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A steady-state analytical solution for MOSFET channel temperature estimation Rajput, Harish C. 2011

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A Steady-state Analytical Solution for MOSFET Channel Temperature Estimation HeatMOS© by Harish C. Rajput  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) April 2011 © Harish C. Rajput 2011  Abstract A steady state analytical solution for MOSFET (metal oxide semiconductor field effect transistor) channel temperature estimation has been derived and the analytical model has been used to develop a software tool called HeatMOS©. HeatMOS© estimates the MOSFET channel temperature based on information from the device layout and an industry standard BSIM3 compact model. The steady state solution is an approximation for the channel temperature distribution along its length. The HeatMOS© model has been designed to be integrated into a VLSI CAD flow to predict the steady state temperature of a full micro-chip. An equivalent M-network model for steady state temperature can be extended for each MOSFET device in a complete micro-chip. In future work, HeatMOS© can be combined with the models of interconnect to develop a full micro-chip thermal analysis software tool.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Acronyms  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xv  1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Motivation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.3  Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  1.4  Scope and Organization  14  . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents 2  3  Heating in Integrated Circuits . . . . . . . . . . . . . . . . . . . . .  16  2.1  Electrostatic Analogy of Heat Conduction . . . . . . . . . . . . .  18  2.2  Heat Equation  . . . . . . . . . . . . . . . . . . . . . . . . . . .  20  2.3  Heat Generation in MOSFETs . . . . . . . . . . . . . . . . . . .  22  2.4  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  Thermal Model and Analytical Solution  . . . . . . . . . . . . . . .  25  3.1  BSIM3 MOSFET Model . . . . . . . . . . . . . . . . . . . . . .  25  3.2  MOSFET Channel . . . . . . . . . . . . . . . . . . . . . . . . .  27  3.2.1  Channel Geometry for Thermal Model  . . . . . . . . . .  32  3.2.2  Electrical Resistance . . . . . . . . . . . . . . . . . . . .  34  3.2.3  Channel Height Models . . . . . . . . . . . . . . . . . .  37  3.2.4  Channel Length Models . . . . . . . . . . . . . . . . . .  39  3.2.5  Thermal Conductance . . . . . . . . . . . . . . . . . . .  40  3.3  Solution of the Heat Equation  . . . . . . . . . . . . . . . . . . .  42  3.3.1  Region 1: Non-uniform cross-sectional area  . . . . . . .  42  3.3.2  Region 2: Uniform cross-sectional area . . . . . . . . . .  44 iv  Table of Contents  4  5  3.3.3  Boundary Conditions and Complete Solution . . . . . . .  47  3.3.4  The Average Channel Temperature  . . . . . . . . . . . .  51  3.3.5  The Maximum Channel Temperature . . . . . . . . . . .  51  3.3.6  The Channel Temperature at Pinch-off  . . . . . . . . . .  53  3.4  M-Network Representation of the Channel  . . . . . . . . . . . .  53  3.5  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  58  HeatMOS©  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59  4.1  About HeatMOS© . . . . . . . . . . . . . . . . . . . . . . . . .  59  4.2  Integration in the VLSI CAD Flow  . . . . . . . . . . . . . . . .  61  Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . .  63  5.1  Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  65  5.1.1  Test Scenario using Externally Supplied Drain Current . .  67  5.1.2  Test Scenario for a Smaller MOSFET Device . . . . . . .  69  5.1.3  Test Scenario for the Case When the Source, Drain, and  5.2  the Reference Temperatures are Same . . . . . . . . . . .  69  Saturation Mode of Operation . . . . . . . . . . . . . . . . . . .  72  v  Table of Contents 5.2.1  Test Scenario When Gate and Drain are at Same Voltage  5.2.2  Test Scenario for the Case of Different Gate and Drain Voltages  5.2.3  Test Scenario for the Case of High Gate and Drain Voltages  5.2.4  Test Scenario for a Large MOSFET Device with High Drain Current  5.2.5  . . . . . . . . . . . . . . . . . . . . . . . . . .  76  76  . . . . . . . . . . . . . . . . . . . . . . . . . .  79  Linear/Ohmic Mode of Operation . . . . . . . . . . . . . . . . .  79  5.3.1  Test Scenario for the Case of Different Temperatures at the Source and the Drain  5.3.2  . . . . . . . . . . . . . . . . . . .  Average and Maximum Temperature Distributions 5.4.1  . . . . . . . .  81 84  Channel Temperature Distribution with Drain Voltage and Drain Current  5.4.2  81  Test Scenario for a Larger Device in the Ohmic Operating Region . . . . . . . . . . . . . . . . . . . . . . . . . . .  5.4  74  Test Scenario for Different Source/Drain Temperatures and Voltages  5.3  . . . . . . . . . . . . . . . . . . . . . . . . . .  72  . . . . . . . . . . . . . . . . . . . . . . .  85  Channel Temperature Distribution with Effective Gate Voltage and Drain Current . . . . . . . . . . . . . . . . . . .  85  5.5  Discussion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  90  5.6  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  92 vi  Table of Contents 6  Conclusions and Future Directions  . . . . . . . . . . . . . . . . . .  94  6.1  Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .  94  6.2  Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  6.2.1  Fundamental Limitations  . . . . . . . . . . . . . . . . .  95  6.2.2  Limitations of First Implementation . . . . . . . . . . . .  96  Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  97  6.3  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100  Appendices  A BSIM3 Model Parameters  B SPICE Netlist  . . . . . . . . . . . . . . . . . . . . . . . 106  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109  C Numerical Solution of the Heat Equation . . . . . . . . . . . . . . . 112  D HeatMOS© Development . . . . . . . . . . . . . . . . . . . . . . . . 124 D.1 Development Environment . . . . . . . . . . . . . . . . . . . . . 124 D.2 A Sample HeatMOS© Execution  . . . . . . . . . . . . . . . . . 124 vii  List of Tables 2.1  Thermal and electrostatic analogies  . . . . . . . . . . . . . . . .  19  2.2  Thermal and electrical conductivities (300 K) . . . . . . . . . . .  19  2.3  Thermal and electrical versions of heat transfer or capacitance . .  20  A.1 BSIM3 Parameters used in HeatMOS© . . . . . . . . . . . . . . 107 A.2 Other parameters used in HeatMOS© . . . . . . . . . . . . . . . 108  viii  List of Figures 1.1  Transistor thermal resistance versus CMOS technology scaling . .  3  1.2  Impact of technology scaling on temperature increase . . . . . . .  5  1.3  SoC power consumption trends  . . . . . . . . . . . . . . . . . .  11  1.4  A typical micro-chip layout . . . . . . . . . . . . . . . . . . . . .  12  1.5  Interconnect-transistor network . . . . . . . . . . . . . . . . . . .  13  2.1  Conservation of heat (energy) within a differential slice of a resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21  2.2  Cross-sectional view of an nMOS transistor . . . . . . . . . . . .  23  3.1  A typical I-V characteristic of an nMOS transistor . . . . . . . . .  28  3.2  A comparison for I-V characteristics of an nMOS transistor  . . .  30  3.3  Simplified 2-D cross-section of an nMOS transistor . . . . . . . .  31  3.4  3D view of the channel geometry for thermal model . . . . . . . .  34 ix  List of Figures 3.5  3D view of region 1 of the channel . . . . . . . . . . . . . . . . .  42  3.6  3D view of the region 2 of the channel . . . . . . . . . . . . . . .  45  3.7  Resistor M-network used for EMR verification . . . . . . . . . .  55  4.1  VLSI CAD flow . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  5.1  An nMOS layout . . . . . . . . . . . . . . . . . . . . . . . . . .  64  5.2  I-V characteristics of nMOS device under test produced by HeatMOS©  5.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  SourceTemp = 565K | DrainTemp = 567K | Re f Temp = 456K | Vgs = 1.0V | Vds = 1.5V | Ids = 2mA | v7 − nmos.spice . . . . . . .  5.4  73  SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.8V | Vds = 2.2V | v8 − nmos.spice . . . . . . . . . . . . .  5.8  71  SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice . . . . . . . . . . . . .  5.7  70  SourceTemp = 10K | DrainTemp = 10K | Re f Temp = 10K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice . . . . . . . . . . . . . . . .  5.6  68  SourceTemp = 565K | DrainTemp = 567K | Re f Temp = 456K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice . . . . . . . . . . . . .  5.5  66  75  SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 5.0V | Vds = 5.0V | v8 − nmos.spice . . . . . . . . . . . . .  77 x  List of Figures 5.9  SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 5V | Vds = 5V | v7 − nmos.spice . . . . . . . . . . . . . . .  78  5.10 SourceTemp = 318K | DrainTemp = 324K | Re f Temp = 298K | Vgs = 1.2V | Vds = 1.8V | v8 − nmos.spice . . . . . . . . . . . . .  80  5.11 SourceTemp = 300K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.0V | Vds = 0.1V | v8 − nmos.spice . . . . . . . . . . . . .  82  5.12 SourceTemp = 318K | DrainTemp = 324K | Re f Temp = 300K | Vgs = 1.0V | Vds = 0.1V | v7 − nmos.spice . . . . . . . . . . . . .  83  5.13 The average channel temperature versus the drain current and the drain voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . .  86  5.14 The maximum channel temperature versus the drain current and the drain voltage . . . . . . . . . . . . . . . . . . . . . . . . . .  87  5.15 The maximum channel temperature versus the drain current and the drain voltage . . . . . . . . . . . . . . . . . . . . . . . . . .  88  5.16 The maximum channel temperature versus the drain current and the effective gate voltage . . . . . . . . . . . . . . . . . . . . . .  89  5.17 A typical cross-section of an HEMT device . . . . . . . . . . . .  91  5.18 Experimental channel temperature of HEMT device  92  6.1  . . . . . . .  FinFET typical layout and schematic cross-sectional structures (Hisamoto et al., 2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  99 xi  List of Figures D.1 A sample HeatMOS run part 1 . . . . . . . . . . . . . . . . . . . 125 D.2 A sample HeatMOS run part 2 . . . . . . . . . . . . . . . . . . . 126  xii  List of Acronyms 1D  One Dimensional  2D  Two Dimensional  3D  Three Dimensional  µm  Micro-meter  ASIC  Application Specific Integrated Circuit  CAD  Computer-Aided Design  CMOS  Complementary Metal-oxide Semiconductor  EMR  Electro-migration Reliability  GDSII  Graphic Database Information Interchange  HEMT  High Electron Mobility Transistor  IC  Integrated Circuit  K  Kelvin  mA  Mili-ampere  xiii  List of Acronyms  MOS  Metal-oxide Semiconductor  nm  Nano-meter  SoC  System-on-a-Chip  SOI  Silicon on Insulator  V  Volt  VLSI  Very Large Scale Integration  xiv  Acknowledgements I would like to thank Prof. Andrew Labun for the supervising my research and his guidance. The concept of using thermal and electrostatic analogies shown in chapter 2 are based on an unpublished paper by Prof. Labun. I am very thankful for the support that Prof. Andrew has given me. I would like to thank Prof. Carolyn Labun for reviewing my english style and her suggestions to improve my presentation and writing skills. I would like to thank Prof. Thomas Johnson for reviewing the research work and providing further guidance to complete the thesis. I was a member of the Multiphysics research group led by Prof. Labun and I would like to thank other group members including Karan, Ska-hiish, Jason Liao, and Sadegh as well. I would like to thank Prof. Abbas Milani with whom I worked on a different project. I would like to thank my graduate supervising committee members, Prof. Andrew Labun, Prof. Thomas Johnson, Prof. Stephen O’ Leary, and Prof. Kian Mehravaran. xv  Acknowledgements I would like to thank Prof. Deobrah Roberts, Associate Director Graduate Studies and Research. I would like to thank my external examiner Prof. Robert Sobot from University of Western Ontario for reviewing my thesis. Finally, I would like to thank everyone (including department of graduate studies staff, faculty of engineering staff, other faculty members, and fellow students) who have contributed towards this research work indirectly.  xvi  Chapter 1  Introduction 1.1  Background  Verifying the reliability of complex integrated circuits has become an increasingly complex design challenge as device geometries in fabrication processes continue to shrink (Alam et al., 2007). The low cost of fabricating integrated circuits (ICs) using CMOS (complementary metal-oxide semiconductor) continues to be the dominant fabrication technology. Micro-chips fabricated using CMOS technology are very complex and complete systems (SoC - system-on-a-chip) are common in the semiconductor industry. Predicting the operating temperature of these micro-chips should be done as early as possible to minimize the expense of resolving thermal issues late in the design phase.  The ITRS (International Technology Roadmap for Semiconductors) predicts that electro-thermal reliability will continue to present significant challenges as device geometries continue to shrink below 22 nm (ITR, 2009). Due to the high temperatures that can arise in current CMOS technologies, it has become extremely important to accurately estimate the temperatures within the micro-chip for EMR 1  1.1. Background (electromigration reliability) verification. Electromigration is a term applied to the transport of mass in metal and is accelerated by the Joule heating, the heat generated by current flowing through interconnects and devices (Black, 1969). Metal atoms are pushed by strong dc electron currents, sometimes resulting in voids in the interconnect (causing open circuits) or extruding through dielectric (causing short circuits). The Joule melting resulting from extremely high electrical current has been discussed in Fantini and Morandi (1985). There is a need to model the microchip temperature at a detailed level, during post-layout reliability verification, and to identify the specific metal interconnects most at risk for EMR. If certain parts of the micro-chip are heated beyond their reliability limit, it may cause a breakdown of the functionality and reliability in the whole system.  