Device Physics of Silicon Solar Cells

J. O. Schumacher and W. Wettling: "Device physics of silicon solar cells", in Photoconversion of Solar Energy; Vol. 3, edited by M. Archer (Imperial College Press, London, 2000, ISBN 1-860-94161-3).

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CHAPTER 2

DEVICE PHYSICS OF SILICON SOLAR CELLS J. O. SCHUMACHER AND W. WETTLING Fraunhofer Institute for Solar Energy Systems ISE

Oltmannsstr. 5, D–79100 Freiburg, Germany [email protected]

2 Hjj

No, ’tis not so deep as a well, nor so wide as a church-door; but ’tis enough, ’twill serve.

Romeo and Juliet, William Shakespeare. 2.1 Introduction As shown in Chapter 1, a semiconductor solar cell is based on a simple p-n junction. An elementary description of cell performance can therefore be given in terms of a very simple model based on the Shockley diode equation in the dark and under illumination. This model is sufficient for understanding the basic mechanisms in the cell and roughly predicting the performance parameters of a solar cell. For some types of cells that perform far below their theoretical efficiency limit, this basic description may be adequate. However, for advanced solar cells such as high-efficiency monocrystalline silicon (c-Si) or gallium arsenide (GaAs) cells, which have been developed almost to their theoretical upper limit, these simple models are not sufficient to understand the subtleties of the device physics. Indeed, in the past few years improved methods of solar cell modelling have added immensely to a better understanding and performance of high efficiency cells.

Unfortunately, detailed solar cell models are too complicated to be handled by analytical mathematical methods. One has to use numerical techniques that may be complex and time consuming. Therefore in a typical R&D laboratory, simple and detailed device models are used in parallel. The choice of model depends on the problems that have to be solved.

In this chapter the device physics of solar cells is presented in several steps of increasing complexity. A schematic diagram representing the structure of the sections is shown in Fig. 2.1. Starting from the fundamental equations that describe semiconductor devices (Section 2.2), solutions are first discussed for the most simple cell model: the device equations are solved for a simple p-n junction cell consisting of an emitter and a base, each with a constant doping profile, with no boundaries taken into account (Section 2.3). In this most simple model, the ideal current-voltage


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characteristic of a solar cell is obtained. In Section 2.3 a basic skeleton of equations governing the device physics of solar cells is presented. A thorough derivation of the ideal p-n junction characteristics as presented in Section 2.3 is given in a paper by Archer (1996).

In Section 2.4 the most critical assumptions used in the derivation of the current-voltage characteristics are discussed and the ideal solar cell model is extended to include the front and rear surfaces and a diffused emitter. For these models the device equations can still be solved analytically. The contents of Sections 2.2, 2.3 and 2.4 can be found in standard textbooks on solar cell physics (e.g. Hovel, 1975; Green, 1982).

The semiconductor device equations can be solved with higher accuracy by applying numerical methods, to which we turn in Section 2.5, first for a one-dimensional model (Section 2.5.2). In modern high-efficiency solar cells, two- and three-dimensional features like point contacts and selective emitters have to be included in the calculation. Accordingly 2D- and 3D-numerical models must be used. These models are introduced in Section 2.5.3. In this section optical reflection and absorption in a high-efficiency silicon solar cell, calculated by means of ray tracing simulation, are also discussed. Furthermore, front side texturisation is taken into account and the optical carrier generation rate in high efficiency silicon solar cells is modelled.


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Section 2.3: Simple p-n junction no surface, constant doping, only emitter and base

Section 2.4: p-n junction with doping profile and front surface

Section 2.5.2: p-n junction including - space charge region - back surface field - series and shunt resistance

Section 2.5.3:

include texturing

include 2(3)D featureslike point contacts and selective emitter

Figure 2.1 From top to bottom: solar cell models of increasing complexity as they are analysed in this chapter.


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2.2 Semiconductor device equations Five equations describe the behaviour of charge carriers in semiconductors under the influence of an electric field and/or light, both of which cause deviations from thermal equilibrium conditions. These equations are therefore called the basic equations for semiconductor device operation. In the following they are simplified to one dimension.

The Poisson equation relates the static electric field E to the space-charge density ρ

2

2o s

d ( ) d ( ) ( )dd

x xxx

φ = − = −Eε ε

xρ (2.1)

where φ is the electrostatic potential, εo is the permittivity of free space and εs is the static relative permittivity of the medium. The electron current density ie and the hole current density ih are given by eqs. 2.2 and 2.3 d ( )( ) ( ) ( )

de e en xi x qD qu n x x

x= + + E (2.2)

d ( )( ) ( ) ( )dh h hp xi x q D qu p x x

x= − + E (2.3)

where n and p are electron and hole densities, ue and uh are the electron and hole mobilities and De and Dh are the electron and hole diffusion constants. The first terms on the right hand side of eqs. 2.2 and 2.3 are diffusion currents driven by a concentration gradient, and the second terms are drift currents driven by the electric field E.

The divergence of the current density i is related to the recombination and generation rates of charge carriers by the continuity equation. The electron and hole continuity equations may be written as

0)()()(1 =+−+ xgxr

dxxdi

q eee (2.4)

0)()()(1 =+−− xgxr

dxxdi

q hhh (2.5)

where r(x) and g(x) are the position-dependent volume recombination and photogeneration rates, respectively.


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Substitution of the current densities eqs. 2.2 and 2.3 into the continuity equations 2.4 and 2.5 gives a coupled set of differential equations, the transport equations

2

2 ( ) ( ) 0e e e e ed n dn dD u nu r x g x

dx dxdx+ + − + =EE (2.6)

2

2 ( ) ( ) 0 .h h h h hd p dp dD u pu r x g x

dx dxdx− − − + =EE (2.7)

The electron and hole transport equations 2.6 and 2.7 are coupled by the electric

field E. The coupled set of differential equations 2.1, 2.6 and 2.7 can be solved with different degrees of accuracy. The most basic approach will be discussed in the next section.


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2.3 The p-n junction model of Shockley 2.3.1 The p-n junction at equilibrium

For a dark unbiased p-n junction, thermal equilibrium conditions are fulfilled. In this section the density of mobile charge carriers, the electric field and the electrostatic potential at thermal equilibrium are discussed. For now, we assume the most simple case, i.e. a constant doping in the p- and n-regions with an abrupt doping step at the transition. This is a good model for a p-n junction grown by epitaxy. If an n-region is formed by diffusion of donor atoms from the surface into a p-doped material, the model is too simple and must be refined.

The Fermi levels Fµ~ of two separated p- and n-doped semicond-uctors are different, as shown in Fig. 2.2a.

If the two materials are brought into contact, the Fermi levels become identical. Figure 2.2b shows a p-n junction which is in equilibrium, so that it has a constant Fermi level Fµ~ achieved by diffusion of mobile charge carriers from one side of the junction to the other. This causes band bending of the conduction band-edge energy Uc

(a)

(b)

(c)

UUc

p

Uvp

p-type n-type

U

Uvn

Ucn

Ucp

Uvp

Uvn

-qVD

ρ

-

+

x

Ucn

µF

µF

µF

Figure 2.2 (a) Conduction and valence band-edgeenergies of separated p- and n-typesemiconductors; (b) band bending of the p-njunction. The Fermi level is constant atequilibrium; (c) space-charge density ρ across thejunction.
and the valence band-edge energy Uv.
A transient diffusion current of electrons from the n-doped to the p-doped semiconductor leads to a positively charged region in the n-type semiconductor, while hole diffusion from the p-doped to the n-doped semiconductor causes a negative space-charge in the p-type region. The space-charge region is almost completely depleted of mobile charge carriers, so that n and p are negligibly small compared to the donor and acceptor densities there. The resulting electric field produces a drift


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force that opposes the diffusion force. The diffusion and drift forces are equal at equilibrium conditions.

In the quasineutral regions that lie beyond the space-charge region the donor and acceptor charges are compensated by electrons and holes, so the space-charge density is zero. For low-injection conditions, i.e. when the majority carriers are the dominant carrier type and at room temperature the majority carrier concentrations n

);( ppnn nppn >>>>n and pp in the quasineutral regions are given by the density of ionised

dopants

(2.8) Dn Nn ≈

(2.9) .Ap Np ≈

For eqs. 2.8 and 2.9 to be valid it is assumed that the dopants are fully ionised. The validity of this assumption will be discussed in Section 2.4.2. For a non-degenerate semiconductor at thermal equilibrium the free carrier concentrations are given by the Boltzmann expressions

exp c Fc

Un N

kTµ−⎛= −⎜

⎝ ⎠⎞⎟ (2.10)

⎟⎠⎞

⎜⎝⎛ −−=

kTUNp vF

vµ~exp (2.11)

where Nc and Nv are the effective densities of states of the conduction band and of the valence band, respectively.

For thermal equilibrium conditions the semiconductor mass-action law holds, and using eqs. 2.10 and 2.11 this can be written as 2 ( ) ( )in n x p x=

2 exp gi c v

Un np N N

kT⎛ ⎞

= = −⎜ ⎟⎝ ⎠

(2.12)

where is the energy gap ( ) between the conduction and valence band edges.

gU vc UU −

A simple model for the space-charge region, the exhaustion region approximation, was introduced by Schottky assuming rectangular charge density distributions as shown in Fig. 2.3a.


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The space-charge region is depleted of mobile charge carriers. Hence the space-charge density here is given by

The space-charge region is depleted of mobile charge carriers. Hence the space-charge density here is given by

(a)

- Wp

q ND

ρ(x)

Wn x

-q NA

E(x)(b) - Wp Wn

x

(c) φ(x)

φp(x)

φn(x)

p-QNR p-DR n-DR n-QNR

-Wp

VD

Wn

x

x=0 Figure 2.3 (a) Space-charge density ρ(x);(b) electric field E(x); (c) inner potential φ(x)across a p-n junction.

)0()( ≤≤−−= xWqNx pA )0()( ≤≤−−= xWqNx pAρ (2.13) .)0()( nD WxqNx ≤≤=ρ (2.14)

The electric field in the space-charge

region can be found by integrating Poisson’s equation, eq. 2.1, from x = 0 to the edges of the depletion region with the constant charge densities eqs. 2.13 and 2.14.

