Advances in
`Solar Energy
`An Annual Review of
`Research and Development
`
`Volume&
`
`Edited by
`Karl W. Boer
`University of Delaware
`Newark, Delaware
`
`er will bring delivery of each new
`ual shipment. For further informa-
`
`AMERICAN SOLAR ENERGY SOCIETY, INC.
`Boulder, Colorado• Newark, Delaware
`and
`PLENUM PRESS
`New York • London
`
`HANWHA 1058
`
`

`

`The Library of Congress has cataloged this title as follows:
`
`Advances in solar energy.-Vol. 1 (1982)-
`Society, c1983-
`v. ill.; 27 cm.
`Annual.
`ISSN 0731-8618 = Advances in solar energy.
`
`-New York: American Solar Energy
`
`1. Solar energy-Periodicals.
`TJ809.S38
`
`I. American Solar Energy Society.
`621.47'06-dc19
`
`85-646250
`AARC 2 MARC-S
`
`Library of Congress
`
`[8603]
`
`..
`
`ISBN 0-306-43727-9
`
`© 1990 Plenum Press, New York
`A Division of Plenum Publishing Corporation
`233 Spring Street, New York, N.Y. 10013
`
`All rights reserved
`
`No part of this book may be reproduced, stored in a retrieval system, or transmitted
`in any form or by any means, electronic, mechanical, photocopying, microfilming,
`recording, or otherwise, without written permission from the Publisher
`
`Printed in the United States of America
`
`

`

`CHAPTER 4
`
`High-Efficiency Silicon Solar Cells
`Richard M. Swanson and Ronald A. Sinton
`
`4.1 Introduction
`In the last five years silicon solar cells have undergone significant evolution
`resulting in greatly improved efficiencies. As an illustration, Figure 4.1 plots the
`highest reported silicon concentrator cell efficiency versus year. Also shown for
`comparison are gallium-arsenide concentrator cell results. Silicon one-sun cells have
`undergone a similar, but less dramatic, improvement.
`Besides the normal progress obtainable through continued design refinement, the
`major factors enabling this improvement are:
`• incorporation of light trapping,
`• reduction of recombination through improved material quality, cleaner process(cid:173)
`ing, surface passivation and reduced contact coverage fraction, and
`• improvements in understanding of the fundamental physical processes important
`in cell operation such as recombina.tion and carrier transport in heavily doped
`regions.
`Early workers tended to view solar cells as p-n junctions which happen to have
`light-induced generation. This led to simple models of cell operation emphasizing
`diffusion of carriers in low-level injected doped regions (Hovel, 1975). Cell designs
`used thick, rather highly doped wafers. The importance of reducing recombination,
`such as by having high minority carrier lifetime, was recognized early; however, the
`physics of recombination was not sufficiently studied to know what was limiting the
`lifetime, and how high a lifetime could reasonably be attained in various parts of
`the cell.
`An apt analogy to the importance of controlling and reducing recombination is
`provided by a leaky bucket being continually filled with a faucet (Schwartz, 1983).
`The level of water represents the density of electron-hole pairs and the incoming
`stream represents the incident photons. One wishes to attain the highest water level
`(highest carrier density and, hence, voltage) but it is found that every time a leak
`(source of recombination) is plugged the water rises only ·a little until another leak
`takes over. So it is in solar cells; in order to achieve significant increases in output
`voltage it is necessary to examine every part of the cell for potential recombination
`paths and take steps to block those paths. This requires that the internal flow of
`carriers must be well modeled and understood. Our ability to do this continues to
`improve. At this point it seems only fair to reveal that one can now state with
`
`Advances in Solar Energy, Vol. 6
`Edited by K. W. Boer. Plenum Press. New York. 1990
`
`427
`
`

