surface-plasmon enhanced near-bandgap light absorption in silicon photovoltaics

Delivered by Ingenta toRice University Fondren Library

IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE

Copyright copy 2008 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol 5 2096ndash2101 2008

Surface-Plasmon Enhanced Near-Bandgap LightAbsorption in Silicon Photovoltaics

Lu Hu Xiaoyuan Chen and Gang Chenlowast

Massachusetts Institute of Technology Cambridge MA 02139 USA

An extended Mie scattering theory is used in this paper to analyze the surface-plasmon enhancedlight absorption in silicon for photovoltaic applications The calculation results show that the opti-cal absorption in silicon can be enhanced up to 50 times at resonance frequency by embeddedspherical silver nanoparticles due to the local field enhancement by surface plasmons The analysisreveals that the surface-plasmon field is concentrated in a spherical shell that encloses the particleThe enhancement reaches maximum when the thickness of the shell is 026 times the radius ofthe particle The maximum absorption enhancement is found to be induced by silver particles withintermediate radii The enhancement with larger particles is limited by the retardation effect whilethe smaller-particle assisted absorption is hampered by electron scattering on the particle surface

Keywords Surface Plasmon Solar Photovoltaics Silver Nanoparticle Mie Scattering

1 INTRODUCTION

One key challenge for silicon-based solar cells is theweak absorption of long-wavelength photons near thebandgap (11 eV) due to the indirect bandgap of silicon12

A large fraction of the AM 15 solar spectrum falls intoa regime (07 mndash11 m) where silicon does not absorblight well3 The capture of these long-wavelength photonsimposes a particular problem to the thin-film silicon solarcells For this reason thin-film silicon solar cells oftenincorporate some forms of light trapping mechanisms

As the optical absorption rate in a lossy mediumdepends on the optical path length and the local electricfield intensity two different light trapping strategies havebeen developed The most widely used strategy for lighttrapping is applying complex textures on the back or thefront surfaces of the cells4 The surface textures diffusethe incoming light and enhance the absorption by increas-ing the optical path length of the photons It was not untilrecently that another light trapping strategymdashincreasingthe local field intensity via surface-plasmons using metalnanoparticlesmdashstarted to receive attentions5ndash8 Plasmonsare collective oscillations of the conduction band elec-trons excited by incident electromagnetic waves9 Surfaceplasmons are bounded to the interface when the dielec-tric constants of the two sides satisfy certain conditionsdepending on the geometry10 These localized surface

lowastAuthor to whom correspondence should be addressed

modes are supported in noble metals such as gold andsilver It has long been observed that surface plasmonscan lead to significant electric field enhancement near thesurface and thus increase the rate of any processes thatdepend on the field intensity One well-known exampleis the surface-plasmon enhanced Raman scattering wherethe Raman signals proportional to the fourth-power ofthe electric field can be increased many orders of magni-tude using metallic nanoparticles such as silver11 Similarmechanisms can be applied to photovoltaic cells for lighttrapping Instead of using surface textures noble metalparticles are either placed on the surface as in the caseof siliocn56 or embedded in the active regions as in thecase of polymer-based solar cells7 and dye-sensitized solarcells8 as localized photon absorption centers Experimen-tal results show considerable increase of photo currentsespecially in the long wavelength regime5ndash7

Despite the reported enhancement to our best knowl-edge detailed analyses of the surface-plasmon enhancedoptical absorption are lacking for photovoltaic applica-tions Past research efforts rely on either simple dipolemodels or numerical simulations Catchpole et al useda dipole scattering model which does not apply to largeparticles12 Derkacs et al performed finite-element elec-tromagnetic simulations for a single gold particle at afixed wavelength5 Rand et al analyzed the field enhance-ment using two-dimensional finite-element simulations7 Inthis paper we take an analytical approach and study theabsorption enhancement by silver nanoparticles embedded

