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Energy Express

  • Editor: Bernard Kippelen
  • Vol. 18, Iss. S4 — Nov. 8, 2010
  • pp: A544–A553
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Lower bound of energy dissipation in optical excitation transfer via optical near-field interactions

Makoto Naruse, Hirokazu Hori, Kiyoshi Kobayashi, Petter Holmström, Lars Thylén, and Motoichi Ohtsu  »View Author Affiliations


Optics Express, Vol. 18, Issue S4, pp. A544-A553 (2010)
http://dx.doi.org/10.1364/OE.18.00A544


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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 104 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 CO2 production [1

1. ITU-T Focus Group on ICTs and Climate Change, http://www.itu.int/ITU-T/focusgroups/climate/index.html.

]. Various approaches have been intensively studied regarding energy efficiency, ranging from analysis of fundamental physical processes [2

2. L. B. Kish, “Moore's law and the energy requirement of computing versus performance,” IEE Proc., Circ. Devices Syst. 151(2), 190–194 (2004). [CrossRef]

,3

3. J. Gea-Banacloche, “Minimum energy requirements for quantum computation,” Phys. Rev. Lett. 89(21), 217901 (2002). [CrossRef] [PubMed]

] to system-level smart energy management [4

4. The Green Grid, http://www.thegreengrid.org/.

]. 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

5. R. S. Tucker, R. Parthiban, J. Baliga, K. Hinton, R. W. A. Ayre, and W. V. Sorin, “Evolution of WDM Optical IP Networks: A Cost and Energy Perspective,” J. Lightwave Technol. 27(3), 243–252 (2009). [CrossRef]

,6

6. K. Sato and H. Hasegawa, “Prospects and Challenges of Multi-Layer Optical Networks,” IEICE Trans. Commun, E 90-B, 1890–1902 (2007). [CrossRef]

].

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

9. T. Kawazoe, K. Kobayashi, J. Lim, Y. Narita, and M. Ohtsu, “Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy,” Phys. Rev. Lett. 88(6), 067404 (2002). [CrossRef] [PubMed]

], CdSe [10

10. M. Naruse, T. Kawazoe, R. Ohta, W. Nomura, and M. Ohtsu, “Optimal mixture of randomly dispersed quantum dots for optical excitation transfer via optical near-field interactions,” Phys. Rev. B 80(12), 125325 (2009). [CrossRef]

], and CdTe [11

11. T. Franzl, T. A. Klar, S. Schietinger, A. L. Rogach, and J. Feldmann, “Exciton Recycling in Graded Gap Nanocrystal Structures,” Nano Lett. 4(9), 1599–1603 (2004). [CrossRef]

]. Geometry-controlled nanostructures are also seeing rapid progress, such as in size- and density-controlled InAs QDs [12

12. J. H. Lee, Zh. M. Wang, B. L. Liang, K. A. Sablon, N. W. Strom, and G. J. Salamo, “Size and density control of InAs quantum dot ensembles on self-assembled nanostructured templates,” Semicond. Sci. Technol. 21(12), 1547–1551 (2006). [CrossRef]

], stacked InAs QDs [13

13. K. Akahane, N. Yamamoto, and M. Tsuchiya, “Highly stacked quantum-dot laser fabricated using a strain compensation technique,” Appl. Phys. Lett. 93(4), 041121 (2008). [CrossRef]

], ring-shaped QDs [14

14. T. Mano and N. Koguchi, “Nanometer-scale GaAs ring structure grown by droplet epitaxy,” J. Cryst. Growth 278(1-4), 108–112 (2005). [CrossRef]

], and ZnO nanorods [15

15. W. I. Park, G.-C. Yi, M. Y. Kim, and S. J. Pennycook, “Quantum Confinement Observed in ZnO/ZnMgO Nanorod Heterostructures,” Adv. Mater. 15(6), 526–529 (2003). [CrossRef]

]. 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

7. M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, Boca Raton, 2008).

]. Based on these principles and enabling technologies, a wide range of device operations have been demonstrated, such as logic gates [16

16. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]

,17

17. T. Yatsui, S. Sangu, T. Kawazoe, M. Ohtsu, S. J. An, J. Yoo, and G.-C. Yi, “Nanophotonic switch using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 90(22), 223110 (2007). [CrossRef]

