quantum electrodynamics of resonance energy transfer in … · 2016. 3. 17. · quantum...
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Quantum electrodynamics of resonance energy transfer in nanowire systems∗
Dilusha Weeraddana, Malin Premaratne†
Advanced Computing and Simulation Laboratory (AχL),Department of Electrical and Computer Systems Engineering,
Monash University, Clayton, Victoria, 3800, Australia
David L. Andrews‡
School of Chemistry, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, United Kingdom(Dated: February 5, 2016)
Nonradiative resonance energy transfer (RET) provides the ability to transfer excitation energybetween contiguous nanowires (NWs) with high efficiency under certain conditions. Nevertheless,the well established Forster formalism commonly used to represent RET was developed for energytransfer primarily between molecular blocks (i.e. from one molecule, or part of a molecule, toanother). Although deviations from Forster theory for functional blocks such as NWs have beenpreviously studied, the role of the relative distance, orientation of transition dipole moment pairs,and the passively interacting matter on electronic energy transfer, are to a large extent unknown.Thus, a comprehensive theory that models RET in NWs is required. In this context, analyticalinsights to give a deeper and more intuitive understanding of the distance and orientation dependenceof RET in NWs is presented within the framework of quantum electrodynamics. Additionally,the influence of an included intermediary on the rate of excitation energy transfer is illustrated,embracing indirect energy transfer rate and quantum interference. The results deliver equationsthat afford new intuitions into the behavior of virtual photons. In particular, results indicate thatRET efficiency in a NW system can be explicitly expedited or inhibited by a neighbouring mediator,depending on the relative spacing and orientation of NWs.
PACS numbers: 31.30.J-,78.70.-g,33.50.Hv,78.67.De,78.67.Uh,78.67.Hc
I. INTRODUCTION
Radiationless near-field transportation of energy froma donor particle, initially in its excited electronic state, toan acceptor in its ground state, is of considerable interestfor diverse applications in science and engineering. Be-yond wavefunction overlap, a compelling photo-physicalprocess known as resonance energy transfer (RET) gainscontrol. RET, also often known as electronic energytransfer (EET) [1], has been extensively exploited inartificial light harvesting antenna devices [2–4], spasers[5, 6] and especially in biology as a spectroscopic ruler tostudy conformational dynamics [7]. The study of RETin nanostructures has recently envisaged various prospec-tive applications ranging from solar cell systems [8–10]to optical switching [11–14].
Nanotechnology offers means to study and fabricatenanostructures with large aspect ratios and small diame-ters, commonly termed nanowires (NWs) [15, 16]. Thesehave been the focus of extensive research during the lastfew decades [17–20]. Their length is sufficiently large foreasy manipulation as building blocks in fabricating su-perstructures. Electronic interactions between NWs atthe nanoscale elevate the properties of a superstructure.Here, too, one of the important mechanisms for strong
∗† [email protected],[email protected]‡ [email protected]
interaction is the RET, which results from Coulomb in-teraction between excitons confined in NWs [21–23].
Numerous studies have previously been reported onresonance energy transfer mechanism and related effectsin systems consisted of NWs [24–26]. In many respects,the process of RET is well illustrated by semiclassical the-ories of radiationless energy transfer [26]. In this formof representation, the phenomenon is commonly consid-ered as a first order perturbative process. However, inreality, RET processes are fully quantum mechanical innature and they are formally described within the frame-work of quantum electrodynamics (QED) [27–33]. Here,the direct energy transfer emerges as a second-order pro-cess, mediated by the intermolecular propagation of vir-tual photons. The higher order reflects taking due ac-count of causality and retardation; the electronic decayof one component and the resultant excitation of an-other, at a different point in space, cannot be simulta-neous. Moreover, indirect energy transfer emerges via avicinal neighbouring object occurs as a fourth-order pro-cess [34]. Therefore, QED has been widely applied toelectronic coupling between the donor and acceptor overall distances, producing a unified theory that reconcilesboth RET and radiative energy transfer as the short- andlong-range asymptotics of one mechanism [35, 36].
The main purpose of this paper is to investigate RETin NW systems analytically, by developing a comprehen-sive quantum electrodynamical analysis including state-sequence methodology. Here, we extend our recent anal-ysis [37], by relaxing uniaxial constraint imposed on thecoupling photon. Of particular interest to this study is
2
the distance and orientational dependence of a pair ofNWs, embedded within another vicinal NW. Interest-ingly, a path is established towards a formalism that willallow the identification of specific attributes to expediteor inhibit electronic energy transfer, providing a detailedpicture and understanding of RET in NWs. In Sec. IIof this paper, the background molecular QED theory ofRET is reviewed for both direct and third-body modifiedenergy transfer. The direct and indirect RET for NWsare presented in Sec. III and Sec. IV respectively. Re-sults are discussed in Sec.V followed by the conclusionsin Sec. VI.
