Master's Thesis

Chapter IIntroduction to Inductance

The impact of inductance has been a critical issue in the printed circuit board (PCB) design for quite some time. Since a PCB has both nearly perfect dielectric and ground plane, resistive losses in conductors as well as dielectrics are reasonably ignored. The inductance and capacitance are the major concerns in terms of the on-board signal transmission. In addition, the dimension of a PCB is relatively large compared with the signal wavelength, especially in the radio-frequency (RF) and microwave regime. For example, a 1 GHz signal has a wavelength of 30 cm in free space while a typical PCB can easily have one of its dimensions larger than several inches. This means the board has to be treated as a distributed system where the transmission line theory instead of KCL and KVL applies.

The significance of inductance includes increased transmission delay, signal reflection and ringing, inductive coupling, and digital switching noise due to AC voltage drop.

However, inductance has been largely ignored in an on-chip environment, where resistive together with capacitive effects are bigger concerns. In addition, the chip size is no larger than several millimeters, which is tiny enough compared with signal wavelengths.

With the operating frequencies looming into the gigahertz range, inductance is quickly coming into play. On one hand, the parasitic inductance of on-chip interconnects gives rise to signal delay and crosstalk between different signal paths. On the other hand, spiral inductors have been specifically designed and integrated onto a chip to achieve enhanced system integrity. Therefore there is a dramatically increased demand of IC designers for an accurate inductance model, both analytical and computational.MAXWELL EQUATIONS

Maxwells Equations are the complete set of laws for time-varying electromagnetic phenomena.

The four physical terms that describe electromagnetic fields are the electric field , electric displacement , magnetic field , and magnetic intensity . and are analogous in nature in that they both give the force on a moving charge given by,

(1)where is the velocity of the charge. and are analogous because they are independent of material properties and correspond to the space free charge and current respectively. and are related to and through the electric and magnetic polarization of the media material,

(2)

(3)

where is the electric permittivity and is the magnetic permeability.

The first one of Maxwell Equations is derived from Faradays Law, which states that a time-varying magnetic field induces electric fields. The integral form of Faradays Law is

(4)where is an arbitrary surface and is the edge of on which the magnetic field is integrated. Although the integral of electric field along a closed loop has the unit of voltage, it is different from the voltage defined for static fields, which is equal to the potential difference between two points and is independent of the path connecting the two points. The loop integral of the electric field induced by the time-varying magnetic field is defined as the electromotive force (emf) of that loop,

(5)Applying Stokes Theorem, the large-scale form of Faradays Law can be transferred to the differential form

(6)The second one of Maxwells Equations is based on Amperes Law, which describes that both conducting currents and time-varying electric fields generate magnetic fields. The integral form of Amperes Law is

(7)where is an arbitrary surface and is the edge of . Applying Stokes Theorem, the differential form of Amperes Law is

(8)The time derivative of the electric displacement has the same unit as current density that is so-called displacement current.

The third one of Maxwells Equations is based on Gausss Law, which states that the total flux of electric displacement from an arbitrary volume is equal to the net charge enclosed in that volume. The integral form of Gausss Law is

(9)where is the enclosed surface of volume and is the charge density. Applying the divergence theorem, the differential form of Gausss Law is

(10)The fourth one of Maxwells Equations is validated from the fact that there is no magnetic charge existing in nature. Thus the magnetic field lines are always closed and the net flux of magnetic fields through any closed surface is always zero,

(11)

(12)Writing the four equations together, the Maxwells Equations are

(13)In order to solve Maxwells Equations as a set of differential equations, proper boundary conditions need to be applied to get unique solutions. At the surface of two different media, the tangential electrical field and normal magnetic field are continuous

(14)

(15)The difference between the normal electrical displacements is equal to the surface charge density

(16)The difference between the tangential magnetic intensity is equal to the surface current density

(17)

For perfect dielectrics, there are no surface charges or currents, thus

(18)

(19)For perfect conductors under DC conditions, there are no electric field inside the conductor because the internal electric field built up by the surface charges cancels the external electric fields and therefore the net electric field is zero

(20)If the magnetic field is static without varying with time, it will penetrate the perfect conductor. For most conductors, the relative permeability is close to one.

For perfect conductors under AC conditions, both the electric and the magnetic fields inside the conductor are zero

(21)

(22)

INDUCTANCE DEFINITION

Inductance can be defined in several ways that are inherently consistent.

From the energy point of view, the inductance of a device describes the magnetic energy storage capability of the device. The time-average energy stored in the magnetic field is given by

(23)where is the current flowing through the device. Thus the energy definition of inductance is given by

(24)

Although the energy definition is the most fundamental definition of inductance, a more popular definition of inductance is through the magnetic flux leakage that is given by

(25)where is the magnetic flux expressed as

(26)

It is seen that the energy and flux definition are linked by the magnetic field. Therefore the inductance of a device can be calculated from computing the field pattern associated with the device.

From Faradays Law, voltage is linked to the magnetic flux by

(27)By substituting Equation (27) into Equation (25), the AC voltage drop across a device is proportional to the time derivative of the current

(28)This is commonly used in the circuit theory and the directions of and are defined in Figure 1

Figure 1 Voltage and Current Direction in Inductance Definition

The inductance definition gives insight of the impact of inductance on an electric network:

1. Current flowing through a conductor creates a magnetic field;

2. A time-varying current generates a time-varying magnetic field, which induces electric fields;

3. The induced electric field exerts forces on the electrons in the conductor carrying the current and causes emf.

The induced electric field from a conductor can affect not only the electron movement of the conductor itself, but also another conductor nearby. This leads to the separation of inductance definition into self-inductance and mutual inductance.

The self-inductance of a conductor describes the effects of the electromagnetic field generated by the conductor on itself. For a real conductor instead of an ideally filamentary one, it is convenient to further separate the definition of self-inductance into internal self-inductance and external self-inductance. The external one is due to the magnetic flux leakage from the inductor to the external surrounding. The internal one arises from the magnetic energy stored inside the conductor.

The overall classification of inductance is summarized in Figure 2.

Figure 2 Inductance Classification

INTERNAL SELF-INDUCTANCE

When applying the flux leakage definition to the internal of the conductor, the inconvenience arises from the difficulty of distinguishing the flux area, especially.

To gain better understanding of internal inductance, the magnetic energy definition of inductance is used. When a conductor carries a current, magnetic field is generated both inside and outside the conductor. Thus some of the magnetic energy is stored inside the conductor, which gives rise to the internal inductance. For those conductors that are not filamentary, the internal inductance exists. For example, the internal inductance of an infinitely long straight thick wire exists while the external one does not.Although one can try to solve the field pattern inside a given conductor to calculate the energy and thus internal inductance, a more efficient way to solve the problem is by using the definition of internal impedance per unit length, which is given by [2]

(29)where is the electric field on the conductor surface and is the total current flowing through the conductor.

The question is what is for a given ?

From Ohms Law, electric field is directly related to the conducting current and material conductivity

(30)Thus the internal impedance of a conductor per unit length is

(31)where the current is replaced by the integration of the current density over the conductor cross-section.

Since the conductivity of a practical conductor cannot be infinite, the internal impedance of a conductor includes not only internal inductance, but also internal resistance. In another word, the real part of represents internal resistance and the imaginary part of corresponds to internal inductance as shown in Figure 3.

Figure 3 Internal Impedance including Resistance and Inductance

The current distribution on the conductor cross-section is not uniform as long as the current varies with time. This non-uniform distribution is due to the skin effect, which is a frequency dependent phenomenon. As illustrated in Figure 4, the skin effect can be explained physically combining Faradays Law and Amperes Law.

Figure 4 Physical Explanation of Skin Effect

The conducting current generates magnetic field that is given by Amperes Law

(32)A time varying results in a time varying that induces an electric field given by Faradays Law

(33)The induced electric field causes a displacement current that in turn adds to the magnetic fields

(34)It is evident from Figure 4 that the induced electric field points to the direction that tends to cancel the conducting electric fields at the center of the conductor and reinforce it at the surface. From Ohms Law, therefore, the current will crowd at the conductor surface and void at the center. The skin effect becomes more significant at increased frequencies.

From the carrier transportation point of view, the current density can be written as

(35)where is the electron charge magnitude, is the carrier density, and is the carrier velocity. For good conductors like metals, the non-uniform distribution of current density as a result of skin effect is mainly due to the non-uniform distribution of the carrier velocity rather than the carrier density. This is because the conductivity of metals is so large that it is a good approximation that the electron density is uniform throughout the conductor. Thus skin effect in good conductors can also be viewed as the non-uniform distribution of the electron velocity, which is higher at the conductor surface than the center.