], interconnects [18

18. M. Naruse, T. Kawazoe, S. Sangu, K. Kobayashi, and M. Ohtsu, “Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting,” Opt. Express 14(1), 306–313 (2006). [CrossRef] [PubMed]

], energy concentration [11

11. T. Franzl, T. A. Klar, S. Schietinger, A. L. Rogach, and J. Feldmann, “Exciton Recycling in Graded Gap Nanocrystal Structures,” Nano Lett. 4(9), 1599–1603 (2004). [CrossRef]

,19

19. M. Naruse, T. Miyazaki, F. Kubota, T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Nanometric summation architecture based on optical near-field interaction between quantum dots,” Opt. Lett. 30(2), 201–203 (2005). [CrossRef] [PubMed]

], 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

16. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]

,18

18. M. Naruse, T. Kawazoe, S. Sangu, K. Kobayashi, and M. Ohtsu, “Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting,” Opt. Express 14(1), 306–313 (2006). [CrossRef] [PubMed]

]. 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

20. H. Hori, “Electronic and Electromagnetic Properties in Nanometer Scales,” in Optical and Electronic Process of Nano-Matters, M. Ohtsu, ed. (Kluwer Academic, 2001), pp. 1–55.

]. 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

21. P. Kocher, J. Jaffe, and B. Jun, “Introduction to Differential Power Analysis and Related Attacks,” http://www.cryptography.com/resources/whitepapers/DPATechInfo.pdf.

], whereas it is hard in optical excitation transfer since energy dissipation occurs at the destination QDs [22

22. M. Naruse, H. Hori, K. Kobayashi, and M. Ohtsu, “Tamper resistance in optical excitation transfer based on optical near-field interactions,” Opt. Lett. 32(12), 1761–1763 (2007). [CrossRef] [PubMed]

]. 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
H^int=d3ri,j=e,hψ^i(r)er·E(r)ψ^j(r),
(1)
where e represents a charge, ψ^i(r) and ψ^j(r) are respectively creation and annihilation operators of either an electron (i, j=e) or a hole (i, j=h) at r, and E(r) is the electric field [23

23. H. Haug, and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2004).

]. In usual light–matter interactions, E(r) 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

7. M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, Boca Raton, 2008).

].

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 2a, which we call QDS and QDL, respectively, as shown in Fig. 1(a)
Fig. 1 (a) Optical near-field interaction between a smaller quantum dot (QDS) and a larger one (QDL). The input light is given by external propagating light at an optical frequency ωext. (b) State transition diagram of the two-dot system.
. Suppose that the energy eigenvalues for the quantized exciton energy level specified by quantum numbers (nx, ny,nz) in a QD with side length L are given by
E(nx,ny,nz)=EB+2π22ML2(nx2+ny2+nz2),
(2)
where EB is the transition energy of the bulk exciton, and 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 QDS and that of quantum number (2,1,1) for QDL. Hereafter, the (1,1,1)-level of QDS is denoted by ES, and the (2,1,1)-level of QDL is called EL2. There is an optical near-field interaction, which is denoted by USL2, due to the steep electric field in the vicinity of QDS. Therefore, excitons in S can move to L 2 in QDL. Note that such a transfer is prohibited in propagating light since the (2,1,1)-level in QDL contains an even number. In QDL, the exciton undergoes intersublevel energy relaxation due to exciton–phonon coupling, denoted by Γ, which is faster than the near-field interaction [9

9. T. Kawazoe, K. Kobayashi, J. Lim, Y. Narita, and M. Ohtsu, “Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy,” Phys. Rev. Lett. 88(6), 067404 (2002). [CrossRef] [PubMed]

,24

24. S. Sangu, K. Kobayashi, A. Shojiguchi, T. Kawazoe, and M. Ohtsu, “Excitation energy transfer and population dynamics in a quantum dot system induced by optical near-field interaction,” J. Appl. Phys. 93(5), 2937–2945 (2003). [CrossRef]

], and so the exciton relaxes to the (1,1,1)-level of QDL, which is called EL1 hereafter. We should note that the intersublevel relaxation determines the uni-directional exciton transfer from QDS to QDL. Also, we assume far-field input light irradiation at the optical frequency ωext.

