Solar Energy Materials & Solar Cells 100 (2012) 48–52

Contents lists available at ScienceDirect

Solar Energy Materials & Solar Cells

0927-02

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/solmat

Determination of minority carrier diffusion length from distance dependenceof lateral photocurrent for side-on illumination

A.K. Sharma a,n, S.N. Singh a, Nandan S. Bisht b, H.C. Kandpal b, Zahid H. Khan c

a Physics of Energy Harvesting Division, National Physical Laboratory (CSIR), Dr. K.S. Krishnan Road, New Delhi 110012, Indiab Quantum Phenomenon and Applications Division, National Physical Laboratory (CSIR), Dr. K.S. Krishnan Road, New Delhi 110012, Indiac Department of Physics, Jamia Millia Islamia, Jamia Nagar, New Delhi 110025, India

a r t i c l e i n f o

Article history:

Received 31 August 2010

Received in revised form

24 March 2011

Accepted 14 April 2011Available online 6 May 2011

Keywords:

Diffusion length

Side-on illumination

Crystalline Si wafer

Photocurrent generation

48/$ - see front matter & 2011 Elsevier B.V. A

016/j.solmat.2011.04.027

esponding author. Tel.: þ91 11 45608379; fa

ail address: anamics.ak@gmail.com (A.K. Shar

a b s t r a c t

A method of measurement of diffusion length L in p type c-Si wafers based on the lateral collection of

minority carriers is reported. In this method, wafer requires a p–pþ junction on entire back surface and

an nþ–p interface on a part of the front surface leaving the rest part as bare for illumination. A photo-

current Isc is generated when a rectangular area of a part of the bare front surface in the vicinity of the

nþ–p interface is illuminated with a laser beam. The magnitude of Isc varies with the normal distance d

between the electron collecting nþ–p interface and the nearest edge of the illuminated region. The

slope F of the normalized Isc vs. d curve is used to determine a parameter sinh�1y, which has a linear

relation with d. The reciprocal of the slope of sinh�1y vs. d curve in the linear region gives the diffusion

length L. The value of L is less susceptible to error due to the effect of Sf of bare silicon surface if the

linear region of sinh�1y vs. d curve lies in the region of smaller d values. The present method has an

advantage over the other methods in that it does not depend on the intensity of the illumination and

absorption coefficient of Si. Additionally, method has no limitation in terms of wafer thickness to

diffusion length ratio and is applicable to all practical L values.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The cheap availability of solar photovoltaics energy is foremostquest for the present day R&D in the field of photovoltaics. Theapproaches to make solar photovoltaics cost effective lie in betterutilization of available material with improved highly efficientsolar cell designs [1]. In case of dominated c-Si wafer based solarcell, the wafer, as base material, is much more important for highefficiency solar cell production. The quality of wafer used in solarcell production can be judged well with the measurement ofminority carrier diffusion length L in Si wafer. It is also very muchdesirable to measure L under the condition of illumination priorto fabrication of p–n junction to avoid heating of wafers duringthe junction fabrication at temperature above 850 1C.

Surface photovoltage (SPV) method [2,3] and Photocurrentgeneration (PCG) method [4] are used for direct measurement of L

in Si wafers. The former uses a naturally existing depletion layerat the surface to generate photovoltage on illumination. The SPVgenerated with monochromatic radiation of different wave-lengths of the given intensity is measured and L is determinedfrom the plot of the surface photovoltage vs. the absorption

ll rights reserved.

x: þ91 11 45609310.

ma).