The thermal trends for the CMOS technology are demonstrated by recent data presented by the ITRS. As device sizes are reduced to nano-scale gate lengths the power density will continue to rise. Although the power density increases, the material properties of silicon, dielectric layers and metal layers remain constant and the effective thermal conductivity per unit of power will decrease. As a result the effective thermal resistance will increase per microwave of power (Semenov et al., 2003) (Figure 1.1).  Several factors contribute to the increase in power density as the device geometries shrink. First, the geometry is shrinking faster than the scaling in the supply voltage which directly contributes to an increase in power density. Second, the high frequency performance of CMOS devices improves as device geometry shrinks which means that the average clock rate in digital circuits is expected to increase. 2  1.1. Background  4  Thermal resistance  Thermal resistance (C/uW)  3.5  3  2.5  2  1.5  1 0.1  0.15  0.2 0.25 CMOS technology generation (um)  0.3  0.35  Figure 1.1: Transistor thermal resistance versus CMOS technology scaling  3  1.1. Background Dynamic switching results in power losses that are proportional to ΛCV 2 f , where C is the capacitance, f is the clock frequency, Λ is the activity factor (0 ≤ Λ ≤ 1), and V is the applied voltage (Sedra and Smith, 2007; Weste and Harris, 2004). Parallel efforts are made in reducing capacitance (i.e. device size), voltage, and activity factor (i.e. introduction of power management subsystem), at the same time dissipation is proportional to the square of the voltage, and voltage scaling has not kept pace with device scaling. These factors have been analyzed by the ITRS and projections on the relative increase in the operating temperature of devices are shown in Figure 1.2 (Semenov et al., 2003). The junction temperature rise will grow very rapidly and the importance of analyzing thermal profiles and the impact on EMR will become increasingly important.  The major contribution to the rising micro-chip temperatures is due to device and interconnect heating. The heat generated due to electric current flow is called Joule heating. If there is a way to estimate the temperature of a micro-chip from its layout, before manufacturing, we would be able to change the design to rectify the design issues arising from the thermal hot spots. Temperature estimation in the design phase of the integrated circuits will improve the confidence of designing for reliability and reduce the overall design time of a production micro-chip. The influence of the temperature on failure mechanisms which occur in the temperature range of -55°C to 125°C has been discussed by Pecht et al. (1992). Computer modeling of metallization failure of the integrated circuits due to electromigration has been discussed in Kirchheim and Kaeber (1991) and Sukharav et al. (2009).  There are some industrial solutions for estimating interconnects temperature. Soft4  1.1. Background  Normalized temperature increase of CMOS chip  4  W/L ratio is constant, W and L are scaled  3.5  3  2.5  2  1.5  1 0.1  0.15  0.2 0.25 CMOS technology generation (um)  0.3  0.35  Figure 1.2: Impact of technology scaling on temperature increase  5  1.1. Background ware tools to estimate the micro-chip temperature at the package level are discussed by Melamed et al. (2009) and Oprins et al. (2009). Most of these tools use numerical approaches for temperature estimation which are slow and prohibitive for designs with millions of transistors.  There are two main sources of heat in integrated circuits. These are: the dynamic power generated by digital switching signals and the Joule heating generated by the resistance in the interconnects and the devices. Most of the power dissipation in large micro-chips is generated by digital switching signals where the dynamic power is proportional to fV 2 , where f is the switching frequency and V is the applied voltage. On the other hand, Joule heat is proportional to I 2 , where I is the electric current. In today’s complex integrated circuits, switching frequencies are in the Giga Hertz (GHz) range while typical interconnect current and device currents are in the milliampere range. Several approaches have been proposed to analyze the thermal performance of complex micro-chips. Analytical treatments of the 1-D steady state heat equation in IC interconnect have been presented previously in the context of interconnect temperature scaling analysis by Ajami et al. (2005), Im et al. (2002), and Chuan et al. (2009). An analytical solution of the interconnect heating and temperature estimation has been presented by Labun and Jagjitkumar (2008).  The focus of this research work is to develop a model for estimating device temperature based on device layout and steady state bias voltages.  6  1.2. Motivation  1.2  Motivation  This research work is motivated by the need to have design tools to predict the steady state spatial variation of temperature on the integrated circuit in the design phase before the micro-chip is fabricated. A standard integrated design flow process primarily focuses on verifying the electrical performance of the design over process, temperature and voltage changes. However, design tools to model and predict the temperature behavior are not readily available. If efficient temperature modeling tools can be developed, then designers will have the ability to improve the reliability of designs in the early design phase rather than waiting for experimental results from a failure analysis.  Given the complexity of three-dimensional (3D) structures in complex integrated circuits, it is very important to have an efficient way to predict the temperature characteristics of the micro-chip. Brute force numerical solutions of the heat equation are computationally prohibitive for large IC’s with millions of devices and multiple layers of interconnect. Therefore, a key focus of this research work is to investigate efficient solutions to estimate thermal characteristics that are scalable to very large and complex micro-chips.  In this research project, analytic models are developed to construct a thermal net list for the complete micro-chip. The net list is analogous to an electrical net list for a circuit. In this approach, a thermal circuit model is developed which can be used to predict the temperature.  7  1.2. Motivation The Multiphysics research group at UBC has developed a thermal modeling software tool called Therminator3D. This tool can be used to simulate the overall temperature profile of a micro-chip providing thermal models are developed for the interconnects and devices. Thermal models for interconnect have already been developed by other researchers (Labun and Jagjitkumar, 2008). On the other hand, thermal models for devices is much more limited and a tractable model that can be incorporated in a thermal simulation of the micro-chip is required. The goal of this research project is to address this limitation and develop an analytic model for CMOS devices.  The thermal effects in the MOSFET (Metal-Oxide Semiconductor Field Effect Transistor) are complex and multidimensional because the generated heat has spatial as well as time dependent behavior. In order to accurately estimate the temperature due to the generated heat, one must solve the differential heat equation. The heat is generated due to electric current flow and can be estimated by Joule heating. There can be different ways to solve the differential equation (e.g. finite element, finite difference (Chuan et al., 2009) and (Kuball et al., 2007), analytical), but in order to estimate the temperature rapidly so that it can be feasible to estimate the temperature of the full micro-chip during the design, analytic solutions are preferred. There has been some work done in this direction and the analytical approach has been successfully implemented in order to estimate interconnect temperature (Labun and Jagjitkumar, 2008). However the authors did not analyze the device temperature which is the primary focus of this research. Figure 1.1 shows that the device thermal resistance increases as the technology scales down (Semenov et al., 2003), which in turn leads to the higher temperatures. 8  1.2. Motivation The most effective way to meet the reliability target for each technology node is to incorporate design-for-reliability at the start of each technology generation. The most effective way to meet the reliability of each technology node is to have complete built-in-reliability and design-for-reliability solutions available at the start of development of each technology generation. This requires an in-depth understanding of the physics of each failure mechanism and the development of powerful and practical reliability engineering tools. High temperatures not only cause significant changes in the delay characteristics for both the transistors and the interconnects but also degrade micro-chip reliability. With technology scaling, micro-chip power densities are increasing that has led to high operating temperatures and the large thermal gradients, thereby resulting in serious reliability concerns (Tsai and Kang, 2000). Scaling the technology results in higher transistor density and higher clock frequencies. Higher clock frequencies will increases dynamic switching power because the dissipation in switching driving a capacitive load is proportional to fV 2 , where f is the clock frequency and V is the applied voltage (Weste and Harris, 2004).  As device technology scales down, there is an exponential increase in the temperature  T between the MOSFET junction and the ambient temperature (Semenov  et al., 2003). For example, as shown in Figure 1.2,  T has increased by a fac-  tor of three as CMOS gate lengths have reduced from 0.35 um to 0.13 um. The rapid increase in the thermal density with a reduction in scaling has increased the importance of analyzing the temperature. High temperatures or uneven temperature distributions may result not only in the reliability issues, but also in the timing failures of micro-chips. To resolve these issues, high-quality, accurate thermal 9  1.3. Research Goals analysis, and thermally oriented placement optimizations are essential prior to fabrication (Tsai et al., 2006).  Other predictions in terms of the thermal trends in the nano-scale CMOS devices includes the total dissipation of the complex system-on-a-chip (SoC) circuits. ITRS data include for SoC dissipation is shown in Figure 1.3. The data shows that in ten years micro-chips could dissipate as much as 400 watts of heat energy(ITR, 2009). These trends demonstrate that the significant advances in terms of thermal management will be essential to keep junction temperatures within a reasonable operating temperature range. EMR failures become very significant at junction temperatures over 125 C.  1.3  Research Goals  Figure 1.4 shows a typical layout of a micro-chip. The devices are connected via interconnects. The interconnects can be represented by an equivalent resistor network for EMR verification. Each net in the interconnects network can be represented by equivalent pi-network (Labun and Jagjitkumar, 2008) for thermal analysis, and interconnect network models are required for an interconnect metallization and CMOS devices. Therefore an interconnect-transistor resistor network can be used to do a full micro-chip thermal analysis as shown in Figure 1.5. The transistor in this representation may be considered as a special type of resistor. One of the objectives of this research work is to find an equivalent resistor network representation of a transistor which can be integrated in interconnect-transistor resistor network for rapid full micro-chip thermal analysis. 10  1.3. Research Goals  500  SoC power consumption  450  Power consumption (W)  400  350  300  250  200  150  100  50 2008  2010  2012  2014 2016 2018 SoC power consumption trends  2020  2022  2024  Figure 1.3: SoC power consumption trends  11  1.3. Research Goals  Figure 1.4: A typical micro-chip layout  12  1.3. Research Goals  Figure 1.5: Interconnect-transistor network  13  1.4. Scope and Organization The main objective of this research work presented in this thesis is to model the heating behavior of a transistor and to find an analytical solution for the temperature estimation. The research includes the following:  • To develop a mathematical model for the Joule heating due to switching drain current in a MOSFET channel. • To find an analytical equation which solves the heat equation. • To estimate the temperature profile of the channel. • To develop a software package for estimating the channel temperature from the extracted device layout information and using information from a compact model. In this work, the SPICE BSIM3 format was chosen because of its relative simplicity. • To develop an equivalent resistor network representation of the channel which can be integrated into the interconnect-transistor resistor network for full micro-chip thermal analysis. • To validate the analytical solution using a numerical solution.  