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 QDS and QDL are denoted by USLi (i=1,2), and the radiative relaxation rates from ES and EL1 are respectively given by γS and γL. Then, letting the (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

25. H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer-Verlag, Berlin, 1999).

]
dρ(t)dt=i[Hint+Hext(t),ρ(t)]+γS2(2Sρ(t)SSSρ(t)ρ(t)SS)+Γ2(2L2ρ(t)L2L2L2ρ(t)ρ(t)L2L2)+γL2(2L1ρ(t)L1L1L1ρ(t)ρ(t)L1L1),
(3)
where the interaction Hamiltonian Hint is given by
Hint=(00000000000USL1ei(ΩSΩL1)0000000USL2ei(ΩSΩL2)00000USL1ei(ΩSΩL1)USL2ei(ΩSΩL2)0000000000USL2ei(ΩSΩL2)USL1ei(ΩSΩL1)00000USL2ei(ΩSΩL2)0000000USL1ei(ΩSΩL1)00000000000),
(4)
where ΩS, ΩL1, and ΩL2 respectively indicate eigenenergy levels associated with ES, EL1, and EL2. The matrices S, L1, and L2 respectively are creation operators that create excitations in ES, EL2, and EL1, defined by
S=(0001000000000100000000100000000000000001000000000000000000000000),L2=(0000000000100000000000000000000000000000000000100000000000000000),L1=(0100000000000000000010000000010000000000000000000000000100000000),
(5)
and S, L 1, and L 2 in Eq. (4) are respectively annihilation operators given by the transposes of the matrices of Eq. (5). Hext indicates the Hamiltonian representing the interaction between the external input light at frequency ωext and the quantum dot system, given by
Hext(t)=gate(t)×[(exp(i(ΩSωext)S+exp(i(ΩSωext)S)+(exp(i(ΩL1ωext))L1+exp(i(ΩL1ωext))L1)],
(6)
where 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 ES in QDS, and to the (1,1,1)-level EL1 in QDL, 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).

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)
Fig. 2 Two representative quantum dot systems: (a) System A, where the inter-dot interaction is strong (100 ps), and (b) System B, where the interaction is negligible (10,000 ps). (c) Yukawa-type screened potential of an optical near-field interaction between two QDs as a function of the inter-dot distance.
, two quantum dots are closely located in the region where the optical excitation transfer from QDS to QDL occurs. We assume USL21 of 100 ps in System A, denoted by UA1 in Fig. 2(a), which is close to typical values of optical near-field interactions experimentally observed in CuCl QDs (130 ps) [16

16. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]

], ZnO quantum-well structures (130 ps) [17

17. T. Yatsui, S. Sangu, T. Kawazoe, M. Ohtsu, S. J. An, J. Yoo, and G.-C. Yi, “Nanophotonic switch using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 90(22), 223110 (2007). [CrossRef]

], ZnO QDs (144 ps) [26

26. T. Yatsui, H. Jeong, and M. Ohtsu, “Controlling the energy transfer between near-field optically coupled ZnO quantum dots,” Appl. Phys. B 93(1), 199–202 (2008). [CrossRef]

], and CdSe QDs (135 ps) [27

27. W. Nomura, T. Yatsui, T. Kawazoe, and M. Ohtsu, “The observation of dissipated optical energy transfer between CdSe quantum dots,” J. Nanophoton. 1(1), 1–8 (2007). [CrossRef]

]. The intersublevel relaxation time, due to exciton–phonon coupling, is in the 1–10 ps range [9

9. T. Kawazoe, K. Kobayashi, J. Lim, Y. Narita, and M. Ohtsu, “Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy,” Phys. Rev. Lett. 88(6), 067404 (2002). [CrossRef] [PubMed]

,24

24. S. Sangu, K. Kobayashi, A. Shojiguchi, T. Kawazoe, and M. Ohtsu, “Excitation energy transfer and population dynamics in a quantum dot system induced by optical near-field interaction,” J. Appl. Phys. 93(5), 2937–2945 (2003). [CrossRef]

,28

28. W. Nomura, T. Yatsui, T. Kawazoe, M. Naruse, and M. Ohtsu, “Structural dependency of optical excitation transfer via optical near-field interactions between semiconductor quantum dots,” Appl. Phys. B 100(1), 181–187 (2010). [CrossRef]

], and here we assume Γ1=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 QDS and QDL are negligible, and thus the optical excitation transfer from QDS to QDL does not occur, and the radiation from QDL should normally be zero. We assume USL21=10,000 ps for System B, denoted by UB1 in Fig. 2(b), indicating effectively no interactions between the quantum dots.

