Comprehensive modeling of the lithographic errors in laser direct write

Laser direct write (LDW) is a maskless lithography method that transfers computer-aided design patterns into photoresist (PR) pixel-by-pixel via direct laser beam exposure. Compared to another popular maskless lithography method, electron beam lithography (EBL), LDW has much faster write speed and lower cost for larger feature sizes and is easier to operate.¹ Meanwhile, it compromises on the resolution, which is at the scale of optical wavelength.¹ As a result, when the minimum feature size is larger than the dimension of laser wavelength (typically several hundred nanometers), LDW is a better choice than EBL. When using LDW, fewer issues need to be considered due to the more localized effect of the laser beam on the resist than that of the EBL. However, in LDW, poor fidelity of the pattern transferred into the resist similar to that in EBL may still occur. As can be seen in Fig. 1, the critical dimension (CD) of the actual pattern on a positive photoresist [shown in Fig. 1(a)] could be smaller or larger than the actual design if the patterns get under- or overexposed [shown in Fig. 1(b)], respectively. This pattern CD dependence on the dose is known as the exposure latitude. With the understanding of the exposure latitude, dimensional accuracy can be obtained by using the correct dose-to-size or using an arbitrary dose with the appropriate prebiased pattern size, as shown in Fig. 1(b). Moreover, the understanding of the exposure latitude can be used to minimize CD inconsistency caused by variations such as photoresist conditions or laser beam focal point between runs, which is discussed in Sec. III C. This study aims to model the exposure latitude for positive photoresist written by LDW.

In this study, the exposure latitude is defined as the pattern’s CD bias (denoted as $B$⁠) versus the exposure power (denoted as $P$⁠). The CD bias is the actual measured CD in resist minus the designed CD. The exposure power is a substitutive term of the exposure dose. It is the actual power of the laser beam that reaches the surface of the resist, which is proportional to the exposure dose and is computed from the power of the laser generator $Plaser$⁠, the optical filters $F$ on the light path, and the dose factor DF assigned by the acoustic-optical grating of the LDW tool, as shown in Eq. (1). Therefore, mathematically, the exposure latitude is expressed as a function $B(P)$⁠, as shown in Eq. (2),

This function $B(P)$ is assumed to be not decided by the contrast curve of the resist, but only decided by the exposure intensity distribution (EID) of the laser beam, as will be explained in Sec. II A. The concept of EID is borrowed from EBL.² 

In LDW, the laser beam exposes pixel-by-pixel, whose spot (pixel) exposure exhibits a Gaussian-like energy intensity distribution.³ Similar to EBL,² the intensity from all pixel spots adds up to become the overall EID, as shown in Fig. 2(a). The exposure latitude is decided by the EID at the edge of the pattern. As can be seen in Fig. 2(b), the photoresist has an intensity-to-clear. Once the intensity at a specific point reaches this threshold, the resist at this point will be completely removed. The pattern CD is the distance between the intersections of the EID curve and the dose-to-clear intensity line. As the exposure power increases, the EID curve is pulled up, expanding the two intersections point and increasing the pattern CD.

By applying this model, some qualitative properties of the exposure latitude can be predicted. If the pattern CD is much larger than the laser beam width, then the exposure latitude does not strongly depend on the size of the pattern. This is because the EID contribution of the laser beam shots from within the pattern can be considered negligible to the EID at the edge of the pattern [as shown in Fig. 2(a)]. As a result, the EID profile at the edge of the pattern almost remains unchanged with increasing pattern size. Also, if the length between the boundaries of two nearby patterns (pattern gap) is much larger than the laser beam width, the local pattern density or the pattern gap does not affect the exposure latitude because of the same reason. However, as the pattern size shrinks to submicrometer, the exposure latitude curve as well as the EID at the pattern edge are no longer independent of pattern sizes and pattern gaps. This is because the beam width is comparable to the pattern size, which means the EID at the edge of the patterns with different sizes no longer coincides. Yet the exposure latitude of the patterns in these cases can still be understood by the experimental measurements and comprehensive modeling of the EID, which will be shown in Sec. II B.

The challenge of modeling the EID is the fact that the EID cannot be directly measured. This study proposes a method to determine the EID from the experimental exposure latitude via defining a dimensionless standard relative EID function $f(x)$⁠. The actual EID, denoted as $I(x)$ with the unit of measure as $mW/cm2$⁠, is expressed as

where $x$ is the position (⁠$x=0$ corresponds to the pattern edge), $P$ is the exposure power with the unit of measure as $mW$⁠, and $F(x)$ is a distribution function with the unit of measure as $cm−2$⁠. $Ic$ is the intensity-to-clear of the resist. When exposing with the power-to-size $Pts$⁠, the intensity at the pattern edge is merely enough to remove the resist. Therefore,

The dimensionless standard relative EID function $f(x)$ is defined as

This way, $x=0$ is taken as a reference point, and $f(x=0)$ is normalized to be 1. Applying this normalization, the EIDs under different process conditions using different exposure powers $P$ are comparable to each other. Substituting $P$ with $Pts$ in Eq. (3) and then dividing the two sides of the equation with $Ic$⁠, yields

Equation (6) together with Eq. (5) yields

Equation (7) clearly shows that $f(x)$ is the EID function in the relative intensity unit of $I/Ic$ when exposing with the power-to-size $Pts$⁠.

