Model enhancement strategies

The lack of an accurate and scalable model for on-chip inductors becomes one of the most challenging problems for Si-based RF IC design. The existing models suffer two major drawbacks in terms of accuracy for limited bandwidth and poor scalability.

Many reference publications reported improvement on the commonly adopted π-model by modification on the equivalent circuit schematics. However, limited band width to few gigahertz remains an issue for most of the modified π-models. A two π-model was proposed to improve the accuracy of R(ω) and L(ω) beyond self-resonance frequency. Unfortunately, this two π-model suffers a singular point above resonance. Besides, the complicated circuit topology with double element number will lead to difficulty in parameter extraction and greater time consumption in circuit simulation. Recent work using modified T-model demonstrated promising improvement in broadband accuracy and suggested the advantage of T-model over π-model. However, the scalability of model’s major concern was not presented. To solve the mentioned issues, a new T-model was proposed and developed in this work.

This T-model is proposed to realize two primary features, i.e., broadband accuracy and scalability. The T-model is composed of two RLC networks to account for spiral coils, lossy substrate, and their mutual interaction. Four physical elements, Rs Ls Rp

and Cp are incorporated to describe the spiral coils above Si substrate and other elements. All the physical elements are constants independent of frequencies and can

be expressed by a close form circuit analysis on the proposed T-model. Parameter extraction and optimization can be conducted with an initial guess extracted by approximation valid for specified frequency range.

All the model parameters manifest themselves with predictable scalability w.r.t. coil numbers and physical nature. A parameter extraction flow has been established to enable automatic parameter extraction and optimization that is easy to be adopted by existing circuit simulators like Agilent ADS or parameter extractor such as Agilent IC-Cap. The model accuracy over broadband is validated by good agreement with the measured S-parameters, L(W), Re(Zin(W)), and Q(W) up to 20GHz that this scalable inductor model can effectively improve RF circuit simulation accuracy in broad bandwidth and facilitate the design optimization using on-chip inductors.

2.4 Fundamental of quality factor for an inductor

For an ideal inductor free from energy loss due to parasitic resistance and substrate coupling effect, the magnetic energy stored can be given by (2.1),

1 ²

L 2

E = L i _L

(2.1) Where

i _L

is the instantaneous current through the inductor.

From (2.1), the peak magnetic energy stored in an inductor in sinusoidal steady state is given by,

2 2

inductor 2

1

2 2

L

peak L

E L I V

ω L

= =

(2.2)

Where

I _L

and

V _L

correspond to the peak current through and the peak voltage across the inductor.

The quality factor (Q) of an inductor is a measure of the performance of the elements defined for a sinusoidal excitation and given by,

energy stored energy stored 2 energy loss per cycle average power loss

Q = π = ω

(2.3)

The above definition is quite general which causes some confusion. However, in the case of an inductor, energy stored refers to the net peak magnetic energy.

To illustrate the determination of Q, consider an ideal inductor in series with a resistor in Figure 2.11. This models an inductor with resistance in the winding.

Figure 2.11 Inductor with a series resistance

Since the current in both elements is equal, we use the equation for the peak magnetic energy in terms of current given in (2.2) to write,

2

2

peak magnetic energy stored 2 energy loss per cycle

1 2 2

1 2 2

=

= =

=

s s

s s s s

Q

I L

L I R R

where π

π ω

τ

τ π ω

(2.4)

Where τ is the period of the sinusoidal excitation

Note that the quality factor of an inductor with a lossy winding increases with frequency. Also note that as the resistance in the inductor decreases, the quality of the inductor increases and in the limit Q becomes infinite since there is no loss. Using the above procedure, the quality factor of another pure lossy inductor can be determined.

We repeat the detail in the following.

Figure 2.12 Inductor with a parallel resistance.

Since the voltage in both elements is equal, we use the equation for the peak magnetic energy in terms of voltage given in (2.2) to write,

2

2 2

2

peak magnetic energy stored 2 energy loss per cycle

2 2

2

=

=

=

p p

p p p

p

Q

V L V

R R

L π

π ω ω τ

ω

(2.5)

Where τ is the period of the sinusoidal excitation

The definition of quality factor is general in the sense that it does not specify what stores or dissipates the energy. The subtle distinction between an inductor and an LC tank Q lies in the intended form of energy storage. For example, only the magnetic energy stored is of interest and any electric energy stored because of some inevitable parasitic capacitance in a real inductor is counterproductive. Therefore, the Q of an inductor is proportional to the net magnetic energy stored and is given by,

peak magnetic energy stored 2 energy loss per cycle

peak magnetic energy stored-peak electric energy 2

energy loss per cycle

inductor

Q π

π

=

=

(2.6)

