Design, Modeling, and Analysis of a 3-D Spiral Inductor With Magnetic Thin-Films for PwrSoC/PwrSiP DC-DC Converters

A solution architecture for monolithic system-on-chip (SoC) power conversion is in high demand to enable modern electronics with a reduced footprint and increased functionality. A promising solution is to reduce the microinductor size by using novel magnetically-enhanced 3-D design topologies. This work presents the design, modeling, and analysis of a 3-D spiral inductor with magnetic thin-films for power supply applications in the frequency range of 3–30 MHz. A closed-form analytical expression is derived for the inductance, including both the air- and magnetic-core contributions. To validate the air-core inductance model, we implement a 3-D spiral inductor on PCB. The theoretical calculation of air-core inductance is in good agreement with experimental data. To validate the inductance model of the magnetic-core, a 3-D spiral inductor is modeled with Ansys Maxwell electromagnetic field simulation software. A winding AC resistance model is additionally presented. We perform a design space exploration (DSE) to investigate the significance of the 3-D spiral inductor structure. Two important performance parameters are discussed: dc quality factor <inline-formula> <tex-math notation="LaTeX">$(Q_{dc})$ </tex-math></inline-formula> and ac quality factor <inline-formula> <tex-math notation="LaTeX">$(Q_{ac})$ </tex-math></inline-formula>. Also, a 3-D spiral inductor structure with magnetic thin-films is characterized in Ansys Maxwell to estimate its potential, and a novel fabrication method is proposed to implement this inductor. The measured relative permeability (<inline-formula> <tex-math notation="LaTeX">$\mu _{r}$ </tex-math></inline-formula>) and the magnetic loss tangent (<inline-formula> <tex-math notation="LaTeX">$tan~\delta $ </tex-math></inline-formula>) of Co-Zr-Ta-B magnetic thin-films, developed in-house, are used to simulate the proposed structure. The promising results of the DSE can be easily extended to improve the performance of other 3-D inductor topologies, such as the solenoid and the toroid. The numerical simulations reveal that the 3-D spiral inductor with magnetic thin-films has the potential to demonstrate a figure-of-merit (FOM) that is significantly higher than traditional inductors.


