3D Simulation of Base Carrier Transport Effects in Back Side Point Contact Silicon Solar Cells

Chapter 6

THREE-DIMENSIONAL SIMULATION OF BASE

CARRIER TRANSPORT EFFECTS IN BACK SIDE POINT

CONTACT SILICON SOLAR CELLS

K. Kotsovos and K. Misiakos Institute of Microelectronics, NCSR Demokritos, Attiki, Greece

ABSTRACT

This work presents a theoretical investigation of rear junction point contact silicon solar

cells through three-dimensional numerical simulation based on the solution of minority

and majority carrier transport equations in the base of the cell. The device series

resistance is evaluated through the simulated current-voltage (IV) curves under AM1.5

illumination conditions and its dependence on back contact geometry is examined.

Results are presented which show the influence of the majority carrier transport in the

base to the solar cell performance. A comparison is also performed with two other similar

types of point contact solar cells, one with the emitter located on the front surface and the

other on both surfaces, as well as with a conventional solar cell structure.

I. INTRODUCTION

Rear point contact (locally diffused) silicon solar cells with backside p/n junctions are

structures which have already shown their promising potential in solar energy production,

reaching very high conversion efficiency (27.5%) under concentrated illumination [1].

Although these devices were ideal for concentrator applications due to low series resistance

and surface recombination losses, they have some additional interesting advantages,

compared to typical solar cells designed for one-sun operating conditions. Specifically, since

the metallization grid lies entirely on the back surface, there is no shading loss on the

illuminated surface of the solar cell, while the interconnection of individual cells into modules

Institute of Microelectronics, NCSR Demokritos, P. O Box 60228 153 10 Aghia Paraskevi, Attiki, Greece Tel:

(+30)2106503113, Fax: (+30)2106511723, E-mail: [email protected]

K. Kotsovos and K. Misiakos 2

is more easily implemented. However, this optimized solar cell design was considered to be

too complex for use at low concentrations, so a simplified structure was proposed by Sinton

et al. [2], suitable for cost-effective production. Therefore, SunPower Corporation has

developed a process for that purpose, providing solar cells fabricated on high quality FZ

substrates with efficiencies greater than 20% under normal sunlight [3]. The choice of high

quality material is necessary for this type of solar cells, since the photogenerated carriers need

to reach the back surface in order to be collected.

Results of theoretical simulations regarding the back contact structure have already been

published in the literature. A 3D model based on the solution of semiconductor transport

equations using a variational approach has been developed by Swanson [4-5], which was

applied in order to optimize the back point contact solar cell design under concentrated

illumination. An optimization of the interdigitated back contact cell was performed by Chin et

al. [6], while the simulated efficiency limit of this cell was calculated by Ohtsuka et al. [7] by

3D simulations. Epitaxial layer transfer has also been proposed as an alternative way to

produce back contact solar cells [8], where this method is used to create thin silicon films on

foreign substrates and a two-dimensional model was applied for this case.

The purpose of this work is a theoretical investigation of back junction point contact solar

cells by means of numerical three-dimensional simulation based on the solution of minority

and majority carrier transport equations in the base of the cell. The method is based on the

transformation in x and y dimensions of the basic partial differential equations through 2D

Fast Fourier Transform (FFT). In Fourier space these equations become algebraic in Kx and

Ky (the transformed x, y variables), thus reducing to ordinary differential equations with

respect to z, that can be solved in analytical form. The basic assumption for such a problem

reformulation is planar geometry and low injection. The solution of the transport equations

under illumination conditions provides the device IV characteristics and solar cell’s series

resistance is extracted. This model was previously used [9] to simulate a structure similar to

the PERL [10-11] solar cell, a device that is consisted of an emitter covering the front

illuminated surface and point contacts in the back surface. The same method was later applied

for the simulation of the double junction solar cell [12], a device with an additional emitter in

the back surface. Since the back junction point contact solar cell and the previous two types

of solar cells, are high efficiency structures, a direct comparison among them is performed.

The influence of back contact size and spacing in solar cell performance is discussed in detail.

The following section presents a description of the mathematical model of our method as

applied on the point contact structures under consideration. The third section includes our

simulation results and discussion.

II. MATHEMATICAL MODEL- SIMULATION ALGORITHM

II.1. Assumptions –Device Geometry

The base of the solar cells is considered to be under low injection conditions and assumed

as homogeneous with thickness w, while the junctions are infinitesimally shallow.

Photogeneration in the emitter regions is considered negligible, while their ohmic losses are

neglected. We set as x, y the directions parallel to the junction while z is the perpendicular

Three-Dimensional Simulation of Base Carrier Transport Effects… 3

one. The geometry of the simulated devices is shown in Figure 1. The base contacts on the

back surface are assumed as squares with side length d, while the period length, or field

length, of the repeated pattern as shown in Figure 1(a) is l. Figure 1(b) illustrates the structure

of the back junction solar cell, figure 1(c), 1(d) the corresponding point back junction and

single front emitter devices and figure 1(e) the double junction solar cell. The dimensions of

the back point junction and base contact of the cell as shown in figure 1(c) are the same. In

addition, zero front surface reflectance is assumed, while light trapping is similar to

pyramidal texture scheme.

x y

z

d

d

d

l

l/4

l/4

3l/4

3l/4

l

w

w

a

b

c

d e

Emitter

Emitter

Emitter Emitter

Oxide Emitter

Base

Base

Contact

Contact

Contact Contact

Oxide

Oxide

Figure 1. a) Three dimensional back surface geometry of the simulated front junction, back junction and

double junction devices b) Back junction structure, c) Back point junction structure (the locations of the

diffused regions are shown in the insert) d) front junction structure e) Double junction structure. This

pattern is repeated periodically in x and y directions with a period length l


II. 2. Minority Carrier Continuity Equation and Boundary Conditions

The minority carrier continuity equation for a p-type base under low-level injection and

steady state is given by

nn D

zyxG

L

zyxnzyxn

),,(),,(),,(

2

2

(1)

where n(x,y,z) is the minority carrier concentration, Ln is the corresponding diffusion

length, Dn the diffusion constant and G(x,y,z) the local generation rate.

II. 2.1 Boundary Conditions (Back Junction Structure)

The following relation describes the boundary condition at the front of the oxide

passivated solar cell’s surface as shown on figure 1(b)

)0,,(),,(

),,( 1 yxneSdz

zyxdneDwyxJ wznn

(2)

where Jn(x,y,w) is the minority carrier diffusion current in the back surface and S1 is the

recombination velocity in the area covered by the oxide.

