The interaction of radio-frequency fields with dielectric materials at macroscopic to mesoscopic scales.
In order to contrast decoupled spin response with dielectric dipole response in Sec. 10.2, we will develop the well-known statistical approach of noninteracting paramagnetism. In a paramagnetic material, the net magnetic moment is the sum of individual moments in an applied field. If the spin moments are [[sigma].sub.[+ or -]] = [+ or -][micro] and the probability density of the spin being up or down in an applied field is pi, then the net magnetic moment is (4)
< m > = -[summation over (i)] [[sigma].sub.i] [P.sub.i] ([sigma]), (120)
where the probabilities of being in the low energy (-) or high (+) energy states are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (121)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (122)
Therefore, for N spins and when [[vector].[micro]] * B/[k.sub.B]T [much less than] 1
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (123)
In the case of isotropy < m > = N[[micro].sup.2]*B/3[k.sub.B]T. So we obtain the same form as in the case of noninteracting dielectrics in Eq. (107).
12.1.2 Paramagnetic Response With Angular Momentum J
For atoms with angular momentum J with 2J + 1 discrete energy levels, the average magnetization can be expressed in terms of the Brillioun equation [B.sub.J] (4)
< m > = NgJ[[micro].sub.B][B.sub.J](x), (124)
where x = gJ[[micro].sub.B]B/[k.sub.B]T and [B.sub.J](x) = (2J + 1)/2J coth([(2J + 1)x]/2J) - (1/2J) coth(x/2J) and g is the g-factor given by the Lande equation.
12.2 Magneto-Dielectric Response: Magneto-Electric, Ferroelectric, Ferroic, and Chiral Response
Researchers have found that in magneto-electric, ferroic, and chiral materials the application of magnetic fields can produce a dielectric response and the application of an electric field can produce a magnetic response (see for example (121)). These cross coupling behaviors can be found to occur in specific material lattices, layered thin films, or by constructing composite materials. An origin of the intrinsic magneto-electric effect is from the strain-induced distortion of the spin lattice upon the application of an electric field. When a strong electric field is applied to a magneto-electric material such as chromium oxide, the lattice is slightly distorted, which changes the magnetic moment and therefore the magnetic response. Extrinsic effects can be produced by layering appropriate magnetic, ferroelectric, and dielectric materials in such a way that an applied electric field modifies the magnetic response and a magnetic field modifies the electric response. Chiral materials can be constructed by embedding conducting spirals into a dielectric matrix. In artificial magneto-electric materials the calculated permittivity and permeability may be effective rather than intrinsic properties. The constitutive relations for the induction and displacement fields are not always simple and can contain cross coupling between fields. For example, [~.D]([omega]) = [[[left and right arrow].[alpha]].sub.1] * [~.E]([omega]) + [[[left and right arrow].[alpha]].sub.2] * [~.B]([omega]), where [[[left and right arrow].[alpha]].sub.i] are constitutive parameters.
13. Electromagnetic-Driven Material Resonances in Materials at RF Frequencies
At the relatively longer wavelengths of RF frequencies, (1 x [10.sup.4] m to 1 mm), only a few classes of intrinsic resonances can be observed. Bulk geometric resonances, standing waves, and higher-mode resonances can occur at any frequency when an inclusion has a dimension that is approximately equal to an integral multiple of one-half wavelength in the material. These geometrical resonances are sometimes misinterpreted as intrinsic material resonances. Most of the intrinsic resonant behavior in the microwave through millimeter frequency bands are due to cooperative ferromagnetic and ferrite spin-related resonances, antiferromagnetic resonances, microwave atomic transitions, plasmons and plasmon-like resonances, and polaritons at metal-dielectric interfaces. Atoms such as cesium have transition resonances in the microwave band. Large molecules can also be made to resonate under the application of high RF frequencies and THz frequencies. NIM commonly use non-intrinsic splitring structure resonances together with plasma resonances to achieve unique electromagnetic response. At optical frequencies, individual molecules or nanoparticles can sometimes be resonated directly or through the use of plasmons.
Water has a strong relaxation in the gigahertz frequency range and water vapor has an absorption peak in the gigahertz range, liquid water has no dielectric resonances in the microwave range. The resonances of the water molecule occur at infrared frequencies at a wavelength around 9 m. In magnetic materials, ferromagnetic spin resonances occur in the megahertz to gigahertz to yielding MMW bands. Antiferromagnetic resonances can occur at millimeter frequencies. Gases such as oxygen with a permanent magnetic moment can absorb millimeter waves (122). In the frequency region from 22 to 180 GHz, water-vapor absorption is caused by the weak electric dipole rotational transition at 22 GHz, and a stronger transition occurs at around 183 GHz (123).
If high-frequency fields are applied to ferrite materials, there are relaxations in the megahertz frequencies, and in the megahertz to MMW frequencies there are spin resonances (119), (121), (124), (125).
14. Artificial Materials: Plasmons, Super-Lensing, NIM, and Cloaking Response
The term metamaterial refers to artificial structures that can achieve behaviors not observed in nature. NIMs are a class of metamaterials where there are simultaneous resonances in the permittivity and permeability. Many artificial materials are formed from arrays of periodic unit cells formed from dielectric, magnetic, and metal components, and when subjected to applied fields, achieve interesting EM response. Examples of periodic structures are NIM that utilize simultaneous electric and magnetic resonances (126). Metafilms, band filters, cloaking devices, and photonic structures all use artificial materials. Artificial materials are also used to obtain enhanced lensing and anomalous refraction and other behaviors (65), (126-131). A very good overview is given in (128). In the literature NIM materials are commonly assumed to possess an intrinsic negative permittivity and permeability. However, the resonator dimensions and relevant length scales used to achieve this behavior may not be very much smaller than a wavelength of the applied field (132). Therefore, the continuous media requirement for defining the permittivity and permeability becomes blurred. The mapping of continuous media properties onto metamaterial behavior can at times cause paradoxes and inconsistencies (69), (133-137). However, the measured EM scattering response in NIM is achieved, whether or not an effective permittivity and permeability can be consistently defined. Because of the inhomogeneity in the media, the permittivity and permeability in some of these applications are effective parameters and spatially dispersive and not the intrinsic properties that Veselago assumed for a material (26), (138). In some metamaterials and metafilms where the ratio of the particle size to the wavelength is not small, boundary transition layers are typically included in the model so that the terminology of effective permittivity and permeability can be used. In Sec. 4.6, we described the criterion of defining a polarization by a Taylor series expansion of the charge density. The problem of whether these composite materials can be described in terms of a negative index is complicated by the issues described above. The measured permittivity tensor is an intrinsic property and should not depend on the field application or the sample boundaries, if the electrodynamic problem is modeled correctly.
Pendry (127) introduced the idea of constructing a lens from metamaterials that could achieve enhanced imaging that is not constrained by the diffraction limit. It should be noted that microwave near-field probes also have the capability of subwavelength imaging by using the near field around a probe tip (see Sec. 16).
14.1 Veselago's Argument for NIM Materials With Both [[epsilon]'.sub.r(eff)] < 0 and [[micro]'.sub.r(eff)] < 0
In this section we overview the theory behind NIM (26). The real parts of the permittivity and permeability can be negative over a band of frequencies during resonances. Of course, to maintain energy conservation in any passive material, the loss-factor part of the permittivity and permeability must always be positive. This behavior has only been recently exploited to achieve complex field behavior (26), (62), (67), (88), (127), (139).
Polarization resonance is usually modeled by a damped harmonic-oscillator equation. The simple harmonic-oscillator equation for the polarization [~.P]([omega]) for single-pole relaxation can be written as Eq. (58). For a time-harmonic-field approximation, the effective dielectric susceptibility has the form
[x.sub.d]([omega]) = [x.sub.0]/[[-[[omega].sup.2]/[[omega].sub.0.sup.2] + i[omega][tau] + 1]. (125)
The real part of the susceptibility can be negative around the resonance frequency (see Fig. 7). A similar equation can apply for a resonance in a split-ring or other resonator to obtain a negative real part of the permeability.
In most electromagnetic material applications the plane-wave propagation vector and group velocities are in the same direction. Backward waves are formed when the group velocity and phase velocity are in opposite directions. This can be produced when the real parts of the permittivity and permeability are simultaneously negative. When this occurs, the refractive index is negative, n = [square root of (-[absolute value of [[epsilon]'.sub.r]])][square root of (-[absolute value of [[micro]'.sub.r]])] = i[square root of ([absolute value of [[epsilon]'.sub.r]])]i [square root of ([absolute value of [[micro]'.sub.r]])] = -[square root of ([absolute value of [[epsilon]'.sub.r]][absolute value of [[micro]'.sub.r]])]. Because of this result, researchers have argued that this accounts for the anomalous refraction of waves through NIMs, reverse Cherenkov radiation, and reverse Doppler effect, etc.
Snell's law for the reflection of an interface between a normal dielectric and an NIM satisfies [[theta].sub.inc] = [[theta].sub.reflection], but the refracted angle in NIM is [[theta].sub.trans] = sgn([n.sub.NIM])[sin.sup.-1]([[n.sup.norm]/[absolute value of [n.sub.NIM]]]sin[[theta].sub.inc]) (140). In addition, the TEM wave impedance of plane waves for NIM is Z = [square root of ([[micro].sub.0]/[[epsilon].sub.0])][square root of ([-[absolute value of [[micro]'.sub.r]] - [absolute value of i[[micro]".sub.r]]]/[- [[epsilon]'.sub.r]] - i[[epsilon]".sub.r]]])]. If only the real part of the the permittivity or the permeability are negative, then damped field behavior is attained.
These periodic artificial materials do produce interesting and potentially useful scattering behavior; however since they often involve resonances in structures that contain metals, they are lossy (62). There has been debate in the literature over how to interpret the observed NIM behavior, and some researchers believe the results can be explained in terms of surface waves rather than invoking NIM concepts (137).
The approach used to realize a negative effective magnetic permeability is different from that for obtaining a negative effective [epsilon]'.sub.r]. Generally, split-ring resonators are used to obtain negative [[micro]'.sub.r], but recently there has been research into the use of TM and TE resonant modes in dielectric cubes (69) or ferrite spheres to achieve negative properties (62), (141). Dielectric, metallic, ferrite, or layered dielectricmetallic inclusions such as spheres can be used to achieve geometric or coupled resonances and therefore simultaneous negative effective [epsilon]' and [micro]' (62). A commonly used approach to obtain a negative permittivity is to drive the charges in a wire or free charge in a semiconductor or plasma near resonance. Dielectric resonance response occurs in semiconductors in the terahertz to infrared range and in superconductors in the millimeter range. The real part of the permittivity for a plasma, according to the high-frequency Drude model, can be negative ([epsilon] = [[epsilon].sub.0](1 - [[omega].sub.p.sup.2] / [[omega].sup.2])).
