Article pubs.acs.org/JPCC
Cite This: J. Phys. Chem. C 2018, 122, 4607−4614
Entropy Current through a Strongly Correlated Plexcitonic Nanojunction A. Goker* Department of Physics, Bilecik University, 11210 Gülümbe, Bilecik, Turkey ABSTRACT: We study the entropy current through an excitonic Coulomb blockaded two-level quantum emitter attached to plasmonic metal nanoparticles subjected to a temperature gradient by invoking the noncrossing approximation. We find that boosting the plasmon−exciton coupling enhances the entropy current at any temperature for an infinitesimal bias and temperature gradient, whereas increasing the ambient temperature rapidly suppresses it. Introducing finite temperature gradient serves to increase the entropy current. On the other hand, increasing the bias voltage causes the entropy current to increase slightly around the Kondo temperature for low plasmon−exciton coupling values. Finally, we determine that the entropy current always stays finite at elevated voltage bias values for any nonvanishing plasmon−exciton coupling. We elucidate on the microscopic origin of these intriguing results.
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mechanics.13 However, the development of a full quantum theory that incorporates spin-dependent features and Coulomb interaction was still missing. Toward this end, exquisite Green function techniques have been employed to show that the strong-coupling regime is dominated by only the dipolar plasmon mode.14 Exciton transfer through various configurations of quantum emitters in the vicinity of a metal nanoparticle has also been investigated subsequently with the same method.15,16 Ambient temperatures above the Kondo temperature of the discrete level with the higher energy have been demonstrated to gradually suppress the Fano resonance in the optical absorption spectrum of a plexcitonic nanojunction by invoking the timedependent noncrossing approximation.17 Subsequent investigations employing the same technique unambiguously showed that the conductance through such a device in Coulomb blockade can be enhanced at low temperatures with the aid of the Kondo effect.18 We will extend these studies in this paper by investigating the entropy current flowing through such a plexcitonic nanojunction subjected to a temperature gradient and voltage bias. Entropy flow is a fundamental concept that characterizes the ability of a quantum nanodevice to harvest thermal energy from optical energy. This is especially a crucial aspect for photovoltaic devices which are usually envisaged as heat engines.19 Work is extracted from the heat flow from the hot reservoir to the cold one. Therefore, it is of critical importance to understand the behavior of the entropy current to be able to improve the efficiency of the photovoltaic devices. In the present model, we will account for the Coulomb interaction and spin effects explicitly for the first time. We will also interpret our results
INTRODUCTION Tailoring the objects at the nanoscale in a precise fashion has long been sought after as a direct path to develop new state-of-the art optoelectronic devices which are expected to be miniaturized according to the Moore law.1 New experimental techniques acquired from nanotechnology greatly aided in tuning the surface plasmon resonance of metal nanoparticles, which is an invaluable tool for focusing the optical energy into spatial regions smaller than the wavelength of illumination.2 Giant field enhancements around the metal nanoparticles arising from these plasmon resonances3 facilitate coupling with various quantum emitters in their vicinity. Hybrid light-matter states originating from the strong coupling of plasmons with electronic excitations such as excitons have been dubbed as plexcitons. They are widely believed to hold great promise for future tunable molecular optoelectronic devices4 and plasmonic switches.5 Metal nanoparticles possessing plasmon resonances exhibit extinction coefficients several orders of magnitude larger than quantum dots and molecular dyes.6 This feature makes them ideal candidates to utilize the concentrated energy of the plasmon in photovoltaic applications.7−9 Demonstration of the plasmon to exciton energy transfer10 also presents a unique opportunity to convert the strongly localized near field of metal nanoparticles into useful energy in next-generation electronic devices. Plexcitons, which have the lightest effective mass observed so far, exhibit thermalization and cooling at room temperature which can pave the way for the quantum condensation of these bosonic quasiparticles.11 It is also thought that plexcitons can be quite useful to achieve polariton lasing at the nanoscale owing to their low thresholds. Previous theoretical studies involving plexcitonic nanojunctions range from semiclassical treatments12 to hybrid schemes combining a discrete interaction model with quantum © 2018 American Chemical Society
Received: January 3, 2018 Revised: February 12, 2018 Published: February 12, 2018 4607
DOI: 10.1021/acs.jpcc.8b00057 J. Phys. Chem. C 2018, 122, 4607−4614
Article
The Journal of Physical Chemistry C making use of a nonlinear generalization of the well-known Wiedemann−Franz law.
