Optimizing computational efficiency by multiple truncation of spatial harmonics
Methods and systems for solving measurement models of complex device structures with reduced computational effort and memory requirements are presented. The computational efficiency of electromagnetic simulation algorithms based on truncated spatial harmonic series is improved for periodic targets that exhibit a fundamental spatial period and one or more approximate periods that are integer fractions of the fundamental spatial period. Spatial harmonics are classified according to each distinct period of the target exhibiting multiple periodicity. A distinct truncation order is selected for each group of spatial harmonics. This approach produces optimal, sparse truncation order sampling patterns, and ensures that only harmonics with significant contributions to the approximation of the target are selected for computation. Metrology systems employing these techniques are configured to measure process parameters and structural and material characteristics associated with different semiconductor fabrication processes.
1. A measurement system comprising:
an illumination source configured to provide an amount of illumination light to a periodic metrology target having multiple spatial periods including a fundamental spatial period and one or more approximate spatial periods in a first direction of the periodic metrology target, wherein each of the one or more approximate spatial periods is an integer fraction of the fundamental spatial period;
a detector configured to receive an amount of collected light from the periodic metrology target in response to the amount of illumination light and generate a plurality of measured signals; and
one or more computing systems configured to:
receive an indication of the multiple spatial periods of the periodic metrology target;
classify a plurality of spatial harmonics associated with the periodic metrology target into a plurality of different groups according to each of the multiple spatial periods of the periodic metrology target, wherein each group of spatial harmonics has a different harmonic spacing;
select a truncation order associated with each of the plurality of different groups of spatial harmonics;
estimate a value of one or more parameters of interest of the periodic metrology target based on a fitting of a measurement model, simulated at the selected truncation orders, to the measured signals; and
communicate an indication of the value of the one or more parameters of interest to a semiconductor fabrication tool that causes the semiconductor fabrication tool to adjust one or more parameters of a fabrication process of the semiconductor fabrication tool.
2. The measurement system of claim 1 , wherein the measurement model, simulated at the selected truncation orders, includes an electromagnetic simulator utilizing a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.
3. The measurement system of claim 1 , wherein the indication of the multiple spatial periods of the metrology target is received from a user.
4. The measurement system of claim 1 , further comprising:
determining each of the multiple spatial periods of the periodic metrology target.
5. The measurement system of claim 4 , wherein the determining of each of the multiple spatial periods of the periodic metrology target involves:
computing a Fourier series of spatial harmonics of a dielectric permittivity of the periodic metrology target geometry to a relatively high truncation order; and
analyzing the amplitudes of each of the spatial harmonics of the Fourier series.
6. The measurement system of claim 4 , wherein the determining of each of the multiple spatial periods of the periodic metrology target involves an analysis of the geometry of the periodic etrology target.
7. The measurement system of claim 1 , wherein the selecting of the truncation order associated with each of the plurality of different groups of spatial harmonics involves:
determining an amplitude of each of the spatial harmonics of each of the plurality of different groups; and
selecting the truncation order associated with each of the plurality of different groups based on a rate of decay of the amplitude of the spatial harmonics of each of the plurality of different groups.
8. The measurement system of claim 1 , wherein the selecting of the truncation order associated with each of the plurality of different groups of spatial harmonics involves:
performing a separate convergence test for each of the plurality of different groups of spatial harmonics using an electromagnetic simulator that utilizes a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.
9. The measurement system of claim 1 , wherein the periodic metrology target having multiple spatial periods in the first direction of the target, also includes multiple spatial periods in a second direction, different from the first direction, wherein the multiple spatial periods in the second direction includes a fundamental spatial period and one or more approximate spatial periods in the second direction of the periodic metrology target, wherein each of the one or more approximate spatial periods in the second direction is an integer fraction of the fundamental spatial period in the second direction.
10. The measurement system of claim 1 , wherein a different truncation order is associated with each of the plurality of different groups of spatial harmonics.
11. The measurement system of claim 2 , wherein the electromagnetic simulator is a Fourier based electromagnetic simulator.
12. The measurement system of claim 1 , wherein the periodic metrology target includes a first periodic target having a relatively small pitch divided into groups by a second periodic pattern having a relatively large pitch.
