288 7.6  High-Throughput Techniques

Microfabrication consists of multiple sequential stages (sometimes several tens of indi­

vidual steps) of manufacture involving treatment of the surface of a solid substrate through

either controllably removing specific parts of the surface or adding to it. The substrate in

question is often silicon based, stemming from the original application for integrated circuits,

such as pure silicon and doped variants that include electron (n-​type, using typical dopants

of antimony, arsenic, and phosphorus) and electron hole (p-​type, using typical dopants of

aluminum, born, and gallium) donor atoms. Compounds of silicon such as silicon nitride and

silicon dioxide are also commonly used. The latter (glass) also has valuable optical transmit­

tance properties at visible light wavelengths. To generate micropatterned surfaces, a lift-​off

process is often used that, unlike surface removal methods, is additive with respect to the

substrate surface. Lift-​off is a method that uses a sacrificial material to creating topological

surface patterns on a target material.

Surface removal techniques include chemical etching, which uses a strong acid or base

that dissolves solvent accessible surface features, and focused ablation of the substrate using a

FIB (see Chapter 5). Chemical etching is often used as part of photolithography. In photolith­

ography, the substrate is first spin-​coated with a photoresist. A photoresist is a light-​sensitive

material bound to a substrate surface, which can generate surface patterns by controllable

exposure of light and chemical etching using an appropriate photoresist developer. They are

typically viscous liquids prior to setting; a small amount of liquid photoresist is applied on the

center of the substrate that is then centrifuged by spin-​coating the surface controllably in a

thin layer of photoresist. We can estimate the height h(t) of the photoresist after a time t from

spinning by using the Navier–​Stokes equation assuming laminar flow during the spinning,

resulting in equating frictional drag and centripetal forces on an incremental segment of

photoresist at a distance r from the spinning axis with radial speed component vr and height

z above the wafer surface:

(7.9)

η

ρω

∂

∂

=

2

2

2

v

z

r

r

where the wafer is spun at angular frequency ω, η is the viscosity, and ρ is the density.

Assuming no photoresist is created or destroyed indicates

∂

∂^{= −∂(}

)

∂

h

t

r

rQ

r

1

where the flow rate by volume, Q, is given by

(7.10)

Q

v d

h t

r

z

=

( )

∫

0

Assuming zero slip and zero sheer boundary conditions results in

(7.11)

h t

h

h

t

( ) =

( )

+ ( )



^



^













0

2

0

4

3

2

2

ρω

η

Here, we assume an initial uniform thickness of h(0). At long spin times, this approximates to

(7.12)

h t

t

( ) ≈

3

4

2

η

ρω