2096 J Comput Theor Nanosci 2008 Vol 5 No 11 1546-1955200852096006 doi101166jctn20081103


IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE

Hu et al Surface-Plasmon Enhanced Near-Bandgap Light Absorption in Silicon Photovoltaics

in silicon using an extended Mie scattering theory for asphere embedded in absorbing media1314 The extendedMie scattering theory takes into account the near field com-ponents without making any far field approximations15

which ensures the localized surface modes can be repre-sented in the model A single particle analysis is performedas it reveals fundamental mechanisms of the absorptionenhancement although in real applications multiple metalnanoparticles are deployed instead of a single particle Inphotovoltaic applications the metal articles are usually ran-domly distributed and the primary enhancement mecha-nism is the localized surface modes which are less affectedby the incoherent scattered light from the randomly spacedneighbors Therefore the results obtained from the para-metric study of a single particle can serve as the first-orderinput parameters for the design of more complex struc-tures involving multiple particles In our calculations wechoose silver nanoparticles as the photon absorption cen-ters because the plasma frequency of silver is the lowestamong noble metals16 which is ideal for solar photovoltaicapplications Our calculation shows that with the presenceof a silver nanoparticle the net absorption enhancement insilicon can be up to 50 times in the long wavelength regimeThe dependence of absorption enhancement on the particlesizes shows a non-monotonic trend Maximum enhance-ment is found for particles with intermediate sizes Largerparticles lead to lower enhancement due to the retardationeffect of the electromagnetic waves Particles with radiismaller than the optimized radius give smaller enhance-ment which is due to the size effects caused by boundaryscattering of electrons In addition to the extended Mie the-ory we present simplified models for absorption analysisbased on dipole models which is found to be good approx-imations for particles with small radii Our analysis alsosuggests that the absorption by the silver nanoparticle itselfis significant which should be addressed in the future

2 EXTENDED MIE SCATTERING THEORY

Figure 1 shows a schematic illustration of the presentanalysis The gray sphere represents a silver particle witha radius of a Region 1 is inside the silver sphere andregion 2 is in the host absorbing medium ie silicon Theangle is the zenith angle and is the azimuth angleAn imaginary sphere with a radius of R is also drawn aswe will analyze the absorption in silicon within this radiusA plane wave is propagating in the positive z directionWithout the loss of generality we choose the electric fieldof the plane wave to be polarized in the x direction Theelectric and magnetic fields of the incident plane wave aregiven by

Ei = E0e

jk2r cosex (1)

Hi =

k2

2

E0ejk2r cosey (2)

ki

x

y

a

R

r

θ

Φ

1

2

z

Fig 1 Schematic drawing of a silver particle (region 1) embedded insilicon (region 2) The gray sphere is the silver particle which is enclosedby an imaginary spherical shell

where E0 is the amplitude of the electric field at z= 0k2 the wave vector of the incident plane wave r theradial coordinate the angular frequency 2 the per-meability j the imaginary unit ex the unit vector in thex direction and ey the unit vector in the y directionThe wave vector is given by k2 = 2N2 where N2 isthe optical constant of silicon and is the wavelength invacuum Since neither silver nor silicon is magnetic mate-rial we have 1 = 2 = 0 where subscripts 1 2 and 0denote regions 1 2 and vacuum Following procedures inRef [13] the scattered field can be expanded into vectorspherical harmonics

Es =

sumn=1

En(jan

N3e1nminusbn

M3o1n

)(3)

Hs =

k2

0

sumn=1

En(jbn

N3o1n+an

M3e1n

)(4)

whereM and

N are vector spherical harmonics En =

jnE02n+1nn+1 an and bn are the expansion coef-ficients and the subscripts e and o denote even and oddThe definitions of the vector spherical harmonics and thederivations of the expansion coefficients are not given heresince Ref [13] contains detailed information Note thatthe scattered field and the expansion coefficients derivedin Ref [13] can be directly applied to the absorbinghost media by treating the optical constant of the hostmedia as complex number We can now define the fieldenhancement by

r= E2

r

Eir

= Eir+Es

r

Eir

(5)

where r gt a

J Comput Theor Nanosci 5 2096ndash2101 2008 2097


IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE

Surface-Plasmon Enhanced Near-Bandgap Light Absorption in Silicon Photovoltaics Hu et al

The rate of energy absorption in a sphere with radiusR is obtained by integrating the Poynting vector over thesurface of the sphere14

