Research Article Strongly Localized Image States of ...

Research ArticleStrongly Localized Image States of Spherical Graphitic Particles

Godfrey Gumbs1 Antonios Balassis2 Andrii Iurov1 and Paula Fekete3

1 Department of Physics and Astronomy Hunter College at the City University of New York 695 Park AvenueNew York NY 10065 USA

2 Physics Department Fordham University 441 East Fordham Road Bronx NY 10458-5198 USA3Department of Physics and Nuclear Engineering US Military Academy 698 Mills Road West Point NY 10996 USA

Correspondence should be addressed to Andrii Iurov theoristphysicsgmailcom

Received 9 August 2013 Accepted 21 October 2013 Published 22 January 2014

Academic Editors Y P Kalmykov V Yukalov and A Zorko

Copyright copy 2014 Godfrey Gumbs et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the localization of charged particles by the image potential of spherical shells such as fullerene buckyballs Thesespherical image states exist within surface potentials formed by the competition between the attractive image potential and therepulsive centripetal force arising from the angular motionThe image potential has a power law rather than a logarithmic behaviorThis leads to fundamental differences in the nature of the effective potential for the two geometries Our calculations have shownthat the captured charge is more strongly localized closest to the surface for fullerenes than for cylindrical nanotube

1 Introduction

The experimental and theoretical study of carbon is currentlyone of themost prevailing research areas in condensedmatterphysics Forms of carbon include several allotropes such asgraphene and graphite as well as the fullerenes which coverany molecule composed entirely of carbon in the form ofa hollow sphere ellipsoid or tube Like graphite fullerenesare composed of stacked graphene sheets of linked hexagonalrings For these the carbon atoms form strong covalent bondsthrough hybridized sp2 atomic orbitals between three nearestneighbors in a planar or nearly planar configuration

Mass spectrometry experiments showed strong peakscorresponding to molecules with the exact mass of sixtycarbon atoms and other carbon clusters such as C

70 C76

and up to C94

[1 2] Spherical fullerenes well known asldquobuckyballsrdquo (C

60) were prepared in 1985 by Kroto et al

[3] The structure was also identified about five years earlierby Iijima [4] from an electron microscope image where itformed the core of a multishell fullerene or ldquobuckyonionrdquoSince then fullerenes have been found to exist naturally [5]More recently fullerenes have been detected in outer space[6] As a matter of fact the discovery of fullerenes greatlyexpanded the number of known carbon allotropes which

until recently were limited to graphite diamond and amor-phous carbon such as soot and charcoal Both buckyballs andcarbon nanotubes also referred to as buckytubes have beenthe focus of intense investigation for their unique chemistryaswell as their technological applications inmaterials scienceelectronics and nanotechnology [7]

Recently the image states of metallic carbon nanotubes[8] and double-wall nonmetallic nanotubes [9 10]were inves-tigated Experimental work [11] includes photoionization [12]and time-resolved photoimaging of image-potential statesin carbon nanotubes [13] There has been general interest[14] in these structures because of electronic control onthe nanoscale using image states This has led to wide-ranging potential applications including field ionization ofcold atoms near carbon nanotubes [15] and chemisorption offluorine atoms on the surface of carbon nanotubes [16] Theimportant role played by the centripetal term in determiningthe total potential of a captured charged particle orbitingabout C

60 was demonstrated by McCune et al in [12] The

local density approximation was used in the calculations ofphotoionization in that study

Here we calculate the nature of the image-potential statesin a spherical electron gas (SEG) confined to the surface of abuckyball in a similar fashion as in the case of a nanotube

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 726303 6 pageshttpdxdoiorg1011552014726303

2 The Scientific World Journal

In Figure 1 we show a schematic of a charged particlelocalized in a spherical image state For the semi-infinitemetalvacuum interface [17] related image-potential stateshave been given a considerable amount of theoretical atten-tion over the years Additionally these states have beenobserved for pyrolytic graphite [18] and metal-supportedgraphene [19] Silkin et al [20] further highlighted the impor-tance of image states in graphene [21] by concluding that theinterlayer state in graphite is formed by the hybridizationof the lowest image-potential state in graphene in a similarway as it occurs in bilayer graphene [22 23] The significanceof image states was also discussed by Rinke et al in [24]We mention these facts to show why we were motivated tostudy image states for the spherical geometry Furthermorethe significance of the role played by the image-potentialhas led to the observation that for planar layered materialsstrongly dispersive interlayer states are present However theeigenstates for a spherical shell are nondispersive and so tooare the collective plasma modes [25ndash29] This difference initself leads to unique and interesting properties which wehave found for the image potential

We will consider the image states outside the SEG inthe following However our method may be extended in astraightforwardmanner to the case when the image states areinside the shell A relevant discussion of image states for C

60

with multiply charged anions [30 31] was recently given in[32] Here we only deal with the case of a singly chargedparticle In our formalism for obtaining a spherical imagestate we consider a spherical shell of radius 119877 whose centeris at the origin The background dielectric constant is 120598

