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Enhanced carrier collection at grain-boundary barriers in solar cells made from large grain polycrystalline material
Solid-SIarr Eiccfmnics Vol. 22. pp. 651-658 0 Pergamon Press Ltd.. 1959. Printed in Great Britain
ENHANCED CARRIER COLLECTION AT GRAIN-BOUNDARY BARRIERS IN SOLAR CELLS MADE FROM LARGE GRAIN POLYCRYSTALLINE MATERIAL HERBERT F. MATAR~‘S ISSEC,P.O.Box 49177,Los Angeles,CA 90049,U.S.A. (Received4 August 1978;in revised fon 4 &ember
1978)
Ah&met-The case of the solar cell in polycrystalline material is investigated speciftcally in view of the electronic properties of grain-boundaries. Material in thin tihn form, ribbons or cast ingots display a preponderance of vertically oriented grains. This and the fact that grain-boundaries are excellent photo-electric converters, when correctly contacted, is the basis for a new diksion- and contacting scheme. It qahes use of the grain-boundaries as vertical junctions, enhancing the photo-current. The argument is substantiated by a discussion of the electronic features of grain-boundaries under these conditions and the enhancement of the photovoltage due to the frequency extension of the photoresponse.
a, capture cross sectionfor holes NOFATION A, lower limit of wavelength spacing between occupied interface states A. upper limit of wavelength Burgers vector Q%,,trqJ2kn = equilibrium barrier height spacing between dangling bond levels (available 264~ eVr - eV,, = ~5- I$,,= change in barrier height states) dislocation spacing electron charge INTRODUCTION barrier height on either grain-boundary side dislocation level Recent efforts to lower the S/W-Iigureof solar cells have Fermi level lead to the application of polycrystalliie material. Quasi Fermi levels for electrons/holes Numerous methods have been developed to grow espeelectron energy at lower edge of conduction band electron energy at upper edge of valence band cially silicon in sheet form, as ribbon material (edge grain-boundary tilling factor = c/a defined growth), as thin films on foreign substrates etc. vertical junction +rea and to use this defective material directly or after some horizontal junction area recrystallization procedure. (Annealing, Laser or Eleclight generated current tron-beam recrystallization and the like). recombination current saturation current The way such material is used, is basically the same as horizontal resp. vertical current in the case of perfect Czochralski-respectively Floatingtotal junction current Zone grown crystal material, i.e. no particular steps are injection current taken to include the grain boundaries and their barrier electron current density hole current density (also vertical current density) layers into the structure for carrier collection. In most horizontal current density cases of n on p-cells, the grain boundary layers @‘) are minority carrier ditfusion length electronically floating and reduce the number of coUeceffective state densities in conduction resp. valence ted carriers by bulk recombination or, in the case of p on band number of photons incident in interval A + dA n solar cells, the grain boundaries represent e5cient carrier density at point x shunts between top and bottom electrodes. reducing the carrier (electron) concentration efficiency. carrier (hole) concentration In the case of Schottky barriertop contacts, the opendonor density circuit vohage is less effected by the grain-boundariesbut acceptor density interface state density enhanced bulk recombination leads to decreased shortoccupied grain-boundary interface states circuit current values. equilibrium number of grainboundaryinterface states In all these cases as in those involving devices where dislocationpipe radius I efficient carrier recombination within a junction is interface recombination velocity required (Light emitter diodes; Lasers) defects and grain external voltage carrier velocity boundaries are a major hurdle to performance when they barrier voltage are statistically distributed within the neighbourhood of absorption coefficient the active junction. FJFh = vertical/horizontal junction area The early study of carrier recombination at lineage grain boundary tilt angle boundaries (small angle grain boundaries) showed111that capture cross section for electrons 651
652
H. F.
hhTAi&
these structures act as hole sinks in n- and p-material. (In the latter case within a doping range ~10” cme3)[21. This is due to the higher electrondensity around the p-type core of the grain boundary or the space charge pipe. It is well known that dislocations (one-dimensional) and grain-boundaries (two-dimensional) are most influential when they are oriented vertically to the active junction. This is due to the tendency of dislocations to form overlapping line charges. Much of their electronic behaviour is dependent upon the number of charges (misfit angle between the grains); the ratio of filled to available states and the doping environment which defines the extent of their space charge[2]. Dislocations as recombination centers are often considered as simple cylinders or walls which attract carriers Fii. 1. Energy diagram of n-type semiconductor with grain boundaryat x = 0. Neutral interface state density qo.Barriersat and enhance bulk-recombination. It this case the both sides. q51=eV, and & = eV, are e.qualwithout illumination effective cross section is simply a function of dislocation or bias voltage. Space chargeradiusis R. density, their radius and of the carrier diffusion length[3]. While some conclusions with respect to the influence of a dislocation on the recombination process, for holes and electrons, the application of a Shockleye.g. in electroluminescence, can be drawn, the actual Read-Hag type recombination process leads to a physics of the dislocation and especially of the grain recombination current equality: jC = j”. The total recomboundary, requires more detailed definition as to the bination current in this case is: extent of the space charge (the core radius itself is much smaller), the number of available states, their dependence on the external potential and the conduction parallel to the line charges[2]. This especially in cases of parallel orientation of grain-boundaries as is the case in large grain polycrystals of limited thickness or in thin film form. We will show that grain-boundaries can also be used in where e equals the electron charge; N,= the interface a positive way if they are included into the actual collecstate density and u the carrier velocity. tion process for the optically injected charge carriers. As a consequence of a,, = a, also n(o) =p(o) (Fig. This is also known to be true for III-V-light emitting 1). In other words, J, is the maximum recombination structures, when the difference between Q and &discurrent shifting VD and establishing the condition n(o) = locations and their respective environment is taken into p(0) at x = 0. account[2,4]. It can also be shown that the interface recombination Therefore the problem of solar cell efficiency is more velocity S(w) can be expressed as: complex in III-V-compound material when poiycrystal1 line layers are used[S], because of the dependence of the electronic effects, especially the recombination process, [6] (‘) on dislocation type, doping environment of the dislocation and its orientation with regard to the active with exp(eVdkT) being the grain boundary enhancejunction plane. ment factor. Estimates of the lifetime reduction as a function of 1. FZLECTRONICPROCESSESAT GRAIN BOUNDAUIES interface state density and grain size give reasonable values when isolated grains are assumed interfering at Electronic processes around dislocations under optical illumination have been studied in the past and con- random with the carrier flow[6]. As a physical structure, clusions have been drawn with respect to the population the dislocation or the two-dimensional grain boundary of interface states and minority carrier lifetime[61. This can have varied properties with respect to the current was carried out for the isolated dislocations. In this case, flow. Figure 2 shows the basic structure and its dimenthe excess carriers generated by illumination shift the sions as it appears within the host lattice. As has been shown(2], the outer space charge extension or the pipe quasi-Fermi levels Ef, and E,,, according to: radius R may reach values in the 10 p or up to 1OOp (1) range. It can therefore have a decisive influence on n = NCexp [(Ef, - EdkTl carrier flow when it is connected to one side of a junction, e.g. when its p-type core is connected to the (2) P = N. expWY - Ef,MTl p-type side. If dislocations are formed between two grains, we have the planar structure of a grain boundary. see Fig. 1. Figure 3 shows such a grain boundary plane formed at Assuming equal capture cross sections (CT”= up = u)
Carrier collectionat grain-boundaryharriersin solar cells
whole structure can be included in a contacting or charge collecting scheme.
