Analytical modeling of the radial pn junction nanowire solar cellsNouran M. Ali, Nageh K. Allam, Ashraf M. Abdel Haleem, and Nadia H. Rafat

Citation: Journal of Applied Physics 116, 024308 (2014); doi: 10.1063/1.4886596 View online: http://dx.doi.org/10.1063/1.4886596 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/2?ver=pdfcov Published by the AIP Publishing

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Analytical modeling of the radial pn junction nanowire solar cells

Nouran M. Ali,1 Nageh K. Allam,2 Ashraf M. Abdel Haleem,3,a) and Nadia H. Rafat11Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt2Energy Materials Laboratory, Department of Physics, School of Sciences and Engineering,The American University in Cairo, New Cairo 11835, Egypt3Department of Engineering Mathematics and Physics, Faculty of Engineering, Fayoum University, Egypt

(Received 16 May 2014; accepted 18 June 2014; published online 11 July 2014)

In photovoltaic solar cells, radial p-n junctions have been considered a very promising structure to

improve the carrier collection efficiency and accordingly the conversion efficiency. In the present

study, the semiconductor equations, namely Poisson’s and continuity equations for a cylindrical

p-n junction solar cell, have been solved analytically. The analytical model is based on Green’s

function theory to calculate the current density, open circuit voltage, fill factor, and conversion effi-

ciency. The model has been used to simulate p-n and p-i-n silicon radial solar cells. The validity

and accuracy of the present simulator were confirmed through a comparison with previously pub-

lished experimental and numerical reports. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4886596]

I. INTRODUCTION

Recently, radial structures have attracted great interest

to enhance the solar energy conversion. Planar junction solar

cells must have a minority carrier diffusion length long

enough to allow for effective collection of photo-generated

carriers over the entire depth of light absorption. For the

materials that have low diffusion lengths relative to the opti-

cal thickness, radial p-n junction nanorod geometry with ra-

dius in nano-size has the potential to improve the cell

efficiency due to the independence between carrier transport

and light absorption directions.

In silicon solar cells, improvement in External

Quantum Efficiency (EQE) and absorption efficiency was

observed experimentally when using Si Nanowire Solar

Cells (NW SCs).1 Additionally, when the performance of

this NW SCs has been calculated using numerical simula-

tions and compared to conventional planar p-i-n thin-film Si

solar cells, NW SCs achieve nearly 80% efficiency

enhancement.2

Besides, Cd (Se, Te) nanorod arrays were observed to

have improved fill factors relative to their planar counter-

parts. Furthermore, the spectral response data indicate that

the nanorod array electrodes behave as if they had a much

longer minority carrier collection length than planar elec-

trodes that is made using identical materials fabrication

processes.3 It was shown by numerical simulation that the

device design using nanowire CdS can yield a substantial

(�25%) enhancement in the power conversion efficiency of

this cell.4

It is worth mentioning that during the year 2013 several

studies were done concerning the vertically aligned NW ra-

dial p-n and p-i-n junctions. For instance, Si NWs were

investigated using the commercially available well known

packages, namely COMSOL multiphysics and 3-D TCAD

numerical simulator. COMSOL has been used to optimize

the length and the doping level of the Si-NWs array.5 The 3-

D TCAD numerical simulator has been utilized to study the

performance of the 10 lm, in long, crystalline amorphous sil-

icon core-shell (c-Si/a-Si/AZO/Glass) NWs solar cells that

reached photogenerated current up to 22.94 mA/cm2 and

conversion efficiency of 13.95%.6

Furthermore, silicon radial p-n and p-i-n junction NWs

arrays were experimentally fabricated by low temperature

epitaxial growth process using silane-based chemical vapor

deposition and they achieved solar energy conversion effi-

ciency of 10% under AM 1.5 G illumination.7 Additionally,

Si-NWs arrays with excellent light trapping property were

fabricated by metal-assisted chemical etching technique and

achieved a short circuit current density of 37.13.8

Concerning the NWs contacts, wrap-around top Ag contacts

for radial junction Si-NWs solar cells were fabricated and

obtained a fill factor of 65.4%, which is higher than that of

conventional top contacts.9

Although many numerical and experimental comparative

studies were performed to prove the enhancement achieved

when applying nanorod geometry in organic cells and inor-

ganic cells, analytical studies were rarely discussed.10–12 In

2005, Kayes and Atwater showed analytically that extremely

large efficiency gains from 1.5% to 11% are possible by

applying the radial p-n junction nanorod geometry.13

Even though the analytical model for cell parameters is

not enough to know the detailed parameters at each point

inside the cell such as carrier concentrations and current den-

sity, it is a sufficient and a quick tool to calculate the impor-

tant parameters of the photovoltaics (PV) solar cells such as

Isc, Voc, FF, and efficiency.