1.4  Scope and Organization  The scope of this thesis is limited to the research objectives, problem definition, solution and comparative studies with numerical solution of the heat equation using the finite difference methods. The thesis treats the heating problems in integrated 14  1.4. Scope and Organization circuits and its solution is based on an analytical approach. Chapter 2 provides the background of the heating in the integrated circuits along with the problem definition of the MOSFET channel heating. Chapter 3 provides the mathematical model and the analytical solution for MOSFET channel temperature estimation. Chapter 4 describes HeatMOS©, a software package that has been developed through this research, implementing the analytical solution to estimate the channel temperature. Chapter 5 shows the results of HeatMOS© which are compared with numerical solutions using a finite difference solver. The conclusions and the future directions for this research are discussed in Chapter 6. The appendices provide other relevant informations such as different device parameters, SPICE netlist, and a pseudo code for the numerical solver.  15  Chapter 2  Heating in Integrated Circuits The spatial variation of temperature across an integrated circuit (IC) is caused by charge carrier flow through the devices and the interconnects. The temperature variation in ICs is predominantly due to device heating and package level heat transport and is a subject of various chip-scale modeling approaches (Li et al., 2006). In the case of MOSFETs, the applied voltages at it its terminals create the longitudinal and transverse electric fields and charge carriers move from the source end to the drain end via drift and diffusion mechanisms (Park et al., 1991). Since the channel is a conducting path for the electric current, it can be considered as a resistor and the electric current flow in the channel leads to the Joule heating. The Joule heating causes the rise in the channel temperature.  Heat conduction in solids is governed by two principle mechanisms: diffusive heat conduction by electrons and heat conduction by lattice waves whose quanta are represented by phonons (Flik et al., 1992). Heat conduction by phonons in semiconductors is reviewed by Abeles(Abeles, 1963). In silicon the heat conduction is dominated by phonons (Sinha and Goodson, 2006). The hot electrons near the drain end transfer some energy to phonons and this creates a transient, non16  Chapter 2. Heating in Integrated Circuits equilibrium, high temperature "hot spot" that effectively leads to higher thermal resistance (Sinha and Goodson, 2006). Thermal conductivity analysis and conductivity measurements for silicon are provided by (Morris and Hust, 1961). The detailed thermal analysis at the dimensions less than a phonon mean free path (~100 nm) requires the transient solution of the Boltzmann transport equation (BTE), that may not be feasible for large, netlistbased, full micro-chip thermal analysis. The phonon transport of energy can be described by the BTE , which can capture the sub-continuum effects particularly important for short impulsive events such as ESD (electrostatic discharge). These sub-continuum effects are not modeled by the diffusion model of energy transfer. These sub-continuum effects should be accounted for during the transient analysis but in a steady state analysis the time-average temperature given by the diffusion model are appropriate. The BTE for the phonons can be written as:  e”eq − e” ∂ e” = −ν · ∇e” + + qelectron−phonon ∂t τ phonon  (2.1)  where e” is the phonon energy per unit volume per unit solid angle, ν is the phonon velocity, e”eq is the phonon equilibrium energy density, τ phonon is the phonon scattering rate, qelectron−phonon is the phonon energy absorption by hot electron.  The effective thermal conductivity of a complex silicon FET structure is technologyspecific and could be obtained by the detailed modeling, but this is beyond the 17  2.1. Electrostatic Analogy of Heat Conduction scope of the thesis, which is aimed at a very compact model for netlist-based analysis. Our research is therefore directly applicable for the technology nodes above 180 nm (using the BSIM3 device model). The extension of this approach toward nanoscale technologies is discussed briefly in the Future Work.  2.1  Electrostatic Analogy of Heat Conduction  The analytical methods developed in this research to model heat conduction in the devices are based on analogous relationships between the form of equations for the heat conduction and the electrostatics. The fundamental analogies between the heat and the electrostatics are shown by comparison of the governing equations in Table 2.1. The temperature T in a conduction problem in the presence of a heat sources φ is governed by the analogy to Poisson’s equation (Guass’ Law) in the electrostatics. Similarly, the heat flow is analogous to the current flow in the electric circuits. In these equations, λ is the thermal conductivity in W/mK, ε is the permittivity in F/m, φ is the heat energy density in W/m3 , ρ is the electric charge density in C/m3 , T is the temperature in K, V is the electric potential in V, q is the heat flux in W/m2 , J is the electric current density in C/m2 s. Use of both analogies solves the heating problem (Labun, 2009).  Table 2.2 continues the analogy between electrical and thermal models by comparing material properties of thermal conductivity and electrical conductivity (Lide, 2004). Metals are good conductors of both heat and electrical current and dielectrics are poor conductors, although the ratio of conductance’s is only two orders 18  2.1. Electrostatic Analogy of Heat Conduction  Table 2.1: Thermal and electrostatic analogies → − → − ∇ · (λ ∇ T ) = −φ  → − → − ∇ · (ε ∇V ) = −ρ  → − → − q = −λ ∇ T  → − → − J = −σ ∇V  Table 2.2: Thermal and electrical conductivities (300 K) Material  λ (W m−1 K −1 )  σ (Ω−1 m−1 )  Cu  398  5.7 × 107  Al  237  3.7  W  174  1×107  Si (intrinsic)  83.5  10−3  Si (doped)  60.5  10−3  SiO2  1.4  10−13  of magnitude for heat. The tabulated values also show that the range in thermal conductivity properties is much smaller than the range in electrical conductivity. For example, the thermal conductivity of copper is four times of magnitude better than silicon, while the the electrical conductivity of copper is 10 orders of magnitude greater than silicon (Lide, 2004).  The analogy with Possion’s equation leads to a capacitance/heat transfer coefficient analogy through the application of the divergence theorem to Poisson’s equation. Table 2.3 Row 1 shows the identical construction of integral equations which in the 19  2.2. Heat Equation  Table 2.3: Thermal and electrical versions of heat transfer or capacitance ¸ ´ → − → − → − → − ∇ (λ ∇ T )dΩ = (λ ∇ T )d A = φVolume Area  Volume  1 RT  =  ¸ → ´ → − → − → − − ∇ (ε V )dΩ = (ε ∇V )d A = QVolume Volume  φ T  Area  C = VQ  electrical case are solved to obtain capacitance. The analogous thermal model for heat transfer shows that the reciprocal of the thermal resistance (RT ) is equal to the ratio of the heat density divided by the temperature difference across the material.  The relationship between the heat conduction and the electrostatics is completed using Ohm’s law to relate the current flow with the heat flux. The Joule heating in a wire gives I 2 R = φVolume . The heat is conducted through the dielectrics according to the Ohm’s Law analogy. Thus, capacitance tables (in homogeneous dielectrics) can be read as thermal resistance tables by multiplying by a constant factor.  2.2  Heat Equation  Before developing a heat model for a MOSFET channel, a model for heating in a resistor is derived first. Consider a resistor of uniform width and composition as shown in Figure 2.1. The resistor temperature variation over its cross-section is neglected. For metal interconnects embedded in a dielectric, this can be readily justified, since (by Table 2.2) the thermal conductivity within the resistor is much greater than the thermal conductivity of the surrounding insulator. In MOSFETs 20  2.2. Heat Equation  Figure 2.1: Conservation of heat (energy) within a differential slice of a resistor  the electric current is also confined to a well-defined region (the channel). Heat flow is confined by a poor thermal conductor above (the SiO2 gate dielectric) and also below, in SOI devices. However, even in bulk MOSFETs, the useful approximation of a uniform temperature over the channel cross-section is not too far off the mark because of the localization of the heat source.  With reference to Figure 2.1, the heat per unit length, mCp T (x) x in a differential slice of a resistor segment of length  x, where T (x) is the temperature along the  resistor’s length, m is the mass per unit length, and Cp is the heat capacity. The time rate of change of the heat is then given by (Labun and Jagjitkumar, 2008)  mCp  ∂T ∂t  x = F(x +  x) − F(x) + φ (x) x − f (x) x  (2.2)  21  2.3. Heat Generation in MOSFETs Where F(x) is the diffusive heat flux. The heat flux is oriented along the resistor’s length and is related to the temperature gradient by Fourier’s law of heat conduction, including the “longitudinal” thermal conductance:  F(x) = Glong  ∂ T (x) ∂x  (2.3)  2 R ; R is the electrical resistance The Joule heat per unit length is given by φ = Irms  per unit length, assumed constant within a resistor; Irms is the time-averaged rms current. The function f (x) in equation (2.2) models the heat flux through the sides of the resistor to the surrounding environment at a temperature T re f . If there is diffusive heat flux through the sides of the resistor to the local environment at some uniform, constant temperature T re f (established, for example, by a full-chip temperature simulation including device heating), then one may define a “lateral” thermal conductance Glat per unit length  f (x) = Glat T (x) − T re f  2.3  (2.4)  Heat Generation in MOSFETs  When the transistor is on and is in operation, the charge carriers move from the source to the drain and in this process these loose energy due to physical reasons such as collisions, scattering, phonon-electron interaction etc. The dissipated  22  2.3. Heat Generation in MOSFETs  Figure 2.2: Cross-sectional view of an nMOS transistor  energy is released in the form of heat and leads to an increase in the device temperature. A cross-sectional view of a MOS transistor has been shown in Figure 2.2. The transistor is said to be in operation when there is a channel formed by applying appropriate voltages across substrate, source, drain, and gate terminals. The channel can be considered as a resistor of variable cross-sectional area, and the heat generated in the channel can be approximated by Joule heating similar to a resistor. Therefore, heat generation is concentrated in the region of the highest resistance, the pinched-off region of the channel near the drain end.  23  2.4. Summary  2.4  Summary  The heat generation in transistors has been presented with a focus on transistor channel heating. The basic equation for Joule heating has been explained, and it was shown that the heat equation is analogous to the solution of the Poisson equation in electrostatics. The basic heat equation for the Joule heating has been explained.  24  Chapter 3  Thermal Model and Analytical Solution The one-dimensional heat equation (2.2) is applied to the thermal model of the MOSFET and solved analytically to give a closed-form expression for the channel temperature from the source to the drain. The thermal model uses the BSIM3 parameters.  3.1  BSIM3 MOSFET Model  BSIM3 is a physics-based, accurate, scalable, robust and predictive industry standard MOSFET SPICE model for circuit simulation and CMOS technology development. It was developed by the BSIM Research Group in the Department of Electrical Engineering and Computer Sciences (EECS) at the University of California, Berkeley. BSIM3 is the industry-standard MOSFET model. The third iteration of BSIM3 Version 3 (commonly abbreviated as BSIM3v3), was established by SEMATECH as the first industry-wide standard model for MOSFETs in 25  3.1. BSIM3 MOSFET Model December of 1996. The development of BSIM3 is based on Poisson’s equation using the gradual channel approximation and coherent quasi 2D analysis, taking into account the effects of device geometry and process parameters. BSIM3 is appropriate for MOSFETs for 180 nm MOSFET technology and larger. BSIM3 models the following physical characteristics in MOSFET devices (Liu et al., 1999):  • Short and narrow channel effects on threshold voltage. • Non uniform doping effects. • Mobility reduction due to vertical field. • Bulk charge effect. • Velocity Saturation. • Drain-induced barrier lowering (DIBL). • Channel length modulation (CLM). • Substrate current induced body effect (SCBE). • Sub-threshold conduction. • Source/Drain parasitic resistances.  