There are a number of metrology issues related to NIM. These include the problem of whether the field behavior should be modeled as the result of negative intrinsic permittivity and permeability and negative index or instead be treated as a scattering problem. This problem is related to the wave length of the applied fields versus the parameters of the embedded resonators. Although the scatterers are generally smaller than a wavelength of the applied field, they are not always significantly smaller. When the lattice spacing a between particles satisfies (142) 0 [less than or equal to] [absolute value of [square root of ([epsilon]'.sub.r][[micro]'.sub.r])]] [omega]a/c [less than or equal to] 1, then effective properties can be defined (62). Even within these bounds the properties are not intrinsic permittivity and permeability as defined previously and are spatially dispersive. A second issue is the determination of the NIM specimen length and boundaries to be used to model the array of macroscopic scatterers (see (69) and references therein for an analysis of this problem). Another area of debate is where in the resonance region is a permittivity and permeability well defined.
14.2 Plasmonic Behavior
At the interface between a dielectric and metal an EM wave can excite a quasiparticle called a surface polariton (see Fig. 13). Plasmons are charge-density waves of electron gases in plasmas, metals, or semiconductors. Surface polariton plasmons travel on the interface between a dielectric and a conductor, analogous to the propagation of the Sommerfeld surface wave on a conductor/dielectric interface. Plasma polaritons decay exponentially away from the surface. The effective wavelengths of plasmons are much shorter than that of the incident EM field and therefore plasmons can propagate through structures where the incident radiation could not propagate through. This effect has been used in photonics and in microwave circuits through the use of metamaterials. For example, thin metal films can be embedded in dielectrics to form dielectric waveguides. Plasmonics is commonly used for imaging where the fields are used to obtain a sub-wavelength increase in resolution of 10 to 100 times. Colors in stained glass and metals are related to the plasma resonance frequency, due to the preferred reflection and absorption of specific wavelengths. High-temperature superconductors also have plasmonic behavior and a negative [[epsilon]'.sub.r] due to the complex conductivity (91). If small metallic particles are subjected to EM radiation of the proper wavelength, they can confine EM energy and resonate as surface plasma resonators. Plasmonic resonances have also been used to clean carbon nanotubes and enhance other chemical reactions by thermal or nonthermal activation. Plasmons have been excited in metamaterials by use of a negative permeability rather than negative permittivity (143).
Maxwell's equations with no source-current densities can be used to obtain
-[[micro].sub.0][[[partial derivative].sup.2]D]/[[partial derivative][t.sup.2]] = [nabla] x [nabla] x E. (126)
If E [varies] [e.sup.i[omega]t-ikz], the dispersion relation is
k (k * E) - [k.sup.2]E = - [[epsilon].sub.r](k, [omega])[[[omega].sup.2]/[c.sup.2]] E. (127)
For transverse plane waves k * E = 0, and therefore [k.sup.2] = [[epsilon].sub.r] (k, [omega]) [[[omega].sup.2]/[c.sup.2]]. For longitudinal waves [[epsilon].sub.r] (k, [omega]) = 0 (144) (this condition [epsilon]([omega]) = 0) also implies the Lyddane-Sachs-Teller relation (102) for the ratio of the longitudinal to transverse phonon frequencies that satisfies [[omega].sub.L.sup.2]/[[omega].sub.T.sup.2] = [[epsilon].sub.s]/[[epsilon].sub.[infinity]].
From Eq.(62), in the time domain for the case of no loss, and P(t) = - Nex(t), where N is the density of electrons, we obtain the equation for a harmonic oscillator for bulk longitudinal plasmon oscillations, [[d.sup.2]x]/[d[t.sup.2]] = - [[omega].sub.p.sup.2]x. The permittivity of a plasmon can be modeled as
[epsilon]([omega]) = [[omega].sub.0](1 - [[omega].sub.p.sup.2]/[[omega].sup.2]). (128)
Below the plasmon frequency [[omega].sub.p] = [square root of (N[e.sup.2]/[[epsilon].sub.0]m)], plasma is attenuative and follows the skin-depth formulas of a metal.
Above the plasma frequency, the real part of the permittivity becomes negative.
Surface plasmon polaritons (144) can travel at the interface of a metal and dielectric to produce surface wave guiding. Plasmonic surface waves have fields that decay rapidly from the surface interface. For example, for a 1 m excitation wavelength, the waves can travel over 1 cm, leading to the possibility of applications in microelectronics. Surface plasmonic EM waves can be squeezed into regions much smaller than allowed by the diffraction limit. Obtaining the negative effective [[epsilon]'.sub.rp] for plasmons in the megahertz through MMW range would require the use of NIM. Some applications of plasmonic behavior can also be tuned by a dc external magnetic field, and the applied magnetic field produces a plasmon with a tensorial permittivity.
For surface plasmons, the effective wavelengths of the plasmons can be much less than that of the exciting EM fields due to the difference in sign of the permittivities in a metal and dielectric. For example, for a plasmon at an interface between a metal and a dielectric substrate, if the permittivity of the plasmon is [[epsilon]'.sub.rp] and that of the substrate is [[epsilon]'.sub.rd], then the dispersion relation is k = 2[pi]/[lambda] = ([omega]/c) [square root of ([[[epsilon].sub.rd][[epsilon].sub.rp]]/([[epsilon].sub.rd] + [[epsilon].sub.rp]))] (144). When R([[epsilon]'.sub.rp]) < 0 and R([[epsilon]'.sub.rp]) is slightly larger than [[epsilon]'.sub.rd], then we see that the wavelength becomes very short in comparison to that of the applied field. This is also attained by application of laser light to nanoparticles to obtain a resonant state. However, this can also happen in coupled microwave resonant structures.
14.3 Transmission Through Subwavelength Apertures
Under certain conditions, electromagnetic radiation has been observed to pass through subwavelength apertures (145-147). In extraordinary optical (EOT) or millimeter wave (EMT) transmission, free-space EM waves impinging on a metal plate with small holes transmits more energy than would be expected by a traditional analysis (148). At optical frequencies, this transmission is mediated by surface plasmons. At MW and MMW frequencies, plasmons are not formed on homogeneous conducting metal plates. However, plasmon-like behavior can be formed by an appropriate selection of holes, metal plate thickness, or corrugations to produce a behavior that simulates surface plasmons. These plasmons-like features that are sometimes referred by the jargon "spoof plasmons", can be the origin of extraordinary transmission through the holes in metal plates at MW to MMW frequencies.
14.4 Behaviors in Structures Where [[epsilon]'.sub.r(eff)] [right arrow] 0
There are applications where a material is constructed in such a way so that the real part of the "effective" permittivity is close to 0 (ENZ) (see Fig. 7 as [[epsilon]'.sub.r] [right arrow] 0). This is closely related to plasmon-like behavior. In this case, the EM behavior simulates static behavior in that [nabla] x H = 0 and [nabla] x E = i[omega][micro]H, which implies [[nabla].sup.2]E = 0. In this case, the phase velocity approaches infinity and the guided wavelength becomes infinite, which is analogous to cutoff in a waveguide ([lambda].sub.c]) (47). This type of behavior can be achieved for a waveguide near cutoff. The equation for the guided wavelength in a waveguide is
[lambda].sub.g] = 2[pi]/[square root of ([[epsilon].sub.r] [[micro].sub.r][[[omega].sup.2]/[c.sup.2]] - [(2[pi]/[[lambda].sub.c]).sup.2])] = [lambda]/[square root of (1 - [([lambda]/[[lambda].sub.c]).sup.2]))], (129)
where [[lambda].sub.c] is the cutoff wavelength of the guide. Due to the long effective wavelength near cutoff, the phase of the wavefront changes minimally. Because the effective permittivity goes through zero near resonance, we can think of ENZ as a resonance condition similar to the propagation cutoff in a waveguide when there is resonance in the transverse plane. This type of behavior is achieved, for example, if we have a low-loss dielectric of length L that completely fills the cross section of a waveguide (see Fig. 24). Near the cutoff frequency the material could be thought of as having an effective permittivity [[epsilon]'.sub.r(eff)] [approximately equal to] 0. This same behavior is reminiscent of a cavity, because as the transmission attains a maximum, reflection is a minimum, and the reactance goes to 0 near resonance. The [[epsilon]'.sub.(eff)] in this model violates the condition for an intrinsic permittivity since the applied field wavelength ([lambda] = 1/f [square root of ([epsilon][micro])]), must be much larger than the feature size. It has been argued that in ENZ, unlike in a normal wire, the displacement current dominates over the charge current in transporting the EM waves (146). There could be analogous effective permeability going to zero [[micro]'.sub.r(eff)] [right arrow] 0 (MNZ) behavior.
14.5 Modeling Electrical Properties to Produce Cloaking Behavior
Recently, there have been many research papers that examine the possibility of using the electrical properties of artificial materials to control the scattering from an object in such a way as to make the object appear invisible to the applied EM field (129), (130), (149). This is distinct from radar-absorbing materials, where the applied field is absorbed by ferrites or layered, lossy materials. Research in this area uses the method of transformation optics (149), (150) to determine the material properties that produce the desired field behavior. In order to exhibit a typical cloaking property, Shivola (151) derived simple equations for a dielectric-layered sphere that are assigned permittivities to produce a nearly zero effective polarizability. Recently, complex arrangements of non-resonant metamaterials have been designed by inverse optical modeling to fabricate broadband electromagnetic cloaks (129), (152).
15. Macroscopic to Mesoscopic Heating and Electromagnetic-Assisted Reactions
15.1 Overview of EM Heating
15.1.1 Dielectric and Magnetic Heating
In EM wave interactions with materials, some of the applied energy is converted into heat. The heating that takes place with the application of high-frequency fields is due to photon-phonon processes modeled by the friction caused by particle collisions and resistance to dipole rotation. Over the RF spectrum, heating may be volumetric at low frequencies and confined to surfaces at high frequencies. Volumetric heating is due to the field that penetrates into the material producing dissipation through the movement of free ions and the rotation of dipolar molecules. Nanocomposites can be heated volumetrically by RF EM fields, lasers, and terahertz applicators. Since the skin depth is long at low frequencies, the heating of nanoparticles is not efficient. In the microwave band the heating of very small particles in a host material is limited by the loss and density of particles in the material, the power level of the source, and the diffusion of heat to the surroundings. Plasmon resonances in the infrared to visible frequencies can be used to locally heat particles (153). At high frequencies, heat may be absorbed locally in particles in slow modes where there may be a time lag for heat to dissipate into the phonon bath when the fields are removed.
