby to the construction of internal electric field profiles. Theconcentrations of trapped electrons (nt) and holes (pt), are defined, as a function of the concentrations offree holes and electrons, as the integral over the mobility gap (Eg) of the DOS multiplied by theprobability of occupation f(E):

nt�n; p� �ZEc

Ev

fA�E�gA�E� dE (9)

pt�n; p� �ZEc

Ev

�1ÿ fD�E��gD�E� dE: (10)

According to the Shockley±Read±Hall model, in steady state conditions the electron and the holerecombination rates must be equal. The two expressions for hole and electron recombination rate (Rp

and Rn) are equivalent and equal to:

Rp�n; p� � Rn�n; p��Cv�N�npÿ n2i �ZEc

Ev

gD�E� nC � p� Nv exp�Ev ÿ E�

kT

� �� CNc exp

�E ÿ Ec�kT

� �� �ÿ1

� gA�E� n� Cp� CNv exp�Ev ÿ E�

kT

� �� Nc exp

�E ÿ Ec�kT

� �� �ÿ1

dE; (11)

where ni is the intrinsic carrier concentration and v the thermal velocity of the carriers.

2.3. The photogeneration of carriers

The illumination of a semiconductor with photons having energy exceeding the energy gap causesthe electrons in the valence band to be excited into the conduction band, leading to the creation of freeelectron±hole pairs. Some of the generated carriers will fall trapped or will recombine through thelocalised states. The illumination will anyway move the system away from its thermal equilibrium andwill create a concentration of photogenerated carriers, able to contribute to transport phenomena. Themagnitude of the light absorption of a material is described in terms of an optical absorption coefficient�(�), function of the wavelength (�) of the incident light.�(�) can be derived from the knowledge of g(E). A photon with energy �h! can be absorbed if it can

excite an electron from an occupied state with energy E to an unoccupied state with energy E��h!.Thus, the absorption coefficient can be expressed as [1]:

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 385

���� � K

�h!

Z1ÿ1jM�E�j2g�E�g�E � �h!�f �E��1ÿ f �E � �h!�� dE (12)

where f is the probability occupation function, K is a constant and M(E) is the matrix element of themomentum between the states at energies E and E��h!. As the range of photon energies which is ofinterest in amorphous silicon is limited to 1±3 eV (that is to the neighbourhood of the value of themobility gap Eg), M can be considered constant, and �(�) can be deduced just from g(E).

It is evident from Eq. (12) that �(�) measures the joint density of states, independent of the spatialextent of these states except through the matrix elements M(E). Consequently, the optical absorptionedge does not need to be directly related to the mobility gap. Indeed, there is reason to believe thattransitions between localised and extended states should have matrix elements of the same order ofmagnitude as those connecting two extended states; thus no discontinuity in �(�) must be expected atthe mobility gap.

The usual procedure to determine �(�) has been established by Tauc [46]. Based on experimentalconsiderations and assuming that the densities of states just beyond the mobility edges can be expressedin power law form, Tauc's result is that the expected variation of �(�) with the photon energy, near theminimum energy for absorption Eg, is given by:

���h!�1=2 � B��h!ÿ Eg� (13)

where B is a constant of proportionality that must be related to the specific characteristic of thematerial.

If a photon flux �0(�) is incident on the surface of an absorbing semiconductor, it propagates withinthe material with an exponential decay, in agreement with the following relation:

���; x� � �0��� exp�ÿ����x� (14)

where x is the distance calculated from the incident surface.The photogeneration rate of carriers within the material is then determined as:

G��; x� � ÿ @���; x�@x

� �����0��� exp�ÿ����x�: (15)

A transmission factor P, taking into account for back reflection of light, can also be introduced in theprevious expression to get a more realistic formula for the photogeneration rate:

G��; x� � �����0���1ÿ P

fexp�ÿ����x� � P exp�����x�g: (16)

Finally, the photogeneration caused by a complex illumination spectrum is then obtained byintegrating over the wavelength.

The above described light generation model is based on the so-called kinematic theory of lightabsorption. The photons flux is assumed to decay exponentially, while the absorption coefficient iswavelength dependent. Different models report on the need for taking light interference effects intoaccount [5]. Such dynamic models involve the wave nature of light. Anyway, it was shown that thekinematic model does not yield solar cell electrical properties that differ greatly from those of the

386 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

dynamic model [32]. As this study is aimed at the simulation of macroscopic properties ofphotodevices, such effects can be neglected without a significant loss of information on the effectivedevices performance.

2.4. The basic semiconductor equations

The equations forming the mathematical model to analyse an arbitrary semiconductor device,intended as a self-contained structure under quite general operating conditions are known as the basicsemiconductor equations and are a very general formulation of the electric transport properties within asemiconductor.

Summarising, the basic semiconductor equations are the continuity equation for electrons and holes:

@n

@t� 1

qdiv Jn ÿ Rn � G (17)

@p

@t� 1

qdiv Jp ÿ Rp � G: (18)

The Poisson's equation:

div grad V � ÿ ��

(19)

and the phenomenological transport equations:

Jn � ÿqn�n grad V � qDn grad n (20)

Jp � ÿqp�p grad V ÿ qDp grad p (21)

where t is the time, Jn and Jp the electron and hole density of current, Rn and Rp the recombination rateof electrons and holes (cf. Eq. (11)), G the electron±hole pair generation rate (cf. Eq. (16)), V theelectric potential, � the electric charge density, and � the static dielectric constant. �n, �p and Dn, Dp arethe band mobilities and the diffusion coefficients of electrons and holes, respectively.