BSIM3 uses a group device extraction strategy for parameter extraction. This requires measured data from devices with different geometries. All devices are measured under the same bias conditions. The BSIM3 device parameters for a 180 nm CMOS technology have been used in this research. The model parameters are given in appendix A. 26  3.2. MOSFET Channel  3.2  MOSFET Channel  A simplified model of the MOSFET channel accounts for many characteristics of MOSFET devices and is sufficient for the compact thermal model developed in this thesis. The channel is a cloud of free charge carriers beneath the gate surface which conducts electric current. In the case of nMOS, the electron cloud forms the channel. The electron density is greater at the source end compared to the drain end and decreases towards the drain end (Masu and Tsubouchi, 1994). An example of the typical IV characteristics of an nMOS transistor is shown in Figure 3.1 (Weste and Harris, 2004). The electric current increases as we increase the drain voltage and after a certain voltage, the current becomes constant and the channel is said to be pinched off. Initially when there is no drain voltage, the electrons are attracted towards the positive gate voltage (enhancement mode nchannel MOSFET) and it is clear that the Joule heat constitutes only a small part of the micro-chip power consumption as compared to the power consumption due to switching activity. When the gate voltage is greater than the threshold voltage and the drain voltage is less than the gate voltage, the transistor is turned on, and a channel has been created which allows current to flow between the drain and the source. The MOSFET operates like a resistor controlled by the gate voltage relative to both the source and the drain voltages. Now, if the drain voltage is increased and becomes greater than the gate voltage, the transistor is said to be in the saturation or the active mode of operation. Since the drain voltage is higher than the gate voltage, the electrons spread out, and conduction is through a broad, three-dimensional channel and the current extends away from the interface. The onset of this region is also known as pinch-off to indicate the lack of channel region 27  3.2. MOSFET Channel  Figure 3.1: A typical I-V characteristic of an nMOS transistor  near the drain. The drain current is now weakly dependent upon drain voltage and controlled primarily by the gate–source voltage.  Since the channel provides a conducting path for electric current, it can be considered as a resistor. The cross-sectional area of this resistor is higher at the source end and continuously decreases towards the drain end. When the gate voltage is greater than the threshold voltage, Vgs > Vth and Vds ≥ (Vgs − Vth ), where Vds is the drain voltage, the channel depth at drain end decreases to almost zero, and the channel is said to be pinched-off (Weste and Harris, 2004). Beyond the pinch-off, the depletion region around the drain end provides a path for the drain current to flow. In the channel thermal model the region between the pinch-off point and the drain end 28  3.2. MOSFET Channel can be considered as a region of uniform cross-section. The conductivity of this channel resistor depends on complex physical parameters including area, doping concentration, mobility of charge carriers etc. The channel is controlled between the oxide layer and the substrate layer in vertical direction, and controlled between the source well and the drain well in horizontal direction as shown in Figure 3.3.  Let us consider the case of an nMOS transistor. Initially, when the gate voltage Vgs is less than the threshold voltage Vth , and the drain voltage is at the same voltage as the source, an inversion layer is formed beneath the gate, and the channel is said to exist between the source and the drain. The inversion layer can be assumed to have a uniform cross-sectional area from the source end to the drain end of the transistor. Now, if the gate voltage is increased beyond the threshold voltage of the transistor, a sufficient number of electrons accumulate in the inversion layer to conduct electric current. The electric current increases with the increase in the gate voltage. Since the drain and the source are at same voltage at this point, the height of the inversion layer, and hence the channel remain uniform (Weste and Harris, 2004). Now, if the drain voltage is increased, the voltage causes an electric current to flow between the source and the drain. The electric current flowing in the channel Ids will be proportional to the effective gate voltage Vgs − Vth , and to the voltage Vds which causes Ids to flow. Under these conditions the channel is said to be in the linear or ohmic region. As VDS continues to increase, the channel height near the drain end starts decreasing. However, in the linear mode of operation, the channel height at the drain end is approximately the same as that at the source end (HP  0.99HO ). If Vds > (Vgs −Vth ), Ids no longer increases linearly  and becomes constant. The transistor is said to operate in the saturation region or in 29  3.2. MOSFET Channel  Figure 3.2: A comparison for I-V characteristics of an nMOS transistor  30  3.2. MOSFET Channel  Figure 3.3: Simplified 2-D cross-section of an nMOS transistor the on state. Other second order effects in the BSIM3 model such as drain-induced barrier lowering (DIBL), substrate current induced body effects (SCBE) have not been incorporated in HeatMOS© to calculate effective drain current, which are typically modeled in industrial quality SPICE simulators. A comparison of the the I-V characteristics of simplified device model used in HeatMOS© with the full BSIM3 model (ngspice simulator) is shown in Figure 3.2. The saturation current calculation equation will be explained in section 5.2.  31  3.2. MOSFET Channel  3.2.1  Channel Geometry for Thermal Model  Figure 3.3 shows the channel region in a MOSFET. In the saturation region of operation the channel shrinks near the drain end and the drain current becomes almost constant. The drain current flows through the depletion region at the drain end. For the thermal model in saturation mode the channel is modeled as two separate regions.  • Region 1: Region 1 is a channel of constant width We f f and non-uniform height which tapers from the source end to the pinch-off point (Figure 3.4). This region is approximated by an exponential function of the form y = Ae−αx where where y is the vertical height and x is the horizontal length of the channel. An exponential function is selected for the model to lead to tractable analytic results and differs from the linear model used for the gradual channel approximation used to develop the electrical model for MOSFETs. However, the parameters A and α in the exponential model are determined from the BSIM3 model parameters and are matched for the operating point of the device. If L1 is the length of the channel between the source end and the pinch-off point, HO is the height of the channel at the source end and HP is the height of the channel at the pinch-off point then  x = 0 ⇒ y = HO 32  3.2. MOSFET Channel  x = L1 ⇒ y = HP  HP = HO e−αL1 ⇒ α =  −  y = HO e  HO 1 ln L1 HP  HO 1 L1 ln HP  x  (3.1)  • Region 2: Region 2 models the thermal conduction through the depletion region of the pinched off channel. In this region, the channel cross-section is uniform and has width We f f and height HP . The channel length in region 2 is denoted as L2 .  If L is the total length of the channel then  L = L1 + L2  Figure 3.4 shows the 3D view of channel formed beneath the gate.  33  3.2. MOSFET Channel  Figure 3.4: 3D view of the channel geometry for thermal model  3.2.2  Electrical Resistance  The conduction channel formed beneath the oxide layer is of non-uniform crosssectional area in nature from the source end to the pinch-off point and has uniform cross-sectional area from the pinch-off point to the drain end.  The resistance of non-uniform cross-sectional area of region 1 can be approximated by integrating the resistance of a differential slice over the length from 0 to L1 :  ˆL1 R1 =  ρeαx ρ dx = eαL1 − 1 We f f HO We f f HO α  (3.2)  0  α=  1 HO ln L1 HP  (3.3) 34  3.2. MOSFET Channel  Le f f = L1 + L2  (3.4)  where, ρ = Electrical resistivity HO = Height of channel at source end HP = Height of channel at pinch-off point We f f = Effective channel width L1 = Length of channel between source end and pinch-off point L2 = Length of channel between pinch-off point and drain end Le f f = Effective channel length  The effective channel length and width can be approximated by (Liu et al., 1999)  Le f f = LDrawn − 2dL  (3.5)  We f f = WDrawn − 2dW  (3.6)  dW = Wint +  Ww Wwl Wl + W + W W W wn ln ln L W L W wn  (3.7) 35  3.2. MOSFET Channel  dL = Lint +  Lw Lwl Ll + L + L L L wn ln ln L W L W wn  (3.8)  where, LDrawn = Layout drawn length of transistor. WDrawn = Layout drawn width of transistor. Wint , Lint ,Wl ,Ww ,Wwl ,Wwn , Ll , Lw , Lwn , Lwl = BSIM3 parameters.  The electrical resistivity of the channel can be approximated as  ρ=  1 We f f Cox µe f f  (3.9)  where, Cox = Capacitance of oxide layer. µe f f = Effective mobility of charge carriers.  Region 2 has a uniform cross-sectional area. This can be considered as a uniform bar of length L2 , width We f f , and height HP . Its resistance is given by  R2 =  ρL2 We f f HP  (3.10)  36  3.2. MOSFET Channel  3.2.3  Channel Height Models  Initially when the channel starts taking its shape an inversion charge layer is formed beneath the oxide layer. The charge accumulates beneath the gate and forms an approximately uniform layer of a certain height and width from the source end to the drain end. It can also be approximated that when the channel takes on its nonuniform shape in the saturation region of operation, the drain end of the channel shrinks, while the source end of the channel remains same. The drain end of the channel shrinks to the height, HP , the height at the pinch-off point. The height of the channel at the source end can be approximated by the thickness of the inversion layer (Masu and Tsubouchi, 1994).  The height of the channel at the source end is  HO = tinv =  φs = 2  kT  (3.11)  εεo 2qNA φs  kT NA ln q ni  (3.12)  where, k = Boltzmann’s constant T = Temperature ε = Dielectric constant 37  3.2. MOSFET Channel εo = Permittivity of free space q = Electron charge NA = Substrate doping concentration ni = Intrinsic silicon concentration φs = Surface potential  We can approximate the height of the channel at the pinch off point HP by the following method.  R1 + R2 =  VDS Isat  (3.13)  ρ eαL1 − 1 ρL2 VDS + = We f f HO α We f f HP Isat Under the approximation HP <<< HO ,  HP =  ρL2 Isat VDSWe f f  (3.14)  where Isat = Electric current in saturation region of operation VDS = Drain-Source voltage 38  3.2. MOSFET Channel The height of the channel at the drain end is the same as at the pinch-off point, HP , as it is assumed that the channel becomes uniform in nature from the pinch-off point to the drain end.  3.2.4  Channel Length Models  The channel length between the pinch-off point and the drain can be calculated using the channel length modulation approximation (Park et al., 1991).  The length of the channel between the pinch-off point to the drain is  L2 =  VDS −V DS EL  (3.15)  where EL =  qNa (VDS −V DS ) 2εεO  V DS  −vsat Le f f /µe f f  vsat = Saturation velocity of charge carriers  So, the channel length between the source end and the pinch-off point is  L1 = Le f f − L2  (3.16)  39  3.2. MOSFET Channel  3.2.5  Thermal Conductance  There are two kinds of thermal conductances: lateral thermal conductance and longitudinal thermal conductance. The conduction channel is formed beneath the oxide layer. The substrate can be approximated as having a uniform background temperature T re f . The lateral thermal conductance must be taken to both the gate electrode above and a thermal reference somewhere in the substrate below. These may be combined into a single lateral conductance to a single, averaged reference temperature. Given the relative short channel length, when compared to the distant reference temperature (e.g substrate), the small variation in the height of the channel compared to that great distance is negligible. Therefore, the variation of thermal conductance along the height of the channel can be considered negligible. The lateral thermal conductance (Glat ) can be considered uniform along the length of the channel. For complex device structures a uniform, effective lateral thermal conductivity could be calibrated from detailed 2D or even 3D thermal conduction simulations.  The longitudinal thermal conductance (Glong ) vary along the length of the channel and is of concern. Since the height of the channel is not uniform, a tractable solution to the heat equation arises if we can drive an expression of average thermal conductance in a similar manner to the electrical resistance of the channel. This weights the thermal conductance according to the localization of the heat source.  The average longitudinal thermal conductance is  40  3.2. MOSFET Channel  Glong avg  =  1  ˆLe f f (3.17)  G(x)dx  Le f f 0  where Le f f is the effective channel length.  The longitudinal thermal conductance along the length is  Glong (x) = GO eγx  (3.18)  where γ = − Le1f f ln  GO GL  GO = Thermal conductance per unit length at source end (i.e. when x = 0 ⇒ Glong (x = 0) = GO ). GL = Thermal conductance per unit length at drain end (i.e. when x = L ⇒ Glong (x = L) = GL ).  The solution to the equation 3.17 is  Glong avg =  GO eγLe f f − 1 γLe f f  If λ is the thermal conductivity of channel, then GO =  (3.19)  HOWe f f λ  and GL =  HPWe f f λ  . 41  3.3. Solution of the Heat Equation  2.2  Figure 3.5: 3D view of region 1 of the channel  3.3  Solution of the Heat Equation  The basic heat equation 2.2 is solved for both regions in the following sections.  3.3.1  Region 1: Non-uniform cross-sectional area  Figure 3.5 shows region 1 of the channel.  For a differential length  x of the channel,  2 φ (x) x = IDS R(x) x  42  3.3. Solution of the Heat Equation  R(x) x =  ρ x αx e We f f HO  where IDS is the drain current.  The heat flux can be written as  F(x) = Glong avg  F(x +  x) = Glong avg  ∂ T (x) ∂x  ∂ T (x + ∂x  x)  f (x) = Glat T (x) − T re f  In the limit when  x → 0, the basic heat equation 2.2 takes the form of the 2nd  order non-homogeneous differential equation.  Glong avg  2 ρ IDS d2T lat re f − G T (x) − T eαx = 0 + dx2 We f f HO  (3.20)  If the temperature function for region 1 is T1 (x), equation 3.20 can be written as  Glong avg  2 ρ IDS d 2 T1 lat re f eαx1 = 0 − G T (x ) − T + 1 1 We f f HO dx12  (3.21) 43  3.3. Solution of the Heat Equation The solution of non-homogeneous differential equation 3.21 can be written as  T1 (x1 ) = C1 eξ1 x1 +C2 e−ξ1 x1 + T re f −  Ψ2 eαx1 α 2 − ξ12  (3.22)  where  ξ1 =  Ψ=  Glat Glong avg  2 ρ Isat  We f f HO Glong avg  K1 =  Ψ2 α 2 − ξ12  C1 and C2 are constants whose values will be determined.  3.3.2  Region 2: Uniform cross-sectional area  Figure 3.6 shows the 3D view of region 2 of the channel. The total resistance in region 2 is denoted as R2 .  