The history of practical RF heating started in the era when radar was being developed. There are stories of where engineers sometimes heated their coffee by placing it near antennas. Also there are reports a researcher working on a magnetron that noticed that the candy bar in his pocket had melted when he was near the high-frequency source.
In a microwave oven, water and bound water are heated by the movement of free charge and non-resonant rotation (154). Because the water molecules at these frequencies cannot react in concert with the field, energy is transferred from the field energy into kinetic energy of the molecules in the material. In dielectric materials at low frequencies, as frequencies increase into the HF band, the rotations of the molecules tend to lag the electric field, and this causes the electric field to have a component in phase with the current. This is especially true in liquids with hydrogen bonding, where the rotational motion of the bonding is retarded by the interconnections to other molecules. This causes energy in the electric and magnetic fields to be converted into thermal energy (155). Some polymer molecules that have low friction, such as glycerol in solution, tend to rotate without significant molecule-molecule interactions and therefore produce little thermal energy.
The power dissipated in a bulk lossy material in a time-harmonic field is
P ([omega], T) = 1/2[omega][integral]([epsilon]"([omega], T) [[absolute value of [~.E]([omega])].sup.2] + [micro]"([omega], T) [[absolute value of [~.H]([omega])].sup.2])dV. (130)
The total entropy produced per unit time at a temperature T is P([omega], T) / T. Equation (130) is modified for very frequency-dispersive materials (116). Dielectric losses in ohmic conduction and Joule heating originate in the frictional energy created by charges and dipoles that are doing work against nonconservative restoring forces. Magnetic losses include eddy currents, hysteresis losses, and spin-lattice relaxation. Some of the allocated heating frequencies are given in Table 3.
Table 3. Heating Frequencies Frequency (MHz) Wavelength (cm) 13.56 2200 433.92 69 914 33 2450 12
Heating originates from dielectric and magnetic loss and the strength of the fields. For magnetic materials the losses relate to [micro]" and [[sigma].sub.dc]. In high-frequency fields, magnetic materials will be heated by both dielectric and magnetic mechanisms (104), (156). If applicators are designed to subject the material to only magnetic or electric fields, then the heating will be related only to magnetic or dielectric effects, respectively.
When studying dielectric heating we need to also model the heat transport during the heating process. This is accomplished by use of the power dissipated as a source in the heat equation (157). The transport of heat through a material is modeled by the thermal diffusivity [[alpha].sub.h] = [kappa]/[[rho].sub.d][c.sub.p], where [[rho].sub.d] is the density and [kappa] is the thermal conductivity. In order to model localized heating, it is necessary to solve the Fourier heat equation and Maxwell's equations with appropriate boundary conditions. The macroscopic heat transfer equation is
[[rho].sub.d][c.sub.p][[[partial derivative]T]/[[partial derivative]t]] = [nabla] * [[left and right arrow].[kappa]] * [nabla]T + [1/2][epsilon][omega]"[[absolute value of E].sup.2], (131)
where [[left and right arrow].[kappa]] is the thermal conductivity dyadic. The mass density is [[rho].sub.d] and the specific heat is [c.sub.p]. For nanosystems, the heat transfer is more complicated and may require modeling phonon interactions. Also, the above heat transfer expression is only approximate for nanoscale materials. The temperature rise obtained by application of EM energy to a material can be estimated by use of the power dissipation relation in Eq. (130). When the temperature is changed by [DELTA]T, the thermal energy-density increase is [Q.sub.h] = [[rho].sub.d][c.sub.p][DELTA]T. The power dissipated per unit volume by an electric field interacting with a lossy dielectric material is [P.sub.d] = (1/2)[sigma][[absolute value of E].sup.2], where [sigma] is the conductivity. Therefore, the temperature rise in a specimen with density [[rho].sub.m] through heating with a power [P.sub.d] for a time [DELTA]t is
[DELTA]T = [[sigma][[absolute value of E].sup.2][DELTA]t]/[2[[rho].sub.m][c.sub.p]] (132)
The heating rate is determined by the field strength, frequency, and the loss factor. From the equations for the skin depth [[delta].sub.s] [approximately equal to] (1/[omega])[square root of ([micro]'[epsilon]'/2)] [square root of ([square root of ((1 + [tan.sup.2][delta]))] - 1)] we see that fields at lower frequencies will penetrate more deeply ([[delta].sub.s] [right arrow] [2.sub.c][square root of ([epsilon]')]/[omega][square root of ([[micro]'.sub.r])] [[epsilon]".sub.r]). In order to obtain the same dissipative power densities as those at higher frequencies, the electric field strength at a lower frequency would have to increase. For example, to obtain the same power densities at two different frequencies we must have ([[epsilon]".sub.1]([[epsilon].sub.1])[[epsilon].sub.1]/[[absolute value of [E.sub.2]].sup.2]/[[absolute value of [E.sub.1]].sup.2]).
The unique volumetric heating capability by EM fields over broader ranges of frequencies should stimulate further applications in areas such as recycling, enhanced oil recovery, and as an aid to reactions.
15.1.2 Electromagnetic-Assisted Reactions
When RF waves are applied to assist a chemical reaction, or polymer curing, the observed rate enhancement is due primarily to the effects of microscopic and volumetric heating. Because chemical reaction rates proceed in an Arrenhius form [tau] [infinity] exp (E/[k.sub.B]T), small temperature increases can produce large reductions in reaction times. The kinetics of chemical reaction rates is commonly modeled by the Eyring equation,
[k.sub.eyr] = [[k.sub.B]T/h][e.sup.-[DELTA]/RT], (133)
where h is Planck's constant, [DELTA]G = [DELTA]H - T[DELTA]S is the Gibb's free energy, H is Helmholtz's free energy, [DELTA]S denotes changes in entropy, and R is the gas constant. A plot of ln ([k.sub.eyr]/T) = - [DELTA]H/RT + [DELTA]S/R + ln ([k.sub.B]/h) versus 1/T can yield [DELTA]S, [DELTA]H, and possibly [k.sub.B]/h.
One would expect that heat transfer by conduction would have the same effect on reactions as microwave heating, but this is not always found to be true. Part of the reason for this is that thermal conduction requires strong temperature gradients, whereas volumetric heating does not require temperature gradients. Because it does not depend on thermal conduction, an entire volume can obtain nearly the same temperature simultaneously without appreciable temperature gradients. In addition, some researchers speculate on non-thermal microwave effects that are due to the electric field interacting with molecules in specific ways that modify the activation energy through changes in the entropy (158), (159). Avenues that have been proposed for nonthermal reactions may be related to dielectric breakdown that causes plasma of photons to be emitted, causing photo-reactions. Another avenue is related to the intense local fields that can develop near corners or sharp bends in materials or molecules that cause dielectric breakdown.
Typical energies of microwave through x-ray photons are summarized in Table 4. Covalent bonds such as C-C and C-O bonds have activation energies of nearly 360 kJ/mol, C-C and O-H bonds are in the vicinity of 400 kJ/mol, and hydrogen bonds are around 4 to 42 kJ/mol. Microwaves are from 300 MHz to 30 GHz and have photon energies from 0.0001 to 0.11 kJ/mol. Therefore, microwave photon bond-breaking events are rare. Nonthermal microwave effects, therefore, are not likely due to the direct inter-action of microwave photons with molecules and, if they occur at all, and must have secondary origins such as the generation of intense local fields that produce localized dielectric breakdown or possibly EM-induced changes in the entropy. Most of the effects seen in microwave heating are thermal effects due to the volumetric heating of high-frequency fields (160).
Table 4. Radiation Classes and Approximate Photon Energies Type Frequency (Hz) Photon energy (J) [gamma]-rays 3 x [10.sup.20] 1.9 x [10.sup.-13] X-rays 3 x [10.sup.16] 1.9 x [10.sup.-14] Ultraviolet 1 x [10.sup.15] 6.4 x [10.sup.-19] Visible light 6 x [10.sup.14] 4.0 x [10.sup.-19] Infrared light 3 x [10.sup.12] 2.0 x [10.sup.-22] Microwave 2 x [10.sup.9] 6.0 x [10.sup.-25] High frequency (HF) 1 x [10.sup.6] 6.4 x [10.sup.-28]
Microwave heating can result in superheating where the liquid can become heated above the typical boiling point. For example, in microwave heating, water can be heated above its boiling temperature. This is due to the fact that in traditional heating, bubbles form to produce boiling, whereas in microwave heating the water may become superheated before it boils.
15.2 Heat Transfer in Nanoscale Circuits
In microelectronic circuits, higher current densities can cause phonon heating of thin interconnects that can cause circuit failure. This heating is related to both the broad phonon thermal bath and possibly slow thermal modes where thermal energy can be localized to nanoscale regions (161), (162). New transistors will have an increased surface-to-volume ratio and, therefore, the power densities could increase. This, combined with the reduced thermal conductance of the low conductivity materials and thermal contact resistance at material interfaces, could lead to heat transport limitations (162), (163).
15.3 Heating of Nanoparticles
When a large number of metallic, dielectric, or magnetic micrometer or nanometer particles in a host media are subjected to high-strength RF EM fields, energy is dissipated. This type of EM heating has been utilized in applications that use small metallic particles, carbon black, or palladium dispersed in a material to act as chemical-reaction initiators and for selective heating in enhanced drug delivery or tumor suppression (164), (165). Understanding the total heat-transfer process in the EM heating of microscopic particles is important. A number of researchers have found that, due to the thermal conduction of heat from nanoparticles and the small volumes involved and the large skin depths of RF fields, the nanoparticles rapidly thermalize with the phonon bath and do not achieve temperatures that deviate drastically from the rest of the medium (166). Only when there is an appropriate density of particles, is heating enhanced. There have recently been reports that thermal energy can accumulate in nanoscale to molecular regions in slow modes, and it can take seconds to thermalize with the surrounding heat bath (166-169). In such situations, regions may be unevenly heated by field application. However, thermal conduction will tend to smooth the temperature profile within a characteristic relaxation time. Lasers can selectively heat micrometer-size particles and by use of plasmonics lasers can heat conducting nanoscale particles.
15.4 Macroscopic and Microscopic High-Frequency Thermal Run-Away
The dielectric loss and thermal conductivity of a material may possess a temperature dependence so that the loss increases as temperature increases (170). This is due to material decomposition that produces ions as the temperature increases and results in more loss. Thermal run away can lead quickly to intense heating of materials and dielectric breakdown. The temperature dependence of thermal run away has been modeled with the dielectric loss factor as [[epsilon]".sub.r] = [[alpha].sub.0] + [[alpha].sub.1] (T - [T.sub.0]) + [[alpha].sub.2] [(T - [T.sub.0]).sup.2], where [T.sub.0] is a reference temperature and [[alpha].sub.i] are constants (171).