I
h-l -lo-' em
R
r.
As shown in the Appendix, the space charge pipe radius of a grain boundary depends on the angle of mistit 0 and on the bulk doping. The dislocation spacing is:
t--
423 \
2. PHUlOVOL.TAlCEEpEeFsATGRAlNBOUNDAlUES
m
,--;--
653
.+_-+/’
-_L
l”llW
cure
rtm A
+
-
mobile slectronr
+
mobile holes
+
1
D = b/2 cosec 812
(6)
where b equals Burgers vector. The density of interface states is in the range 10” < 1/P < IO” cm-*
(7)
for angles of misfit 8 between 4 and 10” (pure edge dislocations). In a typical case, a misfit angle of a few degrees leads to 10’2cm-2 as area state density. This figure results from the fact that there is a filling factor for the line charge: f = c/a = 0.05-O.1
Fig. 2. Detailed form of a space charge cyclinderaround a dislocation with p-type inner core aad space charge pipe. Dimensions for normal doping environment (10’6-10”cm-3 m-type).
03
(in the temperature range from 50 to 300°K and for Fermi statistics. See Ref. [2], p. 215). f is the fraction of occupied states; c the spacing between dangling bonds (available states) and a the spacing between occupied sites. (Fig. 3). There is also another fill-factor f perpendicular to the pipes:
(8) and (9) lead to the area state density:
(10) (see Ref. [2], p. 218). For the above range for the till-factor, we have Nos
Fig. 3. Dislocation plane between two crystalline grains. c = distance between dangling bonds, a = distance between filled levels, 8 = tilt angle. 2R = diiter of space charge pipe (see text). D = dislocation distance; D = (b/2). coscc#2 for B> 1”; dopingrange IO”-IO” cmb3.
= 1-4 X 1Or2cmm2
as the number of tilled states in the grain boundary sheet [2,61. We now derive the light generated current due to carrier injection into the grain boundary space charge pipe. Injection causes a current across the junction: Z, = const exp (-Adk7’).
the mating interface of two grains misoriented by an angle 8. If the dislocation distance D = b/2 cosec(e/2) (b = Burgers vector) is small or 1”< 8’ < 25” (medium tilt angle, see Ref. [2]), one can assume that for normal doping ranges the dislocation distance equals the space charge radius R( see Appendix A). Thus a coherent layer with overlapping space charges is formed. The p-type cores can be connected by a p-type layer and thereby the
(11)
(12)
As before (Fa. I), there is a change in equilibrium barrier height and A4=EF-Ek=eV.
(13)
eVI - eV2 = eV, (V, = external voltage either due to injection or to bias) (see Fig. 4). As IL, the light generated
654
H. F. MATAM?
introducing the distribution function, (17) leads to the expressions: v,=$-[~[l+(l+$$“3 +~[1+(1+~)“2]-1]
- - __
-EF
v*=&[~[l+(l+5$)“*] -~[l+(l+~)“*]-‘~.
Fig. 4. Grain boundary barrier under photon injection (illuminaNeutral number of states= qo. Neutral barrier height q&=
tion).
nqo2/2kn (k = dielectric constant, n = doping range). Lit generated bias: eV, = eVt-eV, = A& Light generated additional states:
It is apparent that injection, e.g. into the left side of the n-p-n layer (Fig. 4) causes A4 to increase: AC#J = eV,eVlo= eV. and therefore also q is increased: q = qd2 [ 1 t (1 t (Ad&))“*] and more interface states become available (up to V, = kT/e). For V, %=kT/e and VI B kT/e, the variation of available number of states n, with V, is given by the function:
an,o= 2n, & “* w q. eve
(>I
4=qd2[l+(l+$yJ.
(19)
&‘I2
fT
T;T
em
( >
+$[V~~*exp(--~)~~-’ current is: Zi- I, with I, = saturation current, e.g. for the left n/p junction: zL =
const e-=“tfkT ee”z/*T _
1,
= Zs(emANkT- I).
(14)
(15)
(Ref. 121,p. 360). Assuming also that V2* kZ’/e, the second term in brackets can be neglected. As the incremental photovoltage is AV
Thus
P
with (13) photon injection generates a voltage: V.=$ln(l+ZJZS).
(16)
The shifted electrostatic potential due to photon injection corresponds to an external voltage V, = Ade. However, we now consider the fact that the equilibrium number of occupied states q. changesalso, as can be shown (Ref. [2], pp. 288-295). The number q of states occupied is higher due to the deviation from thermal equilibrium caused by virtue of V.. The ratio of actually occupied states to their equilibrium number q. is: (17) & = ?rqo2/2kn= barrier height in equilibrium. The expressions for the barrier height on either side of the space charge cylinder are: (Fig. 4) VI==
k (
2?rq 7+q
enV= * H
k is the dielectric constant and n the charge density; and
(20)
&,a”‘0 e
av,
(21)
it results from (20) and (21) that the photovoltage is linearly dependent on V, This has been substantiated by measurements, e.g. on germanium bicrystals[2,7,8] and is similar in silicon and III-V semiconductors with the complications discussed[2,4]. The fact that the photovoltage is enhanced by the barrier dependent number of available interface states is a feature of grain boundary layers when correctly contacted. Examples are given in Figs. 5 and 6. In Fig. 5, the frequency extension as compared to a normal p-n-junction (same material) appears clearly in the long wave-length region while in Fig. 6 the short optical wavelength extension is shown. We conclude that it is possible to arrange the diffusion and contacting scheme in such a way that the majority of the widely separated grain boundary planes in large grain polycrystalline material contributes to the resultant photo-voltage. 3. GRAIN BOUNDARY PLANES AS VFlyTlcAL JUNCTIONS
In most cases of crystal growth or layer growth from a substrate, during growth of thin films and ribbons, especially under application of a temperature gradient, vertical to the growing surface, the grain boundaries are oriented vertically across the layer. This has significance for the possible use of grain boundary barriers as vertical
Carriercollection at grain-boundarybarriersin solar cells
A
1 PHomN
EIklmY-ifi
I1
Fig. 5. Photoresponse vs photon energy under the same conditions and on the same bulk material (Ge) for a p/n junction (diffused)and a grain boundary(see Ref. [2]).
