## Abstract

We theoretically analyzed the lower bound of energy dissipation required for optical excitation transfer from smaller quantum dots to larger ones via optical near-field interactions. The coherent interaction between two quantum dots via optical near-fields results in unidirectional excitation transfer by an energy dissipation process occurring in the larger dot. We investigated the lower bound of this energy dissipation, or the intersublevel energy difference at the larger dot, when the excitation appearing in the larger dot originated from the excitation transfer via optical near-field interactions. We demonstrate that the energy dissipation could be as low as 25 μeV. Compared with the bit flip energy of an electrically wired device, this is about 10^{4} times more energy efficient. The achievable integration density of nanophotonic devices is also analyzed based on the energy dissipation and the error ratio while assuming a Yukawa-type potential for the optical near-field interactions.

© 2010 OSA

## 1. Introduction

Energy efficiency is an issue of increasing importance in today’s information and communications technology (ICT) in order to abate CO_{2} production [1]. Various approaches have been intensively studied regarding energy efficiency, ranging from analysis of fundamental physical processes [2,3] to system-level smart energy management [4]. Energy efficiency involving optical processes has also been considered in terms of, for example, how we can fully exploit the low-loss, wavelength-multiplexed, high-bandwidth nature of optical communications for efficient energy usage [5,6].

The fundamental physical attributes of photons exploited so far are, however, typically just the ability to utilize fully optical end-to-end connections and the multiplexing nature of propagating light. One of the primary objectives of this paper is to note another unique attribute available on the nanometer scale, that is, optical near-field interactions [7]. In related issues, the limit on the integration density determined by power dissipation and signal-to-noise ratio was studied by Thylén et al. in systems composed of metal and quantum dots [8]. However, the fundamental physical mechanisms studied so far are based on a dipole coupling, which is significantly different from the optical excitation transfer via optical near-field interactions discussed in this paper. Here we theoretically demonstrate that the energy dissipation in optical excitation transfer via optical near-field interactions could be as low as 25 μeV, which is about 10^{4} times smaller than the bit flip energy in a conventional electrically wired device.

As discussed in detail later, optical excitations could be transferred from smaller quantum dots (QDs) to larger ones via optical near-field interactions that allow transitions even to conventionally electric-dipole forbidden energy levels. Such optical excitation transfers have been experimentally demonstrated in various materials, such as CuCl [9], CdSe [10], and CdTe [11]. Geometry-controlled nanostructures are also seeing rapid progress, such as in size- and density-controlled InAs QDs [12], stacked InAs QDs [13], ring-shaped QDs [14], and ZnO nanorods [15]. Theoretical foundations have been constructed, such as the dressed photon model that unifies photons and material excitations on the nanometer scale, validating the non-zero transition probabilities even for conventionally electric-dipole forbidden energy levels [7]. Based on these principles and enabling technologies, a wide range of device operations have been demonstrated, such as logic gates [16,17], interconnects [18], energy concentration [11,19], and so forth. In considering a bit flip in nanophotonic logic gates, we need to combine multiple optical excitation transfers from smaller QDs to larger ones among multiple quantum dots; for instance, three QDs are needed in the case of an AND gate [16,18]. Therefore, this paper, dealing with the energy dissipation in optical excitation transfer composed of two dots, constitutes a foundation for nanophotonic devices in general.

The energy dissipation in such an optical excitation transfer from a smaller QD to a large one is the energy relaxation processes that occur at the destination QD (namely, the larger QD). In other words, the coherent interaction between two QDs via optical near-fields results in a unidirectional excitation transfer from a smaller QD to a larger one by the energy dissipation occurring in the larger QD [20]. In electrically wired devices, the dissipation occurring in external circuits is crucial in completing signal transfer. Such a fundamental principle also impacts tamper resistance against non-invasive attacks; tampering of information could easily be possible by monitoring power consumption patterns in electrically wired devices [21], whereas it is hard in optical excitation transfer since energy dissipation occurs at the destination QDs [22]. In this paper, we quantitatively investigate how such energy dissipation at the destination QD could be minimized. Here, it should be noted that the primary objective of this paper is to reveal the theoretical limitations of the principle provided by optical excitation transfer. We start our discussion with a theoretical model for cubic quantum dots and assume typical values for the inter-dots distance and interaction time, and so on. However we can extend the theoretical investigation into the parameter range where one of the ideal dots corresponds practically to a coupled quantum dot system in which inter-dot electron transfer takes place between energy level with a separation much smaller than those feasible by a single quantum dot.

