Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY NANOSCIENCE AND NANOTECHNOLOGY
Pedro Ludwig Hernández Martínez
Alexander Govorov
Hilmi Volkan Demir
Understanding and
Modeling Förster-type
Resonance Energy
Transfer (FRET)
FRET from Single Donor
to Single Acceptor and
Assemblies of Acceptors,
Vol. 2
123
SpringerBriefs in Applied Sciences
and Technology
Nanoscience and Nanotechnology
Series editor
Hilmi Volkan Demir, Nanyang Technological University, Singapore, Singapore
Nanoscienceandnanotechnologyoffermeanstoassembleandstudysuperstructures,
composed of nanocomponents such as nanocrystals and biomolecules, exhibiting
interestinguniqueproperties.Also,nanoscienceandnanotechnologyenablewaysto
makeandexploredesign-basedartificialstructuresthatdonotexistinnaturesuchas
metamaterials and metasurfaces. Furthermore, nanoscience and nanotechnology
allow us to make and understand tightly confined quasi-zero-dimensional to
two-dimensionalquantumstructuressuchasnanoplateletsandgraphenewithunique
electronic structures. For example, today by using a biomolecular linker, one can
assemblecrystallinenanoparticlesandnanowiresintocomplexsurfacesorcomposite
structureswithnewelectronicandopticalproperties.Theuniquepropertiesofthese
superstructures result from the chemical composition and physical arrangement of
such nanocomponents (e.g., semiconductor nanocrystals, metal nanoparticles, and
biomolecules).Interactionsbetweentheseelements(donorandacceptor)mayfurther
enhancesuchpropertiesoftheresultinghybridsuperstructures.Oneoftheimportant
mechanisms is excitonics (enabled through energy transfer of exciton-exciton
coupling) and another one is plasmonics (enabled by plasmon-exciton coupling).
Also, in such nanoengineered structures, the light-material interactions at the
nanoscalecanbemodifiedandenhanced,givingrisetonanophotoniceffects.
These emerging topics of energy transfer, plasmonics, metastructuring and the
likehavenowreachedalevelofwide-scaleuseandpopularitythattheyarenolonger
the topics of a specialist, but now span the interests of all “end-users” of the new
findings in these topics including those parties in biology, medicine, materials
scienceandengineerings.Manytechnicalbooksandreportshavebeenpublishedon
individual topics in the specialized fields, and the existing literature have been
typicallywritteninaspecializedmannerforthoseinthefieldofinterest(e.g.,foronly
the physicists, only the chemists, etc.). However, currently there is no brief series
available,whichcoversthesetopicsinawayunitingallfieldsofinterestincluding
physics,chemistry,materialscience,biology,medicine,engineering,andtheothers.
The proposed new series in “Nanoscience and Nanotechnology” uniquely
supports this cross-sectional platform spanning all of these fields. The proposed
briefs series is intended to target a diverse readership and to serve as an important
reference for both the specialized and general audience. This is not possible to
achieveundertheseriesofanengineeringfield(forexample,electricalengineering)
or under the series of a technical field (for example, physics and applied physics),
whichwouldhavebeenveryintimidatingforbiologists,medicaldoctors,materials
scientists, etc.
TheBriefsinNANOSCIENCEANDNANOTECHNOLOGYthusoffersagreat
potential by itself, which will be interesting both for the specialists and the
non-specialists.
More information about this series at http://www.springer.com/series/11713
á í
Pedro Ludwig Hern ndez Mart nez
Alexander Govorov
Hilmi Volkan Demir
Understanding and Modeling
ö
F rster-type Resonance
Energy Transfer (FRET)
FRET from Single Donor to Single Acceptor
and Assemblies of Acceptors, Vol. 2
123
PedroLudwig Hernández Martínez Hilmi VolkanDemir
Schoolof Physical andMathematical Department ofElectrical andElectronics
Sciences, LUMINOUS! Centreof Engineering, Departmentof Physics, and
Excellencefor Semiconductor Lighting UNAM—National Nanotechnology
andDisplays,TPI—The Institute of ResearchCentreandInstituteofMaterials
Photonics ScienceandNanotechnology
NanyangTechnological University Bilkent University
Singapore Ankara
Singapore Turkey
Alexander Govorov and
Department ofPhysics andAstronomy
Schoolof Electrical andElectronic
OhioUniversity
Engineering, Schoolof Physical and
Athens, OH
MathematicalSciences, LUMINOUS!
