Основы наноелектроники / Основы наноэлектроники / ИДЗ / Книги и монографии / Наноматериалы, методы, идеи. Сборник научных статей, 2007, c.206

The elliptical polarization enables bending about the the corner with little losses. The optical phase affords a way of breaking the symmetry so as to funnel the energy into one or the other of the T-junction branches. To check the sensitivity of the control results to deviation from perfect regularity, we introduce random size and location deviation of 2 nm for each particle. The sensitivity of the results to the particle size was checked by varying the NP’s radii in the 25 to 30 nm range. Simulations show that, while the details of Fig. 1 are not invariant, the extent of control is not affected by either deviation from perfect regularity of small size modifications.

Fig. 1. Phase-polarization control of EM energy transport through the T-structure depicted schematically in the inset. Logarithm of the ratio W1/W2 of the time averaged EM energies in the upper (D1) and lower (D2) ends of a T-structure (see inset) as a function of the laser polarization parameters, and . The incident pulse is centered at a plasmonic resonance wavelength, λ=300 nm.

To quantify the effect of losses, we have computed the power transmission coefficient, defined as the ratio of the time-averaged EM

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energy at the center of the NP closest to the incident source NP to the time-averaged EM energy at detection points, D1 and D2. We found a power transmission coefficient of 2.6 (that is about 50 times smaller than the controlled branching ratio shown in Fig. 1). Whereas the control ratio, W1/W2, improves with increasing duration of the incident pulse, the power transmission coefficient is at most essentially independent on the pulse duration.

We found that the phase-polarization control is equally effective in larger T-structure consisting of two interacting linear chains of NPs of 900 nm length each. The results of Fig. 1 suggest a potential nano-scale plasmonic switch or inverter with an operational time comparable to the incident pulse duration of tens of fs.

Fig. 2. Phase-polarization control of EM energy transport through the X-structure depicted schematically in the inset. Logarithm of the ratio W1/W2 of the time averaged EM energies in the upper (D1) and lower (D2) detectors of the X-structure (see inset) as a function of

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incident sources, LS1 and LS2. The incident pulses are centered at the plasmon resonance, λ=435 nm.

Figure 2 suggests the potential of a different coherent control concept, based on the interference between two EM waves propagating towards the array intersection. The two sources strategy is exemplified here by an X-structure, as shown in the inset of Fig. 2, where LS1 and LS2 indicate the location of the two phase-locked laser sources. The main frame of Fig. 2 presents (on a logarithmic scale) the ratio of the time averaged EM energies, W1/W2, detected in two symmetrically

placed detectors, D1 and D2. We set 1 2 1/ 2 , corresponding to equal magnitudes of the x- and y-components of both sources, and vary1 and 2 (or, equivalently, the relative phases of LS1 and LS2 in the x-

field components). Similar to the case of the T-junction, the branching ratio can be varied over a significant range by choice of the relative phases of the excitation sources. Phase-polarization control of plasmonic devices is a general effect that we found also in other constructs and which was not recognized before.

4. Conclusion

We proposed the extension of the tools of optical control to manipulate the transport of EM energy in the nano-scale and to derive new insights into nano-plasmonic physics. The phase and polarization properties of laser sources are shown to provide interesting tools in nano-plasmonics, capable of driving plasmon excitations to desired spatial locations, potentially making nano-scale optical switches and inverters.

Potential applications are many and varied. For instance, colloidal NPs and synthetic nanomaterials (such as dented and striped multi-layer nanorods) with engineered optical properties have attracted much interest in recent years [10], due to the rapid progress of fabrication techniques such as template electrochemical synthesis physical/chemical vapor deposition, and chemical lithography [3]. Controllability over the properties of such systems translates into control over their dielectric constant and therefore over the EM wave dispersion and plasmon resonances. This suggests an opportunity for coherent control strategies, with which a variety of desired targets (e.g., minimization of losses, enhancement of a given mode, sensitivity to a given species etc.) could

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be potentially attained by adjusting the width and length of different layers of individual NPs, and selecting different materials ranging from noble metals to insulating solids.