The energy dissipation in the optical excitation transfer from QDS to QDL is the intersublevel relaxation in QDL given by Δ=EL2EL1. Therefore, the issue is to derive the minimum of Δ. When this energy difference is too small, the input light may directly couple to L 1, resulting in output radiation from QDL, even in System B. In other words, we would not be able to recognize the origin of the output radiation from QDL if it involves the optical excitation transfer from QDS to QDL in System A, or it directly couples to 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.

In the case of (i), there is nearly zero population in System B from EL1, which is the expected proper behavior of the system since there are no interactions between the quantum dots. The radiation from QDS is observed with its radiation decay rate (γS). In System A, on the other hand, populations from EL1 do appear. Note that the population involving the output energy level EL1 is only 0.17 when the input pulse is terminated (t = 150 ps), whereas the population involving ES at t = 150 ps is 0.81. Therefore, the increased population from EL1 after t = 150 ps is due to the optical excitation transfer from QDS to QDL. In the case of (iii), due to the small energy difference, the input light directly couples with L 1; therefore, both System A and System B yield higher populations from EL1, 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 QDS and QDL.

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)
Fig. 4 (a) Steady-state population involving energy level EL1 in System A (squares) and System B as a function of the energy dissipation. For System B, three different cases are shown, with UB-1 of 500, 1,000, and 10,000 ps respectively indicated by , , and marks. (b) Energy dissipation as a function of error ratio regarding optical excitation transfer and classical electrically wired device (more specifically a CMOS logic gate) based on Ref [2]. The energy dissipation of optical excitation transfer is about 104 times lower than that in classical electrically wired devices. (c) As the optical near-field interaction time of System B decreases, the lower bound of the error ratio increases, indicating that the performance could be degraded with increasing integration density. The error ratio is evaluated as the number of independent functional blocks within an area of 1 μm2.
summarizes the steady state output populations involving energy level EL1 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 QDL), the signal should come from QDS in its proximity (as in the case of System A), not from QDS far from QDL (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 ωext irradiates QDS. With SNR, the error ratio (PE), or equivalently Bit Error Rate (BER), is derived by the formula PE=(1/2)erfc(SNR/22) where erfc(x)=2/πxexp(x2)dx, called the complementary error function [30

30. S. Haykin, Communication Systems (John Wiley & Sons, New York, 1983).

]. 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

2. L. B. Kish, “Moore's law and the energy requirement of computing versus performance,” IEE Proc., Circ. Devices Syst. 151(2), 190–194 (2004). [CrossRef]

], the minimum energy dissipation (Ed) in classical electrically wired devices (specifically, energy dissipation required for a single bit flip in a CMOS logic gate) is given by
Ed=kBTln(32PE)
(8)
which is indicated by the squares in Fig. 4(b). For example, when the error ratio is 10−6, 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 104 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 (UB1) 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 UB1= 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 UB1 increases (namely, weaker inter-dot interaction), as demonstrated by the triangular and square marks (2) to (6) in Fig. 4(b).

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

18. M. Naruse, T. Kawazoe, S. Sangu, K. Kobayashi, and M. Ohtsu, “Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting,” Opt. Express 14(1), 306–313 (2006). [CrossRef] [PubMed]

], 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

31. M. Naruse, T. Inoue, and H. Hori, “Analysis and Synthesis of Hierarchy in Optical Near-Field Interactions at the Nanoscale Based on Angular Spectrum,” Jpn. J. Appl. Phys. 46(No. 9A), 6095–6103 (2007). [CrossRef]

] 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 QDL represents a theoretical model of a coupled quantum dot system such as a pair of quantum dots. It exerts optical near-field interactions with QDS 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

32. K. Ohmori, K. Kodama, T. Muranaka, Y. Nabetani, and T. Matsumoto, “Tunneling of spin polarized excitons in ZnCdSe and ZnCdMnSe coupled double quantum wells,” Phys. Status Solidi 7(6), 1642–1644 (2010). [CrossRef]

], which can be applicable to quantum dot systems [33

33. J. Seufert, G. Bacher, H. Schömig, A. Forchel, L. Hansen, G. Schmidt, and L. W. Molenkamp, “Spin injection into a single self-assembled quantum dot,” Phys. Rev. B 69(3), 035311 (2004). [CrossRef]

]. 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

16. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]

]. In a future study, we will investigate the required energy for nanophotonic devices in general based on the results shown in this paper.

