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ISSN : 1225-0562(Print)
ISSN : 2287-7258(Online)
Korean Journal of Materials Research Vol.30 No.10 pp.509-514
DOI : https://doi.org/10.3740/MRSK.2020.30.10.509

# Simulation on Optimum Doping Levels in Si Solar Cells

Kwang Su Choe†
Dept. of Electronic Materials Engineering, College of Engineering, The University of Suwon, Hwaseong-Shi, Gyeonggi-Do 18323, Republic of Korea
Corresponding author E-Mail : kschoe@suwon.ac.kr (K. S. Choe, Univ. Suwon)
October 16, 2020 September 11, 2020 October 5, 2020

## Abstract

The two key variables of an Si solar cell, i.e., emitter (n-type window layer) and base (p-type substrate) doping levels or concentrations, are studied using Medici, a 2-dimensional semiconductor device simulation tool. The substrate is ptype and 150 μm thick, the pn junction is 2 μm from the front surface, and the cell is lit on the front surface. The doping concentration ranges from 1 × 1010 cm−3 to 1 × 1020 cm−3 for both emitter and base, resulting in a matrix of 11 by 11 or a total of 121 data points. With respect to increasing donor concentration (Nd) in the emitter, the open-circuit voltage (Voc) is little affected throughout, and the short-circuit current (Isc) is affected only at a very high levels of Nd, exceeding 1 × 1019 cm−3, dropping abruptly by about 12%, i.e., from Isc = 6.05 × 10−9 A·μm−1, at Nd = 1 × 1019 cm−3 to Isc = 5.35 × 10−9 A·μm−1 at Nd = 1 × 1020 cm−3, likely due to minority-carrier, or hole, recombination at the very high doping level. With respect to increasing acceptor concentration (Na) in the base, Isc is little affected throughout, but Voc increases steadily, i.e, from Voc = 0.29 V at Na = 1 × 1012 cm−3 to 0.69 V at Na = 1 × 1018 cm−3. On average, with an order increase in Na, Voc increases by about 0.07 V, likely due to narrowing of the depletion layer and lowering of the carrier recombination at the pn junction. At the maximum output power (Pmax), a peak value of 3.25 × 10−2 W· cm−2 or 32.5 mW· cm−2 is observed at the doping combination of Nd = 1 × 1019 cm−3, a level at which Si is degenerate (being metal-like), and Na = 1 × 1017 cm−3, and minimum values of near zero are observed at very low levels of Nd ≤ 1 × 1013 cm−3. This wide variation in Pmax, even within a given kind of solar cell, indicates that selecting an optimal combination of donor and acceptor doping concentrations is likely most important in solar cell engineering.

## 초록

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 1. Introduction

In the sense that the photovoltaic power generation is driven by the potential energy of electrons elevated by absorbing sun light, it is akin to hydroelectric power generation where the energy source is the potential energy of water molecules elevated by absorbing heat from the same sun light. A solar cell, or a pn junction, where the photovoltaic effect takes place should thus share many common physical attributes with a hydroelectric dam or a waterfall. In another words, the attributes that make a waterfall great should also apply to a pn junction. The pn junction being invisible to naked eyes, this waterfall analogy is very useful in understanding and visualizing the workings of a pn junction and has been used in the past by the present author.1)

To maximize the use of potential energy, a waterfall must have a high elevation and a stiff drop, or respectively a pn junction must have a high barrier height and a narrow depletion layer. To maximize flow, a waterfall needs hard bedrocks preventing water seepage, or a pn junction needs defect-free materials preventing carrier loss by recombination. All these three attributes concerning the pn junction, i.e., barrier height, depletion layer, and carrier recombination, are affected by the doping levels in the p- and n-type regions. In theory, a higher doping level, in both the p- and n-type regions, leads to a higher barrier height and a narrower depletion layer, but not a lower carrier recombination. Thus, an optimum doping level or a combination of doping levels needs to be defined to satisfy these diverging requirements.

An issue key to understanding proper doping is solar cell design. Currently, the prevailing Si solar cell design is front-lit or front-junction where the pn junction is formed very near the front surface that faces the sun. In its fabrication, the substrate (base) is usually doped ptype, and on it a thin layer of n-type window layer (emitter) is formed by thermal diffusion. For electrodes, thin grid lines of cathode, usually of Al or Ag, are deposited on the n-type window layer, and a full anode layer, usually of Al, is deposited on the rear (substrate). In this solar cell structure, the relative levels of doping concentration in the p- and n-type regions are closely linked with the relative thicknesses of the p- and n-type regions. That is, the p-type region which is at a higher energy level needs to be relatively much thicker, so that more light absorption, or free electron-hole pair generation, takes place in that region as much as possible. So, the substrate itself is usually chosen as p-type. On this substrate, the n-type window layer is formed by doping to the degenerate level, i.e., above 1×1018 cm−3, so that it can be made as thin as possible and, at the same time, does not have any carrier depletion.

