SPIE Handbook of Microlithography, Micromachining and Microfabrication
Volume 1: Microlithography
Section 2.4 Proximity Effect
The net result of the electron scattering discussed in the previous section is that the dose delivered by the electron beam tool is not confined to the shapes that the tool writes, resulting in pattern specific linewidth variations known as the proximity effect. For example, a narrow line between two large exposed areas may receive so many scattered electrons that it can actually develop away (in positive resist) while a small isolated feature may lose so much of its dose due to scattering that it develops incompletely. Fig. 2.13 shows an example of what happens to a test pattern when proximity effects are not corrected. 
Many different schemes have been devised to minimize the proximity effect. If a pattern has fairly uniform density and linewidth, all that may be required is to adjust the overall dose until the patterns come out the proper size. This method typically works well for isolated transistor gate structures. Using higher contrast resists can help minimize the linewidth variations. Multilevel resists, in which a thin top layer is sensitive to electrons and the pattern developed in it is transferred by dry etching into a thicker underlying layer, reduce the forward scattering effect, at the cost of an increase in process complexity.
Higher beam voltages, from 50 kV to 100 kV or more, also minimize forward scattering, although in some cases this can increase the backscattering. When writing on very thin membranes such as used for x-ray masks, higher voltages reduce the backscatter contribution as well since the majority of electrons pass completely through the membrane. 
Conversely, by going to very low beam energies, where the electron range is smaller than the minimum feature size, the proximity effect can be eliminated.  The penalty is that the thickness of a single layer resist must also be less than the minimum feature size so that the electrons can expose the entire film thickness. The electron-optical design is much harder for low voltage systems since the electrons are more difficult to focus into a small spot and are more sensitive to stray electrostatic and magnetic fields. However, this is the current approach in optical maskmaking, where a 10 kV beam is used to expose 0.3 um thick resist with 1 um minimum features on a 5 mask. In more advanced studies, a 1.5 kV beam has been used to expose 70 nm thick resist with 0.15 um minimum features.  A technique that can be used in conjunction with this approach in order to increase the usable range of electron energy is to place a layer with a high atomic number, such as tungsten, underneath the resist. This has the effect of further limiting the range of the backscattered electrons.
The most common technique of proximity correction is dose modulation, where each individual shape in the pattern is assigned a dose such that (in theory) the shape prints at its correct size. The calculations needed to solve the shape-to-shape interactions are computationally very time consuming. Although the actual effect of electron scattering is to increase the dose received by large areas, for practical reasons proximity correction is normally thought of in terms of the large areas receiving a base dose of unity, with the smaller and/or isolated features receiving a larger dose to compensate.
Several different algorithms have been used. In the self-consistent technique, the effect of each shape on all other shapes within the scattering range of the electrons is calculated. The solution can be found by solving a large number of simultaneous equations;  unfortunately, this approach becomes unwieldy as the number of shapes increases and their size decreases. An alternative is to define a grid and compute the interaction of the pattern shapes with the grid and vice versa;  however, the accuracy and flexibility of this technique may be limited. An optimal solution may also be arrived at by an iterative approach.  Finally, neural network techniques have been applied to the problem of proximity correction;  while not an attractive technique when implemented on a digital computer, it might be advantageous if specialized neural network processors become a commercial reality. Many of the algorithms in use assume that the energy distribution has a double Gaussian distribution as discussed in Sec. 2.3.
A computationally similar approach to dose modulation is pattern biasing. [38-39] In this approach, the extra dose that dense patterns receive is compensated for by slightly reducing their size. This technique has the advantage that it can be implemented on EBL systems that are not capable of dose modulation. However, the technique does not have the dynamic range that dose modulation has; patterns that contain both very isolated features and very dense features will have reduced process latitude compared to when dose modulation is used, since the isolated features will be under-dosed while the dense features will be overdosed. Pattern biasing cannot be applied to features with dimensions close to the scale of the pixel spacing of the e-beam system.
A third technique for proximity correction, GHOST, has the advantage of not requiring any computation at all. The inverse tone of the pattern is written with a defocused beam designed to mimic the shape of the backscatter distribution (Fig. 2.14). The dose of the GHOST pattern, ee / (1 + ee), is also set to match the large area backscatter dose. After the defocussed inverse image is written, the pattern will have a roughly uniform background dose. GHOST is perhaps an underutilized technique; under ideal conditions it can give superb linewidth control.  Its disadvantages are the extra data preparation and writing time, a slight to moderate loss of contrast in the resist image, and a slight loss in minimum resolution compared to dose modulation due to the fact that GHOST does not properly correct for forward scattering.
A number of companies for some time have had proprietary software for proximity correction.  [42-43] Just recently, commercial proximity packages have become available, or are about to become available. [44-45] At present, these are limited in their accuracy, speed, and data volume capability; while excellent for correcting small research patterns, they may have difficulties with complex chips. Finally, several packages have been developed at university and government laboratories, some of which might be available to an adventurous user with excessive amounts of free time.  
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This material is based upon work supported by the National Science Foundation under Grant No. ECCS-1542081. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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