In amorphous silicon the electric charge density is defined as:

� � q�pÿ n� pt ÿ nt � N�D ÿ NÿA � (22)

where q is the elementary electric charge. p and n are the free electron and hole concentrations, NÿA andN�D the ionised shallow donors and acceptors concentrations and nt and pt are the trapped electron andhole concentrations.

When the band structure of a solid becomes non-uniform, and material parameters like the dielectricconstant, the electron affinity (�) and the energy gap (Eg) become position dependent within thestructure, forces in addition to those from the macroscopic electric field act on the carriers, and the non-uniform DOS modifies carrier diffusion. In order to take into account these effects, the basicsemiconductor equations must be modified and new terms are to be considered. The non-uniformmaterial composition can be described by inserting a position-dependent dielectric constant intoPoisson's equation and two band parameters into the transport equations.

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 387

The constitutive relations for the flux density (D) and the hole and electron current densities Jp and Jn

are [25]:

D � ÿ"s"0 grad V (23)

Jp � ÿpq�p grad �V ÿ Vp� ÿ kT�p grad p (24)

Jn � ÿnq�n grad�V ÿ Vn� � kT�n grad n: (25)

The non-uniform material composition is described by the position-dependent dielectric constant andby the two band parameters Vp and Vn.

For semiconductors with parabolic bands, the two band parameters become [24]:

qVp � ÿ��ÿ �r� ÿ �Eg ÿ Egr� � kT logNv

Nvr

� �� kT log

F1=2��v�e�v

� �(26)

qVn � ÿ��ÿ �r� � kT logNc

Ncr

� �� kT log

F1=2��c�e�c

� �(27)

where �V and �C are related to the quasi-Fermi levels for holes and electrons (EFpand EFn

) as:

�c � EFnÿ Ec� �=kT (28)

�v � ÿ EFpÿ Ev

ÿ �=kT (29)

and F1/2 is the Fermi±Dirac integral of order 1/2. The last terms in Eqs. (26) and (27) are due to theFermi±Dirac statistics influence on the Einstein relation and reduce to zero when Boltzmann statisticsare assumed. The subscript `r' refers to the values of the various parameters at a reference locationwithin the heterostructure.

2.5. Boundary conditions

The solution to this system of partial differential equations has to be obtained in a bounded domain,representing the device geometry and appropriated boundary conditions must be imposed on such adomain. The domain and its frontier must be physically investigated and characterised to let the modeldescribe a situation with a physical possibility of existence. If only steady state solutions are searched,the time derivatives of the free carrier concentrations in equations are null and we obtain a system ofthree elliptic equations.

At a voltage controlled contact, Dirichlet boundary conditions can be imposed for the potential. Thedifference of potential at the contacts is given by the sum of the junction built-in potential (Vbi), theSchottky barrier height (VS) and the externally applied bias (Va):

V � Vbi � VS ÿ Va: (30)

While Vbi and Va represent a difference of potential that can be referenced to some arbitrary point, VS

expresses an effective property of the contact itself. The Schottky barrier height depends on the contactquality; it is generally located around 0.5±1.0 V and reduces to zero in an ideally ohmic contact. Thebuilt-in difference of potential in a semiconductor junction is calculated from the carrier concentrations

388 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

in thermodynamic equilibrium. These depend on the DOS distribution and on the eventual dopingconcentration and obey the solution of the system:

R�n; p� � 0 ��n; p� � 0: (31)

Indicating respectively with a and b the two sides of a semiconductor junction, and with (n0,a, p0,a) and(n0,b, p0,b) the solution of the system Eq. (31) obtained for a and b, the built-in voltage is expressed as [25]:

qVbi � kT

2log

n0;ap0;b

n0;bp0;a

� �ÿ ��a ÿ �b� ÿ 1

2�Eg;a ÿ Eg;b� � kT

2log

Nv;aNc;b

Nc;aNv;a

� �: (32)

The boundary values for the carrier concentrations (n,p) in an ideally ohmic contact are equal to theircorrespondent thermodynamic equilibrium values (n0,p0). However, more general boundary conditions forthe continuity equations can be given by modelling the current densities at the contact through theintroduction of the surface recombination velocities at the semiconductor±contact interface. In this way weobtain mixed boundary conditions [12]:

Jn � v � ÿqSn�nÿ n0� (33)

Jp � v � qSp�pÿ p0� (34)

where Sn and Sp are the surface recombination velocities, respectively, for electron and holes and v is a unitvector perpendicular to the contact.

3. The numerical simulation

The solution to the system of partial differential equations defined by the basic semiconductorequations cannot, in general, be found explicitly, and the problem must be approached from a numericalpoint of view. The application of numerical techniques introduces a new set of approximations andmost numerical methods give answers that are only approximations to the desired true solution. Theerror introduced by the finite discretization of the definition domain and of the differential operatorsentering in the semiconductor equations can be controlled, but its explicit estimation is a quitecomplicated task. Moreover, a compromise must be found between the requested precision of thesolution and the computer time needed for reaching it. Such a choice depends, among other things, onthe available computer resources, and it must be related to the effective purpose that one has in mind.The solution of similar problems, obtained with the same numerical technique will have, in general, thesame degree of approximation and a comparison among them will then be possible.