The relative EID function when exposed with an arbitrary exposure power $Pi$⁠, as shown in Fig. 3(b), is

It should be noted that any point $Pk$ can be taken as a reference point and the resulting $f(x)$ would be the relative EID function when exposing with the exposure power $Pk$⁠. The new $f(x)$ obtained from a new reference point $Pk$ is only a constant factor different from the $f(x)$ defined in Eq. (5). With the definition of $f(x)$⁠, the experimental exposure latitude, $Bi(Pi)$⁠, can be converted to the experimental relative EID function $f(xi)$⁠. As shown in Fig. 3(b), at the exposure power $Pi$⁠, the pattern has a bias $Bi$⁠; therefore, at $xi=Bi/2$⁠, $I(xi)=Ic$⁠. Substituting the values of $xi$ and $I(xi)$ into Eq. (3) yields

Comparing Eq. (9) with Eq. (5) yields

Equation (10) is the equation that convert $Bi(Pi)$ to $f(xi)$⁠. Figure 3 shows the data point conversion between the exposure latitude curve and the relative EID function. In principle, this normalized standard relative EID function $f(x)$ is the property of the optical system and should be independent of various conditions such as the exposure power, resist thickness, resist type, and substrate type. The relative EID obtained in this way is used to do the modeling.

To model the EID, the laser beam is assumed to be a Gaussian beam with the wavelength $λ$ and the waist width $w0$⁠, as shown in Fig. 4(a). The intensity distribution of the Gaussian beam in 3D space is shown in the following equation:⁴ 

where $z=0$ indicates the focal plane. $w(z)$ is the beam width at the height z,

$zR$ is the depth of focus (DOF),

When $z$ is within the range of $zR$⁠, the 2D distribution $I(r)$ does not change much. Therefore, assuming that $zR$ is much larger than the resist thickness, and the resist film is near the focal plane of the laser, the EID contribution from a single shot of the beam is a 2D Gaussian distribution function,

where $wx$ and $wy$ are the beam widths in x and y directions, which might have different values as the actual laser beam might not be the perfect Gaussian beam but an ellipse shaped beam.⁵ The overall relative EID function is assumed to be the superposition of the contribution from all laser beam shots,

These assumptions are the baseline for the modeling. Simple line patterns can be used to fit the experimental EID. With the fitting beam width, EID of more complex patterns can be calculated. In addition, process inconsistency can be modeled.

For a long line pattern whose length is along the y-direction and width is along the x-direction, with the beam width $w$ [as shown in Fig. 4(b)] and the pattern pitch $a$ which is smaller than the beam width $w$⁠, the EID contribution from a single column can be approximated as an infinite integration over y-direction, which gives a 1D Gaussian distribution function,

The overall intensity distribution is the superposition of this 1D Gaussian distribution function from multiple columns,

where $xi$ is the position of each column. Another assumption is made. The width of the pattern is assumed to expand by $Δx$ in the development process, as shown in Fig. 4(c). The fitting EID equation is modified by adding an additional term $Δx$⁠,

Equations (16) and (18) are the equations used to fit the experimental relative EID function of line patterns.

When changing to a new recipe with a new resist thickness, resist type, or substrate type, the pattern can be sized accordingly using the EID model without doing a series of exposure tests. First, the relative EID function $f(x)$ for any specific patterns can be calculated using the fitted parameters $A0$⁠, $wx$⁠, $wy$⁠, $Δx$ and following Eqs. (14) and (15). Subsequently, the new recipe is run with an arbitrary testing exposure power $P1$⁠, and the pattern CD bias is measured and donated as $B1$⁠. Considering Eq. (10) with the calculated $f(x)$⁠, the power-to-size can be directly obtained,

Similarly, the corresponding power-to-size for the patterns with a prebias $B0$ is

Even with the same protocol and process parameters, different runs of LDW lithography may yield slightly different results. This process inconsistency can also be modeled by the relative EID function. Two kinds of inconsistencies among different runs are considered.

