- Email: [email protected]
Model calculations for metal-insulator-semiconductor solar cells
Solid-Slatr
Elecwonm.
1911. Vol. 20. pp. 741-751.
Pergamon Press.
Printed in Great Britam
MODEL CALCULATIONS FOR METAL-INSULATOR-SEMICONDUCTOR SOLAR CELLS? LARRY C. OLSEN Department of Materials Science and Engineering, Joint Center for Graduate Study, 100 Sprout Road, Richland, WA 99352,U.S.A. (Receiued 12 January 1977;in revised form 1 March 1977) Abstract-An analytical approach to calculating MIS solar cell properties has been formulated and utilized to study solar cell efficiency as a function of interfacial layer thickness, various interfacial film parameters and band gap. Three models are considered regarding interface state recombination kinetics. Calculations are presented for the case of interface states being in equilibrium with the metal (Model (I), in equilibrium with the majority carriers of the semiconductor (Model II) and in equilibrium with the minority carriers (Model III). It is found that in all three cases, the efficiency of low barrier height, Schottky barrier cells can be increased very significantly. For example, it is shown that for a band gap of 1.5eV and a barrier height of 0.5 eV, it appears possible to increase the cell efficiency from essentially zero to 12%. If the barrier height is 1.0eV, an efficiency of over 20% is possible. It is determined, however, that MIS solar cell performance is limited by leakage currents due to minority carrier diffusion back into the bulk. As a result, the upper limit of performance is defined by that for a homojunction. These calculations identify ranges of surface state density and interfacial barrier heights necessary for a good MIS solar cell.
NOTATION
n
Richardson constant for electrons (holes) using transverse (to plane of barrier) effective mass constant (2ee,N&eiZ) density of interface states, cm-’ eV’ charge of a proton, coulombs location of conduction band at surface, eV Fermi level for holes, eV Fermi level for electrons, eV Fermi level for metal, eV electron band gap, eV location of valance band at surface electron flux in direction as shown in Fig. 2, and due to carrier generation in the depletion region, cm-* see-’ electron flux in direction as shown in Fig. 2, and due to carrier generation in the base region, cm-* see-’ electron flux due to electrons tunneling through the interfacial region in the direction as shown in Fig. 2 hole flux in direction as shown in Fig. 2 photon flux per unit wavelength, cm-’ nm-’ current density mA/cm* minority carrier leakage current component, mA/cm’ current due to collection of carriers generated in the depletion region, mA/cm* current due to carrier generation in the base region, mA/cm2 maximum possible current supplied by a solar cell, mA/cm’ open circuit voltage voltage at maximum power current at maximum power saturation current associated with minority carrier diffusion in base region, mA/cm2 recombination current at surface, mA/cm* saturation current associated with thermionic emission of majority carriers, mA/cm* tunneling current due to electrons, mA/cm’ tunneling current due to holes, mA/cm’ Boltzmann constant, eV “K-’ relative dielectric constant minority carrier diffusion length, wrn
b
N.4 No
N, NA
Q RE RH R, T
TA V V,
V, w a Q* E 6 E, 8 A 40 2 4)RP 6” ? 7 X XC0 X= x.0
tWork supported under NSF Grants AER 75-20501and ENG 74-20444. 741
X”
n-value in J-V relationship electron concentration in p-type region under equilibrium condition, cm-’ acceptor density, cm-’ total incident photon flux, cm-’ total number of surface states, cm-’ incident photon flux of one wavelength, cmm2 collection efficiency for solar spectrum of Schottky barrier rate of recombination of electrons into surface states, cm-’ see-’ rate of recombination of surface state electrons into holes, cm-’ set-’ reflection coefficient for photons of wavelength h temperature, “K transmission coefficient for photons of wavelength A through metal load voltage change in potential across interfacial layer, volts change in potential across depletion layer, volts depletion layer width, cm e&D&, optical absorption coefficient, cm-’ distance of electron quasi Fermi level below conduction band, eV dielectric constant of interfacial layer material, f/cm dielectric constant of semiconductor, f/cm interfacial layer thickness, A Ez-4BP-~,eV level above valence band to which surface state levels are occupied in isolated semiconductor, eV E,+,y-&, eV Schottky barrier height for p-type device, eV effective barrier height metal work function, eV solar cell power conversion efficiency, % minority carrier life-time, set electron affinity, eV interfacial barrier height above conduction band with no photocurrent flowing, eV interfacial barrier height with finite photocurrent, eV interfacial barrier height below valence band with no photocurrent flowing, eV interfacial barrier height with finite photocurrent, eV
742
L.
c.
I. INTRODUCTION
MIS cells refer to Schottky barrier devices with a nonconducting interfacial layer between the metal and semiconductor. Several recent experimental papers indicate that the addition of such an interfacial layer improves solar cell performance [l-5]. The clearest evidence of such an effect is provided by the work reported in Ref. [S] on GaAs cells. MIS structures were fabricated by depositing gold onto n-type GaAs after various times of exposure to air to grow an interfacial oxide. Significant improvements in efficiency were observed as the exposure time was increased. There also have been several recent theoretical papers concerning the photovoltaic characteristics of MIS cells[bll]. Each paper has emphasized different aspects of the problem. In Ref. [6], a MIS system is treated in detail by numerical methods for one particular operating mode, namely, that of a minority carrier diode. The barrier height and interface voltage distribution are considered in a fairly complete manner in Ref. [7] for the case of interface states being in equilibrium with either majority carriers or the metal. It was also assumed that the interfacial barrier has a negligible effect on minority carrier transport. The effect of interfacial charge on barrier height was considered in Ref. [9]. The present author reported on calculations of MIS solar cells in Ref. [lo] for the case of interface states being in equilibrium with majority carriers. Finally, the effects of interface states being in equilibrium with minority carriers was examined to a small degree in Ref. [I 11.
OLSEN
The purpose of this paper is to provide a practical analytical approach to calculating MIS solar cell performance, and to present the potential performance of such cells. A model is developed which accounts for three possible situations for the interface states in a MIS cell under illumination, namely, with the interface states in equilibrium with the metal, the semiconductor majority carriers or minority carriers. It is assumed that the charged impurity concentration in the semiconductor is ? 10” cmm3. Although this assumption presents some constraints on the analysis, it allows a straight forward analysis of cell performance to be carried out. It is also assumed that the interfacial region is chargefree. Effects of such charge could easily be added. Effects of such charge were not included because of the number of device configurations increases tremendously if it is included. In addition, the main effects of interfacial charge are clear. The impact of interfacial charge on results presented here is discussed in a later section. Finally, the analysis and calculations are carried out for a p-MIS system. All the models considered can be applied to a n-type material, by simply inverting the band structure and reversing the roles of electrons and holes. The analytical approach to MIS solar cell calculations is presented in Section 2. Calculations of solar cell characteristics for three models regarding interface state behavior are given in Section 3. Discussion of the results is presented in Section 4, and conclusions are considered in Section 5.
Fig. I. Equilibrium electron band diagram for p-MIS cell
743
Model calculations for metal-insulator-semiconductor solar cells 2. ANALYTICAL AFTROACH
Consider a p-type semiconductor coupled to a metal.
The equilibrium band diagram is shown in Fig. 1. xc0 and xWOrefer to equilibrium values for the interfacial barrier height. Figure 2 shows a band diagram for a cell under illumination. The location of the surface state Fermi level EFs depends on recombination kinetics. This will be considered below. The analytical approach will be discussed in four parts. First, features of a MIS band diagram will be considered. The voltage distribution across the interface and depletion region is then considered. Current-voltage characteristics are examined and finally, an outline of the computational procedure is presented. 2.1 MIS band configuration There are features of the band diagram in Fig. 1 which need elaboration. xc0 and xaO are referred to as interfacial film barrier heights. The “0” subscript refers to the value when no voltage exists across a load. xc0 and xuo will be treated as variable parameters. 4sP can be related to interface parameters. Using the abrupt junction approximation, and following Sze [ 121,
interface states in cm-*eV-’ and ei is the dielectric constant of the interfacial layer material. & is defined such that if the semiconductor surface is in a vacuum, surface states will be filled up to 40 in order to achieve short-range charge neutrality. & is set equal to E,/3 in this study. Ds is assumed to be constant over the band gap. More specifically, D, can be viewed as the average value of interface state density over the region that the surface state Fermi-level (EF~) exists during cell operation. This region will depend on the model assumed regarding interface state kinetics. In writing I$B~according to (l), it is assumed that C=
2eesN& 7
Ei
a 1.
This condition is generally satisfied if NA < 10” cmm3. The interfacial barrier heights xc0 and xuo are often assumed to have values determined for thick interfacial layers (several hundred A). Thus, at a SiOJSi interface, one might assume X=O= 3.2eV and xuo= 3.7eV. However, Card and Rhoderick’s[ 111 analysis of MIS silicon diodes suggest numbers for xc0 to be more like 0.1 $BP = Y&IO•t (1 - r)&l (I) to 0.2 eV. Film thicknesses of interest are on the order of 10-40A. As a result, use of bulk properties for such a d~ao=Eg+x-$~m (2) film should be done cautiously. As a final note, the abrupt junction approximation 7=(1+(y)-’ (3) implies that the minority carrier concentration is less eSD, than Na in the depletion region. If n, is the surface electron concentration, we must have n,i: NA. Thus, in order for the abrupt junction approximation to be valid, where 40 is a surface state level, D, is the density of E ~0.085 eV if Na - 10’8cm--3, E ~0.144eV if Na(y=-
FDEPLand FTE refer to electron Fig. 2. p-MIS solar cell under illuminationand supplyingcurrent to a load. F DIP~, fluxes, and FTH refers to “hole” flux. The J’s with the same subscripts designate the corresponding current densities.