1for

0 lt 119903 lt 119877 and 1205982for 119903 gt 119877 An electron gas is confined to the

surface of the sphere If a charge 119876 is located at (1199030 1205790 1206010) in

spherical coordinates then for 1199030gt 119877 the total electrostatic

potential is given by Φtot = Φext + Φind where Φext is theexternal potential due to the point particle and Φind is theinduced potential When 119903 lt 119877 we express the total potentialas follows

Φ(1)

tot (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1119860119871119903119871119884119871119872

(Ω)

Φext (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1

119903119871

gt

119903119871+1

gt

119884lowast

119871119872(Ω0) 119884119871119872

(Ω)

Φ(2)

ind (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1119861119871119903minus(119871+1)

119884119871119872

(Ω)

(1)

where 119896 = (41205871205980)minus1 with 120598

0the permittivity of free space

Also 119884119871119872(Ω) is a spherical harmonic and Ω is a solid angle

For 119903 gt 119877 the total potential is

Φ(2)

tot (119903 120579 120601)

= 4120587119896119876sum

119871119872

1

2119871 + 1

times [119903119871

lt

119903119871+1

gt

119884lowast

119871119872(Ω0) + 119861119871119903minus(119871+1)

]119884119871119872

(Ω)

(2)

Figure 1 Schematic illustration of a charged particle captured bythe image potential and orbiting around a buckyball The radiusof the orbit is determined by the dielectric constant within andsurrounding the shell as well as the angular momentum quantumnumber of the captured particles For semiconducting shells thelocalization is strong and the radius of the stable orbit can be a fewnanometers The localization is weak for metallic shells

On the surface of the sphere the boundary condi-tions are Φ(1)tot (119877 120579 120601) = Φ

(2)

tot (119877 120579 120601) and [1205981Φ(1)

tot (119903 Ω 120596)minus

1205982Φ(2)

tot (119903 Ω 120596)]|1015840

119903=119877= 4120587119896120590(119877 120579 120601 120596) where 120590(119877 120579 120601 120596) is

the induced surface charge density on the spherical shellExpanding120590 in terms of spherical harmonics and using linearresponse theory we find

120590 (119877 120579 120601 120596) = minus21198961198761198902

1198772sum

119871119872

1

2119871 + 1119860119871119877119871Π119871(120596) 119884119871119872

(Ω)

(3)

In this equationΠ119871(120596) is the SEG polarization function for 119871

an integer and given in terms of the Wigner 3-119895 symbol [2527]

Π119871(120596) = sum

ℓℓ1015840

1198910(119864ℓ) minus 1198910(119864ℓ1015840)

ℏ120596 + 119864ℓ1015840 minus 119864ℓ

(2ℓ + 1) (2ℓ1015840+ 1)

times (119897 1198971015840119871

0 0 0)

2

(4)

where 1198910(119864) = 1(1 + exp[(119864 minus 120583)119896

119861119879]) is the Fermi-

Dirac distribution function 119896119861is Boltzmannrsquos constant and

120583 is the chemical potential 119864ℓ= ℏ2ℓ(ℓ + 1)(2119898

lowast1198772) with

ℓ = 0 1 2 and 119898lowast is the electron effective mass The

induced potential Φ(2)ind(r 120596) outside the spherical shell maybe calculated to be

Φ(2)

ind (r 120596)

= 4120587119896119876sum

119871

[1205982

120576119871(120596 = 0)

minus1

2119871 + 1]1198772119871+1

119903119871+1

0

times1

119903119871+1sum

119872

119884lowast

119871119872(Ω0) 119884119871119872

(Ω)

(5)

The Scientific World Journal 3

where 120576119871(120596) = 119871(120598

1+ 1205982) + 120598

2+ (2119890

2119877)Π119871(120596) is the

dielectric function of the SEG The force on a charge 119876 atr0= (1199030 1206010 1205790) is along the radial direction and can be found

using 119865(1199030) = minus119876120597Φ

(2)

ind(r)120597119903|r0

yielding

119865 (1199030) = 119896119876

2sum

119871

(119871 + 1) [(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1]1198772119871+1

1199032119871+3

0

(6)

The interaction potential energy Uim(1199030) may now be calcu-lated from

Uim (1199030) equiv sum119871

U(119871)

im (1199030)

=1

2119876Φind (1199030 1205790 1206010)

=1198961198762

21199030

sum

119871

[(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1](119877

1199030

)

2119871+1

(7)

The effective potential is the sum of the image potential andthe centrifugal term and is given by [8 9]

119881(119871)

eff (1199030 120579) = U

(119871)

im (1199030) +

ℏ2(1198712+ 119871 minus (14))

2119872lowast(1199030sin 1205790)2

(8)

showing that 119881(119871)eff is not spherically symmetric In thisnotation 119872lowast is the effective mass of the captured chargedparticle in an orbital state with angular momentum quantumnumber 119871