This gives

0

( ) ( ) ( 0)Ap p

s

qNx W x W xε ε

= − + − ≤ ≤E (2.15)

0

( ) ( ) (0 )Dn n

s

qNx W x x Wε ε

= − − ≤ ≤E . (2.16)

Figure 2.3b shows the linear dependence of the electric field on position in the

space charge region. The electric field vanishes outside this region. Defining the Debye length LD as


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2/1

20

⎟⎟⎠

⎞⎜⎜⎝

⎛=

A

spD Nq

kTL

εε (2.17)

2/1

20

⎟⎟⎠

⎞⎜⎜⎝

⎛=

D

snD Nq

kTL

εε (2.18)

the electric field can be expressed as

2( ) ( ) ( 0)( )

thp pp

D

Vx W x W x

L= − + − ≤ ≤E (2.19)

2( ) ( ) (0 )( )

thnn

D

Vnx W x x W

L= − − ≤ ≤E (2.20)

where (2.21) qkTVth /= is the so-called thermal voltage.

The Debye length is a characteristic length of the p-n junction. At thermal equilibrium the depletion-layer widths of abrupt junctions are about 8LD for silicon. For typical doping densities higher than 1016 cm–3, the Debye length for silicon is less than ~40 nm.

The electrostatic potential in the depletion region is found by integrating eqs. 2.15 and 2.16 along x to be

)0()2(2

)(0

≤≤−+= xWxWxqNx pps

Ap εε

φ (2.22)

.)0()2(2

)(0

nns

Dn WxxWxqNx ≤<−=

εεφ (2.23)

At the depletion region edge and in the quasineutral regions beyond, the potential

takes the constant values for the p-type region and )(−∞pφ )(∞nφ for the n-type region, respectively.

The potential difference at the depletion region edges is denoted with )()()()( ppnnpnj WWV −−=−∞−∞= φφφφ . (2.24)


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For equilibrium conditions jV is called the diffusion voltage or built-in potential D of the p-n junction (Fig. 2.3). Substituting V )( nn Wφ and into eq. 2.24,

we find )( pp W−φ

( ).2

22

0pAnD

sD WNWNqV +=

εε (2.25)

The interface as a whole must be electrically neutral so

(2.26) .nDpA WNWN =

Using eqs. 2.25 and 2.26 the depletion region widths for equilibrium conditions follow as

2/1

0 /2⎟⎟⎠

⎞⎜⎜⎝

⎛+

=DA

DADsn NN

NNq

VW εε (2.27)

./22/1

0⎟⎟⎠

⎞⎜⎜⎝

⎛+

=DA

ADDsp NN

NNq

VW εε (2.28)

2.3.2 The junction under bias in the dark At thermal equilibrium the diffusion current of one carrier type is compensated by a drift current of the same carrier type so the net current flow vanishes. The diffusion of electrons from the n-doped region to the p-doped region can be expressed as a recombination current: electrons recombine with holes in the p-doped region creating a current density ie,rec. Similarly the drift current of the electrons from the p-doped to the n-doped region is supplied by thermally generated electrons in the p-region, creating a current density ie,gen. When the junction is unbiased (2.29) 0)0()0( ,, ==−= jagenejarece ViVi is valid.

An externally applied bias voltage Vja disturbs equilibrium conditions and shifts the potential barrier across the p-n junction.1 A forward bias Vja > 0 decreases the

1 For simplicity we do not account for a voltage drop due to the series resistance of a solar cell here; Vja

denotes the portion of the applied voltage that appears across the junction.


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potential barrier and the recombination current ie,rec(Vja) increases with the Boltzmann factor exp (qVja / kT)

, ,( ) ( 0) exp . (2.30) jae rec ja e rec ja

qVi V i V

kT⎛ ⎞

= = ⎜ ⎟⎝ ⎠

The rate of thermally generated charge carriers ie,gen is not influenced by the

external voltage, therefore . (2.31) )0()( ,, == jagenejagene ViVi

The analogous equations for holes are (2.32) )0()0( ,, === jagenhjarech ViVi

, ,( ) ( 0)exp jah rec ja h rec ja

qVi V i V

kT⎛ ⎞

= = ⎜ ⎟⎝ ⎠

(2.33)

(2.34) .)0()( ,, == jagenhjagenh ViVi

The external electron and hole current densities are given by (2.35) generecee iii ,, −= . (2.36) genhrechh iii ,, −=

The net current density is the sum of electron and hole currents . (2.37) he iii +=

Using eqs. 2.30 to 2.37 the net current density is therefore given by

o( ) exp 1jaja

qVi V i

kT⎡ ⎤⎛ ⎞

= −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(2.38)

where io is the saturation current density of the p-n junction, given by . (2.39) o , ,e gen h geni i i= +


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For forward bias, the current increases exponentially with the applied voltage. For high reverse voltages Vja < 0 the Boltzmann factor in eq. 2.38 can be neglected and the external current corresponds to the flux of thermally generated charge carriers as follows . (2.40) ( )o , ,e gen h geni i i i≈ − = − +

Equation 2.38 is known as the Shockley equation (Shockley, 1949). Note that it was derived without considering any special semiconductor characteristics such as charge carrier lifetime. The current-voltage behaviour of this idealised p-n junction is simply governed by the recombination current as a function of the Boltzmann factor

( )kTqVja /exp . The Shockley equation describes the ideal rectifier diode and is thus a fundamental equation for microelectronic device physics.

For a quantitative discussion of the idealised p-n junction the charge carrier densities at non-equilibrium have to be calculated. The device works under non-equilibrium conditions if charge carriers are either injected by an applied voltage or optically generated. For this purpose, we can derive boundary conditions for the charge carrier densities at the junction edges –Wp and Wn, respectively, as follows: at thermal equilibrium the charge carrier densities are given by the Boltzmann distribution eqs. 2.10 and 2.11 based on the energy difference between the band-edge energies Uc and Uv and the Fermi energy Fµ~ (Fig. 2.2). A similar dependence can be stated for non-equilibrium conditions by introducing separate Fermi energies for electrons and holes, the quasi-Fermi energy levels eµ~ and hµ~ . These are defined so that replacement of the single equilibrium Fermi energy Fµ~ in the equilibrium expressions on the left side of Table 2.1 by the quasi-Fermi energy levels eµ~ and

hµ~ yields the non-equilibrium carrier densities on the right hand side of Table 2.1.


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Table 2.1 Charge carrier density for equilibrium and non-equilibrium

Equilibrium Non-equilibrium

o exp c Fc

Un N

kTµ−⎛ ⎞= −⎜

⎝ ⎠⎟ (2.41) ⎟

⎠

⎞⎜⎝

⎛ −−=

kTU

Nn ecc

µ~exp (2.42)

o exp F vv

Up N

kTµ −⎛ ⎞= −⎜

⎝ ⎠⎟ (2.43) ⎟

⎠

⎞⎜⎝

⎛ −−=

kTU

Np vhv

µ~exp (2.44)

2o o exp c vi c v

U Un p n N N

kT−⎛ ⎞= = −⎜ ⎟

⎝ ⎠

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

kTU

NN gvc exp (2.45)

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ −−=

kTkTUUNNpn ehvc

vcµµ ~~

expexp

o o exp h en pkT

µ µ−⎛= ⎜⎝ ⎠

⎞⎟ (2.46)

Left: The charge carrier densities for equilibrium are given by the difference of the band-edge energy and the Fermi energy. Right: For non-equilibrium conditions the quasi-Fermi levels eµ~ and hµ~ are introduced.

At the edges of the space-charge regions the minority and majority carrier quasi-Fermi levels are separated by the bias qVja imposed across the p-n junction ( ) ( )ja h p e nqV W Wµ µ= − − . (2.47)

Here, we assumed the validity of the Boltzmann approximation for the charge carriers (Table 2.1) and low-injection conditions. More general expressions for the separation of carrier quasi-Fermi levels were derived by Marshak and van Vliet (1980). With respect to our assumptions, the separation of the quasi-Fermi levels Vja is related to the potential difference Vj across the junction (eq. 2.24) by

. (2.48) jDja VVV −=

Under low-injection conditions, the majority carrier concentrations are unperturbed throughout the quasineutral regions, so that o( )p p pp W p− = (2.49)

(2.50) o( )n n nn W n= where o

pp and are the majority carrier concentrations in the quasineutral p- and n-type regions, respectively.

onn


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Using eqs. 2.46 to 2.50 we find the boundary conditions for the minority carrier concentrations at the edges of the space-charge regions as

o( ) exp jap p p

qVn W n

kT⎛ ⎞

− = ⎜ ⎟⎝ ⎠

(2.51)

o( ) exp jan n n

qVp W p

kT⎛ ⎞

− = ⎜ ⎟⎝ ⎠

. (2.52)

With these boundary conditions the transport equations can be solved to find

quantitative expressions for the saturation current density of the p-n junction. 2.3.3 The superposition principle The transport equations for electrons and holes eqs. 2.6 and 2.7 contain the electric field E, thus forming a coupled set of differential equations for the carrier concentrations n and p. If the spatial dependence of the electric field and the carrier concentrations is known the current densities can be calculated with the help of eqs. 2.2 and 2.3.

With the approximations discussed in Section 2.3.1 it was found that the electric field in the quasineutral region vanishes, and therefore the transport equations 2.6 and 2.7 decouple in these regions. In this case the carrier transport is purely diffusive and the minority carrier concentrations can be calculated separately for both quasineutral regions. Under low-injection conditions the perturbation of the majority carrier concentration due to generation and recombination processes can be neglected. The recombination rate of minority carriers is then proportional to the excess minority carrier concentration (n – no) on the p side and (p – po) on the n side. Therefore the minority carrier recombination rate r is given by

o

ee

n nrτ−= (2.53)

o

hh

p prτ−= (2.54)

where eτ and hτ are the minority carrier lifetimes of electrons and holes, respectively.