Using Green’s function in the present model reduces the

need for uniform generation assumption used in conventional

analyses.14 It depends on calculating the Green’s function

a)Author to whom correspondence should be addressed. Electronic mail:

ama05@fayoum.edu.eg. Present address: Institute of Science and

Technology Research, Chubu University, 1200 Matsumoto-cho, Kasugai,

Aichi 487-8501, Japan.

0021-8979/2014/116(2)/024308/7/$30.00 VC 2014 AIP Publishing LLC116, 024308-1

JOURNAL OF APPLIED PHYSICS 116, 024308 (2014)

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that satisfies the required equations and boundaries but with

point source, then the actual generation source can be

applied to get a general solution.

II. THEORY/CALCULATION

In our analysis, we considered a radial structure with p-

type core and n-type outer shell, as shown in Figure 1. The

following assumptions are considered in the analysis: (1) the

ratio between the rod length and diameter is very high, (2)

recombination of carriers is due to bulk traps and surface

traps, (3) the doping is low and uniform, (4) carrier transport

is only in the radial direction, and (5) incident light is normal

to the top surface with zero reflectance. The energy band dia-

gram at thermal equilibrium is shown in Figure 2. The solar

cell total current, I, takes the form

I ¼ IdarkðVÞ– Isc; (1)

where Isc is the short circuit current and Idark (V) is the volt-

age dependent dark current.

The continuity equation for electrons in the p-type

region takes the form

@n

@t¼ GL – Un þ

1

qr � Jeð Þ; (2)

where GL ¼Ð

aFo exp ð�azÞ dk represents the generation

rate of carriers due to incident light assuming zero reflec-

tance of light at the upper surface, where a is the absorption

coefficient, Fo is the incident flux density. Un ¼ Dnsn; repre-

sents the net recombination rate of the minority carriers in

the quasi neutral region assuming low-level of injection.15

Dn is the excess carriers’ concentration due to illumination

and sn is the electron recombination time. Je ¼ qDern; rep-

resents the electron current density assuming diffusion only

and De is the electron diffusion constant.

Using cylindrical coordinates and assuming no depend-

ence on u and z, Eq. (2) for excess electrons in the p type

region at steady state can be written in the form

@2

@r2þ 1

r

@

@r–

1

L2n

!Dn ¼ �g01: (3)

Here, Ln ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsnDe Þ

pis the electron diffusion length

and g01 ¼ GDe

. It is worthy to mention that Eq. (3) represents

an inhomogeneous Helmholtz equation.

Similarly, the continuity equation for excess holes in the

n-type region can be written as

@2

@r2þ 1

r

@

@r–

1

L2p

!Dp ¼ �g02; (4)

where g02 ¼ GDh

and Dh is the hole diffusion constant.

On the other hand, the voltage dependant parameters

Dr1, Dr2, Dr3, and r4 are calculated by solving Poisson’s

equation in all regions, as stated in Appendix A.

A. Dark current calculations

Idark ¼ Ied þ Ih

d þ Ir; (5)

Idark ¼ Ieo þ Ih

o

� �exp

V

Vt

� �� 1

� �þ Idep

o expV

2Vt

� �� 1

� �;

(6)

where Vt is the thermal potential, Ieo; Ih

o are the electrons and

holes reverse saturation current, respectively, and Idepo is the

depletion region recombination current. It is approximately

defined, from the planar structure,16 as

Idepo ¼ q ni2p Dr2 þ Dr3ð Þ2L

2ffiffiffiffiffisnp

sp

� � : (7)

The boundary conditions for Dn and Dp are

FIG. 1. The structure of radial p-n junction, where L is the nanorod length,

Rp is the radius of the metallurgical junction, R is the total radius of the rod,

and Rn¼R�Rp.

FIG. 2. The energy band diagram at thermal equilibrium, where Dr1, Dr2,

Dr3, and r4 represent the width of quasi-neutral n-type region, the width of

the depletion n-type region, the width of the depletion p-type region, and the

width of quasi-neutral p-type region, respectively.