744
L. C.
10’7cm--3 or E 3 0.203 eV if Na - lOi cm-3, where E defines the location of the electron quasi Fermi level. It will be assumed E 5 0.12 eV in this work. A more general treatment must, of course, take into account the effect of the minority carrier density on the potential in the interface region. 2.2 Voltage distribution across interface and depletion regions As photocurrent is produced and a voltage V is developed across a load, the potential across the interfacial region and depletion regions are affected. These changes are defined as Vi and V.,, respectively, with a sign convention defined by Fig. 2. Vi and V, are related by
OLSEN
the semiconductor to the metal, and JrH the current due to holes tunneling from the semiconductor to the metal. It will be understood that actual positive charge flows from left to right. JTE and JTH are, of course, minority and majority current terms. The majority and minority carrier currents will be treated separately. 2.3.1 Majority carrier current. Neglecting shunting conductance and series resistance effects, JTH is strictly due to holes flowing from the semiconductor to the metal. JTH is given by the usual result for Schottky barriers, except for a factor accounting for tunneling through the interfacial layer. Using the WKB approximation as described by Card and Rhoderick[l3], J.rH = J, (e”s”’ ~ ] )
v = vi + v,.
(II)
(6) where
The values of Vi and V, depend on the change in the amount of charge stored in the interface states as a result of a voltage drop V existing across a load. Fonash developed a general expression relating V and V, for a n-type MIS system[7]. Using the same procedure in the case of a p-type device, V can be written as V = V, + a[4,,
- (Em -Em)]
+ c”2[(vh~)“2 - (Vbr - VY’I = v, + a [l#wP- (Em - E”S)I
(7)
where E,, refers to valence band edge at the surface and it is assumed that C Q 1. Clearly, Vi can be identified as
Vi = (Y[~EZP -(Em - &,)I.
(8)
The voltage distribution therefore depends on the location of EFS, or the recombination kinetics involving the interface states and the metal and semiconductor. Three cases will be considered in this paper. Model I will be based on assuming the interface states are in equilibrium with the metal. Model II is based on the interface states being in equilibrium with the majority carriers. Model III involves the interface states being in equilibrium with the minority carriers. As V, and VS are established, xc and x,, are affected. These interfacial barriers are utilized in expressions for tunneling currents based on the WKB approximation. Using the average height to define xc and xv, xc =x4+-
V, 2
(9)
V, X” = xc>0-_ 2 . 2.3 Current-voltage characteristics Referring to Fig. 2, the current-voltage are given by .I=&-JTH
J, = A*,T’exp
[-$&HI
exp [-xZ “‘61
(12)
X~ and 6 are to be written in units of eV and A, respectively, in the exponential term of (12). AqH is the Richardson constant for holes with their momentum transverse to the barrier. 2.3.2 Minority carrier current. Photogenerated minority carrier current comes from two sources, diffusion from the bulk (JDLFF) as a result of electronhole production in that region, and as a result of electron-hole production in the depletion region (JDEPI.). It will be assumed that under steady state conditions, recombination currents at the interface can be neglected. The validity of this assumption is examined in Appendix B. Under these circumstances, requirement of continuity of current through the interfacial layer dictates that J-r,. =J,,,,
(13)
AJ,,, 12,.
I,,,, can be determined by solving the minority carrier transport problem in the same manner as done for a n/p cell, or Schottky barrier. Assuming that E is known, the boundary condition at x = I+’is chosen to be Sn(x = w) = n,(,(eA’l“ e’s”’
- I)
(14)
where ;\=E,-&z-e.
(15)
Assuming a semi-infinite, frontwall device coupled to a monochromatic beam of photons, and an exponential absorption of photons, the solution to the minority carrier transport problem in the bulk region leads to
JoeFr = eTh(l - K)N, characteristics
( 16) where
(IO)
where &E is the current due to electrons tunneling from
J0 = en,o(L/7). R, is the reflection coefficient of the cell. r, the trans-
Modelcalculationsfor metal-insulator-semiconductor solar cells mission coefficient (through the metal layer), a, the photon absorption coefficient, L the minority carrier diffusion length and T the minority carrier life-time. The first term in (16) is the photocurrent. The second term represents diffusion of electrons opposite the generation current direction. This term is normally neglected in Schottky barrier analysis. In MIS devices, it can be enhanced due to accumulation of minority carriers at the interface. MIS diodes based on the dominance of the minority carrier diffusion current have been considered in Ref. [6]. In calculations presented here, JOis expressed in terms of E8 by using the expression originally utilized by Loferski[lS], namely, Jo- 1.5X 101’e-E~kT*. cm
(17)
Assuming all electron-hole pairs created in the depletion region are collected as current, JDEP,_is given by JDWL= e(1 -&)ThN,,(l
-e-OLAw)
(18)
where w is the depletion layer width. &,IFF and JDEP~can be combined as JDIFFtJDEPL=
q,(ehr,)-J,(eA'kTeVJ'eT-1)
(19)
where Qk is the spectral collection efficiency of the related Schottky barrier (i.e. with S = 0). Finally, for a spectrum of photons, J D,FF+ JDEPI.= QJmax- Jo(eA’kTevJk7- 1)
(20)
where .lnlax=e)=N* h and Q is the total cell collection efficiency. It has been assumed that the location of the minority carrier quasi Fermi level, or E, is known. To determine E, JTE is identified as the electron tunneling current. Thus, following Ref. [13], JZ = A*7ET* e-@’ F(EFi-+ EFM)
(21)
where F(EK-+ EFM) involves an integral of the difference in occupation probabilities between an electron in the semiconductor and metal. The assumption that E F 0.12 allows (20) to be simplified to JTZ
=
Ahe
-_x,W .,kT [e-
_
e-G?,-+.,,-V,,/kT
1.
(22)
In factoring out A&, we haye also assumed that the effective mass of electrons in the metal and semiconductor are equal. Utilizing (13), (20) and (22), an equation is obtained which can be used to determine E. In particular, we find A&T
Z --xc1/28-.,kT e
-e
[e
=
-GS-OBp-ViVkTj
QJmax- Jo[eA’kTeVJkT-11.
(23)
745
By definition (eqn 15) A is a function of e. In general, V, is also a function of e. 2.3.3 Solar cell current-voltage characteristics. The expressions for Jm and JTH can be substituted into (10) to obtain J= QJ,,,ax-JO[eWkTeV~‘kT-l]-J,[eVs’kT- l] (24) = QJmax- JDE- JTH. Thus, the current supplied by a MIS cell can be interpreted as the photocurrent, QJmax,minus two “leakage” currents, one due to minority carriers (&) and one due to majority carriers (JTH). QJmax is actually the shortcircuit current of the related Schottky barrier (S = 0). In the case of Models I and II, the short-circuit current for a MIS cell is given by J,, = QJmax - Jo[eA’kT - l]
(25)
For Model III, however, V, can be finite when V=O; thus, J,, is given by (24) with V = 0. It is convenient to refer to a MIS cell as a majority carrier cell or minority carrier cell, depending on which leakage term dominates. It should be noted that the dominate leakage current may be different in the case of dark or illuminated characteristics. Clearly, when these terms are used here characteristics under illumination are appropriate. Equation (24) allows one to readily determine the potential advantage of MIS cells. If an interfacial layer can be inserted in a metal-semiconductor system such that the majority carrier tunneling current is significantly reduced, but yet the minority carrier tunneling current is not, then current can be delivered at a higher voltage than in the case of S = 0. The minority carriers can tunnel through the interfacial layer easier than majority carriers because n, %ps. The performance is limited by minority carrier diffusion back into the bulk. As a result, the limiting efficiency for MIS cells is equivalent to that for homojunctions. 2.4 Computational procedure Calculations have been carried out in the following manner -in this work: (1) Select E,, dsO, Q, xeO,xvO,S, DJK and the solar spectrum. J,,,, is determined from knowledge of the solar spectrum and E8 (e.g. from Ref. 15). (2) Calculate y, ~BP and JO. (3) Select Model I, II or III. (4) Determine J vs V. For a given voltage, calculate in the following order: Vi, V,, xc, x”, E, J, and J. (5) Maximum power is found by determining the maximum value of the product JV. 3.Ml.9SOLAR CELL CALCULATIONS
Calculations of solar cell performance are presented in the following sections for Models I, II and III. An AM1 solar spectrum and Q = 1.0 are always assumed. For each model, results are presented in two ways. First, calculated cell properties for -a particular band gap are
146
L.
c.