The numerical formulation of our mathematical problem consists in the partition of the domaindefining the device geometry into a finite number of subdomains where the solution can beapproximated with the desired accuracy. Once effectuated the discretization of the domain where thesolution of the basic semiconductor equation is sought, the differential operators are replaced at each ofthe mesh points by finite difference equations where only the nearest neighbouring points are taken intoaccount and the resulting systems of algebraic equations are solved by means of numerical iterativetechniques. The finite differences approach used in this work is the Sharfetter and Gummel's [37]scheme, widely used in semiconductor simulations. It consists in applying the five points finitedifference operators directly to the basic semiconductor equations. The components of the electric field,

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 389

the current density vectors and the carrier mobilities are assumed constant in each of the mesh intervals.Under this assumption the transport equations are treated as ordinary linear differential equations,defined on each of the mesh interval, in p and n. Doing so, the components of the current densityvectors are directly related to the carrier concentrations calculated at the major mesh points.

The numerical reduction of the basic semiconductor equations expressed by Eqs. (17)±(21) involvesthe computation of the recombination rate and the concentration of trapped carriers at each point of thegrid mesh. This calculation involves, through Eqs. (9)±(11), the evaluation of some integrals over someenergy intervals. Even if the DOS is previously defined as a simple smooth function, the integrals aretoo complicated to be evaluated explicitly because of the occupation function terms defined by Eqs. (7)and (8) and a 32-points Gauss±Legendre formula can be successfully used.

The dependent variables (V,n,p) entering in the basic semiconductor equations assume, along thestructure to be described, values of different orders of magnitude. The solution to this system ofequations involves the solution of some system of linear equations and the significant difference amongthe numerical values assumed by the dependent variables is reflected in the size of the coefficientmatrix of the linear systems that are to be solved. If the elements of the coefficient matrix vary greatlyin size, then it is likely that significant errors will be introduced, and the propagation of rounding errorswill be worsened. To avoid or at least to contain this problem, the basic semiconductor equations havebeen scaled by the so-called Markowich factors [28].

The physical model described above, together with its numerical reduction, has been implemented ina computer program dedicated to the simulation of the electrical properties of photodevices based onthe a/mc-Si:H technology. Such a program, called ASCA (Amorphous Silicon Solar Cell Analysis) runson standard personal computers in DOS environment and it has been written in FORTRAN 90. The systemformed by the numerical discretization of the basic semiconductor equations is solved by means ofpublic domain numerical libraries for the solution of system of non-linear (differential) equations. ASCA isimplemented with different libraries dedicated to the solution of systems of non-linear equations (MINPACK

[30], DAFNE [2]) and of systems of mixed non-linear algebraic±differential equations (ODEPACK [20]).Non-linearity is in practice a difficult problem; even when a solution is found, it is only a local

solution that may not correspond to the searched one. Each one of the different methods is consideredto be the best applicable in a particular situation but not in all of them. The DAFNE algorithm has beenfound to work very well in illumination and short circuit condition. If the application of an external biashas to be simulated, the most effective numerical solver has been found to be the one belonging to theMINPACK collection. Even if causing a sensible increase of the computation time, the ODEPACK routinescan be confidently used to find asymptotic solutions with a high possibility of success in most of the cases.

ASCA permits to define the structure to be simulated as a multi-layer structure. The material of eachlayer can be defined as a one- or two-phase one. For each material used in the simulation one can definethe dielectric constant, carrier mobility, energy gap extension, electron affinity, absorption coefficient,the effective density of ionised dopant and the density of states function. The input set used in oursimulation is reported in Table 1. The characterisation of the material is made through a friendlygraphic user interface. Other graphic user interfaces are used to characterise the device geometry, thecontact types and the grid mesh of discretization. Two-dimensional structures can be simulated but onlyon a rectangular domain.

Once the materials are characterised and the device structure defined, ASCA calculates thecorresponding thermodynamic equilibrium carrier concentrations and the internal potential profile, todefine the boundary condition to be inferred to the basic semiconductor equations. Therefore, one

390 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

should define the external condition in which the device operation should be simulated: illuminationand/or applied bias. The numerical solvers are then called to find the solution. When this is achieved, aset of output data is saved on the hard disk in an ASCII format suitable to be imported by most of thecomputer graphic applications. During the iteration process to solve the basic semiconductor equations,ASCA produce runtime outputs that are visualised in graphical form on the screen.

The outputs that ASCA produces are: the J±V characteristic curve, free and trapped carrierconcentrations, potential distribution, electric field profile, space charge density, recombination andphotogeneration rate profiles, current density profiles (hole and electron currents, diffusion, drift anddisplacement currents), density of states inside the mobility gap, probability occupation function of themidgap states, time derivative of the carrier concentration, and residuals of Poisson's equation.

4. Applications and results

4.1. The a-Si:H p-i-n photovoltaic cell

The a-Si:H p-i-n photovoltaic cell has been analysed in thermodynamic equilibrium, illuminated bydifferent monochromatic illuminations and by the AM1.5 spectrum. This analysis, performed in onedimension, outlines the influence of the i-layer DOS and thickness on the terminal characteristics of thedevice.

