A Geometrically Scalable Lumped Model for Spiral Inductors in Radio Frequency GaN Technology on Silicon

Abstract

This paper presents a lumped scalable model for spiral inductors in a radio frequency (RF) gallium nitride (GaN) technology on silicon substrate. The model has been developed by exploiting electromagnetic (EM) simulations of geometrically scaled spiral inductors. To this aim, the technology substrate, i.e., the metal back-end-of-line along with dielectric and semiconductor layers of the adopted GaN process, has been validated by means of experimental data and then used to define the EM simulator set-up for the spiral inductors. The proposed model adopts a simple π-topology with only seven lumped components and predicts inductor performance in terms of inductance, quality factor (Q-factor) and self-resonance frequency (SRF) for a large range of geometrical parameters of the spiral (i.e., number of turns, metal width, inner diameter).

1. Introduction

Radio frequency (RF) integrated circuits (ICs) extensively employ inductive components (i.e., inductors and transformers) to implement crucial functionality, such as 50-ohm impedance matching at the input/output chip terminals, impedance matching in multi-stage amplifiers, tuned resonant load in the RF amplifiers, LC tank in voltage-controlled oscillators, LC filters, etc. [1]. For this reason, modern RF silicon technologies adopt a proper back-end-of-line (BEOL) with thick metals and thick intermetal oxides to reduce the resistive losses and the parasitic capacitances, respectively [2]. Typically, RF silicon technologies provide general purpose inductor devices into the process design kit (PDK). However, scalable models or stack-up descriptions of electromagnetic (EM) simulations are required for a customized inductor design [3,4].

In the last few years, GaN-based High-Electron-Mobility Transistors (HEMTs) have demonstrated excellent performance for high-power applications thanks to the wide bandgap and the crystal structure of GaN, which allow achieving both high voltage and high currents in a wide temperature operative range. Despite the vast literature available for GaN-based technologies, very few papers deal with inductive components (i.e., inductors and transformers) in RF GaN processes, either on silicon [5,6] or on SiC substrates [7,8,9]. Moreover, to the authors’ knowledge, very scarce literature is available for inductor modeling in GaN technologies since only in [6] has a lumped scalable model been proposed, whereas [9] presents an equivalent circuit extracted for a single geometry. Therefore, unlike silicon-based inductors and transformers [10,11,12,13,14,15,16,17,18], there is a lack of knowledge on the analysis, design and modeling of inductive devices in GaN technologies. This study aims to partially cover this gap.

This paper is organized as follows. Section 2 briefly describes the adopted RF GaN technology on a Si substrate. Section 3 summarizes the physical phenomena taking place in integrated spiral inductors with the aim of providing the reader with the main design guidelines. The proposed scalable lumped model for spiral inductors in RF GaN technology is detailed in Section 4, while its validation is reported in Section 5. Finally, the main conclusions are summarized in Section 6.

2. RF GaN Technology

The adopted technology is an RF GaN process fabricated on a thin silicon (Si) substrate, which has been developed by STMicroelectronics mainly for sub-6 GHz band power applications [19,20,21,22]. The process features 50 V HEMTs with a gate length of 0.5 µm. Since the technology has been specifically addressed to RF applications, transistors have been optimized for normally-on depletion operation, thus exploiting the total two-dimensional electron gas (2DEG) density at the AlGaN/GaN heterointerface. The technology also provides resistors and high-quality-factor (Q-factor) metal–insulator–metal (MIM) capacitors for the integration of biasing and matching networks, respectively, while allowing for the fabrication of polygonal/circular spiral inductors and interleaved/stacked transformers that are essential for advanced RF ICs [23,24,25,26,27,28,29]. To this aim, the technology BEOL provides three AlCu metals, two thick layers at the top of the stack, which are used for the inductor coil and the underpass, along with a lower thinner one to access the transistor terminals. The upper metal layers, namely MTL3 and MTL2, have equal thickness to avoid Q-factor degradation on one side and a bottleneck for the maximum current due to the underpass connection on the other side. Moreover, the two thick metals are suitable for the implementation of high-performance integrated transformers, especially for staked configurations. For the sake of completeness, Figure 1 shows the Scanning Electron Microscope (SEM) microphotograph of the upper metal layers (i.e., MTL3 and MTL2) as they are connected in the underpass region of a spiral inductor.

The adopted RF GaN technology enables the design and implementation of watt-level sub-6 GHz power amplifiers (PAs) for base station applications. The final target is replacing the traditional micromodule approach based on discrete components with a fully integrated system, thus achieving a significant reduction in cost and size. This study has been developed in the framework of the development of the above-described RF GaN technology with the aim of driving the implementation of a geometrically scaled library of spiral inductors for RF applications, while providing a reliable design tool for inductor customization into the PDK of the technology.