`

`428 High-Efficiency Silicon Solar Cells
`
`30
`
`l
`t
`rfi 20
`[5 u: u.
`
`w
`
`-LIMIT-
`
`...
`. .
`
`A
`
`. .
`.
`
`A
`
`•
`
`. .
`
`A
`
`.
`• .
`
`A
`
`A
`
`a
`
`A
`
`. .
`. .
`
`I
`
`A GaAs
`• SI
`- - - -N + PP +-----+-NEW STRUCTURES-
`10 '---.L.-----'---......_ _ _ _.__ _ _ . _ _~ - -~ - - -
`88
`86
`84
`76
`78
`82
`80
`
`YEAR
`Figure 4.1: Highest reported sili con and gallium arsenide concentrator cell efficiencies
`versus date.
`
`some certainty that the practical limit efficiency for one-sun silicon solar cells at
`room temperature is between 24 and 25 percent, and that for concentrator cells
`lies bet.ween 30 and 31 percent. An implementation of a concentrator system with
`a restricted acceptance angle for the incident light may extend these efficiencies
`somewhat towards the limit of 36-37% (Campbell, 1986). Thus, barring some
`unforeseen breakthrough. laboratory cell efficiencies are rapidly approaching their
`limits. Nevertheless, much work remains to consolidate these recent improvements
`int.a practical, product.ion-worthy designs.
`This article discusses these results. Particular emphasis is directed toward
`concentrator cells which operate in high-inject.ion conditions, as this is where the
`bulk of the authors' experience lies. Most of the material presented in this chapter,
`however, is eq11ally applicable to flat-plate cells. Concentrator cells have historically
`led the drive to higher efficiency because concentrator economics can tolerate the
`complex processing and high-quality materials that are necessary to achieve those
`high efficiencies. Many of the high-efficiency concepts which first appeared on
`concentrator cells, such as passivated emitters, have subsequently found their way
`into the entire range of solar cell applications including one-sun cells.
`Section 2.3.4D presents a simplified view of basic cell operation while Section 4.3
`contains the details of the relevant. semiconductor device physics. Finally, Section 4.4
`discusses the design of high-efficiency cells.
`
`4.2 Basic Cell Operation
`
`We assume that the reader is familiar with the fundamentals of both semicon(cid:173)
`ductor device operation and solar cells. Excellent texts on the basics of solar cell
`operation can be found in references (Green, 1982), (Fahrenbruch and Bube, 1983),
`and more particularly Green, 1987). Rather than trying to re-do what has been done
`
`

`

`4 .2 Basic Cell Operation
`
`429
`
`in these texts, we present here a simplified viewpoint of device operation which has
`been central to our thinking.
`The analysis of silicon solar cells is often couched in terms of the current density
`equations, drift plus diffusion, and the continuity equation. Such an approach ob(cid:173)
`scures the physics of operation, however, because the operation of a well-designed
`solar cell is controlled by generation and recombination. The need for current trans(cid:173)
`port is responsible only for certain second-order losses in the cell. This Section
`presents an analysis of cell operation in terms of an integral formulation of the con(cid:173)
`tinuity equation, which brings forth the balance between generation and recombina(cid:173)
`tion as determining the output of current, and quasi-Fermi potentials as determining
`the output of voltage.
`
`4.2.1 The Semiconductor Device Equations
`The standard equations used to model steady-state carrier transport in silicon
`devices, when heavy doping effects can be neglected, are:
`1. current transport equations
`
`( 4.1)
`
`(4.2)
`
`(4.3)
`
`( 4.4)
`
`(4.5)
`
`(4.6)
`
`(4.7)
`
`Jn = -qµnnV'l/;; + qDn Vn = -qµnn"9¢n
`J~ = -qµpp'9'¢; - qDp '9p = -qµPp'9 </Jp
`2. continuity equations
`
`'9 • fn = q(r - 9ph)
`'9 • J~ = -q(r - 9ph)
`
`3. Poisson's equation
`
`4. carrier density equations
`
`p = n;eq( ¢p -,.t,; )/kT.
`The symbols used have their standard meanings as given below:
`r = Tth - 9th
`
`net thermal recombination rate per unit volume
`(Here, Tth is the recombination rate and
`9th is thermal generation rate.)
`photogeneration rate per unit volume
`electron current density
`hole current density
`electron concentration
`hole concentration
`electron and hole mobilities
`
`9ph
`Jn
`J~
`n
`p
`µn, µp
`
`

