Solar Cells, 17 (1986) 85-118
`
`85
`
`POINT-CONTACT SOLAR CELLS: MODELING AND EXPERIMENT
`
`R. M. SWANSON
`
`Department of Electrical Engineering, Stanford University, Stanford, CA 94305 (U.S.A.)
`
`(Received August 25, 1985; accepted August 26, 1985)
`
`Summary
`
`A new typeofsilicon solar cell designed for high concentration applica-
`tions, the point-contact solar cell (PCSC), is discussed. It is predicted that
`the PCSC is capable of an efficiency of 28% at the design point of 500X
`geometric concentration and 60°C cell temperature. This paper discusses
`the modeling of this device and presents recent experimental results which
`have obtained 23% efficiency at 200X and 21% at 500X, both at a tempera-
`ture of 24 °C.
`
`1. Introduction
`
`Silicon solar cells designed for high concentration present difficult
`design constraints. High base doping density is needed for low seriesresist-
`ance, yet high doping levels reduce minority carrier diffusion length and
`reduce the benefit of a back surface field. Similar problems appear in the
`emitter and grid. These constraints make it difficult to envision a conven-
`tional concentrator cell that has over 22% efficiency at high intensities [1].
`Various approaches have been attempted to circumvent some of the
`problems with conventional cells [2]. In the etched vertical multi-junction
`cell (EVMJ), for example, deep grooves are etched in the front to allow the
`front surface grid to be much thicker, and hence of lowerresistance [3]. In
`this manner the top junction is also brought closer to where electron-hole
`pairs are generated in the base, improving collection and decreasing series
`resistance. The EVMJ, however, suffers from a very large contacted emitter
`area and hence low output voltage. Another unconventional design is the
`interdigitated back-contact cell [4]. In this cell both the n- and p-type diffu-
`sions are interleaved on the back surface and contacted with two interdigitated
`comb-shaped metal bus systems. The cell is usually made of lightly doped
`silicon and operates in high injection. Following ref. 2 the advantages of this
`cell are as follows.
`1. There is no shading of the front by the contact grid.
`
`0379-6787/86/$3.50
`
`© Elsevier Sequoia/Printed in The Netherlands
`
`HANWHA1051
`
`HANWHA 1051
`
`

`

`86
`
`2. The metal contacts on the rear may cover nearly the entire cell and
`be quite thick, thus reducingseries resistance.
`3. The emitter diffusions may be optimized for low dark current because
`there is no needfor lateral flow to the contact regions and thereis very little
`generation in them because they are on the back.
`4. The illuminated front surface does not contain any diffusions and as
`a result may have a very low surface recombination velocity. This yields good
`blue response.
`5. The free carriers are generated in a lightly doped base and hence long
`diffusion lengths can be expected.
`The IBC cell also has some drawbacks, the most serious of which is
`mounting difficulty; it is necessary to remove both types ofcarriers as well
`as the waste heat from the back side. Generally, these efforts have yielded
`disappointing results with efficiencies less than that currently achieved with
`highly refined conventionalstructures [5].
`It has long been recognized that an important path to improved effici-
`ency is through reducing carrier recombination. There have been attempts to
`reduce the emitter component of the dark current, and thus increase output
`voltage, by decreasing the contact coverage fraction both for the front
`and back contact [6]. Nevertheless, these cells also have had performance
`inferior to conventional concentrator cells. An apt analogy to controlling
`and reducing recombination is provided by a leaky bucket being continually
`filled with a faucet [7]. 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 onlyalittle until another leak is found. 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 flows of
`carriers must be well modeled and understood.
`The point-contact solar cell has been proposed and has shown promising
`performance [8]. It represents one attempt to re-think cell design for higher
`performance. Much attention has been given to reducing recombination as
`well as maximizing photocurrent. In Section 2 the structure of the PCSCis
`presented and in Section 3 its operation is discussed to illustrate the design
`philosophy. Section 4 presents a model of PCSC operation that has been
`used to both design experimental devices and assess its mature potential.
`Finally, in Section 5 some experimental results are shown.
`
`2. Structure of the point-contactsolar cell
`
`The structure of a point-contact solar cell is shown in Fig. 1. The top
`diffused junction that covers the entire surface of a conventional solar cell
`has been eliminated and the p and n diffusions are relegated to an array of
`
`

