Data analysis sheet for residual strain measurements with more detailed calculations of the combined standard uncertainty, u_cer

Figure RS.2.1.  Top view of fixed-fixed beam used to measure residual strain.

To obtain the following measurements, consult ASTM standard test method E 2245 entitled
"Standard Test Method for Residual Strain Measurements of Thin, Reflecting Films
Using an Optical Interferometer" and NISTIR 7291 entitled "MEMS Length and Strain
Round Robin Results with Uncertainty Analysis."

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date data taken (optional) = / /
identifying words (optional)   =   
instrument used (optional)   =   
fabrication facility/process (optional)   =   
test chip name (optional)   =   
test chip number (optional)   =   
filename of 3-D data set (optional)   =   
filename of 2-D data traces (optional) =
    
    
 

Table 1 - Preliminary ESTIMATES			Description
1	material =	Poly1        Poly2       stacked Poly1 and Poly2    SiO2       SiC-2       SiC-3     other	material
2	t =	μm	beam thickness
3	design length =	μm	design length
4	design width =	μm	design width (needed for test structure identification purposes only)
5	which beam?	first      second third     fourth   other	which fixed-fixed beam on the test chip, where "first" corresponds to the topmost fixed-fixed beam in the column or array that has the specified length?
6	magnification =	×	magnification
7	orientation =	0°      90°    other	orientation of the fixed-fixed beam on the chip
8	calx =		x-calibration factor (for the given magnification)
9	interx =	μm	interferometer's maximum field of view (for the given magnification)
10	σxcal =	μm	one sigma uncertainty in a ruler measurement (for the given magnification)
11	xres =	μm	uncalibrated resolution of the interferometer in the x-direction
12	calz =		z-calibration factor (for the given magnification)
13	cert =	μm	certified value of physical step height used for calibration
14	σcert =	μm	certified one sigma uncertainty of the certified physical step height used for calibration
15	zrepeat(shs) =	μm	uncalibrated maximum range of the six calibration measurements taken before the data session or after the data session (whichever is larger)
16		greater than      less than      (Therefore, ameanz = ).	Is the mean value of the six calibration measurements used to obtain zrepeat(shs) greater than or less than the mean value of all twelve calibration measurements?
17	zdrift =	μm	uncalibrated drift in the calibration data (i.e., the absolute value of the mean value of the six calibration measurements taken before the data session minus the mean value after the data session)
18	zperc =	%	over the instrument's total scan range, the maximum percent deviation from linearity, as quoted by the instrument manufacturer (typically less than 3%)
19	zres =	μm	calibrated resolution of the interferometer in the z-direction
20	Rtave =	μm	calibrated peak-to-valley roughness of a flat and leveled surface of the sample material calculated to be the average of three or more measurements, each measurement of which is taken from a different 2-D data trace
21	Rave =	μm	calibrated surface roughness of a flat and leveled surface of the sample material calculated to be the average of three or more measurements, each measurement of which is taken from a different 2-D data trace
22	aligned?	Yes      No	alignment ensured ?
23	leveled?	Yes      No	data leveled ?
24	stiction?	Yes      No	Is this fixed-fixed beam exhibiting stiction ?   (If it is exhibiting stiction, do not fill out the remainder of this form.)

                                          
 
 

Table 2 - INPUTS (uncalibrated values from Trace "a" or "e")			Notes
25	x1max (i.e., x1upper) =	μm
26	x1min (i.e., x1lower)  =	μm	(x1min > x1max)
27	x2min (i.e., x2lower)  =	μm	(x2min > x1min)
28	x2max (i.e., x2upper) =	μm	(x2max > x2min)

                     

Table 3 - INPUTS (uncalibrated values from Trace "b")			Notes
29	x1F = μm	z1F = μm	(x1ave < x1F  calx)
30	x2F = μm	z2F = μm	(inflection point) ( x1F < x2F < x3F )
31	x3F = μm	z3F = μm	(most deflected point) ( x1S = x3F ; z1S = z3F )
32	x2S = μm	z2S = μm	(inflection point)
33	x3S = μm	z3S = μm	( x3S  calx < x2ave ) ( x1S < x2S < x3S )

