SWNT Photovoltaic Cell

8/9/2019 SWNT Photovoltaic Cell

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IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 8, NO. 3, MAY 2009 303

Assessment of Optical Absorption in CarbonNanotube Photovoltaic Device

by Electromagnetic TheoryChangxin Chen, Liu Yang, Yang Lu, Gaobiao Xiao, and Yafei Zhang

AbstractAn electromagnetic (EM) scattering model is builtfor a kind of single-walled carbon nanotube (SWCNT) photo-voltaic device excited by light. In this model, the exciting lightis treated as classical EM wave with a very high frequency, and theSWCNTs in the device were treated as a lossy dielectric cylin-der with frequency-dependent complex permittivity. Based on theEM scattering model, the FoldyLax multiple-scattering equationfor the SWCNT cylinders can be derived, and then, the absorbedpower of SWCNTs can be estimated. We also use EM simulationsoftwarehigh frequency structure simulator (HFSS)to extract

the optical absorption of SWCNTs, and then the property of opti-cal absorption of the device is studied more carefully; and the EMscattering model is also validated through HFSS simulation. Fromthe results, some advices are given for the design of such kind ofdevice.

Index TermsElectromagnetic (EM) scattering, high frequencystructure simulator (HFSS), optical absorption, photovoltaic de-vice, single-walled carbon nanotube (SWCNT).

I. INTRODUCTION

SEMICONDUCTING single-walled carbon nanotubes

(SWCNTs) are potentially an attractive material for photo-

voltaic applications [1] due to their many unique structure andelectrical properties. They have a tunable direct bandgap, which

Manuscript received August 5, 2008; revised November 1, 2008. First pub-lished December 12, 2008; current version published May 6, 2009. This workwas supported in part by the National Natural Science Foundation of Chinaunder Grant 60807008, by Shanghai-Applied Materials Research and Develop-ment Fund under Grant 08520741500, by Specialized Research Fund for theDoctoral Program of Higher Education (SRFDP) under Grant 200802481028,by Shanghai Science and Technology under Grant 0752nm015, by the Na-tional Natural Science Foundation of China under Grant 50730008, and by theNational Basic Research Program of China under Grant 2006CB300406. Thereview of this paper was arranged by Associate Editor H. Misawa.

C. Chen and Y. Zhang are with the National Key Laboratory ofNano/Microfabrication Technology, Key Laboratory for Thin Film and Mi-

crofabrication of the Ministry of Education, Research Institute of Micro/NanoScience and Technology, Shanghai Jiao Tong University, Shanghai 200240,China (e-mail: [email protected]; [email protected]).

Y. Lu was with the National Key Laboratory of Nano/Micro FabricationTechnology, Key Laboratory for Thin Film and Microfabrication of the Min-istry of Education, Research Institute of Micro/Nano Science and Technology,Shanghai Jiao Tong University, Shanghai 200240, China. He is nowwith RsicshInstrument, Shanghai 201201, China.

L. Yang was with the School of Electronic, Information and Electrical En-gineering, Shanghai Jiao Tong University, Shanghai 200240, China. He isnow with Louisiana State University, Baton Rouge, LA 70803 USA (e-mail:[email protected]).

G. Xiao is with the School of Electronic, Information and Electrical En-gineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNANO.2008.2010493

Fig. 1. Structure of CNT photovoltaic device. A directed array of monolayer

SWCNTs was nanowelded onto palladium (Pd) and aluminum (Al) electrodesthat are patterned on silicon wafers with a thermally oxidized layer. An Alelectrode is also sputtered on the back of Si substrate, through which the gatevoltage can be applied to Si substrate.

show strong photoabsorption [2][4] and photoresponse [5][9]

from ultraviolet to IR radiation. They possess high carrier mo-

bility [9] and low transport scattering [11], attributing to their

unique 1-D and almost defect-free structure. Previous study had

attempted to fabricate SWCNT films into photoelectrochemical

solar cells [12]. However,in this cell, the device structure hadnot

been well designed and SWCNTs were not separatedly aligned,

which cause the inefficient separation and collection of pho-

toexcited carriers. Thus, the obtained incident photo-to-current

conversion efficiency is low. Recently, a novel SWCNT-based

solar photovoltaic microcell had been reported [1]. As shown

in Fig. 1, in this microcell, an array of carbon nanotubes are

well aligned between metal electrodes, which are placed on the

insulating substrate. The array can be composed of individual

SWCNTs or individual bundles of SWCNTs since SWCNTs are

prone to self-assemble into bundles [13], [14]. Theelectrode ma-

terial and structure parameters can be adjusted to optimize the

device performance.