`

`a* Bussbar
`
`87
`
` 120A SiO,
`
`
`
`he
`
`Fig. 1. Cross-sectional structure of the PCSC.
`
`small points on the back surface. The back surface metalization is in the
`form of two interdigitated comb structures; one set of fingers contacting the
`n diffusions (becoming the negative contact) and the other contacting the p
`diffusions (becoming the positive contact). The structure is thus similar to
`the interdigitated back contact cell in that it has both electrical contacts on
`the back side [4]. In the PCSC, however,rather than having alternating n and
`p fingers, the contact metal touches thesilicon only in an array of points on
`the back surface. The contact areas contain small diffused regions which
`alternate between n-type and p-type in a checkerboard fashion. The top
`surface and the regions between contacts on the bottom are covered with
`SiO, for surface passivation. The major advantage of restricting the contact
`coverage to small points is, as described in Section 3, that it increases the cell
`output voltage.
`The base material is high resistivity float-zone silicon and is nominally
`80 um thick. As will be discussed, various thicknesses have been explored,
`both by modeling and by experiment, and it is found that the cells must be
`quite thin in order to maintain good quantum efficiency at high intensities.
`In its fully-developed proposed form, thecell will be texturized on the
`top surface and an anti-reflection coating applied. Neither of these features
`has been incorporated into the current experimental cells discussed in Section
`5. In addition, a mounting scheme has been proposed that allowsthe cell to
`be soldered down to a header, much as a conventional cell is, but also has
`not yet been incorporated in the devices reported here [9].
`
`3. Operation
`
`In order to obtain optimum performance,a solar cell must (a) absorb
`as much light as possible in electron-hole production, (b) transport the
`largest possible fraction of these electrons and holes to their respective
`
`

`

`88
`
`terminals and (c) do so at the highest possible terminal voltage. The way the
`PCSC optimizes each of these processes will be discussed in turn.
`
`3.1. Optimizing electron-hole production
`It is often felt that state-of-the-art silicon cells generate about as much
`current as theoretically possible. In fact, significant advancesare still available
`through the use of light trapping and back surface reflectors [10]. Once a
`photon enters the cell it must produce an electron-hole pair to generate
`current. Silicon is only weakly absorbing at energies just above the bandgap
`so photons in this region have a significant chance of reaching the back of
`the cell. In cells with alloyed aluminum back contacts most of these photons
`will be parasitically absorbed there. By making the back reflective (prefer-
`ably highly reflective) many of these photonswill be reflected toward the
`front and have an additional chance to produce electron-hole pairs. It has
`been found that it is much easier to make the undoped, uncontacted regions
`reflective than the contact regions [11]. Thus the small contact coverage
`fraction of the PCSCis desired from this point of view. In addition, as shown
`in ref. 10, if one of the surfacesis slightly texturized, with tilt angles greater
`than 16° or so, most of the photonsreturned toward the top will be beyond
`the critical angle for escape and will once again head toward the back. In
`effect, the weakly absorbed photonsare trapped within thecell.
`In addition, because the point-contact cell has its metal on the back side
`there is no grid obscuration.
`
`3.2. Optimizing current collection
`Once electron-hole pairs are generated, electrons must end up at the
`n-type contact and holes at the p-type contact to appear as terminal current.
`The standard approach usedhereis to increase the minority carrier diffusion
`length by reducing impurities and defects as much as possible. When this
`length becomes much greater than the cell thickness the collection fraction
`approaches a maximum, butit is not necessarily unity. In a conventional cell
`electron-hole pairs created near the back have a large probability of being
`absorbed at the back contact. One can define a local collection efficiency
`which is the probability that an electron-hole pair produced at a particular
`point in the cell will be collected. Thus the local quantum efficiency is small
`near the back of a conventional cell. Even the presence of a back surface
`field does little if the cell has sufficient base doping to provide adequately
`low resistance for 500X operation (i.e. around 0.2 92 cm) [12]. If the cell
`base is lightly doped (or undoped) a built-in field at the back can trap the
`carriers, resulting in near unity collection efficiency. This is especially true
`if the contacts are shrunk to points, as in the PCSC, provided that the
`surface between contacts has sufficiently low recombination velocity (less
`than about 20 cms'). Unfortunately, lightly doped bases are not very suit-
`able for conventional high concentration cells because of high series resistance.
`More correctly, the problem is actually one of loss of conductivity modula-
`tion near the back of the cell [13]. A significant advantage can thus accrue
`
`