`

`430 High-Efficiency Silicon Solar Cells
`
`"
`
`DEVICE
`SURFACE
`
`Dn, DP
`<Pn, ¢p
`1/.•;
`t
`
`Figure 4.2: A general two terminal device.
`
`electron and hole diffusivities
`electron and hole quasi-Fermi potentials
`pot.ent.ial referenced to the intrinsic level
`st.atic dielectric permitivity
`magnitude of electron charge
`ionized doping density
`
`Using Equations (4.1} through ('4.4), the electron and hole current densities
`can be eliminated. The resulting equations, along with Poisson's Equation (4.5),
`define three coupled non-linear differential equations in three unknowns, n, p, and
`1/.•;. Equations (4.6) and
`(4.7) are superfluous but will prove useful in relating
`terrn.jnal voltage to carrier densities. Given appropriate boundary conditions these
`equations can be solved -
`in principle. The search for reasonable analytic solutions
`to these equations is hopeless: however; either some simplifications must be made
`or numerical techniques used. Several approximate analytic approaches will be
`presented below and in Section 4.4.
`
`4.2.2 An Integral Approach to the Terminal Current
`Once a solution for n, p. and if;; is found, either numerically or through judicious
`analytic approximation, the terminal current can be found in either one of two ways;
`by a differential or an integral method.
`
`4.2.2A Differential Method
`Consider the general two terrn.jnal device, shown jn Figure 4.2, with an imaginary
`surface S that surrounds terminal 1. The terminal currents are i1 and i2 = -i1. In
`the steady-state, i 1 may be found by
`
`( 4.8)
`where n is an outward unit vector normal to S and Jn and Jp are given by
`Equations (4.1) and (4.2). Any surface can be used to calculate i1 that surrounds
`cont.act 1, but if one doesn't use e:ract solutions for 1: and J~ ( or equivalently, for
`n, p and 1/.•;) then different surfaces may give different answers.
`
`

`

`4 .2 Basic Cell Operation
`
`431
`
`4.2.2B Integral Method
`Using this method, one integrates the continuity Equations (4.3) and (4.4), over
`the device volume,
`
`J v '\7 • 1: dv = q J v ( r - 9ph) dv
`J '\7,lpdv=-qf (r-gph)dv.
`
`V
`V
`Next, Gauss' divergence theorem is used to convert the left-hand side to surface
`integrals over the device's surface, Stat·
`
`( 4.9)
`
`(4.10)
`
`(4.11)
`
`( 4.12)
`
`j i,, · n dS = q J ( r - 9ph) dv
`j JP · h dS = - q J ( r - 9ph) dv
`
`V
`
`V
`
`S1c,
`
`S, 0 ,
`
`The surface integration is divided into three regions; 5 1 (contact 1), S2 (contact
`2), and 5 (remainder of device) giving
`
`I,
`:I
`,.
`:,
`
`At contact 1 the current is
`
`( 4.13)
`
`(4.14)
`
`( 4.15)
`The minus sign results because the normal vector n is outward while the current
`is referenced inward. Now, solving for - J51
`J~ · iidS in Equation (4.14), inserting
`this into Equation (4.15), and using Equation (4.12) gives,
`
`(4.16)
`
`Let us suppose that contact 1 is p-type and contact 2 is n-type so that Jp and Jn
`are minority carrier current densities in the integrations in Equation (4.16). Since,
`in a solar cell, positive current out of the p-type contact corresponds to positive
`output power it is convenient to define the terminal current I = - ii = iz . Then
`Equation (4.16) can be written in the suggestive form
`
`J = I ph - Jb,rec -
`
`J,,rec -
`
`fconl,rec = Iph - Irec
`
`where
`
`free = Jb,rec + J,,rec + Iconl,rec
`
`(4.17)
`
`(4.18)
`
`