We now turn to numerical calculation of the effectivepotential and its comparison with nanotubes In Figure 2(a)we calculated 119881(119871)eff (1199030 120579) as a function of 119903

0 for chosen 119871 119877

For the background dielectric constant we chose 1205981= 24

corresponding to graphite and 1205982= 1 for the surrounding

medium The electron effective mass used in calculating thepolarization function Π

119871in (4) was 119898 = 025119898

119890where 119898

119890is

the bare electron mass the Fermi energy is 119864119865= 06 eV and

the orbiting particle effective mass is119872lowast = 119898119890

Figure 2(b) shows how the peak values of the effectivepotential depend on radius Of course the height of the peakis linked to the localization of the particle in orbit In Figure 3the ground and three lowest excited state wave functionsare plotted for the effective potential 119881(119871)eff when 119871 = 2 inFigure 2(a)

The value of the angular momentum quantum number119871 as well as the curvature of the surface of these complexcarbon structures clearly plays a crucial role in shaping theeffective potential Generally the form for the 119871th term maybe expressed as 119881(119871)eff (1199030) = minus120572

119871119903minus2(119871+1)

0+ 120573119871119903minus2

0 where 120572

119871

and 120573119871are due to the image potential and centrifugal force

respectively The coefficient 120572119871is always positive whereas 120573

119871

is only negative for 119871 = 0 This power-law behavior ensuresthat no matter what values the two coefficients may havethe image term dominates the centrifugal term leading toa 119897 local maximum in the effective potential This is unlikethe behavior for a cylindrical nanotube where the image

term is logarithmic due to the linear charge distribution andmay be dominated by the 119903minus2

0centrifugal term leading to

a local minimum instead Consequently the capturing andlocalization of a charged particle by the image potential ofspherical conductors and dielectrics are fundamentally dif-ferent from those for a cylindrical nanotube For the sphereas shown in Figure 3 the wave function is more localizedaround the spherical shell within its effective potential thatis the wave function is not as extended Additionally theconfinement of the charged particle is close to the sphericalsurface

The choice for the radius does render some crucialchanges in 119881

(119871)

eff to make a difference in the location andheight of the peak However only the higher-lying localizedstates are affected In Figure 2 we show how the peakheight changes as 119871 is varied Additional numerical resultscorresponding to 120598

1≫ 1205982have shown that a spherical

metallic shell has a reduced potential peak for confiningthe captured charge Thus the spherical metallic shell isnot as susceptible for particle confinement in highly excitedstates as the metallic nanotube [8ndash10] This indicates thatthe dimensionality plays a nontrivial role in formation ofimage states and their spatial extension near the surface ofthe nanometer-size graphitic structure This direct crossoverfrom a one-dimensional to a three-dimensional regime is notdetermined by polarization effects for the structure in themetallic limit since in the limit 120598

1rarr infin in (6) the Π

119871(120596)

term makes no contributionThe difference is due entirely tothe geometrical shape in the metallic regime where graphiticplasmons fail to develop For finite values of 120598

1 we encounter

the regime where excited particles contribute through thepolarization function Π

119871(120596) defined in (4) The behavior

of plasmon excitation as a function of angular momentumquantumnumber 119871 for fullerenes resembles in all respects thelong wavelength (119902 rarr 0) limit of carbon nanotubes [33ndash35]Furthermore in the case of the low-frequency 120587-plasmons incarbon nanotubes a surface mode may develop for large 119902due to the difference in the values for the dielectric constantswithin the graphitic structure and the surrounding medium[36]

Since the polarization function Π119871(120596) vanishes

identically for 119871 = 0 the attractive part of the effectivepotential is only significantly modified by screening for afast-rotating external charge This behavior at zero angularmomentum differs from tubular-shaped image states forsingle-walled carbon nanotubes which are formed in apotential isolated from the tube [8] The large angularmomentum image states for spheres may be probedby femtosecond time-resolved photoemission [13] Ourformalism shows that considering photoionization fromvarious levels of C

60 the Coulomb interaction between an

external charge and its image is screened by the staticallystretched SEG through the dielectric function 120598

119871(120596 = 0)

The polarization of the medium Π119871which is driven by

the electrostatic interaction is generated by particle-holetransitions across the Fermi surface The polarization alsodetermines the Ruderman-Kittel-Kasuya-Yosida (RKKY)


L = 3

L = 2

L = 0

L = 1

R = 1nm1205981 = 24

r0R

Veff(ke2R

)

11 15 25 30

015

minus005

minus015

minus010

(a)

1205981 = 24

L = 3

L = 2

L = 4

R (nm)

Peak

V(

)eff

(ke2R)

025

020

005

14 16 18 20

(b)

Figure 2 (a) The effective potential 119881eff between a charged particle and a spherical shell is shown for a number of angular momenta 119871 Theradius of the sphere is 119877 = 1 nm and we chose 120598