The transport equations 2.6 and 2.7 simplify to the following decoupled linear differential equations for the quasineutral regions if the minority carrier lifetimes do not depend on the carrier concentrations


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2 o

2 ( ) 0e ee

d n n nD gdx τ

−− + =x (2.55)

2 o

2 ( ) 0h hh

d p p pD gdx τ

−− + x = . (2.56)

In addition it is assumed that the photogeneration rates ge(x) and gh(x) are

independent of the carrier concentrations and depend only on distance x from the illuminated surface. For dark conditions [g(x) = 0] eqs. 2.55 and 2.56 are then homogeneous differential equations. The carrier concentrations under illumination are found by adding particular solutions of the inhomogeneous differential equations to the general homogeneous solutions. For electrons in the p-type region this may be expressed as

(2.57) Lt Dk phn n n= + where nDk is the solution for dark conditions, nph is the photogenerated electron density and nLt is the electron concentration under illumination. This implies that the illuminated I–V curve can be found simply by adding the photogenerated current to the dark I–V curve (shifting approximation)—the model cell is said to exhibit superposition.

This is shown schematically in Fig. 2.5, where the dark diode I–V characteristic is shifted from the first quadrant to the fourth quadrant by adding the photogenerated current In a circuit diagram as shown in Fig. 2.8 the superposition of currents means that the diode and the photogenerated current flow is parallel.

.scLt ii −=

2.3.4 Carrier density solutions for dark conditions The Shockley equation 2.38, for the current-voltage characteristic of a p-n junction was found in Section 2.3.2 with the help of a qualitative discussion. A quantitative expression for the saturation current density io in the Shockley equation can be calculated by solving the diffusive carrier transport equations in the quasineutral regions.

If the superposition principle discussed in Section 2.3.3 applies, the carrier density solutions in the quasineutral regions can be found by solving eqs. 2.55 and 2.56 separately. Under dark conditions eqs. 2.55 and 2.56 reduce respectively to


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2 o

2 0 ( )ee

d n n nDdx τ

−− = ≤ − px W (2.58)

2 o

2 0 ( ).hh

d p p pDdx τ

−− = ≥ nx W (2.59)

The general solution of eq. 2.58 can be expressed as

o cosh sinhe e

x xn n A BL L

⎛ ⎞ ⎛ ⎞− = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (2.60)

where the distance eee DL τ≡ (2.61) is the diffusion length of electrons in the p-doped side. The diffusion length is the average length a minority carrier can diffuse between generation and recombination, i.e. during its lifetime τ.

If it is assumed that there is no recombination loss at the surface of the p-QNR, the boundary condition

0=−∞→xdxdn (2.62)

applies. Applying eqs. 2.62 and 2.60 it is found that A = B so the concentration of electrons in the p-QNR is given by

o exp .e

xn n AL

⎛ ⎞− = ⎜ ⎟

⎝ ⎠ (2.63)

The boundary condition eq. 2.51 yields

o exp 1 exp .ja p

e

qV WA n

kT L⎡ ⎤⎛ ⎞ ⎛

= −⎢ ⎥⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝⎣ ⎦

⎞⎟⎠

(2.64)

Note that the exponential excess carrier concentration decays (eq. 2.63) from position

in the p-QNR with the diffusion length LpWx −= e as characteristic length.


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The electric field in the quasineutral region vanishes because of the exhaustion region approximation, so the current flow is driven by diffusion only. At the edge of the p-doped depletion region the current density is given by

( )o( )pe p e x W

di W qD n ndx =−− = − (2.65)

and from eqs. 2.63 and 2.64 we find

o

( ) exp 1 .jaee p

e

qVq D ni W

L kT⎡ ⎤⎛ ⎞

− = −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(2.66)

Similar arguments give the excess hole concentration in the n-QNR as

o o exp 1 expja n

h

qV W xp p p

kT L⎡ ⎤⎛ ⎞ ⎛ −

− = −⎢ ⎥⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝⎣ ⎦

⎞⎟⎠

(2.67)

and the current density of the diffusive hole carrier flow at x = Wn as

o

( ) exp 1 .jahh n

h

qVq D pi W

L kT⎡ ⎤⎛ ⎞

= −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(2.68)

To derive analytical expressions for the electron and hole carrier densities in the n-

doped depletion region it is assumed in this Section that the recombination loss in the depletion region can be neglected, i.e. r = 0 for .p nW x W− ≤ ≤

The transport equation 2.6 then reduces to

2

2 ( ) 0.e e ed n dn dD u x nu

dx dxdx+ + EE = (2.69)

Using Schottky’s model for the space-charge region discussed in Section 2.3.1 and the Nernst–Einstein relation

ee uq

kTD = (2.70)


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the electric field can be expressed in terms of the characteristic Debye length LDn

(eq. 2.18) of the space-charge region

( ) ( ) .01)()(1222

2

=+−−nD

nnD L

xndxdnxW

Ldxnd (2.71)

At x = Wn the carrier concentrations must match the solutions for the quasineutral

regions, and therefore (2.72) o( )nn W n=

o( ) exp .jan

qVp W p

kT⎛ ⎞

= ⎜ ⎟⎝ ⎠

(2.73)

Under low-injection conditions the majority carrier concentration gradient

vanishes at x = Wn, i.e.

.0== nWxdx

dn (2.74)

The boundary condition for the hole gradient at the depletion region edge follows

from eq. 2.67 as

o

exp 1 .n

jax W

h

qVdp pdx L kT=

⎡ ⎤⎛ ⎞= − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦ (2.75)

Integrating eq. 2.71 from Wn to x using the boundary condition eq. 2.74 we obtain

2

( ) ( ) 0.( )

nnD

x Wdn x n xdx L

−+ = (2.76)

The differential equation for the hole carrier density in the n-doped depletion

region can be found by using the boundary condition eq. 2.75. The same steps are used as for the derivation of eq. 2.76, leading to

o

2

( ) ( ) exp 1 .( )

jann

hD

qVx Wdp x pp xdx L kTL

⎡ ⎤⎛ ⎞−− = − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦ (2.77)


19

The electron carrier density in the n-doped depletion region is found by integrating eq. 2.76 subject to the boundary condition eq. 2.72, thus

2

o2

( )( ) exp (0 ) .

2( )n

nnD

x Wn x n x W

L⎛ ⎞−

= − ≤ ≤⎜ ⎟⎝ ⎠

(2.78)

Integrating eq. 2.77 using the boundary condition eq. 2.73 gives the hole carrier

density in the n-doped depletion region as

1/ 2

2o

2

( ) exp 1 erf exp2 2

( )exp (0 ).

2 ( )

nja janD

nh D

nnn

D

q V qVW xLp xkT L kTL

x Wp x W

L

π ⎫⎧ ⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞−⎪ ⎪⎛ ⎞= − +⎜ ⎟⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎝ ⎠⎩ ⎭⎛ ⎞−

≤ ≤⎜ ⎟⎝ ⎠

×

(2.79)

The Debye length LD introduced in eq. 2.17 and 2.18 is several orders of

magnitude smaller than typical diffusion lengths Le, Lh. Thus the first term in eq. 2.79 can be neglected and this equation reduces to

2

o2

( )( ) exp exp (0 ).

2( )ja n

nnD

qV x Wp x p x W

kT L⎡ ⎤⎛ ⎞ −

= ≤⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

≤ (2.80)

The depletion-region concentration profiles predicted by eqs. 2.78 and 2.80 are

illustrated in Fig. 2.4. These equations are sometimes known as the quasiequilibrium expressions. They indicate that, although there is a net carrier flux across the biased junction, the carrier concentrations are, to a very good degree of approximation, still related to those at the depletion region edges by Boltzmann expressions. Physically this is because the net hole and electron currents in the depletion regions arise from the small difference between very large opposed drift and diffusion forces. These are exactly balanced at equilibrium (Section 2.2) and only slightly unbalanced when the junction is subjected to moderate bias, so the carrier profile remains quasistatic, that is, solely determined by the local electric potential and independent of any transport properties, such as carrier mobility (Archer et al., 1996).

We can now find an expression for the saturation current density io. Since the hole and electron currents are constant across the dark, biased junction, it follows that all majority carriers injected into the junction at one depletion region edge must emerge as minority carriers from the other depletion region edge. We can therefore calculate


20

the saturationcurrents emerg

where

Comparing

using also eqsbe written as

p-type n-typedepletionregion

log

(car

rier c

once

ntra

tion)

-Hp Hn-Wp Wn0

n0 exp(qVja / kT)

p0 exp(qVja / kT)

p=NAn=ND

n(x)

p(x)

x

n0

p0

Figure 2.4 Electron and hole carrier concentrations across the dark forward biased p-n junction. Arrows indicate the points where the boundary conditions 2.51 and 2.52 apply.

current from eqs. 2.66 and 2.68, as the sum of the minority carrier ing from the depletion edges into the quasineutral regions, which gives

o( ) ( ) ( ) exp 1jaja e p h n

qVi V i W i W i

kT⎡ ⎤⎛ ⎞

= − + = −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(2.81)

o o

oe h

e h

qD n qD pi

L L= + . (2.82)

Shockley’s current–voltage characteristic eq. 2.38 with eq. 2.81 and . 2.8, 2.9 and 2.12, the saturation current density of the p-n junction can

2o .e h

ie A h D

D Di qn

L N L N⎛ ⎞

= +⎜⎝ ⎠

⎟ (2.83)


21

2.3.5 The illuminated I–V curve To find the carrier concentration under illumination we use the superposition principle (Section 2.3.3). The electron concentration in the illuminated p-type quasi-neutral region is found by solving eqs. 2.55 and 2.56, where the position-dependent photogeneration rate ge(x) has now to be considered. For mathematical simplicity the case of spatially homogeneous generation g of electron hole pairs is assumed (see Green, 1982). This means we assume weakly absorbing material so photogeneration of charge carriers does not fall off with x. This assumption simplifies the treatment and does not alter the essential conclusions. From the superposition principle we obtain

( )Lt o2 Lt

2 2 0 ( )ee p

e

D n nd nD gdx L

−− + = ≤x W− (2.84)

and ph Lt Dn n n= − k . (2.85) Substracting eq. 2.58 from eq. 2.84 gives

ph2 ph

2 2 0 ( )ee

e

D nd nD gdx L

− + = ≤ − px W . (2.86)

The photogenerated charge carrier density at the edge of the space-charge region

is considered to be negligible, and therefore . (2.87) ph ( ) 0pn W− =

This approximation is discussed in Section 2.5.2. Assuming no recombination loss at the surface of the p-type region the boundary condition eq. 2.62 holds also for nph, i.e.

