Principles of Ellipsometry
*This article is a continuation of the link below (hereafter referred to as Part 1, Part 2), and the equations (1) to (12) shown in this article refer to the equations in Part 1, Part 2.
Part 1:
Part 2:
So far, we have expanded Snell's law and Fresnel's formula to complex numbers, thereby broadening their scope of application to absorbing media. Applying this principle makes it possible to measure thin film thickness and evaluate optical constants (refractive index, extinction coefficient, etc.), composition ratio, surface roughness, crystallinity, etc. We will now derive the basic principles of ellipsometry, the analytical technology that makes this possible.
・Derivation of the definition formula of ellipsometry
First, we will examine the range of values below that we used in Part 2 .
From (11)-② (in Part 2),
If n2 > n1 , then
Therefore, from (11)-①-3 (in Part 2),
Also, from (11)-①-3 (in Part 2) and (13)-②-1,
Based on (13)-②-1,2,3 and ξi < 0 (∵(11)-③-5 (in Part 2)), when n2 > n1 , (12)-②-1 and (12)-③-1 (in Part 2) can be expressed as follows using the amplitude and argument :
Here, the blue part in the formula indicates that the case distinction has been incorporated into the formula, where the value of θ+ξr is zero when it is less than π/2, and is -π when it is greater than π/2.
And, from (11)-③-5,7 and (11)-④-5,7 (in Part 2), ξr and ξi can be expressed as functions of θ.
By applying this result, the Fresnel amplitude reflection coefficient ratio ρ defined in ellipsometry can be expressed as follows using the phase difference Δ and the amplitude ratio angle ψ (these are called ellipsometry parameters):
However, the domain of the phase difference Δ is set to 0 ≤ Δ ≤ 2π . If the value of (13)-④-2 is outside this range, simply add 2mπ (m is an integer) and select the value of m so that it falls within the domain.
For example, when n=1.44 and κ=5.23, ψ and Δ are calculated using θ as a variable as follows:
・Description of electric field vectors for p- and s-polarized light in absorbing media
Let us now review the electromagnetic wave equation. If the electric field vectors of the p- and s-polarized incident light are Eip and Eis respectively, the electric field amplitude vectors of the p- and s-polarized incident light are EAip and EAis respectively, and the wave vector of the incident light is ki , then from the previous chapter (2)-① (in Part 1), these relationships can be expressed as follows:
Here, the unit vector of the electric field amplitude vector of p-polarized and s-polarized incident light is defined as follows:
In this case, (13)-⑤-1,2 can be rewritten as follows:
Similarly, if the electric field vectors of the p- and s-polarized reflected light are Erp and Ers respectively, the electric field amplitude vectors of the p- and s-polarized reflected light are EArp and EArs respectively, and the wave number vector of the reflected light is kr, then from the previous chapter (2)-① (in Part 1), these relationships can be expressed as follows:
Here, the unit vector of the electric field amplitude vector of p-polarized and s-polarized reflected light is defined as follows:
In this case, from (12)-②-3 and (12)-③-3 (in Part 2), (13)-⑧-1,2 can be rewritten as follows:
・Description and illustration of electric field vectors of p- and s-polarized light in ellipsometry
In ellipsometry, the incident light is set as follows.
In this case, the following holds true from (12)-②-3, (12)-③-3 (in Part 2), and (13)-④-1.
(13)-⑦-1,2 can be rewritten as follows by applying (13)-⑪.
This is linearly polarized light tilted at π /4 (45 degrees) with respect to the p-axis (or s-axis). This state can be achieved by using a linearly polarized laser in monochromatic ellipsometry, or by passing white light through a polarizer in spectroscopic ellipsometry.
Also, (13)-⑩-1,2 can be rewritten as follows by applying (13)-③-1,2, (13)-④-2, and (13)-⑪.
Therefore, when viewed on the time axis, p-polarized light leads Δ/ω ahead of s-polarized light.
Based on the above, ψ and Δ can be illustrated on the time axis as follows:
Here, the electric field of the incident light is shown with the absolute value normalized to 1 in addition to the condition (13)-⑪.
In this way, a phase difference Δ occurs upon reflection of linearly polarized incident light, and the reflected light generally becomes elliptically polarized light.
The polarization changes depending on the value of Δ as shown in the diagram below. When Δ=0,π, the light is linearly polarized, and otherwise it is elliptically polarized, but the direction of rotation (clockwise and counterclockwise) is reversed before and after Δ=0,π.
Furthermore, (13)-⑭-1a and 2a can also be expressed as follows:
Therefore, when viewed on the spatial axis, p-polarized light is delayed by Δ/kr relative to s-polarized light.
Therefore, ψ and Δ can be illustrated on the spatial axis as follows:
Here again, the electric field of the incident light is normalized to 1 in absolute value in addition to the condition (13)-⑪.