II. EXCITATION ENERGY TRANSFER: QEDPERSPECTIVE
A. System Hamiltonian
It is appropriate to begin with the generic quantumenergy operator for a system comprising of a number ofparticles and the radiation field, described by multipo-lar formulation of molecular QED. This Hamiltonian isexpressible as follows [28]:
Htotal =∑ξ
Hint(ξ) +∑ξ
Hmat(ξ) +Hrad (1)
where Hint(ξ) is the interaction Hamiltonian of the par-ticle ξ with the radiation field, Hmat(ξ) is the matterHamiltonian of the particle ξ, and Hrad represents thesecond-quantized radiation field Hamiltonian of the inci-dent light field. The Hint(ξ) describes interactions suchas absorption and emission, (equally for either real orvirtual photons), and it is described using the multipolarHamiltonian in the dipole approximation [28]
Hint = −µ(ξ).E(Rξ) (2)
where the interaction Hamiltonian compromises contri-butions for each species ξ located at Rξ, the µ(ξ) is theelectric-dipole moment operator and E(Rξ) is the oper-ator for the electric displacement field at the specifiedlocation Rξ.
The transfer of energy beyond significant wavefunctionoverlap, generally entails a mechanism known as RET,mainly associated with electric dipole-electric dipole(E1-E1) coupling. In this event, an excited donor(D) emits excitation energy which is transferred to anacceptor (A); the donor falls back to its ground statewhile the acceptor is excited corresponds to the process
Dα +A0 −→ D0 +Aβ .
Here, superscripts denote donor and acceptor states.In the QED theory, this process is mediated by avirtual photon, coupling the donor decay and acceptor
Y
α 0
p
α β
0 β
p
0 0
Y
β0β0
α 0α 0
(b) (c)
pp
X
X
(a)
FIG. 1. State-sequence diagram (a) for direct RET. In eachbox, the state of donor D is represented by the symbol on theleft and the state of acceptor A by the symbol on the right,p denotes virtual photon. The Feynman diagrams show thetime-order: (b) (X)→(Y); (c) (Y)→(X).
excitation through its creation and subsequent annihi-lation. The quantum amplitude for RET is based ona Schrodinger state vector representation of quantumdynamics, where matrix element for RET is representedas a sum of differently time-ordered contributions [28].However, one can apply alternative formulations, forexample, in terms of a density matrix in Liouville space[38]. The state sequence representation of the two timeordered contributions is depicted in Fig.1 (a) and thecorresponding Feynman diagrams [39] are depictedin Fig.1 (b),(c). As shown in the figures, tracing theupper pathway; the virtual photon is created at D andannihilated at A. The lower path depicts the case wherevirtual photon is created at A and annihilated at D,as is also consistent with the time-energy uncertaintybasis for conventional time orderings. As both pathslead to the same final state, calculation of the full RETquantum amplitude requires their summation.
B. Direct energy transfer and the influence of anincluded intermediary
The standard starting point for the development of theQED formalism of RET is the Fermi’s golden rule rateequation. For a system proceeding from its initial stateI to final state F , the transfer rate (probability per unit
3
W
X
Y
Z
α 0 0
p
0 0 0
p
α r 0
q
α r 0
q
α 0 β
0 r 0
p q
0 0 0
p q
0 0 β
p q
α 0 0
p q
α r β
α r β
X
Y
Z
W
Y
Z
Z
W
q
0 0 0
q
0 r 0
p
0 r 0
p
α 0 β
Y
Y
X
W
ZX
Z
X
W
Z
X
Y
W
W
Z
Y
X
W
X
0 0 β
Y
Radiation
D M A
I
R
S
T
F
FIG. 2. State-sequence diagram for the indirect RET. Timeprogresses left to right, each of the 16 boxes representing oneof the possible overall states of the system in one of the fivestages I, R, S, T, F , p, q denote virtual photons
time) is explicitly given by,
Γtran =2π
}|MFI |2ρ (3)
where ρ represents the density of final states, and MFI isthe matrix element connecting the initial and final statesof the system (sometimes loosely termed the quantumamplitude of the energy transfer process), which has theperturbation expansion
MFI = 〈F |Hint|I〉+∑R
〈F |Hint|R〉〈R|Hint|I〉EI − ER
+
∑R,S
〈F |Hint|R〉〈R|Hint|S〉〈S|Hint|I〉(EI − ER)(EI − ES)
+
∑R,S,T
〈F |Hint|T 〉〈T |Hint|S〉〈S|Hint|R〉〈R|Hint|I〉(EI − ER)(EI − ES)(EI − ET )
+ ..
(4)
where I,F are initial, final states respectively andR,S,T denote intermediate states. If ζ = I,F,R,S,T,then Eζ denotes the corresponding eigenenergy.
For the case of direct energy transfer, electronic energytransfer from a donor to an acceptor in the absence of asurrounding medium is calculated from the second termin the time-dependent perturbation series given in Eq.4. Thus, the general formula for the direct interactionbetween two nano-particles can be expressed as
MdFI =
µ0αn (D)µβ0m (A)
2V ε0
∑a
E∗an(RD)Eam(RA)p
k − p−
Ean(RD)E∗am(RA)p
k + p
(5)
Here a concise notation for the transition dipole mo-ments is introduced, e.g. µ0α(D) ≡ 〈D0|µ(D)|Dα〉, anda = {p, λ} represents the photonic modes, V is an arbi-trary quantization volume and i, j are cartesian coordi-nates; also p is the corresponding photon wave numberwhich need not be equal to k.
