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High-yield, high-performance memory design
Trent McConaghy - Solido Design Automation
11/5/2012 10:33 AM EST
Two solido and TSMC case studies
Two solido and TSMC case studies
Below we describe two memory circuit examples, which illustrate the benefits of combining TSMC’s statistical device models with the Solido HSMC tool.
The first example is on a 6 transistor bitcell, using statistical device models from the TSMC 28nm PDK. With 6 devices, it has 60 local process variables. As a reference, we generated 1M Monte Carlo samples, then simulated them and measured read current (cell_i). We plotted those in a normal quantile (NQ) plot, shown below. Each dot is a Monte Carlo sample point.
NQ plots make it easier to see the tails of a distribution. In a NQ plot, the x-axis is the circuit output and the y-axis is the cumulative distribution function (CDF) scaled exponentially. In a circuit with linear response of output to process variables, the NQ curve will be linear – a straight line of dots from the bottom left to the top right. Nonlinear responses give rise to nonlinear NQ curves. In the bitcell NQ plot, the bend in the middle of the curve indicates a quadratic response in that region. The sharp dropoff in the bottom left shows that for process points in a certain region, the whole circuit shuts off, for a current of 0. The curve’s shape clearly indicates that any method assuming a linear response will be extremely inaccurate, and even a quadratic response will suffer.
On the same bitcell read current problem, we then ran Solido HSMC, using 100M generated Monte Carlo samples. In 5500 simulations, it accurately found the tail of the read current distribution, shown as the red dots in the upper right corner of the NQ plot. With this tail, the designer can estimate yield (given a target spec), or a spec (given a target yield). Traditional Monte Carlo sampling would have needed 100M simulations to get the same tail information that HSMC got in 5500 simulations. Furthermore, each of the red dots is an actual point in process variable space, which can be used as a corner to design against. This makes it easy to do rapid design iterations: the designer can test a design point by doing just one simulation on the statistical corner to measure read current; while implicitly designing against the target yield.
The second case is a sense amp delay, having 15 devices and 150 process variables, also using statistical device models from the TSMC 28nm PDK. Once again, we generated and simulated 1M Monte Carlo samples for reference. The NQ plot is shown below. The three vertical “stripes” of points indicate three distinct sets of values for delay -- a trimodal distribution. The jumps between the strips indicate discontinuities: a small step in process variable space sometimes leads to a giant shift in performance.
Such strong nonlinearities will make linear and quadratic models fail completely; in this case they would completely miss the mode at the far right at delay of about 1.6 ns. The NQ plot below also illustrates Solido HSMC results (in red) on 150-variable sense amp, where Solido HSMC used 100M generated Monte Carlo samples. In <10K simulations, Solido HSMC effectively found the tails of each distribution.
Summary
This article reviewed the performance and yield challenges that memory designers face at 28nm, and the market pressures driving those challenges. It then describes how Solido assisted to provide a path for memory designers to rapidly and accurately estimate the yields of their memory designs, using TSMC statistical models and PDKs combined with Solido’s High-Sigma Monte Carlo (HSMC) tool.
I wish to thank Bob Mullen, Technical Manager at TSMC for helping organize this article and apply some of the Solido solutions to TSMC.
Two solido and TSMC case studies
Below we describe two memory circuit examples, which illustrate the benefits of combining TSMC’s statistical device models with the Solido HSMC tool.
The first example is on a 6 transistor bitcell, using statistical device models from the TSMC 28nm PDK. With 6 devices, it has 60 local process variables. As a reference, we generated 1M Monte Carlo samples, then simulated them and measured read current (cell_i). We plotted those in a normal quantile (NQ) plot, shown below. Each dot is a Monte Carlo sample point.
NQ plots make it easier to see the tails of a distribution. In a NQ plot, the x-axis is the circuit output and the y-axis is the cumulative distribution function (CDF) scaled exponentially. In a circuit with linear response of output to process variables, the NQ curve will be linear – a straight line of dots from the bottom left to the top right. Nonlinear responses give rise to nonlinear NQ curves. In the bitcell NQ plot, the bend in the middle of the curve indicates a quadratic response in that region. The sharp dropoff in the bottom left shows that for process points in a certain region, the whole circuit shuts off, for a current of 0. The curve’s shape clearly indicates that any method assuming a linear response will be extremely inaccurate, and even a quadratic response will suffer.
On the same bitcell read current problem, we then ran Solido HSMC, using 100M generated Monte Carlo samples. In 5500 simulations, it accurately found the tail of the read current distribution, shown as the red dots in the upper right corner of the NQ plot. With this tail, the designer can estimate yield (given a target spec), or a spec (given a target yield). Traditional Monte Carlo sampling would have needed 100M simulations to get the same tail information that HSMC got in 5500 simulations. Furthermore, each of the red dots is an actual point in process variable space, which can be used as a corner to design against. This makes it easy to do rapid design iterations: the designer can test a design point by doing just one simulation on the statistical corner to measure read current; while implicitly designing against the target yield.
The second case is a sense amp delay, having 15 devices and 150 process variables, also using statistical device models from the TSMC 28nm PDK. Once again, we generated and simulated 1M Monte Carlo samples for reference. The NQ plot is shown below. The three vertical “stripes” of points indicate three distinct sets of values for delay -- a trimodal distribution. The jumps between the strips indicate discontinuities: a small step in process variable space sometimes leads to a giant shift in performance.
Such strong nonlinearities will make linear and quadratic models fail completely; in this case they would completely miss the mode at the far right at delay of about 1.6 ns. The NQ plot below also illustrates Solido HSMC results (in red) on 150-variable sense amp, where Solido HSMC used 100M generated Monte Carlo samples. In <10K simulations, Solido HSMC effectively found the tails of each distribution.
This article reviewed the performance and yield challenges that memory designers face at 28nm, and the market pressures driving those challenges. It then describes how Solido assisted to provide a path for memory designers to rapidly and accurately estimate the yields of their memory designs, using TSMC statistical models and PDKs combined with Solido’s High-Sigma Monte Carlo (HSMC) tool.
I wish to thank Bob Mullen, Technical Manager at TSMC for helping organize this article and apply some of the Solido solutions to TSMC.
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