Table 1

Set of parameters used to simulate an a-mc-Si:H p-i-n junction

Parameter Symbol Unit p-layer i-layer n-layer

a-Sia-Si c-Si

a-Si

Layer thickness mm 0.03 1.0 1.0 0.05

Holes band mobility �p cm2 Vÿ1 sÿ1 1.0 1.0 10 1.0

Electron band mobility �n cm2 Vÿ1 sÿ1 10.0 10.0 100 10.0

Ionised acceptor density NÿA cmÿ3 1018 0.0 0.0 0.0

Ionised donor density N�D cmÿ3 0.0 0.0 0.0 1018

Electron affinity � eV 0.1 0.1 0.4 0.1

Energy gap Eg eV 1.72 1.72 1.2 1.72

Ratio of a charged/neutral trap cross sections C ± 100

DOS at the top of the valence band Nv eVÿ1 cmÿ3 1021

DOS at the bottom of the conduction band Nc eVÿ1 cmÿ3 1021

Energy position exponential band tails minimuma Em eV

Energy slope acceptor-like band tail EA eV 0.057 0.048 0.057

Energy slope donor-like band tail ED eV 0.082 0.068 0.082

Number of states in the donor-like Gaussian NgD cmÿ3 1017 1016 109 1018

Number of states in the acceptor-like Gaussian NgA cmÿ3 1018 1016 109 1017

Position of the donor-like Gaussian peaka EgD eV 1.15 1.15 0.9 1.15

Position of the acceptor-like Gaussian peakb EgA eV 1.15 1.15 0.9 1.15

Standard deviation of the donor-like Gaussian �D eV 0.15 0.15 0.10 0.15

Standard deviation of the acceptor-like Gaussian �A eV 0.15 0.15 0.10 0.15

a Measured from the conduction band edge.b Measured from the valence band edge.

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 391

A photovoltaic cell is normally designed for working under the illumination of a complex solarspectrum, and it is normally tested for its effective efficiency under the AM1.5 spectrum illumination.Anyway, a solar cell system can be used as a power supply for a broad range of applications and a solarcell is not always designated to work in external environments. The relevant device output parametersare the short circuit current, the open circuit voltage and the power developed by the device per unitarea. Depending on the particular application to which the cell is designated, a tuning of the requiredoptimised output to a particular monochromatic incident radiation, defined by its photon energy (Ep),could be needed.

The energy conversion efficiency (�) of the solar cell is defined as:

� � JscVoc

Pinc

FF (35)

where Jsc is the short circuit current density, Voc the open circuit potential, Pinc the incident powerdensity and FF is the fill factor, which is defined by the relation:

FF � Pm

JscVoc

(36)

where Pm is the maximum power developed by the cell.Most of the recombination takes place in the region near the i±n interface when Ep is low (uniform

absorption), and near the p±i interface when Ep is high (strong absorption). The electron component ofthe current dominates the transport processes inside the structure, which is drift dominated. Anyway,diffusion currents have been observed within the i-layer when its thickness increases up to 0.6 mm.These diffusion currents lead to a saturation of the total current density and limits further enhancementof the short circuit current that could be obtained with the better absorption of a low Ep radiation by athick i-layer.

Fig. 1 displays the short circuit current density generated by the cell as a function of Ep for a set ofp-i-n a-Si:H photovoltaic cell with different i-layer thicknesses. Jsc increases together with Ep until itreaches its maximum value and then begins to decrease. An enhancement of the i-layer thickness

Fig. 1. Short circuit current density, as a function of the photon energy of the incident radiation, for a set of p-i-n a-Si:H

photovoltaic cells with different i-layer thicknesses. Incident power density�40 mW cmÿ2.

392 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

causes the current, once it has reached its maximum value, to decrease for high values of Ep, while forlow values of Ep, the current reaches its maximum and remains constant. In the former case (high Ep)the decreasing of the current can be directly ascribed to a worsening of the generation±recombinationbalance. The latter case (low Ep) is caused by a superposition of two opposite effects: the betterabsorption of the radiation by a thick i-layer and the lowering of the bulk electric field. The value of Ep

at which the maximum Jsc is reached depends on the i-layer thickness and when it increases, theoptimum Ep moves towards the low energy values. Such a value is located between 2.1 and 2.3 eV. Thehighest current obtained in the simulations is about 18 mA cmÿ2 and it is produced by a cell with ani-layer thickness of 0.8 mm and illuminated by a radiation with Ep�2.1 eV.

Fig. 2 shows the fill factor, as a function of the i-layer thickness, for a p-i-n a-Si:H photovoltaic cellilluminated with different Ep. The value of FF ranges between 0.6 and 0.75 and increases together withEp. Such an enhancement is more evident in the structures with a thicker i-layer. When the i-layerthickness increases, the generation±recombination balance causes a general decrease of FF, thisdecrease being more pronounced when the cell is illuminated with a low Ep radiation.

