Rapid measurement method for ultra-thin film optical constant

Abstract

The invention discloses a rapid measurement method for an ultra-thin film optical constant, which includes following steps: S1: using a p-light amplitude reflection coefficient r.sub.p and an s-light amplitude reflection coefficient r.sub.s of an incident light irradiating to an ultra-thin film to be measured to express an amplitude reflection coefficient ratio ρ of the ultra-thin film: $ρ = \frac{r_{p}}{r_{s}};$
S2: performing a second-order Taylor expansion to $ρ = \frac{r_{p}}{r_{s}}$
at d.sub.f=0 while taking 2πd.sub.f/λ as a variable to obtain a second-order approximation form; S3: performing merging, simplifying and substituting processing to the second-order approximation form for transforming the same into a one-variable quartic equation; S4: solving the one-variable quartic equation to obtain a plurality of solutions of the optical constant of the ultra-thin film, and obtaining a correct solution through conditional judgment, so as to achieve the rapid measurement for the ultra-thin film optical constant.

Claims

1. A rapid measurement method for an ultra-thin film optical constant, comprising: S1: irradiating an incident light to an ultra-thin film disposed on a substrate, wherein the ultra-thin film is located between a source of the incident light and the substrate, controlling instruments to measure an incident angle of the incident light and a refraction angle of the incident light transmitted to the substrate, calculating, a p-light amplitude reflection coefficient r.sub.p and an s-light amplitude reflection coefficient r.sub.s of the incident light irradiating to the ultra-thin film, and then obtaining a measured amplitude reflection coefficient ratio ρ of the ultra-thin film: $ρ = \frac{r_{p}}{r_{s}};$ S2: performing an approximation form of ρ to satisfy a following equation: $ρ \approx ρ_{0} + i ρ^{'} \frac{2 π d_{f}}{λ} + (ρ_{1}^{′′} + ρ_{2}^{′′}) {(\frac{2 π d_{f}}{λ})}^{2}$ where ρ.sub.0 is an amplitude reflection coefficient ratio of the substrate, d.sub.f is a thickness of the ultra-thin film, λ is a wavelength of the incident light, and ρ′, ρ.sub.1“and ρ.sub.2” are all coefficients; S3: transforming the approximation form of ρ into a one-variable quartic equation; and S4: controlling the instruments to measure measured-ellipsometric parameters of the ultra-thin film, and then completing the rapid measurement for the ultra-thin film optical constant to be measured by obtaining a correct solution of the one-variable quartic equation defined by the measured amplitude reflection coefficient ratio ρ, wherein the ultra-thin film optical constant to be measured satisfies following conditions: the correct solution is the optical constant of the ultra-thin film to be measured; according to a degree of fitness, ellipsometric parameters of the correct solution is consistent with the measured-ellipsometric parameters of the ultra-thin film, wherein the ellipsometric parameters of the correct solution is calculated by a Fresnel equation; the correct solution is one of a plurality of solutions of the one-variable quartic equation; and the correct solution satisfies a conditional judgment, wherein the conditional judgment comprises: excluding optical constant solutions of the one-variable quartic equation that do not satisfy physical conditions.

2. The rapid measurement method the ultra-thin film optical constant as claimed in claim 1, wherein the p-light amplitude reflection coefficient r.sub.p is calculated according to a following equation: $r_{p} = \frac{n_{s u b} \cos α_{inc} - n_{0} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} (n_{f}^{2} \cos α_{inc} \cos α_{t r a} - n_{0} n_{s u b} + n_{0} n_{s u b}^{3} \sin^{2} α_{t r a} / n_{f}^{2})}{n_{s u b} \cos α_{inc} + n_{0} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} (n_{f}^{2} \cos α_{inc} \cos α_{t r a} + n_{0} n_{s u b} - n_{0} n_{s u b}^{3} \sin^{2} α_{t r a} / n_{f}^{2})}$ wherein n.sub.sub is an optical constant of the substrate used by the ultra-thin film, α.sub.inc is the incident angle of the incident light, no is an optical constant of a surrounding medium of the ultra-thin film, α.sub.tra is the refraction angle of the incident light transmitted to the substrate, d.sub.f is a thickness of the ultra-thin film, λ is a wavelength of the incident light, and n.sub.f is an optical constant of the ultra-thin film to be measured.

3. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 1, wherein the s-light amplitude reflection coefficient r.sub.s is calculated according to a following equation: $r_{s} = \frac{\begin{matrix} n_{0} \cos α_{inc} - n_{s u b} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} \\ (n_{0} n_{s u b} \cos α_{i n c} \cos α_{t r a} + n_{s u b}^{2} \cos α_{i n c}^{2} - n_{f}^{2}) \end{matrix}}{\begin{matrix} n_{0} \cos α_{i n c} + n_{s u b} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} \\ (n_{0} n_{s u b} \cos α_{i n c} \cos α_{t r a} - n_{s u b}^{2} \cos α_{i n c}^{2} + n_{f}^{2}) \end{matrix}}$ where n.sub.sub is an optical constant of the substrate used by the ultra-thin film, α.sub.inc is the incident angle of the incident light, no is an optical constant of a surrounding medium of the ultra-thin film, α.sub.tra is the refraction angle of the incident light transmitted to the substrate, d.sub.f is a thickness of the ultra-thin film, λ is a wavelength of the incident light, and n.sub.f is an optical constant of the ultra-thin film to be measured.

4. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 1, wherein ρ′ is calculated according to a following equation: $ρ^{'} = - 2 \frac{n_{0}}{n_{s u b}^{2} - n_{0}^{2}} \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} \frac{(n_{f}^{2} - n_{0}^{2}) .Math. (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}$ where n.sub.0 is an optical constant of a surrounding medium of the ultra-thin film, a.sub.inc is the incident angle of the incident light, of is an optical constant of the ultra-thin film to be measured, n.sub.sub is an optical constant of the substrate used by the ultra-thin film, and α.sub.tra is the refraction angle of the incident light transmitted to the substrate.

5. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 1, wherein ρ.sub.1″ is calculated according to a following equation: $ρ_{1}^{″} = - 2 \frac{n_{0} n_{s u b}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos^{2} α_{i n c}}{\cos^{3} (α_{i n c} - α_{tra})} {(\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}})}^{2}$ where n.sub.0 is an optical constant of a surrounding medium of the ultra-thin film, α.sub.inc is the incident angle of the incident light, of is an optical constant of the ultra-thin film to be measured, n.sub.sub is an optical constant of the substrate used by the ultra-thin film, and α.sub.tra is the refraction angle of the incident light transmitted to the substrate.

6. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 1, wherein ρ.sub.2″ is calculated according to a following equation: $ρ_{2}^{″} = - 2 \frac{n_{0}}{n_{sub}} \frac{\cos α_{i n c}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}) (\frac{(n_{f}^{2} - n_{0}^{2}) n_{s u b}^{4} + (n_{f}^{2} - n_{s u b}^{2}) n_{f}^{2} n_{0}^{2}}{n_{f}^{2}})$ where n.sub.0 is an optical constant of a surrounding medium of the ultra-thin film, α.sub.inc is the incident angle of the incident light, n.sub.f is an optical constant of the ultra-thin film to be measured, n.sub.sub is an optical constant of the substrate used by the ultra-thin film, and α.sub.tra is the refraction angle of the incident light transmitted to the substrate.

7. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 1, wherein the step S3 of performing merging, simplifying and substituting processing to the second-order approximation form for transforming the same into the one-variable quartic equation comprises: S31: letting: $A = - 2 \frac{n_{0}}{n_{s u b}^{2} - n_{0}^{2}} \frac{\sin^{2} α_{i n c} .Math. \cos α_{inc}}{\cos^{2} (α_{inc} - α_{t r a})} B = - 2 \frac{n_{0} n_{s u b}}{{(n_{s u b} - n_{0})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos^{2} α_{i n c}}{\cos^{3} (α_{i n c} - α_{t r a})} C = - 2 \frac{n_{0}}{n_{s u b}} \frac{\cos α_{i n c}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} D = \frac{2 π d_{f}}{λ}$ S32: simplifying the second-order approximation form into: $ρ = ρ_{0} + i A D \frac{(n_{f}^{2} - n_{0}^{2}) .Math. (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}} + B {D^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}})}^{2} + {CD}^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}) \frac{(n_{f}^{2} - n_{0}^{2}) n_{s u b}^{4} + (n_{f}^{2} - n_{s u b}^{2}) n_{f}^{2} n_{0}^{2}}{n_{f}^{2}}$ S33: expanding and simplifying the equation of the step S32 to obtain a one-variable quartic equation related to n.sub.f.sup.2:
para4.Math.N.sub.f.sup.4+para3.Math.N.sub.f.sup.3+para2.Math.N.sub.f.sup.2+para1.Math.N.sub.f+para0=0 where N.sub.f=n.sub.f.sup.2, para0, para1, para2, para3 and para4 are respectively a constant term, a one-order term coefficient, a quadratic term coefficient, a cubic term coefficient and a quartic term coefficient.

8. The rapid measurement method for the ultra-thin film optical constant as claimed in claim 7, wherein the constant term para0, the one-order term coefficient para1, the quadratic term coefficient para2, the cubic term coefficient para3 and the quartic term coefficient para4 are represented as:
para0=BD.sup.2n.sub.0.sup.4n.sub.sub.sup.4−CD.sup.2n.sub.0.sup.4n.sub.sub.sup.6
para1=iADn.sub.0.sup.2n.sub.sub.sup.2−2BD.sup.2n.sub.0.sup.4n.sub.sub.sup.2−2BD.sup.2n.sub.0.sup.2n.sub.sub.sup.4+2CD.sup.2n.sub.0.sup.2n.sub.sub.sup.6
para2=−iADn.sub.0.sup.2+BD.sup.2n.sub.0.sup.4−iADn.sub.sub.sup.2+4BD.sup.2n.sub.0.sup.2n.sub.sub.sup.2+2CD.sup.2n.sub.0.sup.4n.sub.sub.sup.2+BD.sup.2n.sub.sub.sup.4−CD.sup.2n.sub.0.sup.2n.sub.sub.sup.4−CD.sup.2n.sub.sub.sup.6−ρ+ρ.sub.0
para3=iAD−2BD.sup.2n.sub.0.sup.2−CD.sup.2n.sub.0.sup.4−2BD.sup.2n.sub.sub.sup.2−2CD.sup.2n.sub.0.sup.2n.sub.sub.sup.2+CD.sup.2n.sub.sub.sup.4
para4=BD.sup.2+CD.sup.2n.sub.0.sup.2.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The accompanying drawings are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification. The drawings illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention.

(2) FIG. 1 is a flowchart illustrating a rapid measurement method for an ultra-thin film optical constant according to an embodiment of the invention.

(3) FIG. 2 illustrates spectrum curves of optical constants in 250-1690 nm waveband of a silicon substrate according to an embodiment of the invention, where (a) is reflective index, and (b) is extinction coefficient.

(4) FIG. 3 is a schematic diagram of an optical model of a GaAs film sample on the silicon substrate according to an embodiment of the invention.

(5) FIG. 4 illustrates spectrum curves of ellipsometric parameters in 250-1690 nm waveband of the silicon substrate according to an embodiment of the invention, where (a) is amplitude ratio, and (b) is phase difference.

(6) FIG. 5 illustrates spectrum curves of GaAs film ellipsometric parameters in 250-1690 nm waveband on the silicon substrate according to an embodiment of the invention, where (a) is amplitude ratio, and (b) is phase difference.

(7) FIG. 6 illustrates spectrum curves of calculated GaAs optical constants (two sets) in 250-1690 nm waveband according to an embodiment of the invention, where (a) is reflective index, and (b) is extinction coefficient.