The exchange of an additional virtual photon with anincluded intermediary is the lowest-order coupling pro-cess that promotes RET to third-body-mediated RET[40–42]. In our previous work, we have described howthe four distinct matter-radiation interaction events (W,X, Y, Z in Fig. 2) modify the direct RET rate to third-body-modified RET [37], Fig. 2 illustrates the systemsevolution through all five stages in one state-sequencediagram. Essentially, at each event, one particle under-goes a transition between states 0, α, r, β and one photonis either created or annihilated by giving rise to 4!=24Feynman diagrams. Therefore, the coupling matrix el-ement for the case of third-body-mediated is calculatedfrom the fourth term in the time dependent perturbationseries stated in Eq. 4
M iFI = −
∑a
∑b
( }cp2V ε0
)( }cq2V ε0
)µ0αn (D)µβ0m (A)αkl(M ; k)
×{Ean(RD)E∗ak(RM )Ebm(RA)E∗bl(RM )
(p− k)(q − k)+Ean(RD)E∗ak(RM )E∗bm(RA)Ebl(RM )
(p− k)(q + k)+
E∗an(RD)Eak(RM )Ebm(RA)E∗bl(RM )
(p+ k)(q − k)+E∗an(RD)Eak(RM )E∗bm(RA)Ebl(RM )
(p+ k)(q + k)
}(6)
αkl(M ; k) =∑r
µr0k (M)µ0rl (M)
{ 1
Er0 − }ck+
1
Er0 + }ck
}(7)
4
Donor (D)ħk
θDA
R
θD
θA
D
A
Donor transition
dipole plane
Acceptor transition
dipole plane
Acceptor (A)
(a)
(b)
FIG. 3. Schematics for the direct resonance energy transfer(a) NW to NW; (b) the orientational factors in Eq.(17), θDand θA are the angles formed by the donor and acceptor tran-sition dipole moments with respect to the displacement vectorR, and θDA, the angle between the two transition dipole mo-ments.
Here, also, αkl(M ; k) is the dynamic polarizability ofparticle M [43, 44].
The total matrix element is given by the sum of secondand fourth terms in Eq. (4), and the transfer rate is thenseen to be a sum of three terms, namely
Γtotaltran =2π
}|Md
FI +M iFI |2ρ
=2π
}[|Md
FI |2 + |M iFI |2 + 2ReM
d
FIMiFI ]ρ
(8)
where the third term is a quantum interference contri-bution to the rate arising from both direct and indirect
mechanisms.
III. DIRECT COUPLING OF TWO NANOWIRESYSTEM
We consider a system which comprises of the radiationfield, and two NWs of length L separated by a distance R(center to center separation) as depicted in Fig. 3 (a). Incontrast to our previous work [37], the significance is onremoval of the directional constraint on the coupling pho-ton. Owing to the cylindrical symmetry of NWs, it is con-venient to model EM waves using Hankel function of or-der n [25, 45, 46]: e(λ)(p)
∑nHn(pR)einα, where e(λ)(p)
is the polarization vector R and α are the radial and an-gular coordinates respectively. Directly substituting intoEq. (5) and converting the discrete summation over vir-tual photon wave vector [25, 46],
∑p ⇒
∫A
(2π)2 d2p to
an integral yields
MdFI =
µ0αn (D)µβ0m (A)
8π2Lε0(−∇2δnm +∇n∇m) (9)∫ ∞
0
∫ 2π
0
H(1)0 (pR)
k − p− H
(2)0 (pR)
k + pdφdp
Expanding the Hankel function, performing contourintegration and by the residue theorem
MdFI =
µ0αn (D)µβ0m (A)
2πLR2ε0(−∇2δnm +∇n∇m) (10)∮
c
ikRY0(c) + c2J0(kR)
(kR− c)(kR+ c)dc
MdFI =
µ0αn (D)µβ0m (A)
4Lε0(−∇2δnm +∇n∇m) (11)
{Y0(kR)− iJ0(kR)}
where Y0(kR), J0(kR) are zero order second and firstkind of Bessel functions respectively.
