This work presents a theoretical study of using the interference of multiple counter-propagating evanescent waves as a lithography technique to print periodic two dimensional features. The formulation of the three dimensional Cartesian space expression of an evanescent wave is presented. In this work, the evanescent wave is generated by the total internal reflection of a plane wave at the interface between a incident dielectric material and a weakly absorbing transmission medium. The influences of polarization, incident angle and the phase shifting of the incident plane waves on the evanescent wave interference are studied. Numerical simulation results suggest that this technique enables fabrication of periodic two dimensional features with resolution less than one third the wavelength of the irradiation source.
©2007 Optical Society of America
Optical lithography is an important high precision technology with which sub-micron features are fabricated. With appropriate configuration, the optical lithography offers patterning of high resolution features. One commonly used optical lithography scheme, which is the optical projection lithography (OPL). The resolution achievable by conventional OPL is however fundamentally limited by diffraction. To overcome the diffraction limit, wavefront engineering and liquid immersion have been developed as resolution enhancement techniques. The wavefront engineering for higher resolution encompasses off-axis illumination, polarization, phase-shifting and optical proximity correction . The minimum half-pitch resolution size,CDhalf-pitch, achievable by an OPL technology is theoretically determined by CDhalf-pitch = k 1 λ/NA where λ is the vaccum wavelength of the irradiation source, NA is the numerical aperture of the projection optics of the system and k 1 is a constant value indication of the effectiveness of the wavefront engineering techniques that have been implemented. An effective wavefront engineering technique would result in a smaller k 1 value. With higher resolution requirement, the complexity and therefore cost for implementing these waterfront engineering techniques would increase. Even with the most complex wavefront engineering techniques, it is theoretically impossible to achieve features with dimensions less than a quarter of the wavelength of the irradiation source. With the limiting capability of the wavefront engineering techniques to achieve higher resolution, much work has been done to develop the liquid immersion technique in recent years. A number of literatures have already reported the resolution enhancement capability of the liquid immersion technique for optical lithography [2–6]. In fact, some semiconductor manufacturing companies have already begun fabrication of more advanced devices with the liquid immersion steppers. Though the liquid immersion technique is new for optical lithography purpose, it has been used for many years for the purpose of microscopy where oil is used as the immersion liquid. The liquid immersion technique enhances resolution by a factor equivalent to the refractive index of the immersion liquid.
The irradiation sources that are currently used for optical lithography are in UV to DUV range. At these wavelengths, the highest refractive index among suitable liquids is about 1.6. On the other hand, the refractive index of solids usually can be higher than this. Hence the solid immersion technique would theoretically achieve higher resolution than the liquid immersion technique. There are two possible methods to apply the solid immersion technique to optical lithography purpose. The first possible method is to use a hemispherical solid immersion lens (SIL) [7–11]. The hemispherical SIL focuses a beam, incident on its top hemispherical surface, from the range of low angle to high angle. The incidence energy on the bottom planar surface of the SIL consists of two parts. One part of the incidence energy is a homogenous cone of energy that would propagate out of the SIL. This homogeneous cone of energy is from the portion of the beam focused at low incident angle. The other part of the incidence energy is an annular ring of evanescent waves (EWs), around the homogeneous cone of energy. This annular ring of EWs arises from the total internal reflection of the portion of the beam at high incident angles which is larger than the critical angle. The resultant high resolution spot at the imaging plane is formed by both the waves that propagate out of the SIL and the EWs that propagate on the bottom surface of the SIL. The EWs define the sharpest features of the focused spot [8, 12]. Due to this focusing principle of the SIL, optical lithography with the SIL would require it to be implemented as a direct write method such that features are patterned individually. Therefore, a complex system would be required for the realization of a parallel write system via the use of microlens array . Another possible method of applying the solid immersion technique is to use the interference of the EWs. This can be achieved with two possible configurations- using total internal reflection (TIR) in an incident medium of high refractive index and using sub-wavelength mask . In both cases, a recording medium such as photoresist would have to be placed in close proximity to either the high refractive index medium or the sub-wavelength mask to capture the evanescent interference field, the amplitude of which decays exponentially with distance away from the surface of the medium or the mask. This technique has been studied as early as in the late 1960s by Stetson [15, 16] for the purpose of holography recording using 638nm wavelength lasers. In recent years, there is a renewed interest in this technique. A number of recent literatures have reported on this technique using UV and DUV lasers as irradiation sources, thereby allowing features of even higher resolution to be achieved [17–21]. These literatures have reported the fabrication of periodic lines features using the interference of two counter-propagating EWs.