3. Integrated Inductor Physics: Design Guidelines

The design of integrated inductors is aimed at increasing both the Q-factor and the self-resonance frequency (SRF) of the component by minimizing the energy losses and the parasitic capacitances, respectively. Energy dissipation occurs into both the metal windings and the underneath layers (i.e., oxides, insulators, diffused semiconductors, etc.). On the other hand, the operative frequency range of the inductor is limited by the interwinding capacitances and especially by the area/perimeter capacitances of the spiral toward the substrate.

Figure 2 depicts the main EM phenomena taking place in an integrated spiral inductor [1]. The losses in the metal layers are mainly due to the current crowding, i.e., the non-uniform distribution of the current in a conductor, which is particularly exacerbated by the peculiar structure of a spiral inductor. Indeed, the current crowding is higher on the internal sides of the spiral where the magnetic field is at its maximum. This phenomenon is the result of the superimposition of the skin and proximity effects, and rises with the increase in the operative frequency, thus resulting in the frequency dependency of the inductor resistance [30]. Therefore, thick and highly conductive metal layers are mandatory to reduce the inductor series resistance, especially in the lower RF bands, since the effectiveness progressively degrades at higher frequencies. On the other hand, the use of “hollow” spirals is a common rule for high-frequency inductive components to avoid the losses in the inner turns [30,31].

Another significant impact on the performance of an integrated inductor is also due to the substrate losses. Indeed, both electric and magnetic fields are significant in the layers underneath the inductor. Specifically, two different mechanisms take place, as represented in Figure 2 using blue and green arrows. The vertical currents (blue arrows) injected into the substrate are the displacement currents (i.e., due to the electrical field and therefore related to the capacitive effects), while the horizontal currents (green arrows) flowing into the conductive substrate layers are magnetically induced currents (i.e., due to the time-varying electrical field according to the Faraday–Lenz law). Both electrically and magnetically induced currents increase at radio frequencies and their effect can be dominant with respect to the ohmic losses into the spiral, especially for large inductors. Specifically, in GaN technology, the bidimensional electron gas in the AlGaN/GaN heterojunction must be properly neutralized below the inductor footprint to avoid the flowing of magnetically induced eddy currents and consequent energy losses [22].

An important role in enlarging the operative frequency range of an integrated inductor or equally in increasing the maximum inductance value that can be used at a given working frequency is played by the minimization of the parasitic capacitances. Typically, the main contributions are due to the electrical coupling with the substrate, especially when its thickness is reduced for thermal reasons, such as in the adopted GaN technology. On the other hand, a thinned substrate helps in reducing the losses due to the displacement currents flowing toward the ground. The main parasitic capacitances are related to the overlap regions between the spiral and the underpass, especially for large-width metal inductors. When the metal spacing is higher than a few microns, the fringing winding capacitances are lower, but their contribution could become significant for inductors with a high perimeter/area ratio (i.e., low w, high n and d_IN).

4. Geometrically Scalable Lumped Model Description

The model has been developed to predict the electrical performance of circular inductors for a large range of geometrical parameters to comply with circuit designer demands in RF IC optimization. A circular shape is preferred since it guarantees a better Q-factor performance than squared or polygonal shapes [1]. Although in nanoscale CMOS, the circular shape is forbidden [2,27,28,29], in the adopted GaN process, circular spirals can be manufactured without any process problems, which do occur in other technologies with wider lithography [16,32,33]. Due to the lack of a sufficiently high number of available fabricated inductors, model parameter extraction has been carried out by using 2D EM simulations of geometrically scaled circular inductors in the adopted GaN technology. To increase the accuracy of the simulations, the process substrate (i.e., the metal back-end-of-line along with dielectric and semiconductor layers of adopted GaN technology) has been validated by means of experimental data of microstrips fabricated in the three metal layers (i.e., MTL1, MTL2 and MTL3).

Figure 3 shows the micrograph of a MTL1 microstrip fabricated on the adopted GaN technology on a 60-µm thick silicon substrate. To achieve an accurate estimation of both the inductance and Q-factor, a frequency dispersity model of the silicon substrate has also been included in the EM simulations. Specifically, the Svensson/Djordjevic model has been used [34]. The comparisons between measured and EM-simulated scattering parameters, S₂₁ and S_11, are reported in Figure 4 for the sake of completeness. The EM simulated S-parameters are in good agreement with the measured ones over a wide frequency range, especially for S₂₁. The errors are quite low and largely acceptable for our purpose.

As an example, a three-turn circular spiral inductor with the main geometrical parameters (i.e., the metal width, w, the inner diameter, d_IN, and the metal spacing, s) is shown in Figure 5.