16. Overview of High-Frequency Nanoscale Measurement Methods
In the past few decades, a number of methods have been developed to manipulate single molecules and dipoles. Methods have been implemented to move, orient, and manipulate nanowires, viruses, and proteins that are several orders of magnitude smaller than cells. These methods allow the researcher to study the electrical and mechanical properties of biological components in isolation. Molecules and cells can be manipulated and measured in applied fields using dielectrophoresis, microwave scanning probes, atomic force microscopy, acoustic devices, and optical and magnetic tweezers. Some of the methods use magnetic or electric fields or acoustic fields, others use the EM field radiation pressure, and others use electrostatic and van der Waals forces of attraction (139), (172). Microfluidic cells together with dc to terahertz EM fields are commonly used to study microliter to picoliter volumes of fluids that contain nanoparticles (173-175). Surface acoustic waves (SAW) and bulk acoustic waves (BAW) can be used to drive and enhance microfluidic processes. Since there is a difference of wave velocities in a SAW substrate and the fluid, acoustic waves can be transferred into the fluid, to obtain high fluid velocities for separation, pumping, and mixing.
Due to symmetry and charge neutrality, a polarizable particle in a uniform electric field will experience no net force. If a material with a permanent or induced dipole is immersed in an electric field gradient, then a dielectrophoretic force on the dipole is formed, as indicated in Fig. 15 (176). In a nonuniformelectric field, the force on a dipole moment p is F = (p * [nabla])E.
From this the following equation for the dielectrophoresis force on a small sphere of radius r of permitivity [[epsilon].sub.p] in a background with permittivity [[epsilon].sub.m] has be derived (177), (178)
[F.sub.DEF] = 2[pi][[epsilon].sub.m][r.sub.3]R([[[epsilon].sub.p] - [[epsilon].sub.m]]/[[[epsilon].sub.p] + 2[[epsilon].sub.m]]) [nabla][[absolute value of E].sup.2]. (134)
This force tends to align the molecule along the field gradient. The force is positive if [[epsilon].sub.p] > [[epsilon].sub.m]. For dispersive materials, the attraction or repulsive force can be varied by the frequency. Dielectrophoresis is commonly used to stretch, align, move, and determine force constants of biomolecules such as single-stranded and double-stranded DNA and proteins (179). Dielectrophoresis can also be used to separate cells or molecules in a stream of particles in solution. Usually, dielectrophoretic manipulation is achieved through microfabricated electrodes deposited on chips. For dispersive materials, where the permittivity changes over the frequency band of interest, there is a cross-over frequency where there is no force on the molecule. The approximate force, due to diffusion forces from particle gradients on a particle with a dimension d, is [F.sub.b] = [k.sub.B]T/d. For micrometer particles, the dielectrophoretic field gradients required to overcome this force is not large. However, for nanoscale particles this field gradient is much larger.
Spherical particles can be made to rotate through electrorotation methods (177). This motion is produced by a rotating electric field phase around a particle. The dipole induced in the particle experiences a net torque due to the dielectric loss that allows the dipole formation to lag the rotating field, as shown in Fig. 16. The net torque is given by N = p x E. For particles [[epsilon].sub.p] in a matrix [[epsilon].sub.m] the torque is (177)
N = 4[pi][[epsilon]'.sub.rm][r.sup.3]R([[[[epsilon].sub.p] - [[epsilon].sub.m]]/[[[epsilon].sub.p] + 2[[epsilon].sub.m]]]E([omega]) x E * ([omega])). (135)
Optical tweezing originates from the EM field gradient obtained from a laser source that produces a field differential and results in a force on particles. This effect is similar to dielectrophoresis. The strength of the radiation pressure on particles is a function of the size of the particles and the wavelength of the laser light (180). Molecules can also be studied by magnetic tweezers with magnetic-field gradients. By attaching magnetic particles to molecules it is possible to stretch molecules and determine force constants. Opto-plasmonic tweezers use radiation from resonant electrons to create patterned electric fields that can be used through dielectrophoresis to orient nanoscale objects.
Atomic force microscopy (AFM) is based on cantilevers. In AFM the force between the probe tip and the specimen is used to measure forces in the micronewton range. An AFM probe typically has cantilever lengths of 0.2 mm and a width of around 50 [micro]m. An AFM can operate in the contact mode, noncontact, or tapping mode. Force information of the interaction of the tip with a material is obtained by means of cantilever bending, twisting, and, in the noncontact mode, by resonance of the cantilever.
In the microwave range, near-field microwave scanning probes are commonly used. These probes have proved valuable to measure the permittivity and imaging on a surface of a thin film at subwavelength resolution. These needle probes usually use near-field microwaves that are created by a resonator above the probe, as shown in Fig. 17. A shift in resonance frequency is then related to the material properties under test through software based on a theoretical model. Therefore, most of these probes are limited to resonant frequencies of the cavity. Continuous-wave methods based on microstrip tips have also been applied.
16.1 Properties and Measurement of Dielectric Nanomaterials
Nanomaterials could consist of composites of nanoparticles dispersed in a matrix or isolated particles. A mixture of conducting nanoparticles dispersed into a matrix sometimes yields interesting dielectric behavior (23), (181). Lewis has noted that the interface between the nanoparticle and matrix produces unique properties in nanocomposites (23). Interfaces and surface charges are a dominant parameter governing the permittivity and loss in nanocomposites (23), (181), (182). Double layers (Sec. 8) near the particle surface can strongly influence the properties (23). In addition, conductivity in some nanoparticles can achieve ballistic transport.
In order to model a single dielectric nanoparticle in an applied field the local field can be calculated, as summarized in Sec. 4.3. Kuhn et al. (59) studied the local field around nanoparticles, and they found that use of the macroscopic field for modeling of a sphere containing nanoparticles was not valid at below 100 nm. In order to model small groups of nanoparticles, they found that the effects of the interface required the use of local fields rather than the macroscopic field.
When individual nanoparticles are subjected to EM fields, the question arises of whether it is possible to define a permittivity of the nanoparticle or whether an ensemble of particles is required. Whether permittivity of a nanoparticle is well defined depends on the number of dipole moments within the particle. If we use the analogy of a gas, we assume that the large number of gas molecules together with the vacuum around the particles constitutes a bulk permittivity. This permittivity does not apply to the individual gas molecules, but rather to the bulk volume. When individual nanoparticles contain thousands of dipoles, according to criteria of permittivity developed in Sec. 4.6, long-wavelength fields would allow defining a permittivity of the particle and a macroscopic field. However, such a permittivity would be spatially varying due to interfacial effects, and the definition would break down when there are insufficient particles to perform an ensemble average (59).
16.2 Electrical Properties and the Measurement of Nanowires
Nanowires are effectively one-dimensional entities that consist of a string of atoms or molecules with a diameter of approximately [10.sup.-9] meters. Nanowires may be made of Ti[O.sub.2], Si[O.sub.2], platinum, semiconducting compounds such as gallium nitride and silicon, single (SWNT) or multi-wall (MWNT) carbon nanotubes, and inorganic and organic strings of molecules such as DNA (183-188). Because they are effectively ordered in one dimension, they can form a variety of structures such as rigid lines, spirals, or zigzag pattern. Carbon nanotubes that have lengths in the millimeters have been constructed (189).
At these dimensions, quantum-mechanical effects cannot be totally neglected. For example, the electrons are confined laterally, which influences the available energy states like a particle in a one-dimensional box. This causes the electron transport to be quantized and therefore the conductance is also quantized (2[e.sup.2]/h). The impedance of nanoconductors is on the order of the quantum resistance h/[e.sup.2], which is 25 k[OMEGA]. For SWNTs, due to band-structure degeneracy and spin, this is reduced to 6 k[OMEGA]. The ratio of the free-space impedance to the quantum impedance is two times the fine structure constant 2[alpha]. This high impedance is difficult to probe with 50 [OMEGA] systems (190), and depositing a number of them in parallel has been used to minimize the mismatch (191).
The resistance of a SWNT depends on the diameter and chirality. The chirality is related to the tube having either metallic or semiconducting properties. For device applications such as nanotransistors, the nanowires need to be either doped or intrinsic semiconductors. Semiconducting nanowires can be connected to form p-n junctions and transistors (192).
Many nanowires have a permanent dipole moment. Due to the torque in an electric field, the dipole will tend to align with the field, particularly for metallic and semiconducting nanotubes (193).
16.3 Charge Transport and Length Scales
Electrical conduction through nanowires is strongly influenced by their small diameter. This constriction limits the mean free path of conduction electrons (88), (194). For example in bulk copper the mean free path is 40 nm, but nanowires may be only 1 to 10 nm in diameter, which is much less than a mean free path and results in constriction of the current flow.
Carbon nanotubes can obtain ballistic charge transport. Ballistic transport is associated with carrier flow without scattering. This occurs in metallic nanowires when the diameter becomes close to the Fermi wavelength in the metal. The electron mean-free path for a relaxation time [[tau].sub.e] is [l.sub.e] = v[[tau].sub.e], and if [l.sub.e] is much larger than the length of the wire, then it is said to exhibit ballistic transport. Carbon nanotubes can act as antennas and can have plasmonic resonances in the low terahertz range.
The Landauer-Buttiker model of ballistic transport was developed for one-dimensional conduction of spinless/noninteracting electrons (195), (196). This model has been applied to nanowires.
Graphene has shown promise for construction of transistors due to its high conductivity, but is hampered by defects. The very high carrier mobility of graphene makes it a candidate for very high speed radiofrequency electronics (197).
