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Data Analysis Sheet YM.1

Data analysis sheet for determining the Young's modulus value of a thin film layer from the resonance frequency of a single-layered cantilever


a)                                                             b)
Figure YM.1.1.  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."

IDENTIFYING INFORMATION:



 

Preliminary INPUTS

Description
1 mag= × magnification
2 mat=

composition of the thin film layer
3* ρ= g/cm3 density of the thin film layer
4 σr= g/cm3 the one sigma uncertainty of the value of r
5* μ= ´10-5 Ns/m2 the viscosity of the ambient surrounding the cantilever
6* W= mm suspended width
7 σW= mm the one sigma uncertainty of the value of W
8* t= mm thickness of the thin film layer
9 σthick= mm the one sigma uncertainty of the value of t
10* Einit= GPa an initial estimate for the Young's modulus value of the thin film layer
* The five starred entries in this table are required inputs for the calculations in the Preliminary Estimates Table.

 

Cantilever INPUTS

Description
1

name=

 cantilever name (optional)
2 orient=

 

orientation of the cantilever
3* Lcan= mm suspended cantilever length
4 whichcan=

 

 

 

 

which cantilever on the test chip?
5 σL= mm the one sigma uncertainty of the value of Lcan
6 fresol= Hz the frequency resolution for the given set of measurement conditions
7 fdamped1= kHz the first damped resonance frequency measurement
8 fdamped2= kHz the second damped resonance frequency measurement
9 fdamped3= kHz the third damped resonance frequency measurement
* The starred entry in this table is a required input for the calculations in the Preliminary Estimates Table.

 

Fixed-Fixed Beam INPUTS

(for 2nd and 3rd uc calculation)

Description
1 name2= fixed-fixed beam name (optional)
2 orient2=

orientation of the fixed-fixed beam
3* Lffb=  mm suspended fixed-fixed beam length
4 whichffb=

 

 

 

 

which fixed-fixed beam on the test chip?
5 fffb=  kHz the average 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.

 

Optional INPUTS

For residual stress calculations:

Description
1 εr= ´10-6

the residual strain of the thin film layer

(Compressive residual strain can be found using ASTM E 2245 and Data Sheet RS.1 or RS.2.)
2 uer= ´10-6

the combined standard uncertainty value for residual strain

(For compressive residual strain, uer can be found using Data Sheet RS.1 or RS.2.)
For stress gradient calculations:

 
3 sg= m-1

the strain gradient of the thin film layer

(can be found using ASTM E 2246 and Data Sheet SG.1 or SG.2)
4 usg= m-1

the combined standard uncertainty value for strain gradient

(can be found using Data Sheet SG.1 or SG.2)


                                  

                                  

Preliminary ESTIMATES*

Description
1 fcaninit= kHz

= SQRT[Einit t2 / (38.330 ρ  Lcan4)]

(the estimated resonance frequency of the cantilever)
2 fffbinithi= kHz

= SQRT[Einit t2 / (0.946 ρ Lffb4)]

(the estimated upper bound for the resonance frequency of the fixed-fixed beam)
3 fffbinitlo= kHz

= SQRT[Einit t2 / (4.864 ρ Lffb4)]

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

= W t2 SQRT(ρ Einit) / (24 m Lcan2)

(the estimated Q-factor)
5 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:

Frequency calculations:

Description
1 fdampedave= kHz

= AVE [fdamped1, fdamped2, fdamped3]

(the average damped resonance frequency of the cantilever)
2 fundamped1= kHz

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

(the undamped resonance frequency calculated from the cantilever's first damped resonance frequency measurement)
3 fundamped2= kHz

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

(the undamped resonance frequency calculated from the cantilever's second damped resonance frequency measurement)
4 fundamped3= kHz

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

(the undamped resonance frequency calculated from the cantilever's third damped resonance frequency measurement)
5

fcan=

kHz

= fundampedave =AVE [fundamped1, fundamped2, fundamped3]

(the average undamped resonance frequency of the cantilever)
6 σfreq=

= STDEV (fundamped1, fundamped2, fundamped3)

(the one sigma uncertainty of the value of fcan)

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 fdamped1, fdamped2, and fdamped3 in the second table are displacement
                resonance frequencies.)

           b.  E = 38.330 ρ fdampedave2 Lcan4 / t2  =  GPa
               (Use this value if fdamped1, fdamped2, and fdamped3 in the second table are velocity
                resonance frequencies.)

