Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty

• US 7,286,247 B2
• Filed: 02/22/2005
• Issued: 10/23/2007
• Est. Priority Date: 12/20/2002
• Status: Expired due to Fees

• Current Assignee: International Business Machines Corporation
• Original Assignee: International Business Machines Corporation
• Inventors: Archie, Charles N., Sendelbach, Matthew J., Banke, G. William Jr.
• Primary Examiner(s): Pham; Hoa Q.
• Application Number:
  US11/062,668
• Publication Number:
  US 20050197772A1
• Time in Patent Office: 973 Days
• Field of Search: 356/625, 356/635, 356/636, 3562371-2375, 700/97, 700/121
• US Class Current: 356/625
• CPC Class Codes: G01B 21/045 Correction of measurements G01B9/02055 takes precedence
• CPC Class Codes: G03F 7/70483 Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
• CPC Class Codes: G03F 7/70616 Monitoring the printed patterns
• CPC Class Codes: H01L 22/12 for structural parameters, e.g. thickness, line width, refractive index, temperature, warp, bond strength, defects, optical inspection, electrical measurement of structural dimensions, metallurgic measurement of diffusions electrical measurement of diffusions H01L22/14

Abstract

Methods and related program product for assessing and optimizing metrology instruments by determining a total measurement uncertainty (TMU) based on precision and accuracy. The TMU is calculated based on a linear regression analysis and removing a reference measuring system uncertainty (U_RMS) from a net residual error. The TMU provides an objective and more accurate representation of whether a measurement system under test has an ability to sense true product variation. The invention also includes a method for determining an uncertainty of the TMU.

Claims

1. A method for assessing a measurement system under test (MSUT), the method comprising the steps of:

(a) providing a substrate having a plurality of structures;

(b) measuring a dimension of the plurality of structures using a reference measurement system (RMS) to generate a first data set, and calculating an RMS uncertainty (U_RMS) from the first data set, where the RMS uncertainty (U_RMS) is defined as one of

(i) an RMS precision;

(ii) an independently determined RMS total measurement uncertainty (TMU_RMS); and

(iii) V_URMS=V_ST+V_AG, wherein V_URMS is U_RMS expressed as a variance, V_ST is a short term precision variance, and V_AG is an across grating variance;

(c) measuring the dimension of the plurality of structures using the MSUT to generate a second data set, and calculating a precision of the MSUT from the second data set;

(d) conducting a linear regression analysis of the first and second data sets to determine a corrected precision of the MSUT and a net residual error;

(e) determining a total measurement uncertainty (TMU) for the MSUT by removing the RMS uncertainty (U_RMS) from the net residual error; and

(f) outputting the TMU to a system capable of optimizing the MSUT.

2. The method of claim 1, wherein the plurality of structures represent variations in a semiconductor process.

3. The method of claim 1, wherein the dimension includes at least one of line width, depth, height, sidewall angle and top corner rounding.

4. The method of claim 1, wherein the TMU for the MSUT is determined according to the formula:

TMU=√

{square root over (D²−

U_RMS²)}

where D is the net residual error.

5. The method of claim 1, wherein the linear regression is calculated using a Mandel linear regression wherein a ratio variable I is defined according to the formula:

$\mathrm{\lambda <br><br>}=\frac{U_{\mathrm{RMS}}^2}{U_{\mathrm{MSUT}}^2}$

where U_MSUT is as an MSUT uncertainty defined as one of the corrected precision of the MSUT and the TMU for the MSUT.

6. The method of claim 5, wherein, in the case that the TMU for the MSUT is substantially different than the MSUT uncertainty (U_MSUT) after step (e), steps (d) and (e) are repeated using the TMU for the MSUT as the MSUT uncertainty (U_MSUT) in determining the ratio variable I.

7. The method of claim 5, wherein the TMU for the MSUT is determined according to the formula:

TMU=√

{square root over (D_M²−

U_RMS²)}

where D_M is the Mandel net residual error.

8. The method of claim 1, wherein V_URMS further includes at least one of a multiple reference system variance (V_MRMS) and a long term precision variance V_LT according to the equation

V_URMS=V_ST+V_AG+V_MRMS+V_LT.

9. The method of claim 1, further comprising the step of determining an uncertainty in the total measurement uncertainty (TMU), the method comprising the steps of:

determining a confidence limit for a Mandel net residual error (D_M); and

determining a confidence limit for the total measurement uncertainty (σ

_TMU) according to the equation

σ

_TMU=√

{square root over (D_M²−

U_RMS²)},

where U_RMS is a measurement uncertainty of the reference measurement system.