A key metric for the device is the power conversion efficiency

conv , which can be expressed as

conv =(Im Vm )

Pm=

(FFIsc Vo c )

Pm(1)

where Im and Vm are the output current and voltage of CNTwhen the power generation (Im Vm ) is at maximum, Pm is theincident power, Isc and Vo c are short-circuit current and open-circuit voltage of CNT, respectively, and FF is the fill factor

indicating the power delivery capability of the photovoltaic de-

vice, which is determined by the manufacturing process of the

photovoltaic device, such as the chirality of the CNTs, contact

resistance between CNTs and metal electrode, etc. In practi-

cal working process, the semiconducting CNTs of the device

first need to absorb the photovoltaic energy to generate the

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304 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 8, NO. 3, MAY 2009

electron-hole pairs, then convert it to electric current; thus, the

optical absorption efficiency of the device is also a very im-

portant factor to influence the power conversion efficiency. The

optical efficiency is expressed as

ab =PabPm

(2)

where Pab is the absorbed optical power. The optical absorptionefficiency ab is determined by not only the semiconductingcharacteristic of SWCNTs, but also the geometry parameter of

the device, such as the length and diameter of SWCNTs, the

distance of the nearby two SWCNTs in the parallel array, height

of the metal electrode, etc. Thus, it is necessary and important

to evaluate the influence of geometry size to ab .However, because the SWCNTs have a thickness of only

about several nanometers, only a very small fraction of inci-

dent power can be absorbed by SWCNTs. It is hard to ob-

tain the accurate value of absorbed optical power for SWCNTs

through experimental measurement. Thus,theoretical prediction

and computer simulation are useful for tackling this problem.

Since light is a kind of electromagnetic (EM) wave with very

high frequency, about several hundred terahertzs, and the optical

properties of semiconducting SWCNTs can be represented by

anisotropic, frequency-dependent, and complex dielectric func-

tion [15]; thus, EM theory and computational technology can be

used to extract the parameter of optical absorption of SWCNTs,

and then the influence of geometry size of device to ab can beanalyzed.

In this paper, an EM scattering model is built for the photo-

voltaic device, then the FoldyLaxmultiple-scattering equations

are derived according to the model, electric field in the SWCNTs

can be obtained by solving the equations, and the absorbed op-tical power can then be evaluated according to the electric field

distribution; next, the EM numerical software high frequency

structure simulator (HFSS) is used to simulate the EM scat-

tering properties for the device model. This paper is organized

as follows. In Section II, a convenient simplified EM scatter-

ing model is built for the device, and the FoldyLax multiple-

scattering equations are derived and solved. In Section III, the

software HFSS was used to validate the scattering model and the

optical absorption properties of the device is investigated more

carefully. Finally, concluding remarks are made in Section IV.

II. EM SCATTERING MODEL FOR CNT PHOTOVOLTAIC DEVICE

The optical absorption of carbon nanotubes in nature involves

the electronphoton interaction and dynamics of excitons. There

are several works using the first principle calculation to eval-

uate the optical response of carbon nanotubes [21], [22]. The

approach used in these works is solving quantum mechanical

equations to define electronic and photonic coefficients [23].

However, only incident field is considered in these calculations.

The scattered fields by electrode and nanotube array are ne-

glected. For our device, the span of the carbon nanotubes is in

the micrometer scale, which is comparable to the wavelength

of light. This means that the propagation and distribution of

optical field will be greatly influenced by the detailed device

Fig. 2. Significance of the equivalent dielectric function of CNTs.

structure. To improve the performance of CNT photoelectric

devices, this factor is important and need to be considered. In

order to carefully evaluate all scattered fields, an EM scattering

model needs to be built for our CNT photovoltaic device. With

the EM scattering model, the influence of the device structures

on the absorption of CNTs can be well reflected. In the model,

the carbon nanotube is treated as lossy dielectric cylinder, which

hides the internal quantum process behind a priori knowledge

of different dielectric functions. This approximation enables us

to carry out calculation on the level of classical EMs.In EM theory, a materials response to EM field is character-

ized by its dielectric function. Previous several works have stud-

ied the dielectric function of SWCNTs, which are frequency-

and polarization-dependent [15], [24][26]. In the EM scatter-

ing model for the device shown in Fig. 1, the SWCNTs are

treated as lossy dielectric cylinders with a frequency-dependent

and complex permittivity [15] ofp = r () + ii (); it shouldbe noted that the equivalent dielectric function p of CNT playsa key role in this EM modeling process. The significance of

p is shown in Fig. 2. The two contact electrodes are treatedas electric conductor; the exciting light is treated as common

EM wave with frequency as high as visible light, about severalhundred terahertzs.

In order to estimate the optical absorption of the SWCNT in

the photovoltaic device, we need to calculate the internal electric

filed of SWCNT. For convenience, a simplified model for the

device shown in Fig. 1 will then be built for EM scattering

analysis.

According to the structure of the photovoltaic device, some

simplification for the device can be made as follows.

1) Ignore the effect brought by the multilayered substrate

media.

2) The two contact metal electrodes are treated as infinite

plane waveguide.

3) The SWCNT array is treated as infinite periodic array.

Then, we can get a simplified model for EM scattering anal-

ysis, as shown in Fig. 3. After the previous simplification, the

problem becomes multiple scatteringby lossydielectric cylinder

array placed inside a parallel plate waveguide. The problem is

conveniently formulated in terms of the dyadic Greens function

of vector cylindrical waves and expressed in terms of waveguide

modal solutions [16]. In the following, a procedure is presented

for formulating FoldyLax multiple-scattering equations [17]

of the problem using dyadic Greens function.