WabsR=E020k22

Reklowast2

sumn=1

2n+1

times jlowastn

primenminus jn primelowast

n + jbn primelowastn n+ jblowastnn primelowast

n

+ jan2 primennminus jbn2n primelowast

n

minus janlowastn

primenminus jalowastn prime

nlowastn$

(6)

where n and n are the Riccati-Bessel functions and primen

and primen are their derivatives The symbol Re means thereal component The superscript lowast denotes the complexconjugate Note that the absorption rate given in Eq (6)differs from the standard Mie scattering theory13 in termsof additional cross terms due to the fact that the hostmedium is absorbing For non-absorbing host media thecross terms vanish and Eq (6) goes back to the standardform By energy balance the absorption rate in the siliconspherical shell (a lt r lt R) is then given by

ampWabs =WabsRminusWabsa (7)

Note that the absorption by the silver sphere is excludedbecause the photon energy absorbed in the sphere convertsto heat and does not generate electricity

To obtain the absorption enhancement the absorptioninside homogeneous silicon host medium (without thepresence of a silver particle) needs to be calculated Inhomogeneous medium the scattering field is zero and thetotal field is identical to the incident field The absorptioninside of a spherical region with radius r is obtained byintegrating the Poyting vector over the sphere surface

WabsSi =int 2

0

int 0

12

ReEitimes

Hlowasti middotminusr r2 sincosdd

= Rek2E02r2

0

[cosh2kprimeprime2rkprimeprime2r

minus sinh2kprimeprime2r2kprimeprime2r2

](8)

where kprimeprime2 is the imaginary part of the incident wave vectorThe absorption enhancement factor is defined as the ratioof Eqs (7) and (8)

(Ra= ampWabsRa

WabsSiR(9)

The optical constants are important input parametersfor computing the electromagnetic fields For the calcu-lation of the enhancement factor we take the experimen-tally measured optical constants of silicon from Ref [17]At room temperature the mean free path of electronsin bulk silver is around 40 nm9 Since in this work thediameter of the silver particles of interest is on the sameorder or smaller than the mean free path of the elec-trons the collision of the electrons with the particle surface

becomes important as an additional relaxation process18

The dielectric function of small silver particles should thusbe modified to take account for the free electron boundaryscattering which is expressed as19

)= )bulk+2p

2 + j+bulkminus 2

p

2 + j+ (10)

where +bulk is the damping constant of bulk silver + is themodified damping constant )bulk is bulk dielectric functionof silver from Ref [17] and p is the plasma frequencyFor silver p = 140times1015 rads16 The last two terms inEq (10) represent the modification to the free electron partin the dielectric function The second term cancels the freeelectron contribution in the bulk medium while the lastterm includes electron collisions in bulk materials as wellas with the surface by modifying the damping constant

+ = +bulk +AFa (11)

where A = 1 and 1+bulk = 31 times 10minus15 s1619 and F =139times106 ms is the Fermi velocity of electrons in silver9

Equation (10) ensures that experimental data on the dielec-tric constant which includes other optical transitions canbe used directly

3 CALCULATION RESULTS ANDDISCUSSION

The electric field of surface plasmons is known to con-centrate only near the surface and decays rapidly whenmoving away from the surface which naturally raises thequestion regarding the net absorption enhancement Toquantify the effects of field concentration on the absorp-tion the enhancement factors as defined in Eq (9) areplotted in Figure 2 in log scale for silver particles with dif-ferent radii The wavelength of the incident plane wave is

1 1001

1

10

100

Ra

a = 10 nm

a = 5 nm

a = 20 nm

a = 100 nm

a = 200 nm

Ra

λ = 08 microm

Abs

orpt

ion

Enh

ance

men

t Fac

tor

Fig 2 The absorption enhancement factors of silver particles with dif-ferent sizes The wavelength of the incident wave is 08 m The radiusof the particle is a and the outer shell has a radius of R as shown in theinsert Note R is normalized to a

2098 J Comput Theor Nanosci 5 2096ndash2101 2008


IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


set at 08 m which is the plasmon resonance wavelengthof small silver particles embedded in silicon as we willshow later Note the radius R of the outer shell is normal-ized to the radius of the sphere At R= a the enhancementfactors for all particles are zero (not shown in the figuredue to the log scale) because the imaginary silicon spher-ical shell has zero thickness When R increases to a largevalue the near field effect diminishes and the enhance-ment factors for all cases approach 1 In the intermediaterange of R the enhancement factors reach maximum forall particles except the 200 nm one which increases mono-tonically with R and never exceeds 1 As shown in thefigure smaller particles lead to larger enhancement How-ever the dependence of the enhancement factor on the par-ticle radius does not follow a monotonic trend The 10 nmparticle gives the maximum enhancement while the larger20 nm and smaller 5 nm particles yield less enhancementThe peaks for the 5 nm and 10 nm particles are both atR = 126a which suggests that the field enhancement isconcentrated in a spherical shell with a thickness of 026a