At the back surface, the boundary condition at the diffused contacts is also expressed by

the minority carrier diffusion current, which depends on the surface recombination velocity in

that area:

),,(),,(

),,( 2 wyxneSdz

zyxdneDwyxJ wznn

(3)

where Jn(x,y,w) is the minority carrier diffusion current in the back surface and S2 is the

recombination velocity in the diffused contact areas, which is assumed as constant and given

by the following expression

2

0

2

i

AC

en

NJS

(4)

where J0C is the saturation current density in the diffused contacts, ni the intrinsic carrier

concentration of the semiconductor and NA the base doping.

The rest of the back surface area is covered by the junction, and the boundary condition

is:

1

),,(exp),,(

2

KT

wyxVVe

N

nwyxn

dropB

A

i

(5)


where VB is the junction bias voltage and Vdrop is the voltage drop caused by the majority

carrier flow through the base series resistance. This voltage drop is initially set to zero.

II. 2.2 Boundary Conditions (Back Point Junction Structure)

The front surface of this structure, shown on figure 1(c) is covered by oxide, so relation

(2) gives the expression of the boundary condition in that region. Expressions (3) and (5)

define the boundary conditions in the back diffused contacts and the junction area

respectively.

The rest of the back surface is oxide passivated, so the boundary condition is defined by

the minority carrier diffusion current

),,(),,(

),,( 3 wyxneSdz

zyxdneDwyxJ wznn

(6)

where S3 is the recombination velocity in the area covered by the back oxide.

II. 2. 3. Boundary Conditions (Front Junction Structure)

The emitter of the front junction solar cell, which is illustrated in figure 1(d) covers the

whole illuminated surface, so the boundary condition inside the junction is

1

)0,,(exp)0,,(

2

KT

yxVVe

N

nyxn

dropB

A

i

(7)

At the back surface, the minority carrier diffusion current is determined by the surface

recombination velocity, where in the diffused base contact regions is defined by relation (4),

while in the oxide passivated surface has a constant value (S1). Therefore, the general form of

the boundary condition at the back surface may be written as

),,(),(),,(

),,( wyxnyxeSdz

zyxdneDwyxJ wznn

(8)

II. 2. 4. Boundary Conditions (Double Junction Structure)

The double junction structure (fig. 1(e)) is consisted of an emitter covering the whole

front surface (as in the front junction device), so the boundary condition in that area is given

by (7). In a similar way, the conditions in the back surface are expressed by relations (3) and

(5) of section II.2.1. The front and back emitters of this device are biased with same voltage

VB.

II.3. Majority Carrier Voltage Drop Equation

The solution of continuity equation (1) may be used to obtain the voltage drop caused by

the majority carrier flow. We begin from the current density relation for the majority carriers


peDpEeJ ppp (9)

Charge neutrality in the semiconductor is assumed, so it follows that δp(x,y,z)=δn(x,y,z).

Since the cell is operated under low injection, pNA, where NA is the base doping. Using these

assumptions and with the aid of (1), we differentiate (9)

nnAp

np

nn

p

Ap

n

nn

pApp

pApppApp

D

G

L

n

N

DDE

zGn

D

DENG

n

D

G

L

neDENeJ

neDENeJpeDENeJ

2

2

22

)(

The comparison of this equation with (1), gives a more compact expression

nDDN

V pn

Ap

22 )(1

(10)

where Dn, Dp are the diffusion constants for electron and holes respectively and μp is the

hole mobility. The solution of this equation provides the voltage drop due to majority carrier

flow and is subjected to the boundary conditions given in the next subsection.

II. 3.1. Boundary Conditions (Back Junction Structure)

Since there is no total current flow in the oxide covering the whole front surface of the

back junction solar cell, the majority carrier current value is exactly the opposite of the

minority carrier equivalent:

)0,,()0,,( 1 yxneSyxJ p (11)

At the back surface, the diffused back contact areas are considered as the ground

terminal, so the majority carrier voltage drop is zero:

0)w,y,x(Vdrop (12)

At the rest of the back surface area, covered by the rear junction, the majority carrier

current is given by

1

)),,((exp),,( 0

KT

wyxVVeJwyxJ

dropB

p

(13)


where J0 is the emitter saturation current density. Expressions (11) and (13) can be

converted as boundary conditions for the electric field E if we make use of (9) with the

following way:

Ap

n

n

p

p

n

n

p

App

pAppppp

Ne

JD

DJ

EJD

DENeJ

neDENeJpeDpEeJ

(14)

The minority carrier current density Jn, which is required in expression (14) is obtained

from the solution of the continuity equation described in section II. 2.

II. 3.2. Boundary Conditions (Back Point Junction Structure)

The front surface of this structure is covered by oxide as in the case of the back junction

structure, so expression (11) defines the boundary condition in that region. Relations (12),

(13) also describe the boundary conditions inside the back diffused contact and junction areas

respectively, while the rest of the back surface is covered by oxide, so the following condition

holds

),,(),,( 3 wyxneSwyxJ p (15)

As reported on the previous subsection, the majority carrier expressions may be

converted to electric field boundary conditions by making use of (14).

II. 3.3. Boundary Conditions (Front Junction Structure)

The boundary condition for the majority current at the front surface and inside the

junction is written as:

1

))0,,((exp)0,,( 0

KT

yxVVeJyxJ

dropB

p

(16)

where J0 is the saturation current density in the emitter. The expression of the boundary

condition in the oxide-covered part of the back surface is also given by

),,(),,( 1 wyxneSwyxJ p (17)

The diffused back contact areas are considered as the ground terminal, where the majority

carrier voltage drop is given by (12).


II. 3.3. Boundary Conditions (Double Junction Structure)

Since the front emitter of this structure covers the whole surface, relation (16) of the

previous subsection defines the boundary condition in that area. In a similar way, back

surface boundary conditions are expressed by relations (12) and (13) of section II.3.1.

II. 4. Algorithm Description

In this section we will provide a description of the algorithm, which is used to obtain a

numerical solution of the problem formulated in the previous subsections. The derived

expressions from the solutions of the minority carrier diffusion equation and the majority

carrier voltage drop equation are given in appendices B and C respectively.

II. 4. 1. Diffusion Equation Solution Algorithm

(1) The algorithm starts with an initial guess for the minority carrier concentrations in

front and back surface areas not covered by the emitters, while in the junction

regions the corresponding boundary conditions (depending on the investigated

structure) that define the minority carrier concentrations are applied. Majority

carrier voltage drop is initially set to zero.

(2) A two-dimensional Fast Fourier Transform (FFT) with respect to x, y is performed

to both surface concentrations and the minority carrier current density in Fourier

space is calculated by differentiating the general solution of the diffusion equation

with respect to z (appendix B).