Number the cylinders in Fig. 3. The distance between cylin-

der j, j = 1, 2, . . . , and cylinder 0 is jD, and the positive and

negative symbols of j represent the two opposite directions of



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CHEN et al.: ASSESSMENT OF OPTICAL ABSORPTION IN CARBON NANOTUBE PHOTOVOLTAIC DEVICE BY ELECTROMAGNETIC THEORY 305

Fig. 3. Simplified model for the CNT photovoltaic device, a parallel periodicdielectric SWCNT cylinder array placed inside an infinite PEC waveguide. Thediameter of each cylinder is a, the length of each cylinder is d, and the periodicdistance of cylinder array is D. Each cylinder is of permittivity p ().

cylinder j, j = 1, 2, . . . , displaced from cylinder 0. The FoldyLax multiple-scattering equation states that the field exciting

cylinder l is the sum of incident wave and scattered waves fromall cylinder j, except j = l.

Since the SWCNT array is treated as infinite periodic cylinder

array, each cylinder has the same exciting wave and internalfield; therefore, it is enough to calculate the internal field of

cylinder 0 in the array. To determine the exciting field of cylinder

0 and scattered field from other cylinders, then formulate the

FoldyLax multiple-scattering equation for cylinder 0, we can

use the following procedure, which is trivial and can be found

in [15] and [16].

1) Write down expression of the internal field for cylinder

j = 0 in terms of waveguide modal solutions expressedwith vector cylindrical waves with unknown coefficients.

2) Use Greens function to find the scattered field from cylin-

derj. TheGreens function is expressed in terms of waveg-uide modal solutions expressed with vector cylindrical

waves. It should be noticed that the vector cylindrical

waves are centered at rj , which is the coordinate of thecenter of cylinder ofj.

3) Use vector translation addition theorem [18] to express

the vector cylindrical waves centered at rj .4) Equate exciting field of cylinder 0 to the incoming

wave that includes incident and scattered fields from all

cylinders j, j = 1, 2, . . . , from step 3, then obtain self-consistent FoldyLax multiple-scattering equation.

In waveguide, magnetic dyadic Greens function is more

widely used; here, we first calculate the internal magnetic field

of cylinder step by step, and the internal electric field can be

obtained readily from the internal magnetic field.

Fig. 4. Translation addition theorem in the cylindrical coordinate system.

Step 1): In the coordinate system shown in Fig. 4, internal

field of cylinder j has the following form:

Hjint (r ) =

+n=

kz

cTMn RgH

TMn

k , kz , j , z + d

2

+ cTEn RgHT En

k , kz , j , z + d

2

(3)

jp Ejint (r )

= jp+

n=

kz

cTMn RgE

T Mn

kp , kz , j , z + d

2

+ cTEn RgETEn

kp , kz , j , z + d

2

(4)

where kp =

k2p k2z , kp =

0 0 r ,and p is theequiva-

lent dielectric function of SWCNT cylinder; and cTMn , and cTEn

are unknown internal field coefficients to be determined self-

consistently.

In (3) and (4), the symbols of RgHTM (E)n and RgE

T M (E)n

represent the waveguide modal solutions in terms of cylindrical

waves (see the Appendix) for magnetic field and electric field.

Step 2): The scattered field from cylinder j can be obtained

from the internal field expression by applying Huygens principle

Hs(j )

(r ) =

d/ 2d/ 2

dz2

0

dj a jp [ Eint (r)

G(r , r ) + j Hint (r) G(r , r )] (5)

where a is the radius of SWCNT cylinder, and j , j are po-

lar coordinates with center at j (Fig. 4). In (5), G is the dyadicGreens function between two perfect conductors. It is conve-

nient to expand the dyadic Greens function using waveguide

modal solutions RgHTM (E)

n and RgETM (E)

n ; then, we have



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for z > z

G(r ,

r)

= 4d

n ,l

(1)n + lk2l

fl (1 + e2j kz (z

+d/ 2) )ej k z (z+d/ 2)

HTMnkl , kz l , j , z d2

Rgmn (kl , kz l , j )ejn

j4d

n, l

(1)n+ lk2l

fl (1 + e2j k z (z

+d/ 2) )ej kz (z+ d/ 2)

HTEnkl , kz l , j , z

d

2

Rgnn (kl , kz l , j )ejn

(6)

and for z < z

G(r ,

r)

=4d

n, l

(1)n + lk2l

fl (1+e2j kz (z d/ 2) )ej kz (z

d/ 2) d

HTMnkl , kz l , j , z +

d

2

Rgmn(kl , kz l , j)ejn

j4d

n, l

(1)n+ lk2l

fl (1 + e2j kz (z d/ 2) )ej kz (z

d/ 2)

HTEn

kl , kz l , j , z +d

2

Rgnn (kl , kz l , j )ejn

(7)

where the symbols of HTMn and H

T En represent the waveg-

uide modal solutions in terms of vector cylindrical waves, and

Rgmn and Rgnn represent vector cylindrical waves (see theAppendix); fl is

fl =

1

2, l = 0

1, l = 1, 2, . . . .