MdFI =
µ0αn (D)µβ0m (A)
4Lε0
[kδnm
{− Y2(kR) +
Y1(kR)
kR
}− k{Y1(kR)
(δnm − RnRmR
)+ (12)
kRnRm
(− Y2(kR) +
Y1(kR)
kR
)}− ikδnm
{− J2(kR) +
J1(kR)
kR
}+ ik
{J1(kR)
(δnm − RnRmR
)+
kRnRm
(− J2(kR) +
J1(kR)
kR
)}]
We now deploy the asymptotic series for 0 < kR�√n+
1 and n 6= 0 [47]
Yn(kR) ∼ −Γ(n)
π
( 2
kR
)n+
1
Γ(n+ 1)
(kR2
)ncot(nπ)
(13)
5
Jn(kR) ∼ 1
Γ(n+ 1)
(kR2
)n(14)
By applying near-field limits 0 < kR � 1 on Eq.12yields
MdFI =
µ0αn (D)µβ0m (A)
4Lε0
{Γ(1)
π
( 2
kR
)(δnm − RnRmR
)+kRnRm
(− Γ(2)
π
( 2
kR
)2+
Γ(1)
πkR
( 2
kR
))}(15)
where Γ(n) is the standard Gamma function which isexpressed in
∫∞0tn−1e−tdt = (n− 1)Γ(n− 1). Therefore,
Eq.15 becomes
MdFI =
µ0αn (D)µβ0m (A)
2πLR2ε0(δnm − 2RnRm)
=κDA|µ0α
n (D)||µβ0m (A)|2πLR2ε0
(16)
where |κDA|2 is an orientation factor. |κDA|2 describesthe influence of the relative orientations of the transitiondipole moments of the donor and acceptor NWs, as givenby
κDA = µ0αn (D)(δnm − 2RnRm)µβ0m (A)
= cos(θDA)− 2 cos(θD) cos(θA)(17)
where θD is the angle between donor and separation vec-tor (R), and θA is the angle between acceptor and R.θDA is the angle between donor and acceptor NWs ( Fig.3 (b)).
The energy transfer rate can be obtained from Eq. (3)and emerges in the following simple form:
Γdtran =|κDA|2|µ0α
n (D)|2|µβ0m (A)|2ρ2πL2R4ε02}
(18)
A. Distance dependance
From Eq. (18), it is observed that the direct energytransfer in NWs exhibits an inverse fourth power de-pendence on the separation distance. This is becausequantum amplitude is inversely proportional to thespacing between donor and acceptor (Md
FI ∝ R−2), andthe energy transfer rate is proportional to the squaremodulus of the quantum amplitude of RET process(Γtran ∝ |Md
FI |2). The plots of Eq. (16), Eq. (18)are shown in Fig. 4 and Fig. 5 respectively. In thedevelopment of the plots, the following values were used[48]: |µ0α(D)| = |µβ0(A)| = 5×10−30 C m ; ρ = 2×1025
J−1. Fig. 4 (a) shows the functional dependence ofresonant dipole-dipole interaction (RDDI) for various
values of the donor-acceptor separation distance (R).It is observed that due to the behaviour of the virtualphoton propagation in a 2D realm, the RDDI decreaseswith R by following an inverse square power dependence,resulting in a gradual decline of the energy transferefficiency with the distance as illustrated in Fig. 5 (a).Fig. 4 (b) and Fig. 5 (b) compares the NW-to-NWcoupling and transfer efficiency with quantum dot (QD)to QD as a function of R, where distance dependanceof latter is sharper than former, signifying the highphysical loses in the transmission medium due to thespherical symmetry of the QD.
B. Orientational dependence
The relative orientation of the donor and acceptor, andtheir individual orientations with respect to relative sep-aration vector (see Fig. 3 (b)), influence the RDDI inEq.(16) in a variety of different ways, as depicted in Fig.4 (c). Here, we explore the variation of RDDI at differentθD values ranging from 0 to π
2 for fixed θDA (at 0 andπ), θA (at 0). The coupling matrix element reaches itspeak when the donor the is orthogonal to the displace-ment vector (θD = π
2 ), θDA = 0 and also when θD = 0,θDA = π. Further, it becomes minimum when θD = π
2 ,θDA = π and also θD = 0, θDA = 0. Therefore, thetransfer rate strongly depends on the orientation factor.It is clear from the Eq.(17) that κ2 does not does notchange if following operations are performed [29]:
1. flip the donor transition moment (µ0α(D) →−µ0α(D)),
2. flip the acceptor transition moment (µβ0(A) →−µβ0(A)),
3. allow the donor and acceptor to trade places (R→−R),
4. interchange the donor and acceptor transition mo-ments (µ0α(D)↔ µβ0(A)).
The angles θDA, θD, θA are dependent on one another.Therefore, we consider three cases to study the orienta-tional dependence of RET rate.
1. case 1: θDA = 0, θD = 0, θA = 0; κDA = −1, (Fig.6 (a))
2. case 2: θDA = 0, θD = π2 , θA = π
2 ; κDA = 1, (Fig.6 (b))
3. case 3: θDA = π2 , θD = π
2 , θA = π2 ; κDA = 0, (Fig.
6 (c))
As can be anticipated, the orientation factor becomesmost favourable when the transition dipole moments are
6
2
6
10
14
18
4 8 12 16 20
−18
−12
x 10
QD-QD
NW-NW
R (nm) R (nm)
(a) (b)
M (
ev
)
log
(M)
4 8 12 16 20
−1
0
1
x 10−7
10 30 50 70 90
−7
θD (degrees)
(c)
M (
ev
)
θDA =0
θDA =π
θD
DA
θDDA
(d)
θDA =0
θDA =π
FIG. 4. RDDI strength in direct RET: (a) as a function of the relative distance (R) between D and A; (b) NW to NW andQD to QD with respect to R; (c) variation as a function of θD when θDA = 0 and θDA = π; (d) corresponding schematics forthe relative orientations depicted in (c).