024308-2 Ali et al. J. Appl. Phys. 116, 024308 (2014)

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Dnjr¼0 is finite; and Dnjr¼r4¼ npo exp

V

Vt

� �� 1

� �; (8)

Dpjr¼RpþDr2¼ pno exp

V

Vt

� �� 1

� �and

Dh@

@rDpð Þjr¼R ¼ �SpDp j r¼Rð Þ: (9)

Here, Sp is the holes surface recombination velocity, npo

and pno are the minority carrier concentrations at thermal

equilibrium, and Dn and Dp represent the excess carrier con-

centrations due to the bias only.

Solving the homogeneous equations (3) and (4) with

g01¼ 0 and g02¼ 0, we get expressions for the concentration

and the current density as follows:

Dn ¼ npo expV

Vt

� �� 1

� � Ior

Ln

� �

Ior4

Ln

� � ; (10)

where Io is the modified Bessel function of order zero of the

first kind.

Jed ¼ qDe

@Dn

@r

����r¼r4

; (11)

Ied ¼

ð ðJe

d r dudz

¼ 2 pr4

L

LnqDenpo exp

V

Vt

� �� 1

� �I�1 b1ð ÞIo b1ð Þ

!; (12)

where I�1 is the modified Bessel function of order �1 of the

first kind.

DP ¼ pno expV

Vt

� �� 1

� �AIo

r

Lp

� �þ BKo

r

Lp

� �� ; (13)

where Ko is the modified Bessel function of order zero of the

second kind.

Jhd ¼ �qDh

@Dp

@r

����r¼RpþDr2

; (14)

Ihd ¼

ð ðJh

d r dudz ¼ �2 p Rp þ Dr2ð Þ

� L

LpqDhpno exp

V

Vt

� �� 1

� �AI�1 b2ð Þ � BK�1 b2ð Þ½ �:

�(15)

With b1 ¼r4

Ln; b2 ¼

Rp þ Dr2

Lp; b3 ¼

R

Lp; (16)

A ¼K�1 b3ð Þ �

SLp

DhKo b3ð Þ

I�1 b3ð ÞKo b2ð Þ þ K�1 b3ð ÞIo b2ð Þ þSLp

DhIo b3ð ÞKo b2ð Þ � Io b2ð ÞKo b3ð Þ� �

8>><>>:

9>>=>>;; (17)

B ¼ 1

Ko b2ð Þ� AIo b2ð Þ

Ko b2ð Þ: (18)

B. Short circuit current calculations

Isc ¼ððIe

sc ðkÞ þ IhscðkÞ þ Idep

sc ðkÞÞ d k; (19)

where Idepsc (k) is the contribution of the depletion region to

the light-generated current. It was calculated by assuming

that all absorbed photons in the depletion region generate

carriers that are all collected.13

Idepsc ðk Þ ¼ qFoðkÞð1� exp ð�aLÞÞððRp þ Dr2Þ2 � r2

4Þ :(20)

To get Iesc and Ih

sc, we have to solve the continuity equations

(3) and (4), at V¼ 0.

Using the Green’s function theory, we defined a Green’s

function which satisfies the same equation, but from a point

source located at r0, so the equation becomes homogeneous

when r 6¼ r0.So, the equations in the p-type region would be as fol-

lows17 (see Appendix B):

@2

@r2þ 1

r

@

@r–

1

L2n

!G1 r; r0ð Þ ¼ 1

rd r � r0ð Þ; (21)

Dn ¼ð

r0 �g01ð ÞG1 r; r0ð Þdr0

þ r0Dn r0ð Þ @G1

@r0

����surf ace

– r0G1

@Dn r0ð Þ@r0

����surf ace

: (22)

According to the Dirichlete boundary conditions (B.C)

Dnjr¼r4 ¼ 0, where Dn represents the excess carrier concen-

trations due to illumination only with zero bias, the Green’s

function can derived as shown in Appendix C

G1 r; r0ð Þ ¼ Ko b1ð ÞIo b1ð Þ

� Ior <

Ln

� �Io

r >

Ln

� �–

Io b1ð ÞKo b1ð Þ

Kor >

Ln

� �( )" #:

(23)

Here r< and r> are the smaller and larger of r and r0.

