Subsections

• 2.4.1 Introduction
• 2.4.2 Imaging Basics
• 2.4.3 Optical Proximity Effect

2.4 Mask Generation

2.4.1 Introduction

A fundamental requirement for almost all useful semiconductor devices is the definition of patterned elements. The main stream technology choice for patterning has been optical lithography. Up to the early 70's lithography was done as a contact printing process in which blue or near UV light was passed through a photo mask directly onto a photo resist coated semiconductor substrate [30].This shadow imaging process has been described in many research publications and handbooks [31], [32]. Beginning in the early 80's a new class of projection exposure tools, known as steppers, was introduced [33]. For the first time the pattern definition imaging on the semiconductor wafers was performed one chip at a time in a step-and-repeat fashion. Most stepper systems employed a reduction projection lens to ease the fabrication difficulty of the photo mask and to improve the overall precision and accuracy of the overlay of patterns on the wafer. Even more recently a combination of the earlier scanning approach with the step-and-repeat approach was created [34]. The step-and-scan approach has spread rapidly throughout the lithography tool industry, and is used for critical layers (like gate, metallization and contact layers) at the 250nm node and below [35]. Until the mid 90's all optical photo masks have been chrome on glass (called COG-photo masks) [36], also called binary photo masks. Starting from the 350nm node significant innovations in binary masks such as OPC (optical proximity correction) [37] and AAPSM (alternating aperture phase shift masks) [38] were introduced which improved the resolution capability of binary photo masks. A second approach besides the binary photo masks then emerged as EAPSM (embedded attenuated phase shift masks) [39] also called HTM (half tone masks).

2.4.2 Imaging Basics

Lithography is based on replicating the pattern on a photo mask into resist covered wafers. In an ideal case without degradation in the imaging process, a simple copy of the mask pattern would result, as shown in Figure 2.4 a.

Figure 2.4: Basic imaging characteristics (a) Ideal shadow imaging; (b) Diffraction-broadened projection imaging

However, in a projection process the imaging is always subject to degradation from diffraction and from imperfections in the projection system. An example of the image from a diffraction-limited projection system is shown in Figure 2.4 b. The spreading of the image profile results from the wave nature of light, and it is this property that limits the resolution capability of optical imaging systems. In an imaging lens system with a circular aperture of radius $\tilde{r_0}$ and imaging distance $D'$, the image intensity resulting from a point source can be described by an expression containing a first order BESSEL function,

$$I'(x)=I(0) \left(2\frac{J_1(x)}{x}\right)^2$$ (2.1)

where $x=\frac{2\pi\tilde{r_0}r'}{D'\lambda}$ and $r'$ is the distance in the image plane from the geometrical image point. $\lambda$ is the wavelength of the monochromatic light source. A detailed deduction of this expression is given in Appendix E.2. The fraction $\frac{\tilde{r_0}}{D'}$ given by

$$\frac{\tilde{r_0}}{D'} = \tan \theta \cong \sin \theta \qquad \theta \ll$$ (2.2)

equals the numerical aperture $NA$ defined by

$$NA \equiv n \sin \theta$$ (2.3)

with n as the refractive index of the medium behind the aperture or lens. Therefore the expression for $x$ can be further simplified to $x=2\pi\rho\frac{NA}{\lambda}$. For air as medium ( $n \approx 1$) a simplified description of $NA$ is given in Figure 2.5 as $NA=\sin \theta$

Figure 2.5: Geometric situation in simple projection optical system

This light intensity distribution is known as the AIRY pattern, after G.B. AIRY who first derived it in 1835 [40]. In addition to the general shape of the curve, shown in Figure 2.6, the first zero value is of interest. At about $x = 0.61 2\pi$ occurs a intensity minimum and an intensity maximum at $x=0.82 2\pi$.

