Simulation based study of thermal effects on periodic nanostructures fabricated by laser interference

Materials Science in Semiconductor Processing 17 (2014) 119–123

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Simulation based study of thermal effects on periodic nanostructures fabricated by laser interference V. Velmurugan n, J.P. Raina Center for Nanotechnology Research VIT University, Vellore, India

a r t i c l e i n f o

abstract

Available online 5 October 2013

Laser interference based direct writing is a potential solution for wide range of fabrication of nanostructure. This paper deals with two dimensional heat conduction analysis and the simulation of the same on silicon and germanium surfaces. Simulation of direct writing of patterns with 193 nm, 27 nsec, single pulse laser source with power varied from 800 mJ/cm2 to 1100 mJ/cm2 are presented. The rise in the temperature on the surfaces as well as beneath the surface is analyzed on the basis of the results. We also report about the dependency of the thermal diffusion length on the homogeneity of the formed structures. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Periodic structures Thermal diffusion length Laser interference Homogenous structures Numerical simulation Silicon Germanium

1. Introduction Recently there has been growing interest for direct writing (DW) methods in the metallurgical processing, microelectronics industry, MEMS fabrication, microfluidics etc., [1] as the photo-resist based lithography is reaching its limits. DW offers advantages of depositing or transferring pattern on round, flexible, inflatable, irregular and 3D structures [2]. Exhaustive literature survey is presented [2,3] on various types of DW techniques like droplet based, tip based and energy based with their probable applications. Conventional lithography processes in semiconductor device fabrication involve in large quantities of material wastages and are often energy intensive [1]. Laser based DW finds itself as a suitable candidate in particular to the semiconductor industry which emphasizes on the shorter line widths, high volume manufacturing, and low cost. Laser based DW are essentially mask-less pattern transfer technique on a wider area onto the wafer. This overcomes the normal drawbacks that are faced due to line edge roughness, wave front distortion, parallax alignment, etc. Laser Interference Lithography (LIL) based DW is a scheme that uses two or more coherent light sources to create pattern

of definite line spacing on the substrate. It has wider option for mix and match approach with other conventional and non conventional lithography techniques [4]. Direct writing on surfaces to create structures without smudging can be analyzed in terms of threshold fluence [5], pulse shape [6], inward flux of molten material [7], thermal diffusion length, etc. In this paper we report thermal simulation studies of direct laser interference lithography to analyze the topographical effects, heat distribution pattern, effect of period and the intensity for silicon and germanium based on thermal diffusion length. The simulation results are compared with the recent experimental works reported by Tavera et al. [7], Peng et al. [8], Lasagni et al.[9] and Fukushima et al. [10].

2. Direct writing based on laser interference Interference fringes are formed in the intersecting region of two (Fig. 1) or more monochromatic optical plane waves. For a two beam, transverse electrically (TE) polarized light, the intensity distribution is given by I ¼ 4jEj2 cos 2 ðkx sin θÞ

ð1Þ

n

Corresponding author. Tel.: þ 91 416 2202412; fax: þ 91 416 2243092. E-mail addresses: [email protected], [email protected] (V. Velmurugan). 1369-8001/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mssp.2013.09.007

k is the wave number, E-Intensity of the individual beam. E1 ¼E2 ¼E

120

V. Velmurugan, J.P. Raina / Materials Science in Semiconductor Processing 17 (2014) 119–123

The resulting standing waves will have a period (P) which is given by K dλ P¼ 2nSinθ

ð2Þ

K d is the scaling factor for a particular technology node, λ is the wavelength of the source in nm, θ is half angle of incidence of the interfering beams, n is the refractive index of the medium. In general, the intensity pattern for a multi beam interference is
 I ¼ ∑Ei 2 þ ∑i o j 2Ei Ej Cos ðki  kj Þr þðϕi  ϕj Þ

ð3Þ

where Ei, Ej; ki, kj; and ϕi , ϕj are the field amplitudes, wave vectors and phases of two interfering beams respectively, ϕi , ϕj is the shift in the phase of the ith and jth field. The control in the phase change produces tunable lattices or structures [11]. The way to reduce the period could be by reduction in the wavelength, having a very high numerical aperture path length in the optical column and a small variation in the angle of incidence. The experimental conditions like temperature, humidity, mechanical vibrations, and air pressure will also have an effect variation of the patterns [8]. The recent ITRS road map [12] forecasts that interference lithography could be a possible option for fabricatingo 16 nm DRAM/MPU flash memory. The transition from 157 nm to 13.5 nm (Table 1) are constrained due to poor conversion efficiency of the EUV sources, reflective coating by multilayer optical elements and debris mitigation [13]. Since there is large change in the peak intensity due to interference, a high intensity source can be used for direct surface restructuring. The advancement in the laser technology delivers systems with shorter wavelength and narrow line widths (Δλ) and hence a very good longitudinal coherence length (λ2/Δλ) which enables in fabrication of finer structures and wide area patterning [4]. Well established examples of LIL applications can be found elsewhere [14–19]. E1

E2

Surface for patterning Fig. 1. Schematic representation of interference lithography.