 

Table 4 - INPUTS (uncalibrated values from Trace "c")			Notes
34	x1F = μm	z1F = μm	(x1ave < x1F  calx)
35	x2F = μm	z2F = μm	(inflection point) ( x1F < x2F < x3F )
36	x3F = μm	z3F = μm	(most deflected point) ( x1S = x3F ; z1S = z3F )
37	x2S = μm	z2S = μm	(inflection point)
38	x3S = μm	z3S = μm	( x3S  calx < x2ave ) ( x1S < x2S < x3S )

                    

Table 5 - INPUTS (uncalibrated values from Trace "d")			Notes
39	x1F = μm	z1F = μm	(x1ave < x1F  calx)
40	x2F = μm	z2F = μm	(inflection point) ( x1F < x2F < x3F )
41	x3F = μm	z3F = μm	(most deflected point) ( x1S = x3F ; z1S = z3F )
42	x2S = μm	z2S = μm	(inflection point)
43	x3S = μm	z3S = μm	( x3S  calx < x2ave ) ( x1S < x2S < x3S )

                                      


OUTPUTS (calibrated values):
            x1_ave =  μm            x2_ave =  μm
             L      =  μm
                         L_max = ( x2_max − x1_max ) cal_x
                         L_min = ( x2_min − x1_min ) cal_x
                         u_LL  =  ( L_max − L_min ) / 6 =  μm
                         u_Lxcal  = ( σ_xcal / inter_x ) ( L / cal_x ) = μm
                         u_Lxres  = x_res cal_x / 1.732 =  μm
             u_cL   =  SQRT[u_LL² + u_Lxcal² + u_Lxres²]  =  μm
             s        =               from Trace "c"
                         s = 1       (for downward bending fixed-fixed beams)
                         s = −1     (for upward bending fixed-fixed beams)

                             A_F   =  μm      from Trace "b"
                           w_1F   =             from Trace "b"
                             A_S   =  μm      from Trace "b"
                           w_3S   =               from Trace "b"
             x_eF   =  μm            from Trace "b"
             x_eS   =  μm            from Trace "b"
            ε_r0    =  × 10^-6       from Trace "b"
            ε_r     =    × 10^-6       from Trace "b"

                             A_F   =  μm      from Trace "c"
                           w_1F   =             from Trace "c"
                             A_S   =  μm      from Trace "c"
                           w_3S   =               from Trace "c"
             x_eF   =  μm             from Trace "c"
             x_eS   =  μm             from Trace "c"
             ε_r0   =  × 10^-6        from Trace "c"
             ε_r    =    × 10^-6        from Trace "c"                (USE THIS VALUE)
                         u_Rave    =    × 10^-6      from Trace "c"
                         u_noise    =    × 10^-6      from Trace "c"
                         u_W    =    × 10^-6          from two or three traces
                         u_xcal    =    × 10^-6        from Trace "c"
                         u_L    =    × 10^-6           from Trace "c"
                         u_cert    =    × 10^-6        from Trace "c"
                         u_repeat(shs)    =    × 10^-6        from Trace "c"
                         u_drift    =    × 10^-6        from Trace "c"
                         u_linear    =    × 10^-6        from Trace "c"
                         u_zres    =    × 10^-6        from Trace "c"
                         u_xres    =    × 10^-6        from Trace "c"
                         u_xresL    =    × 10^-6        from Trace "c"
             u_cer  = SQRT[u_Rave² + u_noise² + u_W² + u_xcal² + u_L² + u_cert² + u_repeat(shs)² + u_drift² + u_linear² + u_zres² + u_xres² + u_xresL²]
                        (Each of the standard uncertainty components is obtained using a Type B analysis.)
             u_cer  =    × 10^-6        from two or three traces

                             A_F   =  μm      from Trace "d"
                           w_1F   =             from Trace "d"
                             A_S   =  μm      from Trace "d"
                           w_3S   =               from Trace "d"
             x_eF   =  μm             from Trace "d"
             x_eS   =  μm             from Trace "d"
            ε_r0    =  × 10^-6        from Trace "d"
            ε_r     =    × 10^-6        from Trace "d"

Report the results as follows:  Since it can be assumed that the estimated values of the uncertainty
components are approximately uniformly or Gaussianly distributed with approximate combined standard
uncertainty u_cer, the residual strain is believed to lie in the interval e_r ± u_cer (expansion factor k=1)
representing a level of confidence of approximately 68 %. 