presented in detail. Then, maximum power conversion efficiency (71)is plotted vs band gap for 4,0 = E,/3, 2E,/3 and E,. For a detailed study, EX = 1.5eV was chosen because this value occurs near the maximum of the 7 vs E, curves for all three models. To conserve space, and to allow a direct comparison of results, plots for all three models are given in each figure. 3.1 Model I: Interface states in equilibrium with metal In this case, the surface states are in communication with the metal and EFs is pinned to EFM. Thus, the net charge stored in the surface states never changes. Since E FM -E,, = &,p, eqn (8) indicates that Vi = 0. Therefore, for this model MODEL, I
EFs = EF.M v, = 0 v, = v
(25)
,y<= ,y<0 ,yL= ,yvo. It should be noted that if the C-term is not neglected in (7), that one finds V, = V/n, with the n-value being slightly greater than one. This result was determined both by Card and Rhoderick[l3] and Fonash[7]. 3.1.1 Calculations for E, = 1.5 eV. Calculations for solar cell performance for E, = 1.5 eV are presented in Figs. 3-5. As previously indicated, results for Models I1 and III are also shown in these figures. Figure 3 gives 7 vs S for x~~=x~,,~= 0.2eV, and 2.0eV. As indicated, D,/Ki has been set equal to 10” cm-‘eV-’ for Model I. This choice for Ds/Ki is discussed below. Consider Fig. 3. The values of efficiency for 8 = 0
OLSEN
correspond to an ideal Schottky barrier. Thus, as indicated in Fig. 3, 7 =O for &so = E,/3 and 10.1% for &” = 2E,/3 in the case of S = 0. The addition of an interfacial layer decreases the majority carrier leakage current resulting in increased efficiency. As 6 is increased, the minority carrier quasi Fermi level approaches the conduction band, and the minority carrier leakage current (JDE) increases until the system is a minority carrier cell. The transition occurs approximately at the maximum of the 7 vs 6 curve. As 8 is further increased, JD8 increases and 7 decreases. Fairly extreme values for xc (I and x,,, have been chosen in Fig. 3. ,i<,, = I,,,= 3.OeV gives rise to a sharply peaked 7 vs S curve, while xc (I= x,,,, = 0.2 eV results in a broader curve. This difference is related to the required value for 6 in order that the transition to a minority carrier cell occurs. An increase in xL,ocauses a decrease in the value of 8 for which the transition occurs. It is significant that the power conversion efficiency for a MIS cell is larger than the corresponding Schottky barrier. In the case of a~,, = E,/3. 9 = 0 for 6 = 0. But with an interfacial film, 7 can be increased to ==5%. In the case of $Ro = 2EJ3, 7 can be increased from 10 to 15 or 16%. Figure 4 describes 7 vs 6 for xc (I= 0 and xL,, = 2.0. It is clear that to take advantage of the MIS concept, one prefers a small interfacial barrier for minority carrier flow, and a large one for majority carriers. Tailoring the interfacial layer in this manner allows one to obtain a minority carrier cell at relatively low values of 6. A large band gap semiconductor with a similar value for electron affinity may serve as an effective interfacial layer to
&J =s
--
(approximate) ------
I I
Fig.3. MIS solar cell efficiency vs S for Models I. II and III. and assuming /$ = l.SeV. x~,,=x~,)= O.?eV and 2.0eV. For &,, = E,,E is between 0.05 and 0.1 eV for maximum power conditions. Thus, the results are slightly overestimated. (Compare the approximate results for ball = E, with the exact result for do,,,,= XJ3 in Fig. -1.1 E, = I.5 eV (-_) Model I with DJK, = IO” crn~-*eV ‘; AMI spectrum (--_). Model II with D,/K, = 0.5 x IO’?cm-‘eV ‘: Q = 1.0 (---I. Model III with D,/K, = IO” cm ’ eV ‘: \, ,, 1,,,.
141
Model calculations for metal-insulator-semiconductor solar cells
25 I
lppoximate,
‘~I----_ \
all cases)
2+‘90=23Eg \
I
IO
I
I
/
15
20
25
Fig. 4. MIS solar cell efficiency vs 6 for Models I, II and III, and assuming ER= 1.5eV, ~~0= 0 and xUO= 2.0 eV. For $SO= E8, e is between 0.05 and 0.1 eV for maximum power conditions. Thus, the results are slightly over-estimated. (Compare the approximate results for 4 B. = Eg with the exact result for &n, = 2E,/3.) E, = 1.5eV t---), Model I with D,/I(, = 10” cm-‘eV_r; AM1 spectrum (--), Model II with D,/Ki = 0.5x IO” cm-* eV_‘; Q = 1.0 (---), Model III with D,/Ki = 10” cm-’ eV-‘; xc0= 0, x,,”= 2.0 eV.
25 -
0
a-0, &=g, -_
10’0
I
/
I
I
10'2
IO" Ds/$,
,
\ \
\
10'3
cm-* eV_’
Fig. 5. MIS solar cell efficiency vs DJKi for ModelsI, II and III, and assumingE, = 1.5eV, xc0= 0 and xUO= 2.0 eV. E, = 1.5eV (-_), Model I; AMI spectrum (--), Mode1II; Q = 1.0 (---), Model III; ,ycO= 0, xUO= 2.0 eV.
achieve small xc,,, but large xVO.For example, ZnS would appear to be a good choice for several semiconductors. This choice of xc0 and xUohas a significant impact on the results for q vs 6. As shown in Fig. 4, q increased to -9% and 20.5% for +B~= EJ3 and 2E,/3, respectively. The efficiency is then essentially constant with S since xCo= 0, and is not a function of voltage. It is significant SSE Vol. 20. No. 9-B
to note that essentially ideal homojunction behavior is obtained in the case of c#Q~=2E,/3, and reasonable efficiencies are achieved by adding an interfacial layer in the case of &o = EJ3. The point at which JTIl = JDE is indicated in Fig. 4. For values of 6 greater than the transition point, the device becomes a minority carrier cell.
748
L. C. OLSEN
Figure 5 shows the effect of varying Ds/Ki when xco=O and xUo= 2.0 eV. Maximum efficiency obtained for an optimum value of 8 is plotted vs DJKi. Consider I#J~~= 2E,/3. As DJKi increases, a increases which causes &P to decrease more rapidly with 6. As a result, the maximum efficiency decreases. In the case of &0 = E,/3, 4BP is a constant because of the choice $0 = E,/3. In general, however, the maximum efficiency will decrease with larger values of 0,/K, for Model I. I-
/
I
IO
15
us band gap. Figures 6 and 7 show TJvs to an optimum value of 6. Results are shown for the three choices of 4~~~.Efficiency vs band gap is also indicated for a Schottky barrier (8 = 0). Figure 6 corresponds to xc11= X”O= 2.0 eV, and Fig. 7 to the more “tailored” configuration, xc0 = 0 and x,,,) = 2.0 eV. As indicated by Fig. 6, addition of an interfacial layer with xc0 = xUO= 2.0 increases the maximum efficiency 3.1.2 Eficiency
Es for all three models. Each value of TJcorresponds
/
I
20 5.
1
25
+J
Fig. 6. MIS solar cell efficiency vs band gap for Models I. II and III, with xc ,, = x,,, z 2.0 eV. .4M1 spectrum (-1, Model I with D,/K, = IO” cm-‘eV_‘; Q = 1.0 (--) Model II with DJK, =05x IO”eV ’ cm-‘: x,,,= k,,, = Z.Oe\ (---) Model III with 0,/K, = IO” eV ’ cm ‘: (-_) Schottky barrier (8 = 0).
Eg , ev Fig. 7. MIS solarcell efficiency vs band gap for Models I, II and III, with x, jj = Oandx,,,, = 2.0 eV. AM1 spectrum (----_). Model I with 0,/K, = 10” cm-‘eV_‘; Q = 1.0 (--), Model II with D,/K, = 0.5 x IO” eV ’ cm ‘: X,o = 0. X,,) = 2.0 eV (---), Model III with 0,/K, = IO” eVml cm ?: (-). Schottky barrier (r~= 0).
149
Modelcalculationsfor metal-insulator-semiconductor solar cells tremendously for all values of Eg. The results are even more significant in the case of xc0 = 0 and ,+o = 2.0 eV. Results for 4~9 = 2E,/3 are very close to that for ideal homojunctions. In the case of ~BO= EJ3, values of efficiency >9% are possible, while for S = 0 only 1% is possible. 3.2 Model ZZ: Interface states in equilibrium with ma-
states in equilibrium
with
minority carriers
This model is probably the most valid of the three, particularly for 6 > 10A. This model is characterized by EFS = EFN. As a result, MODEL ZZZ EFS = EFN vi= -(YA
jority carriers
This model is characterized by EF~ = EFH. Thus, MODEL ZZ
3.3 Model ZZZ: Interface
V,=V+(wA xc =X=0-aAl
EFS = EFH vi = CYV,
xu =
V, = V/n n=lta xc =,,+Gv n-l xu = xuo-- 2n V. The parameter a can be written as a = (O.l808)(D,/Ki)S
(27)
if D, is in units of 1013crnm2eV_‘, ZG is the relative dielectric constant and S is in A. Thus, when the interface states are in equilibrium with the majority carriers and C < 1, diode characteristics are described by n-values which can be significantly greater than 1. The interfacial barrier heights are functions of voltage in the cases of Models II and III. In the case of Model II, xc increases with V while xWdecreases. It should be noted at this point, that in order for Model II to be valid for cells under illumination, the capture cross-section for the majority carriers must be several orders of magnitude larger than that for minority carriers. In particular, it appears necessary to have aPlan > 103.This subject is considered in Appendix C. 3.2.1 Calculations for E, = 1.5 eV. Figures 3-5 show results for Model II. As indicated in Fig. 5, cell efficiency increases with D,/Ki for Model III. The n-value increases more rapidly with S for larger values of DJK. This is particularly true as D,/ZG increases above 10’ cm-* eV. DS/Ki has been set equal to 0.5 x 10” cm-’ set-’ for Model II calculations. This value appears to be realistic, Values of D,/Ki used for Models I and III are also quite realistic. Referring to Figs. 3 and 4, the peak in the efficiency vs film thickness curve is higher for Model II than the other two models. However, the key result is the same for all models; namely, efficiencies of Schottky barriers can be greatly enhanced by adding an insulating fiIm between the metal and semiconductor. 3.2.2 Efficiency us band gap. Efficiency vs band gap is shown in Figs. 6 and 7 for 4~9 = E,/3 and 2E,/3. The possible efficiencies at a given band gap are usually highest for Model II. This is particularly true for $JBO = E,/3, xc0 = 0 and xv0 = 2.0 eV.