16.4 Distributed Parameters and Quantized Aspects
A high-frequency nanocircuit model may need to include the quantum capacitance and kinetic and magnetic inductance in addition to the classical parameters. The magnetic inductance per unit length for a nanowire [micro] of permeability of diameter d and a distance s over a ground plane is given by (189)
[L.sub.M] = [[micro]/2[pi]][[cosh.sup.-1](2s/d) [approximately equal to] [[micro]/2[pi]]ln[s/d], typically1(pH/[micro]m). The kinetic inductance due to quantum effects is is related to the Fermi velocity [v.sub.F], [L.sub.k] = h/[2[e.sup.2][v.sub.F]], typically, 16 (nH [micro]m). At gigahertz frequencies, the kinetic inductance is not a dominant contribution to the transmission line properties (189). The electrostatic capacitance between a wire and ground plane in a medium with permittivity [epsilon] is [C.sub.ES] = [2[pi][epsilon]]/[[cosh.sup.-1](2s/d)], typically, 50 (aF/[micro]m). The quantum capacitance is [C.sub.Q] = 8[e.sup.2]/h[v.sub.F], typically, 400 (aF/[micro]m). The electrostatic capacitance is found to dominate over the quantum capacitance at gigahertz frequencies. At terahertz frequencies and above they are of the same order of magnitude, and both should be included in calculations for nanowires. Burke notes that the resistance and classical capacitance dominates over the quantum inductance and capacitance and are not important contributions at gigahertz frequencies, but may be important at terahertz frequencies (189). The wave velocity in nanowires is approximated by
[v.sub.F] [approximately equal to] 1/[square root of ([L.sub.K][C.sub.Q])] (136)
The quantum characteristic impedance is
Z = [square root of (L/C)] = h/[2[e.sup.2]]. (137)
If the noninteracting electrostatic and quantum impedance are combined, we have
Z = [square root of (([L.sub.K] + [L.sub.M])(1/[C.sub.Q] + 1/[C.sub.ES]))] (138)
Whereas the free-space impedance is 377 [OMEGA], the quantum capacitance and inductance of carbon nanotubes yields an impedance of approximately 12.5 k[OMEGA].
The resistivity of nanowires and copper are generally of the same order of magnitude. The ballistic transport properties at small scales represents an advantage; however, the resistance is still quite high. Copper interconnects have less resistance until the conductor sizes drop below about 100 nm; currently the microelectronic industry uses conductors of smaller size. This is an origin of heating (14), (198). Because the classical resistance is calculated from R/L = [rho]/A, where [rho] is resistivity, L is length, and A is the cross-sectional area, the small area of a SWNT limits the current and increases the resistance per unit length and the impedance. Due to the high impedance of nanowires, single nanowires have distinct disadvantages; for example, carbon nanotubes may have impedances on the order of [10.sup.4] [OMEGA]. Bundles of parallel nanowires could form an interconnect (191). Tselev et al. (191) performed measurements on bundles of carbon nanotubes that were attached to sharp metal tips by dielectrophoresis on silicon substrates. Electron-beam lithography was used to attach conductors to the tubes. High-frequency inductance measurements from 10 MHz to 67 GHz showed that the inductance was nearly independent of frequency. In modeling nanoscale antennas made from nanowires, the skin depth as well as the resistance are important parameters (189).
17. Random Fields, Noise, and Fluctuation-Dissipation Relations
17.1 Electric Polarization and Thermal Fluctuations
As transmission lines approach dimensions of tens of nanometers with smaller currents, thermal fluctuations in charge motion can produce small voltages that can become a significant source of noise (199). The random components of charge currents, due to brownian motion of charges, produce persistent weak random EM fields in materials and produces a flow of noise power in transmission lines. These fields contribute to the field felt by the device. Random fields also are important in radiative transfer in blackbody and non-blackbody processes.
Thermal fluctuations in the dipole moments in dielectric and magnetic materials influence the polarization and are summarized in the well known fluctuation-dissipation relationships. These relationships are satisfied for equilibrium situations. Equilibrium is a state where the entropy is a maximum and macroscopic quantities such as temperature, pressure, and local fields are well defined. Fluctuation-dissipation relationships can be obtained from the linear-response formalism (Sec. 4.4) that yields the susceptibility in terms of the Fourier transform of the associated correlation functions. By use of Eq. (30), an expression can be written for the susceptibility in terms of the polarization
[[[left and right arrow].x"].sub.e]([omega]) = [[integral].sub.0.sup.[infinity]][[[left and right arrow].f].sub.e](t)sin([omega]t)dt = -[V/[[k.sub.B]T]][[integral].sub.0.sup.[infinity]]d/dt(< P(0)P(t) >)sin([omega]t)dt = [[[omega]V]/[2[k.sub.B]T]] [[integral].sub.0.sup.[infinity]] < P(0)P(t) > sin([omega]t)dt. (139)
Equation (139) is a fluctuation-dissipation relationship that is independent of the applied field. In this approach, if the correlation function is known, then the material properties can be calculated. However, in practice most material properties are measured through applied fields. The interpretation of this relationship is that the random microscopic electric fields in a polarizable lossy medium produce fluctuations in the polarization and thereby induces loss in the decay to equilibrium. These fluctuations can be related to entropy production (44), (61). We can obtain an analogous relation for the real part of the susceptibility by use of Eq. (29). This relation relates the real part of the susceptibility to fluctuations
[[[left and right arrow].x'].sub.e]([omega]) = [[integral].sub.0.sup.[infinity]][[[left and right arrow].f].sub.e](t)cos([omega]t)dt = [V/[[k.sub.B]T]][[integral].sub.0.sup.[infinity]]d/dt(< P(0)P(t) >)cos([omega]t)dt = [[[omega]V]/[[k.sub.B]T]] [[integral].sub.0.sup.[infinity]] < P(0)P(t) > sin([omega]t)dt. (140)
17.2 Magnetic Moment Thermal Fluctuations
Magnetic-moment fluctuations with respect to signal-to-noise limitations are important to magnetic-storage technology (200). This noise can also be modeled by fluctuation-dissipation relations for magnetic response. The linear fluctuation-dissipation relation for the magnetic loss component can be derived in a way similar to the electric response:
[[[left and right arrow].x"].sub.m]([omega]) = [[integral].sub.0.sup.[infinity]][[[left and right arrow].f].sub.e](t)sin([omega]t)dt = - [[V[[micro].sub.0]/[[k.sub.B]T]][[integral].sub.0.sup.[infinity]]d/dt(< M(0)M(t) >)sin([omega]t)dt = [[[omega]V[[micro].sub.0]]/[2[k.sub.B]T]] [[integral].sub.0.sup.[infinity]] < M(0)M(t) > sin([omega]t)dt. (139)
17.3 Thermal Fields and Noise
Due to thermal fluctuations, brownian motion of charges produce random EM fields and noise. In noise processes the induced current density can be related to microscopic displacement [[vector].D] and induction fields [[vector].B].
The cross-spectral density of random fields is defined as (18)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (142)
The relationship to the time-harmonic correlation function for the field components is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (143)
Thermally induced fields can be spatially correlated (17) and can be modeled to first order as
< [[vector].D]([omega], r)[[vector].D] * ([omega], r') > = [[2i[THETA]([omega], T)]/[omega]]([left and right arrow]. [epsilon] - [left and right arrow].[epsilon] *)[delta](r - r'), (144)
< [[vector].B]([omega], r)[[vector].B] * ([omega], r') > = [[2i[THETA]([omega], T)]/[omega]]([left and right arrow]. [micro] - [left and right arrow].[micro] *)[delta](r - r'), (145)
< [[vector].B]([omega], r)[[vector].D] * ([omega], r') > = 0, (146)
Where [THETA]([omega], T) = (h[omega]/2) coth(h[omega]/2[k.sub.B]T). [THETA] [right arrow] [k.sub.B]T for [k.sub.B]T[much greater than] h[omega].
The voltage V and current I in a microscopic transmission line with distributed noise sources [v.sub.n] and [i.sub.n] that are caused by random fields can be modeled by coupled differential equations as shown in (199).
A special case of Eq. (144) is the well-known Nyquist noise relation for voltage fluctuations from a resistance R over a bandwidth [DELTA]f is
< [v.sup.2] > = 4[k.sub.B]TR[DELTA]f. (147)
17.4 Fluctuations and Entropy
In thermal equilibrium macroscopic objects have a well-defined temperature, but in addition there are equilibrium temperature fluctuations. When the particle numbers in a system decrease, the thermodynamic quantities such as temperature and internal energy, have a less precise meaning than in a large-scale system (61), (201). In nanosystems, fluctuations in particle energy, momentum, and local EM fields can be large enough to affect measurements. These fluctuations translate into fluctuations in the measured EM fields, internal energy, temperature, and heat transfer. A system that is far from thermal equilibrium or very small may not have a well-defined temperature, macroscopic internal energy, or specific heat (199), (202), (203). When the applied driving fields are removed, some polymers and some spin systems have relaxation times of seconds to hours until they decay from a nonequilibrium state to an equilibrium state. In these types of nonequilibrium relaxation processes, equilibrium parameters such as temperature have only a fuzzy meaning. Fluctuation-dissipation relations that are used to define transport coefficients in equilibrium do not apply out of equilibrium.
Nanosystems operate in the region between quantummechanical and macroscopic description and between equilibrium and nonequilibrium states. Whereas Johnson noise is related to fluctuations in equilibrium voltages, there is a need for theoretical work that yields results that compare well to measurements in this transition region. As an example, Hanggai et al. showed that the theoretical bulk definitions for specific heat and entropy in some nanosystems break down in the high or low temperature limits (204). Noise also occurs in nonequilibrium systems and the theoretical foundations are not as well developed as in thermal equilibrium.
17.4.2 Fluctuations and Entropy Production
For reliable operation, microelectronic interconnects require a stable thermal environment because thermal fluctuations could potentially damage an interconnect or nanotransistor (205). An understanding of thermodynamics at the nanoscale and the merging of electromagnetism and non-equilibrium thermodynamics is important for modeling small systems of molecules. Modeling of thermal fluctuations can be achieved by relating Nyquist noise to fluctuations in thermal energy. Another approach away from equilibrium is to use the concept of entropy production (44). Entropy can be increased either by adding heat to a material, [DELTA]S = [DELTA][Q.sub.h] / T, or by spontaneous processes in the relaxation of a system from nonequilibrium to an equilibrium state. In EM interaction with materials, we can produce entropy either through the dissipation of the fields in the material or by relaxation processes. Relaxation processes are usually spontaneous processes from nonequilibrium into an equilibrium state.
The entropy is defined as S = [k.sub.B]ln(W), where W is the number of accessible states. Entropy is a cornerstone of thermodynamics and non-equilibrium thermodynamics. In thermodynamics the free energy is defined in terms of the internal energy U as F = -[k.sub.B]TlnZ, where Z is the partition function. The entropy is also defined in terms of the free energy as
S(T) = -[[partial derivative]F]/[[partial derivative]T]. (148)
In thermodynamics, temperature is defined as
[[partial derivative]S(T)]/[[partial derivative]U] = 1/T. (149)
A very general evolution relation for the macroscopic entropy production rate [SIGMA](t) in terms of microscopic entropy production rate s(t) was derived from first principles by use of a statistical-mechanical theory (19), (44), (61), (89), (206):
[SIGMA](t) = [1/[k.sub.B]][[integral].sub.0.sup.t] < s(t)T(t, [tau])(1- P(t))s([tau]) > d[tau], (150)
where s(t) satisfies (< s(t) > = 0), [SIGMA](t) is the net macroscopic entropy production in the system, and T and P are evolution operators and projection operators, respectively. The Johnson noise formula is a special case of Eq. (150) near equilibrium, when [SIGMA](t) = [I.sup.2]R/T and s (t) = (1/2)Iv(t)/T, (with < v(t) > = 0) is a fluctuating voltage variable, and I is a bias current.