USA
Centreof Excellence for Semiconductor
LightingandDisplays,TPI—TheInstitute
ofPhotonics
NanyangTechnological University
Singapore
Singapore
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISSN 2196-1670 ISSN 2196-1689 (electronic)
Nanoscience andNanotechnology
ISBN978-981-10-1871-8 ISBN978-981-10-1873-2 (eBook)
DOI 10.1007/978-981-10-1873-2
LibraryofCongressControlNumber:2016943801
©TheAuthor(s)2017
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Contents
1 Applying Förster-Type Nonradiative Energy Transfer Formalism
to Nanostructures with Various Directionalities: Dipole Electric
Potential of Exciton and Dielectric Environment ... .... ..... .... 1
1.1 Spherical Geometry: Nanoparticle Case .... .... .... ..... .... 1
1.2 Cylindrical Geometry: Nanowire Case . .... .... .... ..... .... 2
1.3 Planar Geometry: Quantum Well Case. .... .... .... ..... .... 5
Reference .. .... .... .... ..... .... .... .... .... .... ..... .... 8
2 Förster-Type Nonradiative Energy Transfer Rates for
Nanostructures with Various Dimensionalities . .... .... ..... .... 9
2.1 Cases of Förster-Type Energy Transfer to an Nanoparticle:
NP → NP, NW → NP, and QW → NP... .... .... ..... .... 10
2.2 Cases of Förster-Type Energy Transfer to an Nanowire:
NP → NW, NW → NW, and QW → NW. .... .... ..... .... 14
2.3 Cases of Förster-Type Energy Transfer to a Quantum Well:
NP → QW, NW → QW, and QW → QW. .... .... ..... .... 18
2.4 Example: Energy Transfer Between Nanoparticles
and Nanowires .. .... ..... .... .... .... .... .... ..... .... 21
2.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 23
References.. .... .... .... ..... .... .... .... .... .... ..... .... 25
3 Nonradiative Energy Transfer in Assembly of Nanostructures . .... 27
3.1 Energy Transfer Rates for Nanoparticle, Nanowire,
or Quantum Well to 1D Nanoparticle Assembly . .... ..... .... 29
3.2 Energy Transfer Rates for Nanoparticle, Nanowire,
or Quantum Well to 2D Nanoparticle Assembly . .... ..... .... 30
3.3 Energy Transfer Rates for Nanoparticle, Nanowire,
or Quantum Well to 3D Nanoparticle Assembly . .... ..... .... 32
3.4 Energy Transfer Rates for Nanoparticle, Nanowire,
or Quantum Well to 1D Nanowire Assembly.... .... ..... .... 33
v
vi Contents
3.5 Energy Transfer Rates for Nanoparticle, Nanowire,
or Quantum Well to 2D Nanowire Assembly.... .... ..... .... 35
3.6 Summary .. .... .... ..... .... .... .... .... .... ..... .... 36
References.. .... .... .... ..... .... .... .... .... .... ..... .... 37
Appendix A... .... .... .... ..... .... .... .... .... .... ..... .... 39
Chapter 1
ö
Applying F rster-Type Nonradiative
Energy Transfer Formalism
to Nanostructures with Various
Directionalities: Dipole Electric Potential
of Exciton and Dielectric Environment
In this chapter, we present analytical equations for the exciton electric potential
inside and outside a nanostructure; including analytical expressions, for the long
distanceapproximation,whicharederivedfortheoutsideelectricpotential.Finally,
theeffectivedielectricconstantexpressions,forthislimit,areobtained.Thischapter
is reprinted (adapted) with permission from Ref. [1]. Copyright 2013 American
Chemical Society.