024308-3 Ali et al. J. Appl. Phys. 116, 024308 (2014)

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Dn r; kð Þ ¼ �g01LnKo b1ð ÞIo b1ð Þ

rI1

r

Ln

� �Io

r

Ln

� �–

Io b1ð ÞKo b1ð Þ

K1

r

Ln

� � !( )þ r4Io

r

Ln

� �I1 b1ð Þ þ

Io b1ð ÞKo b1ð Þ

K1 b1ð Þ� �( )2

4

� rIor

Ln

� �I1

r

Ln

� �þ Io b1ð Þ

Ko b1ð ÞK1

r

Ln

� � !( )3775; (24)

Iesc kð Þ ¼ 2pr4 �qFo kð Þ

� �1� exp �aLð Þ� �

r4I1 b1ð ÞKo b1ð ÞIo b1ð Þ

I�1 b1ð Þ� �

þ r4I1 b1ð ÞK�1 b1ð Þð Þ�

: (25)

On the other hand, the B.C in the n-type region would be:

G2ðr; r0Þjr¼RpþDr2¼ 0 (Dirichlete boundary condition) and

@G2 r;r0ð Þ@r0 jr0¼R ¼ 1

R (Neuman boundary conditions)17,18

G2 r; r0ð Þ ¼ const� n1 r <ð Þn2 r >ð Þ

¼ const Ior <

Lp

� �þ c1Ko

r <

Lp

� �� �

� Ior >

Lp

� �c2Ko

r >

Lp

� �� �; (26)

Dp r; kð Þ ¼ �g02ð ÞðR

RpþDr2

r0G2 r; r0ð Þdr0

þ Dp Rð Þ 1þ SpR

DhG2 r; r0ð Þjr0¼R

� �; (27)

Ihsc ¼ �qDh2 pðRp þ Dr2ÞðF1 þ F2Þ: (28)

Here,

F1 ¼ ð�Foð1� e�aLÞÞconstðI�1ðb2Þ� c1K�1ðb2ÞÞðRðI1ðb3Þ � c2K1ðb3ÞÞ� ðRp þ Dr2ÞðI1ðb2Þ � c2K1ðb2ÞÞÞ; (29)

F2 ¼ Dp Rð Þ SpR

LpDhconst I�1 b2ð Þ � c1K�1 b2ð Þð Þ

� Io b3ð Þ þ c2Ko b3ð Þð Þ; (30)

c1 ¼ �Io b2ð ÞKo b2ð Þ

; (31)

c2 ¼ �I�1 b3ð ÞK�1 b3ð Þ

; (32)

const ¼ 1

c1 � c2

; (33)

Dp Rð Þ ¼ � F3;RDh

Sp R G2 R;Rð Þ ; (34)

F3;R ¼ð

r0ð�Foð1� e�aLÞÞG2ðr; r0Þdr0 jr¼R

¼ const� Lpð�Foð1� e�aLÞÞðIoðb3Þþ c2Koðb3ÞÞ ðRðI1ðb3Þ � c1K1ðb3ÞÞ� ðRp þ Dr2ÞðI1ðb2Þ � c1K1ðb2ÞÞÞ: (35)

From the above analysis, we calculate the short circuit

current, the I–V relation, the open-circuit voltage Voc,

the maximum output power density Pmax, the fill factor

ðFF ¼ Pmax

JscVocÞ; and the conversion efficiency ðg ¼ Pmax

PinÞ, where

Pin is the overall incident power density.

III. RESULTS AND DISCUSSION

To check the validity of our analysis, a comparison was

done between our results and previously published works.

The parameters of Si are listed in Table I, when the cell is

illuminated by AM1.5 G standard spectrum of ASTM G-173

global tilt solar spectrum.19 In all comparisons, we defined

the error asjour analysis�publishedj

published � 100%

TABLE I. Si cell parameters.

T (K) 300 Eg (eV) 1:17� ð4:73� ð10ð�4ÞÞ � T2=ðT þ 636Þ (Ref. 20)

Na (cm�3) 1017 Nc (cm�3) 6:2� ð1015Þ � T32

Nd (cm�3) 1017 Nv (cm�3) 3:5� ð1015Þ � T32

er 11.7 le (cm2/Vs) (De¼Vt le) 68:5þ 1414� 68:5ð Þ1þ Nd

9:2�1016

�0:711

0@

1A (Ref. 21)

ni (cm�3) 1010 lh (cm2/Vs) (Dh¼Vt lh) 44:9þ 470:5� 44:9ð Þ

1þ Na2:23�1017

�0:719

0@

1A

024308-4 Ali et al. J. Appl. Phys. 116, 024308 (2014)

