On Miniaturizing On-Chip Microstrip Inductors Using Embedded Radiating Dipoles

As RF mixed-signal and patch-antenna-equipped SOC devices are becoming the dominant products worldwide, concerns over the large real-estate consumption by the spiral inductors (including those for microstrip antennas and impedance-matching inductances), as well as their generally Q-low (quality factor) performance, are now being discussed more than ever. Liao et al. have recently addressed the Q-low issue via using location-selective proton beam bombardment, whereby Q-improvements of 100%– 300% were evidenced. That success, nevertheless, is at times tarnished by some undesirable features, that is, the explosive rises of inductances near certain frequencies, which practically cut short the Q-enhancement and were identified to be due to resonant interactions between the inductor-propagating EM wave and the proton-caused defect dipoles. In this paper, however, the authors attempt to turn this resonance-caused undesirability to favor by proposing a new way to greatly shrink down the needed inductor size through dipoles engineering.


INTRODUCTION
As RF mixed-signal and patch-antenna-equipped RF SOC (system-on-chip) devices are becoming the dominant products worldwide, concerns over the large real-estate consumption by the desired spiral inductors, as well as their generally Q-low performance, are now being discussed more than ever.In fact, owing to these concerns, RF circuit designers have already resorted their impedance-matching and signalfiltering needs to RC type of work, rather than LC style of design, even though the latter, in many situations, may render superior circuit performance and design flexibility.The inductor Q-low problem has recently been attacked by Liao et al. (see, e.g., [1,2]) via using location-selective proton beam bombardment, and Q-value improvements of 100%-300% were shown to be readily achievable (see the obtained inductance spectral behavior in Figure 1 and Q-improvement in Figure 2).However, the real-estate consumption issue has remained unresolved.
Economically, it will be much celebrated if commonly needed inductance values can be realized on inductors of fairly smaller physical sizes.On the other hand, occasionally, high-inductance-value on-chip inductors are convenient to have for some special applications.Unfortunately, however, it is also well known that, other than topology, the inductance value is entirely determined by the electrical properties (including skin effects) of the spiral metal line and the underlying substrates such as silicon [3].In this paper, the authors attempt to explore the possibility of greatly enhancing the inductance value of an ordinary spiral inductor by means of dipoles engineering.
This innovative idea actually came from the first author's aforementioned experience with proton-enhanced device isolation and inductor Q-improvement [1,2].Although implementing the new idea itself no longer relies on any proton beam effort but more likely relies on VLSI or ULSI (ultra-large-scale integration), modern nanotechnology, or other new methods to emerge.That is, in spite of numerous evidenced successes of proton bombardment treatment in both the device isolation and the inductor Q-improvement [1,2] (see Figures 1 and 2) on already-manufactured mixedmode IC wafers (prior to packaging), it was also found that occasionally there showed several undesirable and puzzling features in the obtained inductance spectral behaviors for some large inductors post the proton irradiation.Among others, there are explosive rises of inductance (and the relative dielectric coefficient) near certain frequency (or frequencies) [4,5] (see Figures 3 and 4).A theory identifying the cause to be resonant interaction between the inductor EM wave and the proton-created defect dipoles (or displaced charge imbalance) has explained the observation quantitatively [4,5], should the electromagnetic mass of electron [6]    be taken into account [4,5].Its relevant importance to the current work is manifested through the high frequency part of the inductance L, L = L (0) + L (ω) , where L (0) and L (ω)  are the low-frequency and high-frequency components, respectively.Namely, it was found analytically [4,5] that the high-frequency component of the inductance (L (ω) ) of the inductor sitting on the now proton-treated substrate is proportional to the new defect-dipoles-modified relative dielectric coefficient ( ε r ) of the substrate (a more elaborate explanation is offered for this paper at the end of this section, or  see [4,5]).Accordingly, the proton-conditioned inductance L also explodes near the resonance frequency of ε r .The resonance of ε r will be elaborated in the following section.It thus occurred to the authors that if such observed explosive spikes in inductance can be arranged in a preprogrammed fashion so that an on-chip inductor is virtually cushioned frequency-wise by an array of dipole resonators of incrementally separated resonant frequencies, then the resultant inductance can be made very large, possibly to the extent inaccessible by any traditional VLSI means.
Prior to entering the main issue, a more elaborate explanation of the proposed theory [4,5] relating the dipolemodified substrate dielectric coefficient ( ε r ) and the highfrequency part inductance (L (ω) ) of the inductor on the former substrate is now due.Other than the already presented somewhat complicated mathematical demonstration involving solution of the system equations associated with a quasi-TEM approximation [4,5], here, rather, an explanation based on more intuitive physical arguments is given.First of all, a quasi-TEM wave traveling along the inductor metal line and its underlying substrate would excite dipoles (embedded within the substrate) of natural resonance frequency matching the wave frequency, in a fashion much like the resonant excitation among tuning forks.This newly established energy exchange channel then dominates the spectral redistribution of the inductor EM wave energy pattern by concentrating considerable proportion of the EM wave energy to around the resonance frequency, which is effectively the work of a trapping action imposed on an EM wave by a high ε r (or high-refractive-index) material.In other words, the reaction of the now resonantly excited dipoles (of a total polarization moment) would drag and slowdown the EM wave to facilitate such efficient energy exchange around the resonance frequency, and this effectively would enhance the magnetic flux linkage (in azimuthal direction with respect to the wave propagation vector), and thus inductance (L), around the resonance frequency.Secondly, as shown in [4,5], the scaling of the observed explosive rise in the inductance spectral behavior (see Figure 3) is around ω 4 , that is, with L (0) the low-frequency part of the inductance, the high-frequency component of the inductance is [4, 5] (quite unlike the relatively smooth rise of ordinarily evidenced inductance spectral behavior near the self-resonance frequency), which is the typical characteristic of dipole radiation oscillations [7].Additionally, effort of computer simulation on such inductor-dipole coupling is being planned, and the results will be published elsewhere.To this point, revelation of the cause of the explosive rise of substrate relative dielectric coefficient ( ε r ) is now in order.

