Data analysis sheet for strain gradient measurements for use with the MEMS 5-in-1 RMs

Figure SG.3.1.  Top view of a 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."

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

Table 1 - Preliminary ESTIMATES			Description
1	temp	°C	temperature during measurement (should be held constant)
2	relative humidity	%	relative humidity during measurement (if not known, enter -1)
3	material	Poly1        Poly2       stacked Poly1 and Poly2  SiO2       SiC-2       SiC-3     other	material
4	design length	μm	design length
5	design width	μm	design width (needed for test structure identification purposes only)
6	which cantilever?	first          second     third         fourth   fifth      sixth     other	indicates which cantilever on the test chip, where "first" corresponds to the topmost cantilever in the column or array that has the specified length?
7	magnification	×	magnification
8	orientation	0°          90°        180°      270°      other	orientation of the cantilever on the chip
9	calx		x-calibration factor (for the given magnification)
10	rulerx	μm	maximum field of view in the x-direction for the given magnification (as measured on the screen of the interferometric microscope)
11	σxcal	μm	one sigma uncertainty in a ruler measurement (for the given magnification)
12	xres	μm	uncalibrated resolution of the interferometric microscope in the x-direction (for the given magnification)
13	caly		y-calibration factor (for the given magnification)
14	calz		z-calibration factor (for the given magnification)
15	cert	μm	certified value of physical step height standard used for calibration
16	σcert	μm	certified one sigma uncertainty of the certified physical step height standard used for calibration
17	σ6same	μm	maximum of two uncalibrated values (σsame1 and σsame2) where σsame1 is the standard deviation of six measurements taken at the same location on the physical step height standard before the data session and σsame2 is the standard deviation of six measurements taken at this same location after the data session
18	z6same	μm	uncalibrated average of the six calibration measurements used to calculate σ6same
19	zdrift	μm	uncalibrated drift in the calibration data [i.e., the uncalibrated positive difference between the average of the six measurements taken before the data session at the same location on the physical step height standard and the average of the six measurements taken after the data session at this same location]
20	zlin	%	maximum relative deviation from linearity over the instrument's total scan range, as quoted by the instrument manufacturer (typically less than 3 %)
21	zres	μm	calibrated resolution of the interferometer in the z-direction
22	srepeat(samp)	%	relative strain gradient repeatability standard deviation as obtained from cantilever test structures fabricated in a process similar to that used to fabricate the sample
23	sgcorrection	m-1	strain gradient correction term for the given design length of the cantilever
24	Rtave	μm	calibrated peak-to-valley roughness of a flat and leveled surface of the sample material and calculated as the average of three or more measurements, each measurement of which is taken from a different 2D data trace
25	Rave	μm	calibrated surface roughness of a flat and leveled surface of the sample material and calculated as the average of three or more measurements, each measurement of which is taken from a different 2D data trace
26	aligned?	Yes      No	alignment ensured ?
27	leveled?	Yes      No	data leveled ?
28	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 Traces a and e)*,**,***
		Trace a inputs:	Trace e inputs:
29	x1uppert =	μm	μm
30	n1t =	μm (for informational purposes only)	μm (for informational purposes only)
31	yt =	μm	μm

^*Where x1_uppert is the uncalibrated x-value that most appropriately locates the upper point of
transitional Edge 1 using Trace "t"
^**The values for n1_t indicate the data point uncertainty associated with the chosen value for x1_uppert with
the subscript "t" referring to the data trace.  In other words, if it is easy to identify one point that accurately
locates the upper point of transitional Edge 1, the maximum uncertainty associated with the identification
of this point is n1_tx_rescal_x, where n1_t=1.
^***Where y_t is the uncalibrated y-value associated with Trace "t" such that y_a > y_e

Table 3 - INPUTS (uncalibrated values from Traces b, c, and d)			Notes
Trace b inputs:
32	x1 = μm	z1 = μm	(x1ave < x1 )
33	x2 = μm	z2 = μm	(x1ave < x2 )
34	x3 = μm	z3 = μm	(x1ave < x3 )
Trace c inputs:
35	x1 = μm	z1 = μm	(x1ave < x1 )
36	x2 = μm	z2 = μm	(x1ave < x2 )
37	x3 = μm	z3 = μm	(x1ave < x3 )
Trace d inputs:
38	x1 = μm	z1 = μm	(x1ave < x1 )
39	x2 = μm	z2 = μm	(x1ave < x2 )
40	x3 = μm	z3 = μm	(x1ave < x3 )

                                     

                                  
 

Table 4 - OUTPUTS			Notes
41	x1ave =	μm	= (x1uppera + x1uppere) / 2
42	α =	radians   °	= tan-1 [Δx calx / (Δy caly)] Δx =x1uppera - x1uppere Δy =ya -  ye
43	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)
44	f =	μm	= x1ave calx

 