At the wavelength of 08 m the distribution of thenormalized field or field enhancement (given by Eq (5))outside the 100 nm 10 nm and 5 nm (radius) particles atthe y= 0 plane is shown in Figures 3(andashc) Note 08 m isthe wavelength in vacuum and it is around 022 m in sil-icon The field pattern around the 100 nm particle clearlyshows the retardation effects of the electric field whichis anticipated because the wavelength is smaller than theparticle diameter such that the interference of the scatteredlight is strong As shown in the figure the field enhance-ment for the 100 nm particle is rather limited and scattersaround the particle surface which is mainly due to theinterference effects Compared to the 100 nm particle thefields around the 10 nm and 5 nm particles show a typicalpattern of a dipole field and have much higher enhance-ment factor The electric field is concentrated around thetop and bottom of the particles where the vector of theelectric field of the incident wave has the maximum com-ponent along the surface normal

An interesting observation from Figures 3(b and c) isthat the fields around the small particles (with radii of5 nm and 10 nm) share similar distribution patterns thatdiffer only in magnitude Since for the 5 nm and 10 nmparticles the wavelength is more than 10 times larger thanthe particle diameters the dipole model is considered tobe a reasonable approximation The field of an oscillatingdipole is given by20

E = ejk2r

4)2)0

k2

2er timesptimes err

+ 3er er middotpminusp$

times(

1r3

minus jk2

r2

)(12)

where )0 is the permittivity of vacuum )2 is the rel-ative permittivity of silicon and er is unit vector in

the r direction The dipole moment of the particle isexpressed as

p = pex = 4)0)2E0a

3 )1 minus)2

)1 +2)2

ex (13)

where )1 is the relative permittivity of the silver parti-cle Since a the field near the particle surface isdominated by the near field terms and Eq (12) can be

a = 100 nm

z

x1

2

3

4

5

(a)

0

5

10

15

20

a = 10 nm

z

x

(b)

2

4

6

8

10

12

14

a = 5 nm

z

x

(c)

Fig 3 The field distributions outside the silver particles at the y = 0plane The field distributions are normalized to the incident plane wave asdefined in Eq (5) (a) 100 nm particle (b) 10 nm particle (c) 5 nm particle



IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


simplified as

E = 1

4)2)0r33er er middotpminus

p$ (14)

Substituting Eq (13) into Eq (14) leads to

E =∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣radic

3 sin2 cos2+1(a

r

)3

(15)

It is obvious that the distribution pro-file of the electric field only depends onradic4 sin2 + cos2 cos2+ sin2ar3 which explains

the similarity between Figures 3(b) and (c) The magnitudeof the electric field is given by E0)1 minus )2)1 + 2)2which depends on the radius of the particle since )2 is afunction of a due to the electron scattering at the particlesurface Figure 4 shows )1minus)2)1+2)2 for selectiveparticles with small radii where the dipole approximationis expected to hold with acceptable accuracy All thecurves show peaks around 08 m which is the resonanceplasmon wavelength when the denominator ()1 + 2)2)approaches zero

Furthermore using the dipole approximation theabsorption in the silicon shell (Eq (7)) has a closed ana-lytical form

ampWabsRa =int 2

0

int 0

int Ra

12)primeprime2 E2r2 sin dr d d

= )primeprime2∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣2

a6

(1a3

minus 1R3

)(16)

where )primeprime2 is the imaginary part of the dielectric functionof silicon Taylor expansion of Eq (8) yields

WabsSiR=4Rek2k

primeprime2 E02R3

30+Okprimeprime2R3 (17)

02 04 06 08 10 120

2

4

6

8

10

12

Wavelength (microm)

a = 2 nma = 5 nma = 10 nm

|(ε1

ndashε 2

)(ε

1+

2ε2)

|

Fig 4 The magnitude of )1 minus)2)1 +2)2 The peaks for all threeparticles are centered at around 08 m