22
Modern electronic devices demand increased functionality 23 coupled with device miniaturization. This highlights the chal-24 lenge of power delivery to these highly integrated devices, 25 where traditional approaches to point-of-load power con- 26 version (the converter is placed in proximity to the load 27 The associate editor coordinating the review of this manuscript and approving it for publication was Valentine Novosad. to provide a precise voltage) can no longer be applied. 28 In the case of point-of-load power conversion [1], [2], each 29 switching regulator requires discrete passive components 30 (i.e., inductor and capacitor). Specifically, the inductor is the 31 pain point due to its large footprint and height profile. The 32 ideal solution here is to have the power conversion circuitry 33 integrated directly with the load (e.g., CPU, SOC); that is, the 34 load and power supply are monolithically integrated. In other 35 words, we need to realize a system-on-chip (SoC) [see Fig. 1] 36 inductance density over a conventional 2-D inductor [11], 71 [12]. The structural difference between 2-D and 3-D induc-72 tors is illustrated in Fig. 3. Three-dimensional (3-D) induc-73 tors can be classified as in-substrate (also referred to as 74 in-chip or in-silicon) inductors and on-substrate (also referred 75 to as on-chip or on-silicon) inductors. Although in-silicon 76 inductors are challenging to fabricate, this type of inductor 77 makes use of unused substrate volume, resulting in a low 78 profile inductor, and hence, improved volumetric inductance 79 density. As the inductor footprint reduces, the DC ratio of 80 inductance to resistance (L dc /R dc ), also known as the DC 81 quality factor (Q dc ) [13], [14], decreases significantly [15]. 82 Inductors employed in PwrSiP [16], [17] and PwrSoC [4], 83 [17] carry current which is composed of AC (ripple current) 84 and DC components. DC resistance (DCR) is a major con-85 tributor to copper loss due to a large DC component in the 86 inductor current, which indicates that there is a necessity to 87 design inductors that exhibit lower DCR (R dc ) and in turn 88 higher Q dc . 89 As discussed in [19], the DCR of 3-D toroidal inductors 90 can be reduced by the following ways: (a) connecting vertical 91 conductors in parallel [see Fig. 4] and (b) increasing the 92 winding cross-section. However, the traditional 3-D inductor 93 structure such as a toroid allows a limited number of parallel 94 connections and a small increase in the winding cross-section 95 due to the footprint constraint. As a result, the improvement 96 in DCR is not satisfactory. Also, the parallel connection of 97 vertical conductors requires a trapezoidal-shaped bottom and 98 top radial conductors [see Fig. 4(b)]. A trapezoidal-shaped 99 conductor does not have a closed-form analytical solution 100 for the inductance, which makes it challenging to design 101 the inductor. The characterization of non-standard structures 102 such as trapezoidal-shaped conductors in finite element anal-103 ysis (FEA) takes a significant amount of time and memory. 104 Another disadvantage of parallel connection is the increase 105 of parasitic capacitance. increase, and (b) the second approach increases the interwind-114 ing capacitance. 115 In [15], [21], and [22], the study has been carried out to 116 optimize the Q dc of on-chip inductors. These are magnetic-117 core inductors. The Q dc of magnetic-core inductors is bet-118 ter than air-core inductors due to enhanced inductance 119 contributed by the magnetic-core. However, the integra-120 tion of a magnetic-core has following disadvantages: (a) it 121 involves core loss [23] (b) it is challenging to integrate the 122 core [24], [25], and (c) it is difficult to introduce anisotropy 123 into the core [26], [14]. In the previous research on 3-D 124 inductors [5], [27] [see Fig. 5], a magnetic-core is integrated 125 to improve the inductance density. However, these inductors 126 are not suitable to operate at high-frequency (HF)   127 as the core is solid (non-laminated); they suffer from low 128 Q ac (2πfL ac /R ac ) at HF due to large eddy currents. In [24], 129 even though the laminated core is embedded, the inductor 130 occupies a large footprint. This implies the necessity of a 3-D 131 spiral inductor structure where it is relatively easy to integrate 132 a thin core and induce anisotropy while maintaining high Q ac 133 and Q dc . 134 VOLUME 10, 2022  [31], and 153 (c) removing the silicon core partially or completely [19] [see sented. In addition, analytical equations are presented for the 174 FIGURE 7. Substrate loss reduction in 3-D TSV inductors by means of (a) microchannel shields [12] ( 2015 IEEE) (b) a complete removal of silicon core [19] ( 2018 IEEE). resistance and inductance of three-quarter and full turns with 175 circular and rectangular cross sections, 3) a novel fabrication 176 method is proposed for a magnetically-enhanced 3-D spiral 177 microinductor with radially oriented magnetic materials in a 178 unique quadrant topology, 4) we discuss the advantages of 179 using single-layer thin-films and a high resistivity substrate, 180 as well as the significance of working in the frequency range 181 of 3-30 MHz and the potential impact of EMI in the given fre-182 quency range. Furthermore, Table 1 summarizes the benefits 183 of a 3-D spiral inductor with thin-films as compared to a 2-D 184 spiral inductor with thin-films.

186
In this work, the traditional spiral inductor structures such 187 as the planar and the two-layer are modified to improve 188 Q dc and Q ac . The goal is to maximize volume coverage, 189 that is, increase the dimensions of the winding conductors 190 to lower the DCR. Also, the distance between the opposite 191 current carrying vertical conductors is increased to reduce the 192 number of conductors, which in turn reduces DCR without 193 significantly affecting the air-core energy storage density of 194 the inductor. The winding conductors are arranged so that the 195 direction of currents in all the windings are the same, which 196 contributes to positive mutual inductance. Also, this strategic 197 arrangement of conductors reduces substrate loss as most of 198 the magnetic field lines due to vertical conductors cancel 199 each other inside the substrate except the strong H -fields near 200 the vertical conductors, details are discussed in section III. 201 Another important reason for the reduction in the substrate 202 loss is the magnetic field direction of vertical conductors is 203 parallel to the substrate plane. Thin-film magnetic-cores with 204 a thickness less than the skin depth are integrated to improve 205 the inductance density. Unlike 3-D solenoidal and toroidal 206 inductors (winding surrounds the core), the role of conductor 207 and core is switched in 3-D spiral inductor structures with 208 magnetic thin-films (core surrounds the winding) to optimize 209 the performance. This novel approach offers a low reluctance 210   Table 1.