,y"~ +
a Al2
where A = E8 - &W - E.This is a rather surprising result. Since EFS = EFN, additional negative charge can be stored at the interface. As a result, the change in interface potential will be characterized by Vi V. It is found, therefore, if DS/Ki - lOI cm-’ eV_’ making a 1, the potential solar cell performance is low due to enhanced leakage currents. However, if DS/Ki = 10” cm-* eV_‘, then Vi = 0 and V, = V. Under these circumstances, results similar to Models I and II are obtained. 3.3.1 Calculations for E, = 1.5 eV. Results for Model III with Eg = 1.5 eV are also shown in Figs. 3-5. The effect of increasing D,/Ki is depicted in Fig. 5. As the density of interface states increases, larger negative potential changes occur across the interface with increases values for V,. The resulting enhanced leakage currents lower cell efficiencies. For DSIKi = 1Ol3cm-* eV_‘, the maximum value for n occurs at S = 0; that is, no benefit is obtained from the interfacial layer. It is clear, however, that if D,/Ki is 7 10” cm-’ eV_’ in the case of Model III, increased efficiencies are possible with a MIS solar cell configuration. This constraint is the same for both Models I and III. Since there are so many interdependent quantities in these relatively simple MIS models, it is a challenge to convey the detailed changes in device parameters as 8 is varied. Table 1 gives a tabulation of several important quantities for a Model III cell with E8 = 1.5eV, I#JBO = 0.5 and $Q~= l.OeV. In both cases, E is always >O.l2eV. As indicated, JDE is relatively small for 6 = 0, and then gradually increases until JDE=.&J for 8 = 12 A for both cases. The main cause of n declining after going through a maximum is the continually increasing of JDE. The increased backward diffusion of minority carriers occurs because of the lower tunneling probabilities resulting from increased S. The n vs 6 curves in Figs 3 and 4 learly establish the potential for increased cell efficiency for MIS cells if Model III is appropriate. Calculated performance of cells with interfacial films having xc0 = 0 and xv0 = 2.0 eV are particularly interesting. Note that as S is increased, n decreases very slowly. The broad peak in n results because increased 6 does not enhance xc, and since a is low, +B~ does not decrease very significantly. This broad peak could obviously be very important for practical cells.
I>. c.
750
OLSFh
Table I. Model III cltlculations for E, 2 1.5 eV. Parameter values: xc ,) = x, ,, = 0.5 eV, DJK, = IO” eV
6 (A)
4HP (eV)
biW
fi
(eV)
(eV)
ImA/cm?
J,Ht
Jrrf
(mA/cm’)
J,<
(mA/cm’)
VI),
(V)
.I
0 3 6
0.500 0.500
0.500 0.555 0.610
0.51 I 0.457 0.402
Ii.0 13.5 7.7h
2.0 x 10 ” 4.6 1.4xx IO (:
21.5 X.9 27.3
0.012 0.045 0.093
14.5 14.0 19.7
2
9
0.500
0.665
0.348
6.47
0.078
27.5
0.143
21
0.500 0.500 0.500
0.777 0.72 I 0.833 0.890
0.247 0.296 0.202 0. I66
11. 2.Y3 I42 5.4x IO i 2.3x IO -’
27.4 27.5 26.4 19.2
0 3 6 9 I2 1’;
I .oo 0.997 0.995 0.992 0.989 w#;
I.oo I .05 I.10 I.16 I.21 ;:2;
0.487 0.453 0.402 0.347 0.293 ;:2;;
21
0.982
1.3;
0.132
I.hl I .‘Y I .3! I.20 0.636 0.024 3.7 x IO J 6.1 x IO ‘>
27.5 27.5 27.5 27.5 27.5 21.5 ‘7.5 27.5
2
2 2 ‘i _$
ii
6.33 2.24 12.3 IY.5 1.5x IO (,I x IO I’ 1.F x IO -I 0.014 0.5I8 I .30 I .42 I .hO
3.3.2 Eficiency L’S hand gap. As indicated in Figs. 6 and 7, calculated solar cell performance vs E_ for Models I and III is very similar for the selected value of 0,/K,. In both cases, the effective n-value of the cells is -1.0. 4. DlSCUSSION
Four topics are discussed to supplement previous discussion, namely, assumptions, effects of interfacial charge, potential impact of an interfacial film on shunting conductance, and the relative value3 of JIJrr and .7ffi, when &O.l2eV. Plotted results in this work are compatible with this constraint, except the case of &O = E, in Figs. 3 and 4. The error in these cases is still small,
:;;I
‘“O (mA/cm’)
?I
’ cm ’ (2,
21.0
0.0068 0.030 0.061 0.106
0.093 0.40 I.14 2.10
0.192 0.183 0.191 0.186
21.0 22.3 15.2 8.03
0.109 0.136 0.076 0.045
2.87 2.16 1.09 0.344
0.488 0.540 0.592 0.642 0.682 0.69 I 0.688 0.685
25.9 25.7 26.1 26.3 26.4 26.2 26.1 25.9
0.415 0.470 0.515 0.562 0.596 0.561 0.503 0.446
IO.1 I I.4 12.7 13.9 14.8 13.8 12.4 10.9
when the approximate results obtained for C/W,) = E, are compared to the exact results for &(, = 2E,/3 in Fig. 4. Inclusion of the effects of fixed interfacial charge simply alters the expression for C#W.These effects were not included here since it would increase the number of parameters to consider, and would not alter the main results. For example, even if 4Bp is increased as a result of fixed interfacial charge, cell performance is limited by the minority carrier diffusion into the bulk. One potential improvement in solar cell properties as a result of an interfacial film may be increased values of shunt resistance (L). This is particularly true for polycrystalline solar cells. Shunting conductance is probably due to regions of low barrier height where majority carriers can easily flow, or due to metallic-like shorting paths. In either case, an insulating layer would probably help. A film with xCo= 0 and large xUO(for p-MIS cells) would be particularly valuable in this regard, since relatively large film thicknesses can be utilized. As a final point of discussion, consider the case for &Cl = E, and S = 0. This situation probably has never been, nor ever will be, realized in practice. Nevertheless, it should be noted that the ideal performance of such a device is equivalent to that of a homojunction, and not that normally attributed to a “Schottky barrier” with J, = A*T’ exp (-EJkT). This interesting circumstance has resulted because of the assumed experession for Jo, namely. I,, = 1.5 x IO” x exp (-E,/kT) mA/cm’. If we assume A* = 1.2 x IO’ mA/“K’, then .L = 1.08 X 10”‘~ exp (-E,/kT) mA/cm’. Thus, whether or not an ideal Schottky barrier with 4B0 = E, is characterized as a majority carrier, or minority carrier cell depends on the expression assumed for Jo. The theoretical efficiency determined for the two types of cells are nearly the same, however. a5 indicated
5. CONCLUSIONS MIS solar cell performance has been considered, assuming three likely models for interface state behavior. In all cases, the addition of an insulating layer between
751
Model calculations for metal-insulator-semiconductor solar cells
the metal and semiconductor enhances the power conversion efficiency. This conclusion is true whether the film is characterized by xc0 = x,,~= 0.2 or 2.0 eV. In the case of E, = 1.5eV and E,/3, efficiencies are increased from =O to 4 to 6% for xc0 = xso. If a more tailored film is used with xc0 = 0 and x,,~= 2.0 eV, then TJ approaches 9% for Models I and III, and 12.5% for Model II. If 2E,/3, the power conversion efficiency increases to essentially that of an ideal homojunction. The primary conclusion is that the efficiency of Schottky barriers with C#Q~Z2E,/3 can be greatly increased by adding an interfacial layer. The limiting performance is that of a homojunction, since minority carrier diffusion back into the bulk becomes the dominant leakage current as the film thickness is increased. A key parameter is the value of a/K,. If the interface states are in equilibrium with the metal (Model I), or if they are in equilibrium with the minority carriers of the semiconductor (Model III), then D,/Ki must be & 10” cm-’ eV_’ m order that these increased efficiencies be possible. If the interface states are in equilibrium with the majority carriers, enhanced performance of a Schottky barrier is possible regardless of the value of 0,/K,. However, if Ds/Ki can be > 10” cm-* eV_‘, then a more significant effect results. It would appear that under illumination, the appropriate model is most likely Model III. Many previous discussions of MIS cells has been based on cells having n-values significantly greater than one (Model II). It is encouraging to find that even in the case of Model III, the MIS concept will probably be extremely valuable. $JBO
&SO
14. Joseph J. Loferski, JAP 27, 777 (1956). 15. Harold J. Hovel, Solar Cells, p. 38. Semiconductors and Semimetals Series, Academic Press, New York (1975).
=
=
Acknowledgement-The author wishes to acknowledge the assistance of Mr. Robert C. Bohara in developing the computer code used for the calculations presented here. BF#EBENCFS
1. W. A. Andersen, A. E. Delahoy and R. A. Milano, JAP 45, 31913(1974). 2. D. R. Lillington and W. G. Townsend, Appl. Phys. Let. 28,97 (1976). 3. E. .I. Charlson and J. C. Lien, .JAF’46, 3981 (1975). 4. J. P. Ponpon and P. Siffert, FAP 47, 3248 (1976). 5. R. J. Stirn and Y. C. M. Yeh, Appl. Phys. Let. 27, 95 (1975). 6. M. A. Green, F. D. King and J. Shewchun, Solid-St. Electron. 17, 551 (1974). 7. Stephen J. Fonash, JAP 46, 1286(1975). 8. L. C. Olsen and R. Bohara, Proc. 11th IEEE Photouoltaic Specialists Conj. 381 (1975). 9. D. L. Pulfrey, IEEE Trans. Electron Deu. 587 (1976). 10. L. C. Olsen, Proc. 12th IEEE Photoooltaic Specialist Conj. (1976). 11. H. C. Card and E. S. Yang, Appl. Phys. Let. 29, 52 (1976). 12. S. M. Sze, Physics of Semiconductor Devices. Wiley-Interscience, New York (1969). 13. H. C. Card and E. H. Rhoderick, .I.Phys. D: Appl. Phys. 4,1589 (1971).
APPENDIXA
Conditions for neglecring recombination current Neglecting recombination at the interface implies that the recombination current (JKE) must be 4.l,,,, where J,,, is the maximum possible photocurrent for a given band gap. In Ref. 1161JR~ is given as JRE= eN,u”u,[(l -fh
-fn,l
(Al)
where N, refers to the total number of surface states, n, refers to the surface concentration of electrons and
For simplicity, let us assume all the surface states are located at the quasi Fermi level. Substituting typical values for parameters, J.QE- 4 x 10”’N.,u,,emr’lr7mA/cm’.
