Method and apparatus for calculating electromagnetic scattering properties of finite periodic structures

A method, for implementation in a lithography or metrology process, of reconstructing an approximate structure of a finite periodic structure on a target portion of a semiconductor substrate from a detected electromagnetic scattering property arising from illumination of the finite periodic structure by radiation, the method comprising:

• estimating at least one structural parameter of the finite periodic structure on the target portion of the semiconductor substrate;
• determining at least one model electromagnetic scattering property from the at least one structural parameter, the determining comprising:
• determining a single-cell contrast current density within a unit-cell supporting domain of a single unit cell of the finite collection of unit cells;
• determining a scattered electric field outside the finite collection of unit cells, arising from the finite periodic structure by integrating, over the single unit cell's supporting domain, a Green's function with the determined single-cell contrast current density, wherein the Green's function is obtained for observation points outside the finite collection of unit cells by summation across the finite collection of unit cells;
• and
• determining an electromagnetic scattering property of the finite periodic structure using the scattered electric field;
• comparing the detected electromagnetic scattering property to the at least one model electromagnetic scattering property;
• determining an approximate structure of the finite periodic structure based on the result of the comparison;
• storing the determined electromagnetic scattering property for subsequent adjustment of parameter of the lithography or metrology process for the semiconductor substrate or another semiconductor substrate;
• and
• transmitting the determined electromagnetic scattering property to a lithographic apparatus or a metrology tool in order to adjust the parameter and improve accuracy of the lithographic apparatus or the metrology tool.

An inspection apparatus of a lithographic apparatus or a metrology tool, the inspection apparatus comprising:

• an illumination system configured to illuminate a finite periodic structure on a target portion of a semiconductor substrate with radiation;
• a detection system configured to detect an electromagnetic scattering property arising from the illumination of the finite periodic structure on the target portion of the semiconductor substrate;
• and
• a processor configured to:
• estimate at least one structural parameter based on the detection;
• determine at least one model electromagnetic scattering property from the at least one structural parameter, the determining comprising:
• determining a single-cell contrast current density within a unit-cell supporting domain of a single unit cell of a finite collection of unit cells;
• determining a scattered electric field outside the finite collection of unit cells, arising from the finite periodic structure by integrating, over the single unit cell's supporting domain, a Green's function with the determined single-cell contrast current density, wherein the Green's function is obtained for observation points outside the finite collection of unit cells by summation across the finite collection of unit cells;
• and
• determining an electromagnetic scattering property of the finite periodic structure using the scattered electric field;
• compare the detected electromagnetic scattering property to the at least one model electromagnetic scattering property;
• determine an approximate structure of the finite periodic structure based on the result of the comparison;
• store the determined electromagnetic scattering property for subsequent adjustment of a parameter of a lithography or metrology process for the semiconductor substrate or another semiconductor substrate;
• and
• transmit the determined electromagnetic scattering property to the lithographic apparatus or the metrology tool in order to adjust the parameter and improve accuracy of the lithographic apparatus or the metrology tool.

A computer program product containing one or more sequences of machine-readable instructions for determining electromagnetic scattering properties of a finite periodic structure on a target portion of a semiconductor substrate, the instructions being adapted to cause one or more processors in a lithographic apparatus or a metrology tool to perform operations comprising:

• determining a single-cell contrast current density within a unit-cell supporting domain of a single unit cell of a finite collection of unit cells;
• determining a scattered electric field outside the finite collection of unit cells, arising from the finite periodic structure by integrating, over the single unit cell's supporting domain, a Green's function with the determined single-cell contrast current density, wherein the Green's function is obtained for observation points outside the finite collection of unit cells by summation across the finite collection of unit cells;
• determining an electromagnetic scattering property of the finite periodic structure using the scattered electric field;
• storing the determined electromagnetic scattering property for subsequent adjustment of a parameter of a lithography or metrology process for the semiconductor substrate or another semiconductor substrate;
• and
• transmitting the determined electromagnetic scattering property to the lithographic apparatus or the metrology tool in order to adjust the parameter and improve accuracy of the lithographic apparatus or the metrology tool.