The extreme case will be a superconductive conductor, whose conductivity is infinity. All the current will flow at the surface of the conductor. There will be only surface current and no body current.

Since the carrier density is assumed to be uniform, there will be no normal electric fields perpendicular to the conductor surface, but only tangential ones along the current path. Choosing the tangential direction to be , the electric field can be expressed as

(36)The non-uniform distribution of the electric field can be solved through Maxwells Equations. The first and second equations are rewritten below

(6)

(8)By taking curl on both sides, Equation (6) becomes

(37)Substitute Equation (8) into Equation (37),

(38)From Gausss Law,

(39)The first term in Equation (38) can be expressed as

(40)For most of the practical conductors, it is a good assumption that there is no gradient of both the charge density and the material permittivity . Therefore Equation (38) can be simplified as

(41)The phasor form of this equation becomes a complex Helmholtz equation

(42)This is a partial differential equation and its solution depends on the boundary conditions.

A useful parameter called skin depth is defined to describe quantitatively the skin effect, which is defined as

(43)where is the angular frequency of the fields and is the conductor conductivity. The skin depth is derived as the depth at which the magnetic field can penetrate a conductor. It is also consistent with the depth beneath the surface of a conductor at which the current mainly flows.

EXTERNAL SELF-INDUCTANCE

A current flowing through a conductor generates magnetic field in its surroundings, which gives rise to the external inductance of the conductor. In order to calculate the external inductance using the flux definition, a finite flux area has to be properly defined. Since a current always flows through a closed path, the surface area can then be chosen as enclosed by the current loop that is illustrated in Figure 5.

Figure 5 Surface Area Enclosed by Current Loop

Therefore the rigorous definition of external inductance is referred to the inductance of a conductor loop. The physical explanation gives further insight of the loop inductance concept. Consider a current loop as shown in Figure 6.

Figure 6 Physical Explanation of Loop Inductance

The loop can be differentiated into many infinitesimal current elements. Each current segment generates circular lines surround itself given by

(32)Under the time-varying condition, the induced field is given by

(33)The induced electric field tends to point to the direction of opposing the change of the conducting current. As shown in Figure 6, if the current increases, the induced electric field points to the opposite direction of .

It is seen that the lines are closed, corresponding to the electromotive force (emf). For most of the cases in integrated circuits, the circuit operating frequency is so low that the displacement current is much smaller than the conducting current and therefore can be ignored.

The loop concept of the external inductance can be further illustrated by considering a straight wire with infinite length. Since the wire by itself does not construct a complete loop, there is no external inductance associated with the infinitely long wire.

The inductance of a filamentary conductor is given by

(44)where is the flux area bounded by the conductor. In reality, however, a conductor will have an arbitrary cross-section. The concept of average flux [12] is used to account for the conductor cross-section. The average flux is defined as

(45)where is the area of the conductor cross-section as illustrated in Figure 7.

Figure 7 Area of Conductor Cross-Section

As shown in Figure 8, the average flux can be understood by replacing the thick conductor loop with a filamentary loop that is located somewhere in between the inner and outer edges of the thick conductor. The average flux area of the thick conductor equals the area bounded by the filamentary loop.

Figure 8 Illustration of Average Flux

After properly defining the loop, it is necessary to solve the fields. Amperes Law states that the static magnetic field generated by a small current element in an unbounded, homogeneous, and isotropic media is

(46)where the vectors are illustrated in Figure 9.

Figure 9 Coordinates for Calculating the Magnetic Field from a Current Element

By summing all the magnetic field generated by all the current elements, the total magnetic field generated by a complete filamentary current loop is

(47)where is the current path.

The average flux of the current loop with arbitrary conductor cross-section is

(48)And the external self-inductance of a current loop is give by

(49)For low frequencies, the current distribution on the conductor cross-section can be approximated to be uniform. Therefore Equation (49) can be simplified to

(50)MUTUAL INDUCTANCE

Mutual inductance describes the magnetic coupling between two conductors. It refers to the interference between either two current loops or two segments on the same current loop. Similar to the external self-inductance definition, mutual inductance can also be explained by Faradays and Amperes Law. Figure 10 shows two coupled current loops that are labeled as and . Loop carries a conducting current and generates magnetic fields in the space. Some of the lines generated by may cut loop and generate electromotive forces (emf) on . The emf either enhances or impedes the current flow on loop .

Figure 10 Physical Explanation of Mutual Inductance between Current Loops

Mutual inductance defined by the magnetic flux leakage is given by

(51)The mutual flux between and is

(52)where is the surface bounded by loop . This expression is derived from two loops. It, however, does not seem to apply to the mutual inductance between conductor segments for the difficulty of defining the flux area. To solve this problem, the vector magnetic potential is used instead of the magnetic flux to calculate the mutual inductance, which avoids the use of flux area.

The magnetic vector potential is defined as

(53)From Amperes Law, the magnetic field generated by a conducting current is

(47)Combining Equation (53) and (47), vector magnetic potential generated by a current loop is

(54)By replacing the current with current density, can be expressed as

(55)where is the volume of the conductor.

The magnetic flux can be expressed by , applying Stokes theorem

(56)Thus the average mutual flux between two conductor loops is

(57)or

(58)where and are the cross section area of loop and respectively as illustrated in Figure 11, and are the infinitesimal segments of loop and respectively, and is the distance vector from to as shown in Figure 12.

Figure 11 Illustration of Conductor Cross-Sections

Figure 12 Coordinate Illustration to Calculate Mutual Inductance between Conductor Loops

The mutual inductance between two conductor loops is

(59)This equation is the general form of mutual inductance computation by taking into account the non-uniform distribution of the current density in the conductor. The dot product in this equation implies that the mutual inductance between two orthogonal loops is zero. The mutual inductance is positive when the currents in the two loops flow in the same direction, and vice versa.

Although this equation is derived from conductor loops, it can be easily modified for conductor segments by applying the virtual loop concept [12]. Consider two straight conductor segments, not necessarily coplanar, as shown in Figure 13.

Figure 13 Virtual Loops of Conductor Segments

The virtual loop is defined for segment by adding two straight edges and that are perpendicular to segment and extend to infinity. The virtual loop is closed at infinity. Thus the mutual inductance between segment and the virtual loop is the summation of the mutual inductance between segment and the four segments of the virtual loop

(60)

and are both zeros because and are orthogonal to . is also zero because of the infinite distance from to infinity. Therefore

(61)In this way, Equation (59) can be modified to calculate the mutual inductance between two conductor segments

where and donate the starting points of segment and respectively, and represent the ending points on segment and .Thesis OverviewIn this chapter, the concept of inductance has been explained in detail. Three inherently consistent definitions of inductance are given in different aspects: magnetic energy storage, magnetic flux leakage, and voltage-current relationship. The classification of inductance provides a more insightful understanding of the inductive mechanism, including internal self-inductance, external self-inductance, and mutual inductance.

Starting with Maxwells Equations, analytical expressions are derived for the three kinds of inductance, which serve as the guidelines for inductance calculation of specific cases.The internal self-inductance is caused by the skin effect, which is calculated by solving the complex Helmholtz equation. The external self-impedance can be computed by using the flux leakage definition. The mutual inductance is modeled by introducing the magnetic vector potential to revise the flux definition. It changes the surface integral of the magnetic field into the loop integral of the magnetic vector potential to calculate flux. In this way, one can develop the partial mutual inductance idea to compute the mutual inductance between two conductor segments instead of conductor loops.In the next chapters, the inductance classification and analytical model are applied to an on-chip environment to characterize interconnects and integrated inductors. The analytical model is revised to include the semiconductor substrate losses and the skin effect. And the computer simulation gives numerical solutions of the analytical model applied to on-chip interconnects and inductors.Chapter IICharacterizations of On-Chip Interconnects

As the integrated circuit evolves towards faster operating speed and higher level of system integration, the on-chip interconnection network becomes more complicated. On-chip interconnects are carrying signals with higher frequencies and extending into larger dimensions. There is a growing demand of high-frequency circuit designers to accurately characterize on-chip interconnects. A distributed equivalent circuit model has been used to model on-chip interconnects in terms of distributed impedance and admittance of the interconnect. The line inductance and resistance are closely related to the signal delay and attenuation. An on-chip interconnect defers from an ideal microstrip line because of the presence of the lossy semiconductor substrate. High frequency effects such as the skin effect should also be considered to model the interconnect. Internal self-Impedance of On-Chip InterconnectsFrom the discussion of the internal self-inductance in Chapter I, the internal impedance of an on-chip interconnect per unit length is given by

(31)

where is the current density on the interconnect cross-section, is the current density at the interconnect surface, and is the interconnect conductivity.

