Problem $1.$ You juat got a new position at a company called SemiSensing, where you are now a

junior engineer in the chemical vapor depoaition $/$ epitaxy division. Because you have a degree

in Chemical Engineering from UCR, they are going to ask you to modify the setup within the first

six months with the possibility of promotion when the division head retires. This first week you

are asaigned to shadow the other engineer and get a feel for the process. They make sensors

out of p$-$type semiconductors $($also known as transiators$).$ SemiSensor builds the chip itself, by

doping B onto$/$into a Si substrate. A layer of $($elemental$)$ boron acoumulates on the

mirror$-$polished silicon substrate in a small cell and diffuses into the Si lattice, replacing some Si

with $B$ and thus oreating the desired material properties. The application is deteotion of

acetaminophen in blood, something that SemiSenaing has promised to deliver to Regia

Perennial, a large health system in your region.

$($a$)$ The boron reaches ita maximum concentration within the ailicon lattice at the surface, where

boron is continually deposited from the gas phase. We are concerned with its time$-$dependent

diffusion into the wafer. The "solubility" of boron in the ailicon lattice $($the value reached at the

gas$/$solid boundary$)$ is $0\mathrm{.}83029M.$ Convert this concentration $($given in $M,$ or mol$/$L$,$ which is

used normally for liquids$),$ to atoms of elemental boron per $c{m}^{{?}^{{?}^{?}}}3$ of Si lattice. You ahould get a

high number. Round to $1$ significant figure. This is your "surface" boundary oondition for boron

concentration.

b$)$ The wafer needs an adequate concentration of boron impurity throughout the first $0\mathrm{.}2m$ of

depth from the surface. Reaching a larger concentration near the aurface is not the iasue. The

issue is$,$ boron is alow to diffuse and you need to know exactly how long to wait to ensure the

quality of the chip, which is inserted into the cell with no pre$-$exiating boron. The minimum

concentration at $0\mathrm{.}2m$ is $2e19$ atoms B per $c{m}^{{?}^{{?}^{?}}}3Si.$ The density of the Si lattice is $1e22$ atoms

per $c{m}^{{?}^{{?}^{?}}}3.$ Identify the variables and boundary conditions given and write out the appropriate

equation.

c$)$ You need to make two batches of chips per $1\mathrm{.}5$ hours. Half an hour is set aside for 'aetup.$\text{'}$

leaving you with about $1$ hour of actual diffusion time. How long does boron have to diffuse into

the silicon in each batch?

d$)$ You have a few choices when it comea to the operating temperature of the aetup, and the

choice atrongly influences the diffuaivity of $B$ in $Si.$ The diffuaivity is given by

DAB$=D,O^{**}$exp$(-Q\frac{O}{RT}),$ where $DO=0\mathrm{.}019c{m}^{{?}^{{?}^{?}}}\frac{2}{s}$ and $Q0=2\mathrm{.}7\frac{J}{m}ol.$ Limiting the diffusion

time to $30$ minutes, at what temperature $($in Kelvin$)$ ahould you operate the reactor to achieve the

minimum concentration, at a depth of $0\mathrm{.}2m,$ of $2e19$ atoma $B$ per $c{m}^{{?}^{{?}^{?}}}3Si?$

e$)$ Conaider the implications of inadequate sensor accuracy. Discusa the sources of error in this

manufacturing process and the potential ramifications of certifying faulty aensors. What kinds

of statistica will you include with the calibration certificate that your sensor ships with? $($Short

answer$)$