(A3)
Typically, c = 0.2 to 0.3 eV. Thus, JRE- lO’g”N, mA/cm*.
(A4)
Thus, for JRE< 1mA/cm* requires that cr,,N,z IO_‘.
(AS)
In the case of Models I and III, N, = IO” cm-*. In those cases, o. must be e10m’6cm’ in order that JR~ can be neglected. In the case of Model II, N, = lOI cmm2,and gn must be c lo-” cm*. APPENDIX B
Conditions for validity of Model II
In order for Model II to be valid, the recombination rates of electrons residing in surface states with holes (RH) must be much greater than the rate at which electrons in the conduction band recombine with surface states (RE). Considering recombination rates for a single level RH
W,Psf
,>1,
G=u~unns(l-j)
@I)
Assuming j - f,
A typical value for (Et V,,,,/n - I#JB~)is = -0.2 eV. Assuming (m:/m:) - 5, F
= :
F
(2 x
IO_‘).
”
Thus, as one would expect, the hole capture cross section must be several orders of magnitude greater than that for electrons before RH/RE,> 1.
Elecwonm.
1911. Vol. 20. pp. 741-751.
Pergamon Press.
Printed in Great Britam
MODEL CALCULATIONS FOR METAL-INSULATOR-SEMICONDUCTOR SOLAR CELLS? LARRY C. OLSEN Department of Materials Science and Engineering, Joint Center for Graduate Study, 100 Sprout Road, Richland, WA 99352,U.S.A. (Receiued 12 January 1977;in revised form 1 March 1977) Abstract-An analytical approach to calculating MIS solar cell properties has been formulated and utilized to study solar cell efficiency as a function of interfacial layer thickness, various interfacial film parameters and band gap. Three models are considered regarding interface state recombination kinetics. Calculations are presented for the case of interface states being in equilibrium with the metal (Model (I), in equilibrium with the majority carriers of the semiconductor (Model II) and in equilibrium with the minority carriers (Model III). It is found that in all three cases, the efficiency of low barrier height, Schottky barrier cells can be increased very significantly. For example, it is shown that for a band gap of 1.5eV and a barrier height of 0.5 eV, it appears possible to increase the cell efficiency from essentially zero to 12%. If the barrier height is 1.0eV, an efficiency of over 20% is possible. It is determined, however, that MIS solar cell performance is limited by leakage currents due to minority carrier diffusion back into the bulk. As a result, the upper limit of performance is defined by that for a homojunction. These calculations identify ranges of surface state density and interfacial barrier heights necessary for a good MIS solar cell.
NOTATION
n
Richardson constant for electrons (holes) using transverse (to plane of barrier) effective mass constant (2ee,N&eiZ) density of interface states, cm-’ eV’ charge of a proton, coulombs location of conduction band at surface, eV Fermi level for holes, eV Fermi level for electrons, eV Fermi level for metal, eV electron band gap, eV location of valance band at surface electron flux in direction as shown in Fig. 2, and due to carrier generation in the depletion region, cm-* see-’ electron flux in direction as shown in Fig. 2, and due to carrier generation in the base region, cm-* see-’ electron flux due to electrons tunneling through the interfacial region in the direction as shown in Fig. 2 hole flux in direction as shown in Fig. 2 photon flux per unit wavelength, cm-’ nm-’ current density mA/cm* minority carrier leakage current component, mA/cm’ current due to collection of carriers generated in the depletion region, mA/cm* current due to carrier generation in the base region, mA/cm2 maximum possible current supplied by a solar cell, mA/cm’ open circuit voltage voltage at maximum power current at maximum power saturation current associated with minority carrier diffusion in base region, mA/cm2 recombination current at surface, mA/cm* saturation current associated with thermionic emission of majority carriers, mA/cm* tunneling current due to electrons, mA/cm’ tunneling current due to holes, mA/cm’ Boltzmann constant, eV “K-’ relative dielectric constant minority carrier diffusion length, wrn
b
N.4 No
N, NA
Q RE RH R, T
TA V V,
V, w a Q* E 6 E, 8 A 40 2 4)RP 6” ? 7 X XC0 X= x.0
tWork supported under NSF Grants AER 75-20501and ENG 74-20444. 741
X”
n-value in J-V relationship electron concentration in p-type region under equilibrium condition, cm-’ acceptor density, cm-’ total incident photon flux, cm-’ total number of surface states, cm-’ incident photon flux of one wavelength, cmm2 collection efficiency for solar spectrum of Schottky barrier rate of recombination of electrons into surface states, cm-’ see-’ rate of recombination of surface state electrons into holes, cm-’ set-’ reflection coefficient for photons of wavelength h temperature, “K transmission coefficient for photons of wavelength A through metal load voltage change in potential across interfacial layer, volts change in potential across depletion layer, volts depletion layer width, cm e&D&, optical absorption coefficient, cm-’ distance of electron quasi Fermi level below conduction band, eV dielectric constant of interfacial layer material, f/cm dielectric constant of semiconductor, f/cm interfacial layer thickness, A Ez-4BP-~,eV level above valence band to which surface state levels are occupied in isolated semiconductor, eV E,+,y-&, eV Schottky barrier height for p-type device, eV effective barrier height metal work function, eV solar cell power conversion efficiency, % minority carrier life-time, set electron affinity, eV interfacial barrier height above conduction band with no photocurrent flowing, eV interfacial barrier height with finite photocurrent, eV interfacial barrier height below valence band with no photocurrent flowing, eV interfacial barrier height with finite photocurrent, eV
742
L.
c.
I. INTRODUCTION
MIS cells refer to Schottky barrier devices with a nonconducting interfacial layer between the metal and semiconductor. Several recent experimental papers indicate that the addition of such an interfacial layer improves solar cell performance [l-5]. The clearest evidence of such an effect is provided by the work reported in Ref. [S] on GaAs cells. MIS structures were fabricated by depositing gold onto n-type GaAs after various times of exposure to air to grow an interfacial oxide. Significant improvements in efficiency were observed as the exposure time was increased. There also have been several recent theoretical papers concerning the photovoltaic characteristics of MIS cells[bll]. Each paper has emphasized different aspects of the problem. In Ref. [6], a MIS system is treated in detail by numerical methods for one particular operating mode, namely, that of a minority carrier diode. The barrier height and interface voltage distribution are considered in a fairly complete manner in Ref. [7] for the case of interface states being in equilibrium with either majority carriers or the metal. It was also assumed that the interfacial barrier has a negligible effect on minority carrier transport. The effect of interfacial charge on barrier height was considered in Ref. [9]. The present author reported on calculations of MIS solar cells in Ref. [lo] for the case of interface states being in equilibrium with majority carriers. Finally, the effects of interface states being in equilibrium with minority carriers was examined to a small degree in Ref. [I 11.
OLSEN
The purpose of this paper is to provide a practical analytical approach to calculating MIS solar cell performance, and to present the potential performance of such cells. A model is developed which accounts for three possible situations for the interface states in a MIS cell under illumination, namely, with the interface states in equilibrium with the metal, the semiconductor majority carriers or minority carriers. It is assumed that the charged impurity concentration in the semiconductor is ? 10” cmm3. Although this assumption presents some constraints on the analysis, it allows a straight forward analysis of cell performance to be carried out. It is also assumed that the interfacial region is chargefree. Effects of such charge could easily be added. Effects of such charge were not included because of the number of device configurations increases tremendously if it is included. In addition, the main effects of interfacial charge are clear. The impact of interfacial charge on results presented here is discussed in a later section. Finally, the analysis and calculations are carried out for a p-MIS system. All the models considered can be applied to a n-type material, by simply inverting the band structure and reversing the roles of electrons and holes. The analytical approach to MIS solar cell calculations is presented in Section 2. Calculations of solar cell characteristics for three models regarding interface state behavior are given in Section 3. Discussion of the results is presented in Section 4, and conclusions are considered in Section 5.
Fig. I. Equilibrium electron band diagram for p-MIS cell
743
Model calculations for metal-insulator-semiconductor solar cells 2. ANALYTICAL AFTROACH
Consider a p-type semiconductor coupled to a metal.
The equilibrium band diagram is shown in Fig. 1. xc0 and xWOrefer to equilibrium values for the interfacial barrier height. Figure 2 shows a band diagram for a cell under illumination. The location of the surface state Fermi level EFs depends on recombination kinetics. This will be considered below. The analytical approach will be discussed in four parts. First, features of a MIS band diagram will be considered. The voltage distribution across the interface and depletion region is then considered. Current-voltage characteristics are examined and finally, an outline of the computational procedure is presented. 2.1 MIS band configuration There are features of the band diagram in Fig. 1 which need elaboration. xc0 and xaO are referred to as interfacial film barrier heights. The “0” subscript refers to the value when no voltage exists across a load. xc0 and xuo will be treated as variable parameters. 4sP can be related to interface parameters. Using the abrupt junction approximation, and following Sze [ 121,
interface states in cm-*eV-’ and ei is the dielectric constant of the interfacial layer material. & is defined such that if the semiconductor surface is in a vacuum, surface states will be filled up to 40 in order to achieve short-range charge neutrality. & is set equal to E,/3 in this study. Ds is assumed to be constant over the band gap. More specifically, D, can be viewed as the average value of interface state density over the region that the surface state Fermi-level (EF~) exists during cell operation. This region will depend on the model assumed regarding interface state kinetics. In writing I$B~according to (l), it is assumed that C=
2eesN& 7
Ei
a 1.
