Corresponding author: Nikita N. Balan ( nbalan@niime.ru ) © 2020 Nikita N. Balan, Vladidmir V. Ivanov, Alexey V. Kuzovkov, Evgenia V. Sokolova, Evgeniy S. Shamin.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Balan NN, Ivanov VV, Kuzovkov AV, Sokolova EV, Shamin ES (2020) Basic approaches to simulation of resist mask formation in computational lithography. Modern Electronic Materials 6(1): 37-45. https://doi.org/10.3897/j.moem.6.1.55056
|
Main currently used resist mask formation models and problems solved have been overviewed. Stages of "full physical simulation" have been briefly analyzed based on physicochemical principles for conventional diazonapthoquinone (DNQ) photoresists and chemically enhanced ones. We have considered the concepts of the main currently used compact models predicting resist mask contours for full-scale product topologies, i.e., VT5 (Variable Threshold 5) and CM1 (Compact Model 1). Computation examples have been provided for full and compact resist mask formation models. Full resist mask formation simulation has allowed us to optimize the lithographic stack for a new process. Optimal thickness ratios have been found for the binary anti-reflecting layers used in water immersion lithography. VT5 compact model calibration has allowed us to solve the problem of optimal calibration structure sampling for maximal coverage of optical image parameters space while employing the minimal number of structures. This problem has been solved using cluster analysis. Clustering has been implemented using the k-means method. The optimum sampling is 300 to 350 structures, the rms error being 1.4 nm which is slightly greater than the process noise for 100 nm structures. The use of SEM contours for VT5 model calibration allows us to reduce the rms error to 1.18 nm for 40 structures.
photoresist, projection photolithography, VT5, CM1, k-means clustering method.
The resolution of projection photolithography (i.e., the minimum half-period size of printed structures) can be determined using the Rayleigh ratio [
,
where λ is the wavelength, NA = nsinθ is the numerical aperture of the projection objective (θ is the aperture angle and n is the refractive index of the medium) and k1 is the coefficient determined by process parameters. In the classical Rayleigh formula for the resolution of optical systems this coefficient is 0.61 [
Correct use of RET allows reducing k1 to 0.28 or even smaller thus allowing printing of elements far smaller than the wavelength value.
The most important section of computational lithography is resist mask formation simulation. This simulation is always combined with one of the above listed k1 coefficient reduction methods and may have the following implementations:
This work provides an overview of the main currently used resist mask formation models with some examples of computations for existing and new processes.
"Full" simulation describes all photomask formation stages (Fig.
The main process considered in the full simulation is latent image formation in the photoresist layer during exposure.
2D intensity distribution generated in wafer plane upon template illumination by an extended spatially incoherent radiation source (so-called "air" image) could be computed with the Hopkins method [
,
where is the so-called transmission cross-coefficient (TCC) depending only on the properties of the optical system and the illuminator shape,
;
is the pupil function, is the normalized illuminator function and Tm (fx) = F{Ei (x)tm (x)} is the Fourier image of the product of the incident radiation amplitude and the template transmittance function (generally, it have complex value). For simplicity we consider the 1D case here.
The main advantage of the Hopkins method is that the transmission cross-coefficient TCC (a four-dimensional matrix in a discrete 2D case) only needs to be determined once following which one can compute intensity distributions for completely different source images using the already computed TCC thus achieving great time-saving.
The next stage is to determine the spatial light intensity distribution in the photoresist film. It can be computed using a number of approximations the simplest of which is intensity representation in the following form:
I (x,z) = II (x)IS (z),
where II (x) is the intensity of the "air" image determined e.g. with the Hopkins method and IS (z) is the intensity distribution through film depth computed using the Fresnel coefficients [
Then it is necessary to simulate the formation of a latent image in the photoresist layer during its exposure. To this end we use the Grothus-Draper Law (the First Law of Photochemistry) according to which only the absorbed portion of incident light can initiate a chemical reaction in the material. In turn, light absorption is described by the Bouguer-Lambert Law describing the attenuation of light intensity during radiation propagation in an absorbing medium:
I (z) = I0 exp(-αz),
where I0 is the incident radiation intensity, z is the coordinate inside the material (with the origin on the photoresist film surface) and α is the absorption coefficient [
It is important to understand that each component of the light-sensitive composition has its unique molar absorption ratio and that α generally depends on the concentrations of the components. Typical positive photoresists intended for operation at 436 and 365 nm wavelengths contain novolac resin and a diazonaphtoquinone (DNQ) based photosensitive component (Fig.