For a differential length  x of the channel  44  3.3. Solution of the Heat Equation  Figure 3.6: 3D view of the region 2 of the channel  2 φ (x) x = IDS  R2 L2  x  The heat flux can be written as  F(x) = Glong avg  F(x +  x) = Glong avg  ∂ T (x) ∂x  ∂ T (x + ∂x  x)  f (x) = Glat T (x) − T re f 45  3.3. Solution of the Heat Equation In the limit when  x → 0, the basic heat equation 2.2 takes the form of the second-  order non-homogeneous differential equation 3.23.  Glong avg  d2T 2 − Glat T (x) − T re f + IDS R2 = 0 dx2  (3.23)  If the temperature function of region 2 is T2 (x), the equation 3.23 can be written as  Glong avg  d 2 T2 2 − Glat T2 (x2 ) − T re f + IDS R2 = 0 dx22  (3.24)  The solution of differential equation 3.24 can be written as  T2 (x2 ) = D1 eξ2 x2 + D2 e−ξ2 x2 +  β ξ22  (3.25)  where  ξ2 =  β =−  Glat 2 Glong avg  Glat 2  2 IDS  Gavg  Glong avg  T re f − long  K2 =  R2  β ξ22 46  3.3. Solution of the Heat Equation D1 and D2 are constants whose values will be determined using the boundary conditions.  3.3.3  Boundary Conditions and Complete Solution  The channel is between the source end and the drain end. The end point temperatures are determined by the boundary conditions. If the temperature of the source end is TO and the drain end is TL , then the two boundary conditions can be written as  T1 (x1 ) |x1 =0 = TO  (3.26)  T2 (x2 ) |x2 =L2 = TL  (3.27)  Since the channel is one entity, the pinch-off point is a virtual point. The temperature at the pinch-off point as determined by equation 3.22 should be the same as it is determined by equation 3.25. Further the heat flux leaving region 1 enters into region 2. These facts give us two more conditions:  T1 (x1 ) |x1 =L1 = T2 (x2 ) |x2 =0  Glong avg  dT1 dT2 |x =L = Glong |x =0 avg dx1 1 1 dx2 2  (3.28)  (3.29) 47  3.3. Solution of the Heat Equation Using equations 3.22, 3.25, 3.26, 3.27, 3.28, and 3.29, the unknown coefficients, C1 , C2 , D1 , and D2 can be calculated. The values of these coefficients are given below.  Ψ2 α 2 − ξ12  (3.30)  β ξ22  (3.31)  Ψ2 β = D1 + D2 + 2 2 2 α − ξ1 ξ2  (3.32)  TO = C1 +C2 + T re f −  TL = D1 eξ2 L2 + D2 e−ξ2 L2 +  C1 eξ1 L1 +C2 e−ξ1 L1 + T re f −  Glong C1 ξ1 eξ1 L1 −C2 ξ1 e−ξ1 L1 − α avg  Ψ2 eαL1 α 2 − ξ12  = Glong avg (D1 ξ2 − D2 ξ1 ) (3.33)  C1 = C1 −C1  (3.34)  D1 = D1 − D1  (3.35)  where  48  3.3. Solution of the Heat Equation  ξ2 + ξ2 e2ξ2 L2 C1 =  eξ1 L1 − e−ξ1 L1  1 + e2ξ2 L2 ξ2 − eξ1 L1 + e−ξ1 L1  1 − e2ξ2 L2  ×  TL eξ2 L2 − (TO − T re f )e−ξ1 L1 − T re f + Ψ2 (eξ1 L1 − e−ξ1 L1 ) + α 2 − ξ12 β (1 − eξ2 L2 ) ξ22  eξ1 L1 + e−ξ1 L1 D1 =  eξ1 L1 − e−ξ1 L1  1 + e2ξ2 L2 ξ2 − eξ1 L1 + e−ξ1 L1  1 − e2ξ2 L2  ×  TL eξ2 L2 − (TO − T re f )e−ξ1 L1 − T re f + Ψ2 (eξ1 L1 − e−ξ1 L1 ) + α 2 − ξ12 β (1 − eξ2 L2 ) ξ22  1 − e2ξ1 L1 C1 =  eξ1 L1 − e−ξ1 L1  1 + e2ξ2 L2 ξ2 − eξ1 L1 + e−ξ1 L1  1 − e2ξ2 L2  ×  (TO − T re f )ξ1 e−ξ1 L1 − TL eξ2 L2 ξ2 + Ψ2 (αeαL1 + ξ1 e−αL1 ) + α 2 − ξ12 β ξ2 eξ2 L2 ξ22  49  3.3. Solution of the Heat Equation  eξ1 L1 − e−ξ1 L1 D1 =  eξ1 L1 − e−ξ1 L1  1 + e2ξ2 L2 ξ2 − eξ1 L1 + e−ξ1 L1  1 − e2ξ2 L2  ×  (TO − T re f )ξ1 e−ξ1 L1 − TL eξ2 L2 ξ2 + Ψ2 (αeαL1 + ξ1 e−αL1 ) + α 2 − ξ12 β ξ2 eξ2 L2 ξ22  C2 = TO −C1 − T re f +  Ψ2 α 2 − ξ12  D2 = TL eξ2 L2 − D1 e2ξ2 L2 −  β ξ2 L2 e ξ22  The temperature of the channel along its length from the source end to the drain end can be written as  T (x) =     T1 (x1 ), 0 ≤ x1 ≤ L1 f or0 ≤ x ≤ L1  (3.36)    T2 (x2 ), 0 ≤ x2 ≤ L2 f orL1 ≤ x ≤ L where  L = L1 + L2  (3.37) 50  3.3. Solution of the Heat Equation  3.3.4  The Average Channel Temperature  The average temperature of the channel can be found by integrating the temperature function over the length and by dividing the total length.    ˆL1 ˆL2 1 1 1 T1 (x1 )dx1 + T2 (x2 )dx2  Tavg = 2 L1 L2 0  1 1 Tavg = 2 L1  0  ˆL1 C1 eξ1 x1 +C2 e−ξ1 x1 + T re f − K1 eα1 x1 dx1 + 0  1 L2  ˆL2 D1 eξ2 x2 + D2 e−ξ2 x2 + K2 dx2 0  Tavg =  3.3.5  1 2L1  C2 −ξ1 L1 K1 α1 L1 C1 ξ1 L1 e −1 − e − 1 + L1 T re f − e −1 + ξ1 ξ1 α1 D2 −ξ2 L2 1 D1 ξ2 L2 −1 − − 1 + K2 L2 e e 2L2 ξ2 ξ2  The Maximum Channel Temperature  If T1max is the maximum temperature given by the temperature function T1 (x) and T2max is the maximum temperature given by the temperature function T2 (x),  dT1 = 0 ⇒ x1 = x1max dx1 51  3.3. Solution of the Heat Equation  T1max = T1 (x1 ) |x1 =x1max  dT2 = 0 ⇒ x2 = x2max dx2  T2max = T2 (x2 ) |x2 =x2max  where 0 ≤ x1max ≤ L1 and 0 ≤ x2max ≤ L2 .  The maximum channel temperature Tmax at a distance xmax along the length of the channel from the source end to the drain end is the greatest of T1max and T2max .  If T1max ≥ T2max ⇒ Tmax = T1max and xmax = x1max . If T2max > T1max ⇒ Tmax = T2max and xmax = L1 + x2max . For x2max :  max  max  D1 ξ2 eξ2 x2 − D2 ξ2 e−ξ2 x2 = 0  x2max =  D2 1 ln 2ξ2 D1  For x1max : 52  3.4. M-Network Representation of the Channel  max  max  max  C1 ξ1 eξ1 x1 −C2 ξ1 e−ξ1 x1 − K1 α1 eα1 x1 = 0  The above equation for x1max does not have a closed form solution and a numerical solution is needed. It is not feasible to calculate x1max under these conditions. However, the maximum temperature along the length can be calculated via a simple technique. The channel is subdivided into small lengths and at each point T (x). After calculating all the values at each point, the maximum channel temperature can be found.  3.3.6  The Channel Temperature at Pinch-off  The channel temperature at pinch-off point Tpinch is given as  Tpinch = T1 (x1 ) |x1 =L1  3.4  M-Network Representation of the Channel  The analytical solution gives promising results. HeatMOS© is able to do a thermal analysis of the transistor channel and estimates temperature along its length. The approach can be used for a complete micro-chip electro-thermal reliability verification. In a complete micro-chip EMR verification methodology, each of the resistors that forms a net is represented by an equivalent interconnect network (Labun and 53  3.4. M-Network Representation of the Channel Jagjitkumar, 2008) and (Alam et al., 2007). Based on this research work an equivalent M-network model is proposed to model channel temperature. The M-network model is shown in Figure 3.7 and can be used to integrate the channel temperature into a full micro-chip thermal analysis. The channel has two regions with conductance η1 and η2 corresponding to region 1 and 2 of the channel respectively. The current sources S1 and S2 correspond to the source end and the drain end of the channel, respectively. The currents at the nodes of the M-network are given by FO (at the source end) and FL (at the drain end). The temperatures at the source and the drain end are TO and TL respectively. The temperature at the pinch-off point Tpinch (the junction of region 1 and region 2 in the channel) corresponds to a voltage source at pinch-off point. The source end and drain end are the ports of a M-network. The additional conductances θ1 and θ2 , correspond to the source end and the drain end respectively. These conductances represent a conduction path to ground that is not usually important electrically (and so for electrical analysis, θ1 , θ2 = 0) but will be important thermally (and so for thermal analysis, θ1 , θ2 > 0).  The heat currents, FO and FL , can be given by  dT1 |x=0 dx  (3.38)  dT2 | x = L2 dx  (3.39)  FO = Glong avg  FL = Glong avg  Kirchhoff’s current law can be applied to the both sides of the M-network of Figure 54  3.4. M-Network Representation of the Channel  Figure 3.7: Resistor M-network used for EMR verification  3.7.  FO = η1 Tpinch − (η1 + θ1 )TO − S1  (3.40)  FL = (η2 + θ2 )TL − η2 Tpinch + S2  (3.41)  where  Tpinch = C1 eξ1 L1 +C2 e−ξ1 L1 + T re f − K1 eα1 L1  Equations 3.38, 3.39 can be solved and compared with equations 3.40, 3.41 to find 55  3.4. M-Network Representation of the Channel out the values of η1 ,η2 , θ1 , θ2 , S1 , and S2 .  η1 = Glong avg 2ξ1  θ1 = Glong avg  S1 = Glong avg  4ξ1 ξ2 −ξ1 L1 e + Glong avg ξ1 − η1 M  2ξ2 e−ξ1 L1 + 1 + e−ξ1 L1 M  2ξ1 η1 ξ1 L1 e − e−ξ1 L1 γ1 − γ1 − K1 (α1 + ξ1 ) − T re f ξ1 − M M T re f e−ξ1 L1 + T re f + K1 e−ξ1 L1 − eα1 L1  η2 = −  eξ1 L1 eξ1 L1 − e−ξ1 L1  Me−ξ1 L1 ξ2 L2 −ξ1 L1 + Glong e avg e 2ξ2  eξ1 L1 + e−ξ1 L1  +ξ1 eξ1 L1 − e−ξ1 L1  θ2 = η2    2ξ2 eξ2 L2 eξ1 L1 − e−ξ1 L1   M  −1      − Glong avg  2ξ2 2ξ2 L2 e M  e−ξ1 L1 + ξ2 eξ1 L1 − e−ξ1 L1  eξ1 L1 + − ξ2  56  3.4. M-Network Representation of the Channel  ξ2 K2 + S2 = Glong avg  2ξ2 eξ2 L2 γ2 + T re f e−ξ1 L1 − 1 − K1 e−ξ1 L1 M η2 ξ1 L1 −eα1 L1 − e − e−ξ1 L1 γ1 M  where  M = eξ1 L1 − e−ξ1 L1  γ1 = ξ2 1 + e2ξ2 L2 eξ2 L2  1 + e2ξ2 L2 ξ2 − eξ1 L1 + e−ξ1 L1  1 − e2ξ2 L2  −T re f e−ξ1 L1 + 1 + K1 eα1 L1 − e−ξ1 L1 + K2 (1− −T re f ξ1 e−ξ1 L1 + K1 α1 eα1 L1 + ξ1 e−ξ1 L1  + 1 − e2ξ2 L2  +K2 ξ2 eξ2 L2  γ2 = T re f K1  e−ξ1 L1 − 1  eξ1 L1 + e−ξ1 L1 + ξ1 e−ξ1 L1 eξ1 L1 − e−ξ1 L1  eα1 L1 − e−ξ1 L1  +  eξ1 L1 + e−ξ1 L1 − α1 eα1 L1 + ξ1 e−ξ1 L1 +  eξ1 L1 − e−ξ1 L1  + K2  1 − eξ2 L2  eξ1 L1 + e−ξ1 L1 −  ξ2 eξ2 L2 eξ1 L1 − e−ξ1 L1 57  3.5. Summary  3.5  Summary  An analytic solution for estimating channel temperature in a MOSFET device has been derived. The solution is composed of two sets of equations corresponding to the tapered channel region and the pinch-off region. Equations were also derived for estimating the average channel temperature and the temperature at the pinchoff point. The analytic equations are used to calculate equivalent model values in a M-network that can be incorporated in top level micro-chip simulations to predict the spatial variation of temperature across the micro-chip and assist designers in identifying potential EMR failure points. The foundation for the development of the software to estimate temperature of MOSFET channel has been set.  58  Chapter 4  HeatMOS© 4.1  About HeatMOS©  HeatMOS© is a software tool which solves the heat equation for MOSFET devices analytically and estimates the channel temperature profile. The HeatMOS© software tool is developed as part of this research project. The program uses physical device parameters extracted from BSIM3 models (Liu et al., 1999). HeatMOS© reads the device geometry information from an extracted layout in the SPICE format. It calculates saturation electric current and other required variables for the analytical model. HeatMOS© then calculates the channel temperature using the analytical model and the outputs temperature distribution along the channel length.  The input variables required for HeatMOS are listed below:  1. SourceTemp: Source end temperature. 2. DrainTemp: Drain end Temperature. 3. RefTemp: Reference temperature of the medium of the channel. 59  4.1. About HeatMOS© 4. Vgs: Gate voltage. 5. Vds: Drain voltage. 6. Vbs: Substrate bias voltage. 7. Ids: Drain current; this is optional parameter, if not given will be calculated (device is in saturation region). 8. Netlist: Extracted spice netlist from layout.  HeatMOS© outputs:  1. The channel temperature distribution along the length of the channel and a text file containing the temperatures at different points along the channel length. 2. The drain current. 3. The average thermal conductance. 4. The average channel temperature. 5. The maximum channel temperature. 6. The temperature at pinch-off point. 7. The effective length and width of the channel.  A sample output of of HeatMOS has been provided in Appendix D.  60  4.2. Integration in the VLSI CAD Flow  4.2  Integration in the VLSI CAD Flow  Figure 4.1 shows a typical post-logic synthesis VLSI (Very-large-scale Integration) CAD (Computer-aided Design) flow. Reliability verification is necessary for nanoscale technologies to ensure the reliability of integrated circuit. As can be seen from the figure, thermal verification is needed after physical verification. HeatMOS© can be integrated with interconnect temperature estimation CAD tools (e.g. Therminator (Labun and Jagjitkumar, 2008)) to do a complete micro-chip thermal verification. If the integrated circuit design does not pass reliability verification, there is still a chance to modify the layout of the circuit in its design phase.  61  4.2. Integration in the VLSI CAD Flow  Figure 4.1: VLSI CAD flow  62  Chapter 5  Results and Discussions HeatMOS solves the heat equation analytically. For a comparative analysis of the results of HeatMOS, the heat equation for both regions of the channel has been solved using the finite difference approach (numerical approach). The finite difference approach is based on Taylor’s approximation (Strikwerda, 2004) for 2nd order derivatives given by equation 5.1  f (x − h) − 2 f (x) + f (x + h) f (4) (v) 2 = f (x) + h h2 12  (5.1)  where, h is a small interval and f (x) is the second derivative of function f (x).  The second term on the right hand side of equation 5.1 is the remainder term and can be considered as an approximation error. The detailed finite-difference method and its pseudo code is given in appendix C.  63  Chapter 5. Results and Discussions  Figure 5.1: An nMOS layout  64  5.1. Verification If we apply voltages to the terminals of this nMOS transistor, an electric current will flow in the channel. In a complex micro-chip, the source and the drain of this transistor will be connected to other nets via vias and contacts. The temperatures of these vias can be estimated by interconnect estimations tools such as Therminator3D (Labun and Jagjitkumar, 2008). If we neglect the distance between the source and the drain ends of the channel and the contacts, the source and the drain ends will be approximately at the same temperature as the contacts connecting these vias.  