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Modify the input data, given the information supplied in any flagged statement below, if applicable, then recalculate:

1.		Please fill out the entire form.
2.		The value for t should be between 0.000 μm and 10.000 μm.
3.		The value for the design length should be between 0 μm and 1000 μm.
4.		The measured value for L is more than 3ucL from the design length.
5.		The value for the design width should be between 0 μm and 60 μm.
6.		Is the magnification appropriate given the design length ?
7.		Magnifications at or less than 2.5× shall not be used.
8.		Is 0.95 < calx < 1.05 but not equal to "1"?  If not, recheck your x-calibration.
9.		The value for interx should be between 0 μm and 1500 μm.
10.		The value for σxcal should be between 0 μm and 4 μm.
11.		The value for xres should be between 0 μm and 2.00 μm.
12.		Is 0.95 < calz < 1.05 but not equal to "1"?  If not, recheck your z-calibration.
13.		The value for cert should be greater than 0 μm and less than 25 μm.
14.		The value for σcert should be between 0 μm and 0.100 μm.
15.		The value for zrepeat(shs) should be between 0 μm and 0.070 μm.
16.		The value for zdrift should be between 0 μm and 0.010 μm.
17.		The value for zperc should be between 0 % and 3 %.
18.		The value for zres should be greater than 0 μm and less than or equal to 0.005 μm.
19.		The value for Rtave should be between 0 μm and 0.100 μm and greater than Rave.
20.		The value for Rave should be between 0 μm and 0.020 μm.
21.		Alignment has not been ensured.
22.		Data has not been leveled.
23.		x1min should be greater than x1max.
24.		x2min should be greater than x1min.
25.		x2max should be greater than x2min.
26.		The calibrated values for x1min and x1max are greater than 10 μm apart.
27.		The calibrated values for x2min and x2max are greater than 10 μm apart.
28.		In Traces "b," "c," and "d," the value for s is not the same.
29.		x1ave should be < (x1F calx) in all traces.
30.		(x3S calx) should be < x2ave in all traces.
31.		In all traces, make sure ( x1F < x2F < x3F ).
32.		In all traces, make sure ( x1S < x2S < x3S ).
33.		For Trace "b," | [(x2F calx) − xeF ] | = μm.  This should be < 5 μm. If it is not, choose (x2F, z2F) such that (x2F calx) is closer to xeF = μm.
34.		For Trace "b," | [(x2S calx) − xeS ] | =  μm.  This should be < 5 μm. If it is not, choose (x2S, z2S) such that (x2S calx) is closer to xeS = μm.
35.		For Trace "c," | [(x2F calx) − xeF ] | = μm.  This should be < 5 μm. If it is not, choose (x2F, z2F) such that (x2F calx) is closer to xeF = μm.
36.		For Trace "c," | [(x2S calx) − xeS ] | =  μm.  This should be < 5 μm. If it is not, choose (x2S, z2S) such that (x2S calx) is closer to xeS = μm.
37.		For Trace "d," | [(x2F calx) − xeF ] | =  μm.  This should be < 5 μm. If it is not, choose (x2F, z2F) such that (x2F calx) is closer to xeF = μm.
38.		For Trace "d," | [(x2S calx) − xeS ] | =  μm.  This should be < 5 μm. If it is not, choose (x2S, z2S) such that (x2S calx) is closer to xeS = μm.

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Email questions or comments to [email protected].

NIST is an agency of the U.S. Commerce Department.
The Semiconductor and Dimensional Metrology Division is within the Physical Measurement Laboratory.
The MEMS Measurement Science and Standards Project is within the Nanoscale Metrology Group.

Date created: 12/4/2000
Last updated: 4/26/2013