The current distribution on the interconnect cross-section is given by solving the complex Helmholtz equation

(42)The shape of the interconnect cross-section can be approximated to be rectangular. Applying the rectangular coordinates as shown in Figure 14 where is the direction of the current path,

Figure 14 Coordinates Illustration to Calculate Skin Effect of On-Chip Interconnects

Equation (42) can be separated as

(62)

(63)

(64)For good conductors, the normal electric fields and are negligible compared with . The propagation form of can be written as

(65)where gives the current distribution on the conductor cross-section. Substituting Equation (65) into (64) gives

(66)This a two dimensional problem.

For on-chip interconnects, the thickness of different metal layers is listed in Appendix II in terms of various CMOS processes [37]. It is seen that the thickness of the metal interconnects ranges roughly from 0.5 to 1.

Taking the 0.25 (m Aluminum process as an example, the skin depths of different metal layers versus frequency are compared in Figure 15. It is seen that in the typical radio-frequency range from 1GHz to 10GHz, the skin depth varies around several microns. Since higher metal layers have better conductivity, they have smaller skin depth than lower layers.

Figure 15 Skin Depth of Different Metal Layers in TSMC 0.25 (m Process

The nonuniform of current distribution on a conductor cross-section has to be considered when the dimension of the conductor cross-section is comparable to twice the skin depth at specified frequencies [13].

It is evidence from Appendix II and Figure 15 that when the operating frequency is below 10GHz, twice the skin depth is larger than the thickness of the corresponding metal layers. Thus the skin effect can be neglected on the thickness dimension of on-chip interconnects by assuming there is no current density variation. And the skin effect is only considered on the width dimension. This leads to the one dimensional approximation of Equation (66), which for on-chip interconnects can be written as

(67)where

The homogeneous solution of Equation (67) can be written as

(68)By choosing the coordinates to have the origin located at the center of the conductor as illustrated in Figure 14, becomes an even function in terms of

(69)where is a constant.

If the current density at the interconnect edge is assumed to be unity, the normalized current density is given by

(70)where is the width of the interconnect.

Since is a complex number, both the electric field and the current density inside the interconnect are also complex. This means the current flow on the interconnect cross-section will have a spatial dependent phase. As an example, the current density on the cross-section of a 4 (m wide interconnect is plotted in Figure 16. The interconnect is on the Metal 5 layer in a 0.25 (m CMOS process.

(a) Real Part of the Current Density

(b) Imaginary Part of the Current Density

(c) Magnitude of the Current DensityFigure 16 Current Density on the Interconnect Cross-Section at Different FrequenciesIt is evidence from Figure 16 that the skin effect becomes more significant at increased frequencies. At 1GHz, the magnitude of the current density in the middle of the interconnect is about 90% of that at the edge. However, when the frequency goes up to 10GHz, nearly all the current crowds to the edge of the interconnect.As seen from Figure 16 (a), the real part of the current density remains positive below 5GHz. However, as the frequency increases, the real part of the current density becomes negative in the middle of the interconnect. Since the plotted current density is normalized to the surface one, it means the current in the middle begins to flow in the opposite direction as to the surface current at increased frequencies. Such phase difference is due to the increased displacement current at high frequencies.

The internal impedance of an on-chip interconnect per unit length can be calculated by taking Equation (70) into (31), which gives

(71)or

(72)Again, taking as an example the interconnect on the Metal 5 layer in a 0.25 (m CMOS process, the internal inductance and resistance per unit length versus frequency with different line widths are plotted in Figure 17.

(a) Internal Inductance of the Interconnect

(b) Internal Resistance of the InterconnectFigure 17 Interconnect Internal Impedance per unit length versus Frequency

It is evident from Figure 17 (b) that the interconnect internal resistance increases with the frequency. This is because at higher frequencies more current crowds at the interconnect edges and thus less area of the interconnect cross-section contributes to the current conduction, which leads to the increase of the internal resistance. At low frequencies, the current distribution on the interconnect cross-section is close to uniform and thus wider interconnects have smaller internal resistance. However, at high frequencies, the current distribution is dominated by the skin effects and the interconnect width tends to have less effects on the internal resistance.It is seen from Figure 17 (a) that the interconnect internal inductance decreases with the frequency. This is mainly because at higher frequencies, the total current carried by the interconnect decreases for an increased internal resistance. This results in less magnetic energy stored inside the interconnect and thus less internal inductance.

Wider interconnects have more internal inductance and less internal resistance than narrower interconnects. This is because wider interconnects have bigger area of cross-section and thus more internal volume to store magnetic energy and conduct current. However, at higher frequencies, the interconnect width shows less effects on the internal impedance. This is mainly due to the skin effect that at high frequencies all current flow on the interconnect edges and the width do not really matter.

The internal impedance of interconnects with less width shows less frequency-dependence in that the skin effect is less important than in the wider interconnects.ON-CHIP microstrip systemAs shown in Figure 18, an on-chip interconnect is a planar metal or poly trace on a silicon substrate with a silicon dioxide layer in between as the dielectric insulator. Sometimes, there is also a metal plate beneath the silicon substrate, which is either the back metallization of the silicon wafer or the IC package that contains the silicon die.

Figure 18 On-Chip Interconnects

The characterizations of an on-chip interconnect are determined by not only the properties of metal (or poly) traces, but the whole metal-insulator-substrate-metal system. Such system can be classified as a microstrip system. The metal (or poly) trace is the signal path and the silicon substrate together with the metal plate behaves like the signal return path.

When there is a current flow in the metal trace, the same amount of current will flows in the return path but in the opposite direction. Only in this way can a signal transmit (or propagate). If the silicon substrate is a perfect dielectric without a conductive plane beneath it, there will be no return path for the current and the signal will not transmit or propagate along the metal trace.The distribution of the return current is strongly dependent on the frequency. At low frequencies as shown in Figure 19, the skin depth of the substrate is much larger than the substrate thickness. The magnetic field generated by the signal current will penetrate both the insulator and the substrate. It will be terminated at the surface of the metal plate. By assuming that the metal plate has perfect conductivity, there is no magnetic field inside the metal plate. The return current mainly flows on the surface of the metal plate given by

(73)

At high frequencies as shown in Figure 20, the magnetic field generated by the signal induces eddy current in the substrate, which shields the magnetic field penetration. Therefore some of the magnetic fields will not penetrate the substrate and will be terminated in the bulk of the substrate.

Figure 19 Return Current Distribution of On-Chip Interconnects at Low Frequencies

Figure 20 Return Current Distribution of On-Chip Interconnects at High Frequencies

The depth at which the magnetic field can penetrate the substrate depends on the substrate properties and the frequency, which is given by the skin depth

(43)It is seen from Equation (43) that the higher frequency and the higher substrate conductivity, the less the skin depth. This implies that more magnetic fields will be terminated in the substrate and therefore more return current flows in the substrate. If the frequency and the substrate conductivity are so high that the skin depth of the substrate is smaller than the substrate thickness, the majority of the return current will flow in the substrate instead of the metal plate. In this condition, the signal propagation on the interconnect will not be in quasi-TEM model any more, but in a so-called slow mode [14].External self-Impedance OF ON-CHIP INTERCONNECTS

For an ideal microstrip system with perfect dielectrics and ground planes, the line will be lossless. However, for an on-chip interconnect with a semi-conductive semiconductor substrate, there will be both external self-inductance and self-resistance. The external self-inductance arises from the magnetic flux leakage from microstrip line. The external self-resistance is a result of the low conductivity of the substrate.In order to calculate the external self-impedance of an on-chip chip interconnect, one needs to solve the field pattern associated with the system, which requires the solution of the current distribution. However, the distribution of the return current in the substrate and the metal plate is complicated that requires solving Maxwells Equations. A more time-efficient way of solving the problem is the magnetic image approach.First, consider an ideal microstrip system consists of a perfectly conductive microstrip line and an infinite large ground plane with perfect conductivity as shown in Figure 21.

Figure 21 Image Theory in Ideal Microstrip System

The magnetic field generated by the signal current is terminated at the surface of the perfect ground plane and extends to infinity above the ground plane. The boundary conditions on the surface of the ground plane are given by

(21)

(74)

(22)

(75)The total magnetic field generated by this microstrip system above the ideal ground plane is contributed by both the signal current and the surface current on the ground plane.

The boundary conditions can be satisfied by replacing the whole ground plane with an image current, which mirrors the signal current on the other side of the ground plane. The image carries the same amount of current as the signal but points in the opposite direction, as shown in Figure 21. In this way, the contribution of the return current to the total magnetic field equals that of the image current. By taking the magnetic image approach, one can solve the field pattern associated with an ideal microstrip system by simply summing the magnetic fields generated by the signal current and its image.