(8) FIG. 7 illustrates spectrum curves of ellipsometric parameters obtained through inversion of two sets of optical constant solutions and ellipsometric parameters obtained through measurement in 250-1690 nm waveband according to an embodiment of the invention, where (a) is amplitude ratio, and (b) is phase difference.

(9) FIG. 8 illustrates spectrum curves of calculated GaAs optical constants (unique correct solution) in 250-1690 nm waveband according to an embodiment of the invention, where (a) is reflective index, and (b) is extinction coefficient.

DESCRIPTION OF THE EMBODIMENTS

(10) Reference will now be made in detail to the present preferred embodiments of the invention, examples of which are illustrated in the accompanying drawings. It should be understood that the specific embodiments described herein are merely used for explaining the invention and are not intended to be limiting of the invention. Furthermore, the technical features involved in the various embodiments of the invention described below may be combined with each other as long as they do not conflict with each other.

(11) As shown in FIG. 1, an embodiment of the invention provides a rapid measurement method for an ultra-thin film optical constant, which includes following steps:

(12) S1: obtaining a p-light amplitude reflection coefficient r.sub.p and an s-light amplitude reflection coefficient r.sub.s of a light source (incident light) incident to an ultra-thin film to be measured, and using r.sub.p and r.sub.s to express an amplitude reflection coefficient ratio

(13) $ρ = \frac{r_{p}}{r_{s}} .$

(14) To be specific, the p-light amplitude reflection coefficient r.sub.p is calculated according to a following equation:

(15) $r_{p} = \frac{\begin{matrix} n_{s u b} \cos α_{inc} - n_{0} \cos α_{tra} + i \frac{2 π d_{f}}{λ} \\ (n_{f}^{2} \cos α_{inc} \cos α_{tra} - n_{0} n_{s u b} + n_{0} n_{s u b}^{3} \sin^{2} α_{t r a} / n_{f}^{2}) \end{matrix}}{\begin{matrix} n_{s u b} \cos α_{inc} + n_{0} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} \\ (n_{f}^{2} \cos α_{inc} \cos α_{t r a} + n_{0} n_{s u b} - n_{0} n_{s u b}^{3} \sin^{2} α_{tra} / n_{f}^{2}) \end{matrix}}$

(16) Where n.sub.sub is an optical constant of a substrate used by the ultra-thin film (with a thickness smaller than 10 nm), α.sub.inc is an incident angle of the incident light, n.sub.0 is an optical constant of a surrounding medium of the ultra-thin film, α.sub.tra is a refraction angle of the incident light transmitted to the substrate (a refraction angle of the incident light on the substrate), d.sub.f is a thickness of the ultra-thin film, λ is a wavelength of the incident light, and n.sub.f is an optical constant of the ultra-thin film.

(17) Further, the s-light amplitude reflection coefficient r.sub.s is calculated according to a following equation:

(18) $r_{s} = \frac{\begin{matrix} n_{0} \cos α_{inc} - n_{s u b} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} \\ (n_{0} n_{s u b} \cos α_{i n c} \cos α_{t r a} + n_{s u b}^{2} \cos α_{i n c}^{2} - n_{f}^{2}) \end{matrix}}{\begin{matrix} n_{0} \cos α_{i n c} + n_{s u b} \cos α_{t r a} + i \frac{2 π d_{f}}{λ} \\ (n_{0} n_{s u b} \cos α_{i n c} \cos α_{t r a} - n_{s u b}^{2} \cos α_{i n c}^{2} + n_{f}^{2}) \end{matrix}}$

(19) S2: performing a second-order Taylor expansion to

(20) $ρ = \frac{r_{p}}{r_{s}}$
at d.sub.f=0 while taking 2πd.sub.f/λ as a variable to obtain a second-order approximation form:

(21) $ρ \approx ρ_{0} + i ρ^{'} \frac{2 π d_{f}}{λ} + (ρ_{1}^{″} + ρ_{2}) {(\frac{2 π d_{f}}{λ})}^{2}$

(22) Where ρ.sub.0 is an amplitude reflection coefficient ratio of the bare substrate, d.sub.f is a thickness of the ultra-thin film, λ is a wavelength of the incident light, and ρ′, ρ.sub.1″ and ρ.sub.2″ are respectively coefficients.

