date data taken (optional) = / /
identifying words (optional)  = 
instrument(s) used (optional)  = 
fabrication facility/process (optional)  = 
test chip name (optional)  = 
test chip number (optional)  = 
filename of 3-D data set from test structure #1 (optional) =
filename of 3-D data set from test structure #2 (optional) =
filename of 3-D data set from test structure #3 before etch (optional) =
filename of 3-D data set from test structure #3 after etch (optional) =  

Preliminary INPUTS
Data Set Prelims			Description
1	mag =	×	magnification
2	orient =	0°      90°   other	orientation of the test structure on the test chip
3	align =	Yes          No	alignment ensured ?
4	level =	Yes       No	data leveled ?
5	cert =	μm	certified value of physical step height used for calibration
6	σcert =	μm	certified one sigma uncertainty of the certified physical step height used for calibration
7	zrepeat(shs) =	μm	maximum uncalibrated range of the six calibration measurements taken before the data session at the same location on the physical step height or after the data session at the same location on the physical step height (whichever is larger)
8	z6 =	μm	the uncalibrated average of the six calibration measurements from which zrepeat(shs) is found
9	zdrift =	μm	uncalibrated drift in the calibration data (i.e., the uncalibrated positive difference between the average of the six calibration measurements taken before the data session at the same location on the physical step height and the average of the six calibration measurements taken after the data session at the same location on the physical step height)
10	calz =		the z-calibration factor = the certified value of the physical step height divided by the average of the twelve calibration measurements taken at the same location on the physical step height
11	zperc =	%	if applicable, over the instrument's total scan range, the maximum percent deviation from linearity, as quoted by the instrument manufacturer (typically less than 3%)
12	crystal =	%	the estimated one sigma uncertainty for the crystal lattice percentages
13	sigma =		= 1 for a one sigma uncertainty analysis of the thicknesses = 2 for a two sigma uncertainty analysis of the thicknesses (recommended) = 10 to determine if the 3uc error bars touch or overlap

                                      

                                    

 

TABLE 1a - Calibrated Step Height Measurements (in Micrometers)^*,**

#	Step	Height	uWstep	ucert	urepeat(shs)	udrift	ulinear	ucW
1	step1AB
2	step1CD
3	step1EF
4	step1GH
5	step1rA
6	step1rD
7	step1rE
8	step2rA
9	step2AB
10	step2BC
11	step2CD
12	step2BD
13	step3AB(0)
14	step3AB(n)-
15	step3BC(0)
16	step3CD(0)

^* Supply inputs to the columns labeled "Height" and "u_Wstep."
^** Where u_cert = |stepN_XY| σ_cert / cert
and u_repeat(shs) = |stepN_XY| z_repeat(shs) / [2 (1.732) z₆]
and u_drift = |stepN_XY| (z_drift cal_z) / [2 (1.732) cert]
and u_linear = |stepN_XY| z_perc / (1.732)
and u_cW = (u_Wstep² + u_cert² + u_repeat(shs)² + u_drift² + u_linear²)^1/2
where each standard uncertainty component is obtained using a Type B analysis,
except for u_Wstep, which uses a Type A analysis.

TABLE 1b - Calibrated Step Height Measurements from Table 1a (in Micrometers) with Additional Uncertainty Components^*,**,***
[where σ_rough = μm^****]

#	Step	Height	σplatNX	σplatNY	ubasic	uWstep	uLstep	ures*****	uc
1	step1AB
2	step1CD
3	step1EF
4	step1GH
5	step1rA
6	step1rD
7	step1rE
8	step2rA
9	step2AB
10	step2BC
11	step2CD
12	step2BD
13	step3AB(0)
14	step3AB(n)-
15	step3BC(0)
16	step3CD(0)

