5 Scaling in EM

Prof. Subhasis Ghosh

2.2.1 Scaling in Electricity and Magnetism

2.2.2 Scaling in optics

2.2.1     Scaling in Electricity and Electromagnetism

Electricity is the main source of power and driver for applications such as electrostatic and piezoelectric actuation, thermal resistance heating, electrodynamic pumping, and electromechanical transduction. As in mechanical systems, it is possible to come up with scaling laws in electrical systems in terms of their characteristic length L. If it assumed that over a length d is,

=-(2.32)

that voltage V is constant and  electric field

This means electric field scales as L−1. The resistance R of a conductor is, R = , where l is the length of the conductor, A is the area of cross section and ρ is the resistivity of the material. If resistivity is constant, then R scales as L−1. Using Ohm’s law, the current, I is I = R. That means I scales as L1 and so current density J = AI will scale as L−1. However, in very high electric field, Ohm’s law loses validity because the material’s resistivity can vary with the electric field. The power dissipated by an electrical element, W, is defined by Joule’s law,

W = RI2,

which scales as L1. So, the power dissipated per unit area will scale as parallel plates capacitor of area A, separated by a distance, d,

C = ε0εrA (2.33)

L−1. In case of capacitance of (2.34)

Hence, capacitance scales as L1. ε0 is the permittivity of free space (8.85 × 10−12m−3kg−1s4A2) and εr is the relative permittivity of the dielectric material between the plates and scales as L0.

The characteristics of a electrical circuit is governed by the basic units, capacitor C and resistor R and their combination. For example, the time constant of a circuit is dependent on the product RC and governs the behavior of the circuit.

∝ × −1 ∝ 0    (2.35)

Hence the  time constant of circuit  is independent of scaling.

The  breakdown voltage in a parallel plate capacitor can be varied  with gap between the plates, an effect known as  Paschen effect.  Paschen showed that the breakdown characteristics of a gap in a paralle plate configuration are a function of the product of the gas pressure and the gap length, usually  written as V= f(pd ), where p is the pressure and d is the gap. For air, and gaps on the order of a millimeter, the breakdown is roughly a linear function of the gap length,

V = 30pd + 1.35 kV(2.36)

where d is in centimeters, and p is in atmospheres. Though this empirical relation is quite valid in linear region, but the complete understanding of the mechanism of breakdown is still lacking. There has been extensive research to provide a theoretical basis for this law. Above linear empirical relation breaks down often and following empirical relation has been obtained [Ref.5] by fitting experimental data.

=    1    (2.37)

(   )−ln[ln(1+ )]

For our purpose, we assume that that the largest voltage we are able to safely apply, V, is proportional to L. The expression for electrostatic force, Fe, between parallel plates scales as

=—(2.37)

2.2.2     Scaling in optics

Scaling in optics is important both for small systems as well as devices used for investigating small systems, such as microscopes. In micro/nanoelectronics, small devices can only be fabricated using photolithographic or electron beam lithographic techniques, in which photosensitive materials are selectively etched using light or electron beam.

Fig. 2.3  Whenlight with wavelength, λ, pass through a small and a large opening, the width of each

opening is represented by the characteristic dimension   , light   diverges at an angle,

≈ . If is much larger than the wavelength, the wave hardly diverges at all.

If light  of wavelength, λ, shines upon an object with a characteristic dimension, L, the reflected wave diverges at an angle, ϴ ≈ Lλ. This is also the case if light passes through an opening with characteristic dimension, L as shown in Figure 2.3. That means that

ϴ ∝1/L(2.38)

so ϴ scales as L−1. If the opening is much larger than the wavelength, the waves hardly diverge at all. The scaling in optics is very important in visualization of objects in micro and nanoscale. In case microscopic object, the light diffracts and spreads out. The more the amount of this diffracted light that can be collected in the lens of the microscope, the more information can be acquired for creating the image.

Generally lithography using light or electron beam is used to fabricate artificial nanostructure. Hence, the scaling is extremely important in deciding the capability of different lithographic techniques. In case of photolithography, light is focused through a lens onto a surface to “write” a predesigned patterns. A thin polymer film is spread over a surface of sample on which the artificial nanostructure has to be created and wherever the light hits, the polymer hardens. When the unexposed polymer is washed away in a solvent, the hardened features remain. The exposed and unexposed areas gives rise to particular pattern. The minimum diameter of the spot size, or irradiated zone, is also the smallest feature one can write. The characteristic dimension, L, of this feature is given by

L=2λ/π(NA)(2.39)

Here, λ is the wavelength of light and NA is the numerical aperture of the system, that decides how sharp the angle of divergence would be. Therefore,

L ∝ λ   (2.40)

The resolution, ∆, of optical elements in microscopes (including your eyes) is the smallest distinguishable distance between two objects. It is determined in part by the diameter of the circular aperture because the size of this aperture dictates how much light can be collected. This diameter scales proportionally with L. Therefore ∆ scales as L−1. The focal length of a lens, FL, is given by the lens-maker’s formula:

Here n is the refractive index of the lens, and and are the radii of the curvature of the lens front and back surfaces, respectively. If a lens retains its shape while scaling up or down, the radii must vary proportionally. This means that the lens focal length scales as L1. However, it is to be noted that these scaling laws hold true as long as the dimensions of the optical components remain much larger than the wavelength of the light(L ≫ λ). This explains the necessity of using lower wavelength light for creating smaller and smaller patterns/structure.