-I
In this case the grain boundaries are included in the collection process for the majority carriers (holes) and do not act as sensitive photovoltaic barriers (see Chap. 4). The argument concerning the positive influence of the grain-boundaries as recombination centers is based upon the well known sheet conductance (p-type core) of the space charge layer, surrounding the dislocation array[2]. Different from [6], we assume a continuous flow of excess carriers along the space charge pipes, contributing essentially to the resulting efficiency when the correct contacting scheme is applied (see Fig. 7). A p’diffused or ion-implanted top layer establishes contact to the p-type core of the grain boundaries and the n’-side (backside) layer forms isolating junctions, blocking carrier flow from the space charge pipes (p’) into the base contact. In cast layers or thin films of polycrystalline material, grown without special precautions concerning gradient freezing, the grain boundaries may of course deviate from vertical planes and therefore will act more at random, than shown in Fig. 7. However, as the entire top layer is p+. all grains ending on the surface are contacted and act as vertical multijunctions [ 111. In reasonably pure material, the bulk lifetime between the grains can be assumed to be normal and carriers are collected by the grain boundaries to a depth, consistent with Ln. Especially in the case of high density photon injection (concentration) with attendant increase in carrier lifetime, grain boundaries improve the carrier collection. The collected number of carriers at a depth x is:
xN(h) e -a(*)Xe-“$ .a
1.0
1.2
1.4
1.6
1.8-h
1
Fig. 6. Photoresponse vs optical wavelength for a bicrystal at two differenttemperatures.A at T = 300°K;0 at T = 77°K.
junctions and as extensions of the active areas. In addition, a significant increase ifi optical response may result, as will be discussed. This proposition[9] grew out from the availability of large grain size polycrystalline layers of silicon [ IO] and GaAs [5] with grain boundaries oriented mainly vertically to the surface of the layers and separated by monocrystalline domains in the 10-1OOj~m range. As the grain distance d is comparable to or exceeds the minority carrier lifetime (d s L), each grain boundary can act as a vertical collector if it is contacted correctly and electronically included into a p+ surface layer, while the bottom layer is separated electronically by an $+-layer, making contact only to the base or n-type material, separating the grains. The reverse case of p-type bulk material and an n+ top junction with separating barriers to the grain boundaries at the top, cannot yield a vertical extension of the active junctions through the grains.
655
dx
(22)
where A, + A0 equals the wavelength limits (absorption curve); a(A) the absorption coefficient; N(A) the number of incident photons in the interval A +dA and & the minority carrier diiusion length. The computerization of eqn (22) for normal vertical junctions has shown that the short-circuit current density
Fii. 7. Proposedcontactingscheme for polycrystallinesamples. Grain boundariesare contacted by top p+ diffused layer. Base contact is separatedoff by n+ diffusedlayer. Reverse situationin the case of n-type cores or in the case of III-V material.
H. F. MATARB
656
is increased by about 20% as compared to the horizontal junction-case only. The open circuit voltage on the other hand is reduced by approx. 10%. In the case of grain-boundary junctions, A,+& is extended, but also the photovoltage is increased, as we have seen. This compensates for the usual loss in V, when Z,, is increased. The total junction current can be written: ZtO,= Z,,+ Z, = j,,F,, + j,,F, (2% where I,,, 1. equals the currents in horizontal and vertical junctions; jh, jv the horizontal and vertical current densities; Fhr F, the horizontal and vertical junction areas. With ~3= pjFh, we have: (24) or for a unit surface area, the total reverse saturation current amounts to: ilor = Ml + B).
(25)
Consequently, the overall current density is enhanced. The open circuit voltage V, decreases however, as seen from the diode characteristic: L, = id1 + /3Nexp(qV/akT) - 11- Lt.
(26)
Here:
WI
(27)
jJch and L are the short circuit photocurrent densities of the illuminated cell in the horizontal and vertical junctions. One can see from eqn (26) that the voltage is reduced as /3 increases. In the case of grain boundaries, the enhanced spectral sensitivity will compensate for this loss, as seen from (22). The extension of the sensitivity is active in both the long-wavelength and short-wavelength ranges (Figs. 5 and 6). This is also typical for silicon bicrystals. (See discussion of photoelectric effects on grain boundaries in diierent materiais[2, U-151). If we assume a factor of 5 as a conservative enhancement of the photovoltage in the short- and long wavelength sections of the integrated response (Figs. 5 and 6) and assume that this extension amounts to l/IO of the total output only, the overall efficiency is increased by 50%. The capture cross section of the grain boundary depends on the actual space charge, barrier height,
4 FRACTICAL
RENAZATIONS
In most practical cases, structure Fii. 7 is convenient to produce. It has the two fold advantage of the contribution of the grain boundary barriers to the enhancement of the photovoltage V, on account of the activity of the interface states according to eqn (19) and the effect of the vertical junction according to eqns (22) respectively (24) and (26). For a given material with a defined geometry, computerization of results is possible when the frequency extension of a single grain boundary plane is known or has been measured. Due to random arrangement of the grains in thin films, ribbons and cast material, exact predictions may be difficult, but the structure proposed should enhance efficiency in all cases. The foregoing considerations are also valid for III-V solar cells, but with the additional diiculty that for wider gap materials the grain boundaries may be p-type or n-type, depending on their orientation within the lattice, grain boundary orientation with respect to the active junction and doping environment (see Ref. [2], p. 168 and 382). Depending on the bulk doping, the a( Ga) dislocation is an acceptor in n-type environment and a donor in p-type material. @dislocations behave similarly [4,17]. In the case of n-type grain boundary cores, the doping should be chosen opposite to Fig. 7, top junction n on p and p+ at the base layer. When thin films are deposited on insulator substrates, the alternating surface contact scheme (Fig. ll), can be applied. If it would be possible to arrange the grain boundary layers in such a fashion that they would alternate with the N-type material (Fig. 9) with contacts to the ends of the crystal slab and light falling mainly onto one side of + r,
/-
Pf
!
“+
0
P+
INSULATOR BASE
Fig. 8. Alternatingp+-n+-layers on bulk n-type thin film solar cell with grain boundaries.P-type cores contact all p+ layers and have blocking layers to II+ regions. hv
quasi-Fermi level and bias. Capture rates are of the order of 4 x IO”see-’ cm’ and capture cross sections are 400 set-‘/dislocation.
Fig. 9. Alternating p-n layers due to grain boundary barriers. Case of anomalousphotovoltage.
Carrier collection at grain-boundary barriers in solar cells
the junctions, the case of anomalously high produced. According to eqn (16)
V,
could be
(29) where ZIei is the short circuit photo-current density of ith-junction and Z,, th’e saturation current density of ith-junction[l6]. Recently solar cells have been fabricated from inexpensive, low-grade, non crystalline silicon with a domain structure of sufficient distance between grains (d> L). This material is produced by casting ingots, subject to directional cooling[l8]. Ingots and wafers cut from this material have large central regions where the grain boundaries are running perpendicularly to the surfaces, due to the gradient freezing process. Improvements in thermal set-up will undoubtedly lead to a better yield of good material. Measurements on solar cells, produced from such polycrystalline material have resulted in unexpectedly high efficiencies, in some cases over 12% AhfO[ 191. These results were achieved on p-type base material, as reported by H. Fischer (AEG-Telefunken) on Wacker-Heliotronic ingots. The reasons for the use of p-type base material are two-fold. Firstly, a relatively simple purification process leads to p-type material (residual Boron) and secondly, it was intended to minimise the electrical activity of the grain boundaries [20]. But since the grain boundaries are electrically connected to the p’-base layer, they represent active hole sinks within the generally lesser doped base material and can enhance efficiency in this role. However, if the starting material could be purified sufficiently to convert it to n-type of, e.g. 5 x 10”cm3 doping, the scheme of Fig. 7 could be applied, with the result of the full use of the grain boundaries as vertical junctions with enhanced photoresponse. CONCUISWNS
The electronic features of grain-boundaries are assessed in view of the use of polycrystalline material in photovoltaic (solar)-cells. After discussion of the dependency of some basic properties as, recombination velocity and state density from grain-boundary parameters, the photovoltaic properties are considered, based on known relations governing barrier height, state density and external voltage. It is shown that the positive differential of state density vs bias, results in an improved photosensitivity, suggesting the use of the grain-boundary barriers as vertical junctions. This case is discussed along the lines considered by other authors and yields only a 20% increase in output, while the correct use of the grainboundaries as photoioltaic converters could result in a substantial increase in output. Some practical results from tests on polycrystalline wafers are discussed and a more efficient use of the grain-boundary posed.