1= 24 120598

2= 1 In (b) the height of the peak in the effective potential appearing in (a) is

plotted as a function of the radius

n = 3n = 4

n = 2n = 1

02

01

11 24

minus01

minus02

r0R

11 24

r0R

Ψn L=2(r)

02

minus02

Figure 3 The wave functions for the ground state (119899 = 1) and firstthree excited states (119899 = 2 3 4) are plotted for the effective potential119881eff between a charged particle and a spherical shell when 119871 = 2 Wechose 120598

1= 24 120598

1= 1 and 119877 = 1 nm

interaction energy between two magnetic impurities as wellas the induced spin density due to a magnetic impurity

Increasing radius the position of the peak moves closerto the sphere as 119877minus1(2119871) In the absolute units the position ofthe peak depends as 119903(peak)

0∽ 1198771minus1(2119871) The typical distances

from the surface are between 13119877 and 15119877 Figure 2(b)demonstrates how the potential peaks (corresponding to thelocal maximum for the 119881eff) depend on the radius of thebuckyball for various angular momentum quantum numbers119871 Clearly we see that the potential peak decreases withincreased radius leading us to conclude that confinement isthe strongest for smaller buckyballs and particles with largeangular momentum For the nanotube increasing 119871 leads toa reduced local minimum in the effective potential and theability to localize the charge [8] Approximately the curvesmay be fitted analytically to ∽1119877 for all considered values of119871 However we found that a better fit for 119871 gt 5 would be ofthe form 119888

1119877 + 119888

21198772 where 119888

1 1198882are constants

For increased 1205981 that is the metallic limit with 120598

1≫ 1205982

we have 120598119871⋍ 1205981119871 so that the coefficient [(2119871 + 1)(120598

2120598119871) minus

1] rarr minus1 In the case of dielectric constant 1205981sim 24 for

the buckyball the above-mentioned coefficients lie withinthe range from minus04 to minus09 and decrease with increasing 119871Consequently for the transition to the metallic limit thesecoefficients are more affected for states with large 119871 So wemay conclude that 120598

1has little effect on the position and

height of the peak in the effective potential In fact for themetallic case the peak is observed to be slightly further awayfrom the center (very little difference sim137119877 compared tosim131119877 for 119877 = 1 nm) The height of the peak is only slightlydecreased in the metallic case (024 compared to 027 in thecase of fullerenes) These numbers are provided for fixed 119871and the unit of energy is the same as that in Figure 2

Regarding the wave functions and density plots Figures 3and 4 demonstrate the wave function of a bounded electrontrapped between the infinite hard wall of the sphere and thepotential peak First we note that we obtained qualitativelysimilar behavior for different values of 119871 so the electronstates corresponding to the potentials with different angularmomenta are almost the same We clearly see that theelectron wave functions are not exactly localized in theldquopotential wellrdquo due to the asymmetry of the boundaryconditions that is infinitely high wall on the left and theeffective potential profile on the right-hand side The wavefunctions corresponding to 119871 = 0 are extremely delocalizeddue to the relatively shallow potential The fact that theeffective potential is not spherically symmetric means thatfor arbitrary angle 120579 we must solve a three-dimensionalSchrodinger equation However for trajectories parallel tothe 119909 minus 119910 plane for the constant angle 120579 the problem reducesto a quasi-one-dimensional Schrodinger equation involvingthe radial coordinate In our calculations we set 120579 = 1205872 sothat the captured charge is moving in the equatorial planeIn this case the centrifugal term is weakest compared to theimage potential but it still affords us the opportunity to seeits effect on localization

The density plots in Figure 4 show that the innermostring is substantially brighter than the outer rings This is


n = 2n = 1

3nm 3nm

(a)

n = 3 n = 4

3nm 3nm

(b)

Figure 4 Probability density plots for |Ψ119871119899(1199030)|21199032

0when 119871 = 2 and 119899 = 1 2 (a) as well as 119899 = 3 4 (b) where 119899 labels the eigenstates for

a spherical shell of radius 119877 = 10 A The wave function Ψ119871119899(1199030) is a solution of the one-dimensional Schrodinger equation with effective

potential 119881eff(1199030 120579 = 1205872) shown in Figure 2 We chose 1205981= 24 inside the ball whose outline is shown as a thin circle and 120598

2= 1 in the

surrounding medium

a consequence of the presence of the 119903minus20

factor in the electronprobability function In contrast the corresponding plotsfor the nanotube [10] do not have the innermost ring somuch brighter than the outer rings because of the fact thatthe density function in that case depends on the inversedistance of the charge from the center of the cylinder insteadThis is another unusual specific feature of the consideredgeometry and indicates that the captured charge is morestrongly localized for the spherical shell closest to the surfacefor fullerenes than for cylindrical nanotube The lowestbound states for Figure 3 are in the range from minus10 to aboutminus100meV with the first few excited states lying very close tothe ground state energy

