Recombination in the space-charge region of Schottky barrier solar cells

sdu-we Ps

U~maicS vol. 23. pp. 41-4l Press Ltd., Mm. Printed in Great Britain

RECOMBINATIOti IN THE SPACE-CHARGE REGION OF SCHO’lTKY BARRIER SOLAR CELLS P. PANAYOTATOS and H. C. CARD Departmentof Ekctrical Eng&ring and ComputerScience, Cohunbii University, New York, NY 10027, U.S.A. (ReCn’wd19 February 1979; in rwidform

18 June 1979)

Abslmt-The expression for the recombiition rate as a function of voltage, of ilhuninationlevel, and of position in the spacetharg6 region of the semiconductoris derived analytically. The recombinationcurrent density is derived by numericalintegrationof the above expression. The results show good agr66mentwith experim6ntfor the typical Au-nSi near-idealSchottky barrkr solar cells, and the comparisonprovides informationon the uncov6ringof deep recombinationcenters by the hok quasi-Fermilevel underincreasingillumination.It is found that the priacipaleffect of recombinationunder illuminatkn is the reduction of the photocurrent.A rather surprisingbut gratifyingresult is that, once the above effect is taken into account by using short-circuitcurrents rPtherthan photocurr6nts,the r6maining(vohage d6pendent)e&t of recombinationis extremelyclose to th6 On6 in the dark, providedthe increase in “uucovertd” recombinationcenters with ilhuninationis taken into account. NUMTION

have shown that generation-recombinationcurrents form a sign&ant part of the total current in such devices. This can account for the deviations of the n-factor from unity in the expression for current density:

Richardsonconstant A crnm2 (deg K)-’ electric skid, V cm-’ silicon permittivityfaradcm-r energy of conduction,vaknc6 band edg6, eV metal Fermi level, eV quasi-Fermikvels for electrons, holes, eV &on bandgap,eV uot6nM barrias for electrons, hoks. V btalcurrentdensity,A cm-* ekctron, hde currentdensities, A cm-* J *c k I4 n n(x), P(X) N, Nv

J = Mexp (qV/nkT)- I].

short-circuitcurrentdensity, A cm-*

Bohxmaus constant. eV dettK-r mobility for holesci V-r Pi diode “n” value, (iiity factor) fr66 ekctron, hole concentrations,crne3 efkctive densities of states in conduction, vaknc6 ban& of si, cmT3 Nd donor concentration,crne3 intrinsiccarrierconc6ntration,cm-) $ trap density, cm-’ 4 ekctron charge, coul 0, G captur6cross section for holes, electrons, cm5 T absOlutetempernnue,degK U(x, v) r6combiition rate cm-) s-’ V forwardbii volu&, v VdO diiusion potential,V VOCopen-circuitvohag6, v thermalvelocity of carrkrs, cm s-’ &h W depletion-regionwidth, cm distance in semiconductorfrom metal-semiconX ductor contact, cm tNlRODUCTK)N

Since the introduction of Shockley-Read-HaNSRH) statistics for the generation-recombinationprocesses of holes and electrons through forbidden gap centers[l, 21, and the pioneering work of Sah ef al.[3], a number of very elaborate papers on p-n junction depletion-region. recombination have appeared in the titerature[4-6] explanting deviations of experimental devices from the simple theory. Furthermore, Yu and Snow[7l have extended this work to metal-semiconductordiodes and SE

Vol. 23. No.

I--c

(1)

For the simpk therm&k theory, n=l and JO= AP exp (- &&I’). Rhode&k has shown[8] that the addition of two exponentially volmge+e&m currents, one with an n-value of one and another with an n-value of two (representinga pure recombmationcomponent) can appear as an exponential dependence with an n-valut between one and two over several decades of current. All of the above analyses, however, are based on the assumption that the quasi-Fermi levels for electrons and holes are essentially parallel throughout the depletion region. This is far from being the case for metal-semiconductorcontacts under illuminationlevels appropriate to solar cell applications.Optical bii, even at very low ikunination levels, has a dramatic e&t on the shape of the minority-carrier quasi-Fermi level, enhancing the effect of depktion-region recombination currents on the characteristics accordingly (see, for example, the hole quasi-Fermilevel &,,, Fig. 1). The growinginterest in Schottky barrkr and MIS solar cells has stimulated the publication of several noteworthy papers on the mechanismsinvolved and the possible models for solar cell analysisf9-121.We feel, however, that depletion-regionrecombiition processes, as they appear under illumination, have not received proper attention and that the effects of recombiition on solar cell characteristics justify a study dedicated to them. We have, in the past, indicated[l3-151 that experimental results show that the sum of the majority-carrier current density and the recombinationcurrent density (1. and J, in Fig. 1) which oppose the photogenerated current does not coincide with the diode current in the