(23) To be specific, ρ′ is calculated according to a following equation:

(24) $ρ^{'} = - 2 \frac{n_{0}}{n_{s u b}^{2} - n_{0}^{2}} \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} \frac{(n_{f}^{2} - n_{0}^{2}) .Math. (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}$

(25) Where n.sub.0 is an optical constant of the surrounding medium of the ultra-thin film, α.sub.inc is an incident angle of the incident light, n.sub.f is an optical constant of the ultra-thin film, n.sub.sub is an optical constant of the substrate used by the ultra-thin film, and α.sub.tra is a refraction angle of the incident light transmitted to the substrate.

(26) Further, ρ.sub.1″ is calculated according to a following equation:

(27) $ρ_{1}^{″} = - 2 \frac{n_{0} n_{s u b}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos^{2} α_{i n c}}{\cos^{3} (α_{i n c} - α_{tra})} {(\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}})}^{2}$

(28) Further, ρ.sub.2″ is calculated according to a following equation:

(29) 0 $ρ_{2}^{″} = - 2 \frac{n_{0}}{n_{s u b}} \frac{\cos α_{i n c}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}) \frac{(n_{f}^{2} - n_{0}^{2}) n_{s u b}^{4} + (n_{f}^{2} - n_{s u b}^{2}) n_{f}^{2} n_{0}^{2}}{n_{f}^{2}}$

(30) S3: performing merging, simplifying and substituting processing to the second-order approximation form to transform the same into a one-variable quartic equation, which includes following sub-steps:

(31) S31: letting:

(32) $A = - 2 \frac{n_{0}}{n_{s u b}^{2} - n_{0}^{2}} \frac{\sin^{2} α_{i n c} .Math. \cos α_{inc}}{\cos^{2} (α_{inc} - α_{t r a})} B = - 2 \frac{n_{0} n_{s u b}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos^{2} α_{i n c}}{\cos^{3} (α_{i n c} - α_{t r a})} C = - 2 \frac{n_{0}}{n_{s u b}} \frac{\cos α_{i n c}}{{(n_{s u b}^{2} - n_{0}^{2})}^{2}} .Math. \frac{\sin^{2} α_{i n c} .Math. \cos α_{i n c}}{\cos^{2} (α_{i n c} - α_{t r a})} D = \frac{2 π d_{f}}{λ}$

(33) S32: introducing the above A-D to the second-order approximation form to obtain:

(34) $ρ = ρ_{0} + i A D \frac{(n_{f}^{2} - n_{0}^{2}) .Math. (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}} + B {D^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}})}^{2} + {CD}^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}}) \frac{(n_{f}^{2} - n_{0}^{2}) n_{s u b}^{4} + (n_{f}^{2} - n_{s u b}^{2}) n_{f}^{2} n_{0}^{2}}{n_{f}^{2}}$

(35) S33: expanding and simplifying the equation of the step S32 to obtain a one-variable quartic equation related to n.sub.f.sup.2: para4.Math.N.sub.f.sup.4+para3.Math.N.sub.f.sup.3+para2.Math.N.sub.f.sup.2+para1.Math.N.sub.f+para0=0, specifically:

(36) Letting:

(37) $N_{f} = n_{f}^{2} and ρ = ρ_{0} + i A D \frac{(n_{f}^{2} - n_{0}^{2}) .Math. (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}} + B {D^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{s u b}^{2})}{n_{f}^{2}})}^{2} + C D^{2} (\frac{(n_{f}^{2} - n_{0}^{2}) (n_{f}^{2} - n_{0}^{2})}{n_{f}^{2}}) \frac{(n_{f}^{2} - n_{0}^{2}) n_{s u b}^{4} + (n_{f}^{2} - n_{s u b}^{2}) n_{f}^{2} n_{0}^{2}}{n_{f}^{2}}$
simplifying
to obtain:
para4.Math.N.sub.f.sup.4+para3.Math.N.sub.f.sup.3+para2.Math.N.sub.f.sup.2+para1.Math.N.sub.f+para0=0