There are closed-form expressions available to calculate the external self-inductance of an ideal microstrip system [1]. For an ideal microstrip line as shown in Figure 22, its external self-inductance is given by [17]

Figure 22 Ideal Microstrip Line

(76)

where is the width of the microstrip line and is the vertical distance between the microstrip line and the ground plane.

However, Equations (76) cannot accurately model an on-chip interconnect mainly because of the lossy semiconductor substrate. In order to account for the lossy semiconductor substrate, the complex image method [16] is used.The complex image method is similar to the magnetic image approach for the ideal microstrip system. The difference is that for the complex image method, the vertical distance between the signal current and its image is complex. This is due to the lossy nature of the substrate.

The discussion of the complex image method can be divided into two conditions [14]:1. The skin depth of the substrate is much smaller than the substrate thickness and the signal propagates in the skin mode;

2. The skin depth of the substrate is larger than the substrate thickness and the signal propagates in the quasi-TEM mode.

First, when the skin depth of the substrate is much smaller than the substrate thickness, the majority of the return current flows near the surface of the substrate instead of the metal plate. As illustrated in Figure 23, all the return current in the substrate can be replace by an image current.

Figure 23 Illustration of Complex Image Method when the Skin Depth of the Substrate is Much Smaller than the Substrate ThicknessThe vertical distance between the signal current and its image is given by solving the Greens Function [15]

(77)where is the oxide thickness and is the skin depth of the lossy substrate rewritten as

(43)Here is the substrate permeability and is the substrate conductivity. The magnetic permeability of silicon is very close to that of free space, which equals H/m. The conductivity of the silicon substrate is directly related to the doping level [5].It is worth clarifying that a complex distance does not have a physical representation, rather a computational convenience [16].

Second, when the skin depth of the substrate is much larger than the substrate thickness, the return current will flow inside the metal plate as well as the substrate. As illustrated in Figure 24, the return current in the metal plate as well as the substrate can be replaced by an image current.

Figure 24 Illustration of Complex Image Method when the Skin Depth of the Substrate is Much Smaller than the Substrate ThicknessThe vertical distance between the signal current and its image is given by [15]

(78)

where is the oxide thickness, the thickness of the substrate, and is the skin depth of the lossy substrate.

By taking the complex image approach, the external self-impedance of an on-chip interconnect as shown in Figure 18 equals the external inductance of an ideal microstrip line as shown in Figure 22, where the vertical distance between the microstrip line and the ideal ground plane is given by

(79)There are two expressions for as given by Equation (77) and (78), depending on the relationship between the substrate thickness and the substrate skin depth.The skin depth of a silicon substrate with different doping level is plotted in Figure 25.

Figure 25 Skin Depth of Silicon Substrate at Different Doping Levels versus Frequency Recall the external self-inductance of an ideal microstrip line is given by

(76)

Applying Equation (76) for on-chip interconnects by applying the complex image method, becomes a complex number that gives a complex inductance. By examining the flux definition of the external self-inductance, a complex inductance can by explained by a complex flux given by [15]

(80)where the term appears as the complex inductance.

What physically gives rise to the external self-resistance? The answer is the lossy substrate. Since the silicon substrate serves as part of the current return path with a poor conductivity (about ten thousand times smaller than the conductivity of metal), the return current in the substrate suffers from resistive losses and therefore limits the signal current.Figure 26 plots the self inductance and resistance of an on-chip interconnect on the Metal 5 layer in a 0.25 (m CMOS process.

(a) External Self-Inductance of an On-Chip Interconnect

(b) External Self-Resistance of an On-Chip Interconnect

Figure 26 External Self-Impedance of an On-Chip Interconnect [Equation (76)]The thickness of silicon substrate is 250 (m [36]. The oxide thickness between the metal strip and the substrate is about 4 (m. The width of the interconnect is taken to be 4 (m. The result agrees with the full-wave solution given by ADS Momentum [15].

It is seen from Figure 26 that the substrate conductivity will greatly affect the external self-impedance of on-chip interconnects. Figure 26 (a) shows that external self-inductance decreases with increased substrate conductivity. This is because a higher conductive substrate has a smaller skin depth. Therefore fewer magnetic fields can penetrate the substrate and more return current is induced inside the substrate. This implies more return current will flow in the substrate instead of the metal ground plane. The external self-inductance is proportional to the flux area bounded by the signal current and the return current. The largest flux area and external self-inductance is achieved if all return current flows in the metal ground plane as shown in Figure 19. However, if more return current flows in the substrate as shown in Figure 20, the average vertical distance between the signal current and the total return current becomes smaller and thus the average flux becomes smaller. This is why higher substrate conductivity results in smaller external self-inductance. This also explains that when the substrate conductivity is very low, the external self-inductance shows little frequency dependence because the metal ground plane is the main return path. However, when the substrate conductivity becomes higher, the external self-inductance shows more frequency dependence.Similar analysis can be applied to external self-resistance, which increases with increased substrate conductivity. The smallest external self-resistance is achieved when the substrate has the lowest conductivity. And the whole system approaches an ideal microstrip system.MUTUAL Impedance BETWEEN ON-CHIP INTERCONNECTS

Between two coupled on-chip interconnects, there exist mutual inductance. The typical layout of on-chip interconnects are either vertical or horizontal. If the two interconnects are perpendicular to each other, they are not inductive coupled and the mutual inductance is zero. If they are parallel to each other, the mutual inductance is maximized.

Consider two parallel interconnects and as shown in Figure 27.

Figure 27 Parallel On-Chip Interconnects

The mutual inductance between p and q is defined as

(84)where is the voltage across the two ends of p induced by q. The low frequency expression of mutual inductance between two conductors and is given by [12]

(85)where and are the area of the conductor cross-section of k and m respectively, and are the starting points, and are the end points, and and are infinitesimal conductor segments. To account for high frequency effects and the lossy semiconductor substrate, Equation (85) has to be modified.Since the thickness of on-chip interconnects is smaller than double the skin depth below 10GHz, one can assume no frequency dependence of the current distribution on the thickness dimension of the on-chip interconnects, which can then be approximated as filaments as shown in Figure 28. The effects of the lossy substrate can be modeled by taking the complex image approach [16].

As shown in Figure 28, p and q are two thin parallel on-chip interconnects. q carries a current. The return current of q in the substrate and the metal ground plane is modeled by replacing the substrate and the metal ground plane with a complex image q.

Figure 28 Illustration of Mutual Inductance Calculation between Parallel On-Chip InterconnectsThe induced voltage on p as a result of the current carried by q is contributed by both the signal current on q and its image current, which is given by

(86)

The mutual inductance between and is the summation and . They are opposite in sign because the image current flows in the opposite direction of the signal current. By expanding Equation (85), is given by

(87)where and are the width of p and q respectively and is given by the complex image method. Similarly, is expressed as

(88)

The total mutual inductance between the two parallel on-chip interconnects p and q is given by

(89)

The integration can be numerically calculated by finite discretization. As illustrated in Figure 29, conductor p can be discretized into N1-by-N2 small segments, where N1 is the total number of mesh points on the x direction and N2 is the total number of mesh points on the y direction. Similar discretizations can be done on q and q. The differential mutual inductance is calculated for each two segments, and the total mutual inductance is the summation of all the differential ones.

Figure 29 Illustration of Discretization to Calculate Mutual Inductance between Parallel On-Chip Interconnects

Since both the current distribution on q and q and the distance between q and q are complex, the mutual inductance is complex, too. This complex inductance can again be interpreted into the combination of mutual inductance and mutual resistance.

(90)

The mutual inductance represents the magnetic coupling of the two interconnects through the magnetic field. The mutual resistance can be explained by the resistive coupling between the two interconnect through the substrate. The substrate provides a resistive path between the return current of both p and q.The simulation results have been compared with published results and agree well with the full-wave solution given by ADS Momentum [15]. Here, the mutual impedance between two parallel interconnects in a 0.25 (m CMOS process is studied.

Figure 30 plots the mutual impedance at different substrate doping levels versus the frequency. The two interconnects are both on Metal 5 layer and 4 (m wide. The edge-to-edge spacing between them is taken to be 2um. The thickness of the substrate is 250 (m and the oxide thickness is 3.96 (m [36]. The interconnect conductivity is S/m [36].

(a) Mutual Inductance

(b) Mutual Resistance

Figure 30 Mutual Impedance between Coupled On-Chip Interconnects versus Substrate Conductivity

From Figure 30, it is seen that the mutual inductance decreases and the mutual resistance increases at increased substrate conductivity. Both of them show more frequency dependence at higher substrate doping levels.

