Conference Paper

NREL/CP-520-45002

February 2009

Computer Simulations of Edge 

Effects in a Small-Area Mesa N-P 

Junction Diode 

Preprint

J. Appel and B. Sopori

National Renewable Energy Laboratory

N.M. Ravindra

New Jersey Institute of Technology

Presented at the Materials Research Society (MRS) 

Fall Meeting 2008

Boston, Massachusetts

December 1-5, 2008
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1

Computer Simulation of Edge Effects in a Small-Area Mesa N-P Junction Diode

Jesse Appel1, Bhushan Sopori1 and N.M. Ravindra2

1

National Renewable Energy Laboratory, Golden, CO 80401, USA.

2

New Jersey Institute of Technology, Newark, NJ 07102, USA.

ABSTRACT

The influence of edges on the performance of small-area solar cells is determined using a 

modified commercial, finite-element software package. The n+

/p mesa device is modeled as 

having a sub-oxide layer on the edges, which acquires positive charges that result in 

development of an electric field within the device. Our computer simulations include generation/ 

recombination at the diode edges as well as the influence of light on the recombination 

characteristics of the edges. We present a description of our model, dark and illuminated 

characteristics of devices with various surface charge concentrations, and the dynamics of carrier 

generation/recombination. The influence of edge geometry on diode performance is determined.

INTRODUCTION

Use of mesa diode arrays, fabricated by chemical etching, is a valuable technique for detailed 

characterization of photovoltaic properties of single-crystalline and multicrystalline silicon 

wafers. The chemical etching process creates a sub-oxide, which is rich in hydrogen and 

passivates the edges. The diode array consists of 100-mil-diameter devices that are etched 3 to 4 

microns deep and have a blanket back Al contact common to all devices. Each device has a front 

Al contact, 10 mils in diameter, for ease in automatic probing. The diodes are electrically 

isolated from one another and can be probed to measure the current density versus voltage curves 

under dark and illuminated conditions [1,2]. We find that the degree of passivation depends 

strongly on the process conditions. To understand the details of the edge passivation, we 

modeled an n-p junction diode using a commercial, finite-element software package. These 

simulations have led to a determination of the self-consistent solution to the continuity equations 

for electrons and holes using the steady-state drift-diffusion model for carrier dynamics coupled 

with electric potential determined from Poisson’s equation [3]. The purpose of these simulations 

is to determine the influence of edge conditions on the overall performance of mesa diodes under 

dark and illuminated conditions. In particular, we examine the effect of edge shape on the I-V 

characteristics of the diode.

The underlying mechanisms of bulk and surface recombination have been well established

for crystalline silicon semiconductor devices [4-7]. We have applied them to our mesa device 

using the COMSOL software. Our simulations show that the space-charge region becomes 

extended along the vertical edge of the mesa diode due to the fixed positive surface charge. At 

the intersection of the vertical edge and step, a strong electric field is produced because it has a 

small convex radius of curvature [8]. Depending on the sharpness of this intersection, the entire 

device can become significantly shunted. Simulations have been performed with a sharp corner 

and a smooth curve at the intersection of the vertical edge and the step. The use of a smooth 
2

curved transition results in significantly lower dark current density versus voltage and a greater 

open-circuit voltage and fill factor under illumination. Yet, even with a curved transition, the 

space charge region can extend approximately 100 microns into a 199.5-micron-thick mesa 

diode and have a bulk recombination rate that is two orders of magnitude greater than the rest of 

the device at low forward biases.

SIMULATION RESULTS

The diode configuration used for modeling is illustrated in Figure 1. Each diode is 100 mils 

in diameter, etched 3.5 mils deep, and has a concentric front metallization of 10 mils. The 

surface of the diode is coated with an antireflection layer having an interface charge of 1x1011

cm-2

. In the current simulation, we assume that the edge of the diode has a thin oxide, and the 

charge density of the oxide is varied from 1x1011cm-2 to 5x1011cm-2. The junction depth is 

assumed to be 0.5 μm.

The mesa diode was simulated under dark and illuminated conditions. Figure 1 shows a 

cross section of the device. Detailed simulations have shown that charge loss can occurs via a 

number of mechanisms, which include: carrier flow from the diode to the edge, recombination by 

the field produced at the edge, which propagates radially into the diode and through the thickness 

of the diode, and recombination at the etched surface. 

