PML Banner
 Data Analysis Sheet RS.3

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

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

Figure RS.3.1.  Top view of a 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."


                                          
 
                                      
 
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) =
    
    
 

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       
      
   
             
    
     

material
4 t μm beam thickness
5 design length μm

design length
6 design width μm design width
(needed for test structure identification purposes only)
7 which beam?



 		 indicates 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? 
8 magnification × magnification
9 orientation     
   
 		 orientation of the fixed-fixed beam on the chip
10 calx x-calibration factor (for the given magnification)
11 rulerx μm maximum field of view in the x-direction for the given magnification (as measured on the screen of the interferometric microscope)
12 σxcal μm one sigma uncertainty in a ruler measurement (for the given magnification)
13 xres μm uncalibrated resolution of the interferometric microscope in the x-direction (for the given magnification)
14 caly y-calibration factor (for the given magnification)
15 calz z-calibration factor (for the given magnification)
16 cert μm certified value of physical step height standard used for calibration
17 σcert μm certified one sigma uncertainty of the certified physical step height standard used for calibration
18 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
19 z6same μm uncalibrated average of the six calibration measurements used to calculate σ6same  
20 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]
21 zlin % maximum relative deviation from linearity over the instrument's total scan range, as quoted by the instrument manufacturer (typically less than 3 %)
22 zres μm calibrated resolution of the interferometer in the z-direction
23 σLrepeat(samp)' μm in-plane length repeatability standard deviation (for the given magnification of the given interferometric microscope) as obtained for the same or a similar type of measurement and taken on test structures with transitional edges that face each other
24 σrepeat(samp) relative residual strain repeatability standard deviation obtained from fixed-fixed beam test structures fabricated in a process similar to that used to fabricate the sample
25 δεrcorrection relative residual strain correction term
26 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
27 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
28 Loffset μm in-plane length correction term for the given type of in-plane length measurement taken on similar structures when using similar calculations and for a given magnification of a given interferometric microscope
29 aligned? Yes      No alignment ensured?
30 leveled? Yes      No data leveled?
31 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 Traces a', a, e, and e')

Notes*,**,***

Trace a' inputs:
32 x1uppera' = μm n1a' = 1 < n1a' < 4
33 x2uppera' =   μm  n2a' =

1 < n2a' < 4
(x2uppera' > x1uppera')

34 ya' =   μm   
 

Trace a inputs:
35 x1uppera = μm n1a = 1 < n1a < 4
36 x2uppera =   μm  n2a = 1 < n2a < 4
(x2uppera > x1uppera)
 

Trace e inputs:
37 x1uppere = μm n1e = 1 < n1e < 4
38 x2uppere =   μm  n2e = 1 < n2e < 4
(x2uppere > x1uppere)
 

Trace e' inputs:
39 x1uppere' = μm n1e' = 1 < n1e' < 4
40 x2uppere' =   μm  n2e' = 1 < n2e' < 4
(x2uppere' > x1uppere')
41 ye' =   μm  ya' > ye'
*Where xuppert is the uncalibrated x-value that most appropriately locates the upper corner of the
transitional edge (Edge 1 or Edge 2) using Trace "t"
**The values for n1t and n2t indicate the data point uncertainties associated with the chosen value for 
xuppert 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 corner of the transitional edge, the maximum uncertainty associated with
the identification of this point is ntxrescalx, where nt=1.
*Where ya' and ye' are the uncalibrated y-values associated with Traces a' and e', respectively.

Table 3 - INPUTS (uncalibrated values from Trace b)

Notes
42 x1F = μm z1F = μm (x1ave < x1F )
43 x2F = μm z2F = μm (inflection point)
( x1F < x2F < x3F )
44 x3F = μm z3F = μm (most deflected point)
( x1S = x3F ; z1S = z3F )
45 x2S = μm z2S = μm (inflection point)
46 x3S = μm  z3S = μm ( x3S  < x2ave )
( x1S < x2S < x3S )

 

Table 4 - INPUTS (uncalibrated values from Trace c)