41

P. PANAYOTATOS and H. C. Clw,

J, = /W(X) dEf, (x)/dx

(2)

The hole and electron concentrations p(x) and n(x) are given respectively by[ 17):

(3)

n(x) = Ncexp I- (EC(x)- &.(x))/kTl.

(4)

and

LE” -;p?\

~0) = NVexp[ - (Efp(xl - Edx))/kTl

(a)

EC

If JP is taken to be independentof position and equal to /,, for every x, which is equivalent to assumingthat the recombinationcurrent density is much less than J,, from (2) and (3). for any given value of L, both &,(x) and p(x) can be calculatedfor all x, knowingp(O).Once n(x) is also evaluated the recombination current density J, can be calculated.The above assumptionis later justified for recombination rates encountered in crystalline sib-

Fig. I. Electron energy band diagramsfor a Au-n-type Si Schottky-barrier solar cell drawnto scale. &,,I, &.z are hnk con, but is expected not to hold in cases where the quasi-Fermi levels for 1, - 1.88r&/c&, 44.4OmA/cm’.recombiin rates are sign&ant as for example in the

(a)Shortcircuitco&ions, (b) V= 0.1V.

dark. Apart froma possible reduction in barrier height under illummatiorr, which has a comparatively small effect for near-ideal contacts[l4], the above deviations are attriited to the enhanced recombination.

case of polycrystalline silicon. Thus, setting J, = J,, in (2), substituting (3) into (2), rearrangingand integrating fromOtoxweobtaim

In this work we derive analytically an expression for = exp( - & (xl/kT)- 1 (5) the recombination rate in the space-charge region of a Schottky barrier solar cell, as a fuuction of position in wherethexeroleveiofeuergyistakenatthemetaiFermi the semiconductor, of applied voltage and &nination. level G= = 0 = Q(O) and cupis takeu to be independentof From this expression we perform calculations that lead position, and therefore of the electric 5eld g(x). to the recombination current density vs voltage and Substituting the expression for J%(x) from the &uitWion curves, which are then compamd to the depletion approximation: experimentalresults. Valuableinsighton the distribution and rok of forbUen+p reco&&& centers is o..-x< w w)=-4+h+4‘&(~-2W-) derived from the above comparisons. * &

q2NM

TIIUXY xrw (6) =-&esThe electron energy-band diagram for a Schottkybarrier soiar cell under illuminatedconditions, both for short circuit conditions and under forward bias, are the integral shown, drawn to scale, in Fii. 1. The shape of the band edges and the quasi-Fermi levels are calculated from I = exp ( - EJxMkT) dx 0 eqns (6) and (9) below. The parameters (barrier height, d8usion potential) correspond to those determmed experimentallyfor a Au-u-type Si Schottky barrier solar cell. .T”and J,, are the majority (electron) and minority (hole) current densities. J. is given by (1) above with I = exP (qb/kT) [ exp I- q2Nd(x2 - 2 Wx)/24kT] dx. I=&. J, is the recombination current density. Recombination through metal-semiconductor interface states is ignored, since, for near-ideal structures, inter- getting W-x = y in order to complete the square we face states are in equilibriumwith the metal rather than obtain: with the semiconductor[l6] so that the capture of a hole .&

I

Ix

is followed by a capture of an electron from the metal, I = exp (q#JkT + #Ndw5/2e,kT) j:_X rather than from the conduction band of the semicon‘exp ( - q2Ndy212r,kT) dy ductor as it would be for a recombinationevent.