Fig. 3 reports the energy conversion efficiency, as a function of Ep, for a set of p-i-n a-Si:Hphotovoltaic cells with different i-layer thickness. The optimum � is obtained with a cell with a thinneri-layer and illuminated by a radiation with a higher value of Ep than the ones where the maximum Jsc isobtained. The best � are obtained by a cell with an i-layer 0.4±0.5 mm thick illuminated by a radiationwith photon energy of 2.2±2.3 eV. A cell with a thin i-layer reaches its optimum performance whenilluminated by a radiation with a high photon energy. A cell with a thick i-layer behaves better when itis illuminated by a radiation with a low photon energy. � has no dependence on the power density of theillumination.

The Voc of the cell ranges between 0.75 and 0.9 V. It does not depend significantly on the i-layerthickness. Increasing Ep, the Voc shows a strong enhancement until Ep is lower than 2.2 eV. Once it hasreached its maximum value, it stabilises and no further enhancement can be obtained by increasing Ep.The Voc produced by the cell under AM1.5 illumination decreases slightly with the increasing of thei-layer thickness in the range 0.2±0.6 mm and than stabilises.

Under AM1.5 illumination, the different DOS does not cause any significant difference in the Jsc ifthe i-layer is thinner than 0.4 mm. The current enhances with the i-layer thickness. In the structure with

Fig. 2. Fill factor, as a function of the i-layer thickness, for a p-i-n a-Si:H photovoltaic cell illuminated by radiation with

different photon energies. Incident power density�40 mW cmÿ2.

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 393

the highest i-layer DOS, it reaches a maximum at about 0.8 mm and then decreases. In the other casesJsc remains virtually independent on the i-layer thickness when this is larger than 0.8 mm, showing thatlosses due to bulk recombination are still negligible in this range.

Fig. 4 displays the plot of the Voc, FF and � under AM1.5 illumination of a set of a-Si:H p-i-n solarcells with different i-layer DOS as a function of the i-layer thickness. The increase of the i-layer DOSof one order of magnitude causes a decrease in the Voc of about 3±4%. FF decreases significantly whenincreasing the i-layer thickness and its rate of decrease is enhanced by the enhancement of the i-layerDOS. � is significantly reduced by an increasing DOS. The general reduction of the efficiency is due tothe deterioration of FF and Voc. The localisation of this worst behaviour at photon energies higher than2.3 eV is due to the correspondent lowering of Jsc. Under AM1.5 illumination the optimum � has befound in a structure with an i-layer 0.4±0.5 mm thick. The effect of an increasing DOS is to lower �. Anincrease by one order of magnitude in the i-layer DOS leads to a decrease of about 20% in the optimum� foreseen by the simulation. The more the i-layer is thick the more � is reduced by the increase of theDOS. In a 1.2 mm i-layer structure, � reduces from 9.8% to 6.5%, when the i-layer DOS is enhanced byone order of magnitude, in a 0.5 mm i-layer structure � reduces from 10.2% to 8.7%. The values of Jsc,FF, Voc and � obtained by our simulation are in the typical output range of a-Si:H solar cells. Also, theirdependence on the i-layer characteristics hereby shown has been experimentally observed during the80's [8].

The different input used to define the DOS regards only the density of deep states. These areintroduced as a measure of the average defect density, in contrast with the band tails that are introducedas a measure of the disorder in the atomic lattice. Hence, they can be also interpreted as a representationof the (light-induced) degradation of the material. The simulation shows that when the i-layer is thickerthan about 0.4 mm the degradation of the cell efficiency under AM1.5 illumination is more pronounced.The real information coming from the simulation is about the existence of this value, about its locationin a range that is still useful and possible for a-Si:H thin film technology and mainly about itsexplanation in terms of recombination±generation balance and drift-diffusion mechanism. If thisexplanation is considered together with the results obtained about the effect of an increasing DOS on

Fig. 3. Energy conversion efficiency, as a function of the photon energy of the incident radiation, for a set of p-i-n a-Si:H

photovoltaic cells with different i-layer thicknesses. Incident power density�40 mW cmÿ2.

394 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

the energy conversion efficiency when the cell is illuminated by a monochromatic radiation, thepassage to the basic concept of a-Si:H tandem cells comes straightforward.

4.2. The �c-Si:H p-i-n junction

Recent progress has demonstrated that microcrystalline hydrogenated silicon is a very attractivematerial for the active layer of thin film solar cells. Efficiencies up to 7±8% have been demonstrated onentirely microcrystalline p-i-n solar cells with no sign of light induced degradation. The crucial point inmodelling a device based on mc-Si:H material is to define the role played by the boundary regions

Fig. 4. Open circuit voltage (a), fill factor (b) and energy conversion efficiency (c) under AM1.5 illumination of a set of

a-Si:H p-i-n solar cells with different i-layer DOS as a function of the i-layer thickness. DOS 1: NgA � N

gd � 1016cmÿ3. DOS 2:

NgA � N

gd � 5� 1016cmÿ3. DOS 3: N

gA � N

gd � 1017cmÿ3.

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 395

between the crystalline grains and the amorphous matrix. In these transition regions it is supposed that ahigh concentration of defects in the lattice structure is localised. If the distance between two adjacentgrains is large enough these regions could be treated similarly to a heterojunction (cf. Eqs. (23)±(29)).The difference between the energy gaps and the intrinsic carrier concentration causes the presence oflocal electric field in this transition regions. Moreover, due to the high density of defects, a high densityof trapped charge is localised near the grain boundaries and additional local electric fields, appear insuch regions. These local fields influence markedly the electrical properties of the intrinsic materialused in the p-i-n device. Here, they do not contribute to the total built-in potential developed by thejunction, but only to its distribution within the structure (mainly in the bulk of the intrinsic layer, wherethe strength of the junction built-in electric field is lower).

