CHAPTER 5 CONCLUSIONS AND FUTURE WORK
A.3 Model for Membrane Acoustics
In this section we develop a model to obtain the frequency response in a pump-probe setup if the transient response from a single pump pulse is a signal (S0(t)). We develop the final in-phase and out-of phase expressions
for the voltages read by the lock-in amplifier which are indendent of the specific character of the single pulse resposne S0(t). The method developed
herein is generic and may be applied to electronic, thermal and acoustic measurements. Figure A.2 provides an overview of the measurement. A series of pump pulses modulated by an envelope function trigger a response S0(t) that is the cumulative contributions of all pulses. The pump and probe
as follows Lpump(t) = EL TP ∞ X n=−∞ rect [θL] where θL = t − (αL+ nTL) TP Lprobe(t) = EP TP ∞ X n=−∞ rect [θP] where θP = t − (αL+ nTL+ td) TP . (A.9) td is the delay time between pump and probe pulses. EL and EP are the
energies per pulse for pump and probe beams respectively. I the limit of short pulses, we can approximate the pulsed by delta functions as
Lpump(t) ∼ EL ∞ X n=−∞ δ [t − (αL+ nTL)] Lprobe(t) ∼ EP ∞ X n=−∞ δ [t − (αL+ nTL+ td)]. (A.10)
The modulation envelope for a square wave modulation becomes M (t) = M0+ ∆M (t) M (t) = M0+ ∆M0 ∞ X n=−∞ rect [θM] where θM = t − (αM + nTM) pTM M (t) = ( M0+ ∆M (t) −pT2M + αM + nTM < t < pT2M + αM + nTM M0 else (A.11) The total response signal form a series of pump pulses becomes
S (t) = S0(t) ∗ (Lpump(t) × M (t)) . (A.12)
The reflectance is then
R (t) = R0+ ∆R0(t) + ∆R (t) R (t) = R0+ dR dS × S0(t) ∗ (Lpump(t) × M0) + dR dS × S0(t) ∗ (Lpump(t) × ∆M (t)) (A.13)
dR/dS is the change in reflectance with respect to the measured signal. The reflected intensity is therefore
I (t) = R (t) × Lprobe(t) I (t) = R0 × Lprobe(t) + dR dS × [S0(t) ∗ (Lpump(t) × M0)] × Lprobe(t) + dR dS × [S0(t) ∗ (Lpump(t) × ∆M (t))] × Lprobe(t) I (t) = I0(t) + ∆I0(t) + ∆I (t) (A.14) In frequency domain this becomes
˜ I (f ) = ˜I0(f ) + ∆ ˜I0(f ) + ∆ ˜I (f ) ˜ I (f ) = R0 × ˜Lprobe(f ) + dR dS × h ˜S0(f ) × ˜Lpump(f ) × M0i ∗ ˜Lprobe(f ) +dR dS × h ˜S0(f ) × ˜Lpump(f ) ∗ ∆ ˜M (f )i ∗ ˜Lprobe(f ) (A.15) The lock-in amplifier measures the rms voltage of the
Vrms(t) =
Q (f ) √
2 × RP (λ) × I (t) . (A.16) RP(λ) is the responsivity of the photodetector at the wavelength λ of the
reflected probe beam. Q (f ) is the effective quality factor of the electron- ics connecting the photodetector to the lock-in amplifier at the modulation frequency fM of the pump beam. Finally, the in-phase and out-of-phase volt-
the probe are Vin(fM) = Q (fM) √ 2 × RP(λ) × 1 2 dR dS × p∆M0× ELEP T2 L × sinc (p) × ∞ X k=−∞ ˜ S0 k TL + 1 TM exphi2π td TM i + ˜S0 k TL − 1 TM exph−i2π td TM i exp i2πktd TL Vout(fM) = Q (fM) √ 2 × RP(λ) × 1 2 dR dS × p∆M0× ELEP T2 L × sinc (p) × ∞ X k=−∞ ˜ S0 k TL + 1 TM exphi2π td TM i − ˜S0 k TL − 1 TM exph−i2π td TM i exp i2πktd TL (A.17) This concludes our derivation of the frequency response. We have used this model to simulate the acoustic response at negative delay times for long-lived acoustic modes. We assumed a single pulse acoustic response signal of the form S0(t) = A0exp −t − tac τ sin [2πf0(t − tac)] (A.18)
Our fits of the negative delay time data yielded damping times that are similar to the ones obtained through linear regression.