`

`89
`
`for the PCSC, but to realize this advantage, the modeling in Section 4 shows
`that the minority carrier diffusion length must be quite large (greater than
`about 800 um), the surface recombination velocity must be very low (less
`than about 10 cm s“‘), and thecell must be thin (less than 100 um).
`Obviously the so-called dead layer loss sometimes experienced in con-
`ventional cells when light is absorbed in the highly doped diffused regions
`is eliminated in the PCSC.
`In order to quantify these effects, a ray tracing program was written
`that incorporates thestatistical optics theory of Yablonovich and Cody [10].
`This program was then used to explore the effect of various cell parameters
`on the photocurrent. The results of this exercise are shown in Tables 1 and
`2. The first line of Table 1 gives a baseline case, that of a typical cell 350 um
`thick and doped to 10!7 cm™ with boron. This cell, as do all the rest, has a
`two layer anti-reflection coating consisting of 540 A of TiO, and 1020 A of
`MgF.. The current loss dueto reflection is about 3% with this coating. Read-
`ing across the first row of Table 1 it is seen that this cell has no texturizing,
`a minority carrier diffusion length of 200 um, a back surface recombination
`
`TABLE 1
`
`Computed short-circuit currents for conventional cells with a thickness of 350 um and
`base doping of 10!7 em™?
`
`Texturized
`
`Laie (um)
`
`s(ems')
`
`R
`
`Jee (mA cm”)
`
`
`Active
`Total
`
`
`% Ine.
`
`200
`1000
`200
`200
`200
`
`10°
`108
`10°
`10°
`10°
`
`0
`0
`0.95
`0
`0.95
`
`36.56
`87.34
`37.00
`38.94
`39.72
`
`23.64
`34.35
`34,04
`35.82
`36.54
`
`0
`2.1
`1.2
`6.5
`8.6
`
`No
`No
`No
`Yes
`Yes
`
`TABLE 2
`
`Computed short circuit currents for a thin cell (thickness, 100 um) with a lightly doped
`base
`
`
`Texturized
`
`Laye(um)
`
` s(cems?)
`
`R
`
`Jee (MA cm™*)
`
`
`Active
`Total
`
`
`% Inc.
`
`No
`No
`No
`No
`Yes
`Yes
`Yes
`Yes
`Yes
`
`—5.7
`31.71
`34,47
`0
`10°
`1000
`—4.3
`32.20
`35.00
`0.95
`10°
`1000
`1.9
`34.28
`37.26
`0
`10
`1000
`5.0
`35.34
`38.41
`0.95
`10
`1000
`1.2
`34.04
`37.00
`0
`10°
`1000
`5.7
`35.57
`38.66
`0.95
`10°
`1000
`7.5
`36.16
`39.30
`0
`10
`1000
`17.0
`39.36
`42.78
`0.95
`10
`1000
`
`
`
`
`
`10 0.95 42.78 42.781000 27.2
`
`

