Chapter 3 EM Theory and Simulation for Broadband Inductor Design
3.3 HFSS simulation condition setup…
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.
1
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]
0
(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
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
y
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.
0
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,
t
( )ω =δ(1− −δt)
teff e (3.11) The dc series resistance, Rdc, 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 inteff 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 explains why the current crowding was created and the dramatic increase of effective
resistance.
I I
B eddy
B eddy B
B
I eddy
Figure 3.12 Eddy currents generated according to Lenz’s law, with a direction in phase with excitation current on the inner edge (near the center of spiral coils) but opposite to the excitation current on the outer edge. The phenomenon explains current crowding caused by proximity effect in multi-turn spiral inductors.
Applying the mentioned theory in a multi-turn spiral inductor as shown in Fig. 3.13, it can facilitate an understanding of the proximity effect. The eddy current opposite to the excitation current on the edge near the adjacent coil leads to non-uniform current distribution revealed by EM simulation in Fig. 3.11. Note that proximity effect becomes significant at sufficiently high frequency and dominates skin effect as mentioned previously.
current
I
B
.
Ieddy
Beddy
Figure 3.13 Eddy currents generated according to Lenz’s law, with a direction in phase with excitation current on the inner edge (near the center of spiral coils) but opposite to the excitation current on the outer edge. The phenomenon explains current crowding caused by proximity effect in multi-turn spiral inductors.
In the following, analytical models will be derived for calculating the effective resistance associated with eddy currents due to proximity effect denoted as Reddy, and the introduced excess power loss defined as Peddy.
I I w s w
Figure 3.14 Cross section of a metal trace showing the normal magnetic field B x y , ( , ) and eddy current flowing round the edges within a width of wed.
Figure 3.14 shows the eddy current induced in a metal trace due to magnetic fields from adjacent conducting wires where B x y is the magnetic flux density normal to the ( , ) metal segment. EM simulations show that the distribution of current induced by an external magnetic field occurs along the width of the metal trace, and not across its thickness. Eddy current introduced in a metal trace due to magnetic fields from neighboring wires follows Lenz’s law and leads to current crowding effect, so called proximity effect. Therefore, we assume that the eddy current flows near the edges of the metal only within an effective width of wed. Though B x y varies within the segment, ( , ) for the sake of a simple analytical formula, it is reasonable to assume a constant B , which is approximately the value at the middle of the metal trace along the center line. B is calculated from the excitation current in all other parallel metal wires in the inductor. Note that the contribution of eddy currents in another segment toward B will roughly cancel since they flow in opposite directions at the two edges of a segment. Assume that there is on phase delay between the excitation currents in different traces, which is acceptable for the inductor size and frequency range studied.
Based on Ampere’s law, the magnetic field at a distance from an infinite wire conducting a current
r I can be expressed as
0
2 μ
= πI
B r (3.15) The magnetic field B introduced from the neighboring metal wire can be derived as
( ) 0 In equation (3.16), w is the width of the metal wire, s is the distance between two adjacent wires
According to Faraday’s law of electromagnetic induction, an electromagnetic field (emf) can be induced by the time varying magnetic flux introduced from adjacent metal wire addressed in (3.15)~(3.16). The emf was generated around a close loop in the conductor and created an electric field in the loop, given by (3.17)
Concurrently, the emf acting on the conducting material generated a current around the loop, namely eddy current. The induced eddy current corresponds to the electric field following ohm’s law, expressed in (3.18). In accordance with Faraday’s law, the eddy currents dissipate energy and create a magnetic field that tends to oppose the change in the magnetic flux
Following (3.17) and the cross section of winding metal with induced magnetic field in Fig. 3.15, the electrical field and eddy current can be derived as follows.
According to ohm s law
J E Bx
Figure 3.15 Cross section of a metal trace showing the normal magnetic field B x y , ( , ) and eddy current flowing within a width wed round the edges
The eddy currents introduced from adjacent metal wires, due to proximity effect become another source of power dissipation and Q degradation in spiral inductors. The power dissipation due to proximity effect induced eddy currents can be calculated as follows. First, specify the eddy currents flow within a finite width of around the edges of a metal trace, as shown in Fig. 3.15. The power dissipation is treated as the joule heating associated with eddy currents confined in around edges of the metal trace in two sides, and the corresponding resistance defined as . Based on the
wed
Peddy
Ieddy wed
Reddy
argument, the formulas are derived in the following. Taking B i derived in
2 2 2
where I is the excitation current in the metal wire
Define R as the effective resistance due to eddy current
2 2
Assume that the distribution of eddy currents is analogous to that of excitation currents due to skin effect. Then is defined as a skin depth given by (3.11). Then the effective resistance due to eddy currents
I wed
Reddy can be further derived as follows.