An added benefit in having p-type substrate is that the minority carriers are electrons that diffuse faster than holes, resulting in less minority-carrier recombination in the substrate. Had the substrate been n-type, the minoritycarrier holes would be more prone to recombination due to their slower speed. A drawback in using the p-type substrate is light-induced degradation (LID) known to exist in the p-type Czochralski (CZ) silicon associated with boron-oxygen complexes.2) The Si substrate, the ptype region, is usually doped to a much lesser degree than the n-type layer, mainly to prevent carrier depletion in the latter layer and also to minimize minority-carrier recombination within itself.

In regards to energy conversion efficiencies, for frontlit Si solar cells, the highest efficiency reported to date for an actual cell is 24.7 % by a method known as PERL (Passivated Emitter, Rear Locally-diffused),3) while the more typical value accepted in laboratory-scale solar cells is about 20 %, or 20 mW·cm−2 under AM1.5.4) In production, the efficiency further drops to 17 ~ 18 %, which calls for improvement. Together with substrate and window layer thicknesses, the respective doping levels are two key variables directly influencing the solar cell performance - the energy conversion efficiency and ultimately the power output. The extent of their influence is one that needs to be studied comprehensively.

Typical doping levels used in the industry is approximately 1.5×1016 cm−3 (1 Ω·cm) for the base and 3 × 1019 cm−3 peak (100 Ω/sq) for the emitter. These values, derived experimentally, are not always necessarily optimal for evolving Si substrate designs. In the field of computer simulation, optimization has been generally done utilizing PC1D, a widely used 1-dimensional solar cell software. A recent PC1D study indicates an optimal base doping range of Na = 1.84×1017 - 2.68×1017 cm−3 at a fixed emitter doping level of Nd = 7.5×1018 cm−3 for 300 ~ 320 μm thick Si substrates.5) Due to limited doping ranges reported thus far, a more expanded study is being called for.

In this work, a computer simulation study was conducted in lieu of laboratory experiments by using a 2- dimensional semiconductor device simulation software package called Medici.6) Despite limitations of computer simulation, the results will show by means of quantitative trends in key performance factors, such as short-circuit current, open-circuit voltage, and maximum power output, the most desirable combination of p- and n-type doping concentrations in Si solar cells.

## 2. Experimental Procedure

In computer simulation, the computer is a virtual laboratory. That is, the solar cell fabrication is moved from a real laboratory into the computer. When an experiment was to be repeated many times to the extent that it is not practically possible in a real laboratory, the mathematical modeling and computer simulation is a viable alternative. A graphic version of a solar cell is made in the virtual space step by step, emulating the fabrication steps in a real laboratory, and an essential variable or variables can be tested and investigated in iterative manner. In this study of Si solar cells, two essential variables are emitter (n-type window layer) and base (p-type substrate) doping concentrations.

For simulation, the 2-dimensional version of Medici, a well-known semiconductor device simulation software tool, was utilized. As in any device simulation, the Medici simulation consists of four major steps. First, the solar cell structure is formed by a grid or mesh, sort of like constructing a modeled building using match sticks. The regions near the front and back surfaces and at the pn junction where active photo-generation and/or carrier transport occur are finely meshed, and the bulk region deep in the body of the solar cell, where the electrical activities are relatively steady and not as much spacedependant, is coarsely meshed to save computation time. Each type of the p- and n-type regions is then doped to a desired level. Second, the physical models, expressed in differential equations, relating to the operation of the solar cell and the mathematical methods of solving them are each specified. Third, the solutions are reached by iterative numerical calculations. Fourth, the simulation results or the outputs are accessed in texts, plots or figures.

For the simulation, a standard 2-dimensional back-lit grid structure in the Medici operational manual6) is modified to fit the front-lit design. In this front-lit structure, the substrate is 150 μm thick and doped p-type, and the window layer is 2 μm thick and doped n-type. Thus, the location of the pn junction is 2 μm below the front surface. Laterally, only a 10μm segment of the substrate is defined to conserve the simulation time. The front electrode (cathode) is 1μm wide and centered on this 10 μm segment, and the back electrode (anode) covers the whole rear surface. The said grid structures had been used successfully and illustrated in earlier studies.7,8)

The doping concentration, the experimental variable, ranges from 1×1010 cm−3 to 1×1020 cm−3 for both the substrate and the window layer. Thus, a matrix of 11 by 11, or a total of 121 data points, was experimented in the simulation study.