Notes
47 x1F = μm z1F = μm (x1ave < x1F )
48 x2F = μm z2F = μm (inflection point)
( x1F < x2F < x3F )
49 x3F = μm z3F = μm (most deflected point)
( x1S = x3F ; z1S = z3F )
50 x2S = μm z2S = μm (inflection point)
51 x3S = μm z3S = μm ( x3S < x2ave )
( x1S < x2S < x3S )

                    

Table 5 - INPUTS (uncalibrated values from Trace d)

Notes
52 x1F = μm z1F = μm  (x1ave < x1F )
53 x2F = μm z2F = μm (inflection point)
( x1F < x2F < x3F )
54 x3F = μm z3F = μm (most deflected point)
 ( x1S = x3F ; z1S = z3F )
55 x2S = μm z2S = μm (inflection point)
56 x3S = μm z3S = μm ( x3S < x2ave )
( x1S < x2S < x3S )


                                          
 
                                      
 

Table 6 - OUTPUTS (for in-plane length)

Notes
57 x1ave = μm = (x1uppera' + x1uppera + x1uppere+ x1uppere' ) / 4
58 x2ave = μm = (x2uppera' + x2uppera + x2uppere+ x2uppere' ) / 4
 
59 Lmeasa' =

μm

 = (x2uppera' - x1uppera' ) calx
60 Lmeasa =

μm

 = (x2uppera - x1uppera ) calx
61 Lmease =

μm

 = (x2uppere - x1uppere ) calx
62 Lmease' =

μm

 = (x2uppere' - x1uppere' ) calx
63 Lmeas =

μm

 = (Lmeasa' + Lmeasa + Lmease + Lmease' ) / 4
64 α = ° = tan–1[Δx calx / (Δy caly )]
where Δy = ya' – ye'  and
if  (n1a' + n1e' ) < (n2a' + n2e' ) then
Δx = Δx1 = x1uppera' - x1uppere'
if  (n1a' + n1e' ) > (n2a' + n2e' ) then
 Δx = Δx2 = x2uppera' - x2uppere'
65 f =

μm

 f = x1ave calx
66 l =

μm

 l = (x2ave calx- f) cos(α) + f
 
67 Laligned =

μm

 = aligned length
= l - f
68 L =

μm

 = Laligned + Loffset
 
69 v1end =

μm

 = one endpoint along the v-axis
(the axis parallel to the length of the fixed-fixed beam)
= f - Loffset / 2
70 v2end =

μm

 = the other endpoint along the v-axis
= l + Loffset / 2
                             

Uncertainty Outputs (for in-plane length):
71 uLL  =  μm = ( LmaxL − LminL ) / 6
LminL= Lmeasmin cos(α)+Loffset
Lmeasmin = (Lmeasmina'+Lmeasmina
                       +Lmeasmine+Lmeasmine')/4
Lmeasmint = Lmeast−(n1t+n2t) xres calx

LmaxL= Lmeasmax cos(α)+Loffset
Lmeasmax = (Lmeasmaxa'+Lmeasmaxa
                       +Lmeasmaxe+Lmeasmaxe')/4
Lmeasmaxt = Lmeast+(n1t+n2t) xres calx
72 uLrepeat(L)  =  μm = σrepeat(L) cos(α)
= STDEV(Lmeasa', Lmeasa, Lmease,
                                Lmease') cos(α)
73 uLxcal  =  μm = ( σxcal / rulerx ) Lmeas cos(α)
74 uLalign  = μm = |(Lmaxalign − Lminalign) / (2 SQRT(3))|
where Lmaxalign = Lmeas cos(αmax) + Loffset
and Lminalign = Lmeas cos(αmin) + Loffset
αmin = tan-1 [ Δx calx / (Δy caly)
                       − 2 xres calx / (Δy caly) ]
αmax = tan-1 [ Δx calx / (Δy caly)
                       + 2 xres calx / (Δy caly) ]
75 uLoffset  = μm = |Loffset | / 3
76 uLrepeat(samp) =   μm = σLrepeat(samp)'
 
77 ucL   = μm = SQRT [uLL2 + uLrepeat(L)2 + uLxcal2
      + uLalign2+ uLoffset2 + uLrepeat(samp)2]
(Each of the standard uncertainty components is obtained using a Type B analysis, except for uLrepeat(L) and uLrepeat(samp), which use a Type A analysis.)