ph

0xdndx →−∞ = . (2.88)

The general solution to eq. 2.86 is


22

ph exp expee e

x xn g C DL L

τ⎛ ⎞ ⎛ ⎞

= + + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2.89)

where the constants C and D can be determined from the boundary conditions. Using eq. 2.87 we obtain

exp exp 0p pe

e e

W Wg C D

L Lτ

−⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= . (2.90)

The boundary condition eq. 2.88 implies that D = 0 for a physically meaningful solution; thus

exp pe

e

WC g

Lτ

⎛ ⎞= − ⎜

⎝ ⎠⎟ (2.91)

and the photogenerated carrier density in the p-QNR is found as

ph( ) 1 exp pe

e

W xn x g

Lτ⎧ + ⎫⎛ ⎞⎪ ⎪= −⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

. (2.92)

Substituting eqs. 2.63, 2.64 and 2.92 in eq. 2.85 we obtain the electron

concentration in the p-QNR under illumination as

Lt o oexp exp 1 ( )p jae e

e

W x qVn n g n g x W

L kTτ τ

⎡ ⎤⎡ + ⎤ ⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= + + − − ≤ −⎢ ⎥⎢ ⎥ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎩ ⎭⎣ ⎦

p . (2.93)

The diffusive electron flow in the p-type quasineutral region follows by

differentiating eq. 2.93 with respect to x as

o( ) exp exp 1 exp ( )p ja pee e p

e e e

W x qV W xDi x q n qgL x W

L L kT L⎧ + ⎫⎧ ⎫ +⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪⎪ ⎪= − −⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎩ ⎭

≤ − (2.94)

with a similar expression for the diffusive hole current flow in the n-type quasineutral region


23

o( ) exp exp 1 exp ( )jah n nh h n

h h h

qVD W x W xi x qp qgL x W

L L kT L⎧ ⎫⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎪ ⎪⎪ ⎪= − −⎨ ⎬⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎩ ⎭

≥

W

. (2.95)

The total current density is found by adding the diffusive minority carrier flow at

the edges of the depletion regions as in Section 2.3.4. Including the change in current density qgW arising from the generation of electron-hole pairs in the depletion region of width W = Wn + Wp we obtain . (2.96) total ( ) ( )e p h ni i W i W q g= − + −

Substituting eqs. 2.94 and 2.95 in eq. 2.96 yields the ideal diode equation under illumination

Lttotal o exp 1jaqV

i i ikT

⎡ ⎤⎛ ⎞= −⎢ ⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦−⎥ (2.97)

where io is the saturation current density eq. 2.82 and [ ]Lt

e hi q g W L L= + + (2.98) is the light-generated current.

The most important simplifying assumptions used to obtain the ideal diode equation are:

• the exhaustion region approximation. • low-injection conditions. • the superposition principle. • the cell is wide enough that no surface recombination has to be taken into

account, i.e. and . ne ph• spacially homogeneous photogeneration of electron hole pairs.

HL << HL <<

• no parasitic losses due to series and parallel resistances. With these assumptions we were able to derive simple expressions for the charge

carrier distribution in an illuminated p-n junction. The saturation current density was found from the diffusive current flow at the edges of the QNRs.


24

2.4 Real diode characteristics Though the ideal diode equation 2.97 takes account of the basic physical principles of charge carrier transport that can be found in real devices, silicon solar cells in general can exhibit strong deviations from this ideal characteristic. In the following sections the parameters that characterise a ‘real’ solar cell are discussed. 2.4.1 Solar cell parameters For a practical analysis of solar cell performance the dark and light I–V characteristics shown in Fig. 2.5 are investigated. Prominent parameters of the illuminated I–V characteristics include the open-circuit voltage Voc, the short-circuit current density isc, the maximum power voltage Vmp and the current density for maximum power imp. The maximum power Pmp is given by the product Vmp imp. The efficiency of the cell at the maximum power point is the ratio of output power Pmp to the inci

mp

o

PE

η =

where ηfill is the fill factor

i.e. the ratio of the two rectangular a

By setting itotal in eq. 2.97 to zeroand the saturation current density io

dark

illuminated

V

isc

Voc

imp

i

Vmp Figure 2.5 Typical dark and illuminated solar cell I–V curve. 9 shows the maximum power point.

dent solar power Eo

mp mp oc sc fill

o o

V i V iE E

η= = (2.99)

mp mpfill

sc oc

,i Vi V

η = (2.100)

reas shown in Fig. 2.5. , the relation between the open-circuit voltage Voc is found as


25

Lt

oc tho

ln 1 iV Vi

⎛ ⎞= +⎜

⎝ ⎠⎟ . (2.101)

2.4.2 Assumptions regarding the majority carrier concentration In eqs. 2.8 and 2.9 it was assumed that the majority carrier concentration was equal to the density of the dopants throughout the QNRs. For this to be so, low-injection conditions have to be fulfilled: that is, the perturbation of the majority carrier concentration due to light generation or carrier injection is small and therefore (in n-type material) (2.102) +≈ DNn (in p-type material). (2.103) −≈ ANp In other words, the density of free charge carriers is equal to the density of ionised dopants. This approximation is not valid for illumination by concentrated sunlight or high forward bias voltages.

If, in addition, the energy level of the dopant lies near the relevant band edge, all the dopants will be ionised at room temperature, and in this case (2.104) +≈≈ DD NNn . (2.105) A Ap N N −≈ ≈ 2.4.3 Charge carrier lifetime For the superposition principle to be valid, the minority carrier lifetimes in eqs. 2.53 and 2.54 must be constant, as noted in Section 2.3.3. These lifetimes are determined by the dominant recombination mechanism. For an n-type semiconductor, the upper bound on the hole lifetime, namely the radiative recombination lifetime , is given by

rhτ

1 D

1rh rk N

τ = (2.106)

where is the radiative recombination rate constant. 1

Often the minority carrier lifetime is determined by recombination through traps. For the simple model of recombination centres of a single energy, the Shockley–

rk


26

Read–Hall (SRH) hole carrier lifetime expression in an n-type semiconductor is given by

TT

T

1h

hk Nτ = (2.107)

where NT is the trap concentration and is the hole capture rate constant. T

hkIn other cases, minority carrier lifetimes are determined by Auger recombination.

In this case, the hole lifetime for low-injection conditions is given by

AA 2

D

1h

ck Nτ = (2.108)

where is the Auger band-to-band recombination rate constant. A

ckShockley–Read–Hall and Auger recombination are the two dominant

recombination mechanisms in Si solar cells. The minority carrier lifetime of holes in n-doped silicon is then given by

hτ

T

1 1 1

h h hτ τ τ= + A . (2.109)

The -dependence of the Auger lifetime causes the minimum carrier lifetime

to be smaller in heavily doped regions (e.g. the emitter) than in lightly doped regions (e.g. the base). Also, the doping dependence of leads to a position-dependent lifetime in the emitter region of cells with a doping profile. The saturation current density of emitters with position-dependent carrier lifetimes can be calculated analytically as demonstrated in Section 2.4.8.

2D1/ N

Ahτ

If low-injection conditions are not fulfilled, the recombination rates for electrons, re and rh, depend on both the electron and hole concentrations, i.e. the superposition principle does not apply. This is the case in solar cells working under illumination by concentrated sunlight. The coupled set of differential equations 2.1, 2.6 and 2.7 have then to be solved numerically, as described in Section 2.5.


27

2.4.4 Surface recombination In the discussion of the dark diode characteristics in Section 2.3.4 it was assumed that there was no recombination loss at the cell surfaces. A boundary condition that includes the surface recombination of electrons at the surface of the p-type region (at px H= − ) can be expressed as

( )o

p p

e ex H x

dnD S n ndx =− =−

= −H

(2.110)

where Se is the surface recombination velocity of electrons. Introducing a similar boundary condition for the recombination of holes at the surface of the n-type region and performing the same steps as in Section 2.3.4 gives a modified expression for the saturation current density (compare with eq. 2.82)

o o

o .e p h np

e h

qD n qD pi

L L⎛ ⎞

= Ξ + Ξ⎜⎜⎝ ⎠

n ⎟⎟ (2.111)

The geometric factor Ξn is given by

sinh cosh

cosh sinh

n h h n

h h hn

n h h n

h h h

Q S L QL D LQ S L QL D L

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠Ξ =⎛ ⎞ ⎛ ⎞

+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2.112)

where n n nQ H is the width of the n-doped quasineutral region, Sh is the surface recombination velocity of holes at the surface of the p-doped region, Lh is the hole diffusion length and Dh is the hole diffusion constant as introduced before.

W= −

A similar expression can be found for the geometric factor p. The effect of surface recombination on the saturation current density can be studied by plotting the geometric factor as a function of Q/L for various values of SL/D, as shown in Fig. 2.6. For L < Q, (i.e. Q/L > 1) the surface does not contribute to the saturation current density io.

Ξ

A thin quasineutral region of width Q in combination with a high surface recombination velocity results in a high saturation current density, and consequently a low open-circuit voltage (see eq. 2.101). On the other hand, good surface passivation


28

can reduce io considerably at low values of Q/L. This is particularly important for thin solar cells.

0.01 0.1 1 100.01

0.1

1

10

100

0

1/5

1/20

1/2

21

5

20

8

SL/Dge

omet

ric fa

ctor

Ξ

Q/L

Figure 2.6 Geometric factor as a function of the ratio of the width of the quasineutral region Qto the minority carrier diffusion length L. Each curve represents a fixed ratio SL/D.

2.4.5 Series and shunt resistance

In the real device a series resistance Rs and a shunt (or parallel) resistance Rp may be present. Accounting for series and shunt resistance losses, the illuminated I–V characteristic eq. 2.97 becomes

( ) Lt

o( ) exp 1s s

p

q V iR V iRi V i i

kT R⎡ ⎤−⎛ ⎞ −

= − +⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

− (2.113)

where V is the voltage at the cell terminals.

The series resistance of a solar cell is composed of the resistance of the metal grid, the contact resistance and the base and emitter sheet resistances. Shunt conductive losses can arise from imperfections on the device surface as well as leakage currents across the edge of the cell.