(8)

Using the dyadic Greens functions (6) and (7), the Foldy

Lax multiple-scattering equations can be decoupled amongthe waveguide modes, and therefore can be solved for each

mode separately. Substitute (3) and (4) into (5), we then

have

Hsj = ak

2d

n, l

(1)lk2l

fl

k z

cTE (j )n

P

nk2 kz

kk p aJn (kl a)Jn (k

p a)

+ Qnk2p kz


p

a)

+jcTM (j )n P

k2 k

p

kJn (kl a)J

n (k

p a)

rk2p k

kpJn (k

p a)J

n (kl a)

HTM

n


2

+

jcTE (j )n R

r

kp k2

kJn (kl a)J

n (k

p a)

k2

p k

kpJn (kl a)Jn (k

p a)

+ cTM (j )n

r R

nk2 kz


p a)

+jr Pnk2p kz


p a)

HTEn


2

(9)

where P, Q, and R are

P =

zd/ 2

dz2cos

kz

z +

d

2

cos

kz

z +

d

2

ej kz d

+

d/ 2z

dz2cos

kz

z d

2

cos

kz

z +

d

2

(10)

Q =zd/ 2

dz2sin

kz

z + d2

sin

kz

z + d2

ej kz d

+

d/ 2z

dz2sin

kz

z d

2

sin

kz

z +

d

2

(11)

R =

zd/ 2

dz2cos

kz

z +

d

2

sin

kz

z +

d

2

ej kz d

+

d/ 2z

dz2cos

kz

z d

2

sin

kz

z +

d

2

.

(12)

Step 3): We next use the translation addition theorem (see

Appendix B) to express vector cylindrical waves centered at rjin terms of vector cylindrical waves centered at r0 to deriveexpression of exciting field at cylinder 0 due to the scattered

field from cylinder j (Fig. 4); then, (9) can be rewritten asfollows:

Hsj = ak

2d

n, l

(1)lk2l

fl

n

k z

cT E(j )

nP

nk2 kz


p

a)



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+ Qnk2p kz


p a)

+jcT M (j )n P

k2 k

p

kJn (kl a)J

n (k

p a)

rk2p k

kp Jn (kp a)Jn (kl a)

H(1 )n n (klj )RgHT Mn

kl , kz l , , z +

d

2

+

jcTE (j )n R

r

kp k2

kJn (kl a)J

n (k

p a)

k2

p k

kpJn (kl a)Jn (k

p a)

+ cTM (j )n

r R

nk2 kz


p a)

+jr Pnk2p kz


p a)

H(1 )n n (klj )RgHT En

kl , kz l , , z +

d

2

. (13)

Step 4): To obtain self-consistent multiple-scattering equa-

tion, we expressed the exciting field of cylinder 0 in terms of

waveguide modal solutions (see Appendix A)

Hex (r ) =

+n=

kz

aTMn RgH

TMn

k , kz , , z +

d

2

+ aTEn RgHTEn

k , kz , , z + d

2

(14)

where aT Mn and aTEn are unknown coefficients. In addition, the

exciting field is also the sum of the scattered fields from cylinder

j,j = 0, and incident field, i.e.

Hex (r ) =

j= 0

Hs(j ) (r ) + Hin c (

r ) (15)

where Hs(j ) (r ) is expressed in (13) and Hin c (

r ) is the main

mode in plate waveguide

H

in c

=

n RgH

TM

n

k0 , kz 0 , j , z +d

2

.(16)

The main mode in plate waveguide is TEM mode that has

much similar propagating properties as the incident plane wave,

so it is proper to choose TEM mode in this plate waveguide of

the device model (Fig. 3) as the incident field, and such setting

is also convenient for solving the multiple-scattering equation.

By matching the coefficient of the two different expression

of exciting field (14), (15), and equating the exciting field of

cylinder 0 to its internal field at the boundary between cylin-

der and free space, we can obtain the following relationship

between the exciting field unknown coefficients aTMn and aTEn ,

and internal field coefficients cTMn and cTEn ; additionally, ac-

cording to the boundary of electric field and magnetic field at

the boundary between cylinder 0 and free space, another rela-

tionship between aTMn , aTEn and c

TMn , c

TEn can be obtained (see

Appendix A). From the earlier two relationship, we can get the

FoldyLax multiple-scattering equation for cTMn and cTEn

AM Mn cT Mn (l) + A

M En c

T En (l)

= ak 2d (1)l

k2lfl

n

k z

j=0

H(1 )nn (klj)

cT E(j )n (k

z )

Pnk

2l k

z


p a)

Q nk2

p kz l


p a)

+jc

TM (j )n (k

z )P

k2l k

pl

k

Jn (kl a)Jn (kp a) rk2p kl

kpJn (k

p a)J

n (kl a)

+ (l) (17)

where

(l) =

1, l = 00, l = 0

AEMn cTMn (l) + A

EEn c

TEn (l)