If we denote by x the distance from the metal-semiconductor interface towards the bulk of the- semicon- and by making the change of variables: ductor, the hole (minority) current density is given z = (q2NJ2e,kT)‘“y, by[13]:

Recombinstion in Schottky&uriersolarcells we have finally that;

43

with a. = a, = Q (see tabk of symbols)(10)becomes[ 17:

Z = exp (qf#dkT+ q2N~W2/2ekT)(~(r)lte~ -ertbW-4))

rrw) (7)

where a = (q2N‘,/2c,kT)‘R.

where

Furthermore, since the depletion width W as a function of V is Rivenby[l7): W = (2~ ( Vd, - VYqNs, )In

F&&T

+ Vd,

- VYkT) x{erf(b(v))-erf@(V)-@I

(9)

where

Nv)=(q(V*-

and p=niexp

VlkT)

A’ *2 ( m26&T Inexp(q(h >

=1--

Now since

(8)

by substitutionof (7)and (8) in (5) we obtainthe expression for the hole quasi-Fermilevel as a function of positionand voltage: exp t- J&A

T = IluN,u,,,.

WkT)‘n

and c = q(Nd/2r,kZ’))‘n.

!SubstiMiag (9) and (6) into (3) we obtain the analytical expnssionforp(x,v)aad,since~(x)~~(x)+~Bnd Ej~=qVfordlxaccmbgtotbctbermiogicemission tbeory[81, eqn (4) provides the expression for n(x, V). Then, us& the expressionfor the recombimuionrate [3]:

( %P

>

the cash term above is negligible compared to either exp ((Z&- EMT) or exp ((B - J&)/kT)providcd & is at least 2kT away from either quasi-Fermilevels. In other words, the recombinationcenters are equally efkctive as midsap recombination centers and the cash term can be approximated by unity. This is usually the case under illuminationsince even low kvels of iUumi@ionresult in a big split in enngy between the two quasi-Fermi levels (Fig. 1). Ihen the expression for the recombinationbecomes u=+-nj~)i(p+n+2ni).

(11)

So that the product U(x, v * lm)T Can be Caicuiatcd from (11). The value of l/r, which equals the product of the factors oN,u,~ (see tabk of symbols and assume o= )J cr, =cr.j can be obtained by Wing the experimental (10) results to the theoretical predictions. Cakuktions of p(x), n(x) and V(x) from eqn (3), (4) and (I 1)are plotted in Fig. 2, for r = 1.4x IO4 sec. which is the value of T best fitting the experimental results for the epitaxially grown Si wafer used. From eqn (IQ, the value of the depletion-region recombination current density Z, can be evaluated as:

(I=

Jr&%)=s~

Wx, V,L)dX

(12)

Since the final expression for U as a function of position does not yield to analytical evaluation of the integral in qn (12), the inteRration is performed numerically.Calculationsof I, as a function of voltage and of J,, (i.e. as a function of ikunination) from eqn (12)are plotted in Fig. 5. UEVICE FABRICATION 105 .0

I

I

I

0.2

0.4

0.6

DISTANCE SEMICONDUCTOR Fii.

2. cAlcul6t6d

ConCcntr6tions

66d

?mD

ttills

r6CQmm

0.6

.10’2 1.2

LO

FROM METALCONTACT ” (*m)

ekCtm0

@DtI6d late

s6miconductor.

@olid),

li66),

hoie

(diuhd)

vs distAlICe

in the

The band diagramsof Fii. 1 correspond to aAu on n-Si Schottky bar+ with no intentionaIinterface oxide kyer. These devices have been fabricated as two concentric Au circks on an n-type epi-layer of thickness - 40 p and doping of Nd = 5 x lOI cm-‘, on an tr + layer with an AZ back (ohmic) contact The larger of the two Au circles