The transformation process of an optical image to a latent image (i.e., the photochemical reaction kinetics) is described by Dill’s set of differential equations [
where t is time, M (z,t) is the relative concentration of the photosensitive component, A is the exposure-dependent part of the absorption ratio, B is the constant part of the absorption ratio and C is the photosensitive component decomposition rate. The A, B and C parameters also referred to as Dill’s parameters are determined experimentally [
Simulating DNQ photoresist exposure and development one should take into account the effects caused by preliminary heat treatment and the influence of residual solvent on reaction products diffusion during post-exposure bake. The main purpose of preliminary heat treatment (or post-coating heat treatment (Fig.
Conventional resists become inefficient at wavelengths of 248 nm or less due to strong light absorption by the novolac resin. Therefore the so-called chemically enhanced resists were developed in which photo acid generators (PAG) react with the incident radiation (Fig.
Polymer matrix hydrophobic group decomposition in chemically enhanced resists during post-exposure heat treatment.
Furthermore chemically enhanced resists are designed to contain low concentrations of reaction inhibitors, i.e., hydroxides aimed to neutralize the acid released at low irradiation doses. This allows one to achieve the required threshold characteristic of the resist and decrease its sensitivity to acid diffusion into unexposed areas and its contamination with alkali from the environment.
Exposure of chemically enhanced resists is also described by Dill’s set of equations but the key stage there is simulation of post-exposure bake kinetics which is considerably more chemically complex compared with that for DNQ photoresists.
Then exposed areas should be etched in an alkaline etchant which is usually tetramethylammonium hydrochloride for both types of photoresists. There are multiple exposed photoresist development simulation methods which use quite complex and resource-consuming models based on the cellular automata theory [
D (M) = exp(E1 + E2M + E3M2),
where M (x,z) is the latent image or light-sensitive component concentration and E1, E2 and E3 are experimentally determined coefficients.
The possibility of resist mask profile computation during full simulation allows predicting the generation of standing waves and obtaining the so-called "swing curves", i.e., dependences of a parameter (e.g., critical size or full exposure dose) on photoresist mask thickness, as well as evaluating the so-called "process windows", i.e., regions in the dose vs defocus coordinates in which the resist mask retains process-acceptable geometry.
The main objective of empirical approximation simulation in resist mask contour computation is usually to rapidly simulate a 2D resist mask geometry for full-scale lithographic layer topology during the topological correction of optical proximity distortions. As noted above this simulation is in fact threshold processing of a 2D air image computed using the Hopkins method.
The so-called design-oriented CAD systems implementing this simulation provide a tremendous time advantage in resist mask contour computation as compared with process-oriented ones [
In some cases it is expedient to use compact models for process simulation. For example, they allow illuminator shape optimization [
There are several types of compact models used in 2D resist contour computation for full-scale product topology. The most widely used are the VT5 (Variable Threshold 5) and CM1 (Compact Model 1) models, e.g. implemented in Mentor Graphics Calibre CAD.
The VT5 model has been developed from simpler variable threshold models, e.g. VTR and VTRE, and has the following operation principle. Reference marks are set at topological area contours for which image parameters are determined, e.g. maximum and minimum intensity along the mark, as well as the first and second derivatives of intensity by coordinate for the point the intensity at which is equal to the reference threshold. The photoresist edge position is determined through the threshold value T which in turn depends on the preset air image parameters T = T (Imax, Imin, dI/dx, dI2/dx2). The threshold function T is typically a first or second order polynomial. Photoresist edge position simulation accuracy within the model considered can be increased by using the result of photomask image convolution with a preset set of kernels and applying the threshold function to the resultant distribution. This approach allows for the effect of photoresist components diffusion and etching effects [
Model calibration implies determination of T polynomial coefficients while analyzing results of printing of the so-called test calibration matrix containing a set of simple structures (Fig.
The more accurate CM1 model yields a contour formed by a cross-section at the constant threshold of the R (x,y) surface corresponding to the latent image in the resist [
, (4)
where сi are the model calibration coefficients and Mi (x,y) are the terms obtained by optical image transformations I (x,y):
, (5)
where Gs,p (x,y) is the Gauss-Laguerre function, the k, n and p parameters are user-defined for each term and the b and s parameters are computed during model calibration. The b parameter determines the linear-piecewise transformation of the I (x,y) function and characterizes the pattern of the interaction between acids and alkali in the photoresist layer, the s parameter being the diffusion length of active components.