barriers is pro-
657
Acknowledgements-The
author expresses his gratitude to Dr. Horst Fischer, head of the optoelectronics and solar cell departments of AEG-Telefunken, Hcilbronn, Germany, and his associates for a thorough discussion of their experience with solar cells, produced from polycrystalline wafers made by the Wacker-Heliotronic process of casting and gradient-freexiq. I. F. L. Vogel, W. T. Read and L. C. Lowell, Recombination of holes and electrons at lineage boundaries in germanium. Phys. Rev. 94, 1791(1954). 2. H. F. Matark, Zkfecl Elecrmnics in Semiconductors. WileyInterscience, New York (1971). 3. M. Lax, Junction current and luminescence near a dislocation or surface. 1. Appl. Phys. 49(5), 2796-2810 (1978). 4. A. L. Esquivel, S. Sen and W. N. Lin, Cathodoluminescence and electrical anisotropy from Q and B dislocations in plastically deformed gallium arsenide. I: Appl. Phys. 47(6),25882603(1976). 5. R. J. Stim, High efficiency thin lilm GaAs-solar cells. 1st Interim Rep. Jet Propulsion Lab., No. 730-9, and NTIS Report: P&258,493, JPL 5030-33(1977). 6. H. C. Card and E. S. Yang, Electronic processes at grain boundaries in polycrystalline semiconductors under optical illumination. IEEE Trans. Hecfron Lkv. ED U(4), 397-402 (1977). 7. H. F. Matare and K. S. Cho, Parameter dependence of LF. voltage in optical mixers and bias dependence of photo conductivity in bicrystals. Prtz. Eleclmn. Compounds Conf., pp. 415425. IEEE, Washington, D.C. (1966). 8. H. F. Matark, Dislocation plane devices in optical communications, advance in electronics. Proc. XZZZthZnt. Sci. Gong. Elccrron., Rome, Italy, Vol. I (1966). 9. H. F. Matark, The structure and electronic properties of grain boundaries in semiconductors. Pax. Nat. Workshop on Low Cost Polycrysralline Silicon Solar Ceils. South. Methodist University, Dallas, Texas (1976). IO. T. L. Chu, H. C. Mollenkopf and Shirley S. C. Chu, Deposition and properties of silicon on graphite substrates. 1. Elecrmchem. Sot. 123(l). lob110 (1976). Il. P. Shah, Analysis of vertical multijunction solar cells using a distributed circuit model. Solid-St. Elecfmn. 18. 10994106 (1975). 1 12. M. K. Mukherjee, F. Pfisterer, G. H. Hewig, H. W. Schock and W. H. Bloss. Shaoe of the u-n iunction and crvstalline structure of Cu$-CdS thin film so&r cells. 1. Ap& Phys. 48(4), 1538-1548 (1977). 13. R. K. Mueller, Transient response of grain boundaries and its ;;iation for a novel light sensor. J. Ajpl. Phys. 30(7), 1004 14. W. W. Lindemann and R. K. Mueller, Grain boundary photoresponse. /. Appl. Phys. 31(10), 1746(1960). IS. H. F. Matark, D. C. Cronemeyer and W. W. Baubien, Gebicrystal photoresponse. Solid-St Elec&m. 7,583 (1!%4);see also [21, pp. 349-364. 16. J. I. Pancove, Optical Processes in Semiconductors. Electrical Engineering Series, pp. 327-329. Prentice Hall, Englewood Clifis, New Jersey (1971). 17. H. F. Matark, Light emitting devices, Part I. Methods. In Advances in Electronics and Electron Physics (Edited by L. Marton), Vol. 42. Academic Press, New York (1976).Part II, Vol. 45 (1978). 18. B. Authier, Polycrystalline silicon with columnar structure. In Festk&erpmbleme XVIII (Advances in Solid State Phvsits) (Editkd by J. Treusch), pd. l-17. Vieweg, Braunschwe-ig (1978). 19. H. Fischer and W. Pschunder, Lowzost solar cells based on large-area unconventional silicon. IEEE Tmns. E/e&on Devices, Vol. ED24, No. 4, pp. 438-442 (1977). 20. H. Fischer, Solar cells based on nonsingle crystalline silicon. In Feslko”rperprobleme (Advances in Solid Stare Physics), Vol. XVIII, pp. 19-32 (Edited by J. Treusch). Vieweg, Braunschweig (1978).
H. F. MATA&
658 APPENLM.x
bRcm
The space charge pipe The distribution function of the grain boundary states depends on the average dislocation level ED and the ratio of occupied states q and equilibrium number of states qo:
with l+exp f”=(
w4
EF is the Fermi level. q/q0 is related to the ratio of actual barrier height eV, and barrier height at equilibrium: 4 = ?rqo2/2kN (see Ref. [2], p. 300). q/qo=;[l+(l+~)“‘].
(A3)
Fig. ib. R&us R(cm) of space chargepipe of grainboundaryas a function of tilt angle 0. Nd- N. is parameter.D = dislocation distance.
therefore:
Therefore:
f=l(l+exp~)[l+(l+$$)“*].
2sin(0/2) [lt(l+eVJfk)‘“] TbAN 1 texp(ED-EF)/kT
(A4)
The grain boundary filling factor c/a (spacing of free states/spacing of filled states) is another expression for f (A4). l/a is the state density within the core of the dislocation and is compensated by the space charge.
In
a
simplified
=
set
f=
range) and V, = 0, no external voltage. This function or =
(y)”
[’ sin;e/2)]“2AN-l/*
(A@
(A9
Therefore, the space charge radius R defining the range for the compensating charges is given by:
R
we
(A7)
1 + exp (ED - EF)/kT = 0.1 (for a hormal temperature
R
nk*( Nd - No) = l/u.
calculation,
“* ’
l/2 1 t (1 t eVJ,dcbo)“* 12rc(Nd - Na)[ 1 t exp (ED - EF)kT] 1 ’
is plotted in Fig. 10 for medium tilt angles and different impurity ranges AN from lOI to 10’9cm-3. D is the dislocation spacing according to D = b/2 * sin 812
646) c can also be replaced by the dislocation distance D= b/2 sin (e/2), b equals the Burgers vector, and one has
for pure tilt boundaries. The actual space charge pipe and its inner core with preponderant hole conduction is shown in Fig. 2.