First, as can be seen in Fig. 5(a), the pattern exposed with the exposure power $P1$ has a CD bias $B1$⁠. At $x=B1/2$⁠, $P=P1$⁠, the relative EID function reaches the intensity-to-clear of the resist 1 (relative unit). This intensity of the resist may deviate from 1 by a small amount $Δ$⁠, which induces a pattern CD bias inconsistency $ΔB$⁠. Substituting the value of $P=P1$⁠, $x=(B1+ΔB)/2$⁠, and intensity-to-clear $1−Δ$ into Eq. (8), we get

As mentioned, the desired CD can be obtained by using arbitrary exposure power combined with the corresponding prebias. However, to minimize the CD inconsistency induced by the inconsistency of the intensity-to-clear of the resist, Eq. (21) should be considered.

The second possible inconsistency is the laser beam focus inconsistency. The laser beam might be slightly defocused, especially for a short write head focal length. In this case, the beam width $wx$ in Eq. (14) is enlarged, causing a deviated EID function that leads to a pattern CD error. However, as shown in Fig. 5(b), at a specific point the value of the EID function is almost not affected by the small change in the beam width. This point is known as the isofocal point. The isofocal point can be found by calculating the EID with two slightly different beam widths, $w$ and $w+Δw$⁠. The exposure power and prebias can be adjusted to reach this point to minimize the pattern CD inconsistency.

Exposure tests were conducted to authenticate the theoretical predictions and justify the feasibility of the fitting of EID functions. MicroChem S1800 series photoresist was spun coat on single crystalline 100 silicon wafers. A Heidelberg DWL66+ (355 nm laser) was used to write test patterns on the resist, followed by 90 s development using the MF-319 developer. Two different write heads with the focal length of 10 and 2 mm were used in DWL66+, and the corresponding pixel size is $0.5$ and $0.1μm$⁠, respectively. The test pattern geometry and CDs are shown in Fig. 6. For 10 mm write head (⁠$0.5μm$ pixel size), the test patterns are squares with a CD of $3$–$8μm$⁠; for 2 mm write head (⁠$0.1μm$ pixel size), the test patterns are long lines with $20μm$ length and $0.4$–$1.0μm$ width. Different DFs were assigned to the pattern using beamer by GenISys⁶ to investigate the exposure latitudes. An optical microscope and SEM were used to measure the CDs of the patterns. Detailed process parameters are summarized in Table I.

The exposure latitudes of the patterns with different sizes ranging from 3 to 8 $μm$ are shown in Fig. 7(a). CD bias comparison between patterns with different local pattern densities ranging from 1% to 75% is plotted in Fig. 7(e). As expected, at micrometer scale, the exposure latitude does not have visible dependence on both the pattern size and the pattern density. The error bars of the data are also presented in Figs. 7(a) and 7(e). The uncertainty of the measurements is below $0.03μm$ in both figures. This value is quite beyond the resolution limitation of the optical microscope used in the measurement of the data in Fig. 7(a), so the uncertainties of the data in Fig. 7(a) probably just reflected the noise in the imaging process. As for the uncertainties of the data in Fig. 7(e), which is obtained from the SEM measurement, two other issues are considered the possible sources. First, the roughness of the edge line of the patterns may cause errors when using image processing software to measure the CD of the pattern. Second, the output power of the laser generator may be unstable, printing patterns with nonuniform CDs under the same setup condition.

The exposure latitudes of the square patterns in resists with the thicknesses ranging from 0.5 to 1.5 $μm$ are shown in Fig. 8(a), so do the error bars of the data. The red and blue lines represent the CDs measured along X and Y directions, respectively. These exposure latitudes data $Bi(Pi)$ were converted to the experimental standard relative EID function $f(xi)$ using Eq. (10). The power-to-size values are shown in the boxes and are used as the reference point $Pts$ in Eq. (10). The results are shown in Fig. 8(b). As can be seen, the experimental $f(x)$ from the exposure latitudes of resist with different thicknesses coincide within the range of the measurement uncertainty, which agrees with the prediction in Sec. II B.