## 2. Modeling

We begin with the interaction Hamiltonian between an electron–hole pair and an electric field, which is given by

*e*represents a charge, ${\widehat{\psi}}_{i}^{\u2020}(r)$ and ${\widehat{\psi}}_{j}(r)$ are respectively creation and annihilation operators of either an electron (

*i*,

*j*=

*e*) or a hole (

*i*,

*j*=

*h*) at

**, and**

*r***(**

*E***) is the electric field [23]. In usual light–matter interactions,**

*r***(**

*E***) is a constant since the electric field of diffraction-limited propagating light is homogeneous on the nanometer scale. Therefore, as is well known, we can derive optical selection rules by calculating the dipole transition matrix elements. As a consequence, in the case of cubic quantum dots for instance, transitions to states containing an even quantum number are prohibited. In the case of optical near-field interactions, on the other hand, due to the steep electric field of optical near-fields in the vicinity of nano-scale material, an optical transition that violates conventional optical selection rules is allowed. A detailed physical discussion is found in Ref [7].**

*r*Using near-field interactions, optical excitations in quantum dots can be transferred to neighboring ones. For instance, assume two cubic quantum dots whose side lengths are *a* and $\sqrt{2}a$, which we call QD* _{S}* and QD

*, respectively, as shown in Fig. 1(a) . Suppose that the energy eigenvalues for the quantized exciton energy level specified by quantum numbers (*

_{L}*n*,

_{x}*n*,

_{y}*n*) in a QD with side length

_{z}*L*are given by

*E*is the transition energy of the bulk exciton, and

_{B}*M*is the effective mass of the exciton. According to Eq. (2), there exists a resonance between the level of quantum number (1,1,1) for QD

*and that of quantum number (2,1,1) for QD*

_{S}*. Hereafter, the (1,1,1)-level of QD*

_{L}*is denoted by*

_{S}*E*, and the (2,1,1)-level of QD

_{S}*is called ${E}_{{L}_{2}}$. There is an optical near-field interaction, which is denoted by ${U}_{S{L}_{2}}$, due to the steep electric field in the vicinity of QD*

_{L}*. Therefore, excitons in*

_{S}*S*can move to

*L*

_{2}in QD

*. Note that such a transfer is prohibited in propagating light since the (2,1,1)-level in QD*

_{L}*contains an even number. In QD*

_{L}*, the exciton undergoes intersublevel energy relaxation due to exciton–phonon coupling, denoted by*

_{L}*Γ*, which is faster than the near-field interaction [9,24], and so the exciton relaxes to the (1,1,1)-level of QD

*, which is called ${E}_{{L}_{1}}$ hereafter. We should note that the intersublevel relaxation determines the uni-directional exciton transfer from QD*

_{L}*to QD*

_{S}*. Also, we assume far-field input light irradiation at the optical frequency*

_{L}*ω*.

_{ext}Here we first introduce quantum mechanical modeling of the total system based on a density matrix formalism. There are in total eight states where either zero, one, or two exciton(s) can sit in the energy levels of *S*, *L*
_{1}, and *L*
_{2} in the system, as schematically summarized in the diagram shown in Fig. 1(a). Here, the interactions between QD* _{S}* and QD

*are denoted by ${U}_{S{L}_{i}}$ (*

_{L}*i*=1,2), and the radiative relaxation rates from

*E*and ${E}_{{L}_{1}}$ are respectively given by ${\gamma}_{S}$ and ${\gamma}_{L}$. Then, letting the (

_{S}*i*,

*i*) element of the density matrix correspond to the state denoted by

*i*in Fig. 1(b), the quantum master equation of the total system is given by [25]

*H*is given by

_{int}*S*,

*L*

_{1}, and

*L*

_{2}in Eq. (4) are respectively annihilation operators given by the transposes of the matrices of Eq. (5).

