Problem Formulation for Arch Sim and EM Model

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Problem Formulation for Arch Sim and EM Model 1 Problem Formulation 1.1 System Description The system consists of M modules, each module has L wire segments.

1 Problem Formulation for Arch Sim and EM Model 1 Problem Formulation 1.1 System Description The system consists of M modules, each module has L wire segments. The wire segments are routed in same or different metal layers available for a process technology. Figure 1 shows this system in the form of a tree structure. The system is at the root of the tree whereas the wire segments form the leaf nodes in the tree. The system, in our case, is a functional unit. Table 1 is a glossary of terminology used in the problem description and formulation. Figure 1: System Description as a tree. 1.2 Inputs to the EM Model 1. Foundry technology LEF with T re f, α re f and MT T F re f as in Table 1. The LEF describes design rules for current density limits, J peak,max, J rms,max and J avg,max based on these reference values. 2. The number of years, N, of operation after which probability of failure of the system is to be calculated. 1

2 Table 1: Glossary of terms Term Range Meaning T re f 105 C Foundry reference temperature α re f 0.2 Foundry reference switching activity MT T F re f 10 years Foundry reference Mean Time to Failure MT T F red - Reduced Mean Time to Failure J peak,max Foundry peak current density limit J rms,max J peak α re f Foundry RMS current density limit J J peak α re f Foundry avg current density limit avg,max (same for AC and DC) m i i = 1,2,..M module with index i p i i = 1,2,..M probability of failure of m i s j j = 1,2,..L wire segment with index j α j 0 < α j 1 switching activity of s j W min, j j = 1,2,..L Foundry min. wire width for s j W max, j j = 1,2,..L Foundry max. wire width for s j w 0, j W min, j w 0, j W max, j width of s j after initial place and route w red, j w 0, j w red, j W max, j width of s j when s j MTTF is MT T F red λ j j = 1,2,..K failure rate of s j p j j = 1,2,..L probability of failure of s j J peak, j j = 1,2,..L Peak current density of s j J rms, j j = 1,2,..L RMS current density of s j λ j j = 1,2,..L failure rate of s j T joule,max 5 Max incr in temperature due to Joule-heating T T j T re f + T joule,max s j temperature increase j due to Joule-heating (for AC stress) MT T F j 1/λ j MTTF of s j MT T F EQ, j - Equivalent MTTF of s j at α j and T j P re f 0 P re f 1 probability of failure of system with MT T F re f P red 0 P red 1 probability of failure of system with MT T F red MT T F - MTTF of the system N 1,2,..,MT T F re f Years of operation E a 0.7eV Activation energy of AlCu metal ions B ev /K Boltzmann s constant t ins 0.27µm Inter-layer dielectric (ILD) thickness K ins 28W/m K Thermal conductivity of ILD t m 0.125µm for M1, M2 Thickness of metal wire W m 0.07µm for M1, M2 Minimum width of metal wire ρ K 1 Resistivity of metal wire at 300K β K 1 Temperature coefficient of ρ ρ(t j ) ρ 0 (1 + βt joule,max ) Resistivity of metal wire at T j W e f f W m t ins Effective width for 2D heat conduction [2] t F ins tmwmρ(t j ) T h K ins W e f f Thermal factor 3. Switching activity of each wire segment, α j ( j = 1,2,..,L) 1.3 Model Assumptions The EM model assumes the system has been placed and routed within a given die area and has met timing constraints, i.e., has no paths with a negative slack. 1.4 Model Constraints 1. Temperature of a wire segment, T j can increase at most by T joule,max due to Joule-heating [2] under AC stress. T joule,max = 5K for most designs. 2. P red P re f 2