222
In this paper, we propose a reduced frequency of operation  Table 4], which means that the example converter 244 can operate in CCM if the proposed inductor is employed for 245 energy storage. Larger inductance results in lower ripple and 246 hence reduction in RMS value of inductance current, which 247 leads to higher efficiency. In other words, the 3-D spiral 248 inductor with magnetic thin-films can serve as an energy 249 storage element for dc-dc converters while maintaining high 250 FOM.
A limitation of the 3-D spiral inductor structure is that the 253 fluxes are not confined, which leads to magnetic interference 254 VOLUME 10, 2022 (radiated EMI) with other circuits. If the stray fields (external 255 magnetic field) that it produces can be mitigated, it could be 256 a promising candidate for integrated power supply applica-257 tions. A brief introduction to the concept of EMI and mitiga-258 tion methods is provided below.

259
There are two types of EMI: conducted and radiated.

260
Radiated EMI is a concern only above 30 MHz  [44]. The maximum frequency that we 270 propose in this paper is 30 MHz. As a result, the 3-D inductor 271 with thin-films can be employed in switching converters 272 without imposing potential harmful effects of radiated EMI.

273
The paper is organized as follows: after the introduction,   The internal inductance of a circular pillar (L circ,in ) [45] and 305 a rectangular pillar (L rect,in ) [46] are given by (2) and (3), 306 FIGURE 9. Impact of magnetic-core conductor radius on L dc , R dc , and Q dc .  (2) 309 where l cp is the length of the circular pillar, l rp , α, and β 314 are the length, width, and thickness of the rectangular pillar, 315 respectively and µ o (4π × 10 −7 H /m) is the permeability of 316 free space. from previous publications [5], [17], [47]. From the graphical 322 study, we see that L rect,in reaches a maximum value when 323 the width (α) becomes equal to the thickness (β). Therefore, 324 we will use square conductors (pillars and interconnects) as

342
Observation 1: The internal inductance of a circular pillar 343 is independent of the radius. On the other hand, the internal 344 inductance of a rectangular pillar is a maximum when the 345 width (α) is equal to the thickness (β), i.e., a square pillar.

347
Here we study the Q dc of pillars with rectangular and circular 348 cross-sections. Simplified analytical formulas are considered 349 to develop an easy understanding of the relation between Q dc 350 and the conductor dimensions. However, the exact formulas 351 are used for presenting data in graphical form. The exact 352 formulas are given in Appendix I. 353

354
Simplified formula for the self-partial inductance (L circ ) [48] 355 and resistance (R circ ) of circular cross section pillars are given 356 by (4) and (5), respectively, where r is the radius. From (4) and (5) section for the study for the reason mentioned earlier. values were used for these and the following percentages.

393
Observation 3: The Q dc of a rectangular pillar is 394 27.36-29.15% higher than a circular pillar for approximately 395 the same footprint and length. The comparisons are made 396 at the lowest and highest Q dc values, which occurs at the 397 smallest and largest of pillar volumes, respectively. In this 398 study, the smallest volume corresponds to r = 5 µm, l cp = 399 200 µm and the largest volume corresponds to r = 200 µm, 400 l cp = 500 µm for circular conductors. Similarly, the smallest 401 volume corresponds to α = β = 10 µm, l rp = 200 µm and 402 largest volume corresponds to α = β = 400 um, l rp = 500 µm 403 for rectangular conductors. To estimate the Q dc , we need to compute the inductance 412 as well as the resistance. The inductance of two parallel 413 connected circular pillars, L pac , is half of the sum of the self-414 partial inductance of a single circular pillar, given by (4) and 415 the mutual partial inductance between pillars, given by (8). 416 The expression for L pac is given by (9). The net resistance of 417 parallel circular pillars R pac is half that of the single circular 418 pillar, which is given by (10). The plot of Q dc versus radius 419 is shown in Fig. 15(a).
where l is the length of a conductor (a pillar or an intercon-422 nect) and p is the center-to-center distance between conduc-423 tors (pillars or interconnects).  The gap between two parallel pillars is assumed to be 5 µm, 427 which is determined from the critical dimension (CD) [50].