10. The method of claim 1, further comprising the step of estimating an uncertainty in an estimated total measurement uncertainty (TMU), comprising the steps of:

making a plurality of measurements;

calculating a Mandel net residual error (D_M) and a reference measurement system uncertainty (σ

_RMS) for each measurement;

calculating a variance for each D_M and σ

_RMS;

constructing a Chi-squared probability distribution function (χ

²-pdf) for the D_M variances and the σ

_RMS variances;

calculating a χ

²-pdf for the TMU according to the equation

pdf_TMU=χ

_DM2²{circle around (×

)}χ

<sub>σ

RMS2</sub>²,

wherein pdf_TMU is the TMU χ

²-pdf, χ

_DM2² is the D_M variance χ

²-pdf, χ

<sub>σ

RMS2</sub>² is the σ

_RMS variance χ

²-pdf, and {circle around (×

)} is a convolution operation; and

determining a TMU range from pdf_TMU, wherein the TMU range determines the uncertainty in the estimated TMU.

11. The method of claim 1, wherein each of D_M and U_RMS is determined by at least about 40 measurements and wherein

s_TMU is represented as s_TMU±

d_TMU;

U_RMS is represented as U_RMS±

d_RMS;

D_M is represented as D_M±

d_Mandel; and

the equation σ

_TMU=√

{square root over (D_M²−

U_RMS² )} is represented as

${\mathrm{\delta <br><br>}}_{\mathrm{TMU}}=\sqrt{\frac{D_M^2}{{\mathrm{\sigma <br><br>}}_{\mathrm{TMU}}^2}<br><br>{\mathrm{\delta <br><br>}}_{\mathrm{Mandel}}^2+\frac{U_{R<br><br><br><br>\mathrm{MS}}^2}{{\mathrm{\sigma <br><br>}}_{\mathrm{TMU}}^2}<br><br>{\mathrm{\delta <br><br>}}_{R<br><br><br><br>\mathrm{MS}}^2}.$

12. A method for optimizing a measurement system under test (MSUT), the method comprising the steps of:

(a) providing a plurality of structures;

(b) measuring a dimension of the plurality of structures according to a measurement parameter using a reference measurement system (RMS) to generate a first data set, and calculating an RMS uncertainty (U_RMS) from the first data set, where the RMS uncertainty (U_RMS) is defined as one of

(i) an RMS precision;

(ii) an independently determined RMS total measurement uncertainty (TMU_RMS); and

(iii) V_URMS=V_ST+V_AG, wherein V_URMS is U_RMS expressed as a variance, V_ST is a short term precision variance, and V_AG is an across grating variance;

(c) measuring the dimension of the plurality of structures according to the measurement parameter using the MSUT to generate a second data set, and calculating a precision of the MSUT from the second data set;

(d) conducting a linear regression analysis of the first and second data sets to determine a corrected precision of the MSUT and a net residual error;

(e) determining a total measurement uncertainty (TMU) for the MSUT by removing the RMS uncertainty (U_RMS) from the net residual error;

(f) repeating steps (c) to (e) for at least one other measurement parameter;

(g) outputting the TMU to a system capable of optimizing the MSUT; and

(h) optimizing the MSUT by determining an optimal measurement parameter based on a minimal total measurement uncertainty.

13. The method of claim 12, further comprising the step of selecting a set of measurement parameters to be evaluated.

14. The method of claim 12, wherein the MSUT is an SEM and a measurement parameter includes at least one of:

a data smoothing amount, an algorithm setting, a beam landing energy, a current, an edge detection algorithm and a scan rate.

15. The method of claim 12, wherein the MSUT is a scatterometer and a measurement parameter includes at least one of:

a spectra averaging timeframe, a spectra wavelength range, an angle of incidence and area of measurement, a density of selected wavelengths and a number of adjustable characteristics in a theoretical model.

16. The method of claim 12, wherein the MSUT is an AFM and a measurement parameter includes at least one of:

a number of scans, a timeframe between scans, a scanning speed, a data smoothing amount and area of measurement, and a tip shape.

17. The method of claim 12, wherein the plurality of structures represent variations in a semiconductor process.

18. The method of claim 12, wherein the dimension includes at least one of line width, depth, height, sidewall angle and top corner rounding.

19. The method of claim 12, wherein V_URMS further includes at least one of a multiple reference system variance (V_MRMS) and a long term precision variance V_LT according to the equation

V_URMS=V_ST+V_AG+V_MRMS+V_LT.

20. The method of claim 12, wherein a total measurement uncertainty (TMU) for the MSUT is determined according to the formula:

TMU=√

{square root over (D²−

U_RMS²)}

where D is the net residual error.

21. The method of claim 12, wherein the linear regression is calculated using a Mandel linear regression wherein a ratio variable I is defined according to the formula:

$\mathrm{\lambda <br><br>}=\frac{U_{R<br><br><br><br>\mathrm{MS}}^2<br><br>}{U_{\mathrm{MSUT}}^2}$

where U_MSUT is as an MSUT uncertainty defined as one of the corrected precision of the MSUT and the TMU for the MSUT.