= ak 2d

(1)lk2l

fl

n

k z

j=0

H(1 )nn (kl

j )

jcTE (j )n (k

z )R

r

kp k2l

kJn (kl a)J

n (k

p a)

k2

p klkp

Jn (kl a)Jn (kp a)

+ cTM (j )n (kz )

Rr nk2

l kzkk p a

Jn (kl a)Jn (kp a) +jr Pnk2p kz l


p a)

(18)

where AM Mn , AM En , A

EMn , and A

EEn are the coupling coeffi-

cients (see Appendix A), and j = |j D|.It is convenient to solve (17) and (18) by iteration. Let

cTEn = cTE(1st)n (l) + c

TE (H)n (19)

cT Mn =TM(1st)n (l) + c

TM (H)n (20)

where superscript 1st denotes first-order solution, which repre-

sents the internal field of cylinder 0 excited directly by incident

field and h denotes higher order solution, which represents theinternal field of cylinder 0 excited by the scattered field by

cylinder j, j = 0.In this problem, since the polarization of incident electric field

is parallel to the axis of the cylinder, and thus the coefficient cTEndiminished, and (17) and (18) become more concise.

For considering the influence of geometry structure to

absorbed power, first-order and second-order solutions of

internal field coefficients are sufficient. By substituting (19)

and (20) into multiple-scattering equations (17) and (18),

and matching the coefficients before (l), we can obtain the



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Fig. 5. Absorbedpower ofeach SWCNTfor parallelpolarized incidentlightasD varies. The parameters are f = 300THz, p = (20 + i20)0 , d = 500 nm,and a = 15 nm.

first-order solutions of the internal field. The second-order

solutions for the internal field coefficients are obtained by

solving (17) and (18) with the first-order solutions substituted

into the summation items of (17) and (18). Then, the internal

electric field can be obtained by the superposition of electric

waveguide model solutions with coefficient cTEn

Eint (r ) =

+

l

cT Mn RgE

T Mn

kp , kz , , z +

d

2

(21)

where cTMn = cTM(1st)n + c

TM( 2nd)n , then the absorbed power is

Pa =

Vc n t |E2

int |dv

=

d/ 2d/ 2

dz

a0

d

20

d(0 Img(r )|Eint (r )|2 )

(22)

where = 0 Img(r ) is the conductivity of dielectric cylin-ders for SWCNTs and Vcn t is the volume of single SWCNT.

A MATLAB program is developed to solve the self-consistent

FoldyLaxmultiple-scattering equations and estimate the power

absorption. In the program, we choosethe simulation parameters

of the device as follows.

1) The radius of the SWCNT a = 30 nm, the length of theSWCNT d =500 nm.

2) The frequency of the incident wave f=300 THz.3) The equivalent dielectric function of SWCNT

r = 20 + i20 .4) The periodic distance of the parallel array in the device

model (Fig. 3) D changes from 300 nm to two or threetimes of the incident wavelength, then the influence of the

value ofD to the power absorption is investigated.The absorbed power Pa of each single SWCNT in different

periodic distance D is shown in Fig. 5 and Table I. From thefigure, we find that there are strong peaks of Pa , where D =n

, n = 1, 2, . . ., and the value of Pa is sensitive to D near

these peaks.

TABLE IABSORBED POWER OF EACH SWCNT AS D VARIES

Fig. 6. HFSS simulation model for the CNT photovoltaic device. The top faceof the whole box is assigned radiation-incident boundary condition, and the sideface is assigned masterslave periodic boundary condition; the geometry size

and dielectric constant of each component of the device could be adjusted ifneeded.

III. RESULTS OF HFSS SIMULATIONS AND DISCUSSION

In this section, a more careful investigation is made for

the influence of geometry structure of device to the absorbed

power Pa of SWCNTs using EM simulation software HFSS,and the scattering model of the device is also validated. The

HFSS simulation model for the device is shown in Fig. 6. From

the EM simulation of this model, we can get the following

results.

A. Influence of the Polarization of Incident Light on Pa

In Section II, we developed a MATLAB program to compute

the absorbed power of single SWCNT in the device when the

electric field of incident light is parallel to the tube axes; how-

ever, when the electric field of the light is perpendicular to the

tube axis, the scattered field will not be isotropic and will be

angle-dependent, and the internal field in SWCNT cylinder will

also become irregular; thus, it will be much harder to compute

Pa based on FoldyLax multiple-scattering equation. In thispart, we use HFSS to extract the absorbed power for both par-

allelly and perpendicularly polarized incident light, and some

analyses are also made.



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Fig. 7. Absorbed power of each SWCNT for parallel polarized incident light.The curve in (a)(c) shows the relationship between Pa and p () at differentvalue of a. The parameters are f = 300THz, p = (20 + i20)0 , and d =500 nm .