1.1 Spherical Geometry: Nanoparticle Case
The electric potential for an exciton in the a-direction ða¼x;y;zÞ, illustrated in
Fig. 1.1a, is given by
(cid:1) (cid:3) (cid:1) (cid:3)
ed a^(cid:2)r 2ðe (cid:3)e Þ r3
Uin ¼ exc 1þ NP 0 ð1:1Þ
a e r3 e þ2e R3
NP NP 0 NP
(cid:1) (cid:3)(cid:1) (cid:3)
ed 3e r(cid:2)a^
Uout ¼ exc NP ð1:2Þ
a e e þ2e r3
NP NP 0
where e and e are the nanoparticle (NP) and medium dielectric constants,
NP 0
respectively. The electric potential is the same in any direction because of the
spherical symmetry of the NP. In the long distance approximation the outside
electric potential can be written as
©TheAuthor(s)2017 1
P.L.HernándezMartínezetal.,UnderstandingandModelingFörster-type
ResonanceEnergyTransfer(FRET),NanoscienceandNanotechnology,
DOI10.1007/978-981-10-1873-2_1
2 1 ApplyingFörster-TypeNonradiativeEnergy…
(cid:1) (cid:3)
ed r(cid:2)a^
Uout ¼ exc ð1:3Þ
a e r3
eff
where e is the effective dielectric constant given by
eff
e þ2e
e ¼ NP 0 ð1:4Þ
eff
3
1.2 Cylindrical Geometry: Nanowire Case
In this case, the electric potential for an a-exciton ða¼x;y;zÞ, illustrated in
Fig. 1.1a, is
(a) (b)
a.u) 5.0x10-5 OTouttasli dEele Ecltericct rPico Pteontteianlt iNalP N (Pa. u(a).u)
P ( 4.0x10-5
N
ntial 3.0x10-5
e
ot
P 2.0x10-5
c
ectri 1.0x10-5
El
Total 0.00 1 2 3 4 5 6
z (nm)
(c) (d)
NW (a.u) 45..00xx1100--55 OTouttasli dEele Ecltericct rPico Pteontteianlt iNalW N W(a. u(a).u) QW (a.u) 45..00xx1100--55 OTouttasli dEele Ecltericct rPico Pteontteianlt iQalW Q W(a. u(a).u)
Total Electric Potential 231...000xxx1110000.0---5550 1 2 3 4 5 6 Total Electric Potential 231...000xxx1110000.0---5550 5 10 15 20 25
z (nm) z (nm)
Fig.1.1 aSchematicofanexcitoninanNP,anNW,andaQW.Redcirclerepresentsanexciton
inthea-direction.R istheNP(NW)radius.L istheQWcappinglayerthickness.b,c,
NP(NW) QW
anddElectricpotentialalongthe“z”axisforaz-exciton.Totalandlongdistanceapproximation
electricpotentialforthez-excitoninside:banNP;canNW;anddaQW[Reprinted(adapted)
withpermissionfromRef.[1](Copyright2013AmericanChemicalSociety)]
1.2 CylindricalGeometry:NanowireCase 3
Z
X (cid:4) (cid:5)
Uian ¼Uaþ eimue(cid:3)ikyAamðkÞImðjkjqÞ dk ð1:5Þ
m
Z
X (cid:4) (cid:5)
Uoaut ¼Uaþ eimue(cid:3)ikyBamðkÞKmðjkjqÞ dk ð1:6Þ
m
whereImðjkjqÞandKmðjkjqÞarethemodifiedBesselfunctionsoforderm,andUa
is the a-exciton electric potential. After applying the boundary conditions at the
surface of the nanowire (NW), the coefficients Aa and Ba are
m m
(cid:1) (cid:3)
K ðjkjR Þ
AaðkÞ¼ m NW BaðkÞ ð1:7Þ
m I ðjkjR Þ m
m NW
2ðe (cid:3)e ÞgaðjkjÞ
BaðkÞ¼ (cid:6) j(cid:7)kj 0 NW m ð1:8Þ
m
eNW KImmððjjkkjjRRNNWWÞÞ ImðjkjRNWÞþe0KmðjkjRNWÞ
where I ðjkjR Þ; K ðjkjR Þ, and gaðjkjÞ are defined as
m NW m NW m
ImðjkjRNWÞ¼Im(cid:3)1ðjkjRNWÞþImþ1ðjkjRNWÞ ð1:9Þ
KmðjkjRNWÞ¼Km(cid:3)1ðjkjRNWÞþKmþ1ðjkjRNWÞ ð1:10Þ
Z2p Z1 (cid:8) (cid:9)
gaðjkjÞ¼ 1 dudye(cid:3)imueiky @Ua ð1:11Þ
m ð2pÞ2 0 (cid:3)1 @q q¼RNW
For an exciton in the y-direction (along the cylinder axis), the coefficient By
m
becomes
0 1
(cid:1) (cid:3)
ByðkÞ¼ edexc ðe (cid:3)e Þ i jkj@(cid:6) 1 (cid:7) A ð1:12Þ
0 eNW NW 0 p KK01ððjjkkjjRRNNWWÞÞII10ððjjkkjjRRNNWWÞÞ eNWþe0
with an electric potential given by
(cid:1) (cid:3) Z
(cid:4) (cid:5)
ed y
Uout ¼ exc þ e(cid:3)ikyByðkÞK ðjkjqÞ dk ð1:13Þ
y eNW ðq2þy2Þ32 0 0
In the long distance approximation, the coefficient By and the outside electric
m
potential are simplified as