0

10

20

30

40

50

02

a = 2 nm

a = 5 nm

a = 10 nm

a = 20 nm

a = 40 nm

a = 100 nm

Enh

ance

men

t fac

tor

Wavelength (microm)

1210080604

Fig 5 The wavelength dependent enhancement factors for particleswith different radii For all cases R is chosen to be 126a Theenhancement factor induced by a 10 nm particle is 50 at the resonancewavelength

The absorption enhancement ratio is then given by

(Ra = 30)primeprime2 )1 minus)2)1 +2)22

4Rek2kprimeprime2R

3

times[(a

R

)3

minus(a

R

)6](18)

which reaches maximum when Ra= 3radic

2 ie R= 126aas confirmed in Figure 2 for the 5 nm and 10 nm casesAs for the larger particles retardation effects push themaximums to smaller R values

We have studied the field enhancement at a fixed wave-length heretofore Figure 5 shows the wavelength depen-dent absorption enhancement with R = 126a As seenfrom the figure the absorption is enhanced up to 50 timeswith the presence of the 10 nm silver particle Decreaseof the radius from 10 nm to 5 nm and 2 nm leads todramatic reduction in absorption A similar trend is seenwhen increasing the radius of the particles where the larger20 nm 40 nm and 100 nm particles give less absorp-tion Nevertheless the enhancement for all particles ismost significant in the wavelength regime between 07 mand 11 m where the silicon does not absorb light ade-quately As shown in the figure the smaller 2 nm 5 nm and10 nm particles have the maximum absorption at around08 m with slight shifting of the peaks while the larger20 nm 40 nm and 100 nm particles have multiple absorp-tion peaks shifted to longer wavelengths For the smallparticles the slight shifting of the absorption peaks fromthe plasmon wavelength comes from the combined effectsof weak retardation and additional wavelength dependent)primeprime2Rek2k

primeprime2 term in Eq (18) for the larger particles the

shifting of the absorption peak is due to the retardationeffect which causes a more pronounced variation from theresonance wavelength of 08 m The results suggest that a



IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a

Fig 6 The absorption in a 10 nm silver particle The absorption lossis significant in the long-wavelength regime

wide spectrum range can be covered by carefully selectingand mixing particles with different radii

As revealed in our analysis the absorption enhance-ment is caused by the field magnification due to the sur-face plasmons However the surface waves extend to bothsides of the interface Therefore the unwanted absorptioninside the silver particles is inevitably increased along withthe absorption enhancement in silicon Figure 6 shows thewasted absorption in a 10 nm particle and the absorption inthe outside shell (R= 126a) The absorption is normalizedto WabsSi (126a) For wavelengths larger than 05 m thesilver particle absorbs significantly more light than siliconespecially in the long wavelength range which presentsa problem for the overall optical absorption efficiencyOne possible solution is to use the corendashshell structure21

which has a very thin layer of metal that the correspondingabsorption loss may be alleviated

4 SUMMARY

In summary we have studied the surface-plasmonenhanced absorption in silicon medium Both extendedMie theory and dipole approximation have been used forthe calculations The field concentration effects of the sur-face plasmon are examined The enhancement is found tobe mostly confined in a spherical shell surrounding theparticle and the optimal thickness of the shell is 026aThe spectral analysis shows that the long-wavelength opti-cal absorption in silicon can be enhanced significantlyby embedded silver nanoparticles Enhancement up to50 times is obtained with a 10 nm silver particle while

smaller and larger particles yield less enhancement Thecalculation reveals that the energy loss inside the silverparticles may present a problem for efficient energy uti-lization which could be potentially addressed by usingnovel nanostructures We want to point out that our cal-culation is based on local field enhancement induced by asingle particle Particlendashparticle interaction can potentiallylead to even higher enhancement Anisotropic scatteringfrom particles may also give additional enhancement byrandomizing the incident solar light

Acknowledgments We would like to thankMr Vincent Berube for helpful comments This workis partially supported by DOE (Grant No DE-FG02-02ER45977) and MASDAR

