Skin effect suppression for millimeter-wave frequencies through manipulation of permeability

A new device strategy for suppressing unwanted skin effect in coplanar transmission lines at frequencies beyond 20 GHz is presented. Utilization of the anti-resonance point as a source of imaginary permeability to suppress the skin effect is proposed. This idea is backed up by Kramers-Kronig and published experiments. The principle is independent of Landau–Lifshitz-Gilbert magnetic precession damping. The present work contributes both a greater understanding of skin effect manipulation and a method for obtaining zero skin effect using known laminate materials.


Introduction
Wireless communication frequencies are steadily shifting upward with the introduction of 5G. A variety of 5G links will experience a massive shift in frequencies from a few GHz to ∼24-40 GHz, and another set of physical effects to be incorporated in circuits [1]. In particular, skin depth will drop to dimensions that restrict the available wire cross section area for carrier transport in integrated circuits. The resulting increase in resistance cannot be ignored, and solutions are needed to suppress skin effect. Otherwise, information transfer rate will remain severely limited. Possibly for the first time, complex permeability μ, as opposed to the more common complex permittivity ò, is used as a method of skin effect suppression. Relation to experiments [2] and Landau-Lifshitz-Gilbert damping are presented.
The fundamental physics background for inspecting transmission line wire materials begins by discussing electromagnetic (EM) wave behavior as a function of òμ product. This product can be manipulated through various physical phenomena. Nominally ò manipulation is considered, but μ manipulation can have the same effect. Only μ manipulation is proposed to be used in skin effect suppression in this article. Caution is necessary here: it is tempting to consider Landau-Lifshitz damping in precession as a direct mechanism for this, but it does not change magnetization magnitude and turns out to be irrelevant. This complicated relation is discussed in detail in appendix B.
Incorporating ferromagnetic laminate layers is one method of changing the μ(ω) function near 28 GHz (which is a commonly-used 5G frequency and used here as an example). However, this only works when the magnetic field within the material is parallel to the layers. Thus, it is necessary to exaggerate the cross section aspect ratio such that both electric and magnetic field lines are dominated by one direction only. This results in the coplanar structure, which can be and has been built for experiment in the following figures. Proof of concept experiments show that the μ(ω) function shape does influence the skin depth and the total wire resistance. More detailed μ manipulation through ferromagnetic laminates will be discussed in future work.
òμ product as measure of damping Skin effect is a phenomenon that restricts current transport through conductors at AC frequencies [3,4] as in figure 1(d). Note that this is in a different direction from signal propagation, which is often perpendicular as in figures 1(a)-(c). For frequencies ∼28 GHz, because of the frequency dependence in d = wsm , j 1 this restriction becomes nm scale (d ∼350 nm for copper). This is a problem for current carrying wires like those found in transmission lines for microelectronics. The resistance of said wires increases dramatically if the usable cross section is reduced to a small fraction of the physical cross section. It is therefore vital to find a way to suppress skin effect without reducing the operating frequency. Experiments at UT Austin have used gridded conduction fibers to effectively reduce the thickness of the conductor and thus nullify the influence of skin effect, as in Ref. [5]. The key to finding such a method starts with an analogy to electromagnetic wave propagation through a conductor at high AC frequencies [6,7].
The relevant physical conditions for figure 1 can be written as follows.
