Differential excitation - Inductor Theory and Inductor Measurement Technique

Chapter 2 Inductor Theory and Inductor Measurement Technique

2.6 Differential excitation

On the other hand, if the inductor will be used in a differential configuration, that is, neither port is at AC ground potential, a different approach is required. The one-port differential S-parameter (S_d) is defined in equation (2.24) [11] .

11 22 12 21 For differential excitation, the input impedance is given by equation (2.25) by using the result of equation (2.24) [11].

11 22 12 21 0 To understand the differential excitation it’s helpful to look at the lumped equivalent-circuit model in Fig. 2.16 and Fig. 2.17. For a differential excitation, the signal is applied between the two ports and the differential input impedance Z is the parallel _d combination of two substrate parasitic networks and winding network itself. At lower frequencies, the input impedance in either the single-ended or differential connections is approximately the same, but as the frequency increases, substrate parasitic C_o and R _o come into play.

Ls Rs

Cox1 Cox2

Csub1 Rsub1

V

diff

Port 1 Port 2

Csub2

Rsub2

Figure 2.16 A lumped element π-model for a differential excitation.

Rp//(Ro1+Ro2)

1 2

o o

C C C C C

+ ⋅ +

Ls Rs

Co1 Ro1

V

diff

Port 1 Port 2

Co2

Ro2 Virtual GND

.

V

diff

Port 1 Port 2

Zdiff

Figure 2.17 An equivalent circuit conversion for a lumpled element π model under a differential excitation to a parallel RLC network

For differential excitation, these parasitics have a higher impedance at a given frequency than in the single-ended connection. This reduces the real part and increases the reactive component of the input impedance. Therefore, the inductor Q is improved when driven differentially, and the self-resonant frequency increases due to the reduction in the effective parasitic capacitance from (C_p +C_o₁) to [C_p+C C_o1 _o2/

(

C_o1+C_o2

)

The differential quality factor Q_d and self-resonant frequency ω_d can be derived based on circuit analysis in Fig. 2.17, and the results given in (2.26) ~ (2.28).

1 2

Chapter 3 EM Theory and Simulation for Broadband Inductor Design

Wireless communication emerges as one of the fastest growing area in modern microelectronic industry. The strong demand for mobile communications and wireless data or voice transmission becomes a key driving force for high frequency IC technology development and creates a new market for global semiconductor manufacturing. Moreover, it has fueled a drastic competition in production and marketing for lower cost, easy portability, and enhanced functionalities. To realize system on a single chip (SoC), some effort has been tried on different technologies like CMOS, SiGe HBT, or BiCMOS, or maybe a heterogeneous integration. The latter two, i.e. SiGe and BiCMOS may offer better performance in terms of active device speed, bandwidth or current drivability.

Unfortunately, both SiGe and BiCMOS reveal the penalties of process complexity and higher cost, as compared to CMOS. Furthermore, the aggressive advancement of CMOS technology in recent years to nanoscale era escalates active device speed, such as f_T and fmax to above 100 GHz and makes itself the most competitive technology in aspects of cost, performance, and integration level, etc. Due to the mentioned advantages, RF CMOS technology is becoming the main stream of choice for manufacturing wireless communication. It explains the focus of our choice in this research for on-Si-chip inductors.

The most critical challenges encountered for high frequency circuits and products built on bulk Si CMOS technology are two folds, one is the degraded quality factor (Q) and self-resonant frequency (fSR) in passive devices like inductors, due to worse substrate loss compared to GaAs, and another one is lack of an accurate model for RF CMOS circuit simulation in which substrate loss plays a key role, particularly at very high frequency.

Although off-chip bonding wires generally adopted in conventional board design can provide relatively higher Q as compared with on-chip inductors, they sometimes suffer much larger variations in inductance value originated from mechanical process. Regarding the problem of lacking an accurate inductor model for RF circuit simulation, one of alternative solutions is a table look-up method based on inductor test-key’s data base.

However, this kind of trial-and-error approach is time consuming and not suitable for advanced development in an aggressive time frame. What is worse, the table look-up method causes resource wasted and restricts RF designers in a limited database with little room for tuning and optimization. Another approach is by using EM simulators; however this kind of numerical simulation generally requires extensive computation time as well as memories and cannot fit circuit simulation, which always demands a fast turn-around cycle. One more drawback with EM simulation is the need of an extensive calibration over material and process parameters for ensuring accuracy. The sensitivity of 3D EM simulation to wave injection methods and guard-ring layout, particularly significant for very small inductors in broadband design is identified and introduces difficulty in this approach.