`

`90
`
`velocity of 10° cm s™!, and a back surface reflectance of zero. The predicted
`active area photocurrent is 36.56 mA cm~ at 100 mW cm™? AM 1.5 illumina-
`tion. Assuming an 8% grid coverage fraction the cell short circuit current
`will be 33.64 mA cm’. The last column gives the percent improvement over
`this base case. In the next row it is seen that increasing the diffusion length
`to an improbable 1000 um gives only a 2.1% improvement. Including a back
`surface reflector (third row) yields 1.2%. These meager improvements might
`make one pessimistic about the possibility for significant gains. The reason
`that there is so little benefit is that the back region of thecell, as discussed
`previously, has a very low local collection efficiency. Light reflected from
`the back will most probably generate electron-hole pairs near the back where
`the collection efficiency is low. Texturizing the cell produces some improve-
`ment; mainly because the reflection loss is reduced and the light travels a
`longer path through thecell. The predicted 836 mA cm”for a texturized cell
`is in line with the best observed current for conventional concentratorcells.
`Table 2 considers the case of a lightly doped, thin (100 um) cell. It will
`be shownin Section 4, that lightly doped cells must be thin to prevent loss
`of conductivity modulation.
`In the lightly doped cells it is possible to
`achieve very long diffusion lengths, as will be discussed in Section 5. As the
`first two lines show,significant current is lost due to the reduced cell thick-
`ness. In a lightly doped cell, however, back surface fields have the effect of
`reducing the apparent recombination velocity at the back face. It is seen that
`by combining a back surface reflector with a back surface field one can
`obtain performance comparable to that of a conventional thick cell. Truely
`impressive gains are realized by also adding texturizing. In fact, by examining
`the last rows of Table 2 it is clear that there is a synergistic effect among
`texturizing, back surface reflectors, and back surfacefields(i.e. low effective
`surface recombination velocity). The effect of incorporating all of them is
`greater than the sum of their separate contributions. This is simply because
`light trapping caused by texturizing and back surface reflectors requires high
`local collection efficiency at all points in the cell (not just at the front) to be
`maximally effective.
`Finally, by putting the contacts on the back, as in the point-contact
`cell, grid obscuration is eliminated. The resulting current, shown in the last
`row of Table 2 at 42.78 mA cm”~, is 27% greater than that of the baseline
`cell. This accounts for the major portion of the increased performance of the
`point-contactcell.
`The carrier density in the back surface contact devices that are made
`on undoped material decreases in going from front to back because the
`carriers must diffuse to the contact areas. This will be discussed fully in
`Section 4. In order to maintain sufficient conductivity modulation at the
`back it is necessary to make thecell rather thin, around 100 pm as in Table
`2. The point-contact cell, therefore, relies rather heavily on light trapping
`to produce increased absorption. There is concern, however, whether the
`standard texturizing procedure, which produces regular pyramidal facets,
`has sufficient randomization to promote light trapping.
`
`

`

`91
`
`To address this issue the following experiment was performed to com-
`pare the measured and calculated optical absorption versus photon energy
`in two wafers. The first wafer was polished on both sides and the second
`received a standard texturizing treatment on both sides. Both wafers had
`approximately 1000 A of SiO, on each face. The total absorptance of the
`wafers was found by measuring the reflectance and transmittance versus
`energy and subtracting their sum from unity. By referring to Fig. 2 it can be
`seen from the symmetry of the problem that the absorptance in this caseis
`equivalent to that of a wafer half as thick with a perfect reflector on the
`back. The measured results are shown in Fig. 3 along with the calculated
`absorption. Calculating the absorption in the polished wafer is, of course,
`straightforward. For the texturized case, the theory of Yablonovich and
`Cody [10] was used. The agreementis quite satisfactory indicating that the
`currents calculated using this theory should be obtained in practice.
`(Texturizing has not been implemented in complete cells to date.) Note that
`the absorption edge of the texturized wafer is shifted around 0.1eV lower
`than that of the polished wafer. There is even significant optical absorption
`apparent below the band gap energy. This is because light trapping has
`enhanced the absorption sufficiently to see the weak photon-absorption-
`with-phonon-absorption process.
`
`3.3. Optimizing output voltage
`Finding ways of increasing the output voltage is difficult due to many
`complex interacting factors. As a rough estimate, the output voltage V is
`
`he=
`
`\ é1 i b1
`
`\ | 1
`
`o
`
`wa
`
`“
`
`PHOTON ENERGY (eV)
`
`ww
`
`1.2
`
`i 14
`
`1.5
`
`o=ooTo
`
`ABSORPTANCE
`
`1
`
`Fig. 2. Illustration that the absorption path length in a texturized cell with a perfect
`reflector on the back is equivalent to that of a cell twice as thick and texturized on both
`sides.
`
`Fig. 3. Measured and calculated total absorptance vs. photon energy: continuous curve,
`untexturized, 120 um; dashed curve, texturized, 80 um:; solid circles, theory.
`
`