An inductor is said to be self-resonant when the peak magnetic and electric energies are equal. Therefore, Q of an inductor vanishes to zero at the self-resonant frequency. At frequencies above the self-resonant, no net magnetic energy is available from an inductor to any external circuit. In contrast, for an LC tank, the Q is defined at the resonant frequency

ω _o

, and the energy stored term in the wxpression for Q given by (2.3) is the sum of the average magnetic and electric energy. Since at resonance the average magnetic and electric energies are equal, so we have,

average magnetic energy + average electric energy 2 energy loss per cycle

peak magnetic energy peak electric energy

2 2

energy loss per cycle energy loss per cycle

o

The average magnetic or electric energy at resonance for sinusoidal excitation is

2 2

1 1

4 L I ^L = 4 C V ^c

which are half the peak magnetic energy given by (2.2) Lets look at the parallel RLC circuit of figure 2.5 to clarify its inductor and tank Q.

Figure 2.13 Parallel RLC circuit.

The quality factor of the inductor is calculated as follows,

2

peak magnetic energy - peak electric energy 2 energy loss per cycle

1 1

where the resonant frequency

⁰ 1

p p

ω = L C

_.

Here

^p

p

R

ω L

accounts for the magnetic energy stored and ohmic loss of the parallel resistance in figure 2.4. The second term in equation 2.8 is the self-resonance factor describing the reduction in Q due to the increase in the peak electric energy with

frequency and the vanishing of Q at the self-resonant frequency. In the parallel RLC circuit, VL = VC = VP which is depicted in the figure 2.5. Note that in each quarter cycle, when energy is being stored in the inductor, it is being released from the capacitor and vice versa. As ω increases, the magnitude of decreases while the magnitude of

increases until they become equal at the resonant-frequency ω

I L

I C

0, so that an equal

amount of energy is being transferred back and forth between the inductor and capacitor. At this frequency, given by equation 2.8 is zero. As ω increases above ω

inductor

Q

0, the magnitude of becomes increasingly more negative. That is, as the previous mention, no net magnetic energy is available from an inductor to any external circuit at frequency above

I L

ω o

. The inductor is capacitive in nature, and given by (2.8) is negative. Now using (2.7) to calculate the tank Q we have

inductor

energy loss per cyle V

Also, note that the same result can be derived using the ratio of the resonant-frequency to -3 dB bandwidth as follows,

p

tan

3

0

f= 1

2 L

1 2

− =

=

= =

o

p

k

dB f f

p

p p

p

p p

C p

Q f

BW

f R

L R C

R C C

π

ω π

(2.10)

(2.9) and (2.10) are the same as we expect.

Both Q definitions discussed above are important, and their applications are determined by the intended function in a circuit. While evaluating the quality of on-chip inductors as a single element, the definition of inductor quality given by (2.6) is more appropriate. However, if the inductor is being used in a tank, the definition given by (2.7) is more appropriate.

Figure 2.6 shows a real inductor can be replaced by a parallel RLC circuit of π-model.

Figure 2.14 Alternative method for determining the Q in real inductors.

In contrast with (2.8), it can be easily determined that the real inductor quality

factor of a parallel RLC circuit is given by the negative of the ratio of the imaginary part to the real part of the input admittance, namely the ratio of the imaginary part to the real part of the input impedance. The above statements are summarized in (2.11) and are appropriate for determining the Q of inductors from simulation or measurement results.

{ }

{ } { } { }

tan

Im Im

Re Re

in in

k

in in

Z Y

Q = Z = − Y

(2.11)

Chapter 3

Broadband and Scalable On-chip Inductor Model

3.1 Broadband accuracy for on-chip inductors

In silicon-based radio-frequency (RF) integrated circuits (ICs), on chip spiral inductor are widely used due to their low cost and ease of process integration. As a necessary tool for circuit design, equivalent circuit models of spiral inductors, using lumped RLC elements, efficiently represent their electrical performance for circuit simulation with other design components. Compared with the generic 3D electromagnetic field solver (e.q., HFSS) or other 2.5D electromagnetic field solver (e.q., ADS Momentum), a lumped equivalent-circuit model dramatically reduces computation time and supports rapid performance optimization. On the other hand, model inaccuracy, which stems from the complexity of on-chip inductor structures and high-frequency phenomena, presents one of the most challenging problems for RF IC designers.

Current equivalent-circuit approaches simply represent the inductor as a lumped circuit and π-model is one of examples. π-model includes series metal resistance and inductance, feedthrough capacitance, dielectric isolation, and substrate effects. A physical model is proposed to capture the high-frequency behavior as shown in Fig.