Based on the consideration, the equivalent circuit model was selected as the most appropriate approach to fit application in RF circuit simulation and design. The equivalent circuit model simply represent the inductor as a lumped element circuit and π-model is the most popular one due to its simplicity and easy implementation in circuit simulators.

examples. A typical π-model includes series metal resistance and inductance, coupling capacitance between port-1 and port-2 and that between spiral metals and substrate underneath, and substrate effects. A physical model is proposed to capture the high-frequency behavior as shown in Fig. 3.1. Therefore, the spiral inductor was built on Si-substrate where the high-frequency behavior is complicated due to semi-conducting

substrate nature. The conventional π-model reveals limitation in broadband accuracy due to some neglected effects such as eddy current on metal and substrate. In order to overcome this disadvantage, 3D EM simulation was performed using HFSS to investigate the lossy substrate effect.

Figure 3.1 The device cross section and representative π-model for an inductor integrated on Si substrate.

3.1 EM Simulation tool and simulation method

In a conventional approach, the development of an equivalent model depends on model parameter extraction from measured data. Unfortunately, the availability of measured data is determined by the fabrication of test key, which generally requires a long

cycle time of around 3 months or more for advanced processes like 130nm or 90nm technologies. Besides, limited quota for test chip tape-out in advanced processes due to extremely high fabrication cost is another difficulty encountered for academic community compared to semiconductor industry. To overcome the mentioned restrictions and difficulties, EM simulation is recognized as a promising method in consideration of time, cost, flexibility in test structure design, and optimization. Besides, EM simulation can help verify the de-embedding methods, which are particularly critical for very small inductors.

Currently, there are many commercialized or proprietary EM simulation tools, which can calculate high frequency characteristics of RF components, transmission lines (TML), or package, etc. Sonnet, Microwave office, Ansoft HFSS, and Agilent ADS Momentum are popular and frequency used tools among the commercialized tools. ADS Momentum is a planar full-wave EM solver that can simulate electric and magnetic fields in the, conductors, dielectric and substrate. This kind of 2.5D EM wave simulators can save computation time to a certain extent as compared with truly 3D simulators like HFSS.

However, some critical problems, such as degraded accuracy and failure in convergence may happen in this kind of tools. To overcome the critical issues, particularly worse in very small inductors for our study, HFSS is of our choice in this research, even though an extensive computation time and memories are required. It is desirable to achieve high frequency parameters, such as S-parameters and derived Y- as well as Z-parameters with sufficient accuracy for inductors on Si substrate, over a broadband of frequencies. To ensure the accuracy over high frequencies, a number of key mechanisms, such as skin effect, proximity effect, and substrate coupling effect must be adequately implemented in the simulation tools. The ultimate goal is to predict S-parameters with guaranteed accuracy up to self-resonance frequency and beyond. Subsequently, the key performance parameters like quality factor Q, maximum Q (Q_max), f_m corresponding to Q_max, and self-resonance

frequency (fSR) can be predicted before chip fabrication. In this way, the EM simulation results can serve as measured data for model parameter extraction and optimization aimed for an equivalent circuit model build-up.

3.2 HFSS for 3D EM simulation

As mentioned previously, an extensive calibration is indispensable for general EM simulators to attaining required accuracy. The calibration should cover parameters related to material, process, layout, topology, and substrate. Unfortunately, the calibration done on HFSS and ADS momentum in previous work is limited to inductors with sufficiently large size (L>1nH) but reveals increasing deviation in very small inductors for broadband design targeting frequency up to 70 GHz in this topic. Through a careful study, it is identified that simulation environment setup in terms of RF signal injection direction and guard ring (GR) layout or metal placement has a critical effect on simulation accuracy, particularly for very small inductors ((L<1nH).

Figure 3.2 Vertical mode with RF signal injection at top metal (M8) and grounded ring (GR) at M1.

Fig. 3.2 illustrates the simulation environment setup, which was suggested in HFSS user manual and commonly used in on-chip inductor simulation. The RF signal was applied in a vertical way through the vias, then reaching top metal (M8) and the guard ring (GR) was placed in the bottom metal (M1). This kind of setup can enable acceptable accuracy over higher frequencies but over-estimates the real part of input impedance Re(Z_in) for miniaturized inductor (L< 1nH) in lower frequencies.

Figure 3.3 Horizontal mode with RF signal injection and grounded ring (GR), both at top metal (M8).

Trying to solve the mentioned problem identified from vertical mode setup in Fig. 3.2, a different one namely horizontal mode with RF signal injection in a horizontal way through the top metal (M8) and GR at the same plane, i.e. M8 was proposed as shown in Fig. 3.3. In this setup, RF signal injection approaches the real condition but the GR at M8 is inconsistent with the practice in real layout. The inconsistency introduced in 3D EM simulation causes a dramatic deviation from measured data in terms of L=Im(Zin)/ω, Re(Zin), Rs=Re(-1/Y21), and Q=Im(Zin)/ Re(Zin), as shown in Fig. 3.5. This horizontal setup with GR at M1 under-estimates Re(Z_in) and over-estimates Q.

