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

Figure SG.2.1.  Top view of cantilever test structure used to measure strain gradient.

To obtain the following measurements, consult ASTM standard test method E 2246 entitled
"Standard Test Method for Strain Gradient Measurements of Thin, Reflecting Films
Using an Optical Interferometer" and NISTIR 7291 entitled "MEMS Length and Strain
Round Robin Results with Uncertainty Analysis."

---

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	design length =	µm	design length
3	which cantilever?	First     Second     Third     Fourth Fifth Other	which cantilever on the test chip ?
4	magnification =	×	magnification
5	orientation =	0°          90°        180°      270°      Other	orientation of the cantilever on the chip
6	calx =		x-calibration factor (for the given magnification)
7	interx =	µm	interferometer's maximum field of view (for the given magnification)
8	σxcal =	µm	one sigma uncertainty in a ruler measurement (for the given magnification)
9	xres =	µm	resolution of the interferometer in the x-direction
10	calz =		z-calibration factor (for the given magnification)
11	cert =	µm	certified value of physical step height used for calibration
12	σcert =	µm	certified one sigma uncertainty of the certified physical step height used for calibration
13	zrepeat =	µm	maximum range of the six calibration measurements taken before the data session or after the data session (whichever is larger)
14		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?
15	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)
16	zperc =	%	over the instrument's total scan range, the maximum percent deviation from linearity, as quoted by the instrument manufacturer (typically less than 3%)
17	zres =	µm	resolution of the interferometer in the z-direction
18	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
19	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
20	aligned?	Yes      No	alignment ensured ?
21	leveled?	Yes      No	data leveled ?
22	stiction?	Yes      No	Is this cantilever 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
23	x1max (i.e., x1upper) =	µm
24	x1min (i.e., x1lower) =	µm	(x1min > x1max)

Table 3 - INPUTS (uncalibrated values from Trace "b")			Notes
25	x1 = µm	z1 = µm	(x1ave < x1 * calx)
26	x2 = µm	z2 = µm	(x1ave < x2 * calx)
27	x3 = µm	z3 = µm	(x1ave < x3 * calx)

Table 4 - INPUTS (uncalibrated values from Trace "c")			Notes
28	x1 = µm	z1 = µm	(x1ave < x1 * calx)
29	x2 = µm	z2 = µm	(x1ave < x2 * calx)
30	x3 = µm	z3 = µm	(x1ave < x3 * calx)

Table 5 - INPUTS (uncalibrated values from Trace "d")			Notes
31	x1 = µm	z1 = µm	(x1ave < x1 * calx)
32	x2 = µm	z2 = µm	(x1ave < x2 * calx)
33	x3 = µm	z3 = µm	(x1ave < x3 * calx)

                                  


OUTPUTS (calibrated values):
           x1_ave =  µm
            s =               from Trace "c"
                    s = 1 (for downward bending cantilevers or
                              if data was taken from the bottom of an upward bending cantilever)
                    s = −1 (for upward bending cantilevers unless
                                data was taken from the bottom of an upward bending cantilever)

                      R_int =  µm    from Trace "b"
                        a   =  µm    from Trace "b"
                        b   =  µm    from Trace "b"
            s_g =  m⁻¹    from Trace "b"

                      R_int =  µm    from Trace "c"
                        a   =  µm    from Trace "c"
                        b   =  µm    from Trace "c"
             s_g =  m⁻¹    from Trace "c"                              (USE THIS VALUE)
                      u_W =  m⁻¹       from two or three traces
                      u_Rave =  m⁻¹   from Trace "c"
                      u_noise =  m⁻¹   from Trace "c"
                      u_xcal =  m⁻¹     from Trace "c"
                      u_cert =  m⁻¹     from Trace "c"
                      u_repeat =  m⁻¹     from Trace "c"
                      u_drift =  m⁻¹     from Trace "c"
                      u_linear =  m⁻¹     from Trace "c"
                      u_zres =  m⁻¹     from Trace "c"
                      u_xres =  m⁻¹     from Trace "c"
             u_c =  SQRT[u_W² + u_Rave² + u_noise² + u_xcal² + u_cert² + u_repeat² + u_drift² + u_linear² + u_zres² + u_xres²]
             u_c =  m⁻¹   from two or three traces

                      R_int =  µm    from Trace "d"
                        a   =  µm    from Trace "d"
                        b   =  µm    from Trace "d"
            s_g =  m⁻¹    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 strain
gradient is believed to lie in the interval s_g ± u_c with a level of confidence of approximately 68 %
assuming a Gaussian distribution. 

---

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 the design length should be between 0 µm and 1000 µm.
3.		Is the magnification appropriate given the design length ?
4.		Magnifications at or less than 2.5× shall not be used.
5.		Is 0.95 < calx < 1.05 but not equal to "1"?  If not, recheck your x-calibration.
6.		The value for interx should be between 0 µm and 1500 µm.
7.		The value for σxcal should be between 0 µm and 4 µm.
8.		The value for xres should be between 0 µm and 2.00 µm.
9.		Is 0.95 < calz < 1.05 but not equal to "1"?  If not, recheck your z-calibration.
10.		The value for cert should be greater than 0 µm and less than 25 µm.
11.		The value for σcert should be between 0 µm and 0.100 µm.
12.		The value for zrepeat should be between 0 µm and 0.070 µm.
13.		The value for zdrift should be between 0 µm and 0.010 µm.
14.		The value for zperc should be between 0 % and 3 %.
15.		The value for zres should be greater than 0 µm and less than or equal to 0.005 µm.
16.		The value for Rtave should be between 0 µm and 0.100 µm and greater than Rave.
17.		The value for Rave should be between 0 µm and 0.020 µm.
18.		Alignment has not been ensured.
19.		Data has not been leveled.
20.		x1min should be greater than x1max.
21.		The calibrated values for x1min and x1max are greater than 10 µm apart.
22.		In Trace "b," the calibrated values of x1, x2, and x3 should be > x1ave.
23.		In Trace "c," the calibrated values of x1, x2, and x3 should be > x1ave.
24.		In Trace "d," the calibrated values of x1, x2, and x3 should be > x1ave.
25.		In Traces "b," "c," and "d," the value for s is not the same.

---

Return to Main MEMS Calculator Page.

Email questions or comments to mems-support@nist.gov.

NIST is an agency of the U.S. Commerce Department
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