The device under consideration is a p-i-n structure where the doped layers are standard p and n typeamorphous silicon whose thickness is fixed at 0.1 mm. The mc-Si:H intrinsic layer is then defined bysquare crystalline grains, whose size is 0.3 mm, regularly distributed and imbedded in the a-Si:H matrix.There we considered that the distance separating the grains is in the range of 0.1 mm. This analysis,performed in one and two dimensions, outlines the influence of the local electric fields at the crystallinegrain boundaries on the transport properties within the device.

The photogeneration process is mainly localised inside the crystalline grains, where the red andinfrared radiations of the AM1.5 spectrum are better absorbed. This causes high gradients in the carriersconcentration at the crystalline±amorphous transition regions. In these regions, localised is a highdensity of electric charge and high peaks of the electric field directed towards the bulk of the crystallinegrains. The intensity of such peaks depends on Ep, enhancing when the radiation moves towards the redand it is comparable to the junction built-in field. The local fields at the amorphous±crystallinetransition regions are not influenced by the application of an external applied bias and result in alumped distribution of the potential profile. Thus, are created preferential transport paths for thecarriers, depending on the grains distribution and size.

The recombination process is mainly localised in the amorphous regions, where a high density oflocalised states is assumed. Even if the profile of the recombination rate profile depends on the natureof the incident radiation, the region where the recombination appears to be more critical is the bulk ofthe intrinsic layer, regardless of the value of Ep.

Figs. 5 and 6 show the transverse electron and hole current profiles within a mc-Si:H solar cell underAM1.5 illumination in short circuit condition. The transverse transport mechanism is mainlyconcentrated in the crystalline grains. The transverse transport of holes is preferentially made in thebulk of the grains while electrons move in the crystalline regions close to the grain borders. Theconduction within the amorphous region is very poor, and it contributes to the transport only byallowing a percolation of the carriers through the crystalline grains. The percolation of holes is mainlydirected along the transverse direction, while the electron percolation is primarily directed along thelateral direction. Such a lateral percolation of electrons results in a leakage current that can reduce thefill factor of the solar cell. The simulated collected short circuit current obtained by the simulationunder AM1.5 illumination is about 18 mA cmÿ2.

Fig. 7 shows the potential profile within the device cell under AM1.5 illumination in open circuitcondition. The local fields observed at the grain boundaries do not affect the Voc. Percolation of carriersbetween grains remains theoretically possible and a small photocurrent can be expected even when anexternal bias higher than the open circuit voltage is applied. This effect is confirmed by experimentalresults about the photocurrent measurement of mc-Si p-i-n solar cells [49].

396 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

4.3. The a-Si:H p-i-n junction under local illumination

The non-uniformity of light causes the appearance of a gradient in the carriers concentrationsbetween the illuminated and the dark zones. Carriers start to flow in accordance with such gradients,and when equilibrium is reached, the lateral diffusion process is counterbalanced by the appearance of alateral component of the electric field vector in addition to the usual transverse component. The lateralfields act as a gate for the lateral flow of the free carriers and small lateral currents appears at thetransition regions between the illuminated and dark regions. This lateral effect was first reported bySchottky [39] but only many years later rediscovered [47] and intensively investigated [23,51,10].

In short circuit conditions a channel of current between the two collecting contacts in correspondenceto the irradiated region is generated. The lateral currents are to be considered as leak on the transversecurrent flow that reach the collecting contacts.

Fig. 5. Transverse hole current profile within a mc-Si:H solar cell under AM1.5 illumination in short circuit condition.

Fig. 6. Transverse electron current profile within a mc-Si:H solar cell under AM1.5 illumination in short circuit condition.

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 397

In open circuit conditions, in correspondence of the incident light appears a lateral component of theelectric field well distributed along the entire intrinsic layer, higher in the p-layer and almost null in then-layer. Fig. 8 depicts the potential profile within an a-Si:H p-i-n junction in open circuit conditionilluminated by a light spot. The perturbation of the potential caused by the local illumination reaches, inthe p-layer, regions of the structure that are well away from the illuminated region. The potential decayis abrupt at the illuminate±dark transition region, and it becomes almost linear in the far dark regions.

Fig. 9 shows the lateral current density profile within the junction in open circuit condition andilluminated by a light spot. The lateral currents in the intrinsic layer are very low and they are mainlylocalised in the doped layers and can be ascribed to the drift caused by the lateral electric field. The

Fig. 7. Potential profile within a mc-Si:H solar cell under AM1.5 illumination in open circuit condition.

Fig. 8. Potential profile within an a-Si:H p-i-n junction in open circuit condition illuminated by a light spot (Ep�2.0 eV;

Pinc�30 mW cmÿ2).

398 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

lateral current density is about 4 mA cmÿ2 at the transition illuminated±dark regions(Pinc�30 mW cmÿ2, Ep�2.0 eV). It gradually reduces to less than 0.1 mA cmÿ2 at 0.5 cm far fromthe illuminated region.