3 1.5 Model Objective Reduce MTTF under given inputs and constraints. 1.6 Model Outputs 1. Reduced MTTF, MT T F red. 2. Probability of failure of the system, P re f, P red and MTTF of the system, MT T F. 2 Model Development 2.1 Model with Foundry Guardband Algorithm 1 Probability of Failure of System Procedure ProbFail [1] For each module m i, i = 1,..,M [2] For each wire segment, s j, j = 1,..,L [3] J rms, j α j J peak,max ; [4] T j T re f + Jrms, 2 j F T h; ( Jrms,max [5] MT T F EQ, j MT T F re f J rms, j [6] λ j 1/MT T F EQ, j ; ) 2 e Ea B (1/T j 1/T re f ) ; [7] p j 1 e λ j N ; //because MTTF is exponentially distributed L [8] p i 1 (1 p j ); j=1 [9] if MT T F re f then M [10] P re f 1 (1 p i ); i=1 [11] MT T F N loge(1 P re f ) [12] else M [13] P red 1 (1 p i ); i=1 [14] MT T F N loge(1 P red ) In lines [1] and [2] of Algorithm 1, we iterate through all the M modules in the system and L wire segments in a module respectively. In line [3], we calculate the RMS current density for each wire segment from the peak AC density defined by the foundry for the metal layer to which the wire segment belongs. In line [4], we calculate the effective temperature on the wire segment after accounting for temperature rise due to Joule-heating. In line [5], we calculate the equivalent MTTF for the wire segment with its equivalent temperature as T j and RMS current density as J rms, j using Black s Law. In line [6], we calculate the failure rate of the wire segment which is the inverse of its MTTF calculated in line [5]. In line [7], we calculate the probability of failure of the wire segment using the fact that reliability of a wire after N years of operation follows an exponential distribution [1]. In line [8], we calculate the probability of failure of the module from the probability of failure of all wire segments that it contains [3]. In line [9], we check if ProbFail has been invoked with MT T F re f. In line [10], we calculate the probability of failure of the system after N years from the probability of failure of all the modules it contains, when the procedure is invoked with MT T F re f. In line [11], we calculate the overall MTTF of the system from its probability of failure calculated in line [10]. In line [13], we calculate the probability of failure of the system after N years from the probability of failure of all the 3

4 modules it contains, when the procedure is invoked with MT T F red. In line [14], we calculate the overall MTTF of the system from its probability of failure calculated in line [13]. 2.2 Model with MTTF Optimization Table 2 explains the tunable knobs for MTTF and their potential impact. Table 2: MTTF knobs Knob Impact J peak,max Affects J rms,max and J avg,max. Increasing J rms,max will cause increase in temperature due to Joule-heating and exponentially decrease MTTF. Wire Width Increasing wire width decreases J peak,max and wire delay but reduces TDDB. [Any other reliability downsides??] Driver Size Increasing driver size increases J peak,max and reduces gate delay and slew. MTTF Reducing the guardband on lifetime of operation can allow higher J peak,max or reduced wire widths. In this formulation, we minimize MT T F re f such that P red P re f. We cannot reduce switching activity since it is a function of the workload. We cannot reduce T re f since it is a foundry parameter. We can only increase J peak,max by sizing up drivers and reducing wire widths subject to temperature increase due to Joule-heating be T joule,max. One possible Algorithm is shown in 2. Algorithm 2 MTTF Optimization Procedure MTTFOpt [1] For each wire segment, s j, j = 1,..,LM 1 T [2] J peak, j joule,max α 2 F j T h [3] J if peak, j J 1 then peak,max [4] For w red, j = W max, j W min, j [5] Increase Driver size (gate width) until peak current is J peak, j w red, j [6] Perform STA to ensure no hold or setup violations. Go to Step [5] if there is a failure and re-size depending on failure. [7] J peak, j J peak, j w red, j w 0, j [8] MT T F red MT T F re f ( Jpeak,max minj peak, j L j=0 [9] Call Procedure ProbFail with MT T F red [10] if P red P re f then [11] return MT T F red [12] MT T F red MT T F re f [13] return MT T F red ) 2 e Ea B (1/(Tre f +T joule,max ) 1/T re f ) In line [1] of Algorithm 2, we iterate through all the L wire segments across M modules, LM. In line [2], we calculate the peak current density of the wire segment at its switching activity α j and maximum temperature due to Joule-heating as T joule,max. It is assumed that this peak current density is with the wire width after place and route, w 0, j. In line [3], we check if the peak current density calculated in line [2] is greater than foundry peak current density limit. If this is true, then in line [4], we choose an optimized wire width by iterating from the foundry maximum wire width to the foundry minimum wire width. In line [5], we increase the driver size so that the peak current is as close to the product of the peak current density and optimized wire width. In line [6], we perform STA to ensure no timing violations. If there are violations, go back to Step [5] and re-size drivers differently depending on the failure. In line [7], we re-calculate the peak current density of the wire segment based on the optimized value chosen, w red, j. In line [8], we calculate the reduced MTTF value of the system using the minimum peak current density 4

5 across all the wire segments. In line [9], we calculate the probability of failure of the system with the reduced MTTF, P red. In line [10], we check if the reduced probability of failure is less than the reference probability of failure at MT T F re f after N years. In line [12], if the condition in line [9] is true, then return this reduced MTTF. If not, then return the reference MTTF in line [13]. References [1] D. J. Smith, Reliability, Maintainability and Risk 6th edition. Butterworth Heinemann Publishers, Woburn, MA, [2] K. Banerjee and A. Mehrotra, Coupled Analysis of Electromigration Reliability and Performance in ULSI Signal Nets, Proc. ICCAD, 2001, pp [3] E. Karl, D. Blaauw, D. Sylvester and T. Mudge, Reliability Modeling and Management in Dynamic Microprocessor-Based Systems, Proc. DAC, 2006, pp

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