`

`I i
`
`, I
`I
`'
`I I
`I
`l I
`I
`I
`
`I
`
`432 High-Efficiency Si!icon Sola·r Cells
`
`and
`lph = qfvgp1,dv
`h,rec = q J v r dv
`I,,rec = f 5 J~ • n dS
`Iconl,rec = fs2 J~' ndS -
`
`photogeneration current
`bulk recombination
`surface recombination
`contact recombination.
`
`( 4.19)
`
`fs1 ln • iidS
`
`In other words, if one properly accounts for recombination, one can say that the
`output current equals the photogeneration current minus the total recombination
`current.*
`It should be stressed that Equation (4.17) is an exact result of the
`continuity equation.
`The position of the surface over which one integrates to get the contact recombi(cid:173)
`nation is completely arbitrary, as long as it surrounds the contact. If the integration
`surface is moved away from the metal-sem..iconductor contact, then the volume in
`the region between the integration surface and the metal must not be included in
`the recombination integral. This a convenient maneuver when the region around
`the contact is heavily doped and low-level injected. Consider, for example, a met.al
`contact. to a heavily doped n-type region. If the integration surface is moved in from
`the surface to elim..inate this region, the hole current at the new surface is then com(cid:173)
`prised of hole recombination in the doped layer plus the hole surface recombination.
`(This is of course what would have been calculated using by including the region in
`the original volume.) This current is then easily calculated, using conventional low(cid:173)
`level-injection theory, as a function of the hole density at the edge of the integration
`region.
`The integral approach has the advantage that any errors in n and p tend to
`be averaged out when performing the integrations. It also does not depend on
`knowing Vn and Vp. This is an advantage because most analytical approximations.
`or numerical calculation techniques, yield better results for n and p than for their
`gradients. For solar cells, using Equation (4.17) approaches the essence of what is
`desired more closely than the differential method which requires knowledge of the
`carrier fluxes throughout the device.
`
`4.2.3 The Terminal Voltage
`
`A typical advanced solar cell has highly doped regions near the contacts (for
`ohm..ic contact and reduction of contact recombination), and a more lightly doped
`base. This is illustrated in the band diagram in Figure 4.3. The horizontal
`coordinate on this diagram, position, is not necessarily a spatially straight line from
`
`* Sometimes one sees the terminal current written as the short-circuit. current minus
`the "dark current." The short circuit current will be the photogeneration -current minus
`whatever recombination occurs at short circuit .. In t.his case the dark current is then any
`additional recombination obtaining when the device is not at short-circuit. In the simplest
`case of low-level inject ion, the equations defining the carrier density are linear and the dark
`current becomes independent of light intensity and equal to the recombination of the cell
`in the dark at the same junction potential. Such a cell is said to obey the "superposition
`principle." We will see below that narrow base cells also obey the superposition principle.
`In all other cases such a separation cannot be rigorously made. Equation ( 4.17) remains
`true, however.
`
`

`

`4 .2 Basic Cell Operation
`
`433
`
`CONDUCTION
`BAND
`MINIMUM
`
`I
`
`..J
`
`< ll ♦
`
`I Do~;D I METAL
`METAL I DO:ED I
`REGION r- -----, REGION
`
`VALENCE
`BAND
`MINIMUM
`
`•
`
`BASE
`
`Figure 4.3: Solar cell band diagram.
`
`front to back, but just represents a continuous physical path from the p+ to n+
`cont.act areas. The terminal volt.age (less any drop in the contacts and metal bus)
`is </Jp at the p+ contact minus <Pn at the n+ contact.
`To a very good approximation </Jn can be assumed constant through the n+
`region, its space charge region. and into the edge of the quasi-neutral base near the
`n+ contact at position 2. The same applies to </Jp in the p+ region up to position
`1. This is because these regions are heavily doped and hence have an abundance of
`majority carriers. Referring to Figure 4.3 it is easily seen that the output voltage is
`
`where
`
`l-'jn = 1/;; -<!>n = J.:T/qln(n/n;) [evaluated at point 2]
`
`l·jp = </Jp- 1/;; = kT/qln(p/n;) [evaluated at point l]
`
`Ve [contact voltage]
`
`Fm [metal grid voltage].
`
`(4.20)
`
`(4.21)
`
`( 4.22)
`
`(4.23)
`
`(4.24)
`
`( 4.25)
`
`In normal operation, VB, Ve, and Vm are all negative and represent the base,
`contact and metal resistive loss respectively. Detailed methods of calculating Ve and
`Vm are available in the literature (Basore, 1984 ).
`The importance of Equations (4.20) throll·~h (4.25) lies in the need to have
`solutions for n, p, and 1/.•; only in the base senlilc,nduct.or region. The details of what
`happens in the n+ and p+ semiconductor regions are unimportant for determining
`voltage. This is fortunate for, as discussed below, what happens in these regions
`is difficult to model because Equations (4.1) through (4.i) do not apply in very
`highly doped regions. It should be kept in nlind that the above voltage and current
`
`