18. Dielectric Response of Crystalline, Semiconductors, and Polymer Materials
18.1 Losses in Classes of Single Crystals and Amorphous Materials
A class of dielectric single-crystal materials have very low loss, especially at low temperatures. The low loss is related to the crystal order, lack of free charge, and the low number of defects. Anomalously low values of the dielectric loss in single-crystal alumina at low temperatures were reported in 1981 (14), (207). In this study, dielectric resonators were used to measure the loss tangent because cavity resonators do not have the required precision for very-low-loss materials. Since then, there has been a large body of research (208), (209) performed with dielectric resonators that supports these results. Braginsky et al. (207) showed that the upper bound for loss in high-quality sapphire was 1.5 x [10.sup.-9] at 3 GHz and at T = 2 K. These reports were supported by Strayer et al. (210). These results are also consistent with the measurements by Krupka et al. (209), who used a whispering-gallery mode device to measure losses. Very low loss is obtained in sapphire, diamond, single-crystal quartz, MGO, and silicon. Low loss resonators have been studied at candidates for frequency standards.
The whispering-gallery mode technique is a particularly accurate way of measuring the loss tangent of materials with low loss (14). These researchers claim that the loss tangent for many crystals follows roughly a [f.sup.2] dependence at low temperatures.
In nonpolar materials, dielectric loss originates from the interaction of phonons or crystal oscillations with the applied electric field. In the absence of an applied electric field, the lattice vibrates nearly harmonically and there is little phonon-phonon interaction. The electric-field interaction modifies the harmonic elastic constant and thereby introduces an anharmonic potential term. The anharmonic interaction allows phonon-phonon interaction and thereby introduces loss (73). Some of the scattering of phonons by other phonons is manifested as loss.
The loss in many crystals is due to photon quanta of the electric field interacting with phonons vibrating in the lattice, thereby creating a phonon in another branch. Dielectric losses originate from the electric field interaction with phonons together with two-, three-, and four-phonon scattering and Umklapp process (73). The three- and four-quantum loss corresponds to transitions between states of the different branches. Crystals with a center of symmetry have been found to generally have lower loss than ones with noncentrosymmetry. The temperature dependence also depends on the crystal symmetry. For example, a symmetric molecule such as sapphire has much lower loss than noncentrosymmetric ferroelectric crystals such as strontium barium titanate. Quasi-Debye losses correspond to transitions, which take place between the same branch. In centro-symmetric crystals three- and four-quantum processes are dominant. In noncentro-symmetric crystals the three-quantum and quasi-Debye processes dominate.
Gurevich and Tagantsev (73) studied the loss tangent for cubic and rhombohedric symmetries for temperatures far below the Debye temperature [T.sub.D] = 1047 K. For these materials, the loss tangent can be modeled as
tan [delta] = [[[omega].sup.2][([k.sub.B]T).sup.4]]/[[epsilon][rho][v.sup.5]h[([k.sub.B][T.sub.D]).sup.2]], (151)
where [epsilon] is permittivity, [rho] is density, v is speed of sound in air. For hexagonal crystals, without a center of symmetry,
tan [delta] = [[omega][([k.sub.B]T).sup.3]]/[[epsilon][micro][v.sup.5]h], (152)
and with symmetry,
tan [delta] = [[omega][([k.sub.B]T).sup.5]]/[[epsilon][micro][v.sup.5][h.sup.2][([k.sub.B][T.sub.D]).sup.2]]
For many dielectric materials with low loss, Gurevich showed that there is a universal frequency response of the form tan [delta] [infinity] [omega].
The loss tangent in the microwave band of many low-loss ceramics, fused silica, and many plastics and some glasses increases nearly linearly as frequency increases (211). For materials where the loss tangent increases linearly with frequency, we can interpolate and possibly extrapolate microwave loss-tangent measurement data from one frequency range to another (Fig. 6). This approach is, of course, limited. This behavior can be understood in terms of Gurevich's relaxation models (73) or by the moment expansion in (212).
This behavior is in contrast to the model of Jonscher (213) who has stated that x"/x' is nearly constant with frequency in many disordered solids.
18.2 Electric Properties of Semiconductors
Excellent reviews of the dielectric properties of semiconductors in the microwave range have been given by Jonscher and others (14), (213-217). The dc conductivities of semiconductors are related to holes and free charge. In the gigahertz region, the total loss in most semiconductors decreases significantly since the effects of the dc conductivity decreases; however, the dielectric component of loss increases. For gallium arsenide and gallium nitride the conductivity is relatively low. Figure 18 shows measurement results on the permittivity of high-resistivity gallium arsenide as a function of frequency. These measurements were made by a mode-filtered T[E.sub.01] X-band cavity. Silicon semiconductors can exhibit low to high loss depending on the level of dopants in the material. There are Schottky barriers at the interface between semiconductors and metals and at p-n junctions that produce losses.
The conductivities of semiconductors at low frequencies fall between those of metals and dielectrics. The theory of conductivity of semiconductors begins with an examination of the phenomena in intrinsic (undoped) samples. At temperatures above 0 K, the kinetic (thermal) energy becomes sufficient to excite valence band electrons into the conduction band, where an applied field can act upon them to produce a current. As these electrons move into the conduction band, holes are created in the valence band that effectively become another source of current. The total expression for the conductivity includes contributions from both electrons and holes and is given by [[sigma].sub.dc] = q (n[[micro].sub.n] + p[[[micro].sub.p]), where q is charge, n is the electron density, p is the hole density, and [[micro].sub.n] and [[micro].sub.p] are the electron mobility and hole mobility, respectively.
In intrinsic semiconductors, the number of charge carriers produced through thermal excitation is relatively small, but [[sigma].sub.dc] can be significantly increased by doping the material with small amounts of impurity atoms. These additional carriers require much less thermal energy in order to contribute to [[sigma].sub.dc]. This results in more carriers becoming available as the temperature increases, until ionization of all the impurity atoms is complete.
For temperatures above the full ionization range of the dopants, [[sigma].sub.dc] is increasingly dominated by [[micro].sub.n] and [[micro].sub.p]. In semiconductors such as silicon, the mobility of the charge carriers decreases as the temperature increases, due primarily to the incoherent scattering of the carriers with the vibrating lattice. At a temperature [T.sub.i], intrinsic effects begin to contribute additional charge carriers beyond the maximum contributions of the impurity atoms, and [[sigma].sub.dc] begins to increase again (215), (216), (219-222).
19. Overview of the Interaction of RF Fields With BiologicalMaterials
19.1 RF Electrical Properties of Cells, Amino Acids, Peptides, and Proteins
In this section, we will overview the dielectric relaxation of cells, membranes, proteins, amino acids, and peptides (97), (223-229). This research area is very large and we summarize only the most basic concepts as they relate to RF fields.
Dielectric response of biological tissues to applied RF fields is related to membrane and cell boundaries, molecular dipoles, together with associated ionic fluids and counterions (230). The ionic solution produces low-frequency losses that are very high. As a consequence of these mobile charge carriers, counterions adhere to molecular surfaces, interface charge causes Maxwell-Wagner capacitances, and electrode polarization is formed at electrode interfaces. All of these processes can yield a very high effective [[epsilon].sub.r]' at low frequencies. Some of the effects of the electrodes can be corrected for by use of standard techniques (230), (231) (Sec. 8).
Some biological tissues exhibit an [alpha] relaxation in the 100 Hz to 1 kHz region due to dipoles and Maxwell-Wagner interface polarization, another [beta] relaxation in the megahertz region due to bound water, and [gamma] relaxation in the microwave region due to the relaxation of water and water that is weakly bond.
Amino acids contain carboxyl (COOH) groups, amide (N[H.sub.2]) groups, and side groups. The side groups and the dipole moment of the amino and carboxyl groups determine most of the low-frequency dielectric properties of the acid. Some of the side groups are polar, while others are nonpolar. When ionized, the amino and carboxyl groups have positive and negative charges, respectively. This charge separation forms a permanent dipole (Fig. 5). [alpha] amino acids have an amino group and carboxyl group on the same carbon denoted [C.sub.[alpha]] and [alpha]-amino acids have a dipole moment of 15 to 17 debyes (D) (1 debye equals 3.33 x [10.sup.-30] coulomb-meter). [beta] amino acids have a C[H.sub.2] group between the amino and carboxyl groups, which produces a large charge separation and therefore a dipole moment on the order of 20 D. For a very good overview see Pethig (223). Peptides are formed from condensed amino acids. A peptide consists of a collection of amino acids connected by peptide bonds. Peptide bonds provide connections to amino acids through the CO-NH bond by means of the water molecule as a bridge. The peptide unit has a dipole moment on the order of 3.7 D. Chains of amino acids are called polyamino acids or polypeptides. These are terminated by an amide group on one end and a carboxyl group on the other side. Typical dipole moments for polypeptides are on the order of 1000 D.
Table 5. Approximate dipole moments Material Dipole Moment (D) [H.sub.2]O 1.85 CO 0.12 NaCl 9.00 Typical protein 500 Peptide 3.7 Amino acid 20 Polypeptide 1000
Polyamino acids can be either in the helical or random-coil phase. In the helical state, C = O bonds are linked by hydrogen bonds to NH groups. The helix can either be right-handed or left-handed; however, the right-handed helix is more stable. Generally, polyamino acids have permanent dipole moments and dielectric relaxation frequencies in the kilohertz region (232).
The origin of relaxation in proteins has been debated over the years. Proteins are known to be composed of polyamino acids with permanent dipole moments, but they also have free and loosely bound protons. These protons bind loosely to the carboxyl and amino groups. Kirkwood et al. hypothesized that much of the observed relaxation behavior of proteins is due to movement of these nearly free protons in the applied field or the polarization of counterion sheaths around molecules (233). Strong protonic conductivity has also been observed in DNA. At present, the consensus is that polar side chains and both permanent dipoles and the proton-induced polarization contribute to dielectric relaxation of proteins.
In the literature three dielectric relaxations in proteins have been identified (231). These are similar to that in DNA. The first is the [gamma] relaxation in the 10 kHz to 1 MHz region and is due to rotation of the protein side chains. The second minor [beta] relaxation occurs in the 100 MHz to 5 GHz range and is thought to be due to bound water. The third [gamma] relaxation is around 5 GHz to 25 GHz and is due to semi-free water.