APPENDIX B
SUPPLEMENTARY INFORMATION FOR
CHAPTER 2
B.1
Instrumentation and Uncertainties in Thermal
Conductivity Measurements
A DC current, Idc ∼ 0-3 µA from a constant current source (Keithley 6221)
provided the heating power. AC currents, Iac ∼ 150 250 nA from two
different lock-in amplifiers (SR830) measured coil resistances. Two addi- tional lock-in amplifiers that were phase-locked to the previous two provided voltages across the heating and sensing coils respectively. A fifth lock-in measured the voltage across one of the supporting legs and the heating coil in order to determine the heating power supplied to the device. We used the of f set and expand functions on the SR830 to decrease truncation and digitization errors in our measurement. These allow measurements of coil resistances with 0.5 µΩ resolution.
Thermal fluctuations outside the cryostat are a major source of noise in measuring the change in coil resistance, R. The voltage source of the lock-in amplifier, combined with a high precision resistor (20 MΩ) provided the con- stant ac current in our measurement. Both suffer from amplitude instability due to thermal fluctuations. The resulting ac current has an amplitude drift of around 70 ppm/oC. This instability in the ac current induces a random noise in the resistance measurement. Its standard deviation is around 250 mΩ that corresponds to the lower limit of measurable change in resistance. While thermal fluctuations affect the measurement of change in resistance R, other sources of noise affect the measurement of the zero current resistance, R0 (R at Idc = 0) at each temperature point T0. Pressure fluctuations in the
helium Dewar, during the measurement, bring the entire chip out of thermal equilibrium and induce fluctuations in R0. We installed a low-pressure valve
(1 psi) on the venting line of the Dewar in order to reduce this effect. Another source of error in measuring R0 relates to the long temperature transients of
the measurement. The suspended platforms are well insulated from the chip and this leads to a settling time of ∼ 4 hours at low bath temperatures (T0
∼ 5 K). This settling time increases with temperature, reaching around 10 hours at 300 K. These two factors contribute an uncertainty of 1-2 Ω in R0.
Finally, uncertainty in measuring the diameters and lengths of the nanowires through a combination of SEM and TEM is 10%.
B.2
Supplementary Figures
Figure B.1: SEM micrographs of silicon nanowires MAC-etched with ρ =75 % and different M (H2O) . We can reduce the etch rate independently of
rho with water dilution. The silicon nanowires remain straight and sharp all the way to the tips as the etch rate reduces by three orders of magnitude.
Figure B.2: An SEM image of the measurement platform showing a suspended nanowire between the heating and cooling islands. The inset shows a high resolution image of the suspended nanowire.
Figure B.3: Height profile (TEM trace), autocorrelation function and Fourier power spectrum for a nanowire that is 100 nm in diameter and has an RMS height σ = 1.0 nm and a correlation length lc = 4 nm. For
comparison, we show the height profile, autocorrelation function and Fourier power spectrum of random Gaussian and exponential profiles having the same RMS height and correlation length. To generate the random profiles, we used a moving average numerical technique described in [7]. a is the lattice constant of Silicon.
Figure B.4: Temperature dependence of thermal conductivity for single nanowires with different diameters φ, RMS roughness heights σ and correlation length lc. The exponent n is the temperature dependence of κ
as obtained from the fits to the low temperature data using a modified Holland-Callaway model. Roughness and thermal conductivity data is from [38].
