3.1.4 Modeling the Effect of Kelvin Voltage Taps on Electrically-Certified Linewidth Reference Materials
Statistical Engineering Division, ITL
Transducer Systems Division, Renishaw plc
Semiconductor Electronics Division, EEEL
As in most types of manufacturing, reference materials are used with integrated circuits (IC's) to ensure that the measurement processes used for quality control and manufacturing process characterization are accurate. Linewidth measurement procedures, which measure the width of Al `wires' incorporated in a chip to connect circuit components, are one type of procedure commonly validated using reference materials.
The systems used to measure linewidth in production include a variety of scanning beam and scanning probe systems. Certification of reference materials can be done with accurate, laboratory versions of scanning systems, or by electrical methods. Electrical methods have the advantages of being relatively precise and insensitive to many of the sources of error affecting the other systems.
Unlike many other types of manufacturing, the reference materials used with IC's must have test structures which simulate circuit features added to the material. In the case of linewidth, test structures consist of a length of conductor called a bridge that is connected to voltage taps at each end which link it to pads where the electrical measurement instrumentation is connected.
Until recently, designs for electrically-certifiable linewidth test structures were limited by the need to keep voltage taps narrow relative to the linewidth. Wide taps effectively shorten the bridge length by slightly increasing the surface area of the conductor, increasing its measured linewidth. New test structures designed at NIST, however, eliminate the need for restrictive design rules by correcting for the effective shortening of the bridge, denoted L. The new test structures increase the spatial resolution of the reference material measurements, but require relatively complex nonlinear analysis to determine linewidths.
To ensure that the results of linewidth data analysis from a monocrystalline Si reference material were accurate, finite element calculations of L were made for confirmation. Because the computations are time consuming and are needed for test structures of many different linewidth sizes, development of a simple model relating L and the linewidth was undertaken, rather than direct comparison of measured and calculated results. L also depends on the tapwidth and a measure of the incomplete etching of the corners between the line and taps, called the facet size, so those factors were also included.
Two considerations dictated the choice of the experiment design. First, the form of the model was not known, but scientific knowledge suggested L should vary smoothly and slowly with the varying inputs. The second point was the expense of the finite element calculations. As a result of these considerations, a variant of the central composite design was chosen. This design allows estimation of a full quadratic model in three factors and requires only fifteen data points.
The initial fit of the full model to the data and subsequent graphical residual analysis suggested that the model fit well except for one outlier. Omitting the outlier and refitting the model verified that it fit the remaining data well and could be simplified even further by dropping two interaction terms.
In this situation, selection of an appropriate model hinges on the interpretation of the outlier. The finite element calculations were verified to be correct, however, and extensive additional data collection was not an option. As a compromise, the outlier was omitted from the analysis and the scope of the model was limited to an appropriate region.
Because of the outlier, and the fact that the data is used primarily to estimate parameters rather than to detect lack of fit with this design, it seemed prudent to test the model with some independent data. A comparison of predictions from the model to finite element responses for five random test points is shown in the accompanying figure. From the figure it is clear the model predicts the values of L observed at the test points well, evidence that the limited-scope model is reasonable.
The L predictions from the model matched the experimental results reasonably well, directly confirming the benefit of the L correction and helping verify the suitability of test structures replicated in monocrystalline Si as measurement references.
Figure 4: Linewidth test structure in monocrystalline silicon reference material (left). Predictions from the limited-scope model for L versus finite element `data' (right).
Date created: 7/20/2001