`

`434 High-Efficiency Sdicon Solar Cells
`
`equations are valid regardless of whether the base is high-levi:!l injected (i.e., has a
`nunority carrier density greater than the doping density) or not.
`Notice that if one neglects the base, contact and metal voltage drops, and
`assumes that n and p are constant throughout the base then Equations (4.21)
`through ( 4.25) imply that
`
`'
`1,~T
`11 :::: -
`ln(pn/n1 ).
`q
`b this case, </>n and <pp are constant throughout the base and
`
`(4.26)
`
`( 4.27)
`the terminal voltage is
`These equations describe the essence of device voltage -
`just the separation of quasi-Fermi potentials.* Equivalently, the pn product appears
`as the major determinant of voltage.
`Transport losses, which are hopefully small in a high-efficiency cell, will decrease
`the voltage below this ideal as shown by Equation ( 4.20 ). If the cell can be well
`modeled as having a series resistance, R., so that VB +Ve + Fm :::: -JR., then the
`above equations imply
`
`( 4 .28)
`
`In many cases this is a reasonable approximation; however, in cells operating in
`high-level injection, a more rigorous calculation of the base voltage is required. This
`will be covered in subsequent sections.
`
`4.2.4 Carrier Photogeneration
`
`A solar cell's first function is to absorb light to produce electron-hole pairs.
`From Equation (4.17) it is immediately apparent that to rnaxinuze efficiency one
`must design the cell to produce as many electron-hole pairs as possible. The number
`of electron-hole pairs produced in t.he silicon by a given amount and spectrum of
`incident light is simply a function of:
`• how many of the photons are transmitted through the cell front surface into the
`bulk, and
`• the path length that t.hese photons can travel once in the bulk silicon before
`leaving or being absorbed in a way which does not produce electron-hole pairs.
`Silicon is weakly absorbing since it has an indirect bandgap. Thus, for practical
`solar cells, a su bst.ant.ial volume of silicon is usually devoted simply to absorbing
`light. In virtually all cells, this volume of silicon is the most lightly clop ed region.
`The amount of light absorbed depends upon the cell thickness and path that the
`light takes through the cell. The details of the optimization to maximize this
`photogeneration are discussed in Section 4.4, Cell Design Engineering. Current,
`st.ate-of-the-art silicon cells incorporating light-trapping produce about 42 mA of
`current for each 100 mW of incident sunlight (AM 1.5 direct spectrum).
`
`* Thermodynamically, this is equivalent to saying t.hat t.he terminal voltage is t.he differ(cid:173)
`ence of the electron and hole electromotive force.
`
`