Most materials are described by the left two image of figure 1(e) in terms of electromagnetic (EM) wave interaction. If òμ > 0, the EM wave velocity is real and the wave can propagate through the material. In this case, the electric (E) field is allowed to exist throughout the volume of the material. If òμ < 0 , the wave velocity is imaginary and the wave is forced into an evanescent mode. Previous theory work has considered negative permeability materials as a possible method of skin effect suppression, as in Ref. [8]. Any oscillating E field in the lateral direction can only exist where the exponential decay is not significant yet. The third image of figure 1(e) describes òμ < 0 as well, but | | em » 0 such that the decay length is as long as the material thickness. In this case, the EM wave is still evanescent, but the E field can exist through a greater volume of the material. This is one method of relaxing the constraints on the E field region within the material. The rightmost image of figure 1(e) introduces a hybrid state between evanescent and propagating called damped oscillation. In this case, òμ is complex, and some of the magnitude is spent on propagation rather than exponential decay. This is another method of relaxing E field constraints, and trades one difficulty for another. To achieve (c), either ò or μ must be close to zero. To achieve (d), either ò or μ must be imaginary. The above discussion applies to EM wave propagation. The skin effect, the main focus of this paper, pairs carrier current with magnetic field, i.e. σ with μ, as opposed to displacement E field with magnetic field as above. The same arguments apply regarding the products wsm j and òμ. If wsm > j 0 and real, the resulting current J is allowed to exist throughout the material. If wsm < j 0 and real, J can only exist near the incident surface. If wsm » j 0 or is complex, we have the in between case of damped oscillation. Since all materials considered for transmission lines have real σ, the phase of wsm j depends entirely on the phase of μ. This implies that the best condition for suppressing skin effect, i.e. allowing J to exist throughout the material and not just at the surface, requires imaginary μ. The second best condition is for | | m » 0. Skin effect becomes unavoidable for significantly large real μ values, whether positive or negative. The main problem with 28 GHz is the high resistive losses within the metal plates of the transmission line resulting from this high frequency. The available area for device current to flow in can be restricted to within 0.25 microns of the surface, as in equation (8). For standard microscale coplanar transmission line plate dimensions, this constitutes an 88% decrease in available cross-section area as compared to the physical plate cross-section area. This work proposes to eliminate this source of loss through complex permeability materials. Normally such strange magnetic properties can only be found for a narrow window of frequencies, and so we need Kramers-Kronig (relating real part of inductance to imaginary part of inductance) to help predict where these windows might exist for a given ferromagnetic laminate material used as a transmission line plate.

Sources of complex ò and μ
Now let us focus on how to achieve the aforementioned complex ò values. It is worth noting that the Kramers-Kronig relation between the real and imaginary parts of a complex function do not specify whether the real is even and imaginary is complex or vice versa. As seen from appendix A, the mathematical complex function α(ω) does specify real α as even and imag a as odd, but ( ) a w can be related to  ( ) w or ( ) m w by any integer power of I [3]. Thus we can have, for example, real  as odd, as can be seen in figure 2.
For a standard semiconductor, the  value at DC is a real positive value. Complex  is defined for AC operation. At near DC frequencies, only the real  value changes and there is no change to imag  = 0. One of the first major physical effects to change the  phase must therefore involve some kind of resistance, or equivalently energy absorption. A quick way to justify this intuition is to inspect the impedance of the material as / w j C 1 : an imaginary  component implies a real part to the impedance and thus an effective resistance. (e) If real òμ > 0, the EM wave propagates. If real òμ < 0, the EM wave is evanescent. If real òμ < 0 with a decay length longer than the material thickness, the EM wave appears to survive through the material. If complex òμ, the EM wave undergoes damped oscillation. Figure 2(a) focuses on a lower frequency effect that involves absorption: photon absorption in a semiconductor [9,10]. When the frequency is of sufficient magnitude such that w E , g  the semiconductor suddenly gains the ability to absorb the incoming EM wave by exciting an electron from valence to conduction band [11]. This results in a sudden step function increase to imag  . During the same frequency window, real  hardly changes because polarization is unaffected as a physical mechanism. This pair of functions observes Kramers-Kronig relationships in terms of even and odd functions. Imag  , as a step function, is odd. Real  , as a constant function, is even. Because we consider complex  and real m, This same principle that applies for permittivity ò should apply for permeability μ in the resonance and surrounding antiresonance (AR) point at ∼28 GHz for a ferromagnetic material such as the laminate material of this paper.
Such an argument can also be made for systems involving electrolytes [12][13][14][15][16]. Figure 2(b) focuses on a higher frequency effect, higher than the 28 GHz focus of this paper, that involves absorption: plasma frequency in a conductor. When the frequency is so high that screening in the semiconductor is too slow to respond, the conductor can no longer polarize in response to an incoming E field. This results in real  crossing the horizontal axis, i.e. real  = 0 and the system sees spontaneous E fields in the conductor. During the same frequency window, imag  hardly changes because the conductor can still absorb the incoming EM wave. This also observes the Kramers-Kronig even-odd relationships. Imag  , as a constant function, is even. Real  , which crosses real  = 0 and can therefore be seen as a linear function, is odd.