 

Table 7 - OUTPUTS (for residual strain)

Notes
  Points Trace b Trace c Trace d  
78 g = μm μm μm g = (x1Ft calx- f) cos(α) + f
79 h = μm μm μm h = (x2Ft calx- f) cos(α) + f
80 i = μm μm μm i = (x3Ft calx- f) cos(α) + f
  = (x1St calx- f) cos(α) + f
81 j = μm μm μm j = (x2St calx- f) cos(α) + f
82 k = μm μm μm k = (x3St calx- f) cos(α) + f
 
83 s =          
s = 1  (for downward bending fixed-fixed beams)
s = −1 (for upward bending fixed-fixed beams)		 from Trace c
84 AF = μm μm μm use for plotting
85 w1F = use for plotting
86 AS = μm μm μm use for plotting
87 w3S = use for plotting
88 veF = μm μm μm v-value of first inflection point
89 veS = μm μm μm v-value of second inflection point
90 εr0  = ×10-6  ×10-6 ×10-6  residual strain assuming a zero, axial-compressive, critical force
91 εrt   = ×10-6  ×10-6 ×10-6 residual strain assuming a non-zero, axial-compressive, critical force
 
92 εr  =  × 10-6 = average residual strain value from Traces b, c, and d
(USE THIS VALUE)

 

Table 8 - Preliminary uncertainty OUTPUTS (for residual strain)
93 uW =   × 10-6     
    Trace b Trace c Trace d
94 uLt =  ×10-6 ×10-6 ×10-6
95 uzrest =   ×10-6 ×10-6 ×10-6
96 uxcalt =   ×10-6 ×10-6 ×10-6
97 uxrest =   ×10-6 ×10-6 ×10-6
98 uRavet =   ×10-6 ×10-6 ×10-6
99 unoiset =   ×10-6 ×10-6 ×10-6
100 ucertt =   ×10-6 ×10-6 ×10-6
101 urepeat(shs)t =       ×10-6 ×10-6 ×10-6
102 udriftt =   ×10-6 ×10-6 ×10-6
103 ulineart =      ×10-6 ×10-6 ×10-6
104 ucorrectiont =     ×10-6 ×10-6 ×10-6
105 urepeat(samp)t = ×10-6 ×10-6 ×10-6
         
106 ucεrt =       ×10-6  ×10-6  ×10-6 

 

Table 9 - Uncertainty OUTPUTS (for residual strain)
 Averaging the values from Traces b, c, and d, where applicable
107 uW =      × 10-6  due to variations across width of beam (using data from Traces b, c, and d)
108 uL =   × 10-6  due to the measurement uncertainty of L,
but not including uLxcal
109 uzres =    × 10-6  due to the resolution of the interferometer in the z-direction
110 uxcal =    × 10-6  due to the calibration in the x-direction
111 uxres =    × 10-6  due to the resolution of the interferometric microscope in the x-direction
112 uRave =    × 10-6  due to the sample's surface roughness
113 unoise =    × 10-6  due to interferometric noise
114 ucert =    × 10-6  due to the uncertainty of the value of the physical step height standard
115 urepeat(shs) =        × 10-6  due to the repeatability of a measurement taken on the physical step height standard
116 udrift =    × 10-6  due to the amount of drift during the data session
117 ulinear =       × 10-6  due to the deviation from linearity of the data scan
118 ucorrection =       × 10-6  due to the uncertainty of the relative residual strain correction term
119 urepeat(samp) =  × 10-6  due to the uncertainty of residual strain repeatability measurements
       
120 ucεr =        × 10-6  = (ucεrb + ucεrc + ucεrd) / 3
ucer  = SQRT[uW2 + uL2 + uzres2 + uxcal2 + uxres2 + uRave2 + unoise2 + ucert2
+ urepeat(shs)2 + udrift2 + ulinear2 + 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.)
121 ucer =       × 10-6  combined standard  uncertainty
122 2ucer =  Uer =    × 10-6       expanded uncertainty
123 3ucer =    × 10-6     
124 εr − Uer = × 10-6    a lower bound for εr
125 εr + Uer = × 10-6    an upper bound for εr

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 ucer, the residual strain is believed to lie in the interval er ± ucer (expansion factor
k=1) representing a level of confidence of approximately 68 %. 