The objective of this work is to extend the application of EWs interference to be used for patterning of periodic two dimensional features using the interference of four counter-propagating EWs. For this purpose, it is important to analyze the propagation of optical waves in a three dimensional space, taking into account the polarization states and their orientation with respect to each other. In practice, it is only possible to control the polarization state of the incident beam but not that of an EW. In this context, this manuscript focuses on (a) the formulation of correlation of the polarization state between that of an incident beam and its corresponding TIR-generated EWs, and (b) the spatial field structure of the EWs under stationary interference for different orientation and polarization state of the incident waves .
2. Theoretical analysis
The concept of generating evanescent wave by TIR in a two dimensional space is illustrated by Fig. 1(a).
A plane wave ψ n propagates in a homogeneous, dielectric medium that has a real refractive index value of ni. This dielectric medium is in contact with an optically less dense material, of which the complex refractive index value is n̂t =nt(1 + ikt). When the incident angle, θi of ψn at the interface is larger than the critical angle such that sin θi > nt/ni , TIR occurs and an evanescent wave, ψen , is generated. The corresponding lateral Goos-Hanchen shift is indicated by GHL . The concept of generating evanescent wave by TIR in a three dimensional space is illustrated by Fig. 1(b).
The incident angle of the plane wave at the interface between the dielectric medium and the optically less dense material is given by θi. The corresponding azimuthal direction is given by θi. Here, we assume that the transverse Goos-Hanchen shift, GHT , is negligible. This is a valid assumption for the purpose of this study since the transverse Goos-Hanchen shift is small after a single reflection . Hence, the azimuthal direction of the evanescent wave would be the same as that of the incident wave. The expression for the resultant evanescent wave, ψe, suppressing the time-dependence, is given as follows ,
where k t = 2πnt/λ(sinθtcosϕiî + sinθt cosθiĴ + cosθtk̂)and r = xî + yĵ + zk̂ define the unit wave vector of the plane wave and position vector in the transmitted medium respectively. The expression in the Eq. (1) shows that the evanescent wave propagates at the interface(x-y plane) between the incident and transmitted medium. The amplitude of the evanescent wave decays exponentially in the z direction. The terms nti and E e(x,y,z) refers the ratio nt/ni and the electric field vector of the evanescent wave in Cartesian space respectively. The electric field vector of the resultant evanescent wave in Cartesian space can be correlated to that of the incident wave as follows,
The polarization state of the incident wave is described by E in(s,p,Ϛ) in the (s,p,Ϛ) coordinate system which is defined such that EϚ = 0, given as follows,
where Es and Ep refer to the amplitudes of s and p polarization components for the incident wave.
The matrix T is the transmission coefficient matrix given as,
where ts and tp refer to the Fresnel transmission coefficients for the s and p-polarization component of the incident wave. For the case where the transmitted medium has a complex refractive value, the transmission coefficients for s and p-polarization can be derived from the Fresnel equations by setting n̂t cos θt = ut + ivt to be given as , for s-polarization,
and for p-polarization,
The terms τs and τp represent the amplitude transmission factor for the s and p-polarization respectively. The terms χs and χp are the corresponding phase shifts. The transmitted medium considered in this work is a weakly absorbing medium, such that ntkt ≪1. Therefore, the linear (over kt) approximation, given the condition sin2 , of the series expansion of Eq. (7) will result in the following simplified expressions for ut and vt ,
The matrix M describes the transformation of the Jones Vector from the (s, p,Ϛ) coordinate system into the global Cartesian coordinate system,
The angle θ e can be expressed in terms of the polar angle of the incident wave using the Snell’s law and for the condition sin θ i > nti such that,
The matrix R describes the azimuthal direction of the evanescent wave in three dimensional Cartesian space. Based on the previous assumption that transverse Goos-Hanchen shift is negligible, R is as follows,
From the above equations, the electric vector of the evanescent wave shown by Eq. (2) can therefore be re-expressed as,
When the incident wave is s polarization, the resultant electric vector is,
When the incident wave is p polarized, the resultant electric vector is,
Appropriate matrix transformation operation can be applied to both Eq. (13) and Eq. (14) to find the corresponding polarization state of the EWs. It was theoretically shown that s-polarization incident beam would result in s-polarization EWs and p-polarization incident beams would result in elliptical polarization EWs .