According to above inference, the total power dissipation and the corresponding series resistance under high frequency, namely Rac can be revised to contain skin effect
and proximity effect.
where R is the resis ce incorporating skin effect given in
R R
is the effective resistance due to proximity effect, written in (3.30)
(1 )
then the series resis ce under high frequency R containing both skin effect and proximity effect can be ressed as
R t
3.4.2 Substrate loss – Capacitive coupling and Inductive eddy currents
In addition to the conductor losses contributed from skin effect and proximity addressed in section 3.4.1, substrate losses is one more important factor responsible energy loss and Q degradation in on-chip inductors. According to Maxwell equations for EM analysis, there are two kinds of different mechanisms responsible for the substrate losses.One is electric loss caused by capacitive coupling effect, and another one is the magnetic loss due to eddy currents induced by the magnetic coupling, both between the metal coils and the semiconducting substrate underneath. The capacitive coupling effect can be understood by Gauss law and the magnetic coupling can be explained by Faraday and Lenz’s law as addressed previously (section 3.4.1) for conductor losses. Note that substrate losses tend to increase dramatically with increasing frequency and may dominate proximity effect as the primary factor responsible for fSR as well as Q degradation. This
subject and associated challenges stimulate our motivation in this work to overcome substrate losses and conductor losses for achieving broadband inductors with fSR up to 70 GHz.
Fig. 3.16 presents analysis of input impedance Re(Zin)for conventional symmetric inductor by using full wave EM simulation and simplified model incorporating proximity effect. The comparison with measurement indicates the simplified model with proximity effect only under-estimates Re(Zin), due to inappropriate neglect of substrate loss effects.
As for a quadruple semi-symmetric inductor shown in Fig. 3.17, a new symmetric inductor realized in this work for broadband design, the comparison with measurement indicates the simplified model with proximity effect can fit Re(Zin)of this new symmetric inductor.
Through a careful analysis and comparison on the inductor layouts and geometries, it can be identified that conventional symmetric inductors suffer worse substrate coupling introduced energy loss, due to larger capacitive coupling and magnetic coupling (eddy currents on the substrate). It accounts for the effective suppression of substrate losses in this new symmetric inductor and the mechanism contributing significant improvement in fSR. Furthermore, EM simulation revealed a trade-off between proximity effect and substrate coupling in symmetric inductors with different layouts. For the conventional fully symmetric inductor with multi coils, the substrate coupling effect dominates proximity at higher frequency. As for the new symmetric inductors design for broadband design, proximity effect dominated even up to 100 GHz.
Figure 3.16 Analysis of input impedance Re(Zin)for conventional symmetric inductor by full wave EM simulation and simplified model incorporating proximity effect. The comparison with measurement indicates the simplified model with proximity effect only under-estimates Re(Zin).
Figure 3.17 Analysis of input impedance Re(Zin)for quadruple semi-symmetric inductor (new design for broadband applications) by full wave EM simulation and analytical model incorporating proximity effect. The comparison with measurement indicates the simplified model with proximity effect can fit Re(Zin)of quadruple semi-symmetric inductor.
Chapter 4
Broadband and symmetric Inductor Design
With the advancement of Si CMOS technology and its advantages in high speed, high integration level, and low cost, Si RF CMOS becomes a vital technology to realize a single-chip communication integrated circuit. Going with the development path, on-chip inductors become a key component in RF integrated circuits such as LNA, VCO, filters, and impedance matching networks, and will determine RF circuit performance in terms of gain, power, and noise. However, the on-chip inductor design faces several challenges, such as broadband and high-Q, as well as area estate. The major difficulty comes from the energy loss associated with low resistivity Si substrate. The lossy substrate introduces challenges, not only in performance optimization but also in simulation and modeling.
In recent years, symmetric inductors were proposed to replace widely used spiral inductors for achieving higher Q and broader band at a smaller area. Moreover, two-port symmetry as an intrinsic property of symmetric inductors is an important feature desired for widely used differential circuits for low noise and high gain design
In this thesis, new symmetric inductors with features of broadband and high-Q will be design with performance target of fSR ≧ 70 GHz and Qmax ≧ 15 to enable applications in V-band microwave circuits. 3D EM simulation was performed to guide layout design for achieving the target performance. Some critical issues emerging from this practice on small inductor design in terms of EM simulation, ulta-high frequency measurement, and de-embedding methods will be discussed.
4.1 Symmetric inductor design and fabrication
As mentioned previously, on-chip inductors become a critically important component in RF circuits such as VCO, LNA, and impedance matching networks. For conventional inductors, quality factor Q and self-resonance frequency fSR are well known parameters affecting RF circuit performance. As for differential circuit topology widely used in analog and RF circuits, the differential excitation (i.e., voltages and currents of two signals are 180° out of phase but with the same magnitude) has become an important operation mode of choice. It is because that differential mode operations can provide better noise immunity and higher gain. Taking this advantage for broadband and low noise design, symmetric inductors with new layouts and geometries were designed and fabricated in this work.
Figure 4.1 The layout of a conventional differential inductor.
First, the conventional symmetric inductor illustrated in Fig. 4.1 was designed for
First, the conventional symmetric inductor illustrated in Fig. 4.1 was designed for