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 104 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

34. H. Imahori, “Giant Multiporphyrin Arrays as Artificial Light-Harvesting Antennas,” J. Phys. Chem. B 108(20), 6130–6143 (2004). [CrossRef]

,35

35. H. Tamura, J.-M. Mallet, M. Oheim, and I. Burghardt, “Ab Initio Study of Excitation Energy Transfer between Quantum Dots and Dye Molecules,” J. Phys. Chem. C 113(18), 7548–7552 (2009). [CrossRef]

] 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 104 times superior to today’s electrical computers [36

36. V. P. Carey and A. J. Shah, “The Exergy Cost of Information Processing: A Comparison of Computer-Based Technologies and Biological Systems,” J. Electron. Packag. 128(4), 346–352 (2006). [CrossRef]

]. 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.

References and links

1.

ITU-T Focus Group on ICTs and Climate Change, http://www.itu.int/ITU-T/focusgroups/climate/index.html.

2.

L. B. Kish, “Moore's law and the energy requirement of computing versus performance,” IEE Proc., Circ. Devices Syst. 151(2), 190–194 (2004). [CrossRef]

3.

J. Gea-Banacloche, “Minimum energy requirements for quantum computation,” Phys. Rev. Lett. 89(21), 217901 (2002). [CrossRef] [PubMed]

4.

The Green Grid, http://www.thegreengrid.org/.

5.

R. S. Tucker, R. Parthiban, J. Baliga, K. Hinton, R. W. A. Ayre, and W. V. Sorin, “Evolution of WDM Optical IP Networks: A Cost and Energy Perspective,” J. Lightwave Technol. 27(3), 243–252 (2009). [CrossRef]

6.

K. Sato and H. Hasegawa, “Prospects and Challenges of Multi-Layer Optical Networks,” IEICE Trans. Commun, E 90-B, 1890–1902 (2007). [CrossRef]

7.

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, Boca Raton, 2008).

8.

L. Thylén, P. Holmström, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on Integration as Determined by Power Dissipation and Signal-to-Noise Ratio in Loss-Compensated Photonic Integrated Circuits Based on Metal/Quantum-Dot Materials,” IEEE J. Quantum Electron. 46(4), 518–524 (2010). [CrossRef]

9.

T. Kawazoe, K. Kobayashi, J. Lim, Y. Narita, and M. Ohtsu, “Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy,” Phys. Rev. Lett. 88(6), 067404 (2002). [CrossRef] [PubMed]

10.

M. Naruse, T. Kawazoe, R. Ohta, W. Nomura, and M. Ohtsu, “Optimal mixture of randomly dispersed quantum dots for optical excitation transfer via optical near-field interactions,” Phys. Rev. B 80(12), 125325 (2009). [CrossRef]

11.

T. Franzl, T. A. Klar, S. Schietinger, A. L. Rogach, and J. Feldmann, “Exciton Recycling in Graded Gap Nanocrystal Structures,” Nano Lett. 4(9), 1599–1603 (2004). [CrossRef]

12.

J. H. Lee, Zh. M. Wang, B. L. Liang, K. A. Sablon, N. W. Strom, and G. J. Salamo, “Size and density control of InAs quantum dot ensembles on self-assembled nanostructured templates,” Semicond. Sci. Technol. 21(12), 1547–1551 (2006). [CrossRef]

13.

K. Akahane, N. Yamamoto, and M. Tsuchiya, “Highly stacked quantum-dot laser fabricated using a strain compensation technique,” Appl. Phys. Lett. 93(4), 041121 (2008). [CrossRef]

14.

T. Mano and N. Koguchi, “Nanometer-scale GaAs ring structure grown by droplet epitaxy,” J. Cryst. Growth 278(1-4), 108–112 (2005). [CrossRef]

15.

W. I. Park, G.-C. Yi, M. Y. Kim, and S. J. Pennycook, “Quantum Confinement Observed in ZnO/ZnMgO Nanorod Heterostructures,” Adv. Mater. 15(6), 526–529 (2003). [CrossRef]

16.

