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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
kk p aJn (kl a)Jn (k
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
kl , kz l , j , 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
kk p aJn (kl a)Jn (k
p a)
+jr Pnk2p kz
kk p aJn (kl a)Jn (k
p a)
HTEn
kl , kz l , j , z +d
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
kk p aJn (kl a)Jn (k
p
a)
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+ Qnk2p kz
kk p aJn (kl a)Jn (k
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
kk p aJn (kl a)Jn (k
p a)
+jr Pnk2p kz
kk p aJn (kl a)Jn (k
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
kk p aJn (kl a)Jn (k
p a)
Q nk2
p kz l
kk p aJn (kl a)Jn (k
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
kk p aJn (kl a)Jn (k
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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310 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 8, NO. 3, MAY 2009
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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CHEN et al.: ASSESSMENT OF OPTICAL ABSORPTION IN CARBON NANOTUBE PHOTOVOLTAIC DEVICE BY ELECTROMAGNETIC THEORY 311
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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312 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 8, NO. 3, MAY 2009
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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CHEN et al.: ASSESSMENT OF OPTICAL ABSORPTION IN CARBON NANOTUBE PHOTOVOLTAIC DEVICE BY ELECTROMAGNETIC THEORY 313
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.