(3) An inverse FFT is then applied to each of the transformed current densities to obtain

the real current densities at the areas not covered by the junctions, while at the

regions covered by oxide or the back contacts the current densities are acquired

from the boundary conditions.

(4) Subsequent Fast Fourier Transforms are used in order to calculate the new carrier

concentrations as functions of the transformed current densities )0,,(~

yxn kkJ ,

),,(~

wkkJ yxn (Appendix B).

(5) Inverse Fast Fourier Transforms are performed to the previously obtained carrier

concentrations to calculate the new estimated ones in real space.

(6) The solution is set as a mixture of the previously calculated and the newly obtained

minority carrier distributions with a defined percentage. If this solution fulfills the

convergence condition, the results are stored in order to proceed with the voltage

drop equation, else calculations are repeated from step 2.

II. 4. 2. Voltage Drop Equation Solution Algorithm

The results of the solution of the minority carrier diffusion equation are required in order

to obtain the majority carrier voltage drop. Therefore, the corresponding procedure for the

case of equation (10), which follows the one referred to the previous section, is described

through the following steps:


(1) An initial guess for the majority carrier voltage drop (Vdrop) on both surfaces is used.

This is considered as zero. This estimate also fulfills the boundary condition at the

back-diffused contacts.

(2) Two-dimensional Fast Fourier Transforms with respect to x, y are performed to the

voltage distributions and the electric field distributions in Fourier space

)0,,(~

yx kkE , ),,(~

wkkE yx are calculated by differentiating the general solution of

equation (10) with respect to z (appendix C).

(3) An inverse FFT is then applied to each of the transformed electric field distributions

to obtain the corresponding values in real space, while at the regions covered by

oxide or the junctions the current densities are acquired from the boundary

conditions.

(4) In this step, Fast Fourier Transforms are used in order to obtain the electric field

distributions in Fourier space and the new transformed voltage

distributions )0,,(~

yx kkV , ),,(~

wkkV yx as functions of

)0,,(~

yx kkE , ),,(~

wkkE yx are calculated (appendix C).

(5) Inverse Fast Fourier Transforms are performed to the previously obtained voltage

distributions to calculate the new estimated equivalents in real space.

(6) The solution is set as a mixture of the previously calculated and the newly obtained

one in a similar way as that referred to in the previous subsection. If this solution

fulfills the convergence condition, the results are stored in order to be inserted in the

boundary conditions of the minority carrier diffusion equation, else calculations are

repeated from step 2.

II. 4. 3. Solution of the Final Coupled Problem

The separate solutions of the differential equations (1) and (10) are not necessarily self-

consistent for all working conditions, since the boundary conditions that define the minority

carrier diffusion equation depend on the calculated majority carrier voltage distribution in a

non-linear way. Therefore self-consistency should be achieved by following a proper iterative

procedure described as following:

(1) Calculation of the solution of equation (1) by following the steps described in

section II. 4.1. The calculations are performed under the assumption of zero voltage

drop.

(2) Numerical solution of equation (10) using the minority carrier currents and density

distributions as obtained previously.

(3) New solution of the minority carrier diffusion equation taking into account the

voltage distributions calculated from the previous step.

(4) The procedure continues between steps 2 and 3 until self-consistency is achieved.


III. SIMULATION RESULTS

The substrate of the simulated solar cells is considered as monocrystalline silicon, with

base doping density NA=1016cm-3. We assume that the oxide-covered surfaces have ideal

passivation properties, so there are no recombination losses at these areas. Therefore, the

recombination velocity at these surfaces is zero while in the diffused contact regions is

calculated from the relation (4), where we assume that the recombination current in these

regions is J0c=10-12 A/cm2. The emitter saturation current value of all devices is the same (10-

13 A/cm2), while for all acquired results of the following sections III.1-III.4, a base diffusion

length Ln of 800μm is assumed. The mobilities for minority and majority carriers are taken

from Klaasen [13]. The simulated illumination is considered as the global AM1.5 sun

spectrum [14] normalized to 100mW/cm2, where light trapping similar to the pyramidal

textured scheme is assumed (Appendix A). The back surface contact has reflective

characteristics with reflectivity R=95%.

The simulation program, which is based on the algorithm of section II. 4 is used to

calculate the IV characteristic of the cell and from that curve the maximum power, the short

circuit current, the open circuit voltage and the series resistance of the cell are obtained. The

series resistance of the cell is calculated for each point of the curve using the following

relation

I

II

I

e

KTVV

R sc

sc

oc

s

ln

(18)

The value of Rs which is obtained by (18) is caused by the current crowding effect at the

back point contacts, since inside these regions current density values are large [9]. Such an

effect induces a voltage drop which rises fast near the base contact edges. This effect is more

intense in the back point junction structure, where near the back point emitter as shown in

figure 2 an additional voltage drop is induced. The maximum voltage drop value is reached

inside the emitter area, where it remains constant. In this structure d/l equals 0.2, while the

period of the repeated pattern and the base thickness is 400μm.


80

160

240

320

400

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

80

160

240

320

400

Vo

lta

ge

Dro

p (

mV

)

y (m

)x (m)

Figure 2. Majority carrier voltage drop at the back surface of a back point junction structure with period

length 400 μm and 80 μm back diffused junction sidelength. The cell is operated in the maximum

power point (576 mV). Base thickness w is 400μm and Ln=800μm

III. 1. Short Circuit Current (Jsc)

Figure 3 illustrates the short circuit current of the back junction structure as a function of

the back base contact size and its spacing as parameter, assuming base thickness of 200μm.

As expected, the reduction of the back contact area results to an increased photocurrent since

the surface recombination velocity in this area (S2) is high. Jsc is also improved when the back

contact spacing is smaller since in this case current crowding is reduced and carriers are

collected more efficiently.

The short circuit current of the back point junction structure is shown on figure 4. In this

case, the increase of the back contact size is beneficial to the device photocurrent in contrast

to the previously analyzed structure. A dramatic reduction in Jsc is also observed for the

largest contact spacing (400μm), since in this case the required path for the collection of

photogenerated carriers is significantly increased, thus the base minority carrier diffusion

length should be higher for a more efficient current collection.

The next figure shows a comparison of the Jsc of the four different structures of figure 1

when back contact spacing is 400μm. The right hand axis is the current normalized to the

corresponding typical solar cell structure where the back contact covers the whole back

surface (1D case). Figure 6 is the same plot calculated for the smallest back contact spacing

of 50μm. Since all devices are illuminated on the front side, most carriers are generated near

the front surface, thus the single, double junction as a well as the typical solar cell device

show an improved Jsc compared to both back junction structures. The reduction of the back

contact spacing to 50μm results to a Jsc increase of these devices, especially in the point

junction one, while the corresponding short circuit current of single and double junction


devices remains almost unaffected from that change in l [9, 12]. A greater diffusion length

would significantly improve the carrier collection ability of back contact devices, as will be

discussed later.

