grable functions L(R) back onto itself in a way that preserves its structure exactly, it can be interpreted as simply a relabelling operator that takes every function x E L2(R) and gives it a new name Vx. This relabelling is equivalent to changing the frame of reference or to changing bases. Two operators, A and B, are defined as unitarily equivalent if they are equivalent modulo a change of basis, that is, if B = U-IAU for some unitary transformation U WSI. Given a linear operator Q on L2(R), solution of the eigenequation (5) yields the eigenfunctions {e,Q(r)) and the eigenvalues {&Q} of Q, both of which are indexed by the parameter q. If Q is unitary, then the eigenfunctions form an orthonormal basis for L'(R). In this case, the basis expansion onto these eigenfunctions yields another unitary operator, which we will refer to as the Q-Fourier transform F,. The forward transform of a signal x E Lz(R) is given by Note that F, is invariant (up to a phase shift) to the operator Q; that is, l(F~Qx)(d/= I(Fpx)(dl. The eigenfunctions of the time operator e:(u) = @'pru yield the usual Fourier transform FT = F, while the eigenfunctions of the frequency operator, e,'(u) = S(u-t), yield the (trivial) signal transform (FFx)(t) = x(t). The eigenfunctions of the dilation operator on L2(R+)are the 'hyperbolic chirp' functions = u-1/2e.?2ack" , 21 >0 (7) Because these functions are by definition invariant to dilations, the Mellin transform Fo in eqn. 1 is likewise invariant. Scale transform: The derivation of the scale transform follows directly from the unitary equivalence of the time, frequency, and dilation operators. The key observation is as follows. The transform FT based on the time operator eigenfunctions indicates the frequency content of a signal x, because it IS covariant to frequency shifts Fx of the signal. Similarly, the transform FF based on the frequency operator eigenfunctions indicates the time content of a signal, because it is covariant to time shifts Tx of the signal. Therefore, the transform FHcovariant to the scale operator will be based on the eigenfunctions of an operator H that is dual to the scale operator in the same sense that the time and frequency operators are dual. Fortunately, the time and dilation operators are unitary equivalent, allowing the direct computation of H. The time operator (on L'(R)) and the dilation operator (on L2(R+))are related by where E - : L2(R+) Dd = E-'TdE (8) L'(R) is the unitary exponential axis warping and to is an arbitrary positive reference time. Because the dual operator to time shift is frequency shift, the dual operator to dilaThe eigenfunctions of H are given by tion is therefore H, = E-IF~E. e,"(.) = (€-'e:) (U) = eU/'6(u - e U t o ) , U >o the FF transform, the functions e,"(u) in eqn. 10 play the role of pure scale functions having no spread about their inherent scale U. The scale transform can he applied to measure scale content in any domain, so long as the variable t of the function x(t) is interpreted appropriately. One particularly enlightening interpretation considers t as a spatial variable in an imaging system, with x(t) the distribution of an object in the direction perpendicular to the image plane; then o represents a zoom parameter and the spread of Sx indicates the amount of focus change required to successively bring all of the object into focus at the image plane. While S is defined only for one-sided signals in L2(R+), signals in Lz(R) can he handled by computing separate scale transforms along the positive and negative time axes. Alternatively, we can consider two values + & of the reference time parameter. Conclusions: Scale is a relative quantity ('this thing is twice as large as that thing'), and so we should anticipate difficulties with a transform indicating purely the scale content of a signal. The scale transform derived here is simply a warped version of fhe signal being analysed, and thus it is instantaneous in the sense that its value (Sx)(o) at point U depends only on the signal value x(e't,) at point e"t,. This is contrary to most other popular signal transforms, including the Fourier and Mellin transforms, which use integral averages over the totality of the signal time axis to compute a single value of the transform. However, the instantaneity of the scale transform is not completely heretical, because the concept of scale is bound closely to the physical domain where it is applied. Finally, we note that the concept of duality introduced in the scale transform section can be related to the concept of orthogonality of operators developed in detail in [5]. This approach is used to generalise the results of this Letter to other operators in [61. Acknowledgments: This work was supported by an NSERCNATO postdoctoral fellowship and was completed while the author was with the Laboratoire de Physique, Ecole Normale Superieure de Lyon, France. 