*H*indicates the Hamiltonian representing the interaction between the external input light at frequency

_{ext}*ω*and the quantum dot system, given by

_{ext}*gate*(

*t*) specifies the duration and the amplitude of the external input light. Also, note that the input light could couple to the (1,1,1)-level

*E*in QD

_{S}*, and to the (1,1,1)-level ${E}_{{L}_{1}}$ in QD*

_{S}*, because those levels are electric dipole-allowed energy levels. Setting the initial condition as an empty state, and giving the external input light in Eq. (6), the time evolution of the population is obtained by solving the master equation given by Eq. (3).*

_{L}## 3. Lower bound of energy dissipation in the optical excitation transfer

Then we introduce two different system setups to investigate the minimum energy dissipation in the optical excitation transfer modeled above. In the first one, System A in Fig. 2(a)
, two quantum dots are closely located in the region where the optical excitation transfer from QD* _{S}* to QD

*occurs. We assume ${U}_{S{L}_{2}}^{-1}$ of 100 ps in System A, denoted by ${U}_{\text{A}}^{-1}$ in Fig. 2(a), which is close to typical values of optical near-field interactions experimentally observed in CuCl QDs (130 ps) [16], ZnO quantum-well structures (130 ps) [17], ZnO QDs (144 ps) [26], and CdSe QDs (135 ps) [27]. The intersublevel relaxation time, due to exciton–phonon coupling, is in the 1–10 ps range [9,24,28], and here we assume ${\Gamma}^{-1}=10ps$. In System B on the other hand, shown in Fig. 2(b), the two quantum dots are located far away from each other. Therefore the interactions between QD*

_{L}*and QD*

_{S}*are negligible, and thus the optical excitation transfer from QD*

_{L}*to QD*

_{S}*does not occur, and the radiation from QD*

_{L}*should normally be zero. We assume ${U}_{S{L}_{2}}^{-1}=$10,000 ps for System B, denoted by ${U}_{\text{B}}^{-1}$ in Fig. 2(b), indicating effectively no interactions between the quantum dots.*

_{L}One remark here is that the particular value of the inter-dot interaction time of System A and System B is related to the distance between two quantum dots. The optical near-field interaction between two nanoparticles is known to be expressed as a screened potential using a Yukawa function, given by

where*r*is the distance between the two [29]. In this representation, the optical near-field is localized around nanoparticles, and its decay length is equivalent to the particle size. Here it should be noted that the inter-dot distance of System B indicates how closely independent functional elements can be located. In other words, the interaction time of System B is correlated with the

*integration density*of the total system. In order to analyze such spatial density dependences, we assume that ${U}_{S{L}_{2}}^{-1}$ values of 100 ps and 10,000 ps correspond to the inter-dot distances of 50 and 500 nm, respectively. Here, the stronger interaction (100 ps) has been assumed, as already mentioned, based on a typical interaction time between closely separated quantum dots [7]. We also assume that the negligible magnitude of the interaction (10,000 ps) corresponds to a situation when the inter-dot distance is around optical wavelengths. Figure 2(c) shows the Yukawa-type potential curve given by Eq. (7). Further discussion on the integration density will be given at the end of this paper.

The energy dissipation in the optical excitation transfer from QD* _{S}* to QD

*is the intersublevel relaxation in QD*

_{L}*given by $\Delta ={E}_{{L}_{2}}-{E}_{{L}_{1}}$. Therefore, the issue is to derive the minimum of*

_{L}*Δ*. When this energy difference is too small, the input light may directly couple to

*L*

_{1}, resulting in output radiation from QD

*, even in System B. In other words, we would not be able to recognize the origin of the output radiation from QD*

_{L}*if it involves the optical excitation transfer from QD*

_{L}*to QD*

_{S}*in System A, or it directly couples to*

_{L}*L*

_{1}in System B. Therefore, the

*intended*proper system behavior is to observe higher populations from

*L*

_{2}in System A while at the same time observing lower populations from

*L*

_{2}in System B.