`

`92
`
`=~ in(P5)— Vie
`
`kT
`
`q
`
`/pn
`
`rn
`
`(1)
`
`where kT'/q is the thermal voltage, p is the hole density and n the electron
`density in the base, n;
`is the intrinsic carrier concentration and V,,, is the
`total resistive loss, including base region drop. A high pn product is needed
`to obtain high junction voltage. In the lightly doped (orintrinsic) base cell,
`a high pn product is also needed for another reason. Here onerelies on main-
`taining sufficient conductivity modulation to limit the base’s resistive voltage
`drop. Unfortunately, however, a high pn product results in a large recombi-
`nation (or dark) current. Recombination current subtracts from thecollected
`photocurrent and thus reduces the terminal current. At the maximum power
`point the balance between loss of current due to recombination and operating
`at a high voltage is optimized.
`Carrier recombination can occur in four regions of the PCSC: (1) the
`bulk silicon (which is largely mediated through defects and impurities),
`(2) the surfaces of the silicon on the front and between the contact diffusions
`on the back (passivation with SiO, reduces this component), (3) the diffused
`junction regions and (4) the metal-silicon contacts. Research into improving
`the recombination related parameters of solar cells [11] has resulted in cells
`where the recombination is dominated entirely by the diffused junction
`regions. This is the major impetus for reducing the coverage fraction of the
`diffused regions. In fact, however, modeling has shown that sufficient carrier
`density in the base to produce low series resistance can be achieved only by
`reducing this coverage fraction, as in the PCSCstructure [1].
`In actuality, the situation is complicated by the fact that in the PCSC
`carriers must diffuse into the contact regions. This causes a drop in carrier
`density and junction voltage at the contact. Also, as the current converges
`into the contact regions, a sort of spreading resistance loss results. Clearly,
`the contacts cannot be arbitrarily small because of these effects. For a given
`contact size there will be an optimum contact spacing. These issues are
`discussed in the next section.
`
`4, Modeling point-contactsolar cells
`
`In order to properly design the point-contact cell, as well as to assess
`the potential of the concept, a three-dimensional cell model is needed. Exact
`numerical solution of the semiconductor transport equations in the complex,
`three-dimensional geometry of the point-contact cell would, at best, be
`expensive. On the other hand, a simplified one-dimensional analytic solution
`would not address the essential three-dimensional aspect of the problem and
`could not tackle the issues of optimum contact size and spacing. The diffi-
`culty is compounded by thefact that the cell operates in high level injection
`so that the superposition principle is not applicable. A compromise method
`of modeling this cell
`is presented here which has proven tractable and
`
`