20 July 1993 Electronics Letters Online No: 19931073 R. G. Baraniuk (Department of Electrical and Computer Engineering, Rice University, PO Box 1892, Houston, TX 77251-1892, USA) 0 IEE 1993 References o., and FLANDRIN, P.: 'A general class extending wavelet transforms', IEEE Trans. Signal Process., 1992, 40,(7), pp. 17461757 COHEN, L : 'The scale representation', to be published in lEEE Trans. Signal Process., December 1993, 41 PAPANDREAOU, A., HLAWATSCH. F , and BOUDREAUX-BARTELS. G F : 'The hyperbolic class of quadratic time-frequency representations. Part I: Constant-Q warping, the hyperbolic paradigm, properties and members', IEEE Trans. Signal Process. December 1993,41 BARANIUK. R G , and JONES, D L.: 'Warped wavelet bases: Unitary equivalence and signal processing', F'roc. IEEE Int. Conf. Acoustics, Speech and Signal Processing - ICASSP '93, 1993 BARANIUK. R.G , and IONES, D.L.: 'Unitary equivalence: A new twist on signal processing', submitted to lEEE Trans. Signal Process., 1993 BARANIUK. R. G: 'Integral transforms covariant to operators'. Preprint, 1993 RIOUL, (10) The expansion of eqn. 6 onto these eigenfunctions yields the scale transform FH = S given in eqn. 3 for signals in L2(R+).This transform clearly possesses the dilation covariance property eqn. 4. Interpretation: Although almost disappointingly simple, the transform of eqn. 3 has the attributes of a scale indicating transform. Just as the value x(t) of the time function at the point t indicates the signal content at the reference point t = 0 when the signal is translated by -t seconds, the value (Sx)(o) of the scale transform at the point U indicates the signal content at the reference point t = ta when the signal is dilated by the factor 4 . Furthermore, by analogy to the pure frequency functions efT(u) = f12pru of the FT Fourier transform and the pure time functions e:(u) = S(u - t) of 1676 Interfacial polarisation in AI-Y,O,-Si0,-Si capacitor C.H. Ling Indexing terms: Capacitors, Dielectric materials ~ The variation by a factor of 2 in the observed permittivity of yttrium oxide film is explained in terms of interfacial polarisation ELECTRONICS LE77ERS I 76th September 7993 Vol. 29 No. 79 (ii) Steady-state response to sinusoidal voltage excitation: The admittance of the double-layer capacitor to an AC excitation exp j o t is given by Consider the double-layer structure as a single-layer of relative permittivity E ~ .The admittance is Y = jWE’C0 (4) where complex permittivity E* = E’-jE’’. CO = Edd is the capacitanceiarea of a layer of total thickness d = d,+d2.Comparing eqns. 3 and 4 leads to effective permittivity of the double-layer, given by For w = 0, eqn. 5 reduces to Device structureifabrication e-h E‘ - (1 - Permittivity + d)(k-’ + dp2) 61 + anod (1 + dP)2 -. (6) Eqn. 6 agrees with the limiting cases: E’ -E, as d 0 and E’ - E , as d m. E ’ / Eis ~ not very sensitive to k, except when k-’ . dn*. Assuming k = 4, eqn. 6 is plotted in Fig. 2 for different resistivity ratios, p = lo”. Over the thickness range 1 c d < 1000, the curves merge for n < 4. From eqn. 6, as the thickness ratio becomes very large, E’/E~ 1 for all values of p. E’/E,peaks when -+ A1-Y,03-A1 13.5 AI-Y,O,-Si m-s + ox 11.8 AI-Y,O,-SIO,-SI m-s + ox 17-20 151 AI-Y,O,-Si e-b 13 I61 AI-Y ,O,-?-Si e-h 18-27 [7] + ox -+ I d = d,, = - Discussion: The observed permittivity lies in the range 12-27. Taking E~ = E(Y,O,) . 13 gives E’/&,a maximum value of -2. Fig. 2 shows a broad range of (p,d) values for which the relation 1 < E’/E, < 2 is easily satisfied. For a given d, &’/E,generally increases with decreasing p; for a given p, E’/E’increases initially but decreases For the Y,O,/SiO, capacitor, generally, we have d >> for d > dnx. 1, and n < 0. Good quality SiO, has resistivity > 10“ Qcm, whereas the resistivity of Y,O, is smaller. Thus variations in resistivities and thickness can result in different observed permittivity. 