PML Banner
 
Data Analysis Sheet YM.3

Data analysis sheet for determining the Young's modulus value of a thin film layer
for use with the MEMS 5-in-1 RMs

a)
b)           
Figure YM.3.1.  For CMOS cantilever a) a design rendition and b) a cross section

To obtain the following measurements, consult SEMI standard test method MS4 entitled
"Test Method for Young's Modulus Measurements of Thin, Reflecting Films Based on the
Frequency of Beams in Resonance."



                                  


                                  

date (optional) = / /






comments (optional) =
 

Table 1 - Preliminary INPUTS

Description

1 temp the temperature during measurement (should be held constant)
2 relative
humidity
% the relative humidity during measurement (if not known, enter -1)
3 × the magnification
4 mat

  

 

the composition of the thin film layer
5* ρ g/cm3 the density of the thin film layer
6 σρ g/cm3 the one sigma uncertainty of the value of ρ
7* μ ×10-5 Ns/m2 the viscosity of the ambient surrounding the cantilever
8* W μm the suspended beam width
9* t μm the thickness of the thin film layer
(as found using Data Sheet T.1 or Data Sheet T.3)
10 σthick μm the one sigma uncertainty of the value of t
(as found using Data Sheet T.1 or Data Sheet T.3)
11 dgap μm the gap depth (distance between the bottom of the suspended beam and the underlying layer)
12* Einit GPa the initial estimate for the Young's modulus value of the thin film layer
13 finstrument MHz used for calibrating the time base of the instrument:  the frequency setting for the calibration measurements (or the manufacturer's specification for the clock frequency)
14 fmeter  MHz used for calibrating the time base of the instrument:  the calibrated average frequency of the calibration measurements (or the calibrated average clock frequency) taken with a frequency meter
15 smeter  Hz used for calibrating the time base of the instrument:  the standard deviation of the frequency measurements taken with the frequency meter
16 ucertf  Hz used for calibrating the time base of the instrument:  the certified uncertainty of the frequency measurements as specified on the frequency meter's certificate
* The five starred entries in this table are required inputs for the calculations in the Preliminary Estimates Table.

 

Table 2 - Cantilever INPUTS

Description

17

name

the cantilever name (optional)
18

    

   

 

 

 

the orientation of the cantilever
19* Lcan μm the suspended cantilever length
20

    

    

  

     

    
   

indicates which cantilever on the test chip, where "first" corresponds to the topmost cantilever in the column or array that has the specified length?
21 σL μm the one sigma uncertainty of the value of Lcan
22 fresol Hz the uncalibrated frequency resolution for the given set of measurement conditions
23 fmeas1 kHz the first uncalibrated, damped resonance frequency measurement (or the first uncalibrated, undamped resonance frequency measurement, for example, if the measurements were performed in a vacuum)
24 fmeas2 kHz the second uncalibrated, damped resonance frequency measurement (or the second uncalibrated, undamped resonance frequency measurement, for example, if the measurements were performed in a vacuum)
25 fmeas3 kHz the third uncalibrated, damped resonance frequency measurement (or the third uncalibrated, undamped resonance frequency measurement, for example, if the measurements were performed in a vacuum)
26 fcorrection kHz the correction term for the cantilever's resonance frequency
27 σsupport kHz the uncertainty in the cantilever's resonance frequency due to a non-ideal support (or attachment conditions)
28 σcantilever kHz the uncertainty in the cantilever's resonance frequency due to geometry and/or composition deviations from the ideal
* The starred entry in this table is a required input for the calculations in the Preliminary Estimates Table.

 

Table 3 - Fixed-Fixed Beam INPUTS

(if cantilever not available)

Description

29 name2 the fixed-fixed beam name (optional)
30

       

     

  

the orientation of the fixed-fixed beam
31* Lffb μm the suspended fixed-fixed beam length
32

    

  

     

   

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?
33 fffb kHz the average uncalibrated resonance frequency of the fixed-fixed beam
* The starred entry in this table is a required input for the calculations in the Preliminary Estimates Table.

 

Table 4 - Optional INPUTS

For residual stress calculations:

Description

34 εr ×10-6

the residual strain of the thin film layer

(as found using ASTM E 2245 and Data Sheet RS.3 for compressive residual strain)

35 ucεr ×10-6

the combined standard uncertainty value for residual strain

(as found using Data Sheet RS.3 for compressive residual strain)

For stress gradient calculations:

 

36 sg m-1

the strain gradient of the thin film layer

(as found using ASTM E 2246 and Data Sheet SG.3)

37 ucsg m-1

the combined standard uncertainty value for strain gradient

(as found using Data Sheet SG.3)


                                  

                                  

Table 5 - Preliminary ESTIMATES*

Description

38 fcaninit kHz

= SQRT[Einit t2 / (38.330 ρ  Lcan4)]

(the estimated resonance frequency of the cantilever)

39 fffbinithi kHz

= SQRT[Einit t2 / (0.946 ρ Lffb4)]

(the estimated upper bound for the resonance frequency of the fixed-fixed beam)

40 fffbinitlo kHz

= SQRT[Einit t2 / (4.864 ρ Lffb4)]

(the estimated lower bound for the resonance frequency of the fixed-fixed beam)

41 Q

= W t2 SQRT(ρ Einit) / (24 μ Lcan2)

(the estimated Q-factor)

42 pdiff %

={1-SQRT[1-1 / (4 Q2)]}×100 % should be < 2 %

(the estimated percent difference between the damped and undamped resonance frequency of the cantilever)

* The seven starred inputs in the first three tables are required for the calculations in this table.