coefficient. The method has a severe limitation in that it is validonly if the thickness of the wafer is 3 times or larger than thediffusion length [5,6]. The PCG method [4] requires formation of apþ (nþ) accumulation layer on one side of the p (n) wafer and annþ (pþ) inversion layer on the other side. The wafer is illumi-nated with monochromatic radiation from accumulation layerside and the diffusion length is determined from the slope of thephotocurrent density vs. illumination intensity characteristics.The method is applicable for determining L smaller or larger thanthe wafer thickness and is valid as long as ratio of wafer thicknessto diffusion length is larger than 0.6. The PCG method [4] issensitive to the quality of accumulation layer, which is formed bydepositing a semitransparent metal layer on the wafer, whichmay degrade on exposure to air and lead to determining ofsmaller values of L [7]. Recently a Laser Beam Induced Current(LBIC) method [8] has been reported for determining L in Siwafers. The method was applied under low level injection condi-tion and was found to be valid if thickness of the wafer was largerthan 4L. For wafer thickness less than 4L it was found to giveerroneous results.

In this work, we present a method for the measurement of L inc-Si wafer, which is based on lateral collection of photo-generatedcarriers due to side-on illumination with a monochromatic band gapradiation like in the LBIC method. Normalized experimental shortcircuit current density Jsc is plotted as a function of distance d and

A.K. Sharma et al. / Solar Energy Materials & Solar Cells 100 (2012) 48–52 49

the change of the slope of the curve is utilized to determine thediffusion length L. The value of L determined using PCG method [7]is also presented for comparison.

2. Experimental

Two chemically and mechanically polished Cz Si wafers, 50 mmdiameter, /1 0 0S orientation, 400710 mm thickness, p type, Bdoped and of 1.0 O cm resistivity were taken. Thereafter an pþ

accumulation layer was created on one side of the wafer bydepositing a �50 nm semitransparent layer of Pd and an nþ

inversion layer on half portion of opposite side of the wafer bydepositing an �80 nm Al layer using thermal evaporation undervacuum. As shown in Fig. 1, the part of the wafer covered with metalson both side acquired a structure akin to nþ–p–pþ and was capableof generating a photocurrent on illumination at a bare portion ofwafer on front or from the Pd side on back and was thus suitable formeasurement of L by both the side-on illumination being reported inthis work and the PCG method. We shall present in the following thedetails of L measurement for one specimen P-1 using the side-onillumination method and compare it with the PCG method.

2.1. Diffusion length measurement using PCG method

The diffusion length measurement was first done using PCGmethod [4]. For this the both side covered portion of the waferwas illuminated from Pd covered back side of the wafer with abeam of circular aperture area 1.0 cm2 of laser beam used and theIsc was measured for various intensities in 25–40 mW cm�2

range. The short circuit current density Jsc was (computed bydividing Isc with the illuminated area) plotted against inputintensity Pin on a linear scale and z, the slope of the straight linethat passes through the origin, was measured. The L was deter-mined from the slope z and the reflectivity rl of Pd coveredsurface. The care was taken to store the wafers in vacuum beforeand soon after the measurement of Jsc–Pin characteristics toensure a very low recombination velocity of minority carriers atthe Pd coated back surface of the wafer [7].

2.2. Isc–d measurement using side-on illumination

The specimen prepared for measurement of L with side-onillumination method was mounted on a gold plated brass jig withits nþ side and the bare silicon surface on top and Pd coated surfaceat the bottom. A diode laser beam with fixed intensity 40 mW cm�2

(l¼789, DL 100-TOPTICA) was passed through a beam expander, acollimator and a rectangular aperture of dimension 10 mm�1.5 mm. It illuminated the bare silicon surface normally as shown

Fig. 1. Schematic diagram of the side-on illumination structure.

in Fig. 1. The jig was kept in a micropositioner (STANDA-USA) andwas moved towards the illuminated rectangle in step of 20 mm tovary the normal distance d between the induced nþ–p interface at Aand the nearest edge B of the illuminated rectangle. The steady stateshort circuit photo current Isc between the front and back metalcontacts was measured using a Keithley model 2420 source-meter.

The intensity of illumination was measured in cases of bothPCG and side-on illumination methods using a reference siliconsolar cell supplied by PV Measurements Inc. USA. The reflectivityRl of Pd covered back side, were measured using a spectro-photometer SHIMADZU model UV 3101 PC.