In conclusion we note that our calculations have shownthat the bound state energies of charged particles localizedaround nanosized spherical shells such as buckyballs may beadjusted by varying the radius 119877 of the shell In our paperwe used an electron gas model for the electron energiesHowever we may incorporate a more realistic energy bandstructure into the polarization function through a formfactor by making use of the results presented in [37] Thiswould also account for the prescribed number of electronson the fullerene We note that the peak potential decreasesaccording to a power-law function with increasing radiusThis property allows for considerable manipulation of acaptured external electron and its release to a source of holesfor recombination followed by the release of a single photonwhose frequency and polarization are linked to those ofthe electron This single-photon source may have variablefrequency with a broad range of applications in quantumcomputation where the message is encoded in the numberof photons transmitted from node to node in an all-opticalnetwork Gate operations are performed by the nodes basedon quantum interference effects between photons whichcannot be identified as being different The low frequencyphotons could be in the infrared which is the most use-ful range for telecommunications Another more generalpractical application and technological use of such uniquequantum states would be to quantum optical metrology ofhigh accuracy and absolute optical measurements

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Contract no FA 9453-07-C-0207 of AFRL

References

[1] E Osawa ldquoE Superaromaticityrdquo Kagaku vol 25 pp 854ndash8631970

[2] J F Anacleto and M A Quilliam ldquoLiquid chromatogra-phymass spectrometry investigation of the reversed-phaseseparation of fullerenes and their derivativesrdquoAnalytical Chem-istry vol 65 no 17 pp 2236ndash2242 1993

[3] H W Kroto J R Heath S C OrsquoBrien R F Curl and R ESmalley ldquoC

60 buckminsterfullerenerdquoNature vol 318 no 6042

pp 162ndash163 1985[4] S Iijima ldquoDirect observation of the tetrahedral bonding

in graphitized carbon black by high resolution electronmicroscopyrdquo Journal of Crystal Growth vol 50 no 3 pp 675ndash683 1980

[5] P R Buseck S J Tsipursky and RHettich ldquoFullerenes from thegeological environmentrdquo Science vol 257 no 5067 pp 215ndash2171992

[6] J Cami J Bernard-Salas E Peeters and S EMalek ldquoDetectionof C60andC

70in a young planetary nebulardquo Science vol 329 no

5996 pp 1180ndash1182 2010[7] C A Poland R Duffin I Kinloch et al ldquoCarbon nanotubes

introduced into the abdominal cavity of mice show asbestos-like pathogenicity in a pilot studyrdquo Nature Nanotechnology vol3 pp 423ndash428 2008

[8] B E Granger P Kral H R Sadeghpour and M ShapiroldquoHighly extended image states around nanotubesrdquo PhysicalReview Letters vol 89 no 13 Article ID 135506 4 pages 2002

[9] G Gumbs A Balassis and P Fekete ldquoImage potential for adouble-walled cylindrical nanotuberdquo Physical Review B vol 73no 7 Article ID 075411 2006


[10] S Segui C Celedon Lopez G A Bocan J L Gervasoni andN R Arista ldquoTubular image states general formulation andproperties for metallic and nonmetallic nanotubesrdquo PhysicalReview B vol 85 no 23 Article ID 235441 7 pages 2012

[11] K Schouteden A Volodin D A Muzychenko et al ldquoProbingquantized image-potential states at supported carbon nan-otubesrdquoNanotechnology vol 21 no 48 Article ID 485401 2010

[12] M A McCune M E Madjet and H S Chakraborty ldquoUniquerole of orbital angular momentum in subshell-resolved pho-toionization of C

60rdquo Journal of Physics B vol 41 no 20 Article

ID 201003 2008[13] M Zamkov N Woody S Bing et al ldquoTime-resolved pho-

toimaging of image-potential states in carbon nanotubesrdquoPhysical Review Letters vol 93 no 15 Article ID 156803 2004

[14] S Segui G A Bocan N R Arista and J L GervasonildquoPhotoionization of image states around metallic nanotubesrdquoJournal of Physics Conference Series vol 194 Article ID 1320132009

[15] A Goodsell T Ristroph J A Golovchenko and L V HauldquoField ionization of cold atoms near the wall of a single carbonnanotuberdquo Physical Review Letters vol 104 no 13 Article ID133002 2010

[16] V A Margulis and E E Muryumin ldquoChemisorption of singlefluorine atoms on the surface of zigzag single-walled carbonnanotubes a model calculationrdquo Physica B Condensed Mattervol 390 no 1-2 pp 134ndash142 2007

[17] P M Echenique and J B Pendry ldquoThe existence and detectionof Rydberg states at surfacesrdquo Journal of Physics C vol 11 no 10p 2065 1978