In open circuit conditions, the extension of the lateral redistribution of potential profile caused by thelocal illumination increases with the value of Pinc, tending to a saturation configuration. Such aconfiguration does not depend on the photon energy, but the incident power density needed to reach isreduced by increasing Ep.

Experimental evidences of the appearance of lateral currents and electric fields can be found in [29].

5. Conclusions

We presented a modelling and simulation study of amorphous and microcrystalline siliconphotodevices. The physical model and its mathematical formulation was extensively described. Itsnumerical reduction has been also discussed together with the presentation of a computer programdedicated to the simulation of the electrical behaviour of such devices, which represents the naturalapplication of these models. This computer program, called ASCA, has the capability of simulating theinternal electric behaviour of multi-layer structures, homojunctions and heterojunctions under simple orcomplex spectra illumination and externally applied bias. This simulator permits the analysis ofstructures defined by one or two-dimensional geometries. Based on a theoretical model, the simulationpermits the analysis of the internal electrical behaviour of the cell allowing a comparison among thedifferent internal configurations determined by a change in the input set. The considerations that theinternal analysis permits lead to a better understanding of the more evident macroscopic characteristicsand the physics of the device.

The program has been successfully applied to the study of the a/mc-Si:H p-i-n junction underdifferent conditions of illumination. The obtained results are here summarised.

Fig. 9. Detail of the lateral current density profile within an a-Si:H p-i-n junction in open circuit condition illuminated by a

light spot (Ep�2.0 eV; Pinc�30 mW cmÿ2).

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 399

(a) The a-Si:H p-i-n solar cell: The simulated behaviour of the cell under different monochromaticradiations and the solar spectrum AM1.5 reveal that there exists a strong correlation among the basicoutput of the device (Jsc, Voc, FF, �), the characteristics of the i-layer (thickness and DOS), and theintensity and photon energy of the incident radiation. Depending on the final application of theprojected device, these relations should be taken into account when an effective information about theradiation photon energy (i.e. the colour of the light) is sought. The efficiency of the conversion ofsunlight into electricity foreseen by the simulator is about 8±11%. The simulation shows that when thei-layer is thicker than about 0.4 mm the degradation of the cell efficiency under AM1.5 illumination ismore pronounced.

(b) The �c-Si:H solar cell: The interest in the simulation of the mc-Si:H p-i-n junction resides in theoutlining of the effects caused by the highly non-uniform light absorption along the junction and of thepresence, intensity and distribution along the junction of the local electric field caused by theamorphous±crystalline interfaces. These local fields govern the internal transport process, and they areshown not to be influenced by the application of an external bias, and they show a very smalldependence on the energy photon radiation. They can be possibly related to an anomalous behaviour ofthe photocurrent and the spectral response observed in the microcrystalline p-i-n photovoltaic cells.

(c) The a-Si:H p-i-n junction under local illumination: This analysis, performed in two dimensions,outlines the appearance of lateral components of the electric field and current densities vectors withinthe device. While in short circuit the lateral currents are to be considered as leak on the transversecurrent flow that reach the collecting contacts, in open circuit conditions the perturbation of thepotential profile caused by the local illumination reaches, in the p-layer, regions of the structure that arevery far away from the illuminated region. The potential decay is abrupt at the illuminate±darktransition region and it becomes almost linear in the far dark regions.

Acknowledgements

This work has been financially supported by the Portuguese program PRAXIS XXI, which isgratefully acknowledged.

References

[1] D. Adler, in: D. Adler, B.B. Schwartz, M.C. Steele (Eds.), Physical Properties of Amorphous Materials, Plenum Press,

New York, 1985.

[2] F. Aluffi Pentini, V. Parisi, F. Zirilli, ACM-TMS 10(3) (1984) 299.

[3] J.M. Asensi, Celulas solares de silicio amorfo: Obtencion, caracterizacion y modelizacion, Ph.D. Thesis, University of

Barcelona, 1994.

[4] R. Bruggemann, C. Main, G.H. Bauer, Material Research Society Symposium Proceeding, Pittsburgh, PA, vol. 258,

1992, p. 729.

[5] W. den Boer, R.M. van Strijp, Solar Cells 8 (1983) 169.

[6] R. BruÈggeman, C.D. Abel, G.H. Bauer, Proceedings of the 11th Photovoltaic Solar Energy Conference, Montreux,

Switzerland, 1992, p. 676.

[7] J. Bruns, Die Entwicklung eines numerishen Simulationmodells fur a-Si:H Solarzellen und seine Anwendung zur

analyse experimentell ermittelter Spektralcharakteristiken, Ph.D. Thesis, Technische Univertsitaet, Berlin, 1993.

[8] D.E. Carlson, Semiconductors and Semimetals D 21 (1984) 7.

400 A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401

[9] D.E. Carlson, C.R. Wronsky, Applied Physics Letters 33 (1976) 935.

[10] W.P. Connors, IEEE Transaction on Electron Devices ED-18 (1971) 591.

[11] A. De Mari, Solid State Electrons 11 (1968) 33.

[12] S.J. Fonash, Solar Cell Device Physics, Academic Press, New York, 1981.

[13] J.L. Gray, IEEE Transactions on Electron Devices 36 (1989) 906.