Figure 3.4 Horizontal mode with RF signal injection at top metal (M8) and ground ring (GR) at M1.

To overcome the deviation revealed by the first version of horizontal mode in Fig. 3.3, a modification was done in terms of 3D topology for RF signal injection and GR placement to fit miniaturized inductors. Fig. 3.4 presented an improved horizontal mode built up following the actual layout in which the ground ring (GR) is implemented with the ground metal (M1) and RF signal is injected through the top metal (M8). As shown in Fig.

3.5, this improved horizontal mode with GR at M1 can fix the deviation suffered by the previous one with GR at M8 and achieve a much better match with measurement in the specified key parameters (L, Re(Z_in), Rs, and Q) over ultra high frequency to 110 GHz.

0 20 40 60 80 100 120

75 The horizontal mode GR at M1

Figure 3.5 Comparison between measurement and HFSS simulation using horizontal injection mode with different guard ring (GR) layouts, at M1 and M8 respectively. (a) Im(Zin)/ω (b) Re(Zin) (c) Rs=Re(-1/Y21), and (d) Q=Im(Zin)/ Re(Zin) .

3.3 HFSS simulation condition setup

In a real process, spiral inductors adopted in this work were fabricated by 0.13um back end technology with eight layers of Cu and FGS as the inter-metal dielectric (IMD).

However, it is a difficulty task using a EM simulator like HFSS to build a 3D structure, which exactly follows the 3D topology consisting of multi-layer metal (Cu), IMD, and vias. The major problems are excessive computation time as well as memories, and the introduced failure in convergence.

Figure 3.6 Inductor structure setup for HFSS simulation (a) a 3D structure incorporating

strates the 3D structure and 2D layout of a spiral inductor built for HFSS simu

constant

spiral inductor and guard ring (2) a 2D layout from a projection of the original 3D structure in (a).

Fig. 3.6 illu

lation. In a real process such as 0.13um back-end technology for fabricating on-chip inductors, the 3D structure incorporates 8 layers of metals (Cu) and composite dielectric layers in the IMD. To simply the problem for EM simulation, an effective dielectric

, r eff

ε corresponding to the composite dielectric layers, was derived from eqs. (3.1)

~ (3.2), based on a simple theory of series capacitance.

(a) (b)

Figure 3.7 (a) Multi-layer metals and inter-metal dielectrics with various dielectric constants and thicknesses and Effective oxide dielectric constant and thickness (b) effective dielectric constant representing the composite dielectric layers.

3.4 Electromagnetic theory for inductor analysis

For a spiral inductor with current flowing through the metal coils, a magnetic flux denoted as can be generated through the area enclosed by the winding coils. The magnetic flux is defined as a product of the average magnetic field B times the normal component of the area that it penetrate and given by (3.3). Notes that the average magnetic field B is also named as magnetic flux density.

ΦB According to Biot-Savart law, the magnetic field B corresponding to a close loop of metal

conducting with current of I can be calculated by (3.4) [4]

(3.4) indicates that the magnetic field B is proportional to current I flowing in the metal coil, and then the magnetic flux derived from the product of B and area, given in (3.3) follows the proportional relation with respect to I. The proportional coefficient between

and I is denoted as inductance L in (3.5) ΦB

ΦB

(3.5)

B L I

Φ = i

If there are N turns of coil in a spiral inductor, the total magnetic flux and the corresponding inductance are expressed as [4]

Λ L

Λ = ⋅Φ = ⋅N _B L I L I

= Λ (3.6)

For spiral inductors applied in RF circuits operating in high frequencies up to GHz, the effective sheet resistance in the winding metals will increase dramatically above its DC value, namely skin effect. As a result, the conductor losses in the winding metals include not only the joule heating from DC resistance at sufficiently low frequency but also skin effect introduced excess losses, which increases with increasing frequency. The skin effect imposed on a spiral inductor under high frequency operation leads to increase of input impedance Re(Zin) and Q degradation. Besides skin effect, proximity effect is one more important mechanism responsible for increase of effective resistance and Q degradation, which becomes significant for spiral inductors with multi-turn coils. The DC resistance can be easily calculated by the product of metal sheet resistance (static state) and the square number defined by total length over width, as written in (3.7). However, the frequency dependent resistance due to skin effect and proximity is difficult to analyze precisely and sometimes requires EM simulation for a quantitative assessment or prediction.