The influence of the series and shunt components on the illuminated and dark I–V characteristics, according to eq. 2.113, is plotted in Fig. 2.7. The dark I–V curves in the lower half are shown on a logarithmic scale; the ideal I–V characteristic eq. 2.97 gives a straight line on this scale. For high current densities the deviation from


29

0.1 0.2 0.3 0.4 0.5 0.6 0.7-50

-40

-30

-20

-10

0

(d)(c)

(b)(a)

013

510

Rp = 20 Ωcm2

i [m

A c

m-2

]

voltage [V]

0.1 0.2 0.3 0.4 0.5 0.6 0.71E-9

1E-7

1E-5

1E-3

0.1

10

106105

104Rp = 103 Ωcm2

5

20Rs = 0 Ωcm2

i [m

A c

m-2

]

voltage [V]

0.1 0.2 0.3 0.4 0.5 0.6 0.7-50

-40

-30

-20

-10

0

i [m

A c

m-2

] voltage [V]

Rp = 20 Ωcm2

30

100

106

0.1 0.2 0.3 0.4 0.5 0.6 0.71E-9

1E-7

1E-5

1E-3

0.1

10

i [m

A c

m-2

]

voltage [V]

Figure 2.7 Influence of series resistance Rs and parallel resistance Rp on the I–V characteristic: (a) illuminated, Rs varied; (b) illuminated, Rp varied; (c) dark, Rs varied; (d) dark, Rp varied.

straight line behaviour is caused by the series resistance, while low shunt resistances cause deviations from the ideal I–V characteristic for small current densities. 2.4.6 Non-ideal dark current components It has so far been assumed that there is no recombination loss in the depletion region. However, in real solar cells, depletion-region recombination represents a substantial loss mechanism. An analytical expression for the ‘space-charge layer recombination current’ was first given by Sah et al. (1957), for the simplified case of a single recombination centre located within the forbidden gap. Traps located in the vicinity of the gap give a dominating contribution to the Shockley–Read–Hall recombination rate. It was further assumed that the recombination rate is constant across the space-charge region. The resulting recombination rate can be expressed as a recombination current iDR in the depletion region


30

DR o2 exp 12qVi ikT

⎡ ⎤⎛ ⎞= ⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦. (2.114)

Adding this space-charge layer recombination current to eq. 2.113 gives the ‘two

diode model’ expression

( ) ( ) Lto1 o2( ) exp 1 exp 1 .

2s s s

p

q V iR q V iR V iRi V i i i

kT kT R⎡ ⎤ ⎡ ⎤− −⎛ ⎞ ⎛ ⎞ −

= − + − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

− (2.115)

Sah also derived an analytical expression for iDR with less rigid assumptions as stated above (see Fahrenbruch, 1983); for instance, Sah’s expression is still valid for unequal electron and hole lifetimes. In general, the recombination current in the depletion region is a function of the applied voltage and is not necessarily of monoexponential form.

A non-ideal I–V characteristic can further be caused by an injection-level-dependent surface recombination current. In the boundary condition eq. 2.110 the surface recombination velocity S was assumed to be independent of the minority carrier concentration. However, Aberle et al. (1993) found that the origin of the non-ideal diode behaviour of high-efficiency silicon solar cells is a surface recombination velocity at the rear Si/SiO2 interface that strongly depends on the minority carrier concentration.

In practice, most measured I–V curves of solar cells can be approximated by several exponential regions in the dark forward I–V characteristic revealing the presence of several dark current components. One can take this behaviour into account by empirically introducing the ideality factors β1 and β2 so that eq. 2.115 can be expressed as

( ) ( ) Lto1 o2

1 2

( ) exp 1 exp 1 .s s s

p

q V iR q V iR V iRi V i i i

kT kT Rβ β⎡ ⎤⎛ ⎞ ⎡ ⎤− −⎛ ⎞ −

= − + − +⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦− (2.116)

Fig. 2.8 shows the equivalent circuit of eq. 2.116, consisting of two diodes with different ideality factors β1 and β2, the light-generated current iLt and the series and parallel resistances Rs and Rp.

By varying io1, io2 β1, β2, Rp, and Rs, a wide range of experimentally observed I–V curves can be fitted. As outlined above, different recombination components may be lumped in numerical fit values of the model parameters io1, io2 β1 and β2 of eq. 2.116.


31

Therefore a unique assignment of fit results corresponding to the physical origin of one recombination mechanism is in general not possible.

Vja VRp

Rs

i01 i02

β1 β2

iLt

Figure 2.8 Equivalent circuit of a solar cell described by the two-diode model, eq. 2.116.

2.4.7 Photogeneration In Section 2.3.5 an expression for the I–V curve of a p-n junction under illumination was derived under the assumption of a spatially homogeneous generation rate. This is rather unrealistic, since good solar cells must absorb the incident light strongly. For illumination with the solar spectrum the spatially dependent generation rate of electron-hole pairs can be calculated with respect to a wavelength-dependent absorption coefficient ∀(8).

A photon of wavelength 8 penetrating the surface of the solar cell is absorbed with the probability ∀(8) per unit length. Each absorbed photon creates one electron-hole pair.2 Thus the generation rate g8(x) of electron-hole pairs per unit volume with respect to wavelength 8 is given by

( ) ( ) ( )g x jλ α λ= xλ (2.117) where j8(x) is the spectral photon flux at depth x. For steady-state conditions the continuity equation for photons gives (2.118) div ( )j g xλ λ= − or

d

( ) ( )dj

j xxλ

λα λ= − . (2.119)

2 For photon energies exceeding twice the bandgap energy Ug it is possible to create two electron-hole pairs by impact ionisation. However, the number of photons with energies exceeding 2Ug can be neglected for silicon solar cells illuminated by terrestrial sunlight.


32

This leads to an exponential decay of the photon flux [ xjxj )(exp)0()( ]λαλλ −= . (2.120) From eq. 2.117 the generation rate is found as [ ]( ) (0) ( ) exp ( )g x j xλ λ α λ α λ= − . (2.121) The total generation rate of electron-hole pairs per unit volume can be found by integrating eq. 2.121 over wavelength ( ) ( )dg x g xλ λ= ∫ . (2.122) The electron and hole current densities can still be calculated analytically for the generation rate expression eq. 2.122, following the same steps as described in Section 2.3. The full current expressions can be found in Sze (1981). 2.4.8 Accounting for position-dependent doping profiles We have so far assumed the dopant concentrations to be constant through the n- and p-type regions, but this is an oversimplification. The emitter of a silicon solar cell has a spatially dependent donor profile with an error or gaussian function shape since it is processed by diffusion of phosphorus atoms into a p- (boron) doped wafer.

Where the dopant concentr-ation varies with depth x, the assumption of a constant mobility (Section 2.3) is not in general justified. For example, ionised impurities in the semiconductor cause carrier scattering, which significantly affects the mobility. In more heavily doped regions of the semiconductor the average time between collisions of the charge carriers with ionised impurity atoms decreases and thus the

p-type

log |ND-NA|

n-type

-Hp -Wpx=0

Wn Hn

x

Figure 2.9 Schematic doping profile of a solar cell withan n-type emitter diffused into a p-doped substrate.


33

mobility decreases. Additionally, the Auger lifetime in emitters with a non-uniform doping profile becomes position-dependent due to eq. 2.108.

An analytical approach to the calculation of the emitter saturation current density in semiconductors with position-dependent doping profiles has been given by Park et al. (1986). Consider a p-n junction formed by diffusing n-type impurities of concentration ND(x) into a uniform p-type substrate, as illustrated in Fig. 2.9.

To find the emitter saturation current density, the hole recombination current density in the n-QNR is calculated. Using eqs. 2.5 and 2.54, the hole continuity equation for dark conditions can be written as

od ( ) ( )( )

d (h

hh

i p x p xqr x qx xτ

−= − = −)

. (2.123)

At thermal equilibrium the hole drift current and hole diffusion current balance each

other. Thus, a quasi-electric field for the diffusion component can be expressed as

d ln ( )d D

kT N xq x

=E . (2.124)

Substituting this hole quasi-field into eq. 2.3 gives

d( ) ( ) ln ( ) .d dh h D

pi x qD x p N xx x

d⎡ ⎤= − +⎢ ⎥⎣ ⎦ (2.125)

The first term on the right side of eq. 2.125 is the drift current and the second term is the diffusion current. Note that the effective diffusion constant Dh(x) of the holes is now position-dependent

D( ) ( ) .(h hkTD x u N xq

= ) (2.126)

The boundary condition for the hole concentration at the edge of the space-charge

region is given by eq. 2.52. The boundary condition for the recombination of holes at the emitter surface at x = Hn is

( )o( )n n

h n h hx Hi H qS p p qS p

== − ≈

x H=. (2.127)

Integrating eqs. 2.125 and 2.123 from Wn to x gives


34

oo

1 1( ) ( ) 1 d ´ ( )( ) ( )

n

x

hW

p x p x x i xq D x p x

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦∫ (2.128)

and

( )( ) ( ) d ´( )n

nh

x

W

p xi x i W q xxτ

= − ∫ (2.129)

where i´(x) and p´(x) are the normalised hole current and hole density expressions

( )

( )exp ( / ) 1

h

ja

i xi x

qV kT=

− (2.130)

o( ) ( )( )

exp ( / ) 1ja

p x p xp xqV kT

−=−

. (2.131)

The second term on the right hand side of eq. 2.128 contains the current flow in

the emitter region, which causes the hole concentration to differ from the local equilibrium value po(x). Substituting the hole current density expression eq. 2.129 into eq. 2.128 gives an integral equation for the hole density. Successive substitution of eqs. 2.128 and 2.129 into each other yields an iterative scheme for the calculation of the hole carrier distribution. Depending on the number of iteration steps, different approximation orders for p´(x) can be gained. The innermost integral of the approximation is taken over the equilibrium hole carrier distribution, which can be found from eqs. 2.45 and 2.104 as

2

o

D

( )( )in

p xN x

= . (2.132)

The emitter saturation current density ioe is composed of the surface and bulk

components . (2.133) o bulk( ) ( ) ´e n ni i W i H i= = +

Using eq. 2.127 and the calculated hole density p´(x) we find

o( )( ) d( )

n

n

e h nh

H

W

p xi qS p H qxτ

= + ∫ x . (2.134)


35

The integral in eq. 2.134 is taken over the volume recombination rate in the quasineutral emitter region and represents the recombination current in this region. Here the minority carrier lifetime )(xhτ is given by eq. 2.109.