Both experimental and theoretical work on nanoplasmonics have focused on the case of nanostructures in vacuum or rare gases. One might expect, however, optical nonlinearities of the embedding media to lead to interesting phenomena. Second harmonic generation, selffocusing and four-wave mixing induced by a Kerr medium, for instance, can play a significant role in EM energy enhancement, localization and transport. Other potential applications of phase and polarization control in nanoplasmonics include optimization of the EM field enhancement in single molecular spectroscopy, where properly shaped NPs could noticeably improve the sensitivity, and design of laser-assisted 3D atom probe experiments.

5. References

1.W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, p. 824, 2003.

2.S. Link and M. A. El-Sayed, “Optical properties and ultrafast dynamics of metallic nanocrystals,” Annu. Rev. Phys. Chem. 54, p. 331, 2003.

3.E. Hutter and J. H. Fendler, “Exploitation of localized surface plasmon resonance,” Adv. Mat. 16, p. 1685, 2004.

4.G. P. Wiederrecht, “Near-field optical imaging of noble metal nanoparticles,” Eur. Phys. J. Appl. Phys. 28, p. 3, 2004.

5.M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, p. 1331, 1998.

6.T. Klar, M. Perner, S. Grosse, G. V. Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, p. 4249, 1998.

7.G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. Leipzig 25, p. 377, 1908.

8.U. Kreibeg and M. Vollmer, Optical Properties of Metal Clusters, Springer, Berlin, 1995.

9.S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics - a route to nanoscale optical devices,” Adv. Mat. 13, p. 1501, 2001.

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10.“Nanoscale materials special issue,” In Acc. Chem. Res. 32, pp. 387—454, 1999.

11.S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in

12.metal/dielectric structures,” J. Appl. Phys. 98, p. 011101, 2005.

13.S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics, John Wiley and Sons, 2000.

14.M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes, Hoboken, N.J.: Wiley-Interscience, 2003.

15.H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?,” Science 288, p. 824, 2000.

16.R. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, p. 1500, 1992.

17.T. Brixner and G. Gerber, “Quantum control of gas-phase and liquid-phase femtochemistry,” ChemPhysChem 4, p. 418, 2003.

18.D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. on Ant. and Prop. 45, p. 343, 1997.

19.M. J. Buckley, “Linear array synthesis using a hybrid genetic algorithm,” Proc. IEEE Ant. Prop. Soc. Int. Symp. , p. 584, 1996.

20.K. Aygun, D. S. Weile, and E. Michielssen, “Design of multilayered strip gratings by genetic algorithm,” Microwave Opt. Tech. Lett. 42, p. 81, 1997.

21.Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, 2000.

22.F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.

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interest to increase thermal conductance in the micro and nanodevices [1, 6, 7]. In this paper study thermal transport in nonmetallic systems, in which heat is transported by thermal excitations only. In addition to practical importance of such studies, thermal transport in nanowires is interesting from fundamental point of view. Recent theoretical [8] and experimental [9] findings proved the existence of the quantum of thermal conductance in ballistic regime, which is similar to the quantum of electronic conductance. The state of experimental and theoretical understanding of thermal transport in nanoscale systems is comprehensively discussed in the review [10]. Recently D, Li et al [11] reported an accurate measurement of lattice thermal conductivity in silicon nanowires for a wide range of temperatures and wire diameters. They demonstrated the significant influence of the system size not only on the magnitude of the thermal conductivity coefficient, but also on its temperature dependence. It is well known that for large enough diameters of the wire and diffusive phonon - boundary scattering, thermal conductivity coefficient at low temperatures is proportional to

T3 . But for small values of the wire diameters experiment [11] shows clear crossover from cubic to near linear dependence on the temperature. In the present paper we consider one particular relaxation mechanism which can explain the observed crossover.

Recently Mingo [12] carried out an accurate numerical study of thermal conductance of silicon nanowires to explain the decrease of the thermal conductivity coefficient with wire diameter observed in the experiment. He assumed that all the effects can be explained by the reconstruction of the phonon dispersion, where realistic phonon modes obtained from MD simulations were applied to general expression of the thermal conductivity coefficient. His numerical analysis shows excellent quantitative agreement with the experiment [11] for large enough diameters at high temperatures. As the system size becomes smaller, the approach fails to describe a sharp decrease of thermal conductance as well as qualitative change of its temperature dependence. This is likely because the Matheissen's rule has been used for evaluation of the phonon lifetime, which has rather restricted range of applicability (see e.g. [13] and references therein).