3. Theoretical

Consider Fig. 2, where point A (x¼0), marks the nþ–p interface(Al–Si) an induced nþ–p junction that can collect photogeneratedelectrons from the bare Si portion. Let us illuminate the bare pregion with a visible monochromatic light (such as the laser lightof l¼789 nm used experimentally in this case) that has anabsorption depth of �10 mm or so in silicon. The illuminatedarea is a rectangle of length ‘a’ and width ‘b’. The side ‘a’ is parallelto the induced nþ–p junction interface at A the intensity ofillumination is uniform throughout the illuminated rectangle.Typically, a¼10 mm and b¼1.5 mm, and ‘b’ is substantially morethan 3 times of the expected value of L in the wafer.

The intensity profile of the illuminated rectangle is a stepfunction, where the intensity is uniform across the length ‘a’ andthe width ‘b’ of the rectangle. Here, we consider movement ofphotogenerated carriers in the p-region in one direction (i.e.x-direction) only, i.e. from B to A. Fig. 2 also shows the intensityprofile of illumination, which is nearly step function (or comple-mentary error function) in nature and is represented as

I¼I0

2erfc

d�x

b

� �

where I0 is the maximum intensity of the illumination within theilluminated rectangle region and I is the intensity at a distance x

from point A and b is an arbitrary constant describing the shape ofthe intensity distribution in the x-direction. The value of b is verysmall in comparison with d and x and ensures I¼0 in the un-illuminated 0oxod region and a steep change in the value of I

near x¼d such that I¼ I0 for x4d.As the monochromatic light is incident normally to the wafer

surface the carriers will be generated under the illuminated area

Fig. 2. Schematic of intensity of illumination in the bare silicon region. Photo-

generated carriers are generated under semi-infinite illumination rectangle and

move towards the collection point A (x¼0) by diffusion from point B (x¼d) and

point C (x¼dþg) within the illumination rectangle where intensity profile is

uniform.

A.K. Sharma et al. / Solar Energy Materials & Solar Cells 100 (2012) 48–5250

along the wafer thickness due to absorption of the light and thusperpendicular to the x-direction. Since the pþ accumulation layerat the bottom surface provides Back Surface field (BSF) therecombination at the back (z¼t) where z is the direction alongwafer thickness t is negligible. For simplification we also ignorerecombination of minority carriers at the front surface. Thereforewe assume that under steady state, the photogenerated carrierswill undergo recombination only in the p-region and not at thefront and back surfaces while moving in the x-direction frompoint B to A and in doing so are collected at the induced nþ–pjunction interface at A. We shall show later that the flow ofphotogenerated carriers along the wafer thickness would notaffect the final results significantly.

The one dimensional current continuity equation for minoritycarriers (electrons in the p-region) between the nþ–p interface atx¼0 and the illuminated rectangle under steady state can bewritten as

1

q

dJn

dx¼

n�np

t�GðxÞ: ð1Þ

In Eq. (1) Jn is the minority carrier current density, n is theminority carrier density, np is the equilibrium minority carrierdensity and t is lifetime of minority carriers. G(x) is the photo-generation rate of minority carriers and is defined as

GðxÞ ¼G0

2erfc

d�x

b

� �, ð2Þ

where

G0 ¼nphð1�RlÞð1�e�altÞ

t:

The nature of G(x) follows from the intensity profile

I¼I0

2erfc

d�x

b

� �,

where G0 corresponds to I0. In the above expressions, G0 isaverage photogeneration rate at x4d along the wafer thicknesst, nph is the photon flux of the incident monochromatic light of awavelength l, Rl is the reflectance of illuminated surface and al isthe absorption coefficient of incident light in Si.