Nucleic acids are high-molecular mass polymers formed of pyrimidine and purine bases, a sugar, and phosphoric-acid backbone. Nucleic acids are built up of nucleotide units, which are composed of sugar, base, and phosphate groups in helical conformation. Nucleotides are linked by three phosphates groups, which are designated [alpha], [beta], and [gamma]. The phosphate groups are linked through the pyrophosphate bond. The individual nucleotides are joined together by groups of phosphates that form the phosphodiester bond between the 3' and 5' carbon atoms of sugars. These phosphate groups are acidic. Polynucleotides have a hydroxyl group at one end and a phosphate group on the other end. Nucleosides are subunits of nucleotides and contain a base and a sugar. The bond between the sugar and base is called the glycosidic bond. The base can rotate only in the possible orientations about the glycosidic bond.
Watson and Crick concluded through x-ray diffraction studies that the structure of DNA is in the form of a double-stranded helix. In addition to x-ray structure experiments on DNA, information has been gleaned through nuclear magnetic resonance (NMR) experiments. Types A and B DNA are in the form of right-handed helices. Type Z DNA is in a left-handed conformation. There is a Type B to Z transition between conformations. A transition from Type A to Type B DNA occurs when DNA is dissolved in a solvent (234). The Watson-Crick conception of DNA as a uniform helix is an approximation. In reality, DNA exists in many conformations and may contain inhomogeneities such as attached proteins. In general, double-stranded DNA is not a rigid rod, but rather a meandering chain. Once formed, even though the individual bonds composing DNA are weak, the molecule as a whole is very stable. The helical form of the DNA molecule produces major and minor grooves in the outer outer surface of the molecule. There are also boundwater molecules in the grooves. Many interactions between proteins or protons with DNA occur in these grooves.
The helix is formed from two strands. The bases in adjacent strands combine by hydrogen bonding, an electrostatic interaction with a pyrimidine on one side and purine on the other. In DNA, the purine adenine (A) pairs with the pyrimidine thymine (T). The purine guanine (G) pairs with the pyrimidine cytosine (C). A hydrogen bond is formed between a covalently bonded donor hydrogen atom that is positively charged and a negatively charged acceptor atom. The A-T base pair associates by two hydrogen bonds, whereas C-G base pairs associate by three hydrogen bonds. The base-pair sequence is the carrier of genetic information. The genetic code is formed of a sequence of three base pairs, which determine a type of amino acid. For example, the sequence of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] determines an amino acid sequence of phenylalanine-lysine-lysine-alanine.
The DNA molecule has a net negative charge due to the phosphate backbone. When dissolved in a cation solution, some of the charge of the molecule is neutralized by cations. The double-stranded DNA molecule is generally thought to have little intrinsic permanent dipole moment. This is because the two strands that compose the helix are oriented so that the dipole moment of one strand cancels the other. However, when DNA is dissolved in a solvent, such as saline solution, an induced dipole moment forms due to reorganization of charge into a layer around the molecule called the counterion sheath.
The interaction of the counterions with biomolecules has been a subject of intensive research over the years. Some of the counterions bind to the phosphate backbone with a weak covalent bond. Other counterions are more loosely bound and some may penetrate into the major and minor grooves of DNA (235). Ions are assumed to be bound near charges in the DNA molecule, so that a double layer forms. The ions attracted to the charged DNA molecule forms a counterion sheath that shields some of the charge of the DNA. The counterion sheath around a DNA molecule is composed of cations such as Na or Mg, which are attracted to the backbone negative phosphate charges. These charges are somewhat mobile and oscillate about phosphate charge centers in an applied electric field. A portion of these counterions is condensed near the surface of the molecule, whereas the vast majority are diffusely bound. Double-stranded DNA possesses a large induced dipole moment on the order of thousands of debye, due to the counterion atmosphere. This fact is gleaned from dielectric relaxation studies, birefringence, and dichroism experiments (236), and other light-scattering experiments (237). The induced dipole moment [[vector].[micro]] in an electric field E is defined in terms of the polarizability [[vector].[micro]] = [alpha]E.
Because the individual strands of double-stranded DNA are antiparallel and the molecule is symmetrical, the transverse dipole moments should cancel. However, a number of researchers have measured a small permanent dipole moment for DNA (238). In alternating fields, the symmetry of the molecule may be deformed slightly to produce a small permanent dipole moment (231). Another origin of the small permanent dipole moment is attached charged ligands such as proteins or multivalent cations (239). These ligands produce a net dipole moment on the DNA molecule by breaking the symmetry. The question of how much of the relaxation of the DNA molecule is due to induced dipolemoment versus permanent moment has been studied by Hogan et al. (236). The response of permanent vs. induced dipole moment differs in terms of field strength. The potential energy of a permanent dipole moment at an angle [theta] to the electric field is U = - [micro] E cos [theta], whereas the induced dipole moment in the electric field is quadratic, U = - ([DELTA][alpha]/2)[E.sup.2] [cos.sup.2] [theta], where [DELTA][alpha] is the difference in polarizability along anisotropy axes of the molecule. Experiments indicate that the majority of the moment was induced rather than permanent. Charge transport through DNA can be ballistic.
19.2 Dielectric Properties of BoundWater and Polyelectrolytes
Knowledge of the permittivity of the water near the surface of a biomolecule is useful for modeling. The region close to a biomolecule in water has a relatively low real part of permittivity and a fixed charge. The region far from the molecule has a permittivity close to that of water. Lamm and Pack (240) studied the variation of permittivity in the grooves, near the surface, and far away from the DNA molecule. The effective permittivity depends on solvent concentration, distance from the molecule, the effects of the boundary, and dielectric field-saturation. The variation of permittivity with position significantly alters the predictions for the electric potential in the groove regions. Model predictions depend crucially on knowing the dielectric constant of water. Numerical modeling of the DNA molecule depends critically on the permittivity of water. When the permittivity of water varies in space, numerical models indicate that small ions such as hydrogen can penetrate into the minor and major grooves (235), (241). These predictions are not obtained for models that use spatially independent permittivity for water. From modeling results it was found that the real part of the effective permittivity around the DNA molecule varies as a function of distance from the center of the molecule and as a function of solvent concentration in moles per liter (mol/l) (240).
The molecular structure of water is not simple. Besides the basic [H.sub.2]O triad structure of the water molecule, there are also complicated hydrogen-bonded networks created by dipole-dipole interactions that form hydroxyl O[H.sup.-] and hydronium [H.sub.3][O.sup.+] ions. The dielectric constant of water at low frequencies is about 80, whereas biological water contains ions, which affect both the real and imaginary parts of the permittivity. Water bound in proteins and DNA has a decreased permittivity. This is due to constraints on the movement of the molecules when they are attached to biomaterials.
19.3 Response of DNA and Other Biomolecules in Electric Driving Fields
The low-frequency response of DNA is due primarily to longitudinal polarization of the diffuse counterion sheath that surrounds the molecule. This occurs at frequencies in the range of 1 to 100 Hz. Another relaxation occurs in the megahertz region due to movement of condensed counterions bound to individual phosphate groups. Dielectric data on human tissue is given in Figs. 19 and 20. A number of researchers have studied dielectric relaxation of both denatured and helical conformation DNA molecules in electrolyte solutions both as a function of frequency and applied field strength. Single-stranded DNA exhibited less dielectric relaxation than double-stranded DNA (98), (243-246). Takashima concluded that denatured DNA tended to coil and thereby decrease the effective length and therefore the dipole moment. Furthermore, a high electric field strength affects DNA conductivity in two ways (244). First, it promotes an increased dissociation of the molecule and thereby increases conductivity. Second, it promotes an orientation field effect where alignment of polyions increases conductivity.
There are many other types of motion of the DNA molecule when subjected to mechanical or millimeter or terahertz electrical driving fields. For example, propeller twist occurs when two adjacent bases in a pair twist in opposite directions. Another motion is the breather mode where two bases oscillate in opposition as hydrogen bonds are compressed and expanded. The Lippincott-Schroder and Lennard-Jones potentials are commonly used for modeling these motions. These modes resonate at wavelengths in the millimeter region; however, relaxation damping prevents direct observation. Other static or dynamic motions of the base pairs of the DNA molecule are roll, twist, and slide.
Single-stranded DNA, in its stretched state, possesses a dipole moment oriented more or less transverse to the axis. The phosphate group produces a permanent transverse dipolemoment of about 20 D per 0.34 nm base-pair section. The Debye (D) is a unit of dipole moment and has a value of 3.336 x [10.sup.-30] C m. Because the typical DNA molecule contains thousands of base pairs, the net dipole moment can be significant. However, as the molecule coils or the base pairs twist, the dipole moment decreases. If single strands of DNA were rigid, since there is a transverse dipole moment, and relaxation would occur in the megahertz to gigahertz frequencies.
19.4 Dynamics of Polarization Relaxation in Biomaterials
In order to study relaxation of polypeptides and DNA in solution, we first consider the simplest model of a dipolar rigid rod.
The torque on an electric dipole moment p is
N = p x E. (154)
For cases where the dipole moment is perpendicular to the rod axis, rotations about the major axis can occur. The longitudinal rotation relaxation time for a molecule of length L is given in (247). The relaxation time varies with the molecule length. Major axis rotation could occur if the molecule had a transverse dipole moment; for example, in a single strand of DNA.
When the dipole moment is parallel to the major axis, end-over-end rotation may occur. This is the type of relaxation at low frequencies that occurs with the induced dipole moment in the counterion sheath or a permanent dipole moment parallel to the longitudinal axis of the molecule. The relaxation time varies as [L.sup.3]. Because length of the molecule and molecular mass are related, the responses for the two relaxations depend on molecular mass. Also, the model presented in this section assumes the rod is rigid. In reality, DNA is not rigid, so a statistical theory of relaxation needs to be applied (247-249).
Takashima (98) and Sakamoto et al. (243) have derived a more comprehensive theory for counterion relaxation and found that the relaxation time varies in proportion to the square of the length of the molecule (249), (250). Most experimental evidence indicates a [L.sup.2] dependence. This is in contrast to the rigid-rod model where the relaxation time varies as [L.sup.3].
19.5 Counterion Interaction With DNA and Proteins
The real and imaginary parts of permittivity depend on the concentration and type of cations (250). As the concentration of the solvent increases, more of the phosphate charge is neutralized and the dielectric increment (difference between the permittivity of the mixture and solvent by itself) decreases.