In Fig. 7(a)(c), we plot theabsorbedpower of singleSWCNT

for a parallelly polarized incident light for different SWCNT

diameters and dielectric constant. From these three figures, it is

not difficult to find that the absorbed power Pa does not change

Fig. 8. Absorbed power of each SWCNT for perpendicularly polarized in-cident light. The curves show the relationship between Pa and p when theradius of SWCNT a is 5, 10, and 15 nm. The parameters are f = 300THz,

p = (20 + i20)0 , and d = 500 nm .

much when the real part of dielectric function changes as the

imaginary part of dielectric function is fixed.

Actually, such effect is caused by the long-thin structure of

SWCNT. To explain this effect, we can treat the single SWCNT

as a long-thin time-harmonic electric dipole inducted by the

electric field of incident light. Such dipole does not give an in-

tensive radiation because of the long distance between the center

of the separated positive and negative polarized electric charge.

Thus, the real part of dielectric function does not influence Pa

obviously. This effect is also called polarization effect [20].From Fig. 7(a)(c), we can also find that when the diameter

of SWCNT decreases, Pa is less dependent on the real part ofdielectric function; this is because the polarization effect will

become more influential as the ratio of diameter to length of

SWCNT decreases.

In Fig. 8, we plot the absorbed power of single SWCNT for

perpendicular polarized incident light for different SWCNT di-

ameters and dielectric function. From Fig. 8, we can find that Pafor perpendicular polarized incident light is about only 1% of

the Pa for parallel polarized incident light. This phenomena canalso be explained through polarization effect, i.e., the distance

between the center of the separated positive and negative po-

larized electric charge of the dipole for perpendicular polarized

incident light is much less than Pa for parallel polarized inci-dent light, and then the radiation of the dipole for perpendicular

incident light becomes more intensive; thus, Pa decreases a lot.A more visualized understanding of polarization effect can be

got through Fig. 9(a) and (b), which are the distribution of elec-

tric field near SWCNT in parallel and perpendicular polarized

conditions, respectively. Comparing these two figures, we can

find that electric field strength near SWCNT in perpendicular

polarized condition is much less than that in parallel polarized

condition; thus, the absorbed power of SWCNT will also be

much less in perpendicular polarized condition. The simulation

results and the theoretical prediction agree well.



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Fig. 9. Image of scattering field near SWCNT for (a) parallel and (b) perpen-dicular polarized incident light. The parameters are d = 500 nm , a = 15 nm,and D = 300 nm.

B. Influence of the Period of CNT Array on Pa

From thepolarizationanalysisin Section III-A, we have found

that there is a better absorption efficiency of SWCNT in the

device in parallelly polarized condition. So, it is more important

to study the influence of geometry structure of the device to

power absorption in parallel polarized condition. In this section,

the influence of the period D of CNT array to power absorptionis studied through HFSS simulation.

As D increases, it will cost much more to compute the Pathrough full-wave simulation. In Table II, we provide the value

of Pa as D varies from 300 to 1000 nm; in this simulation,the geometry structure of the device model is consistent with

Table I. Comparing the data in Table II with that in Table I, we

can find that Pa in these two tables varies in a similar way asD increases though the values ofPa are not equivalent exactly,

and there are two peaks for Pa when D = in these two tables.

TABLE IIHFSS SIMULATION RESULTS

Fig. 10. Absorbed power for parallel polarized incident light. The curvesshow the relationship between Pa and height of electrode. The parameters aref = 300 and 600 THz, p = (20 + i20)0 , d = 500 nm, and D = 300 nm.

C. Influence of the Height of Electrode on Pa

In this section, the influence of electrode height to power

absorption is taken into account. In Fig. 10, we plot Pa forparallel polarized incident light as a function of the height of

electrode in two different incident wavelengths of 1000 and

500 nm. From Fig. 10, we can find that there is a peak of power

absorption when the height of electrode is near 150 nm. Such

effect can be explained as follows.

Fig. 11(a) shows the electric field distribution when the elec-

trode height is as low as the diameter of SWCNT, and Fig. 11(b)

shows the electric field distribution when the electrode height

is high enough. In these two figures, we find that the wave near

SWCNT is approximately plane wave, and now let us go back

to Fig. 9(a), where the electrode height is between the electrode

height in Fig. 11(a) and (b), where we can find that the elec-

tric field strength near SWCNT is intensified. This is because

in Fig. 9(a), there are some high-order modes of electric field

excited by the edge of electrode when the device is exposed to

the incident light, and these high-order modes will surely give

a positive effect to the power absorption of SWCNT. Thus, it is

totally reasonable that a peak ofPa will appear at a proper valueof the electrode height.

According to HFSS simulation results and the relevant expla-

nation presented in Section III-AC, the following short con-

clusions could be made.



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Fig. 11. Image of scattering field near SWCNT when the height of electrodeis (a) 30 and (b) 500 nm for parallel polarized incident light. The parameters aref = 300 THz, p = (20 + i20)0 , D = 300 nm, and a = 15 nm.

1) The absorbed power Pa is mainly dependent on the imag-inary part of dielectric function, and the real part of di-

electric function influences Pa much more when the in-cident light is perpendicularly polarized than when paral-

lelly polarized; additionally, the power absorption in par-

allelly polarized condition is much higher than in per-

pendicularly polarized condition. This phenomenon is

called polarization effect. Because of this effect, this

photovoltaic device could act as an optical polarization

detector.

