PX4131/PXT301 Physical Constants.
Physical Constants.
PX4131/PXT301Physical Constants.
2
Useful formulae. Note that many of these equations are specific to their particular applications.
Variables may mean different things depending on the context.
= + ×
() = (0) −
2
+
= exp (
2
)
=
(∗ − )
0
= −
//
∗
=
Δ//
Δ∗
=
= 2//
2 1 +
1 −
sin =
=
−1
=
0.6
= 0.9 (
)
3
2
1 =
2 =
2
= {exp (
) − 1}
Γ =
√2
=
Γ
22
()
= −()
= exp (−
)
() = 0 exp(−)
=
1
2
3
=
Δ
=
1.2
√
=
ln
ln
= |log (
2
1
)|
−1
=
2 sin
() =
1
(1 + )
(
1
2
exp [−
2
2
] +
2
exp [−
2
2
])
< > =
2
( + )2
Δ
=
< >
4
√
8
=
0
≈ (
)
4
PX4131/PXT301
3
1. GaAs InP
Crystal structure zinc-blende zinc-blende
Lattice constant (nm) 0.565 0.587
Poisson’s ratio 0.31 0.36
(a) Molecular beam epitaxy can be used to grow thin films of a semiconductor. Briefly explain
the molecular beam epitaxy procedure. Briefly outline the advantages of molecular beam
epitaxy over other epitaxial growth methods such as MOCVD.
[3]
(b) What is the definition of the critical thickness when growing an epitaxial layer?
[1]
(c) Refer to the table above.
(i) Briefly explain what a zinc-blende structure is.
[1]
(ii) Calculate the length of the Ga-As bond.
[1]
(iii) Calculate the length of the In-P bond.
[1]
(d) What is the lattice mismatch between InP and GaAs?
[1]
(e) In the epitaxial growth of InP on a GaAs substrate, the indium K-cell is maintained at a
temperature of 1000°C. A calibration procedure measures the mass loss as Δ = 30 mg
over a duration Δ of 6 hours when the circular aperture is opened to a diameter d of
2.0 mm. Note: the atomic mass of indium is = 114.8 g/mol.
(i) Given that the total mass loss is Δ = ΓΔ (where Γ is the effusion flux and is the
mass of one indium atom), show that the partial pressure of indium in the K-cell is
given by
=
4Δ
2Δ
√
2
where = 8.314 J/mol/K is the universal gas constant. Use this formula to estimate
the partial pressure of indium in the K-cell.
[5]
(ii) Estimate the indium atom flux for a sample placed 8 cm away. What are the
assumptions involved in your calculation?
[3]
(iii) Assuming pseudomorphic growth, estimate the growth rate.
[6]
(iv) Comment on this growth rate. How would you increase the growth rate? What are
the limits?
[3]
PX4131/PXT301
4
2. (a) Define what is meant by the sensitivity of a resist used in electron beam lithography.
Estimate the dwell time or exposure time for a pattern with a total exposed area of 3 cm2 if
the beam current is 0.2 A and the resist sensitivity is 25 C/cm2.
[3]
(b) A certain device requires a minimum feature size of 13 nm. If the electron beam has a
diameter at the focus of 5 nm and an energy of 30 keV, what is the maximum resist
thickness that can be used?
[6]
(c) The point scattering function describing the two dominant electron scattering processes in
a resist is
() =
1
(1 + )
(
1
2
exp [−
2
2
] +
2
exp [−
2
2
])
For a certain resist/substrate combination, the parameters in the above function are =
93 nm, = 1.306 m and = 1.4.
Name and briefly describe the two processes.
Which of the two processes contributes the most to the proximity effect?
[5]
(d) A device requires two identical small features to be written in close proximity, with a
300 nm centre-to-centre separation.
The proximity effect correction can be framed as a set of simultaneous equations in matrix
form:
(
1
2
) = (
11 12
21 22
) (
1
2
)
1,2 is the intended exposure dose to each feature for correct writing.
1,2 is the corrected exposure dose to adjust for the proximity effect.
are the proximity interaction matrix elements.
Assuming each of the features is written in a point-like exposure, give an expression for,
and calculate the values of, 11, 12, 21 and 22.
[5]
(e) By solving the simultaneous equation in part (d), or otherwise, show that the dose factor
(relative to the uncorrected dose 0) required for each of the features to correct for the
proximity effect is given by
1
0
=
11
11 + 12
Use your values of 11 and 12 from above to give a value for the proximity effect corrected
dose factors.