Table 5 - OUTPUTS					Notes
		Trace b	Trace c	Trace d
45	g =	μm	μm	μm	= (x1t calx- f) cosα + f
46	h =	μm	μm	μm	= (x2t calx- f) cosα + f
47	i =	μm	μm	μm	= (x3t calx- f) cosα + f
		Trace b	Trace c	Trace d
48	Rint =	μm	μm	μm	Rint is the radius of the circle describing the shape of the topmost surface of the cantilever and (m, n) are the coordinates of the origin of that circle.  These values are used to plot the function with the data.
49	m =	μm	μm	μm
50	n =	μm	μm	μm
51	sgt =	m−1	m−1	m−1	strain gradient
52	sg =	m−1			average strain gradient value from Traces b, c, and d (USE THIS VALUE)

Table 6 - Preliminary Uncertainty OUTPUTS
53	uW =	m−1
		Trace b	Trace c	Trace d
54	uRavet =	m−1	m−1	m−1
55	unoiset =	m−1	m−1	m−1
56	uxcalt =	m−1	m−1	m−1
57	ucertt =	m−1	m−1	m−1
58	urepeat(shs)t =	m−1	m−1	m−1
59	udriftt =	m−1	m−1	m−1
60	ulineart =	m−1	m−1	m−1
61	uzrest =	m−1	m−1	m−1
62	uxrest =	m−1	m−1	m−1
63	ucorrectiont =	m−1	m−1	m−1
64	urepeat(samp)t =	m−1	m−1	m−1
65	ucsgt =	m−1	m−1	m−1

 

Table 7 - Uncertainty OUTPUTS Averaging the values from Traces b, c, and d, where applicable
66	uW =	m−1	due to variations across width of beam
67	uRave =	m−1	due to sample's surface roughness
68	unoise =	m−1	due to interferometric noise
69	uxcal =	m−1	due to calibration in the x-direction
70	ucert =	m−1	due to uncertainty of the value of the physical step height standard
71	urepeat(shs) =	m−1	due to the repeatability of a measurement taken on the physical step height standard
72	udrift =	m−1	due to the amount of drift during the data session
73	ulinear =	m−1	due to the deviation from linearity of the data scan
74	uzres =	m−1	due to the resolution of the interferometer in the z-direction
75	uxres =	m−1	due to the resolution of the interferometric microscope in the x-direction
76	ucorrection =	m−1	due to the uncertainty of the correction term
77	urepeat(samp) =	m−1	due to the uncertainty of strain gradient repeatability measurements
78	ucsgave =	m−1	= (ucsgb + ucsgc + ucsgd) / 3
ucsg =  SQRT[uW2 + uRave2 + unoise2 + uxcal2 + ucert2 + urepeat(shs)2 + udrift2 + ulinear2 + uzres2 + uxres2 + ucorrection2 + urepeat(samp)2] (Each of the standard uncertainty components is obtained using a Type B analysis, except for uW and urepeat(samp), which use a Type A analysis.)
79	ucsg =	m−1	combined standard uncertainty
80	2ucsg = Usg =	m−1	expanded uncertainty
81	3ucsg =	m−1
82	sg − Usg =	m−1	a lower bound for sg
83	sg + Usg =	m−1	an upper bound for sg

Report the results as follows:  If it is assumed that the estimated values of the uncertainty
components are approximately Gaussianly distributed with approximate combined standard
uncertainty u_csg, the strain gradient is believed to lie in the interval s_g ± u_csg (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 temp should be between 19.4 °C and 21.6 °C, inclusive.
3.		The value for relative humidity (if known) should be between 0 % and 60 %, inclusive.
4.		The value for the design length should be between 0 μm and 1000 μm.
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. Is 0.95 < caly < 1.05 but not equal to "1"?  If not, recheck your y-calibration.
9.		The value for rulerx 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 σ6same should be between 0 μm and 0.200 μm.
16.		The value for z6same should be between (cert - 0.150 μm) /calz and (cert + 0.150 μm) calz.
17.		The value for zdrift should be between 0 μm and 0.100 μm.
18.		The value for zlin should be between 0 % and 5 %.
19.		The value for zres should be greater than 0 μm and less than or equal to 0.005 μm.
20.		The value for srepeat(samp) should be greater than 0 % and less than or equal to 25 %.
21.		The values for sgt should be greater than 0.0 m−1, so increase sgcorrection.
22.		The value for Rtave should be between 0 μm and 0.500 μm and greater than Rave.
23.		The value for Rave should be between 0 μm and 0.050 μm.
24.		Alignment has not been ensured.
25.		Data has not been leveled.
26.		The cantilever is exhibiting stiction.
27.		ya should be greater than ye.
28.		n1t should be between 1 and 4, inclusive.
29.		α should be between -2° and 2°.
30.		In Trace b, the values of x1, x2, and x3 should be > x1ave.
31.		In Trace c, the values of x1, x2, and x3 should be > x1ave.
32.		In Trace d, the values of x1, x2, and x3 should be > x1ave.
33.		In Traces b, c, and d, the value for s is not the same.

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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