Correction of thermal expansion in a lithographic device

A method for writing a pattern on a substrate in a charged particle lithographic apparatus using a scanning exposure by means of a charged particle beam, wherein the beam is directed to a sequence of exposure positions on a surface of the substrate, and at each exposure position the beam is used to write structures on the substrate within a beam range around the respective exposure position according to a respective pattern portion which represents a corresponding sub-region of the pattern to be written, the method comprising the following steps performed with regard to a respective exposure position:

• determining, based on the exposure position, the power of the beam imparted to the substrate at the exposure position;
• calculating heating of the substrate generated by the beam during an exposure duration associated with the exposure position, and calculating, for a plurality of locations defined in a predetermined array over the surface of the substrate, the amount of thermal diffusion and radiative cooling due to thermal emission of the substrate, wherein thermal diffusion is calculated using the inhomogeneous heat equation including a source-sink function which includes thermal emissivity calculated as proportional, by a common constant of proportionality, to one of:
• the difference of fourth powers of the substrate temperature and ambient temperature;
• and the difference of the substrate temperature to an ambient temperature;
• calculating, for a plurality of locations defined in a predetermined array over the surface of the substrate, the positional change of the substrate resulting from thermal expansion based on the results of the previous step;
• calculating a displacement distance, said displacement distance describing said positional change at the exposure position, and
• applying a correction with respect to the exposure position and/or the pattern portion associated with the exposure position using the displacement distance,
• continuing writing structures on the substrate by means of the beam according to the exposure position and pattern portion thus corrected,
• wherein the above steps are carried out for each of a sequence of exposure durations, each of said exposure durations covering a time interval associated with one or more subsequent exposure positions, wherein the calculations for a respective time interval are done with respect to the results of the calculations obtained for the time interval respectively preceding the respective time interval,
• wherein said common constant of proportionality of thermal emissivity is determined beforehand by writing markers on a test substrate at several different stages of a writing process, measuring deformation positions of said writing markers, and performing a best-fit calculation of the constant of proportionality to the deformation positions thus measured;
• and
• wherein the substrate is held at two or more fixed mounting positions.

A method for writing a pattern on a substrate in a charged particle lithographic apparatus using a scanning exposure by means of a charged particle beam, wherein the beam is directed to a sequence of exposure positions on a surface of the substrate, and at each exposure position the beam is used to write structures on the substrate within a beam range around the respective exposure position according to a respective pattern portion which represents a corresponding sub-region of the pattern to be written, the method comprising the following steps performed with regard to a respective exposure position:

• determining, based on the exposure position, the power of the beam imparted to the substrate at the exposure position;
• calculating heating of the substrate generated by the beam during an exposure duration associated with the exposure position, and calculating, for a plurality of locations defined in a predetermined array over the surface of the substrate, the amount of thermal diffusion and radiative cooling due to thermal emission of the substrate, wherein thermal diffusion is calculated using the inhomogeneous heat equation including a source-sink function which includes thermal emissivity calculated as proportional, by a common constant of proportionality, to one of:
• the difference of fourth powers of the substrate temperature and ambient temperature;
• and the difference of the substrate temperature to an ambient temperature;
• calculating, for a plurality of locations defined in a predetermined array over the surface of the substrate, the positional change of the substrate resulting from thermal expansion based on the results of the previous step;
• calculating a displacement distance, said displacement distance describing said positional change at the exposure position, and
• applying a correction with respect to the exposure position and/or the pattern portion associated with the exposure position using the displacement distance,
• continuing writing structures on the substrate by means of the beam according to the exposure position and pattern portion thus corrected,
• wherein the above steps are carried out for each of a sequence of exposure durations, each of said exposure durations covering a time interval associated with one or more subsequent exposure positions, wherein the calculations for a respective time interval are done with respect to the results of the calculations obtained for the time interval respectively preceding the respective time interval,
• wherein said common constant of proportionality is determined by fitting to substrate temperature measurements performed beforehand on a test substrate undergoing a test writing process in the same charged particle lithographic apparatus where thereafter the pattern is written on the substrate, the test substrate and the test writing process being representative for the substrate and the pattern written on the substrate;
• and
• wherein the substrate is held at two or more fixed mounting positions.