(38) Where, para0 is a constant term:
para0=BD.sup.2n.sub.0.sup.4n.sub.sub.sup.4−CD.sup.2n.sub.0.sup.4n.sub.sub.sup.6

(39) para1 is a one-order term coefficient:
para1=iADn.sub.0.sup.2n.sub.sub.sup.2−2BD.sup.2n.sub.0.sup.4n.sub.sub.sup.2−2BD.sup.2n.sub.0.sup.2n.sub.sub.sup.4+2CD.sup.2n.sub.0.sup.2n.sub.sub.sup.6

(40) para2 is a quadratic term coefficient:
para2=−iADn.sub.0.sup.2+BD.sup.2n.sub.0.sup.4−iADn.sub.sub.sup.2+4BD.sup.2n.sub.0.sup.2n.sub.sub.sup.2+2CD.sup.2n.sub.0.sup.4n.sub.sub.sup.2+BD.sup.2n.sub.sub.sup.4−CD.sup.2n.sub.0.sup.2n.sub.sub.sup.4−CD.sup.2n.sub.sub.sup.6−ρ+ρ.sub.0.sup.4

(41) para3 is a cubic term coefficient:
para3=iAD−2BD.sup.2n.sub.0.sup.2−CD.sup.2n.sub.0.sup.4−2BD.sup.2n.sub.sub.sup.2−2CD.sup.2n.sub.0.sup.2n.sub.sub.sup.2+CD.sup.2n.sub.sub.sup.4

(42) para4 is a quartic term coefficient:
para4=BD.sup.2+CD.sup.2n.sub.0.sup.2

(43) S4: solving the one-variable quartic equation para4.Math.N.sub.f.sup.4+para3.Math.N.sub.f.sup.3+para2.Math.N.sub.f.sup.2+para1.Math.N.sub.f+para0=0 to obtain 8 solutions of the optical constant of the ultra-thin film, and obtaining a correct solution through conditional judgment, where the correct solution is the optical constant of the ultra-thin film to be measured, so as to achieve the rapid measurement for the ultra-thin film optical constant. To be specific, a Ferrari method is adopted to get 4 solutions for the one-variable quartic equation related to N.sub.f, and after the square root, 8 solutions of n.sub.f is obtained.

(44) To be specific, the conditional judgment is performed in a following method to obtain the correct solution:

(45) S41: excluding optical constant solutions according to physical conditions:

(46) A refractive index n and an extinction coefficient k are required to satisfy following physical conditions: n=Re(n.sub.f)>0 and k=−Im(n.sub.f)>0, where Re(n.sub.f) represents a real part of the optical constant n.sub.f, Im(n.sub.f) represents an imaginary part of the optical constant n.sub.f, the corresponding refractive index n (n=Re (n.sub.f)) and the extinction coefficient k (k=−Im (n.sub.f)) are obtained according to real parts and imaginary parts of each of the optical constant solutions, and then most of the 8 solutions are excluded according to aforementioned physical conditions;

(47) S42: excluding the remaining optical constant solutions by using a Fresnel equation:

(48) The remaining optical constant solutions are introduced into the Fresnel equation to calculate ellipsometric parameters of the ultra-thin film, and determining which one is the correct solution according to a degree of fitness between the calculated ellipsometric parameters and ellipsometric parameters of the ultra-thin film obtained through the measurement, so as to obtain the unique correct solution. To be specific, the ellipsometric parameters of the ultra-thin film obtained through calculation and the ellipsometric parameters of the ultra-thin film obtained through the measurement are compared, and the solution corresponding to the ellipsometric parameter with the highest degree of fitness is the correct solution.