This condition is generally satisfied if NA < 10” cmm3. The interfacial barrier heights xc0 and xuo are often assumed to have values determined for thick interfacial layers (several hundred A). Thus, at a SiOJSi interface, one might assume X=O= 3.2eV and xuo= 3.7eV. However, Card and Rhoderick’s[ 111 analysis of MIS silicon diodes suggest numbers for xc0 to be more like 0.1 $BP = Y&IO•t (1 - r)&l (I) to 0.2 eV. Film thicknesses of interest are on the order of 10-40A. As a result, use of bulk properties for such a d~ao=Eg+x-$~m (2) film should be done cautiously. As a final note, the abrupt junction approximation 7=(1+(y)-’ (3) implies that the minority carrier concentration is less eSD, than Na in the depletion region. If n, is the surface electron concentration, we must have n,i: NA. Thus, in order for the abrupt junction approximation to be valid, where 40 is a surface state level, D, is the density of E ~0.085 eV if Na - 10’8cm--3, E ~0.144eV if Na(y=-
FDEPLand FTE refer to electron Fig. 2. p-MIS solar cell under illuminationand supplyingcurrent to a load. F DIP~, fluxes, and FTH refers to “hole” flux. The J’s with the same subscripts designate the corresponding current densities.
744
L. C.
10’7cm--3 or E 3 0.203 eV if Na - lOi cm-3, where E defines the location of the electron quasi Fermi level. It will be assumed E 5 0.12 eV in this work. A more general treatment must, of course, take into account the effect of the minority carrier density on the potential in the interface region. 2.2 Voltage distribution across interface and depletion regions As photocurrent is produced and a voltage V is developed across a load, the potential across the interfacial region and depletion regions are affected. These changes are defined as Vi and V.,, respectively, with a sign convention defined by Fig. 2. Vi and V, are related by
OLSEN
the semiconductor to the metal, and JrH the current due to holes tunneling from the semiconductor to the metal. It will be understood that actual positive charge flows from left to right. JTE and JTH are, of course, minority and majority current terms. The majority and minority carrier currents will be treated separately. 2.3.1 Majority carrier current. Neglecting shunting conductance and series resistance effects, JTH is strictly due to holes flowing from the semiconductor to the metal. JTH is given by the usual result for Schottky barriers, except for a factor accounting for tunneling through the interfacial layer. Using the WKB approximation as described by Card and Rhoderick[l3], J.rH = J, (e”s”’ ~ ] )
v = vi + v,.
(II)
(6) where
The values of Vi and V, depend on the change in the amount of charge stored in the interface states as a result of a voltage drop V existing across a load. Fonash developed a general expression relating V and V, for a n-type MIS system[7]. Using the same procedure in the case of a p-type device, V can be written as V = V, + a[4,,
- (Em -Em)]
+ c”2[(vh~)“2 - (Vbr - VY’I = v, + a [l#wP- (Em - E”S)I
(7)
where E,, refers to valence band edge at the surface and it is assumed that C Q 1. Clearly, Vi can be identified as
Vi = (Y[~EZP -(Em - &,)I.
(8)
The voltage distribution therefore depends on the location of EFS, or the recombination kinetics involving the interface states and the metal and semiconductor. Three cases will be considered in this paper. Model I will be based on assuming the interface states are in equilibrium with the metal. Model II is based on the interface states being in equilibrium with the majority carriers. Model III involves the interface states being in equilibrium with the minority carriers. As V, and VS are established, xc and x,, are affected. These interfacial barriers are utilized in expressions for tunneling currents based on the WKB approximation. Using the average height to define xc and xv, xc =x4+-
V, 2
(9)
V, X” = xc>0-_ 2 . 2.3 Current-voltage characteristics Referring to Fig. 2, the current-voltage are given by .I=&-JTH
J, = A*,T’exp
[-$&HI
exp [-xZ “‘61
(12)
X~ and 6 are to be written in units of eV and A, respectively, in the exponential term of (12). AqH is the Richardson constant for holes with their momentum transverse to the barrier. 2.3.2 Minority carrier current. Photogenerated minority carrier current comes from two sources, diffusion from the bulk (JDLFF) as a result of electronhole production in that region, and as a result of electron-hole production in the depletion region (JDEPI.). It will be assumed that under steady state conditions, recombination currents at the interface can be neglected. The validity of this assumption is examined in Appendix B. Under these circumstances, requirement of continuity of current through the interfacial layer dictates that J-r,. =J,,,,
(13)
AJ,,, 12,.
I,,,, can be determined by solving the minority carrier transport problem in the same manner as done for a n/p cell, or Schottky barrier. Assuming that E is known, the boundary condition at x = I+’is chosen to be Sn(x = w) = n,(,(eA’l“ e’s”’
- I)
(14)
where ;\=E,-&z-e.
(15)
Assuming a semi-infinite, frontwall device coupled to a monochromatic beam of photons, and an exponential absorption of photons, the solution to the minority carrier transport problem in the bulk region leads to
JoeFr = eTh(l - K)N, characteristics
( 16) where
(IO)
where &E is the current due to electrons tunneling from
J0 = en,o(L/7). R, is the reflection coefficient of the cell. r, the trans-
Modelcalculationsfor metal-insulator-semiconductor solar cells mission coefficient (through the metal layer), a, the photon absorption coefficient, L the minority carrier diffusion length and T the minority carrier life-time. The first term in (16) is the photocurrent. The second term represents diffusion of electrons opposite the generation current direction. This term is normally neglected in Schottky barrier analysis. In MIS devices, it can be enhanced due to accumulation of minority carriers at the interface. MIS diodes based on the dominance of the minority carrier diffusion current have been considered in Ref. [6]. In calculations presented here, JOis expressed in terms of E8 by using the expression originally utilized by Loferski[lS], namely, Jo- 1.5X 101’e-E~kT*. cm
(17)
Assuming all electron-hole pairs created in the depletion region are collected as current, JDEP,_is given by JDWL= e(1 -&)ThN,,(l
-e-OLAw)
(18)
where w is the depletion layer width. &,IFF and JDEP~can be combined as JDIFFtJDEPL=
q,(ehr,)-J,(eA'kTeVJ'eT-1)
(19)
where Qk is the spectral collection efficiency of the related Schottky barrier (i.e. with S = 0). Finally, for a spectrum of photons, J D,FF+ JDEPI.= QJmax- Jo(eA’kTevJk7- 1)
(20)
where .lnlax=e)=N* h and Q is the total cell collection efficiency. It has been assumed that the location of the minority carrier quasi Fermi level, or E, is known. To determine E, JTE is identified as the electron tunneling current. Thus, following Ref. [13], JZ = A*7ET* e-@’ F(EFi-+ EFM)
(21)
where F(EK-+ EFM) involves an integral of the difference in occupation probabilities between an electron in the semiconductor and metal. The assumption that E F 0.12 allows (20) to be simplified to JTZ
=
Ahe
-_x,W .,kT [e-
_
e-G?,-+.,,-V,,/kT
1.
(22)
In factoring out A&, we haye also assumed that the effective mass of electrons in the metal and semiconductor are equal. Utilizing (13), (20) and (22), an equation is obtained which can be used to determine E. In particular, we find A&T
Z --xc1/28-.,kT e
-e
[e
=
-GS-OBp-ViVkTj
QJmax- Jo[eA’kTeVJkT-11.
(23)
745
By definition (eqn 15) A is a function of e. In general, V, is also a function of e. 2.3.3 Solar cell current-voltage characteristics. The expressions for Jm and JTH can be substituted into (10) to obtain J= QJ,,,ax-JO[eWkTeV~‘kT-l]-J,[eVs’kT- l] (24) = QJmax- JDE- JTH. Thus, the current supplied by a MIS cell can be interpreted as the photocurrent, QJmax,minus two “leakage” currents, one due to minority carriers (&) and one due to majority carriers (JTH). QJmax is actually the shortcircuit current of the related Schottky barrier (S = 0). In the case of Models I and II, the short-circuit current for a MIS cell is given by J,, = QJmax - Jo[eA’kT - l]
(25)
For Model III, however, V, can be finite when V=O; thus, J,, is given by (24) with V = 0. It is convenient to refer to a MIS cell as a majority carrier cell or minority carrier cell, depending on which leakage term dominates. It should be noted that the dominate leakage current may be different in the case of dark or illuminated characteristics. Clearly, when these terms are used here characteristics under illumination are appropriate. Equation (24) allows one to readily determine the potential advantage of MIS cells. If an interfacial layer can be inserted in a metal-semiconductor system such that the majority carrier tunneling current is significantly reduced, but yet the minority carrier tunneling current is not, then current can be delivered at a higher voltage than in the case of S = 0. The minority carriers can tunnel through the interfacial layer easier than majority carriers because n, %ps. The performance is limited by minority carrier diffusion back into the bulk. As a result, the limiting efficiency for MIS cells is equivalent to that for homojunctions. 2.4 Computational procedure Calculations have been carried out in the following manner -in this work: (1) Select E,, dsO, Q, xeO,xvO,S, DJK and the solar spectrum. J,,,, is determined from knowledge of the solar spectrum and E8 (e.g. from Ref. 15). (2) Calculate y, ~BP and JO. (3) Select Model I, II or III. (4) Determine J vs V. For a given voltage, calculate in the following order: Vi, V,, xc, x”, E, J, and J. (5) Maximum power is found by determining the maximum value of the product JV. 3.Ml.9SOLAR CELL CALCULATIONS
Calculations of solar cell performance are presented in the following sections for Models I, II and III. An AM1 solar spectrum and Q = 1.0 are always assumed. For each model, results are presented in two ways. First, calculated cell properties for -a particular band gap are
146
L.
c.