P. PANAYOWIW and H. c. CluD

The experimental results of this study are shown in FUS. 3, 4 and 6 and are representative of approximatelya dozen samples. Eight current density vs voltage(J-V) characteristics for various levels of illumination, ranging from dark (.L = 0) to a value of J,, = 44.4mA/cm2were obtained experimentally for the same device. Illumination was provided by a tungsten lamp and the ilhuninationlevel was controlled by varying the distance of the lamp from the device. Each of the above characteristics are then plotted as log (J - .L) vs voltagealongwith the log/ vs V characteristic of the device in the dark (typical results shown in Fis. 3). An increase in “n” value in the expression for the (electron) current density as given by eqn (1) with J = J., as well as a small decrease in barrier height are observed for the curves corresponding to illuminated conditions, in agreement with previously reported resuhs[l9,15,14,12].This difference in barrier -/ // height for open circuit conditions is in agreement with ! , I I 1 I I the shit observed from C-V measurementsin the dark 0 0.30 O.lO a20 and under illuminatedconditions. Assumingthe recomFORWARD BIAS VOLTAGE V (VOLTS) bination in the dark to be negligiblecompared to the Fw3. iogp-J,Jvs Vpbtuakilluminued~(J,= recombination under illuminatedconditions (as verified 7.75mAfal?) (open cifcks) and log1 vs v ph for tbc same experimentally),a line parallel to the dark characteristic &viceinthedark@olidcircks).Rl&cdlinei!3shifted%Irlr” is drawn from the (&, V,) point and its difference from chamcte&ticpassingthrough(log1, V,) (seetext). the characteristic under illumination is obtained. This diderence represents the voltage-dependentpart of the (area-O.2 cm21is a semitransparent Au film of thick- recombination current density under illumination.The ness -68 A which has been found to be optimum for nsuiting experimental curves of recombination current solar cells with Au electrodes[ 18,191and which provides density Jk vs voltage are shown in Fa. 4 for various the actual metal-semiconductor Schottky+arrier con- levels of illumination. Calculationsbased upon eqn (12)and shown in Fig. 5. tact.ThesmalkrAucirckisofthickness-1OOOAand is simply a dot (area -0.05 cm2) that provides for a give the dependence of J, on /, and V predicted by the model for the above illuminationlevels. The recomcontacttotheAufilm.ThechemWprepa&~ofthe Si waft was such(l9] as to minim& the in&facial bination current density consists of two parts: the zero(oxide) layer, whkh is the&ore assumed to be of the bias recombinationcurrent density (LO) and the voltage order of 10A as expected of a device exposed to air for dependent part (JL( Vj). Since experimentalplots of the 24 hr [20]. kind shown in Fig. 3 in which IJ -&,I is plotted have

I

I

I

I

I

’

(6)

’

(7)

10 -

lf6’

z z

(5)

-

(4)

a (3)

.

i_:

.

1;

. (2)

(1)

I 0

I I I 0.06 0.16 FORWARD BIAS VOLTAGE

I

1 0.24 v (VOLTS)

I

Fig. 4. Experimentally obtained voltage-dependentpart of the recombinationcurrent density vs voltagefor various illuminationlevels. (See Fig. 5 for correspondingvalues of IS=.)Curves are terminatedwell before V = V,.. in every case. By constructioncurves would bend down and go lo zero at V = V,,..

45

Recombinstionin Schottky-barrier solarcells

(2) (3): (4) (5) (6) (7) (6):

: 3.06 7.75 : 15.00 : 21.44 : 31.25 : 38.12 44.40

0.24 FORWARD BIAS VOLTAGE

0.40

a32 V (VOLTS)

densityvs voltagefor the illumkption kvels (valuesof I,) of Pii 4.

Fw 5. Calculated recombinntion current

already

omitted the dlect of xero bias recombination (which at&& only J,), the two components of the recombination cutrent density have to be compared separotdy. cakuktmg U(&O,JX) for the various kvels of iIluminatknandinmgratkgoverx,weobtainthevaluesof which are plotted vs 3, (solid line, Pig. 6). IExperimmtrlly the values of I,, are obtained by sub tracting the difference of the total current density .I at V--lVandofthedark/atthesamevoltagefromJ.,. This is just&d as follows: The width of the depktion-region at V- - 1V (W = 2.0pm) is double the depletion-region width at xero bias (W=l.Okm). The shape of the recombination rate vs distance curve, on the other hand,