The CM1 model provides good accuracy in comparison with VT5 for 65 nm or less processes. One reason is that CM1 indirectly uses simulation on the basis of physical fundamentals.
In this section we give an application example of "full" resist mask formation simulation on the basis of physical principles in lithographic stack layer thickness optimization for a new process.
When lithographic stack layers are not matched by their thickness and refractive indices, the overall stack reflection coefficient varies quasi-periodically in a range of 20–60 % depending on initial resist film thickness variation. Therefore the effective energy absorbed in the resist layer varies in approximately the same range and hence the critical structure size should also meet this limit. The curve describing this dependence is referred to as swing curve [
.
In our example we simulated the lithographic process at a 193 nm wavelength in an immersion medium having a refractive index of 1.44 with a phase attenuating photomask having a dark field transmittance coefficient of 0.06 and a phase rotation angle of 180 deg. The photomask was dark-field. The illuminator parameters were as follows: Quasar, Cross orientation, cut angle = 34 deg, σext = 0.9 and σint = 0.8. We used a periodical linear array of 100 nm spaced 50 nm wide bright dashes. Figure
CD swing curves and resist profiles (120 nm height, 21.298 mJ/cm2 dose): (a) non-optimized lithographic stack; (b) optimized lithographic stack.
Varying the thicknesses of the bottom anti-reflecting layers with simultaneous computation and analysis of the resultant CD swing curves allows selecting the optimal lithographic stack parameters.
The best thicknesses of the bottom anti-reflecting layers were hB1 = 22 nm and hB2 = 26 nm. The respective stack CD swing curve and the 120 nm resist mask profile for a 21.398 mJ/cm2 dose are shown in Fig.
Lithographic stack | Notation | Not optimal | Optimal |
---|---|---|---|
Immersion | nI | 1.44 | 1.44 |
BARC1 | hB1, nm | 16 | 22 |
nB1 | 1.9064 | 1.9064 | |
kB1 | 0.6711 | 0.6711 | |
BARC2 | hB2, nm | 23 | 26 |
nB2 | 1.7021 | 1.7021 | |
kB2 | 0.196 | 0.196 | |
Resist | hR, nm | 120 | 120 |
nR | 1.706 | 1.706 | |
kR | 0.00922 | 0.00922 | |
Immersion protective coating | hI, nm | 90 | 90 |
nI | 1.53 | 1.53 |
The most time-consuming stage of compact model calibration process is measurement data acquisition. For non-automated measurements, data acquisition may take dozens of hours and therefore OPC model development and the whole OPC solution become quite time-consuming. The necessity to reduce the number of measurements imposes the problem of optimal set of structures to be measured. As noted above the optimal sampling should maximally cover the optical image parameters space while employing the minimal number of structures.
The test calibration pad contained 3300 periodical linear structures sized 60 to 200 nm and spaced 140 to 1200 nm. The aim of the tests was to determine the optimal number of structures providing for an acceptable model accuracy the latter being evaluated by comparing simulation results with measurement data for 50 differently shaped structures that were not used for calibration. The accuracy criterion was chosen to be the rms error of simulation results.
The wavelength was λ = 193 nm, NA = 0.75. The illuminator parameters were as follows: Quasar, Cross orientation, σint = 0.5, σext = 0.8 and cut angle = 30 deg (Fig.
The shape of the illuminator (Quasar, σint = 0.5, σext = 0.8, cut angle = 30 deg) used in the lithographic process for which the VT5 model was calibrated.
The tests were performed for different data arrays that included 40 to 360 structures with a step of 40. Each test structure was represented by a point in the image parameters space (Fig.
For each of the grouped data sets we calibrated the VT5 model with Mentor Graphics Calibre CAD and analyzed its accuracy. The results are presented in Fig.
As we expected, sets containing 40 or 80 structures yielded a large error which decreased monotonically with an increase in the number of measurements. At above 280 structures the error stopped decreasing and reached an almost constant level of 1.4 nm. We therefore could conclude that the optimal calibration set is 300 to 350 test structures if sampling is performed using cluster analysis. The use of SEM contours for VT5 model calibration allows one to reduce the rms error to 1.18 nm for 40 structures. Simulation results are presented in Fig.
This work describes two existing approaches to photomask formation simulation and demonstrates their applicability for projection photolithography and OPC recipe development. Full resist mask formation simulation allowed us to optimize the lithographic stack for a new process. VT5 model calibration with different data arrays showed that cluster analysis is a powerful sampling tool. The optimal sampling size made up at 300 to 350 structures.