428
It should be mentioned that the equation in [50] is a first 429 approximation. As reported in [51], the CD is a function of   (11) and (12), respectively. The plot of Q dc versus length 437 and width is shown in Fig. 15(b).

438
L par = 0.5 L rect (l rp , α, β) + M (l rp , p) 440 Observation 4: The Q dc of parallel pillars is greater than a 441 single pillar. This is true for circular and rectangular pillars. The schematic of a three-quarter-turn with circular pil-450 lars is shown in Fig. 12(a). The inductance of a three-451 quarter-turn with circular pillars, L 0.75TCP , is equal to the 452 sum of the self-inductances of the pillars plus the self-453 inductance of the interconnect minus twice the mutual 454 inductance between the pillars and is given by (13). The 455 mutual inductance between a pillar and an interconnect is 456 assumed to be zero as the magnetic fluxes produced by them 457 are orthogonal to each other. The DCR of a three-quarter-turn 458 with circular pillars, R 0.75TCP , is given by (14). Variation of 459 Q dc of a three-quarter-turn with circular pillars versus pitch 460 for four different lengths of pillars is shown in Fig. 16.

465
The schematic of a three-quarter-turn with rectangular pil-466 lars is shown in Fig. 12(b). Similar to the previous case, pillars is shown in Fig. 17.  The sign of mutual inductance is negative as the currents flow 507 in opposite directions. The DCR of a full-turn with circular 508 pillars, R TCP , is given by (18). Variation of Q dc of a full-turn 509 with circular pillars versus pitch for four different lengths is 510 shown in Fig. 18.
The schematic of a full-turn with rectangular pillars is shown 516 in Fig. 12(d). Similar to the previous case, the inductance 517 (L TRP ) and resistance (R TRP ) for a full-turn with rectangular 518 pillars are given by (19) and (20), respectively. Variation of 519 Q dc of a full-turn with rectangular pillars versus pitch for four 520 different lengths is shown in Fig. 19.   The results of the above study demonstrate that the Q dc of  Fig. 20.

580
(ii) The pillars of the windings are placed along x and 581 y axes. The distance between two consecutive pillars 582 along the axes is S, as shown in Fig. 22.

583
(iii) Energy stored in the proposed inductor is in two forms: 584 air-core energy and magnetic-core energy. Intercon-585 nects contribute significantly to the air-core energy 586 while contributing negligibly small to the magnetic-587 core energy. On the other hand, pillars contribute signif-588 icantly to the magnetic-core energy while contributing 589 negligibly small to the air-core energy. To simplify 590 the analysis, we can consider an ideal situation where 591 interconnects contribute 100% to the air-core energy, 592 and pillars contribute 100% to the magnetic energy.

593
(iv) The energy storage volume can be segmented into four 594 quadrants, and the magnetic field is along the radial 595 direction, illustrated in Fig. 21. The magnetic fields due 596 to pillars in opposite quadrants nearly cancel each other; 597 that is, the magnetic fields in the first and third quad-598 rants, as well as the second and fourth quadrants, are 599 in opposite directions. The H -fields which are imme-600 diately around the copper pillars do not get completely 601 cancelled. Consequently, the B − field lines due to the 602 pillars can be enhanced by wrapping high permeability 603 magnetic materials around the pillars, which contributes 604 to the inductance. Also, wrapping the thin-films imme-605 diately around the pillars offers a low reluctance path, 606 We derive an analytical expression for the winding inductance 630 (air-core inductance) of the 3-D spiral inductor structure. 631 We make use of the basic formulas given in Appendix A. 632 The inductance of this inductor structure is the summation 633 of the self-inductances of all N -windings and the mutual 634 inductances between all pairs of windings. The physical 635 structure used to compute the inductance is shown in Fig. 22. 636 To simplify the process of analytical modelling, the windings 637 are assumed to be completely closed, and each winding is 638 excited separately [54]. The current direction in all windings 639 is the same: either clockwise or anticlockwise.