3. Thermal simulation Absorption of laser by materials results in heating, melting, vaporization, plasma formation etc [20]. The extent of these effects depend on the thermo-physical characteristics like adsorbity, thermal conductivity, specific heat, density, latent heat, relaxation time etc., and the laser parameters like intensity, wavelength, spatial and temporal coherence, angle of incidence, polarization, illumination time, pulse repetition rate etc. Following are the assumptions made for the thermal simulation of the laser based interference direct writing method [9] (i) No radiation is lost from the surface. (ii) When the material is melting there is no convection due to gravitational or Electromagnetic effects. (iii) Material is homogeneous. (iv) Initial temperature of the material is constant. The basic heat conduction equation is given by. y

ρCp

∂T e δa ∇ðk∇T Þ ¼ IðtÞð1  RðtÞÞ ∂t δa

ð4Þ

ρ is the material density, t is time of exposure, Cp is the specific heat capacity, k is the thermal conductivity, y is the depth, δa is the penetration depth, R is the reflectivity of the surface and is dependent on n and κ which are strong functions of wavelength and temperature. The right hand side of Eq. (4) is the heat distribution (q(x)) due to laser source. For an interference pattern, the heat distribution term [5] is given by ! ðt t p Þ2 IðxÞ q ðxÞ ¼ α pffiffiffiffiffiffiexp  ð1 RÞe  yα ð5Þ 2s2 s 2π where τp s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ln 2 I(x) is the energy per unit area created by the interference pattern, tp is the pulse time and α is the absorption co-efficient. The interference patterns produce a periodic heating through local photo-thermal interaction mechanism. The temperature can be increased in concentrated regions even above the boiling point [3]. The initial conditions of the problem can be written as T ðy; 0Þ  T 0 ¼ 0 f or 0 r y r1; t

Table 1 Semiconductor technology nodes, sources and the variation in the NA [13]. λ (nm)

Source

NA variation

Kd variation

P variation (m)

248 193 157 13.5

KrF excimer laser ArF excimer laser F2 excimer laser Plasma based

0.35–0.82 0.6–1.34 0.85–0.93 0.1–0.3

0.6–0.4 0.4–0.3 0.4–0.3 0.8–0.5

1.7  10  7–7.56  10  8 8.04 1  10  8–3.6  10  8 4.617  10  8–4.22  10  8 3.375  10  8–1.125  10  8

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Table 2 Data used for simulation. Ge

Wavelength λ (nm) Pulse width tp (ns) Solid Density ρs (Kg/m3) Liquid Density ρl (Kg/m3) Melting temperature Tm(K) Latent heat of fusion Lv (J/g) Thermal conductivity of solid Ks(W/mK) Thermal conductivity of Liquid Kl(W/mK) Sp Heat capacity of solid Cps(J/(Kg*K)) Sp Heat capacity of liquid Cpl (J/(Kg*K))

193 27 2320 2500 1690 1650 148 200 710 680

193 27 5323 5600 1211 466.5 60 43 321.4 380

δ ¼ 0 for t 4t p The solution [19] of the heat conduction equation results in solution containing the change in the temperature (ΔT) along the depth of the material and is given in two regions as shown below. During heating (0 ototp) ! H y ΔT ðy; t Þt o tp ¼ ð4αt Þ1=2 ie ð6Þ k ð4αt Þ1=2

y ð4αðt  t p ÞÞ1=2

1000 800 600

0

1

2

3

4

5

6

TIME (Sec)

∂Tð0; tÞ ¼ δH δ ¼ 1 for 0 r t r t p ∂y

ðt  t p Þ1=2 ie

1200

200

T0 is the initial temperature on the surface. The boundary condition for the surface is

During cooling (t4tp) " ! 2Hα1=2 1=2 y t ie ΔT ðy; tÞt 4 tp ¼ k ð4αtÞ1=2

1400

400

!# ð7Þ

H is the heat transfer co-efficient. The function “ie” is defined as  1  ie ðxÞ ¼ pffiffiffi exp ð  x2 Þ  x ð1 erf ðxÞÞ π and erf(x) is the complementary error function of each element of x.

7 x 10-8

Fig. 2. Variation of the temperature on silicon surface for various fluencies. 3500

TEMPERATURE(K)

Si

TEMPERATURE(K)

1600

Parameter

k

121

3000 2500 2000 1500 1000 500 0

0

0.2

0.4

0.6

TIME (Sec)

0.8

1

1.2 x 10-7

Fig. 3. Temporal variation of temperature on surface of silicon for three consecutive pulses.