The skin effect frequency window falls in between figures 2(a), (b) and is outlined in figure 2(c). Unlike the previous examples, the parameter in focus is m instead of  . If the goal was to affect EM waves, any engineering could have been focused on  to influence m. However, because the goal is to affect skin effect, the engineering must focus on m to influence wsm j . Despite the change in material parameter, the same observations through Kramers-Kronig can be made for skin effect. Because we consider complex m and real  , Around 28 GHz, there is a possible resonance of magnetic fields given a special laminate material of Cu and some ferromagnetic metal. For the scope of this paper, an alloy of FeCo is used as the ferromagnetic metal. Real m is an odd function with equal but opposite peaks surrounding the central resonance frequency w . R Imag m is an even function with a negative peak exactly on the w . R Putting real m through equation (2) above recovers imag m and vice versa if equation (2) is split into its own real and imaginary parts.
The real m shape and its negative peak is most interesting for skin effect suppression. At the negative peak, the frequency is labelled as the anti-resonance (AR) frequency w .
AR The average value of real m hovers around the vacuum permeability m . 0 If the negative peak is sufficiently deep, real m at w AR can become negative. Recalling the conclusions in figure 1(a), total skin effect suppression is possible if real m = 0 and the appropriate imag m exists. Therefore if it is possible to engineer the negative peak such that real m = 0 for as wide a frequency window as possible surrounding w , AR it is safe to say the skin effect would be suppressed within this window. This may be achieved within a laminate material that incorporates layers of a ferromagnetic material (with negative m) and layers of a normal metal which has positive m m = 0 (and is chosen typically to be copper so that the conductivity is improved). For layer thicknesses that are smaller than the skin depth in either material, waves will respond to the spatial average m of the laminate incorporating both materials, as well as spatially averaged conductivity. The frequency at which the average m is near zero can be tailored by adjusting the layer thicknesses of Cu and FeCo.
Incorporating laminate material for complex μ We can start from the two rot based Maxwell's equations.
To be complete, the Maxwell's equations for skin effect must be applied inside the metal, i.e. the laminate ferromagnetic material [2,17]. They cannot be applied to the vacuum. Therefore, within the metal, all electric fields go to zero while electrons can transport current. Thus, respectively, displacement current is zero while conduction current is finite. This is in contrast to the vacuum, where electric fields are finite and electrons do not exist to transport current.  Alternatively, we can use Therefore, the skin depth common to both  E and  H becomes Let us consider Cu bulk to explain why we need FeCo laminate layers to engineer m. Commonly for Cu, s =´S For evaluation of the skin depth expression, the dimensional interaction between s and m is such that This leaves sec to be eliminated by s and /m 1 2 for the skin depth itself.
That is, at 28 GHz frequencies, the skin effect is expected to restrict the current carrying region to within m 0.25 m of the nearest surface. A practical coplanar structure with dimensions is shown in figure 3(a). If we use bulk Cu and do not consider m engineering, we run into severely increased wire resistance as in figure 3 2 This is an 88% reduction in available cross section for current transport, leading a roughly 8x increase in resistance.
As a direct result of the high 28 GHz frequency, there is additional and severe loss, due to a violently decreased area available to device current, that we call skin effect. This additional skin effect loss is area based and not material coefficient based like the standard resistivity losses of the dielectric and the plate. To combat skin effect loss, we consider unusual magnetic properties also near 28 GHz that involve complex permeability values, as in figure 2. We want to find an exact frequency window, and thus depend on the Kramers-Kronig relations which relate real and complex parts of any parameter including permeability. In addition to the skin effect contributions to loss, there will be resistivity as well as additional contributions due to surface or interface scattering, grain boundary scattering, and roughness scattering. These contributions are generally much smaller than the skin effect contributions for relatively thick metals at millimeter-wave frequencies.

