Measuring critical dimensions of a semiconductor structure

An apparatus for optical metrology for measuring a critical dimension of a grating layer, the apparatus comprising:

• a polarized illuminator configured to illuminate an area of a geometrical structure of dispersive materials with incident polarized electromagnetic radiation;
• a detector for measuring spectral components of the incident electromagnetic radiation reflected from the area;
• and
• a data processing system comprising
• computer-readable code configured to determine parameter values that minimize an objective function J which depends on differences between the measured spectral components and computed spectral components based on a parameterized model of the geometrical structure, wherein the objective function J comprises
• 1
• 2
• ⁢
• (
• α
• -
• α
• 0
• )
• 2
• +
• 1
• 2
• ⁢
• (
• β
• -
• β
• 0
• )
• 2
• where α0 and β0 are the measured spectral components, and α and β are the computed spectral components, and
• computer-readable code configured to compute ∇pJ, where p is a parameterized vector of the parameter values, wherein the parameter values include at least ƒ and h, where ƒ is a critical dimension fraction comprising the critical dimension divided by a pitch of the grating layer, and h is a height of the grating layer, wherein the grating layer comprises a non-uniform index of refraction.

A method for estimating a critical dimension of a grating layer from spectroscopic measurements, the method comprising:

• using an illuminator of a scatterometer apparatus to illuminate an area of a geometrical structure of dispersive materials with incident electromagnetic radiation;
• measuring spectral components of the incident electromagnetic radiation reflected from the area using a detector of the scatterometer apparatus;
• using a computer for the scatterometer apparatus to compute a solution to state equations driven by a function representing the incident electromagnetic radiation, wherein the state equations are derived from Maxwell's equations with dispersive materials;
• using the computer for the scatterometer apparatus to compute a solution to an adjoint to the state equations, wherein the adjoint to the state equations has a source function which is a derivative of a measurement function as a function of the solution of the state equations;
• and
• using the computer for the scatterometer apparatus to determine parameter values for the semiconductor structure that minimize an objective function J which depends on differences between the measured spectral components and computed spectral components based on a parameterized geometrical model of the geometrical structure,
• wherein steps for determining the parameter values that minimize the objective function include:
• computing ∇pJ where p is a parameterized vector of the parameter values, wherein the parameter values comprise ƒ and h, where h is a height of the grating layer, and ƒ is a critical dimension fraction comprising the critical dimension divided by a pitch of the grating layer.