The oxide-on-silicon stack of the last section (oxide index

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Polarisation and anisotropic media

What an Instrument Can See, Ch The boundary as a two-port

Problem

n_1=1.457

, silicon

\tilde n_2=3.882+0.019i

, vacuum wavelength

\lambda_0=632.8\,\mathrm{nm}

, incidence

70{}^\circ

) is now measured after a longer growth run than the

100.0\,\mathrm{nm}

worked example. The ellipsometer reads

(\Psi,\Delta)=(30.4{}^\circ,\ 263.7{}^\circ)

in this chapter's sign convention. Growth-rate monitoring during deposition places the oxide thickness

d

between

150\,\mathrm{nm}

and

250\,\mathrm{nm}

, a window narrower than the

284\,\mathrm{nm}

period of the

(\Psi,\Delta)

pattern. The two interface coefficients depend only on the angle and the indices, so they are the same at every trial thickness. Air to oxide:

r_{1s}=-0.5300,\quad r_{1p}=+0.2106.

Oxide to silicon:

\begin{aligned} r_{2s}&=-0.5437-0.0018i,\\ r_{2p}&=-0.3546-0.0020i.\end{aligned}

Only the round-trip phase changes with thickness, through

\beta=n_1k_0d\cos\theta_1

(the chapter's layer phase,

k_0=2\pi/\lambda_0

) with

\cos\theta_1=0.7642

from Snell's law; the last section's slab formula then gives, for each polarisation

x=s,p

r_x^{\mathrm{slab}}=\frac{r_{1x}+r_{2x}\,e^{2i\beta}}{1+r_{1x}r_{2x}\,e^{2i\beta}},

and

\rho=r_p^{\mathrm{slab}}/r_s^{\mathrm{slab}}=\tan\Psi\,e^{i\Delta}

as before. ((a)) Evaluate

(\Psi,\Delta)

from this model at the two trial thicknesses

d=200.0\,\mathrm{nm}

and

d=220.0\,\mathrm{nm}

. ((b)) Interpolate between these two trial values to estimate the thickness that would give the measured

\Psi=30.4{}^\circ

, then check that thickness against the measured

\Delta

. ((c)) State the oxide's thickness, and explain why the

284\,\mathrm{nm}

periodicity leaves no second possible answer once the growth-rate estimate is used.