When there are m numbers of waves incident at the interface, with incident angles larger than the critical angle, the resultant EWs interfere with each other. As a result, the summation of the electric vector ψet, and the corresponding EW interference intensity, I e, at the interface are given respectively as,
Here, we present a numerical study of using four counter-propagating EWs interference, as illustrated by Fig. 2, to pattern two dimensional features.
Four beams, each a plane wave, propagate in an optically dense dielectric medium. The incident angles and azimuthal orientation of the four beams are appropriately configured such that four evanescent waves counter propagate towards each other on the x-y plane, which represents the interface between the dielectric medium and an optically less dense, and weakly absorbing medium (i.e. photoresist in this work). We assume that the two mediums are in full contact with each other. As shown in Fig. 2, both ψ 1 and ψ 3 are on the plane of incidence A while ψ 2 and ψ 4 are on the plane of incidence B. The beams lying on the same plane of incidence are assigned the following properties:
- (a) They have the same incidence angle at the interface i.e. both ψ 1 and ψ 3 have incidence angle value of θA and ψ 2 and ψ 4 have incidence angle value of θB .
- (b) Since the incident beams on the same plane of incidence have an azimuthal angular spacing of 180°, the azimuthal direction of the incident beams can thus be defined by the azimuthal angular spacing between the planes of incidence.
In this work, the irradiation source to be employed has a free-space wavelength (λ) of 355 nm. We assume that the incident medium is a homogeneous glass with ni,=1.65. A thick layer of photoresist with a complex refractive index value of 1.58+i0.02 is used as the optically rarer and weakly absorbing medium in contact with the glass. As such, the critical angle is 73.25°.
The interfacial interference intensity distribution (at z=0 plane) and the penetration profile are studied for variation of the optical parameters given as follow,
- (a) s and p-polarized incidence plane waves which would generate EWs of s and elliptical polarization.
- (b) The incidence angles (θA or θB) of the incident beams.
- (c) Initial phase (ϵ) of the incident beams.
The amplitude of the electric vector at the interface for each incident beam is taken to be unity and ϕdiff is fixed at 90°.The figures presented in this work indicate the level of normalized intensity from white (maximum normalized intensity) to black (zero intensity). It is assumed that the spatial profile of the intensity distribution in the resist would result in similar surface profiles at certain intensity threshold for different exposure times.
3. Results and discussion
3.1 Influence of polarization
Two polarization states of the EWs, namely s-polarization and elliptical polarization, are studied for the influence on the intensity distribution of EWs interference at the interface. Two configurations are used for this study. One configuration comprises of four s-polarization incident beams which would generate four counter-propagating s-polarization EWs. The other configuration comprises of four p-polarization incident beams which would generate four counter-propagating elliptical polarization EWs. The orientations of the incident beams for both configurations are given as θA=θB= 75°. Shown in Fig. 3 are the resultant normalized intensity distributions of the s-polarization EWs and elliptical-polarization EWs interference at the z=0 plane respectively. The intensity is normazed by the highest intensity value at the interface.
The simulation results show that the different polarization state of the EWs would influence the geometry of the features (size and shape), the aspect ratio and the array orientation of the features.
The s-polarization EW interference produced alternating columns of bright and dark circular features. If it is assumed that the photoresist has a normalized intensity threshold of 0.3, the size of each feature would be about 50nm diameter. In the same context, the aspect ratio of the features for the s-polarization case would be about 16 features per λ2 area. For the case where p-polarization incident beams are used, the resultant elliptical-polarization EWs interference produced bright elliptical features orientated in a hexagonal array. Each of the bright elliptical features is well resolved from each other as they are separated by zero intensity regions. Each elliptical features is about 125nm by 100nm and the aspect ratio is about 5 features per λ 2 area.
The Fig. 4 shows the normalized intensity profile of the s-polarization and elliptical-polarization EW interference in the second medium. The intensity values are normalized by the highest intensity value at the interface.