`

`4 .2 Basic Cell Operation
`
`435
`
`4.2.5 Sources of Recombination
`
`In this sect.ion we explore the impact of various types of recombination on the
`terminal characteristics. The phenomenological equations describing the common
`recombination processes are presented. Detailed discussion of the experimental
`determination of the various coefficients involved is delayed until Section. 4.3, The
`Physical Details of Recombination and Transport.
`To calculate terminal current the various recombination terms in Equation ( 4.17)
`need t.o be evaluated. Various regions of the cell require different approaches.
`
`4.2 .5A Diffusion Region Recombination
`Part of the integration in Equation (4.17) for the bulk recombination involves
`the n+ and p+ diffused regions where no solution, as yet., exists. The same applies
`to contact recombination. This problem can be side-stepped by defining the surface
`of t.he device to skirt the highly doped regions.* The new surface of integration
`is then just inside the neutral base under the contact regions. It is easy t.o see in
`a physical manner why this is possible because t.he net hole Cl}rrent entering the
`n-type diffusion area, for example, either recombines within the diffusion region or
`at the cont.act. Thus t.he same result is obtained by either evaluating the hole current
`at. the edge of t.he space charge region or calculating the net recombination in the
`diffused area plus cont.act. (The photogenerat.ion in the diffused regions under the
`contacts is being neglected.)
`It will be shown in Section 4.3 that the minority carrier hole injection current
`density into a highly doped n-t.ype region can always be written, provided the doped
`region is everywhere low-level injected, as
`
`( 4. 29)
`
`where Jon is a temperature-dependent constant, t.o be called the diffusion saturation
`current, and p and n are evaluated at the internal edge of the neutral highly doped
`region ( del Alamo, 1984). The pn product will be constant to within a very small
`error across the space charge region, so the pn product. at the edge of the space
`charge region in the neutral base may be used instead. Equation ( 4.29) may still
`be used to calculate the injection current; however, any recombination in the space
`charge region will be thereby neglected if this region is also excluded from the
`volume recombinat.ion calculation. In high efficiency cells the space charge region
`recombination is normally very small.
`Jo11 can be considered in two ways; it can be calculated using some particular
`model of transport in heavily doped regions or it can simply be measured for the
`diffusion employed and taken as an experimental parameter.
`Similarly
`
`Jn = Jop ( :~ - 1)
`
`(4 .30)
`
`for the electron current density entering a p-type diffused region.
`
`* As mentioned above, the location of this surface is arbitrary.
`
`

`

`-136 High-Efficiency Silicon Solar Cells
`
`4.2.5B Recombination in the Base
`Recombination in the base of silicon solar cells can occur through a variety of
`mechanisms. Radiative recombination is the inverse of the phot.ogeneration process
`responsible for the operation of solar cells and hence must occur to some degree.
`Other, non-radiative mechanisms usually dominate in silicon. A wide variety of
`material defects, hot.h intrinsic and extrinsic ( or impurity related} can catalyze the
`recombination process. The three-particle process known as Auger recombination,
`which is discussed more fully in Sect.ion 4.3, becomes important at high carrier
`densities.
`Radiative recombination.The simplest to model, if least important, form of
`recombination is radiative recombination. The process is proportional to the excess
`pn product:
`
`r = B(pn - n;)
`(4.31)
`where B is called the radiative rate coefficient. B has an experimental value of
`9.5 x 10- 15 cm3 / s (Schlangenotto, 1974). Direct-gap semiconductors such as GaAs
`t.ypically have radiative coefficients that are several thousand times larger. The
`weak, phonon-rnediat.ed radiative process in silicon means that other recombination
`paths usually dontinate.
`Defect mediated recombination. Any type of crystalline defect can poten(cid:173)
`tially act as a catalyst. for electron-hole recombination. The physical processes
`involved in this recombination are not well understood. The carriers must lose
`their energy through some mechanism. This could be radiative or non-radiative;
`t.he non-radiative process is usually thought to involve energy exchange with lattice
`distortions around the defect. Due t.o the lack of physical understanding, defect(cid:173)
`mediated recombination is usually modeled using t.he phenomenological approach
`of Shockley, Read and Hall (Sze, 1981). Under )ow-level inject.ion conditions, this
`gives a recombir,ation rate proportional to the excess minority carrier density;
`
`( 4.32)
`
`n - no
`r= -- -
`T
`where no is the electron density in equilibrium and Tis the low-level lifetime. l'vlore
`complicated expressions are obtained if one includes the effects of majority carrier
`capture.
`Much of the early work on single crystal silicon cells was concerned with increas(cid:173)
`ing the base lifetime, T, through reduction of crystalline defects. The use of high(cid:173)
`quality, float-zone silicon combined with the incorporation of state-of-the-art senti(cid:173)
`conductor proce~sing has reduced defect levels in laboratory cells to the point that
`hulk, defect-mediat.ed recombination is no longer the dominant loss. Recombination
`in low-cost. commercial cells, particularly poly-crystalline cells, is st.ill dominated
`by defect-mediated recombination, although progress continues. Recent results in
`hydrogen passivation (Schrnidt, 1982) and phosphorus gettering (Narayanan, 1986)
`have allowed improved cell performance.
`This work will concentrate on results achievable in high-quality silicon and,
`hence, defect mediated recombination will play a small role. One type of recom(cid:173)
`bination that is intrinsic to semiconductors, and hence cannot he reduced through
`improved technology, is Auger recombination.
`
`