presented in detail. Then, maximum power conversion efficiency (71)is plotted vs band gap for 4,0 = E,/3, 2E,/3 and E,. For a detailed study, EX = 1.5eV was chosen because this value occurs near the maximum of the 7 vs E, curves for all three models. To conserve space, and to allow a direct comparison of results, plots for all three models are given in each figure. 3.1 Model I: Interface states in equilibrium with metal In this case, the surface states are in communication with the metal and EFs is pinned to EFM. Thus, the net charge stored in the surface states never changes. Since E FM -E,, = &,p, eqn (8) indicates that Vi = 0. Therefore, for this model MODEL, I
EFs = EF.M v, = 0 v, = v
(25)
,y<= ,y<0 ,yL= ,yvo. It should be noted that if the C-term is not neglected in (7), that one finds V, = V/n, with the n-value being slightly greater than one. This result was determined both by Card and Rhoderick[l3] and Fonash[7]. 3.1.1 Calculations for E, = 1.5 eV. Calculations for solar cell performance for E, = 1.5 eV are presented in Figs. 3-5. As previously indicated, results for Models I1 and III are also shown in these figures. Figure 3 gives 7 vs S for x~~=x~,,~= 0.2eV, and 2.0eV. As indicated, D,/Ki has been set equal to 10” cm-‘eV-’ for Model I. This choice for Ds/Ki is discussed below. Consider Fig. 3. The values of efficiency for 8 = 0
OLSEN
correspond to an ideal Schottky barrier. Thus, as indicated in Fig. 3, 7 =O for &so = E,/3 and 10.1% for &” = 2E,/3 in the case of S = 0. The addition of an interfacial layer decreases the majority carrier leakage current resulting in increased efficiency. As 6 is increased, the minority carrier quasi Fermi level approaches the conduction band, and the minority carrier leakage current (JDE) increases until the system is a minority carrier cell. The transition occurs approximately at the maximum of the 7 vs 6 curve. As 8 is further increased, JD8 increases and 7 decreases. Fairly extreme values for xc (I and x,,, have been chosen in Fig. 3. ,i<,, = I,,,= 3.OeV gives rise to a sharply peaked 7 vs S curve, while xc (I= x,,,, = 0.2 eV results in a broader curve. This difference is related to the required value for 6 in order that the transition to a minority carrier cell occurs. An increase in xL,ocauses a decrease in the value of 8 for which the transition occurs. It is significant that the power conversion efficiency for a MIS cell is larger than the corresponding Schottky barrier. In the case of a~,, = E,/3. 9 = 0 for 6 = 0. But with an interfacial film, 7 can be increased to ==5%. In the case of $Ro = 2EJ3, 7 can be increased from 10 to 15 or 16%. Figure 4 describes 7 vs 6 for xc (I= 0 and xL,, = 2.0. It is clear that to take advantage of the MIS concept, one prefers a small interfacial barrier for minority carrier flow, and a large one for majority carriers. Tailoring the interfacial layer in this manner allows one to obtain a minority carrier cell at relatively low values of 6. A large band gap semiconductor with a similar value for electron affinity may serve as an effective interfacial layer to
&J =s
--
(approximate) ------
I I
Fig.3. MIS solar cell efficiency vs S for Models I. II and III. and assuming /$ = l.SeV. x~,,=x~,)= O.?eV and 2.0eV. For &,, = E,,E is between 0.05 and 0.1 eV for maximum power conditions. Thus, the results are slightly overestimated. (Compare the approximate results for ball = E, with the exact result for do,,,,= XJ3 in Fig. -1.1 E, = I.5 eV (-_) Model I with DJK, = IO” crn~-*eV ‘; AMI spectrum (--_). Model II with D,/K, = 0.5 x IO’?cm-‘eV ‘: Q = 1.0 (---I. Model III with D,/K, = IO” cm ’ eV ‘: \, ,, 1,,,.
141
Model calculations for metal-insulator-semiconductor solar cells
25 I
lppoximate,
‘~I----_ \
all cases)
2+‘90=23Eg \
I
IO
I
I
/
15
20
25
Fig. 4. MIS solar cell efficiency vs 6 for Models I, II and III, and assuming ER= 1.5eV, ~~0= 0 and xUO= 2.0 eV. For $SO= E8, e is between 0.05 and 0.1 eV for maximum power conditions. Thus, the results are slightly over-estimated. (Compare the approximate results for 4 B. = Eg with the exact result for &n, = 2E,/3.) E, = 1.5eV t---), Model I with D,/I(, = 10” cm-‘eV_r; AM1 spectrum (--), Model II with D,/Ki = 0.5x IO” cm-* eV_‘; Q = 1.0 (---), Model III with D,/Ki = 10” cm-’ eV-‘; xc0= 0, x,,”= 2.0 eV.
25 -
0
a-0, &=g, -_
10’0
I
/
I
I
10'2
IO" Ds/$,
,
\ \
\
10'3
cm-* eV_’
Fig. 5. MIS solar cell efficiency vs DJKi for ModelsI, II and III, and assumingE, = 1.5eV, xc0= 0 and xUO= 2.0 eV. E, = 1.5eV (-_), Model I; AMI spectrum (--), Mode1II; Q = 1.0 (---), Model III; ,ycO= 0, xUO= 2.0 eV.
achieve small xc,,, but large xVO.For example, ZnS would appear to be a good choice for several semiconductors. This choice of xc0 and xUohas a significant impact on the results for q vs 6. As shown in Fig. 4, q increased to -9% and 20.5% for +B~= EJ3 and 2E,/3, respectively. The efficiency is then essentially constant with S since xCo= 0, and is not a function of voltage. It is significant SSE Vol. 20. No. 9-B
to note that essentially ideal homojunction behavior is obtained in the case of c#Q~=2E,/3, and reasonable efficiencies are achieved by adding an interfacial layer in the case of &o = EJ3. The point at which JTIl = JDE is indicated in Fig. 4. For values of 6 greater than the transition point, the device becomes a minority carrier cell.
748
L. C. OLSEN
Figure 5 shows the effect of varying Ds/Ki when xco=O and xUo= 2.0 eV. Maximum efficiency obtained for an optimum value of 8 is plotted vs DJKi. Consider I#J~~= 2E,/3. As DJKi increases, a increases which causes &P to decrease more rapidly with 6. As a result, the maximum efficiency decreases. In the case of &0 = E,/3, 4BP is a constant because of the choice $0 = E,/3. In general, however, the maximum efficiency will decrease with larger values of 0,/K, for Model I. I-
/
I
IO
15
us band gap. Figures 6 and 7 show TJvs to an optimum value of 6. Results are shown for the three choices of 4~~~.Efficiency vs band gap is also indicated for a Schottky barrier (8 = 0). Figure 6 corresponds to xc11= X”O= 2.0 eV, and Fig. 7 to the more “tailored” configuration, xc0 = 0 and x,,,) = 2.0 eV. As indicated by Fig. 6, addition of an interfacial layer with xc0 = xUO= 2.0 increases the maximum efficiency 3.1.2 Eficiency
Es for all three models. Each value of TJcorresponds
/
I
20 5.
1
25
+J
Fig. 6. MIS solar cell efficiency vs band gap for Models I. II and III, with xc ,, = x,,, z 2.0 eV. .4M1 spectrum (-1, Model I with D,/K, = IO” cm-‘eV_‘; Q = 1.0 (--) Model II with DJK, =05x IO”eV ’ cm-‘: x,,,= k,,, = Z.Oe\ (---) Model III with 0,/K, = IO” eV ’ cm ‘: (-_) Schottky barrier (8 = 0).
Eg , ev Fig. 7. MIS solarcell efficiency vs band gap for Models I, II and III, with x, jj = Oandx,,,, = 2.0 eV. AM1 spectrum (----_). Model I with 0,/K, = 10” cm-‘eV_‘; Q = 1.0 (--), Model II with D,/K, = 0.5 x IO” eV ’ cm ‘: X,o = 0. X,,) = 2.0 eV (---), Model III with 0,/K, = IO” eVml cm ?: (-). Schottky barrier (r~= 0).
149
Modelcalculationsfor metal-insulator-semiconductor solar cells tremendously for all values of Eg. The results are even more significant in the case of xc0 = 0 and ,+o = 2.0 eV. Results for 4~9 = 2E,/3 are very close to that for ideal homojunctions. In the case of ~BO= EJ3, values of efficiency >9% are possible, while for S = 0 only 1% is possible. 3.2 Model ZZ: Interface states in equilibrium with ma-
states in equilibrium
with
minority carriers
This model is probably the most valid of the three, particularly for 6 > 10A. This model is characterized by EFS = EFN. As a result, MODEL ZZZ EFS = EFN vi= -(YA
jority carriers
This model is characterized by EF~ = EFH. Thus, MODEL ZZ
3.3 Model ZZZ: Interface
V,=V+(wA xc =X=0-aAl
EFS = EFH vi = CYV,
xu =
V, = V/n n=lta xc =,,+Gv n-l xu = xuo-- 2n V. The parameter a can be written as a = (O.l808)(D,/Ki)S
(27)
if D, is in units of 1013crnm2eV_‘, ZG is the relative dielectric constant and S is in A. Thus, when the interface states are in equilibrium with the majority carriers and C < 1, diode characteristics are described by n-values which can be significantly greater than 1. The interfacial barrier heights are functions of voltage in the cases of Models II and III. In the case of Model II, xc increases with V while xWdecreases. It should be noted at this point, that in order for Model II to be valid for cells under illumination, the capture cross-section for the majority carriers must be several orders of magnitude larger than that for minority carriers. In particular, it appears necessary to have aPlan > 103.This subject is considered in Appendix C. 3.2.1 Calculations for E, = 1.5 eV. Figures 3-5 show results for Model II. As indicated in Fig. 5, cell efficiency increases with D,/Ki for Model III. The n-value increases more rapidly with S for larger values of DJK. This is particularly true as D,/ZG increases above 10’ cm-* eV. DS/Ki has been set equal to 0.5 x 10” cm-’ set-’ for Model II calculations. This value appears to be realistic, Values of D,/Ki used for Models I and III are also quite realistic. Referring to Figs. 3 and 4, the peak in the efficiency vs film thickness curve is higher for Model II than the other two models. However, the key result is the same for all models; namely, efficiencies of Schottky barriers can be greatly enhanced by adding an insulating fiIm between the metal and semiconductor. 3.2.2 Efficiency us band gap. Efficiency vs band gap is shown in Figs. 6 and 7 for 4~9 = E,/3 and 2E,/3. The possible efficiencies at a given band gap are usually highest for Model II. This is particularly true for $JBO = E,/3, xc0 = 0 and xv0 = 2.0 eV.