4 8 12 16 20
0
4
8
12
16
10
15
20
25
0
1
2
3
4x 10
-1
4 8 12 16 20 10 30 50 70 90
R (nm) R (nm) θD (degrees)
(a) (b) (c)
En
ere
gy
tra
nsf
er
rate
(n
s-1
log
(En
ere
gy
tra
nsf
er
rate
)
QD-QD
NW-NW5 nmR =
6 nmR =
7 nmR =
8 nmR =
En
ere
gy
tra
nsf
er
rate
(n
s-1)
)
FIG. 5. Direct RET rate: (a) as a function of the relative distance (R) between D and A; (b) NW to NW and QD to QD withrespect to R; (c) as a function of θD for four different relative distances (5 nm, 6 nm, 7 nm, 8 nm) between D and A.
7
θDA = π/2
D
θD=θA=0 A
θDA = 0
D
A
D
A
(a) (b)
(c)
θDA = 0
θD=θA=π/2
θD=θA= π/2
κDA = -1 κDA =1
κDA = 0
FIG. 6. Direct RET orientation factor in Eq.(17): (a) θDA =0, θD = 0, θA = 0, κDA = −1; (b) θDA = 0, θD = π
2, θA = π
2,
κDA = 1; (c) θDA = π2, θD = π
2, θA = π
2, κDA = 0
parallel (or anti-parallel) to one another and to the dis-placement vector. This is classic Forster behaviour. Ad-ditionally, the coupling may also be prohibited betweenNWs by arranging them such that the transition dipolemoments are mutually orthogonal. We further investi-gate this factor by keeping θDA, θA fixed at 0 and varyingθD from 0 to π
2 . Interestingly, the orientation dependenceof rate becomes weaker as the separation distance be-tween donor and acceptor decreases, which is illustratedin Fig.5 (c). This is similar to the numerous studies re-ported for molecules and QDs discussed in [49, 50], withone important difference. Due to the cylindrical sym-metry and the physical nature of the exchanged photonvirtue in the 2D geometry, the orientation factor variesfrom 0 ≤ κ2 ≤ 1.
IV. NANOWIRE TO NANOWIRE RET IN THEVICINITY OF ANOTHER NANOWIRE
We now insert a third NW, M , which acts as a bridgespecies between D and A as shown in Fig.7 (a). M isengaged in relaying energy between donor and acceptor,but otherwise remains unchanged overall. Direct substi-tution into Eq. (6) yields
M iFI = −µ
0αn (D)µβ0m (A)
(2V ε0)2
∑p
∑q
e(λ)n (p)e(λ)l (p)e(λ)m (q)e
(λ)k (q)αkl(M ; k)p2q2 ×
{H(2)0 (pR′)H
(1)0 (qR′′)
(p− k)(q − k)+H
(2)0 (pR′)H
(2)0 (qR′′)
(p− k)(q + k)+H
(1)0 (pR′)H
(1)0 (qR′′)
(p+ k)(q − k)+H
(1)0 (pR′)H
(2)0 (qR′′)
(p+ k)(q + k)
}(19)
where R′ = RM −RD and R′′ = RA −RM , so that R = RA −RD.
M iFI = −µ
0αn (D)µβ0m (A)
(2V ε0)2
∑p
∑q
e(λ)n (p)e(λ)l (p)e(λ)m (q)e
(λ)k (q)αkl(M ; k)p2q2 ×
(H(2)0 (pR′)
p− k+H
(1)0 (pR′)
p+ k
)×(H(1)
0 (qR′′)
q − k+H
(2)0 (qR′′)
q + k
)(20)
By converting the discrete summation over p, q to an integral yields
M iFI = −µ
0αn (D)µβ0m (A)
(8π2Lε0)2(−∇2δnl +∇n∇l)(−∇2δmk +∇m∇k)
∫ ∞0
∫ 2π
0
∫ ∞0
∫ 2π
0
αkl(M ; k)×
(H(2)0 (pR′)
p− k+H
(1)0 (pR′)
p+ k
)×(H(1)
0 (qR′′)
q − k+H
(2)0 (qR′′)
q + k
)dqdφdpdϕ (21)
In a similar manner as the previous case in Sec. (III)the quantum amplitude, on performing contour integra-
tion and using the residue theorem twice over two virtualphotons, becomes
8
M iFI = −µ
0α(D)µβ0(A)αkl(M ; k)
(4Lε0)2
[kδnl
{− Y2(kR′) +
Y1(kR′)
kR
}− k{Y1(kR′)
(δnl − R′nR′lR′
)+ kR′nR
′l
(− Y2(kR′)
+Y1(kR′)
kR′
)}− ikδnl
{− J2(kR′) +
J1(kR′)
kR′
}+ ik
{J1(kR′)
(δnl − R′nR′lR′
)+ kR′nR
′l
(− J2(kR′) +
J1(kR′)
kR′
)}][kδmk
{− Y2(kR′′) +
Y1(kR′′)
kR′′
}− k{Y1(kR′′)
(δmk − R′′mR′′kR
)+ kR′′mR
′′k
(− Y2(kR′′)
+Y1(kR′′)
kR′′
)}− ikδmk
{− J2(kR′′) +
J1(kR′′)
kR′′
}+ ik
{J1(kR′′)
(δmk − R′′j R′′kR′′
)+ kR′′mR
′′k
(− J2(kR′′) +
J1(kR′′)
kR′′
)}](22)
The Eq.(22) for matrix element can be simplified byimposing near-field limits and using Eq.(13), (14)
M iFI = −µ
0α(D)µβ0(A)
(2πLR′R′′ε0)2[(δnl − 2R′nR
′l)(δmk − 2R′′mR
′′k)
αkl(M ; k)]
(23)
