Correction of thermal expansion in a lithographic device
A pattern writing method for charged-particle lithography apparatuses using an improved correction for thermal distortion of the substrate includes determining an exposure position where the beam impinges on the substrate and the power of the beam at the exposure position; calculating heating of the substrate at the exposure position, and calculating, for a plurality of locations over the substrate, and the thermal diffusion and radiative cooling; calculating, for the same or a reduced plurality of locations on the substrate, the positional change of the substrate due to thermal expansion; determining a displacement distance which compensates the positional change at the exposure position, updating the structure to be written by shifting the exposure position of the beam by said displacement distance, and writing the updated structures on the substrate with the beam. These steps are repeated as a function of time and/or varying exposure position of the beam substrate position.
1 . 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;
calculating, for a plurality of locations defined in a predetermined array over the surface of the substrate, the positional change of the substrate and mechanical strain in 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; and
wherein calculating mechanical strain in the substrate includes effects of mechanical stress introduced by external forces including holding forces applied to the substrate at two or more fixed mounting positions.
2 . The method of claim 1 , wherein the step of applying a correction to the pattern portion associated with the exposure position using the displacement distance comprises at least one of
(i) shifting the exposure position of the beam by a first displacement to obtain an updated exposure position, and
(ii) recalculating the pattern portion by shifting structures contained in the pattern portion by a second displacement to obtain an updated pattern portion,
where the displacement distance results from said at least one of the first and second displacements, followed by writing structures on the substrate by means of the beam according to the exposure position and pattern portion thus updated.
3 . The method of claim 1 , wherein in the step of calculating heating of the substrate generated by the beam on the substrate, the energy deposited by the beam during a time interval associated with one or more subsequent exposure positions is modeled as being deposited by a sequence of heating spots, each heating spot having a heat insertion distribution according to a predetermined spatial distribution and being centered at one of a sequence of deposition positions representing the average of the beam position over a respective subinterval of the time interval, said predetermined spatial distribution having a width which is wider than the actual beam range at the substrate.
4 . The method of claim 1 , wherein in the step of calculating the heating of the substrate by the beam, the rate of heating is calculated as being proportional to a predetermined beam power, by a constant of proportionality.
5 . The method of claim 4 , wherein said constant of proportionality 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.
6 . The method of claim 4 , wherein said 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.
7 . The method of claim 1 , wherein at least one parameter which relates to a mechanical or thermomechanical property of the substrate is determined using a test substrate by performing 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, measuring quantities enabling to determine said at least one parameters, and calculating said at least one parameter from the quantities thus measured.
8 . The method of claim 1 , wherein the steps of claim 1 are performed in real-time during a process for writing the pattern on the substrate using actual exposure positions, pattern and current density values registered by the exposure control system.
9 . The method of claim 1 , wherein the substrate is exposed using a stripe-scanning writing method, writing structures on the substrate stripe by stripe, and the calculating steps of claim 1 are performed for a number of consecutive durations which each correspond to a respective portion of each stripe before writing the structures of the pattern portions that belong to the respective stripe or portion of the stripe.
10 . The method of claim 1 , wherein the step of calculating heating of the substrate generated by the beam on the substrate at the exposure position also includes prior heating of the substrate generated by the beam during its path on the substrate prior to said exposure position.
11 . The method of claim 1 , wherein the step of calculating the amount of thermal diffusion and radiative cooling is repeated for a sequence of exposure durations, which respectively comprise a number of subsequent exposure positions extending over at least a first distance which is larger than the width of a beam range on the substrate surface.
12 . The method of claim 11 , wherein the step of calculating the positional change of the substrate is repeated for a sequence of secondary durations which has a lower update ratio than the sequence of exposure durations, wherein said secondary durations respectively comprise a number of subsequent exposure positions extending over at least a second distance which is larger than said first distance.
13 . The method of claim 1 , wherein the amount of positional change at each of the two or more fixed mounting positions is a predetermined value.
14 . The method of claim 13 , wherein the predetermined value of said amount of positional change is zero positional change.
15 . 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.
16 . 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.