0.1 0.2 0.3 0.4 0.539.4

39.6

39.8

40.0

40.2

40.4

40.6

Jsc (

mA

/cm

2)

d/l

d=400m

d=200m

d=50m

Figure 3. Short circuit current (Jsc) of the back junction structure as a function of back contact size for

different contact spacing l. Base thickness is 200μm and Ln=800μm

0.1 0.2 0.3 0.4 0.526

28

30

32

34

36

38

40

Jsc(m

A/c

m2)

d/l

d=400m

d=200m

d=50m

Figure 4. Short circuit current versus back contact size of the back point junction solar cell. The other

parameters are the same as of figure 3


0.1 0.2 0.3 0.4 0.526

28

30

32

34

36

38

40

42

0.64

0.69

0.74

0.79

0.83

0.88

0.93

0.98

1.03

3D

to 1

D r

atio

Jsc(m

A/c

m2)

d/l

front junction device

double junction device

back junction device

point junction device

Figure 5. Short circuit current of all different point contact structures versus back contact size for a

given contact spacing of 400μm, device thickness 200μm and Ln=800μm. The right axis is the current

normalized to the corresponding value of the conventional 1D structure

0.1 0.2 0.3 0.4 0.538.0

38.4

38.8

39.2

39.6

40.0

40.4

40.8

41.2

41.6

42.0

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

3

D t

o 1

D r

atio

Jsc(m

A/c

m2)

d/l





Figure 6. Short circuit current of all different point contact structures versus back contact size for a

given contact spacing of 50μm. The other parameters are the same as of the previous figure

III. 2. Open Circuit Voltage (Voc)

Figure 7 demonstrates the open circuit voltage of all different structures as a function of

the back contact size. The graph is referred to a specific contact size of 50μm, but it is also


valid for the other ones since our simulations have shown that the influence of back contact

spacing l on Voc is negligible, while base thickness is 200μm, as in the previous section. We

observe that in contrast to the short circuit current, the back point junction device has the

highest open circuit voltage compared to the other solar cell structures.

0.1 0.2 0.3 0.4 0.5644

648

652

656

660

664

668

672

676

680

684

1.015

1.021

1.027

1.034

1.040

1.046

1.052

1.059

1.065

1.071

1.078

3D

to

1D

ra

tio

Vo

c(m

V)

d/l

front junction dev.

double junction dev.

back junction dev.

point junction dev.

Figure 7. Open circuit voltage (Voc) of all different point contact structures versus back contact size.

The right axis is the current normalized to the corresponding value of the conventional 1D structure.

Base thickness is 200μm and Ln=800μm

The improved open circuit voltage may be attributed to the reduced surface

recombination of the point junction structure, since the minimization of the area of the

diffused regions is required to maximize the voltage [5], so Voc is improved at a faster rate

compared to the other devices when the d/l ratio is reduced. The Voc of the front junction solar

cell follow the corresponding point junction equivalent due to the low back surface

recombination, while the back junction structure Voc values are slightly lower compared to the

front junction cell. Finally, the double junction device has the lowest open circuit voltage of

all point contact structures as expected, due to recombination in both emitters.

III. 3. Base Series Resistance (Rs)

Figure 8 shows the base series resistance of the back junction structure near the cell

maximum power point as a function of the back base contact size and its spacing as parameter

when base thickness is 200μm. This graph shows that decreasing contact size leads to greater

series resistance that grows dramatically for the smallest back contact area coverage fraction

due to the current crowding at the back contact, as already reported for the front [9, 15,16]

and double junction rear point contact solar cells [12]. In addition, the reduction of the back

contact spacing limits the series resistance considerably.

The current crowding effect is more evident in figure 9, where the series resistance of the

back point junction structure is shown. It can be observed that there is an almost ten-fold


increase to the Rs value for the smallest d/l ratio when the back contact spacing is changed

from 50μm to 400μm. A rapid reduction of Rs is also observed when the back-diffused

coverage ratio is increased.

0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

Rs (

Oh

m.c

m2)

d/l

l=400m

l=200m

l=50m

Figure 8. Base series resistance of the back junction structure near the cell maximum power point as a

function of the back base contact size and its spacing as parameter. Base thickness is 200μm and

Ln=800μm

0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Rs (

Oh

m.c

m2)

d/l

l=400m

l=200m

l=50m

Figure 9. Base series resistance of the point back junction structure near the cell maximum power point

as a function of the back base contact size and its spacing as parameter. Base thickness is 200μm and

Ln=800μm

Figure 10 shows the series resistance dependence on the d/l ratio of the four different

structures when the base contact spacing is 400μm. The simulations are performed for devices

with different thickness, 200μm and 400μm respectively.


0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.00

3.27

6.54

9.80

13.07

16.34

19.61

22.88

3D

to

1D

ra

tio

Rs (

Oh

m.c

m2)

d/l





A

0.1 0.2 0.3 0.4 0.50.000

0.049

0.098

0.148

0.197

0.246

0.295

0.344

0.00

1.50

3.00

4.50

6.00

7.50

9.00

10.50

3D

to

1D

ra

tio

Rs (

Oh

m.c

m2)

d/l





B

Figure 10. Series resistance of the four different structures versus d/l ratio when the base contact

spacing is 400μm, Ln=800μm and different base thickness: (A) 200μm and (B) 400μm. The right hand

axis is the series resistance normalized to the corresponding value of the typical solar cell structure (1D

case)

The right hand axis is the series resistance normalized to the corresponding value of the

typical solar cell structure (1D case). In this case the back junction device exhibits the lowest

Rs, while the corresponding series resistance of the double junction structure is slightly

higher. On the contrary, the single front and point junction devices exhibit the highest Rs

values. The series resistance of the front junction structure is significantly influenced from the

base thickness in contrast to the rest of the point contact cells, except for the case of the

smallest contact coverage fraction. Therefore, it can be concluded that the series resistance of

the front junction cell is significantly influenced by the majority carrier flow in the vertical (z)

direction. In most cases the conventional structure (1D) has the lowest series resistance,

except for the largest d/l ratio, where the back and double junction structure Rs values are

smaller.


The situation is different in figure 11, which is the same plot as of figure 10 where the

back contact spacing is reduced to 50μm.