`

`93
`
`accurate. The approach is based on emphasizing the accurate determination
`of the total recombination current, rather than carrier densities and fluxes,
`and proceedsas follows. The model solves the semiconductor transport using
`a variational approach to obtain the base carrier density. Knowing this, the
`terminal voltage is calculated by assuming constant quasi-Fermi levels across
`the n-i and p-i space charge regions. Finally, the terminal current is found
`by subtracting the total recombination current from the photo-production
`current. In this section the modelis presented and exercized to determine
`the optimum point-contact cell design. The potential efficiency of mature
`point-contact cells is then calculated. The details of this model are described
`in more detail in ref. 1.
`
`4.1, Semiconductor device equations
`The standard equations used to model steady-state carrier transport in
`silicon devices, when heavy doping effects can be neglected, are as follows.
`1. Current transport equations:
`I, = — quan; + aDaiin
`J, =— quypVV; — aD, Vp
`2. Continuity equations:
`$d = ae—an)
`Vd, =—a(r —&pn)
`8. Poisson’s equation:
`
`(2)
`(3)
`
`(a
`(5)
`
`Vn =—< (p + Np —n—Ng)
`
`4. Carrier density equations:
`n= ny edibn)/ kT
`p=n, atdp— Vi/RT
`
`(6)
`
`(7)
`(8)
`
`The symbols used have their standard meanings, defined as follows:
`r, net recombinationrate per unit volume; &pn, Photogeneration rate per unit
`volume; Jy, electron current density; Jp, hole current density; n, electron
`concentration; p, hole concentration; uz, Mp, electron and hole mobilities;
`D,, Dy, electron and hole diffusivities; ¢,, 6), electron and hole quasi-Fermi
`levels; Y,, potential at the intrinsic level; e, dielectric constant; q, electron
`charge.
`Using eqns. (2)-(5) the electron and hole current densities can be
`eliminated. The resulting equations, along with Poisson’s eqn. (6), define
`three coupled non-linear differential equations in three unknowns, n, p and
`W;. Equations (7) and (8) are superfluous but will prove useful in relating
`terminal voltage to carrier densities. Given appropriate boundary conditions
`these equations can,
`in principle, be solved. The search for reasonable
`analytic solutions to these equations is hopeless and some simplifications
`
`

`

`94
`
`must be made. An approach that has proven tractable and accurate for the
`point-contact cell is presented below.
`
`4,2. Integral approach to the terminal current
`Once a solution for n, p and y; is found, the terminal current can be
`found in either of two ways — by a differential or an integral method.
`
`4.2.1. Differential method
`Consider the general
`two-terminal device shown in Fig. 4 with an
`imaginary surface S that surrounds terminal 1. The terminal currents are
`i, and i, = —i,. In the steady-state, i, may be found by
`
`i, = Joi. +J,) nds
`
`s
`
`(9)
`
`where 7 is an outward unit vector normal to S, and J, and J, are given by
`eqns. (2) and (3). Any surface can be used to calculate i, that surrounds
`contact 1, but if exact solutions for J, and J, (or equivalently, for n, p and
`wW,) are not used then different surfaces may give different answers.
`
`4.2.2. Integral method
`Using this method, one integrates the continuity eqns. (4) and (5), over
`the device volume
`
`(10)
`
`(11)
`
`[idea Jeep») a0
`
`V
`
`Vv
`
`[i-dav=—¢ [ee son) av
`
`V
`
`Vv
`
`
`
`
`DEVICE
`SURFACE
`
`Fig. 4. General two-terminal semiconductor device.
`
`

`

`Next the divergence theorem is used to convert the left-hand side to sutface
`integrals over the device’s surface, Sto¢
`
`95
`
`f Gp-tas=a fr—epndav
`
`Stot
`
`Vv
`
`| 5,-ias=—a Jee son) ao
`
`(12)
`
`(13)
`
`ys
`Stot
`The surface integration is divided into three regions: S, (contact 1), S,
`(contact 2) and S (remainder of device) giving
`
`Stot
`
`8,
`
`| f.-tas = fi, -aas+ fd, fe fo “ads
`f J, ias = ie -aas+ fi, “nas + [J, “ads
`
`S,
`
`Stot
`
`8,
`
`8,
`
`§
`
`At contact 1 the current is
`
`fom [as|i
`
`(14)
`(15)
`
`(16)
`
`The minussign results because the normal vector n is outward while the
`current is referenced inward. Now,solving for — fs, J. *ndS in eqn. (15),
`inserting this into eqn. (16), and using eqn.(13) gives
`
`=| d,-nas—fJ,-nas+ fiat frav—a fanav
`
`8,
`
`5,
`
`8
`
`Vv
`
`Vv
`
`(17)
`
`Let us suppose that contact 1 is p-type and contact 2 is n-type so that
`J, and J, are minority carrier currents in the integrations in eqn. (17).
`Since, in a solar cell, positive current out of the p-type contact corresponds
`to positive output power it is convenient to define the termimal cuftent
`I = —i, =i. Then eqn. (17) can be written in the suggestive form
`
`i= lon ~L-res = Teevee = Toonteee
`
`:
`
`(18)
`
`where
`
`nh=@ /pn dv
`
`photoproduction current
`
`