As it was noted in Ref. [14], decrease of the temperature increases the characteristic phonon wavelength and reduces the scattering probability

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at the boundary surface. This leads to a modification of the phonon spectrum. For ideal wires it is represented by a set of branches with energies proportional to ID momentum directed along the wire. So the standard theory of thermal conductance in dielectrics and semiconductors has to be modified to account for low dimensionality effects as well as phonon spectrum modification at low temperatures. Thus, to understand thoroughly the physical processes occurring inside the nanowires with decreasing sizes, we need an analytical theory to account for different mechanisms explicitly, such as dispersion reconstruction and restricted geometry.

To approach the problem we consider low enough temperatures, where the quasiparticles states of "acoustic" branches are thermally populated ( 0 when p 0). The corresponding acoustic branches have the following dispersion relations [14, 15]:

corresponding momentum, a is the wire diameter, u1 and u2 are the characteristic velocities. The first expression in Eq. (1) is the phonon dispersion and the second expression is the dispersion of flexural mode. The nature of flexural modes conies from the fact they are analogous to bending modes of classical elasticity theory, or the antisymmetric Lamb waves of a free plate [16]. Appearance of such a mode is just a direct consequence of restricted geometry, and under some conditions it can be considered as the only size effect in thermal transport properties. Strictly speaking there are two modes for each type of dispersion, but we do not account for them separately since their contributions are qualitatively the same. Consequently, we have to solve the kinetic problem for two-component gas of quasiparticles. It is well known [13, 17-20] that using simple Callaway formula, to estimate the thermal conductivity coefficient in two-component systems sometimes leads to serious confusions. Simple summation of the relaxation rates, as is done in the majority of theoretical works, is questionable under many physical conditions. The dependence of the kinetic coefficients on different relaxation times is much more complicated in reality. Accurate method for calculation of the diffusion coefficient in two-component

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gas of quasiparticles was proposed in Ref. [20]. Here we extend this formalism to thermal conductance problem.

To start, we consider a system of two types of quasiparticles. Their kinetics is described by equations for corresponding distribution functions fi :

where Cij(fi,fj) is the collision integral of thermal excitations, and Ci3 (fi ) is the collision integral describing the scattering processes

between quasiparticles and scatterers. In general the latter includes all the processes leading to the non-conservation of the total energy. Vi = d i /di i is the group velocity of the corresponding thermal excitation.

The main purpose of our theory is to obtain analytic expressions of thermal conductance, which are applicable to quasiparticles with arbitrary dispersion relations. In other words, the explicit dispersion relations in Eq. (1) are needed only at the last stage when calculating corresponding relaxation times and thermodynamic quantities. As usual, we seek a perturbative solution of the system (2) in the form

represents small deviation from the equilibrium. The perturbation term can be conveniently chosen to be fi gi fi(0) / i with gi the new

target functions. After the standard linearization procedure, Eq. (2) can be written in the following matrix form:

The 2D collision matrix С can be decomposed into a sum of three terms,

corresponding to different relaxation mechanisms: C J S U where

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quasiparticles, and U Uij Ci3 ij describes all the other relaxation

mechanisms, which do not conserve total energy of the quasiparticle system. U includes scattering on defects, boundaries, umklapp processes etc. [21] Here dj represent linearized collision operators.

Let us define the scalar product of two-dimensional braand ket-vectors as follows [20]:

where (фk | and | k ) are the correspondent one-component vectors, dГ

is the element of phase volume. Under this condition, the collision operator С becomes hermitian. System (4) is the system of nonuniform linear integral equations. According to the general theory of integral

equations the target solution |g must be orthogonal to the solution of

corresponding uniform equations C | uni . It is therefore convenient to

write the formal solution of (4) so that the orthogonality condition is imposed explicitly in the solution. For this purpose we define the projection operator Vn onto the subspace orthogonal to the vector \фиПг),

Vn = 1 — Vc, Vc = \Фипг)(Фипг\- As a resg uni ult, the formal solution of the system (4) can be written in the form

The heat flux density due to the thermal excitations of different types is

given by the expression Q k 1,2 k k fkdГk Using relation (3) and definition of scalar product (5), Q can be rewritten as Q=(фк|g). On the other hand, the effective thermal conductivity coefficient is defined

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