On physical models, the Auger and concentrationdependent Shockley-Read-Hall recombination models and the concentration-dependent mobility model are chosen. On equation solving, the 2-carrier Newtonian iteration is chosen as the mathematical method in solving the governing differential and continuity equations by finite element analysis. The information on thermal processing conditions, which is important in fabrication as it affects the pn junction formation, is not an input variable in Medici which is a device-level simulator.

The photo-generation rate, electron-hole-pairs·cm−3·s−1, is an exponential function of distance y from the surface and can be expressed as

$G p h o t o n = F L U X exp [ − y Y . C H A R ] 10 − 4 × Y . C H A R$
(1)

where FLUX is defined as the photon flux (photons·cm−2· s−1) and Y.CHAR is defined as the absorption distance (μm).6) The photon absorption is strongly frequencydependent. That is, higher the frequency, or shorter the wavelength, the stronger the photon absorption and therefore the shorter the absorption distance. For the present simulation experiments, the wavelength of 0.6 μm, the peak point in the AM1.5 spectrum having an absorption distance of 2 μm, is chosen for the incident photons. The photon flux, directed normal to the surface, is set at 4×1017cm−2·sec−1. For these absorption distance and photon flux, the Eq. (1) can be rewritten as

$G p h o t o n = 2 × 10 21 e − y 2 μ m c m − 3 − sec − 1$
(2)

Finally, after the simulation, the output is plotted in terms of the photovoltaic equation as expressed in

$I = I o ( e q V k T − 1 ) − I s c$
(3)

From the I-V curve plotted, the open-circuit voltage, VOC, is extracted from where the curve intersects the xaxis, and the short-circuit current, ISC, is extracted from where the curve intersects the y-axis. The cell power as defined by the maximum power rectangle in a loaded circuit is then extracted.

## 3. Results and Discussion

The grid structure of the front-lit Si solar cell formed in Medici is illustrated in Fig. 1. The front and back electrodes and the sun are added to the figure. As the absorption distance, i.e., the distance at which the intensity of the light is decreased to 1/e of the initial value, is 2 μm for the 0.6 μm wavelength light used in the simulation, the 63 %, 86 %, and 95 % of the incident photons would be absorbed within one, two and three absorption distances or 2 μm, 4 μm, and 6 μm from the front surface, respectively. Thus, the most of light absorption, or free electron-hole pair generation, takes place within the distance of 6 μm from the front surface, or mostly near the pn junction. Since the generation sites are not very far from the pn junction, the minority-carrier recombination should not be significant except at the pn junction where the two minority carriers meet. In this regard, the grid in Fig. 1 is dense near the front surface and more so at the pn junction.

Effects of n-type emitter doping or donor concentration (Nd) and p-type base doping or acceptor concentration (Na) viewed separately are illustrated in Figs. 2(a) and 2(b), respectively. With regard to increasing donor concentration, for a fixed acceptor concentration, Fig. 2(a) shows that the open-circuit voltage (Voc) is little affected throughout and that the short-circuit current (Isc) is affected only at a very high donor concentration exceeding 1×1019 cm−3, where it drops abruptly by about 12%, i.e., from Isc = 6.05×10−9 A·μm−1 at Nd = 1×1019 cm−3 to Isc = 5.35×10−9 A·μm−1 at Nd = 1×1020 cm−3. This indicates that the minority-carrier, or hole, recombination in the emitter layer is not an issue up until the very high doping level. The near constancy in open-circuit voltage indicates that the depletion region at the pn junction is mostly all formed in the p-type base and as such any additional donor doping in the n-type emitter has little effect.

With regard to increasing acceptor concentration, for a fixed donor concentration, Fig. 2(b) shows that the shortcircuit current is little affected throughout, but the opencircuit voltage increases steadily. i.e., from Voc = 0.29 V at Na = 1×1012 cm−3 to Voc = 0.69 at Na = 1×1018 cm−3. On average, per an order increase in acceptor concentration, the open-circuit voltage increases by about 0.07 V, indicating the narrowing of the depletion layer and the lowering of the carrier recombination at the pn junction.

The overall effect of donor and acceptor doping on the maximum output power (Pmax) is illustrated in Fig. 3. Of the 121 data points in the 11×11 matrix experimented, no viable solution was reached when Na ≥ Nd or Na ≥ 1×1019 cm−3, i.e., situations where the 2 μm n-type layer is likely becoming depleted of carriers, and the good data points collected are plotted in the figure. In this threedimensional illustration resembling a graded hill, a ridge leading up to the peak is visible along the donor concentration of 1×1019 cm−3, a level at which Si is degenerate (being metal-like). The peak is reached when the acceptor concentration is increased to 1×1017 cm−3. The maximum output power at this peak point of Nd = 1×1019 cm−3 and Na = 1×1017 cm−3 is 3.25×10−2 W·cm−2 or 32.5 mW·cm−2. The maximum output power is near zero near the base of the hill where Nd ≤ 1×1013 cm−3. This elevation from zero to 32.5 mW·cm−2 in the maximum output power indicates that a wide range of power output is possible even for a given kind of solar cell and that selecting an optimal combination of donor and acceptor doping concentrations is likely most important in solar cell engineering.

