Wafer exposure method using wafer models and wafer fabrication assembly

1. A wafer exposure method, comprising:

obtaining critical dimension values from wafer structures at predefined measurement sites;

obtaining position-dependent process parameters of an exposure process used for forming the wafer structures;

determining, from the critical dimension values at the measurement sites, coefficients of a preset model and coefficients of at least one further model, wherein each further model differs in at least one term from the preset model and from other further models, and wherein the preset model and the at least one further model approximate the critical dimension values across a main surface of a wafer substrate in a closed mathematical form, the position-dependent process parameters and/or correction values of the position-dependent process parameters as an algebraic function of two position coordinates, wherein the at least one term is a product of a coefficient and the algebraic function of at least one of the position coordinates;

determining residuals between approximated critical dimension values obtained from the preset model and the at least one further model and the critical dimension values obtained at the measurement sites; and

selecting, among the preset model and the at least one further model, an updated model, wherein the selection is based on a criterion weighting the residuals optionally with the number of terms of the preset model and the at least one further model and/or an order of terms of the preset model and the at least one further model.

2. The wafer exposure method according to claim 1 , further comprising:

outputting information descriptive for terms of the updated model.

3. The wafer exposure method according to claim 1 , further comprising:

updating the position-dependent process parameters on the basis of the updated model.

4. The wafer exposure method according to claim 1 , wherein

the selection of the updated model is based on a trade-off that weights the number of terms of the preset model and the at least one further model against a total amount of the residuals.

5. The wafer exposure method according to claim 1 , wherein

the selection of the updated model is based on at least one of a Bayesian information criterion and an Akaike information criterion.

6. The wafer exposure method according to claim 1 , wherein

the models include at least one of Zernike polynomials and Legendre polynomials.

7. The wafer exposure method according to claim 1 , wherein

the at least one further model includes all Zernike polynomials up to a maximum order.

8. The wafer exposure method according to claim 1 , further comprising:

determining a quality index for each coefficient of the updated model, wherein the quality index gives information of a degree of dependency of the coefficient from variations of the critical dimension values.

9. The exposure method according to claim 1 , wherein

the selection is based on a criterion that weights a distribution of the residuals.

10. The wafer exposure method according to claim 1 , wherein

the critical dimension values comprise overlay discrepancy values,

the process parameters used in the exposure process comprise position offsets, and

each model comprises a wafer overlay model approximating the overlay discrepancy values, the position offsets and/or correction values of the position offsets as a function of the position coordinates.

11. The wafer exposure method according to claim 1 , wherein

the critical dimension values comprise critical dimensions of the wafer structures,

the process parameters used in the exposure process comprise exposure parameters, and

each model comprises a wafer exposure model approximating the critical dimensions, the exposure parameters and/or correction values of the exposure parameters as a function of the position coordinates.

12. The wafer exposure method according to claim 1 , wherein

the wafer structures comprise photoresist structures.

13. The wafer exposure method according to claim 12 , further comprising:

exposing a first photoresist layer formed on a first wafer substrate, wherein the exposure uses the position-dependent exposure parameters; and

developing the exposed first photoresist layer, wherein the patterned photoresist structure is obtained from the exposed photoresist layer.

14. The wafer exposure method according to claim 13 , further comprising:

exposing a second photoresist layer formed on a second wafer substrate, wherein the exposure uses position-dependent exposure parameters obtained from an updated wafer exposure model.

15. A wafer fabrication assembly, comprising:

an exposure tool assembly configured to expose photoresist layers coating wafer substrates to an exposure beam using position-dependent process parameters and to define wafer structures on the basis of the exposed photoresist layers;

a processor unit data-connected with the exposure tool assembly, wherein the processer unit is configured to:

receive and/or hold position-dependent process parameters used in the exposure tool assembly for defining the wafer structures,

receive critical dimension values of the wafer structures at predefined measurement sites,

determine, from the critical dimension values at the predefined measurement sites, coefficients of a preset model and of at least one further model, wherein each further model differs in at least one term from the preset model and from other further models, and wherein the preset model and the at least one further model approximate the critical dimension values across a main surface of one of the wafer substrates in a closed mathematical form, the position-dependent process parameters, and/or correction values of the position-dependent process parameters as an algebraic function of at least two position coordinates, wherein the at least one term is a product of a coefficient and the algebraic function of at least one of the position coordinates;

determine residuals between approximated critical dimension values obtained from the preset model and the at least one further model and the critical dimension values obtained at the measurement sites; and

select, among the preset model and the at least one further model, an updated model, wherein the selection is based on a criterion weighting the residuals optionally with the number of terms of the preset model and the at least one further model and/or an order of terms of the preset model and the at least one further model.

16. The wafer fabrication assembly according to claim 15 , wherein

the processor unit ( 200 ) is further configured to output information descriptive for terms of the updated model.

17. The wafer fabrication assembly according to claim 15 , wherein

the processor unit is further configured to update the position-dependent process parameters on the basis of the updated model and to output the updated position-dependent process parameters to the exposure tool assembly.

18. The wafer fabrication assembly according to claim 17 , further comprising

a metrology unit configured to obtain the critical dimension values of the wafer structures at the measurement sites and to output the measured critical dimension values to the processor unit.

19. The wafer fabrication assembly according to claim 15 , wherein

the critical dimension values comprise overlay discrepancy values,

the correction parameters used in the exposure process comprise position offsets, and

each model comprises a wafer overlay model approximating the overlay discrepancy values, the position offsets and/or correction values of the position offsets as a function of the position coordinates.

20. The wafer fabrication assembly according to claim 15 , wherein

the critical dimension values comprise critical dimension values of the wafer structures,

the correction parameters used in the exposure process comprise exposure parameters, and

each model comprises a wafer exposure model approximating the critical dimension values, the exposure parameters and/or correction values of the exposure parameters as a function of the position coordinates.