[18] J Lehmann M Merschdorf A Thon S Voll and W Pfeif-fer ldquoProperties and dynamics of the image potential stateson graphite investigated by multiphoton photoemission spec-troscopyrdquo Physical Review B vol 60 no 24 pp 17037ndash170451999

[19] I Kinoshita D Ino K Nagata K Watanabe N Takagiand Y Matsumoto ldquoAnomalous quenching of electronic statesof nanographene on Pt(111) by deuterium edge terminationrdquoPhysical Review B vol 65 no 24 Article ID 241402(R) 4 pages2002

[20] V M Silkin J Zhao F Guinea E V Chulkov P M Echeniqueand H Petek ldquoImage potential states in graphenerdquo PhysicalReview B vol 80 no 12 Article ID 121408(R) 2009

[21] G Gumbs D Huang and P M Echenique ldquoComparingthe image potentials for intercalated graphene with a two-dimensional electron gas with and without a gated gratingrdquoPhysical ReviewB vol 79 no 3 Article ID035410 6 pages 2009

[22] J ZhaoM Feng J Yang andH Petek ldquoThe superatom states offullerenes and their hybridization into the nearly free electronbands of fulleritesrdquo ACS Nano vol 3 no 4 pp 853ndash864 2009

[23] L-J Hou and Z L Miskovic ldquoImage force on a chargedprojectile moving over a two-dimensional strongly coupledYukawa systemrdquo Physical Review E vol 77 no 4 Article ID046401 7 pages 2008

[24] P Rinke K Delaney P Garcıa-Gonzalez and R W GodbyldquoImage states in metal clustersrdquo Physical Review A vol 70Article ID 063201 5 pages 2004

[25] T Inaoka ldquoMultipole excitations of an electron gas constrainedto a spherical surfacerdquo Surface Science vol 273 no 1-2 pp 191ndash204 1992

[26] P Longe ldquoExcitations in a spherical two-dimensional electrongasmdashapplication to the C

60moleculerdquo Solid State Communica-

tions vol 97 no 10 pp 857ndash861 1996

[27] J Tempere I F Silvera and J T Devreese ldquoMany-bodyproperties of a spherical two-dimensional electron gasrdquoPhysicalReview B vol 65 no 19 Article ID 195418 9 pages 2002

[28] J Tempere I F Silvera and J T Devreese ldquoMultielectronbubbles in helium as a paradigm for studying electrons onsurfaces with curvaturerdquo Surface Science Reports vol 62 no 5pp 159ndash217 2007

[29] C Yannouleas E N Bogachek and U Landman ldquoCollectiveexcitations of multishell carbon microstructures multishellfullerenes and coaxial nanotubesrdquo Physical Review B vol 53 no15 pp 10225ndash10236 1996

[30] C Yannouleas and U Landman ldquoStabilized-jellium descriptionof neutral and multiply charged fullerenes C

60rdquo Chemical

Physics Letters vol 217 no 3 pp 175ndash185 1994[31] X-B Wang and L-S Wang ldquoObservation of negative electron-

binding energy in a moleculerdquoNature vol 400 no 6747 p 2451999

[32] V K Voora L S Cederbaum and K D Jordan ldquoExistence ofa correlation bound s-type anion state of C

60rdquo The Journal of

Physical Chemistry Letters vol 4 no 6 pp 849ndash853 2013[33] G Gumbs and G R Aızin ldquoCollective excitations in a linear

periodic array of cylindrical nanotubesrdquo Physical Review B vol65 no 19 Article ID 195407 6 pages 2002

[34] M F Lin and K W-K Shung ldquoElementary excitations incylindrical tubulesrdquo Physical Review B vol 47 no 11 pp 6617ndash6624 1993

[35] M F Lin and K W-K Shung ldquoMagnetoplasmons and persis-tent currents in cylindrical tubulesrdquo Physical Review B vol 48no 8 pp 5567ndash5571 1993

[36] A Bahari and A Mohamadi ldquoDependence of plasmon excita-tion energy on filler material in interaction of charged particlewith filled nanotubesrdquo Nuclear Instruments and Methods inPhysics Research Section B vol 268 no 20 pp 3331ndash3334 2010

[37] A K Belyaev A S Tiukanov A I Toropkin V K Ivanov R GPolozkov andAV Solovrsquoyov ldquoPhotoabsorption of the fullereneC60and its positive ionsrdquo Physica Scripta vol 80 no 4 Article

ID 048121 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


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In Figure 1 we show a schematic of a charged particlelocalized in a spherical image state For the semi-infinitemetalvacuum interface [17] related image-potential stateshave been given a considerable amount of theoretical atten-tion over the years Additionally these states have beenobserved for pyrolytic graphite [18] and metal-supportedgraphene [19] Silkin et al [20] further highlighted the impor-tance of image states in graphene [21] by concluding that theinterlayer state in graphite is formed by the hybridizationof the lowest image-potential state in graphene in a similarway as it occurs in bilayer graphene [22 23] The significanceof image states was also discussed by Rinke et al in [24]We mention these facts to show why we were motivated tostudy image states for the spherical geometry Furthermorethe significance of the role played by the image-potentialhas led to the observation that for planar layered materialsstrongly dispersive interlayer states are present However theeigenstates for a spherical shell are nondispersive and so tooare the collective plasma modes [25ndash29] This difference initself leads to unique and interesting properties which wehave found for the image potential