Therefore, in the presence of a neighbouring mediator,the matrix element for the mechanism of RET is dulymodified to Eq.(23), which can also be written in thefollowing form
M iFI = −|µ
0α(D)||µβ0(A)|κDMκMA
(2πLR′R′′ε0)2|αkl(M ; k)| (24)
where κDM , κMA are expressed as
κDM = µ0α(D)(δnl − 2R′nR′l)µ
r0(M)
= cos(θDM )− 2 cos(θ′D) cos(θM )(25)
κMA = µ0r(M)(δmk − 2R′′mR′′k)µβ0(A)
= cos(θMA)− 2 cos(θ′M ) cos(θ′A)(26)
in which θ′D is the angle between µ(D) and D-M sepa-ration vector (R′), and θ′A is the angle between µ(A) andR′′. θDM is the angle between µ(D) and µ(M), θMA isthe angle between µ(M) and µ(A). θM , θ
′M are angles
formed by µ(M) with respect to R′ and R′′ (Fig. 7 (b)).The application of the Fermi’s golden rule now gives
rise to the following expression for the transfer rate
Γitran =|µ0α(D)|2|µβ0(A)|2||κDM |2|κMA|2αkl(M ; k)|2ρ
8π}(L2R′2R′′2ε02)2
(27)
Ideally, the result in Eq. (27) should be used to eval-uate the indirect contribution to the RET rate mediatedby neighbouring body M in near field regime. Undoubt-edly, the key factors are the position, orientation, andthe trace polarizability of the passive NW.
D
M
A
ħk
θD
θDM
θMAθA
θM
D
M
A
Donor transition
dipole plane
Acceptor transition
dipole plane
Passive NW transition
dipole plane
R /
R //
(b)
(a)
FIG. 7. Schematics for the third-body modified resonanceenergy transfer of (a) NW to NW; (b) the orientational factorsin Eq.(25), (26), θD, the angle between the donor and thedisplacement vector, θA, the angle between the acceptor andthe displacement vector. θDM , is the angle between the D,Mtransition dipole moments and θMA, the angle between theM,A transition dipole moments
In the development of the plots in Sec. (IV), the fol-lowing values were used [48]: |µ0α(D)| = |µβ0(A)| =5×10−30 C m; ρ = 2×1025 J−1. Again, αkl(M ; k) is theground state dynamic polarizability of the passive nanos-tructure. Polarizability values are related to refractive in-dex by the Clausius- Mossotti equation, and an in-depthanalysis of relevance to the current application has beencarried out in [41, 43, 44]. Therefore, in the developmentof the graphs, we assume |αkl(M ; k)| of the included in-termediary takes a typical value of 25× 10−35J−1C2 m2
[42].
It is important to note that, values of R,R′, R′′ < 1nm will generally signify the possibility of wave functionoverlap. In this regime, an alternative mode of energytransfer occurs, called the Dexter mechanism, and it ispredicted to have an essentially exponential dependenceon separation distance. In the present study, we restrictourselves to separations where RET processes are me-
9
4 8 12 16 200
2
4
6x 10
−3
4 8 12 16 200
5
10
15
10 30 50 70 90
0
1
2
3
4
En
ere
gy
tra
nsf
er
rate
(n
s-1
R (nm)/ R (nm)
/
Dir
ect
RE
T t
o In
dir
ect
RE
T r
ati
o
En
ere
gy
tra
nsf
er
rate
(n
s-1
θM (degrees)
3 nmR =/
5 nmR =/
10 nmR =/
(a) (b) (c)
) )
FIG. 8. Indirect RET rate: (a) as a function of the relative distance between D and M , (R′), keeping R constant at 20 nm;(b) as a ratio of direct RET with respect to R′, keeping R constant at 20 nm; (c) as a function of θM for three different relativedistances (3 nm, 5 nm, 10 nm) between D and M .
diated only via Coulombic interactions. Therefore, thevalues of quantum amplitude and energy transfer ratein this region should not be regarded as physically sig-nificant (also in Dexter zone, expressions presented inour work are less meaningful as the neighbouring mattercould no longer be regarded as an electronically separateentities).