^* Supply inputs for "σ_platNX," "σ_platNY," "u_res," and "σ_rough."
^** The values for "Height" and "u_Wstep" are taken directly from Table 1a when the "Calculate and Verify" button is pushed.
^*** Where u_basic = (u_cert² + u_repeat(shs)² + u_drift² + u_linear²)^1/2
and u_Lstep = [(σ_platNX - σ_rough)² + (σ_platNY - σ_rough)²]^1/2
and u_c = (u_basic² + u_Wstep² + u_Lstep² + u_res²)^1/2
where each standard uncertainty component is obtained using a Type B analysis,
except for u_Wstep, which uses a Type A analysis.
^****Where σ_rough is the smallest of all the values for σ_platNX and σ_platNY
^***** Non-zero data can be added to the column labeled "u_res" to obtain E_n values less than or equal to 1.0 in Tables 5 and 6.

TABLE 2 - Oxide Thickness Values From Capacitances^*,**
[with σ_ε = (aF/μm)^*** and σ_resCa = (aF/μm²)^***]

#	Thickness Designation	Ca	σCa***	ures	t	uc
		(aF/μm2)	(aF/μm2)	(μm)	(μm)	(μm)
1	tfox(p1/sub)elec
2	tthin(p1/aan)elec
3	tfox(p2/sub)elec
4	tthin(p2/aan)elec
5	tthin(p2/p1)elec
6	[tfox,m1(pmd/sub)+tpmd(m1/fox)]elec
7	tpmd(m1/aan)elec
8	tpmd(m1/p1)elec
9	tpmd(m1/p2)elec
10	[tfox,m2(pmd/sub)+tpmd(imd/fox)
	+timd(m2/pmd)]elec
11	[tpmd(imd/aan) +timd(m2/pmd)]elec
12	[tpmd(imd/p1) +timd(m2/pmd)]elec
13	[tpmd(imd/p2) +timd(m2/pmd)]elec
14	timd(m2/m1)elec

^* Supply inputs for "C_a," "σ_Ca," "σ_ε," and "σ_resCa."
^** Where t = ε_SiO2 / C_a with ε_SiO2 = 34.5 aF/μm
and u_c = (u_Ca² + u_ε² + u_res²)^1/2
with u_Ca = [ε_SiO2 / (C_a + σ_Ca) - ε_SiO2 / (C_a - σ_Ca)] / 2
and u_ε  = [(ε_SiO2 + 3σ_ε) / C_a - (ε_SiO2 - 3σ_ε) / C_a ] / [2 (1.732)] 
and u_res = σ_resCa t / C_a
where each standard uncertainty component is obtained using a Type B analysis.
^*** Non-zero data can be added to "σ_resCa" to obtain E_n values less than or equal to 1.0
in Tables 5 and 6.  Alternatively or in addition, the value(s) for σ_ε and/or σ_Ca can be modified. 

 

TABLE 3 - Thickness Values For The Interconnects^*,**

#	Symbol	Rs	σRs***	ρ	σρ***	ures***	t	uc
		(Ω/□)	(Ω/□)	(Ω-μm)	(Ω-μm)	(μm)	(μm)	(μm)
1	t(p1)elec
2	t(p2)elec
3	t(m1)elec
4	t(m2)elec

^* Supply inputs to the columns labeled "R_s," "σ_Rs," "ρ," "σ_ρ," and "u_res."
^** Where t = ρ / R_s
and u_c = (u_Rs² + u_ρ² + u_res²)^1/2
with u_Rs = [ρ  / (R_s + σ_Rs) - ρ  / (R_s - σ_Rs)] / 2
and u_ρ  = [(ρ + 3σ_ρ) / R_s - (ρ - 3σ_ρ) / R_s ] / [2 (1.732)]
where each standard uncertainty component is obtained using a Type B analysis.
^*** Non-zero data can be added to the column labeled "u_res" to obtain E_n values less than or equal
to 1.0 in Tables 5 and 6.  Alternatively or in addition, the values for σ_Rs and/or σ_ρ can be modified. 