2) Pa varies in a periodic way (Fig. 5), and there are peakswhen D is an integer multiple of the incident wavelength.In addition, the value of Pa is sensitive to the value of Dnear these peaks. Thus, this photovoltaic device could also

act as an optical wavelength detector.

Fig. 12. Absorbed powerof each SWCNT forparallelpolarized incident light.The curves show the relationship between Pa and volume of each equivalentcylinder. The parameters are f = 300 THz, p = (20 + i20) 0 , D = 300 nm,

and a = 15 nm.

3) Pa is also influenced by the height of electrode, and thereis a peak of Pa at a proper value of electrode height.Thus, the photovoltaic device can achieve a better optical

absorption by adjustment of electrode height.

4) In Fig. 12, we plot the absorbed power Pa as a func-tion of the diameter of SWCNT in different dielectric

constants, and there is an approximately linear relation-

ship between Pa and square of diameter. It implies thatthe electric field strength in SWCNT should be approx-

imately uniform, which means that the exciting field toSWCNT is approximately uniform. Thus, it is reasonable

to choose TEM mode in plate waveguide as Hin c in(15) inSection II.

IV. CONCLUSION

In this paper, we develop an EM scattering model to an-

alyze the optical absorption of dispersively aligned CNT ar-

ray in photovoltaic device. The optical absorption of CNTs

in the device depends on the characteristic of CNTs, the po-

larization of incident light, and the geometry structure of the

device, such as the alignment period of CNT array and the

height of metal electrodes; through EM modeling and numer-

ical simulation, the optical absorption of the device is studied

carefully. According to the simulation results, some advices

are also proposed for the design of such kind of photovoltaic

device.

A more accurate model may include both quantum mechan-

ical treatment of carbon nanotubes and classical description of

scattered optical field. To do this, one needs to couple quantum

mechanical equations self-consistently with classical electrody-

namics equations. This proposes additional complexity in mod-

eling work and simulation techniques, but will be definitely an

interesting topic in future researches.



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APPENDIX A

VECTOR CYLINDRICAL WAVES, WAVEGUIDE MODAL

SOLUTIONS, AND COUPLING COEFFICIENTS

Vector cylindrical waves are useful when considering scat-

tering from one cylinder to another in multiple-scattering prob-

lem. The expressions of vector cylindrical waves and waveguide

modal solutions in terms of vector cylindrical waves used in thispaper are listed here.

For the scalar wave equation, the solution in a cylindrical

coordinate systemr = (,,z) that is regular at origin is

Rgn (k , kz ,r ) = Jn (k )e

ik z z + in , n = 0,1,2, . . .(23)

where Jn is a Bessel function of order n and Rg stands forregular. The outgoing cylindrical wave is

n (k , kz ,r ) = H(1 )n (k )e

ik z z + in (24)

where H(1 )n is a Hankel function of the first kind. The vector

cylindrical wave functions are Ln , Nn , and Mn , where

RgLn (k , kz ,r ) = n (k , kz , r ) (25)

RgMn (k , kz ,r ) =

1

k RgNn (k , kz , r ) (26)

RgNn (k , kz ,r ) = [z n (k , kz , r )]. (27)

When there is no Rg sign, then the Besell function is to be

replaced by a Hankel function of the first kind. Note that RgMnand RgNn satisfy the vector wave equation, while RgLn is curlfree and can only contribute to the field in the source region;

thus, RgMn and RgNn are enough to expand the field whiletackling the problem in Section II, and the expression is

RgMn (k , kz ,

r ) = Rgmn (k , kz ,

)eik z z +in (28)

RgNn (k , kz ,r ) = Rgnn (k , kz ,

)eik z z +in (29)

with

Rgmn (k , kz ,) =

in

Jn (k ) k Jn (k ) (30)

Rgnn (k , kz ,) =

ik kzk

Jn (k ) nkzk

k

Jn (k ) + zk2k

Jn (k ). (31)

The magnetic field wave and electric field modal solutions

are defined as follows:

RgHTEn (k , kz ,

, z) =

1

2[eik z a Rgnn (k , kz ,

)

eik z a Rgnn (k ,kz , )]ein (32)

RgHTMn (k , kz ,

, z) =

i2

[eik z a Rgmn (k , kz ,)

+ eik z a Rgmn (k ,kz , )]ein (33)

RgETEn (k , kz ,

, z) =

i

RgHTEn (k , kz , , z) (34)

RgE

TM

n (k , kz ,

, z) =

i

RgHTM

n (k , kz ,

, z) (35)

where is the intrinsic impedance in dielectric media.According to the boundary conditions of the continuity of

Eand Eat = a, we have the coupling equation[19] between exciting field coefficients aT Mn , a