                                  


OUTPUTS:

Table 6 - Frequency calculations:

Description

43 calf = fmeter / finstrument
(the calibration factor for a frequency measurement)
44 fmeasave kHz

= AVE [fmeas1, fmeas2, fmeas3]calf

(the average calibrated damped resonance frequency of the cantilever, fdampedave, or the average calibrated undamped resonance frequency of the cantilever if, for example, the measurements were performed in a vacuum)

45 fundamped1 kHz

= fdamped1 / SQRT[1-1/(4Q2)] where fdamped1=fmeas1(calf)

(the first calibrated undamped resonance frequency calculated from the cantilever's first damped resonance frequency measurement, if applicable)

46 fundamped2 kHz

= fdamped2 / SQRT[1-1/(4Q2)] where fdamped2=fmeas2(calf)

(the second calibrated undamped resonance frequency calculated from the cantilever's second damped resonance frequency measurement, if applicable)

47 fundamped3 kHz

= fdamped3 / SQRT[1-1/(4Q2)] where fdamped3=fmeas3(calf)

(the third calibrated undamped resonance frequency calculated from the cantilever's third damped resonance frequency measurement, if applicable)

48

fundampedave

kHz

= AVE [fundamped1, fundamped2, fundamped3]

(the average calibrated undamped resonance frequency of the cantilever assuming fmeas1, fmeas2, and fmeas3 from the second table are damped resonance frequencies)

49 σfundamped

= STDEV (fundamped1, fundamped2, fundamped3)

(the one sigma uncertainty of the value of fundampedave assuming fmeas1, fmeas2, and fmeas3 from the second table are damped resonance frequencies)

50 fcan = fundampedave + fcorrection
(the modified resonance frequency of the cantilever for use if fmeas1, fmeas2, and fmeas3 from the second table are damped resonance frequencies)
51 fmeasavenew = fmeasave + fcorrection
(the modified resonance frequency of the cantilever for use if fmeas1, fmeas2, and fmeas3 from the second table are undamped resonance frequencies)

1.   Young's modulus calculation (as obtained from the cantilever assuming clamped-free boundary
      conditions)
:
           a. 
E = 38.330 ρ fcan2 Lcan4 / t2 GPa  
              
(Use this value if fmeas1, fmeas2, and fmeas3 in the second table are damped
                resonance frequencies.)

          
b. 
E = 38.330 ρ fmeasavenew2 Lcan4 / t GPa
               (Use this value if fmeas1, fmeas2, and fmeas3 in the second table are undamped
                resonance frequencies.)

           c. 
ucE = σE = E SQRT[(σ
ρ/ρ)2 + 4(σfcan/fcan)2  + 16(σL/Lcan)2 + 4(σthick/t)2] = * σE / E = *   σfcan/fcan = SQRT[(σfundamped/fcan)2 + (σfresol/fcan)2 
                                                            
+ (σfreqcal/fcan)2+ (σsupport/fcan)2 + (σcantilever/fcan)2],
                                σfresol = fresol calf / [2SQRT(3)],
                    
and
     σfreqcal = fundampedave [SQRT(σmeter2+ ucertf2) / fmeter]
                                        

                                            σ
ρ/ρ =             Type B   
                                      
σthick/t =             Type B       
                                      σ
L/Lcan =              Type B           
                    
      σfundamped/fcan = *           Type A                            
                                 σ
fresol/fcan =             Type B                             
                          
     σfreqcal/fcan       Type B
                              σ
support/fcan       Type B
                            σ
cantilever/fcan       Type B                         
               *assumes fmeas1, fmeas2, and fmeas3 in the second table are damped resonance frequencies

                
UE = 2ucE =   GPa 
(expanded uncertainty)
                 3ucE =   GPa

          
d.   E - UE = GPa    (a lower bound for E)
                
E + UE = GPa    (an upper bound for E)
               
(assuming fmeas1, fmeas2, and fmeas3 in the second table are damped resonance frequencies)

           e.   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 ucE
, the Young's modulus value is believed to lie in the interval E ± ucE
                
(expansion factor k=1) representing a level of confidence of approximately 68 %.     

2.  Young's modulus calculation (as obtained from a fixed-fixed beam...not recommended):
           a.
  Esimple  =  4.864 ρ ( fffb calf )2 Lffb4 / t2  = GPa
                            (as obtained from the fixed-fixed beam assuming simply-
                             supported boundary conditions for both supports)

   
        b.  Eclamped = 0.946 ρ ( fffb calf )2 Lffb4 / t2 = GPa
                            (as obtained from the fixed-fixed beam assuming
                             clamped-clamped boundary conditions)

           c. 
E = (Esimple + Eclamped) / 2 =
(use this value, if must)

           d. 
uE = (Esimple - Eclamped) / 6 =
     (as obtained from a Type B analysis)

           e.   Report the results as follows:  If it is assumed that the estimated value of the standard
                 uncertainty,
uE, is approximately Gaussianly distributed, the Young's modulus value is
                 believed to lie in the interval E ± uE
(expansion factor k=1) representing a level of
                 confidence of approximately 68 %.
  