A. Distance dependance of indirect RET rate
From the matrix element for indirect RET given in Eq.(23), it can be seen that the RDDI depends on relativeposition of all three particles, exhibiting inverse squarepower dependence on each of the donor and acceptor dis-tances relative to body M . Moreover, compared to thedirect interaction of two NWs derived in Eq. (16), cre-ation and annihilation of two virtual photons reduce thequantum amplitude of the indirect interaction, display-ing a (R′)−4(R′′)−4 distance dependence of indirect RETrate (see Eq.27). Furthermore, indirect RET rate grad-ually decreases as a function of displacement of A fromM and D (keeping M at the midpoint of the D-A axis).In addition, indirect transfer rate is less distinguishablewhen M is located in the center of the D-A axis, as shownin Fig.8 (a).
We now focus on the important possibility of alteringthe energy transfer rate by indirect energy transfer, byanalysing the ratio of the, direct to indirect transferrates. This is depicted in Fig.8 (b) for various R′ valuesat fixed R. It can be observed that the influence ofthe included intermediary NW, (M), becomes moreprominent when it is situated close to either donor oracceptor.
B. Orientational dependance of indirect RET rate
The relative orientation of the D,M,A, and theirindividual orientations with respect to relative separa-tion vectors, strongly influence the quantum amplitude(M i
FI ∝ |κDM ||κMA|) and the indirect energy transferrate (Γitran ∝ |κDM |2|κMA|2). We now conduct a com-prehensive analysis of how the transfer rate varies as afunction of θM for various R′ and R values. As illus-trated in Fig.8 (c), the orientation dependence of ratebecomes substantial when M gets closer to either theacceptor or donor. The orientation factor is less signif-icant when the passive NW is placed at the center ofthe donor-acceptor displacement vector. Moreover, anal-ogous to the direct energy transfer, orientational factorbecomes stronger when the acceptor is placed near to thedonor.
The angles given in Eq. (25) and Eq. (26) are de-pendent on one another. Once again we identify threenon-trivial cases to study the orientational dependenceof third-body modified energy transfer.
1. case 1: θDM = θMA = 0, θ′D = θ′A = θM = θ′M = 0;κDM = κMA = −1, (Fig. 9 (a))
2. case 2: θDM = θMA = 0, θ′D = θ′A = 0, θM = θ′M =π3 ; κDM = κMA = 0, (Fig. 9 (b))
3. case 3: θDM = θMA = π3 , θ′D = θ′A = θM = θ′M =
π3 ; κDM = κMA = 0, (Fig. 9 (c))
In all three cases θDA = θD = θA = 0.The first case corresponds to situations when the tran-
sition dipole moments are parallel (or antiparallel) toeach other and separation vectors, leading to |κDM |2 =|κMA|2 = 1. These geometrical configuration of threeNWs gives highest indirect RET rate. In the secondcase, we keep all angles unaltered except θM , θ
′M , here the
dipole moment of M formes angle π3 with respect to both
10
θ’D=θ’A=θM=θ’M=0
A
θDA=θDM=θMA= 0
D
(a) (b)
(c)
κDM =κMA= -1
M
θ’D=θ’A=0
A
D
M
θM =θ’M= π/3θDA=θDM=θMA= 0
θDM =θMA = π/3
M
A
θM = θ’M = π/3
D
θ’D = θ’A = π/3
κDM = κMA = 0
D,M,A transition
dipole planesD,M,A transition
dipole planes
M transition
dipole plane
D,A transition
dipole planes when,
θDA=0,
θD = θA = 0
κDM =κMA= 0
FIG. 9. Indirect RET orientation factor in Eq. (25) and Eq. (26): (a) M lies in the D-A separation vector, and all D,M,Aare parallel to each other; (b) M lies in the D-A separation vector, θM = θ′M = π
3, and θDM = θMA = 0; (c) θDA = π
3,
θM = θ′M = θ′D = θ′A = π3
. Note that in all cases θDA = θD = θA = 0
R′,R′′, leading to a prohibited indirect energy transferrate. The case three is similar to the case two, except theplane of M makes an angle of π
3 with respect to donorand acceptor transition dipole planes. This configurationof the NW system makes indirect RET rate contributionto the total energy transfer rate to be vanished.
C. Quantum interference contribution to thetransfer rate
From the matrix elements for direct, and indirecttransfer, it is straightforward to calculate the third termof Eq. (8), that arises from the interference of these twoprocesses.
Γinttran =4πρ
}Re{Md
FIMiFI}
= −|µ0α(D)|2|µβ0(A)|2κDAκDMκMA|αkl(M ; k)|ρ
2π2L3R2R′2R′′2ε03}(28)
The quantum interference, Eq. (28), displays the in-verse square power dependence on each relative displace-ment component (R,R′, R′′). Fig. 10 (a) shows the in-terference dependence as a function of the position of thethird NW. As can be anticipated, it is clear from the plotthat the quantum interference acquires higher values forlower or higher values of R′, delivering minimum directto interference ratio when M is at the center of the D-Adisplacement, as illustrated in Fig. 10 (b).