Dipole resonance theory
When a host material is programmed with numerous dipoles of one kind, say, by normal front side proton bombardment (to create defect charge distribution-dipoles, as previously accidentally found) or better, by embedding of a selected type of molecular dipoles, an anisotropy is imposed upon the host substrate and manifested through its dielectric coefficient ε (say, in y direction, upward perpendicular to the inductor spiral).Let us pay attention to the y-dimension to which dipoles are assumed to dominantly align.Beneath the spiral inductor, each dipole is subjected to a time-varying electric field of the EM wave propagating along the spiral inductor and penetrating down to the substrate.The latter causes a time-varying displacement r to the bound electron relative to its positive counterpart within each dipole.The corresponding equation of motion for the bound electron is where m is the mass of electron, K is dipole restoring force constant, e is absolute electron charge, ν is a phenomenal electron collision frequency, and E = E + P b /(3ε 0 ) with E the applied electric field, P b the electric dipole moment per unit volume of dielectric, and ε 0 the vacuum electric permittivity.Note that it is this electric vector E that has to be used in dense media, that is, on the right-hand side of (2) in place of E (see [8] and references therein).Defining Ω 2 = K/m, as the squared dipole natural radian frequency, (2) can be rewritten in terms of the displacement vector r along the y direction as where Z ≡ ε y /ε 0 − 1 ≡ ε r − 1 and ε y /ε 0 ≡ ε r .Further manipulation of (4) gives The explosive behavior of the real part of ε r can be seen in ( 5) as ξ approaches 1, while y is very small.The limit value is usually obtained through numerical means (see Figure 4).Recall that the high frequency component of the inductance L (ω) of the inductor above such dipole-engineered substrate is proportional to ε r and thus the inductance L explodes near the resonance frequency of ε r (as proved in [4,5]).

Programmed matter with multiple resonances of dipoles
When a host material is embedded with more than one kind of dipoles, the overall modified dielectric equation (4), can be described by the well-known Clausius-Mossotti equation [8]: Frequency (Hz) The real part of dielectric constant Figure 5: Absorption coefficient and relative dielectric coefficient for a medium with more than one resonance (though the resonances are not very close to each other as proposed).
where f j is the number fraction of the jth type dipoles among all, and thus j f j = 1.It is to be noted that, normally, the real and imaginary parts of ε r are mathematically related by the well-known Kramers-Kronig integral relation (see, e.g., [9]).Namely, dispersion and absorption are intimately related.In the situations of interest here, however, the interrelationship between real and imaginary parts of ε r is already fully explicit as shown by ( 6) and therefore, effort for direct application of the more implicit Kramers-Kronig relation is saved.The important feature of interest here is that the imaginary part of ε r can be confined near the resonance frequencies, while the real part contributes at all frequencies near and below the resonances (see Figure 5 with arbitrary ordinate scale).Namely, away from the resonance frequencies, the real part of relative dielectric coefficient (i.e., ε y ≡ Re( ε r )) (or refractive index n = (ε y ) 1/2 for nonmagnetic materials) is constant and the medium is approximately nondispersive and nonabsorptive.More importantly, each resonance contributes a constant value to the refractive index at all frequencies smaller than the lowest resonance frequency [9] (see Figure 5 in arbitrary scale).Note that, for the overall relative dielectric coefficient, the final resonant curve similar to that in Figure 5 cannot be obtained by simply adding together each resonant curve associated with one dipole kind.It has to be secured rigorously through solving (6) numerically.Now, as our proposal, if these resonance frequencies Ω j 's can be arranged in such a way that they are very close to each other, then the distinctive feature of sharp peaks in Figure 5 will no longer be discernable, and correspondingly, ε y below and away from the resonances can be significantly enhanced.Accordingly, inductance of the spiral inductor residing on such a cushion of predefined dipoles can also be greatly enhanced, since L ∝ ε y as proved in [4,5] and explained in Section 1.

