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

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/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    SiC-2       SiC-3     other	material
2	t =	µm	beam thickness
3	design length =	µm	design length
4	which beam?	First Second Third Other	which fixed-fixed beam on the test chip ?
5	magnification =	×	magnification
6	orientation =	0°      90°    Other	orientation of the fixed-fixed beam on the chip
7	calx =		x-calibration factor (for the given magnification)
8	interx =	µm	interferometer's maximum field of view (for the given magnification)
9	σxcal =	µm	one sigma uncertainty in a ruler measurement (for the given magnification)
10	xres =	µm	resolution of the interferometer in the x-direction
11	calz =		z-calibration factor (for the given magnification)
12	cert =	µm	certified value of physical step height used for calibration
13	σcert =	µm	certified one sigma uncertainty of the certified physical step height used for calibration
14	zrepeat =	µm	maximum range of the six calibration measurements taken before the data session or after the data session (whichever is larger)
15		greater than      less than      (Therefore, ameanz = ).	Is the mean value of the six calibration measurements used to obtain zrepeat greater than or less than the mean value of all twelve calibration measurements ?
16	zdrift =	µm	drift in the calibrated 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)
17	zperc =	%	over the instrument's total scan range, the maximum percent deviation from linearity, as quoted by the instrument manufacturer (typically less than 3%)
18	zres =	µm	resolution of the interferometer in the z-direction
19	Rtave =	µm	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
20	Rave =	µm	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
21	aligned?	Yes      No	alignment ensured ?
22	leveled?	Yes      No	data leveled ?
23	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
24	x1max (i.e., x1upper) =	µm
25	x1min (i.e., x1lower)  =	µm	(x1min > x1max)
26	x2min (i.e., x2lower)  =	µm	(x2min > x1min)
27	x2max (i.e., x2upper) =	µm	(x2max > x2min)

                     

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

 

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

                    

Table 5 - INPUTS (uncalibrated values from Trace "d")			Notes
38	x1F = µm	z1F = µm	(x1ave < x1F * calx)
39	x2F = µm	z2F = µm	(inflection point) ( x1F < x2F < x3F )
40	x3F = µm	z3F = µm	(most deflected point) ( x1S = x3F ; z1S = z3F )
41	x2S = µm	z2S = µm	(inflection point)
42	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 ) * calx
                         L_min = ( x2_min − x1_min ) * calx
                         u_LL  =  ( L_max − L_min ) / 6 =  µm
                         u_Lxcal  = ( σ_xcal / interx ) * ( L / calx ) = µm
                         u_Lxres  = x_res* calx / 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    =    × 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_c    = SQRT[u_Rave² + u_noise² + u_W² + u_xcal² + u_L² + u_cert² + u_repeat² + u_drift² + u_linear² + u_zres² + u_xres² + u_xresL²]
             u_c    =    × 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 possible estimated values are either
approximately uniformly distributed or Gaussian with approximate standard deviation u_c, the residual
strain is believed to lie in the interval e_r ± u_c with a level of confidence of approximately 68 %
assuming a Gaussian distribution. 

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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.		Is the magnification appropriate given the design length ?
6.		Magnifications at or less than 2.5× shall not be used.
7.		Is 0.95 < calx < 1.05 but not equal to "1"?  If not, recheck your x-calibration.
8.		The value for interx should be between 0 µm and 1500 µm.
9.		The value for σxcal should be between 0 µm and 4 µm.
10.		The value for xres should be between 0 µm and 2.00 µm.
11.		Is 0.95 < calz < 1.05 but not equal to "1"?  If not, recheck your z-calibration.
12.		The value for cert should be greater than 0 µm and less than 25 µm.
13.		The value for σcert should be between 0 µm and 0.100 µm.
14.		The value for zrepeat should be between 0 µm and 0.070 µm.
15.		The value for zdrift should be between 0 µm and 0.010 µm.
16.		The value for zperc should be between 0 % and 3 %.
17.		The value for zres should be greater than 0 µm and less than or equal to 0.005 µm.
18.		The value for Rtave should be between 0 µm and 0.100 µm and greater than Rave.
19.		The value for Rave should be between 0 µm and 0.020 µm.
20.		Alignment has not been ensured.
21.		Data has not been leveled.
22.		x1min should be greater than x1max.
23.		x2min should be greater than x1min.
24.		x2max should be greater than x2min.
25.		The calibrated values for x1min and x1max are greater than 10 µm apart.
26.		The calibrated values for x2min and x2max are greater than 10 µm apart.
27.		In Traces "b," "c," and "d," the value for s is not the same.
28.		x1ave should be < (x1F * calx) in all traces.
29.		(x3S * calx) should be < x2ave in all traces.
30.		In all traces, make sure ( x1F < x2F < x3F ).
31.		In all traces, make sure ( x1S < x2S < x3S ).
32.		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.
33.		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.
34.		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.
35.		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.
36.		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.
37.		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 mems-support@nist.gov.

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The Semiconductor Electronics Division is within the Electronics and Electrical Engineering Laboratory.
The MNT Project (http://www.eeel.nist.gov/812/MNT/index.html) is within the Enabling Devices and ICs Group.

Date created: 12/4/2000
Last updated: 6/2/2009