From the centre of each bright feature, where the intensity is the highest, the normalized intensity decay profile along the z direction into the depth of the second medium is studied. The Fig. 5 illustrates the difference in the intensity decay profile for both s-polarized EWs interference as well as elliptical polarized EW interference. The intensity values are normalized by the highest intensity at interface (z=0) of the elliptical polarization EWs interference.
The result shows that the elliptical polarization EW interference results in higher intensity at the interface as well as through the depth of the second medium as compared to the s-polarization EW interference. The reason could be that elliptical polarization EWs have electric field in the z component which enhances the penetration depth of the EWs interference into the second medium If photoresist with normalized intensity threshold of 0.3 is used, the result would therefore suggest that the elliptical polarization EWs interference has larger penetration depth and hence allow the use of thicker resist coating. This is an advantage in the context of lithography as it is technically challenging to achieve thin resist coating (in the nanometer range).
3.2 Influence of incidence angle
In conventional laser interference lithography, the period (Λ) of the interference fringes created by two beams symmetrical about the optical axis is dependent on the incident angle (θi) of the incident beams such that Λ = λ/(2 sin θi). We have found in our study that increasing the value of θi beyond 75° does not reduce the pitch size of the interference patterns. This is because the EWs propagate along the plane at which interference occurs. Therefore, the periodicity of EWs interference pattern can be approximated to be Λ = λ/(2n) where n refers to the real refractive index value of the dielectric medium in which the EWs are generated.
However, our simulation results suggest that larger incidence angle adversely affects the penetration depth of the EWs interference into the second medium. From the centre of each bright feature, the through-depth intensity decay profiles at the centre of each bright feature are studied, for both s-polarization and elliptical-polarization EW interference. The results are shown in Fig. 6. For both cases, the through-depth intensity values are normalized by the intensity at the interface.
In both cases, when the incidence angle is increased beyond 75°, the intensity at the interface and the extinction depth (depth at which the normalized intensity values fall to zero) decrease. When the value of θi. is increased beyond 80°, the intensity value at the interface increases again. When the values of θi. is at 82.5° and 85°, there is no significant difference between the corresponding through-depth intensity profile. The results which have been presented so far are obtained with the incident beams having no phase difference between them. The following section presents a study of the effect of phase difference between the incident beams on the EW interference.
3.3 Effect of phase shifting of incident beams
When there is an initial phase for the incident beams, the Eq. (3) is to be expressed in a more general form given as,
where ϵ refers to the initial phase.
In this study, the initial phase of the incident beams is assigned such that there is only phase difference between adjacent incident beams, while incident beams lying on the same plane of incidence remains in phase. This is to avoid possible destructive interference between the incident beam and the reflected beam if there is a coincidence of the point of incidence and point of reflection at the interface. The phase difference between adjacent incident beams would be denoted by the term δϵ . All the incident angles are maintained at 75°.
In the first case, the initial phases of ψ 1 and ψ 3 are shifted by π radian with respect to that of ψ 2 and ψ 4 (δϵ = π between the adjacent incident beams). The result of the intensity distribution at the interface for s-polarization EWs interference is the same as that in Fig. 3 (a). This is because the the counter-propagation EWs in the same plane of incidence are in phase. The electric field of adjacent s-polarization EWs are perpendicular to each other therefore there is no interference between them. For the elliptical-polarization EW interference, the corresponding result is shown in the Fig. 7.
The result suggests that a δϵ = π between the adjacent incident beams would cause a possible shift of the entire array either along the y-axis or the x-axis. On the other hand, there is no significant influence on the aspect ratio as well as the geometry of individual feature.
When the initial phases of ψ 1 and ψ 3 are shifted by π/2 with respect to that of ψ 2 and ψ 4 (δϵ = π/2 between the adjacent incident beams), there is a significant change in the intensity distribution at the interface for both s and elliptical polarization EW interference. The corresponding of the normalized intensity distribution plots, at the interface, is shown in Fig. 8.