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]

17.

T. Yatsui, S. Sangu, T. Kawazoe, M. Ohtsu, S. J. An, J. Yoo, and G.-C. Yi, “Nanophotonic switch using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 90(22), 223110 (2007). [CrossRef]

18.

M. Naruse, T. Kawazoe, S. Sangu, K. Kobayashi, and M. Ohtsu, “Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting,” Opt. Express 14(1), 306–313 (2006). [CrossRef] [PubMed]

19.

M. Naruse, T. Miyazaki, F. Kubota, T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Nanometric summation architecture based on optical near-field interaction between quantum dots,” Opt. Lett. 30(2), 201–203 (2005). [CrossRef] [PubMed]

20.

H. Hori, “Electronic and Electromagnetic Properties in Nanometer Scales,” in Optical and Electronic Process of Nano-Matters, M. Ohtsu, ed. (Kluwer Academic, 2001), pp. 1–55.

21.

P. Kocher, J. Jaffe, and B. Jun, “Introduction to Differential Power Analysis and Related Attacks,” http://www.cryptography.com/resources/whitepapers/DPATechInfo.pdf.

22.

M. Naruse, H. Hori, K. Kobayashi, and M. Ohtsu, “Tamper resistance in optical excitation transfer based on optical near-field interactions,” Opt. Lett. 32(12), 1761–1763 (2007). [CrossRef] [PubMed]

23.

H. Haug, and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2004).

24.

S. Sangu, K. Kobayashi, A. Shojiguchi, T. Kawazoe, and M. Ohtsu, “Excitation energy transfer and population dynamics in a quantum dot system induced by optical near-field interaction,” J. Appl. Phys. 93(5), 2937–2945 (2003). [CrossRef]

25.

H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer-Verlag, Berlin, 1999).

26.

T. Yatsui, H. Jeong, and M. Ohtsu, “Controlling the energy transfer between near-field optically coupled ZnO quantum dots,” Appl. Phys. B 93(1), 199–202 (2008). [CrossRef]

27.

W. Nomura, T. Yatsui, T. Kawazoe, and M. Ohtsu, “The observation of dissipated optical energy transfer between CdSe quantum dots,” J. Nanophoton. 1(1), 1–8 (2007). [CrossRef]

28.

W. Nomura, T. Yatsui, T. Kawazoe, M. Naruse, and M. Ohtsu, “Structural dependency of optical excitation transfer via optical near-field interactions between semiconductor quantum dots,” Appl. Phys. B 100(1), 181–187 (2010). [CrossRef]

29.

M. Ohtsu, and K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).

30.

S. Haykin, Communication Systems (John Wiley & Sons, New York, 1983).

31.

M. Naruse, T. Inoue, and H. Hori, “Analysis and Synthesis of Hierarchy in Optical Near-Field Interactions at the Nanoscale Based on Angular Spectrum,” Jpn. J. Appl. Phys. 46(No. 9A), 6095–6103 (2007). [CrossRef]

32.

K. Ohmori, K. Kodama, T. Muranaka, Y. Nabetani, and T. Matsumoto, “Tunneling of spin polarized excitons in ZnCdSe and ZnCdMnSe coupled double quantum wells,” Phys. Status Solidi 7(6), 1642–1644 (2010). [CrossRef]

33.

J. Seufert, G. Bacher, H. Schömig, A. Forchel, L. Hansen, G. Schmidt, and L. W. Molenkamp, “Spin injection into a single self-assembled quantum dot,” Phys. Rev. B 69(3), 035311 (2004). [CrossRef]

34.

H. Imahori, “Giant Multiporphyrin Arrays as Artificial Light-Harvesting Antennas,” J. Phys. Chem. B 108(20), 6130–6143 (2004). [CrossRef]

35.

H. Tamura, J.-M. Mallet, M. Oheim, and I. Burghardt, “Ab Initio Study of Excitation Energy Transfer between Quantum Dots and Dye Molecules,” J. Phys. Chem. C 113(18), 7548–7552 (2009). [CrossRef]

36.