ENHANCED CARRIER COLLECTION AT GRAIN-BOUNDARY BARRIERS IN SOLAR CELLS MADE FROM LARGE GRAIN POLYCRYSTALLINE MATERIAL HERBERT F. MATAR~‘S ISSEC,P.O.Box 49177,Los Angeles,CA 90049,U.S.A. (Received4 August 1978;in revised fon 4 &ember
1978)
Ah&met-The case of the solar cell in polycrystalline material is investigated speciftcally in view of the electronic properties of grain-boundaries. Material in thin tihn form, ribbons or cast ingots display a preponderance of vertically oriented grains. This and the fact that grain-boundaries are excellent photo-electric converters, when correctly contacted, is the basis for a new diksion- and contacting scheme. It qahes use of the grain-boundaries as vertical junctions, enhancing the photo-current. The argument is substantiated by a discussion of the electronic features of grain-boundaries under these conditions and the enhancement of the photovoltage due to the frequency extension of the photoresponse.
a, capture cross sectionfor holes NOFATION A, lower limit of wavelength spacing between occupied interface states A. upper limit of wavelength Burgers vector Q%,,trqJ2kn = equilibrium barrier height spacing between dangling bond levels (available 264~ eVr - eV,, = ~5- I$,,= change in barrier height states) dislocation spacing electron charge INTRODUCTION barrier height on either grain-boundary side dislocation level Recent efforts to lower the S/W-Iigureof solar cells have Fermi level lead to the application of polycrystalliie material. Quasi Fermi levels for electrons/holes Numerous methods have been developed to grow espeelectron energy at lower edge of conduction band electron energy at upper edge of valence band cially silicon in sheet form, as ribbon material (edge grain-boundary tilling factor = c/a defined growth), as thin films on foreign substrates etc. vertical junction +rea and to use this defective material directly or after some horizontal junction area recrystallization procedure. (Annealing, Laser or Eleclight generated current tron-beam recrystallization and the like). recombination current saturation current The way such material is used, is basically the same as horizontal resp. vertical current in the case of perfect Czochralski-respectively Floatingtotal junction current Zone grown crystal material, i.e. no particular steps are injection current taken to include the grain boundaries and their barrier electron current density hole current density (also vertical current density) layers into the structure for carrier collection. In most horizontal current density cases of n on p-cells, the grain boundary layers @‘) are minority carrier ditfusion length electronically floating and reduce the number of coUeceffective state densities in conduction resp. valence ted carriers by bulk recombination or, in the case of p on band number of photons incident in interval A + dA n solar cells, the grain boundaries represent e5cient carrier density at point x shunts between top and bottom electrodes. reducing the carrier (electron) concentration efficiency. carrier (hole) concentration In the case of Schottky barriertop contacts, the opendonor density circuit vohage is less effected by the grain-boundariesbut acceptor density interface state density enhanced bulk recombination leads to decreased shortoccupied grain-boundary interface states circuit current values. equilibrium number of grainboundaryinterface states In all these cases as in those involving devices where dislocationpipe radius I efficient carrier recombination within a junction is interface recombination velocity required (Light emitter diodes; Lasers) defects and grain external voltage carrier velocity boundaries are a major hurdle to performance when they barrier voltage are statistically distributed within the neighbourhood of absorption coefficient the active junction. FJFh = vertical/horizontal junction area The early study of carrier recombination at lineage grain boundary tilt angle boundaries (small angle grain boundaries) showed111that capture cross section for electrons 651
652
H. F.
hhTAi&
these structures act as hole sinks in n- and p-material. (In the latter case within a doping range ~10” cme3)[21. This is due to the higher electrondensity around the p-type core of the grain boundary or the space charge pipe. It is well known that dislocations (one-dimensional) and grain-boundaries (two-dimensional) are most influential when they are oriented vertically to the active junction. This is due to the tendency of dislocations to form overlapping line charges. Much of their electronic behaviour is dependent upon the number of charges (misfit angle between the grains); the ratio of filled to available states and the doping environment which defines the extent of their space charge[2]. Dislocations as recombination centers are often considered as simple cylinders or walls which attract carriers Fii. 1. Energy diagram of n-type semiconductor with grain boundaryat x = 0. Neutral interface state density qo.Barriersat and enhance bulk-recombination. It this case the both sides. q51=eV, and & = eV, are e.qualwithout illumination effective cross section is simply a function of dislocation or bias voltage. Space chargeradiusis R. density, their radius and of the carrier diffusion length[3]. While some conclusions with respect to the influence of a dislocation on the recombination process, for holes and electrons, the application of a Shockleye.g. in electroluminescence, can be drawn, the actual Read-Hag type recombination process leads to a physics of the dislocation and especially of the grain recombination current equality: jC = j”. The total recomboundary, requires more detailed definition as to the bination current in this case is: extent of the space charge (the core radius itself is much smaller), the number of available states, their dependence on the external potential and the conduction parallel to the line charges[2]. This especially in cases of parallel orientation of grain-boundaries as is the case in large grain polycrystals of limited thickness or in thin film form. We will show that grain-boundaries can also be used in where e equals the electron charge; N,= the interface a positive way if they are included into the actual collecstate density and u the carrier velocity. tion process for the optically injected charge carriers. As a consequence of a,, = a, also n(o) =p(o) (Fig. This is also known to be true for III-V-light emitting 1). In other words, J, is the maximum recombination structures, when the difference between Q and &discurrent shifting VD and establishing the condition n(o) = locations and their respective environment is taken into p(0) at x = 0. account[2,4]. It can also be shown that the interface recombination Therefore the problem of solar cell efficiency is more velocity S(w) can be expressed as: complex in III-V-compound material when poiycrystal1 line layers are used[S], because of the dependence of the electronic effects, especially the recombination process, [6] (‘) on dislocation type, doping environment of the dislocation and its orientation with regard to the active with exp(eVdkT) being the grain boundary enhancejunction plane. ment factor. Estimates of the lifetime reduction as a function of 1. FZLECTRONICPROCESSESAT GRAIN BOUNDAUIES interface state density and grain size give reasonable values when isolated grains are assumed interfering at Electronic processes around dislocations under optical illumination have been studied in the past and con- random with the carrier flow[6]. As a physical structure, clusions have been drawn with respect to the population the dislocation or the two-dimensional grain boundary of interface states and minority carrier lifetime[61. This can have varied properties with respect to the current was carried out for the isolated dislocations. In this case, flow. Figure 2 shows the basic structure and its dimenthe excess carriers generated by illumination shift the sions as it appears within the host lattice. As has been shown(2], the outer space charge extension or the pipe quasi-Fermi levels Ef, and E,,, according to: radius R may reach values in the 10 p or up to 1OOp (1) range. It can therefore have a decisive influence on n = NCexp [(Ef, - EdkTl carrier flow when it is connected to one side of a junction, e.g. when its p-type core is connected to the (2) P = N. expWY - Ef,MTl p-type side. If dislocations are formed between two grains, we have the planar structure of a grain boundary. see Fig. 1. Figure 3 shows such a grain boundary plane formed at Assuming equal capture cross sections (CT”= up = u)
Carrier collectionat grain-boundaryharriersin solar cells
whole structure can be included in a contacting or charge collecting scheme.