Many types of cations compounds have been used in DNA solvents; for example, NaCl, LiCl, AgN[O.sub.2], CuC[l.sub.2], MnC[l.sub.2], MgC [l.sub.2], arginines, protamine, dyes, lysine, histones, and divalent metals such as Pb, Cd, Ni, Zn, and Hg (243), (251-253). The simple inorganic-monovalent cations bind to the DNA molecule near the phosphate backbone to form both a condensed and diffuse sheath. There is evidence that strong concentrations of divalent metal cations destabilize the DNA helix (254). Sakamoto et al. (252) found that the dielectric increment decreased for divalent cations.
On the other hand, histones and protamines tightly bind in the major groove of the DNA molecule. They produce stability in the double helix by neutralizing some of the phosphate charge. Dyes can attach to DNA, neutralize charge, and thereby decrease dielectric increment.
20. Methods for Modeling Electromagnetic Interactions With Biomolecules, Nanoprobes, and Nanowires
Modeling methods for EM interactions with materials include solving mode-match solutions to Maxwell's equations, finite-element and molecular dynamics simulations, and finite-difference time-domain models. Finite-element modeling software can solve Maxwell's equations for complicated geometries and small-scale systems.
Traditionally, mode-match solutions to Maxwell's equations meant solving Maxwell's equations in each region and then matching the modal field components at the interfaces and requiring, by the boundary conditions, all the tangential electric fields go to zero on conductors. On the nanoscale, the microwave and millimeter wavelengths are much larger than the feature size; the skin depths are usually larger than the device being measured. Therefore modes must be defined both outside the nanowire and inside the wire and matched at the interface. Also, the role of the near field is more important.
The EM model for a specific problem must capture the important physics such as skin depth, ballistic transport, conductor resistance, and quantized capacitance, without including all of the microstructural content. Modeling nanoscale electromagnetics is particularly difficult in that quantum effects cannot always be neglected; however, the EM field in these models is usually treated classically. In the case of near-field probes the skin depths are usually larger than the wire dimensions, and therefore the fields then need to be determined in both the wire and in the space surrounding the wire. Sommerfeld and Goubau surface waves and plasmons propagate at the interface of dielectric and finite conductivity metals and need to be taken into account in modeling probe interactions. The probe-material EM communication is often transmitted by the near field.
Recently, simulators for molecular dynamics have advanced to the stage where bonding, electrostatic interactions, and heat transfer can be modeled, and some now are beginning to include EM interactions.
21. Metrology Issues
21.1 Effects of Higher Modes in Transmission-Line Measurements
In this section we describe various common difficulties encountered in measurements of permittivity and permeability using transmission lines.
The definition of dielectric permittivity becomes blurred when the particle size in a material is no longer much smaller than a wavelength. To illustrate this problem, consider the permittivity from a transmission-line measurement of a PTFE specimen, which was reduced using the common Nicolson-Ross method (13) as shown in Fig. 21. Typical scattering parameters are shown in Fig. 22. The permittivity obtained from the scattering data is plotted as a function of frequency. The intrinsic relative permittivity is seen to be roughly 2.05, the commonly accepted value. However, when dimensional or Fabry-Perot resonances (see example in Fig. 22) across the sample occur at multiples of onehalf wavelength, the specimen exhibits a geometrical standing-wave behavior at frequencies corresponding to n[lambda]/2 across the sample. So if the sample is treated as a single particle at these standing wave frequencies, then the "effective" permittivity from this algorithm is no longer the intrinsic property of the material, but rather an artifact of geometric resonances across the sample. Geometrical resonances are sometimes used by metamaterial researchers to obtain effective negative permittivities and permeabilities that produce negative index response.
Homogeneous solid or liquid dielectric and magnetic materials have few intrinsic material resonances in the RF frequencies. The intrinsic resonances that do occur are primarily antiferromagnetic, ferromagnetic, water vapor and oxygen absorption bands, surface wave and plasma resonances, and atomic transitions. Dielectric resonances or standing waves that occur in solid and liquid dielectrics in RF frequencies are usually either a) geometric resonances of the fundamental mode across the specimen, b) an artifact of a higher mode that resonates across the length of the specimen, c) resonances or standing waves across the measurement fixture, or d) due to surface waves near interfaces between materials.
In the measurement of inhomogeneous materials in a transmission line or samples with a small air gap between the material and the fixture, higher modes may be produced and resonate across the specimen length in the measurement fixture. For example, in a coaxial line, the T[E.sub.0n] or T[E.sub.11] mode may resonate across the specimen in a coaxial line measurement. These higher modes do not propagate in the air-filled waveguide since they are evanescent, but may propagate in the material-filled guide. Because these modes are not generally included in the field model, they produce a nonphysical geometric-based resonance in the reduced permittivity data, as shown in Fig. 25. These higher modes usually have low power and are caused by slight material or machining inhomogeneities. When these modes do propagate and resonate across the length of the specimen, it may appear as if the molecules in the material are under going intrinsic resonance, but this is not happening. In such cases, if the numerical model used for the data reduction uses only the fundamental mode, then the results obtained do not represent the permittivity of the material, but rather a related fixture specific geometric resonance of a higher mode (Fig. 25). These resonances are distinct from the fundamental-mode resonances obtained when the Nicolson-Ross-Weir reduction method is used (11) in transmission lines for materials at frequencies corresponding to n [[lambda].sub.g]/2, where n is an integer and [[lambda].sub.g] is the guided wavelength, as indicated in Fig. 21. The fundamental-mode resonances are modeled in the transmission-line theory and do not produce undue problems. However, in magnetic materials where there are both a permeability and permittivity, half wave geometric resonances and produce instabilities in the reduction algorithms (12).
21.2 Behavior of the Real Part of the Permittivity in Relaxation Response
For linear, homogeneous materials that are relaxing at RF frequencies, the permittivity decreases as frequency increases. The permittivity increases only near tails of intrinsic material resonances that only occur for frequencies in the high gigahertz region and above. To show this, we will analyze the prediction of the DRT permittivity model (90), (212).
We know that the behavior of the orientational polarization of most materials in time-dependent fields can, as a good approximation at low frequencies, be characterized with a distribution of relaxation times (53). Typical numerical values of dielectric relaxation times in liquids are from 0.1 [micro]s to 1 ps.
We consider a description that has a distribution function y([tau]), giving the probability distribution of relaxation times in the interval ([tau], [tau] + d[tau]). The DRT model is summarized in Eq. (69). There are fundamental constraints on the distribution y([tau]). It is nonnegative everywhere, y([tau]) [greater than or equal to] 0 on [tau] [member of] [0, [infinity]), and it is normalized,
[[integral].sub.0.sup.[infinity]]y([tau])d[tau] = 1. (155)
From Eq. (69) we have
[d[epsilon]'([omega])]/[d[omega]] = -2([[epsilon].sub.s] - [[epsilon].sub.[infinity]])[omega][[integral].sub.0.sup.[infinity]][[[[tau].sup.2]y([tau])]/(1 + [[omega].sup.2][[tau].sup.2])]d[tau] < 0. (156)
This shows that [epsilon]' is a decreasing function for all positive [omega] where the DRT model is valid (low RF frequencies), with a maximum only at [omega] = 0. The result of Eq. (156) holds for any distribution function y([tau]). This model assumes there is only a relaxational response. If resonant behavior occurs at millimeter to terahertz frequencies, then the real part of the permittivity will show a slow increase as it approaches the resonance. In the regions of relaxation response, the real part of the permittivity is a decreasing function of frequency. Therefore, [epsilon]'([omega]) attains a minimum at some frequency between relaxation and the beginning of resonance.
22. Permittivity Mixture Equations
We can readily estimate the permittivity of a mixture of a number of distinct materials. The effective permittivity of a mixture [[epsilon].sub.eff] of constituents with permittivities ei and volume fractions [[theta].sub.i] can be approximated in various ways. The Bruggeman equation (256) is useful for binary mixtures:
[[theta].sub.1][[[epsilon]'.sub.eff] - [[epsilon]'.sub.1]]/[[[epsilon]'.sub.1] + 2[[epsilon]'.sub.eff]] = [[theta].sub.2][[[epsilon]'.sub.1] - [[epsilon]'.sub.eff]]/[[[epsilon]'.sub.1] + 2[[epsilon]'.sub.eff]] (157)
or the Maxwell-Garnett mixture equation (256) can be used:
[[[epsilon]'.sub.eff] - [[epsilon]'.sub.2]]/[[[epsilon]'.sub.eff] - 2[[epsilon]'.sub.2]] = [[theta].sub.1][[[epsilon]'.sub.1] - [[epsilon]'.sub.2]]/[[[epsilon]'.sub.1] + 2[[epsilon]'.sub.2]], (158)
where [[epsilon]'.sub.1] is the permittivity of the matrix and [[epsilon]'.sub.2] is the permittivity of the filler (257). The formula by Lichtenecker is for a powerlaw dependence of the real part of the permittivity for -1 [less than or equal to] k [less than or equal to] 1, and where the volume fractions of the inclusions and host are [v.sub.p] and [v.sub.m]:
[[epsilon].sup.k] = [v.sub.p][[epsilon].sub.p.sup.k] + [v.sub.m][[epsilon].sub.m.sup.k]. (159)
This equation has successfully modeled composites with random inclusions embedded into a host. An approximation to this is
ln[epsilon] = [V.sub.p]ln[[epsilon].sub.p] + [V.sub.m]ln[[epsilon].sub.m.]. (160)
The broad area of RF dielectric electromagnetic interactions with solid and liquid materials from the macroscale down to the nanoscale materials was overviewed. The goal was to give a researcher a broad overview and access to references in the various areas. The paper studied the categories of electromagnetic fields, relaxation, resonance, susceptibility, linear response, interface phenomena, plasmons, the concepts of permittivity and permeability, and relaxation times. Topics of current research interest, such as plasmonic behavior, negative-index behavior, noise, heating, nanoscale materials, wave cloaking, polariton surface waves, biomaterials, and other topics were covered. The definition and limitations of the concept of permittivity in materials was discussed. We emphasized that the permittivity and permeability are well defined when the applied field has a wavelength much longer than the effective particle size in the material and when multiple scattering between inclusions is minimal as the wave propagates through the material. In addition, the use of the concept of permittivity requires an ensemble of particles that each have dielectric response.
We acknowledge discussions with team members of the Innovation Measurements Science Program at NIST: Detection of Corrosion in Steel-Reinforced Concrete by Antiferromagnetic Resonance, discussions with Nick Paulter of OLES, and Pavel Kabos and many other over the years.
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About the authors: James Baker-Jarvis is a physicist and a Project Leader and Sung Kim is an electrical engineer and a Guest Researcher and are both in the Electromagnetics Division of the NIST Physical Measurement Laboratory. The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce.
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|Title Annotation:||p. 31-60|
|Publication:||Journal of Research of the National Institute of Standards and Technology|
|Date:||Jan 1, 2012|
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