640
The dc winding inductance L w,dc of the novel inductor with 641 N windings can be expressed as follows: The first term, L w i , corresponds to the self-inductance of 644 the ith winding. The second term, M w i −w j , represents the 645 mutual inductance between the ith and jth windings.

646
The self-inductance of the ith winding is given by Similarly, the mutual inductance between the ith and jth 650 windings is The basic idea of the computations of the self and mutual

667
(ii) A thin-film magnetic-core is not present.

668
(iii) The current is distributed uniformly in the conductor; 669 that is, the skin and proximity effects are not present.

670
Next, we consider high-frequency effects, such as the skin 671 effect, on the inductance. The proximity effect, which is one 672 of the high-frequency effects, is not considered for the reason 673 explained in the next section. Due to the skin effect, the 674 high-frequency current tends to be crowded in an annulus 675 (thickness equal to the skin depth) at the surface of the 676 conductor. Consequently, no internal current is linked by the 677 field. As f → ∞, the internal inductance vanishes (i.e., 678 L in → 0). The inductance value with the skin effect is given 679 by the following equation: where L in is the internal inductance of the proposed inductor. 682 The expression for L in is given by In the frequency range of 3-30 MHz, the skin depth is very 685 high, and hence, the effect of skin effect on the winding 686 inductance is negligibly small. Consequently, the AC winding 687 inductance is approximately equal to DC winding inductance, 688 that is, L w,ac ≈ L w,dc .

689
Equation (21) does not consider the inductance contribu-690 tion from thin-film magnetic-cores. The integration of mag-691 netic thin-films into the proposed inductor structure and the 692 analytical model for the inductance due to the magnetic thin-693 films are discussed in section IV. 694 VOLUME 10, 2022 where f is the operating frequency, ρ is the resistivity, and 718 µ is the permeability. The permeability can be written as the 719 product of the relative permeability µ r and the permeability 720 of free space, µ o = 4π × 10 −7 H /m: µ = µ o µ r .

721
The high frequency resistance of pillars and interconnects, 722 derived in [56], is given by (29 and (30   The main loss mechanisms involved in magnetic thin-films 776 are hysteresis loss, eddy current loss, and anomalous loss [5]. 777 The eddy current loss can be classified as conduction eddy 778 current loss and displacement eddy current loss. Multi-layer 779 film stacks (multi-layer magnetic thin-films) are the source 780 of displacement eddy current loss. In the case of single-781 layer magnetic thin-film, displacement eddy current loss is 782 absent. Consequently, the losses associated with the single-783 layer magnetic thin-film are hysteresis loss, conduction eddy 784 current loss, and anomalous loss.

785
The thickness of each layer used in 3-D spiral inductor 786 structure is 1 µm, which is much less than one skin depth 787 at 30 MHz. The skin depth of Co − Zr − Ta − B is 5 µm 788 at 30 MHz. The measured relative permeability considered 789 for the skin depth calculation is 500, and the resistivity of 790 Co − Zr − Ta − B is 115 µ · cm.

792
The inductance due to magnetic thin-films can be derived 793 using a basic magnetic circuit approach. The flux φ mag in the 794 thin-film magnetic-core is given by words, (36) can be employed to find the inductance due 813 to thin-film magnetic-cores without appreciable error in the 814 frequency range of 3-30 MHz; that is, L mag,ac ≈ L mag,dc .

816
To verify the proposed model for winding inductance, an air-817 core 3-D spiral inductor is implemented on PCB and electri-818 cally characterized with an HP4285A precision LCR meter 819 (75 kHz-30 MHz). The specifications and validation results 820 are presented in Appendix IV.