References

1 M A Green Prog Photovolt Res Appl 7 317 (1999)2 J Zhao A Wang P Altermatt and M A Green Appl Phys Lett

66 3636 (1995)3 Air Mass 15 Spectra American Society for Testing and Materials

httprredcnrealgovsolarspectraam154 K Yamamoto A A Nakajima M Yoshimi T Sawada S Fukuda

T Suezaki M Ichikawa Y Koi M Goto T Meguro T MatsudaM Kondo T Sasaki and Y Tawada Sol Energy 77 939(2004)

5 D Derkacs S H Lim P Matheu W Mar and E T Yu Appl PhysLett 89 093103 (2006)

6 S Pillai K R Catchpole T Trupke and M A Green J ApplPhys 101 093105 (2007)

7 B P Rand P Peumans and S R Forrest J Appl Phys 96 7519(2004)

8 G Zhao H Kozuka and T Yoko Thin Solid Films 277 147 (1996)9 C Kittel Introduction of Solid State Physics Wiley New York

(1996)10 H Raether Surface Plasmons on Smooth and Rough Surfaces and

on Gratings Springer-Verlag Berlin (1988)11 R K Chang and T E Furtak (eds) Surface Enhanced Raman Scat-

tering Plenum New York (1982)12 K R Catchpole and S J Pillai J Appl Phys 100 044504

(2006)13 C F Bohren and D R Huffman Absorption and Scattering of Light

by Small Particles Wiley New York (1983)14 I W Sudiarta and P Chylek J Opt Soc Am A 18 1275 (2001)15 W C Mundy J A Roux and A M Smith J Opt Soc Am 64

1593 (1974)16 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)17 E D Palik (ed) Handbook of Optical Constants of Solids Aca-

demic Orlando (1985)18 W T Doyle Phys Rev 111 1067 (1958)19 C G Granqvist and O Hunderi Phys Rev B 16 3513 (1977)20 J D Jackson Classical Electrodynamics Academic New York

(1998)21 E Prodan C Radloff N J Halas and P Nordlander Science 302

419 (2003)

Received 2 February 2008 Accepted 3 March 2008



IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


in silicon using an extended Mie scattering theory for asphere embedded in absorbing media1314 The extendedMie scattering theory takes into account the near field com-ponents without making any far field approximations15

which ensures the localized surface modes can be repre-sented in the model A single particle analysis is performedas it reveals fundamental mechanisms of the absorptionenhancement although in real applications multiple metalnanoparticles are deployed instead of a single particle Inphotovoltaic applications the metal articles are usually ran-domly distributed and the primary enhancement mecha-nism is the localized surface modes which are less affectedby the incoherent scattered light from the randomly spacedneighbors Therefore the results obtained from the para-metric study of a single particle can serve as the first-orderinput parameters for the design of more complex struc-tures involving multiple particles In our calculations wechoose silver nanoparticles as the photon absorption cen-ters because the plasma frequency of silver is the lowestamong noble metals16 which is ideal for solar photovoltaicapplications Our calculation shows that with the presenceof a silver nanoparticle the net absorption enhancement insilicon can be up to 50 times in the long wavelength regimeThe dependence of absorption enhancement on the particlesizes shows a non-monotonic trend Maximum enhance-ment is found for particles with intermediate sizes Largerparticles lead to lower enhancement due to the retardationeffect of the electromagnetic waves Particles with radiismaller than the optimized radius give smaller enhance-ment which is due to the size effects caused by boundaryscattering of electrons In addition to the extended Mie the-ory we present simplified models for absorption analysisbased on dipole models which is found to be good approx-imations for particles with small radii Our analysis alsosuggests that the absorption by the silver nanoparticle itselfis significant which should be addressed in the future

2 EXTENDED MIE SCATTERING THEORY

Figure 1 shows a schematic illustration of the presentanalysis The gray sphere represents a silver particle witha radius of a Region 1 is inside the silver sphere andregion 2 is in the host absorbing medium ie silicon Theangle is the zenith angle and is the azimuth angleAn imaginary sphere with a radius of R is also drawn aswe will analyze the absorption in silicon within this radiusA plane wave is propagating in the positive z directionWithout the loss of generality we choose the electric fieldof the plane wave to be polarized in the x direction Theelectric and magnetic fields of the incident plane wave aregiven by

Ei = E0e

jk2r cosex (1)