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 t should be between 0.000 μm and 10.000 μm.
5. The value for the design length should be between 0 μm and 1000 μm. 
6. The measured value for L is more than 3ucL from the design length.
7. The value for the design width should be between 0 μm and 60 μm.
8. Is the magnification appropriate given the design length ?
9. Magnifications at or less than 2.5× shall not be used.
10. 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.
11. The value for rulerx should be between 0 μm and 1500 μm.
12. The value for σxcal should be between 0 μm and 4 μm.
13. The value for xres should be between 0 μm and 2.00 μm.
14. Is 0.95 < calz < 1.05 but not equal to "1"?  If not, recheck your z-calibration.
15. The value for cert should be greater than 0 μm and less than 25 μm.
16. The value for σcert should be between 0 μm and 0.100 μm.
17. The value for σ6same should be between 0 μm and 0.200 μm.
18. The value for z6same should be between (cert-0.150 μm)/calz and (cert+0.150 μm)/calz.
19. The value for zdrift should be between 0 μm and 0.100 μm.
20. The value for zlin should be between 0 % and 5 %.
21. The value for zres should be greater than 0 μm and less than or equal to 0.005 μm.
22. The value for sLrepeat(samp)' should be greater than or equal to 0 μm and less than or equal to 5 μm.
23. The value for srepeat(samp) should be greater than 0 % and less than or equal to 20 %.
24. The value for δεrcorrection should be between −0.3 and 0.3.
25. The value for Rtave should be between 0 μm and 0.500 μm and greater than Rave.
26. The value for Rave should be between 0 μm and 0.050 μm.
27. The value for Loffset should be between −20.0 μm and 20.0 μm, inclusive.
28. Alignment has not been ensured.
29. Data has not been leveled.
30. The fixed-fixed beam is exhibiting stiction.
31. x2uppert should be greater than x1uppert.
32. ya' should be greater than ye'.
33. n1t and n2t should be between 1 and 4, inclusive. 
34. α should be between −2° and 2°.
35. In Traces b, c, and d, the value for s is not the same.
36. x1ave should be < x1F in all traces.
37. x3S should be < x2ave in all traces.
38. In all traces, make sure ( x1F < x2F < x3F ).
39. In all traces, make sure ( x1S < x2S < x3S ).
40. For Trace b, | [h − veF ] | = μm.  This should be < 5 μm.
If it is not, choose (x2F, z2F) such that h is closer to veF = μm.
I.e., such that x2F is closer to [(veF−f)/cosα + f ] /calx= μm.
41. For Trace b, | [j − veS ] | =  μm.  This should be < 5 μm.
If it is not, choose (x2S, z2S) such that j is closer to veS = μm.
 I.e., such that x2S is closer to [(veS−f)/cosα + f ] /calx= μm.
42. For Trace c, | [h − veF ] | = μm.  This should be < 5 μm.
If it is not, choose (x2F, z2F) such that h is closer to veF = μm.
 I.e., such that x2F is closer to [(veF−f)/cosα + f ] /calx= μm.
43. For Trace c, | [j − veS ] | =  μm.  This should be < 5 μm.
If it is not, choose (x2S, z2S) such that j is closer to veS = μm.
 I.e., such that x2S is closer to [(veS−f)/cosα + f ] /calx= μm.
44. For Trace d, | [h − veF ] | =  μm.  This should be < 5 μm.
If it is not, choose (x2F, z2F) such that h is closer to veF = μm.
 I.e., such that x2F is closer to [(veF−f)/cosα + f ] /calx= μm.
45. For Trace d, | [j − veS ] | =  μm.  This should be < 5 μm.
If it is not, choose (x2S, z2S) such that j is closer to veS = μm.
 I.e., such that x2S is closer to [(veS−f)/cosα + f ] /calx= μm.
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 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