3.4.1 Conductor loss – Skin effect and Proximity effect

For a conductor operating at sufficiently low frequency, its DC resistance can be

calculated by (3.7), in which the sheet resistance refers to the process parameters dependent on each individual metal layer,

Rsh

: tan /

: :

dc sh

R R

w where

R metal sheet resis ce in unit of metal total length

w metal width

= ×

Ω (3.7)

Regarding the frequency dependence of effective trace resistance under high frequency to several GHz and beyond, EM simulation was performed to identify the current distribution and investigate the underlying mechanisms. Fig. 3.8 presents the current distribution across the conductor and reveals non-uniform distribution with an obvious current crowding near the conductor surface. The phenomenon is known as skin effect.

Figure 3.8 EM (HFSS) simulation reveals skin effect apparent in the conductors in which non-uniform current distribution and current crowding near surface will lead to increase of effective resistance in conductors under high frequency operation.

δ

δ t

w

Figure 3.9 The cross section and skin depth definition in a metal line for the analysis of non-uniform current distribution under high frequency operation

For an analysis and modeling of the skin effect, a skin depth δ was specified in Fig. 3.9 in a metal line and used to model the non-uniform current distribution across the metal given by (3.8), in which the current density decay from the maximum value at surfaceJ₀ to the center according to an exponentially decreasing function. The skin depth δ was modeled as shown in (3.9), in which the higher frequency or the higher metal conductivityσ will lead to thinner skin depth and then aggravated skin effect. Taking Cu as an example, its metal conductivity σ =5.531 10× ⁷ / m will result in a skin depth δ of around 2.1μm at 1.0 GHz and as thin as 0.93 μm at 5.0 GHz, which is much thinner than metal width in general applications, including logic, analog, and RF circuits.

( )= 0× ⁻^δ

J y J e (3.8) Skin effect is defined by skin depth δ in which most of the current is localized at high frequency. The skin depth (δ) is calculated by equation (3.9). Show the electric field decays and reaches surface electric field 1

e times of largest thickness.

δ 2

= ωμ σ (3.9)

The current ( I ) is obtained by integrating the current density over the wire cross-sectional area. Since only varies in the y direction (Fig.3.8),

J where is the physical thickness of the wire. The frequency dependent thickness appearing in the last term of (3.10) is defined as an effective thickness,

( )ω =δ(1− ⁻^δ^t)

teff e (3.11) The dc series resistance, R_dc, free from frequency dependence can be expressed as

=σ =

dc sh

R R

A w (3.12) where A is the area of the cross-section, and R is the resistance per unnit area. The _sh frequency dependent series resistance, due to skin effect can be derived based on R and _dc frequency dependence int_eff as follows

Note that the increase of effective series resistance, due to skin effect approaches an asymptote following a square root function of the frequency given in (3.14). In comparison, proximity effect presents a strong function of frequency and may lead to resistance increase at a higher than linear rate and dominate Q degradation at higher frequency. In the following, 3D EM simulation was carried out to investigate the proximity effect on current distribution in a multi-turn spiral inductor under increasing frequency and its impact on effective resistance. For very low frequency at 0.01 GHz as

shown Fig. 3.10, the current flow in the metal trace presents a uniform distribution.

However, for increasing frequency to 10 GHz, as shown in Fig. 3.11, the EM simulation reveals dramatically non-uniform current distribution in the multi-turn metal coils.

Proximity effect originated from EM field coupling between adjacent coils is proposed as the mechanism besides skin effect, responsible for current re-distribution and increase of resistance at sufficiently high frequency.

Figure 3.10 Current density distribution in the multi-turn metal coils of an inductor, at very low frequency 0.01 GHz. The EM simulation done by HFSS indicates a perfect uniform distribution in the metal trace, free from current crowding effect.

Figure 3.11 Current density distribution in the multi-turn metal coils of an inductor, at very high frequency, 10 GHz. The EM simulation done by HFSS indicates a dramatic non-uniform current distribution in the metal trace, accounting for proximity effect.

Even though the proximity effect induced current crowding has been known for a long time, and the general mechanism was cited and interpreted in couple of literatures.

However, quite few works were done to realize a quantitative prediction without resort to numerical simulation. In this thesis, an analytical model will be derived to calculate and predict the frequency dependent resistance associated with proximity effect and facilitate an accurate prediction of Q at very high frequency.

The basic mechanism underlying the current crowding originated from proximity effect can be understood through an illustration in Fig. 3.12. The magnetic field B in the adjacent coil penetrates the target metal trace in the direction normal to the surface.

According to Lenz’s law, eddy currents will be created in the target metal trace with a direction to generate a magnetic flux opposite to that introduced from the adjacent coil. As a result, the generated eddy currents add to the excitation current on the inner edge (near the center of spiral coils) while subtract from the excitation current on the outer edge. It

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