REFERENCES
[1] A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of rough silicon nanowires,” Nature, vol. 451, no. 7175, pp. 163–167, Jan 2008. [Online]. Available: http://dx.doi.org/10.1038/nature06381
[2] J.-K. Yu, S. Mitrovic, D. Tham, J. Varghese, and J. R. Heath, “Reduction of thermal conductivity in phononic nanomesh structures,” Nat Nano, vol. 5, no. 10, pp. 718–721, Oct 2010. [Online]. Available: http://dx.doi.org/10.1038/nnano.2010.149
[3] K. Esfarjani, G. Chen, and H. T. Stokes, “Heat trans- port in silicon from first-principles calculations,” Phys. Rev. B, vol. 84, p. 085204, Aug 2011. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.84.085204
[4] K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. McGaughey, and J. A. Malen, “Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermore- flectance,” Nature communications, vol. 4, p. 1640, 2013.
[5] W. N. Ashcroft and D. N. Mermin, Solid State Physics. Belmont: Brooks/Cole, Cengage Learning, 1976.
[6] R. E. Peierls, Quantum Theory of Solids. New York: Oxford University Press, 2001.
[7] J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces. New York: Adam Hilger, 1991.
[8] B. A. Auld, Acoustic Fields and Waves in Solids. New York: John Wiley and Sons, 1973.
[9] G. Chen, Nanoscale Energy Transport and Conversion. New York: Oxford University Press, 2005.
[10] S. Wei and M. Y. Chou, “Phonon dispersions of sili- con and germanium from first-principles calculations,” Phys.
Rev. B, vol. 50, pp. 2221–2226, 1994. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.50.2221
[11] G. Dolling, in Proceedings of the Symposium on Inelastic Scattering of Neutrons in Solids and Liquids (2nd edition), vol. 2. Vienna: IAEA, 1963, p. 37.
[12] G. Nilsson and G. Nelin, “Study of the homology between silicon and germanium by thermal-neutron spectrometry,” Phys. Rev. B, vol. 6, pp. 3777–3786, 1972. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.6.3777
[13] R. L. Liboff, Kinetic Theory, 3rd ed. New York: Springer-Verlag, 2003.
[14] J. Bernhard, “The relation between relaxation time, mean free path, collision time and drift velocity pitfalls and a proposal for an approach illustrating the essentials,” European Journal of Physics, vol. 30, no. 1, p. 1, 2009. [Online]. Available: http://stacks.iop.org/0143- 0807/30/i=1/a=001
[15] P. G. Klemens, “The thermal conductivity of dielectric solids at low temperatures (theoretical),” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 208, no. 1092, pp. 108–133, 1951. [Online]. Available: http://www.jstor.org/stable/98761
[16] M. G. Holland, “Analysis of lattice thermal conductivity,” Phys. Rev., vol. 132, pp. 2461–2471, 1963. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRev.132.2461
[17] H. M. G. and L. J. Neuringer, in Proceedings of the International Con- ference on the Physics of Semiconductors, vol. 2. Exeter: The Institute of Physics and Physical Society, 1962, p. 474.
[18] C. J. Glassbrenner and G. A. Slack, “Thermal conductivity of silicon and germanium from 3k to the melting point,” Phys. Rev., vol. 134, pp. A1058–A1069, 1964. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRev.134.A1058
[19] Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev. Mod. Phys., vol. 71, pp. S306–S312, 1999. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.71.S306
[20] K. Schwab, E. Henriksen, J. Worlock, and M. L. Roukes, “Measurement of the quantum of thermal conductance,” Nature, vol. 404, no. 6781, pp. 974–977, 2000.
[21] M. V. Simkin and G. D. Mahan, “Minimum thermal conductivity of su- perlattices,” Phys. Rev. Lett., vol. 84, pp. 927–930, Jan 2000. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.84.927 [22] K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. H.
McGaughey, and J. A. Malen, “Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance,” Nat Commun, vol. 4, p. 1640, Mar 2013. [Online]. Available: http://dx.doi.org/10.1038/ncomms2630
[23] A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K. A. Nelson, and G. Chen, “Thermal conductivity spectroscopy technique to measure phonon mean free paths,” Phys. Rev. Lett., vol. 107, p. 095901, Aug 2011. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.107.095901
[24] Y. S. Ju and K. E. Goodson, “Phonon scattering in silicon films with thickness of order 100 nm,” Applied Physics Let- ters, vol. 74, no. 20, pp. 3005–3007, 1999. [Online]. Available: http://scitation.aip.org/content/aip/journal/apl/74/20/10.1063/1.123994 [25] G. Chen, “Size and interface effects on thermal conductivity of
superlattices and periodic thin-film structures,” Journal of Heat Transfer, vol. 119, no. 2, pp. 220–229, May 1997. [Online]. Available: http://dx.doi.org/10.1115/1.2824212
[26] A. Maznev, “Boundary scattering of phonons: Specularity of a ran- domly rough surface in the small-perturbation limit,” Physical Review B, vol. 91, no. 13, p. 134306, 2015.