Figure 2.6: Light intensity distribution from a point source projected through a circular imaging lens. The variable x on the horizontal axis is defined in the text

Resolution is defined as the ability to distinguish components of an object or a group of objects. The resolution capability of astronomical telescopes was studied in detail by LORD RAYLEIGH in the 19th century [41]. He defined the limit of resolution for a telescope as the angular separation between two stars when the peak of the AIRY intensity pattern from one star coincided with the first minimum of the AIRY intensity pattern for the other star. This leads to the well-known RAYLEIGH condition for angular resolution

$$NA = \sin \theta = 0.61 \frac{\lambda}{\tilde{r_0}}$$ (2.4)

where $\tilde{r_0}$ is again the radius of the imaging objective aperture. A sketch of the RAYLEIGH resolution condition is shown in Figure 2.7. Note that the intensity at the midpoint between the image peaks is reduced to about 78% of the peak intensity, which provides discernible separation, but not with high contrast between the bright and dark regions.

Figure 2.7: RAYLEIGH criterion for resolution of two point images in (a) 1D and (b) 2D. Scale is the same as in Figure 2.6

(a)

(b)

While the analogy of astronomical imaging to photo lithography is not completely quantitative, some key observations can be made. There is a limit to resolution for any given optical projection system, and it is not possible to resolve arbitrarily small or closely spaced features. It is also apparent that the resolution can be improved by using a smaller wavelength of the exposure light, and the resolution can be improved by making the projection system aperture larger.
In practical lithography the RAYLEIGH condition is typically restructured into the ``RAYLEIGH equation''

$${resolution}=k_1\frac{\lambda}{{NA}}$$ (2.5)

where ${NA}$ is the numerical aperture of the projection system and $k_1$ is a constant in the order of 0.4-0.8. There is no rigorous optical definition for the constant $k_1$, and it is generally used as a qualitative descriptor of the overall lithography process capability. More details on the RAYLEIGH equation are given in Table 2.1. Table 2.2 shows the numeric aperture, the resolution, and the depth of focus for the most important wavelengths in lithography.

Table 2.1: Definitions for important wavelength nodes in lithography

Definitions
$\lambda$(g-line)	436 nm
$\lambda$(i-line)	365 nm
$\lambda$(KrF)	248 nm
$\lambda$(ArF)	193 nm
$\lambda$($F_2$)	157 nm
Resolution
RAYLEIGH Resolution	$R = k_1\frac{\lambda}{NA}$
Traditional	$k_1 = 0.8$
Advanced	$k_1 = 0.4 - 0.6$
Depth of Focus
RAYLEIGH Depth of Focus	$DoF = k_2 \frac{\lambda}{NA^2}$
Traditional	$k_2 = 1.0$

Table 2.2: Examples for typical lens configurations ($k_1=0.6$) for deep sub micron technology nodes

Wavelength	NA	Resolution[$\mu m$]	DoF [$\mu m$]
i-line	0.63	0.35	0.92
KrF	0.60	0.25	0.69
KrF	0.70	0.21	0.51
ArF	0.70	0.17	0.39
$F_2$	0.70	0.13	0.32

This common description of resolution capability is closely related to the AIRY pattern described above. In particular, the first minimum of the AIRY pattern occurs at about $r'=0.61 \frac{\lambda}{NA}$, and the first maximum occurs at $r'=0.82 \frac{\lambda}{NA}$. The qualitative agreement with the usual range of $k_1$ is apparent.

2.4.3 Optical Proximity Effect

High performance optical projection imaging for lithography is strongly impacted by diffraction effects as noted in several previous sections. One result of this behavior is that individual pattern features do not image independently, but rather they interact with neighboring pattern features. A detailed analysis of the projection imaging process, for example, the analysis described in the paper by Hopkins [42], considers contributions from every portion of the reticle object and every portion of the projection optics in determining the exact image at the wafer plane. A simple heuristic argument considers the extended diffraction structure of the AIRY function comprising of the additional local maxima in the intensity distribution. Overlap of the diffraction peaks with adjacent pattern features leads to increased or decreased exposure intensity at any point in the image, compared to a purely geometrical image model.