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A study has been made on the effects of scanning electron microscope parameters on the accuracy in
measuring the linear dimensions in microtechnology and nanotechnology. Definitions are given of the
errors with which these parameters should be known for using such microscopes in such technologies.
Key words: scanning electron microscope, relief-structure linear dimensions.
Scanning electron microscopes (SEM) are widely used [1–4]. Their technical and economic parameters are governed
by the characteristics of the electron probe, of which the most important are the geometrical ones: focused electron beam size
(diameter), convergence and divergence angles, and focal depth. That information is important in SEM design and upgrading
and also in using the microscope in research and industry.
Knowledge of the electron-probe geometrical characteristics is not an independent purpose. One needs a correct
understanding of the physical significance of the parameters characterizing narrow electron beams and their effects on the
SEM parameters and the focusing systems, as this enables one to build microscopes with new properties and focusing sys-
tems, which provide marked reduction in the electron-probe dimensions.
However, exact measurement of probe parameters is not restricted to the needs of SEM upgrading. The probe dimen-
sions have a large effect on the determination of microstructure linear dimensions [5–7], particularly in the nanometer range.
We consider what probe parameters and what measurement accuracy are required at present and will be necessary
in the near future for measuring the linear dimensions of microstructure and nanostructure elements.
Relief Structures Used in Microtechnology and Nanotechnology. These have a fairly complicated profile (Fig. 1),
and the details of electron-probe interaction with such a surface [8] (relation between structure and probe parameters) indi-
cate that there are four main groups of structure:
1) rectangular, which are not usually encountered. They have been made especially for use as standard measures [9]
for calibrating SEM [10] and are characterized by ϕ < ϕ
d
/2,where ϕ
d
is the convergence-divergence angle of the electron probe;
2) trapezoidal with small angles of inclination in the side walls, which are the basic form of structures:
d > s = htanϕ; (1)
Measurement Techniques, Vol. 51, No. 6, 2008
ACCURACY IN LINEAR DIMENSIONS
MEASUREMENT IN SCANNING ELECTRON
MICROSCOPES IN MICROTECHNOLOGY
AND NANOTECHNOLOGY
Yu. A. Novikov,
1
A. V. Rakov,
1
and P. A. Todua
2
UDC 537.533
Translated from Izmeritel’naya Tekhnika, No. 6, pp. 15–18, June, 2008. Original article submitted February 8, 2008.
0543-1972/08/5106-0599
©
2008 Springer Science+Business Media, Inc.
599
1
Prokhorov General Physics Institute, Russian Academy of Sciences; e-mail: nya@kapella.gpi.ru.
2
Surface and Vacuum Properties Research Center; e-mail: fgupnicp@mail.ru.
3) trapezoidal with large inclinations, which are fairly often used in that area and are characterized by d << s = htanϕ;
the most important point for these structures is their use for calibrating scanning electron and atomic-force microscopes [11];
4) trapezoidal with negative slopes of the side walls (ϕ < 0). These structures are encountered fairly rarely and are
not used to calibrate SEM.
Measuring Microstructure and Nanostructure Linear Dimensions.
Recent fundamental researches [8] have served to define the positions of the reference points on video signals
obtained in slow secondary electron collection. Figures 2a,3a, and 4a give the actual shapes of the signals, while Figs. 2b,
3b, and 4b show schemes for the signals and the reference points selected on them. These points correspond to signal maxi-
ma or are points of intersection between straight lines approximating individual signal parts (the base level of the signal and
the flanks). Figures 2b,3b, and 4b also show the reference segments (distances between certain reference points). The sizes
of the segments are linearly related to the sizes of the relief structures:
• for rectangular structures,
l = L/M – 2δ; (2)
l = G/M + d; (3)
• for structures with small angles of inclination of the side walls,
t = T/M; (4)
l
p
= (u
p
+ b
p
)/2 = L
p
/M; (5)
l
t
= (u
t
+ b
t
)/2 = L
t
/M; (6)
u
p
= (2L
p
– G
p
)/M + d; (7)
b
p
= G
p
/M – d; (8)
u
t
= (2L
t
– G
t
)/M – d; (9)
b
t
= G
t
/M + d; (10)
• for structures with large angles of inclination of the side walls,
t = T/M; (11)
s = S/M; (12)
d = D/M; (13)
600
Fig. 1. Scheme for trapezoidal stepped structure
with characteristic parameters.
u
p
= L
p
/M + d; (14)
b
p
= G
p
/M – d; (15)
u
t
= L
t
/M – d; (16)
b
t
= G
t
/M + d. (17)
The parameters of the linear equations have a clear-cut physical significance, which has been considered in [8].