TEn , and internal

field coefficients cTMn , cTEn

aTM

n

= AM M

n

cTM

n

+ AM E

n

cTE

n

(36)

aTEn = AEMn c

TMn + A

EEn c

TEn (37)

where the coupling coefficients are

AM Mn =j a

2

kp J

n (kp a)Hn (k a)

k2

p

kJn (kp a)H

n (k a)

(38)

AM En =j a

2

nkzkp a

Jn (kp a)Hn (k a)

nkz k

2

pkp k2 a

Jn (kp a)Hn (k a)

(39)

AEMn =j a

2

nkzka

Jn (kp a)Hn (k a)

nkz k2

p

kk2 aJn (kp a)Hn (k a)

(40)

AEEn =j a

2

kpk

kp Jn (kp a)Hn (k a)

kk2

p

kp kJn (kp a)H

n (k a)

. (41)

APPENDIX B

VECTOR CYLINDRICAL WAVES TRANSLATION

ADDITION THEOREM

The vector cylindrical waves centered atrj can be expressed

in terms of vector cylindrical waves centered atr l . This vector

translation addition theorem is applied to derive the expression

of exciting field at cylinder l due to the scattered field from cylin-der j in Section II. We make use of scalar translation additiontheorem [18], and use Fig. 4

RgMm (k , kz ,r rj )

=+

n=Jnm (k |j l |)eik z (zj z l )in (nm )j l

RgMn (k , kz , r r l ) (42)RgNm (k , kz ,

r rj )

=+

n=Jnm (k |j l |)eik z (zj z l )in (nm )j l

RgNn (k , kz , r r l ). (43)For outgoing vector cylindrical waves, similar relations hold;

however, there are two expressions depending on the relative



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magnitudes of|j l | and | l |. Since we need the excitingfield of cylinder l with

r slightly outside cylinder l, we have

Mm (k , kz ,r rj )

=

+

n =

H(1 )nm (k |j l |)eik z (zj z l )in (nm )j l

RgMn (k , kz , r r l ) (44)Nm (k , kz ,

r rj )

=+

n =H

(1 )nm (k |j l |)eik z (zj z l )in (nm )j l

RgNn (k , kz , r r l ) (45)for |j l | > | l |.

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Changxin Chen received the Ph.D. degree in micro-electronics and solid-state electronics from ShanghaiJiao Tong University, Shanghai, China, in 2007.

He is currently an Assistant Professor with theNational Key Laboratory of Nano/MicrofabricationTechnology, Key Laboratory for Thin Film and Mi-crofabricationof the Ministry of Education,ResearchInstitute of Micro/Nano Science and Technology,Shanghai Jiao Tong University. His current researchinterests include carbon nanotube electronics and op-toelectronics, novel semiconductor or nanodevices,

and nanofabrication and nanoassembly. He is the author or coauthor of morethan 20 papers in scientific journals. He is the holder of several patents in themicroelectronics area.

Liu Yang received the B.S. degree in electronic en-gineering and mathematics, and the M.S. degree inelectromagnetics from Shanghai Jiao Tong Univer-sity, Shanghai, China, in 2005and 2008, respectively.He is currently working toward the Ph.D. degree inelectrical engineering at Louisiana State University,Baton Rouge.

His current research interests include computa-tional electromagnetics, nanophotonics, and theoryand simulation of plasmonic devices.



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Yang Lu received the B.S. degree in electrical andcomputer engineering and the M.S. degree in micro-electronics and solid-state electronics from ShanghaiJiao Tong University, Shanghai, China, in 2005 and2008, respectively.

He is currently a Device Engineer with Rsicsh In-strument, Shanghai. His current research interests in-clude the theory andmodelingof nanoscaleelectronicdevices, especially carbon-nanotube-based field ef-fect transistors, photonic devices, and developmentof algorithms for semiconductor characterization

instruments.

Gaobiao Xiao, photograph and biography not available at the time ofpublication.

Yafei Zhang received the B.Sc., M.S., and Ph.D. de-grees in condensed physics from Lanzhou Universityof China, Lanzhou, China, in 1982, 1986, and 1994,respectively.

From 1996 to 1998, he was a Research Scien-tist and a Visiting Professor at the Centre of Super-Diamond and Advanced Films, City University ofHong Kong. From 1998 to 2001, he was a SeniorResearch Scientist at Japan National Institute for Re-search in Inorganic Materials. In 2001, he joinedthe Shanghai Jiao Tong University, Shanghai, China,

where he established a research team in the field of nanomaterials and nano-electronics, and is currently the Chair Professor of nanomaterials and nanoelec-tronics in the National Key Laboratory of Nano/Microfabrication Technology,Key Laboratory for Thin Film and Microfabrication of the Ministry of Edu-cation, Research Institute of Micro/Nano Science and Technology. His currentresearch interestsincludenanoelectronic devicesby nanomaterialsarrangement,single-wall carbon nanotubes and related sensors, nanoireland single electronicdevices,mutibarrier electron tuning devices,SiC nanowhiskers and related func-tional devices, Si nanowires, and photocatalytic films of TiO2. He is the authoror coauthor of more than 150 papers. He is the holder of 30 patents.

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SWNT Photovoltaic Cell