Georgia Institute of Technology JAD EE 6458 Homework II Gigascale Integration Assigned: February 12, 2007 SPRING2007 Due: February 23, 2007 Homework II: Fundamental Limits 1. Please define the term “Fundamental Limit.” 2. Voltage Distributions Due to Thermal Fluctuations In class, we discussed that the mean-square voltage due to thermal movement of electrons is given by: V2= kT 4R(!f ) Using concepts of entropy, can you determine the voltage distribution that has the highest entropy UNDER the constraint of a constant variance? 3. Entropy of Wire Length Distributions Using the analysis in class as a guide, can you use arguments of statistical mechanics to describe the most probable wire length distribution in a netlist. Assume that the distribution is constrained by the total interconnect length (i.e. total energy) and the number of wires in a GSI system. Also, assume that the length of the wires is measured by a discrete number of gate pitches. Furthermore, use the following expression for the average wire length: Lavg=p ! 0.5p! N !p ! 0.56 N p + 0.5( )+ Np!p ! 1 + 4p !0.52 p + 0.5( )p p ! 1( )"#$%&'"#$%&'Np !0.5!2p ! 1 + 22 p ! 12p p ! 1( )2p ! 3( )!p ! 0.5( )6p N+ 1 !p ! 0.5( )Np ! 1( )"#$%&' where p is an empirical constant called the “Rent’s exponent” (assume a value of 2/3) and N is the number of logic gates in a netlist. Also assume that the number of point-to-point connections (i.e. wires) in a given system is equal to the average fanout (assume 2.5) times the total number of logic gates. Can you calculate the Lagrangian Multipliers based on these constraints? Using excel, make a plot of your wire length distribution systems with 1 million gates, 100 million gates, 1 billion gates, and 100 billion gates, respectively. 4. Quantum Mechanical Limits Calculate the intersection in the power delay plane of the thermodynamic limit and the quantum limit for 300oK, 77oK , and 3oK? In light of the ultra-violet catastrophe, do any of these points have significant error?5. de Broglie relationship and Electron Wavelength (!=hp) To understand when we must include quantum effects in our switching devices, one rule of thumb is to compare the electron wavelength to the dimensions of the device. When they are comparable then quantum mechanical uncertainty and tunneling could emerge. Calculate the electron wavelength for the following conditions. Assume that the rest mass of an electron is 9.11x10-31 kg, the saturation velocity of an electron in silicon is roughly 107 cm/s, and Planck's constant is 6.626x 10-34 J-s. Also assume that the "effective mass" of a electron in a silicon crystal is a third of the rest mass. a. What is the electron wavelength for a single electron traveling at the saturation velocity? b. What is the electron wavelength for a single electron traveling at a tenth of the saturation velocity? c. What if we could hypothetically control the electron movement (without lattice collisions, phonon scattering, etc) such that in a 90nm channel the electron transit time took an entire microsecond. What would be the electron wavelength? If we tried to measure the kinetic energy of this electron within a time resolution of 1 microsecond, what is the uncertainty in the energy of the electron according to quantum mechanics? How does this energy uncertainty compare to the classical kinetic energy of the electron? If we were to measure the momentum of the particle with an uncertainty that is 10% of the classical momentum value that you have calculated, what is the uncertainty in the position of this slow moving electron? 6. Single Electron Transistors Please read the paper entitled “Amplifying Quantum Signals With the Single-Electron Transistor” by Michael Devoret and Robert Schoelkopf and answer the following questions. a. Draw a picture of the SET device described in the paper. Indicate where the quantum tunneling occurs on your drawing. b. What is coulomb blockage? c. What is quantum shot noise? 7. Energy Dissipation Limits Using basic circuit theory for a lumped RC circuit with an ideal voltage source with a finite rise time (call it Trise), calculate the TOTAL AMOUNT OF ENERGY DISSIPATED by the resistance for the following conditions. a. Assume Trise= 0 (i.e. a step input response) b. Assume Trise=! c. AssumeTrise=!RC d. Assume Trise! " 8. Can you suggest another fundamental limit on GSI not discussed in