As seen from Figure 28, the total mutual inductance consists of two terms: one is the mutual inductance between p and q, the other is the one between p and q. As seen from Equation (85), mutual inductance is reverse proportional to the spacing between the two interconnects. Since the spacing between p and q is much smaller than that between p and q, dominates the total mutual inductance. Since the current on q and q are in the opposite direction, will subtract to get the total mutual inductance as shown in Equation (89). Therefore, if the spacing between p and q is constant, the further q is away from p, the smaller and the larger the total mutual inductance. When the substrate conductivity is very low, the majority of the return current of the interconnect flows in the metal ground plane. The distance between the signal current and the return current is close to twice the substrate thickness. The total mutual inductance is maximized. However, at increased substrate conductivity, more return current flows in the substrate instead of the metal ground plane, which leads to reduced vertical distance between q and q. This implies a larger and a smaller total mutual inductance.

Similar analysis can be applied for the mutual resistance. If the substrate conductivity is low, the return current of q will be mainly the surface current on the metal ground plane. Since the vertical distance between the metal ground plane and p is relatively large, the return current will induce little current on p. Therefore the mutual resistive coupling is small. However, if the substrate conductivity is high, more return current will flow near the substrate surface. Since they are very close to p with a thin insulator in between, they will induce significant amount of current on p and the mutual resistive coupling is strong.Chapter IIICharacterizations of ON-CHIP INDUCTORS

With the appearance of 0.25 (m, 0.18 (m, and recent 0.13 (m CMOS technologies, a complete radio-frequency system operating in the gigahertz range can be integrated on silicon with a conventional CMOS process including both active and passive components, such as a wireless transceiver. In monolithic radio-frequency analog integrated circuits, the on-chip inductor plays an important role. It is widely used in low noise amplifiers, mixers, and voltage-controlled oscillators. Integrating inductors on chip greatly increases the system integrity and reduces the packaging parasitic effects. However, because of its large occupation of chip area and significant magnetic energy leakage, on-chip inductors also affect the performance of high-frequency integrated circuits through electromagnetic coupling. Therefore, it is of great importance to accurately characterize on-chip inductors.

inductance of Rectangular On-Chip Inductors

Conventional on-chip inductor design follows the pattern of metal spirals. The metal is typically chosen to be on the top metal layer in order to reduce the parasitic capacitance between the metal and the silicon substrate. As an example, Figure 31 shows a rectangular on-chip inductor. Aside from the rectangular shape, circular and octagonal spirals are also commonly used.

Figure 31 On-Chip Rectangular Spiral Inductor

In order to model the inductance of the entire structure, a rectangular spiral is decomposed into segments and the total impedance is the summation of the self-impedance of each segment and the mutual impedance between each two of them. The decomposition of a rectangular spiral is illustrated in Figure 32. N is the number of spiral turns. The total number of segments is 4N. Similar decomposition can also be applied to octagonal and circular spirals.

Figure 32 Illustration of the Decomposition of an N-Turn Rectangular Spiral

The total AC voltage drop across the two ports of the spiral is the summation of the voltage drop on each segment

(91)According to the inductance definition, the voltage drop on each segment is contributed by both the self-inductance and the mutual inductance between the segment and every other segments

(92)Therefore, a matrix of voltage-current relationship can be established given by

(93)or

(94)where the inductance matrix is given by

(95)In this matrix, the diagonal elements are the self-inductance of each segment and the off-diagonal elements are the mutual inductance between each two segments. Applying Equation (91), the total inductance of the rectangular spiral is the summation of all the elements of the inductance matrix.

(96)

Since the inductor is an on-chip component, the inductance of the spiral given by Equation (96) is a complex inductance, which includes both inductance and resistance.Modeling impedance of on-chip inductors

The inductance of an on-chip inductor depends on several factors, such as total spiral length, number of turns, line spacing, line width, and substrate conductivity. These factors are illustrated in Figure 33. The effects of these factors are studied by computer simulation.

Figure 33 Illustration of Spiral Parameters

Figure 34 plots the impedance of a rectangular spiral with different total spiral lengths versus frequency, while keeping constant the number of turns, the line spacing, the line width, and the substrate conductivity. The process is chosen to be a 0.25 (m CMOS process. The spiral is on the Metal 5 layer. The number of turns is set to be 4. The line width is set to be 8 (m and line spacing is set to be 2 (m. The oxide thickness is 3.96 (m and the substrate thickness is 250 (m [36]. The metal conductivity is S/m [36]. The substrate doping level is assumed to be cm-3.

(a) Inductance

(b) Resistance

Figure 34 Spiral Inductor Impedance with Different Total Spiral Lengths versus FrequencyIt is seen from Figure 34 that both the inductance and the resistance of the spiral are roughly proportional to the total spiral length. Therefore, increasing the total length not only increases the inductance, but also gives rise to the resistive loss and thus decreases the inductor quality factor.

Figure 35 plots the impedance of a spiral inductor with different numbers of turns versus frequency, while keeping constant the total spiral length, the line spacing, the line width, and the substrate conductivity. The spiral parameters are the same as the previous analysis except that the total length is set to be 3000 (m and the number of turns varies.

(a) Inductance

(b) Resistance

Figure 35 Spiral Inductor Impedance with Different Number of Turns versus FrequencyIt is seen evident from Figure 35 that increasing the number of turns can effectively increase the spiral inductance as well as slightly decrease the resistance. This is because increasing the number of turns can effectively increases the total magnetic flux of the spiral [4].

Figure 36 plots the impedance of a spiral inductor with different line spacing versus frequency, while keeping constant the total spiral length, the number of turns, the line width, and the substrate conductivity. The spiral parameters are the same as the previous analysis except that the line spacing varies.

(a) Inductance

(b) Resistance

Figure 36 Spiral Inductor Impedance with Different Line Spacing versus Frequency

It is seen from Figure 36 that decreasing the line spacing is an effective way to increase the spiral inductance without having much effects on the resistance. This is mainly because decreasing the line spacing increases the mutual inductance between the adjacent segments and thus the total inductance. The smallest line spacing is limited by the process design rule.

Figure 37 plots the impedance of a spiral inductor with different substrate doping levels versus frequency, while keeping constant the total spiral length, the number of turns, the line spacing, and the line width. The spiral parameters are the same as the previous analysis except that the substrate conductivity varies.

(a) Inductance

(b) Resistance

Figure 37 Spiral Inductor Impedance with Different Substrate Doping Levels versus FrequencyIt is seen from Figure 37 that when the substrate doping level is low (around cm-3), the inductance increases slightly at higher frequencies. This is because the skin effect in the substrate can be neglected. As a result of the skin effect, the current crowds at the edge of the conductor at higher frequencies. This implies a decreased spacing between the current elements in the adjacent segments and therefore an increased mutual inductance. However, if the substrate doping level is high (above cm-3), the spiral inductance decreases with increased frequencies, because both the self-inductance of each segment and the mutual inductance between each two segments decreases at higher substrate doping levels, as explained in the previous chapter. Similar analysis can be applied for the spiral resistance, which increases at higher substrate conductivity.Multi-Layer On-Chip Inductors

There are several drawbacks of the conventional spiral inductors. First, they occupy large chip areas. Taken as an example, Figure 38 shows the die photo of a 1.5 GHz CMOS low noise amplifier [20], where two on-chip inductors (square spirals) take about two thirds of the die area.

Figure 38 Die Photo of A 1.5V 1.5 GHz CMOS Low Noise Amplifier

Even though, it is hard to fit an inductor of more than 10nH on a chip. The inductance value is greatly limited for on-chip inductors. As a result, intermediate and low frequency applications have to turn to off-chip alternatives. Second, because of the lossy semiconductor substrate on which an on-chip inductor is laid out, there are significant resistive as well as capacitive losses associated with on-chip inductors. Therefore the quality factor of an on-chip inductor is much lower than their off-chip counterparts. For example, the quality factor of on-chip inductors whose inductance is several nanohenries will not go beyond ten in the lower gigahertz range [29].

Achieving high quality factor is the key goal of the on-chip inductor design. The inductor quality factor is directly related to the noise performance of RF circuits, such as the phase noise of a LC tank voltage-controlled oscillator [21]. An effective way to increase the quality factor of an on-chip inductor without decreasing the inductance is the implementation of multi-layer on-chip inductors. The concept of multi-layer inductors is illustrated in Figure 39.