Figure 1. Cross section of modeled mesa diode with bulk and surface conditions

Dark Condition

The dark current density versus voltage plots of the simulations are shown in Figure 2. The 

current density increases with increasing surface charge. The initial increase in dark current 

density is due predominantly to recombination near the mesa’s edge. The greatest increase 

occurs at low voltages and is due to resistive shunting, which is related to the distortion of the 

electric field at the edge of the mesa. One of the most important results of the simulations 

performed on the mesa diode is the calculation of a high point in the electric field at the corner of 

the vertical edge and the step. The resultant current density versus voltage curves for the 
3

simulations in Figure 2 were performed with a corner that was a right angle with a sharp point. 

To determine if this feature of the mesa diode was responsible for the increase in dark current 

density, similar simulations were run with a curved transition rather than a sharp corner. A 

small, curved piece of p-type semiconductor was added at the corner of the vertical edge of the 

mesa and the step. It was defined as a second-order Bezier curve with a height and length of 

3.5 μm. This height was chosen because the junction is 3.5 μm above the step. The bulk 

1.E-09

1.E-07

1.E-05

1.E-03

1.E-01

1.E+01

1.E+03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Voltage (volts)

Current Density (mA/cm^2)

1x10^11

3x10^11

5x10^11

Edge and Step Surface Charge [cm^-2]

Figure 2. Dark current density vs. voltage for the mesa diode at different vertical edge and step 

surface charges

recombination rate and electric field for simulations with the vertical edge and step surface 

charges of 5x1011cm-2 are shown for sharp and curved corners in Figure 3. The addition of a 

curved piece at the mesa’s edge clearly reduces the bulk recombination rate in this region. 

However, the curved piece has a smaller radius of curvature than the flat continuous sections 

away from it. Therefore, even with an additional curved piece, the bulk recombination rate 

Sharp Corner: Surface charge on vertical edge and 

step = 5x1011 cm-2

Scale

[m-3s

-1

]

Curved Transition: Surface charge on vertical 

edge and step = 5x1011 cm-2

Figure 3. Comparison of bulk recombination rate (logarithmic) and electric field at 0.1 volt 

forward bias for a mesa diode with sharp and curved corners with a surface charge of 5x1011 

cm-2 under dark conditions.
4

and electric field in this region are greater than the rest of the device except for the junction, 

where they are almost equal. Consequently, the results of the dark simulations of the mesa diode 

indicate that the distortion of the electric field at the corner of the junction and the vertical edge, 

and the step and the vertical edge in conjunction with the inversion layer on the p-side adjacent 

to the junction, results in an extension of the space-charge region deep into the bulk of the

device. This leads to an increase in the recombination rate and the dark current, particularly at 

low forward biases.

A comparison of the dark-current density versus voltage for sharp and curved corners is 

shown in Figure 4 for a surface charge of 5x1011

cm-2. By increasing the radius of curvature of 

corner at the step and vertical edge, the resistive shunting caused by excessive bulk 

recombination can be reduced significantly. For a surface charge of 5x1011cm-2, the reduction 

of dark current density is almost four orders of magnitude at low forward biases.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Voltage (volts)

Current Density (mA/cm^2) 

Sharp Edge

Curved Transition

Figure 4. Current density vs. voltage plots of the comparison of sharp and curved transitions

between the step and the vertical lines with a surface charge of 5x1011cm-2 under dark 

conditions.

Illuminated Condition

Simulations of the mesa diode were also performed under illuminated conditions, in which 

only the vertical-edge surface charge was changed from 1x1011cm-2 to 5x1011cm-2. The 

resulting illuminated current density versus voltage curves are shown in Figure 5. The opencircuit voltage and fill factor become reduced as the vertical-edge surface charge is increased 

from 1x1011cm-2 to 5x1011cm-2. The loss in Voc and FF are significant, and the mechanism that 

causes them as the charge increases is a result of increased recombination due to the increased 

electric field caused by the sharp corner at the edge of the mesa. 

To determine if the sharp corner at the edge of mesa was the cause of the reduction in opencircuit voltage and fill factor, simulations under illumination were performed with a curved piece 

at the corner of the vertical edge and the step. It had exactly the same dimensions as the one 

used for the dark simulations. Also, the simulations using a sharp corner were repeated with the 

surface charge of the vertical edge and the step set to the same value.
5

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Voltage (volts)

Current Density (mA/cm^2)

1x10^11

2x10^11

3x10^11

5x10^11

Vertical Edge 

Surface Charge 

Note: Curves for 1x10^11 

and 2x10^11 cm^-2 surface 

charges are almost the same.