[14] H.K. Gummel, IEEE Transactions on Electron Devices ED-11 (1964) 455.

[15] M. Hack, S. Guha, M. Shur, Physical Review B 30 (1984) 6991.

[16] M. Hack, M. Shur, Journal of Applied Physics 55 (1983) 5858.

[17] M. Hack, M. Shur, Journal of Applied Physics 58 (1985) 997.

[18] M. Hack, J. Shaw, Journal of Applied Physics 68 (1990) 5337.

[19] M. Hack, J. Shaw, Material Research Society Symposyum Proceedings 219 (1991) 315.

[20] C. Hindmarsh, IMACS Trans. Scient. Comp. 1 (1982) 55.

[21] W.J. Kopetzky, Numerische modellierung einer a-Si:H Heterostruktur-Solarzelle im ausgeheilten und gealterten Zustand,

Ph.D. Thesis, Technical University of Muenchen, 1992.

[22] N.E. Kusack, The Physics of Structurally Disordered Matter, Arrowsmith, Bristol, 1987.

[23] G. Luckovsky, Journal of Applied Physics 31 (1960) 1088.

[24] M.S. Lundstrom, R.J. Shuelke, IEEE Transaction on Electron Devices ED-30 (1983) 1151.

[25] M.S. Lundstrom, R.J. Schwartz, J.L. Gray, Solid State Electronics 24 (1981) 195.

[26] A. Madan, M.P. Shaw, The Pysics and Application of Amorphous Semiconductors, Academic Press, London, 1988.

[27] J. Maier, P. Torres, R. Platz, S. Dubail, U. Kroll, J.A. Selvan, N. Pellaton-Vaucher, C. Hof, D. Fisher, H. Keppner, A.

Shah, K.D. Ufert, P. Giannoules, J. Koeler, in: Amorphous Silicon Technology'1996, Material Research Society

Symposium Proceeding, vol. 420, 1996, p. 3.

[28] P.A. Markowich, C.A. Ringhofer, E. Langer, S. Selberherr, Report 2482, MRC, University of Wisconsin, 1983.

[29] R. Martins, E. Fortunato, Optical Materials 5 (1996) 137.

[30] J. MoreÂ, B. Garbow, K. Hillstrom, Argonne Nat. Lab. Rep. ANL-80-74, 1980.

[31] H. Okamoto, H. Kida, S. Nonmura, Y. Hamakawa, Solar Cells 8 (1983) 317.

[32] P. Popovic, J. Furlan, W. Kusian, Solar Energy Materials and Solar Cells 34 (1994) 393.

[33] P. Popovic, E. Bassanese, J Furlan, F. Smole, Proceedings of the 22nd International Conference on Microelectronics,

Ljubljana, Slovenia, 1994.

[34] A. Poruba, Z. Remes, J. Fric, M. Vanecek, J. Meier, P. Torres, N. Beck, N. Wyrsch, A. Shah, Proceedings of the 14th

Photovoltaic Solar Energy Conference, Barcelona, 1997, p. 2105.

[35] F.A. Rubinelli, J.K. Arch, S.J. Fonash, Journal of Applied Physics 72 (1992) 1621.

[36] F. Shapiro, Journal of Applied Physics 70 (1991) 4001.

[37] D.L. Sharfetter, H.K. Gummel, IEEE Trans. Electron Devices ED-16 (1969) 64.

[38] W. Shockley, W.T. Read, Physical Review 87 (1952) 835.

[39] W. Schottky, Physik Zeitung 31 (1930) 913.

[40] P. Sichanugrist, M. Kongai, K. Takahashi, Journal of Applied Physics 55 (1984) 1155.

[41] J.G. Simmons, G.W. Taylor, Physical Review B 4 (1971) 502.

[42] F. Smole, J. Furlan, Journal of Applied Physics 72 (1992) 5964.

[43] W.E. Spear, P.G. Le Comber, Philosophical Magazine 33 (1976) 671.

[44] D.L. Staebler, C.R. Wronsky, Applied Physics Letters 31 (1977) 92.

[45] G.A. Swartz, Journal of Applied Physics 53 (1982) 712.

[46] J. Tauc, in: F. Abels (Ed.), Optical Properties of Solids, North-Holland, Amsterdam, 1970.

[47] J.T. Wallmark, Proceedings of the IRE 43 (1956) 474.

[48] S. Veprek, V. Marecek, Solid State Electronics 11 (1968) 683.

[49] M. Vieira, Mac,arico, A.E. Morgado, R. Schwarz, Journal of Non-Crystalline Solids 227 (1998) 1311.

[50] M.B. Von der Linden, R.E.I. Schropp, W.G.J.H.M. van Sark, M. Zeman, J.W. Metselaar, Proceedings of the 11th

Photovoltaic Solar Energy Conference, Montreux, Switzerland, 1992, p. 647.

[51] H.J. Woltring, IEEE Transaction on Electron Devices 8 (1975) 581.

[52] C.R. Wronsky, B. Abels, T. Tiejde, G.D. Cody, Solid State Communications 44 (1982) 1423.

[53] N. Wyrsch, P. Torres, J. Meier, A. Shah, Journal of Non-Crystalline Solids 227±230 (1998).

A. Fantoni et al. / Mathematics and Computers in Simulation 49 (1999) 381±401 401