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The present analysis was compared with previously

reported experimental study of Si-NWs.8 In that study, the

nanowire radius equals 100 nm with n-type shell thickness of

10 nm and length of 1 lm. The used doping concentrations Na

and Nd were equal to 2.68� 1020cm�3 and 1.5� 1016cm�3,

respectively, under AM1.5 G illumination. As shown in

Figure 3, the analysis achieves good agreement, with an error

of Jsc and Voc equal 3.7% and 1.7%, respectively. Basically,

the calculations assume ideal solar cell with zero series resist-

ance, infinite shunt resistance, and zero reflection that surely

have other non-ideal values in such experimental work. On

the other hand, the non-ideal parameters were assumed; series

resistance equals Rs¼ 20 X, shunt resistance equals Rsh¼ 20

TX, surface recombination velocity equals Sp¼ 3� 103 cm/s,

and reflectivity of normal incident light between air/SiO2

Ref ¼ n2�n1

n2þn1

� 2.22 In this case, error of Jsc decreased to 0.03%

and that of Voc decreased from to 1.3%.

Another comparison was accomplished with previously

published numerical study of radial Si p-i-n junction that

used a simulator built on COMSOL Multiphysics.5 We used

total radius, p-type radius, intrinsic thickness, and n-type

thickness equal 120 nm, 80 nm, 20 nm, and 20 nm, respec-

tively. The length of the NW equals the diffusion length, and

the doping concentration equals 1018 cm�3. The J-V charac-

teristic under dark conditions for our analytical work and the

previously published numerical work is shown in Figure 4.

A good match is achieved between both results. Figure 5

shows the J-V characteristic under AM1.5 G illumination

with a good match between our results and the published

one. The difference in Jsc and Voc is calculated to be 4.1%

and 0.5%, respectively.

IV. CONCLUSION

In the present study, an analytical model for the radial p-n

junction nanowires solar cells was successfully established.

We solved analytically the semiconductor equations, namely

Poisson’s equation and the continuity equations for nanorods

p-n junction PV solar cells. This analytical solution is imple-

mented using Green’s function theory that helps in decreasing

the dependence on many assumptions used in previous con-

ventional studies.

Afterwards, a detailed study for Si-NWs solar cells was

presented in a good agreement with previously published ex-

perimental and numerical results.

A detailed study of different nanorods materials under

different conditions of operation could take place that

reflects the advantage of our analytical models to character-

ize any radial p-n junction solar cell. In further work, this

study can be extended using different ways of contacts/semi-

conductor connections on the advances in fabrication of such

nanorods PV cells.

APPENDIX A: SOLVING POISSON’S EQUATION

Assuming homojunction case, the Poisson’s equation

will be as follows:

r2 w ¼ � q�; (A1)

FIG. 3. J-V characteristics of Si-NWs solar cell resulted from the present

simulator without (solid) and with fitting parameters (dots) compared with

experimentally fabricated Si-NWs (*) under AM1.5 G illumination.

FIG. 4. J-V characteristics of Si-NWs solar cell resulted from the present

simulator (solid) and COMSOL simulator (*) under dark condition.

FIG. 5. J-V characteristics of Si-NWs solar cell resulted from the present

simulator (solid) and COMSOL simulator (*) under AM1.5 G illumination.

024308-5 Ali et al. J. Appl. Phys. 116, 024308 (2014)

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where w is the potential, q is the charge density in each

region, and e is the permittivity of the material of the rod.

At r4< r<Rp (in the p-type depletion region),

r2 w ¼ � q �Nað Þ�

: (A2)

So, we obtain

wp ¼ qNa

2�

R2p � r2

4

2� r2

4lnRp

r4

!: (A3)

Similarly in n-type region, we obtain

wn ¼ qNd

2�

R2p � Rp þ Dr2ð Þ2

2� Rp þ Dr2ð Þ2ln

Rp

Rp þ Dr2

!:

(A4)

Then, we can get the radii from the relation

wo ¼ wp þ wn ¼ Vb � V; (A5)

where Vb is the built-in potential which, for non-degenerate

semiconductors, equals ¼ Vt ln NdNa

ni2 ;15

APPENDIX B: DERIVING THE CONCENTRATIONEXPRESSION FROM GREEN’S FUNCTION THEORY

@2

@r2þ 1

r

@

@r–

1

L2n

!G1 r; r0ð Þ ¼ 1

rd r � r0ð Þ; (B1)