Resistance measurements from skin depth affected by laminate material
In most materials near the 28 GHz frequency, conductivity s stays constant. In the skin depth expression, aside from the factor of w itself, only m depends on w. This makes the manipulation of skin depth a chiefly magnetic material property engineering process as in d wsm = j 1 . Skin depth is measured indirectly because of its influence on the total resistance of the transmission line conductor. Therefore, in addition to calculating a singular skin depth at a given frequency, it is possible to connect the discussion to resistance profiles as a function of frequency. Therefore, the plot in figure 4(a) shows the skin depth as a function of frequency. For the ferromagnetic laminate material, we can clearly see a valley and a peak. The valley is due to the positive antiresonance point, where m reaches its most positive value near the resonance frequency. This leads to nearly zero d, which means the skin effect is strongest and most undesirable. The peak is due to the negative antiresonance point. This leads to a maximized d value, meaning we have the desired suppressed skin effect. These changes to m will only take effect if (1) the material has ferromagnetic layers that are (2) correctly aligned (parallel) to the magnetic field.
FeCo FeCo FeCo Figure 3. (a) A schematic of the coplanar transmission line using laminate ferromagnetic materials as the conductor. At lower (< 1 GHz) frequencies, the coplanar lines can use their full cross section to support incoming current. The laminate material does not affect the magnetic field between the coplanar lines, which is the standard transmission line operation. Instead, it affects the magnetic field penetrating within the lines, which is related to the skin effect. (b) The red areas are current carrying regions unaffected by skin effect and should not be confused with the voltage wave of figure 1. The inset shows a particular snapshot of the wave where the central plane happens to be positive and the ground planes negative. The E field strength is denoted by the thickness of the field lines. At ∼28 GHz frequencies, the lines are subject to skin effect, which restricts current to a very thin surface layer and effectively decreases cross section area, thus decreasing current. In this case, the aspect ratio is wide as in figure 4(c). The insulating substrate suppresses the E field within it but does not otherwise change the E field shape. These result in figure 4(b): resistance as a function of frequency. There are three sets of conditions. For bulk Cu, the d shape from figure 4(a) suggests a monotonic increase in R, which is consistent with the trend shown aside from the jagged data near 15 GHz. For misaligned ferromagnetic laminate, i.e. NFM with H parallel to x-axis, we should see a similar d shape to bulk Cu, and thus we see another monotonic increase in R. For correctly aligned ferromagnetic laminate, i.e. NFM with H parallel to z-axis, the valley and peak d shape should take effect. We do see a corresponding peak and valley to R for this material alone. Other ferromagnetic laminate materials experimentally investigated for skin effect suppression include Al/NiFe, Cu/Co, and Cu/Ni, as in Refs. [18][19][20].

Transmission line geometric aspect ratio
To connect the m property discussion to a real world experiment, it is necessary to clarify the physical structure of the experimental device as it relates to the magnetic field. Just as we understood m through considering  first, let us examine the magnetic field by first considering the electric field. One of the most immediate concerns for device design is to specify the dominant electric field direction near the transmission lines [21,22]. If we know the AC electric field direction, we deterministically know the AC magnetic field direction. This represents the H field rather than the B field. This H field must oscillate in-plane with respect to the laminate ferromagnetic material layers. Otherwise the laminate material cannot affect m in the H field oscillation direction. The laminate material must affect m to suppress skin effect.
For the E field lines drawn in figure 5, the main rules are: Figure 4. Data taken from Ref. [2]. Experimental measurements displaying the effect of Re μ = 0 in a material, causing a peak in the decay length and thus a reduction in lateral resistance through suppression of skin effect. In this case, due to the Kramers-Kronig effect, Im μ≠0, and the EM wave ends up propagating rather than evanescent.
That is, the E field inside a conductor must be zero, and the conductor carriers must screen any external E field that tries to penetrate the conductor surface. Because the parallel E fields must be equal inside and outside the conductor and the E field inside the conductor must be zero, the E field outside the conductor must also be zero. This does not affect the perpendicular D field: inside the conductor, zero E field and infinite  results in a finite D field that can match the outside D field. These principles are most correct for a DC scenario, and they will apply while the frequency involved is below plasma frequency for the material.