3.1. Herein, 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 substrate. In order to overcome this disadvantage, 3D EM

simulation was done using HFSS to investigate the lossy substrate effect. Following the HFSS simulation results, a new T-model has been developed to accurately simulate the broadband characteristics of on-Si-chip spiral inductors, up to 20 GHz.

Figure 3.1 Conventional π-model

3.1.1 Simulation tool and simulation method

Some electromagnetic (EM) field simulators are used, like sonnet, microwave office, HFSS and ADS Momentum to predict the component characteristics such as S-parametera, quality factor, and self-resonant frequency. However, we found that the simulation time of HFSS for 3D is slower than the others. Because it can estimate the magnetic substrate eddy current effect, we can obtain more accurate S-parameter.

ADS Momentum EM simulation is a planar full-wave EM solver that can calculate the fields in the substrate and the dielectric and spend less time, but this simulation tools for 2.5D is less accurate than HFSS. Thus, the capacitance between the spiral windings and the eddy current in the windings are not modeled. The advantage of

these EM simulators is that they can report their simulation results in S-parameters.

These results can then be numerically fitted to the circuit model. But in general, it is desirable to simulate circuits with these components by directly using the S-parameters extracted from the EM simulator or measured from the instruments.

This is because a number of the component values in this circuit model vary with frequency due to the skin effect, substrate loss and so on.

For the mentioned reason, the fast and adequately accurate simulation program is strongly demanded. In order to predict the frequencies corresponding to Qmax and self-resonance (fSR), the amount of the parasitic capacitance should be predicted accurately. Due to the requirement, we select HFSS for EM simulation and analysis in this work.

Figure 3.2 layer stackup simulation by HFSS

Spiral inductors were fabricated by 0.13um back end technology with eight layers of Cu and low-k inter-metal dielectric (k=3.0). The top metal of 3μm Cu was used to implement the spiral coils of width fixed at 15μm and inter-coil space at 2μm. The inner radius is 60μm and outer radius is determined by different coil numbers N=2.5, 3.5, 4.5, 5.5 for this topic. The physical inductance achieved at sufficiently low frequency are around 1.96~8.66nH corresponding to coil numbers N=2.5~5.5.

S-parameters were measured by using Agilent network analyzer up to 20 GHz and de-embedding was carefully done to extract the truly intrinsic characteristics for model parameter extraction and scalable model build up. In Figure 3.2, it is clear that HFSS simulation environment is a solid structure. In HFSS simulation window, it can’t simulate 0.13um back end technology with eight layers of Cu and low-k inter-metal dielectric (k=3.0), so we must make some modifications for simulation setup.

Figure 3.3 effective oxide dielectric constant equivalents from M1 to M2

From Figure 3.3, we give an example for dielectric constant equivalent from Metal-1 to Metal-2. In 0.13um back end technology, the inter-metal dielectrics is a complex layer structure of various dielectric constants. In order to simplify these layers, we make two

series capacitances be equal to one capacitance. We use above theory to extend complex type and show the formula as follows

= 1

= ∑ ⁿ

eff i

i

D d

(3.1)

1

,

1

−

=

⎛ ⎞

= ×⎜

⎝ ∑ ^{n i} ⎠

r eff eff

i ri

D d

ε ε ^⎟

(3.2)

Where ε

r

is relative permittivity and di is thickness

In the layout of the inductor, to prevent flux radiation to cause flux degradation in the center area, we generally plot ground ring to protect flux radiation. As shown in Fig 3.4, in order to simulate ground ring by HFSS, we could setup ground ring material for PEC to decrease the loss. Adopting the described simulation method, we will discuss T-model build-up for single-end spiral inductor in the next section.

Figure 3.4 Ground ring setup by HFSS

3.1.2 Conductor and substrate loss effect – model and theory

Figure 3.5 Layout of convention single-end spiral inductor

There are several sources of loss in a single-end inductor. The DC resistance of single-end inductor is easily calculated as the product of this sheet resistance and the number of squares in the strip. However, at higher frequencies the resistance of the strip increases due to the skin effect and current crowding. Moreover, substrate losses increase with frequency due to the dissipative currents that flow in the silicon substrate. According to Maxwell equation, there are tow different mechanisms that cause the induction of these loss effects. One is the capacitive coupling between the strip and the substrate induces display current, namely electric substrate losses. The other is the magnetic is the magnetic coupling caused by the time varying magnetic field linked to the strip induces eddy currents under the strip and in the inner turns of the strip, namely magnetic substrate losses. From (3.3) and (3.4) of Maxwell equation, we can show above theory.