 

TABLE 4 - Crystal Lattice Percentages

#	Approach	%tab (%)	%tbe (%)
1	Electrical*
2	Physical**

^*  A prediction for the ideal case
^**  As calculated from step1_AB in Table 1a and t_{fox(p1/sub)elec} and t_{thin(p1/aan)elec} in Table 2


TABLE 5 - Calculated Thickness Values for the Given Thicknesses^*,**

#	Thickness Designation	tphys (μm)	uc,phys (μm)	telec (μm)	uc,elec (μm)	En***
							En < 1.0 ?
1	tfox(p1/sub)
2	tthin(p1/aan)
3	t(p1)
4	t(p1')
5	tfox(p2/sub)
6	tthin(p2/aan)
7	tthin(p2/p1)
8	t(p2)
9****	tfox,m1(pmd/sub)
10	tpmd(m1/fox)
11	tpmd(m1/aan)
12	tpmd(m1/p1)
13	tpmd(m1/p2)
14	tpmd(imd/fox)
15	tpmd(imd/aan)
16	tpmd(imd/p1)
17	tpmd(imd/p2)
18	tfox,m1(pmd/sub)+tpmd(m1/fox)
19	t(m1)
20****	tfox,m2(pmd/sub)
21	tpmd(imd/aan)+timd(m2/pmd)
22	tpmd(imd/p1)+timd(m2/pmd)
23	tpmd(imd/p2)+timd(m2/pmd)
24	timd(m2/pmd)
25	timd(m2/m1)
26	tfox,m2(pmd/sub)+tpmd(imd/fox)
	+timd(m2/pmd)
27	t(m2)
28	timd(gl/pmd)
29	timd(gl/m1)
30	t(gl)
31	t(ni)

^* For use in calculations, choose the thickness (either physical or electrical) with the lower value of u_c.
^** This analysis assumes that [t_pmd(imd/aan)+t_imd(m2/pmd)]_phys=[t_pmd(imd/aan)+t_imd(m2/pmd)]_elec
and t_{pmd(m1/p1)phys}=t_{pmd(m1/p1)elec}.
^*** Where E_n = |t_phys - t_elec| / [sigma (u_c,phys² + u_c,elec²)^1/2]    if sigma ¹ 10
and E_n = |t_phys - t_elec| / (3u_c,phys+3u_c,elec)    if sigma = 10.  For the above data, sigma = .
If sigma = 2 (recommended), then 95 % of the data in Tables 5 and 6 should have E_n values less than or equal to 1.0.
^**** If t_fox(pmd/sub) = t_{fox,m1(pmd/sub)} = t_{fox,m2(pmd/sub)}, then for the sample data t_phys for #9 is the
thickness to use in calculations due to it having the lower value of u_c.

 

TABLE 6 - Calculated Thickness Values For The Virtual Oxides^*

#	Thickness Designation	tphys (μm)	uc,phys (μm)	telec (μm)	uc,elec (μm)	En**
							En < 1.0 ?
1	tthin,be(p1/aan)
2	tthin,ab(p1/aan)
3	tthin,be(p2/aan)
4	tthin,ab(p2/aan)
5	tthin,be(p2/p1)
6	tthin,ab(p2/p1)
7	tfox,be(p1/sub)
8	tfox,ab(p1/sub)
9	tfox,ab(p2/sub)
10	tfox,ab,m1(pmd/sub)
11	tfox,ab,m2(pmd/sub)

^* For use in calculations, choose the thickness (either physical or electrical) with the lower value of u_c.
^** Where E_n = |t_phys - t_elec| / [sigma (u_c,phys² + u_c,elec²)^1/2]   if sigma ¹ 10
and E_n = |t_phys - t_elec| / (3u_c,phys+3u_c,elec)   if sigma = 10.  For the above data, sigma = .
If sigma = 2 (recommended), then 95 % of the data in Tables 5 and 6 should have E_n values less than or equal to 1.0.