variation of temperature is similar to that of metals [6]. The temperature on the surface increases with the increase in the irradiation time (Fig. 2) and reaches the maximum at the pulse duration time. For silicon a fluence of 1000 mJ/cm2 (melting threshold) induces the process of melting, whereas germanium has relatively lower threshold of 680 mJ/cm2. In case of low energy (500 mJ/cm2) multiple pulses are needed to reach the melting threshold (Fig. 3). Once the pulse is extinguished the surface starts cooling rapidly till the temperature is close to the melting point and starts solidifying. The tendency of the graph could be justified by the change in the thermal diffusion length (lT ) which is dependent on the thermal diffusivity (D). qffiffiffiffiffiffiffiffi ð8Þ lT ¼ Dτp

4. Results and discussions Thermal simulations were carried out for silicon and germanium. The data (Table 2) were drawn based on the commercial lasers used in optical lithography. 4.1. Effect of fluence The importance of this study is to know the temporal evolution of melting during the irradiation of the laser on the material with the help of Eqs. (6) and (7). The reflectivity of the surface is varied from 59% to 65% based on the surface temperature [22]. In the x–z plane the heat distribution follows Beer-lambert's law. The temporal

Thermal diffusivity is directly proportional to the thermal conductivity and for semiconductors, the thermal conductivity increases very rapidly at low temperatures (20 K) and reduces at faster rate (1/T) to the melting point [21]. Below Debye temperature (645 K for Si, 360 K for Ge) the thermal conductivity due to electron's contribution is very negligible, and at slightly higher temperature more carriers are produced and this is of the same order of the magnitude as the phonon conduction. In silicon this effect is not significant for temperatures less than 800 K. The calculated thermal diffusion length varies from 1.55746  10  6 m to 1.78227  10  6 m for silicon and for germanium it varies from 9.73097  10  7 m to 7.38636  10  7 m.

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TEMPERATURE (k)-->

1200 1000 800 600 400 200 0

0

1

2

3

4

5

6

7

INCIDENT TIME (sec)-->

x 10-8

Fig. 4. Temperature profile along the depth for a fluence of 700 mJ/cm2.

Fig. 5. Variation in the intensity on the surface of the Si for an energy of 4 mJ. producing interference in a area of 2  2 μm2. Table 3 Comparison of wavelength to the minimum period achieved on different. sample surfaces.

Table 4 Thermal diffusion length of different materials. Material

Material

λ (nm)

PCal (nm)

PExp

Error(%)e

a

Si Cu SSa Al Si GaAs Si(111) SiO2 CARb e a b

355 355 355 355 308 308 308 308 13.5

500 410 410 410 154 154 154 154 6.75

833 nm 2 mm 1.3 nm 2 mm 400 nm 250 nm 200 nm 250 nm 25 nm

39.97 79.5 68.4 79.5 61.5 38.2 23 38.4 73

Thermal diffusion length (mm)

Reference [7] [9] [9] [9] [8] [8] [8] [8] [10]

Absolute error between PCal and PExp. Stainless Steel Chemically Amplified Resist

The temperature on the surface reaches the maximum (Fig. 4) at the pulse time and for the depths below the surface (y40) the maximum temperature does not reach exactly at the pulse time. As we go for higher depths it is found that the time taken to reach the maximum temperature increases. This is due to the fact that the absorption co-efficient is higher on the surface of the silicon than at deeper layers.

Cu Stainless Steela Ala Sib Geb a b

1.09 0.48 0.96 1.55 73

Taken from [6]. Calculated using Eq. (8).

4.2. Effect of period The simulation is extended to the interference of two beams. The angle between the interfering beam is varied to get patterns of different periods according to the Eq. (2). Fig. 5 shows that the thermal gradient(M/m) is close to unity for shorter periods. Such a narrow gradient will make the line structure to melt of both sides and results in the loss of homogeneity. The recent experiment by many researchers supports this argument. Table 3 shows the comparison of the calculated minimum period (PCal) that can be obtained by interference of lasers on the different materials with the experimental value(PExp). The period is calculated (PCal) based on the wavelength of the source and the number of beams that makes the interference pattern.

V. Velmurugan, J.P. Raina / Materials Science in Semiconductor Processing 17 (2014) 119–123

These references report structures obtained by the LIL direct writing method and the resolution is much larger than λ/2. This is due to the dominance of thermal diffusion length (Table 4) at shorter wavelengths. So the applications that are based on the narrow structures will have to compromise on the uniformity of the structures obtained by interference based direct writing technique.

5. Conclusions Laser Interference on the surface of silicon and germanium is considered for direct writing and the effect of power and period is analyzed. Semiconducting materials on exposure of laser leads to a high increase in the temperature and as soon as the pulse get extinguished the temperature reduces drastically. The melting threshold is 1000 mJ/cm2 and 680 mJ/cm2 for silicon and germanium respectively. For the regions just below the surfaces the temperature reaches the maximum after the pulse stops. Interference on the surface leads to deformation of the structures that may not be uniform for shorter periods due to large thermal diffusion lengths. Therefore the fabrication of homogenous structures with resolution of λ/2 in case of silicon and germanium is not possible in a thermally uncontrolled environmental condition.

Acknowledgment The authors would like to thank Mr Victor Ishrayelu for his support in developing MATLAB codes.

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