`

`4 .2 Basic Cell Operation
`
`437
`
`Auger recombination.In an Auger recombination event the excess energy of
`an electron-hole pair is given to a third free particle, either an electron or a hole.
`The recombination rate is then
`
`r =
`
`n n p - n 0 p0 ) + Cp(p n - p0 na)-
`2
`,
`2
`2
`C( 2
`Cn is the n-type Auger coefficient and refers to the process where the excess
`energy is given to an electron. This process will dominate in n-type silicon. C P is
`the corresponding p-type Auger coefficient. In material doped to over 1018 cm- 3
`experiments show that Cn = 2.8x 10-31 cm6 /sand Cp = 0.99 x 10-31 cm6 /s (Dziewior
`and Schmid, 1977). When the doping, or carrier density, is less than 1017 cm- 3 , the
`coefficients appear to be about f.our times larger (Sinton and Swanson, 1987a). These
`coefficients will be discussed more fully in Section 4.3.
`
`(4.33)
`
`4.2.5C Surface Recornbinat"ion
`The only remaining portion of the cell where recombination can occur is at
`the free surfaces between contacted areas. Like bulk defect-mediated recombina(cid:173)
`tion, surface recombination is catalyzed through defects. These defects are usually
`thought to be mostly the dangling bonds of surface silicon at.oms not bonded t.o four
`nearest neighbors. The net recombination rate per unit area is u.rnally assumed to
`be proportional to the excess carrier density so that
`
`on p-type surfaces and
`
`J.,rec = l.,rec/A = qs(n - no)
`
`(4.34)
`
`J•,rec = l•,rec/A = qs(p - Po)
`on n-type surfaces. s is the phenomenological surface recombination velocity. It
`is very dependent on the surface preparation method. Clean, bare silicon surfaces
`have sin the range of 103 to 105 cm/s. Surfaces which are "passivated" through the
`growth of silicon dioxide can have s in the range of 1 to 103 cm/s. Improvements
`in surface passivation have been responsible for considerable improvement in cell°
`performance. The details of surface recombination and passivation are discussed
`more fully in Sect.ion 4.3.
`
`( 4.35)
`
`4.2.6 Simplified Device Operation - The Narrow-Base Cell
`
`In this section we use the previous concepts to analyze cell operation in a
`simplified manner. The main simplification is to assume constant electron and hole
`quasi-Fermi potentials, or equivalently, constant pn product. The voltage is then
`given by Equation (4.26).
`This might be called the "narrow-base" approximation because, if the cell is
`sufficiently thick, gradients in <Pn and </>p will necessarily become significant. We
`are thus ignoring any voltage drops in the base region, which will be small in the
`narrow-base approximation (i.e., VB= 0 in Equation(4.20)). It is possible to make a
`first order improvement in this approximation by including resistance losses through
`a series resistance via Equation ( 4.28).
`
`

`

`438 High-Efficiency Silicon Solar Cells
`
`4.2.6A Current-Voltage Characteristics
`To calculate the current we use Equation (4.17) and evaluate all the sources of
`recombination.
`Diffused region recombination. In this case the n-contact diffused region
`recombination current density is given by Equation ( 4.29)
`
`l)
`and the total current is just the area of the n-contact regions, An-cont, so that.
`
`_ J
`J
`n-cont,rec- -
`on
`
`( qV/kT
`e
`
`-
`
`(4.36)
`
`_ 4
`J
`I
`- n-cont on
`n-cont,rec -
`
`( qVf k,T
`£
`
`-
`
`)
`1
`
`.
`
`(4.37)
`
`Note how these equations have the fam..iliar form for the diffusion current in a
`diode. The current. is actually composed of drift and diffusion components if the
`diffusions are non-uniformly doped.
`Similarly
`
`_
`I
`p-cont,rec -
`
`J
`,1
`-""ip-c ont op
`
`( qF/kT
`£
`
`-
`
`)
`
`1
`
`( 4.38 J
`
`for t.he recombination current in the p-t.ype contact diffusion.
`Recombination in the base. To perform t.he remainder of the bulk recom(cid:173)
`bination int.egration the recombination in the base needs to be evaluated. If it. is
`assumed that t.he base is p-type and low-level injected then
`
`and
`
`n - no
`Rn= Rµ = - - (cid:173)
`Tn
`
`(4.39)
`
`I base ,rec = q r n - no dV.
`Jv
`Tn
`Here, T,, includes the aggregate of all hulk recombination mechanisms: defect.
`Auger and radiative. Noting that, under low-level inject.ion,
`
`( 4.40)
`
`gives
`
`(4.41)
`
`- qA iV n7 ( qV/kT
`N
`ATn
`so that base recombination has t.he same voltage dependence as diffused region
`recombination. Here A is t.he area oft.he device and W is the thickness oft.he base.
`We will find that anytime a region is in low-level injection and t.he recombination
`is linear in excess carrier density then the current will have this dependence on
`term..inal voltage. That solar cells oft.en have high-level injection and non-linear
`recombination effects is the main reason for introducing the above analysis.
`
`( 4.42)
`
`lbase,rec -
`
`I:
`
`)
`
`-
`
`1
`
`