B

E t

∇× = − ∂

∂ G G

(3.3)

c

E dS B dS

t

E d t

∇× • = − ∂ •

∂

• = − ∂Φ

∂

∫∫ ∫∫

∫

G G

G G

G G

v A

(3.4) Figure 3.5 show the electric and magnetic substrate losses of single-end spiral inductor. The magnetic field

JJJJG B t ( )

extends around the windings and into the substrate. Faraday’s Law states that this time-varying magnetic field will induce an electric field in the substrate. This field will force an image current to flow in the substrate in opposite direction of the current in the winding directly above it. The magnetic field will not only penetrate into the substrate but also into the other windings of the coil. The effect causes the inner turns of the strip to contribute much more loss to the inductor while having a minimal impact on the actual inductance. This phenomenon is sometimes referred to as current crowding.

For on-chip single-end spiral inductors, the line segments can be treated as microstrip transmission lines. In this case, the high frequency current recedes to the bottom surface of the wire, which is above the ground plane. Please see figure 3.6.

Figure 3.6 cross section for single-end spiral inductor coils

The attenuation of the current density (

J

in

A m / ²

) as a function of distance (y) away from the bottom surface can be represented by the function

The current ( in A) is obtained by integrating over the wire cross-sectional area.

Since only varies in the y direction, can be calculated as

Where is the physical thickness of the wire. The last term in equation 3.6 can be defined as an effective thickness

d

(1 )

d

d eff = δ − e ^{− δ}

(3.8) The dc series resistance, Rdc, can be expressed as

DC sh

R R l

= w

(3.9) The series resistance, Rs, can be expressed as

frequencies. At the higher frequencies, we will include skin effect depended on

frequency in the (3.10).

Regarding to substrate effect, we use 3D simulation tools, for example, HFSS to simulate current flow direction on the substrate surface to verify above theory. The simulated current flow expressed by vectors is shown in figure 3.7

Figure 3.7 simulate eddy current on the substrate surface by HFSS

Figure 3.7 indicates that the eddy current on the Si substrate flows in the opposite direction w.r.t that of spiral coils. According to Faraday’s Law states that this time-varying magnetic field will induce an electric field in the substrate and generate a current on the substrate surface. But in the interior substrate also is generated, we also obtain result from 3D simulation tools by HFSS. From figure 3.8, we find current generated in the interior substrate. This effect also causes Q degeneration of the single-end spiral inductors. In order to decrease magnetic field coupling to substrate, we usually use pattern ground shield at the lower metal and increase Q value.

Figure 3.8 simulate eddy current in the interior substrate surface by HFSS

According to above method, we will present a new T-model developed to accurately simulate the broadband characteristics of single-end spiral inductors. In figure 3.9, we integrate all physic parameters and obtain a compact model. Please

see figure 3.9, and we will use equivalent circuit to analysis in the next section.

Figure 3.9 Simplified illustration of T-model

3.1.3 Varying substrate resistivity effect – model and theory

On-chip passive components are imperative for silicon-based RF IC’s. The detrimental effects of the semi-conducting substrate parasitics on metal-insulator- metal capacitors, bond pad single spiral inductors. However, the basic understanding of the physics behind these effects is still not well known. In the current process technology, heavily doped substrates, also known as “epi” substrates, are routinely employd in CMOS and BICMOS processes while lightly doped (1-30 Ω−cm) substrate are commonly used in bipolar and some CMOS technologies. Typical epi substrate consist of a lightly doped (1-30 Ω−cm) epitaxial layer grown on a degenerately doped (10-20 mΩ−cm) bulk substrate. The substrate effects on the performance of single-end spiral inductors are critical to silicon RF IC’s. Based on ADS Momentum simulation results and physical modeling, we present an extensive study on the substrate parasitic. So we will create a broadband and scalable model developed to accurately simulate on-chip inductors of various dimensions and substrate resistivities.

The 3D eddy current is identified as key element essential to accurately simulate

broadband characteristics. EM simulation using ADS Momentum is conducted to predict the on-chip inductor performance corresponding to wide range of substrate resistivity (ρsi =0.05~1KΩ ). Three operation models such as TEM, slow wave, and eddy current are presented. The model parameters manifest themselves physics-base through relevant correlation with ρsi over three operation modes. The onset of slow-wave mode can be consistently explained by a key element introduced in improve T-model, which accounts for the conductor loss due to eddy current arising from magnetic field coupling through substrate return path. It can facilitate optimization design of on-chip inductors through physics-based model parameters relevant to varying substrate resistivities. We find one reference to explain physic behind, but it is based on measurement result and presented. We use above result to research varying substrate resistivityes. From this reference, the single-end spiral inductors on epi, lightly doped, and quartz substrates are presented. The quartz sample serves as a control for no substrate eddy current can be induced in dielectric.

In Table

Table Summary of spiral inductors from reference paper

From reference paper, inductor Gp8nH is fabricated with a 0.32-Ω/sq aluminum

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