The thermal channel model is verified by running a number of different test scenarios with a typical nMOS device. The layout of the device is shown in Figure 5.1 and has a drawn length of 0.18 µm and a drawn width of 0.40 µm. The full SPICE BSIM3 model for the device is given in Appendix A. HeatMOS includes a utility to approximate the IV characteristics of the device using a simplified device model and the results are shown in Figure 5.2. The IV characteristic at the pinch-off point is not smooth because the curve-fitting spline functions have not be modeled in HeatMOS. The program also support a direct input of the operating device current which can be extracted from full BSIM3 SPICE simulations. This feature is included in the test scenarios described below.  5.1  Verification  Let us consider a few test scenarios to verify that HeatMOS© compiles and provides the results that are consistent with numerical simulations. Te objective of 65  5.1. Verification  0.35 Vgs = 1.5 V Vgs = 1.0 V 0.3  Ids (mA)  0.25  0.2  0.15  0.1  0.05  0 0  0.5  1  1.5  2  2.5  Vds (V)  Figure 5.2: I-V characteristics of nMOS device under test produced by HeatMOS©  66  5.1. Verification these tests is to verify that HeatMOS provides accurate results under different test scenarios and free from run-time errors.  5.1.1  Test Scenario using Externally Supplied Drain Current  The objective of this test is to verify that HeatMOS© can analyze channel temperature using an externally supplied current. Sometimes designer’s may want to analyze the temperature behavior for higher current than calculated from circuit simulations. In this test, a current of 2 mA has been supplied externally by designer, the layout drawn length of device under test is 0.28 µm and the width is 0.40 µm, the applied gate voltage is 1.0 V and the drain voltage is 1.5 V, the source end temperature is set to 565 K, the drain end is set at 567 K, and the reference temperature has been set to 456 K. The results of this test scenario are shown in Figure 5.3. HeatMOS© calculates an average channel temperature of 586.19 K and the maximum channel temperature is 588.68 K at a distance of 0.2115 µm from from the source end. The temperature at the pinch-off point is 588.46 K at a distance 0.2093 µm from the source end. As we can see from the Figure 5.3 the difference between the analytic solution and numerical solution is significant and the peak temperature difference is approximately 4 K. The difference is due to the round-off error in the finite difference approximation and the chosen step size. In this test scenario the gate voltage and drain voltage are different and electric current has been provided externally and not been calculated by HeatMOS©. The accuracy of the numerical solution can be improved by reducing the step size as is shown by the blue line in Figure 5.6. 67  5.1. Verification  590  HeatMOS Finite Difference Finite Difference with smaller step size  Channel temperature T(x) (K)  585  580  575  570  565 0  0.05  0.1 0.15 Effective channel length x (um)  0.2  0.25  Figure 5.3: SourceTemp = 565K | DrainTemp = 567K | Re f Temp = 456K | Vgs = 1.0V | Vds = 1.5V | Ids = 2mA | v7 − nmos.spice  68  5.1. Verification  5.1.2  Test Scenario for a Smaller MOSFET Device  In this test, the layout drawn length and width of the device are 0.18 µm and 0.40 µm, and gate and drain voltages are both 1.5 V. The source end temperature is 565 K, the drain end temperature is 567 K, and the reference temperature has been set at 456 K. The results of this test scenario are shown in Figure 5.4. Results from HeatMOS© show the average channel temperature is 575.87 K and the maximum channel temperature is 570.52 K at a distance of 0.114 µm from the source end. The temperature at the pinch-off point is 570.36 K at a distance 0.109 µm along the length of the channel from the source end. HeatMOS© calculates a saturation current of 0.269 mA.  As we can see from the results of this test the channel becomes hotter towards drain end. It would be interesting to verify HeatMOS© keeping the source, the drain and the reference temperature all at the same time.  5.1.3  Test Scenario for the Case When the Source, Drain, and the Reference Temperatures are Same  In this test, the layout drawn length of the device under test is 0.18 µm and the width is 0.40 µm, the applied gate voltage is 1.5 V, the drain voltage is 1.5 V, and the source, drain, and reference temperatures all are set to 10 K. The results of this test scenario are shown in Figure 5.5. The drain current calculated by HeatMOS© is 0.2699 mA, the average channel temperature is 5.05 K, and the maximum channel temperature is 8.1 K at a distance of 0.1124 µm along the length of the channel 69  5.1. Verification  571  HeatMOS Finite Difference  Channel temperature T(x) (K)  570  569  568  567  566  565 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.4: SourceTemp = 565K | DrainTemp = 567K | Re f Temp = 456K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice  70  5.1. Verification  90  HeatMOS Finite Difference  80  Channel temperature T(x) (K)  70  60  50  40  30  20  10 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.5: SourceTemp = 10K | DrainTemp = 10K | Re f Temp = 10K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice  from the source end. The temperature at the pinch-off point is 7.95 K at a distance of 0.109 µm from the source end.  The above verification tests shows that HeatMOS© is free from runtime errors, bugs, and provides meaningful results. There is a possibility to write other corner test scenarios to find out hidden bugs in the software. Some of the following test scenarios are focused to verify the HeatMOS© results when the device under test operates in different modes of operation.  71  5.2. Saturation Mode of Operation  5.2  Saturation Mode of Operation  In the saturation mode, the MOSFET is on and it conducts current between the source and the drain. The charge carriers flow in the channel and the current becomes constant (IDS = Isat ) and can be approximated by (Weste and Harris, 2004)  Isat =  1 We f f µnCox (VGS −VT h )2 {1 + λ (VDS −V DS)} 2 Le f f  (5.2)  1 V DS = (VGS −VT h ) − λ (VGS −VT h )2 2 where We f f is the effective channel width, Le f f is the effective channel length, µn is the electron mobility, Cox is the gate oxide layer capacitance per unit area, VGS is the applied gate voltage, VT h is the threshold voltage of the nMOS transistor, and λ is the fitting parameters for channel length modulation. In this mode of operation VDS > V DS. The nMOS threshold voltage for the BSIM3 technology file used in this work is 0.39V . The value of λ is 0.20. Let us consider a few test cases in this mode of operation.  5.2.1  Test Scenario When Gate and Drain are at Same Voltage  In this test, the source and drain temperatures have been set to 325 K respectively, and the reference temperature has been set to 300 K. The applied gate voltage is 1.5 V and the drain voltage is 1.5 V. The layout drawn width of the nMOS is 72  5.2. Saturation Mode of Operation  331  HeatMOS Finite Difference  Channel temperature T(x) (K)  330  329  328  327  326  325 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.6: SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.5V | Vds = 1.5V | v8 − nmos.spice  0.40 µm and the the length is 0.18 µm. HeatMOS© calculates a saturation drain current of 0.2699 mA. The results of this test scenario are shown in Figure 5.6. The average channel temperature for this case is 332.50 K and the maximum channel temperature is 330.57 K at a distance of 0.113 µm from the source end. The temperature at the pinch-off point is 330.46 K at a distance 0.109 µm along the length of the channel from the source end.  As we can see from this test that the temperature varies between 325 K to 330.57 K. It would also be interesting to note that in this case the source and the drain has 73  5.2. Saturation Mode of Operation been at the same voltage level. It would be interesting to check the temperature along the length of channel if the gate and the drain voltages are different.  5.2.2  Test Scenario for the Case of Different Gate and Drain Voltages  In this test, the source and the drain end temperatures are set to 325 K respectively, and the reference temperature is set to 300 K. The applied gate voltage is 1.8 V and the drain voltage is 2.2 V. The layout drawn width of nMOS is 0.40 µm and the length is 0.18 µm. HeatMOS© calculates a saturation drain current of 0.4322 mA. The results of this test scenario are shown in Figure 5.7. The average channel temperature for this case is 336.78 K and the maximum channel temperature is 338.12 K at a distance 0.1137 µm from the source end. The temperature at the pinch-off point is 337.81 K at a distance 0.108 µm from the source end.  The temperature difference between the minimum and the maximum temperature in this case is 13.12 K. We can see that upon increasing the gate and drain voltage the maximum channel temperature rises. The channel temperature variation is complex and is not based on one factor (e.g. voltage change), therefore the correlation between voltage and temperature is not explicit. However in simple terms it indicate that upon increasing the gate and drain voltage, the drain current increases and it leads to higher temperature in the channel as the more heat is generated in the channel due to electric current flow. Let’s consider another test scenario with very high voltage on the gate and drain ends.  74  5.2. Saturation Mode of Operation  340  HeatMOS Finite Difference  338  Channel temperature T(x) (K)  336  334  332  330  328  326  324 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.7: SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.8V | Vds = 2.2V | v8 − nmos.spice  75  5.2. Saturation Mode of Operation  5.2.3  Test Scenario for the Case of High Gate and Drain Voltages  In this test, source and drain temperatures are set to 325 K respectively, the reference temperature has been set to 300 K. The applied gate and drain voltages are both 5.0 V. The layout drawn width of the nMOS is 0.18 µm and the the length is 0.40 µm. HeatMOS© calculates a saturation drain current of 4.52 mA. This is a high value for the drain current. The results of this test scenario are shown in Figure 5.6. The average channel temperature for this case is 460.66 K and the maximum channel temperature is 556.00 K at a distance 0.1102 µm from the source end. The temperature at the pinch-off point is 550.26 K at a distance 0.10631 µm from the source end.  As we can see form this test that the maximum temperature difference is 231 K which is a very high value. This indicates that the higher drain current leads to higher temperature which is expected. It would be interesting to see the temperature distribution for the same test on a larger device.  5.2.4  Test Scenario for a Large MOSFET Device with High Drain Current  In this test, the source and the drain end temperatures have been set to 325 K respectively, and the reference temperature has been set to 300 K. The applied gate and drain voltages are both 5.0 V. The layout drawn width of the nMOS is 0.40 µm and the length is 0.28 µm. HeatMOS© calculates the saturation drain current of 2.689 mA. The results of this test scenario are shown in Figure 5.9. The average 76  5.2. Saturation Mode of Operation  600  HeatMOS Finite Difference  Channel temperature T(x) (K)  550  500  450  400  350  300 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.8: SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 5.0V | Vds = 5.0V | v8 − nmos.spice  77  5.2. Saturation Mode of Operation  520  HeatMOS Finite Difference  500  Channel temperature T(x) (K)  480 460 440 420 400 380 360 340 320 0  0.05  0.1 0.15 Effective channel length x (um)  0.2  0.25  Figure 5.9: SourceTemp = 325K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 5V | Vds = 5V | v7 − nmos.spice  channel temperature for this case is 429.03 K and the maximum temperature is 500.50 K at a distance 0.208 µm from the source end. The temperature at the pinch-off point is 499.17 K at a distance 0.206 µm from the source end.  The results show that the maximum channel temperature peaks at 175.5 K above the source and drain terminal temperatures which is about 50 K cooler than the previous case. Clearly the larger device has more area to dissipate heat in the channel. Let’s consider another arbitrary test scenario.  78  5.3. Linear/Ohmic Mode of Operation  5.2.5  Test Scenario for Different Source/Drain Temperatures and Voltages  In this test, the source end temperature is set to 318 K, the drain end temperature is set to 324 K, and the reference temperature is set to 298 K. The applied gate voltage is 1.2 V and the drain voltage is 1.8 V. The layout drawn width of the nMOS is 0.40 µm and the length is 0.18 µm. HeatMOS© calculates a saturation drain current of 0.145 mA. The results of this test scenario are in Figure 5.9. The average channel temperature for this case is 327.91 K and the maximum temperature is 326.97 K at a distance of 0.118 µm from the source end. The temperature at the pinch-off point is 326.58 K at a distance of 0.109 µm from the source end.  5.3  Linear/Ohmic Mode of Operation  When the MOSFET operates in the linear mode, the drain current increases linearly with the drain voltage. Under these conditions the MOSFET behaves as a voltage dependent resistor where the resistance is determined by the gate voltage. The current and the voltage follow the ohmic relationship in this mode of operation. The drain current in this mode of operation is given by (Sedra and Smith, 2007)  IDS = µnCox  We f f 2 2.0(VGS −VT h )VDS −VDS Le f f  (5.3)  where, We f f is the effective channel width, Le f f is the effective channel length, µn is the electron mobility, Cox is the gate oxide layer capacitance per unit area, VGS 79  5.3. Linear/Ohmic Mode of Operation  327  HeatMOS Finite Difference  326  Channel temperature T(x) (K)  325  324  323  322  321  320  319  318 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.10: SourceTemp = 318K | DrainTemp = 324K | Re f Temp = 298K | Vgs = 1.2V | Vds = 1.8V | v8 − nmos.spice  80  5.3. Linear/Ohmic Mode of Operation is the applied gate voltage, VDS is applied drain voltage, and VT h is the threshold voltage of the nMOS. In this mode of operation VDS < V DS.  