SHORT-CIRCUIT

FI. 6. Zerobiasrecombination

remains unaltered for V= - 1V as compared to the curve for V = 0 and is only shifted by an amount equal to W. for all illuminationlevels observed (Fig. 7). We therefore assume that the increase in current at V = - 1V is the amount of recombinationthat was present in the depletion-regionat xero bias, namely L,,. In other words, since the amount of recombination in the entire neutral region (i.e. for x > W = 1.0pm) remains unaltered in both cases, the difference is the amount of recombination in the tirst 1 pm, in the case of V = 0. Since, however, most of the light is absorbed in the fist 10~m of the semiconductor[21],a more accurate ap preach would be to calculate the absorption vs distance for every wavelength[21]and integrate over wavelength and distance in order to evaluate the amount of recombination in the neutral region which is present at zero bias and lost because of the extension of the depletion region at V=- 1V. Without the above calculation the values of J,, that are detcmincd experimentally are somewhat overestimated. The use of the difference of the total current under illuminationfrom the dark current at the same voltage (V= - 1V) guarantees the elimination of any leakageeffects such as edge effects, channels, surface recombination etc. Plotting the above experimental values of Jnco vs J, (solid circks, Fw 6) and comparison with the theoretically calculated line provides for the evaluation of 7. The voltagedcpendent part of the recombinationcurrent density (theoretically obtahkd as .I,( V)- J-) is predicted by the model to very nearly obey an expressionofthesameformasthedarkrecombmationcutrent density: L(v)

= Uexp (NM

- 1)

(13)

where u is a weak function of voltage, for all illumination levels (see curves 1, la, lb in Fu. 9). We can understand this from Fig. 8 as follows: Once the recombination correspondingto the zero-bias case (i.e. J-) is removed for a given illumination level, what is left

CURRENT

DENSITY

JIc (mA/cm*)

current density vs short circuit cement density. Full circks: experdnti

solidline:calculated.

results,

P. PANAYOTATOS and H.

46

0

0.8

0.4

c. hll

1.2

DISTANCE FROM METAL-SEMICONDUCTOR

1.6

2.0

CONTACT X (Pm)

Fig. 7. Calculated recombination rate vs distance for zero bias aad V = - 1 V, for the lowest aad highest illuminationlevels used ill the experimentaimeasurements.

(whichcorresponds to JL( V))is close to the dark U(x). The experimentaUy_obtainedcurves for IL,(V) vs voltage, however, as plotted in Fa. 4, show an increase in J, of qn (13)with ilhmination. The above variation is in agreementwith the model if we consider the chauge in r (i.e. in IV,)with illumidon kvel. The theoretidtycalculateddark recombinationcurrent density vs voltage curves when plotted for decrming 7 (increasing Nt) preserve their shape and are shifted towards higher current densities (Fig. 9). The same shifting is observed

ldSL 0

I

I

I

I

I

0.6 1.0 0.4 0.6 DISTANCE FROM METALSEMICONDUCTOR CONTACT x (/un) 0.2

I 1.2

Fii 8. Calculated recombinationrate vs distance ia the dark (solid lines) sod, yder ihhation (dashed lioes, J,== 44.4 mA/cm’), for variousforward voltages.

for the experimentallyobtained curves of Fe 4, where increasingiUmination is responsiblefor the decrease in 7. The “II” due for the dark theoretically calculated refmmbiuation current density curve and for the experimentalcurves for the higher iUmim&n levels of F,g. 4, are almost identical at a value of 7 - 1.85.The deviation to higher “n” dues at lower ilIumdim levels is attributed to the enhanced error introdufzcdfor such low levels by the neglect of the dark recodina&n_ Thus, taking the above mentioned varkdion in 7 into

-/, , , , , , , , , -’

16’6

0.32 O.lS 0.24 FORWARDBIAS VOLTAGE V (VOLTS)

0.00

0.40

F@9. (l-4) Calculateddark mbination correat densities vs voltage for various values of trap den&&, aad (la, lb) calculated total mhs’Yero&as recombhtion curreat densities undertwolWdsofiliumhhandforthesamevaiueoftrap .~ density (lifetime)as curve 1.

Recombination in Schottky-barrier solarcegs account, the experimentally-obtainedcurves of Fig. 4 are in excellent qualitative agreement with the curves calculated from the model.