182011j Fig. 1 Double-layer capacitor model (i) Transient response to step voltage excitation: Fig. 1 illustrates the double-layer capacitor, characterised by relative permittivity E , , resistivity pLand thickness 4, where i = 1, 2 specify, respectively, layers 1, 2. Each layer is modelled by a capacitance C in parallel with a resistance R. Applying a step voltage V, to the capacitor leads to I 16’ “ 100 , .,, ,...I . . ., ,,..I 10’ 102 thickness ratio d , , , , ,A lo3 Fig. 2 Relative permittivity ratio E ’ / E ~ against thickness ratio d for various resistivity ratio p=lO”; o=O where dielectric relaxation time constants T , = E , E , ~T,=E,E,P,, ~, r=~,(l+dk)/(l+dp),and ratios d=d,/d,, k=El/E2and p=pI/p2.~o is the permittivity of free space. For t > o’,a charge Qf builds up at the interface, given by ELECTRONICS LETTERS 16th September 1993 Vol. 29 The dependence of permittivity on annealing temperature and Y,O, thickness was observed 171. For an 800A film, E < 10 initially and increased to 27 after annealing at 700K. Annealing at higher temperatures caused permittivity to decrease, attributed to.the oxidation of yttrium suboxides to Y,O, and the growth of an interfacial SiO, layer. Annealing also changed the resistivity of the Y,O, films. These results can be explained by the plots in Fig. 2. The initial increase in E’ is attributed to the rapid decrease of d (in the region d > &ax) as SiO, grows. At higher annealing temperatures, Y,O, resistivity and ratio p increase, thus reducing E’, The permittivity of Y,O, has been found to be frequency inde- No. 19 1677 pendent up to l M H z [2,5]. Using typical values: pI . p2 -lOWcm, E ~ 13, . E ~ 3.9 . for Y20,, SO,, respectively, and with d > IO, we find T.T, . 100 < s. T:, 104s. With these values of time constants, eqn. 5 reduces to --E’ El - (1 + d ) k(l+dp) which is independent of frequency. This is the case as long as t2 >> T, and t . t,. Eqn. 8 differs from eqn. 6 by a factor (1 + kdp2)/(1 +dp), which approaches 1 for d = IO, p < lo-’. Although Fig. 2 was based on o = 0. the results are believed to be applicable over the range of frequencies reported [2, &7], and to be of general applicability to other dielectric structures. Conclusion: We have shown that the apparent discrepancy in the reported permittivity of yttrium oxide films can be explained by interfacial polarisation of the Y,O,/SiO, double-layer capacitor. The observed frequency independence is attributed to the large difference in the dielectric relaxation times. Caution should be exercised in exploiting the enhanced permittivity of this composite layer, because of charge storage at the interface and transient behaviour under DC voltage conditions. 0 IEE 1993 19 July 1993 Electronics Letters On-line No.:- 19931086 C H. Ling (Department of Electrical Engineering National University of Singapore Kent Ridge Singapore 0511, Singapore) References ‘The anodic oxidation of yttrium thin film’, J. Electrochem. Soc., 1967, 114, p. 75 TSUTSUMI, T : ‘Dielectric properties of Y z 0 3 thin films prepared by vacuum evaporation’,Jpn. J. Appl. Phys., 1970,9, (7), pp. 735-739 SAYER, M , MARTIN, M S I and HELLICAR, N J : ‘Yttrium oxide fihTls prepared by electron-beam deposition’, Thin Solid Films, 1970, 6, pp. R61-63 GURVICH. M , MANCHANDA. L , and GIBSON, l.M : ‘Study O f thermally oxidised yttrium films on Si’, Appl. Phys. Lett., 1987, 51, (12), pp. 9 19-92 I MANCHANDA. L , and GURVITCH. M : ‘Yttrium oxide/SiO,: a new dielectric structure for VLSVULSI circuits’, IEEE Electron Device Lett., 1988, 9, (4), pp. 18lL182 KALKUR. T s . KWOR. R Y , and PAZ DE ARAUJO. C.A : ‘Yttrium oxide based metal-insulator-semiconductor structures on silicon’, Thin Solid Films, 1989, 170, pp. 185-189 SHARMA. R N , LAKSHMIKUMAR. s T , and RASTOGI. A c : ‘Electrical behaviour of electron-beam-evaporated yttrium oxide thin films on silicon’, Thin Solid Films, 1991, 199, (I), pp. 1-8 VON HIPPEL. A.: ‘Dielectrics and waves’ (MIT Press, 1954) RAIRDEN, J R : Formulation: For a perfectly conducting object, the CFIE can he shown to be [I] --R x HS(J) A. Helaly and H.M. Fahmy Indexing terms: Electromagneticwave scattering, Integral equations The combined-field integral equation (CFIE) is a