[27] C. Eckart, “The scattering of sound from the sea sur- face,” The Journal of the Acoustical Society of America, vol. 25, no. 3, pp. 566–570, 1953. [Online]. Available: http://scitation.aip.org/content/asa/journal/jasa/25/3/10.1121/1.1907123 [28] J. M. Ziman, Electrons and Phonons. New York: Oxford University
Press, 2003.
[29] D. Santamore and M. Cross, “Effect of surface roughness on the uni- versal thermal conductance,” Physical Review B, vol. 63, no. 18, p. 184306, 2001.
[30] J. Sadhu and S. Sinha, “Room-temperature phonon boundary scatter- ing below the casimir limit,” Physical Review B, vol. 84, no. 11, p. 115450, 2011.
[31] L. D. Landau and E. M. Lifshitz, Theory of Elasticity. New York: Pergamon Press, 1959.
[32] H. E. B¨ommel and K. Dransfeld, “Excitation and at- tenuation of hypersonic waves in quartz,” Phys. Rev., vol. 117, pp. 1245–1252, Mar 1960. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRev.117.1245
[33] H. J. Maris, “Interaction of sound waves with thermal phonons in di- electric crystals,” Physical Acoustics, vol. 8, pp. 279–345, 2012.
[34] L. Landau and G. Rumer, “Absorption of sound in solids,” Phys. Z. Sowjetunion, vol. 11, p. 18, 1937.
[35] A. Akhieser, “On the absorption of sound in solids,” J. Phys. (USSR), vol. 1, p. 277, 1939.
[36] D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, “Thermal conductivity of individual silicon nanowires,” Applied Physics Letters, vol. 83, no. 14, pp. 2934–2936, 2003.
[37] R. Chen, A. I. Hochbaum, P. Murphy, J. Moore, P. Yang, and A. Ma- jumdar, “Thermal conductance of thin silicon nanowires,” Physical re- view letters, vol. 101, no. 10, p. 105501, 2008.
[38] J. Lim, K. Hippalgaonkar, S. C. Andrews, A. Majumdar, and P. Yang, “Quantifying surface roughness effects on phonon transport in silicon nanowires,” Nano letters, vol. 12, no. 5, pp. 2475–2482, 2012.
[39] X. Li, “Metal assisted chemical etching for high aspect ratio nanostruc- tures: A review of characteristics and applications in photovoltaics,” Current Opinion in Solid State and Materials Science, vol. 16, no. 2, pp. 71–81, 2012.
[40] J. Carrete, L. J. Gallego, L. M. Varela, and N. Mingo, “Surface rough- ness and thermal conductivity of semiconductor nanowires: Going be- low the casimir limit,” Physical Review B, vol. 84, no. 7, p. 075403, 2011.
[41] D. Donadio and G. Galli, “Atomistic simulations of heat transport in silicon nanowires,” Physical review letters, vol. 102, no. 19, p. 195901, 2009.
[42] J. P. Feser, J. S. Sadhu, B. P. Azeredo, K. H. Hsu, J. Ma, J. Kim, M. Seong, N. X. Fang, X. Li, P. M. Ferreira et al., “Thermal conduc- tivity of silicon nanowire arrays with controlled roughness,” Journal of Applied Physics, vol. 112, no. 11, p. 114306, 2012.
[43] Z. Huang, N. Geyer, P. Werner, J. De Boor, and U. G¨osele, “Metal- assisted chemical etching of silicon: a review,” Advanced materials, vol. 23, no. 2, pp. 285–308, 2011.