V. P. Carey and A. J. Shah, “The Exergy Cost of Information Processing: A Comparison of Computer-Based Technologies and Biological Systems,” J. Electron. Packag. 128(4), 346–352 (2006). [CrossRef]

OCIS Codes
(200.3050) Optics in computing : Information processing
(230.5590) Optical devices : Quantum-well, -wire and -dot devices
(260.2160) Physical optics : Energy transfer
(180.4243) Microscopy : Near-field microscopy

ToC Category:
Energy Transfer

History
Original Manuscript: August 2, 2010
Revised Manuscript: August 31, 2010
Manuscript Accepted: September 23, 2010
Published: October 5, 2010

Citation
Makoto Naruse, Hirokazu Hori, Kiyoshi Kobayashi, Petter Holmström, Lars Thylén, and Motoichi Ohtsu, "Lower bound of energy dissipation in optical excitation transfer via optical near-field interactions," Opt. Express 18, A544-A553 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-S4-A544


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References

  1. ITU-T Focus Group on ICTs and Climate Change, http://www.itu.int/ITU-T/focusgroups/climate/index.html .
  2. L. B. Kish, “Moore's law and the energy requirement of computing versus performance,” IEE Proc., Circ. Devices Syst. 151(2), 190–194 (2004). [CrossRef]
  3. J. Gea-Banacloche, “Minimum energy requirements for quantum computation,” Phys. Rev. Lett. 89(21), 217901 (2002). [CrossRef] [PubMed]
  4. The Green Grid, http://www.thegreengrid.org/ .
  5. R. S. Tucker, R. Parthiban, J. Baliga, K. Hinton, R. W. A. Ayre, and W. V. Sorin, “Evolution of WDM Optical IP Networks: A Cost and Energy Perspective,” J. Lightwave Technol. 27(3), 243–252 (2009). [CrossRef]
  6. K. Sato and H. Hasegawa, “Prospects and Challenges of Multi-Layer Optical Networks,” IEICE Trans. Commun, E 90-B, 1890–1902 (2007). [CrossRef]
  7. M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, Boca Raton, 2008).
  8. L. Thylén, P. Holmström, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on Integration as Determined by Power Dissipation and Signal-to-Noise Ratio in Loss-Compensated Photonic Integrated Circuits Based on Metal/Quantum-Dot Materials,” IEEE J. Quantum Electron. 46(4), 518–524 (2010). [CrossRef]
  9. T. Kawazoe, K. Kobayashi, J. Lim, Y. Narita, and M. Ohtsu, “Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy,” Phys. Rev. Lett. 88(6), 067404 (2002). [CrossRef] [PubMed]
  10. M. Naruse, T. Kawazoe, R. Ohta, W. Nomura, and M. Ohtsu, “Optimal mixture of randomly dispersed quantum dots for optical excitation transfer via optical near-field interactions,” Phys. Rev. B 80(12), 125325 (2009). [CrossRef]
  11. T. Franzl, T. A. Klar, S. Schietinger, A. L. Rogach, and J. Feldmann, “Exciton Recycling in Graded Gap Nanocrystal Structures,” Nano Lett. 4(9), 1599–1603 (2004). [CrossRef]
  12. J. H. Lee, Zh. M. Wang, B. L. Liang, K. A. Sablon, N. W. Strom, and G. J. Salamo, “Size and density control of InAs quantum dot ensembles on self-assembled nanostructured templates,” Semicond. Sci. Technol. 21(12), 1547–1551 (2006). [CrossRef]
  13. K. Akahane, N. Yamamoto, and M. Tsuchiya, “Highly stacked quantum-dot laser fabricated using a strain compensation technique,” Appl. Phys. Lett. 93(4), 041121 (2008). [CrossRef]
  14. T. Mano and N. Koguchi, “Nanometer-scale GaAs ring structure grown by droplet epitaxy,” J. Cryst. Growth 278(1-4), 108–112 (2005). [CrossRef]
  15. W. I. Park, G.-C. Yi, M. Y. Kim, and S. J. Pennycook, “Quantum Confinement Observed in ZnO/ZnMgO Nanorod Heterostructures,” Adv. Mater. 15(6), 526–529 (2003). [CrossRef]
  16. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Demonstration of a nanophotonic switching operation by optical near-field energy transfer,” Appl. Phys. Lett. 82(18), 2957–2959 (2003). [CrossRef]