`

`4 .2 Basic Cell Operation
`
`439
`
`If the base is high-level injected, then t·he electron and hole densities will be
`approx.jmately equal by charge neutrality. This implies that
`
`( 4.43)
`
`or
`
`be
`
`n = nieqV /2kT.
`( 4.44)
`Under high level injection, the def ect-mediat.ed recombination will be shown to
`
`n
`RP= -
`Th/
`where Th/ is the high level lifetime ( usually larger than the low level lifetime).
`Assunung constant quasi-Fermi potentials in this case gives
`
`(4.45)
`
`( 4.46)
`
`_ qAl•Vni qV/ 2 kT
`I
`e
`bnse ,rec -
`Th/
`Note that this results in an "ideality fact.or" of two for the case of defect mediated
`bulk recombination in a high-level injected base. This should not be confused with
`the case of space-charge region recombination which often exhibits an ideality factor
`of near two.
`Radiative recombination ( the recombination of an electron-hole pair to produce
`a photon) is not particularly prevalent in silicon but is proportional, nevertheless, to
`the pn product and hence exhibits an "ideality factor" of one in both high-level and
`low-level injection. In Section 4.3.4 it will be shown that Auger recombination often
`dominates in highly injected bases. In high-level injection Equation ( 4.33) shows
`that the recombination will go as Rp = C.4 n 3 where C' .4 = C'n + C p• This gives a
`recombination current of
`
`base ,rec = q
`3 3qVrkT
`Awe
`I
`-
`.4 ni e
`for an "ideality factor'' of 2/3. *
`Surface recombination. The remaining portion of the device is the insulated
`surface between contact regions. Surface recombinat ion, as we will see, proves to
`be a complicated function of excess carrier densities and is thus harder to model. If
`one assumes, however, that the surface region bet.ween contacts is low-level injected,
`and that the recombination is proportional to the excess minority carrier density,
`then
`
`(4.47)
`
`l surf,rec = qs l (n - n 0 ) dA
`
`or, assuming constant quasi-Fermi potentials again,
`_ qAsurrsn~ ( qV /kT _ i)
`e
`
`I
`surf ,rec -
`
`NA
`
`( 4.48)
`
`(4.49)
`
`where Asurf is the surface area of the device.
`
`* This gives an increase in voltage of 40 mV per decade of current, rat.her than the usual
`60 m V per decade at room temperature.
`
`

`

`440 High-Efficiency Silicon Solar Cells
`
`If the surface is high-level injected, this becomes
`
`qV/2kT
`_
`A
`I
`surf ,rec - qs surfn;e
`(4.50)
`.
`To obtain the overall recombination, the effects of all the above are added as
`appropriate. For example, if the cell is in low-level injection throughout, then, from
`Equation (4.17),
`
`- (1 + J
`..L qWn; ..L qAsurrsn;) ( qV/kT - 1)
`J - I/A - J
`-
`-
`ph
`,
`, N
`N
`(4.51)
`A
`.4 Tn
`Noting that t.he recombination current goes as eqV/,kT with 1 = 1, we say that
`the cell has