[44] N. Geyer, B. Fuhrmann, Z. Huang, J. de Boor, H. S. Leipner, and P. Werner, “Model for the mass transport during metal-assisted chem- ical etching with contiguous metal films as catalysts,” The Journal of Physical Chemistry C, vol. 116, no. 24, pp. 13 446–13 451, 2012. [45] C. Chartier, S. Bastide, and C. L´evy-Cl´ement, “Metal-assisted chemical
etching of silicon in hf–h 2 o 2,” Electrochimica Acta, vol. 53, no. 17, pp. 5509–5516, 2008.
[46] S. Mallat, “A wavelet tour of signal p rocessing (academic p ress, new york 1 999); i,” Daubechies Ten L ectures on Wavelets (SIAM, P hiladelphia, 1 992).[1 0]. J ouault, F. S ´e bille and V. de la Mota, Nucl. P hys. A, vol. 628, no. 1, p. 998.
[47] J. E. Van Nostrand, S. J. Chey, M.-A. Hasan, D. G. Cahill, and J. Greene, “Surface morphology during multilayer epitaxial growth of ge (001),” Physical review letters, vol. 74, no. 7, p. 1127, 1995.
[48] R. Sayles and T. Thomas, “The spatial representation of surface rough- ness by means of the structure function: a practical alternative to cor- relation,” Wear, vol. 42, no. 2, pp. 263–276, 1977.
[49] J. Lapujoulade, “The roughening of metal surfaces,” Surface science reports, vol. 20, no. 4, pp. 195–249, 1994.
[50] L. Shi, D. Li, C. Yu, W. Jang, D. Kim, Z. Yao, P. Kim, and A. Ma- jumdar, “Measuring thermal and thermoelectric properties of one- dimensional nanostructures using a microfabricated device,” Journal of heat transfer, vol. 125, no. 5, pp. 881–888, 2003.
[51] M. Krystek and M. Anton, “A weighted total least-squares algorithm for fitting a straight line,” Measurement Science and Technology, vol. 18, no. 11, p. 3438, 2007. [Online]. Available: http://stacks.iop.org/0957-0233/18/i=11/a=025
[52] H. Casimir, “Note on the conduction of heat in crystals,” Physica, vol. 5, no. 6, pp. 495–500, 1938.
[53] N. Mingo, “Calculation of si nanowire thermal conductivity using com- plete phonon dispersion relations,” Physical Review B, vol. 68, no. 11, p. 113308, 2003.
[54] J. Callaway, “Model for lattice thermal conductivity at low tempera- tures,” Physical Review, vol. 113, no. 4, p. 1046, 1959.
[55] D. Li, “Thermal transport in individual nanowires and nanotubes,” Ph.D. dissertation, University of California, Berkeley, 2002.
[56] P. Murphy and J. Moore, “Coherent phonon scattering effects on ther- mal transport in thin semiconductor nanowires,” Physical Review B, vol. 76, no. 15, p. 155313, 2007.
[57] T. Klitsner and R. O. Pohl, “Phonon scattering at silicon crystal surfaces,” Phys. Rev. B, vol. 36, pp. 6551–6565, Oct 1987. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.36.6551
[58] B. Daly, K. Kang, Y. Wang, and D. G. Cahill, “Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in sili- con,” Physical Review B, vol. 80, no. 17, p. 174112, 2009.
[59] J.-Y. Duquesne and B. Perrin, “Ultrasonic attenuation in a quasicrys- tal studied by picosecond acoustics as a function of temperature and frequency,” Physical Review B, vol. 68, no. 13, p. 134205, 2003.
[60] J. Cuffe, O. Ristow, E. Ch´avez, A. Shchepetov, P.-O. Chapuis, F. Alz- ina, M. Hettich, M. Prunnila, J. Ahopelto, T. Dekorsy et al., “Lifetimes of confined acoustic phonons in ultrathin silicon membranes,” Physical review letters, vol. 110, no. 9, p. 095503, 2013.
[61] A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bar- tels, O. Schecker, R. Waitz, A. Erbe, E. Scheer et al., “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at ghz frequencies with a high-repetition-rate femtosecond laser,” Physical review letters, vol. 106, no. 7, p. 077401, 2011.