The proposed model adopts a simple π-topology, shown in Figure 6, which employs only geometrically scalable circuit parameters. The ideal inductor, L, is used to model the spiral and underpass inductances, as well as the magnetic coupling within the substrate. Two capacitors, C_P1 and C_P2 (whose values are different due to asymmetric inductor layout), are adopted considering the capacitive effects toward the substrate. Moreover, a further capacitor, C_S, is exploited to account for other capacitive effects. Specifically, parasitic capacitances throughout the inductor turns along with the overlap between the spiral and the underpass are considered. Finally, ohmic and substrate losses are modeled by means of frequency-variable resistances, R_S and R_P, respectively.

The inductance value, L, is the sum of three contributions,

$$L=L_{COIL}+L_{UND}+L_{SUB},$$

(1)

where L_COIL and L_UND are the inductances of the spiral and the underpass, respectively, while L_SUB is used to account for the induced magnetic field in the substrate in a simple way [35,36,37]. The first two terms of (1) can be calculated by using well-known closed form expressions available in the literature [38,39], whereas L_SUB has been evaluated by using the following monomial formula:

$$L_{SUB}=-4\pi \alpha n^2{d}_{AVG}^{2}w$$

(2)

where d_AVG is the average inductor diameter and α is a fitting coefficient of around 0.92.

It should be noted that L_SUB is a negative value since a reduction in the total inductance is expected due to the induced magnetic field within the substrate.

Capacitive contributions (i.e., C_S and C_P1,2) determine the inductor SRF and can be extracted from EM data. The following scalable equation has been used for C_S:

$$C_S=\frac{1}{N_{OVER}}\frac{{\epsilon }_M{w}^2}{t_M}p(n)$$

(3)

where N_OVER is the number of overlaps between the spiral and the underpass (i.e., N_OVER = n + 1.5), and ε_M and t_M are the electric permittivity and the thickness of the intermetal (i.e., MTL3-MTL2) dielectric layer, respectively. It is worth noting that ${\epsilon }_M{w}^2/t_M$ is the capacitance of a single overlap between the spiral and the underpass. To improve the geometrical scalability of (3), the fitting function p(n) has also been included, which adopts the second-order polynomial expression reported in (4).

$$p\left(n\right)=0.9(n^2-3n+4\alpha )$$

(4)

Substrate capacitances, C_P1,2, are calculated by means of the following equation:

$$C_{P\mathrm{1,2}}=\frac{C_{SUB}{A}_{COIL}}{2}\left[1+(1+{\alpha )e}^{-\beta w}\right]$$

(5)

where C_SUB is the substrate capacitance per unit area, A_COIL is the coil area and β is an experimental coefficient estimated to be 3.53 × 10⁴. The coil area, A_COIL, is given by the product of the width, w, and the spiral length, l_COIL, which is calculated as follows:

$$l_{COIL}=\pi n\left[d_{in}+w+\left(n-0.5\right)\left(w+s\right)\right]$$

(6)

Since spirals are not symmetric, the capacitances C_P1 and C_P2 must be slightly different to properly model the two different values of the self-resonance frequencies of the inductor, SRF₁ and SRF₂, when terminal 1 or 2 is grounded, respectively. To account for this asymmetry, l_COIL is calculated with Equation (6) for C_P1, while it is adjusted according to (7) for C_P2.

$$l_{COIL2}=\pi n\left[d_{in}+w+\left(\frac{7}{12}n-0.5\right)\left(w+s\right)\right]$$

(7)

It is worth noting that when the metal width, w, increases, the capacitance value given by (5) reduces to its ideal value since the fringing effects tend to be negligible.

The series resistance, R_S, is modeled by using a roughly parabolic law according to Equation (8):

$$R_s=R_{DC}\left[1+\left(\frac{freq}{f_{R0}}-\frac{{freq}^2}{2{f_{R0}f}_{RM}}\right){\left(\frac{f_{R0}}{2f_{RM}}-1\right)}^{-1}\right]$$

(8)

where R_DC is the overall dc resistance of the inductor, which can easily be calculated by using the metal sheet resistance for the number of squares (i.e., l_COIL/w), while f_RM and f_R0 are the frequencies for which R_S assumes its maximum and zero, respectively. Since both f_RM and f_R0 cannot be easily predicted from the inductor geometrical parameters, the following empirical formulas have been used:

$$f_{R0}\approx \left(1-2200\sqrt{w{·d}_{in}}\right)·SRF$$

(9)

$$f_{RM}\approx \frac{0.04·SRF}{w^{0.1}·n^{0.23}·d_{in}^{0.12}}$$

(10)

Finally, substrate losses are modeled by means of R_P that is calculated by using the following expression:

$$R_P=\frac{1}{6}\frac{{\rho }_{Si}{t}_{Si}}{\pi {\left(\frac{d_{OUT}}{2}\right)}^2}\frac{f_0}{freq}$$

(11)

where ρ_Si and t_Si are the resistivity and the thickness of the silicon substrate, d_OUT is the inductor outer diameter and f₀ is equal to 3.5 GHz.