We will consider the image states outside the SEG inthe following However our method may be extended in astraightforwardmanner to the case when the image states areinside the shell A relevant discussion of image states for C

60

with multiply charged anions [30 31] was recently given in[32] Here we only deal with the case of a singly chargedparticle In our formalism for obtaining a spherical imagestate we consider a spherical shell of radius 119877 whose centeris at the origin The background dielectric constant is 120598

1for

0 lt 119903 lt 119877 and 1205982for 119903 gt 119877 An electron gas is confined to the

surface of the sphere If a charge 119876 is located at (1199030 1205790 1206010) in

spherical coordinates then for 1199030gt 119877 the total electrostatic

potential is given by Φtot = Φext + Φind where Φext is theexternal potential due to the point particle and Φind is theinduced potential When 119903 lt 119877 we express the total potentialas follows

Φ(1)

tot (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1119860119871119903119871119884119871119872

(Ω)

Φext (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1

119903119871

gt

119903119871+1

gt

119884lowast

119871119872(Ω0) 119884119871119872

(Ω)

Φ(2)

ind (119903 120579 120601) = 4120587119896119876sum119871119872

1

2119871 + 1119861119871119903minus(119871+1)

119884119871119872

(Ω)

(1)

where 119896 = (41205871205980)minus1 with 120598

0the permittivity of free space

Also 119884119871119872(Ω) is a spherical harmonic and Ω is a solid angle

For 119903 gt 119877 the total potential is

Φ(2)

tot (119903 120579 120601)

= 4120587119896119876sum

119871119872

1

2119871 + 1

times [119903119871

lt

119903119871+1

gt

119884lowast

119871119872(Ω0) + 119861119871119903minus(119871+1)

]119884119871119872

(Ω)

(2)

Figure 1 Schematic illustration of a charged particle captured bythe image potential and orbiting around a buckyball The radiusof the orbit is determined by the dielectric constant within andsurrounding the shell as well as the angular momentum quantumnumber of the captured particles For semiconducting shells thelocalization is strong and the radius of the stable orbit can be a fewnanometers The localization is weak for metallic shells

On the surface of the sphere the boundary condi-tions are Φ(1)tot (119877 120579 120601) = Φ

(2)

tot (119877 120579 120601) and [1205981Φ(1)

tot (119903 Ω 120596)minus

1205982Φ(2)

tot (119903 Ω 120596)]|1015840

119903=119877= 4120587119896120590(119877 120579 120601 120596) where 120590(119877 120579 120601 120596) is

the induced surface charge density on the spherical shellExpanding120590 in terms of spherical harmonics and using linearresponse theory we find

120590 (119877 120579 120601 120596) = minus21198961198761198902

1198772sum

119871119872

1

2119871 + 1119860119871119877119871Π119871(120596) 119884119871119872

(Ω)

(3)

In this equationΠ119871(120596) is the SEG polarization function for 119871

an integer and given in terms of the Wigner 3-119895 symbol [2527]

Π119871(120596) = sum

ℓℓ1015840

1198910(119864ℓ) minus 1198910(119864ℓ1015840)

ℏ120596 + 119864ℓ1015840 minus 119864ℓ

(2ℓ + 1) (2ℓ1015840+ 1)

times (119897 1198971015840119871

0 0 0)

2

(4)

where 1198910(119864) = 1(1 + exp[(119864 minus 120583)119896

119861119879]) is the Fermi-

Dirac distribution function 119896119861is Boltzmannrsquos constant and

120583 is the chemical potential 119864ℓ= ℏ2ℓ(ℓ + 1)(2119898

lowast1198772) with

ℓ = 0 1 2 and 119898lowast is the electron effective mass The

induced potential Φ(2)ind(r 120596) outside the spherical shell maybe calculated to be

Φ(2)

ind (r 120596)

= 4120587119896119876sum

119871

[1205982

120576119871(120596 = 0)

minus1

2119871 + 1]1198772119871+1

119903119871+1

0

times1

119903119871+1sum

119872

119884lowast

119871119872(Ω0) 119884119871119872

(Ω)

(5)


where 120576119871(120596) = 119871(120598

1+ 1205982) + 120598

2+ (2119890

2119877)Π119871(120596) is the


using 119865(1199030) = minus119876120597Φ

(2)

ind(r)120597119903|r0

yielding

119865 (1199030) = 119896119876

2sum

119871

(119871 + 1) [(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1]1198772119871+1

1199032119871+3

0

(6)