Results for number of E_n values less than or equal to 1.0 in Tables 5 and 6 with sigma = :
                            number of "yeses" =    percent "yeses"= %
                               number of "nos" =       percent "nos"= %
     total number of "yeses" and "nos" =                    total = %

Report the results as follows:  Since it can be assumed that the estimated values of the uncertainty
components are approximately uniformly or Gaussianly distributed with approximate combined standard
uncertainty u_c, the thickness is believed to lie in the interval t ± u_c (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.		Is the magnification appropriately greater than 2.5×?
3.		Alignment has not been ensured.
4.		Data has not been leveled.
5.		The value for cert  should be between 0.000 μm and 15.000 μm.
6.		The value for σcert  should be between 0.000 μm and 0.100 μm.
7.		The value for zrepeat(shs)  should be between 0.000 μm and 0.050 μm.
8.		The value for z6  should be between (cert- 0.100 μm)/calz and (cert + 0.100 μm)/calz.
9.		The value for zdrift  should be between 0.000 μm and 0.050 μm, inclusive.
10.		The value for calz should be between 0.900 and 1.100, but not equal to 1.000.
11.		The value for zperc should be between 0.0 % and 3.0 %, inclusive.
12.		The value for crystal should be between 0.0 % and 3.0 %, inclusive.
13.		The value for sigma should be between 1 and 3, inclusive, or equal to 10.
14.		The values for the steps in Table 1a should be between -2.500 μm and 2.500 μm.
15.		The values for the uWsteps in Table 1a should be between 0.0000 μm and 0.0300 μm.
16.		The value of σrough for Table 1b should be between 0.0000 μm and the smallest value for σplatNX and σplatNY, inclusive.
17.		The values for σplatNX and σplatNY in Table 1b should be between 0.0000 μm and 0.0200 μm, inclusive.
18.		The values for ures in Table 1b should be between 0 μm and 0.0500 μm, inclusive.
19.		The value of σε  for Table 2 should be between 0.0 aF/μm and 0.3 aF/μm, inclusive.
20.		The value of σresCa for Table 2 should be between 0 aF/μm2and 3.0 aF/μm2, inclusive.
21.		The values for Ca in Table 2 should be between 1.0 aF/μm2 and 1300.0 aF/μm2.
22.		The values for σCa in Table 2 should be between 0.00 aF/μm2 and 7.00 aF/μm2.
23.		The values of Rs in Table 3 should be between 10.0 Ω/□ and 35.0 Ω/□ for t(p1)elec and t(p2)elec and between 0.0100 Ω/□ and 0.0700 Ω/□ for t(m1)elec and t(m2)elec.
24.		The values of σRs in Table 3 should be between 0.05 Ω/□ and 0.50 Ω/□ for t(p1)elec and t(p2)elec and between 0.0001 Ω/□ and 0.0050 Ω/□ for t(m1)elec and t(m2)elec.
25.		The values of ρ in Table 3 should be between 5.0 Ω-μm and 10.0 Ω-μm for t(p1)elec and t(p2)elec and between 0.020 Ω-μm and 0.040 Ω-μm for t(m1)elec and t(m2)elec.
26.		The values of σρ in Table 3 should be between 0.0 Ω-μm and 0.5 Ω-μm, inclusive, for t(p1)elec and t(p2)elec and between 0.000 Ω-μm and 0.005 Ω-μm, inclusive, for t(m1)elec and t(m2)elec.
27.		The values of ures in Table 3 should be between 0 μm and 0.0500 μm, inclusive.
28.		The physical and electrical crystal lattice percentages in Table 4 are greater than 5 % apart.  Reexamine the data for step1AB in Table 1a and for tfox(p1/sub)elec and tthin(p1/aan)elec in Table 2.
29.		All of the thicknesses in Tables 5 and 6 should be between 0.00 μm and 2.50 μm.
30.		All of the values for uc in Tables 5 and 6 should be between 0.00 μm and 0.30 μm.
31.		There are thicknesses with En values greater than 1.0 in Tables 5 and/or 6 with sigma = .  Is this ok?

---

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: 3/4/2006
Last updated: 4/26/2013