Hi =

k2

2

E0ejk2r cosey (2)

ki

x

y

a

R

r

θ

Φ

1

2

z

Fig 1 Schematic drawing of a silver particle (region 1) embedded insilicon (region 2) The gray sphere is the silver particle which is enclosedby an imaginary spherical shell

where E0 is the amplitude of the electric field at z= 0k2 the wave vector of the incident plane wave r theradial coordinate the angular frequency 2 the per-meability j the imaginary unit ex the unit vector in thex direction and ey the unit vector in the y directionThe wave vector is given by k2 = 2N2 where N2 isthe optical constant of silicon and is the wavelength invacuum Since neither silver nor silicon is magnetic mate-rial we have 1 = 2 = 0 where subscripts 1 2 and 0denote regions 1 2 and vacuum Following procedures inRef [13] the scattered field can be expanded into vectorspherical harmonics

Es =

sumn=1

En(jan

N3e1nminusbn

M3o1n

)(3)

Hs =

k2

0

sumn=1

En(jbn

N3o1n+an

M3e1n

)(4)

whereM and

N are vector spherical harmonics En =

jnE02n+1nn+1 an and bn are the expansion coef-ficients and the subscripts e and o denote even and oddThe definitions of the vector spherical harmonics and thederivations of the expansion coefficients are not given heresince Ref [13] contains detailed information Note thatthe scattered field and the expansion coefficients derivedin Ref [13] can be directly applied to the absorbinghost media by treating the optical constant of the hostmedia as complex number We can now define the fieldenhancement by

r= E2

r

Eir

= Eir+Es

r

Eir

(5)

where r gt a



IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE



WabsR=E020k22

Reklowast2

sumn=1

2n+1

times jlowastn



n


n

minus janlowastn


nlowastn$

(6)






WabsSi =int 2

0

int 0

12

ReEitimes


= Rek2E02r2

0



](8)


(Ra= ampWabsRa

WabsSiR(9)




)= )bulk+2p

2 + j+bulkminus 2

p

2 + j+ (10)


+ = +bulk +AFa (11)





1 1001

1

10

100

Ra

a = 10 nm

a = 5 nm

a = 20 nm

a = 100 nm

a = 200 nm

Ra

λ = 08 microm

Abs

orpt

ion

Enh

ance

men

t Fac

tor




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE





E = ejk2r

4)2)0

k2

2er timesptimes err


times(

1r3

minus jk2

r2

)(12)



p = pex = 4)0)2E0a

3 )1 minus)2

)1 +2)2

ex (13)


a = 100 nm

z

x1

2

3

4

5

(a)

0

5

10

15

20

a = 10 nm

z

x

(b)

2

4

6

8

10

12

14

a = 5 nm

z

x

(c)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


simplified as

E = 1


p$ (14)


E =∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣radic

3 sin2 cos2+1(a

r

)3

(15)




ampWabsRa =int 2

0

int 0

int Ra



)1 minus)2

)1 +2)2

∣∣∣∣2

a6

(1a3

minus 1R3

)(16)


WabsSiR=4Rek2k

primeprime2 E02R3


02 04 06 08 10 120

2

4

6

8

10

12

Wavelength (microm)

a = 2 nma = 5 nma = 10 nm

|(ε1

ndashε 2

)(ε

1+

2ε2)

|


0

10

20

30

40

50

02

a = 2 nm

a = 5 nm

a = 10 nm

a = 20 nm

a = 40 nm

a = 100 nm

Enh

ance

men

t fac

tor

Wavelength (microm)

1210080604




4Rek2kprimeprime2R

3

times[(a

R

)3

minus(a

R

)6](18)








IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a





4 SUMMARY




References

















419 (2003)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE



WabsR=E020k22

Reklowast2

sumn=1

2n+1

times jlowastn



n


n

minus janlowastn


nlowastn$

(6)






WabsSi =int 2

0

int 0

12

ReEitimes


= Rek2E02r2

0



](8)


(Ra= ampWabsRa

WabsSiR(9)




)= )bulk+2p

2 + j+bulkminus 2

p

2 + j+ (10)


+ = +bulk +AFa (11)