Uim (1199030) equiv sum119871

U(119871)

im (1199030)

=1

2119876Φind (1199030 1205790 1206010)

=1198961198762

21199030

sum

119871

[(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1](119877

1199030

)

2119871+1

(7)


119881(119871)

eff (1199030 120579) = U

(119871)

im (1199030) +

ℏ2(1198712+ 119871 minus (14))

2119872lowast(1199030sin 1205790)2

(8)



0 for chosen 119871 119877




119871in (4) was 119898 = 025119898

119890where 119898

119890is





119871119903minus2(119871+1)

0+ 120573119871119903minus2

0 where 120572

119871



119871






(119871)





119871(120596)


1 we encounter








119871(120596 = 0)




L = 3

L = 2

L = 0

L = 1

R = 1nm1205981 = 24

r0R

Veff(ke2R

)

11 15 25 30

015

minus005

minus015

minus010

(a)

1205981 = 24

L = 3

L = 2

L = 4

R (nm)

Peak

V(

)eff

(ke2R)

025

020

005

14 16 18 20

(b)


1= 24 120598



n = 3n = 4

n = 2n = 1

02

01

11 24

minus01

minus02

r0R

11 24

r0R

Ψn L=2(r)

02

minus02


1= 24 120598

1= 1 and 119877 = 1 nm





1119877 + 119888

21198772 where 119888



1≫ 1205982


2120598119871) minus








n = 2n = 1

3nm 3nm

(a)

n = 3 n = 4

3nm 3nm

(b)





2= 1 in the

surrounding medium






Acknowledgment


References








































60rdquo Chemical











ID 048121 2009






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 120576119871(120596) = 119871(120598

1+ 1205982) + 120598

2+ (2119890

2119877)Π119871(120596) is the


using 119865(1199030) = minus119876120597Φ

(2)

ind(r)120597119903|r0

yielding

119865 (1199030) = 119896119876

2sum

119871

(119871 + 1) [(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1]1198772119871+1

1199032119871+3

0

(6)


Uim (1199030) equiv sum119871

U(119871)

im (1199030)

=1

2119876Φind (1199030 1205790 1206010)

=1198961198762

21199030

sum

119871

[(2119871 + 1)1205982

120576119871(120596 = 0)

minus 1](119877

1199030

)

2119871+1

(7)


119881(119871)

eff (1199030 120579) = U

(119871)

im (1199030) +

ℏ2(1198712+ 119871 minus (14))

2119872lowast(1199030sin 1205790)2

(8)



0 for chosen 119871 119877




119871in (4) was 119898 = 025119898

119890where 119898

119890is





119871119903minus2(119871+1)

0+ 120573119871119903minus2

0 where 120572

119871



119871






(119871)





119871(120596)


1 we encounter








119871(120596 = 0)




L = 3

L = 2

L = 0

L = 1

R = 1nm1205981 = 24

r0R

Veff(ke2R

)

11 15 25 30

015

minus005

minus015

minus010

(a)

1205981 = 24

L = 3

L = 2

L = 4

R (nm)

Peak

V(

)eff

(ke2R)

025

020

005

14 16 18 20

(b)


1= 24 120598



n = 3n = 4

n = 2n = 1

02

01

11 24

minus01

minus02

r0R

11 24

r0R

Ψn L=2(r)

02

minus02


1= 24 120598

1= 1 and 119877 = 1 nm





1119877 + 119888

21198772 where 119888



1≫ 1205982


2120598119871) minus








n = 2n = 1

3nm 3nm

(a)

n = 3 n = 4

3nm 3nm

(b)





2= 1 in the

surrounding medium






Acknowledgment


References








































60rdquo Chemical











ID 048121 2009






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




L = 3

L = 2

L = 0

L = 1

R = 1nm1205981 = 24

r0R

Veff(ke2R

)

11 15 25 30

015

minus005

minus015

minus010

(a)

1205981 = 24

L = 3

L = 2

L = 4

R (nm)

Peak

V(

)eff

(ke2R)

025

020

005

14 16 18 20

(b)


1= 24 120598



n = 3n = 4

n = 2n = 1

02

01

11 24

minus01

minus02

r0R

11 24

r0R

Ψn L=2(r)

02

minus02


1= 24 120598

1= 1 and 119877 = 1 nm





1119877 + 119888

21198772 where 119888



1≫ 1205982


2120598119871) minus








n = 2n = 1

3nm 3nm

(a)

n = 3 n = 4

3nm 3nm

(b)





2= 1 in the

surrounding medium






Acknowledgment


References








































60rdquo Chemical











ID 048121 2009






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




n = 2n = 1

3nm 3nm

(a)

n = 3 n = 4

3nm 3nm

(b)





2= 1 in the

surrounding medium






Acknowledgment


References








































60rdquo Chemical











ID 048121 2009






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





























60rdquo Chemical











ID 048121 2009






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Strongly Localized Image States of ...

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