(49) Before the method of the invention is used to measure the ultra-thin film optical constant n.sub.f, following parameters are required to be determined: the surrounding medium of the ultra-thin film n.sub.0, the optical constant of the substrate used by the ultra-thin film n.sub.sub, the incident angle α.sub.inc of the incident light, the amplitude reflection coefficient ratio ρ.sub.0 of the substrate, the amplitude reflection coefficient ρ of the ultra-thin film, the thickness d.sub.f of the ultra-thin film, the wavelength λ of the incident light and the refraction angle α.sub.tra of the incident light transmitted to the substrate.

(50) To be specific, the optical constants of the thin film may vary along with a change of the wavelength, and measured wavelength ranges of different instruments are also different, so that an appropriate measurement wavelength range may be selected according to the measurement requirements and equipment conditions, for example, a single wavelength or spectrum measurement, and then the optical constant n.sub.0 of the surrounding medium and the optical constant n.sub.sub of the substrate used by the ultra-thin film in the measurement wavelength range may be obtained, and in general, the surrounding medium in measurement is air, i.e. n.sub.0=1. The optical constant of the substrate may be measured by an ellipsometer or other instrument, and if optical properties of the substrate are stable, known data in literatures may also be used. There is no specific requirement on the incident angle α.sub.inc of the incident light, which may be selected according to actual needs to ensure validity of the measured data, for example, 60°, 65° or 70°.

(51) Regarding the amplitude reflection coefficient ratio ρ.sub.0 of the substrate, a calculation equation thereof is: ρ.sub.0=tan(ψ.sub.sub)exp(iΔ.sub.sub), and in the actual operation, the ellipsometric parameters of the substrate: an amplitude ratio Ψ.sub.sub and a phase difference Δ.sub.sub are obtained by using a general ellipsometer, an imaging ellipsometer, a Mueller matrix ellipsometer or other instruments capable of obtaining the ellipsometric parameter information of the sample, and after the ellipsometric parameters Ψ.sub.sub and Δ.sub.sub are obtained, the equation ρ.sub.0=tan(Ψ.sub.sub)exp(iΔ.sub.sub) is used to calculate the amplitude reflection coefficient ratio ρ.sub.0 of the substrate.

(52) Regarding the amplitude reflection coefficient ratio ρ of the ultra-thin film in the final one-variable quartic equation, a calculation equation thereof is ρ=tan(Ψ.sub.f)exp(iΔ.sub.f), and in the actual operation, the ellipsometric parameters of the ultra-thin film: an amplitude ratio Ψ.sub.f and a phase difference Δ.sub.f are obtained by using a general ellipsometer, an imaging ellipsometer, a Mueller matrix ellipsometer or other instruments capable of obtaining the ellipsometric parameter information of the sample, and after the ellipsometric parameters Ψ.sub.f and Δ.sub.f are obtained, the equation ρ=tan(Ψ.sub.f)exp(iΔ.sub.f) is used to calculate the amplitude reflection coefficient ratio ρ of the ultra-thin film.

(53) The thickness d.sub.f of the ultra-thin film may be determined according to measurement approaches of AFM, SEM or the like. Regarding the wavelength λ of the incident light, a range thereof is determined according to an actual requirement. To be specific, the method of the invention may implement calculation of the optical constants of two-dimensional materials from an ultraviolet band to an infrared band, and implement calculation of the optical constants of ordinary ultra-thin films (with a thicknesses less than 10 nm) from a partial visible light band to the infrared band can be calculated. The refraction angle α.sub.tra of the incident light transmitted to the substrate is calculated according to a following equation:
n.sub.0 sin(α.sub.inc)=n.sub.sub sin(α.sub.tra)

(54) An embodiments of the invention is provided below, and in the embodiment, a spectroscopic ellipsometer is used to measure a GaAs film with a thickness of 3 nm (d.sub.f=3 nm) on a silicon substrate, and the optical constants of the ultra-thin film are rapidly calculated by introducing the ellipsometric parameters of the substrate and the thin film to an optical constant solving process.

