# Line Edge Roughness (LER)

## Lacerm works well with a wide range of images. The line edge roughness and other parameters of a line/space pattern can be obtained all together by first selecting a region of interest (ROI) then simply hitting the [Calculation] button.

The picture above shows that Lacerm works great with a horizontal line/space pattern. In fact, it can run smoothly and accurately with a pattern of any tilt angle, likes the following image where the orientation of the lines is very much off axis.

During an SEM inspection, more secondary electrons can leave the sample at an edge than they do at a flat surface. The edges are therefore often the brightest areas on the image. An edge detector in turn can mistake these bright areas for the lines while the darker areas for the spaces. As a result, two edges close to each other are generated (picture on the left).

Another great feature of Lacerm is the capability of removing double edges caused by the edge effect. By checking [Remove Double Edge] box before hitting [Calculate] button, the edges can be correctly detected and measured (right picture).

**Tips**

There are two ways to calibrate the scale (if no calibration is carried out, nm/pixel ratio will take the default value of 1). You can directly change the nm/pixel ratio by single clicking on it and inputting a new value, or you can also get the nm/pixel ratio from the known distance of two points (a scale bar) on the image.

**How are the result values calculated?**

Assume that we have detected *n* edges, the number of points on i^{th} edge is *N*_{i} (*i* = 1,...,n) , and Δ*x*_{ij} is the edge residual of point *j* on edge *i*.

In lithography metrology, a set of lines and spaces are often created in the same orientation. For this reason, the orientations of the fitting lines on each edge are considered to be the same in all the following calculations with only one exception when calculating Mean-LER*.

**1) LER**

Line edge rough is defined as 3 times the standard deviation *σ* (LER = 3**σ*).

σ=1N0-1∑j=1N0∆xj-μ2,

where N_{0} is the total number of all points *N*_{0}* = N*_{1}* + N*_{2}* + ... + N*_{n}, and *µ* is the average of Δ*x*_{j}.

**2) Mean LER**

Line edge roughness of the i^{th} edge LER_{i} is defined as 3 times its standard deviation *σ*_{i}. LER_{i} = 3**σ*_{i}*.*

σi=1Ni-1∑j=1Ni∆xij-μi2,

where *µ*_{i} is the average of Δ*x*_{ij}.

Mean LER is then defined as the average of LER_{i} (i = 1,...,n).

**3) Mean LER**^{*}** **

Mean LER* is defined the same way as Mean LER, except that that the orientations of the fitting lines are different and are best fitted to each edge. In a typical line-space pattern, the values of Mean-LER and Mean-LER^{*} are expected to be very close.

**4) Odd LER**

Odd LER is the average of odd-numbered edges, i.e. the ones on the left in the case of vertical lines or the upper in the case of horizontal lines.

**5) Even LER**

Even LER is the average of the right-side edges (or the upper in horizontal lines).

**6) Mean CD**

Mean CD is the average of the linewidths of the lines.

**7) Pitch**

Pitch is the average of the pitches of the lines and spaces.