We first assume pulsed input light irradiation with a duration of 150 ps at 3.4 eV (wavelength 365 nm), and assume that the energy level *S* is resonant with the input light. Also, we assume the radiation lifetime of QD* _{L}* to be ${\gamma}_{L}^{-1}=1ns$ and that of QD

*to be ${\gamma}_{S}^{-1}={2}^{3/2}\times \text{1}~2.83\mathrm{ns}$ since it is inversely proportional to the volume of the QDs [10]. The solid and dashed curves in Fig. 3 respectively represent the evolutions of populations related to the radiation from the energy level of ${E}_{{L}_{1}}$ and*

_{S}*E*for both System A and System B, assuming three different values of

_{S}*Δ*: (i)

*Δ*=2.5 meV, (ii)

*Δ*= 17 μeV, and (iii)

*Δ*= 0.25 μeV.

In the case of (i), there is nearly zero population in System B from ${E}_{{L}_{1}}$, which is the expected proper behavior of the system since there are no interactions between the quantum dots. The radiation from QD* _{S}* is observed with its radiation decay rate (${\gamma}_{S}$). In System A, on the other hand, populations from ${E}_{{L}_{1}}$ do appear. Note that the population involving the output energy level ${E}_{{L}_{1}}$ is only 0.17 when the input pulse is terminated (

*t*= 150 ps), whereas the population involving

*E*at

_{S}*t*= 150 ps is 0.81. Therefore, the increased population from ${E}_{{L}_{1}}$ after

*t*= 150 ps is due to the optical excitation transfer from QD

*to QD*

_{S}*. In the case of (iii), due to the small energy difference, the input light directly couples with*

_{L}*L*

_{1}; therefore, both System A and System B yield higher populations from ${E}_{{L}_{1}}$, which is an unintended system behavior. Finally, in case (ii), the population from

*L*

_{1}in System B is not as large as in case (iii), but it exhibits a non-zero value compared with case (i), indicating that the energy difference Δ = 17 μeV may be around the middle of the intended and unintended system operations involving optical excitation transfer between QD

*and QD*

_{S}*.*

_{L}When we assume a longer duration of the input light, the population converges to a steady state. Radiating a pulse with a duration of 10 ns at the same wavelength (365 nm), Fig. 4(a)
summarizes the steady state output populations involving energy level ${E}_{{L}_{1}}$ evaluated at *t* = 10 ns as a function of the energy dissipation. The intended system behavior, that is, higher output population in System A and lower one in System B, is obtained in the region where energy dissipation is larger than around 25 μeV.

If we treat the population from System A as the amplitude of the “signal” and that from System B as “noise”, the signal-to-noise ratio (SNR) can be evaluated based on the numerical values obtained in Fig. 4(a). To put it another way, from the viewpoint of the destination QD (or QD* _{L}*), the signal should come from QD

*in its proximity (as in the case of System A), not from QD*

_{S}*far from QD*

_{S}*(as in the case of System B); such a picture will aid in understanding the physical meaning of the SNR defined here. Also, here we suppose that the input data are coded in an external system, and the input light at frequency*

_{L}*ω*irradiates QD

_{ext}*. With SNR, the error ratio (*

_{S}*P*), or equivalently Bit Error Rate (BER), is derived by the formula ${P}_{E}=(1/2)erfc(\sqrt{SNR}/2\sqrt{2})$ where $erfc(x)=2/\sqrt{\pi}{\displaystyle {\int}_{x}^{\infty}\mathrm{exp}(-{x}^{2})dx}$, called the complementary error function [30]. The circles in Fig. 4(b) represent the energy dissipation as a function of the error ratio assuming the photon energy used in the above study (3.4 eV). According to Ref [2], the minimum energy dissipation (

_{E}*E*) in classical electrically wired devices (specifically, energy dissipation required for a single bit flip in a CMOS logic gate) is given by

_{d}^{−6}, the minimum

*Δ*in the optical excitation transfer is about 0.024 meV, whereas that of the classical electrical device is about 303 meV; the former is about 10

^{4}times more energy efficient than the latter.