[62] A. Maznev, F. Hofmann, J. Cuffe, J. Eliason, and K. Nelson, “Life- time of high-order thickness resonances of thin silicon membranes,” Ultrasonics, vol. 56, pp. 116–121, 2015.
[63] I. De Wolf, “Micro-raman spectroscopy to study local mechanical stress in silicon integrated circuits,” Semiconductor Science and Technology, vol. 11, no. 2, p. 139, 1996. [Online]. Available: http://stacks.iop.org/0268-1242/11/i=2/a=001
[64] S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy, and O. L. Krivanek, “Surface roughness at the si(100)-sio2 interface,”
Phys. Rev. B, vol. 32, pp. 8171–8186, Dec 1985. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.32.8171
[65] T. Yoshinobu, A. Iwamoto, and H. Iwasaki, “Scaling analysis of sio2/si interface roughness by atomic force microscopy,” Japanese journal of applied physics, vol. 33, no. 1S, p. 383, 1994.
[66] F. Hudert, A. Bruchhausen, D. Issenmann, O. Schecker, R. Waitz, A. Erbe, E. Scheer, T. Dekorsy, A. Mlayah, and J.-R. Huntzinger, “Confined longitudinal acoustic phonon modes in free-standing si mem- branes coherently excited by femtosecond laser pulses,” Physical Re- view B, vol. 79, no. 20, p. 201307, 2009.
[67] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface gener- ation and detection of phonons by picosecond light pulses,” Physical Review B, vol. 34, no. 6, p. 4129, 1986.
[68] M. Grossmann, M. Klingele, P. Scheel, O. Ristow, M. Hettich, C. He, R. Waitz, M. Schubert, A. Bruchhausen, V. Gusev et al., “Femtosec- ond spectroscopy of acoustic frequency combs in the 100-ghz frequency range in al/si membranes,” Physical Review B, vol. 88, no. 20, p. 205202, 2013.
[69] J. Porto, F. Garcia-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Physical review letters, vol. 83, no. 14, p. 2845, 1999.
[70] A. Barbara, P. Qu´emerais, E. Bustarret, and T. Lopez-Rios, “Optical transmission through subwavelength metallic gratings,” Physical Re- view B, vol. 66, no. 16, p. 161403, 2002.
[71] F. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Physical Review B, vol. 66, no. 15, p. 155412, 2002.
[72] F. J. Garcia-Vidal, L. Martin-Moreno, T. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Reviews of Modern Physics, vol. 82, no. 1, p. 729, 2010.
[73] D. Crouse and P. Keshavareddy, “Role of optical and surface plas- mon modes in enhanced transmission and applications,” Optics express, vol. 13, no. 20, pp. 7760–7771, 2005.
[74] H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, “Light transmis- sion through a subwavelength slit: Waveguiding and optical vortices,” Physical Review E, vol. 67, no. 3, p. 036608, 2003.
[75] A. Borisov, F. G. De Abajo, and S. Shabanov, “Role of electromag- netic trapped modes in extraordinary transmission in nanostructured materials,” Physical Review B, vol. 71, no. 7, p. 075408, 2005.
[76] D. Crouse and P. Keshavareddy, “Polarization independent enhanced optical transmission in one-dimensional gratings and device applica- tions,” Optics express, vol. 15, no. 4, pp. 1415–1427, 2007.
[77] Y. Lu, M. H. Cho, Y. Lee, and J. Y. Rhee, “Polarization-independent extraordinary optical transmission in one-dimensional metallic gratings with broad slits,” Applied Physics Letters, vol. 93, no. 6, p. 061102, 2008.
[78] E. Lansey, I. R. Hooper, J. N. Gollub, A. P. Hibbins, and D. T. Crouse, “Light localization, photon sorting, and enhanced absorption in sub- wavelength cavity arrays,” Optics express, vol. 20, no. 22, pp. 24 226– 24 236, 2012.
[79] A. Enemuo, M. Nolan, Y. U. Jung, A. B. Golovin, and D. T. Crouse, “Extraordinary light circulation and concentration of s-and p-polarized phase resonances,” Journal of Applied Physics, vol. 113, no. 1, p. 014907, 2013.