Equation 2.134 can be solved for different combinations of surface recombination velocities Sh and doping profiles ND(x) as demonstrated by King et al. (1990). From the emitter saturation current density a maximum open-circuit voltage can be obtained by applying eq. 2.101.

The surface recombination velocity Sh depends on processing conditions. For high Sh a low minority carrier concentration at the surface p(Hn) is advantageous. This can be achieved with a heavily doped emitter so that the first term in eq. 2.134 is minimised. Figure 2.10b shows an example for an unpassivated emitter surface with Sh =106 cm s–1 (Preu et al., 1998). The highest open-circuit voltage is found for a deep doping profile with a high peak doping concentration. Here, gaussian-shaped doping profiles were assumed.

The second term in eq. 2.134 represents the bulk recombination contribution to the emitter saturation current density. Varying an emitter profile, an optimum profile is found if an increase in the bulk recombination contribution balances the decrease of the surface recombination component. An example can be found in Fig. 2.10a for a surface recombination velocity of Sh = 104 cm s–1. In this case, optimum open circuit voltages are found for peak doping concentrations around 5 × 1019 cm–3.


36

100 200 300 400 500

b)a)

1019 1019

10201020

Voc,max [mV] for Sh = 104 cm/s

665 - 670 661 - 666 657 - 661 653 - 657 648 - 653 644 - 648 639 - 644 635 - 639

junction depth [nm]

peak

dop

ing

conc

entra

tion

[cm

-3]

100 200 300 400 500

Voc,max [mV] for Sh = 106 cm/s

623 - 630 616 - 623 609 - 616 603 - 609 596 - 603 589 - 596 582 - 589 575 - 582

junction depth [nm]

peak

dop

ing

conc

entra

tion

[cm

-3]

Figure 2.10 Maximum achievable open-circuit voltages for emitters with different gaussian doping profile shapes. The graphs were calculated with the Park model discussed in this section. (a) surface recombination velocity Sh = 104 cm s-1; (b) Sh = 106 cm s–1.

More detailed models require numerical calculations, as described in the next section.


37

2.5 Numerical solar cell modelling So far we have treated the solar cell analytically in one dimension. In practice more sophisticated numerical models must be used to describe a ‘real’ solar cell and to allow for quantitative comparison of simulation results with solar cell measurements. More accurate models require two- or three-dimensional solar cell simulations, particularly with regard to the lateral current density flow, which may be substantial, to describe most of the high efficiency solar cell structures.

Numerical simulation techniques have been extensively used to quantify optical and electrical losses for many solar cell structures. Aberle et al. (1994) presented a numerical optimisation study of high-efficiency silicon solar cells with rear point-contact patterns. This cell type shows low minority carrier recombination losses at the rear surface due to the small rear-surface metallisation fraction. This 2D parameter study took lateral current components due to the point contacts into account.

Recombination losses at the cell perimeter of solar cells which either stay embedded in the wafer or are sawn from the wafer can also be described with numerical methods, as described in Section 2.5.3 (Altermatt et al., 1996a).

Distributed resistive losses in the semiconductor material and in the front metal grid of high-efficiency silicon solar cells have been investigated using a combination of device simulation and circuit simulation. Such a combination allows the simulation of complete solar cells, instead of the usually restricted simulation domain, which is kept to a geometrically irreducible minimum. With the inclusion of the whole device domain, the predictions became so precise as to contribute significantly to an increase in the world record efficiency of silicon solar cells (Altermatt et al., 1996b).

Optical and electrical losses of silicon-on-insulator thin-film solar cells with interdigitated front contacts were analysed by Schumacher et al. (1997). These cells have two- and three-dimensional current flow patterns as well as a textured front surface for light trapping. Their performance was modelled by a combination of 3D optical ray tracing with 2D electrical device simulation.

Using ray-tracing programs, the spatially dependent photogeneration rate of cells with textured surfaces can be calculated numerically. For example, mechanically textured silicon solar cells were investigated by means of ray-tracing simulation by Zechner et al. (1998).

Another solar cell structure with pronounced lateral current flow is the rear- contact cell of Swanson (1986). As both the emitter and the base of this cell are contacted at the back side, there are no front-surface shading losses. Furthermore, this cell type allows for simplified module assembly because of the single-sided metallisation. We discuss the numerical simulation of rear-contacted silicon cells in detail in Section 2.5.3 (see also Dicker et al., 1998).


38

To describe current transport in metal–insulator–semiconductor silicon solar cells, a quantum mechanical model describing the tunnelling of charge carriers through the tunnel insulator of the metal-insulator contact is needed. The resulting set of equations cannot be solved analytically without making severe simplifying assumptions. A detailed numerical model utilising two-dimensional device simulation and circuit simulation was presented by Kuhlmann et al. (1999).

2.5.1 Solving the semiconductor device equations numerically For the numerical calculation of solar cell performance, the device equations 2.1, 2.6 and 2.7 are solved at discrete mesh-points in space. A comprehensive treatment of discretisation methods for semiconductor device simulation can be found in Selberherr (1984). In order to solve the Poisson equation and the electron and hole continuity equations numerically, we discuss the box scheme as described by Bürgler (1990) and Heiser (1991). The device volume is discretised, that is, a symmetry element of the device is divided into boxes.

Figure 2.20 shows an example discretisation of a symmetry element of a rear contact cell. A simple discretisation mesh can be constructed out of rectangular boxes. However, a more efficient approach that consumes less computer resources is to allow for triangular boxes. Such triangular boxes can be seen around the enlarged n-contact in Fig. 2.20. A single box Ωi of a two-dimensional discretis-ation mesh for one node i (mesh-point) of the symmetry element is shown in Fig. 2.11. The boxes have to be constructed in such a way as to cover the whole symmetry element. Therefore, box boundaries (dashed lines) are chosen to lie on the perpendicular bisectors of lines between neighbouring nodes.

Ωi

dij

lij

vertex i

vertex j

lij

Figure 2.11 The graph shows part of a discretisation mesh for the electrical device simulation; solid lines are drawn between neighbouring mesh-points. The shaded area represents the volume Ωi of a single box for a triangular discretisation mesh in two dimensions.


39

In the following we will derive the discretised form of the Poisson equation, eq. 2.1. For this purpose, we will use the fact that an electrostatic electric field E is a conservative field.3

More generally, the partial differential equations treated here can be stated as a conservation law

)()( xSx =Γ∇ (2.135) where )(xΓ is a vector field and )(xS is a position-dependent scalar field. To obtain an equation for vertex i, eq. 2.135 is integrated over the volume of box Ωi using Gauss’s theorem [ ] 0)()()()()( =−Γ=−Γ∇ ∫∫∫

ΩΩ∂Ω iii

dVxSxndxdVxSx . (2.136)

The first integral on the right hand side of eq. 2.136 is taken over the boundary i of the box, i.e.

Ω∂)(xn denotes the normal vector of the box boundary. The discretised

form of eq. 2.136 can be written as ∑

≠=−Γ

ijiiijij VSd 0 (2.137)

where Γij is the projection of the vector field )(xΓ onto the edge ijl from node i to node j (Fig. 2.11). Here dij denotes the length of the perpendicular bisector on this edge. Si is the value of the scalar field )(xS at node i. In two dimensions Vi is the area of box Σi, in three dimensions it is the box volume. The sum in eq. 2.137 extends over all nodes j that neighbour node i. Applying eq. 2.137 to the Poisson equation 2.1 yields 0ij ij i i

j id Vρ

≠− =∑E . (2.138)

The electric field E in eq. 2.138 can be expressed as the differential quotient ( ) /ij iji j lφ φ= −E , thus

0)( =+−−−≡ ∑≠

iiiiij

ijij

iji NnpV

ld

F φφ (2.139)

3 This means that Gauss’s theorem is valid, and the surface integral of E over a closed surface is equal to the volume integral over the total charges enclosed by the surface.


40

where jiij φφφ −= is the potential difference along ijl .

A thorough derivation of the discretised electron and hole continuity equations 2.4 and 2.5 is given in a thesis by Heiser (1991). The discretised continuity equations are

( )( ) ( ) 0ijn ni ij j ji i ij i i

j i ij

dF u n B n B V r g

lφ φ

≠

⎡ ⎤≡ − − + − =⎣ ⎦∑ i (2.140)

( )( ) ( ) 0ijp pi ij j ij i ji i i

j i ij

dF u p B p B V r g

lφ φ

≠

⎡ ⎤≡ − − + − =⎣ ⎦∑ i (2.141)

where B is the Bernoulli function

( )(exp ) 1

xB xx

=−

. (2.142)

The mobility is denoted as uij, and is assumed to be constant on the box edge perpendicular to ijl .

To solve the discretised differential equations 2.139–2.141 with a computer, the physical entities have to be scaled. For example, carrier concentrations are scaled by the intrinsic carrier concentration, the electrostatic potentialφ is scaled by the thermal voltage Vth, and the electric field is scaled by Vth /LD, where LD is the Debye length (eq. 2.18). This scaling is essential for the numerical calculation because the potential typically varies by one or two orders of magnitude whereas the carrier densities vary over ten to twenty orders of magnitude.

For the N nodes of the discretisation mesh we obtain 3N partial differential equations from eqs. 2.139–2.141 with the solution variables φ , n and p. These differential equations can be abbreviated as

(2.143)

0),,(

0),,(

0),,(

=

=

=

pnF

pnF

pnF

pi

ni

i

φ

φ

φφ

These equations can be solved by the Newton method. Given the nonlinear system

of equations 2.143 written as

0)( =zF (2.144)


41

the Newton procedure iteratively computes a new solution (2.145) k

ikk

iki zszz δ+=+1

from the old one zk. The update k ks zδ is found as the solution of the equation

)()( ki

kj

k

jkj

ki zFzszzF −=

∂∂∑ δ . (2.146)

To achieve numerical convergence of the Newton iteration, a damping factor

which is determined in each iteration step from the last successful one is used in eq. 2.146.