For s-polarization EW interference, an initial phase difference of π/2 between adjacent incident beams would result in hexagonal array of bright features. The aspect ratio of the features is also found to be significantly higher in comparison to the case where there is no phase difference between adjacent incident beams. With the use of photoresist with a normalized intensity threshold 0.3, the fabrication of periodic circular features, with individual dimension smaller than 50nm may be possible. For the elliptical-polarization EW interference, the corresponding intensity distribution at the interface shows that complex two dimensional patterns with two order periodicity. A complex feature is defined in this work as a feature with length and width such as elliptical. On the other hand, a symmetrical shape such as circle would be considered as a simple feature. A two order periodicity, as defined in this work, describes the complexity of the arrangement of the features. The result in Fig. 9(b) shows that the bright features is aligned orthogonally to its vertical and horizontal adjacent neighbors. The geometry of each bright feature is asymmetrical. Hence, it is possible to fabricate periodic rectangular features which can be useful for the fabrication of advanced microelectronic devices. The effect of phase difference between adjacent incident beams on the penetration depth of the EWs interference is investigated. For this purpose, we study the decay profile of the intensity along the depth of the second medium at the spots of highest intensity. The results for the s-polarization and elliptical polarization EW interference are shown in Fig. 9. The maximum intensity value at the interface from the δϵ=0 configuration is used as the normalization factor for each of the cases.
The results for the s-polarization EWs interference suggest that, though the introduction of π phase difference between adjacent incident beams cause the intensity at the interface and, subsequently, through the depth to be adversely affected. If photoresist of 0.3 normalized intensity threshold is used, a π phase difference between adjacent incidence beams would cause the penetration depth of the s-polarization EW interference to be reduced by more than half. On the other hand, if the phase difference between adjacent incidence beams is π/2 , the reduction of the penetration depth is not as severe. Therefore, it is possible to achieve higher resolution features from s-polarization EWs interference by introducing π/2 phase difference between the adjacent incident beams.
For the elliptical polarization EWs interference, the results showed that the introduction of π phase difference between adjacent incident beams would cause an increase in the intensity at the interface and through the depth of the second medium as compared to the case where there is no phase difference. The reason could be the constructive interference between the z-component electric fields of the EWs. Conversely the π/ 2 phase difference may have caused, to certain extent, destructive interference of the z component of the electric field of the EWs, thereby causing the reduction of the penetration depth.
The EW interference is studied as lithography technique to pattern two dimensional features. A theoretical model of the evanescent wave in a three dimensional Cartesian space is presented. The parameters such as polarization, incidence angle and phase difference between adjacent beams are studied, via simulation means, for their influences on patterning effect at the interface and the corresponding intensity profile through the depth of the second medium. The summary of the results is given as follows,
- (a) The s-polarization and p-polarization incident beam would generate EWs interference with different intensity distribution at the interface and different intensity through the depth of the second medium. The s-polarization incident beams generate EW interference that may fabricate features with smaller dimension and higher aspect ratio than those fabricated by the EW interference by generated by p-polarization incident beams. However, the s-polarization generated EW interference would result in lower intensity at the interface comparatively and therefore intensity through the depth of the second medium.
- (b) The results suggest that the periodicity of the interference pattern does not depend on the incidence angle. However, it is shown that larger incidence angle would result in lower intensity at the interface between the incident medium and the weakly absorbing medium as well as through the depth of the weakly absorbing medium. In the context of lithography, the photoresist structures after development determines the structures of the features to be fabricated. Hence, the thickness of the photoresist coating on the substrate would be dependent on the incidence angle. Due to the technical difficulty in achieving photoresist coating with sub-micron thickness, a smaller incidence angle would be preferred when using this technique for lithography purpose. As the minimum incidence angle has to be larger than the critical angle, an incident medium with high refractive index value would allow thicker photoresist coating.
- (c) A phase difference of π/2 between adjacent incident beams for the given configuration would result in different spatial intensity distribution at the interface as well as different characteristic profile of the intensity through the depth of the second medium for both s-polarization and elliptical polarization EWs interference. When the phase difference between adjacent incident beams is π, there is no change in the spatial intensity distribution at the interface for the s-polarization EWs interference. For the elliptical polarization EWs interference, the resultant spatial intensity distribution at the interface is shifted along either vertically or horizontally.
These results will offer potential promise for patterning sub-100nm periodic two dimensional features as a lithography technique.
We would like to thank Nanyang Technological University-Chartered Semiconductor Manufacturing for the financial support of this work.
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