  17. T. Yatsui, S. Sangu, T. Kawazoe, M. Ohtsu, S. J. An, J. Yoo, and G.-C. Yi, “Nanophotonic switch using ZnO nanorod double-quantum-well structures,” Appl. Phys. Lett. 90(22), 223110 (2007). [CrossRef]
  18. M. Naruse, T. Kawazoe, S. Sangu, K. Kobayashi, and M. Ohtsu, “Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting,” Opt. Express 14(1), 306–313 (2006). [CrossRef] [PubMed]
  19. M. Naruse, T. Miyazaki, F. Kubota, T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, “Nanometric summation architecture based on optical near-field interaction between quantum dots,” Opt. Lett. 30(2), 201–203 (2005). [CrossRef] [PubMed]
  20. H. Hori, “Electronic and Electromagnetic Properties in Nanometer Scales,” in Optical and Electronic Process of Nano-Matters, M. Ohtsu, ed. (Kluwer Academic, 2001), pp. 1–55.
  21. P. Kocher, J. Jaffe, and B. Jun, “Introduction to Differential Power Analysis and Related Attacks,” http://www.cryptography.com/resources/whitepapers/DPATechInfo.pdf .
  22. M. Naruse, H. Hori, K. Kobayashi, and M. Ohtsu, “Tamper resistance in optical excitation transfer based on optical near-field interactions,” Opt. Lett. 32(12), 1761–1763 (2007). [CrossRef] [PubMed]
  23. H. Haug, and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2004).
  24. S. Sangu, K. Kobayashi, A. Shojiguchi, T. Kawazoe, and M. Ohtsu, “Excitation energy transfer and population dynamics in a quantum dot system induced by optical near-field interaction,” J. Appl. Phys. 93(5), 2937–2945 (2003). [CrossRef]
  25. H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer-Verlag, Berlin, 1999).
  26. T. Yatsui, H. Jeong, and M. Ohtsu, “Controlling the energy transfer between near-field optically coupled ZnO quantum dots,” Appl. Phys. B 93(1), 199–202 (2008). [CrossRef]
  27. W. Nomura, T. Yatsui, T. Kawazoe, and M. Ohtsu, “The observation of dissipated optical energy transfer between CdSe quantum dots,” J. Nanophoton. 1(1), 1–8 (2007). [CrossRef]
  28. W. Nomura, T. Yatsui, T. Kawazoe, M. Naruse, and M. Ohtsu, “Structural dependency of optical excitation transfer via optical near-field interactions between semiconductor quantum dots,” Appl. Phys. B 100(1), 181–187 (2010). [CrossRef]
  29. M. Ohtsu, and K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).
  30. S. Haykin, Communication Systems (John Wiley & Sons, New York, 1983).
  31. M. Naruse, T. Inoue, and H. Hori, “Analysis and Synthesis of Hierarchy in Optical Near-Field Interactions at the Nanoscale Based on Angular Spectrum,” Jpn. J. Appl. Phys. 46(No. 9A), 6095–6103 (2007). [CrossRef]
  32. K. Ohmori, K. Kodama, T. Muranaka, Y. Nabetani, and T. Matsumoto, “Tunneling of spin polarized excitons in ZnCdSe and ZnCdMnSe coupled double quantum wells,” Phys. Status Solidi 7(6), 1642–1644 (2010). [CrossRef]
  33. J. Seufert, G. Bacher, H. Schömig, A. Forchel, L. Hansen, G. Schmidt, and L. W. Molenkamp, “Spin injection into a single self-assembled quantum dot,” Phys. Rev. B 69(3), 035311 (2004). [CrossRef]
  34. H. Imahori, “Giant Multiporphyrin Arrays as Artificial Light-Harvesting Antennas,” J. Phys. Chem. B 108(20), 6130–6143 (2004). [CrossRef]
  35. H. Tamura, J.-M. Mallet, M. Oheim, and I. Burghardt, “Ab Initio Study of Excitation Energy Transfer between Quantum Dots and Dye Molecules,” J. Phys. Chem. C 113(18), 7548–7552 (2009). [CrossRef]
  36. V. P. Carey and A. J. Shah, “The Exergy Cost of Information Processing: A Comparison of Computer-Based Technologies and Biological Systems,” J. Electron. Packag. 128(4), 346–352 (2006). [CrossRef]

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