H=
THEORY An excitonic quantum emitter composed of two discrete states coupled to metal nanoparticles possessing plasmon resonances can be accurately modeled with a Hamiltonian such as
+
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H=
∑
εKσc K†σc Kσ +
K ∈ {L,R}, σ
+ +
∑
(VK,g(e)c K†σcg(e)σ
c†Kσ(cKσ)
∑
εpK bK† bK
∑ εαaα†aα α
(Wα ,Kaα†bK + h.c.)
∑
(ΔK fe†σ befgσ bg†bK + h.c.)
K ∈ {L,R}, σ
(4)
We will now resort to a few physically sound approximations to be able to deal with this Hamiltonian. First of all, the incident laser is assumed to be perpendicular to the axis connecting the metal nanoparticles in conjunction with earlier studies.17,21 This amounts to the absorption of the photons of the laser beam by the metal nanoparticles in the same phase. It is also assumed that there is no direct coupling between the incident laser and the quantum emitter. The system is pumped exclusively by a single laser mode ε0 which serves to excite only the dipolar plasmon mode of the metal nanoparticles. Consequently, J term in HE will be dropped in this case. The temperature gradient between the metal nanoparticles can be generated either by adjusting the intensity of the laser beam or varying the size of one of the nanoparticles. We are therefore allowed to consider the dipolar plasmon energy of each nanoparticle to be equal such as εpL = εpL = εp with symmetrical plasmon−exciton and plasmon−laser couplings ΔL = ΔR = Δ and W0,L = W0,R = W0. Time-ordered double-time Green functions will be invoked to study the out of equilibrium transport properties of this model. Pseudofermion and slave boson lesser Green functions are given as Gg(e)(t,t′) = ⟨bg(e)(t)b†g(e)(t′)⟩. In terms of these, we can write the retarded components as
h.c.)
(1)
c†sσ(csσ)
where and create(annihilate) an electron with spin σ in the metal nanoparticles and in each spin degenerate discrete level of the molecule, respectively. These spin degenerate levels are designated by |g⟩ and |e⟩ with an energy difference of |εe − εg|. Moreover, εα stands for the energy of the radiation field with mode α, whereas εpK represents the dipolar plasmon energy in the left (K = L) and right (K = R) metal nanoparticle. The contribution of the quadrupolar plasmon modes will be ignored henceforth because it has been established that it does not affect the optical absorption spectrum20 and thus the quantum transport through such a nanojunction. J stands for the electron tunneling amplitude between the discrete levels |g⟩ and |e⟩ of the quantum emitter. On the other hand, VK,g(e) corresponds to the electron tunneling amplitude between the discrete levels |g(e)⟩ of the metal nanoparticles. The amplitude of the interaction between the radiation field with mode α and the dipolar plasmons of each nanoparticle is represented by Wα,K. ΔK denotes the amplitude of coupling between the exciton within the quantum emitter and the dipolar plasmon modes of the metal nanoparticles. Finally, the amount of Coulomb repulsion energy within each discrete level is accounted for by the Hubbard term U, where nsσ symbolizes the occupation number operator. U term in the Hamiltonian easily overwhelms the other energy scales including the thermal energy in the ambient temperature range where the Kondo resonance is manifested as a result of the strong confinement of the electrons inside each discrete level. This implies that we can assume U → ∞. The slave boson method is a quite convenient approach to deal with this situation. This method relies on defining the ordinary electron operators acting on the discrete levels |g⟩ and |e⟩ as the product of a pseudofermion and a massless boson such as
r > < Gg(e) (t , t ′) = −iθ(t − t ′)[Gg(e) (t , t ′) + Gg(e) (t , t ′)]
≔ −iθ(t − t ′)gg(e)(t , t ′), r > < Bg(e) (t , t ′) = −iθ(t − t ′)[Bg(e) (t , t ′) − Bg(e) (t , t ′)]
≔ −iθ(t − t ′)bg(e)(t , t ′)
(5)
We can obtain these retarded pseudofermion Green functions by solving the following Dyson equations ⎛∂ ⎞ ⎜ + iεg ⎟gg (t , t ′) = − ⎝ ∂t ⎠ −
† cg(e)σ = bg(e) fg(e)σ
∫t′
t
∫t′
t
dt1KL>(t , t1)bg̃ (t , t1)gg (t1, t ′)
dt1 |Δ|2 gẽ (t , t1)bg̃ (t , t1)Bẽ KL(R) (t , t ′) = Γ̅
∫−D 2dπε ρ(ε) 1 +e e β
L(R)ε
βL(R)ε
D
L(R)ε
⎛∂ ⎞ ⎜ + iεg ⎟Gg