0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.00

0.33

0.65

0.98

1.31

1.63

1.96

2.29

2.61

2.94

3.27

3D

to 1

D r

atio

Rs (

Ohm

.cm

2)

d/l





A

0.1 0.2 0.3 0.4 0.50.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.00

0.30

0.61

0.91

1.22

1.52

1.83

2.13

2.44

2.74

3.05





3D

to 1

D r

atio

Rs (

Ohm

.cm

2)

d/l B

Figure 11.Series resistance of the four different structures versus d/l ratio when the base contact spacing

is 50μm, Ln=800μm and different base thickness: (A) 200μm and (B) 400μm. The right hand axis is the

series resistance normalized to the corresponding value of the typical solar cell structure (1D case)

In this case, the back and point junction structures show the smallest Rs values, with the

back junction one having the lowest. The Rs of the double junction solar cell is slightly higher

in almost all cases compared to the previous structures and it is significantly reduced when

the base thickness is changed from 400μm το 200μm. This is an indication that this device is

also influenced by the majority carrier flow in the vertical direction as already reported. On

the contrary, the series resistance of the front junction structure is by far the highest of all,

approaching the limit of the conventional 1D device for large back contact coverage fractions.

This may be attributed to the fact that in the front junction and the conventional solar cell

devices the emitter and back contact are located on different surfaces and minority and


majority carriers move to opposite directions, thus minority carrier flow opposes majority

carrier movement, while in the back and point junction devices, the diffused regions lie in the

same surface, so both minority and majority carriers flow towards the back surface. Therefore

in this case, the reduced series resistance values of the back and point junction structures, is a

clear advantage for concentrator applications, where ohmic losses need to be minimized, due

to the high current generated by the solar cell. The double junction structure is also a good

potential candidate for such applications since its series resistance is also low. It must be

additionally noted, that all structures except from the single front junction one, may reach

much lower Rs values compared to the conventional 1D device.

III. 3. Maximum Efficiency (η)

Figure 12 shows the dependence of the back contact size on the back junction cell’s

maximum efficiency with the contact spacing as a parameter. The plots demonstrate that there

is a significant increase of the efficiency when the contact size decreases, provided that the d/l

ratio is not less than 0.2, while with a further decrease of the back contact spacing shifts the

efficiency to a smaller d/l ratio. According to the discussion of the previous sections, the

reduction of the back contact size results to lower back contact recombination, so the open

circuit voltage and the short circuit current are improved. On the other hand, a minimization

of the back contact coverage fraction results to an intense current crowding effect, where the

series resistance is significantly increased (figure 8), thus limiting the efficiency. The

reduction of the back contact spacing limits this effect, so efficiency is improved.

0.1 0.2 0.3 0.4 0.5

21.6

21.8

22.0

22.2

22.4

22.6

22.8

effic

ien

cy (

%)

d/l

d=400m

d=200m

d=50m

Figure 12 Maximum conversion efficiency of the back junction structure near the cell maximum power

point as a function of the back base contact size and its spacing as parameter. Base thickness is 200μm

and Ln=800μm

Figure 13 illustrates the corresponding efficiency plots of the point back junction solar

cell. In contrast with the previous figure, the efficiency of this structure is improved by

increasing the back-diffused regions coverage, since in this case the photocurrent is enhanced

as shown on figure 4, while series resistance is reduced. However, when back contact spacing


is 50μm the device efficiency is improved when the d/l ratio is reduced from 0.5 to 0.2. In

addition, when the back contact spacing is large (400μm) and the d/l ratio is 0.1 the efficiency

is greatly reduced. This is due to the current crowding effect, which not only increases the

series resistance but also requires a significantly larger minority carrier diffusion length for

efficient carrier collection, as already discussed [5].

Figure 14 compares the efficiency dependence on the d/l ratio of the four different

structures, when the base contact spacing is 400μm. The right hand axis is the efficiency

normalized to the corresponding value of the typical solar cell structure (1D case).

0.1 0.2 0.3 0.4 0.515

16

17

18

19

20

21

22

23

eff

icie

ncy (

%)

d/l

d=400m

d=200m

d=50m

Figure 13 Maximum conversion efficiency of the back point junction structure near the cell maximum

power point as a function of the back base contact size and its spacing as parameter. Base thickness is

200μm and Ln=800μm

0.1 0.2 0.3 0.4 0.515

16

17

18

19

20

21

22

23

0.70

0.74

0.79

0.83

0.88

0.93

0.97

1.02

1.07

3D

to

1D

ratio

effic

ien

cy (

%)

d/l





Figure 14. Efficiency of the four different structures versus d/l ratio when the base contact spacing is

400μm. The right hand axis is the series resistance normalized to the corresponding value of the typical

solar cell structure (1D case). Base thickness is 200μm and Ln=800μm.

The plots show that the efficiency of the back and point junction solar cell devices is

significantly lower due to less efficient carrier collection, since the emitter of both structures


is located on the back surface. As already discussed, the efficiency of the point junction solar

cell is severely limited by current crowding. The highest performing structures are the single

front and double junction solar cells, which have almost the same efficiency.

Figure 15 is the same plot as of figure 10 calculated for reduced the back contact spacing

to 50μm. As expected, the efficiency of all structures (except from the conventional solar cell)

is improved and especially the corresponding point junction one. The best performing

structures are still the single front and double junction solar cells, where the single front

junction cell edges out the corresponding double junction one for small contact sizes. The

efficiency of the back and point junction solar cell devices is still lower, where the back

junction structure has the best efficiency of these two.

0.1 0.2 0.3 0.4 0.5

21.8

22.0

22.2

22.4

22.6

22.8

23.0

23.2

23.4

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

3

D t

o 1

D r

atio

eff

icie

ncy (

%)

d/l





Figure 15. Efficiency of the four different structures versus d/l ratio when the base contact spacing is

50μm. The right hand axis is the series resistance normalized to the corresponding value of the typical

solar cell structure (1D case). Base thickness is 200μm and Ln=800μm.

It should be pointed out, however that no front surface reflection is assumed for all

devices and in reality all devices except from the back and point junction ones have additional

losses due to front grid shadowing, which limits significantly their efficiency. On the other

hand, larger diffusion lengths would greatly improve the performance of back and point

junction solar cells; therefore the influence of this parameter is investigated in the next

section.

III. 4. Minority Carrier Diffusion Length Influence on Device Short Circuit

Current, Open Circuit Voltage and Efficiency

Figure 16 shows the short circuit current of all different structures, including the

conventional 1D solar cell, versus minority carrier base diffusion length (Ln). The calculations

are performed for device thickness 200μm and 400μm.