(55) First, various parameters are obtained, and the spectroscopic ellipsometer is adopted in the embodiment, the measurement wavelength range is Γ=[250, 1690] nm, and the optical constant n.sub.0 of the surrounding medium of the ultra-thin film and the optical constant n.sub.sub of the substrate in the wavelength range of [250, 1690] nm are obtained, and in the measurement, the surrounding medium is air n.sub.0=1, the substrate is made of silicon, optical constants of the silicon is relatively stable, and the optical constants of the silicon substrate are shown in FIG. 2; the measured incident angle α.sub.inc=65° is selected; the refraction angle α.sub.tra(λ) of the incident light transmitted to the silicon substrate is calculated according to the equation: n.sub.0 sin(α.sub.inc)=n.sub.sub(λ)sin(α.sub.tra(λ)); the ellipsometric parameters Ψ.sub.sub(λ) and Δ.sub.sub(λ) of the silicon substrate are measured, and a measured optical model is shown in FIG. 3, the measured ellipsometric parameters of the silicon substrate is as shown in FIG. 4, and after the spectroscopic ellipsometer is adopted to obtain data, the amplitude reflection coefficient ratio ρ.sub.0 of the substrate is calculated: ρ.sub.0=ρ.sub.sub(λ)=tan(Ψ.sub.sub(λ))exp(iΔ.sub.sub(λ)); the ellipsometric parameters Ψ.sub.f(λ) and Δ.sub.f(λ) of the GaAs film are calculated, the spectroscopic ellipsometer is used the same as the previous step, and a measurement result is shown in FIG. 5, and after data is obtained, the amplitude reflection coefficient ratio ρ of the GaAs ultra-thin film is calculated: ρ=ρ.sub.f(λ)=tan(Ψ.sub.f(λ))exp(iΔ.sub.f(λ))

(56) Then, the aforementioned obtained various parameters are introduced into the one-variable quartic equation para4.Math.N.sub.f.sup.4+para3.Math.N.sub.f.sup.3+para2.Math.N.sub.f.sup.2+para1.Math.N.sub.f+para0=0 to calculate the optical constants n.sub.f(λ) of the GaAs film. 8 optical constants of the GaAs film are obtained according to the above calculation process, and positive and negative of the refractive index and of the extinction coefficient are used to exclude 6 of the 8 solutions, and the remained two solutions (i.e. two sets of optical constants) are as shown in FIG. 6, and two sets of ellipsometric parameters of the ultra-thin film are obtained by inversion of the remained two sets of optical constants introduced into the Fresnel equation, and then the calculated ellipsometric parameters are compared with measurement values (i.e. the measured ellipsometric parameters of the ultra-thin film), as shown in FIG. 7, and the solution corresponding to the ellipsometric parameter having the highest degree of fitness with the measurement value is selected as the correct solution of the optical constant of the GaAs film, as shown in FIG. 8.

(57) In the aforementioned embodiment, the ellipsometer is adopted for measurement, for example, a laser ellipsometer is used to implement calculation of ultra-thin film single waveform optical constants. In the aforementioned calculation process, only the GaAs film on the silicon substrate is taken as an example for description, and for other types of ultra-thin films or different types of substrates, rapid measurement of optical constants may also be performed according to the same method.

(58) In summary, in the rapid measurement method for the ultra-thin film optical constant, by performing Taylor second-order expansion to the amplitude reflection coefficient ratio ρ, a second-order equation is adapted to achieve an approximation of the original amplitude reflection coefficient ratio ρ, at the same time, a non-linear equation that is originally unable to obtain analytical solutions is transformed into a one-variable quartic equation, so as to obtain the analytical solutions of the ultra-thin film optical constant to achieve rapid measurement and calculation of the ultra-thin film optical constant.

(59) It will be apparent to those skilled in the art that various modifications and variations can be made to the disclosed embodiments without departing from the scope or spirit of the invention, for example, to use other instruments capable of measuring polarization information, or replacing the type of the ultra-thin film or the substrate, etc. In view of the foregoing, it is intended that the invention covers modifications and variations provided they fall within the scope of the following claims and their equivalents.

Rapid measurement method for ultra-thin film optical constant

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G01N21/9501

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G01N21/211

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G01B11/0641

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G01N2021/213

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G01N21/8422

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International classification

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G01B11/06

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Abstract

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