Let us consider a few test cases in this mode of operation.  5.3.1  Test Scenario for the Case of Different Temperatures at the Source and the Drain  In this test, the source end temperature is set to 300 K, the drain end temperature is set to 325 K, and the reference temperature has been set to 300 K. The applied gate voltage is 1.5 V and the drain voltage is 1.0 V. The layout drawn width of nMOS is 0.40 µm and the length is 0.18 µm. HeatMOS© calculates the linear mode drain current of 0.0245 mA. The results of this test scenario are in Figure 5.11. The average channel temperature for this case is 317.86 K and the maximum channel temperature is 325 K at a distance of 0.146 µm from the source end. The temperature at the pinch-off point is 318.97 K at a distance of 0.110 µm from the source end.  Let’s consider another test scenario with larger device under test.  5.3.2  Test Scenario for a Larger Device in the Ohmic Operating Region  In this test, the source end temperature is set to 318 K, the drain end temperature is set to 324 K, and the reference temperature is set to 300 K. The applied gate voltage 81  5.3. Linear/Ohmic Mode of Operation  325  HeatMOS Finite Difference  Channel temperature T(x) (K)  320  315  310  305  300 0  0.02  0.04  0.06 0.08 0.1 Effective channel length x (um)  0.12  0.14  0.16  Figure 5.11: SourceTemp = 300K | DrainTemp = 325K | Re f Temp = 300K | Vgs = 1.0V | Vds = 0.1V | v8 − nmos.spice  82  5.3. Linear/Ohmic Mode of Operation  324  HeatMOS Finite Difference  Channel temperature T(x) (K)  323  322  321  320  319  318 0  0.05  0.1 0.15 Effective channel length x (um)  0.2  0.25  Figure 5.12: SourceTemp = 318K | DrainTemp = 324K | Re f Temp = 300K | Vgs = 1.0V | Vds = 0.1V | v7 − nmos.spice  is 1.0 V and the drain voltage is 0.1 V. The layout drawn width of the nMOS is 0.40 µm and the length is 0.28 µm. HeatMOS© calculates a drain current of 0.01455 mA. The results of this test scenario are shown in Figure 5.12. The average channel temperature for this case is 325.92 K and the maximum channel temperature is 324 K at a distance of 0.246 µm from the source end. The temperature at the pinch-off point is 323.25 K at a distance of 0.215 µm from the source end.  As we can see from the results that there is discrepancy between the analytical and numerical solutions in a few test scenarios. The two sources of error in finite dif83  5.4. Average and Maximum Temperature Distributions ference method are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation or discretization error. The remainder term of a Taylor’s polynomial is convenient for analyzing the local truncation error. Using the Lagrange form of the remainder from Taylor’s polynomial for f (x0 + h), the error term is  Rn (x0 + h) =  f (n+1) (ξ ) (n+1) (h) , x0 < ξ < x0 + h !(n + 1)  (5.4)  Since the IR drop in the channel is very low in the ohmic region, the thermal effects are not as significant as devices in saturation. Therefore, although the error between the analytic and numeric results is larger in this region the impact on EMR verification is expected to be negligible. The weak-inversion mode is not very important as a very weak current flows through the channel and the rise in the temperature of the channel due to it can be neglected.  5.4  Average and Maximum Temperature Distributions  The average and maximum channel temperature distribution over a range of different values of the drain voltage and gate voltages with corresponding drain current are provided in the following subsections. For these test scenarios the source temperature has been set to 318 K, the drain temperature has been set to 324 K, and the reference temperature has been set to 298 K. The layout drawn channel length of the device under test is 0.18 µm and the channel width is 0.40 µm. 84  5.4. Average and Maximum Temperature Distributions  5.4.1  Channel Temperature Distribution with Drain Voltage and Drain Current  Figure 5.13 shows a plot of the average channel temperature distribution versus the drain current and the effective gate voltage. HeatMOS© is run in a loop. In the first iteration, the gate voltage has been kept constant at 0.5 V and the drain voltage is varied from 0.0 V to 4 V with a step size of 0.5 V. In the second iteration the gate voltage is set to 1.0 V iteration and the drain voltage is varied from 0.0 V to 4.0 V with a step size of 0.5 V. Figure 5.14 shows the maximum channel temperature distribution versus the drain voltage and drain current.  5.4.2  Channel Temperature Distribution with Effective Gate Voltage and Drain Current  In the same manner as described above, the drain voltage can be kept constant and the gate voltage can be varied to get a plot of the average and the maximum channel temperature distribution versus the drain current and the the effective gate voltage, Vgs −Vth . Figure 5.15 shows the average channel temperature distribution versus the drain current and the effective gate voltage, while Figure 5.16 shows the maximum channel temperature distribution versus the drain current and the effective gate voltage. In the first iteration, the drain voltage has been kept constant at 1.0 V and the gate voltage is varied from 0.0 V to 2.0 V with a steps size of 0.05 V. In the second iteration the drain voltage is kept constant at 1.5 V and the gate voltage is varied from 0.0 V to 2.0 V with a step size of 0.05V. 85  5.4. Average and Maximum Temperature Distributions  HeatMOS  344 342 340 338 Tavg (K) 336 334 332 330 328 326 324  0.6  344 342 340 338 336 334 332 330 328 326 324  0.5 0.4 0.3 0  0.5  Ids (mA) 1  0.2 1.5 Vds (V)  2  2.5  0.1 3  3.5  4 0  Figure 5.13: The average channel temperature versus the drain current and the drain voltage  86  5.4. Average and Maximum Temperature Distributions  HeatMOS  355 350 345 340 355  335  350  330  345  325  Tmax (K) 340  0.6  335  0.5  330  0.4  325 320  320  0.3 0  0.5  Ids (mA) 1  0.2 1.5 Vds (V)  2  2.5  0.1 3  3.5  4 0  Figure 5.14: The maximum channel temperature versus the drain current and the drain voltage  87  5.4. Average and Maximum Temperature Distributions  HeatMOS  350 345 340  350  335 345 330 Tvg (K) 340  325 0.6  335 0.5 330 325  0.4 0  0.3 0.2  0.4  0.6  0.2 0.8  Vgs-Vth (V)  1  1.2  Ids (mA)  0.1 1.4  1.6  1.8 0  Figure 5.15: The maximum channel temperature versus the drain current and the drain voltage  88  5.4. Average and Maximum Temperature Distributions  HeatMOS  360 355 350 345 340 335 330 325 320  360 355 350 Tmax (K) 345 340  0.6  335  0.5  330 0.4  325 320  0  0.3 0.2  0.4  0.6  0.2 0.8  Vgs-Vth (V)  1  1.2  Ids (mA)  0.1 1.4  1.6  1.8 0  Figure 5.16: The maximum channel temperature versus the drain current and the effective gate voltage  89  5.5. Discussion  5.5  Discussion  Comparative results of HeatMOS with finite difference simulations validates the analytic model developed in this work. However a comparison of HeatMOS© results with experimental results need to be done in order to validate the accuracy of the results provided by HeatMOS©. A review of published literature shows that very few experimental measurements have been published on the profile of channel temperature in semiconductor devices. The best example that was found were results that were published by Reale for an HEMT device Reale et al. (2007).  If we compare results predicted by HeatMOS with experimental work published in Reale et al. (2007), we find very similar thermal distributions are obtained for the HEMT and the results generated by HeatMOS for MOSFET devices. The HEMT is a device with similar topology to the FET, which would be expected to give a similar channel temperature profile from source to drain. However it is much larger than the current advance technology nodes, so the thermal diffusion model is definitely valid for the HEMT devices. Figure 5.17 shows a cross-sectional view of a HEMT device. The experimental results for an HEMT device given in Reale et al. (2007) are shown in Figure 5.18 . The measured data shows a very similar profile to the results predicted using HeatMOS©. If we look at some experimental results of HEMT-based (High Electron Mobility Transistor) based devices, it can be noticed that HeatMOS© results follows the similar pattern. As can be seen from the Figure 5.18, the temperature is greatest in centre of the channel and toward the drain region (Reale et al., 2007). This is the similar result provided by HeatMOS©. However 90  5.5. Discussion  Figure 5.17: A typical cross-section of an HEMT device  91  5.6. Summary  Figure 5.18: Experimental channel temperature of HEMT device  HEMT-based devices are different than MOSFET in a way that the channel is a junction between two materials with different band gaps instead of a doped region and further work is required to experimentally validate the thermal model.  5.6  Summary  HeatMOS© has been verified under different test scenarios. The results of HeatMOS© have been compared with numerical solutions using the finite difference method. A range of different test scenarios were compared to validate the model in the saturation region and the ohmic region. The test scenarios include changes to source and drain temperature, different gate and drain voltages and different device sizes. In all cases, there was good agreement between the analytic model 92  5.6. Summary and numerical solutions. Finally, the trend in channel temperature predicted by the model was compared with experimental measurements which were found in the literature for an HEMT device. There are very few results available in the literature which report experimental results and this would be a good area for future research. The explanations of the different test scenarios have been provided along with differences with finite difference approach. The average and the maximum channel temperature distributions with the drain current variation and the drain and the gate voltage variations have been provided with explanations. Further, the discussions of HeatMOS© results and experimental results have been provided.  93  Chapter 6  Conclusions and Future Directions 6.1  Contributions  The analytical solution for the MOSFET channel temperature has been developed. The proposed solution for the predicting channel temperature is based on an electrical analogy and can be modeled as an M-network. The proposed solution and HeatMOS© can be integrated with other interconnect thermal analysis software tools to do a complete micro-chip thermal analysis rapidly. The results are comparable with numerical approaches. The advantage of an analytic model is that it will accelerate thermal modeling as compared with a rigorous thermal model using numeric solvers. The analytical thermal model for the device can be used to support EMR verification early in the design phase.  94  6.2. Limitations  6.2  Limitations  The present implementation of HeatMOS© treats the channel as a non-uniform resistor in its simplest form. There are some limitations of HeatMOS for the thermal analysis. The following list of limitations contains two categories: fundamental limitations and limitations of the first implementation of HeatMOS©. The fundamental limitations are somewhat difficult to solve but limitations of the first implementation can be rectified in future versions of the HeatMOS©.  6.2.1  Fundamental Limitations  1. It is limited to steady state thermal solution. The ODE approach is not appropriate for the transient problem because there is no closed form solution. 2. It is currently limited to be used with BSIM3 CMOS model. However extension to BSIM4 model should not be too difficult. 3. It is limited to do the analysis for MOS devices down to channel lengths of 0.18 µm. 4. Heat transport is by diffusion only; no sub-continuum or quantum effects are considered. There is no separate treatment of, say, optical and acoustic phonons. 5. An exponential channel profile with a uniform pinch-off region is assumed for modeling the channel cross-section.  95  6.2. Limitations 6. The effect of the process variation on temperature have not been considered. BSIM3 does not provide any information about the process variation on the temperature.  6.2.2  Limitations of First Implementation  1. It is limited to do an analysis for nMOS devices only in its current form. Only minor modifications are required to support pMOS devices (e.g. carrier mobility). 2. Junction effects are neglected in the analysis. 3. The effects of resistance between the contact and the source and the drain ends are neglected. 4. It is assumed that lateral thermal conductance is constant. 5. The body effect which describes the changes in the threshold voltage with a change in source-bulk potential has not be modeled. 6. The effects of leakage current on temperature distribution have not been modeled. 7. Channel thermal noise has not been considered. 8. The effects of substrate thermal noise have not been modeled.  96  6.3. Future work  6.3  Future work  The analytical approach can be fine-tuned in the future for thermal analysis taking into account the effect of various physical parameters (e.g. junction depth, variation in the lateral thermal conductance etc.). The next improvements in HeatMOS© would be to make it capable of estimating the pMOS channel temperature. This can be easily by including the pMOS device compact model parameters in HeatMOS© calculations and the analytical solution would be same as for the nMOS. The only difference is that in case of pMOS the charge carriers are holes instead of electrons so the physical properties of holes needs to be incorporated instead of electrons.  The effect of varying lateral thermal conductance can be taken into account along with the effect of contact resistance on thermal behavior. For full-chip thermal analysis and for EMR verification (Labun and Jagjitkumar, 2008), the M-network (Figure 3.7) can be connected to full micro-chip interconnects network and an interconnect network temperature estimation tool (e.g. Therminator) can be used to do full-chip thermal analysis.  