`

`96
`
`Loewe @ fr dv
`
`Vv
`
`I, see = [a -nds
`
`s
`
`Teont,ree = [a ‘nds — [a “nds
`
`8,
`
`8,
`
`bulk recombination
`
`surface recombination
`
`contact recombination
`
`(19)
`
`In other words, if one properly accounts for recombination, one can say that
`the output current equals the photoproduction current minus the total
`recombination current. It should be stressed that eqn. (18) is an exact result
`of the continuity equation.
`The integral approach has the advantage that any errors in n and p tend
`to be averagedout when performing the integrations. It also does not depend
`on knowing Yn.and Vp. For solar cells, using eqn. (18) approaches the
`essence of what is desired more closely than the differential method which
`requires knowledge of the carrier fluxes throughout the device. This is the
`approach used here.
`
`4.3. Terminal voltage
`The point-contact cell has highly doped regions near the contacts and a
`lightly doped base. Its band diagram is shown in Fig. 5. The horizontal co-
`ordinate position on this diagram is not a spatially straight line from front to
`back, but just represents a continuous physical path from the p* to n* con-
`tact areas. The terminal voltage (less any drop in the contacts and metal
`bus) is ¢, at the p* contact minus ¢, at the n* contact.
`To a very good approximation @¢, can be assumed constant through the
`n* region, its space charge region, and into the edge of the neutral base near
`
`o
`
`<=
`
`CONDUCTION
`BAND
`MINIMUM
`
`=> +POTENTIAL0.
`
`METAL
`
`ntDOPED |-— BASE ———| DOPED | METAL
`
`REGION
`
`REGION
`
`+
`
`Fig. 5. Point-contact solar cell band diagram,
`
`

`

`the n* contact at position 2. The same applies to ¢, in the p* region up to
`position 1. Referring to Fig. 5 it is easily seen that the output voltage is
`
`97
`
`V = Vin + Vip + Va + Vo + Ven
`where
`
`Vin = Vi — On =RT/q In(n/n,)
`
`(evaluated at point 2)
`
`Vip = > — Wi =RT/q In(p/n;,)
`
`(evaluated at point 1)
`2
`
`Vp = vi(1)—Wi(2) =— fiaWi ral=[ Bal
`
`1
`
`V,
`
`(contact voltage)
`
`V,,
`
`(metal grid voltage)
`
`(20)
`
`(21)
`
`(22)
`
`(23)
`
`(24)
`
`(25)
`
`In normal operation Vg, V, and V,, are all negative and represent the
`base, contact and metal resistive loss respectively. Detailed methods of
`calculating V, and V,, are available in the literature [14].
`The importance of eqns. (20)-(25) lies in the need to have solutionsfor
`n, p and wW; only in the base semiconductor region. The details of what
`happens in the n* and p* semiconductor regions is unimportant. This is
`fortunate for, as discussed below, what happensin these regions is difficult
`to model because eqns. (2)-(8) do not apply in very highly doped regions.
`Notice that if one neglects the base voltage drop and contact drop and
`assumes that n and p are constant throughout the base then eqns. (21)-(25)
`imply that
`
`kT
`V ~— In(pn/n,?) —
`qd
`
`as posited by eqn. (1).
`
`(26)
`
`4.4, High level transport equations in the base
`As seen in the previous section, solutions for n, p and ); are needed in
`the base. For high concentration cells with base doping less than 101° em™?
`the carrier density in the base will be greater than the doping, i.e. the base
`is high level injected. It has been shown, however, that the base will be
`quasi-neutral to a very good approximation [15]. This implies
`
`n+No=p+Ny'
`
`where Na and Nj‘ are the ionized acceptor and donor densities. In high
`level injection one can further neglect the charge on donors and acceptors
`giving
`
`n=p
`
`(27)
`
`