Inspection of Fig. 10 (c) allows one to gain insightinto how the interference term is influenced by the rela-
tive orientation of particles, exhibiting the quantum in-terference variation with respect to θM when θDM = 0, π.In both cases, quantum interference is maximum whenθM = π
3 , and minimum when θM = 0, π2 . For a collineararrangement of the three NWs, the quantum interferencebetween direct and indirect transfer can be negative, de-creasing the total exchange rate.
Thus, the total third-body-modified energy transferrate becomes
Γtotaltran =|µ0α(D)|2|µβ0(A)|2|κ|2ρ
2πL2R4ε02}+
|µ0α(D)|2|µβ0(A)|2|κDM |2|κMA|2|αkl(M ; k)|2ρ8π}L4R′4R′′4ε04
−
|µ0α(D)|2|µβ0(A)|2κDAκDMκMA|αkl(M ; k)|ρ2π2L3R2R′2R′′2ε03}
(29)
The result, Eq. (29) can be interpreted as the rate forRET between donor and acceptor NWs, modified by thepresence of the passively interacting medium.
V. DISCUSSION
We have derived resonance energy transfer rate equa-tions for a system consisting of nanowires of the samedimensionality. The calculations are presented for directRET and the influence of a passive NW on the energytransfer rate. The results have demonstrated the relativespacing between NWs along with the relative orientationof the transition dipoles determine the controllability of
11
4 8 12 16 20
−0.2
−0.1
4 8 12 16 20
0
10 30 50 70 90
−2
−1
0x 10
R (nm)
0
−4
−3
−2
−1
-2x 10
-2E
ne
reg
y tr
an
sfe
r ra
te (
ns-1
En
ere
gy
tra
nsf
er
rate
(n
s-1
Dir
ect
RE
T t
o in
terf
ere
nce
ra
tio
/R (nm)
/θM (degrees)
(a) (b) (c)
θDM =0,π
) )
FIG. 10. Quantum interference: (a) as a function of the relative distance between D and M (R′), keeping R constant at 20 nm;(b) as a ratio of direct RET with respect to R′, keeping R constant at 20 nm; (c) variation as a function of θM when θDM = 0and θDM = π
resonance energy transfer rate. Furthermore, the RETin NW systems displays a slower spatial decay comparesto a system consisted of QDs.
In Sec. III, unmediated RET between two NWs hasbeen studied, applying the second order perturbationtheory. The results have demonstrated the energy trans-fer rate exhibits R−4 distance dependence. The orienta-tion factor, |κ|2 varies from 0 to 1. Moreover, in Sec. IVthird body-mediated RET has been investigated, exploit-ing the fourth order perturbation mechanism. Here, thepassive NW, M , remains in its ground state but is ex-cited in intermediate states. It participates in the trans-fer of excitation as a polarizable body coupled to theelectromagnetic field. Therefore, the energy transfer rateembraces additional contributions associated with indi-rect RET rate and quantum interference. We obtainedanalytical expressions for both indirect energy transferrate and quantum interference contribution to the totalthird-body mediated RET rate. The indirect couplingof resonance energy transfer becomes more significant asthe density of particles M increases. In a nutshell, theseresults suggest that when designing an artificial energytransfer system, optimal configurations for fast transferbetween nearby sites are those where the NW transitiondipole moments and the separation vector are collinear.However, coupling may also be “switched off”betweenparticles by arranging them such that the transitiondipole moments are perpendicular to each other. Thiscan, in effect, enable one to design a nano-antenna sys-tem that are optimized to focus the energy transfer tospecific points. Moreover, an interacting third NW caneffectively enhance or inhibit the RET between donorand acceptor, and this contribution can be significant ifa sufficient number of passive objects (M) are present.
VI. CONCLUSIONS
In this article, the direct RET between two NWs andthe influence of a passively interacting NW have beenstudied using the theory of molecular quantum electro-dynamics. Within the QED context, treatment of reso-nance energy transfer in both cases, exchange of excita-tion is mediated by electromagnetic signals propagatingat the speed of light. Thus, the coupling matrix elementsand rate equations are derived for both cases, and theintricate interplay of the relative distance, orientation,and the effect of the passively interacting medium on thetransfer efficiency have been studied to a greater extent.
Summarizing, the ensuing results demonstrated thepossibility of altering the strength and the directivityof the resonance energy transfer between two NWs bycareful engineering of the spacing, orientation and in-clusion of an additional quantum objects in the vicinity.The analysis thus provides pointers to the means of con-trolling and optimizing the transfer of energy betweendiscrete components, potentially in any multi-nanowiresystem, inviting surface and layer applications. There-fore, this research opens up substantial opportunities todevelop a thorough understanding on well-known RETmechanism in NWs, based on quantum electrodynamics.In particular, the results we have secured should supportthe further improvement of design of biological sensors,organic photovoltaics and light-driven catalysis, opticalswitching, through the achievement of new methods foroptically controlled transmission.
ACKNOWLEDGMENTS
The work of D.W. is supported by the Monash Univer-sity Institute of Graduate Research. The work of M.P. is
12
supported by the Australian Research Council, through its Discovery Grant No. DP140100883.
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