In the present case the drift component of current density Jn

can be ignored and therefore Eq. (1) becomes

d2n

dx2¼

n�np

L2�

c

2erfc

d�x

b

� �, ð3Þ

where L2¼Dnt; Dn is the diffusion coefficient of the minority

carriers and c is a constant defined as c¼ G0=Dn.The solution of Eq. (1) can be written as

n¼ c1eðx=LÞ þc2e �x=Lð ÞþnpþcL2

4e

xL �

dLþ

b2

4L2

� �erf

d

b�

x

b�

b2L

� �" #

þcL2

4e�x

LþdLþ

b2

4L2

� �erf

d

b�

x

bþ

b2L

� �" #þ

cL2

2erfc

d�x

b

� �� �, ð4Þ

where c1 and c2 are constants, whose values can be computedapplying the following boundary conditions:

(i)

(ii)

n¼ np at x¼ 0,

and

dn

dx¼ 0 at x¼ dþg

The condition (i) is attributed to the minority carrier collectionnature of the nþ–p interface present at x¼0. On the other hand,

the condition (ii) signifies that, practically, the photogeneratedcarriers of distance beyond x¼(dþg) do not have to flow towardsthe collection interface.

The short circuit current density Jsc can be obtained using therelation

Jsc ¼ qDndn

dx

����x ¼ 0

: ð5Þ

Eq. (4) helps obtain an expression for dndx 9x ¼ 0, which is then used

in Eq. (5) to obtain an expression for Jsc as

Jsc ¼qG0L

4e

gLþ

b2

4L2

� �erf

d

b�

b2L

� �sech

dþg

L

� �" #

�qG0L

4e�

gLþ

b2

L2

� �erf

d

bþ

b2L

� �sech

dþg

L

� �" #

þqG0L

4e

gLþ

b2

4L2

� �erf

g

bþ

b2L

� �sech

dþg

L

� �" #

�qG0L

4e�

gLþ

b2

4L2

� �erf

g

b�

b2L

� �sech

dþg

L

� �" #

þqG0L

2erfc

d

b

� �tanh

dþg

L

� �� �: ð6Þ

Since b is very small in comparison with g, d and L theconditions; d=bbb=2L, g=bbb=2L, b2=4L2-0 are generally satis-fied in practice.

Then, for d=b43 and g=b43; erf d=b

¼ 1, erf g=b

¼ 1,erfc d=b

¼ 0 and Eq. (6) is simplified to

Jsc ¼ qG0Lsinhðg=LÞ

cosh dþgð Þ=L

" #: ð7Þ

For d-0, Eq. (7) further reduces to

Jsc ¼ qG0Ltanhg

L

� �: ð8Þ

Here, Eq. (8) shows the maximum current density, generatedwhen illumination rectangle just touches the nþ–p interfacecollection point.

The normalized short circuit current density Jscn can beobtained by dividing Eq. (7) by (8) as

Jscn ¼ coshg

L

� �sech

dþg

L

� �: ð9Þ

3.1. Methodology for determination of L using Eq. (9)

Eq. (9) represents the Jscn–d curve. The slope f¼ dJscn=dd ofthis curve can be obtained as

f¼�1

Lcosh

g

L

� �sech

dþg

L

� �tanh

dþg

L

� �: ð10Þ

Putting dF/dd¼0 we obtain the maximum slope Fm which isgiven by

Fm ¼�1

2Lcosh

g

L

� �: ð11Þ

This occurs at a value d which satisfies the condition

sinhdþg

L

� �¼ 1: ð12Þ

Using Eqs. (11) and (12) in (10) we get

ffm

¼ 2sechdþg

L

� �tanh

dþg

L

� �: ð13Þ

Eq. (13) yields a relation

sinh�1y¼d

Lþ

g

L, ð14Þ

Fig. 3. Variation of normalized experimental short circuit density Jscn with

distance d for specimen P-1. The theoretical curve is a polynomial in which the

A.K. Sharma et al. / Solar Energy Materials & Solar Cells 100 (2012) 48–52 51

where

y¼17

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðf=FmÞ

2qf=Fm

: ð15Þ

In certain ranges of d that may be identified by d1odod2,Eq. (14) may show a linear relation between sinh�1y and d. Forsuch cases the slope of the line represented by Eq. (14) is equal tothe reciprocal of L. Here, y has two roots yþ and y� which aregiven by

yþ ¼1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðf=FmÞ

2qf=Fm

, ð16aÞ

and

y� ¼1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðf=FmÞ

2qf=Fm

: ð16bÞ

Thus, in practice, the linear region of sinh�1yþ vs. d orsinh�1y� vs. d curve can be used to determine the diffusionlength L from the measurement of slope of the curve with thed-axis.

experimental data has been fitted.