NUMERICAL SIMULATION RESULTS
Numerical simulation efforts were launched for an on-chip inductor sitting on 5 types of permanent dipoles embedded within a normal Si substrate, and the result is given in Figure 6, in terms of the functional relationship between the dipoles-modified relative dielectric coefficient (ε y ) versus frequency (since high frequency component of the inductance is proportional to ε y ) [4,5].The adopted parameters were total dipole number density N = 1 × 10 14 cm −3 with f j = 0.2 uniformly, where the first type of dipoles has its resonance frequency (Ω 1 /2π) set at 52.19 GHz, and then subsequent resonances incremented by 48.72 MHz were arranged for the rest 4 types of dipoles.The normalized attenuation (damping) coefficients y j (with respect to Ω j ) were all set to be 1× 10 −3 .
It was found that the ordinarily small and roughly constant relative dielectric coefficient (ε y ) of Si, around 12, could then be raised to about 250-500 in the approximately flat regime of around 3-6 GHz (i.e., away from resonances in Figure 6).Thus, using the proposed multi-dipole-cushion approach, the significantly increased real part of the relative dielectric coefficient ε y , together with the favorably small imaginary part ε rI away from the fast-varying region, can lead to super large inductances and extraordinarily high Q values for all on-chip inductors, according to the formula L ∼ L (ω) ∝ ε 1/2 y obtained in [4,5] (see (1)), where L (ω) is the high frequency component of the inductance associated with the spiral inductor lying on the cushion of dipoles.
More importantly, from the economic point of view, this also implies that it is possible to obtain normally needed inductance values of several nanohenries (nH) by on-chip spiral inductors of much smaller physical sizes.For example, for simplicity, if the inductance value of a traditional spiral is roughly proportional to the square root of its spanned area (see [3] for more accurate but complex calculations), then the same inductance may be achieved by a multi-dipolescushioned inductor of ∼50 times smaller.In other words, a traditional inductor spanning 500 μm × 500 μm can now be replaced with the new one spanning merely 70 μm × 70 μm, approximately.
Still, some may argue that the explosive rise of the dielectric coefficient of Si near certain frequencies should not be considered as the real contribution to the inductance of an originally metal inductor on silicon.However, the authors believe that it is what the network analyzer actually measures as a whole (including metal lines and dipoles-embedded substrate underneath) that really counts.In other words, from the widely used standard inductance measuring procedure and machinery, it is known that the network analyzer simply would never be able to distinguish between two pedagogical inductors of identical characteristics but of different topology and contents.Besides, it is how an inductor (in whatever form of practical desirability) will behave that's really relevant, and never its topology.

SUMMARY AND CONCLUSIONS
The verified success of proton bombardment treatment in both the device isolation and the inductor Q-improvement [1,2] (see Figures 1 and 2) on already-manufactured mixedmode IC wafers (prior to packaging) has also uncovered new phenomena, especially the explosive rises of inductance near certain frequency (or frequencies) [4,5] (see Figure 3).A previously proposed theory identified the cause to be resonant interaction between the inductor EM wave and the proton-caused defect electric dipoles [4,5].Based on such understanding, this paper aims at providing a new possibility of greatly enhancing the effectiveness of on-chip inductors via having them cushioned by multiple dipoles of small-step incremented resonance frequencies.These preprogrammed dipoles can be carefully designed permanent dipoles in the form of molecules oriented in the preferred direction(s), or could be achieved by employing other artificial ULSI molecules, such as tiny pn-junctions and Schottky diodes [10].Further, if active biases could be neatly applied upon these dipole molecules, their dipole restoring force constant K j 's and thus the natural resonance frequencies (Ω 2 j = K j /m) can be more actively tuned.Realization of this new concept may likely have to rely on the particle-byparticle construction manner of the coming more advanced nanotechnologies, including the self-assembly and the programmable atoms [11].

Figure 4 :
Figure 4: An example of explosive spectral behavior in the microstrip-perpendicular dielectric coefficient with respect to the normalized frequency (x = ω/Ω), where Ω/2π is the dipole natural frequency.

Employing
P b = N • (−e) • r (N being the volumetric dipole number density) and using the Fourier representation P b ∼ P b e iωt (with i = √ −1), the relation of displacement current D y = ε y E y (with quantities over-barred by ∼ standing for complex amplitudes) gives

Figure 6 :
Figure 6: Great enhancement in real part of the relative dielectric coefficient ε r , which would lead to very large inductance if the inductor size is kept unchanged.