The effects of process variation and leakage currents on temperature distribution can also be modeled. The analytical approach can be extended for double gate FinFET devices temperature estimation. However for nano-scale FinFET devices of below 90 nm geometries, the electron-phonon heat transport needs to be modeled. One idea may be to include the heat transport by phonon-electron interaction for a differential slice of the channel in the channel heat differential equation and solve the resultant differential equation. BTE (Boltzmann’s transport equation) for 97  6.3. Future work phonons may be quite useful to model the phonon’s heat conduction in a differential slice of the channel. The distinguishing characteristic of the FinFET is that the conducting channel is wrapped by a thin silicon “fin”, which forms the gate of the device (Figure 6.1). The heart of FinFET is a thin (∼ 10 nm) Si fin which serves as the body of the MOSFET (Hisamoto et al., 2000). A heavily doped poly-Si film wraps around the fin and makes electrical contact to vertical faces of the fin. The two faces of the fin act as two gates in place. The channel temperature estimation of such a device can be done by first splitting the channel between three regions: 1) region between the source end to left face of the fin 2) region between the left face and the right face of the fin and 3) the region between right face and the drain end of the transistor. With these partitions a similar approach could be taken to find the analytic solution for each region. However the shape of the each region of the FinFET’s channel needs to be considered carefully. Also since the fin has nano-scale dimensions, the nanoscale effects (e.g. phonos heat conduction, thermal noise etc.) need to be considered in order to accurately estimate the channel temperature of FinFET device.  The proposed analytical approach is faster than numerical approaches. HeatMOS© can be modified to do temperature estimation for all transistors in a micro-chip based on its layout information. 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Guoqiang, B. Goplen, H. Qian, Y. Zhan, S. Kang, M. D. F. Wong, and S. S. Sapatnekar. Temperature-aware placement for socs. Proceedings of the IEEE, 94(8):1502 –1518, 2006.  ISSN 0018-9219.  doi:  10.1109/JPROC.2006.879804.  W N. Weste and D. Harris. CMOS VLSI Design: A Circuits and Systems Prespective. Addison Wesley, 3 edition, 2004.  105  Appendix A  BSIM3 Model Parameters The table A.1 provides the list of BSIM3 parameters for nMOS transistor used in HeatMOS©. The table A.2 provides the list of some other parameters used.  106  Appendix A. BSIM3 Model Parameters  Table A.1: BSIM3 Parameters used in HeatMOS© Parameter  Description  Value  Unit  Vth0  Threshold voltage  0.3696386  V  μ0  Mobility  670.0  cm2 /V s  vsat  Saturation velocity  86301.58  m/s  Wl  Coefficient of length dependance for width offset  0.0  mW ln  Wln  Power of length dependence for width offset  0.0  none  Wwn  Power of width dependence for width offset  1.0  none  Ww  Width dependence for width offset  0.0  mW wn  Wwl  Length and width cross term for width offset  0.0  mW wn+W ln  Ll  Length dependence for length offset  0.0  mLln  Lln  Length dependence for length offset  1.0  none  Lw  Width dependence for length offset  0.0  mLwn  Lwn  Width dependence for length offset  0.0  none  Lwl  Length and width cross term for length offset  0.0  mLwn+Lln  Lint  Length offset fitting parameter  1.69494e − 8  m  Tox  Gate oxide thickness  1.5e − 8  Nsub  Substrate doping concentration  6.0e16  cm−3  107  Appendix A. BSIM3 Model Parameters  Table A.2: Other parameters used in HeatMOS© Parameter  Description  Value  Unit  K  Boltzmaan constant  1.38 × 10−23  m2 kgs−2 K −1  εr  Oxide dielectric constant  3.9  none  εo  Permitivity of air  8.85 × 10−12  F/m  108  Appendix B  SPICE Netlist The SPICE codes for the devices under test have been provided below.  109  Appendix B. SPICE Netlist  Program B.1 SPICE netlist "v8-nmos.spice" the nMOS transistor * SPICE3 file created from ../Magic/Work/v1-nmos.ext * technology: scmos .option scale=0.01u M1000 1 2 0 0 N1 w=40 l=18 + ad=200000 pd=1800 as=200000 ps=1800 .include bsim3.lib VDS 3 0 VGS 2 0 VIDS 3 1 .DC VDS 0 2.5 0.05 VGS 0 2.5 1 .control set color0=white set color1=black run plot i(VIDS) .ENDC .END  110  Appendix B. SPICE Netlist  Program B.2 SPICE netlist "v7-nmos.spice" of the nMOS transistor * SPICE3 file created from ../Magic/Work/v1-nmos.ext * technology: scmos .option scale=0.01u M1000 1 2 0 0 N1 w=40 l=28 + ad=200000 pd=1800 as=200000 ps=1800 .include bsim3.lib VDS 3 0 VGS 2 0 VIDS 3 1 .DC VDS 0 2.5 0.05 VGS 0 2.5 1 .control set color0=white set color1=black run plot i(VIDS) .ENDC .END  111  Appendix C  Numerical Solution of the Heat Equation The differential equation 3.20 can be solved by finite difference numerical method.  Let us consider,  T = f (x)  T (x) =     T1 (x1 ) 0 ≤ x1 ≤ L1   T2 (x2 ) 0 ≤ x2 ≤ L2  x2 = x − L 1  L = L1 + L2 112  Appendix C. Numerical Solution of the Heat Equation Function T (x) has a complex nature. For a length from 0 to L1 , it has temperature distribution given by function T1 (x), and for length from L1 to L, it has temperature distribution given by function T2 . Let us consider that the function T1 is evaluated on J1 number of points and T2 is evaluated at J2 number of points. T j1 is T1 (x j ) , the value of temperature function T1 at point numbered x j and T j2 is T2 (x j ), the value of temperature function T2 at point numbered x j , T j is the value of function T at point numbered x j . x1 =  1 J1 ;  x2 =  1 J2 ;  J = J1 + J2 ;  The differential equations 3.20 and 3.23 can be written as  T ” = ξ12 T1 − ξ12 T re f −Ψ 2 eαx ; 0 ≤ x ≤ L1  T ” = ξ22 T2 − β ; L1 ≤ x ≤ L  For each 0 ≤ j ≤ (J − 1), the above equations can be written by using the Taylor’s theorem as following.  1 − 2T 1 + T 1 T j−1 j j+1  x12  = ξ12 T j1 − ξ12 T re f −Ψ 2 eαx j ; 0 ≤ x j ≤ L1  2 − 2T 2 + T 2 T j−1 j j+1  x22  = ξ22 T j2 − β ; L1 ≤ x j ≤ L2 113  Appendix C. Numerical Solution of the Heat Equation For the boundary conditions, T01 = T O; TJ2 = T L.  The above equations can be solved using any matrix solver. The pseudo code of this approach has been given below. This can be solved using other software programs, e.g. Matlab, Octave.  % finite difference method %—————————% ./HeatMOS -SourceTemp 318 -DrainTemp 324 -RefTemp 298 -VGS 1.2 -VDS 1.8 -Netlist v8-nmos.spice %—————————format long; %——————————————————————————————— % Test scenario 1 : Verification test 1 % SourceTemp = 575; DrainTemp = 567; RefTemp = 456; VGS = 1.0; % VDS = 1.5; Ids=2mA; v7-nmos.spice %——————————————————————————————— len1 = 2.093229e-07; len2 = 3.677829e-08; 114  Appendix C. Numerical Solution of the Heat Equation xie1 = 8.791518e+00; psi1 = 5.274110e-10; alpha1 = 1.803726e+07; xie2 = 8.791518e+00; beta2 = 3.790852e+16; TO = 5.650000e+02; TL = 5.670000e+02; TRef = 4.560000e+02; %——————————————————————————————— % Test scenario : Verification test 2 % SourceTemp = 575; DrainTemp = 567; RefTemp = 456; VGS = 1.5; % VDS = 1.5; v7-nMOS.spice %——————————————————————————————— % xie1 = 8.791518e+00; psi1 = 5.274110e-10; alpha1 = 1.803726e+07; % xie2 = 8.791518e+00; beta2 = 3.790852e+16; % TO = 5.650000e+02; TL = 5.670000e+02; TRef = 4.560000e+02; % len1 = 2.093229e-07; len2 = 3.677829e-08; %———————————————————————————————% Test scenario : Verification test 3 % SourceTemp = 565; DrainTemp = 567; RefTemp = 456; VGS = 1.5; % VDS = 1.5; v8-nMOS.spice %———————————————————————————————115  Appendix C. Numerical Solution of the Heat Equation % xie1 = 1.077370e+01; psi1 = 5.808510e-11; alpha1 = 5.279674e+07; % xie2 = 1.077370e+01; beta2 = 7.736773e+15; % TO = 5.650000e+02; TL = 5.670000e+02; TRef = 4.560000e+02; % len1 = 1.095764e-07; len2 = 3.652475e-08; %———————————————————————————————– %———————————————————————————————– % Test scenario : Verification test 4 % SourceTemp = 10; DrainTemp = 10; RefTemp = 10; VGS = 1.5; % VDS = 1.5; v8-nMOS.spice %———————————————————————————————% xie1 = 6.774943e+00; psi1 = 2.029696e-12; alpha1 = 1.793608e+07; % xie2 = 6.774943e+00; beta2 = 1.389513e+17; % TO = 1.000000e+01; TL = 1.000000e+01; TRef = 1.000000e+01; % len1 = 1.095764e-07; len2 = 3.652475e-08; %———————————————————————————————%———————————————————————————————% Test scenario : Saturation test 1 116  Appendix C. Numerical Solution of the Heat Equation % SourceTemp = 325; DrainTemp = 325; RefTemp = 300; VGS = 1.5; % VDS = 1.5; v8-nMOS.spice % ——————————————————————————————– % xie1 = 1.038507e+01; psi1 = 3.964484e-11; alpha1 = 4.897557e+07; % xie2 = 1.038507e+01; beta2 = 1.092628e+16; % TO = 3.250000e+02; TL = 3.250000e+02; TRef = 3.000000e+02; % len1 = 1.095764e-07; len2 = 3.652475e-08; %——————————————————————————————— %——————————————————————————————— % Test scenario : Saturation test 2 % SourceTemp = 325; DrainTemp = 325; RefTemp = 300; VGS = 1.8; % VDS = 2.2; v8-nMOS.spice %——————————————————————————————— % xie1 = 1.027263e+01; psi1 = 6.417728e-11; alpha1 = 4.839836e+07; % xie2 = 1.027263e+01; beta2 = 2.434460e+16; % TO = 3.250000e+02; TL = 3.250000e+02; TRef = 3.000000e+02; % len1 = 1.084309e-07; len2 = 3.767029e-08; 117  Appendix C. Numerical Solution of the Heat Equation %———————————————————————————————%———————————————————————————————% Test scenario : Saturation test 3 % SourceTemp = 325; DrainTemp = 325; RefTemp = 300; VGS = 5.0; % VDS = 5.0; v8-nMOS.spice %———————————————————————————————– % xie1 = 8.673537e+00; psi1 = 7.965811e-10; alpha1 = 3.447079e+07; % xie2 = 8.673537e+00; beta2 = 3.913625e+17; % TO = 3.250000e+02; TL = 3.250000e+02; TRef = 3.000000e+02; % len1 = 1.063123e-07; len2 = 3.978895e-08; %———————————————————————————————– %———————————————————————————————— % Test scenario : Saturation test 4 % SourceTemp = 325; DrainTemp = 325; RefTemp = 300; VGS = 5.0; % VDS = 5.0; v7-nMOS.spice %———————————————————————————————– % xie1 = 9.219493e+00; psi1 = 4.448968e-10; alpha1 = 2.028882e+07; 118  Appendix C. Numerical Solution of the Heat Equation % xie2 = 9.219493e+00; beta2 = 2.621273e+17; % TO = 3.250000e+02; TL = 3.250000e+02; TRef = 3.000000e+02; % len1 = 2.062546e-07; len2 = 3.984658e-08; %———————————————————————————————%———————————————————————————————% Test scenario : Saturation test 5 % SourceTemp = 318; DrainTemp = 324; RefTemp = 298; VGS = 1.2; % VDS = 1.8; v8-nMOS.spice %———————————————————————————————% xie1 = 1.109780e+01; psi1 = 1.988555e-11; alpha1 = 5.635639e+07; % xie2 = 1.109780e+01; beta2 = 8.011858e+15; % TO = 3.180000e+02; TL = 3.240000e+02; TRef = 2.980000e+02; % len1 = 1.090203e-07; len2 = 3.708086e-08; %———————————————————————————————– %———————————————————————————————– % Test scenario : Ohmic test 1 % SourceTemp = 300; DrainTemp = 325; RefTemp = 300; VGS = 1.0; 119  Appendix C. Numerical Solution of the Heat Equation % VDS = 0.1; v8-nMOS.spice %———————————————————————————————— % xie1 = 4.472554e+00; psi1 = 8.362492e-12; alpha1 = 8.520509e+02; % xie2 = 4.472554e+00; beta2 = 7.811362e+10; % TO = 3.000000e+02; TL = 3.250000e+02; TRef = 3.000000e+02; % len1 = 1.173697e-07; len2 = 2.873146e-08; %———————————————————————————————– %———————————————————————————————– % Test scenario : Ohmic test 2 % SourceTemp = 318; DrainTemp = 324; RefTemp = 300; VGS = 1.0; % VDS = 0.1; v7-nMOS.spice %———————————————————————————————— % xie1 = 4.472554e+00; psi1 = 4.964503e-12; alpha1 = 4.640764e+02; % xie2 = 4.472554e+00; beta2 = 2.753005e+10; % TO = 3.180000e+02; TL = 3.240000e+02; TRef = 3.000000e+02; % len1 = 2.154926e-07; len2 = 3.060865e-08; %———————————————————————————————– 120  Appendix C. Numerical Solution of the Heat Equation lengthTotal = len1 + len2; %J1 = 20; %J2 = 10; J1 = 100; J2 = 50; dx1 = len1/J1; dx2 = len2/J2; xt1 = [0:dx1:len1] ; x1 = xt1.’ ; xt2 = [0:dx2:len2] ; x2 = xt2 .’ ; b = zeros(J1+J2, 1); b(1)=TO; b(J1+J2)=TL; for i=2:(J1+J2-1) if i < J1 b(i) = -(dx1^2)*(xie1^2)*TRef - (dx1^2)*(psi1^2)*(exp(alpha1*x1(i))); elseif i > J1 121  Appendix C. Numerical Solution of the Heat Equation b(i) = -(dx2^2)*beta2; else b(i) = 0.0; endif endfor A=sparse(J1+J2, J1+J2); A(1,1) = 1.0; A(J1+J2, J1+J2) = 1.0; for i=2:(J1+J2-1) if i < J1 A(i, [i-1, i, i+1]) = [1.0, -(2.0-(xie1^2)*(dx1^2)), 1.0]; elseif i > J1 A(i, [i-1, i, i+1]) = [1.0, -(2-(dx2^2)*(xie2^2)), 1.0]; else A(i, [i-1, i, i+1]) = [1.0/dx1, -(1.0/dx1 + 1.0/dx2), 1.0/dx2]; endif endfor 122  Appendix C. Numerical Solution of the Heat Equation Y=A\b; for i=1:J1 x(i)=x1(i); endfor for j=1:J2 x(J1+j)=len1.+x2(j); endfor myfile = "chanTempFD.out" ; FILE = fopen(myfile, "w"); for i=1:(J1+J2) fprintf (FILE, "%e\t", x(i)); fprintf (FILE, "%e\n", Y(i)); endfor fclose(FILE); %———————————————————————  123  Appendix D  HeatMOS© Development D.1  Development Environment  HeatMOS© has been developed in “C” programming language. It has been compiled using gnu c-compiler, g++4.0.1 for i686-apple-darwin9 machine. HeatMOS© has been executed on Apple MAC OS X version 10.5.8 machine with Intel Core 2 Duo 1 GHz processor and 1 GB of DDR2 SDRAM. To plot graphs Gnuplot version 4.4 software has been used. Octave version 3.2.3 software has been used for the numerical solution of the channel heat equation.  D.2  A Sample HeatMOS© Execution  A screen snapshot of sample HeatMOS© execution is split in two parts and shown in Figure D.1 and D.2. The top line in Figure D.1 shows the command for HeatMOS© execution. The temperature distribution along the length of the channel in being output in a text file, which can be used to plot graphs using a graph plotting software (e.g. gnuplot).  124  D.2. A Sample HeatMOS© Execution  Figure D.1: A sample HeatMOS run part 1 125  D.2. A Sample HeatMOS© Execution  126  Figure D.2: A sample HeatMOS run part 2  

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