Fig. 4. Variation of slope F of Jscn vs. d curve for specimen P-1. Fm is the maximum

numerical value of F.

4. Result and discussion

The diffusion length measurements on P-1 and P-2 were doneusing PCG method [4] just after the specimens were taken out fromthe vacuum to avoid the degradation of accumulation layer due toexposure of air [7]. For specimen P-1, the input parameters areused as t¼400 mm, Dn¼25 cm2 s�1, l¼789 nm, al¼1000 cm�1,Rl¼0.546 for L measurement using PCG method and calculatedL¼102 mm. Soon after the measurements the specimens were againstored in vacuum. The values of L are listed in Table 1.

Subsequently the specimens were taken out of vacuum andthe present method (side-on illumination method) was used fordetermination of L. The laser light of l¼789 nm of illuminationintensity 40 mW cm�2 was used and the short circuit current Isc

was measured by varying d between 0 to 500 mm. For specimenP-1 the short circuit current Isc varied from a maximum value33.43 mA for d-0, to 1.13 mA for d¼500 mm. All the values of Isc

were normalized with respect to maximum Isc value 33.43 mA.The normalized Isc values are same as the normalized short circuitcurrent density Jsc, referred to earlier as Jscn in Eq. (9) andtherefore we have plotted Jscn values against d values for P-1 inFig. 3. It may be pointed out that the d values are in mm and arenot normalized. These Jscn, d values were fitted into a polynomialof order five

Jscn ¼ B0þB1dþB2d2þB3d3þB4d4þB5d5, ð17Þ

and thus obtained theoretical curve is also plotted in Fig. 3.It can be seen that the experimental data fitted excellentlywell in to the theoretical curve given by Eq. (17). The valuesof constants are B0¼1.00598, B1¼9.87381E-5 mm�1,

Table 1Comparison of diffusion length L measured by side-on illumination method and

PCG method of specimen P-1.

Diffusion length L (lm)

Side-on illumination method PCG method

(in 5odo85 mm range) (in 120odo180 mm range) 102

115 106

B2¼�6.00376E-5 mm�2, B3¼3.04544E-7 mm�3, B4¼�5.84465E-10 mm�4 and B5¼3.98493E-13 mm�5.

Differentiating the theoretical Jscn vs. d curve of Fig. 3 thevalues of F as a function of d were determined. These values of Fare plotted against d in Fig. 4. This curve give the maximum valueof slope Fm¼ .00491 (mm)�1 at d¼101 mm. From the values of Fand Fm the values of (yþ and y�) were determined and thensinh�1(y) vs. d curves were plotted in Fig. 5.

Fig. 5 shows that for y¼y� the sinh�1(y) vs. d curve is linear in5odo85 mm range, whereas, for y¼yþ the sinh�1(y) vs. d curveis linear in 120odo180 mm range. Slope of these curves yieldedthe value of L¼115 mm corresponding to 5odo85 mm range andL¼106 mm corresponding to 120odo180 mm range.

The slightly smaller values of L obtained using Jscn vs. d data forhigher d values (120odo180 mm) may be due to some deleter-ious effect of front surface recombination velocity Sf on themeasured Isc values.

We may assume that the back surface recombination velocitySb is negligibly small due to the presence of the pþ accumulation

Fig. 5. Variation of sinh�1y with distance d for specimen P-1. y is related with Faccording to Eq. (15). The reciprocal of the slope of sinh�1y vs. d curve gives

diffusion length L.