Figure 5. Illuminated current density vs. voltage for the mesa diode at different vertical edge 

charges.

The illuminated current density versus voltage plots for simulations performed with a sharp 

corner and a curved transition with a surface charge of 5x1011

cm-2 are shown in Figure 6. The 

simulation with a curved transition has a greater open-circuit voltage and fill factor. This occurs 

because there is a lower recombination rate in the mesa diodes modeled with a curved transition. 

It should be noted that the short-circuit current (Jsc) values for the simulations shown in Figure 6 

are significantly greater than the Jsc values shown in Figure 5; this is because the surface charge 

was increased on the step adjacent to the mesa’s edge. This results in an inverted surface, which 

significantly reduces the recombination rate in this region, and consequently leads to an 

improved Jsc value. 

0

5

10

15

20

25

30

35

40

45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Voltage (volts)

Current Density (mA/cm^2)

Sharp Edge

Curved Transition

Figure 6. Current density vs. voltage plots of the comparison of sharp and curved transitions 

between the step and the vertical lines with a surface charge of 5x1011

cm-2 under illuminated 

conditions.
6

CONCLUSIONS

The geometric features of the edge of the mesa diode can significantly distort the electric 

field throughout a large portion of the bulk device. The most important feature occurs at the 

intersection of the vertical edge and step because an electric field becomes large near areas

having a convex radius of curvature and can reach extreme values at sharp points [8]. In certain 

cases, the distortion can expand the space charge region over hundreds of microns in all 

directions emanating from the edge. It was shown that changing the interface from a sharp 

perpendicular step to a Bezier curve in the simulations can greatly reduce the dark-current 

density and increase the open-circuit voltage and fill factor. This is important because the 

chemical etch used to delineate the mesa diodes is isotropic and leaves behind a curved 

transition. Even with a curved interface at the vertical edge and step, the edge effect is a 

significant source of recombination in the mesa diode, in which the recombination rate can be 

more than two orders of magnitude greater than the rest of the bulk and this region can extend 

approximately 100 μm into the device from the curved interface. Therefore, the edge effect in a 

mesa diode is a result of the space-charge region extending along the vertical edge because of the 

inverted region on the p-side of the junction, then expanding deep into the bulk device because 

of the strong electric field at the intersection of the vertical edge and the step. This results in a 

significant increase in recombination rate throughout the device, an increase in dark-current 

density, and a reduction in open-circuit voltage and fill factor under illumination. 

ACKNOWLEDGEMENTS

We would like to thank Peter Rupnowski for his help with calculations and many valuable 

discussions.

REFERENCES

1. B.L. Sopori, Applied Physics Letters 52 (20), 1718-1720 (1988).

2. B.L. Sopori, Journal of Applied Physics 64 (10), 5264-5266 (1988).

3. H.K. Gummel, IEEE Transactions on Electron Devices 11, 455-465 (1964).

4. R.N. Hall, Physical Review 87, 387 (1952).

5. W. Shockley and W.T. Read, Jr., Physical Review 87 (5), 835-842 (1952).

6. A.S. Grove, B.E. Deal, E.H. Snow and C.T. Sah, Solid-State Electronics 8, 145-163 (1965).

7. D.J. Fitzgerald and A.S. Grove, Surface Science 9, 347-369 (1968).

8. Raymond A. Serway, Physics for Scientists & Engineers, 3rd ed. (Saunders College

Publishing, Philadelphia, Vol. II, p. 692 (1992).
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Computer Simulations of Edge Effects in a Small-Area Mesa N-P 

Junction Diode: Preprint

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DE-AC36-08-GO28308

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J. Appel, B. Sopori, and N.M. Ravindra

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14. ABSTRACT (Maximum 200 Words)

The influence of edges on the performance of small-area solar cells is determined using a modified commercial, 

finite-element software package. The n+/p mesa device is modeled as having a sub-oxide layer on the edges, which 

acquires positive charges that result in development of an electric field within the device. Our computer simulations 

include generation/ recombination at the diode edges as well as the influence of light on the recombination 

characteristics of the edges. We present a description of our model, dark and illuminated characteristics of devices 

with various surface charge concentrations, and the dynamics of carrier generation/recombination. The influence of 

edge geometry on diode performance is determined.

15. SUBJECT TERMS

PV; computer simulations; edge effects; small-area; solar cells; mesa diodes, n-p junction; surface charge 

concentrations;

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