[80] T. Ongarello, F. Romanato, P. Zilio, and M. Massari, “Polarization in- dependence of extraordinary transmission trough 1d metallic gratings,” Optics express, vol. 19, no. 10, pp. 9426–9433, 2011.
[81] D. Crouse, A. P. Hibbins, and M. J. Lockyear, “Tuning the polarization state of enhanced transmission in gratings,” Applied Physics Letters, vol. 92, no. 19, p. 191105, 2008.
[82] J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature, vol. 416, no. 6876, pp. 61–64, 2002.
[83] A. Drezet, C. Genet, and T. W. Ebbesen, “Miniature plasmonic wave plates,” Physical review letters, vol. 101, no. 4, p. 043902, 2008. [84] I. Mandel, J. N. Gollub, I. Bendoym, and D. T. Crouse, “Theory and
design of a novel integrated polarimetric sensor utilizing a light sorting metamaterial grating,” Sensors Journal, IEEE, vol. 13, no. 2, pp. 618– 625, 2013.
[85] T. Xu, Y.-K. Wu, X. Luo, and L. J. Guo, “Plasmonic nanoresonators for high-resolution colour filtering and spectral imaging,” Nature Com- munications, vol. 1, p. 59, 2010.
[86] P. Fan, K. C. Huang, L. Cao, and M. L. Brongersma, “Redesigning photodetector electrodes as an optical antenna,” Nano letters, vol. 13, no. 2, pp. 392–396, 2013.
[87] D. Crouse and R. Solomon, “Numerical modeling of surface plasmon enhanced silicon on insulator avalanche photodiodes,” Solid-state elec- tronics, vol. 49, no. 10, pp. 1697–1701, 2005.
[88] T. H. Reilly, S.-H. Chang, J. D. Corbman, G. C. Schatz, and K. L. Rowlen, “Quantitative evaluation of plasmon enhanced raman scatter- ing from nanoaperture arrays,” The Journal of Physical Chemistry C, vol. 111, no. 4, pp. 1689–1694, 2007.
[89] D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technol- ogy through the fusion of microfluidics and optics,” Nature, vol. 442, no. 7101, pp. 381–386, 2006.
[90] N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nature materials, vol. 10, no. 8, pp. 631–636, 2011.
[91] J. Kong, Electromagnetic Wave Theory. New York: John Wiley and Sons, 1986.
[92] J. A. Johnson, A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Ke- hoe, C. M. S. Torres, G. Chen, and K. A. Nelson, “Direct measurement of room-temperature nondiffusive thermal transport over micron dis- tances in a silicon membrane,” Physical review letters, vol. 110, no. 2, p. 025901, 2013.
[93] W. C. Chew, Waves and fields in inhomogeneous media. IEEE press New York, 1995, vol. 522.
[94] J. P. Kottmann and O. J. Martin, “Accurate solution of the volume integral equation for high-permittivity scatterers,” Antennas and Prop- agation, IEEE Transactions on, vol. 48, no. 11, pp. 1719–1726, 2000. [95] J.-M. Jin, Theory and computation of electromagnetic fields. John
Wiley & Sons, 2011.
[96] F. Pardo, P. Bouchon, R. Ha¨ıdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Physical review letters, vol. 107, no. 9, p. 093902, 2011.
[97] P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Physical Review B, vol. 26, no. 6, p. 2907, 1982.
[98] M. Gorkunov, E. Podivilov, and B. Sturman, “Transmission and scat- tering properties of subwavelength slits in metals,” Physical Review B, vol. 83, no. 3, p. 035414, 2011.
[99] P. Lalanne, J. Hugonin, S. Astilean, M. Palamaru, and K. M¨oller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” Journal of Optics A: Pure and Applied Optics, vol. 2, no. 1, p. 48, 2000.
[100] A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electronics letters, vol. 36, no. 4, pp. 321–322, 2000.
[101] V. Kravets, F. Schedin, and A. Grigorenko, “Plasmonic blackbody: Al- most complete absorption of light in nanostructured metallic coatings,”