linear combination of the H-field and the E-field integral equations. Previously, the weighting parameter of the E-field equation in the CFIE had been assumed constant along the generating curve of the body of revolution. However, in the Letter it ui shown that the weighting parameter can take a variable distribution along the generating curve or on a part of it only. In the latter case, a reduction in the computational time of 4%50% is achieved. Introduction: It is known theoretically that neither the H-field nor the E-field integral equations has unique solutions for the surface current on a conducting body at frequencies corresponding to the resonance frequencies of the region enclosed by the surface, but the CFIE does have a unique solution. The H-field equation does not have a unique solution for the scattered fields at these resonance frequencies, but the E-field and the combined-field equa- 17 + -aE;, - 17 just inside S (1) - where is the surface normal, E’ and H’ denote the incident electric and magnetic fields, respectively, and q is the wave impedance. The weighting parameter a can, for now, be viewed as an arbitrary real constant having values 0 and 1. From a theoretical point of view, the results should be independent of a but no numerical solution can be expected to assure this property. In fact, a must not be chosen small, otherwise eqn. 1 is too close to the Hfield equation. Similarly, if a is large, eqn. I is not sufficiently far from the E-field equation. Jt is recommended that an a value of the order of 0.2 is best [2]. The details of solvjng eqn. 1 using the method of moments for the surface current J is out of the scope of this Letter and can be found elsewhere[l]. The purpose of the Letter is to show that the weighting parameter a can take variable distributions along not only the generating curve of the body of revolution, but also on a part of it. Applying the method of moments requires dividing the generating curve into a number of segments N. Therefore, the weighting parameter a can be written as 01 = g(i;:) i: l,N (‘4 where ?, is the position of the ith segment and g is the corresponding value of a,which can take a value between 0 and 1. Moreover, g can be constant over all segments or varies from one segment to another. In the following numerical results, three different distributions for a are studied, namely: (i) Gaussian distribution normalised to along the generating curve (ii) Pulse train of height 0.2, as recommended earlier, i.e. a takes a value of 0.2 over the odd (or even) segments and takes zero over the even (or odd) segments, To examine the validity of the above mentioned technique, it is applied to a perfectly conducting sphere. An error function is then defined as where s is the surface of the sphere and HI is the incidenl magnetic field, also J, is the ex&ct electric current [3] and J is the computed approximation to J,. Numerical results: Eqn. 1 is applied to a conducting sphere of electrical radius ka, where k is the propagation constant. The sphere is excited by a plane wave incident in the axial direction. In all computations, the generating curve is divided into 30 segments, equally spaced from the lower pole to the upper pole of the sphere. Extensive numerical calculations for the RMS error in the surface current have been carried out for different distributions for the weighting parameter a. Representative cases at first and sec- ELECTRONICS LETTERS 1 - “:.,(J)=n x “ (iii) Pulses of the same height 0.2 over some selected segments, while a is taken zero over the rest of segments, in other words, the CFJE can be applied to the selected segments and the H-field integral equation is applied to the rest; this results in a large reduction of the computational time required to fill in the elements of the moment matrix compared to the case of applying the CFIE to all segments. Combined-field integral equation 1678 tions do have unique solutions. The numerical solutions of the Hfield and the E-field integral equations degenerate in the vicinity of the internal resonant frequencies for both the surface currents and the scattered fields. The numerical solution of the CFJE does not degenerate at or in the vicinity of the internal resonance frequencies [I]. 16th September 1993 Vol. 29 No. 19

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