I
h-l -lo-' em
R
r.
As shown in the Appendix, the space charge pipe radius of a grain boundary depends on the angle of mistit 0 and on the bulk doping. The dislocation spacing is:
t--
423 \
2. PHUlOVOL.TAlCEEpEeFsATGRAlNBOUNDAlUES
m
,--;--
653
.+_-+/’
-_L
l”llW
cure
rtm A
+
-
mobile slectronr
+
mobile holes
+
1
D = b/2 cosec 812
(6)
where b equals Burgers vector. The density of interface states is in the range 10” < 1/P < IO” cm-*
(7)
for angles of misfit 8 between 4 and 10” (pure edge dislocations). In a typical case, a misfit angle of a few degrees leads to 10’2cm-2 as area state density. This figure results from the fact that there is a filling factor for the line charge: f = c/a = 0.05-O.1
Fig. 2. Detailed form of a space charge cyclinderaround a dislocation with p-type inner core aad space charge pipe. Dimensions for normal doping environment (10’6-10”cm-3 m-type).
03
(in the temperature range from 50 to 300°K and for Fermi statistics. See Ref. [2], p. 215). f is the fraction of occupied states; c the spacing between dangling bonds (available states) and a the spacing between occupied sites. (Fig. 3). There is also another fill-factor f perpendicular to the pipes:
(8) and (9) lead to the area state density:
(10) (see Ref. [2], p. 218). For the above range for the till-factor, we have Nos
Fig. 3. Dislocation plane between two crystalline grains. c = distance between dangling bonds, a = distance between filled levels, 8 = tilt angle. 2R = diiter of space charge pipe (see text). D = dislocation distance; D = (b/2). coscc#2 for B> 1”; dopingrange IO”-IO” cmb3.
= 1-4 X 1Or2cmm2
as the number of tilled states in the grain boundary sheet [2,61. We now derive the light generated current due to carrier injection into the grain boundary space charge pipe. Injection causes a current across the junction: Z, = const exp (-Adk7’).
the mating interface of two grains misoriented by an angle 8. If the dislocation distance D = b/2 cosec(e/2) (b = Burgers vector) is small or 1”< 8’ < 25” (medium tilt angle, see Ref. [2]), one can assume that for normal doping ranges the dislocation distance equals the space charge radius R( see Appendix A). Thus a coherent layer with overlapping space charges is formed. The p-type cores can be connected by a p-type layer and thereby the
(11)
(12)
As before (Fa. I), there is a change in equilibrium barrier height and A4=EF-Ek=eV.
(13)
eVI - eV2 = eV, (V, = external voltage either due to injection or to bias) (see Fig. 4). As IL, the light generated
654
H. F. MATAM?
introducing the distribution function, (17) leads to the expressions: v,=$-[~[l+(l+$$“3 +~[1+(1+~)“2]-1]
- - __
-EF
v*=&[~[l+(l+5$)“*] -~[l+(l+~)“*]-‘~.
Fig. 4. Grain boundary barrier under photon injection (illuminaNeutral number of states= qo. Neutral barrier height q&=
tion).
nqo2/2kn (k = dielectric constant, n = doping range). Lit generated bias: eV, = eVt-eV, = A& Light generated additional states:
It is apparent that injection, e.g. into the left side of the n-p-n layer (Fig. 4) causes A4 to increase: AC#J = eV,eVlo= eV. and therefore also q is increased: q = qd2 [ 1 t (1 t (Ad&))“*] and more interface states become available (up to V, = kT/e). For V, %=kT/e and VI B kT/e, the variation of available number of states n, with V, is given by the function:
an,o= 2n, & “* w q. eve
(>I
4=qd2[l+(l+$yJ.
(19)
&‘I2
fT
T;T
em
( >
+$[V~~*exp(--~)~~-’ current is: Zi- I, with I, = saturation current, e.g. for the left n/p junction: zL =
const e-=“tfkT ee”z/*T _
1,
= Zs(emANkT- I).
(14)
(15)
(Ref. 121,p. 360). Assuming also that V2* kZ’/e, the second term in brackets can be neglected. As the incremental photovoltage is AV
Thus
P
with (13) photon injection generates a voltage: V.=$ln(l+ZJZS).
(16)
The shifted electrostatic potential due to photon injection corresponds to an external voltage V, = Ade. However, we now consider the fact that the equilibrium number of occupied states q. changesalso, as can be shown (Ref. [2], pp. 288-295). The number q of states occupied is higher due to the deviation from thermal equilibrium caused by virtue of V.. The ratio of actually occupied states to their equilibrium number q. is: (17) & = ?rqo2/2kn= barrier height in equilibrium. The expressions for the barrier height on either side of the space charge cylinder are: (Fig. 4) VI==
k (
2?rq 7+q
enV= * H
k is the dielectric constant and n the charge density; and
(20)
&,a”‘0 e
av,
(21)
it results from (20) and (21) that the photovoltage is linearly dependent on V, This has been substantiated by measurements, e.g. on germanium bicrystals[2,7,8] and is similar in silicon and III-V semiconductors with the complications discussed[2,4]. The fact that the photovoltage is enhanced by the barrier dependent number of available interface states is a feature of grain boundary layers when correctly contacted. Examples are given in Figs. 5 and 6. In Fig. 5, the frequency extension as compared to a normal p-n-junction (same material) appears clearly in the long wave-length region while in Fig. 6 the short optical wavelength extension is shown. We conclude that it is possible to arrange the diffusion and contacting scheme in such a way that the majority of the widely separated grain boundary planes in large grain polycrystalline material contributes to the resultant photo-voltage. 3. GRAIN BOUNDARY PLANES AS VFlyTlcAL JUNCTIONS
In most cases of crystal growth or layer growth from a substrate, during growth of thin films and ribbons, especially under application of a temperature gradient, vertical to the growing surface, the grain boundaries are oriented vertically across the layer. This has significance for the possible use of grain boundary barriers as vertical
Carriercollection at grain-boundarybarriersin solar cells
A
1 PHomN
EIklmY-ifi
I1
Fig. 5. Photoresponse vs photon energy under the same conditions and on the same bulk material (Ge) for a p/n junction (diffused)and a grain boundary(see Ref. [2]).