,y"~ +
a Al2
where A = E8 - &W - E.This is a rather surprising result. Since EFS = EFN, additional negative charge can be stored at the interface. As a result, the change in interface potential will be characterized by Vi
I>. c.
750
OLSFh
Table I. Model III cltlculations for E, 2 1.5 eV. Parameter values: xc ,) = x, ,, = 0.5 eV, DJK, = IO” eV
6 (A)
4HP (eV)
biW
fi
(eV)
(eV)
ImA/cm?
J,Ht
Jrrf
(mA/cm’)
J,<
(mA/cm’)
VI),
(V)
.I
0 3 6
0.500 0.500
0.500 0.555 0.610
0.51 I 0.457 0.402
Ii.0 13.5 7.7h
2.0 x 10 ” 4.6 1.4xx IO (:
21.5 X.9 27.3
0.012 0.045 0.093
14.5 14.0 19.7
2
9
0.500
0.665
0.348
6.47
0.078
27.5
0.143
21
0.500 0.500 0.500
0.777 0.72 I 0.833 0.890
0.247 0.296 0.202 0. I66
11. 2.Y3 I42 5.4x IO i 2.3x IO -’
27.4 27.5 26.4 19.2
0 3 6 9 I2 1’;
I .oo 0.997 0.995 0.992 0.989 w#;
I.oo I .05 I.10 I.16 I.21 ;:2;
0.487 0.453 0.402 0.347 0.293 ;:2;;
21
0.982
1.3;
0.132
I.hl I .‘Y I .3! I.20 0.636 0.024 3.7 x IO J 6.1 x IO ‘>
27.5 27.5 27.5 27.5 27.5 21.5 ‘7.5 27.5
2
2 2 ‘i _$
ii
6.33 2.24 12.3 IY.5 1.5x IO (,I x IO I’ 1.F x IO -I 0.014 0.5I8 I .30 I .42 I .hO
3.3.2 Eficiency L’S hand gap. As indicated in Figs. 6 and 7, calculated solar cell performance vs E_ for Models I and III is very similar for the selected value of 0,/K,. In both cases, the effective n-value of the cells is -1.0. 4. DlSCUSSION
Four topics are discussed to supplement previous discussion, namely, assumptions, effects of interfacial charge, potential impact of an interfacial film on shunting conductance, and the relative value3 of JIJrr and .7ffi, when &O.l2eV. Plotted results in this work are compatible with this constraint, except the case of &O = E, in Figs. 3 and 4. The error in these cases is still small,
:;;I
‘“O (mA/cm’)
?I
’ cm ’ (2,
21.0
0.0068 0.030 0.061 0.106
0.093 0.40 I.14 2.10
0.192 0.183 0.191 0.186
21.0 22.3 15.2 8.03
0.109 0.136 0.076 0.045
2.87 2.16 1.09 0.344
0.488 0.540 0.592 0.642 0.682 0.69 I 0.688 0.685
25.9 25.7 26.1 26.3 26.4 26.2 26.1 25.9
0.415 0.470 0.515 0.562 0.596 0.561 0.503 0.446
IO.1 I I.4 12.7 13.9 14.8 13.8 12.4 10.9
when the approximate results obtained for C/W,) = E, are compared to the exact results for &(, = 2E,/3 in Fig. 4. Inclusion of the effects of fixed interfacial charge simply alters the expression for C#W.These effects were not included here since it would increase the number of parameters to consider, and would not alter the main results. For example, even if 4Bp is increased as a result of fixed interfacial charge, cell performance is limited by the minority carrier diffusion into the bulk. One potential improvement in solar cell properties as a result of an interfacial film may be increased values of shunt resistance (L). This is particularly true for polycrystalline solar cells. Shunting conductance is probably due to regions of low barrier height where majority carriers can easily flow, or due to metallic-like shorting paths. In either case, an insulating layer would probably help. A film with xCo= 0 and large xUO(for p-MIS cells) would be particularly valuable in this regard, since relatively large film thicknesses can be utilized. As a final point of discussion, consider the case for &Cl = E, and S = 0. This situation probably has never been, nor ever will be, realized in practice. Nevertheless, it should be noted that the ideal performance of such a device is equivalent to that of a homojunction, and not that normally attributed to a “Schottky barrier” with J, = A*T’ exp (-EJkT). This interesting circumstance has resulted because of the assumed experession for Jo, namely. I,, = 1.5 x IO” x exp (-E,/kT) mA/cm’. If we assume A* = 1.2 x IO’ mA/“K’, then .L = 1.08 X 10”‘~ exp (-E,/kT) mA/cm’. Thus, whether or not an ideal Schottky barrier with 4B0 = E, is characterized as a majority carrier, or minority carrier cell depends on the expression assumed for Jo. The theoretical efficiency determined for the two types of cells are nearly the same, however. a5 indicated
5. CONCLUSIONS MIS solar cell performance has been considered, assuming three likely models for interface state behavior. In all cases, the addition of an insulating layer between
751
Model calculations for metal-insulator-semiconductor solar cells
the metal and semiconductor enhances the power conversion efficiency. This conclusion is true whether the film is characterized by xc0 = x,,~= 0.2 or 2.0 eV. In the case of E, = 1.5eV and E,/3, efficiencies are increased from =O to 4 to 6% for xc0 = xso. If a more tailored film is used with xc0 = 0 and x,,~= 2.0 eV, then TJ approaches 9% for Models I and III, and 12.5% for Model II. If 2E,/3, the power conversion efficiency increases to essentially that of an ideal homojunction. The primary conclusion is that the efficiency of Schottky barriers with C#Q~Z2E,/3 can be greatly increased by adding an interfacial layer. The limiting performance is that of a homojunction, since minority carrier diffusion back into the bulk becomes the dominant leakage current as the film thickness is increased. A key parameter is the value of a/K,. If the interface states are in equilibrium with the metal (Model I), or if they are in equilibrium with the minority carriers of the semiconductor (Model III), then D,/Ki must be & 10” cm-’ eV_’ m order that these increased efficiencies be possible. If the interface states are in equilibrium with the majority carriers, enhanced performance of a Schottky barrier is possible regardless of the value of 0,/K,. However, if Ds/Ki can be > 10” cm-* eV_‘, then a more significant effect results. It would appear that under illumination, the appropriate model is most likely Model III. Many previous discussions of MIS cells has been based on cells having n-values significantly greater than one (Model II). It is encouraging to find that even in the case of Model III, the MIS concept will probably be extremely valuable. $JBO
&SO
14. Joseph J. Loferski, JAP 27, 777 (1956). 15. Harold J. Hovel, Solar Cells, p. 38. Semiconductors and Semimetals Series, Academic Press, New York (1975).
=
=
Acknowledgement-The author wishes to acknowledge the assistance of Mr. Robert C. Bohara in developing the computer code used for the calculations presented here. BF#EBENCFS
1. W. A. Andersen, A. E. Delahoy and R. A. Milano, JAP 45, 31913(1974). 2. D. R. Lillington and W. G. Townsend, Appl. Phys. Let. 28,97 (1976). 3. E. .I. Charlson and J. C. Lien, .JAF’46, 3981 (1975). 4. J. P. Ponpon and P. Siffert, FAP 47, 3248 (1976). 5. R. J. Stirn and Y. C. M. Yeh, Appl. Phys. Let. 27, 95 (1975). 6. M. A. Green, F. D. King and J. Shewchun, Solid-St. Electron. 17, 551 (1974). 7. Stephen J. Fonash, JAP 46, 1286(1975). 8. L. C. Olsen and R. Bohara, Proc. 11th IEEE Photouoltaic Specialists Conj. 381 (1975). 9. D. L. Pulfrey, IEEE Trans. Electron Deu. 587 (1976). 10. L. C. Olsen, Proc. 12th IEEE Photoooltaic Specialist Conj. (1976). 11. H. C. Card and E. S. Yang, Appl. Phys. Let. 29, 52 (1976). 12. S. M. Sze, Physics of Semiconductor Devices. Wiley-Interscience, New York (1969). 13. H. C. Card and E. H. Rhoderick, .I.Phys. D: Appl. Phys. 4,1589 (1971).
APPENDIXA
Conditions for neglecring recombination current Neglecting recombination at the interface implies that the recombination current (JKE) must be 4.l,,,, where J,,, is the maximum possible photocurrent for a given band gap. In Ref. 1161JR~ is given as JRE= eN,u”u,[(l -fh
-fn,l
(Al)
where N, refers to the total number of surface states, n, refers to the surface concentration of electrons and
For simplicity, let us assume all the surface states are located at the quasi Fermi level. Substituting typical values for parameters, J.QE- 4 x 10”’N.,u,,emr’lr7mA/cm’.
(A3)
Typically, c = 0.2 to 0.3 eV. Thus, JRE- lO’g”N, mA/cm*.
(A4)
Thus, for JRE< 1mA/cm* requires that cr,,N,z IO_‘.
(AS)
In the case of Models I and III, N, = IO” cm-*. In those cases, o. must be e10m’6cm’ in order that JR~ can be neglected. In the case of Model II, N, = lOI cmm2,and gn must be c lo-” cm*. APPENDIX B
Conditions for validity of Model II
In order for Model II to be valid, the recombination rates of electrons residing in surface states with holes (RH) must be much greater than the rate at which electrons in the conduction band recombine with surface states (RE). Considering recombination rates for a single level RH
W,Psf
,>1,
G=u~unns(l-j)
@I)
Assuming j - f,
A typical value for (Et V,,,,/n - I#JB~)is = -0.2 eV. Assuming (m:/m:) - 5, F
= :
F
(2 x
IO_‘).
”
Thus, as one would expect, the hole capture cross section must be several orders of magnitude greater than that for electrons before RH/RE,> 1.