400 600 800 1000 1200 1400 160033

34

35

36

37

38

39

40

41

42

Jsc (

mA

/cm

2)

Ln (m)

Front junction device

Double junction device

Back junction device

Point junction device

Conventional device

A

400 600 800 1000 1200 1400 160024

26

28

30

32

34

36

38

40

42

Jsc (

mA

/cm

2)

Ln(m)



dack junction device


conventional device

B

Figure 16. Short circuit current of all different back point contact structures, including the conventional

1D solar cell versus minority carrier base diffusion length (Ln) and different base thickness: (A) 200μm

and (B) 400μm. The simulations are performed for the smallest contact spacing of 50μm for improved

efficiency, while point contact side length in (A) and (B) is set as 16μm and 10μm respectively

The simulations are performed for the smallest contact spacing of 50μm for improved

efficiency, while point contact side length is considered as 16μm and 10μm for figure 16(A)

and 16(B) respectively, as a good balance between back surface recombination and series

resistance. As expected the short circuit current of the double junction solar cell is the highest,

due to the enhanced carrier collection of the back emitter [12,17], which remains almost

constant for the considered diffusion lengths. The Jsc of the single front junction solar cell is

near to the levels of the previous structure when Ln is greater or equal than 800μm, followed

by the conventional 1D structure, where there is negligible photocurrent improvement when

Ln is increased. In contrast to the former structures, the back and point junction solar cells

benefit a lot from the diffusion length increase, since in this case the minimization of bulk

recombination results to greatly improved carrier collection. This is more evident in the case

of the smaller base thickness (200μm).


Figure 17 shows the open circuit voltage of all different structures versus minority carrier

base diffusion length (Ln) in the same manner as of previous figure.

400 600 800 1000 1200 1400 1600625630635640645650655660665670675680685690695700705

Voc(m

V)

Ln (m)





Conventional device

A

400 600 800 1000 1200 1400 1600610

620

630

640

650

660

670

680

front junction dev.

double junction dev.

back junction dev.

point junction dev.

conventional dev.

Vo

c (

mV

)

Ln(m)

B

Figure 17. Open circuit voltage of all different back point contact structures, including the conventional

1D solar cell versus minority carrier base diffusion length (Ln). Other simulation parameters are the

same as of figure 16

As already discussed in section III.2, the limited front and back surface recombination of

the point junction structure, is the cause of the improved open circuit voltage compared to the

other solar cells. However, when Ln and base thickness is 400μm, bulk recombination limits

the Voc of the former as well as of the back junction structure to values lower than the other

structures. On the contrary, for the greatest Ln value of figure 17 the open circuit voltage gain

of the point junction solar cell compared with the corresponding front junction equivalent is

almost 10mV when w=400μm and exceeds 20mV when w=200μm. The Voc of the back

junction structure almost equals or exceeds the corresponding front junction one for diffusion


lengths greater than 800μm and w=200μm, while the voltage of the double junction structure

is significantly lower compared to the three previously referred solar cells due to front and

back emitter recombination. The conventional structure shows the most limited open circuit

voltage, which is more than 35mV lower than the corresponding back point junction one

when Ln is 1600μm and w=400μm, while this difference is increased to 60mV when

w=200μm.

Finally, figure 18 shows the conversion efficiency of all different structures as a function

of the minority carrier base diffusion length (Ln) in the same manner as of figures 16 and 17.

400 600 800 1000 1200 1400 160017.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

Front junction (shading loss)

Double junction (shading loss)

Conventional (shading loss)

e

ffic

ien

cy (

%)

Ln (m)





Conventional device

A

400 600 800 1000 1200 1400 160012

14

16

18

20

22

24

Front junction (shading loss)

Double junction (shading loss)

Conventional (shading loss)

Single junction device




Conventional device

eff

icie

ncy (

%)

Ln (m)

B

Figure 18. Conversion efficiency of all different back point contact structures, including the

conventional 1D solar cell versus minority carrier base diffusion length (Ln). Other simulation

parameters are the same as of figure 16 and 17. The open-symbol colored plots refer to the front, double

and the conventional solar cell structures when a 4% front grid shading loss is taken into account


As expected the point and back junction structures benefit the most from the diffusion

length increase due to more efficient carrier collection, where for diffusion lengths greater

than 800μm their efficiencies are almost equal, while the efficiency of the point junction solar

cell exceeds the corresponding back junction one by a small margin when Ln=1600μm and

w=400μm. In addition when w=400μm, these structures show superior performance compared

to the conventional solar cell, when Ln=1200μm or larger, while for the largest diffusion

length value of the graph, their efficiencies approach the levels of the single front and double

junction ones. This small efficiency premium (approximately 0.5% absolute for the case of

the single front junction cell and 0.34% for the double junction equivalent) is eliminated if

front surface grid shadowing is taken into account, as shown on the graphs where a 4%

shading loss is assumed. In this case, the back and point junction solar cells exhibit the

highest efficiencies when Ln is greater than 1200μm (Ln/w>3). If the device thickness is

reduced to 200μm and base diffusion length is greater than 1200μm (Ln/w>6), the point

junction cell shows the highest efficiency of all structures, neglecting shadowing losses.

When grid shadowing is set to 4%, the back and point junction structures reach higher

efficiencies compared to the single and double junction solar cells for diffusion lengths

greater than 800μm (Ln/w>4). Therefore, the choice of thin, high quality silicon wafers is

absolutely necessary for the fabrication of the back and point junction solar cells. Topsil

produces such FZ wafers with minority carrier lifetimes greater than 1ms for use in the PV

industry [18], while wafers grown under the MCZ method (magnetically confined

Czochralski) that are already used for the fabrication of high efficiency PERL structures [19]

are good candidates as a starting material and they cost less than electronic quality FZ wafers.

IV. CONCLUSION

In this work back junction, point contact (locally diffused) solar cells have been

investigated through 3D simulations and compared with corresponding single front junction,

double junction as well as conventional (1D) solar cell devices. It was shown that the

simulated base series resistance of the back junction structure reached significantly lower

values compared to the single front and double junction devices, especially for small back

contact spacing. The back point junction solar cell reached the highest open circuit voltage

due to reduced surface recombination, although current-crowding effects would severely

affect its efficiency by reducing the solar cell’s photocurrent and increasing the base series

resistance if the diffused areas are too small or too remotely spaced. A proper choice of back

diffused contact spacing and size, would result to low Rs, close to the values of the back

junction structure. Therefore, these cells are preferable for concentrator applications, since

their efficiency would be significantly less affected from resistive losses compared to the

single front junction back point contact solar cell and the conventional device. However, high

quality starting material and relatively thin substrates (Ln/w>4) are required so that these

devices reach efficiencies significantly higher than the conventional 1D device and close to or

higher than the single front or double junction structure. On the other hand, the double

junction solar cell could also be proposed as a very good choice for all applications, since it

performs marginally lower compared to the corresponding front junction one on high quality


substrates, and it has the best efficiency on low quality ones. In addition its simulated base

series resistance reached values near to those of the back junction solar cells.