It should be noted that figure 5 is drawn with a conductor backing, sometimes known as a metal ground plane, while figure 3 is not. The coplanar waveguide works as described in figure 3, but AC noise suppression can be achieved by introducing a large conducting plane [4].
It is important to note that these effects are classical, and the electrons are unlocalized. This is in sharp contrast to Coulomb blockade devices where electrons are connected through tunneling and a Coulomb island, which are necessarily nanoscale. These conditions would allow miniscule effective capacitances such that a single electron can be detected by a macroscopic voltage change (on the order of mV or greater) [23][24][25]. Figure 5(a) shows that for wires with a very wide and short aspect ratio, the E field is predominantly vertical. Because E and H must be perpendicular, this means that within the material, the H field is parallel to the laminate layers as desired. The closest approximation to this configuration is two plates parallel to but displaced from each other. These considerations are consistent with the Poynting vector and the transmission line signal propagation direction.
  E H does point out of the page in figure 5(a), consistent with the wire for the transmission line being infinite into and out of the page.
It is important to keep this highly asymmetric aspect ratio. If the wire cross section is too square, figure 5(b) results. The E field is no longer dominated by a single direction. Nearly equal numbers of lines move in the vertical and horizontal directions. Then the H field has components perpendicular to the laminate layers, nullifying any attempt at m engineering. Figure 5(c) does the same as (a) for a transmission line lying on a substrate. The E field is still predominantly vertical, and the only difference is that the E field below the line is suppressed within any insulating substrate. With the addition of the substrate, this configuration becomes a coplanar structure, and is commonly used in transmission line experiments. Because the E field is still vertical, the H field is within the plane of the laminate layers as in (a). Figure 5(d) shows the case when the aspect ratio is too square as compared to (c). Now that the E field below the line is suppressed by the insulator, the E field may actually be predominantly horizontal. Then the H field is mostly perpendicular to the laminate layers and will face an unengineered m. The skin effect suppression will not be effective in this case.
The substrate material choice must avoid shorting the transmission line and thus must be an insulator. The only restriction on the possible choices is that the permeability m must be vacuum permeability m 0 to reduce the loss tangent and thus minimize the overall attenuation. The dielectric constant  of the substrate will not affect the skin depth of the conductors. Its influence is restricted to coplanar transmission line vacuum properties, such as characteristic impedance or electric field distribution symmetry above and below the signal line.

Relation to Landau-Lifshitz damping
Can m control be described through other means besides a complex m value? Landau-Lifshitz precession is often cited as the definitive way to describe damped oscillation of a spin in a large external magnetic field [26]. However, it does not appear to describe energy loss. Instead, it describes alignment between spin and B field. To explain fully, it is necessary to inspect the mechanism described in equation (11) below from a visual standpoint in figure 6.
In opposition to the relationship between electric  D and  E fields, the input magnetic field is  H and the responses are magnetization  M and magnetic flux density  B. This is despite the first glance appearance of the expression This asymmetry has been covered in previous work [7]. figure 6(a) shows the case for undamped oscillation. Here, the magnetic dipole  M that represents the spin is always perpendicular to the external magnetic field  H , ext which is the bulk of effective magnetic field  H eff in equation (11a). The oscillation parameter g is set to some finite value. This controls the speed of the oscillation and must therefore be related to angular velocity. This undamped oscillation therefore represents the most basic scenario for spin-field interactions. From the beginning, no connection to skin effect is expected here.
damping 11 eff eff Figure 6(b) shows the case for damped oscillation, and if there is any relation to the skin effect, it would be apparent here. In equation (11b), the damping parameter l is no longer zero, which introduces a second direction to the equation of motion. The critical distinction here is whether the magnitude  | | M is changed by the damping term or not. If it is, the damping term is absorbing energy from the spin and dissipating it as heat, which To break down the damping term, it is necessary to use a formula regarding nested cross products. The identity most useful in this case is Because the damping term involves a negative sign, we apply the identity assuming This form, now made of dot products, can safely be reduced to differential form. Note that  H, representing external field, is a constant over time.