1

r

@

@rr@

@r–

1

L2n

� �Dn ¼ �go1; (B2)

@

@r0r0@

@r0Dn –

r0

L2n

Dn ¼ �r0go1: (B3)

Multiply Eq. (B3) by G1(r,r0), then integrate w.r.t. dr0

ðdr0G1 r; r0ð Þ @

@r0r0@

@r0Dn r0ð Þ �

ðdr0 r0 G1 r; r0ð ÞDn r0ð Þ

L2n

¼ð

dr0 r0 �go1ð ÞG1 r; r0ð Þ: (B4)

Integrate the first term by parts, we get

G1 r; r0ð Þ r0@

@r0Dn r0ð Þjsurf ace –

ðdr0

r0@

@r0Dn r0ð Þ

� �@G1 r; r0ð Þ

@r0;

(B5)

G1 r; r0ð Þ r0@

@r0Dn r0ð Þjsurf ace –

r0@G1 r; r0ð Þ@r0

� �Dn r0ð Þjsurf ace

þð

Dn r0ð Þ @@r0

r0@

@r0G1 r; r0ð Þ dr0: (B6)

Substitute the first term from Eq. (B6) into Eq. (B4), we

get

ðDn r0ð Þ @

@r0r0@

@r0G1 r; r0ð Þ � r0G1 r; r0ð Þ

L2n

" #dr0 ¼ Dn rð Þ

¼ð

dr0 r0 �go1ð ÞG1 r; r0ð Þ –G1 r; r0ð Þr0 @Dn r0ð Þ@r0

����surf ace

þ r0@G1 r; r0ð Þ

@r0Dn r0ð Þjsurf ace: (B7)

APPENDIX C: DERIVATION OF GREEN’S FUNCTION INP-TYPE REGION

Assuming that G1(r,r0) is the green’s function that satis-

fies the same equations and boundaries of DnðrÞ, but with

point source located at r0.

@2

@r2þ 1

r

@

@r–

1

L2n

!G1 r; r0ð Þ ¼ 1

rd r � r0ð Þ: (C1)

So, when r 6¼ r0, the equation will be homogeneous, and it

will be at the same form of modified Bessel function equa-

tion of order zero.

Suppose that n1r

Ln

� �is some linear combination of Io

and Ko, which satisfy the boundary conditions for r< r0, and

suppose that n2r

Ln

� �is another linearly independent combina-

tion for r> r0.The symmetry of Green’s function in r and r0 requires that

G1 r; r0ð Þ ¼ n1

r <

Ln

� �n2

r >

Ln

� �: (C2)

Although G1 is continuous at r¼ r0, but the derivative is not.

So, to get the discontinuity of the slope, integrate Eq. (C1)

with respect to r from (r¼ r0 � e) to (r¼ r0 þ e) with e repre-

sents a very small positive value.

dG1

dr

����r0þe

� dG1

dr

����r0�e

¼ 1

r0: (C3)

Also, we can get the slope discontinuity from Eq. (C2) as

follows:

dG1

dr

����r0þe

� dG1

dr

����r0�e

¼ 1

Lnn1n2

0 � n0

1n2

�����r¼r0

¼ 1

LnW n1; n2½ �; (C4)

where n0 xð Þ ¼ dndx and W½n1; n2� is the Wronskian of n1ðxÞ and

n2ðxÞ.From the boundary conditions at the p-type region of

0< r< r4, G1 should be as follows:

G1 r;r0ð Þ¼const Ior<

Ln

� �Io

r>

Ln

� �–

Io b1ð ÞKo b1ð Þ

:Kor>

Ln

� �( )" #:

(C5)

Then, we get the constant value from the slope discontinuity,

and the fact that Wronskian term is proportional to (1/x) for

all values of x, so, we can use the limiting forms of modified

Bessel function for small and large x.17

024308-6 Ali et al. J. Appl. Phys. 116, 024308 (2014)

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41.36.121.6 On: Fri, 11 Jul 2014 14:17:58

W Im xð Þ; Km xð Þ½ � ¼ � 1

x; (C6)

const ¼ Ko b1ð ÞIo b1ð Þ

: (C7)

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41.36.121.6 On: Fri, 11 Jul 2014 14:17:58