REFERENCES

[1] R. A.Sinton, Y. Kwark, J. Y. Gan, R. M. Swanson, IEEE Electron Device Lett., Vol. 7

(10), p.1855, 1986.

[2] R. A. Sinton and R. M. Swanson, IEEE Trans. on Electron Devices, Vol. 37 (10), p.348,

1990.

[3] W. Mullikan, D. Rose, M. J. Cudzinovic, D. M. De Ceuster, K. R. McIntosh, David D.

Smith, and R. M. Swanson, Proceedings of the 19 EPVSEC, Paris, France, p. 387, 2004.

[4] R. M. Swanson, EPRI Rep., AP-2859, 1983.

[5] R. M. Swanson, Solar Cells, Vol. 17, p. 85, 1986.

[6] D. J. Chin and Navon D. H., Solid State Electron., Vol. 24, p.109, 1981.

[7] H. Ohtsuka, Y. Ohkura, T. Uematsu and T. Warabisako, Prog. Photovoltaics: Res. Appl.,

Vol. 2, p. 275, 1994.

[8] Nichiporuk O., Kaminski A., Lemiti M., Fave A. and Skryshevski V, Sol. Energy Mat.

and Sol. Cells, 86, p. 517, 2005.

[9] K. Kotsovos and K. Misiakos, J. Appl. Phys., Vol. 89, p. 2491, 2001.

[10] J. Zhao, A. Wang, P. Altermatt and M. A. Green, Appl. Phys. Lett., Vol. 66 , p. 3646,

1995.

[11] Zhao J., Wang A., και Green M. A., Progr. In Photovoltaics: Res. Appl., Vol. 7, p. 471,

1999.

[12] K. Kotsovos and K. Misiakos, Sol. En. Mat. and Sol. Cells, Vol. 77, p. 209, 2003.

[13] D.B.M. Klaassen, Solid-State Electronics, Vol. 35, p. 953, 1992.

[14] R. Hulstrom R. Bird and C. Riordan, Solar Cells, Vol. 15, p. 365, 1985.

[15] Zhao J., Wang A., and Green M. A., Sol. Energy Mat. and Sol. Cells,Vol. 32, p. 89, 1994.

[16] Catchpole K. R. and Blakers A. W., Sol. Energy Mat. and Sol. Cells, 73, p. 189, 2002.

[17] E. Van Kerschaver, C. Zechner, and J. Dicker, IEEE Trans. El. Devices, Vol. 47 (4) , p.

711, 2000.

[18] Vedde J., Jensen L., Larsen T. and Klausen T., Proceedings of the 19 EPVSEC, Paris,

France, p. 1075, 2004.

[19] Zhao J., Wang A. and Green M. A., Progr. In Photovoltaics: Res. Appl., Vol. 8, p. 549,

2000.

APPENDIX A. LIGHT GENERATION PROFILE MODEL

The surface of the simulated devices is textured as shown in figure A.1. The back surface

is assumed reflective with a constant reflection coefficient Rb. This light-trapping scheme

improves the absorbing properties of the investigated material, since incoming rays enter the

cell with an angle of incidence, which is different than normal, so the material absorption

coefficient αi is increased according to the following relation.


θ

Figure A.1. Assumed light trapping scheme used for the model calculations

sin

i

ieff

(A.1)

where αieff is the effective material absorption coefficient under a given wavelength,

while θ is the angle shown in figure A.1. Assuming that photon flux decays exponentially

with increasing depth and that light is coupled out after performing a double pass across the

cell, then the photon generation rate which is independent from x and y directions for all

wavelengths of the considered AM1.5 spectrum, is written as

)()()(1

1

1

1

)(zGegeRegzG

N

i

i

N

i

i

zw

iieff

w

b

z

iieffieffieffieff

(A.2)

where gi is the number of generated electron-hole pairs for the given wavelength i. By

substituting in (A.2), the generation rate in front and back surface are obtained

1

1

1

1

2)0()1()0(

N

i

i

N

i

i

w

biieff GeRgG ieff

(A.3)

1

1

1

1

)()()(N

i

i

N

i

i

w

b

w

iieff wGeRegwG ieffieff

(A.4)

The differentiation of (A.2) provides the following expression

1

1

1

1

)(2 )()()( N

i

i

N

i

i

zww

b

z

iieff zGeeRegdz

zdGieffieffieff

(A.5)

while the corresponding values for both surfaces are


1

1

1

1

22

0

)0()1()( N

i

i

N

i

i

w

biieff

z

GeRgdz

zdGieff

(A.6)

1

1

1

1

2 )()()( N

i

i

N

i

i

w

b

w

iieff

wz

wGeRegdz

zdGieffieff

(A.7)

The expressions (A.2)-(A.7) will be used in the following sections for the solution of the

transport equations.

APPENDIX B. SOLUTION OF THE MINORITY CARRIER CONTINUITY

EQUATION

Performing a two-dimensional Fourier Transform on equation (1) the following

expression is obtained:

n

yx

n

yx

yx

D

zGzkkn

Lkk

dz

zkknd )(~

),,(~)1

(),,(~

2

22

2

2

(B.1)

where )(~

),,,(~ zGzkkn yx are the Fourier transforms with respect to x, y of

)(),,,( zGzyxn respectively. This is an ordinary differential equation with independent

variable z, which has the following general solution:

n

zR

yx

zR

yxyxD

zGkekkBekkAzkkn

)(~

),(),(),,(~1

11

(B.2)

where 222

11

nyx L

kkR and k1 is determined by differentiating (B.2) with respect

to z twice and equating the result with the right part of (B.1). By performing the necessary

operations and with the use of (A.2) we get

1

122

1 )(

)(~

),(),(),,(~ 11

N

i ieffn

izR

yx

zR

yxyxRD

zGekkBekkAzkkn

(B.3)

The solution defined by (B.3) incorporates the constants A and B, which should be

determined through the boundary conditions. These constants may be expressed as functions

of ),,(~),0,,(~ wkknkkn yxyx in the following way:


wRwR

N

i ieffn

wR

iiyx

wR

yx

yxee

RD

eGwGwkknekkn

kkA11

1

1

1

122

1 )(

)0(~

)(~

),,(~)0,,(~

),(

(B.4)

wRwR

N

i ieffn

i

wR

iwR

yxyx

yxee

aRD

wGeGekknwkkn

kkB11

1

1

1

122

1 )(

)(~

)0(~

)0,,(~),,(~

),(

(B.5)

where )(~

),0(~

wGG are the carrier generation rates in the front and back surface

respectively and are given by (A.3) and (A.4).