Figure 39 Illustration of Multilayer On-Chip Inductors

The multi-layer inductor takes advantage of the multiple metal layers in the conventional CMOS processes. It consists of several stacked N-turn planar spirals. As seen from Figure 39, all planar spirals have the same direction of current flow (either clockwise or counterclockwise) to maximize the magnetic coupling between them. There are several publications on the experimental investigation of such 3D on-chip inductors [22] [23] [24]. chapter iVHigh-speed On-Chip Digital signal Transmission

With the appearance of advanced CMOS processes, traditional digital circuits are operating at faster speed and higher level of system integration. For example, the latest microprocessors (Intel Pentium 4) have a clock speed up to 3.2 GHz with an 800 MHz front-side bus. When the signal switching speeds exceed 1 GHz and the chip densities exceed tens of millions of transistors, the RLC delays due to on-chip interconnects become significant [27].

At high frequencies especially in the radio-frequency regime, a distributed system analysis has to be applied to the on-chip interconnect networks, instead of the traditional lumped analysis. The transmission line theory is necessary to accurately analyze high-speed on-chip signal transmissions. Although the distributed transmission line has been largely applied for on-board situations with relatively large physical dimensions, it is only recently that attention has been drawn into an on-chip environment [26]. On-chip interconnects together with the lossy silicon substrate behave like a transmission line system when carrying high-frequency signals, especially digital pulses with steep rising and falling edges. The on-chip transmission line systems affects signal transmission in various aspects, including signal delay, attenuation, and dispersion.

Transmission line theory

As its name implies, a transmission line is a physical path along which the signal transmits, which consists of two or more parallel conductors. Typical examples of transmission lines are shown in Figure 40.

(a) Parallel Lines

(b) Coaxial Cable

(c) Strip Lines

(d) Microstrip Lines

(e) Coplanar Lines

Figure 40 Cross-Sections of Typical Transmission Lines

What all transmission lines have in common is that they all consist of both the signal path and the return path. This is easily understood because current has to flow in a closed conducting loop. Same amount of current flow in the return path as in the signal path, but in an opposite direction.

It is worth clarifying that the transmission theory applies to all signals, no matter the signal frequency or the line dimension. However, it is mainly used for radio-frequency and microwave regimes instead of low frequency situations where KVL and KCL are more convenient to analyze problems. KVL and KCL are the approximations of Maxwells Equation at low frequencies, assuming the space derivatives are negligible.

There are three fundamental modes of signal transmission: TEM, TE, and TM. For TEM mode, the electric and magnetic fields along the transmission direction are both zero. Taking to be the direction of transmission, the tangential fields are zeros

(98)For TE mode, there is only magnetic field on the direction of transmission,

(99)

For TM mode, there is only electric field on the direction of transmission,

(100)

When the conductors are completely surrounded by a uniform dielectric medium, the principal mode that can exist on a transmission line is the TEM mode. Since a microstrip line is not fully surrounded by a uniform dielectric medium, it does not support a TEM model. However, at low frequencies, the dominant mode on a microstrip line approaches TEM mode and is therefore called quasi-TEM mode.

For a TEM mode, the relationship between electric and magnetic fields is unique, which is given by

(101)where is a constant. Here the plus sign refers to the signal transmission along the direction and the minus sign refers to direction. Since has the unit of ( and relates uniquely the electric and magnetic fields, it is defined as the characteristic impedance of the transmission line. The characteristic impedance is the most fundamental parameter of a transmission line. For TE and TM modes, the relationship between and is more complicated so that it requires solving Maxwells Equations with properly applied boundary conditions.

Although one can solve the field patterns associated with certain mode of signal transmission on a transmission line, it is more straightforward and convenient for integrated circuit designers to model the transmission line with circuit elements: , , , and , which are distributed components that refer to inductance, resistance, capacitance, and conductance per unit length. The classic distributed model of a transmission line is shown in Figure 41.

Figure 41 Distributed Model of Transmission Lines per Unit Length

Physically, represents the electric field between the two conductors of the transmission line. stands for the magnetic field generated by both the conducting and displacement current. correspond to the resistive loss of the two conductors. means the loss in the dielectric.

The speed that a signal transmits on a transmission line is no faster than the speed of light, which is given by

(102)This means even for a superconductive transmission, there is still LC delay of the signal although the RC delay is negligible. The more inductive and capacitive of the transmission line, the more delay of the signal transmission.

The characteristic impedance of a transmission line is given by

(103)For a lossless transmission where and are zeros, is a real number

(104)

Since a realistic transmission line cannot have infinite length, it has to be arbitrarily terminated with a load. Terminating a transmission line gives rise to signal reflection. As shown in Figure 42, when an incident signal reaches the end of the transmission line, part of it transmits into the load and the rest of the signal is reflected and travels backward along the transmission line.

Figure 42 Terminated Transmission Line

In order to quantitate the signal reflection, a reflection coefficient at the impedance discontinuity is defined as

(105)where and are the voltage and current of the incident signal respectively and and are the voltage and current of the reflected signal. depends on the characteristic impedance of the transmission line and the load impedance only, which is given by

(106)When the load impedance is matched to the characteristic impedance

(107)there will be no signal reflection.

Signal reflection happens as long as impedances are mismatched. It is independent of frequencies and line dimensions. Or in another word, it happens for both high frequency and low frequency signals, both short and long lines, although people always ignore it at low frequencies on short lines. Consider a source driving a load through a transmission line as shown in Figure 43.

Figure 43 Source Drives Load through Transmission Line

Once an incident signal is sent onto the transmission line, it will bounce back and forth until it reaches the steady state. The steady state is determined by satisfying the boundary conditions. Given enough time, the magnitude of the voltage at the end of the transmission line is an infinite series given by

(108)This series will finally converge to

(109)which satisfies the boundary conditions. The steady state solution agrees with KVL or KCL. This is why at low frequency KVL and KCL are used instead of the transmission line theory because the convergence happens so fast compared with the time scale so that a transmitted signal reaches the steady state without notice. However, at high frequencies, the convergence time and the signal time scale are comparable and therefore the signal reflection can be observed such as the ringing effects.

On-chip transmission lines

The transmission line theory applies to signals at any frequencies. However, in the low frequency regime when the wavelength of the signal is much larger than the physical dimensions of the transmission line, classic circuit theory are more convenient to analyze problems. A rule of thumb [8] is when the signal wavelength is larger than ten-time the line length, KVL and KCL are good enough to analyze the electric network instead of applying the distributed transmission line theory.

The question is which one to use, distributed transmission line theory or KCL and KVL, in an on-chip environment.

The answer depends on both the signal frequency and the interconnect length. Special cases of interest includes on-chip bus lines and high-speed digital clock trees, where the interconnect lengths are relatively long and the rising and falling time of the digital pulses are extremely short. Therefore signal degradation, dispersion, and reflection will come into play.

On-chip interconnects are essentially microstrip lines. As shown in Figure 44, the oxide layer serves as the dielectric insulator and the lossy substrate together with a metal plate acts like the return path or the ground plane.

Figure 44 On-Chip Interconnects

In order to accurately model the line characterizations, the commercial field solver, Sonnet, has been used for computer simulation.

Sonnet is a high-frequency electromagnetic simulator for 3D planar circuits. Sonnet together with HFSS and ADS Momentum are the three most popular electromagnetic simulation tools for radio-frequency and microwave design. Sonnet differs from HFSS in that Sonnet is limited to planar structures while HFSS is a complete 3D EM solver. Unlike HFSS and Sonnet, ADS Momentum is based on equivalent circuit theories instead of solving the fields. The accuracy and time efficiency of HFSS, Sonnet, and ADS Momentum are compared in Table 1.

Table 1 Comparison of HFSS, Sonnet, and ADS Momentum

HFSSSonnetADS Momentum

AccuracyHighestMediumLowest

Time EfficiencyLowestMediumHighest

Sonnet yields good trade-off between accuracy and time efficiency compared with HFSS and ADS Momentum.

Sonnet treats the simulated structure as an N-port network and outputs the scattering parameters of the network. Scattering parameters can be transformed into impedance matrix, which can then be plugged into EDA (Electronics Design Automation) tools such as SPICE and ADS for electric network simulation.

Taking for example the interconnect on Metal 1 layer in TSMC 0.25 (m CMOS process. The oxide thickness is 3.96 (m and the substrate thickness is 250 (m. The metal conductivity is S/m and the substrate doping level is cm-3, which corresponds to a conductivity of about 500 S/m. The characteristic impedance simulated by Sonnet is plotted in Figure 45.

(a) Real Part

(b) Imaginary Part

Figure 45 Characteristic Impedance of On-Chip Interconnect versus Frequency

It is seen from Figure 45 that the characteristic impedance is a frequency dependent complex number. At low frequencies, is given by

(103)At very high frequencies, the term dominates and is approximated as

(104)It is evident from Figure 45 that at tens of gigahertz, the imaginary part of quickly approaches zero and the real part of shows less frequency dependence than at low frequencies.