-I
In this case the grain boundaries are included in the collection process for the majority carriers (holes) and do not act as sensitive photovoltaic barriers (see Chap. 4). The argument concerning the positive influence of the grain-boundaries as recombination centers is based upon the well known sheet conductance (p-type core) of the space charge layer, surrounding the dislocation array[2]. Different from [6], we assume a continuous flow of excess carriers along the space charge pipes, contributing essentially to the resulting efficiency when the correct contacting scheme is applied (see Fig. 7). A p’diffused or ion-implanted top layer establishes contact to the p-type core of the grain boundaries and the n’-side (backside) layer forms isolating junctions, blocking carrier flow from the space charge pipes (p’) into the base contact. In cast layers or thin films of polycrystalline material, grown without special precautions concerning gradient freezing, the grain boundaries may of course deviate from vertical planes and therefore will act more at random, than shown in Fig. 7. However, as the entire top layer is p+. all grains ending on the surface are contacted and act as vertical multijunctions [ 111. In reasonably pure material, the bulk lifetime between the grains can be assumed to be normal and carriers are collected by the grain boundaries to a depth, consistent with Ln. Especially in the case of high density photon injection (concentration) with attendant increase in carrier lifetime, grain boundaries improve the carrier collection. The collected number of carriers at a depth x is:
xN(h) e -a(*)Xe-“$ .a
1.0
1.2
1.4
1.6
1.8-h
1
Fig. 6. Photoresponse vs optical wavelength for a bicrystal at two differenttemperatures.A at T = 300°K;0 at T = 77°K.
junctions and as extensions of the active areas. In addition, a significant increase ifi optical response may result, as will be discussed. This proposition[9] grew out from the availability of large grain size polycrystalline layers of silicon [ IO] and GaAs [5] with grain boundaries oriented mainly vertically to the surface of the layers and separated by monocrystalline domains in the 10-1OOj~m range. As the grain distance d is comparable to or exceeds the minority carrier lifetime (d s L), each grain boundary can act as a vertical collector if it is contacted correctly and electronically included into a p+ surface layer, while the bottom layer is separated electronically by an $+-layer, making contact only to the base or n-type material, separating the grains. The reverse case of p-type bulk material and an n+ top junction with separating barriers to the grain boundaries at the top, cannot yield a vertical extension of the active junctions through the grains.
655
dx
(22)
where A, + A0 equals the wavelength limits (absorption curve); a(A) the absorption coefficient; N(A) the number of incident photons in the interval A +dA and & the minority carrier diiusion length. The computerization of eqn (22) for normal vertical junctions has shown that the short-circuit current density
Fii. 7. Proposedcontactingscheme for polycrystallinesamples. Grain boundariesare contacted by top p+ diffused layer. Base contact is separatedoff by n+ diffusedlayer. Reverse situationin the case of n-type cores or in the case of III-V material.
H. F. MATARB
656
is increased by about 20% as compared to the horizontal junction-case only. The open circuit voltage on the other hand is reduced by approx. 10%. In the case of grain-boundary junctions, A,+& is extended, but also the photovoltage is increased, as we have seen. This compensates for the usual loss in V, when Z,, is increased. The total junction current can be written: ZtO,= Z,,+ Z, = j,,F,, + j,,F, (2% where I,,, 1. equals the currents in horizontal and vertical junctions; jh, jv the horizontal and vertical current densities; Fhr F, the horizontal and vertical junction areas. With ~3= pjFh, we have: (24) or for a unit surface area, the total reverse saturation current amounts to: ilor = Ml + B).
(25)
Consequently, the overall current density is enhanced. The open circuit voltage V, decreases however, as seen from the diode characteristic: L, = id1 + /3Nexp(qV/akT) - 11- Lt.
(26)
Here:
WI
(27)
jJch and L are the short circuit photocurrent densities of the illuminated cell in the horizontal and vertical junctions. One can see from eqn (26) that the voltage is reduced as /3 increases. In the case of grain boundaries, the enhanced spectral sensitivity will compensate for this loss, as seen from (22). The extension of the sensitivity is active in both the long-wavelength and short-wavelength ranges (Figs. 5 and 6). This is also typical for silicon bicrystals. (See discussion of photoelectric effects on grain boundaries in diierent materiais[2, U-151). If we assume a factor of 5 as a conservative enhancement of the photovoltage in the short- and long wavelength sections of the integrated response (Figs. 5 and 6) and assume that this extension amounts to l/IO of the total output only, the overall efficiency is increased by 50%. The capture cross section of the grain boundary depends on the actual space charge, barrier height,
4 FRACTICAL
RENAZATIONS
In most practical cases, structure Fii. 7 is convenient to produce. It has the two fold advantage of the contribution of the grain boundary barriers to the enhancement of the photovoltage V, on account of the activity of the interface states according to eqn (19) and the effect of the vertical junction according to eqns (22) respectively (24) and (26). For a given material with a defined geometry, computerization of results is possible when the frequency extension of a single grain boundary plane is known or has been measured. Due to random arrangement of the grains in thin films, ribbons and cast material, exact predictions may be difficult, but the structure proposed should enhance efficiency in all cases. The foregoing considerations are also valid for III-V solar cells, but with the additional diiculty that for wider gap materials the grain boundaries may be p-type or n-type, depending on their orientation within the lattice, grain boundary orientation with respect to the active junction and doping environment (see Ref. [2], p. 168 and 382). Depending on the bulk doping, the a( Ga) dislocation is an acceptor in n-type environment and a donor in p-type material. @dislocations behave similarly [4,17]. In the case of n-type grain boundary cores, the doping should be chosen opposite to Fig. 7, top junction n on p and p+ at the base layer. When thin films are deposited on insulator substrates, the alternating surface contact scheme (Fig. ll), can be applied. If it would be possible to arrange the grain boundary layers in such a fashion that they would alternate with the N-type material (Fig. 9) with contacts to the ends of the crystal slab and light falling mainly onto one side of + r,
/-
Pf
!
“+
0
P+
INSULATOR BASE
Fig. 8. Alternatingp+-n+-layers on bulk n-type thin film solar cell with grain boundaries.P-type cores contact all p+ layers and have blocking layers to II+ regions. hv
quasi-Fermi level and bias. Capture rates are of the order of 4 x IO”see-’ cm’ and capture cross sections are 400 set-‘/dislocation.
Fig. 9. Alternating p-n layers due to grain boundary barriers. Case of anomalousphotovoltage.
Carrier collection at grain-boundary barriers in solar cells
the junctions, the case of anomalously high produced. According to eqn (16)
V,
could be
(29) where ZIei is the short circuit photo-current density of ith-junction and Z,, th’e saturation current density of ith-junction[l6]. Recently solar cells have been fabricated from inexpensive, low-grade, non crystalline silicon with a domain structure of sufficient distance between grains (d> L). This material is produced by casting ingots, subject to directional cooling[l8]. Ingots and wafers cut from this material have large central regions where the grain boundaries are running perpendicularly to the surfaces, due to the gradient freezing process. Improvements in thermal set-up will undoubtedly lead to a better yield of good material. Measurements on solar cells, produced from such polycrystalline material have resulted in unexpectedly high efficiencies, in some cases over 12% AhfO[ 191. These results were achieved on p-type base material, as reported by H. Fischer (AEG-Telefunken) on Wacker-Heliotronic ingots. The reasons for the use of p-type base material are two-fold. Firstly, a relatively simple purification process leads to p-type material (residual Boron) and secondly, it was intended to minimise the electrical activity of the grain boundaries [20]. But since the grain boundaries are electrically connected to the p’-base layer, they represent active hole sinks within the generally lesser doped base material and can enhance efficiency in this role. However, if the starting material could be purified sufficiently to convert it to n-type of, e.g. 5 x 10”cm3 doping, the scheme of Fig. 7 could be applied, with the result of the full use of the grain boundaries as vertical junctions with enhanced photoresponse. CONCUISWNS
The electronic features of grain-boundaries are assessed in view of the use of polycrystalline material in photovoltaic (solar)-cells. After discussion of the dependency of some basic properties as, recombination velocity and state density from grain-boundary parameters, the photovoltaic properties are considered, based on known relations governing barrier height, state density and external voltage. It is shown that the positive differential of state density vs bias, results in an improved photosensitivity, suggesting the use of the grain-boundary barriers as vertical junctions. This case is discussed along the lines considered by other authors and yields only a 20% increase in output, while the correct use of the grainboundaries as photoioltaic converters could result in a substantial increase in output. Some practical results from tests on polycrystalline wafers are discussed and a more efficient use of the grain-boundary posed.