The test patterns used to conduct the experimental EID fitting are long line patterns with $0.4$ and $1.0μm$ widths and $20μm$ lengths. The lines are along X and Y directions, as shown in Figs. 9(a) and 9(d). The widths of these lines were measured with SEM. The exposure latitudes of X- and Y-lines are shown in Figs. 9(b) and 9(e). These exposure latitudes data were converted to $f(xi)$ using Eq. (10). The reference point, $Pts$ in Eq. (10), is chosen to be 1.5 mW. The converted $f(xi)$ are shown in Figs. 9(c) and 9(f). Equations (16) and (18) were used to fit the experimental $f(x)$⁠. Due to the high level of versatility of the fitting equations, various combinations of the fitting parameters $(A0′,wx,Δx)$ may all yield good fits, and the least square fit results might not be the most reasonable one. An additional restriction was considered. The power-to-clear values indicated in Figs. 9(b) and 9(e), below which clear patterns (resist completely removed) no longer exist, are converted into the intensity-to-clear values indicated in Figs. 9(c) and 9(f). These intensity-to-clear values should be close to the top of the fitted EID functions. In addition, the fitting parameters of the EID functions of $0.4/1.0μm$ wide X-/Y-lines should be correlated. Four parameters were included in the fitting. The amplitude $A0′$ and the displacement $Δx$ should be the same for both X- and Y-lines, but the beam width can have different values $wy$ for X-lines and $wx$ for Y-lines, as the laser beam intensity may have different distributions along two orthotropic directions. The fitting results are also shown in Figs. 9(c) and 9(f). Three of the EID functions have a good fitting. The fitting of the EID of $0.4μm$ wide Y-line is not very good. This deviation from the prediction of the model that involves patterns within a relatively large area could be caused by the nonuniformity of the resist due to the spin-coat defects.

The exposure tests in submicrometer scale were repeated using the recipe 4 for three times without mount/unmount the write head. The exposure latitudes measured are shown in Fig. 10(a). As shown, the exposure latitudes among these three tests have no cross sections. Therefore, it is reasonable to assume that the beam width $wx$ remained unchanged while the amplitude $A0′$ changed among different runs. With this assumption in addition to the same power-to-clear restriction in Sec. V B1, the fitted EIDs and parameters are shown in Fig. 10(b). The amplitude inconsistency corresponds to the resist intensity-to-clear inconsistency discussed in Sec. III C. Considering Eq. (21), typically, to minimize this inconsistency, it is better to overexpose the pattern with a negative prebias correction.

As a comparison to the resist intensity-to-clear inconsistency, the focus control parameter was adjusted to obtain slightly defocus exposure latitudes and EIDs, which are shown in Figs. 10(c) and 10(d). As shown, the $A0′$ is the same but $wx$ are different for different focus control parameters. It should also be noted that after unmounting and remounting the write head, the same recipe can yield different beam widths $wx$ as well as different development displacement $Δx$⁠. The fitted $wx$ (with $focus=−80$⁠) is 550 nm in Fig. 9(f) but 680 nm in Fig. 10(b), and the fitted $Δx$ is 250 nm in Fig. 9 but 135 nm in Fig. 10. This result indicated that the beam focal point inconsistency discussed in Sec. III C is likely to happen when remounting the write head. A displacement inconsistency at the scale of 100 nm may also occur. Taking the above inconsistencies into consideration, it is recommended to overexpose the pattern to the isofocal point with a negative bias applied to the pattern. The appropriate exposure power and pattern CD prebias value can be subtracted from patterns’ EID function calculated using the fitted parameters $(A0′,wx,Δx)$⁠. It should be noticed, however, that this method of using isofocal dose with a negative bias has a limitation on the feature size of the pattern. As can be seen in Fig. 10(c), the CD bias at the isofocal point is positive $≈450nm$ for an original feature size of $400nm$⁠. If the original feature size becomes smaller, this bias will also become smaller but not go to zero because of the term $Δx$ in Eq. (18) and the energy spread of the laser beam $wx$⁠, which means the pattern with a feature size smaller than the bias value is not printable using this method.

The exposure latitude of the patterns printed by LDW on photoresist is defined as the bias of the CD of the patterns versus the exposure power. This exposure latitude can be explained by assuming an EID of the laser and an intensity-to-clear value of the photoresist. According to this model, qualitatively, when the pattern CD is much larger than the width of the laser beam, the exposure latitude is not greatly influenced by the pattern size and the local pattern density. This observation breaks down when the pattern CD is comparable to the width of the laser beam at submicrometer scale. Quantitatively, the EID function can be converted from the experimental exposure latitude data, and this function can be fitted by adding up the Gaussian intensity distribution of all laser beam shots. Exposure tests on MichroChem S1800 resist on Si wafers confirmed both the qualitative predictions of the model and the feasibility of the numerical fitting of the EID function. Using the EID model, the conditions to achieve dimensional accuracy in resist when using a new recipe can be determined with relative ease. In addition, within the framework of the EID model, the pattern CD inconsistencies among different runs due to the photoresist intensity-to-clear inconsistency and the laser beam focal point inconsistency were also defined and analyzed. Overexposing the pattern to the isofocal point with a negative bias applied to the pattern was found to be able to minimize these pattern CD inconsistencies. The comprehensive EID model presented in this study could be applied to other positive resist and substrate systems in LDW lithography to achieve dimensional accuracy and improve process consistency run-to-run.

This work was performed at the University of Pennsylvania Singh Center for Nanotechnology, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (NSF) (Grant No. NNCI-1542153).