The scattering parameters of the interconnect on Metal 1 layer in TSMC 0.25 (m CMOS process is plotted in Figure 46, which is 1 mm long and 4 (m wide.

Figure 46 S Parameters of On-Chip Interconnect

Once the scattering parameters are known, they are saved as a Touchstone file with the .snp file extension and taken into ADS for simulation. ADS can directly interpret the Touchstone file as a circuit component.

on-chip signal attenuation and dispersion

First, the Sonnet simulation was run on a 1 cm long on-chip interconnect to investigate the signal reflection. As shown in Figure 47, the interconnect is driven by an ideal pulse source. Each box represents a 1 mm long 4 (m wide on-chip interconnect on Metal 1 layer in a TSMC 0.25 (m CMOS process.

Figure 47 Simulation Setup of Signal Reflection

If the interconnect is open ended, the pulse signal transmission and reflection is shown in Figure 48.

Figure 48 Pulse Propagation with Open Ended

It is clearly seen the signal degradation and positive reflection in Figure 48. The amplitude attenuation is about 200 mV every 3 mm. This is mainly due to the resistive loss of the transmission line. Such resistive loss depends on not only the metal conductivity, but also the silicon substrate loss.

Figure 49 shows the situation with short ended, where negative reflection is seen as well as signal degradation.

Figure 49 Pulse Transmission with Short Ended

According to Figure 45, the characteristic impedance of the interconnect approaches ( around 10 GHz. Therefore by terminating the line with a ( resistor, the signal reflection is minimized as seen in Figure 50.

Figure 50 Pulse Transmission with Matched Load

Next, the ideal source and load in Figure 47 are replaced by two inverters to investigate the effect of on-chip interconnects on the digital signal transmission. The simulation setup is shown in Figure 51. A driver inverter drives the receiver inverter through the interconnect. A pulse is generated by the driver inverter, transmits along the interconnect, and tries to switch the receiver inverter.

Figure 51 Simulation Setup for Digital Signal Transmission

In order to achieve a digital pulse with short rising and falling time around 10 ps, the driver inverter has been designed with TSMC 0.25 (m CMOS process as shown in Figure 52. The SPICE BSIM models of both NMOS and PMOS of this process are listed in Appendix III.

Figure 52 Schematic of Driver Inverter

The lengths of both the NMOS and the PMOS are 0.24 (m. The DC power supply (VDD) for this process is 2.5 V. In order to set the inverter switching point to be around half of VDD, the width of the PMOS is set to be 3 times larger than the NMOS. Figure 53 shows the switching characteristics of the inverter where the switching point is about 1.26 V.

Figure 53 Switching Characteristics of the Driver Inverter

By choosing the width of the NMOS to be 50 (m and PMOS 150 (m, the delay of the inverter is minimized. The result of the transient simulation is shown in Figure 54.

Figure 54 Intrinsic Inverter Delay

As read from Figure 54,

This corresponds to tens of gigahertz frequency components of the pulse.

The receiver inverter has the minimum length and width as shown in Figure 55. In this process, the MOSFETs are constructed with standard gate fingers. Each gate finger is 10 (m long. Thus the width of each MOSFET is in multiples of 10 (m.

Figure 55 Schematic of Receiver Inverter

Because of the signal attenuation along the interconnect, there exists a critical length of the interconnect above which the digital pulse will not switch the receiver inverter. ADS simulation shows that the critical length is about 8 mm. As shown in Figure 56, if the interconnect is longer than 8mm, the signal is not able to reach the switching point at the end of the transmission line and therefore cannot switch the next digital stage.

Figure 56 Effects of Interconnect on Digital Gates Driving Capability

High frequency components suffer more from signal attenuation than lower frequency components. This is mainly because the skin effect in the substrate becomes more significant at higher frequencies. The eddy current shields the magnetic field from penetrating further into the substrate. Therefore, the more eddy current induced near the substrate surface, the more return current flows in the substrate instead of the metal plate. This gives rise to the effective resistance of the interconnect and thus the signal attenuation.

ADS simulation compares the signal attenuation on a 1 cm interconnect at different frequencies: 1 GHz, 3 GHz, 5 GHz, 7 GHz, and 9 GHz. As seen from Figure 57, the attenuation of the 1 GHz signal is 20% while that of the 9 GHz signal is about 70%.

Figure 57 Signal Attenuation at Different Frequencies

Another problem with the high-speed digital signal transmission is the signal dispersion. This is because different frequency components have different transmission speeds. Again, the transmission speed of one frequency component is given by

(110)where and are both frequency dependent. From Figure 57, it is seen that the delay of the 1 GHz signal on a 1 cm long interconnect is about 220 ps while the delay of the 9 GHz signal is close to 150 ps. The signal dispersion strongly depends on the substrate doping. Computer simulation [15] shows that at higher substrate doping level, the distributed inductance and capacitance of an on-chip interconnect show more frequency dependence than the substrate with less conductivity.

Chapter V

Electromagnetic Coupling effects

While advanced semiconductor technologies have brought into integrated circuits faster operating speed and higher level of system integration, electromagnetic effects accompany the on-chip evolutions. In an on-chip environment, all circuit components share a common substrate. The substrate is not a perfect dielectric, but a semiconductor. There are significant parasitic resistance, capacitance, and inductance associated with the semiconductor substrate. It thus establishes a complicated connection between circuit components although they are not connected directly. Once the operating frequency of the circuit devices becomes higher, the substrate parasitics becomes more significant and the coupling between devices through substrate becomes stronger.Since achieving high chip density is pursued by VLSI designers, the spacing between the integrated circuit components decreases significantly with advanced technology. For those devices operating at high frequencies and carries high power signals, they will leak large amount of electric and magnetic energy into the surroundings. The electric and magnetic field can directly couple to the nearby devices and induce noise voltage and current.Scattering Parameters

Scattering Parameters, or simply S Parameters, are widely used in radio-frequency and microwave electric network analysis.

Consider an N-port network as shown in Figure 58. is the voltage of the incident signal and is the voltage of the reflected signal.

Figure 58 N-port Network

The S Parameters of such a network are defined as

(111)or

(112)The S matrix defines the relationship between the reflected and the incident signals.

The advantage of using S Parameters lies in the fact that they can be directly measured by instruments, such as a network analyzer. However, voltage and current cannot be measured in a direct manner at microwave frequencies. Instead, what can be measured directly are field and power. Therefore, S Parameters are more straightforward than impedance parameters (or Z Parameters) and admittance parameters (or Y Parameters) to analyze high-frequency electric networks. Z and Y Parameters can be derived from S Parameters, which are given by

(113)

(114)

For a commonly seen two-port network, the S matrix is given by

(115)where the four S parameters are defined as

(116)

is the insertion loss of Port 1 when Port 2 is matched. is the forward gain from Port 1 to Port 2 when Port 2 is matched. is the insertion loss of Port 2 when Port 1 is matched. is the forward gain from Port 2 to Port 1 when Port 1 is matched. For a symmetric network,

(117)

Experiment setup of electromagnetic coupling measurement

In order to experimentally investigate the electromagnetic coupling effects in an on-chip environment, several testing structures have been designed to demonstrate the coupling effects. The testing chip was taped out through MOSIS in an AMI 0.5 (m CMOS process. for this process is 0.3 (m and all feature dimensions are multiples of . The smallest feature size, the MOSFET gate length, is 0.6 (m. This process has 3 metal layers and 2 poly layers. The process is for 5-V single power supply applications.

Each pair of coupled devices is tested as a two port network by connecting each device to a single port. The electromagnetic coupling between them is investigated by measuring the S Parameters of the network.

The experiment is set up as shown in Figure 59.

Figure 59 Experiment Setup for Electromagnetic Coupling Measurement

The devices are first connected to bond pads through on-chip interconnects. For active devices, the electrostatic discharge (ESD) structure has to be used together with the bond pad to avoid breakdown caused by the high electrostatic voltage on the package pin. On the p-type substrate, a p+ guard ring is buried to isolate the coupled devices from other on-chip structures. The die is packaged for on-board manipulation. In this experiment, the IC package is a standard 28-pin leadless chip carrier (LCC) ceramic package. The advantage of using the LCC package instead of the lead dual-inline package (DIP) is the greatly reduced parasitics induced by the bond wire and the package pin. The IC package is soldered onto a printed circuit board. The pins are connected to the SMA (semi-miniature adapter) connectors via on-board copper traces. SMA connectors have the characteristic impedance of 50 if properly terminated. The network analyzer is connected to the SMA connectors through 50 cables. To reduce on-board AC noise and crosstalk, a ground plane has to be plated on the back side of the board.

coupling between N-Wells

In order to investigate the electromagnetic coupling through the substrate, two n-wells have been laid out on a p-type silicon substrate as shown in Figure 60. Each n-well is 28.8 (m long by 28.8 (m wide. The edge-to-edge distance between the two wells is 102.45 (m. For this process, the n-channel low-field mobility is 496.57 cm2/V-s and the p-channel low-field mobility is 151.80 cm2/V-s [36]. This corresponds to the surface doping level of the p-type substrate of about cm-3 and the n-well doping level of about cm-3 [5].