601
Fig. 2. a) Shape of actual SEM signal obtained in slow secondary electron collection
on scanning a slot-type groove; b) scheme for signal with measurable parameters.
Fig. 3. The same as Fig. 2 but for a stepped structure with small angles of side
wall inclination.
Fig. 4. The same as in Figs. 2 and 3 but for a stepped structure with large side wall
inclination angles.
Measuring rectangular-structure linear dimensions. For rectangular grooves, parameter G of the video signal (VS)
(Fig. 2b) is related to the width l of the groove by (2), while the distance L between VS maxima is given by (3). The physi-
cal meanings of the d and δ in these formulas have been determined after lengthy and difficult researches [8].
The effects of probe diameter on the accuracy of measuring the dimensions of rectangular structures (RS) have been
dealt with in some detail in [8, 12–14], and the effects on the accuracy of measurement for RS width can be defined by the
following:
(∆l/l)
2
= (1 – d/l)
2
[(∆G/G)
2
+ (∆M/M)
2
] + (∆d/l)
2
. (18)
Measuring the linear dimensions of trapezoidal structures with small side wall inclination angles. For these struc-
tures (typical structures used in microelectronics), the linear equations relating the parameters of structure and signal have
the form of (4)–(16). The geometrical meanings of the quantities in those equations are indicated by Figs. 1b and 3b. The
parameters in linear equations (7)–(10) are the magnification M of the microscope and the diameter d of the electron probe.
These equations have been checked out by measurement of the sizes of the ridges and grooves in a unit with trapezoidal pro-
file [6, 7] on various SEM and various probe electron energies and probe diameters (including variation by defocusing [7]).
Equations (7)–(10) indicate that the probe diameter is important in measurements of linear dimensions for relief
structures. We may estimate the contribution from the error in probe diameter measurement to the total error in determining
microstructure linear-element sizes. From (7)–(10), we readily get the following expressions:
• for ridges
(19)
(20)
• for grooves
(21)
(22)
Measuring the linear dimensions of trapezoidal structures with large side wall inclination angles. Equations (14)–(17)
define the dimensions of the bottom base of the ridges and grooves, so the error in measuring the base is found from (19) and
(21) and that for the top from
(23)
(24)
Linear-Dimension Measurement Error Analysis. Equation (18) for a rectangular groove is a consequence of (21)
for the trapezoidal case, while (23) and (24) for trapezoidal structures with large side wall slopes correspond to (19) and
(21) for trapezoidal structures with small side-wall slopes, apart from change in the variables and sign in the first bracket.
Therefore, the two expressions have identical analyses.
∆∆∆∆uu du LL MM du
tt t tt t
///(/)/.
()
=−
()()
+
+
()
222
2
2
1
∆∆∆∆uu du LL MM du
pp p pp p
///(/)/;
()
=−
()()
+
+
()
222
2
2
1
∆∆∆
∆∆
u
u
d
u
L
LG
G
LG
M
M
d
u
t
tt
t
tt
t
tt t
=+
−
+
−
+
+
22 2 2
2
2
1
2
22
.