The main question is whether damping  dM and  M are perpendicular, which indicates constant This confirms that Landau-Lifshitz damped precession is indeed an alignment based phenomenon and not a power dissipation phenomenon. Therefore, despite being called damped oscillation, figure 6(b) is a perfect hemisphere because  | | M does not change throughout the damping. Conversely, complex m does involve energy dissipation, which puts it outside the scope of Landau-Lifshitz damping. In other words, skin effect cannot be explained solely by Landau-Lifshitz. Skin effect is related to B field damping. Landau-Lifshitz introduces precession and does not change the magnitude of the B field. Kramers-Kronig searches for imaginary permeability to lift true B field damping. It is necessary to discuss the nature of the unwanted skin effect if we want to be able to remove it. As a source of loss in the transmission line, the skin effect actively reduces the magnitude of both current and magnetic field simultaneously as we observe further into the bulk of the metal plate, away from the surface, as in figure 1. Here, we must apply a strong contrast between loss and the concept of Landau-Lifshitz damping, which points to very different physics. Landau-Lifshitz damping, as a precession phenomenon, does not discuss the loss of any magnitudes, but rather the misalignment of the magnetization and the external magnetic field as in figure 6. It is fundamentally different from the loss of skin effect.

Conclusion
Suppression of the skin effect in conductors is a device design principle for transmission lines in the 24-40 GHz frequency range. If a region of complex permeability can be found that is mostly imaginary, the skin effect will not exist because wave propagation is allowed. A 20 GHz plus device can therefore maximize its device current within this frequency window and avoid losing information. This window, according to Kramers-Kronig, can be found at the anti-resonance point and can be readily found from experimental data involving laminate ferromagnetic materials.
The skin effect damping is largely related to weakening of the B field within the conductor. Thus, to preserve the B field, complex m must be introduced to the conductor. The effect would be very similar to that on an EM wave incident on a material with complex  . Complex m may be achieved through special material structures such as laminates of ferromagnetic layers. In a coplanar transmission line, the layers would be parallel to the conducting plates, but not so tall as to introduce lateral E fields in vacuum. Landau-Lifshitz is irrelevant to the proposed damping of the skin effect as an engineering optimization model. This method can apply to any suitably engineered material, and is distinct from Landau-Lifshitz-Gilbert damping. The recommended structure is a laminate material used as the conductor in any transmission line structure to allow for signal transfer with zero information loss due to skin effect. This proposal affects permeability (magnetic) instead of the usual permittivity (electric) at 5G frequencies. Skin effect suppression would be useful for wiring between the antenna, the power amplifier, and the DSP circuits at 5G frequencies [27,28]. related to causality, but for the purposes of this paper we will not be discussing causality as it relates to information theory. Instead, we will discuss device related aspects as in figure A2.
In fact, the pole may only work as a quasi-delta function if there are no poles in upper half. w has to do with the input wave, while a is the material response to it. Use of Kramers-Kronig therefore implies a complex w in addition to a complex a. Since we use the upper half, the time domain and frequency domain pairs become  We could consider poles in the upper half for a non-causal system, but this implies energy generation somewhere in the material. This disqualifies it from being a passive element of the transmission line. Only an active element can have a spontaneous response before the input begins. However, at this stage, we are not ready to discuss a one-to-one correspondence between this theory and signal processing theory. We merely state that the transmission line materials discussed in this paper should be passive elements, so the  functions mentioned here should be differentiable in the upper half of the complex plane, without poles. For whatever reason,  ( ) t must be a causal function. This treatment is often done for wave packets with a proper time domain function. To break down the paths for the Cauchy theorem, Because this involves the small semicircle where the radius e  0, e q e i e i approaches 1. Please note the distinction between radius e and permittivity  . That is, the nested exponential function loses its q dependence. The remaining q dependence in the denominator disappears due to the Jacobian regardless of e, leaving a constant value. Thus the influence of the small semicircle involves the permittivity function and Cauchy's residue such that