`

`98
`
`From eqns. (2) and (3), and using the Einstein relation D = kTy/q and
`eqn. (27) gives
`J =RT(Up — Hp) IN — (on + My) VY;
`or
`
`(28)
`
`+
`kT ae
`BT >
`J
`
`iv = c =)
`in
`q
`\Mn + Up/ n
`9 (Mn + Up) n
`Equation (29) will be needed to calculate Vg from eqn. (23). Thefirst
`term in eqn. (29) is called the Dember field and the second term is the
`resistive term. Inserting eqn. (29) into eqns. (2) and (3) gives
`
`(29)
`
`
`i aqptae— J
`Mn * Mp
`
`and
`
`(30)
`
`j.=—¢0on +— 2—J (31).
`
`
`
`a a tty
`where J is called the ambipolar diffusion coefficient and is defined as
`kT 2uy
`D=— EnUp|(32)
`Y Un + Up
`If one assumes that D is a constant independent of n, then substituting
`eqn. (30) into the continuity eqn. (4) gives
`(33)
`DY*n =r—&on
`In actuality uw, and wy, depend on n due to carrier-carrier scattering. The
`approach taken here is to evaluate D at the average value of n in the base.
`This will require an iterative approach because n will prove to depend on D.
`At this point, the problem of determining n in the base has been reduced to
`that of solving a “‘Poisson like’’ equation.
`
`4.5, Recombination
`To calculate the terminal current the various recombination terms in
`eqn. (18) need to be evaluated. Various regions of the cell require different
`approaches.
`
`4.5.1. Diffusion region recombination
`Part of the integration in eqn. (18) 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 the device to skirt the highly doped regions. The new
`surface of integration is illustrated in Fig. 6. It is allowable to move the
`surface of integration to this point because the net hole current entering
`the n-type diffusion area, for example, either recombines within the diffu-
`
`

`

`99
`
`{UNDOPED BASE
`CONTACT
`SURFACE
`SURFACE
`h— —e}o—!_2
`
`SILICON
`
`SURFACE
`OF
`INTEGRATION
`
`
`
`suitWe
`
`
`LEELA
`
`Fig. 6. Modified surface of integration.
`
`sion or at the contact. Thus the sameresult is obtained by either evaluating
`the hole current at the edge of the space charge region or calculating the net
`recombination in the diffused area plus contact. (The photogeneration in the
`diffused regionsis being neglected.)
`It has been shown that the minority carrier hole injection into a highly
`doped n-type region can always be written, provided the doped region is
`everywhere low-level injected
`
`Jp = Jon (2 ~ 1)
`
`pn
`
`ni
`
`(34)
`
`where Jo, is a temperature dependent constant, to be called the diffusion
`saturation current, and p and n are evaluated in the neutral base at the edge
`of the space charge region [16]. J), 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 considered an
`experimental parameter. The latter approach is used here.
`In the case of a highly injected base one has
`
`Jy =Jon (5 a 1)
`
`n2
`
`ny
`
`for the hole current density entering an n-type diffusion. Similarly
`-
`n
`
`Jn = Joy (1)
`
`ny
`
`(35)
`
`(36)
`
`for the electron current density entering a p-type diffusion.
`
`4.5.2. Recombination in the base
`To perform the remainder of the bulk recombination integration the
`recombination in the lightly doped, highly injected base needs to be evaluated.
`It will be assumed that
`
`n—-n
`r=—— + B,(n? —n?) + Ca(n?—n9)
`i
`
`(37)
`
`

`

`100
`
`Thefirst term in eqn. (87) accounts for defect mediated bulk recombination.
`Tt
`is the high level defect related recombination lifetime which is usually
`considerably greater than the low levellifetime. It is very processing depend-
`ent and is considered a measured experimental parameter. The second term
`results from radiative recombination. It varies as pn = n? because an electron
`and hole participate simultaneously in the process. B, has the experimental
`value of 2 X 107!5 cm’ s™! [17]. Thefinal term ari