**3. Character size optimization for higher projection throughput**

Character size optimization techniques were studied for character projection (Sugihara, 2006c, 2007b, 2010). We first presented an idea to optimize a character size for higher projection throughput (Sugihara et al., 2006c). We presented a character size optimization by enumerating all possible character sizes and generating a cell library for every given character size (Sugihara et al., 2006c). Next we presented a cell library development methodology in which a character size and a set of cells were optimized at the same time (Sugihara et al., 2007b). We also presented a character size optimization technique for multicolumn-cell projection equipment. In this section, we focus on the first work (Sugihara et al., 2006c) for a simple explanation.

#### **3.1 Character size optimization problem**

In Section 2, the size of characters on CP masks was given and treated as a constant because of the restriction which attributes to character projection equipment. Cells used in a design were, consequently, partitioned to fit the constant size of characters by intuition. In this

$$\begin{array}{rcl} \mathrm{S\_S(x, l\_{\mathrm{char}}, w\_{\mathrm{char}})} & = & \mathrm{EB} \,\mathrm{shots} \,\mathrm{with} \,\mathrm{CP} + \mathrm{EB} \,\mathrm{shots} \,\mathrm{with} \,\mathrm{VSB} \\ & = & \sum\_{l=1}^{C} \mathrm{S\_{CP}}(l\_{\mathrm{char}}, w\_{\mathrm{char}}) \boldsymbol{\tau}\_{l} \boldsymbol{x}\_{l} + \sum\_{l=1}^{C} \mathrm{S\_{VSB}}\_{\mathrm{VSB}\_{l}} \boldsymbol{r}\_{l} (1 - \boldsymbol{x}\_{l}) \\ & = & \sum\_{l=1}^{C} \left\{ \mathrm{S\_{CP}}(l\_{\mathrm{char}}, w\_{\mathrm{char}}) - \mathrm{S\_{VSB}} \right\} \boldsymbol{\tau}\_{l} \boldsymbol{x}\_{l} + \sum\_{l=1}^{C} \mathrm{S\_{VSB}}\_{\mathrm{VSB}\_{l}} \boldsymbol{\tau}\_{l} . \end{array} \tag{5}$$

$$N\_{l=1}^{\mathbb{C}} c\_l (l\_{\text{char}}, w\_{\text{char}}) \mathbf{x}\_l \quad \leq \quad N\_{\text{char}} (l\_{\text{char}}, w\_{\text{char}}, l\_{\text{CP}}, w\_{\text{CP}}, G), \tag{6}$$




Character Projection Lithography for Application-Specific Integrated Circuits 81

effective to reduce the number of EB shots. The experimental results imply that the CP equipment should be capable of modifying the size of characters by customers' demands

 0 2000 4000 6000 8000 10000 12000 14000

Character length [nm]

0 20000 40000 60000 80000 100000 120000 140000 Character width [nm]

Our systematic character size optimization scheme can achieve the lower number of EB shots and can enhance the throughput of the CP equipment. The throughput enhancement of the CP equipment causes the higher production volume of semiconductors devices at a lower cost. It consequently accelerates the application of semiconductor devices to various industrial fields even if their production volumes are small. Likewise, the throughput enhancement of the CP equipment promises a lower cost in developing photomasks if the

**4. Technology mapping technique for character projection equipment** 

Technology mapping techniques were discussed for character projection lithography of a single-column-cell system (Sugihara et al., 2006b, 2007c). A technology mapping technique was also proposed for a multi-column-cell system (Sugihara et al., 2007a). This section mainly discusses a technology mapping technique which reduces projection time of a single-

In the most popular paradigm for logic synthesis, after a technology independent optimization of a set of logic equations, the result is mapped into a feasible circuit which is optimal with respect to area and satisfies a maximum critical-path delay. In this paradigm, the role of *technology mapping* is to finish the synthesis of the circuit by performing the final gate selection from a particular cell library. The role of technology mapping is the actual cell choice to implement the equations — for example, choosing the fastest cells along the critical path, and using the most area-efficient combination of cells off the critical path (Hachtel,

A set of base functions is chosen such as a two-input NAND-gate and an inverter. The logic equations are optimized in a technology-independent manner and are then converted into a graph where each node is restricted to one of the base functions. This graph is called the *subject graph*. The logic function for each library gate is also represented by a graph where

Fig. 6. The number of EB shots for Circuit 3

EB shots

CP method is utilized in developing photomasks in the future.

column-cell system (Sugihara et al., 2006b, 2007c).

**4.1 Review on technology mapping** 

1996).

after it is shipped out to them.

 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06

The numbers of EB shots obtained by our experiments are shown in Table 10. Comparing Case 1 with Case 2, 72.0% of EB shots was reduced in the best. The feasible EB size was utilized in both Cases 1 and 2 and the difference between them was whether or not the character size was optimized. The gap between Cases 1 and 2 implies that the beam size of the CP equipment should be configurable for users so that the throughput of their equipment can be increased. Supposing any size of EBs can be utilized, more reduction of EB shots can be achieved as the numbers in Case 3 in Table 10 show. Comparing Case 1 with Case 3, 75.9% of EB shots was reduced in the best. Comparing Case 2 with Case 3, 39.5% of EB shots were reduced in the best. The gap between Cases 2 and 3 implies that the development effort to seek for a larger size of EBs is capable of reducing 39.5% of EB shots. The gap between Cases 2 and 3 is directive for equipment developers to determine the EB size of their equipment.