In photolithography, the ability to project a highly resolved image of a small feature onto the wafer or any substrate is limited by the wavelength of the light used for lithography, and the ability of the reduction lens system to collect the maximum number of diffraction orders from the prefabricated illuminated mask . The minimum feature size , which is used as the scaling for different lithographic systems is given by

=-(2.42)

where CD is the minimum feature size and k1 is the coefficient that depends on the process related factors. NA is the numerical aperture of the lens used for collecting the diffracted lights from aperture.

Fig. 2.4 Light diffraction in photolithography setup without lens. Exposure system typically produce an image on the wafer using a photomask which blocks light in some areas and allow it pass in some other areas. Exposure systems are classified by the optics that transfers the image from the photo mask to the wafer.

Fig. 2.5 Light diffraction in photolithography setup with lens. NA defines the ability of lens to collect diffracted light. NA=2r0/D, r0 is the radius of the lens and D is the distance of the object from the lens. Lens with larger NA produces sharper image by collecting higher order diffracted light through the aperture.

Fig. 2.4 and Fig. 2.5 show the difference in the image of the aperture on the sample/substrate/wafer with and without collecting lens in photolithography.

To have higher resolution, photolithographic system uses ultraviolet light from gas-discharge lamps which generally emit light across a broad spectrum with several strong peaks in the ultraviolet range. This spectrum is filtered to select a single spectral line. The most popular emission lines at 436 nm (“g-line”), 405 nm (“h-line”) and 365 nm (“i-line”) are used from mercury based lamps. However, with the gradual demand for both higher resolution and higher throughput from semiconductor industry, laser based photolithography such as excimer laser [Ref.7] replacing lamp based lithography. Presently, photolithography has been extended to feature sizes below 50 nm using the 193 nm ArF excimer laser.

Fig. 2.6 Shows how feature size  decreases with the use of   short wavelength lasers.

Advancement of nanotechnology crucially depends on what is the minimum feature can be created on wafer or substrate for fabrication of nanodevices or more generally accessing the nanoworld.

The electron beam lithography provides better resolution and greater accuracy than photo lithography. Electron beam lithography has been used for many years advanced semiconductor device fabrication and nanotechnology. Higher resolution is achieved by the small size of the focused electron beam and the ability to electronically control the electron beam provides higher accuracy over the pattern developed by photolithography. Generally electron beam and optics in scanning electron microscopes is used for electron beam lithography. The current distribution within the beam is Gaussian and spots are overlapped to provide the smoothest features. A focused beam of electrons is used to draw custom features on wafer or substrate covered with an electron-sensitive film called a resist. The chemical changes in resist brought in by electron beam enables selective removal of either the exposed or non-exposed regions of the resist by immersing it in an suitable organic solvent, such as acetone. As in case of photolithography, the purpose of electron beam lithography is to create very small structures in the resist that can subsequently be transferred to the desired material. Limit of the lithography can be extended to sub-10 nm resolution by electron-beam lithography. Recently atomic resolution has been achieved by electron beam lithography.

Fig. 2.7 Schematic presentation of electron bean optics in SEM based electron beam lithography. Electron beam deflection is achieved by the application of electrostatic or electromagnetic fields to move the beam in the desired direction to create certain feature. A differential voltage applied to parallel plates results in a perpendicular field gradient, which deflects the beam off axis. Hence, by varying the electric field, the degree of deflection of the beam can be varied. Similarly, varying the current through a set of deflection coils can also be used to deflect the electron beam off axis. The angular spread Δα in the deflection angle α due to differential voltage of ΔV can be given by

∆   =∆/2 (2.20)

And diffraction limited electron beam size can be given by

= 0.6 (2.21)

where λ is the de Broglie wavength of electron, λ=1.2Vb-1/2, Vb is the accelerating voltage.

Suggested Reading:

1. Nanotechnology: Understanding small systems by Ben Rogers, Jesse Adams and Sumita Pennathur, CRC Press, Taylor & Francis Group, 2015
2. Scaling Laws in the Macro-, micro- and nanoworlds, European Journal of Physics, 22:601-611 by M. Wautelet, 2001.
3. Solid State Physics by N. W. Ashcroft and N. D. Mermin, Cengage Publishing, 1976
4. Condensed Matter Physics by M. P. Marder, Wiley, 2011.