13. The measurement system of claim 2 , wherein the parameters of interest include any of a critical dimension parameter and an overlay parameter.
14. A method for improving electromagnetic computation efficiency in optical metrology, the method comprising:
providing an amount of illumination light from an illumination source of a semiconductor measurement system to a periodic metrology target having multiple spatial periods including a fundamental spatial period and one or more approximate spatial periods in a first direction of the periodic metrology target, wherein each of the one or more approximate spatial periods is an integer fraction of the fundamental spatial period;
receiving an amount of collected light onto a detector of the semiconductor measurement system from the periodic metrology target in response to the amount of illumination light and generating a plurality of measured signals;
receiving an indication of multiple spatial periods of a periodic metrology target including a fundamental spatial period and one or more approximate spatial periods in a first direction of the periodic metrology target, wherein each of the one or more approximate spatial periods is an integer fraction of the fundamental spatial period;
classifying a plurality of spatial harmonics associated with the periodic metrology target into a plurality of different groups according to each of the multiple spatial periods of the periodic metrology target, wherein each group of spatial harmonics has a different harmonic spacing;
selecting a separate truncation order for each of the different groups of spatial harmonics;
estimating a value of one or more parameters of interest of the periodic metrology target based on a fitting of a measurement model, simulated at the selected truncation orders, to the measured signals; and
communicating an indication of the value of the one or more parameters of interest to a semiconductor fabrication tool that causes the semiconductor fabrication tool to adjust one or more parameters of a fabrication process of the semiconductor fabrication tool.
15. The method of claim 14 , wherein the measurement model simulated at the selected truncation orders includes an electromagnetic simulator utilizing a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.
16. The method of claim 14 , further comprising:
determining each of multiple spatial periods of the metrology target, wherein the determining involves:
computing a Fourier series of spatial harmonics of a dielectric permittivity of the periodic metrology target geometry to a relatively high truncation order; and
analyzing the amplitudes of each of the spatial harmonics.
17. The method of claim 14 , wherein the selecting of the separate truncation order associated with each of the plurality of different groups of spatial harmonics involves:
performing a separate convergence test for each of the plurality of different groups of spatial harmonics using an electromagnetic simulator that utilizes a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.
18. A measurement system comprising:
an illumination source configured to provide an amount of illumination light to a periodic metrology target having multiple spatial periods including a fundamental spatial period and one or more approximate spatial periods in a first direction of the periodic metrology target, wherein each of the one or more approximate spatial periods is an integer fraction of the fundamental spatial period;
a detector configured to receive an amount of collected light from the periodic metrology target in response to the amount of illumination light and generate a plurality of measured signals; and
a non-transitory, computer-readable medium, comprising:
code for causing a computing system to receive an indication of multiple spatial periods of a periodic metrology target including a fundamental spatial period and one or more approximate spatial periods in a first direction of the periodic metrology target, wherein each of the one or more approximate spatial periods is an integer fraction of the fundamental spatial period;
code for causing the computing system to classify a plurality of spatial harmonics associated with the periodic metrology target into a plurality of different groups according to each of the multiple spatial periods of the periodic metrology target, wherein each group of spatial harmonics has a different harmonic spacing;
code for causing the computing system to select a separate truncation order for each of the different groups of spatial harmonics
code for causing the computing system to estimate a value of one or more parameters of interest of the periodic metrology target based on a fitting of a measurement model, simulated at the selected truncation orders, to the measured signals; and
code for causing the computing system to communicate an indication of the value of the one or more parameters of interest to a semiconductor fabrication tool that causes the semiconductor fabrication tool to adjust one or more parameters of a fabrication process of the semiconductor fabrication tool.
19. The measurement system of claim 18 , wherein the measurement model simulated at the selected truncation orders includes an electromagnetic simulator utilizing a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.
20. The measurement system of claim 18 , wherein the selecting of the truncation order associated with each of the plurality of different groups of spatial harmonics involves:
performing a separate convergence test for each of the plurality of different groups of spatial harmonics using an electromagnetic simulator that utilizes a Fourier expansion of the periodic metrology target in terms of the plurality of spatial harmonics.