2.   Minimum uc calculation (the smallest of the three values for uc) = uc =    (USE THIS VALUE)
           a.  uc = SQRT(uthick2 + ur2 + uL2 + ufreq2 + ufresol2 + udamp2) =
                       uthick = GPa
                            ur = GPa
                            uL = GPa
                         ufreq = GPa
                       ufresol = GPa
                       udamp = GPa

           b.  uc =  (Esimple - E ) / 3 = GPa
                        Esimple  =  4.864 ρ fffb2 Lffb4 / t2  = GPa
                            (as obtained from the fixed-fixed beam assuming simply-
                             supported boundary conditions for both supports)

           c.  uc = (E - Eclamped) / 3 =  GPa
                        Eclamped = 0.946 ρ fffb2 Lffb4 / t2 = GPa
                            (as obtained from the fixed-fixed beam assuming
                             clamped-clamped boundary conditions)
                                           
                                                                

Optional OUTPUTS

For residual stress:

Description
1 σr= MPa

= E εr

(the residual stress of the thin film layer)
2 uσr= MPa

= SQRT[uE(σr)2 + uεr(σr)2]

(the combined standard uncertainty value for residual stress)
3 uE(σr)= MPa

= [ (E+3uc)|er| - (E-3uc)|er| ] / 6

(the component in the combined standard uncertainty calculation for residual stress that is due to the measurement uncertainty of E)
4 uεr(σr)= MPa

= [ E(|er|+3uer) - E(|er|-3uer) ] / 6

(the component in the combined standard uncertainty calculation for residual stress that is due to the measurement uncertainty of er)
For stress gradient:

 
5 σg= GPa/m

= E sg

(the stress gradient of the thin film layer)
6 uσg= GPa/m

= SQRT[uE(σg)2 + usg(σg)2]

(the combined standard uncertainty value for stress gradient)
7 uE(σg)= GPa/m

= [ (E+3uc)sg - (E-3uc)sg ] / 6

(the component in the combined standard uncertainty calculation for stress gradient that is due to the measurement uncertainty of E)
8 usg(σg)= GPa/m

= [ E(sg+3usg) - E(sg-3usg) ] / 6

(the component in the combined standard uncertainty calculation for stress gradient that is due to the measurement uncertainty of sg)


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. Is the magnification appropriate given Lcan and Lffb?  The value for mag should be greater than or equal to 20×.
3. The value for r should be between 1.00 g/cm3 and 5.00 g/cm3.
4. The value for sr should be between 0.0 g/cm3 and 0.10 g/cm3.
5. The value for m should be between 1.5´10-5 Ns/m2 and 5.0´10-5 Ns/m2.
6. The value for W should be greater than t and less than Lcan and Lffb.
7. The value for sW should be between 0.0 mm and 2.0 mm.
8. The value for t should be between 0.000 mm and 4.000 mm.
9. The value for σthick should be between 0.0 mm and 0.3 mm.
10. The value for Einit should be between 10 GPa and 300 GPa.
11. The values for Lcan and Lffb should be between 0 mm and 1000 mm and Lffb need not equal Lcan.
12.   The value for σL should be between 0.0 mm and 2.0 mm.
13.   The value for fresol should be between 0 Hz and 50 Hz.
14.   The values for fdamped1, fdamped2, and fdamped3 should be between 5.00 kHz and 300.0 kHz.
15.   The value for fffb should be between 10.0 kHz and 1200 kHz.
16.   If inputted, the value for er should be between -100´10-6 and 100´10-6.
17.   If inputted, the value for uer should be between 0.0 and 4.0´10-6.
18.   If inputted, the value for sg should be between 0.0 m-1 and 20.0 m-1.
19.   If inputted, the value for usg should be between 0.0 m-1 and 2.0 m-1.
20.   The value for pdiff should be between 0 % and 2 %.
21.   The value for σfreq should be between 0.0 kHz and 0.5 kHz, inclusive.
22.   The value of E should be within 20 GPa of Einit.
23.   The value of E should be between Eclamped and Esimple.
24.   The values for uthick, ur, uL, ufreq, ufresol, and udamp should be between 0 GPa and 5 GPa, inclusive.
25.   Each value of uc should be between 0 GPa and 30 GPa.
26.   The smallest value of uc should be between 0 GPa and 10 GPa.

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Date created: 6/5/2006
Last updated: 1/7/2008