As mentioned earlier, the performance of System B depends on the distance between the QDs. When the interaction time of System B (${U}_{\text{B}}^{-1}$) gets larger, such as 500 ps, the steady state population involving *L*
_{1} is as indicated by the triangular marks in Fig. 4(a); the population stays higher even with increasing energy dissipation compared with the former case of ${U}_{\text{B}}^{-1}=$ 10,000 ps. This means that the lower bound of the SNR results in a poorer value. In fact, as demonstrated by the triangular marks (1) in Fig. 4(b), the BER cannot be smaller than around 10^{−4}, even with increasing energy dissipation. The lower bound of the BER decreases as the interaction time ${U}_{\text{B}}^{-1}$ increases (namely, weaker inter-dot interaction), as demonstrated by the triangular and square marks (2) to (6) in Fig. 4(b).

Now, suppose that an independent nanophotonic circuit needs a spatial area specified by the square of the inter-dot distance corresponding to *U*
_{B} so that no interference occurs between adjacent circuits; this gives the integration density of nanophotonic circuits in a unit area. When the energy cost paid (namely, the energy dissipation) is 3.4 meV, the BER of the system is evaluated as the square marks in Fig. 4(c) as a function of the number of independent functional blocks within an area of 1 μm^{2}; we can observe that the system likely sacrifices more errors as the integration density increases.

Finally, here we make a few remarks regarding the discussion above. First, we assume arrays of “identical” independent circuits in the above density discussion. Therefore, two circuits need spatial separations given by *U*
_{B} so that unintended behavior does not occur. However, when two adjacent nanophotonic circuits are operated with different optical frequencies so that they can behave independently [18], those two circuits could be located more closely, which would greatly improve the integration density as a whole. Hierarchical properties of optical near-fields [31] would also impact the integration density. Further analysis and design methodologies of complex nanophotonic systems, as well as comparison to electronic devices, will be another issue to pursue in future work. Second, the energy separation in a single destination QD being limited by its size is lying in the range of meV, so that the results of energy separations in μeV range correspond to the cases where the destination dot QD* _{L}* represents a theoretical model of a coupled quantum dot system such as a pair of quantum dots. It exerts optical near-field interactions with QD

*followed by inter-dot electron transfer resulting in optical radiation. In fact, one of the authors’ research group have recently demonstrated a spin-dependent carrier transfer leading to optical radiation between a coupled double quantum wells system composed of magnetic and nonmagnetic semiconductors [32], which can be applicable to quantum dot systems [33]. Third, as mentioned in the introduction, in considering a bit flip in nanophotonic devices, we need to combine multiple optical excitation transfers from smaller dots to larger ones in systems composed of multiple quantum dots, for instance, three dots in the case of an AND gate [16]. In a future study, we will investigate the required energy for nanophotonic devices in general based on the results shown in this paper.*

_{S}## 4. Summary

In summary, we theoretically investigated the lower bound of energy dissipation required for optical excitation transfer from smaller quantum dots to larger ones via optical near-field interactions. A quantum mechanical formulation of quantum dot systems provides systematic and quantitative analysis of the intended and unintended behaviors of optical excitation transfer as a function of the energy dissipation, or intersublevel relaxation, occurring at the destination quantum dot. We demonstrated that the energy dissipation is as low as 25 μeV, which is about 10^{4} times more energy efficient than the bit flip energy in a conventional electrically wired device. We also discussed the integration density of nanophotonic devices by taking account of energy dissipation, bit error rate, and the optical near-field interactions whose spatial nature is characterized by a Yukawa-type potential.

We will also investigate the energy efficiency by comparing this approach to light harvesting antennae observed in nature [34,35] whose physical mechanism is known to be similar to optical excitation transfer between QDs. More generally, it has been known that the energy efficiency in biological systems is 10^{4} times superior to today’s electrical computers [36]. We will also seek how to realize computational systems by combinations of optical excitation transfer, whose elemental energy efficiency could be comparable to that of biological systems.

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