1 1001

1

10

100

Ra

a = 10 nm

a = 5 nm

a = 20 nm

a = 100 nm

a = 200 nm

Ra

λ = 08 microm

Abs

orpt

ion

Enh

ance

men

t Fac

tor




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE





E = ejk2r

4)2)0

k2

2er timesptimes err


times(

1r3

minus jk2

r2

)(12)



p = pex = 4)0)2E0a

3 )1 minus)2

)1 +2)2

ex (13)


a = 100 nm

z

x1

2

3

4

5

(a)

0

5

10

15

20

a = 10 nm

z

x

(b)

2

4

6

8

10

12

14

a = 5 nm

z

x

(c)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


simplified as

E = 1


p$ (14)


E =∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣radic

3 sin2 cos2+1(a

r

)3

(15)




ampWabsRa =int 2

0

int 0

int Ra



)1 minus)2

)1 +2)2

∣∣∣∣2

a6

(1a3

minus 1R3

)(16)


WabsSiR=4Rek2k

primeprime2 E02R3


02 04 06 08 10 120

2

4

6

8

10

12

Wavelength (microm)

a = 2 nma = 5 nma = 10 nm

|(ε1

ndashε 2

)(ε

1+

2ε2)

|


0

10

20

30

40

50

02

a = 2 nm

a = 5 nm

a = 10 nm

a = 20 nm

a = 40 nm

a = 100 nm

Enh

ance

men

t fac

tor

Wavelength (microm)

1210080604




4Rek2kprimeprime2R

3

times[(a

R

)3

minus(a

R

)6](18)








IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a





4 SUMMARY




References

















419 (2003)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE





E = ejk2r

4)2)0

k2

2er timesptimes err


times(

1r3

minus jk2

r2

)(12)



p = pex = 4)0)2E0a

3 )1 minus)2

)1 +2)2

ex (13)


a = 100 nm

z

x1

2

3

4

5

(a)

0

5

10

15

20

a = 10 nm

z

x

(b)

2

4

6

8

10

12

14

a = 5 nm

z

x

(c)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


simplified as

E = 1


p$ (14)


E =∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣radic

3 sin2 cos2+1(a

r

)3

(15)




ampWabsRa =int 2

0

int 0

int Ra



)1 minus)2

)1 +2)2

∣∣∣∣2

a6

(1a3

minus 1R3

)(16)


WabsSiR=4Rek2k

primeprime2 E02R3


02 04 06 08 10 120

2

4

6

8

10

12

Wavelength (microm)

a = 2 nma = 5 nma = 10 nm

|(ε1

ndashε 2

)(ε

1+

2ε2)

|


0

10

20

30

40

50

02

a = 2 nm

a = 5 nm

a = 10 nm

a = 20 nm

a = 40 nm

a = 100 nm

Enh

ance

men

t fac

tor

Wavelength (microm)

1210080604




4Rek2kprimeprime2R

3

times[(a

R

)3

minus(a

R

)6](18)








IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a





4 SUMMARY




References

















419 (2003)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


simplified as

E = 1


p$ (14)


E =∣∣∣∣E0

)1 minus)2

)1 +2)2

∣∣∣∣radic

3 sin2 cos2+1(a

r

)3

(15)




ampWabsRa =int 2

0

int 0

int Ra



)1 minus)2

)1 +2)2

∣∣∣∣2

a6

(1a3

minus 1R3

)(16)


WabsSiR=4Rek2k

primeprime2 E02R3


02 04 06 08 10 120

2

4

6

8

10

12

Wavelength (microm)

a = 2 nma = 5 nma = 10 nm

|(ε1

ndashε 2

)(ε

1+

2ε2)

|


0

10

20

30

40

50

02

a = 2 nm

a = 5 nm

a = 10 nm

a = 20 nm

a = 40 nm

a = 100 nm

Enh

ance

men

t fac

tor

Wavelength (microm)

1210080604




4Rek2kprimeprime2R

3

times[(a

R

)3

minus(a

R

)6](18)








IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a





4 SUMMARY




References

















419 (2003)




IP 9526237249Sat 21 Jul 2012 075855

RESEARCHARTICLE


02 04 06 08 10 1210ndash4

10ndash2

100

102

104

Wavelength (microm)

Nor

mal

ized

abs

orpt

ion

Silicon

Silver

a = 10 nm

R = 126 a





4 SUMMARY




References

















419 (2003)



surface-plasmon enhanced near-bandgap light absorption in silicon photovoltaics

Documents