A.K. Sharma et al. / Solar Energy Materials & Solar Cells 100 (2012) 48–5252

layer at the back surface and following Hull [9] we may assign avalue 2�103 cm/s (and 104 cm/s) to Sf for our chemically andmechanically polished bare silicon surface of P-1. Using this valueof Sf in expression ts ¼ t=Sf

þ 4=Dn

t=p 2

given by Sproul [10]we calculated the value of surface lifetime ts as 46 ms (and 30 ms).The values of other parameters used in calculation of ts weret¼400 mm, Dn¼25 cm2 s�1. This value 46 ms (and 30 ms) of ts

along with teff¼4.48 ms (obtained from the measured values of L

i.e. Leff¼105.82 mm for P-1 in 120odo180 mm range of d)whenin turn used in expression 1=teff

¼ 1=tb

þ 1=ts

gave bulk

lifetime tb¼4.96 ms (and 5.27 ms). This value of tb gave the bulkdiffusion length Lb¼111.4 mm corresponding Sf¼2�103 cm/s andlength Lb¼114.7 mm corresponding to Sf¼104 cm/s. It may benoted that both these values of Lb are very close to the values ofL¼115 mm determined by the present method using the Jscn vs. d

data of 5odo85 mm range. This shows that the effect of Sf in themeasurement of L using the present method has been negligiblein the 5odo85 mm range. The use of an effective surfacepassivation layer on the bare silicon surface may be able toreduce the difference in the values of L obtained using the regionsof smaller and higher d values.

So determined value of L for P-1 using the Jscn vs. d data of5odo85 mm range are also listed in Table 1. The value of L

determined using the PCG method is slightly smaller and this maybe due to the back surface recombination velocity Sb, thoughassumed negligible may have been significantly high. The recentwork [7] on application of PCG method shows that when Sb is notnegligible the value of L determined using PCG method is smallerthan the true value of bulk diffusion length.

The values of L determined with the side-on illuminationmethod were found reasonably close to that determined usingPCG method. The side-on illumination method does not dependon the intensity of the illumination and absorption coefficient ofSi. Additionally, method has no limitation in terms of waferthickness to diffusion length ratio and is applicable to all practicalL values.

5. Conclusion

A method, referred to as the side-on illumination method,based on lateral collection of minority carriers, for determinationof the diffusion length L in p-type c-Si wafer was developed. Inthis method, the a structure, akin to nþ–p–pþ , is formed on thewafer by creating a pþ accumulation layer on the back side andan nþ inversion layer on a half portion of the front side. A suitablemonochromatic light, such as the laser light of l¼789 nm used inthis work, illuminates a rectangular area in the bare silicon regionon the front side and generates a current due to lateral collectionof photogenerated carriers at the (nþ–p) interface. The photocurrent is varied by changing the normal distance d between theilluminated rectangle and the (nþ–p) interface. The slope F of thenormalized Jscn vs. d curve is used to determine a parametersinh�1y, which depends on F. The reciprocal of the slope ofsinh-1y vs. d curve in the linear region gives the diffusion length L.The value of L is less susceptible to error due to the effect of Sf ofbare silicon surface if the linear region of sinh�1y vs. d curve liesin the region of smaller d values. The present method does notrequire the measurement of intensity of illumination.

We applied the side-on illumination method to a specimen P-1and determined L¼115 mm using the 5odo85 mm range. Thephotocurrent generation (PCG) method gave L¼102 mm. Theslightly smaller value of L obtained with PCG method may bedue to the effect of back surface recombination velocity Sb beingignored in PCG method. The side-on illumination method, whenusing smaller d region, may be less susceptible to the errors dueto Sb than the PCG method.

Acknowledgment

The authors thank Dr. M. Kar for reflectivity measurements.A.K. Sharma, Nandan S. Bisht and S. N. Singh acknowledge thefinancial support of Council of Scientific and Industrial Research(CSIR), India, for this study.

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