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."

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filename of 3-D data set (optional)   = 
filename of 2-D data traces (optional) =
     
     
material =  Poly1       Poly2       stacked Poly1 and Poly2 
                 SiC-2       SiC-3   
design length = µm
which cantilever on the round robin test chip ?  First    Second     Third   
magnification = ×
orientation =  0 degree      90 degree
x-calibration factor (for the given magnification) = calx =
interferometer's maximum field of view (for the given magnification) = interx = µm
one sigma uncertainty in a ruler measurement (for the given mag) = σ_xcal = µm
resolution of the interferometer in the x-direction = x_res =  µm
z-calibration factor (for the given magnification) = calz =
certified value of step height standard = cert = µm
certified one sigma uncertainty of certified value of step height standard = σ_cert = µm
maximum range of the six calibration measurements taken before the data session or after the data session (whichever is larger) = z_repeat = µm
Is the mean value of the six calibration measurements used to obtain z_repeat greater than or less than the mean value of all twelve calibration measurements?   greater than     less than     (Therefore, ameanz = .)
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) = z_drift = µm
percent quoted by interferometer manufacturer for the maximum deviation from linearity of the data scan over the total scan range divided by 1 % such that it is unitless = z_perc =
resolution of the interferometer in the z-direction = z_res =  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 = R_tave = 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 = R_ave = µm
alignment ensured ?   Yes      No
data leveled ?   Yes      No
Is this cantilever exhibiting stiction ?   Yes      No
            If it is exhibiting stiction, do not fill out the remainder of this form.

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INPUTS (uncalibrated values from Trace "a" or "e"):
              x1_max (i.e., x1_upper) = µm
              x1_min (i.e., x1_lower) = µm        (x1_min > x1_max)

INPUTS (uncalibrated values from Trace "b")
              x₁ = µm    z₁ = µm   (x1_ave < x₁ * calx)
              x₂ = µm    z₂ = µm   (x1_ave < x₂ * calx)
              x₃ = µm    z₃ = µm   (x1_ave < x₃ * calx)

INPUTS (uncalibrated values from Trace "c")
              x₁ = µm    z₁ = µm   (x1_ave < x₁ * calx)
              x₂ = µm    z₂ = µm   (x1_ave < x₂ * calx)
              x₃ = µm    z₃ = µm   (x1_ave < x₃ * calx)

INPUTS (uncalibrated values from Trace "d")
              x₁ = µm    z₁ = µm   (x1_ave < x₁ * calx)
              x₂ = µm    z₂ = µm   (x1_ave < x₂ * calx)
              x₃ = µm    z₃ = µm   (x1_ave < x₃ * calx)

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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"
                      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"

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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.   For the round robin test chip, the design length should be 400, 450, 500, 550, 600, 650, 700, 750, or 800 mm.
    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 σ_xcal should be between 0 µm and 4 µm.
    8.   The value for x_res should be between 0 µm and 1.57 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 σ_cert should be between 0 µm and 0.100 µm.
  12.   The value for z_repeat should be between 0 µm and 0.070 µm.
  13.   The value for z_drift should be between 0 µm and 0.010 µm.
  14.   The value for z_perc should be between 0 and 3.
  15.   The value for z_res should be greater than 0 µm and less than or equal to 0.005 µm. 
  16.   The value for R_tave should be between 0 µm and 0.100 µm and greater than R_ave.
  17.   The value for R_ave should be between 0 µm and 0.020 µm.
  18.   Alignment has not been ensured.
  19.   Data has not been leveled.
  20.   x1_min should be greater than x1_max.
  21.   The calibrated values for x1_min and x1_max are greater than 10 µm apart.
  22.   In Trace "b," the calibrated values of x₁, x₂, and x₃ should be > x1_ave.
  23.   In Trace "c," the calibrated values of x₁, x₂, and x₃ should be > x1_ave.
  24.   In Trace "d," the calibrated values of x₁, x₂, and x₃ should be > x1_ave.
  25.   In Traces "b," "c," and "d," the value for s is not the same.

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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: 1/11/2008