5. Model Validation

The proposed lumped model has been validated by comparison with the EM data of geometrically scaled inductors whose parameters are listed in

Table 1. Layout parameters of spiral inductors for model validation.

Inductors	1	2	3	4	5	6	7	8	9	10
w [μm]	10	10	20	20	40	40	60	60	100	100
n	3.5	6.5	1.5	3.5	1.5	2.5	1.5	2.5	1.5	1.5
din [μm]	60	60	220	150	120	200	290	240	100	300

. The model can predict the performance of circular inductors with 10-μm metal spacing, a number of turns from 1.5 to 6.5, a metal width from 10 µm to 100 µm, and inner diameters from 60 µm to 300 µm. Specifically, the large variability in the metal width is essential to demonstrate the soundness of the model not only for low-current inductors but also for high-current inductors required in the power amplifiers. The errors in low-frequency inductance, peak Q-factor, Q_MAX, and SRF are summarized in

Table 2. Model errors of for low-frequency inductance, Q_MAX, and SRF.

Errors	Maximum	Minimum
L @ 100 MHz	9.2%	1.7%
QMAX	16.4%	3.6%
SRF	12.3%	0.3%

.

As an example of the very good estimation of inductor performance, comparisons for inductors 2, 6 and 10 are reported in Figure 7a, Figure 7b and Figure 7c, respectively. It is worth noting that since the model covers a very wide range of geometrical parameters in terms of n, w and d_IN, the inductors selected for the comparison in Figure 7 represent very different geometries for a fair validation of the model. The prediction of the SRF for inductor 2 is less accurate than for the other inductors due to the fringing capacitance of the spiral that has a higher perimeter/area ratio. Generally, the model is more accurate with larger values of w. Indeed, the main aim is to guarantee very high accuracy for large-width inductors mainly used in PAs.

Figure 8 depicts the microphotograph of inductor 7 fabricated in the adopted RF GaN technology for on-wafer characterization [15]. Figure 9 compares the proposed model, the EM-simulated data and the experimental data for inductor 7. The agreement between measurements and EM simulations further confirms the soundness of the EM simulation set-up adopted to develop the inductor model, as defined in Section 4. Moreover, the comparison highlights the accuracy of the proposed lumped model with respect to the experimental measurements.

It is interesting to highlight some remarkable differences between this study and a recently published study [6] on a similar topic (i.e., modeling of GaN on Si inductors). Indeed, although a larger number of inductors with different shapes have been analyzed in [6], the model proposed here has been developed to cover a much wider range of the metal trace width. This circumstance is reflected in the explicit dependency of the model equations, specifically (2) and (5), on the layout parameter, w. Moreover, a considerable difference between the two studies concerns the modeling methodology itself. Indeed, a pure analytical approach has been followed in [6], using a lot of circuit elements to account for the several physical phenomena occurring in the metal and substrate layers. On the other hand, the proposed scalable model adopts a much easier equivalent circuit, consisting of only seven lumped components and starting from some theoretical considerations, exploits experimental coefficients for fitting.

It is worth briefly comparing this study with [6,9]. Despite its simplicity and wider geometrical scalability, the proposed model turns out to be broadly comparable to the model in [6] in terms of percentage errors in the main performance parameters. On the other hand, the model in [9] is more complex and accurate, exploiting customized parameter extraction from single inductor measurements, but it is not geometrically scalable. As already mentioned, the key feature of the proposed model is to predict inductor performance for a wide range of geometrical parameters with very good accuracy.

6. Conclusions

A simple yet effective lumped model for circular inductors has been developed in a 0.5-μm RF GaN technology for sub-6 GHz power applications. The model adopts only seven lumped components and is fully geometrically scalable. It demonstrates very good accuracy in the prediction of inductor performance (i.e., inductance, Q-factor and SRF) for a wide range of geometrical parameters. Both the geometrical scalability and network simplicity of the proposed model allow for its use in actual RF IC design. Thanks to its simplicity, the model can be used in other RF GaN technologies provided that proper tuning for the fitting parameters is performed to adapt it to a different BEOL. To the authors’ knowledge, this is the very first model for spiral inductors in RF GaN technology guaranteeing state-of-the-art accuracy for such a wide range of geometrical parameters.