Figure 60 Coupled n-wells

The measured data is plotted in Figure 61. Two peaks of |S21| are observed at 8 GHz and 4 GHz respectively. A maximum |S21| of about -16 dB occurs at 8 GHz. |S21| varies around -20 dB in the frequency range of 2 GHz to 9 GHz.

Figure 61 - |S21| of Coupled N-Wellscoupling between on-chip Spiral inductors

On-chip inductors have relative large physical dimensions and are associated with significant magnetic energy leakage. Since they are typically used in analog radio-frequency circuits and operate in the gigahertz range, the electromagnetic coupling between two spiral inductors needs to be investigated.

Planar coupling between two side-by-side spiral inductors on the same metal layer has been experimentally studied in [28] and a maximum of -25 dB has been observed at 1.5 GHz.

Here, the vertical coupling of two spirals has been investigated. As shown in Figure 62, two identical spirals overlap with one on top of the other. The vertical distance between them is about 1 (m.

Figure 62 Vertically Coupled On-Chip InductorsThe physical dimensions of the spiral are shown in Figure 63. The total number of turns is six. The line width is 12 (m. The line spacing is 3 (m.

Figure 63 Physical Dimensions of the Coupled Spirals in the Transformer

The equivalent circuit model for an on-chip spiral inductor is shown in Figure 64 [31]. and are the series inductance and resistance of the spiral respectively. represents the capacitance between metal traces. is the oxide capacitance from the spiral to the substrate. and models the substrate capacitance and conductance.

Figure 64 Single- Equivalent Circuit Model of On-Chip Spiral Inductors

The equivalent circuit model of the coupled spirals is shown in Figure 65. The coupling is mainly due to the magnetic field coupling represented by the mutual inductance .

Figure 65 Equivalent Circuit Model of Vertically Coupled Spirals

The measured data is plotted in Figure 61. A maximum |S21| of about -11 dB occurs at 3 GHz and 5 GHz. In a wide bandwidth from 2.5 GHz to 5.5 GHz, |S21| remains above -15 dB.

Figure 66 |S21| of Vertically Coupled SpiralsSince the coupling between two vertically coupled spirals is very strong, they can be used as an on-chip transformer, especially for heterogeneous integration applications, where the via through the wafer is hard to fabricate.coupling between on-chip inductors and transistors

In radio-frequency integrated circuits where passive and active devices coexist on the same chip, spiral inductors can couple a significant amount of electromagnetic energy to sensitive transistors [33]. In order to investigate the coupling between on-chip inductors and transistors, on-chip experiment has been implemented. As shown in Figure 67, a spiral inductor and a transistor have been laid out side-by-side.

Figure 67 Coupled Spiral Inductor and TransistorThe physical dimensions of the spiral are shown in Figure 68. The total number of turns is 4.75. The width and spacing of the conductor are 15 (m and 5.1 (m respectively. The spiral is on the Metal 3 layer.

Figure 68 Physical Dimensions of the Spirals coupled to the Transistor

The transistor has a gate length of 0.6 (m and a width of 300 (m. The edge-to-edge spacing between the spiral and the poly gate is 51.45 (m. The experiment compares the coupling effects on two kinds of transistor layout configurations: one has single gate finger and the other has multiple gate fingers, as illustrated in Figure 69.

(a) NMOS with Single Gate Finger

(b) NMOS with 5 Gate FingersFigure 69 Layout of Single and Multiple Gate Finger NMOS TransistorsThe measured data is plotted in Figure 70. Both curves show a maximum |S21| of about -17 dB, although at different frequencies.

Figure 70 |S21| of Coupled Spiral and NMOS Transistor with Single and Multiple Gate FingersThe coupling between spirals and transistors can significantly affect the performance of analog radio-frequency integrated circuits, especially wireless transceivers. In a monolithic wireless transceiver, on-chip inductors are used in both the low noise amplifier at the input stage and the power amplifier at the output stage. Since the low noise amplifier typically has a high power gain of about 20 dB, the active transistors in the amplifier have to be very wide typically up to several hundred microns. Therefore, the electromagnetic coupling between the inductors and the amplifying transistors can form feedbacks either inside the low noise amplifier or between the low noise amplifier and the power amplifier. Such feedback will degrade the noise performance of the low noise amplifier.Digital Switching NoiseThe fast switching of integrated digital circuits can induce large amount switching current into the substrate, which can propagate in the substrate and couple to analog components [9] [34]. In order to investigate the digital switching noise effects, the coupling between an inverter and a transistor has been studied. As shown in Figure 71, a digital inverter and a NMOS transistor have been laid out side by side on a common substrate.

Figure 71 Coupling between Digital Inverter and Transistor

The transistor is 0.6 (m long and 300 (m wide. The schematic of the digital inverter is shown in Figure 72.

Figure 72 Schematic of Digital Inverter as the Digital Switching Noise GeneratorThe induced noise signal on the transistor gate caused by the inverter switching is measured by the oscilloscope and displayed in Figure 73. The top waveform is the input signal into the inverter, which is an 800 MHz sinusoidal signal with the amplitude of 1.5 V. The waveform below is the induced signal on the transistor gate whose amplitude goes up to 180 mV.

Figure 73 Induced Noise Signal on Transistor Gate by Digital SwitchingSince digital switching generates frequency harmonics, the coupling effects will vary with different frequency components. Figure 74 is a 3D bar plotting showing the digital switching noise. The axis is the inverter switching speed. Measurement has been taken at ten different frequencies from 100 MHz to 990 MHz at which the inverter operates. The axis is the frequency at which the coupling effect on the transistor is measured. The axis is the voltage that is induced on the transistor gate by the digital switching of the inverter.

Figure 74 Voltage induced by Digital Switching on the Transistor GateIt is seen from Figure 74 that a maximum induced voltage of about 130 mV occurs at 800 MHz when the inverter is operating at the same frequency. Strong coupling happens mainly at the fundamental and the second harmonic frequency of the inverter switching speed. When the inverter is operating in the frequency range of 600 MHz to 1 GHz, the coupling is maximized.Conclusions and future workThis thesis investigates some of the radio-frequency effects associated with modern high frequency, high chip density integrated circuits.

First, the concept of inductance has been explained in detail. Three inherently consistent definitions of inductance are given in different aspects: magnetic energy storage, magnetic flux leakage, and voltage-current relationship. The classification of inductance provides a more insightful understanding of the inductive mechanism, including internal self-inductance, external self-inductance, and mutual inductance. Physical pictures are used to explain the fundamentals behind each of the inductance classifications. Starting with Maxwells Equations, analytical expressions are also derived for the three kinds of inductance, which serve as the guidelines for inductance calculation for specific cases.The universal inductance definitions and analytical expressions are applied to an on-chip environment by considering semiconductor substrate losses as well as the skin effect to model metal interconnects and integrated spiral inductors.The internal self-impedance of on-chip interconnects is caused by the skin effect. It is calculated by solving the complex Helmholtz equation, which can be simplified by applying the 1D approximation for on-chip interconnects. The external self-impedance of a single on-chip interconnect and the mutual inductance between parallel interconnects are modeled by taking the complex image theory approach and considering the skin effects. The method of modeling on-chip interconnects is extended to characterize integrated inductors by decomposing the spiral into an inductance matrix.Computer simulation gives accurate results compared with the published data simulated by ADS Momentum. Detailed discussion explains the effects of the semiconductor substrate on both the inductance and resistance of on-chip interconnects and inductors. Design optimization of spiral inductors is also provided aided by the computer simulation.The modeling technique presented in this thesis adds the skin effect into the existing complex image approach to characterize on-chip interconnects. It further provides an alternative way of optimizing on-chip inductors, and the design of novel inductive devices (3D inductors). Comparing with full-wave simulation done by commercial software tools, the numerical method developed here not only provides time-efficient solutions, but also gives insight into the factors that determine the inductance and resistance of on-chip interconnects and inductors.

The characterization of on-chip interconnects is used to investigate the high-speed on-chip digital signal transmission. It is found out that the resistance of the interconnect is the main factor that gives rise to the signal attenuation, while the frequency-dependent line inductance causes the signal dispersion and delay.Experimental measurement demonstrates the electromagnetic coupling effects between on-chip high frequency integrated circuit components, including n-wells, spiral inductors, and t