The minority carrier diffusion current can be obtained by differentiating (B.3) with

respect to z

1

122

1

1)(

)(~

),(),(

),,(~),,(~

11

N

i ieffn

izR

yx

zR

yx

yx

n

yxn

RD

zGekkAekkBR

dz

zkknd

eD

zkkJ

(B.6)

After substituting (B.4) and (B.5) in (A.6) we get the final expressions for the current in

both surfaces:

1

122

1

1

122

1

1

)(

)0(~

)(

)(~

2)0(~

))(0,,(~),,(~2)0,,(

~

11

11

11

N

i ieffn

i

wRwR

N

i ieffn

i

wRwR

iwRwR

yxyx

n

yxn

RD

G

ee

RD

wGeeGeekknwkkn

ReD

kkJ

(B.7)

1

122

1

1

122

1

1

)(

)(~

)(

)(~

)0(~

2)0,,(~2))(,,(~

),,(~

11

11

11

N

i ieffn

i

wRwR

N

i ieffn

i

wRwR

i

yx

wRwR

yx

n

yxn

RD

wG

ee

RD

wGeeGkkneewkkn

ReD

wkkJ

(B.8)

where )(~

),0(~

wGG ii are defined in (A.6) and (A.7). The expressions (B.7) and (B.8)

may be used to calculate the minority carrier diffusion currents in Fourier space as a function

of the corresponding surface concentrations. The opposite procedure could be performed by

solving the system of (B.7) and (B.8) to obtain the transformed surface minority carrier

concentrations as a function of the corresponding diffusion currents


1

122

111

1

122

1

1

1

1

122

1

)(

)0(~

)sinh(

1)(

)(~

),,(~

)coth()(

)0(~

)0,,(~

)0,,(~

N

i ieffn

i

N

i ieffn

i

yxn

N

i ieffn

i

yxn

yx

RD

G

wRR

RD

wGwkkJ

wRR

RD

GkkJ

kkn

(B.9)

1

122

1

1

1

1

122

1

11

1

122

1

)(

)(~

)coth()(

)(~

),,(~

)sinh(

1)(

)0(~

)0,,(~

),,(~

N

i ieffn

i

N

i ieffn

i

yxn

N

i ieffn

i

yxn

yx

RD

wGwR

R

RD

wGwkkJ

wRR

RD

GkkJ

wkkn

(B.10)

The expressions (B.7)-(B.10) are used to solve the diffusion equation by application of

the algorithm described in section II.4.1.

APPENDIX C. SOLUTION OF THE MAJORITY CARRIER VOLTAGE

DROP EQUATION

A similar analysis is used for the solution of equation (9), so by performing a two-

dimensional Fourier Transform in (9) and using (B.3) we get

1

122

1

22

2

2

2

1)(

1)(~

),(),(

),,(~),,(

~

11 N

i ieffnn

i

n

zR

yx

zR

yx

Ap

pn

yx

yx

RLD

zG

L

ekkBekkA

N

DD

zkkVRdz

zkkVd

(C.11)

where 22

yx kkR .This ordinary differential equation has the following general

solution when R0

1

122

1

2

3

2

21

11

1)(

1)(~

),(),(

),(),(),,(~

11 N

i ieffnn

i

n

zR

yx

zR

yx

Ap

pn

Rz

yx

Rz

yxyx

RLD

zGc

L

ekkBcekkAc

N

DD

ekkBekkAzkkV

(C.12)

where c1, c2, c3 are constants, which can be calculated by differentiating (C.12) with

respect to z twice and equating the result with the right part of (C.11). Therefore, by

completing these operations the general solution may written in the following form


),,(~),(),(),,(~

11 zkknN

DDekkBekkAzkkV yx

Ap

npRz

yx

Rz

yxyx

(C.13)

The constants A1 and B1 may be expressed as a function of the transformed surface

voltage distributions in a similar manner as that of previous section

RwRw

yx

Rw

yxyx

Rw

yx

yxee

wkknekknkwkkVekkVkkA

)),,(~)0,,(~(),,(~

)0,,(~

),(1

1

(C.14)

RwRw

Rw

yxyx

Rw

yxyx

yxee

ekknwkknkekkVwkkVkkB

))0,,(~),,(~()0,,(~

),,(~

),(1

1

(C.15)

where

Ap

pn

N

DDk

1 . The electric field can be calculated through differentiation of

(C.13) with respect to z, as following

n

yxnRz

yx

Rz

yx

yxRz

yx

Rz

yx

yx

yx

eD

zkkJkekkBekkAR

dz

zkkndkekkBekkAR

dz

zkkVdzkkE

),,(~

,,

),,(~

,,),,(

~

),,(~

111

111

(C.16)

The substitution of (C.14) and (C.15) in (C.16) leads to the following expressions for the

electric field on both surfaces as a function of the corresponding voltage distributions

n

yxn

yxyx

yxyxyx

eD

kkJk

Rw

wkknkwkkVRwkknkkkVRkkE

)0,,(~

sinh

),,(~),,(~

coth)0,,(~)0,,(~

)0,,(~

1

1

1

(C.17)

n

yxn

yxyx

yxyx

yx

eD

wkkJk

RwwkknkwkkVRw

kknkkkVRwkkE

),,(~

)coth(),,(~),,(~

)sinh(

)0,,(~)0,,(~

),,(~

1

1

1

(C.18)


Conversely, the surface voltage distributions may be related to the corresponding electric

fields by using (C.17) and (C.18)

)0,,(~

)sinh(

1

),,(~

),,(~

)coth(

)0,,(~

)0,,(~

)0,,(~

1

1

1

yx

n

yxn

yx

n

yxn

yx

yx

kknkRwR

eD

wkkJkwkkE

RwR

eD

kkJkkkE

kkV

(C.19)

),,(~)coth(

),,(~

),,(~

)sinh(

1

)0,,(~

)0,,(~

),,(~

1

1

1

wkknkRwR

eD

wkkJkwkkE

RwR

eD

kkJkkkE

wkkV

yx

n

yxn

yx

n

yxn

yx

yx

(C.20)

Expressions (C.17)-(C.20) are valid when R0 and may be used to solve equation (9) by

application of the algorithm described in section II.4.2. If R=0 the general solution of (9) is

reduced to the following simple form

),0,0(~)0,0()0,0(),0,0(~

111 znkBzAzV (C.21)

so the previously described procedure may be used to find the required relations for the

electric field on both surfaces depending on the voltage distributions and conversely.

3D Simulation of Base Carrier Transport Effects in Back Side Point Contact Silicon Solar Cells

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Transcript of 3D Simulation of Base Carrier Transport Effects in Back Side Point Contact Silicon Solar Cells