Table 10. EB shots for the four benchmark circuits under three cases

The computing platform for experiment was an Intel Pentium 4 2.4 GHz with 1 GB of main memory. Computation times to obtain the optimal character sizes under the three cases were shown in Table 11. All character size optimization processes were finished within less than two minutes. Note that computation time was not affected by the number of cell instances but by the number of cell objects. All character size optimization processes were done within practical computation time.


Table 11. Computation time to optimize a character size [s]

Fig. 6 shows the numbers of EB shots with various character sizes for Circuit 3. The figure shows that there exists a minimal point of EB shots. The number of EB shots increases rapidly from the minimal point to the point where the size of characters is smaller while it increases gradually from the minimal point to the point where the size of characters is larger. The number of EB shots rises and falls sharply in the neighborhood of the minimal point. The experimental results show that the character size optimization was quite effective to reduce the number of EB shots and to enhance the throughput of the CP equipment.

#### **3.3 Conclusion**

We proposed the character size optimization technique for improving the throughput of the CP equipment by defining a mathematical problem as an ILP problem. We also showed some experimental results by solving mathematical program instances for several benchmark circuits. According to our experiment, 72.0% reduction of EB shots was achieved with a feasible EB size in the best, comparing with the conventional and intuitional character sizing. It was experimentally found that our character size optimization technique was so

Fig. 6. The number of EB shots for Circuit 3

The numbers of EB shots obtained by our experiments are shown in Table 10. Comparing Case 1 with Case 2, 72.0% of EB shots was reduced in the best. The feasible EB size was utilized in both Cases 1 and 2 and the difference between them was whether or not the character size was optimized. The gap between Cases 1 and 2 implies that the beam size of the CP equipment should be configurable for users so that the throughput of their equipment can be increased. Supposing any size of EBs can be utilized, more reduction of EB shots can be achieved as the numbers in Case 3 in Table 10 show. Comparing Case 1 with Case 3, 75.9% of EB shots was reduced in the best. Comparing Case 2 with Case 3, 39.5% of EB shots were reduced in the best. The gap between Cases 2 and 3 implies that the development effort to seek for a larger size of EBs is capable of reducing 39.5% of EB shots. The gap between Cases 2 and 3 is directive for equipment developers to determine the EB

> Circuit 1 Circuit 2 Circuit 3 Circuit 4 Case 1 52,117 41,469 26,913 164,316 Case 2 23,785 21,122 7,710 46,050 Case 3 14,379 15,120 7,710 39,546

The computing platform for experiment was an Intel Pentium 4 2.4 GHz with 1 GB of main memory. Computation times to obtain the optimal character sizes under the three cases were shown in Table 11. All character size optimization processes were finished within less than two minutes. Note that computation time was not affected by the number of cell instances but by the number of cell objects. All character size optimization processes were

> Circuit 1 Circuit 2 Circuit 3 Circuit 4 Case 1 0.00 0.00 0.00 0.00 Case 2 11.86 14.29 60.55 56.28 Case 3 27.05 32.81 96.04 86.12

Fig. 6 shows the numbers of EB shots with various character sizes for Circuit 3. The figure shows that there exists a minimal point of EB shots. The number of EB shots increases rapidly from the minimal point to the point where the size of characters is smaller while it increases gradually from the minimal point to the point where the size of characters is larger. The number of EB shots rises and falls sharply in the neighborhood of the minimal point. The experimental results show that the character size optimization was quite effective to reduce the number of EB shots and to enhance the throughput of the CP equipment.

We proposed the character size optimization technique for improving the throughput of the CP equipment by defining a mathematical problem as an ILP problem. We also showed some experimental results by solving mathematical program instances for several benchmark circuits. According to our experiment, 72.0% reduction of EB shots was achieved with a feasible EB size in the best, comparing with the conventional and intuitional character sizing. It was experimentally found that our character size optimization technique was so

Table 10. EB shots for the four benchmark circuits under three cases

Table 11. Computation time to optimize a character size [s]

size of their equipment.

**3.3 Conclusion** 

done within practical computation time.

effective to reduce the number of EB shots. The experimental results imply that the CP equipment should be capable of modifying the size of characters by customers' demands after it is shipped out to them.

Our systematic character size optimization scheme can achieve the lower number of EB shots and can enhance the throughput of the CP equipment. The throughput enhancement of the CP equipment causes the higher production volume of semiconductors devices at a lower cost. It consequently accelerates the application of semiconductor devices to various industrial fields even if their production volumes are small. Likewise, the throughput enhancement of the CP equipment promises a lower cost in developing photomasks if the CP method is utilized in developing photomasks in the future.