 

Table 7 - Optional OUTPUTS (using E and ucE from the cantilever and assuming fmeas1, fmeas2, and fmeas3 in the second table are damped resonance frequencies)

For residual stress:

Description

52 σr MPa

= E εr

(the residual stress of the thin film layer)

53 ucσr MPa

= |σr| SQRT[(σE / E)2 + (σεr / εr)2]

(the combined standard uncertainty value for residual stress
where σεr is equated with ucεr)

54 σσr / |σr| where σσr is equated with ucσr
55 σE / E  

as obtained from this data sheet

56 σεr / |εr| where σεr is equated with ucεr and where εr and ucεr were obtained from Data Sheet RS.3
57 2ucσr MPa = Uσr
the expanded uncertainty for residual stress
58 3ucσr MPa three times the combined standard uncertainty for residual stress
59 σr-Uσr MPa a lower bound for σr
60 σr+Uσr MPa an upper bound for σr
For stress gradient:

 

61 σg GPa/m

= E sg

(the stress gradient of the thin film layer)

62 ucσg GPa/m

= σg SQRT[(σE / E)2 + (σsg / sg)2]

(the combined standard uncertainty value for stress gradient
where
σsg is equated with ucsg
)

63 σσg / σg where σσg is equated with ucσg
64 σE / E as obtained from this data sheet
65 σsg / sg where σsg is equated with ucsg and where sg and ucsg were obtained from Data Sheet SG.3
66 2ucσg GPa/m = Uσg
the expanded uncertainty for stress gradient
67 3ucσg GPa/m three times the combined standard uncertainty for stress gradient
68 σg-Uσg GPa/m a lower bound for σg
69 σg+Uσg GPa/m an upper bound for σg


Modify the input data, given the information supplied in any flagged statement below, if applicable, then recalculate:
 
1. Please provide inputs to Tables 1 and 2 for calculations using data from a cantilever.
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. If applicable, please provide inputs to Table 3, ρ, W, t, and Einit for calculations using data from a fixed-fixed beam.
5. The value for mag should be greater than or equal to 20×.
6. The value for ρ should be between 1.00 g/cm3 and 5.00 g/cm3.
7. The value for σρ should be between 0.0 g/cm3 and 0.10 g/cm3.
8. The value for μ should be between 0.70×10-5 Ns/m2 and 3.0×10-5 Ns/m2.
9. The value for W should be greater than t and less than Lcan.
10. If Lffb is inputted, the value for W should be greater than t and less than Lffb.
11. The value for t should be between 0.000 μm and 10.000 μm.
12. The value for σthick should be between 0.0 μm and 0.5 μm.
13.   Squeeze film damping expected for the cantilever since dgap < W / 3.
14.   The value for Einit should be between 10 GPa and 300 GPa.
15. The values for σmeter and ucertf should be between 0.0 Hz and 25.0 Hz, inclusive.
16.   The value for Lcan should be between 0 μm and 1000 μm.
17.   The value for σL should be between 0.0 μm and 2.0 μm.
18.   The value for fresol should be between 0 Hz and 50 Hz.
19.   The values for fmeas1, fmeas2, and fmeas3 should be between 5.00 kHz and 300.0 kHz.
20. The value for fcorrection should be between -10 kHz and 10 kHz, inclusive.
21. The values for σsupport and σcantilever should be between 0 kHz and 10 kHz, inclusive.
22.   If inputted, the value for Lffb should be between 0 μm and 1000 μm.
23.   If inputted, the value for fffb should be between 5.0 kHz and 1200 kHz.
24.   If inputted, the value for εr should be between -4500×10-6 and 4500×10-6 and not equal to 0.0.
25.   If inputted, the value for ur should be between 0.0 and 300.0×10-6.
26.   If inputted, the value for sg should be between 0.0 m-1 and 1500.0 m-1.
27.   If inputted, the value for ucsg should be between 0.0 m-1 and 100.0 m-1.
28.   The values for fmeas1, fmeas2, and fmeas3 are not within 20 kHz of fcaninit.
29. If inputted, the value for fffb should be between fffbinitlo and fffbinithi.
30. The value for pdiff should be between 0 % and 2 %.
31. The value for calf should be between 0.9990 and 1.0010.
32. The value for σfundamped should be between 0.0 kHz and 0.5 kHz, inclusive.
33. The value of E obtained from the cantilever should be within 60 GPa of Einit.
34. The value of E obtained from the cantilever should be greater than 2ucE.
35. If applicable, the value of E obtained from the fixed-fixed beam should be within 70 GPa of Einit.
36. If applicable, the value of uE obtained from the fixed-fixed beam should be between 0 GPa and 70 GPa.

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: 6/5/2006
Last updated:
4/26/2013