Of the two doping parameters, the donor doping in the emitter was less robust than the acceptor doping in the base in influencing the output power, as indicated by the gentler slopes on the donor side (Nd) of the 3-dimensional hill in Fig. 3. This is because the emitter itself is much thinner than the base, which in effect allows photongenerated minority carriers (holes) to diffuse out of the emitter faster and be less affected by the presence of doped majority carriers (conduction electrons) nearby.

A note of caution is that the optimal doping combination leading to the maximum power output is not absolute, but relative to other solar cell parameters, such as minoritycarrier recombination lifetime and substrate quality and thickness. In an earlier study, it was shown that an output power loss is expected if electron recombination lifetime is shorter than 50 μs.8) It is also known that the substrate thickness and the required minority-carrier recombination lifetime are nearly linearly related, i.e., the thicker the substrate, the longer the lifetime. These results indicate that the substrate doping should be made lower for lower quality silicon substrates having shorter minority-carrier recombination lifetimes or that it could be made higher for thinner substrates. Thus, the optimal doping combination found in this study should be used as a guide and adjusted according to changing conditions. The experimental verification of the findings made in this study is left for the future study.

## 4. Conclusions

The relationship between two key variables of the Si solar cell, i.e., emitter (n-type window layer) and base (ptype substrate) doping levels or concentrations, was studied by using Medici, a 2-dimensional semiconductor device simulation tool. The doping concentration, the experimental variable, ranged from 1×1010 cm−3 to 1×1020 cm−3 for both the window layer and the substrate. Thus, a matrix of 11 by 11, or a total of 121 data points, was experimented. With respect to increasing donor concentration (Nd) in the n-type window layer (emitter), the open-circuit voltage was little affected throughout, and the short-circuit current was affected only at a very high donor concentration exceeding 1×1019 cm−3, where it dropped abruptly by about 12 %, indicating that the minority carrier, or hole, recombination in the emitter layer was not an issue up until the very high doping level. With respect to increasing acceptor concentration (Na) in the p-type substrate (base), the short-circuit current was little affected throughout, but the open-circuit voltage was greatly affected, i.e., on average increasing by about 0.07 V per an order increase in acceptor concentration, possibly due to narrowing of the depletion layer and the lowering of the carrier recombination at the pn junction. On the maximum output power. a peak value of 3.25×10−2 W·cm−2 or 32.5 mW·cm−2 was observed at the doping combination of Nd = 1×1019 cm−3, a level at which Si is degenerate (being metal-like), and Na = 1×1017 cm−3, while minimum values of near zero were observed at very low doping combinations. This wide variation even for a given kind of solar cell indicates that selecting an optimal combination of donor and acceptor doping concentrations is likely most important in solar cell engineering. The optimal doping values of Nd = 1×1019 cm−3 and Na = 1×1017 cm−3 found in this work for Si substrate thickness of 150 μm and minority-carrier recombination lifetime of 50 μs are left for verification in future experimental studies.

## Acknowledgement

The simulation experiments and the rendering of the graphical work were assisted by undergraduate students, Messrs. Sang-Hoon Lee, Jong-Min Moon, Ji-Min Kim and Misses In-Young Kim, Yeh-Lim Bang, Soo-Hyeon Sohn, Mi-Jin Sohn of the Department of Electronic Materials Engineering of the University of Suwon. The author wishes to thank them for their diligent assistance. This paper was supported by the research grant of the University of Suwon in 2017.

## Figure

The 10 μm × 150 μm grid structure of the front-lit Si solar cell formed by Medici: the pn junction is located at 2 μm below the top (front) surface, the p-type substrate is 150 μm thick, and the top electrode is 1μm wide and centered on the n-type window layer (the x- and y-axis labels indicate distance in microns).

The I-V characteristics of the front-lit Si solar cell: (a) increasing donor concentration (Nd) while acceptor concentration (Na) is fixed and (b) increasing Na while Nd is fixed (the x-axis label indicates substrate voltage in volts, and the y-axis label indicates substrate current in 10−9 A·μm−1).

Maximum output power (Pmax) as a function of donor concentration (Nd) and acceptor concentration (Na). The peak value (solid red dot) is 32.5 mW·cm−2 at Nd = 1×1019 cm−3 and Na = 1×1017 cm−3.

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