barriers is pro-
657
Acknowledgements-The
author expresses his gratitude to Dr. Horst Fischer, head of the optoelectronics and solar cell departments of AEG-Telefunken, Hcilbronn, Germany, and his associates for a thorough discussion of their experience with solar cells, produced from polycrystalline wafers made by the Wacker-Heliotronic process of casting and gradient-freexiq. I. F. L. Vogel, W. T. Read and L. C. Lowell, Recombination of holes and electrons at lineage boundaries in germanium. Phys. Rev. 94, 1791(1954). 2. H. F. Matark, Zkfecl Elecrmnics in Semiconductors. WileyInterscience, New York (1971). 3. M. Lax, Junction current and luminescence near a dislocation or surface. 1. Appl. Phys. 49(5), 2796-2810 (1978). 4. A. L. Esquivel, S. Sen and W. N. Lin, Cathodoluminescence and electrical anisotropy from Q and B dislocations in plastically deformed gallium arsenide. I: Appl. Phys. 47(6),25882603(1976). 5. R. J. Stim, High efficiency thin lilm GaAs-solar cells. 1st Interim Rep. Jet Propulsion Lab., No. 730-9, and NTIS Report: P&258,493, JPL 5030-33(1977). 6. H. C. Card and E. S. Yang, Electronic processes at grain boundaries in polycrystalline semiconductors under optical illumination. IEEE Trans. Hecfron Lkv. ED U(4), 397-402 (1977). 7. H. F. Matare and K. S. Cho, Parameter dependence of LF. voltage in optical mixers and bias dependence of photo conductivity in bicrystals. Prtz. Eleclmn. Compounds Conf., pp. 415425. IEEE, Washington, D.C. (1966). 8. H. F. Matark, Dislocation plane devices in optical communications, advance in electronics. Proc. XZZZthZnt. Sci. Gong. Elccrron., Rome, Italy, Vol. I (1966). 9. H. F. Matark, The structure and electronic properties of grain boundaries in semiconductors. Pax. Nat. Workshop on Low Cost Polycrysralline Silicon Solar Ceils. South. Methodist University, Dallas, Texas (1976). IO. T. L. Chu, H. C. Mollenkopf and Shirley S. C. Chu, Deposition and properties of silicon on graphite substrates. 1. Elecrmchem. Sot. 123(l). lob110 (1976). Il. P. Shah, Analysis of vertical multijunction solar cells using a distributed circuit model. Solid-St. Elecfmn. 18. 10994106 (1975). 1 12. M. K. Mukherjee, F. Pfisterer, G. H. Hewig, H. W. Schock and W. H. Bloss. Shaoe of the u-n iunction and crvstalline structure of Cu$-CdS thin film so&r cells. 1. Ap& Phys. 48(4), 1538-1548 (1977). 13. R. K. Mueller, Transient response of grain boundaries and its ;;iation for a novel light sensor. J. Ajpl. Phys. 30(7), 1004 14. W. W. Lindemann and R. K. Mueller, Grain boundary photoresponse. /. Appl. Phys. 31(10), 1746(1960). IS. H. F. Matark, D. C. Cronemeyer and W. W. Baubien, Gebicrystal photoresponse. Solid-St Elec&m. 7,583 (1!%4);see also [21, pp. 349-364. 16. J. I. Pancove, Optical Processes in Semiconductors. Electrical Engineering Series, pp. 327-329. Prentice Hall, Englewood Clifis, New Jersey (1971). 17. H. F. Matark, Light emitting devices, Part I. Methods. In Advances in Electronics and Electron Physics (Edited by L. Marton), Vol. 42. Academic Press, New York (1976).Part II, Vol. 45 (1978). 18. B. Authier, Polycrystalline silicon with columnar structure. In Festk&erpmbleme XVIII (Advances in Solid State Phvsits) (Editkd by J. Treusch), pd. l-17. Vieweg, Braunschwe-ig (1978). 19. H. Fischer and W. Pschunder, Lowzost solar cells based on large-area unconventional silicon. IEEE Tmns. E/e&on Devices, Vol. ED24, No. 4, pp. 438-442 (1977). 20. H. Fischer, Solar cells based on nonsingle crystalline silicon. In Feslko”rperprobleme (Advances in Solid Stare Physics), Vol. XVIII, pp. 19-32 (Edited by J. Treusch). Vieweg, Braunschweig (1978).
H. F. MATA&
658 APPENLM.x
bRcm
The space charge pipe The distribution function of the grain boundary states depends on the average dislocation level ED and the ratio of occupied states q and equilibrium number of states qo:
with l+exp f”=(
w4
EF is the Fermi level. q/q0 is related to the ratio of actual barrier height eV, and barrier height at equilibrium: 4 = ?rqo2/2kN (see Ref. [2], p. 300). q/qo=;[l+(l+~)“‘].
(A3)
Fig. ib. R&us R(cm) of space chargepipe of grainboundaryas a function of tilt angle 0. Nd- N. is parameter.D = dislocation distance.
therefore:
Therefore:
f=l(l+exp~)[l+(l+$$)“*].
2sin(0/2) [lt(l+eVJfk)‘“] TbAN 1 texp(ED-EF)/kT
(A4)
The grain boundary filling factor c/a (spacing of free states/spacing of filled states) is another expression for f (A4). l/a is the state density within the core of the dislocation and is compensated by the space charge.
In
a
simplified
=
set
f=
range) and V, = 0, no external voltage. This function or =
(y)”
[’ sin;e/2)]“2AN-l/*
(A@
(A9
Therefore, the space charge radius R defining the range for the compensating charges is given by:
R
we
(A7)
1 + exp (ED - EF)/kT = 0.1 (for a hormal temperature
R
nk*( Nd - No) = l/u.
calculation,
“* ’
l/2 1 t (1 t eVJ,dcbo)“* 12rc(Nd - Na)[ 1 t exp (ED - EF)kT] 1 ’
is plotted in Fig. 10 for medium tilt angles and different impurity ranges AN from lOI to 10’9cm-3. D is the dislocation spacing according to D = b/2 * sin 812
646) c can also be replaced by the dislocation distance D= b/2 sin (e/2), b equals the Burgers vector, and one has
for pure tilt boundaries. The actual space charge pipe and its inner core with preponderant hole conduction is shown in Fig. 2.