22. The method of claim 21, wherein, in the case that the TMU for the MSUT is substantially different than the MSUT uncertainty (U_MSUT) after step (e), steps (d) and (e) are repeated using the TMU for the MSUT as the MSUT uncertainty (U_MSUT) in determining the ratio variable λ

.

23. The method of claim 12, further comprising the step of determining an uncertainty in the total measurement uncertainty (TMU), the method comprising the steps of:

determining a confidence limit for a Mandel net residual error (D_M); and

determining a confidence limit for the total measurement uncertainty (σ

_TMU) according to the equation

σ

_TMU=√

{square root over (D_M²−

U_RMS²)},

where U_RMS is a measurement uncertainty of the reference measurement system.

24. The method of claim 12, further comprising the step of estimating an uncertainty in an estimated total measurement uncertainty (TMU), comprising the steps of:

making a plurality of measurements;

calculating a Mandel net residual error (D_M) and a reference measurement system uncertainty (σ

_RMS) for each measurement;

calculating a variance for each D_M and σ

_RMS;

constructing a Chi-squared probability distribution function (χ

²-pdf) for the D_M variances and the σ

_RMS variances;

calculating a χ

²-pdf for the TMU according to the equation

pdf_TMU=χ

_DM2²{circle around (×

)}χ

<sub>σ

RMS2</sub>²,

wherein Pdf_TMU is the TMU χ

²-pdf, χ

_DM2² is the D_M variance χ

²-pdf, χ

<sub>σ

RMS2</sub>² is the σ

_RMS variance χ

²-pdf, and {circle around (×

)} is a convolution operation; and

determining a TMU range from Pdf_TMU, wherein the TMU range determines the uncertainty in the estimated TMU.

25. A method for estimating an uncertainty in an estimated total measurement uncertainty (TMU) comprising the steps of:

making a plurality of measurements;

calculating a Mandel net residual error (D_M) and a reference measurement system uncertainty (σ

_RMS) for each measurement;

calculating a variance for each D_M and σ

_RMS;

constructing a Chi-squared probability distribution function (χ

²-pdf) for the D_M variances and the σ

_RMS variances;

calculating a χ

²-pdf for the TMU according to the equation

pdf_TMU=χ

_DM2²{circle around (×

)}χ

<sub>σ

RMS2</sub>²,

wherein pdf_TMU is the TMU χ

²pdf, χ

_DM2² is the D_M variance χ

²-pdf, χ

<sub>σ

RMS2</sub>² the σ

_RMS variance χ

²-pdf, and {circle around (×

)} is a convolution operation;

determining a TMU range from pdf_TMU, wherein the TMU range determines the uncertainty in the estimated TMU; and

outputting the TMU range to a system capable of optimizing a measurement system under test.

26. The method of claim 25, wherein the σ

_RMS is defined as one of;

an RMS precision;

an independently determined RMS total measurement uncertainty (TMU_RMS); and

V_URMS=V_ST+V_AG, wherein V_URMS expressed as a variance, V_ST is a short term precision variance, and V_AG is an across grating variance.

27. The method of claim 26, wherein V_URMS further includes at least one of a multiple reference system variance (V_MRMS) and a long term precision variance V_LT according to the equation

V_URMS=V_ST+V_AG+V_MRMS+V_LT.

28. The method of claim 25, wherein the plurality of measurements are of structures that represent variations in a semiconductor process.

29. A method for determining an uncertainty in a total measurement uncertainty (TMU), the method comprising the steps of:

determining a confidence limit for a net residual error from a Mandel analysis (D_M) using chi-squared distribution tables;

determining a confidence limit for a total measurement uncertainty (s_TMU) according to the equation

σ

_TMU=√

{square root over (D_M²−

U_RMS²)},

where U_RMS is a measurement uncertainty of a reference measurement system; and

outputting the S_TMU to a system capable of optimizing a measurement system under test.

30. The method of claim 29, wherein each of D_M and U_RMS is determined by at least about 40 measurements and wherein

s_TMU is represented as s_TMU±

d_TMU;

U_RMS is represented as U_RMS±

d_RMS;

D_M is represented as D_M±

d_Mandel; and

the equation σ

_TMU=√

{square root over (D_M²−

U_RMS²)} is represented as

${\mathrm{\delta <br><br>}}_{\mathrm{TMU}}=\sqrt{\frac{D_M^2}{{\mathrm{\sigma <br><br>}}_{\mathrm{TMU}}^2}<br><br>{\mathrm{\delta <br><br>}}_{\mathrm{Mandel}}^2+\frac{U_{R<br><br><br><br>\mathrm{MS}}^2}{{\mathrm{\sigma <br><br>}}_{\mathrm{TMU}}^2}<br><br>{\mathrm{\delta <br><br>}}_{R<br><br><br><br>\mathrm{MS}}^2}.$