10 ≤< ks

Equations 2.137–2.146 can be stated for one, two and three dimensions as discussed in ISE-TCAD (1997). An example of the one-dimensional solution of eq. 2.144 is given in the next section, and the two-dimensional case is treated in Section 2.5.3.

In modern device simulators, different solution methods can be chosen. It is possible to solve all 3N partial differential equations 2.144 together (‘coupled solution’). Another method with less memory demand and faster convergence for low and intermediate injection conditions is the ‘plug-in method’: First the N Poisson equations are solved. The resulting potential 0),,( =pnFi φφ )( ixφ is inserted into the electron continuity equations , and iteration by the Newton method then yields a new electron density distribution

0),,( =pnF ni φ

)( ixn . Both )( ixφ and )( ixn are used to solve the hole continuity equation . 0),,( =pnF p

i φ 2.5.2 A sample cell As an example of how eq. 2.144 may be solved numerically, we will use it to model a p-n junction solar cell with diffused doping profiles in one dimension. The cell model is shown in Fig. 2.12. The base material consists of homogeneously doped p-type silicon of 250 µm thickness. In order to reduce the surface recombination velocity at the rear surface a back-surface field is built into the cell. The doping profiles for the emitter and the back surface field are of gaussian shape (Fig. 2.12b).


42

tmhc2a

wB

icdscat

n-type emitter

light

p-type baseNA = const.

back surface field

Sfront

Sback

antireflection coating

RS

RPVa

ND- NA

x

xj

(a) (b)

Figure 2.12 One-dimensional model of a planar silicon solar cell with gaussian doping profiles: (a)a back surface field and the series and shunt resistances are accounted for; (b) doping densitydistribution. For this simulation a doping profile of gaussian shape with a peak doping density of5 × 1018 cm–3 and a junction depth of 1.4 µm was assumed. A base diffusion length of Lb = 350 µmand a cell thickness of 250 µm was accounted for; the surface recombination velocity was set toS = 800 cm s–1 at the front side and S = 106 cm s–1 at the rear side. The calculated solar cell outputparameters are isc = 37.6 mA cm–2, Voc = 626 mV, ηfill = 73 % and a conversion efficiency ofη = 17.2 %. One-Sun (AM 1.5) illumination was assumed.

Most of the restrictions of the analytical models can be overcome by solving the

ransport equations numerically. The doping dependence of the charge carrier obility mentioned in Section 2.4.8 can be included. Also bandgap narrowing in

eavily doped regions of the cell, arising from many-body interactions of the charge arriers, can be incorporated. Moreover, the coupled set of differential equations .144 can be solved for intermediate- and high-injection conditions, whereas nalytical solutions can be found only for low-injection conditions.

The simulations presented in this Section were performed with the program PC1D, hich is commonly used for one-dimensional solar cell simulations (Rover, 1985; asore, 1997; Basore, 1988).

Figure 2.13 shows the simulated space-charge density, the electric field and the nner potential across the p-n junction for the cell of Fig. 2.12. The curves were alculated for three terminal voltages: V = 0, V = Vmpp, and V = Voc. Note the ifference between the results shown in Figs. 2.3 and 2.13. The numerically simulated pace-charge density distribution is smooth in comparison to the abrupt analytical ase. The highest space-charge density is found for short-circuit conditions. Applying forward bias to the p-n junction injects carriers into the depletion region and reduces he space-charge density.


43

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5(a)

dφ /dx = - (x)

illuminateddark

short circuit maximum power open circuit

V=0 V=Vmp V=Voc

d /dx = ρ/ε0εs

V=Vmp

V=0

spac

e-ch

arge

den

sity

ρ [m

C c

m-3]

-25

-20

-15

-10

-5

0

(b)

V=Vmp

V=0

elec

tric

field

[k

V cm

-1]

-1.0 -0.5 0.0 0.5 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6 (c)

n-DRp-DR n-QNRp-QNR

x=Hn

x=-Wp x=Wn

x [µm]

ele

ctro

stat

ic p

oten

tial φ

[V]

Figure 2.13 (a) space-charge density )(xρ ; (b) electric field )(xε ; (c) inner potential

)(xφ across the p-n junction of the sample cell drawn in Fig. 2.12. Compare with the idealised Schottky exhaustion layer case shown in Fig. 2.3.


44

The electric field in the space-charge region is also reduced by forward bias, as can be seen in Fig. 2.13. In contrast to the abrupt-doping case, the electric field does not vanish in the n-type quasineutral region. The net force on an electron is given by

[ ]ln ( )

( )e

d n xF q x kT

dx= +E (2.147)

where the first term of eq. 2.147 is the drift force and the second term is the diffusive force. For equilibrium conditions the net force on an electron is zero. Substituting the electron density n(x) by the gaussian doping profile

2

2

( )( ) ( ) exp

2n

D D nx H

N x N Hσ

⎡ ⎤−= −⎢ ⎥

⎣ ⎦ (2.148)

where σ is the standard deviation and setting Fe = 0 yields

th 2( ) nx Hx V

σ

−=E . (2.149)

Thus, a doping gradient of gaussian shape produces the linear dependence of the

electric field on position shown in the n-QNR of Fig. 2.13. Figure 2.14 shows the numerically calculated band-edge energies, quasi-Fermi

potentials, carrier densities and current densities for short-circuit conditions. Cell parameters are listed in the caption of Fig. 2.12. The equilibrium carrier densities shown as black curves in Fig. 2.15b can be compared with Fig. 2.4. Again, the electron majority concentration in the quasineutral region is given by the doping profile. The small electron minority concentration gradient in the p-type quasineutral region is due to the long diffusion length of Lb = 350 µm (see eq. 2.63), which is greater than the cell thickness of 250 µm. This is typical for high-performance c-Si cells.


45

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

ene

rgy

[eV]

~

~

(c)

(b)

(a)

dark

illum

inat

ed Conduction band Valence band Electron QFE Hole QFE

µe

µh

Uv

Uc

Short-circuit conditions

Conduction band Valence band Electron QFE Hole QFE

101

103

105

107

109

1011

1013

1015

1017

1019

illuminateddark

Electron density Hole density

n,dark

n,illuminated

p

car

rier d

ensi

ty [c

m–3

]


-250 -200 -150 -100 -50 0-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

illuminateddark

Electron current density Hole current density Total current density

Electron current density Hole current density Total current Density

ie

ih

itotal

x [µm]

cur

rent

den

sity

[A/c

m2 ]

Figure 2.14 A cut through the whole cell for short–circuit conditions. The black curves are calculated for dark conditions, the red curves for illuminated conditions. (a) band-edge energies and quasi-Fermi energies (QFE); (b) electron density and hole density; (c) electron current density, hole current density and total current density.


46

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

ene

rgy

[eV]

~

~

(b)

(a)

dark

illuminated



µh

µe

Uv

Uc

n-QNRn-DRp-DRp-QNRShort-circuit conditions


101

103

105

107

109

1011

1013

1015

1017

1019

car

rier d

ensi

ty [c

m–3

]

(c)

illuminateddark

p,dark

p,illuminated

n

n,dark

n,illuminated

p


-1.0 -0.5 0.0 0.5 1.0-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

x=Wnx= –Wp

illuminateddark

ih

ie

itotal


x [µm]

cur

rent

den

sity

[A/c

m2 ]


Figure 2.15 Enlargement of the emitter region for short–circuit conditions.


47

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

ene

rgy

[eV]

~

µe

(c)

(a)

(b)

(a) Conduction band Valence band Electron QFE Hole QFE


illuminateddark

µh

~

Uv

Uc

Open-circuit conditions

101

103

105

107

109

1011

1013

1015

1017

1019

car

rier d

ensi

ty [c

m–3

]

illuminateddark


n

p


-200 -150 -100 -50 0-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

illum

inat

eddark


total current for illumination

electron current for illumination

hole current for illumination


x [µm]

cur

rent

den

sity

[A c

m–2]

Figure 2.16 A cut through the whole cell is drawn for open-circuit conditions (V = Voc). The black curves are calculated for an applied voltage equal to Voc of the illuminated cell.


48

-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2

ene

rgy

[eV]

~

~

illuminated

dark


µh

µe

Uv

Uc

Open-circuit conditions


101

103

105

107

109

1011

1013

1015

1017

1019

illuminateddark


n

p

car

rier d

ensi

ty [c

m–3

]


-1.0 -0.5 0.0 0.5 1.0-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04(c)

(b)

p-QNR

illum

inat

eddark


x=Hn

ie

ihitotal

n-QNRn-DRp-DR

(a)

x=Wnx= –Wp


x [µm]

cur

rent

den

sity

[A c

m–2

]

Figure 2.17 Enlargement of the emitter region for open-circuit conditions.


49

To solve the differential equations for the carrier densities, it was assumed that the

photogenerated carrier density vanishes at the edge of the depletion region. In Section 2.3.5, this was expressed by the boundary condition (eq. 2.87). That is, in the analytical approximation the space-charge region serves as a perfect sink for minority carriers: minority carriers at one edge of the space charge region are transferred to the opposite edge where they emerge as majority carriers. We can check this boundary condition by looking at Fig. 2.15b, which shows the dark and illuminated charge carrier densities for short-circuit conditions. The numerical simulation reveals that the boundary condition 2.87 is an idealisation for short-circuit conditions. However, this assumption can be justified with a good degree of accuracy for open-circuit conditions; the differences between dark and illuminated minority carrier concentrations at the edges of the depletion region are small for V = Voc, as can be seen in Fig. 2.17b.

0)( =− pph Wn

The quasi-Fermi levels are separated by the simulated open-circuit voltage of 626 mV, as given by eq. 2.47. This can be seen in Fig. 2.17a. Under illumination the electron and hole currents ie and ih are mainly recombination currents at open-circuit conditions. The small total current density depicted by the red dotted line in Fig. 2.16c represents the current feeding the external shunt resistance Rp. 2.5.3 Multi-dimensional device simulation To analyse high-efficiency silicon solar cells the simulations need to be extended to two and three dimensions, which allows the quantification of all relevant loss mechanisms of the cell.