∆∆∆∆bb db GG MM db
tt t t t t
///(/)/;
()
=−
()()
+
+
()
222
2
2
1
∆
∆∆
∆∆
u
u
d
u
L
LG
G
LG
M
M
d
u
p
pp
p
pp
p
pp
p
=−
−
+
−
+
+
22
22
2
2
1
2
22
;
∆∆∆∆bb db GG MM db
pp p p p p
///(/)/;
()
=+
()()
+
+
()
222
2
2
1
602
The error in measuring the distances G
p, t
and L
p, t
on the video signals is very much affected by the noise compo-
nent and by the algorithms for searching for the corresponding reference points (Fig. 3b), but with reasonable constraints
(noise contribution not more than 10% of the signal amplitude) and automatic image signal processing in current experiments
imply that ∆G
p, t
/G
p, t
~ ∆L
p, t
/L
p, t
~ 10
–3
, so on the basis of the condition
b
p, t
, u
p, t
>> d (25)
we get that (19)–(22) can be simplified:
(26)
(27)
Current methods of calibrating SEM allow one to obtain relative errors in the magnification ∆M/M in the range
0.2–0.7% [10], while ∆d usually does not exceed 1–2 nm. Then in the range of dimensions for microstructure elements
b
p, t
, u
p, t
> 10 µm, the error in measuring the probe diameter ∆d can be neglected. Then the errors in measuring the top and
bottom bases of the trapezium are determined only by ∆M/M on calibration:
(28)
In the range 10 µm > b
p, t
, u
p, t
> 1 µm, the contribution from ∆d can be neglected only for small probe diameters
(d < 100 nm). In that case, one uses (27). For large diameters, one must use (26) and (27).
In the range 1 µm > b
p, t
, u
p, t
> 100 nm, one cannot neglect ∆d or ∆M for any probe diameters, and it is necessary
to use (26) and (27). Then the contribution to the error in measuring the probe diameter to the total error in determining the
linear dimensions of relief structures may attain 80%.
For microstructure element sizes b
p, t
, u
p, t
< 100 nm, the contribution from the magnification error on calibration in
(26) and (27) can be neglected. Then we get ∆b
p, t
≈∆u
p, t
≈∆d.
This means that the error in measuring the linear dimensions of relief structures in the nanometer range (less than
100 nm) is completely determined by the error in measuring the electron probe diameter. Also, the microscope can be used
to measure the sizes of microstructure elements in ranges greater than 1 µm for probe diameters up to 100–200 nm, while for
the range less than 100 nm one requires probes of diameter 30 nm and less. Such probe sizes at present occur only in new
microscopes. After 3–5 years of extensive use (e.g., in industry), the probe dimensions increase to 50 nm and more. Then the
SEM cannot provide for measuring linear dimensions in the nanometer range. It is therefore necessary to develop new micro-
scopes with smaller probe dimensions. These microscopes should provide automatic focusing, i.e., maintenance of the probe
size with an error less than 1 nm when the SEM parameters vary widely.
The following feature occurs in measuring linear dimensions of microstructure elements by the use of (4) and (5) [6].
In that case, the probe diameter has no effect on the measurement of sizes for middle lines (Fig. 1) of trapezoidal ridges l
p
and grooves l
t
(with conditions (1) and (25) met). These features have been confirmed in special experiments [6–8] on SEM
working with primary electron energies E ≥ 15 keV in slow secondary electron collection. Then the error in measuring the
middle line of a structure element is determined in the main by the magnification error:
∆l
p, t
/l
p, t
≈∆M/M, (29)
while the probe diameter and the error in measuring it have little effect (apart from conditions (1) and (25)).
For small probe diameters d << l
p, t
, (29) applies also for the nanometer range, but this does not introduce any
advantage, since in microelectronic and nanoelectronic technologies it is necessary to know the linear dimensions of the
microstructures and nanostructures at the top and particularly at the bottom of the ridges and grooves, and such dimensions
can be determined only if the probe diameter is known.
∆∆∆bb uu MM
pt pt pt pt,, ,,
///.≈≈
∆∆∆uu MM du
pt pt pt,, ,
/(/)/.
()
≈+
()
2
2
2
∆∆∆bb MM db
pt pt pt,, ,
/(/)/;
()
≈+
()
2
2
2
603
An exact knowledge of the probe size thus guarantees measurement in the SEM of linear dimensions for rectangular
and trapezoidal structures over a wide range down to tens of nanometers. The error in such measurements is largely determined
by the error in measuring the electron probe size. An important specification for the latest microtechnologies and nanotech-
nologies is thus an SEM with a probe of diameter less than 20 nm.
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