41

tics at high illuminationlevels. A fundamental assumption, explicitly made by Shockley and Read[2], states that the time of readjustment of the captured carrier in the trap is negligiblecompared to the time required for CONCLUSIONS the emission or capture of a carrier. We have found Comparison of the theoretical calculations with the experimentalindicationswhich suggestthat at very high experimental results suggests the presence of recom- illumination levels the availability of free carriers for bination centers distributed in energy in the forbidden recombination may be sufbciently high that the above gap. Uncovering of the recombinationcenters with illu- assumption is not valid and the notion of a recommination is consistent with the form of the minority bination center “dead time” must be introduced. carrier (hole) quasi-Fermi level in its dependence on Finally we point out that the above calculations can position in the space-charge (depletion) region under easily be extended to the case of polycrystaIline Si ilhnuination. Schottky-barrier solar cells, where the recombination The form of the expression for the depletion-region current density is not negligiile compared to the photorecombination rate U(x, V.I.,) does not allow for the current density, by employinga recursive calculation in derivation of a simpk analytical expression for the which the minority carrier current density will be treated recombinationcurrent density as a function of volage. It as a function of position. is, however, theoretically predicted and experimentally observed that the voltagedependent part of the recomAdaowl~s-We wish to thrrnLE. Ghan and T. Pm aed bination current density follows the form of the recom- especially K. K. Ng for moststimulating discussions and conbination current density in the dark once the difference structive criticism. This work was supported by the National in trap density under illuminatonis taken into account. Science Foundation under grant NSF ENG 7~15463. Therefore, even though an expression of the form J~(VIl,e)=J,(~=)[exp(qVI~~T)-ll does not represent the total recombination current density, such 1. R N. Hall, Phys. Reu. 87,387 (1952). an expression does represent the voltage-dependentpart 2. W. Sbockley and W. T. Read Jr., Phys. Reu. 87,835 (1952). of this density. The zero-bias recombination current 3. C. T. Sah, R N. Noyce and W. Shod&y, Pmt. LRE,Is,1228 density, J-, is less than 3.5% of the short-circuit cur(1% 4. s. c. chuo, so&f-St. E II.1069 (1968). rent density J, and is, therefore, negligiile compared to 5. S. Chou, Sdid-St. Bxtnx 14,811 (19fl). I,Furthennore,thetotalrecombmationcurrentden6. A. Nmsbaum, Phys. SM. sd (a). 19,422 (1973). sity, L ( v), is negligibleover most of the characteristics 7. A. Y. C. Yu ad E. H. Snow, 1. AppL Phys. 39,3M8 (M8). comppnd to the total current density, having only a 8. E. Ii. Rhod&k, Metabmimddor Gmtacts. Cbrendm Press, oxford (1978). second-order effect on the till factor, even for devices 9. R. F. McQmt and D. L. Pulfrey, J. Appl. phys. 47, 2113 fabricated on low lifetime epitaxiailygrown Si wafers, as (1976). the ones used in the present study. In solar cell cal- 10. L c. Olsen, sdid-s Bcrmn. m, 741(1977). culations, the effect of J- is automaticagy taken into 11. J. Shewchun, R. Sii aad Id. A. Green, 1. AppL Phys. 4% 765 (1977). account in the short-circuitcurrent density. The stat&d practice of using the form of the dark recombination 12. D. L Puifrey, MIS solar cells: a review. IEEE Truns. Eketron h. 25,1308 (1978). current density for calculationson open&ionunder illu13. H. C. Card, J. Appl. Phys. 4l, 4%4 (1976). mination is just&d, but the magnitude has to be ad- 14. H. C. Card, Solid-St. Bmim. 26,971(1977). justed according to the increase in the pre-exponential 15. P. Panayotatos. H. C. Card and E. S. Yang, Pmt. IEJX (5, 1213(1977). factor J, above due to the increase in N, under &nina16. H. C. Card and E. H. Rhoderick, J. Phys. o:AppL Phys. 4, tion. 1589(1971). The theoretical results are in good agreement with 17. See, for example, s. M. Sze, Physics of sm&o&@r experiment for the typical Au-n-type Si Schottky-barrier Ikuices. Wiley, New York (1969). solar cell, but it should be expected tbat similar 18. A. H. M. Kippman and M. H. Omar, Appl. Phys. Lat. 28, 620 (1976). agreement can be achieved and similarinformationabout recombination centers obtained for solar cells with 19. P. Panayotatos and H. C. Card, 13th IEEE phoroodtoics Sj~~idisrs’ Conf. Record 634 (1978). difFerentgate metals once the appropriate parameters are 20. J. P. Ponpon and P. Si&rt, 13th IEEE Photocdtuics @eciused in expressions (11) and (12). We wish to point out, ulisrs’ Conf. Record. 639 (1978). however, a caution regardingthe use of the SRH statis- 21. H. J. Hovel, Solar Cdfs. Academic Press, New York (1975).