As an example of a 2D–simulation, we shall consider a special high-efficiency solar cell, namely a rear-contacted cell that is also a bifacial cell. The interdigitated contact pattern with alternating n- and p-contacts on the rear cell side is shown in Fig. 2.18a. Random upright pyramids on the illuminated front side of the cell minimise the external reflection losses. Nearly all of the rear side of the cell is covered by the collecting n-type emitter, mainly to prevent minority electrons from recombining at the rear side. A non-contacted emitter at the front surface provides a small front surface recombination velocity. This emitter is called ‘floating emitter’.


50

The simulation steps are set out in Fig. 2.19 (Dicker, 1998).

calcsymtextrandmonintetrac

reflantiangis c

geneacyiel

n++ p+

hν

p+

a) b)

n-contact pad

p-contact pad

floating emitter

contacted emitter

Figure 2.18 Structure of a rear-contact cell. (a) View of the back side of the rear-contact cellshowing the interdigitated contact pattern. (b) Enlargement; with the cell shown upside down, sothe light illuminates the cell from below.

The reflection and absorption properties of textured silicon solar cells can be ulated with a ray tracing program (Schumacher, 1995; Wagner, 1996). An optical metry element suitable for the ray-tracing simulation of the random pyramid ure of the rear contact cell is shown in Fig. 2.19a. Front-side illumination on the om pyramids is approximated by calculating the geometric path of a ochromatic light ray of random incidence on a single upright pyramid. The

nsity of each traced ray is reduced because of light absorption in silicon, and the ing procedure stops if the light intensity is lower than a certain threshold value. Ray tracing allows the calculation of the wavelength- and angle-dependent ection at each interface, including light interference at thin layers that simulate reflection coatings. Rough cell surfaces can be modelled by adding a random le offset to reflected rays. The angular distribution of a large number of traced rays omputed to fit a chosen scattering model for rough surfaces. Applying eq. 2.119 to each (oblique) ray path allows the calculation of the eration rate of charge carriers: the amount of monochromatic light absorbed in h layer shown in Fig. 2.19a is monitored. Solving the integral equation, eq. 2.122, ds the generation rate needed for the electrical device simulation.


51

In a numerical solar cell simulation the 3D–distribution of charge carrier generation arising from the surface texture cannot be handled; spatial resolution within a single pyramid of e.g. 5 µm base length is not compatible with the wide contact spacing of around 500 µm. Resolving the pyramids for the electrical simulation would increase the number of mesh points by a factor of at least 100. Thus, a 1D projection of the 3D generation function calculated by ray tracing is necessary. This is achieved by simply summing up the generated electron-hole pairs in layers of the same distance from the pyramid surfaces.

z

5 µ m

240

µm

RAYN

DESSIS

DESSIS

MDRAW

(a)

(b)

(d)

(c)

Device Simulation

IV

IV

IV

IVIV

IV IV

IV

IV IV

IV

IV

IV

IV

IV

+

-

IV IV

IV IV IV

Circ

uit S

imul

atio

n

Generate Discretisation Mesh

Ray Tracing

For electrical device simulation, the device volume has to be discretised. Figure 2.20 shows the discretisation mesh of a symmetry element of the rear-contact cell, including one n-to-p contact finger distance. A high density of mesh points in device regions with a steep doping gradient is needed for convergence of the iteration scheme discussed in Section 2.5.1. This is shown in Fig. 2.20 by the enlargement of the mesh around the n–contact region. In order to calculate a realistic short-circuit current, the density of mesh points at the illuminated front surface must be very high. Therefore, the discretisation mesh is adapted to the gradient of the generation function.

Figure 2.19 Simulation flow for thenumerical calculation of the optical andelectrical properties of a Si solar cell. (a) asingle upright pyramid of 5 µm width isthe symmetry element for the optical raytracing simulation; (b) the discretisationmesh is generated in the next step; (c) forthe electrical device simulation, theoptical generation profile and thediscretisation mesh calculated in theprevious two steps are used; (d) thecurrent-voltage characteristic from thedevice simulator is used as input for thecircuit simulation, taking into account thevoltage drops along the metal bus bar andthe metal fingers of the solar cell.

In the next simulation step, the box scheme as described in Section 2.5.1 is applied to the discretisation mesh. Solving the discretised differential equations 2.144 for different applied voltages as boundary conditions yields the I–V curve of one elementary diode, including one n-to-p contact finger distance of the device.


52

a) Optical (3D) b) Electrical (2D)

500 µm

n+

200 nm

240 µm

p+

10 µm

n-contact: 1.5 µm

5 µm

15 µm

top

rear

Figure 2.20 Optical and electrical discretisation of the cell volume. (a) The optical properties of random pyramids on the cell front are approximated by the optical symmetry element of a singleupright pyramid. A vertical slice through one pyramid is shown here. (b) The mesh for the electrical simulation was generated with the program MDRAW (ISE-TCAD, 1997). The high refinement in the top region is important for the calculation of the short-circuit current density because of the very high gradient of the optical generation rate in this cell region. For the electrical discretisation meshof the rear-contact cell shown here, the contact regions are resolved as a function of the dopinggradient.


53

Till now, device simulations performed with a symmetry element of the interior

cell region have been described, and resistive metal grid losses and recombination losses at the cell perimeter have been neglected. Recombination losses at the cell perimeter occur mainly because cutting the cells out of the wafer creates recombination centers at the cut face. However, perimeter recombination losses degrade the cell efficiency of cells which remain embedded in the wafer as well. This is due to recombination of minority carriers at highly recombinative wafer regions outside the cell area.

In the following we shall include the resistive losses of the interdigitated metal grid, using the circuit simulation method described by Heiser (1995). Losses due to recombination of charge carriers at the device perimeter can be quantified by introducing I–V curves of the perimeter region into the circuit simulation. The perimeter region represented by a perimeter diode is shown at the top of Fig. 2.21. Half of a p-contact, the collecting emitter and the shaded cell perimeter are included in the perimeter diode.

The recombination velocity at the non-diffused cell surfaces is higher in comparison with the inner cell parts because the n-type diffusions at the front and back side of the cell cease at the inner edge of the metal area mask that defines the active cell size. The effect of this enhanced perimeter recombination on the electron current flow can be seen in Figs. 2.21b and 2.21c. The metal area mask is indicated by the black bars on top of the enlargements. Graph (b) was calculated for short-circuit conditions, for which the current flow pattern reveals mainly vertical carrier transport. The recombination fraction increases towards the perimeter for maximum-power conditions, as seen in graph (c).

The numerical simulation allows us to quantify the perimeter recombination currents and helps to develop strategies for the minimisation of these parasitic losses.


54

gast

n++p+

n-contact p-contact

p+ n++

elementary diode perimeter diode

p-contact

p+ n++

hν

random pyramids(a)

(c)(b)

Figure 2.21 Recombination losses towards the edges can be accounted for by simulation of theperimeter region of the solar cell. The arrows indicate the direction of positive current flow, opposite tothe direction of electron flow in the perimeter region. (a) vertical section through cell including perimeterregion; (b) enlarged view of the edge of an n++-contact at short circuit; (c) the same view undermaximum-power conditions.

Using circuit simulations, the distributed resistance of the interdigitated metal rid can also be accounted for (Altermatt et al., 1996b). The total current flow causes voltage drop along the metal grid. Thus different cell regions, represented by the 2D ymmetry elements in the model, are driven by voltages differing from the voltage at he cell terminals. The losses due to this are known as non-generation losses.


55

The simulated voltage drop along the metal grid is shown in Fig. 2.22.

0 5 10 15 20 0

5101520

1

2

3

4

5

6

7

volta

ge d

rop

[mV]

alon

g m

etal

fing

er [m

m]

along metal busbar [mm]

E perimeter diode

elementary diode

ohmic resistance

metal grid (p)

metal grid (n)

area definitions forthe voltage sources

(a)

(b)

P

P

P

P

P

P

E

E

E

E

P

P

E

E

E

E

E

E

E

E

E

E

E

P

E

EE E

P

E

P

Distributed resistive and

non-generation losses limit the cell efficiency especially for solar cells with high current output, e.g. for

concentrator cells and large area cells.

Figure 2.22 Voltage drop in the metal grid as calculated by a circuitsimulation. (a) Elementary diodes and perimeter diodes are connected in an electrical circuit representing half of the solar cell. (b) Distributed voltage dropover the n-type metal grid. The lowest point of the surface indicates theposition of the contact pad of the metal bus bar.


56

2.6 Concluding remarks This chapter has presented a review of the theoretical treatment of silicon solar cells. Starting from the general semiconductor device equations the theory of the p-n junction solar cell was described, beginning with simplified models and proceeding to more ‘realistic’ cells with design features that can only be calculated with the help of numerical methods. In particular, calculations of solar cell parameters using the one-dimensional simulation program PC1D and the three-dimensional semiconductor device simulation program DESSIS were discussed. Whereas PC1D is widely used in the photovoltaic R&D community, 2D and 3D modelling is performed in only a few laboratories worldwide. This more involved device modelling is usually necessary only for high-efficiency solar cells. It demands a very good knowledge of the semiconductor material parameters, which is available only for single crystal semiconductors. These 2D and 3D simulations are therefore used mainly for crystalline silicon and III–V compound devices.

Solar cells made of silicon, an indirect semiconductor with a large absorption length for light and currents flowing parallel and perpendicular to the surface, have profited most from numerical calculations. On the other hand, the detailed analysis of high-efficiency silicon solar cells has contributed to a better understanding of material parameters.

Process technology, characterisation and device modelling are the three pillars on which progress in solar cell development is based. It is to be expected that a fruitful interaction of these three fields of work will further improve solar cell efficiencies from the present 24 % towards their theoretical limit. Acknowledgements The authors would like to acknowledge the important contributions by Jochen Dicker of Fraunhofer ISE to Section 2.5.3 on numerical solar cell modelling. We appreciate reviews and comments on this text by Anne Kovach-Hebling, Peter Koltay, Dominik Huljic, Jens Sölter, Wilhelm Warta and Sebastian Schäfer of Fraunhofer ISE. Also, we thank Holger Neuhaus and Pietro Altermatt of the Photovoltaics Special Research Centre of the University of New South Wales, Sydney for reading parts of the manuscript and making constructive suggestions. We also wish to acknowledge with gratitude the special assistance we received from the volume editor, Mary Archer. We are further indebted to Elisabeth Schäffer for technical editing of the manuscript.


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Device Physics of Silicon Solar Cells