821
Similarly, to verify the analytical model for the inductance 822 of magnetic thin-films and understand the potential of 3-D 823 spiral inductors with magnetic thin-films, we constructed and 824 characterized (small-signal) in Ansys Maxwell. The specifi-825 cations considered for the simulation are given in Table 1. 826 The resistivity of the silicon considered for the simulation 827 is 1 k · cm. The total DC inductance is the sum of the 828 inductances due to the windings and the magnetic thin-films, 829 and is given by (37).

835
The total loss in any inductor is the sum of the winding 836 loss and core loss [5].  Q ac at low frequencies can be described by ωL ac /R ac . In this 858 expression, ω (2πf ) is the angular frequency.

859
The small-signal performances can be accurately evaluated 860 using the FEA. The performance of an inductor can be eval-861 uated using a figure-of-merit (FOM), which is defined by [5] 862 as, where V is the volume of the inductor. In Table 4, the FOM of 865 the 3-D spiral inductor with magnetic thin-films and induc-866 tors from previously published works are compared.

867
In Table 4, for the computation of Q ac (2πfL ac /R ac ) 868 at 30 MHz, the calculated resistances (FEA solutions) corre-869 sponding to the winding loss, eddy current loss, and hystere-870 sis loss are 188.54 m (R w,ac ), 94.43 m (R e ), and 49.75 m 871 (R h ), respectively. As discussed in section V, R ac is the sum 872 of all these three resistances and is equal to 332.72 m . The 873 hysteresis loss is estimated by using magnetic loss tangent 874 (tan δ) measured on a single-layer CZTB sample of 1 µm 875 thickness; a sample of size 4 mm × 4 mm × 0.7 mm (including 876 the silicon substrate) is considered. The measured tan δ is 877 0.02 at low frequency where eddy current loss is negligibly 878 small. The loss tangent tan δ is plotted in Fig. 24.

880
In this paper, we report the closed-form analytical model for 881 the inductance of a 3-D spiral inductor with magnetic thin-882 films. The analytical expressions for the inductances of the 883 winding and the magnetic thin-films are validated against 884 measurements and FEA solutions, respectively. The values 885 predicted by the inductance models of the winding and the 886 magnetic thin-films agree with the experimental and FEA 887 simulation results to within 5.63% and 5.29%, respectively. 888 Also, the analytical expression for the winding resistance 889 (R w,ac ) is derived. The values predicted by the expression are 890 validated against simulation results in the frequency range of 891 3-30 MHz; the maximum error reported is 5.5%, as illustrated 892 in Table 3.

893
The design space is graphically described. An exten-894 sive analysis is carried out to evaluate the DC performance 895 TABLE 4. Small-signal performance of a 3-d spiral inductor with magnetic thin-films compared with the previously published works to demonstrate its potential for power supply applications.
(i.e., Q dc ) of 3-D inductors with respect to variation of 896 parameters such as winding dimensions, pitch, height, etc.

897
The data is presented in graphical form. With the presented 898 data, the determination of DC performance (i.e., Q dc ) of 3-D 899 inductors with respect to variation of parameters is fairly easy, 900 which helps in design and fabrication to achieve optimum 901 performance. 902 We construct a 3-D spiral inductor with magnetic 903 thin-films in Ansys Maxwell to understand its potential 904 (small-signal performance) when compared with previously 905 published works. We also discuss the loss mechanism asso- Similarly, the exact expression for the self-partial induc-924 tance of a rectangular pillar having length, l rp , width, α, and 925 thickness, β, is given by where l is the length of a conductor (a pillar or an intercon-933 nect) and p is the center to center distance between conductors 934 (pillars or interconnects).

937
The constituent components of (22) are represented in 938 Fig. 25. The signs of the mutual inductances depend on the 939 current directions. Dots and crosses are used to indicate the 940 pillar current directions as shown in Fig. 25.

966
Next, we compare the analytical model, the FEA simula-967 tion, and the measurements in Table 6.

969
The authors would like to thank the members of the Inte-